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