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