@@@@@@@@@@@@@@@@@@@@#@S+#@@@@@@####S%?*+=---;;;;;;;;;----=+*?SS##*+*?S##@@@@@##S
@@#####@@@@@@@@@@@@@@##??@@@@@####S%%*+=---;;;;;;;;;;;;---==+?SS#S++?%##@@@@@##S
@###@@@##@@@@@@@@@@@@#S%+S@@@@#@@#S%**=---;;;;;;;;;;;;;;----=*?SS#?+*%S##@@@@##S
#@@@@@@@@@@@@@@@@@@#%%S#*?@@@#@@#SS?*+=--;;;;;;;;;;;;;;;;---==*%SSS+*?%S##@@@##S
@@@@@@@@@@@@@@@@@@#%%?#@%*#@@#@@SS%?+=--;;;;;;;;;;;;;;;;;---==+*%S#?+?%S####@###
@@@@@@@@@@@@@@@@@S%??*%S#?S@@@@#S%?*+=++++==;-;;;;-+;;;---+=++*??%SS*??S#######S
@@@@@@@@@@@@@@@@@#SS????*?%@#@#S%?********+++++=++*?++=***?*???%?%%S?*?%S######S
@@@@@@@@@@@@@@@@@@@%*?*S*-*##@#%**??%%?*+++****+++++**+*????%SSSSSSS%??%SS#####S
@#####@@@@@@@%??#@@%SS%%%==%#S%*%%SS%*;;::,,,:--;;====-;;--=?S######S?%%%SS#@##S
SSS#@@@@#@@@@#@S?S#?SS%*???#S??%##SS%*--;:,,,,:::,,::,,,--+*%SSS#####S?%%SSS###S
#@@@@@@@@@@@@@@@??#??S*=%%?%?%SS##S%%%?*+=+=-;,;::::,,;;+?%SS##S##S###%%%%SS###S
@@@@@@##@@@@@#@S?*S#SSSS*?%?%SSSSSSSSSS%SSS%?*;::;;:;*?SSSS###########%%%%S%%##S
@@@@@########SS??**%SS%???%SSSSSSSS%SSSSSSSSS%*-:;-=*SSS##############S%%%%%%S#S
@@@#SSSS#####%#%?%%SS?S%??SSSSSSSS#######SSSSS%=;::=S#######@@#########%%%%?%%##
@@##SSSSS###@#%%#@SS%%S?=##SSSS###SSSSSS###%SSS*:;:=SS####@##SSSSSSS###S%%%???%#
@@#SSSS#####@@@#SSS?=?%?*SSSSS#@#####S%%%SSS%S%?::,+S######S%%SS#####@@#%%%%*?*S
#############S?%SS%?-=*%#SSS#@@@#@@@@#SSSSSS%%S?:::*S#####SS?%##@@@@@@@#S%?%?***
##########@@?*%##SS#*-?SSSS#@@@#@@@@@@#S###S%%S%;;;?S#######SS@@@@@@@@@@#S??%?*+
########@@@SS@##%%#@@#S%%S#SS@#SS#@@#S%%%?%S??S%+=+%######SSS###@@@@@@#@##%???*+
###########@###SS#@@@S?*?S##S%%?*?%SS%????*++*SS?+?%##SS%????%S##@@######@#%?***
#########SS##SSSSS@@@%??*S####%*+===+****+=+?%SS%?%S##SS?**??%%SSSSSSSS####S%?**
########@%##SSSSS#@@@##%*%####SS%?=;::::;-+%S%SSS?%S##S##%?++=++*?%%S#######S%**
#########S##SSSSS#@@@@#S**S###SS%?-::;;-=+?%S#S%*++?%S###S%*=-;=*%S##########S%*
######S####SSSSSS#@@@##S%**S###SS?=--::;+?S##S??*+**?%S##SS?+===*%S#######S##S%?
######S##@#SSSSS##@@@#S?S?*?####S%?+;;-+??SS%??%#####S%S##S%*=-+?S######SSSSS%??
#####SS####SSSS###@@#S%?%S?*?S###S%***+?%%%??%S@###@@@#SSS#S??++?S######SSS#S%?*
?%###S##@##SSS##SS#@#S%?%#S???%%S%%?*+*??*???S@#%?%S#@@#S%%S%???%S###SSS%S##S%**
=-*SS###@@#SS##SSS###S%%S##S%??*%?????*???%%###SS%S###@@#S%%%??%%%SS%%%SS###S?*=
==*%%S#@@@##@#SSSS###SS%#S#@#%%?%?%?%??%%%S@####SSS#@##@@#SSS%?%%%%%%%S###SS#?+=
++%?**%S#@S@#SSSSSSSSSSS#S####S%%%%%SS###@@##SS%%S##########SSSSSS###S?#@#S#S%*=
%##S%????%S#SSSSSSSS#SS#SS###S%?**%S#####SS%??***%%%%SSSS######SSSS##?*###SSS%*=
#@@@#S%???%%SSSSSSSS#SS%**S##SS%???S#S%%??**+=====++*??%%SSSSSSS%%SSS*?####%S%?=
###@@@##SS%??%%SSSSS#S+-=%#@@#%S%%%SS%?******+====+*??%%%%%%SSSSSSSSS*%####%S%?+
#####@@@@##S%???SSSS*---+%S###%%%%?%%SSSS%%???????%%%%%SSSSSS#SS%S##S*S###SS%%**
#####@##@@###S%%??S%=;;-*SS###S%??++++S##S%??*?*???%SSS######S???S##S*%SSS%%???*
#####@#S##@@###S%%??+--=?S%S##S%??*==-=%S*=;-:;:;-;-==+?S#@@S**+?S#S%*S%SSSS%??*
####@@#SSSS##@#%%%=---+%S#%%##S%???+---=*****=-;;;;;-=*?S#S%+++*%SSS*?S%SSSS%??*
####@@SSSSSSSSS?*---=*%##SS?S##S%??*=---+**+*?%SS#SS#SS%%%%*++*%SSS%+##SS%%%%?%%
####@#SSSSSSSSS%--=+?S#@#SSS%S#SS%?*+=--=*?*=---=====+*?SS?++*%SSSS??#SSS%%%%%??
####@#SSSSSS#S+;-=*%#@@@#SSSSSS#S%??*+---=?%?+======+*%SS%*+*%SSSS%*SS%%%%??????
###S@#SSSSSS%=;-=+%#S#@@@##S##SS#S%?*+=---=*%%%%???%SS#S%?**%SSSSS*%SS%%%*****??
###S@#SSSS%+;;-=*%#SSSS#@@###SSSS#S%?*+=----=*?%%SSSSS%%***?SSSSS??SSS%%?+*****?
##SS@#SSSS+;--+?S##SSSSS###@S%SSSS#S%?*=-;;;;---==++++++++?%SSSS?+?S%%%%*++++***
##SS##SSSS+-=+?S###SSSSS%*?%SSSSSSS#S%?+--;;;;;;;;;---==+*%SSSS%%*%%%%??*++=++*?
##SS##SSSS===+%####SSSS%*==+%%%%%SS%#S%?+--;;;::;;;;---=*%SSSSS##*??????**+=+**?
##SS#@SSS%=-==?SSSSSSSS?+--+%%%%%%S%###S?+=-;;;;;;;--==*%SSSS#@@S+*??*???*=+*?*?
##SSS##SS?=--=?SSSSS%%S*+=-+%S%?%%%S#@@#S%*+=-----===+?%SSS#@@@#S+***??%%*=+??*?
##SSS##SS?=--+%SSSSS%%%*+==+?SS??%%SS@@@@#S%?*******?%SS##@@@@##S*+**?%%%*++??**
###SSS##S?=-=*SS*=+?%%%?*+=+%SS??%%%S##@@@@##SSSSSSSS###@@@@@###S*=+?%%%??+*+*?*
###SSS##S*---*?=;;;;-+*++==+%%%%%%?%S#######@@#######@@@@@@@####S?++*??%??*+++*?
##SSSSSSS+---=-;;;;;;;;;--=+%S%%%%??%#########@@@@@@@@@@@@#####SS%*++*?%????+***
####SSSS%=-;--;--;;;-;;;;;-+%%%????%?###########@@@@@@@@@####SSSS%%?*????????***
####SSSS*-;-;;---;;;;;---;--+??????%?############@@@@@@####SSSSSSSSS%%?????%%%%?
#####SS*=-;;-;;;---=---**==-=-+????%?S###################SSSSSSSSSSSS%%%???%%%%%
######?-;;;-;;;-+???*++?%%*+==-=?????S@#S####S##########SSSSSS%%%%%%%%%%%???%%%%
#####S=;;;;;;-+?S%%%S%?%????*++=*????%@#SSSSSSSSS###SSSSSS%%%%%%%%%???????????**
#####%=-;;;;-*S#SS%?%%%%%%??%%%?*????%##SSSSSSSSSSSSSSSS%%%%%%%%%%?????******++=
#####S*-;;;-+?S##S%%%%%%%%%%%%%??????%##SSSSSSSSSSSSSS%%%%%%%%%%??????*****+====
######S=---;=?%##SS%%%%%%%SSSS%??????%##SSSSSSSSSSSS%%%%%%%%%%%??***??***+====--
#######%+=-==-=%SSS%%%%%%%%%SS%%?????S###S%SSSSSSS%%%%%%??%%%%?******??*+===----
########?+=----+%%%%????***?%S%%%%???####S%%%SSSS%%%%%???%%%??**+++++**+==------
########S?+=-;-=*??***++++++*?%%%%%??####S%%%%SS%%%%%%%%%????**++===*?*+=---;;;;
#########%*+-;;-=+*+++======+*%%%%%%S####S%%%%%SS%%%%%??????*++===+*?*+=--;;;;;;
#########S%*=-;;--=++==----=+*?%%%S##S###S%%%%SSSSS%???**??*+===++*??*+=--;;;;;;
##########S?+-;;;--===-----==+?S#####S###S??%%%SS%%??***?*+===++*????*+=--;;;;;;
###########%*=-;;;--==-----==+?S#####S###S????%%%%??**?*+===+**??%%??*+==--;;;;;