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