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