%%%%%%%%%%%%%%%%%%%==%%%%==%&&&&&&&&&&&&&&&%=?+!!;!*==%===+*!!!****+?=%=???%&&&&
%%%%%%%%%%%%%%%&&&&&&%%==%&&&&&&&&&&&&&&&$$&&&&&%=*!!=%%%%%?::;!***!*+?%+*!+=&&&
%%%%%%%%%%%%&&&&&&&%%%=%&&&&&$$$$$$$$$$$$$$$$$$$$$$&%%=*:;%&=!;;!::!!**?=!***=&&
%%%%%%%%%%&&&&%=%%%&&%&&&&&&$$$$$$$$$$$$$$$$$$$#$$$$$%=--;?%&&?--~~~-;**+=*+++==
%%%%%%%%%&&&&&==&&&&&&&$$$$$$$$$$$$$$$$$$#$$$$##$####$=:-;*=%&&?-~~~~-;+??%==+%&
==%%%%%%%&&%%%%&&&&&&$&$$$$$$$$$$$$$$$$$$$##########$$$%?%$&$$$$*-~~-:;*=%&%%=&%
???%%=%%&&&%&%%%%%&&$$$$$$$$$$$$$#$$$$$$$$$######$$$$$$$$$$$$$$$=+:-!+*+?%&&%%&&
*???==%%&&%%=%=%%%&&$$$$$$$$$$$$$$$&&%%%&&$$$###$$$&&&&&&$$$####$$$&%?++?=&$%%&%
+???==%&&%%%====%&&$$$$$$$$$$$$$$$&&%%%===%&$$$$$$&&%?=%%&&$#####$$&%?&&==%$&&&*
???===%&%%====%&&&&$$$$$$$$$##$$$$$&%==+?+?%&$$$$$&%=?+*=&&$$$$$###$%?;*=%%&$&%;
??===%&&=====%&$&&&$$$$$$$$###$$$$&%===++??=&$$$$$$&&%%=?=&&&&&&$###$&=:;*=&&&%?
++?=====????%&$$%&&$$$$$$$####$$$$$$&==%%%&&&&&$$$$$#$&$%=???==&&$###$&%::;***??
;*+?==?????=&$$$%&$$$$$$$$###$$$$$$$$$$&&%%%=?=&&$&&$$&$$%%+*+=%&$####$&%;:!?=*;
!*???????=%&&&$&&&$$$$$$$###$$&$$$$&$$$&%=+!;;*+?==%%&&$&&&%+!=%$$#####$&%;:!!+=
*+??????%&&%%&$&&$$$$$$$$###$&&%%&&$$&&=*!:-~~~:;**?=%%%&&$&=?=&$#######$&%!;;:!
+?????=&&&&&$$$$&$$$$$$$####$&=%&&&&&%?!:-~,,,,,:;-!??==%&$&==%&$&&&$####$&%;;;!
??++??%%%&&&&$$$&&$$$$$$####$&%&&&$$&?*:-~,,,...~;;;;;!+?%&%=%%%=??=%&$###$&+:;!
***!*+==%&%%&&$$&&$$$$$#####$$&&&&$$%+!:~~,,.....,,...~;?===%%=+*+++==&$###$&*:;
;;;;!!+=%%=%$$$$&&$$$$$#####$$&&$$$&=*;-~,,,....''''',:!?====+*!*???==&$####$&;-
::::;*+===%&&&&&&&&$$$$######$$$$$$&?!:-~,,,....'''''.-!*+++*+???====%&$#####$=-
:--:!++?+=&&%=%&&&&$$$$######$$$$#$%+!;-~,,,,,,..,,,.,,~:*?=+++==%%%%&$#######$+
:-:;!*++?%%=?+=%&&&&$$$$$$###$$$$#$&%=?!:~~~~~-*?%&%?+*:~:+=%=?=%%&&&$$########&
!;:::-;+=%%??+=%&&&$$$$$$####$$$#######$?;----*=%%=?!:-:!;!?=%%%%%&&&$$$$$$####$
**;:;--*?%=+?===%&&&&$$$$$####$###$&&&$$$=;--:*?===?**;--!!!=%==%%&&&%%%%&&$####
++*!;-:*?=+++??==%%&&&$$$&&$$$###$$&%=%$$&*~,-*?%=?*;;+?!;;;?=&%===?++=%==%&$###
++*!::!*??+++++??+=&&&&&&&&$$######$&&=&$$!,,-!+%&$$&%===*::*?%%==??+*!??+%&$$##
++*::;!!!******++?=%&$$$$$$#########$$$$#%-.,~-;=%=%&%$&?!--;*%&%%?+?%+??+%&$$##
++!--:::--;;;***?==%&&$$$$$#########&=?=$?~,,,~~!???++?!~,,,-!%&&$&?+%%=?%&$$$##
**!:-~-,.,--:**+?+??%&&$$$$$$$$##$$&=?+=&!~,,,,,,~:::-~,...,~!=&&%&%=%%%%&$$$###
+**!:~~,~~,~!**++???=&&&$$$$$####&+!;!*&+-~~~,...,,,,......,~;?%%%%%&&&&&&$$$###
*++**;:-~~,-;**+?++?=%&&$$$$#####%+!;!?$=*;;-~,.......''.,,,~:+==&&$$$$$$$$$$###
!*+++*!;--::-;*+++++==%&&$&$####$&?*!*%$$%=%+:,.........,,~~-:*%%%&&$$$$$$$$####
*******!!!!;::**+***??%&&=+$#####&%?+=$$$&=+%*...,..,,,,,~~~-:*=%&&&$$###$$#####
++++******!;:;**+***+?=%%*!$$####$&=%&$$$?*~,.''..,,,,,~~~~--;*?=&&&&$##########
++++++****;:;***+****+=%?*?$$$$##$&&&$$%+~-~......,,,,~~~~~-:;!?%=%%%$##########
?+++++++**!!*********+=?*+&$$$$##$&&&&%=%=*!:,....,,,,~~~~--:;!++*+?&###$$######
???++++***!!?=???????==??=&&?=$$&%%&&&&&%%%=+*-,,,,,,,~~~~--:;!!!!*%###$$$$$$###
??+++++****?*?==?????????+=%*;?%=+*+***+?+++=%=+:~,,,~~~~~--:;???+&####$$$$$####
??+++++**++%&$$$$$$$$$&$&??+!;!*+*;:-~~--:;;;!!!;-~,~~~~~---:*&&%######$$$$$####
?????+++++++++++++?&$$%+!;!**!;;;;;;;;*==++*!;-~~~~~~~~~---:;+%&%######$$$$$####
=?????++++++++++++%$$%+++?==%=?????=%%&&&&&%?*:-~~~~~~----:;*!+&%$###$$$$&%$##$$
====??????+++++++?$&%=!;*?%&%%?%%%%&&&$&?;-~~~~~~-------::!**-;%%&$$&&==$$&&$$$$
=====???????+++++=&%=?*;:-:;**!=%%=%&&&%!:-~...,~----::;!*++-~:======&&%$$$&$#$$
======???????++++%&=?*!++*;::;;+=??=&&&&?!;-~,,~--:;;!*+?=?:~~-*=+?%=&$$&&$$$$$$
=======?????++++?&=?+!-:*===*;;!???=%%%&%=?+!;;;!*++?====?:~,,-;!!%&&&&$$$$$$&%%
%=======????++++=%?++?*;:;+%%+;;!???===%%&&&%%=%%%%%%%%%+:~,,~--:;%&&$$$%%$$$&&&
%%======???????%%=??%&%=+!!=%%?*!!+???===%&&&&&&&&&&&&%*:~~,,~-~-;&&$$&?+%&$&&%%
%%%=====??????%&%+*+?%=+=?*%&&&%%?++????=%&$$$$$$$$$&=*:~~~~~~~~~;?=%%!!=%&&&%%%
&%%%%=====????=$=+!;;=%*+%?=%=%&&?++???==%&$$$$$$$$&=!:-~~~--,~~-;*+?!-*=%&&$&%%
&&%%%%======??=$%?!;!+=%!=++*;:!==++???=%&&$$$$$$&&=*;:-~~-~~~-~~:*+*-~!?=%&&&&%
&&&&&&%======??%$=*;:::*?;+*!:--:+?+???%&&$$$$$$&&=+!;--~---~:-,~:!!:~~!?=%%&&$$
&&&&&&%%%=====??&$=*:---;;:::-~~-!++???=&&$$$$$&&%?*!:-:-~-::!:::::-~~;?==%%&$$$
$&$&&&%%%%%%==??=&&?!----:---~~~-;?????=&&$$$$$&&=+*!;:-~-;!*::;-~,,~-*???+=&$$$
$&$$&&&%%%%%%===?%$&+;----:-~~~-:!????%$#$&&&&&&%+*!!!;;!+++;:*:~~~~,:*?+;*%%&%%
$&&&&&&&&&%%%%====%$&+;------~--:*???%$$###$%?*+=++??+??==+!*?+-~~~~,~-;;!=?%=+%
$$$&&&&&&&&%%%%%%%=&$&+!:---:--:!??=%$$######$=?=====%%%%?++==!~~---~~~:!**==-?+
$$$$$$&&&&&&%%%%%%%=&$&?!:--:::;*?=&$##########&%%%%%%%%=?+==!-~~::--~-**;?%!-!+
$$$$$$$$$$&&&&&&%%%%%$$&?*;:;;:;*=&############$$&&&&&%===%&?;:~~----~;*;?!=;*?=
$$$$$$$$$$&&&&&&%%&%%$$&=?*!!!:;+$#############$$$&&&%%%==&%?*:~~~~~~:!:+*;=*===
#$$$$$$$$$&&&&&&&&&%&$&%?+**!;;!?$##$$$$$$$#####$$$$&&%==%&=*-~~;-~--::;=::????=
##$$$$$$$$$$&&&&&%%&$$%?**!;;;;!%###$$$&%%$$####$$&&&%=?%&&=!-~:;:::-::=!-;?+*++
###$$$$$$$$$$$&&%%&$$&=+!;;;;;!+$###&&&=+?%%####$&&&%===%&&=!---:!+!::+=::;*::;*
###$$$$$$$$$$$$%=%&$$%?*!;;;;!*=####&%%%??+?=&$$&&=?=+*=&&%?+;--!+?!!!%=*!;!--;!
##$$$$$$$$$$$$$=?=&$&=+!;;;;;!+&####$&%&%?***!=++=?*?++%&&%=*--:;*+*+=%=+:*;-:::
#$$$$$$$$$$$$$$&+!?&%?*!;;;;!*?$###$&&&=!:-:;*=???!!??=%&%%?;;!****+=&&=*-+;:-::
####$$$$$$$$$$$%=??&=+*;;;;;!*=####&%=?!+!~~;!;;;~~~!?%%&&%?+?==+**%%&&?!:?+;!+;