:::::```````````````````````````````````````````````````````````````````````````````````````````````
::::````````````````````````````````````````````````````````````````````````````````````````````````
:::`````````````````````````````````````````````````````````````````````````````````````````````````
::``````````````````````````````````````````````````````````````````````````````````````````````````
````````````````````````````````````````````````````````````````````````````````````````````````````
```````````````````````````````````````````:````````````````````````````````````````````````````````
`````````````````````````````````````:|||I%%%%XUIIUI=:``````````````````````````````````````````````
```````````````````````````````::::UX%&XX&@&@@@@@@&%%XU=:```````````````````````````````````````````
````````````````````````````:=U%%%X%%%%%%XXX%@@@@@@&&&&&&U|`````````````````````````````````````````
:::::::::::````````````````:IX&%%%%&&@@&&@&&&%XXX&&@@@@@@@@&=````````````:```````````:::::::::::::::
::::::::::::::::::::::::`:=%%%%%XX%&&&&&&@@@&&&&%XX&@%&@@@@@&I:```::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::U&&%&%XXUUU%X%%%%%X%%&%&&&&&&%%&&&@@@X:`:::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::U&&%XUUXUUUUUUXXXUI=IIUU%&&&&&&&%X%&#@@%|:::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::IX%%XUUUUIIIIIII==||::=I=UXUX%%&&&&&%&@@@&=::::::::::::::::::::::::::::::::::
::::::::::::::::::::::|UUUUUUII=||:::`````:`:|::::::=IIUX%&&&@###&=:::::::::::::::::::::::::::::::::
::::::::::::::::::::::IUUXUUI|=:`````..````````````.``::==X%&&@##@%=::::::::::::::::::::::::::::::::
||||||||||||||||||||||IXXUUI|:::``````....```.......````::|IUU%@@@%=||||||||||||||||||||||||||||||||
UUUIIUUUUUUUIUUUUIIIUIUXUXU||::::```````.........`````:::::|=IX%@@&UIIIIIIIIIIIIIIUUUIIIIIIIIIIIIIIU
UUUUUUUUUUUUUUUUUUUUUUUXXXI||||::::`````.....`..````::::::||=IU%@@&XIIIIIIUUUUUUUUUUUUUIIIUUUUUUUUUU
UUUUUUUUUUUUUUUUUUUUI==XXU=||||::::`````........````:::::|||=IIX&&%UIUUUUUUUUUUUUUUUUUUUIIUUUUUUUUUU
UUUUUIUUUIIIUUUUUUUI|I|=XI||||::```````````....`````:::::|===IUU%%XIIUUUUUUUIUUUUUUUUUUUUXXXUUUUUUUU
UUIIIIIIIIIUUUIIIII=::=|XIUX%&&%XXUI=||::```````````````:|===UUUXXUUIIIIIIUUUUUUUUUUUUUUUUUUIIIIIIII
IIIIIII=IIIIIII=IIUI``:IU%#@############@&%%%XX%%XXUUUUUUUUU=IUXXUUIIIIII=IIIIIIIIIIIIIIIIIIIIIIIIII
:::```..``:=X%UIIX%X:``=%&&&@@@@@@@@@@@@@@@##@@@@@@@@@@#@@@@&IXIIXUIUUUUUUUUUUUUUUUUUUUUUUUXXXUUIUUX
===|||====IUXXUIIUUI:```I&&&&@@@@@@@@@@@#&I=&###########@@@&&%%II=IUUUUUUUUUUUUXXXXXXXUUUUUUUUIIIUUU
UUUIIIUUUUUUUUUUUUUU|`.`IX%%&&&&&@@@@@@@%|``|@#@@@@@@@@@@@&&&IX=:=IIIIIIIIIIIIIIIIIIIIII==I=========
UUUUUUUUUUUXXXXXXUUXI```=IUX%%%%%%&&&&&I::``:I@@&&&&&&&&&&%%XII:=XXXUUUIIUUUIIIIIIIIIIIIIIIIIIUUUUUU
UXXXXXUUUUUUUUUUUUUUUI====|::||====II=|||:`::|=X%%%%%%%%XXXUI=::IIIIIIIIIIIIIUUUUUUUUUUUUUXXXXXXXXXX
IIIUUUUIIIIUUUUUUUUUUUIII=||:::::::::|||:``::|||::||||==IIIUI:|=IIIIIIIUUUUUUUUUUUXUUXXXUXXUUUUUUUUU
==IIIIIIIIIIIIIIIIIIIIIII|||||::||:::=XXI=|=IXU|:|::::::|=IUI=IIIIIIIIIIII======IIIIIIIIIIIIIIIIII=I
IIIIIIII=II====I=======II=||:||:|::::::|U%%%XU|:::|:::|||=IUUUUUUUUXUUUUIIIIUUUUUUUUUUUXXXXUUUUUUUUI
UUUUUUIIIIIII====IIIIIUXXI||::|:|IUUII===III=|||==|::||||=UUUUUUUUUUUUUUUUUUUUUUUUUUUIIIUUUUUUUUUUUU
UUUUUUIIUUUUUUUUUUXXXXXX%%I=|||::::|====||=IIUUUI=|:|=||=UUIIIIIIUUUUUUUUUIIIIIIIIIIIUUUUIIIIIIIIUUI
IIIIIIUUUUUUUUUIIIIIIIIX#@XI=|||::::||=======||:::|=|||IUIIIIIIIIIIIIIIIII========IIIIIIIIIIIIIIIIII
IIIIIIIIIIIIIIIIII====X@#@&%U=||:::::::|||||||::|||||=IUUIIIII===IIIIII=================IIIII=======
III==IIIIIIIIIII====I%&##IX&&%I=|:::````````::::|||=IXXXU=IIIIIIIIUUIIIIIIIIIIIIIII=====II==========
=II================U&&&#&=|I&&&%U=|:::::::::::||=IUXXXX@@IIIIIII============I======IIIIIIII=======II
II==============IUX&&&&&I|::|X&&&%XUIIIII====IUXXXXXXX@#@XIIIIIIUUUIUUUUUUUIIIIIUUUUUUUUUUUIII===III
IIIIUX%&&&%%&X&&&&&&&&@U=|::`:I%&&&%%%XXXXXXXXXXXXXXX&#@&&UUUUUIIIIIIIIIIIIIIUUUUUUIIIIIII==========
&@&@@@@@@&@#@&&&&&&&&&@=:::````|X%%%%%%XXXXXXXXXUUXX&#@&&&&UIIIIIIIUIIUUUUIIUUUUUUIIIIIIIIUUIIIIIIII
&&&&&&&&@@#&&&&&&&&&&&&= ```````:IXXXXXXXUUUUUUUUUUU@#&&&%&%UIUUUUUIIIIUUUUIUUUIII===IIUUUXXXXXXUUUU
&&&&@&@@@&&&&&&&&&&%&&&=  ````````:IUUUUUUUUUUUI=|=IU@&&%&&%%XXUIIIIIIIIIIIIIIIIIIIIIUUUXXXXXXX%X%XX
%%&&&&&&&&&&&&&&&&&&&&&U. .:``````````::||=II=|::||`%@&%%&&&%%%@&%UI=======IIIIIIIIUUUUUUXXUUUUUXXXX
%&&&&&%%%%%%&%&&&&&&&@@&%&&@&UX%=`````````````:||: =@&%&&&&&&%X%@&&&%XUI===========IIIIIIIIUUUUUIIII
&&&&&%%%%%%&%%%&&&%%&@####@#@@X@@I``````````:|=|. `&&&&&&&%%%%XX%&@@&&%%%XUI============IIIIIIIIIIII
&&&&%%%%%%&&%%%&&&%%@@###@@@@@X##%`````````:||:.  U&&&&&&&&%XX%XX%&&@@&&&%&%%%XUUIII=||=UUUIIIIIII==
&&&%%%%%%%&&%%%&&&&&@##@@#@@@@X@#&`````````````:=U@&&&&&&&%%%XX%%X%%%&&&&&@@&&&&%%U=|::|||||||||==IU
@@&%%%%%%%&%%%%%&&&&&@##@@@@@@X@#&. .```:==IX%&&@%@@&&&&&%%%XX%XX%%X%%%&&&&&&&@@@&&&%XU=|||||||||IUX
@&&&%%%%%%&&%%%%@&%&&@@##@@@@@U@#@` ..`|===IIII=`X&@&&&&&&%X%XX%%%XUUX%%%%%%%&&&&@@@@@&&%XIIIUUUUUUX
@&&&&%%%%%&%%%%%@&%&&@@@#@@@@@X@##| .:=:`:::```|I&&&&&&&&%%%X%%X%%&&U|%&&%%%%%%&&&&&@#@@@@%UUUUUUUUU