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