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