ASCII art by Yakovlevtech

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+*******+++++++++++++++++++++++++:;+**++++;;;;;;;;;::;::::;;;;;+++?%???*?**???%?*%%%%????????%%**?******++++**++;++*++*++++*****************?*******++++++;;;;:::::::,,::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::::,,,,,,,,,,,,,,,,,,::,,,,,,,..,,...........,,,,,,,,,,:,,,,,,:::::;;;;;;;;;;;;;;;;;::::::::::::,,:::::::;;++;+++*?***+;::,,......,,,,,,,,;+++;;::::::::;;::;:;;;;;;;;;;;;;;++;;+;;;;;;;;;;;;:;;:
+*********+++++++++++++++++++****+;;+;;+++;+;;;;;;;;;;;;;;;;;;;++*?*??*??**??????%%%%???????%S?+******+*++++**++++*+++++******???*********?************+++++;;::::::,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,:,,,,,,,..,,............,,,,,,,,,:,,,,,::::::;;;;;;;;;;;;;;;;::;;:,:::::;:,,,:,:::::;++;:;::,,::,.,:;:,,...,::::;::,,:;;;;;;;;;;;;:;;;;;;;;;+;+++++;++++++;;++++;;;;;;;;;;;
+*******++++++++++++++++++++++******+++;++;;;++;::;;::::;:;;;;;+***??**??*???????%%%%??%%%%%%?++*********+******++++++++*???*+?%*++**+******??**********+++;;;;:::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:,,,,:::::::,,,,,,,..,,............,,,,,,,,::,,::::::::::;;;;;:;;;::::;::;;;:::::::,::::,,::,::;;::,::,,,,..,:::::;::,;;;;;;:::::;;;;;;;;;;;;;;;++++++++++++++++++++++++++++++;+;;;;+;
+****+**++++++++++++++++++++++++*+**+++++;;;;;;:,,:;::::;;;;+++++*****?*?????%%??%%????%%%%%*++**???******??*??++***++++*????+??+++**+**?**************++++;;;;::::::,,,,,,,,,,,,,,,,,,,:;;;;;,,,,,,,,,,,,,,,,,,::::;:::::::::;;::,::,,,,,,,,..,..,,...........,,,,,,:::::::::::::::;;;;::;;:::;;;:;;;;;:;:::::::::,::,,;;:::::,:,,......,::;;;;;;;;;;;;;;;+++++++++++++++++*++**+**+*++++++++++++++++++++++++++
+*++++++++++++++++++++++++++++++++++++;++;;;;;,::,:::::;;;;;;;++++******?****?%*?%%??*?%?%?%*+***????****??*???+*?++++++*?*****++**********************+++;;;;;:::::,,,,,,,,,,,,,,,,,,,:+**?S%;,:,;+:,,.,,,.,,,,,,,;:,,,,:,,,::,....,:,,,,,:..,,..,,..,.........,,,,:::::::::::::::;;;;;:;;;;;;;;;;::;;::::::::,:,::::,::::::,,..,::,,,,,,:;++**++++++++++++++++***+++*+++**********+++++++++++*+*******++++++*+
++++++++++++++++++++++++++++++++++;;;;;+;;:;+;::::,,:::;;;;;;;;++++*****?**+**?**?????*?????**+*????*+*********+**+++++******?*+***********************+++;;;;;:::::,,,,,,,,,,,,,,,,,,:::;+;*%?*?%S#%;;*+*+::::::::;+:,,::,,,::,,,,,:?%%%%?*:,,,..,,,.,,..........,,::::::::;::;;;;;;;;;;;;;;;;;;;;;::::::::::,,:,,:::::::::,,,...,:::::;:::;+**********++++++++++++++*++++++**********************?******+**+++
+++++++++++++++++++++++++++++;;++;;+++++;;;;;;:;:::::::;;;;;;;;++++**********??**??**??***%?**+*???***???*?***+++++++++**+**+*+++++++**++++++++++***++++;++;::::::::,,,,,,,,,,,,,,,,,::::::;;+*%%%%SS?+??*?********+++*??*******+*?%%S##S?*+;:,,,.,.,..,,,........,,,,::::::;:;;;;;;;;;;;;;;;;;;:;;:::::::::::,,:,,:::::,,:::,,...........:;;+************+++++++++++++++++++++****************??????**********+
+++++++++++++++++++++++++++;;;;+++++++;;;;;;:;;;::::::::;;;;;;++++**++****?*******?*???***?***+***??*??????***+++++++++**+**++++++++++*+++++++++++++++++;;;;:::::::,,,,,,,,,,,,,,,,,,::::::::;++*+*%SS%********+++++++***?*****??%SS#S%?*;;;;:,:,,,,,...,,......,...,,,,::::::;;;;;;;+++;;;;;;;;:::::::::::::::,,:,::::,,,::::,,,,,,........,:+++***++***+++++++++++++++++++++++**************??*?????*********+
++++++++++++++++++++++++++;;:;;;++;;::::;;;;;;;;;:::::::;;;;;;+++++**++****??***?****??++*+*******%?*????*****+;+++++++****++++**+;;+++++++++++++++*++++;;;;;;:::,,,,,,,,,,,,,,,,,,,,::::::::;;;;++****?***?*?********?*???????%%S%S%*;;;;::::,,,,,,:,..........,,,,,,,,,::::::;;;;;;++++;;;;;;;::::::::,,::,,:,,:,,::::,,,,,,,:,...,:::,.....:+++++++****++++++++++++++++++++*+**************??*?????*******?*+
+++++++++++++++++++;;+;+;;:,,:,::,,.:;;;;::;;;::::::::::;;;;++++++*+++++++***?*??****?%?+++*?*****?**??**?***++++++++++**+++++;+;;;;;;++++;++++++++*+++;;;;;;::,,,,,,,,,,,,,,,,,,,,,::::::::::;;;;;++++++;+*%?%%%??*****+**?%%%??*+;;;;;::::::,,,,,,,,,,,,,,,,.,,,,,,,,,,::::::;;;;;;;;;+++;;;:::::::,,,,,::,,:,,::,,:::,:,,::,,,..,,,,:,,,..,,;++++++*****+++++++++++++++++++++*************????????*********?*
++++++++++++++++++++++;;;;,,:;:,,,:;;;;:::::::::::::::;;;;++++++++*++**++++;+??********?+;+*??***********?**+++;;;+++***+;++++++;;;;;;;+*++*+++++*+++++;;++;:,,,,,,,,,,,,,,,,,,,,,,,::::::::::;;;;;;;;;;;;;;****++;;;;++;;++**+;;;;::::::::::,,,,,,,,,,,,,,,,...,,,,,,,,,,::::;:::;;;;;;;+++;;:::::::,,,,,::,,:,,,,,,:::,,,,,:,.....,::,,,,..,,,;+++++**********++++++++++++*******************??????********??*
+++++++++++++++++;+;;+;:::,,;;::::::::::::::::::::::::;;;;++++**+++++++**+*+;+*****+*****+++*?***+*??????**+++;;;++++++++++;;;;;;;;++++++;;;;++++++++;;;;++:,,,,,,,,,,,,,,,,,,:,,,,:::::::::::;;;;;;;;;::::;::;;;;;;;;+++++;;;:::::::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:,:::::;;;;;;;;;;;;;:,,,:,,,,,,::,,:,,,,,,,:,,,,,,:,........,::,,.,,,:+++++*******************************************?*********????*
+++++++++++;;;;;;;;;:;;:,::,,::::,,,,::,,,:::::::::::;;;;++;+*?++**+++**+****;;++++***??**+++**+++*???%?*++*++;;;++++;;+;;;;++;;;;++;;+;++++++++++++;;;:++:,,,,,,,,:,,,,,,,,,::,,,,::::::::::::::::::::::::::::::::;:::::::::::,,,,,,,,,,,,,,,,,,,,,,,,:,,,,,,,,,,,,,,,,,.,:,,::;:;;;;;+;:::::::,,,,,,,,,,,::,:,,,,,,,:,,,,,,:,.........,.,::,,:,:;+++***********?**?**?************???*****?***?**************+
+++++++;;;;;;;;;;:::,,,:,::,,,,:,::::;:,,:,,:;::;;:::;;++++++**++***++**+*+*****++*+++?%??*++**+*+*%?*?*;+**++;;:++**;;+++;++;++;+;;;;;;+;;++++++++++;;;+;,,:,.,,,,,,,,,,,,,,:,,,:::::;::::::::::::::::::::::::::::::::::::::::,,,,,,,,,,,,,,,,,,::,,.,,,,,,,,,,,,,,,,,,,,,,,,,:::;;;;:;+::::::,,,,,,,,,,,,,,::,,,,,,,,:,.,,,:,............,,,,:::+++++**********?*?????*?*******?????????****************++++++
+++++++;;::;:;;:::::,,::::,,:::,,,,:,,,,,,:;:;;;++++;;+++++++****%#S*;+******+*??*+++;+??**++;++*******+***;;;;:;;+**+++*+;;:;++;;;;;;;+;;;;;++;+;;+;;;;;;,,::,,,,,::,,,,,,,,:,,,:::::;::::::::::::::::::::::::::::::::::::::::,,,,,,,,,,,,,,,,,,::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::;;::;;:::::,:,,,,,,,,,::,,,:,,,,,,,,,,...,,............::,...,;+++++*********?????*?***********???????*******++********++++;
+++++++;;::::;;,,,,,,,:::::::::,,,..,,,,::++*++++***;;;;*+++++**?????******?**+*??++?+;+*****++;+***?****+;;;::;+++*;+++*;;:;+;;;;;;;;;;;;;++++++;;;;;:::;:::,,,,,,::,,,,,,,:,::::::::::::::;:::;::;::::::::::::::::::::,,,,::::,,,,,,,,,,,,,,,,:::,,,,,,,,,,,,,,,,,,,,,,::::,:::,:::;;,:+;::,,,,,,,,,,,,,:::,,,,,,,,,,,,,..,,,,...........,;;:,....:;+************?????***********???*??***+++++++**********+++
++++++++;:,,:;:,,,,::::,:::::::,,,,,;++;+*+++;++;+?+;;;+*+;++***??*?%*+?***???**??++*?*+;+++*?++++*?**+++;;::;;;+*++;+++;;;:++;:;;;;+;;;;;;;;;;;;;;;;;::::::::,,,,,,::,,,,.,::::::::::::::::::::;:;;;;;:::,,::::::::::::,,,:::::,,,,,,,,,,,,,,,,::::,,,:,:,,,,,,,,,,,,,,,,::::::::::::;:::::,,,,,,,,,,,,,,,:::,:,,::,,,,,,,,,,,,............,;;,,;+:,.,;**************??*******************++++++++*****???????*
:;::;;;;:::,:;;,:;;;:::::;::::,,,,:+?%+;+++*+;++;**;:;+**+;+**???%??%%+***++****++*++*?*;;+++*?*;;+*;++;;:;;+++;+**;;+?+;;;:;+;;;;;++;;;;;;;;;;;:;::::::;:::,,:,,,,,,,,:,,..,:::::::::::::::::::;:::;;;;;:::::::::::::::::::::::,,,,,,,,,,,,:,,::::,,,,:::::::::,,,,,,,,:,,::::::::::::::,::,,,:,,,,,::,::::::::,::::,,,,,,,,,,,,............,,,,+++*+,,:+*******************************+++++++++**???????????*
::,:,::,,:::::,,:;;;:;;;;::,::::;+**++;;++*+;;*+;**;:;*?*;;+?**%#%%S%%??*+***???*++*+??**+;;;??+;;++++;::;;++++;+?*:;?*+;::;;:;;;;;;;++;;;;;;;;;;;;;::;::;:,:,.,,,,.,,,,,,,,..,:::::::::::::::::::;:;;;;;;;;;;;;;::;::::::::::::,,,,,,,,:::,:::::::,,,,,::::::::,,,,,,,,:,,::::::::::::::,,:,:,:::::,,,:::::,,:::::::::::::,,,..,................;+;;::+;:+*******************************+++++***?????????????*
;;:;:;;,:;:,,..,;:::::;;:::::::;+***++;;;**;;;*+:++;:+**+;+*????#S%??%S??**+*?*??***+*S?**:;++;;*+::::;;;;*+++++++;;+*++;;;;;;;;;:;;;+;:::;;;;;;;;;;;;::::;:,,:,,,,,,.,,,,,,,..,,:::::::::::::::::;;;;;;;;;;;;;;;:::;::::::::::::,,::,,:::::::::::::,,,,::::::::,,,,,,,,:,,::,,::,::::,,,,,,,:,:::::,,,,;:::,,,,,,,,,:::::,,,....................:;;;;;;*;;*************************************?????????%?????*
;+:::;:.,;,,....:,,,,,,,::::,:;++***++++;**:++*+;++;;***++****%?SSSS%%?+??*;*?**?%?+**?***+;;::;??::;;;;;;*++;++++++++++;;;;;;;;:;*+;:::::::::::;;::::::::;;::::,,,,,,,.,,,,,,..,,,:::::::::::::::;;;;;;;;;::;;;:::::::::::::::::,,,:::;:::::::::::::::::::::;::,,,:,,,,,,,,,,,,,,,:::,,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,:,,,,....................,:;*+;+*;+********************???????*******?????%???????????*
:;:,,:,..::,..,..........,,,,;+++**+++:;+++;++*+;*+:;+**;+?***??%SSS%%?**??*+?**?%%*+*??*+;:;;;;+;:::;;;+++++;;;+**+++;;;;;;:::;;;*+;::::::::::::::::::::;:;;:::;:,,,,:,,,..,::..,,,:::::::::::;:;;;;;;;;;;;:;;::::::::::::::::::::::::::::::::::::::::::::::;::,,::,,,,,,.,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::,:,,,,....................,:;;:.,++;+**+****************???????????????????????????????*
.,...................,,..,:;+*****++*+:;*;;;+++;+*;;;+**;*%**???%%SS%??*+*S?+*?+?%%?*+*%*+:;;;;;;::::;+;;+*++;;;+*?+;;;;;;;;::,:+;;;+::;::::::::::;;;;:::;;:::::;:,,,::::::,.,:,..,,::::::::::;;;;;;;;+;;;+;;++;;;;;;;;;;:::::::::::::::::::::::::::::::::::::::,,,:,,,,,..,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::,,,,......................;*+;:*+;;+++++**************???????????????????????????%??%?
............,,,.....,,,::+*****+?*++**;+*++++++;++;++*?*+*?****??%SS%??***%*+?%++%???***++;+++;;::,:;;++;;++;+*;;?*;;;;++;;;;:::++::;::+;:::::;;;;;;;;;;:::::::::;::::::;;;:.,::,.,,,::::::::::;;;;+;+++++*++***++++;+;;;;;::::::::::::::::::::::::::::::::;;;::,,,,..,,,..,,.,,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::,,,......................,:;*+:;;;;;+++++************?????%%????????????????????%??%?
,...,,.......,,,,,,,:::;************+*+**;;*+;+++;+++?*++?***?**??%S%%?*+*%*+*S?;???%?+***?+++;:;:,:++;;;++++*+;+*+:;+;+++;:;;:::;;::;:::::;;:;;;;;;;;;;:,:;::::::::::;;;;;:,,::..,,::::::::::::;;;;;;++++******+++++++;;;;;:::::;;;:::::::::::;:::::::::::;;;;::,,,..,,,,.:,.:,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::,,,,,,.....................,:*;,:++::++;+++*************???????????****??*????????????
:,.,,,,....,,,.,,,:;+*?+************+++*?*;++;;**;+**?*+****???????%??%?+%%*+?S%+?%?%?****?+++;:;:;;**++++**;+;++;;;++++++:,;;:::;;::;:::;;;:;;;;;;;;;;:,,;:,:::::::::;;;;;:,,:,..,::::,:::::::::::;;++;*********++++++;;;;;::::::;:::::::::::::::::::::::;;:;;::,...,..,..:,.:,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.....................,;;:.;++::+++++++****************??????*********??????????*
,...,,,...,,,,:;;;++*%?++**********??**+**?*;+:?*;**+**+***????**???*?%%+*?*+?%?+?%??**?*+**:++:::;;;;+++;+*+++++;;;+;+;+;::;+;:;;:::::;;+;:::;::;;;;;::,:;::::::::::::;:::,,,,..,,,,,::::::::::::;;;+;;++++**+*+++++++;;;;::::;::::::::::::::::::::::::;:;;;;;;:,.,:,,,,,.,,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,......,,,,,,............................,;+++++++++++*+*****+++********?**************????????*
....,.,,,,:;;;++++********??????+*??%???+;*?++;**+****++?????%?*?????%S%**????%**?%??*?????*++;;::;+;++++;++++++;;*;;+;+;;;;;;;;;;::,,;+;;;;;:;;;;;;;;::,::::::::::::;:::::,,,,,,,,,,,::::::::::::::;;;;;;;;+;;;+;;;;++;;;;::;:;::::::::::::::,:::::;;::::;;;;;;:,.,:,,,..,,,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,...................................,,...,;;+;;+;++**+*+*+**++++++******?*************??????%%??
...,,,::;;;+++++++**+**+**************??*++++;;;;++**+++**?????**????%%%*****???******????*+*++;::;+;++++;+++;+;;++;;;;;;;;;;:;;;;:,,:;+;;;:;;;;;;;;;::::::::::::,::::::::,,,,,,,,,,,:,:::::::::::::::;;;;;;;;;;;;;;;;;;:::::::::::,,:::::,,:,,:::,,::::::::;::::...,,,,..,,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,....................................,:...,::,,,:;+++++++++++++++++++*******++++++++*****??????*

+**************+++++**+++++++++++++++++++++++++++++++++****************+;;::::::::::::::;;;;:::,,,:::::;;:::,,,.................................................................................................................................................................................................................................................................................................
*??***?*?********************************************************??????**+;;;;;;;;;;:::::;;;;;;;:;;;;;+++;:,....................................................................................................................................................................................................................................................................................................
*????*?*******************+*******++++++*************************??????*?*++++++;;;::::::;;++++++****+;:,,.............................................................................................................................................................,,,,,,,,,,,,:::::::::::::::::::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,..,,,..,.....................,............................
*??????***************************++***+**************************?********+++++++;:::::;;+++++***+;:.............................................................................................................................................,,,,,,,,,,,,,,,,,,,:::::::;;;;;;+++;;;;;;;;;;;;;;;;;;;;;;;;::;;:;::;::::,,,,,:,,,,,,,,,::::::::,,,,,,,,,,,,,,,,,..............................................
*??????**********************************************************************++++++;;+;++++++;:,,......................................................................................................................................,,,,,,,,,,:::::::::::::;;;;;;++++++**********????????***?*?********************+++++++++++++++++++++++++++++++;;;;;;;:,,,,,,,,,,,........................................
*?????************************++**********************************************+++++++**++;:,..................................................................................................................................,,,,,,,,::::::::;;;++++++*****************????*********?%%???????*?*******?****************+++++++++++*+++++++++******++++++++++++++;;::,,,,,,,,..................................
*???????*******************+++++++****************************++++++++******++++++;;;:,,.....................................................................................................................,,,,,,...,,.,,,,,:::::;;;;;;++++*************?******?????????????????????%%S%%%%????????????????********************+*+*+++**+++++*+**+++*+*++++**+**++++++;;;::,,,.,,.............................
*??????*?******************+++++++**********************+++++++++++++++++****++;,....................................................................................................................,,,,,,,,:::::::::::;;:;;;;;++++++++++**********??????????????%???%%?%%%%???????????%S#S%%%%%???????????????????????*******************************++**++****++*****+++++;;;:::,,,..........................
*????*?*******************++++++++++*+******************++**+++*+*++*+++++;;;,...........................................................................................................,,,,,,,,:,:::::::;::;;;;;;;;;;;+++++++++++++++**********???????????%%?%???%%%%%%%%%%%???????????%#SSS%%%%??????????????????????********************************+***++****+++++++**++***+++;;:,,........................
*???*********************++++++++++++*******************+++++++*+***+;;:,................................................................................................,,,,,,,,,,::::::::;;;::;;;;;;;;;;;;;+;++++++++++++**********************???????????????????????%?%%????%??????*??%%SSSS%%%%%??????????????????????**********************************************************+++;,,.....................
*???*********************++++++++++++++***************++*++++****+;:,...............................................................................................,,:,::,::::::;;;;;;;:;;+++;;;;;;;;;;+;++++++++++++++++++++*+***********************???????%????????????????????????*?*?*?SSSSS%%%%%?%?%%?????????????????*****************************************************+*+*+*+;::,,,.................
*??**********************++++++++++++++************+++******++::............................................................................................,..,..,,:::::::;;;;;;;;;;;;;;+;++++++;+++++++++++++++++++++++++++++++++++++++++++++********??????????????%%??????????????????****%%SSSSS%%%%%%?%%?%%??????????????************************************************************++;:,,,...............
*??***********************++++++++++++************+****++;:,..................................................................................,,.,,,,,::::::::::::;;;;;;;;;;;;;+++++++;++++++++++;;;;;;+++++++++++++++++;;;;+;;;;;;;;;;;+;;;;+++++++++*****????%???%%?????????????????????****?%%SSSSS%%%%%%%%%%%?%????????????*?***********************************************************++;;::,,............
*?************************++++++++++++************++;:,,........................................................................,,,,,,,,,,,::::::::::;;;;;;;;;;;+;;;;;;;+;+;;+;;++;;++;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;++;;++++++**??????%?????????????????????????****?%SSSS%%%%%%%%%%%%%%%?????????????************************************************************++++;;::,,........
***************************+++++++++++++******++;:,.........................................................................,,,,,:::::::;;;;;;;+++;;+;;+++++++++++++++++++++++;;;;;;;;;;;;;;;;:;::::::::;;;;;;;;;;;;;;;;;;;:::::::::::::::;;;;;;;+++;;+++****??????????????????**???????????****??%%SSS%%%%%%%%%%%%%%%%%???????????????*************************************************************++;;:,,,....
*??*************************++++++++++++++++;,..........................................................,,,.....,,:,,,,,::::::;;;;;;;+;+++++++++++++*+++***+*********+*+++++++;+;;;;;;::::::;:::::::::::;:;;;::::::::::;;::::::::::::::::::;;;;++++++++******?????????????????**?????????????*****?%%%S%%%%%%%%%%%%%%%%%????????????????***************************************************************++;;:::,,
***************************+++++++++++++;,....................................................,,,,,,...,:::::::::::::::;;;;;;;;;;;+++++++++++****************?***********++++++++;;;;;;::::::::::::::::::;;:::::::::::::;:::::::::::::::;;;;+++++*********??*????**??????????****??????*?*????*****??%%%S%%%%%%%%%%%%%%%%?%???????????????*****************************************************************++++;
****************************++++++++:,,....................................................,..,:;;++;:::;;+;;;;;;+;;++++++++++++***+++++++++++*****************??**?**********+**++;;;;;:;::::::::::::::::;:::::::::::::;::::,,::::::;;;;;+++******************?*?**??????????***?*?%??*?**?*********?%%%%%%%%%%%%%%%%%%%?????????????????????*****************************************************************+
+*************************++*++;;,...............................................,,,,,,,,::::::;;++*++;+;;;;++++++++++++++************++++++++**********?*?*??????**???************+++++;;;;::::::::::::::;:::::::::::::;::,,,,:::;::;;++++**********************?**???%%??*?*****???%?****?***********%%%%%%%%%%%%%%%%%%????????????????????????********************************************??*?**************+
+********************+++***+;,................................................,,::::;;;;;;;;;;+;;++++++++++******+++******????*?????***++++++*********?????**?***???****??***********+++++;;;;;:::::::::::;:::::::::::::;::,:,,,::;;;;;;;;;;;;;;;;;;++++**************?????*******?**???**??************??%%%%?%%%%%%%%%%?????????????????????????********************************************??????***********+
+**********************+;,,.............................................,,,::;;;;;;;++++************+***+*+*****+++*????????????????**+++++++**********????????**************++++*****+++++;;;;;;;::::::::;::::::::::::;;;::::::::::::;;;;:::;;:;;;::;;;;;;;;++++++++++**?*****+****????*???????*********?%%%%%%%%%%?%%%?%%?????????????????????????*******************************************??**************+
******************++;:,........................................,,,..:::::::;;+++++*******??????????*****+*****++++*???????%%???????***++++++***********************++++;;;;;;;;;;;;;+++;+++;;;;;;;:::::::::::::::::::::;;;;::::::::::::::::::::::::::::::;;;;;;;;;;;++++++*****++***???????????????*******??%%%%?%%%%%%%?%%?????????????????????????????****************************************?*?????*??*??**+
*?*************+;,....................................,,,:::::::;;;:;++;+;++++++**??????%%%%??????**********+++++**?????????????*****+++++++***+***+++++++++++++++++;;;;;:::::::::::::::::;;;;;;;;:::::::::::::::::::::;;;+;:::::::::::::::::::::::::::::::::::::::;;;;++******+*+****?*?***??????????****????%%%%%%%?%%%%%?????????????????????????????***************************************??*??*******??**+
*??********+::..............................,,,,:::;;;;++++++++++++++++++*****??*?%%%%%%SS%%?????*********++++****??????????????****+*+*++++++++++++*+++++++;;;;;;;;;;:::::::::::::::::::::::;;;;;;;;;::::::::::::::::::;;;;+;;;;;::::::::::::::::::::::::::::::::::::;;+++****++++*??*?***?***????????????????%%%%%%%%%????????????????????????????????*****???********************************????*??*********
*????*+:,.........................,,,,,:,:::;;;;;;+++++****************??????%%%%%%%%%%%SS%??****++*****++++*****??????????????*******+*++*+++++++*+*++++++;;;:::::::::::::::::,:,::::::::::::;;;;;;;;;:::::::::::::::::::;;++++;;;;:::::::::;;;;;;;;:::::,,,,::::::::::;+++****++++************???????????????%%%%%%%%%%????????????????????????????????????**?********************************?***???******??*
*?*;,.....................,,:;;;;;;;;;;;;;+++++++*********????????%??%%%%%SSSSSSSSS%%%%S%%?****+**++++++++******????????*??????*******++++++++*++++*+*++++;;;;;:::::,,:,,,,::::::::::::::::::;;;;;+++++;;;;:;:::;;:::::::::;+++++;;;;;;;;;++++**%?;;++++;++;::::::::::::;;+++++++++*****??*******???????????????%%%%%%?%%%??????????????????????????????*??????????*******************************************?*
,.................,,,,:::;;;;;+++++++++++++****?**????????????%%%%%%%%%%SSSSSSSSSS%%%%%%?******++*+++;;++******??????%%%**??**********+++++++++++++*+*+++;;;;:::,:::;;;++**;;;:;;;+;;;;;;:::;;;;;++++++++;;;;;;;;;;;;:::;;;;;+++;;;++++*********??****?******+++;;::::::;;+++++++++******??********???????????*?%%%%%%%%%%%%?????????????????????????????????????????*******************************************
...........,,,:::;;;;;;;;++++**+************??????????????%%%%%%%%%%%%%%SSSSSSS%SSSS%%%?*******+++;;++++++*?*?????%%%%%?************++++*+++++++*++++++;;;::;;;;;;+*?***?%S?++++++*?*******++++++**+++++++++++++++++;;;;;;;;;++++;+************++**;:;;++*******++;::::::;+++++++*+********?********?????????????%%%%%%?%%%%%%?????????????????????????????????????????***********?*****************************
.....,,::::;++++++++++******???????????????????????????%%%%%%%%%%%%%%SSSSSSSSS%SSS%%%??????*++++;;;;+++*****???%%%%%%?***************++++++**++*+++++;;::;;+********????**???++++++*?**?????*********++++++++****+++++++;;;;;++++;;;++*******+++;;;:;;;;;;;;+++++**+;:::::;++++++**********************???????????%%%%%%%%%%%???????????????????????????????????????????************??*??***********************
:;;;;;;+++++++*********????????????????????????????????%?%%%%%%%%%%SSSSSSSSSSSSS#S%%???%%%?*++;;;;++++****????%%%%%?*???**********+++++++++++++*++++;;;:;+++*?????*****+++++***************************+****+*++***++++++;+;;;++++++++++++++++;;;;:::::::::::::;;;++++;:::;;++++++**+***********?*********??????????%%%%%%%%%?????????????????????????????????????????????*?????*?????**????********************
++++***********??*???????????????????????????????????%%%?%%%%%%%%%SSSSSSSSSSSSS#S%%???%%S%*+;;;;;;;+*****?????%%????*???***********++++++++++++*+++;;;;;;;++**++*++++++;;;;;;;;;;;;;;;+++++++++***************+***+++++++;;;;;;++++++;;;;++;;;;;:::::::::::::::::;;;;;;;;;:;++++++++*++*****???****?********????????%%%%%%%%%%%%??????????????????????????????????????????????????????????**********************
+*******?????????????????????????????????????????????%%%%%%%%%%S%SSSSSSSSSSSSS#S%%%?%%%%??+;;;;+;;;+*?**????????????????**********++++++*++++****++++++++++++;++;;;;;;;;;;;;;;;:::::::;;;;++++++*************+++++++++;;;;;;;;;;;;;;++;;;;;;;;;;:::::::,::,:::::::::;:;;;;;;;+++++++++++****???**************??*?*??%%%%%%%%%%%??%?????????????????????????????????????????????????????????*********************
+**????????????????????????????????????????%%%?????%%%%%%%%%%%%SSSSSSSSSSSS%SSS%SS???%?*++;;;;;;;++****???**?????????????*?*******+++++++++**+++++++++++++;;;;;:;:::::::::::::::::::::;;;;++++++*******+*+**+++++++++;;;;;;;;;;;;;;;;;;;;::;;;;;;::::::,,,,,::::::::::::;;;;;;+++++++++++******?******************??%%%%%%%%%%%%?%%%????????????????????????????????????????????????????????????*************?*+
*???????????????????????????????????????????%%%%%%%%%?%%%%%%%%SSSSSSSSSSS#S%S%?%%%%???*+++;;;;;;++******????????%%??????????******+++++++++++++++++++++++++;;:::::::::::::::::::::::;;;;;;+++++++++***++*+*++++++++;;;;;;;;;;;;;;;:::;;;:::::::::::::::::::::::::::::;:::::;;;;++++++++*********?*****************?%%%%%%%%%%%%%%%%%????????????????????????????????????????????????????????????????????*****??*
*????????????????????????????????????????????%%%%%%%%%%%%%%%SSSSSSSSS##SS%%S%S%S%%?%?*++++;;;;++++*****??????%%%%???%%%?????******+++++++++++++++++;+;+;;;;;;;;::::::::::::::::::::;;;;;;;++++++++++++++*+**+++++;;;;;;;;;;;;;;;;;;;;;::::;:::::::::::::::::::::::::::;;::::;;;++++;++****************************?%??%%%%%%%%%?%%%????%???????????????????????????????????????????????????????????????????***?*
*??????????????????????????????%%%%%????%???%%%%%%%%%%%%%%SSSSSSSSSSSSSSS%%%%S%?????*+++++;;;;++++++**??????%%%%%%%%%%%???***?****+++++++++++++++++;;;;;;;;;;;;;;;;;;::::::::::::;;;;;;;;++++++++++++++++++*++++++;;;;;;;;;;;;;;;;;;;;:;::::;;;;;;;;::::::::::::::::::::::::::;;+;;;;+**?????**********************?????%%%%%%%%%%%%%??????????????????????????????????????????????????????????????????????*?*?*
*?????????????????????????????%%%%%%%%%%??%?%?%%%%%%%%%S%SSSSSSSSSSSS#SSSS%S%%????***+++++;;++++++****????????%%%%%%????%%??******++++*+++++++++++++;::;;;;;++++;;;;;;;;;;;;;;;;;;;;;;;;+++++++++++++++++***++++++++;;;;;;;;;;;;;;:;:;:;:::;;;;:;;;;;::::::::::::::::::::::::::;;;;:;+***??????*?*??****************????%%?%%%%%%%%%%%%?????????????????????????????????????????????????????????????????????*??*
?????????????????????????????%%%%%%%%%%%???%?%%%%%%%%%%SSSSSSSSSS##S#SSSS%%S%?????*+++*+++++++++++*****???????%%%%%%%SSS%%??**?***++++*+++++++++++;;;;;;+;;;;;;;;;;;;;;;;;;;;;;;;;;;;+++++++++++++++++++++*+++++++++++;;;;;;;;;;;;::::::;:;;;;;:::::;:;;::::::::::::::::::::::::;;;::;*?**???????????***************????%%%%%%%%%%%%%%%%??????????????????????*???????????????????????????????????????????*?*??*
?????????????????????????????%%%%%%%%%%%?%%%%%%%%%%%%SSSSSSSSSS#S#####SSS%%S%?%?*+++****+++++++++*******?????%%%%SSSSSSS%%?**??***++;++*+++++++++;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;+++++++++++++++++++++******+++++++++;;;;;;;;;;;:::::::::::;;:::::::::::::::::::::::::::::::::;:;;;:;******?????*??*******++++**?**????%%?%%%%%%%%%%%????????????????????????????????????????????????????????????????????????*
?%%?????????????????????????%%%%%%%%%%%%%%%%%%S%%%%SSSSSSSS######SSSSSSS%%?%%%??++***+++*++++++;+*****??????%%%SSSS##SS%S%?*????**++++++*+++++++;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;+++++++++++;;+++++++++****+++++++++;+;;;;;;;;;;;;:::::::::::::::::::::::::::::::,::::::::::::;;;;;;;;;++****?**?*??********++++*?***???%%%%%%%%%%%%%%%%%?????????????????????????????????????????????????????????????????????*
?%%%???????????????????%????%%%%?%%?%%%%%%%%SSSSSSSSSSSSSS##S#####SSSSSS%%?%%?*++****++**++++**++***?????%%%%SSSS#S#SSS%S%?*????**++++++++++++++;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;++;++++++;;;;;;+++++++*****+++++++;;;;;;;;;;;;;;;:::::::::::::::::::::::::::::,,:,,::::::::;;:;;;;:;;+;;+***?*?*+??***???***++++*?**?%??%%%%%%%%%%%%%%%%%????????????????????????????????????????????????????????????????????*
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?%%%%%%%%%%%%%%%%%%%%%%?%%%%%%%%%%%%SSSSSSSSSSSS####SSS####SS#####SSSSS%?*++**********++**+*****???%??%%%%%S%S%%??????????????????**++++++++;;;;;;;;;;;;;;;;;;;;;;;;;;;;::::::::::::::;+****++++****++++++;;;;;;;;:::::::::::::::;;;;;;;;;;;;;::,,,,,,,,,,,,,,,,,,::::::::::::;;::::::;;;;;;;;;;;++;;++++*???*++++++++***?%%%%%%%%%%%%%%%%%%???????????????????????????????????????????????????%%%%%%%%%%%?????*
*%%?%?%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%SSSSSSSSSSSS###########S%SSSSS%SSS%%?*+**********?**********??????%%%%S#S%?????????%%??????????**+++++++;;;;;;;;;;;;;;;;;;;;;;;::::::::::::::::::;+***++++++++++++++;;;;;;;:::::::::::::::::::::;;;;:;::;;;::,,,,,,,,,,,,,,,,,,::::::::::::::::::::::;;;;;;;;;+;;;++++????++*++++****???%%%%%%%%%%%%%%%????????????????????????????????????????????????????%%%%%%%%%???????*
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??%%?%%%%%%%%%%%%%%%%%%%%%%%%%%%%%SSSSSSSSS############SSSS%%S#SSSS%%??*;*?*+++++*?????******?????????%SS%???%??????????%?*???????****++++++;;;;;;;;;;;;;;;;;;;:::::::::::::::::::::;+++++;;;;;;;;;;;;;;;;;;::::::::::::,,,,,:::::::::::::::::::::,,,,,,,,,,,,,,,,,:::::::::::::::;;::::::::;;;;;;;;;;;;++*???**++++++****???%%%%%%%%????????????????????????????????????????????????????????%??%%%%%??????????+
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?%%%%%%%%%%%%%%%%%S%%S%%S%%%%SSSSSSSSSSSSSS###########SSSSSS##SS%%%???*****+++***????*+**????***???%%%???????*????????????*?????????****++++;;;;;;;;;;;;;;;;;;::::::::::,,,,,::::::;;;;;;;;;:::::::::::;::::::::::,:,,,,,,,,,,::::::::::,,,,,,,,,::::,,,,,,,,,,,,,,::::::::::::::::::::::::;;;;;;;;;;;;;;;;*??++*++++++***?????%????????????????????????????????????????????????????????????????????????*++;::::
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*??%%?%%%%%%%%%%%SSSSSSSSSSSSSSSSSSSSSSS#SSSS####S##SS###SSSSSS%%?%?**+**++**?*?%?********?***????%%?????????????????????????????????****+++;;;;;;;;;;;:;;;;;;::::::::,,,,,::::::;;;;;;;:;;;;;;;;;;;;;::::::::::::,,,,,,,,,,,,::::::,,,,,,,,,,,,,,,:::,,,,,,,,,,,,::::::::::::::::::::;::::;;;;;;;;;;;;;;;++*?*+++++++++***?????????????????????????????????????????????????????????????????%??%?*+;:,,;;;:,,,,,
*??%%?%%%%%%%%%%%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSS#SS####SSS#S#S%%??%?*+++*++**?*???******?*??*????%%%%???????????????????????????????******++;;;;;;;;;:;;;;;;;;:::::::::::::::::::::;;;:::;::;;;;;;;;;;;;;:::::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::,,,,,,,,,,,,::::::::::::::::::::::::;;;;;;+;;;;;;;;;+++**++++++++++**??????????????????????????????????????????????????????????????????*+;::,::::,,,,,,,,,
*??%%%%%%%%%%%%%S%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS###S##S%%???%%%*+++*****???*********?**??????%%%%??*?????*****?????????????????*?******+;;;;;;;;;;;;;;;;;;:::::::::::::::::::::::::::::::;;;::;;;;:;;:::::::::::,,,,,:,,::,,,,,,,,,,,,,,,,,,,,,,:::,,,,,,,,,,,::::::::::,,:::::::::::;;;;;;;+;;;;;;;;;+++**+++;+++**+***?????????????????????????????????????????????????????????????*++;;+;;;::,.,,,:,,::::
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++*???%%%%%%%%%%SSSSSSSSSSSSSS##SSSSSSSSSSSSSS#SSS#SS%%##?*%%%%?%%+;*+****???*?***?**????????????%%%????%??*************??????******??*****+++;;;;;;;;;;;;;;;;;;;:::::::::::::::::::::::::::::::::::::::::;;:::::::::::,,,,,,,:,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::::::::,:::::::::::;;;;;;;;;:::::;;;;++**+++++++++++**??*???????????????????????????????????????????????????**++;;;;;;:::::::;;;;;;:::::::
++**+*???%%%%%%%SSSSSSSSSS##SSSSSS######SSSSS####SSSSSS%*?SS??%?%*;+******??****??**????????????%%%%%????*?****************???**************++++;;;;;;;;;;;;;;;:;:::::::::::::::::::::::::::::::::::::::::::::::::::,::,,,,,,:,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::::::::,:::::;::::::;;;;;;;;::::::;+;+++*+++++;;++++*+*?????????????????????????????????????????????????*+;;;+++;;::::;;;;;;;;;;;;;;;;:;::
+**++;+??%%%%%%%%%SSSSS#SS#SS###########SSSSS##SSS%%S???%#S??%%%+++**+*????***???**?????????????%%%%%??%**??***++************************??**+++++;;;;;;;;;;;;;;::::::::,,:::::::::,:::::,,:::,,:::::::::::::::::::,,:::,,,,,,,,,,,,,,:,:,,,,,,,,,,,,,,,,,,,,,,,,,:::::::::::,:::::;;::::::;;;;;;:::::::;+;;+++++++;;;++++++*????????????????????????????????????????????**+;;;;+;++;;;;;;+;;;;;;;;+;;;;;;;;:;;:
+++++**+**??%%%%%%SSSS##SSSSS##############S###S%S%%%?%SSS?????*;++****??**???****?????????????%%?%??%??************************?**************+++++;;;;;;;;;;::::::::::,,::::::,,,,:,,,,,,,,,,,,,,,,,,::::::::::::,,,:,,,,,,,,:::,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,::::::::::,:::::;;;::::::;;;;;::::::::;;;;+++++;;;++;++****???????????????????????????????????????*+;::;;;++;::;;;;+++;;;;;;;;;+++++;;;;;;:::
+*******++**?%%%%%SSSSS###SS#################S%%S%%S%S%%%??%???+++**?**?**????***??%%*?%??????%%%????????*+****++****************??*******?******++++;;;;;;;:;:::::::,::::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,::::::::::,,,,,,,,,,,,,:::::,,,,,,,,,,,,,,.,,,,,,,,,,,,,,,,,,::::::::::;;:::;:::::::;::;:::::::;;;;;+++++;;;+;;+**??????????????????????????????????????*+;;,,;+;+;:;;;;++;++++++;;;;;;;;++++++;;;;;:::
+********+++++*?%%SSSSSS###################S%%???%SSSS%%%%%??*++***???*+*??**??*???%%*?%?????%%%%%%%???**+++++********++******************?******++++;;;;;;;;;:::::,,,:::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,:,,:::::,,,,,,,,,,,,,,,:::::,,,,,,,,,..........,,,,,,,,,,,::,,::::::::::::;:::::::::::::::::::::;;;;+++;;;;;;;;**+??????????????????????????????????*++;+++++;+;;++;+++++;;+++++++;;;;;;+++++++++;;;;;:
+********++++;;+**?%%SSSSSSSSSSS##########S%%%????S%%%%%S%??*+++**???++????*??*????%%???????%%%%%%%????*+++**+*****+++++++****************??***+*+++++++;;;;;:::::,,,,::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::,,,,,,,,,,,,,,,,,,,:::,,,,,,,,,..........,,,,,,,,,,,:::,,,:::::;;:::::::;;;;;;;;:::::::::::::;;;;;;;;;;++**+***?????????*??????????????????+++;:;;+++;;;++**++++++;;;;;;;;;;;;;;;;++++***++++++;:
+**********++;;:;;;;+*?%SSSSS%SSSSSSSSS##S%%%??SSS??S??%%%?++*+***?*+*????????*??%%%%?%?????%%%%%???******+*++*?**+++++++++*******?**?**?*********++++++;;;;;:::::,,,,:::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:,,,,,,,,,,,,,,,,:::::,,,,,,,.,,...........,,,,,,,,,,::::,:,::::;;;:;;;;;;;;;;;;;:::::::::::::::;;;:::;;+++*****?????????????????????*++;;;;;+**+;+++*******+++++++;;;;;;:::::::;;;++*+******++++;
+**********++;;+++++;;;+*?%%%%%%%%%%%%%S%%%%%??SS??%S***??++*****?**?%?*?*???????S%S%?%?????%%%%%?******+***++**++++++++++****************?*******++++++;;;;;:::::,,,,,,:,,,,,,,,,,,,,,,,,,,::,:,,,,,,:,,,,,,,,:::::::::::::::::::::,,,,,,,.,,.,,........,,,,,,,,,,::,,,::::::;;;;;;;;;;;;;;;;;;::::::::::::::::::::;;++;*+**?*????????????????**++;;+;;+**+***************++++++;;;;;;;;:::::;;;+************+;
+******+*+++;+++++++++++:;+++*??%%SS%%%%%%%???%%%*??+*%********+?*+?%%**?*???%%?*S%%%???%??%%%???******++++**++++++++**+*******?*******?******+***+*+++++;;;;:::::,,,,,,,,,,,,,,,,,,,,,:,,::,,,,,,,,,,,,,,,,,,,::::::::::,::::,,,,:,,,,,,,..,,.,,..,........,,,,,,,:::::::::::::;;;;;;;;;;;;;;;;:;::::::::::,,::::::;;;+;+++*?????%?%%?????++;;;;,,:++;++*??????*********++++++;;;++;;;;::::::;;;;+*+*+***+****;
+***++++*++++++++++++++++++++++++**??%%%%%%???%S%?***%?*++*******+?%??**?*??????%%%??????%??%%*??*******++*****++**++++******???********?***********++*+;+;;:;::::,,,,,,,,,,,,,,,,,,,::;+**+:,,,:;:,,,,:,.,,,,,,,;:,::,,,,::,,,,.,:;::::;:,,,,.,,,..,,,........,:,,::::::::::::::;;;;::;;;;;;;;;;;;;;;;::::::::,::,:;;;;;++++*????%??*++;:,,:;;::::;;+????????********+++++++;;;;;;;;;;;;:::::;;;;;++++++++++++;
+****+++++++++++++++++++++++++++++;;++++*?%????S?*???***++****?++???*?*%**??%???SSS%???????%%*+*?****+**++******+*+++++******%?*********?*********+**+++;;;;;::::::,,,,,,,,,,,,,,,,,,::;+*%S?+++*??+:;;;+::::;:;;+;;;;;;:::;;:::::+???***;:,,,..,,..,.,,.......,,::::::::::::::;:;;;;;:;;;;;;;;;;::;;;:::::::,,:::,:;;;;;;+;+*****+;;+;::...,:;++;+**+***????*******++++++;;;;;;;;;;;;;;;:::::;;;;;;+++++++++++;
+*****++++++++++++++++++++++++++++*++++;:;++++**???****++****?++*???***%*???????%%%%??%????%?++***********?**?++**+++++*?*?**?*+*************+*++**++++++;;;;:::::,,,,,,,,,,,,,.,,,,::::;+?%%%%%SSS%**?**+;++++++++++++++;;++;;;+*?S#S%?*+;:,,,,,,..,.,.......,,,,::::::::;;;;;;;;;;;;;;;;;;;;;;;;;:::::::::,,,,:::::::;;;;;;;;;:::::::::,.:;+++****+++**********+++++;;;;;;;;;;;;;;;;;;:;:;;;;;;;;;;;;;;;;;;;;;
+******+++++++++++++++++++++++++*******+++;;;;:;+++*+++++*++*+;*???**???????%%*?%%%???%???%?++**????****???*??*+?*+++++*****?*+*************+++*++*++++++;:;;::::,,,,,,,,,,,,,,,,,,,:::::;;+*?SSSSS#%??***+++++++++**?******++**?%SSS%?++;:::,,,,,,..........,,,,,,:::::::;;;;;;;;;;;;;;;;;;;;;;;:;;:::::::,,:,,:::::,:::;;;::::,.,:;:;;;;:+;++++++++;;;++;;;;;;;;;;;;;;;;;;;;;;;;;;;::;:;;;;;;;;;;;;;;;;;;;;;;:
+*****++++++++++++++++++++++++++*******+++;;+;:::::::;;;+;;+++*******??*????%%?*???%??????%?*+*???%?*****??*??*+?*+++*+******+++*********+***+++*+++++;;+;:::::::,,,,,,,,,,,,,,,,,,:::::::::;+*?%%SS%?***+++++++++++**********???????*+;;::::,,,,,,.....,....,,,,,,,,:::::;;;;;;;;++;;;;;;;;;::::::::::::,::,:,,:::::,,:,:::::::,..,,,:;;+++;+;;;;;;;;:::;;::::;::;;;;;;;;;;;;;;;;;;;;;+;;;;+;+++;;;++;;;;;;;;;:
+******++++++++++++++++++*++++;;******++++;;;;;;;;::::::;;;;;+**+******???????**?????*????%***+?????**???***?**+**+++*+*****+++++++++++*++++++*+*+*+++;;;;;:::::,,,,,,,,,,,,,,,,,,,:::::::::;;;;+*????*******+**+******??*??*******+;;;:::,:,,,,,,,,,,,,,,...,,,,,,,,,::::;;;;;;;;+++;;+;;;;;::::::::::::,::,:,,::::,,:,,:,,::::,,.,...,,:;;;;;;;;;:::;;;;;;:::;::;;;;;;;;;;;;++++++++++++++++++++++++++;;;;;;;;
+******++++++++++++++++++++++++;+++++;+;;;;;;;:;;;::::;;;:;;::+++++*+***?**?**?*???????**??*?*+????**?????***+++++++++*****++++++++;++++++++++++++***+;;;;;::::,,,,,,,,,,,,,:,,,,,::::::::::;;;;;;++++++++*****++++*+++*****++++;;;::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::::::;;;;;;;+;;;;;;;::::::::,,,::,::,,::,::::,:,,,,:,,,,,,...,,,,,:::;;;;;;;;;;;;;;;;;;;;;+++;+;;;;++++*++++*+++++++++++++++++++++;;;++;
+****+*+++*+++++++++++++++++;;;;;;;++++;;;;;;;::::::::;;;:;;;;;;++++++***??*****???*??********+**?%?*????****+++++++++***+++++++++;;;;++++++++++++*+++;;;+;:,,,,,,,,::,:,,,,,,,,,,::::::::::;;;;;;;;;;;;;;;++++++++++++++++;;;::::::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::::::;;;;;;;;;;;;;;:::::,,,,,,::,::,,,,,,,::,:,:,,,,,..,,,,....,,::;;;;;;;;;;;;;;+++++++++++++++++++++++*****+++++++++++++++***+++++++;
++++++++++++++++++++++++;;;;;;;++++++++;;++;:;;:::::::::;;;;;;+++++++++********?****??*+***?**+**??**??*??***++++++++++*+++++++;;;;++;+++++++++++++++;;;+;::,,,,,,,,,,,,,,,,,,,,,,:::::::::::;;;;;;;;;:;:;;;;;;;;;;;;;;;;;::::,::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::::::;;;;;;;;;;;:;;:,,,,,,,,,,::,::,,:,,,,:,,,,,,,,,,......,,..,.,;++++++*++++++++++++++++++++*+++++++*+*******++*++++***********++++++
+++++++++++++++++++++++;;;;;;:;****+;;;;+;;;:;;:::::::::;;;;++++**++++*+++**?**?*****??++**???+++***?***?**+++++;++++++++++;;;;;;+++++++++++++++++++;;;++:,:,,,,,,:,,,,,,,,,:,,,,:::;;::::::;;:::::::::::::::::::::::::::::::,,,,,,,,,,,,,,,,,,::,,,,,,,,,,,,,,,,,,,,,,,,,::::;;;;;;;;;::::::,,,,,,,,,,,::,:,,:,,,,:,,,,,,,,,..........,...:;+*????*******+++++++++++++++++++++++**+*************????*****+**+*+
++++++++++++++++++++++;;;;;;;;+**++;;+;;;;;;::::::::::::;;;+++++++++++++++++*??*******?*++**??*+++**????**+++++;;+++++++;++;+;;;;;++;;;+;;++++++++++;;:++,,:,,,,,,,,,,,,,:,,::,,::::;;::::::::::::::::::::::::::::::::::::::,,,,,,,,,,,,,,,,,,:::,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::;;;;+;::::::,,,,,,,,,,,,,:::,,,,,,:,,,,,,::,.....,.,,.....,:;+**********+++++++++++++++++++++++++************?*??????*********+
++++++++++++++++++++;++;;;;::;;:;:,:;;;;;;;::;:::::::::;;;++++**;++++++**++;++*+***+**?**++++??*++*??%?+++*+++;;;++++:;+++;++;+;;;;;;;;;;;;++++++++;;;:;;,::,,,,,:::,,,,,,,,:,,:::::;::::::::::::::::::::::::::::,,,:::::::,,,:,,,,,,,,:,,,,,,:::,,,,,,,,,,,,,,,,,,,,,,,::,,:,:::;;;:;;:::,,,,,,,,,,,,::,:,,::,,,,,,,,,,,,,,,......,,,.,...,:;++++*******+++++++++++++++++++++++++***************??????********+
++++++*++++++++;++;;++;;;;:::,,,:;;++++;;;;;;:::::::::;;;;+;+***++++++++***?++++++*+**??**+;;?*++;?%???++**++;;;;++**;++*+;;;;+;;;;;;;;;;;;;;++++;+;;;;;;:::,,,,,,::,,,,,,,:::::::::::::::::::;;;;::::::::::::::::::::,,,::,,:,,,,,,,,,:,,,,,,:::,,,,:::,,,,,,,,,,,,,,,,::::::::::;;::;;::,:,,,,,,,,,,::::::::,,,,,,:,,,,,,,.........,,,...,,:;+++++++****++++++++++++++++++++++***************???????********?+
++++++*++++;;++;;;;++::;;;;;:::;;;;;;;;;;;::::::;;;;;;;;;;;+***+***+;+*+++*****+++++++???**;+++******?*+**++;;:;;++*++;+*;;;;+;;;;;;;;;;;;;;;;;;;;;;;:;;::::::,,,,,,,,,,,,,,:::::::::::::::;::;:;;;;;:::::::::::::::::::::::,,,,,,,,,,,:,,,,,,::::,,,:::::::::,,,,,,,,,,:::::::::::::::::,,:,,:,,,,,,,,,::::::,:,,,:,,,,,,,,..........,,.,,,,,:;++++++********++++++++++++++++++**********+****?*????*********?*
++++++++++;;;;;;;;;;;,,:;;;;;;;;+;:::::::::::::::::;;;;+;+++**+++*%*;+***+**+*??**+++;+?*+*++;++****?**+*+;;;;;;+++*++++;;;;;+;;;;;+;;;;;;;;;;;;;;;;;::::::::,,,,,,,,,,,,,,,,,:::::::::::::;::;::::;;;;:::::::::;;::::::::::,,,,,,,,,,,:::::::::::,,,:::::::::,,,,,,,:,,::::::::::::::,,::,,:,,::,,:,:,::::::::::,,,,,,,,,,,............,,,,,,,;++++++************++++++++++++*******************??********?**?*
+++++++++;;;::;;;:,::,,::;;;;+;::::::::::::::::::::::;;;++++++*+*??*+++*****++*?*?+**+;+++***++++****++++;;:;++;+**+;+**;;;:;+;;;;;+;;;;;;;;;;;;;;;;;::::::,:,,,,,,,,,,,,,,,..,:::::::::::::::;:;:;;;;;;;;;;;;:::;::::::::::::,,,,,,:,,::::::::::::,,,:::::;::,,,,,,,,,,,::::::::;::::,,::,,,,,::,,:,::::,:::::::,,:,,,,,,,,.............,::,,.,:;++++*****************************************************????*
++++++++;;;;::;:::,,,,,,,,::::::::::::::::;;;;;;;:;:;;;;++++++**?%???+++*****++***+***++;*++*+*+;**+;++;;:;;+++;+*?;;**+;;::;;;;;;;;++;:;;;;;;;;;;;;;;:::;;::,,,,,,,.,,,,,,,,..,::::::::::::::::::;;;:;;;;;;;;;:::::::::::::::::,,,::::::::::::::::::::::::;;:,,,:,,,,,,,,,,:,,,:::,,,,,,:,,:::::,,,,::,:,,,,:,,,,,,,:,,,,................,,,..,::,;;+****************???************??*???**??*************???*
+++++++;;;;;;::,,,,,,,,,,,:::::::::::::::;+++++++*+;+;:+*+;++***???%?+*+++********+*?***;;++;?*;:;++;++:;;;++++++*+;;?*;;;:;;;;;:;+;;;::;;;;;;;;;;;;:::::;;::::,,,,,,,,,,,,,,,..,,::::::::::::::::;;;;;;;;:;;:;::::::::::::::::::::::::::::::::::::::::::;::;:,.,:,,,,,,,,,,,,,,,:,,,,,,,,,,,::,:,,,,,,,,,,,,,,,,:::,:,,,,................,,:,,,:;;;::+*************???*?************?????????***+**********++++
+++++++;::::::::,,,,,,:,,,:::::,,::;;;;;++*+;+*++*+:+:;?*+;;****?%%%%****+****?*+**+?%?*+;;;;++:;:;;;;;:;;+++++++++;+*++;;;;;:;;:;+;::::::::::::::;::::::;:;:::::,,,,:,,,,,,::,.,,:::::::::::::::;;;;;;;;;;;;;;;;;;;:::::::::::::::::::::::::::::::::::::;::;:,,,:.,,,,..,,,,,,,,,,,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,:::,,,,,,..................,,,;+;,,;;:;******?****??????****++*****?????*?*****+++*******++++++
;**++;;::,:;;:,,,,,::::::::::,,.,+?%%*;++**;;+*+;+;;;:+?*;;*?**S%%%?%??%++*+??*?***+*???+++;;;:;?+:::;:;;;++++++;++++++;;;;;:::;;;+;::;:::::::::;:;;:::::;:::::;;:,:::::::,.,::..,,:::::::::::::;:;;;+;;+++++;;;;;;;;;;;;::::::::::::::::::::::::::::::;::::;:,.,,.,,,,.,:,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::::,,,,...................,,;;::::;+;++****?*****????*****+++**************+*+++*******++++++
:;;;;;:::,,;;:,:;:::::::::::,::,:+**+++;+*+;;**;;++;:;*?*:;*?**#S%??%%??**+****?****??*?+;;;;;:;?*:::;;;++++++;;;+*+;+;;;;;;;::;+:;+::;:::::::::::;;;;;:::::::::::::::;;;;:,,::..,,::::::::::::;;:;;;+++++++*++++++++;;;;;:::::;;;:::::::::::::::::::::::::::::,,,,,,,,.,:,.:,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::::,,,....................,,::;++;;+;+*************?*******+***********++++++++****?*******+
,:,,:::::::::::;;;;:;;;;:::::;+++**+++;;+*;+++*+;+;;;**?*;+*+*?%S%%%%%**?+;***+%%**+*%??*+;;;;;;+;,,::++;+++;;+;;+*+;;;;;;;;;:,:;;;;:::::::::;;;::;;;;::,:;::::::::::;;;;;;,,::,.,:::,:::::::::::;;;++;+***+*+++++++++;;;;;;:::;;;::::::::::::::::::::::::;;;;:,.,,,,,,,,:,.:,.,,,,,,,,::,,,,,,,,,,,,,,,,,,,,,,.,,,,:::,,,.....................,,:;+;++;;+***************??**************+*+++++++*??**????????*
:::;;,,;;::,.,;;;;:;:;:::;;::;++**++++;+*+;++++;+*;:;*+*++**+*??%SS%%?+*?*;+?*;%%?*++?%?*+;+;;;;;;:,;;++;;++;+*+;*?;;+;;+;;;;;,:;;;:;;:;:::::;+;;:;;;;:,,:;,:::::::::;;;:;:,,:,..,,::,::::::::::::;;+;;+***+++++++++++;;;;;;::;:::::::::::::::::::::::::::;;;:;,.,,,,,,,,,,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::,,,.,...................,,,:,,;+;;***************?*****************+++++**?????????????*
:;++;,,;::,,..,;::::::::::::;++***++*+:++++;+++;++:;++*+;*%****?%#%%%%+*??*+*?*?%%?*;*?***;+++;;:::;;;++;+;+++*;;**;;+;;++::;;::;;:::;:;;;:::;;;;:;;;;:,,:;:::::::::;:::::,,,,,,,,,:,:::,::::::::::;;;;;++++++;;;;;;;;;:;:::::::::::::::::::::::::::::::;:;;::;,.,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,....................,,,,;;;;+*++*********************?????*********??????????????%*
:;;;:..,:,,,,,.,,,,,,,,,,,:;******++*;;*++++++++++;+**?+;?%*????%SS%%?++??+*??**??%?++**??;++;;;:::;;;;;;*+++++;;;;;;;;+*;,:;;;:;;::::;;;;;::;:;;:;;;;::::::::::::::;:::::,.,,,,,,,:::::,::::::::::;;;;;;;;;;;;;;;;;;:::::::::::::::::::::::,:::::::::::::;::::,.,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.,.,,,,.....................,,,;+;:;++;+**++****************??????????*????????%%?????%??%*
:,,,,,..,,,,,,.....,...,;+*++******+++**+;;+;;++++++*??++??*?*??%%SS?*++??+*?S%**??%*++*??++;;;;:::;+;;++;;+;++++:++;;;++:::;;;;;:,,,:;+;;;:::::::;;:;::::::::::::::;::::,,,,,,,,,,,::::,::;::::::::;;;;;;;;;;;;;::::::::::::::::,:::::::,:,,:,::::::::,:::::::,..,,,,,,...,..,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.......,,.......................:++:.:+;;++*+++***************????????????????????????%??????
...........,.......,,,:+****+*******+;+++;++;;+**++****++*******??%???++??+*?%?**????****??;;++::::;+;+++:;+++;+;;++;;;;::;;;;;;;:,,:;;;;;;:::;;:;;;:;::::;::,::,:::;::::,,,,,,,.,,,::,:,,:;::,,::::;;:;;;:;:;:::::::::::::::::,::::,,:,:::,,,,,,,,,:,,,::::,::,..,,,,,,,,...,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,...........,.......................,:,.,++:;;;;;++++*************?????%?????????????????????????
....,......,,,,,,,:+********++****??*+;+***+;;;*++*++**++**??***????%%*+??**%%**??????*?*?*;??;::::;+;+;+;++++;+;+;;;;+;:::::;;;;:,::;;;;;;;::;;:;;::;::::;;::::,::::::::,,,,,,,,,,,,,,:::::::,,:::::::::::::::::::::::,::::,,:,,,:::,,,::,,,,,,,,,,,,,,,,,,,:,,,.,,,,,,,,....,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.......................................:+++;;;;+++++*************????????????????????????%??????
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.,,,,,,,.,,:,:;+;+*+******???****???????*+++;;;+++****++****??**??%%%%**?*??%??%++**?????*++%?:;:::;+++;;;++;;+;;:;;:;;:::::;;;:;::;;;;;;;;::::;::;::::::::::::::::::::,::,,,,,,,,,,,,,::::::,:::::::::::::::::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,..........................................,,,,,,,:;;+*********************??*************?????????
..,,,,.,,,:;++***********************????**++;;;+******+*?*?????*?%%%?*??*?%??%*+***%%???*++%***:::;++++;+++;;++::;::::::::;;;;:::;;;;;;;;::::::::;;::;:;::::::::::,:::,::,,,:,,,,,,,,,:::::::::::::::::::::::;:,,:,,,,,,,,,,,:::,,:,,,,,,,,,,,,,,,,,,,,,,,,,,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.,,,..............................................,,,,::::::+********************************????????%?
..,,,,:;+*******************************??*++++;+*******+**?%?????%%?????????%?++*??%%?%?*+*?*?*;::;;+++;+++;;++;:;::::::::;;;;::;;;:;;;::;;::::::;;;;;;;;:::::::::::::::;:,,:,,,,,,,,::::::::,:::::::::::::::;:,,:,,,,,,,,::,:::,,,:,,,,,,,,,,,,,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.....................................................,,:;++++;;+*********************************????%??*
.,:;;+**********************************??*****++**++***+*?%%%%??????%?*?????%?*+*%??%%%%*+?**?*+;;;+*+;;+++;;++;;;::;:,:;;;;+;;;;;;::;:;;;;;;;;::;;;;;;;;;;:::::::::,:::::::,,,:,,,,,:::::::::::::::::::::::::::::::,,,:,,,:,:::,,,:,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,...,...................................................,:++;,;**+******+++*********************??????%?%*
,;;;++*******************************************++++++++**???%????????*****??**+?%??%%??++**?**++;;+*++;;+;;;;;;;::::::::::;;:::;;;::::;;;;:;;;:::;:;;;;;::::::::,::,:,:::,,,,,,,,,,:,::::,::,,,::::::::::::::::::,,,,,,,,,,,::,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,........................................................,:;;;;;;;+++++++++++++********++++++*****???????*

+**************+++++**+++++++++++++++++++++++++++++++++****************+;;::::::::::::::;::;::,,,,,,:::;;::,,.......................................................................................................................................................,,,,,,:::::::::::::::::::::::::::::::::::::::::,,,,,,,,,,,,,,,,,,,,,,:::::,,,,,,,,,,,..,....................................................
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*????*?*******************++++++++++++******************+++*+++*+++++*+++;:,,......................................................................................,,,,,::::::::::;;;:::;++;;;;;;+++++++++++++++++***********************????***????????????%%????%??????????????????***?%S#SS%%%%%%?%??%?????????????????**************************************************************++;:,,..................
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**************************++++++++++++*****++*****+;;,.................................................................,,,,,,:::::::;;;;+++++++++++++++++++++********++++++++;;;;;;;;:;;:;;;:;;;:::::::;;;::;;;;;;;;;;;:;::::::::::::::::;;;;;;;;;;;;++++**???????????????????**???????????****?%%%%%%%%%%%%%%%%%%%%%%????????????????******************************************************************+++++;;:
***************************+++++++++++++*++***+;:,.....................................................,,....,,,,:,:,,::::::;;;;+;+;+++++++********++****************+++++++;;;;;;;;;::::;:::::::::::::;;;;;:::;:::::::;;:::::::::::::::::;;;;+++*+++****????????????????????***???????????*****??%%%%%%%%%%%%%%%%%%%%??????????????????********************************************************************+**;
****************************+++++++++++++++;,.................................................,,,,...,::;::::::;;;;;;;;;+;+;;+++++++++++++*******?*********?***?*?***+++++;+;++;;;;;::::::::::::::::::::;:::::::::::::;;:::,::::::::::::;;+++***********??**????**??????????***?????**????*??*****??%%%%%%%%%%%%%%%%%????????????????????*************************************************?********************+
*?**?**********************+++++++++++++,...................................................,;+++;;:,,:;;;;;+;;+++++++++++++++*+**+++++++++********?********????**************+++;;;;;::::::::::::::::::;:::::::::::::;;::,,,,::::::;;;+++*************************??%%%????****?????***??????******?%%%%%%%%%%%%%%%%%%???????????????????????********************************************???????**************+
****************************+++++++;:,..............................................,,,,,:::::;;+++++++++++++++++++++++****?*********+++++********************???****************+++++;;;;;;::::::::::::;:::::::::::::;;::,,,,:::;;;+++++*************************??%%%%???******?????**??***********?%%%%%%%%%%%%%%%%???????????????????????????********************************************????***?*********?+
+************************++*++;;:............................................,,,:::;;;;;;;;;++++++++**++*********+++******????*********++*************????******???******?**********++++;+;;;:::::::::::;:::::::::::::;;::,,,,::;;;;;++++++++;;;;;++++***************??*??****+****????*???************??%%%?%%%%%%%%%%%??????????????????????????******************************************?*?????**?*??*?****+
+********************+++***+:..........................................,.,,,::;;;;;++++++***+*****+**++++*******++++*?????%????????**+++++**********?????????*????******??***?******++;;;;;;;;;;::::::::::::::::::::::;;::,,::::::;;::::::::::::::::::;;;;+++++++++++++*******+****?????*????**??*******??%%%%?%%%%%%%%??????????????????????????????***************************************?**?**?****???*?****
+********************+*;:,.........................................,,,::::;;+++++********??????????************+;***??????%????%??***+++++*+****************?******++++++;;;;;;;++++++++++;;;;;;::::::::;::::::::::::;;;;:::::::::::::::::::::::::::::::::::::::;;;;++++******++****????*???????********??%%%%%?%%%%%%????????????????????????????????**************************************?*?**?*********??*?*
+*****************+;:,..................................,,,,:::::::;;;;;;;+++++*?????%%%%%%%%%??%??**********+;++**??*??????%%%?*?***+*+++++*******+++++++++++++++;;;::::::::::::::::::;;;;;;;;;::::::::::::::::::::::;;;;;::::::::::::::::::::::::::::::::::::::::;;++++*****+**?***?******??????????***????%%%%%%%?%%??????????????????????????????????****?******************************?*?*??********?*?*?*
*?*************+:...............................,,,::;;;+;;+;;;;++++++++++++****?%%%S%%%%SS%?????*++***+***+;+****????**??%??%?*****++*++*++*++*++++++++++++;;;;;;;;;;:::::::::::::::::::::::;;;;:::;;:::::::::::::::::;;++;;;;;:;:;:::::::::::::::::::::::::::::::::;;++++***++++***********?????????????????%%%%%%%%%%????????????????????????????????????????******************************?***************?*
*??********;:,...........................,,,:::;;;++++******+++*++**+****???%%%%%%%%%%?%S%%??****++**++**++;+***???????????????******+*++*++*++++++++++++;;;;;::;;;;:::::::::::::::::::::::::;;;;;;;;;;::::::::::::::::::;;++++;;;;;:::::::::;::;::::,,,,,,,:::::::::::;++++**+++++*?**********???????????????%%%%%%%%%??????????????????????????????????????????*****************************?***************?*
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*?*+,..................,,,,:;+++;;;;;++++++++***????????????%%%%%%%%%%%%SSSSSSSSSS%%%%SS%?****++**+++++++*??***?????????*?????*****+++++++++++++*+**++++;;;;:::::,,,,,,,,::;;;+;;:::::::::::;;;;++++++;;;;;;;;;;;;;:::;:;;;;+++;;;;;;++**?????%SS?++*????%%?*+;;::,::::;;+++++++++****?????*******???????????*?%%%%%%%%%%%???????????????????????????????????????????************?*?**************************?*
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....,::;:;;++++++++*******?????????????????????????????%%%%%%%%%%%%S%SSSSSSSSS%SS%%%??????++*++;:;++++****???%%%%%%%?*************+++++*+++***+++++;;::;;+*?%%%??***++++;;;++******************?***+++*********+*+++++++;+;;+++++;;;;;;+++++++;+;;::::::::::::;;;;+++;::::;++++++*++*****?***************??????????%?%%%%%%%??????????????????????????????????????????????????????????**??********************?*
:;;;;;++++++++*******?????????????????????????????????%%%%%%%%%%%%SSS%SSSSSSS%S#S%%????%%?+*+;;;+++++*?**????%%%%%?**??****++*****+++++*++++**+++;;;:::;************++++++++++++++++++++++++++********************++++++;;;;;;++++++++;;;;;;;:::::::::::::::::::;;;;;++;;;;;++++++++++***?????**?*********?*???????%%%%%%%%%??%????????????????????????????????????????????????????????????*******?***********?*
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+*****?*???????????????????????????????????????????%%%%%%%%%%%%%SSSSSSSSSSS%S#S%%%?*%S%%*+;;;;;;;;+??***??????%???????**********+++++++++++++**+*+*++*++++;;;;;;;::::::::::::::::::::;;;;+++++*+******+*++++++++++++;;;;;;;;;;;;:;;;;;;::::;;;;;;;:::,,,,,,,:::::::::::;;;;;;++++++++++++************************?%%%%%%%%%%%???%?????????????????????????????????????????????????????????????????????*****?****
+*?*???????????????????????????????????????????%%%%%%%%%%%%%%%%SSSSSSSSSSSS%#S%%S%*?%?*+++;;;;;;;+*?***???*??????????????*?*?****++++++*+++++++++++;;;++;;;;;:::::::::::::::::::::::;;;;+++++++++****+**+*+++++++;;;;;;;;;;;;;;;::::;;;:;:::::::::::::::,,,,,,::::::::::::;;;+++++;++***************************+?%%%%%?%%%%%%%%%?????????????????????????????????????????????????????????????????????????*?**?*
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*%%?%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%SSSSSSSSSS#SS#########S#S%%SSSS%SS%??*++?*++*+****??*******?*??????%SS%S?????????????%??????????***++++++;;;;;;;;;;;;;;;;;;;::::::::::::::::::::;;++**+;;;;;:;;;;;;;;;;;::::::::::::,:,,,::::::::::::::::::::::::,,,,,,,,,,,,,,,:::::::::::::::;;;::::::;:::;;;;;;;;;;++???***++++++***????%%?%%%%%?????????????????????????????????????????????????????????????????????????*;
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*?%%%?%%%%%%%%%%SSSSSSSSSSSSSSSSSSSS%SSS#S#######S##SS##SSSSS#S%%%%?*++*+++*??????*******??***???%%%%??%???????????????????????????*****+++;;;;;;;;;;;;;;;;;;:::::::,,,::::::::;:;;:;:::::;;;;;;;;;;;;;:::::::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::,,,,,,,,,,:::::::::::::::::::::::::;;;;;+;;;;;;;;;;++*?*++++++++++**??????????????????????????????????????????????????????????????????*+;:,.....,,,,,,,,,
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*???%%%%%%%%%%%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS##SS##%S%??%%%?*++*****????*********?*???????%%%???*???*******???**??*???????*********+;;;;;;;;;;;;::;;:;;:::::::,:::::::::::::::::::::::::::::::::;:;;;;::::::::::,:,,::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::::::::,::::::::::::;;;;;;;+;::::;;;;+++*++++++++++++**??*????????????????????????????????????????????????*?????**+::,:::::::::::::,,,,,::,
+???%%%%%%%%%%%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS#S%S#%???%?S?%*;;+*****???*******???????????%%%%??????*****???*****????????***********+++;;;;;;;;;;;;;;;;;;;:::::,:::::::::::::::::::::::::::::::::::;;;::::::::::,,,,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::::::::,:::::::::::;;;;;;;;;;::::;;;;++++*++++++++++++*??*????????????????????????????????????????????????????*+;::,::;;;;;::::::::::::::::
++??%%%%%%%%%%%%SSSSSSSSSSSSSSSSSSSSSSS#SSSSSSSSSSSSS%S#S*?%?S%?%*;+*+****???*****?*??????*?????%%%%?????**************???????*************+++;;;;;;;;;;;;;;;;;;:::::::::::,::::::::::::::::::::::::::::::::::::::::::,,,,,,,,,,,,,,::,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::::::::,::::::;;::::;;;;;;;;::::::;+;;+++*+++++;+++++++*??????????????????????????????????????????*???????*+;::;:;;;;;;;;;;;;;:;:::;;::::;:
+++???%?%%%%%%%SSSS#SSSSSS##SSSSSSSS##SSSSSSS###SS#SS%S?*%#%?????++****?*??*****??*????????*????%%%???%?************?***********************+++++;;;;;;;;;;;;;;;:::::::,,::::;:::::::::::::,,::::::::::::::::::::::,,:,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::::::::,:::::;;::;:::;;;;;;::::::::;++;+++++++;;;;++++++????????????????%??????????????????????**?????*;;:,,,:;;;++;+;;;;;;;;;;;;;;:;;;::;:
+++*++*?%%%%%%%%SSS#SSSSSSSS#############SSS###SSS%%S???S#%?%%%?++*+**????*****??**????????????%%%%?%?%**??***+++***********************?****+++++;;;;;;;;;;;;;:::::::,,,:;;::::,:,,,,,,,,,,,,,:,,,,,:::::::::::::,,,,,,,,,,,,,:,:,,,,,,,,,,,,,,,,.,,,,,,,,,,,,,,,,:::::::::,::;;:;;:::::::;;;;;:::::::::;;;;++++;;;;;;;++****?????????????????????????????????????????+;::,.,:;;+++++++;++;;;;;;;;+++;;;;;;;;;:
++++*+**???%%%%%%SSSSS##SSSS##############SS###SSS%%??%%SS?????+;++***???**??*****?????????????%%?%%???****************?**********?************+++++;;;;;;;;;;::::::::,,,::::,,,,,,,,,,,,,,,,,,,,,,,,,,,::::::::::,,,,,,,,,,,,:::::,,,,,,,,,,,........,,,,,,,,,,,,,,::::,::::::;::;;::;::::::::::::::::::;;;;;+++;;;;+;;+***??????????????????????????????????????**+:.,,,:;;+++++;;;;++;++;;;;;;;+++++;;;;;;;;:
+*+**+*****?%%%%%%SSSSS#S##SSS############S##S%%%S%%%%S%%?????++++*?**??**????**???%?*????????%%%??%?*???*++*++++*****************?*?***??*******++++;;;;;;;;::::::,,:::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::::,,,,,,,,,,,,,,::::::,,,,,,,,,...........,,,,,,,,,,,,::::,::::::::::;;:::::::::::::::::::;;;;+++++;;;;;+;****????????????*??????????????????????*++:,,,.;+***+++++;;+;;;+++;;;+;;;;++++++;;;;;:::
+*******++++*?%%%%SSSSSS###SSSS############S%%???%SS%S%%%%%??*++****%?++*%?*??*????%??%?????%%%%%%?????**+;*++*******++++*********?*?*****?****++++++;;;;;;;::::::,,,::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::,,,,,,,,,,,,,,,,,,,:::,,,,,,,,...........,,,,,,,,,,,:::::,:::::;::::::::;;;;;;;:::::::::::::;;;;+;;;;;;++**+***???????????????????????????%?*+;:.,:++**+****+++++++;;;++++++;;;;;;;++++++++;;;;::
+********+++++**??%%SSSSS#SSSSSS#########SS%%%???%S%%S%%%%??*++***??*+*???*??*????%%??%?????%%%%%%????*++++*++****++++++*+***********?****?***+**+++++++;;;;:::::,,,,::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::::,,,,,,..............,,,,,,,,,,,:::::,,:::;;;:;;;;;;;;;;;;;::::::::::::::::;;;::;;;+++*****????????????????????????*++;:,;;;+***********+++++++;;++;;;;;;;;;;;;+++++++++++;;:
+********++++;;;+**??%%SSSSSSSSSS########S%%%??SS%?%%??%%?*++****??*+*?**????**??%%%??%????%%%%%????****+**++****++++++++++**************?*******+++++++;;;;:::::,,,,,:::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:,,,,,:,:::::::::::::::::::,,,,,,..,..,........,,,,,,,,,,,,:::,::::::;;;;;;;;;;;;;;;;;:::::::::::::,,::::::;;++;++**?????????????????????*+::,.,;+*?????****+*****++++;++;;;;;;;;:::;;;;++*********+++;
+**********++;;;;++;+**??%%%%%SSSSSSSS#SSS%%??%SS??S%*????++*****?*+%%**????????%%%S??%?????%%%%??******++*+*++++++++++++************************+++++++;;;:::::,,,,,,,,,,,,,,,,,,,,,,,,:::::::,,,,,,,,,,,:,,,::::;::::::::;;::::::,,,,,,..,...,,........,...,,,,,::,,:::::::::;;;;;;;;;;;;;;;:::::::::::::::,,:::::;;;+;+++*?????????%???%??*+;:,.,:+++*?%????*******++***++++;;;;;;;;;;:::::;;;++*********+*+;
+**********+++;+++++;++++*????%%%%%SS%%%%%%???%S??%?+??**++*****?++%%?**????%%?*%%%%??????%%%%*%?******+++***++;**++**********?*******************+**++++;;;;:::::,,,,,,,,,,,,,,,,,,,:;::;:,,,,,,:,,,.,,,.,,,,,,:;,,:,,,,,::,....,,,...,:,,,...,,..,,,........,,,,:::::::::::::::;;;;:;;;;;;;;;;;:;;;:::::::::,:::,,;;;;++++*?????%%%??***;:..,,,..:;+*????????******+++*++++++;;;;;;;;:;:::::;;;;++********+**;
+****+++*+++++++++++++++;;++++*??%%%%%%%%S%*?%%%%?*+?%?*+*+****?+*%%??*?*???????%%%????????%%?*??*****++++***+*+*++++++*****???********??*******+****+++;;;;::::::,,,,,,,,,,,,,,,,,,:;++?SS?;;;;*?+,:::;:,,,:,,,:;:,::::,,::,,,,,:*%???**::,,..,,,.,,,,.......,,:::::::::::::::::;;;;::;;;;;;;;;;;;;;;:::::::,,:::,:;;;++;+++*??*?*+;::,.....,;++++*+;+***????********++++++;;;;;;;;;;;;;::::::;;;;++++++++*+++;
+****++++++++++++++++++++++++++++++**????%%???%%?*???***+****??++???????**?????%S%S??????%?%%++******+*********++++++++****+?%++***************++***+++;;;;;::::::,,,,,,,,,,,,,.,,,,:::;+?%#SSSS##S?*%?%?*******+********++++++**?%#@#S%*;;:,,.,,,....,.......,,,,:::::;::;;;;;;;;;;;;;;;;;;;;;;;;;:::::::::,,,,:::::::;++;+;+++;:,,,:;::,.,;**????%?+++**????******+++++++;;;;;;;;;;;;;:::::;;;;;;++++++++++++;
+*****+++++++++++++++++++++++++++++++++++*?***??*?***+*++***??++*??***??*??????%%S%???????%?+++***?*****???*??++?*++++****?***+*************+++*+*++++;+;:;;:::::,,,,,,,,,,,,,,,,,,,::::;;+*%SS#SSSS%****++++++++++*********+***?%SSSS*+;;::::,,,,....,......,,,.,,::::::;;;;;;;;;;;;;;;;;;;;;;;;:;::::::::,,:,,:::::,,::;;;:::,,,,:;;;+;;:+???*?******+********++++++++;;;;;;;;;;;;;;;;;:;;;;;;;;;;;;;;;;;;;+;;
+*****+++++++++++++++++++++++++*+***+++;;;+;+;++******++**++?+;*??***??*????%?*?%%%???%%%?%*+**??????****??*%?++?*++++****+*?*+*********+**+++++**++++;+;:;::::::,,,,,,,,,,,,,,,,,,::::::::;+*?%%SSS%***+++++++++++*******+++*??%SS%**;;::::,,,,,:,..........,,,.,,,::::::;;;;;;;;;;;;;;;;;;:::::::::::::::,,:,,:::::,,:::::::::,...:::;;+++********+++++++;;::;:;;;;;;;;;;;;;;;;;;;:::;;:;;;;;;;;;;;;;;;;;;;;;:
+******++++++++++++++++++++++++*******++++;;+;:::;+;;+;+++;+++**?****???????%?**%?%???%??%%?+**?*%%?++*?***??*++**++++**+***+++++++***++++***+++*++++;;;;;::::::,,,,,,,,,,,,,,,,,,:::::::::;;;;+*?%%?******+++++++**???**????????*+;;;;::::,,,,,,:,.,,,,,,..,,,,,,,,,:::::;;;;;;;++++;;;;;;;::::::::,,:::,::,,:,:::::,,,,,::::;;;,...,,;+++++++++;;;;;;;;;::::::::::;;:;;;::;;;;;;;;;;;;;;;;;;+;+;;;;;;;;;;;;;;:
+******+++++++++++++++++++++++;+******++;;;++;;;;::::;;;+;;+++***+*****???????**%???*????%?**+*????**?????****++**++++*****++++*++++++++++++++++++*++;;;;;;::::,,,,,,,,,,,,,,,,,,,:::::::::;;;;;;;++*++++??*********++*???****+;;;;:;::::,,,,,,,,,,,,,,,,,,,,..,,,,,,,,::::;;;;;;;;;;;;;;;;;:::::,,,,,::,,::,,:,:::::,,:,,:,,,,,,,,....,;;;;;;;;;::::::;:::;;::::;;;;;;;;;;;;;+++++++++++++++++++++++++++;;;;;;;
+******++++++++++++++++++++++;;;+**++++;;;;;;;:;;;::::;;;:;;;;+++++******?******???*??***?****+????*?????****+++++++++****++++++;;;;;++;;++++++++****;;;;+;:,,,,,,.,:,:,,,,,,,,,,,::::::::::;;;;;;;;;;;;;+****+++++++++++*+;;::::::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::;:;;;;;;;;;;;;;;:::::,,,,,,;:,::,,,,,::::,,,,:,,,,..,,......,:;;;;;;;;;;;:;;;;;;;;;;;++++;++;;;+++++++*+++++++++++++++++++++++;+;;+;
+******++++++++++++++++++++++++;;;;;+++;;:;;++;:::::::;;;:;;;;;;++++++**????****???*%?+*******+**%%*???******+++++++++**++++++++;+;;;;++++**++*+**+++;;+;;:,,,,,,,,,:,,,,,,,:,,,,,::::::::::;;;;;;;;;::::::;;;;;;;;++++;;;:::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::;;;;;;;;;;;;:;;::,,,,,,,,,::,::,,,,,,,:,,:,,,,,,,...,,,,.....,++++;;;;;;;;;;++++++++++++++++++++++*******++++++++++++++++**+++++++;
++++*++++++++++++++++++++;;;;;+++++++++;;;;;;;;,:::::::;;;;;;++++++*++****?***??****??*****??*++*??*****?****+++;;+++*++++++++;;;;+++++++;++++++++++;;;+;:,:,,,,,::,,,,,,,,,:,,,::::::::::::;;:::::::::::::::::::::::::::::::,:,,,,,,,,,,,,,,,:::,,,,:,,,,,,,,,,,,,,,,,,,,,,::;;;;;;+;:;;::::,,,,,,,,,,,::::,,:,,,,:,,,,,,,,,,......,:,,...,+*+*+++++++++++++++++++++++++++++++********+***+++************+*++*+
++++++++++++++++++++++++;;;;;;;****++;;;++;;::;::::::::;;;;;++++*+++++++++**????******?++**???+++****????*++++;;;++++++++++;;;;;;+++++++;+++++++++++;;;+;,,,,,,,,,:,,,,,,,,,:,,:::::;;:::::::::::::::::::::::::::::::::::::::,,,,,,,,,,,,,,,,,:::,,,,,,,,,,,,,,,,,,,,,,,,,,,:::;;;;:;+;::::,:,,,,,,,,,:,,,,,,,:,,,,,:,,,,,,:,....,.....,,...:+****??******+++++++++++++++++++++++++**************?**?*******+**+
++++++++++++++++++++++;+;;;;;;+***+:;;;;;;;;:::::::::::;;;;+++++*+++++++++;+*?**********+++*??**++*?%%?***++;+;;;++;;;;++;;++;;;;+++;;;;;++++++++;+;;;;+:,::,,,,,,:,,,,,,,,,,,,:::::;:::::::::::::::::::::::::::::::::::::::,,,,,,,,,,,,,,,,,:,::,,,,,,,,,,,,,,,,,,,,,,::::::::::;;::;;:::,,,,,,,,,,,,::,:,,::,,,,,,,,,,,,,:,.....,,,,.....,,,;***********++++++++++++++++++++++++***********?***?????*********+
++++++++++++++++++++++;+;;;;;;;;;:::;;;;;;;::;;::::::::;;;++++*+++++++***+++;+*+***+*?*?*+;++??*++?%???+;**+++;;;++*+;;++;;+;;+;;;;;;;;;;;;+;++;;+;;;;:;:,::,,,,,,::,,,,,,::,:::::::::::::::::;:;;:::::::::::::::::,,,,,:,::::,,,,,,,,,,,,,,,::::,,,::,:,,,,,,,,,,,,,,,:::::::::::;:::;;:,,,,,,,,,,,,,:::::::,,,,,:,,,,,,,,,,.......,,,:,...,,;++*********++++++++++++++++++++++++*************?*?????*********+
+++++++++++++++++++;;+;;;;;::,,:::;++++++;;;;;;:::::;;;;;;;;+*?++++;++*+***?++;+++****?*?++;+**+++????*;+**++;;;++*?++++*+;:;+;;;;;;;;;;;;;+++++;;;;::;:::::,,,,,,,,,,,,,,,:::::::::::::::;:::;;;;;::::::::::::::::::,,::::::,,,,,,,,,,,,,,,,:::::,,:::::::::,,,,,,,,,,,:::::::::::::::::,,:,,,,,,,:,,,:::::::,::,,,,,,,,,,,..........,,,,,,,,:;;++++++****+++++++++++++++++++++++*************???????********?+
++++++*+++++;;++;;;++;::;;;;:::::;;;;;;;;;::::::;;;;;;;;;;+++*++***+;+*++***+?*+++++++??***++++***********;;;;:;;+**;+;++;;;++;:;;;;;;;;;;;;;;;;;;;;;::;::::::,,,,,,,,,,,,.,::::::::::::::::::;:::;;;;:;:::;:::;;;:::::::::::,,,,,,,,,::,::::::::,:,,:::::::::,,,,,,,::,:::::::::::::::,,,::,::,,,,:::,::::::::::,,,,,,,,,,,..............,,,,,:;++++++******++++++++++++++++++***************???????********??*
++++++*++++;;;;;;;;+;,,:;;;;;;;+;;;;::::::::::::;::;;;;;++++**+++*%*;+***+**+*??**+++;+**+**++++**+*?**++;;::;;;+*++++++;:;;++;;;;;+;;;;;;;;;;;:;;::::::::::,,,,,,,,,,,,,,,..,::::::::::::::::;:::;;;;;;;:;;:;;::;:::::::::::,,,,,,,,,:::::::::::::,,::::::::,,,,,,,,,:,,,:,:::,:;::::,,:::,,,:::,,,:::::,::::::::,,,,,,,,,,,.............,,....,++*+++***********+++++++++++*******************????********???*
+++++++++++;:;;;;;,::,,:;;;:;;;;::::::::::::::::::::;:;;++++++*+*%?*++++*****+****+**+;++*+**++++****+++;;::+++++**;;*?+;;::;;;;;;;;;;;;;;;;;;;;;;;;;::::;:,:,,,,,,,,,,,,,,,..,:::::::;:::::::::::;;:;;;;;;;;;::::::::::::::::::,,::::::::::::::::::::::::::::,,:,,,,,,,,,,,,,,,:::,,,,,,:,,::,:,,,,:::::,,,,,:,,,,,::,,,,..,.............,::,...,,;++*****************************************************?????
+++++++++;;;:,;;::,,,,,,,:::;:::::::::::::;;;;;;;;;;;;;;++++++**???%?++++*********+**+*;;++++**+;+*+;++:::;;+++;+**:;?+:;;:;;;;;;;;;+;;:;;;;;;;;;;;;;::::;;:::,,,:,,,,,,,,,,,..,,,::::::::::::::::;;;;;;;;;;;;::::::::::::::::::::,::;::::::::::::::::::::;::,,,::,,,,,,,,,,,,,,::,,,,,,,,,,::,,:,,,,,,,,,,,,,,,,,,,,:,,,,.................,::,,::,:,:+**************?****************????******************???*
++++++++;;;;;;::,,,,,,,,,,,::::::::::::,:;+++++++**;;;:+*+;;+***???%%++++*********+*?***;:++;?*::;+++;;:;:++++++*+;;+?+;;;;;;;;::;+;:,::;::::;;;;::::::::;;;::::,,,,,,,,,.,,:,..,,::::::::::::::;:;;;;;;;;:;;:::;;::;::::::::::::::::::::::::::::::::::::::::,,,:,,,,,..,:,,,,,,,,,,,,,,,,,,,,:,:,,,,,,,,,,,,,,,,:::,:,,,,..................,,,,+++;:::;***********?????***********??????????*****++******++++++
+++++;++;::;::::,,,,,::,,,,::::,,::;;;;;;++*;;*+++*:;;;??+;;****?SS%%***+++***?*+**+*%?**;;;++;;+:;;:::;:;*+++++;;+++++;;;;;::;;:++:::::::::::::::::::::;::::::;:,,,:::::,,,::,..,:,:::::::::::;;:;;;;;;;+;++;;;;;;;;;;;;:::::::::::::::::::::::::::::;;:::;::,,,,.,,,,.,:,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::::::,,,....................,:;;:,:;;:;+********?*???*?**********???????******++*******+++++++
;+*+++;;::::;::,,,,::::::::::,,.,;*?S?+++**+:;*+;++;;:+?*+:+?**SS%%%???%*****?*?****+*??*;+;;;:;%*:::;;;;++++;++;+*;++;;;;;;:::;+;+;::;:::::::::;;;;;;;::::::::;:::::;;;;;,.,:,..,,,::::::::::;;;:;;;+++**+**++++++++;;;;;:::::;;::::::::::::::::::::::::::::;,,,.,,,,,.,:,,:,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,::::,,,,....................,::;;++++;;+*******?*?*?*?***+**+************+++++++******+++++++
;;++++::,,,:;:::::::::::::;::,:::;*++++;;**:;+*+;++;;;*?*;:+?**%#S??%%???*+****??*+*????+;;;;;:;*+,:::;;;+++;;;;;+*+;;;;;;;;:::;+:;;::;:::::::;;:;;;;;::,;:::::::;:::;+;;+:,,:,..,::,:::::::::::::;;++;+*****++++++++;;;;;;;:::;;;::::::::::::;:::::::::::;;;::,.,,,,,,.,,.,:,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,.,,,:::::,,,,....................,:;++++++;+**********?*******************+++++++++****???******+
,,,,:;::::::;:::;::::;;:::::::;++**++++:+*+;++*+;++;;****;+*+*??SSS?%%*+?*;+?+*?%**+*?%?*+;;;;;:;;,::;++;;++;+++;*?;;;;;+;;;;:::;;;:;;:;:::::;;;::;;;;:,,;::::::::::;;;;;;:,,:,.,,::,,::::::::::::;;+;;+****++++++++++;;;;;;:::::;:::::::::::::::::::::;:;;:;;:,.,:,,,,,,,,.,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,.,,,,::::,,,,.....................,.::,:++;+********?*********************++++++++*?????*???????*
::::::,:;::,..:;;:;;:;;;::;;::+++**+++;;*+;;++*;;*;:;++*;+**+**?%S#%?%*+??++*?+*%%?*;*%?*+;++++:;:::;;++;;;+;**;;??:;+;;+;::;:::;;;:;;:::;;::;;;::;;;;:,,;::::::::::::::::,,,,,,,,,,,::::::::::::::;;;;;+++++++;;;+++;;;:::::::::::::::::::::::::::::::::;:;:::,.,:,,,.,,,,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,....................,,:,,:;+;+*+************??*??*************+++++**?????????????*
:;:;+;,:;::,..,;:::::::::::::+++**++*+:;*++;+++;+*;:++*+;*%**?*?%#S%%%*+*?*++%?+%%%?;+?**?++++;;:::;;;+;;++++*+:++;;;;;+*;,:;;:;;::::::;+;;:::::;:;;;;:,:::::::::::;::::::,.,,,,,,,:::::,::::;:::::;;;;;;;;;;;;;;;;;;;:::::::::::::::::::::::::::::::::::::::::,.,::,,,,,,,..,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,..,,..,,,,,.....................,,:++::;++;********************?**???********????????????????*
:;;+;,.,:,,,,..,,,,,,,,:,,,:+******+*+;++;+;;+++++;++*?*;*%**???%SSS%?*+*?*+?%?+???%*+**??;++::;:::;+;;;;*;++;+++;;;;:;++;,:;+;;;:,:::+;;;;;;::;;:;;;;:::::::::::::;::::,,,,,,,.,,,::::,,:;;:::::::;;;;;;;;;;;;;;;;;:::::::::::::::::::::::,,::::::::::::::::::,..,,,,,,...,.,,,,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.........,........................:*+,,;+;;*+******************?????????*?????????%%???%%?%??*
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,,.........,........,,,:+*?**+******++;+*+;++;;**;+*+?*++***??**???%%??+*?*+*%%?*????**?*?%;;*+::;:;+;++;:++++:+;+*+;;+;::;;;;;;;:,::;;;:;:;::;;;;;:;::::;;::,::,::::,::,,,,,,,,,,,,,,::,,:::::,,::::::::::::::::::::::,::::,,,,:,,,,,,,:,,,,,,,,,,,,,,,,:,,,:,,,.,,,,,,,,....,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,........................................:++++;;;++++*************????%%??????????????????%%?????
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..,,,,,:;+++****************************??**++++;+*****+++*??????*?%???***????%*++*??%%%%?+***??+;::;+++;+++;;**;:;::;:,:::;;;;:;;;:::;:::;;::;:::;;;;;;;;:::::::::;,,:::::,,,,,,,,,,,:::::,:::,::::::::::::::;:,::::,,,:,,:,,:;:,,:,,,,,,,,,,,,,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,........................................................,;++++:;+++*+++++++**********************?????%??*
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**************************+++++++++++*******************+;;::,,.......................................................................,,,,,::::::::;;;;;;;;;;+++++++++++++++++++++++++++;;;;;;;;;;;;++++++++++++++++;;;;;;;;;;;;;;;;;;;;;;;;;;+++++++++++*??????????????????????????????*?****?%SSS%%%%%%%%%%%%%%%%%?????????????????*******************************************************************+++++;:,
***************************+++++++++++************+++;;:,,...............................................................,,.,,,,,::,::;;;;;;;;;;;++++++++++++********++++++++;;+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;::::;:::::::;;;;;;;;;;;;+++++*????????????????????*???????????****??%%%%%%%%S%%%%%%%%%%%??????????????????*********************************************************************+*++;
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*??************************++++++++++++++;::,................................................,:::,,,,:;;;;;;;;;;;+;+++++++++++++++++++++++***?**?**********??????****++++++++++;;;;;;::::::::::::::::::;;:::::::::::::;:::,::,::::::::;;;+**+****************???***?????????****??????*??*???******?%%%%%%%%%%%%%%%%%%?%?????????????????????*********************************************??*??****************+
+**************************+++++++++;;::,................................................,,,:;+++;:::;;;;;+++++++++++++++***********+++++++*************?******???************+++++;;;;;;::::::::::::::;;:::::::::::::;::,,,,,:::::;;++++**********************?**??%?%%????*****????******???*******?%%%%%%%%%%%%%%%%??????????????????????????******************************************??????*????***********
+**************************+++++;;::,,.........................................,,,,,:::::;;::;+++++++++++++*****+++++*+***???********+++++*************??******??******??*********++++;;;;;::::::::::::;;:::::::::::::;;:,,,,,::;;;;++++++++++++++++++**************??????*******??????*?*************??%%%%?%%%%%%%%%%%??????????????????????????****************************************?**??????****?*?*****+
+**************************++::,,,..........................................,::::;;;;;;;++++++++++*****++*******++++***??????*********+*+**+********??????*??**???******?*********+++++;;;;;;::::::::::;;:::::::::::::;;:,,,,::::;;;;;;;;;;;;::::;;;;;;++++++++++++++*********++**??????*????**********??%%%%%%%%%%%%%%%????????????????????????????*******************************************??****???*??????+
+*******************+*+++;;::........................................,.,::::;;;;++++******************+++******+++***?????%???????****++++**********?**????*****?****+****+++++++++++++;;+;;;;;::::::::::::::::::::::;;;::,:::::::::::::::::::::::::::::;;;;;;;;;;;;++++*****++++*?**???*???????********??%%%%??%%%%%%????????????????????????????????********************************************?***?*??****?*
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+?***??????????????????????????????????????????%%%%%%%%%%%%%%%%SSSSSSSSSSSSSSS%%S%??%?*++;;;;;+;++*****???*?????????????*??*****+++++++++**+++++++++;++++;:;;::::::::::::::::::::::;;;;;++++++++********++++++++;;;;;;;;;;;;;;;:;:::;::;::::::::::::::::::,::,:::::::;::::::;;++++;;+*****************************?%?%%%%%%%%%?%????????????????????????????????????????????????????????????????????????*?*?*?**
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*?%?%%%%%%%%%%%%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS#SSS##SS##SSS%%??%%?+++**+**?????********?*??????%%%?%???????????????**??*??????????******++;;;;;;;;;;;;;;;;;;:::::::,::::::::::::::::::::::::::::::::::;::::::::::,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,::,,,,,,,,,,,::::::::::::::::::::::::;;;;;;;+;;::;;;;;+++**+++++++++++**??*?*?????????????????????????????????????????????????????*++;;::::,,:::::,,,,,,,,:,
*???%%%%%%%%%%%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS##SSS#S%%%??%?%?+++*****????********??*???????%%%?%?????******?**??*??????????*********++;;;;;;;;;;;;;;;;;;::::::,,::::::::::::::::::::::::::::::::::;;;;::::::::::,,:,,::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::::::::::::::::::::::::;;;;;;;;;;::::;;;;+++*++++++++++++**?*????????????????????????????????????????????????????**++;::,::;;;:::::::::::,::::
+????%%%%%%%%%%%S%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS%%????%%?*;++****????*******????????????%%%??????*************????????***********++++;;;;;;;;;;;;;;;;;;::::::,::::::::::::::::::::::::::::::::::::::::::::::,,,,,,:,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,::::::::::::::::::::::::;;;;;;;;::::::;;;;+++*+++++;+++++++*??*???????????????????????????????????????????????*++;+;;::::;;;;;;;:::::::::::::::
++*??%%%%%%%%%SSSSSSSSSSSSSSSSSSSSSSSS##SSSSSSSSS#SSS%%S%?%??%??%*;++*****??********?????*?*????%%%%?%???***************?????***************+++;;;;;;;;;;;;;;;;;::::::,:::::::::::::::::::::::::::::::::::::::::::,,,,,,,,,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,::::::::::::,:::::;;:::::;;;;;;;:::::::;+;++++*++++;;+;++++**?????????????????????????????????????????????*++++;:,::;;;+;;;;;;;;;;;;;;;;;;::;;:
+++**?%%%%%%%%%%SSSSSSSSSSSS#SSSS#SSSSSSSSSSS##S#SSSS%%%%%S%?%???*+****??*?*****?**?????????????%%%%????*************?**************??******++++++;;;;;;;;;;;;;::::::::,,:::::::,,:::::::,::,,:::,,,::::::::::::::,,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::::::::,::::::::;:::::;;;;;;::::::::;;;+;++++++;;;;+++*+*?????????????????????????????????????????*+++;::::::;;+++++;;;;;;;;;;;;;;;;;;;;;::
+*++*?????%%%%%%%SSSSSSSS#SS############S#SSS##SSS%S%%%?%S%????*+++**+???*****?*?**?????????????%%%%????*******+*******************************++++;;;;;;;;;;;::::::,::,,:;::::,,,,,,,,,,,,,,,,,,,,,,,,:::::::::::,,,,,,,,,,,,:,:,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:,::::::::::,:::::;;;:::::;;;;;;::::::::;;;;+++++;;;;;;++****????????????????????????????????????**+;;;+::::::++;+++;;++;;;;;;;;;;+++;;;;;;;;::
+*+***++*??%%%%%%SSSSSS#SSSS############S##SS#S%%S%%%%%%%%????*+;+****??***??***?*?????????????%%??????*********+***********************??*****+++++;;;;;;;;;;:::::::::,,::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::::,,,,,,,,,,,,::::::,,,,,,,,,,...,......,,,,,,,,,,,,,::::,,::::::::;:;::;:::::::::::::::::;;;;++++;;;;;;;++**????????????????????????????????????**++::,,:;;++++++;;;;+++++;;;;;;;;+++++++;;;::::
+*+***+++**?%%%SS%%SSSS###SS###############SS%%%%%S%%%S%%%???*+++**??*?**?????**???%?*???????%%%%??%???**+++*+*++*******************?************++++;;;;;;;;::::::,,:::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::,,,,,,,,,,,,,,,,:::::,,,,,,,,,...........,,,,,,,,,,,::::::,::::::::::::::::::::::::::::::::;;;+++;;;;;;+;******?????????????????????????????***;;;;;;;;;:+***++++++;;+++++++++;;;;;+++++++;;;;;::
+*****++++++*??%%%%SSSSSSS##SS###########SSS%%???%SSS%%%%%%?*++***???****??*???????%?????????%%%%%%????*+++*++*****++++++***************?********++++++;;;;;::::::,,,::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::,,,,,,,,,,,,,,,,,:,:::,,,,,,.,,...........,,,,,,,,,,:::,,,,,:::::;::::::;;;;;;;;:::::::::::::::;;;;;;;;;;++******?????????*???????????***++*++;++:::+***?******+;+++;;+++;;;;;;;;;;;++++++*+++;+;:
+********+++++*??%%%%SSSSSSSSS###########SS%%??%?%%%%%?%%??*+++***??**+???????*???%%???????%%%%%%%???***+++*++?***+++++++++************?*********++++++;;;;:::::::,,::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::::::::::,,,,,,..,,..,........,,,,,,,,,,::,,,:::::::;;;;;;;;;;;;;;;;;::::::::::::::::::::::;+++++****????????????????????*+++;:;+;;***?***********+++++++;;;;;;;::;;;;;++*+****+*++++;
+*********++++;;+***?%%SSSSSSSSSS####S###S%%??%SS%?%%???%?+++****?****?**?????*??%%%??%?????%%%%??*?****++**+***+*+++++++++********?*************+++++++;;;:;::::,,,,,,,:,,,,,,,,,,,,,,,,,,,,,:,,,,,,,,,,,,,,,,,::::::::,:::::::::,,,,,,,..,,..,,........,,,,,,,,,,:,,:::::::::;;;;;;;;;;;;;;;;::::::::::,::,,:::::::;;+;++***???????????????**+++:;+;;;:;*??????***********+++++;;;;;;;;:::::;;;++***********+;
+***********++;;;;;+++*??%%%%%%SSSSSSS#SSS%%??%SS??%?**??*+++*+*****??**?????%?*?%%S??%????%%%?%?******+++***++++++++++++************************+++++++;;;;::::::,,,,,,,,,,,,,,,,,,,::::;;:,,,,,,,,,,,,,,,,,,::::,::::,,,:::,,,,::,,,,,,,,,,..,,..,........,.,,,,,::::::::::::::;;;;;;;;;;;;;;::::::::::::::::::::::;;++;+++*?????%???****+;;;::,..,;***??????********++**++++;;;;;;;;;;:::::;;;;++********+**;
+***********++++;;+++++;++????%%%SSS%SS%SS%???%%%*?**??**++*+****+*?%?**?????%%*%%%%???????%%?*??******+++***++++++++++*****??*********?************+++++;;;::::::,,,,,,,,,,,,,,,,,,,:;;;*+;:;:;;;:,::::,,,:::::;;::::::,:;:,,,,,:;;:::;:,,,,.,,,..,,.........,,,:,:::::::::::::::;;;::;;::;;;;;;;:;;;::::::::::::,,:;;;++++++?????***++;:,,,.,:::;;;:;+*???*?********+++++++;;;;;;;;;;;;::::::;;;;++++++++*+++;
+******+*++++++++++++++++;;+++***?%%%%S%%S%??%%%%*?**%?*+*********????*?*???????%S%%%??????%%*+?****+*++++*****++*++++******??**********************++++;;;;;::::,,,,,,,,,,,,,,,,,,,::++*%%*;+;*??+:;;;+::::::::++::;;;:::;;:,:::+%???**+::,,,,,...,,.,.....,,,,,::::::::::::::::;;;;;::;;;;;;;;:;;;;;:::::,::,::::,:;;;;+++;++*+++;::;;:,..,:;+****%?++***???******+++++++;;;;;;;;;;;;;::::::;;;;;++++++++++++;
+*****+++++++++++++++++++++++++++++***???%%*??%%????****++*******???**??**?????%S%S?????%?%%*++****************+**++++******??++*************++++*++++++;;;;:::::,,,,,,,,,,,,,,,,,,,:::;+*%%%SSS##%???*?+++++++++++*****++++++***?##SS??+;:,,,,,,,..........,,,,,,:::::::;;;;;;;;;;;;;;;;;;;;;;;;;;::;:::::::,,,::::::::;;;;;;;;;:::,::;;;::;+**???*??*++******+++++;++++;;;;;;:;;;;;;;;:::;;:;;;;;;;;;;;;;;;+;;
+*****+*++++++++++++++++++*++++++++++;;++*?****?*??*+**++***?*+**?*?**??*???%???%%%??%%???%?++**???******??*??++?*++++******?*+***********+*+++*+*++++;+;::;::::,,,,,,,,,,,,,,,,,,,::::::;+*?%%%SSS%?*****++++++++********++**??%SS%%?++;::::,,,,,,...........,,,,,,:::::;;;;;;;;;;;;;;;;;;;;::;:;;::::::::,,,,,:::::,,::::::::::,..,:;:;;;;+********+++*++;;:::;;;;;;;;;;:;;;;;;;;:::;:;;;;;;;;;;;;;;;;;;;;;;;:
+*******+++++++++++++++++++++********+++;;;;;;++****+*++**++*+;*??****?*????%?*?%??%?????%%*+**?????******?*??++?*++++*******++*********+*+*++++*+++++;+;:::::::,,,,,,,,,,,,,,,,,,::::::::;;+*??%%%%?***+++++++++++***?*******???%%**+;;::::,,,,,,,.,,,,.....,,,,,,,,:::::;;;;;;;;+;;;;;;;;;::::::::::::,,::,:,,,::::,,,,,:::::;;:,..,:;;;+++++++++;;;;;;;;::::;:;:;:;:;::;::;;;;;;;;;;;;;;;;;++;;;;;;;;;;;;;;;:
+******++++++++++++++++++++++++******++++;;;;;::;;+;;+++*+;+**+********???????**?%?%*?%??%?****????***??*????*++**++++******+++++++++*++++++*+++++*++;;;;;;::::,,,,,,,,,,,,,,,,,,,:::::::;:;;;;+**??**++*****++++*******?*********+;;;;:::,:,,,,,,,,,,,,,,,,,,,,,,,,,,::::::;;;;;;;++;;;;;;;::::::::,,::,,::,,,,,::::,,,,,,,,,,,::,,,..:;++;;;;;;;:::;;::::;::::::;;;;;;;;;;;;++++++++++;;+++++++++++++++;;;;;;;
+******++++++++++++++++++++++;:+*****+++;;;;;;;;;;:::;;;+;;+++**********??????**?????*?*?%*****?*??*????*?***+++++++++**+*++++++++;+++++++++++++++**+;;;;;::::,,,,,,,,,,,,,,,,,,,,:::::::::;;;;;;++++;;+++****+*+**+++******++;;;::::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::::;:;;;;;;;;;;;;;;::::::,,,,,,:,,::,,,:,,:::,:,,::,,,,..,......,;;;;;;;;;:;;::;;;;;;;;;+;+++++;;;++++++++++++++++++++++++++++++++;++++;
+******+++++++++++++++++++++++;;++++;;;;;;;;+;;;;::::::;;;;;;;++++******?*??*?**??*???***?****+?*??*?????****+++++++++**+++++++++;;;;++++++++++++**++;;;+;:,,,,,,,,,:,,,,,,,,,,,,:::::::::::;;;;;;;;;;;;;+++++++++++;;++++;;:::::::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::;:;;;;;;;;;;;;;;;::::,,,,,,,::,,:,,,:,,,::,,,,,,,,,....,,......,;++;;;;;;;;;;;+++++++++++++++++++++**+*****++++++++++++++++**+++++++;
+******++++++++++++++++++;++;+++;++++++;;;;;;;;::::::::;::;;;;;+++**+++***?***?***?*??********+*?%?*??**?****+++++++++*+++++++;+;;+;;;++++++*++++++++;;+;::,,,,,,,,,,,,,,,,,,,,,,:::::::::::;;;;;;;;;:::::;;;;;;;;;;;;;;;::::::::,,,,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,::::::;;;;;;;;;;::;:,,,,,,,,,,,:,,::,,:,,,::,,,,,,,,,.....,,,,....,+++++++++++++++++++++++++++*++++++*********+*++++*+***********+*++*+
+++**++++++++++++++++++++;;;;;;++++++++;++;;;:;::::::::;:;;;;+++++***++*****?*??****??*+***???++*??*??**?****++;;++++++++++;++;;;+++;;+++++++++++++;;:++;,:,,,,,,,,,,,,,,,,,,,,::::::::::::::::::::::::::::::::::::::::::::::,,,,,,,,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,:,,:::;;;;;;;;:;::::,,,,,,,,,,,,::::,,,,,,::,,,,,,:,.........,,,...,;***********++++++++++++++*++++++++++***************?**?**********+
++++++++++++++++++++++++;;;;;;+***+;;;;;;;;;:::::::::::;;;;;++++*+++*++++++****?******?+;+**?*++****???***+++++;;+++++++++;;;;;;;+++;++;+;+++++++;+;;:++:,:,,,,,,,:,,,,,,,,::,,::::;;:::::::::::::::::::::::::::::::::::::::,,:,,,,,,,,,,,,,,,:::,,,,,,,,,,,,,,,,,,,,,,,::,::::::;;::;;:::,,:,,,,,,,,,,:,,:,::,,,,,,:,,,,,,::.....,,,,...,..,,;+**?*?*?*?*++++++++++++++++++++++++***********???*?????*********+
++++++++++++++++++++++++;;;;;;;;;::::;;;;;;;::::::::::::;;;++++++++++++++++++*****+*****++++*?*+++*??%?****+++;;;+*+;;;++;++;;;;;;;;;;;;;;+++++++++;;:;+:,:,,,,,,,:,,,,,,,::,,:::::;;:::::::::::::::::::::::::::::::::::::::,,,,,,,,,,:,,,,,,,::,,,,,,,,,,,,,,,,,,,,,,,:::::::::::;:::;:::,:,,,,,,,,,,:::,:,::,,,,,,,,,,,,,:,......,,,,,....,,,;+*********+++++++++++++++++++++*++*************?*?????*********+
+++++++++++++++++++++++;;;;;::::::;;++;;+;;;:;;:::::::;;;;+++**+++++++++***++++++*++**?**++++***++????*++**++;;;;+**+;+++;;;;+;;;;;;;;;;;;;+;+++;+;;;::;,:;:,,,,,,:,,,,,,,,:::::::::::::::;:::::;::::::::::::::::::::,,::::::::,,,,,,,,,,,,,,:::,,,,::::::,,,,,,,,,,,,,::::::::::::;:::;:,,:,,,,,,,,,,::::::::,,,,,,:,,,,,,,,.........,,,,,,,,,:;++++++****++++++++++++++++++++*++*************???????********?+
++++++++++++++++++++++;;;;;;::::;;+++++;;;;;;;::;;:;;;;;;++++**++*++++**+****++++++*+*??**+;+*+*+*????*++*+;;;;;++**+++++;;:;+;;;;;;;;;;;;;;+;+;;;;;::::::::,,,,,,,,,,,,.,,,::::::::::::::::::;;;::::::::::::::::::::::::::::,,,,,,,,,,:,,,:,:::::,,:::::::::,,,,,,,,:,,::::::::::::::::::,,:,,:,,,,,,::::::::,::,,,,,,,,,,,,.............,,,,:,:++++++*****++++++++++++++++++++**************???????*********?*
++++++*++++;;;+;;;;++:,:;;;;;;;;;;;;;;;:;::::::;;;;:;;;;+;++***++*?*;+**++******+++++;*?***++++*****?*+**+;;;;;;++*+++++;;;;++;;;;;+;;;;;;;;;;;;;;;;::::::::,,,,,,,,,,,,,,,,,:::::::::::::::::;:;;;;;;;::::;::;::;::::::::::::,,,,,,,,,:,:::::::::,,,::::;::::,,,,,,,:,,:::::::::::::,,:,::,,,:::,,:,::::,::::::::,,,,,,,,,,,..............,,..,,;+++++*********+**+++++++++++******************????********???*
++++++++++;;;;+;;;::;,,:;;;;;;;;;::::::::::::::::::;:;;++++++*+++?%*++**+***++*?*++++;+****+++;+******+++;;;;;;;+**+;**+:;:;++;;;;++;;;;;;;;;;;;;;;;:::::::,,,,,,,,,,,,,,,,..,,:::::::::::::::;:::;;:;;;;;;;;;;:;;::::::::::::::,,,,::::::::::::::,::::::;:;::,,,,,,,,,,,:,,:,,,::::,:,,:::,,::::,,,,::::,,:::,:::,,,,,,,,,,,,.............::....,:++++****************************************************?????
+++++++++;;;:,;;;:,,,,,,::::;;;:::::::::::::::::::;;;;;;+++++*++???**++++****+**?*+**+;++*+**++++**++++;;::;+++;+*+;;**;;;::;;;;;;;+;;;;;;;;;;;;;;;;;::::;:::,,,,,,,,,,,,,,,..,:::::::;:::::::::::;;:;;;:;;;;;::::::::::::::::::::::::::::::::::::::::::::::::,,:,,,,,,,,,,,,,,,:::,,,,,,,,,,::::,,,,::,:,,,,,,,,,,,,,,,,,...,.............,;:,,,,,,:;+*******************************???*******************???*
++++++++;;;;;:;:,:,::,,,,,::::::::::::::::;++++++++;;;;;*+;++*+*???%?*+++*+*******+??**+;++++?*;;+++;++;;;;++++++*+:*?++;::;;;;;:+;;;;::;;;;;;;;;;;;:::::;;:::,,,,,,,,,,,.,,,..,,:::::::::::::::::;;;;;;;;;;;::::;::::::::::::::::::::::::::::::::::::::;:;:::,,:,,,,,,..:,,,,,,,,,,,,,,,,,,,::,,,,,,,,,,,,,,,,,,,:,,:,,,,...................:,,:+;::,:;+************????**********????????????***+*******+*++++
++++++;+;;;;;;::,,,,,,,,,,,::::,:::::;::;++*++*++**;;;:+*+;;+***?%%%%***+*+*****+*+*???*;;+;+*;:;;;+;;;;;;+++++++++;**+;;;;;;;;;;++;::::;::::::;:;::;:::;:;:::::,,,,::,,,,,,:,..,,::::::::::::::::;;;;;;;;;;;;;;;;;;;;;;::::::::::::::::::::::::::::::::;:;:::,,,,,,,,,.,:,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::,:,,,....................,,:;;:,;;:;*********???????***********?????******++++******+++++++
+++++++;;::;;:::,,,,:::::,,:::,,,:++??+;++*+;;*+;++;;:;??+;+***?%%%%%?*?+****?*****+*???+;;;+;:;*+:::::;:+++++++;++++++;;;;;:::;;;+;::;::::::::::;;:::::::::::;;:,,:::::::,,::,.,,::::::::::::;;;;;;;;;+;+;++;;;;;;;;;;;;:::::::::::::::::::::::::::::::;:;:;:,,,..,,,,.,:,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::,::,,,....................,,::;:;+*::****?****????*?***++++***********+++++++*******+++++++
;+++++;:::,:;::,,::::::::::::,,,,:+***++;**+;;*+;++;;;+?*;:+?**%#%%%????***+***?*+*+*???++;;;;:+?+,::;;;;++++;+;;+*;++;;;;;;:::;;;+;::;::::::::;::;;;;:::::::::::::::;;;;;,.::,..,::::::::::::::;:;;;+;++*+**+++++++++;;;;::::::;:::::::::::::::::::::::;:;:;:,,..,,,,,.,:.,:,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::,,,,.....................,::**;;*+:+*********??*******+**********++++++++++*****?*******+
::;;;;;:,::,;;::;;:::::::::::,:;;+**+++;;*+:++*+;++;:+***;;****?SS%?%%**?*++***??****???*;;;;;;;+;,,:;+;;+*+;;+;;+*;;;;;;;;;;::;;:;::;:::::::;;;;;;;;;:,,;:::::::::::;;;;;:,::,.,,:::,:::::::::::;;;++;+**+**+++++++++;;;;;;:::;;;:::::::::::::::::::::::;;;:;:,.,:,,,,,,,.,:,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,.,,,,::,,,,,,....................,.:;:;++++*****************************+++++++++**????*???????*
:::,,:::::::,,,:;;;:;;;;:::;::;++**+++;:+*+:++*+;++:;*+**;**+*??%SS%?%*+??++***?%?*++*??*;;;;+;;;:,:;;++;+++;++;;**;;;;;;;;;;,::;;::;::;:::::;;;;:;;;;:,,;::::::::::;;;;;;,,,,,,,,,:,,::::::::::::;;;;;+*++++++++++++;;;;:;;:::::::::::::::::::::::::::::;:;:::,.,:,,,.,,,,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:,,,,,,....................,,,:,,:++;+**************??**************+++*++**?????????????*
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+*************+++++*+*+++++++++++++++++++++++++++++++++****************++;:::::::::::::::;;;::,,,,,:,::::::,,,,.,,........................................................................................................................................................,..,,,,,,,,,,,,,,,::::,::,,,:::::::::::,,,,,.....,,,,,,,,,,,,,::::::::,,...,..........................................................
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+*+****++++*?%%%%%%SSSSSS#SSSS###########S#SS%?%%%SSS%%%%%%%?*++***???*+*%%*??*????%%*%%?????%%%%%%????**+;+*+**********************************++++++;;;;;;;:::::,,,::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::,,,,,,,,,,,,,,,,,:,:::,,,,,,,,,...........,,,,,,,,,,,::::::,:::::;::::::::;;:::::::::::::::::;;+++;;;;;;;;**+*****????????**?????????????????*+;:,..:**??****+*+++++;;;++;+;;;;;;;;++++++++++++;;:
+*+******++++*??%%%%SSSSSS######S#########S%%%??*%S%%%%%S%%?++++**???++????*??*???%%?*%????%%%%%%%????**+++*++****++++++++**********************+++++++;;;;;::::::,,,::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::,,,,,,,,,,,,,,,,,,,,:::,,,,,,.,,............,,,,,,,,,:,,::,,::::::;:;;::;;;;;;;;;;:::::::::::::::;;;;::;;+++********?????????????????????*+;::::;****?******+***+++++;;;;;;;;;;;;;;;+++*******++++;
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+**********+++;++++++++++;:++*??%SS%S%S%?S%???%%%*?+*S?*+*****+?*+%%??****??????%%%%???????%%?*??***+**+++***+*++*++++*******??*************?********+++;;;:::::::,,,,,,,,,,,,,,,,,,:;*+*%?;,,.:+;,.,,::..,,,,,,;;.,,,,,.,::.....:*+;;;*+:,,,..,,..,,..,.,....,,::,:::::::::::::::;;;;::;;;;;;;;:;;;;:::::::,:,:::,,:+;+++++++*??*?*+:::,.....,;++++%?;;**???******+++++++;;+;;;;;;;;;;;::::;;;;;;;++++++++++++;
+******++++++++++++++++++++++++++**??%%%%S%???%%???%?***++**+*?++*%?*?*?**??%???S%S%?????%?%?++******+*++******++*+++++**?*+?%*+*+*+*?*********+**++++++;;;;:::::,,,,,,,,,,,,,,,,,,,::;+*S#S%%%S#S?+?%??*+++++++**+*****+++++++++?##@#S#*;:,,,,,......,,.,.....,,,:::::::::;:::;;;;;;;;;;;;;;;;;;;;;;;:::::,:,,::::::::;;;++;;+++;;,,,:;;;:.,;+*???*?%?+++*****+++++++;;+;;;;;;;;:;;;;:;:::;;;;;;;;;;;;;;;;;;+;;
+****++++++++++++++++++++***+++++;++;;;**%%??*??*??*++**++***?*+*??***??*???????%%%???%?????+++***?******??**?++**++++***??*?*++************+++*+*++++;;;:;;:::::,,,,,,,,,,,,,,,,,,,:::;;+??S##SSS#%??********+*+++*?***********?%S##%*+;;:::,,,,,...........,,,.,::::::::;;;;;;;;;;;;;;;;;;;;:;;;::::::::::,:,,:::::,,,::;;::::,,..,;;;;;;;+*********+**++;:::;;;;;;;;;;;;;;;;;;;;::;:;:;;;;:;;;;;;;;;;;;;;;;;:
+********++++++++++++++++***********+++;;;;;+;+*???****++**+?*;*??****?*????%???%%%???%?%?%*+**?????****??????++?*++++******?++***********+++++***+++;;+;::::::::,,,,,,,,,,,,,.,,,,:::::::;;*?SSSS%S?***++;;+++++++********++**??S#S?*;;;:::,,,,,,,.....,....,,,.,,,::::::;;;;;;;;;;;;;;;;;;;::;;::::::::,::,:,,,::::,,,,,:::::;::...,:::;;++*+++;+;;;;;;;;::::::::;;;::::;;:;;;;;;;;;;;;;;;;;+++;;;;;;;;;;;;;;:
+******++++++++++++++++++**++*********+++;;;;:::;+*+++++**;+*++??****??*????%%**%%????????%****??%%?+**?*??*?**+**++++*******++++++****+++***+++*++++;;;;;::::::,,,,,,,,,,,,,,,,,,,::::::::;;;+*?%%%??***++++++++++*???******?%%%%?+;+;;:::,,,,,,:,.,,,,,,..,,,,,,,,,,::::::;;;;;;++++;;;;;;;:::::::,,::,,:,,:,,,::::,,,,,,,,,::;;:,..,,;;+++;;;;:;;:::;::;:::::::;;;;;;;;;;;;;+++++++;++++++++++++++++++;;;;;;;
+*******+++++++++++++++++++++;;+******++;;;;;;;;;;:::;;++;;++*?********????????*???%?*???%?**+*????**??????***++**++++**+**++++*++++++++++++++++++*++;;;;;:::::,,,,,,,:,,,,,,,,,,,:::::::::;;;;;+++***++**?************???????*;;;;:;::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,::::;;;;;;;;;;;;;;;;:::::::,:,,,:,,::,,:,:,:::,,:,,,,::,,,,,.....:++;;;;;;;::;;;:;;;;;;;;++++;+++++++++++++++++++++++++++++++++++++;+;++;
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+*++++*++++++++++++++++++;;;;;;;+++*+++;;;;+:;;::::::::;;;;;;+++++**+++**???**??****??*+*+*?**+*???*??**?****+++++++++++;+++;++;;;++;++++++++++++++;;;;+;:,,,,,,,:,,,,,,,,,:,,,,,::::::::::;;;;:;;:::::::::::::::::::::::::::,,,,,,,,,,,,,,,,,,:,,,,,:,,,,,,,,,,,,,,,,,,,,,,::;;;;;;;;;:;::::,,,,,,,,,,,,:,::,:,,,,:,,,,,,,:,,,......,:,,...,;***?******+++++++++++++++*++++++++*******++********?**?**********+
+++++++++++++++++++++++++;;;;;;**+*+;;;;+++;:;;::::::::;:;;;++++****+++*+*****??*****?*+++????;+********?**++++;;;+*+++++++;;;;;;+++;+++;;;++;+++++;;;;+:.,:,,,,,,:,,,,,,,,::,,,::::;::::::;;::::::::::::::::::::::::::::::,,,,,,,,,,,,,,,,,,,:::,,,,,,,,,,,,,,,,,,,,,,,,,,,:::;;;;;;+;::::::,,,,,,,,,,:,,,,:,,,,,,,:,,,,,,::.....,,.....,...,:+**?*******++++++++++++++++++++++++***************?????*********+
++++++++++++++++++++++;+;;;;;;;++;:::;;;;;;;:::::::::::;;;;+++++*+++++++++++*??*********++*???**++*?%???*+*+++;;;++;;;+;;+;++;;;;++;;;;+;;++++++++++;:++:,:,,,,,,,:,,,,,,,,::,,::::;;:::::::::::::::::::::::::::::::::::::::,,,,,,,,,,,,,,,,,,:::,,,,,,,,,,,,,,,,,,,,,,::::,:::::;;::;;;:::,,,,,,,,,,,,:::,,::,,,,,,:,,,,,,::,.....,:,,,....,,,;*+********++++++++++++++++++++++++************??*?????*********+
++++++++++++++++;+++;;;+;;;;:::;:::;;;;;+;;:::;::::::::;;;++++*++++++++**++;++***+*+*?***+;++?*+++??%%*+++*++;;;;+*+;;+++;;+;;+;;;;;;+;;;;+;;++;;;+;;;:;,,;,,,,,,::,,,,,,,:,,::::::::::::::::::::::::::::::::::::::::,,,:::::,,,,,,,,,,,,,,,,,::,,,,::::,,,,,,,,,,,,,,,:::::::::::;;::;;:,,,,,,,,,,,,,,:::::::,,,,,,:,,,,,,,,........,,,:,...,,:+++++++***++++++++++++++++++++++++************????????********?+
+++++++++++++++++++;+++;;;;;::::;;+++++;+;;;;;:;:::;;;;;;;+++**++++;+++******;;+++****?***+;+?**++????*;+**++;;;;+**+;+**;;:;+;;;;;;;+;;;;;+++++;;;;;:;:::::,,,,,,:,,,,,.,::::::::::::::::;::;;;;;:::::,::::::::::::::,,,:::::,,,,,,,,,,,,,,,::::,,,,::::::::,,,,,,,,,,,:::::::::::::::::,,:,,,,,,,,,,,,::::::,::,,,,,,,,,,,,...........,..,,,:,:++++++****++++++++++++++++++++++**************??????*********?*
++++++++++++;;;;;;;++:::;;;;;:;;;;;;;;;:;::::::;:;;;;;;;+;+++*++***+;+**++*****+++++++??**+;+++*****?***?*;;;:;;++*+;+;++;;;++;;;;;;;;;;;;;;;;;;;;;;;:;:::::::,,,,,,,,,,,.,:::::::::::::::;;::;;;:;;;;:::::::::::;::::::::::::,,,,,,,,,:,,::,:::,,,,,::::;:::,,,,,,,,:,,:::::::::::::::,:,,,::,:,,,:,::,:::::::::,,,,,,,,,,,,...............,.,,,;+++++*********+**+++++++++++**************??**????********???*
+++++++++++;;;;;;;:;;,,:++;;;;;;;;:::::::::::::::::;;;;++;++**+++*%*;;**+*+*+*??**++;;+?***+++++********++;;;;;;+*+++++;::;;++;;;:;+;;;;;;;;;;;;:;;::::::::,,,,,,,.,,,,,,,,.,:::::::::::::::::::::;;;;;;:;:;;:;:;;::::::::::::,,,,,,,,:::::::::::,::,::::;:;:,,,,,,,,::,::,,:::,::::::,,,::,::::::,,:::::,,:::::::,,,,,,,,,,,,............,,,....,;+*++*************************************?**************?????
++++++++++;;:,;;;;,,,,,::;;;;;;;:::::::::::::::::::::;;;++++++++*%?*+++++*+*++**?*+**+;+**+**++++***++++;;;:;++++*+;;*?+;;::+;;;;;+;;;;;;;;;;;;;;;;:;;::;:,,:,.,,,,.,,,,,,,..,::::::::::::::::::::;;:;;;;;:;;;:::;:::::::::::::::,,::::::::::::::::::::::::;::,,:,,,,,,,,,,,,,,,:::,,,,,,::,::,::,,,,:;::,,:,,:,:,,,:,,,,,,..,.............,;:,.....;+*******************************??**?****?*************???*
+++++++++;;;;::::,,:,,,,,,::::::::::::::::;;;;;;+;;;;;;;++;+++**???%?++++*****+***+****;;*+*+**+++*;;++;::;++*;++?+:+?*;;::;;;;;;;;;++::;;;;;;;;;;;;;::::;;:::,,,,,,,,,,.,,,..,,::::::::::::::::::;;;;;;:;;;;;::::::::::::::::::,::::;:::::::::::::::::::::;::,,::,,,,,,.,,,,,,,,:,,,,,,,,,,,::,:,,,,,:,,,,,,,,,,,,,,,,,,,..................,:,.,++:,,:***********???????***********????????*??***++******+*++++
++++++++;;;;;;:::,,:,,,,,,,::::,:::::::,:;+*++++++*;;;:+*+;++***??*%%*+++*********+*?***::+;+%*::;++;;;:;;+++*+++*;;*?+;;;;;;;;::;+;::::::;;::;;:;:::::::;;:::::,,,,,,,,,,,::..,,,::::::::::::::::;;;;;;;;:;;:::;;:::::::::::::::::::::::::::::::::::::::::::,,,:,,,,,..,,,,,,,,,,,,,,,,,,,,,,::,,,,,,,,,,,,,,,,,:::,:,,,,,....,.............,,.,;+;:;,,;**********??????**********???????*****+++**+****+++++++
+++++;++;:::::::,,,,,::::,,::::,,:;;**+;;++*;;*+;++:;:;?*+;++***?%S%S?*?+****??++***?%?*+:;;++:;+:;;::;:;+*+++++;;+++*+;;;;;::;:;+;:::::::::::::::::::::;:;::::;:,,:::,,,,.,::..,,::::::::::::::;:;;;;;;;;:;;;:;;;;;;;;;:::::::::::::::::::::::::::::::;::;:;:,,:,.,,,..,:,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::,:,,,....................,,:::,:*+:;+********??**?******************++++++++*******+++++++
++++++;:::,:;;:,,,,:::::::::::,,,,;*??;;+**+:;*+;++;;:+?*+;+?**%S%%S???%+****??**+*++???*++;;:,+%+::;:;;;++++++;;+*;++;;;;;;::;;;;+;::;::::::::::;;;:;;::::::::;:::,:::;;;,.::,..,,,::::::::::;;;:;;;;+++*+**++++++++;;;;;::::::::::::::::::::;::::::::;;:;:;:,,,..,,,,.,:.,;.,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,.,,,::::,,,.....................,:+**;;*+:+?****?***????****************++++++++++**??*?*******+
:;;++;;:,:,,;;::;;:::::::::::,,:;+**;++;;**:;+*+;++;;+***;:+*?*%#S??%%???*++***?**+**?*?+;;;;;:+*;,::;;;++++;;;;;+*+;;;;;;;;:::;;:;;,:;::::::;;:;:;;;;::,;::::::;::::;;;;+:,::,..,:::,::::::::::;:;;;+;+*****+++++++++;;;;;;:::;;;:::::::::;::;:::::::::::;;:::,..,,,,,.,:.,:,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,.,,,,::::,,,.....................,,:*+++++;************?****************+++++++++*?*??*????????*
:,,,,;:::;:::,,:;;;::;;;:::::,:+++**++;:+?+:*+*+;++:;****;+*+*??%#%?%%*+?*++***?%***+???*:;;;;;;+:,::;+;;+*+;++;;*?;:;;;;;;;;,:;;;;::::;:::::;;;;:;;;;:,:+::::::::::;;;;;;:,::,.,,::,,::::::::::::;;++;+******++++++++;;;;;;:::::;:::::::::::::::::::::::;;;;;;,.,:,,,,,,,,.,,,:,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,.,,,,::::,,,,.......................,,.:;+:+*?***********???***************+++***??????????????*
:;:,:;,:;:::..,;;:::::;::::;;:;+++*+++;;*+;;++*;;*;:;+**;+**++??%S#%?%*+*?*+*?+*%?**+*??*+++;+;:::,:;+++;;+;;**:;?*:;+:;+;:;;:,:;;::;::::::::;+;::;;;;:,:;::::::::::::::::,,:,.,,.,,,:::::::::::;:;:;;;++++++++;;++;+;;;;;:::;;::::::::::::,,:::::::::::;::;:::,.,:,,,.,,,,..,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,....................::;::::+:+++*?*************??****???********?????????%???????*
:;:;+;,,;:,,,..,:::::::::::::;*+***+*+:;*;+++++;+*;;++**;*%**?*?%SS%%%*+*??+*%*+%%%?++?***++++;::::;;;;;++;++*+;++;;+;;+*;,:;;:;+::::::;+;:::::;:;:;;;:,::::::::,::;:::::,,.,,,,,.,::::,:::::::::::;;;;;;;;;;;;;;;;;;;::::::::::::::::::::::,::::::::::::::::::,.,:,,,.,.,,..,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.,...,,,,,........................:*+,:;+;:**+*****************????????**???????%%?%????%???*
;;;;;:,.,::,,,,.,,,,,,,,,,,:+******+*+;++++;+++++++++*?+;?%**???%SSS%?*+*?*+?%%;????++**?*;++;;::::;+;;;+*;+++++*;:;;:;++:,:;+;;;:::,:+;;;;:::;:;;;;;;,:::::::::,::;:::::,,,,,,,,,,::::,,:;;:::::::;;;;;;;;;;;;;;;;::::::::::::::::::::::::,:::::::::::::;:::::...,,,,,,......,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.......,,........................,++,.;+;;++++***************??????????????????????????????*
::,.,,,..,,,........,,..:;**++*****++++**;;++;;+*;++**?*++?*****??S#%?*;*?*+*%#++??%?+**?*++;;;::::;;;++;;;+;;++;;*+;;;;;:;;;;;;;::,,;+;;;:::::;:;;;;;,::::::,:::::;::::,,,,,,,,,,:,:,:,,:;::::,,::;;;;:;;;;;::::::::::::::::::::::::::,::,:,::,,,,,:,,,::,,::,,..,,,,,,,....,,,,,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.,,....................................,;++;;;;+++*************??????%??????????????????????%?
,,.....,............::,;+****++*****++;+*++;+;;**;+*+**;+**?*??*?*?%%%?;*%*+*%S?+?%%**?*+%?:+*;::::;;;*+;:+++;;+;+*++;++::;;;;;;;:,::;;;:;;:::;:;:;;:::::;;::,::,::::,::,,,,,,,,,,,,,,::,:::::,,,::::::::::::::::::::::::::::,,,,,,,,,,,,:,:,,:,,,,,,,,,,,,,,:,,,.,,,,,,,,,..,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,........................................:+++;;;;;+++************??????????????????????????%???*
.....,.......,,,,,,:?S?**++**+****???*;;+*?*;;;**;**+**++**????*?????%%++?*+?%?**????**??%**?+;;:;:++;;;+;++;;+;;+,;::+::::::;;;;:,:;;;;:;;;::;;:;;::::::;;;::::,:::::::,,,,,,,,,,,,,,::,::::::,:::::::::::::::::::::::,::,,,,,,,::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.,,,,,,,,,,..,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.....................................,,...,:,,:;:,;+****************?????????****???**?????%???*
,,.,,,,....,,,,::,;**********??*+*???%??;;*?;:++*;+*+**+***????*????%%%+***??%?%?+***???%?**%?,:::;+*+;;:;++;+;;;::;;;;:::::;;:;;::;;;;;;;;:::;;:;;::::::::;::::::,:::::,,,,,,,,,,:,,,::::,,:::::::::::::::::::,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.,,,,,,,,,,,,,,.,,,..,,,,,,,,,,,,,,,,,,,,,,,,,,,..,...........................................,,,...,,,;*****************??????*********????????%?
,..,,,,,,..:::;;++***********?**+**??????++*+;;;+;*****++?**???**?%%%S%****????%*+**?%???***?*;;:::;++;;;;+;;;+;:;;;::;::::;;;;:;::;:::;;::::::::;;;::::::::::::::,,:;,,::,,,:,:,,:,,,::;::,,,::::::::::,::::::,,:,,,,,,,,,,,,,::,,:,,,,,,,,,,,,,,,,,,,,,,,,,,.,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.,,,,..............................................,,:,:+;:,..;**+**+*+************************???????%*
,..,,,,..,::;*?**********************???*?**+++;;+*****++***????**%%?%%*????????+***?%%??**+?*%+::,;;++;;++;:;*+::;::::::;;;;:;::;;:;++;;::::::::;;;;;;;;::::::::::,,;,,;:,.,:.,,,,,,,:::::::,::::::::::::::::;,,,,,,,,:,,,:,::::,,:,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,...,.................................................,,:++*;,,:;+++++++++**********************???????%*
...,,,::;+**************?***************???**+++;+****++*+*?%????*?%%???*?????%*+**??%%%%****??++::;++*;;+++;;*+;:;:::,,:;::;+;:;;;::;;::;;;;::::;;;;;;;;;;::::::::;,,;::::,,,,,,,,,,,::::::::,::::::::::::::::::::::,,:,,,:,,:::,,::,:,,,,,,,,,,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.,.......................................................,++++++;;++++++++++*********************?????%?%*
...,:;+*************************************+**++++++++++++??????**????******?%*++*???????++*?**+;:;+++;;;;;;;;+;::::::,::;:;;::;:;:::::;;;;:;;::::;;;;;::::::::::,::,,,:::::,,,,,,,,,,::::,,,,,::::::::::::::::,,,,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.,,,,,,,,.................................................................,:::;;;;+++++++++++++******++++++++****???????*

***************+++*++**++++++++++++++++++++++++++++++++****************++;:::::::::::::::;;;::,,,,,,,::::;::::,,,,..............................................................................................................................................................................................................................................................................................
*???**?*?*************************+******************************??*??***+;;;;;;;;;:::::;;;;;;::::;::;;;;;;:::,,,...............................................................................................................................................................................................................................,..,.,,.,..,....................................................
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*??*?*?*******************+++++++++++******************+++++++++*++++++++++++*++;::::,,,,,,,,...................................................................................................................,,,,,,,,,,,,,,,,:::,:::::::;;::;;++++++++++++++++++++*******************?%%?????**************************+++++++++++++++++++++*+**+++++++******+++++;::::,.....................................
*??**********************++++++++++++*******************+++++++++++++++++++++;:,,,,,,,:,..................................................................................................,.......,,,,,,,,,,,,,,,,,,,,,,,,,,:::::::;;;;:;;++++++++++++****************???????????????????%%%SS%%?%?????????????????*******************+++++************+*************++*+++;;;::,...............................
*??*?*****************+*+++++++++++++****************+++++++++++++++++++;:,,,,,,:,....................................................................................................,,,,,,,,,,,,,,,,,,,,,,,:::::::::::::;;;;;;;;;;+++++++*++++++++********????????????%%%%??????????????%%SSS%%%%%???????????????????????****?*********************************************++*+;;:,,..........................
*?***********************+++++++++++++++*************++****++++**+++;::,,,,,,:,.............................................................................................,,,,,,,,,,,,,,,,,,,,,::::::::::::;;;;;;;;;;;;;;;;;;;;;;;++++++++******??????????%%%%%%%???%%?%%%?????????????????SSSS%%%%%%%??????????????????????*****************************************************+++;;:,......................
*?************************++++++++++++***************+********++;::,,,:,,,...........................................................................................,,,,,,,,,,,,,,,,,,,::::::::::::;;;;;;;;;;;;;;;;;;;;;;++;++++++++++++++**********??**??????%???????%??????????????????**??%SSSS%%%%%%%?????????????????????*********************************************************++;:,...................
*************************+++++;+++++++*************++****+++;:,,,,::,,........................................................................................,,,,,,::::::::::::::::::::::;;;;;;;;;;;;;;;;;;;;;;+++;;+++;++++++++;+;+++++++++++***+++*****??*???????%????????????????????*****?%SSSSSS%%%%%%??%%????????????????*??*****************************************************+**++;::,...............
*?*************************+++++++++++*+********+****++;:::,,::,:,...............................................................................,,,,,,,,,,,,,:::::::::;::::::;;;;;::;;;;:;;++;;;;;;;;;+;;;;;;;;+;++;+;+;+;;;;;;;;;;;;;+++++++++++++++++**????????%????????????????????????*****?%SSSS%%%%%%%%%%%%??????????????????*****************************************************+****+++;:,............
*?**************************++++++++++***********+;;::,,,::,,,....................................................................,,,,,,,,,,,,,,,,,,,:::::::::;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;++++++++++***??????????????????????????????????*****?%%%S%%%%%%%%%%%%%%????????????????**************************************************************++;:,,........
*?**?**********************+++++++++++++++***+;;::,,::,:,.....................................................................,,,,,,,,,,,,,:::::::::::::;;;;+;;+;;;;;;;;;;;;;;+;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;:;;;;;;;;;++;;++++++++*******????????????????????????????????*****??%%%%%%%%%%%%%%%%%%??????????????????***?*********************************************************++;::,,....
+**************************+++++++++++++++;:::,,,:,,...............................................................,,,,,,,,,,:::::::::::::::;;;;;;;;;;;;++++++++*++++++++++++++++;;;;;;;;;;;:::::::::::;;;;;;;;;;;;;;;;;;::::::::::::;;;;;;;;;;;;++++++++++++*??????%????????**???????**????*******??%%%%%%%%%%%%%%%%%%%???????????????????*?************************************************************++;;:,,
+**************************+++++++++++;;::,,::,,................................................,......,,.,,.,,,,,,,,,::::::::::;;;;;;;;;;;+++++++++*+++****+***+******+++*+++++++;;;;;;;;;::::::::::::::;::::::::::::;:::::::::::::;;;;;;;;;;+++++++++++++******??????%???*******?????*?????**?****??%%%%%%%%%%%%%%%%%%%%????????????????*?*?*************************************************************++++;
+**************************+++++++;;:::,,,,,....................................................,.,,,,,,,::::::::::::::;;;;;;;;;;;++++++++++++++*********+*****************++*+++++;;;;;;;;;:::::::::::::::::::::::::::::::::,:::::::;;::::;;;;;;+++++*******??**?****????*********??????*???**?******??%%%%%%%%%%%%%%%%%??????????????????????????************************************************************+
+***********************+*++++;;::,,,,,,,..........................................,,,,,,,,.,,,,:::::::::;;;;;;;;;;;;;+;+++++++++***+++++++++++******???**************************++;;;;;;;;:::::::::::::::::::::::::::::::::,,:::::::::::;;;;;;;;++++++*+++*+++***++****??*********?????**???????*****???%%%%%%%%%%%%%%?%%????????????????????????************************************************************+
*?***********************++;::,,,,,,............................................,,,,,,,,:::::;++**++:::;;++++++;;++++++++***********++++++++++*******?****?****???***********++******+++++;;;;:::::::::::::::::::::::::::::::,,::::::::;;;;;;;;;+++++++++++++;;+++++***??????*******?????**????***?******?%?%S%%%%%%%%%%%??%?????????????????????????*****************************************?****************+
*??******************++;;:,,,:,,,.........................................,,,,,,:;;;;;;;;+++++;+****++++++++++++++********?*???*******++*++++**********************+++++++++++++++++++++++++;;;;:::::::::::::::::::::::::;;::,::::::;;;;;;;;;++++++;;;;+;;;;;++++++*++**?????*+++****????***??????********??%%%%%%%%%%%%%???%%??????????????????????????**************************************??????***********+
*??**************++;;:,,::::,...................................,,,,,,,,,,:::::;+++++********+****++*********++********?????????*??****++*+++**+*****************+++++++++++++++;++;;;;;;;;;;;;;::::::::::::::::::::::::;;;:::::::::;;;;;;;;;:::::::::::::;;;;;;;+;;;;;++*******++++**?*?*??*???????******?*??%%%%%%%%%%%%%??????????????????????????????*************************************?*?????***********
*??**********+++;:::::::,,...........................,,,,,,,,,,,,,,,,,,:::;;;+++++*******??*******++*++****++********??????????????*****++++++***********++++++++++++;;;;;;;;:;;;;;;;++;;;;;;;;;;;;;:::::::::::::::::::::;;;::::::::::::::::::::::::::::::::::::::::;;;;+++*****++++****?******????????*****??%%%%%%%%%%%%%%????????????????????????????????***********************************?*****?******???*
*??*?*****+;;:::::::,...........................,,,,,,,,,::,,::;;;::;+++++++***********???????*****+*****++*********????*???%?????****+*++++++*+*******+++++;;;;;;;;;::::::::::::::::::::;;;;;;;;;;;:::::::::::::::::::::;;;;;;:::::::::::::;;;;;;;;;;::::::::::::::::;;;++++***++++****?******??????????*???*??%%%%%%%%%%%??????????????????????????????????*************************************************?*
*????*+;;;::::::,,........................,,,,,,,:::,,:;:;;;;++++++++*********???????%%%%???????*******+*****+++***?????????%???******+*+++++++++*****++++;;;;;:;::::::::::::::::::::::::::::;;;;;;;;:::;::::::::::::::::;;;;;;;::::::;;;;+++++*+;:;;;+;;++;;:::::::::::;++++***++++*****??******????????????????%%%%%%%%%%%%????????????????????????????????***************************************************
*?*++;:::,,:,..................,..,,,,,,,::,:::;;;;;+++******************?????????%%%%%%%?????****++*+++***+++*?*???????*??????********+++**+++++***+++++;;;;:::::;;;;;+++;;;;;;;;;::;;;:::::;;;;++;;;;;;;;::;;;;::::::::;;;;++;;;;+++++++;;;;;;;::::;;;;;;;;;;;;;:::::::;++++*++++++**?***********????????????????%%%%%%%%%?????????????????????????????????**************************************************+
;::;;::,...................,,,,,::::::::;;;;;++**********???***?**?*?*????%%%%%%%%%%%%%%????***+++++++++*++***?*?%???%??*?****?*******+*+++++++*+*++++++;;;;;;;;;;++++;++++::::::;;;;;;;;;;;;;;;++++++++;;;;;;;;;;;;;;;;;;:;;++++++++++;;;;;;;::;:;;;:;;::::::::;;;:::::::;++++++*+++++**************??????????????%%%%%%%%%?????????????????????????????????**************************************?***********+
:;:,,..............,,,,,,:::;;;++++;++++++++*************??????%%%%%%%%%%%SSSSSSS%%S%%???%*****+++++++++++***?*?????%?????****?***++**+*+++++++*++++++;;;;+***+++;;;;:::::::;;+;;;;;;;;;;;;;;;+++++++++++;;;;;;;;;;;;;;;;;;;;;;+++;;;;;;;;;;+*****?*;+;++;;;+;;:::::::::::;;++++++++++****?**??**********????????*??%%%%%%%%%%?????????????????????????????????????????*******?????*****************************
,,...........,,,,,,:::;;+++++++****************?????????????%%%%%%%%%%SS%%SSSSSSS%S%%%%%??+**++++++;;++++***?????????????*************++++**+++*++++;;;+***+++;;;;;;;++++**??*+++++**+;+++;;;;++++++*+*+++++++++++++;;;;;;;;;;;++++;;;+++++++*++;;++;;;;;;++***++++;::::::;;++++++++++****?????*********?????????*??%%%%%%%%%%??????????????????????????????????????????????????????????************************
......,,,,:,::;;;;;+++++++******************?????????????%%%%%%%%%%%SSSSSSSSSSSS%S%%%%%%%?*?+;+++;;++*++*??*???????????*************++++++*+++++++++++;;+;;::;;+++***??*+*???*;;:;;+*++********++****++++++++++++++++++;;;;;;;;+++++++*****++;;;;::::::::;;;;;;;++**+;:::::;;++++++++++******?**********?*???????*??%%%%%%%%%%????????????????????????????????????????????????????????????????**?***?**********+
,,::;:;++++;++***************????????????????????????????%%%%%%%%%SSSSSSSS#SS#S%SS%?%%%%%?**;++;;;++*++**??????%??????**?*****+*****++++++++++*+**++++;;;;;;+******++++++;;;;;;;++;;++*+*************+++++**+++++++++++++;;;;;;;;;;++;;++++;+;;;;;;:::::::::::::;:;;;;::::::;+++++++++*+********************??**?*????%?%%%%%%%%%????????????????????????????????????????????????????????????**????*************
;+++++++*************?**????????????????????????????%%%%%%%%%%%%S%%SS%SSSSSSSS%%#S?%%??*???*;;;;;++++**??**??????????????**********++++++++++*+++++;;;;;;:;+++++++;++;;;;;;;;;;;;;++++++++++++++******++*++***++++++++;;+;;;;;;;;;;;;;;;;;;;;;;;;:::::::::::,:::::::;;;;;;::;;++++++++++*******?******************????%%%%%%?%%%%%?????????????????????????????????????????????????????????????*???*************
+*****??*********???????????????????????????????????%%%%%%%%%%%%SSSSSSSSSSSSS%%SS%%%??????*;;;+;+++++*???**??????????????*********+++++++++++++++++++++++;;;;;:;;;;;;;;;::::;;;:::;;;;;;;+++++++++****++***+++++++++++;+;;;;;;;;;;;;;+;;;;;;;;;:::,::::::::::::::::::;;;:;;;;;;++++++++*++******??****************????%%%%%%%%%%%%%????????????????????????????????????????????????????????????*??***?*****?****
+**?*???????????????????????????????????????????%??%%%%%%%%%%%%SSSSSSSSSSSSSS%SS%%S%*??**;;+;;;+++++*****???*???%??????????**?****+++++++++++++++++++++;+;;;;;;;;::::::::::::::::::;:;;;;;+++++++++**++*+**+++++++++;;;;;;;;;;;;;;;;;;;:;;;;;;;;;;::::::::::::::::::::::::::;;;;+++++++**+*?***********************?*?%%%%%%%%?%%%%%???????????????????????????????????????????????????????????????*???*????****
*???????????????????????????????????????????????%%%%%%%%%%%%SSSSSSSSSSSSSSSS%SS%%%%??***+++;;++;+;***?**??????????%%%%??????******+++++++++++++++++;;;;;;++;;;;;;::;:::::::::::::::;;;;;;;++++++++++***++*+++++++++++;;;;;;;;;;;;;;:;;;::::::::::;::::::::::::::::::::::::::::;;++;;+++******************************?%%?%%%%%?%%%%?????????????????????????????????????????????????????????????????????????*??*
*??????????????????????????????????????????????%%%%%%%%%%%SSSSSSSSSSSSSSS%%%%%%%%???**++++++;++;+++****??*?????%%%%????%???**?****+++++*+++++++++++;;;;++++;;;;:;;::;::;:::::::::;:;;;;;;+++++++++++++**++*+++++++++;;;;;;;;;;;;:;:::;;;;;:::::::::::::::::::::::::::::::::::::;;;;;;++***??******??*****************????%%%%%%%%%%??%?????????????????????????????????????*???????????????????????????????????*
*?????????????????????????????????????????????%%%%?%%%%%%%%SS%SSSSSSSSSSSSSS%%%%???***+++++++;;++++****??????%%%%%%%%%S%%??*******+++++*++++++++++;;;;;;;;;;;;;;;;;::::::;;:::::;;;;;;;;++++++++++++++++*+++**++++++;;;;;;;;;;;;;;:::;::;::;;::;::;::::::::::::::::::::;:::::::;;;;;;;+****?????*********************????%%%%%%%%%%%%%%????????????????????????????????????????????????????????????????*??*????*
*?????????????????????????????%%?%%????%??%%%%%%%%%%%%%%%SSSSSSSSSSSSSSSSSS%?%%??*****++++++++;++++*****?????%?%%S%%S%%%%%????****+++++++++++++++;;++;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;++++++++++++++++++*+++++++++++;;;;;;;;;;;;;;::::::::::::::::::::::::::::::::::::::::::::::;;;;;:;+****???****??*****************???%%%%%%%%%%%%%%????????????????????????????????????????????????????????????????????????*
*?%?????????????????????????????%%??%%%%%%%%%%%%%%%%%%%%SSSSSSSS#SS#SSSSS%%%%S?***?*++++++++++++++****?*???????%%%%SSSSSS%%??**?***++++++++++++++++++;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;+++++++++++++++++++++*++***++++++++;;;;;;;;;;;::::::::::::::::::::::::::::::::::::::::::::::::;;;;::;+*+*???*?**????**********+****?????%%%%%%%%%%%%????????????????????????????????????????????????????????????????????????*
*?%??????????????????????????%%?%%%%%%%%%%%%%%%%%%%%%SSSSSSSSSS##SSSS#SS%%%%S%**??*+++***++;*+;+++****???????%%SSSSSSSSSS%???**?***++++++++++++++++;;:;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;++;;++++++++++++++++*+++++++++++++;;;;;;;;;;:;;::;:::::::::::::::::::::::::::::::::::::::::::::;;:;;;+*++*?****????**?******++++*?*?????%%%%%%%%%%%%%%?????????????????????????????????????????????????????????????????????*
?%%????????????????????????%%%%%%%%%%%%%?%%%%%SSSSSSSSSSSSSSS##S#SSSS#S%%%%%????**;*?++*+++;+++*++*?**????%%%%SSSSSSSSSSS%???*???**++++++++++++++;;;:;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;++++++++++++++++****++++++++;;;;;;;;;;;;;;;;:::;;:::::::::::::::::::::::::,,,:::::::::::::::;;;;;;++++*******??*?*****++++++*?*?????%%%%?%%%%%%%%%%%???????????????????????????????????????????????????????????????????*
??%?%??????????????????????%%%%%%%%%%%%%%%%%%SSSSSSSSSSSSSSSS##S#SSSSSS%%%????***;**+**++*++*++*+**?*???%%%%%SS%SSSSSSS%%%%%**???***++++++++++++;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;+***+++++++++++++++**++++++;;;;;;;::;;;;;;;;;;::::;:::::::::::::::::::::::,:,,::::::::::;;;::;;;;;;;;++++*+**+**???**??**++++****?????%%%%%%%%%%%%%%%???????????????????????????????????????????????????????????????????*
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?%?%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%SSSSSSSSSS##SS####SS##SSSSSS%SS%S%%?***+********?*+******????????%S%%%%%%%%???????????????????***+++++++;;;;;;;;;;;;;;;;;;;;:;;;:::::::::::::;;;;;;+++++++++++++++++;;;;;;:::::::::::::::::::::;;::;::;;;:::,,,,,,,,,,,,,,,,,,,,::::::::::::::;::::;;;;;;;;;;;+;;;;+++????++++++++****???%%%%%%%%%%%???????????????????????????????????????????????????????%%%%%???%????%?*
???%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%SSSSSSSSS############SSSSSSSSSS%%S%**?++***++****??********???????%%%S?%S%%?????????????????????***+++++;;;;;;;;;;;;;;;;;;;;::::::::::::::::::;::;;+++++;;;;+++++;;;;;;;;;:::::::::::,::::::::::::;;::::::::::,,,,,,,,,,,,,,,,,,,::::::::::::::::;::::;;;;;;;;;;;;;;+++*???*++++++++**???%%?%%%%%%%????????????????????????????????????????????????????????%%???????%?%??*+;
?%???%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%SSSSSSSSS#############SSSS%SSS#S%%%??*++**++++*************????????%S%%%%???????????????????????*****++++;;;;;;;;;;;;;;;;;;:::::::::::::::::::::;;;;++++;;;;;;;;;;;;;;;;;;:::::::::::,,,,::::::::::::::::::::::,,,,,,,,,,,,,,,,,,:::::::::::::::::::::::::;;;;;;;;;;;;;++??****+++++++**????%%%%?%?????????????????????????????????????????????????????????%%?????????***++;;
?????%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%SSSSSSSSSS############SSSSSS#SS%%%???*+***+*++*???******?**????????%%S%????????????%?????????????****++++;;;;;;;;;;;;;;;;;;::::::::::::::::::::::;;;;++;;;:::;;;;;;;;;;;:::::::::::,,,,,,,::::::::::::,,,,,,::::,,,,,,,,,,,,,,,,,::::::::::::::::;;;:::::;;;;;;;;;;;;;;;+*?****++++++***?????%?????????????????????????????????????????????????????????????????%???*+++;++++;
???%%?%%%%%%%%%%%%%%%%%%%%S%%%%SSSSSSSSSSSSSSS#########S#SS#SS#SS%%%?***+**********?*?*+**??**????????%%%???????????????????????????****++++;;;;;;;;;;;;;;;;:;;::::::::,::,,,::::::;;;;;++;;;;;;;;:::;:::::::::::::::,,,,,,,,:::::::::::,,,,,,,,:::::,,,,,,,,,,,,,,,::::::::::::::::::::::::;;;;;;;;;;;;:;++*??*++++++++++***??????????????????????????????????????????????????????????????????%??*++;;;+++++++;
*???%%%%%%%%%%%%%%SSSSSSS%SSSSSSSSSSSSSSSSSSSSSS#####SS##SSSS##SS%%???**+*++****????***+**?**?*?????%%???????????????????????????????***++++;;;;;;;;;;;;;;;;;;;:::::::::,,,,:::::::;;;;;;;;;;;;::::::;:::::::::::::,,,,,,,,,,,::::,:::::::,,,,,,,:::::,,,,,,,,,,,,,,:::::::::::::::::::::::::;;;;;;;;;;;::;++*??*+++++++++***??????????????????????????????????????????????????????????????????*+;;;++++++++;:,,
*???%%%%%%%%%%%%%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS###SS#S#S#S%S%%%??*+++*++***????******?*??***???%%%???????????????????????????????*****++;;;;;;;;;;;;;;;;;;;::::::::::::::::::::;;;;;;;;;;::::::::::::::::::::::,,,,,,,,,,,,,:,,:::,:,,,,,,,,,,::::,,,,,,,,,,,,,::::::::::::::::::::::::::;;;;;;;;;;;;;;;+*?*++++++++++***?*????????????????????????????????????????????????????????????*++++++++**+;;;:,,,,,
*???%%%%%%%%%%%%%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS##S#SSSSSS%?%???*++*****??*???********?**????%%%?????????*???????*???????????????****+++;;;;;;;;;;;;;;;;;;:::::::,:::::::::::;;;;;::::::;;;;;;;:::::::::::::::,,,,,,,,,,,,,,:::::,,,,,,,,,,,,,,::,,,,,,,,,,,,,:::::::::::::::::::::::;;;;;;;;;;;;;;;;;;++**+++++++++++**?????????????????????????????????????????????????????????***+;+++++**+++;,,.,,:::::
+????%%?%%%%%%%%%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS#SSSSSS%%?%?%%??++******???**********????????%%%????????**????***?????????????*?******+++;;;;;;;;;;;;;;;;;;::::::,,::::::::::;::::::::::;;;;;;;;;::::::::::::::,,,,,,,,,,,,:,::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::::::::::::::::::::;;;;;;;;;;;:;;;;;;+++**+++;++++++**???*???????????????????????????????????????????????????**+++++****++;::,,,:::::::::
+**??%%%%%%%%%%%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS%%%%???%%***++******???********???*??????%%%??????*********???*???????????*********++;;;;;;;;;;;;;;;;;;::::::::::::::::::::::::::::::::;;;:::;;;::::::::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::::::::::::::::::::::::;;;;;;;;;;;;;;;;;++***+++;+++++++**??????????????????????????????????????????????????*+++++******+;;::::::::::::::;::
+*+*?%%%%%%%%%%%SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS##SS#SS%%SS??%%%%*?*;+*****?????*****?????????????%%%???*??**************??????*************++++;;;;;;;;;;;;;;;;;::::::,,::::::::::::::::::::::::::::::::::::::::::::,,,,,,,,::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::::::::::::::::::::::::::;;;;;;;;:::::;;;;++++++++;++++++++??????????????????????????????????????????????*++++++*****++;,,:::;;;;;;;;;;;;:;;:
+*++*??%%%%%%%%%%SSSSSSSSSSS####SSSS##SSSSSSS####SSSSS%?%%?%S%?%*?*+++**???*********?????????????%%%%??????**************????*?**************++++;;;;;;;;;;;;;;;;;:::::,,:::::::,:::::::::::::::::::::::::::::::::::::,,,,,,,::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::::::::::::::::::::::;;;;;;;::::::;;;;+++++++++;+++++++*??????????????????????????????????????????*+;;;+******++;:;;;;;;;;;;;;;;;;;;;;;;:
+**++***??%%%%%%%%SSSSS#SSSSSS###########SSS####SSSS%%%%%%%S%?%***++*+*???*??****??*?????????????%%%%????******************************??*****++++;;;;;;;;;;;;;;;:::::::,::::::::::::,,,,:::::::::::::::::::::::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,::::::::::::,:::::::::::::;;;;;;:::::::;;;;+++*++++++;;++*****????????????????????????????????????*+;;;++*****+++;;;;+++;;;;;+;;;++;;+;;;;;;::
+*++**+++*??%%%%%%SSSSS#SSSSSS#############SSSSSSSS%%%%%%%%????**++****??*??**?**?*??????????????%???????******+++***********************?******++++;;;;;;;;;;;;:::::::::::::::::,,,,,,,,::::,,,::::,,:::::::::::::,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::::::::::::::::;::::::;;;::;;::::::;+++++++++;;;;;++++***????????????????????????????????***+++++*****+;:,::;;;+++++;;;;;;;+++++++;;;;;::
+**+**+*+++**?%%%%%%SSSSSS#SSSS############SSSS%%%SS%%%S%%%???**+++*??**??**???**??????????????%%%??%???**+******************************?*******++++;;;;;;;:;;::::::::::::::::::,,,,,,,,,,,,,,,,,,,,,,,:::::::::::,,,,,:,,,,,,,,,::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:,:::::::::,:::;::;::::::;;;;::::::::::;;;;+++++;;;;;++++****????????????????????????????**++;+*????***+;::;:;;;;;+++++;;;;;;;+++++++++++;;;:
+*******+++++*??%%%%SSSSSS################SSS%%??S%%S%%%%%S%?**++**??*****??????????%??%??????%%%%?????***+*++*************************?**********++++;;;;;;;;;::::::::::,,:::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::,,,,,,,,,,,,,,,,,,::,,,,,,,,,,,,.....,,,,,,,,,,,,,,,,::,:,:::::::;;::;;;:::;::;;:::::::::::::;;;;+;;;;;++++****????????????????????????**++++*??????*+;:::;::++++;;;;;;;;;;;;;;;;+++*****++++++;
+**********+++++*??%%SSSSSSSSSSSS########SSS%%%%???%S%%%%%%?**+++*?**?***??????*???%%%???????%%%%?%??*??**+++*****++*********************?********+++++;;;;;:::::::::,,,:,:::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:,,::::,:,,,:,,,,,:,,:::::,,,,,,,,,,..........,,,,,,,,,,,,:::::,:::::::;;::::;;;;;;;:::::::::::::::;;;;;;;;;;+;+*****?????????????*???????*+++;+**?????*+;;;;;+;+**+++++;;;;;;;:::;;;;;;+***+*****+++;
+***********+++;;;+++*%%SSSSSSSSSSS#######S%%??%%??%%%?????**++*****??+*?????????%%%%???????%%%%%%???****+++*++*****++++++***********?*****?******+++++;;;;;;:::::::,,,::,::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:,,:::::::::,::::::::::::::,,,,,,,,,,.,.........,,,,,,,,,,::,,::,:::::::;;:;;::;;;;;;;;::::::::::::::;;;;;::;;+++*++****?????????????**++;;+*??*???*+++;;;;+*****+++++;;;;;;;;;::::::;;;;++***********+
+************+++;;;;;;;;+?%SSS%SSS%SSSSSSSS%%??%S%%??%???*?*++*****??*?***???????%%%%??%????%%????*?****++***++****+++++++************************+++++++;;;;::::::,,,,::,::,,,,,,,,,,::,,,,,,,,,,,,,,,,,,,,:::,::,,::,,,,::,,,,,,,,,,,,,,,,,,.,..........,,,,,,,,,,:,:::::::::::::;;::;;;;;;;;;::::::::::::::::;::::;;;+++++***?????????***+;;;++++++++++;+++++++++++++++++++;;;;;;;;;;:::::::;;;;++++++++*+++;
+***********+++++++++++++;;+**??%%SSS%%%%S%????%S%*????****+******?*?%?***???????%%%%??%?%??%%???*?*****++***+++++++++++***************************++++++;;;::::::,,,,,,,,,,,,,,,,,,,::::;;:,,,,::,,,,,,,,,,,,,,::,,,,,,,,::,,,,,:;;:::::,,,,,............,,,,,,,,,,:::::::::::::;;;;;:;;;;;;;;;:;:::::::::::,,::::::;;;++;+++*?????**+*+;;;;;;+**+++;::;;+***+++++++++++++;+;;;;;;;;;;:;:::;:;;;;;++++++++++++;
+*******+++++++++++++++++++;;++++**??%%%%%S%??%%%???**??*++******?*?%??***?????%?%%%%????%?%???**?******++****+++++++++******?+****************++**+++++;;;;:::::,,,,,,,,,,,,,,,,,,,,,:::+++++++++;:;;;;::::::::;;;;;;;;;;;;;;;;;+***+++;:,,,,,..,............,,,,,,::::::::::;:;;;;;;;;;;;;;;;;;:;;::::::::::,::::::::;+;;++;+****+++****+++++;+;::::;++++++++++++;;;;++;;;;;;;;;;;;;;;:::;;;;;;;;;;;;;++;;;;;;
+*******++++++++++++++++++****+++;;;;;+**%%%???%?*??*****++******+???*??**??????%%S%??????????***************?++**+++++**?*********************+****++++;;;;:::::,,,,,,,,,,,,,,,,,,,,,,:;:;++++++++;;;:;;:::::;;;;;;;;;;;;;;;;;;:;+++;::::,,,,..,,............,,,,,,:::::::::::::::;;;;:;;;;;;;;;::;:::::::,,::,,:::,,:::;+;;;;+++;;+++;+;::::::;;;;;;;;++;;;;;:;;;;;;;;;;;;;;;;;;;:;:;;;;;;;;;;;;;;;;;;;;;;;;;:
+*******++++++++++++++++++******+***++;;;;;++++*?*??*****++*******??*?*??*???%??%%%%??????%???***??*********?*****+++++**?****?*++************+**++*++++;;;::::::,,,,,,,,,,,,,,,,,,,,::;;;**+;;;;**;::::;:,::::::;;::;;;:::;;:::::;+;::,::,,,,..,,,.,........,,,,,,,:::::::::::;::;;;;;:;;;;;::;;;::::::::::,,::,:::,::;:;;;;;;++;:,,,,,::::,,,:;;;::;;;;;::;::::;;;;;::;::;:;;;;;;;;;;;;;;;;;;;;+;;;;;;;;;;;;;:
+*******++++++++++++++++++++++********++++;;;;;;;+*****++*++***+**?****????????*?%%%?*%??%???****???******??***++**+++++**??***++++**********+++*++*++;+;;;;;::::,,,,,,,,,,,,,,,,,,,::::;;*?***?%%S?+*?**;;;;;;;;;*+;+;+;;:;+;;++++*?*?+++::,,,,,,,.,,,,,...,.,,,,,,,:::::::;::;;;;;;;;;;;;;;::;;::::::;::::,,:,,::::::::;;::::::::,,,,;;;+;;;;;;::;;;;;;;;;;::;;;;;;;;;:;;;;;;;+++++++++++++++++++;;+;+;;;;;;;;
+********++++++++++++++++++++++++*****++++;;;;;;:;:::;+++++++**********??????%?*???%?*%??%??*+*?*???*????*??*?*+***++++**********++++*****++++++++++++;;;;::;:::,,,,,,,,,,,,,,,,,,,,::::;;;+*?????%??*******************?*****+*++**???++;::,,,,,,,..,.,,,....,,,,,,,:::::;:;;;;;;;;;;;;;;;;;;::::::::::::::,,,,,,:::,:,,:::::,,,,,...,,:;;:;;;;;;;;;;;;;;;;;;;;;;++++;;;+++++++++++++++++++++++++++++++++;+;;+;
+*******+++++++++++++++++++++++;;+**+++++;;;;;;;;;;;:::;;;;;;++*********?*?????*?????*??????*+**?????**?**?***+++*+++++***++**+++++++++++++++++++**++;;;;;;:::::,,,,,,,,,,,,,,,,,,,,:::::::;;+*??*???*++****+*++++++++*****++++++**?**+;::::,,,,,,,,,,,,,,....,,,,,,,,,::::;;;;;;;;;;;;;;;;;;::::::::::::,::,,::,,:::,,,:,:::,,,,,:,....,::;++;;;;;;;;;;;+++++++++++++++++++++***+****+++++++++++++++++++++++++;
+*********++++++++++++++++++++++++;;++;;;;;;;++;;::::::;;;;;;;;;+++******??????*??????***???**+*???***???*****++++++++**++++++++++++++++++++**+++*++++;;;;::,:,,,,,,,,,,,,,,,,,,,,,:::::::::;;;++**??*+++++++++++++++**+*+++;;;+***+;;;;:::,,,,,,,,,,,,,,,,,,,,,,,,,,,,:::::::;;;;;;+;;;;;;;;:::::,::,,::,,:,,:,,,:::,:,,:,,,,,,..,:,,...,,:+++++++++++++++++++++++++++++++************++++++++*********+*++++*+
+**+++++++++++++++++++++++;;;;;++++++++++;;;;;;:;::::::;;;;;;;;+++***+***?*******??*??****?***+*???**????*****+++++++++++++++++++++++++++++++++++*++++;;;:::::,,,,,,,,,,,,,,,,,,,,,::::::::;;:;;;;++++;;;;++++;++;;+++++++++++++;;:::::::,,,,,,,,,,,,,,.,,,,,,,,,,,,,,,,,::::::;;;;;++;;;;;;:::,:,,,,,,,::,::,,,:,:::,,,,,,,,,,,.,....,,..,,:;**********++++++++++++++++++++++++++****************?**?*********+
+++++++++++++++++++++++++;;;;;;++++++++++;++;;;::::::::::;;;+++++++++++********?****???+*+****++**??*????****++;;+++++++*+++;;;;;;+;;;+;;;+++++++++++;;;;:;:,,,,,,,,,,,,,,,,,,,,,,::::::::::;;;;;;;;;;;;;;++++;;;;;;;;;;;;+;;;;::::::::,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:,:::;;;;;;;;;;;;;:::,,,,,,,,,,::,,:,,,,,::::,,:,,:,,.....,,,......,;++*??*******+++++++++++++++++++++++++*************??????*********+
++++++++++++++++++++++++;;;;;;;;+*+;+;;;;;;;;::::::::::::;;;++++*++**++********?*****?**+++*?**++*??????*****++;;++++++++;++++;;;++;;;;+++++++++++++;;;;;;:,,,,,,,,,,,,,,,,,,,,,,,:::::::::::;;;;;;;;;::::::;;;;;;;;;;;;;;;;:::::,,:,,:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,:,::::::;;;:;;;:::::::,,,,,,,,,::,,::,,,,,,:,,,,,,,,,........,,.....::+**********+++++++++++++++++++++++++++***********??????*********+
++++++++++++++++++++;;++;;;;;;;;+;:;:;;;;;;;;;:::::::::;;;++++++++*+++++++++************+++*??*++***??*****+++;;;;+++++++;++;+;;;+;;;;;+;;;;++++;++;;;;;;::,,,,,,,:,,,,,,,,,:,,:::::::::::::::;;;;::::::::::::::::::::::::::::,,,,,,,,,,,,,,,,,,:,,,,,,,,,,,,,,,,,,,,,,,,:::::::::;;;;;::::::::,,,,,,,,,,::::,,,,,,,,,,,,,,,:,........,.,,...,:;++++++****+++++++++++++++++++++++**+***********???????********?+
+++++++++++++++++++++++;;;;;::::::;++++;;;;;;;;;;::::::;;;+++**+++++++++*+++++***+*******+++*?*+++*????****+++;;;;**+++++;+;;+;;;;;+;+;;;;;+++++++;;;;;;;:,:,,,,,,:,,,,,,,,:::::::::::::::::::;;::::::::::::::::::::::::::::::,,,,,,,,,,,,,,,,::::,,,,,,,,,,,,,,,,,,,,,,,:,,::::;;;::;;:::::,:,,,,,,,,,,:,::::,,,,,,,,,,,,,,:,.......,,,.....,,,;++++++****+++++++++++++++++++++**************???????*********?*
+++++++*++++;;++;;;+;;:::;;;;:::::;;;;;;;;;;;;:;:;;;;;;;;++++**+++++++*++***+++++++++*?***+++****+*????**++++;;;;+++;+;++;;;++;;;;;;;;;;;;;++++++;;;;::+;,,:,,,,,,,,,,,,,,,,::,::::::;::::::::::::::::::::::::::::::::::::::,,,,,,,,,,,,,,,,,,::::,,,,,,,,,,,,,,,,,,,,,,,,::::::;;:;::;:::::,,:,,,,,,,,,:,,::::,,,,,,,,,,,,,,,..........,,....,,:;+++++********+++++++++++++++*******************??**********??*
++++++++++;;;;;;;;;;;:,,;;;;;;;;;;::::::::::::::::::;;;+;++++**++**+;+*+*+*****+++++++*?***+;+*+*+*??**+;+++;;;;;++**+;+++;;;+;;;;;+;;;;;;;;;;;;;;;;;;:;;,::,,,,,,,,,,,,,,,,,::::::::::::::::::::::::::::::::::::::::::::::::,,,,,,,,,,:,,,::,,::::,,,::::,,,,,,,,,,,,,,:::,:::::::::::;::::,,,:,,,,,,:::,,::::,,,,,,,,,,,,,,,...........,,,,...,:+++++****************************************************????*
+++++++++++;;;;+;;:,::.,:;;;;;;;;::::::::::::::::::::;;++++++**+**?*++**+****+***++*+;+***++++++*******++++;;;;;++++++*+++;;;++;;;;;;;;;;;;;;;;;;;;;;;::::::,,,,,,,,,,,,,,,,,,:::::::::::::;:::;;;;::::::::::::::::::::::::::::,,,,,,,,:::,,::::::,,,,:::::::,,,,,,,,,,,,::::::,:::::,:::::,,,,,,,,,,,:,::,::::,:,,,,,,,,,,,,,.............,::,,,,,:;+++***************************?????*******************????*
+++++++++;;;;::;;::::,,,,,,:::::::::::::::::;:;;;;;;;;;;++++++*+*???*+++*+***+****+**+++****+++++**+++*++;;;+;;;+**;++++;::;;+;;;;;;+;;;;;;;;;;;;;;;;::::;::::,,,,,,,,,,,.,,,,,:::::::::::::::::::;;;;;:::;;;;:::::::::::::::::,,,,,,:,,::::::,:::::,:::::::::,,,,,,,,,,,,:,:::::::::::,,:,,,:,::,,,,,,,:,,,,::,::,,,,,,,,,,,,..............,,..,:++;:;+***********???**?*********???????????*******+**********+
+++++++++;;;;;:::,,,:,,,,,,:::::::::::::::;;+;;++++;;;:;++++++*+?%???*++++****+***+*?*++;+*++*+;;+**++;;;;;;;++++++;++++;;;:;+;:;;;+;:;;;;;;;;;;:;::::::::;:,,,,,,,,,,,,,,,,,,,,:::::::::::::::::;;;;;;;;;;;;;::;;::::::::::::::,,,:::::::::::::::::::::::::::,,,,,,,,,,,::,:,,:,:::,::,,,:,,,:,::,,,,:,,:,,,,,::::,,:,,,,,,.,..............,:,...,,:;+;+**********??????*********?????????******++*++*+++++++++
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;;;;;:,.,::,,,..,,,,,,,,:,,::+*+***++;:+*;+;+++;++;;+**+;*?****?%SS%%%*+*?*+*?*+?%??++?**+;++;;:::::;+;+;+;+;+*;;++;;;;;+;:;;;::;;:::;:;;::::;;;::;;;;:,,;:,:::::::::;;;:::,,,,..,:::,,:::;:::::::;;;;;+++++++++;;;;+;:::::::::::::::::::::::::::::::::::::::::,..,,,,,,.,,.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.,,,,,,,,,,,.......................,:::,:+;+;+++++**************??????????????????????????????*
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