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