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