Doll Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠑⢄⡀⣀⢠⡤⠤⢀⡀⠀⣀⠴⠒⠀⠉⠉⠉⠐⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣀⠀⣾⡀⣙⠀⠀⠢⠀⠁⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠉⠁⢄⡠⠂⠀⠀⠀⢠⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢠⠜⢁⠀⠀⠀⢀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠈⠢⢄⣀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠃⠀⠈⠢⣀⠀⠀⠑⣄⡀⠀⢀⡸⠢⣄⣀⣀⡠⠎⠀⠑⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠉⠁⠘⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⠀⠀⠀⠀⠀⠀⢠⠖⠚⠓⠆⠀⠀⠀⠈⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⢰⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⠆⠀⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡏⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⠣⡀⠈⠀⠀⠀⠑⢤⣀⡀⢀⡀⠤⠊⠀⠀⠀⠀⣠⠊⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡴⠢⠦⢀⣀⣀⣀⣀⡠⠤⠘⢏⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠃⠀⠀⠀⣀⠠⠜⢄⡀⠀⠀⠀⣠⢧⡀⠀⢀⣠⠜⠑⠂⠤⢀⡀⠀⠈⠣⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⢰⡋⠀⠀⠀⠀⠈⠉⠉⠉⠁⡀⢉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⠟⢃⡀⠀⠀⠀⠀⠀⢇⠈⠋⠀⢨⠃⠀⠀⠀⢤⡀⠀⢀⡋⠀⠉⠐⠾⠡⠄⠀⠀⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⣷⣄⣴⠀⠀⠀⠀⠑⢄⡴⠁⠀⠀⠀⠀⠘⡄⠑⠋⠉⠃⠰⠀⣀⣀⣀⠀⠀⠀⠀⢈⠇
⠀⠀⠀⠀⡀⠤⠀⠒⠊⠁⠀⠀⡀⠔⠊⠀⠀⡈⠀⠀⠀⠀⣀⡀⣠⠤⠠⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠀⠐⠋
⠀⠀⠀⡎⠀⠀⠀⠀⡠⠴⠂⠉⠀⠀⠀⠀⢀⠃⠀⠀⠀⠘⢄⠀⠁⠀⠠⠁⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠑⠤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠁⠒⠔⠁⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡜⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⢀⡠⠚⢤⡀⠀⠀⠀⠀⡀⠌⠉⠑⠒⠒⠂⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠉⠉⠈⠉⠀⠀⠀⠀⠈⠁⠀⠈⠉⠀⠀⠀⣀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⣀⡠⠖⠓⢄⣀⠀⠀⠀⠀⠀⣠⡎⠁⠈⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠀⠀⠀⠀⠉⠛⠰⡆⠚⠋⠀⠡⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠙⢄⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠈⢢⡀⢀⣠⠗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⠀⠀⣠⡛⣆⠁⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡘⠁⠀⠈⠳⣄⡀⢨⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⢀⡍⠹⣁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡉⠀⠀⡍⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⡉⠉⠉⠉⠉⠁⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡒⠀⠄⠀⠁⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⢀⣀⠀⣀⡀⠼⡆⢄⠀⠂⠀⠀⠲⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠃⠀⠀⢀⠀⠀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠨⡇⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠀⠀⠑⠀⠃
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡀⠚⣄⢀⠀⠠⠂⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠈⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⢀⡴⢏⠈⠉⠉⡆⠀⠀⠀⠀⠀⠀⡀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢲⡄⣀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⢡⡈⠀⡇⠈⠉⠉⠉⠀⠉⠀⠊⠉⠐⠠
⠀⠀⠀⠀⢀⠀⠀⡀⠀⡀⠁⠀⠀⠀⡰⠀⢡⠀⠉⠀⠒⠀⠈⢹⠢⢄⠀⠀⢀⠄⠨⠒⡏⡀⠀⠀⠀⠤⢀⡀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠊⠀⠀⠀⠱⠄⡀⠀⠀⡠⡎⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⠀⠂⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠘⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢉⠒⠢⠄⡇⠀⡀⠈⡄⠀⠀⠀⠀⠀⡀⠔⠈⠀⠆⠀⠀⠀⠀⠒⠂⠉⡐⣄⡀⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⡌⠀⠀⠀⠘⡀⠀⡀⠀⢖⠒⠂⠉⠁⠀⠀⠀⠀⡆⠀⠀⠁⠀⢀⠒⠁⠀⠀⢀⠀⡐⠁⠀⢀⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠌⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠑⠠⢀⡀⠀⢀⡀⠔⠀⠂⠀⠐⠨⠉⠁⠒⠀⠤⣁⢴⢀⠔⠁⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠨⠀⠩⠀⠀⠸⠀⠐⠄⣈⠃⠂⠄⣀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠁⠀⠀⠀⠀⠀⠀⢸⠄⠐⡀⠀⠀⠀⠀⠁⠂⠒⠀⠀⠀⠀⢀⠊⠀⢀⠔⠁⠈⠐⠢⡈⣄⡀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢀⠀⣀⠠⠤⠀⠀⠀⠀⠐⠒⠈⢁⡠⠐⠁⠀⠀⠀⠀⠀⠈⠉⠓⠤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣡⠔⠫⠀⠀⠈⡤⣀⡠⠤⢤⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠑⣤⠊⡰⠁⢀⡀⠄⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⠀⠰⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⡀⢀⡀⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣠⠀⠀⠀⠶⡀⡄⢀⡴⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠬⠀⡁⠀⠀⠚⠐⠃⠈⢉⠀⠦⡄⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⣤⡄⠀⠀⢠⣤⡄⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠙⠁⠀⠀⠈⠋⠁⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⡠⠊⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠀⠀⠀⠀⠀⠀⡀⢕⠒⢀⡀⠒⡤⠀⠀⠀⠀⠀⠀⠈⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⡄⠀⢀⠒⠀⡇⠀⠓⡉⡑⠚⠀⠀⠀⠁⠀⠀⢤⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⠀⠀⠄⠸⠠⠂⠣⠤⠂⠠⠀⠀⢃⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠨⠀⠐⠊⠀⠀⠀⠔⠊⠒⠭⢉⠉⠈⡭⠭⠓⠑⠢⠀⠀⠀⠑⠂⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠈⠀⠲⠁
⠀⠀⠀⠀⠀⠀⣀⡉⡈⠥⠚⢀⠀⠠⡄⢠⠀⣀⠀⠀⠀⠀⡀⠀⡆⡠⠒⠁⡀⠓⠬⠁⢉⣀
⠀⠀⠀⠀⠀⠉⠀⠈⢂⠑⢴⠒⢀⠀⠀⠣⠀⡠⠂⠀⠀⠺⢀⢠⠋⠀⠀⡠⠐⡤⠋⡐⠁⠀⠈
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠆⠀⠂⠀⣠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣄⠀⠀⠀⠰⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠐⢄⠀⠀⡀⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⢀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣤⠶⠶⠤⠤⠤⠴⠶⡲⠶⣦⣤⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡛⠉⢀⠀⣿⣉⠥⠀⠀⠨⠀⠀⠀⠁⣀⠐⠀⠀⢹⣇⡌⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⡻⠛⡛⣮⠽⠧⠴⢆⡀⠉⣁⠀⡀⠈⠤⢀⣀⣹⠛⠉⠓⣟⠻⠳⢦⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⡔⠊⡠⡚⢀⣟⣀⣠⠤⠞⣃⠀⠤⢊⡰⢆⡐⠀⠤⠌⢓⣦⢤⡗⡈⢆⠁⠈⠳⣄
⠀⠀⠀⠀⢠⡞⠁⡠⠂⢠⢊⠈⠠⠃⣼⠉⠙⠲⠶⠶⠒⠚⠉⠀⠀⠈⠉⠉⠉⠉⠀⠀⣷⠈⠄⠣⠈⠢⡘⢷⡀
⠀⠀⠀⠀⣿⠀⡐⠁⡠⢡⠊⢀⠆⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠘⡄⠱⡄⠱⠈⣧
⠀⠀⠀⠀⢨⡏⠀⡰⢁⠂⠀⡌⠀⡄⡇⠀⠀⠀⠀⢀⡴⢦⠀⠀⠀⢠⢎⣷⠀⠀⠀⠀⢸⠰⠀⢻⠀⠙⢢⣇⣿
⠀⠀⠀⠀⠸⣧⣴⠟⡏⠀⢀⡀⠀⢁⢹⠀⠀⠀⠀⠈⠋⠙⠁⠄⢤⠈⠁⠋⠀⠀⠀⠀⡿⠀⡄⢸⠙⣋⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠟⠙⠶⠾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣼⠦⣆⣱⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⣀⠀⠀⠀⠀⠀⠓⠊⠀⠀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⠉⠛⠒⠒⠒⠒⠚⠛⠉⠉⠈⠛⠶⢦⣤⣀⣀⣀⣠⡴⠶⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⡄⠀⢠⠒⠢⡤⠊⢱⠀⣦⣀⠀⠀⣹⠀⠉⠁⠀⠀⠀⠀⠉⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠓⠦⡀⢀⣼⡁⠀⠈⠣⣀⠀⡠⠊⠀⢸⡏⠛⠳⠿⠦⣤⣤⣤⣀⣀⣀⣀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣹⠏⢘⡇⠀⠀⠀⠀⠉⠀⠀⢀⡔⠛⠶⠶⠦⢤⣤⣤⣄⣀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⢀⡼⠃⢀⡾⠙⠓⠶⠤⠐⢖⠂⠁⠀⠈⠒⠂⠀⠠⠄⣤⣤⢜⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢸⡇⠀⢸⡇⠐⡀⠀⠢⠀⢀⠑⠠⡀⠀⠀⠂⣤⣤⣠⣇⣠⡟⠁⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠈⢷⡀⢸⡁⠀⢡⠀⠀⠀⢄⡀⢄⠀⠉⣴⠞⠁⠀⠀⠈⢉⡟⠛⠓⢶⡞⠉⡉⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⡞⢻⣼⠇⢸⠀⠆⠀⢂⣀⣀⡴⢾⡶⠟⣚⣉⣳⣦⣄⡀⠀⢸⠀⠀⠀⡏⠀⠘⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⠟⠁⠀⠀⠀⢸⣀⠴⣤⣾⡀⠀⠀⠀⣙⣿⣍⣉⣀⣤⣬⡉⠛⠾⣤⣄⣘⣇⠀⢀⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠖⠚⠛⠳⣄⢀⡞⠁⠀⡼⠋⢁⠔⠈⠙⣦⠀⠀⠈⠉⠛⠶⠾⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⡾⠀⢠⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣇⠀⢨⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠺⠶⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣠⡤⠶⠶⠤⠤⠤⠶⠶⠶⡶⣤⣄⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡛⠉⠀⠀⣟⡉⠥⠤⠀⠀⠀⠀⠈⢀⢐⡒⠒⠨⣹⣄⡈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⡛⠛⡓⣮⢽⠧⠴⣄⠉⣀⡉⠀⡀⠀⠄⢀⣀⡴⠋⠩⢳⣗⠛⡳⢤⣀
⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠈⡀⠚⠡⣏⣀⣠⠤⣞⣁⡀⠤⢊⡴⣌⡒⠀⠀⠘⣓⡦⣼⠧⠑⠌⠢⡙⢷⡄
⠀⠀⠀⠀⢠⡞⠁⡠⠂⠠⢈⠊⠀⠁⡿⠉⠙⠒⠶⠖⠒⠊⠉⠀⠀⠉⠉⠉⠉⠉⠀⠸⣇⠡⡈⢆⠘⢄⠙⣆
⠀⠀⠀⠀⣿⠀⡔⠁⡐⢡⠃⢠⠃⠄⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢡⠀⢣⠀⢂⢹⡆
⠀⠀⠀⠀⢨⡏⠀⡐⠠⠁⠀⡀⠀⠀⡇⠀⠀⠀⠀⢀⡴⣦⠀⠀⠀⢰⢛⣧⠀⠀⠀⠀⡇⠀⠀⣧⠈⡗⣶⣀⡇
⠀⠀⠀⠀⠸⢧⣼⠟⡇⠀⢠⡁⠀⠇⣳⠀⠀⠀⠀⠈⠊⠛⢀⠤⢄⠈⠀⠁⠀⠀⠀⢠⠇⢰⠀⣏⢙⣹⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠋⠙⠦⠾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠀⠀⠀⠀⠀⢀⣾⢴⣄⣵⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⠀⠀⠀⠀⠈⠒⠁⠀⢀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠈⠉⠛⠒⠒⠒⠒⠛⠋⠉⠁⠉⠓⠶⣤⣄⣀⣀⣀⣤⠴⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢠⠒⠠⡴⠋⢱⠘⣄⡀⠀⠀⡇⠈⠉⠁⠀⠀⠀⠈⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⢦⡀⢀⣾⠀⠀⠈⠢⣀⢀⡴⠃⠀⢻⡙⠛⠲⠷⠤⣤⣤⣤⣀⣀⣀⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⣽⠋⢸⡇⠀⠀⠀⠀⠉⠀⠀⣀⠜⠛⠶⠶⠶⠤⣤⣤⣀⡀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⢀⡾⠁⢠⡞⠙⠓⠶⠀⠒⠖⠀⠁⠀⠁⠐⠂⠀⠀⢠⣤⡤⣋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢸⡁⠀⢸⠃⠰⠀⠈⢂⠀⠀⠐⠤⡀⠁⠒⢀⡤⢄⣼⡀⣰⠏⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠈⢷⠀⢸⠀⠀⢃⠀⠀⠑⢌⠢⢀⠀⢁⡶⠊⠀⠀⠀⠈⢩⠛⠛⠲⣶⠟⢉⠈⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⢠⡟⢳⡼⠃⣿⠀⠄⠈⢄⣀⣠⠴⢷⠲⠓⣋⣉⣳⢦⣄⡀⠀⡏⠀⠀⢰⠃⠀⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠋⠀⠀⠀⠀⣿⣠⠦⣤⣏⠀⠀⠀⠀⣹⢿⣍⣁⣠⣤⣌⡙⠓⠷⣤⣀⣸⡀⠀⡀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠛⠛⢷⣄⢠⠎⠁⢀⡝⠋⡠⠒⠉⠻⣆⠀⠀⠈⠙⠛⠶⠾⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⢀⡏⠀⠰⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢾⣇⠀⠇⠀⠀⢀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠚⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡻⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠊⠉⠉⠉⠑⠢⠄⠒⠉⠉⠙⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⣠⣖⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣱⠋⢹⡧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡟⠉⢳⡵
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣿⣀⡞⠁⠀⢀⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠙⣆⣼⣃⣀⡀
⠀⠀⠀⠀⢀⡴⠚⠉⢀⡠⢨⠏⠀⠀⡠⠋⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠳⡀⠀⡘⢇⠀⠀⠈⠉⠢⣄
⠀⠀⠀⡴⠁⠀⢀⡴⠋⠀⠊⠀⠀⡴⠁⠀⠀⠸⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠘⡄⢀⠘⡆⠀⠀⢆⠀⠈⠱⡀
⠀⢀⠎⠀⠀⠀⠈⠀⠀⡜⠀⠐⣴⠁⠀⠀⢠⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠸⠆⠀⢹⠀⠀⠘⠀⠀⠀⠘⢆
⠀⡎⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⡇⠀⠀⠀⡌⠀⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢂⠀⠀⠀⠀⠀⡇⠀⡀⡇⠀⠀⠀⠀⠀⠀⠘⡄
⠸⠀⠀⠀⠀⠀⠀⠀⢰⡇⠈⢀⡇⠀⠀⠀⡇⠀⠰⠤⠀⠀⠄⢅⣀⠀⢀⠀⠠⠐⠂⠁⠊⠀⠠⠀⢣⠀⠀⠀⠀⡇⠠⢀⣇⠀⠀⠀⠀⠀⠀⠀⢰
⡇⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠸⡇⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⣧⠀⢸⢻
⡇⠀⠀⠀⠀⠀⠀⠄⢸⠘⡄⢀⣹⣤⠎⠀⢠⣴⣿⣿⣿⢷⣦⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⢿⣶⣄⠀⠱⡄⢰⣅⢀⠇⢸⠀⠀⠀⠀⠀⠀⠀⢠
⠸⡀⠀⠀⠀⠀⠀⠀⢸⡆⠘⠇⡀⠀⠀⠀⣿⢹⠋⣿⣷⡄⢻⠀⠀⠀⠀⠀⠀⠀⣼⢩⣿⣷⡄⢻⢹⡇⠀⣱⠇⠈⡏⠀⣼⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠙⢄⠀⠀⠀⠀⠀⠀⠱⡀⡆⠡⠈⡄⠀⠻⠘⢆⡙⠛⣁⠜⠀⠀⠀⠀⠀⠀⠀⠸⢄⡛⠛⣁⠞⠸⠃⠀⡏⡀⢀⡇⡰⠁⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠑⠢⠄⡀⠀⠀⠀⠱⡘⢦⡀⢃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠀⣠⠞⢠⠃⠀⠀⠀⡠⠔⠚⠁
⠀⠀⢀⠆⠀⠀⠈⠳⢠⠀⠀⢱⠀⠉⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠋⠁⠀⠸⠀⠀⡰⠉⠀⠀⠀⠐⢄
⠀⡠⠃⠀⠀⠀⠀⠀⠀⠑⠀⢸⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⣇⠀⠐⠀⠀⠀⠀⠀⠀⠈⢢
⢠⠃⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⢒⠦⠤⠤⠤⠤⠤⠶⠲⢒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢂
⠸⣄⠄⠀⠀⡀⠄⠁⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣅⣴⠚⠋⣩⣹⠉⠛⣢⣀⡗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠓⠄⠀⠀⠀⢂
⠀⢻⠀⠐⠈⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠰⠀⠈⠛⢁⡇⡈⠆⠙⠉⠀⢁⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠈⠐⠀⢸
⠀⡜⠀⠀⠀⢀⡠⠖⡏⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠃⠀⠀⠀⡀⠁⠀⠹⠀⠄⠀⠀⠄⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠙⠓⠤⣀⠀⠀⠀⠘
⢸⡀⠀⢀⠔⠋⠀⠀⡆⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⡜⠀⠀⠈⠀⠃⠘⠂⠀⠂⠃⠀⠀⢸⡀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⢀⠇⠀⠈⠓⣄⠐
⠀⠹⠆⠌⠀⠀⠀⠀⢹⡀⠀⠀⢀⡀⠻⣄⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⣀⠜⠆⣀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠀⡴
⠀⠀⠐⠦⣀⠀⠀⠀⢸⡆⠀⠀⢸⢀⢠⠀⢑⠴⢋⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⠳⢆⠁⣠⠀⡺⠀⠀⠀⣟⠀⠀⠀⢀⡠⠈
⠀⠀⠀⠀⢸⠀⢀⠀⣸⠃⠀⠀⠀⠀⠘⠒⢉⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠑⠚⠀⠀⠀⠀⠀⢿⡀⠀⠀⢸
⢀⡀⠀⣀⡎⣀⣠⠼⠃⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠹⢤⣀⣈⣆
⠀⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢟⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⢷⣤⣀⣀⠠⠴⠒⠒⢢⡤⣀⠀⠀⠀⠀⠀⢀⣠⠴⠯⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣸⠀⠀⢹⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠄⣀⣀⣀⡀⢿⠀⠀⢸⠀⢀⣀⣀⡠⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⣍⡐⠆⠆⣀⣿⠀⠀⢸⣉⠰⠤⠆⣉⠄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢼⠀⠀⢾⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⣀⡤⠋⠀⠀⠘⠦⣀⣀⣀⡤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠖⠒⠲⠶⠦⠤⢄⣠⠔⠢⡴⠉⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢁⠀⠀⣸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡖⠋⠉⠀⠯⠗⠀⠀⠈⣳
⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣀⣰⠀⠀⢲⠒⠚⢧
⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠀⢦⣀⡰⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢠⠃⢰⡀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⡀⣸⠈⢆⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⢀⣠⢼⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢦⡀⠀⠀⠀⠀⠀⡧⢀⡀
⠀⡴⠋⠀⠘⡄⠀⠐⡶⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢶⠂⠀⢠⠃⠀⠙⢦
⡼⠀⠀⠀⠀⢹⡤⢤⡇⠀⠀⠀⠀⣤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣤⠀⠀⠀⠀⢠⡤⢤⡏
⡇⠀⠀⠀⢠⡇⠀⢰⡏⠀⠀⠀⠘⢰⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⡌⠆⠀⠀⠀⢸⠆⠂⢸
⠹⡄⠀⠀⠀⢧⡀⠸⡇⠀⠀⠀⠘⢌⣛⡛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣛⣛⡡⠃⠀⠀⠀⢸⠃⢀⠞⠀⠀⠀⣠
⠀⠈⠳⠤⢤⣤⠼⠛⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠧⠤⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣤⣀⣀⣀⠀⠀⠀⡀⠀⢀⣀⣀⣀⣤⣴⡒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⢠⠏⠹⣄⣈⡭⠟⠻⢭⣉⣠⠏⠈⢧⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⡧⠔⠒⠀⠉⠉⠉⠉⠉⠉⠁⠐⠒⠠⢇⠀⠀⢀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠳⣄⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⠁⠈⢧
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠈⣇
⠀⠀⠀⠀⠀⠀⡞⢁⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⢑⡀
⠀⠀⠀⠀⠀⠀⠀⢹⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡶⠤⢤⣀⡀⢀⣀⣀⣀⣀⡀⣀⣀⣤⠤⠤⠒⠚⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡧⢄⣀⣀⣀⣀⣩⠏⠉⠉⠁⠸⣄⣀⣀⣀⣀⠠⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠪⠥⢒⣒⣒⡒⣿⠀⠀⠀⠀⠀⡿⢒⣒⣒⡢⠭⠒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠑⠠⠤⠤⠤⠚⠁⡇⠀⠀⠀⣌⠈⠲⠤⠤⠤⠔⠊⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡢⠤⠄⠀⠤⢤⡾⠃⠀⠀⠀⠘⢷⠤⠄⠀⠀⠤⢔⡾⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣥⣬⣥⣼⣤⣷⣥⣾⣯⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠐⠒⠒⠈⠉⠉⠉⠉⠉⠉⠉⠁⠒⠒⠂⠤⢄⣀⣠⠔⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠁⠀⠀⠀⠈⣧⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⡞⠁⣠⠞⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠦⠞⠁⠀⠀⠀⠀⣛
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⣰⠏
⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠚⣅
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢥
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠈⢧
⢃⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀
⠘⡀⠀⠀⢀⡠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⠀⠀⠀⣠
⠀⠹⣄⢀⣈⣆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣀⡀⣰⠃
⠀⢠⠟⠁⠀⠹⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⣿⣷⡄⠀⠀⠀⠀⠀⢸⠁⠀⠈⢳⡄
⠀⣷⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⣿⣷⣤⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣤⣾⣿⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡁
⠀⠹⣄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡼⠃
⠀⠀⠈⠙⠒⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠰⠦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠄⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠘⢯⣉⠉⠉⠀⢀⡶⣆⡀⠀⠈⠉⣉⠝⠁⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠈⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⡏⠘⢧⡀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⢠⠞⠁⢱
⠀⠀⠀⠀⠀⠓⠤⠤⠟⠢⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠜⠫⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⢤⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠈⠉⠑⠛⠓⠐⠀⠤⡤⠤⠤⠴⠄⠀⠐⠚⠛⠀⠉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⠀⠀⠀⢀⠇⠀⠀⢰⡀⠀⠀⠀⣀⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠚⠁⠀⠀⠈⠓⣦⠞⠀⠀⠀⠈⢧⡴⠛⠁⠀⠀⠙⢳⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢐⣂⢉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⡁⢈⡀⠠⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⣠⠊⠀⠑⠒⠤⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣎⠁⠶⠀⠙⠀⠀⡏⠀⠛⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠈⢦⠀⠀⠐⠀⡴⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⣟⡣⣀⣩⣓⠛⣍⠢⡈⠑⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣆⠀⣼⠀⠈⠙⢉⡠⠔⠛⢢⡈⠀⡍⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⡆⠀⠈⡖⢋⢁⠤⠔⠚⠉⠁⢯⠀⠛⠖⠤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡰⡄⡰⠏⠀⠀⠀⠀⠀⣐⣠⠤⠧⢴⠗⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡠⠞⠀⡉⠁⡘⡩⠩⠤⠜⠐⠁⠁⠀⠘⢆⣀⣨⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠙⠛⣇⠀⠀⠀⠀⠤⠀⢂⣅⠦⠚⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣀⡭⠥⠤⠢⠞⠒⠉⢁⡠⠔⢪⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠤⢄⠐⣲⣀⣤⠔⠚⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡁⠉⠀⢩⠈⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠘⡀⠄⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢀⡁⢦⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇⠀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢎⢀⠙⠒⠒⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠑⠮⣮⣀⠃⣰⡡⠴⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠤⠤⢁⣽⠻⣞⣋⣀⣀⣼⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣹⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠿⠿⠟⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣤⣥⡀⠀⠀⢴⡞⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⠈⠁⠀⠄⠀⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⡀⠲⣞⡾⠁⢈⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⢷⡆⢴⡶⠟⢻⡦⠼⠋⠻⢋⣭⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣴⠛⠃⠰⠿⠂⠀⠰⠿⠀⣴⡌⢉⣁⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡀⠀⠀⠀⢘⡷⡇⣴⡄⠠⣶⠼⣴⣶⠀⣴⠋⠁⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⣠⠤⠼⠃⠉⠦⠤⠒⠚⣲⣇⡀⢠⣤⠀⢠⣌⠉⣛⡳⣄⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⣆⠀⠀⠀⢨⣉⣀⠤⠤⠴⢾⡙⢟⠁⡛⠃⠀⣠⣀⠀⡁⠠⢝⡵⠃⠀⠀⢀⡿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⢸⣇⡟⢦⣤⢤⡖⠋⠁⠀⠀⠀⢸⣿⡞⠃⠈⢻⡿⠋⣷⡄⢙⠖⠀⠀⠀⢀⡴⠋⠀⠻⠟⠓⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢤⣿⢧⣾⠀⣨⠷⣤⡀⠀⠀⣀⡾⢿⡟⠋⠳⣆⠀⣤⡄⢾⠁⣢⢀⠀⡠⠟⠃⣿⠆⠢⢄⡀⢙⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⣿⣸⣿⣦⡇⠀⠈⣷⠖⣋⡥⠖⠉⠀⠀⠀⢸⣧⡉⠀⠀⢙⡐⠏⠊⠀⣴⣦⠀⠀⠀⠀⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠀⠀⣧⠀⠀⣸⣿⡍⢳⡄⠀⠀⠀⣠⠞⠁⠙⠲⣤⣛⣋⠀⠀⢀⡉⠉⠀⠀⣴⣶⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠄⣿⣧⣿⢀⣟⢀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣿⠞⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠂⠉⠉⠉⠉⠉⠁⠐⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⢰⠓⡄⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣌⢢⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⢇⢣⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠳⣷⠀⡿
⠀⠀⠀⠀⠀⠀⣠⠬⡿⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⠧⣄
⠀⠀⠀⠀⡠⠋⠀⣸⠇⠀⠀⡄⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢂⠀⠀⠸⣇⠀⠑⢆
⠀⠀⠀⣰⠁⠀⠀⢸⠀⠀⣰⠤⠀⠀⠀⠤⠊⠰⢄⣀⣀⣀⠠⠆⠑⠤⠀⠀⠀⠀⠬⢆⠀⠀⠇⠀⠀⠈⣆
⠀⠀⢀⠇⠀⠀⠀⡄⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢀⢀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⡶⠋⠹⠀⠀⠀⠤⢺⣏⣷⣆⠀⠀⠀⠀⠀⠀⣰⣾⣻⡷⢤⠀⠀⠀⡏⠙⣶⠀⠀⠀⠀⡇
⠀⠀⡆⠀⠀⠀⠀⣦⠀⢰⠀⠀⠀⠀⠺⠿⠿⠉⠀⠀⠀⠀⠀⠀⠉⠿⠿⠗⠈⠀⠀⠀⡇⠀⣼⠀⠀⠀⠀⢰
⠀⡸⠀⠀⠀⠀⢠⠏⠙⠒⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠒⠋⠹⡄⠀⠀⠀⠀⢆
⢠⠃⠀⠀⠀⠀⣸⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⣧⠀⠀⠀⠀⠘
⠈⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠊⠁⠀⠀⠀⠀⠀⣿
⠘⢄⡀⠀⠀⢡⣸⡄⠀⠀⠀⠀⠀⠀⣀⣠⠜⠓⢶⡒⠒⠒⣒⠶⠛⠧⣄⡀⠀⠀⠀⠀⠀⠀⢠⣇⡆⠀⠀⢀⣠
⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢀⡤⠒⠉⠀⠁⠀⠠⠀⠀⠘⠁⠀⠀⠀⠀⠈⠀⠉⠒⢤⡀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣠⠎⠉⠢⡀
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠈⠓⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⠚⠁⠀⠀⠀⠈⠢⡀
⠀⢀⡤⠚⠁⠀⠀⠀⠀⣀⠴⠊⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠑⠦⡀⠀⠀⠀⠀⠈⠒⢤⡀
⡔⠁⠀⠀⠀⠀⣀⠴⠊⠁⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠉
⠳⠤⠤⡄⢠⠞⠁⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠈⠳⡄⢠⠤⠤
⠀⠀⠈⠒⠁⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢐⡃⠀⠀⠀⠀⢀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂
⠀⠀⠀⠀⠀⠀⠉⠀⠒⠂⠉⠁⠸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⡖⠒⠂⢹⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠛⠀⠀⠀⢀⠛⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣠⠀⠀⠀⠈⣄⠀⠀⠀⠀⢀⠇

⠀⠀⠀⠀⠀⠀⢀⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠇⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢀⣤⠀⢢⡴⠂⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠇⠀⠀⢀⡄⠑⠀⠀⣠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⠐⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠘⠁
⠀⠀⠀⠠⠀⠀⠀⠀⢀⢀⢀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⣀⣉⣿⠾⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡗⠀⠀⣴⣁⢀⣀
⠀⠀⠀⠀⠈⠙⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡶⠴⣮⠤⠀⠀⠹⣥⣎⣡⠀⠀⠴
⠀⠀⠀⠀⠀⢀⢸⠀⡖⣣⡀⠀⢲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠀⠀⢀⣰⣀⠀⠀⠀⠉⠁⠀⠀⡄
⠀⠀⠀⠀⠀⣄⢸⣷⡏⠻⠹⠶⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠾⡻⣦⠀⠀⢰⣆⠠⢤
⠀⠀⠒⣶⣶⡿⠀⠋⠐⠿⠀⢶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠆⠀⢹⡅⠸⠷⠀⢀⠉
⠀⠻⠿⢿⣿⠷⠃⠀⢌⠀⡀⠈⢃⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⢸⡇⠀⠀⠈⠋⠀⡀⠠⣏⢁⡀⠉
⠀⢠⣄⣼⠏⠀⠀⠀⢶⣬⣿⣧⣤⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠺⡅⠀⠀⣬⠠⠤⠅⠀⠙⢍⡀
⠀⠀⠿⠃⠀⢦⠀⠐⠄⢹⣯⡿⠋⢫⠀⣀⡀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⢁⠀⠀⣠⡷⢖
⠀⠀⠀⠀⠀⠀⠂⠀⢴⡿⠀⠀⠀⠀⢄⠈⠛⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⡀⠀⣋⠀⠀⣿⣲⣴⡷⠦
⠀⠀⠀⠀⠀⠀⠀⢀⡌⢙⡿⡗⢉⠶⡄⠑⠆⠀⢀⣤⣴⡏⢀⣄⡀⢄⠀⠀⠀⠀⢀⣠⡴⠖⠒⢶⣀⠠⢀⠙⢿⣝⢂⣰⠃⡼⡏⢻⢿⣯⠴⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⢀⣭⣩⡆⢠⣤⣞⢶⠿⣿⣿⠃⠠⢤⣬⣙⣿⠀⣰⣾⣿⡿⠓⢀⡀⣸⣿⣄⡀⠈⢹⣟⣄⡈⢠⡅⢠⠈⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣦⣸⣿⣮⠀⣿⣿⣟⣿⣷⣾⣿⢿⠟⣻⣿⣟⠿⣿⡿⣿⣟⣼⠏⡀⠀⠘⣄⡀⠀⠀⠙⠂⣱⠄
⠀⠀⠀⠀⠀⢀⢀⠂⠀⠁⠀⠀⠀⠘⠉⠉⠻⠛⠇⠈⠉⠛⠋⢉⡿⠛⣾⠀⠉⣯⠙⠀⠈⠉⠉⠉⠹⠟⠉⠁⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠘⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠁⠙⠁⠀⠀⠀⠠⣰⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⣤⢀⡀⠀⠀⣀⣀⣤⣀⡀⠀⢸⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⠉⠻⣿⣤⣴⠿⣿⣿⣿⣿⢿⣧⣨⣿⡅⠀⣰⣶⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢄⠲⣦⠩⢾⣿⣿⣿⣿⣿⠳⡄⠹⣿⡿⠿⣋⣿⣽⠻⡶⢘⣿⠛⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡽⢆⣤⣷⣮⣿⡣⠎⡵⣹⣿⣿⠟⢡⠻⣠⢹⢁⡶⠏⣼⡽⡓⣁⠼⡷⠪⣙⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⡝⠻⡷⣼⡌⣾⣿⣷⣶⣾⣿⣷⣿⣭⣷⣯⠾⢿⣿⡧⢠⣟⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠹⣆⠈⣾⣳⣿⣿⣿⡿⠿⠿⢿⣟⣿⣿⠋⠖⠧⢈⢀⣨⢿⣿⡷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⠀⠻⣮⣾⣿⣽⡿⢉⡶⢛⣛⠳⡜⣿⣿⣧⠀⠀⢱⣁⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣸⣬⣿⣷⣸⣿⣻⣿⢿⡇⣷⣇⢿⣧⣴⢇⣿⣿⣿⠿⢧⠉⣹⣗⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠛⢿⡿⣿⣹⣛⠿⡿⣮⣇⢛⢟⣶⣤⣴⣿⣿⣿⠣⠹⡰⠞⠻⣿⡇⣠⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠙⢿⢎⡻⣶⣿⠃⡬⣹⣞⡯⠇⠣⢽⣿⡋⢗⠂⣤⡧⣄⣤⣿⣿⣿⠟⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⣦⢿⢫⠀⠈⢿⡜⣽⣯⡏⢠⡒⠖⡄⢫⢷⣤⡴⣻⣵⣿⣿⣿⣿⣿⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⢎⠉⠈⣈⣎⢯⠠⡒⣻⣟⣿⣧⣘⣾⣿⣂⢼⡿⣑⣾⡽⣿⡿⣿⣿⣿⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⡂⢁⠵⠋⣁⣤⡾⣿⣿⢿⣿⣿⣿⣾⣿⣿⡷⡿⣟⠟⡵⣚⠝⡿⢿⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢀⣾⡓⠅⣠⠋⣰⡾⣯⠷⣛⣭⣭⣍⡻⣿⣿⣷⣿⣽⣿⢳⣎⡞⡛⣫⣼⡇⠈⠱⠆⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⣾⢅⡀⢱⠂⣼⣿⡿⣡⣾⢿⣿⢡⣽⢇⣿⣟⢾⣿⣷⡉⣻⣼⠞⢱⣯⣀⠑⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢨⣯⠅⢐⠇⢐⣿⣾⢡⣿⢛⡿⠛⣳⠶⣏⠷⣭⣾⣿⡁⣹⣿⡿⠟⠚⠉⠛⢳⣄
⠀⠀⠀⠀⠀⠸⣿⠶⢴⠀⢸⣿⣿⢸⣿⠿⣿⢿⣍⣓⣤⣿⡟⢫⣀⢝⣵⠁⣴⠄⢠⠀⠡⠀⣿⠄
⠀⠀⠀⠀⠀⠐⣻⡯⢀⡄⠀⣿⣿⡌⡿⣿⣛⠗⣶⣽⣏⠛⠛⠉⠐⢿⣿⣷⣿⡀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠒⣿⠥⡺⡄⢹⣿⣷⣹⡱⢏⡾⠿⢿⣮⢿⣄⠀⠀⣼⣿⢿⡟⠻⠶⠶⠶⠟
⠀⠀⠀⠀⠀⣀⣀⠰⣿⣴⡟⠶⠙⠁⠛⠍⡝⠀⢂⡆⢸⣥⣿⣤⣾⣿⣳⠯
⠀⠀⠀⢠⣾⣿⣹⣿⣝⠃⠁⠀⠀⢠⠻⣼⠷⣤⣍⣀⣼⣿⣿⣿⡫⠾⠁
⠀⠀⠀⠐⣿⣿⣽⣿⢇⡀⠀⠀⠀⢧⠪⠴⠔⣿⡽⣿⠞⠛⠃⠉
⠀⠀⠀⠀⠀⠉⠉⠙⠛⠷⠿⡴⠦⢿⡆⠀⣰⡏⠀⢠⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣷⣶⣿⣿⠈⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⡵⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠇⠀⠀⠠⡄⡀⠀⢠⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠁⠀⠀⠀⡄⠀⣀⠀⠀⠀⠀⢀⢤⣌⡛⢏⡤⢏⣠⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣵⡔⠀⠀⠘⢁⢾⢠⣤⡀⠐⠀⢸⣷⠈⢿⣾⣧⣬⣇⣘⡷⠀⡖⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢮⣿⣷⠀⠠⠈⢿⣿⣿⣿⡷⠀⠀⠀⢋⡑⣚⠷⢾⠋⢋⡉⠠⠛⢩⡏⢹⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⣿⢿⣧⠀⠻⡞⢷⣻⣿⣥⠀⠀⠀⣼⣿⣍⣤⣀⠁⠻⠲⠀⠀⢠⡍⢏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠒⠰⠁⠈⠉⣼⢿⣷⣤⡈⠀⠈⢉⣉⡀⠴⠊⠉⠉⠈⠲⣍⣻⡇⢤⡀⡤⠠⢕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣵⡋⣙⡠⢸⠛⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⢦⣳⠐⢉⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⠀⢀⠈⠀⠆⠀⠆⠀⠀⠀⢀⣀⡀⢀⡆⠀⠸⢧⠀⠀⢐⣀⣌⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠺⠁⠀⠀⠀⢟⠛⠮⠼⢷⠁⠀⠀⠀⠀⠀⢻⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠓⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠈⢠⠄⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠋⠀⡀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⣀⣄⣀⣀⣠⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠰⠀⠀⠀⢀⡼⠇⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢇⣽⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⠻⠀⣼⠀⢦⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠽⣷⡞⢦⡄⠰⠋⠑⠀⣀⠀⠀⠀⠀⠘⢿⣶⡶⠶⠶⠶⠾⢿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡝⠈⣋⢿⡛⠧⠀⣤⡀⠀⠙⢻⡆⢠⣦⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⠶⢷⡈⠿⣆⣠⡼⢋⡄⠀⠈⠁⠀⠋⠛⠛⠿⢷⣄⣀⠀⠀⢀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡸⠁⢀⠋⡰⠢⣽⠁⠬⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠓⡤⢀⢀⠾⡀⢠⡄⠘⢲⡄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣶⠈⠁⣬⢱⢾⣤⣤⢤⠤⠀⣤⠀⠀⠒⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⢈⡧⠀⡤⠈⠓⠄⢀⣈⠷⣀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣨⠇⢠⡄⠰⠚⢠⠽⣭⡴⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣠⣤⣶⣶⡶⣦⣤⣀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣀⡀⠀⡆
⠀⣀⡑⠐⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡔⠀⠀⠛⠇⠀⢀⣿⣧⠿⠿⠛⠳⠿⢾⣿⣃⠀⠀⠀⠀⠀⠀⣤⡿⠯⢻⠆⠃
⠀⠀⠈⠉⠁⠀⢹⣦⣤⣀⣀⠀⠀⣿⡟⠃⠀⠀⠀⠀⡠⠔⠊⢁⠴⠚⡆⠀⠀⠀⠀⢸⠿⠿⠛⢀⡄⠛⠃⣀⠘⠿⠿⠏⠀⠀⠠⡶⣼⣥⣴⣶⣍⣀⡀⡀
⠀⢀⠀⠀⠐⣄⠀⠈⢿⣿⣿⣧⠀⢻⡇⠘⠀⠀⠀⠀⣠⠤⠚⠉⣠⠋⠀⠀⠀⠀⠀⣀⣀⡤⣠⣾⣿⣿⣿⠿⠧⠐⢐⣶⠂⠀⠀⠀⠀⠈⣳⣴⣋⠉⠋⠛
⠀⠈⠀⠀⠀⣻⣿⣦⣄⣘⣿⣿⡆⢸⣿⢘⣤⣶⣦⠀⠉⠑⠒⠛⠁⣤⣶⣶⣧⡄⠀⠈⠉⢻⣭⣥⣿⢿⣿⣷⣿⠃⡡⠎⠀⠠⢶⡇⠀⠀⣿⣿⣸⠆⢀⡀
⠀⠀⠙⠲⠿⣿⣿⣿⣾⣿⣿⣿⣿⣦⣙⣿⡿⠋⢉⡧⢀⣀⠰⠀⢐⣿⣣⣿⣻⣿⡃⢀⣶⣾⣿⣿⣻⣋⣿⣯⡻⣧⣤⡀⠀⣰⡇⢠⡖⢲⣿⡏⠁⠀⠼⡛
⠀⠀⠀⠀⠀⢀⠀⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⡶⠂⢰⠈⠉⠤⠀⠈⠿⣿⣶⡿⠿⠠⠞⢛⢋⣘⡛⣻⠟⠿⠛⠙⠏⠛⠛⠂⠹⣿⣾⣶⣿⣿⣷⣜⣳⣴⡇
⠀⠀⠀⣀⢴⣿⣦⢀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠒⢤⣄⣀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠸⠻⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⡎⠙⠛⠛⠻⡇
⠀⠘⢿⣿⣿⣿⣷⣿⡷⠂⠀⠤⠖⠂⠠⠄⠀⠀⠀⠀⠀⠉⠟⣿⣿⣿⣿⣤⡐⢢⠀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠆⠀⢠⣤⣄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠈⣿⣿⣿⣿⣿⠁⠀⢠⠀⢰⠐⡄⠀⡆⠀⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠃⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⠸⣿⣿⣴⣶⡄
⠀⠀⠀⠛⠛⠉⠛⠻⠀⠀⠑⢀⡌⠉⢀⠐⠁⠀⢽⣦⡀⠀⠀⠀⣸⣿⣿⣿⡾⠶⠶⡼⢿⡿⠀⠀⠀⢀⣠⣀⣀⡤⠄⡠⠤⠀⠀⠘⣿⡿⠿⠛⠁⠙⠛⠃
⠀⠀⠀⣀⣀⣤⠶⣶⣄⠀⠀⠀⠁⠀⠀⡀⢠⣴⣿⣿⣝⢦⣤⡄⢁⣤⣝⣿⣇⠀⠀⠃⠀⠀⠀⣎⢝⣽⣭⢽⣳⡖⡷⢁⠀⠁⣼⣦⣀⣀⡀
⠀⢠⡾⣽⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠯⣿⡟⣠⣿⣿⣟⣹⣿⣿⠟⠀⠀⠀⠀⠙⢾⣽⣿⣿⣿⠾⠃⠀⠠⣶⡿⣿⣿⡟⠀⢰⣿⣦⠀⠐
⠀⠈⢿⣿⣿⣿⣿⣿⡿⠀⠀⢰⣿⣷⣤⣴⣦⠀⠀⠀⠀⣴⣵⣿⢿⣷⣿⠿⠟⣁⠀⠀⠀⠀⢀⠀⠀⠙⠻⢿⣋⡀⠀⠀⠀⠈⢿⣿⠻⠿⠀⣿⣿⣿
⠀⠀⠀⠙⠻⠿⠿⠟⠀⠀⠀⠈⢿⣿⣿⣿⠟⠀⠀⢠⠞⠛⠉⠀⠀⠀⢠⣶⠿⢻⣷⡀⠀⡀⠀⠀⠀⠐⠿⠿⠿⠛⣀⣀⣤⣤⣌⠀⠀⠀⠀⣿⡿⣿⡄
⠀⠀⢀⣤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⣠⡶⢶⣀⣤⡀⠀⠀⠻⣿⠿⠛⠑⡿⠙⠿⠆⠀⣀⣤⣶⣾⢻⣿⣹⣧⣿⣿⡆⠸⠧⣤⣿⢷⣿⣤⡤⠿
⠀⠀⣦⠘⡇⠈⡆⠀⠀⠀⠀⢰⣄⣀⡄⠀⠀⠀⠀⢻⣷⣾⣿⣬⣿⠀⠀⠀⢀⡴⠃⠀⣿⣿⢿⠿⣿⣯⣫⣻⡿⠟⠋⢹⣟⣂⣿⠆⠀⠀⠈⣿⣏⣽⡏
⠀⠀⠹⡄⢸⠀⣁⠀⠀⠀⠰⢿⣭⣿⣄⠀⠀⠰⠀⠀⠛⡿⠿⠋⠀⢀⠠⢒⣽⡎⠀⠀⠙⠻⠿⠿⠟⠛⡛⠍⡄⠐⠀⠈⠉⠉⠀⠀⠀⠀⣀⠙⣛⠻⠃
⠀⠀⠀⠗⢠⡄⢰⠀⠀⣰⠄⠀⠟⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠎⢡⣴⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⡥⢀⣶⡄⠓⠀⠀⠉⠀⠀⠂⠀⠀⢈⠘⠸⢡⠆
⠀⠀⠀⠈⢾⡤⠂⢀⣼⠏⢀⠀⠀⢸⡅⠀⠀⢚⣿⠷⠂⣀⣀⠀⠀⣿⣿⣿⠁⠠⡄⠀⠀⠀⡀⠀⠀⠐⠀⢠⠉⠁⠄⢈⡀⢰⣦⠀⣀⠀⠀⢸⣿⣿⡖⠂
⠀⠀⠀⠀⠋⠀⠀⢸⣿⠁⠿⠀⣿⢼⣷⡄⠀⠈⠉⢠⠼⠛⢿⣿⣾⣿⣿⠃⠀⠀⣀⣤⢞⣭⣤⣤⠄⠀⢀⠀⠐⠂⠀⣘⣿⣾⣿⣾⠟⡀⠀⠈⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣠⣀⣠⣾⡏⠀⠀⠀⠀⠀⢠⣤⡾⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠛⠉⠁⣀⠤⠄⠀⠀⠀⠙⠛⣻⣿⣾⣿⡛⠛⠀⠀⢻⣿
⠀⠀⣿⣿⣶⣄⡀⠀⠈⠙⠿⠟⠛⠋⢀⣤⣾⣿⡟⠀⠀⠙⠇⣿⣿⣿⡿⠟⠁⠀⠄⠀⠀⣠⠔⠊⡀⠴⠆⡟⠉⠉⠀⠺⠟⢹⣿⠙⠿⠀⠐⠀⠈⡿⠀⠀⡄
⠀⠀⠸⣿⣿⣿⣿⣦⡀⢀⣀⡀⣠⣶⣿⣿⣿⣿⠁⠀⢠⠊⣰⣿⡏⠀⠀⠀⠀⡀⠀⠀⠈⠁⡠⠊⢁⠔⢂⠄⢠⣄⠀⠀⠀⢈⡁⠄⠐⡇⠄⣀⡀
⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠀⠀⠇⢸⣿⣿⡇⠀⢀⣀⣀⡀⠀⠀⠠⣟⠠⠄⠂⠀⠏⢀⡎⠁⡁⠀⠀⠈⠑⢆⡠⠇⡴⠋
⠀⠀⠘⣿⣿⠿⣿⡿⢟⣩⢿⣛⡿⣿⣿⢿⣿⡿⠀⠀⠈⢾⣿⣿⠀⠀⣿⣿⣿⣿⣆⣀⣀⠀⠀⠀⠀⢘⢁⡜⢀⠜⠀⠀⠐⡃⠚⢒⡸⢗⡓⠓⡚⠂
⠀⠀⠀⠈⢋⣴⣿⣶⣾⣛⣟⡋⣶⣶⣿⣮⠉⠁⠀⠀⠀⠈⢿⣿⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠈⡾⠤⠊⠀⠀⠀⠀⢠⡀⠃⠠⠄⠙⢠⡀⠀⢀
⠀⠠⠄⠀⢸⣿⣿⣿⠏⠈⠉⠀⢻⣿⣿⣿⠀⠀⠐⡿⠃⠀⠘⡟⠀⠀⠀⠈⠁⠸⣿⣿⣿⣿⠃⠀⠀⠈⠀⠀⠀⠀⠒⠀⠀⠈⠈⠀⠢⡖⠀⠈⠁⠀⢜⣸⡁
⠀⠀⠀⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠙⠻⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣄⡀
⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡄
⢩⣕⣗⢨⡗⠀⠀⠤⠒⣴⣿⣿⣿⣿⣿⢾⠇⢀⣀⣤⣤⣀⣨⠄
⣶⡈⠉⣉⡃⠀⡀⢼⣷⣿⣿⣿⣿⣿⡖⠁⣰⣾⣿⣿⣷⣾⣿⡿⡆
⣿⠃⠀⠀⠀⣲⡿⣀⣿⣿⠏⠉⠉⠉⠀⢠⡾⣿⣿⣿⣿⣿⣿⣿⣴⡄
⠋⠀⠀⠀⠐⣳⣎⣺⡿⠷⠀⠀⠀⠀⢀⡾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⢄
⠀⠀⠀⠀⠀⢸⠟⢀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣾⣷⣾⣿⣷
⠀⠀⠀⠀⠐⠂⠀⠈⠁⠀⠀⠀⠀⣼⠟⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣷⣾⡻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠄⠀⣿⣿⣿⣿⣿
⠀⠀⠀⢰⠇⠀⠀⣤⡐⠀⠀⠀⢰⣿⣾⣿⣿⢃⣸⡿⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⣄⠀⣤⣿⣿⣿⣿⣿⡀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣸⣟⣡⣄⠀⠀⣾⢿⣿⣿⡇⣤⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⢻⣆⡀⠀⠀⠀⠀⠀⢠⣦⣀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⡨⠂⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡽⠟⠒⢠⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⠛⠀⠛⢿⠿⡖⢁⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⡿⠃⠀⠀⠘⣯⠀⢨⣽⣿⣿⣿⣿⣿⣿⢿⣿⡆⠀⣀⠁⠒⣾⣷⣾⣶⣷⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⣾⢙⣷⣇⣻⣿⣿⣧⠼⠛⠉⠿⣯⡵⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀
⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠳⠲⠟⢛⣿⣿⣿⡉⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢙⣷
⣿⠛⠁⣿⠂⠀⣀⣴⡶⠦⡀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⡆⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣠⠀⢀⠀⠀⠻⣿⣷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⠀⠙⠳⢦⣥⣀⠀⠈⠻⢧⣍⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠠⢤⣀⡀⠀⠀⠀⠉⠐⢶⣄⠀⠁⠰⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣖⡐⠀⠀⠀⠀⠉⠙⠓⠆⢢⣄⠀⠀⠀⢴⣤⢠⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠛⠿⠿⠿⠷⠶⣶⣤⣤⣀⡀⠈⠁⡲⢦⣜⡲⣿⣶⣷⣶⣬⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣤⣤⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⢙⣳⣮⣿⣿⣿⣾⣿⣿⠟⠋⢡⠟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠈⠙⠉⠃⠚⠛⠂⠶⠒⠠⠀⣈⣛⣿⣿⣿⣿⡿⠛⠁⣘⣠⡴⠾⡛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣁⣠⣤⡤⣤⣤⣤⠄⣠⣀⠄⣀⢀⣀⣸⣿⣿⣿⣿⣿⢫⡄⢈⡽⠛⢁⣠⣀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠈⠀⠀⠀⠀⠀⣀⣀⣈⣻⣿⣿⠿⠿⢯⡽⢤⣶⠋⠀⠀⠼⢿⣿⠙⢆
⠀⠀⠀⠀⠀⠀⢀⡠⠀⡀⡀⠠⠀⠀⠀⠀⡀⡉⠤⠮⠉⠽⢿⣿⡿⠡⠶⣶⣤⡹⡟⣷⡄⠀⠀⠀⠀⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⢠⠒⠛⠁⠐⠀⠐⠀⠀⠀⠀⠀⡀⢀⣴⣴⣾⣿⣧⡴⠿⠿⢉⡼⠃⠘⠷⠀⠀⠀⠀⠀⠀⠄⠈⢇
⠀⠀⠀⠀⠀⠀⠋⠀⠀⡄⢂⣀⡀⠀⠂⣀⠤⠞⠀⠁⠀⢉⣩⣽⣿⣿⡗⢲⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠋⠉⠈⠁⠀⠀⡀⢠⡄⠀⠚⠜⠛⣛⣫⣿⠇⠈⠟⣿⣇⠀⠀⠀⠀⠀⠀⠰⠖⠶⠃
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣶⡯⠓⠋⠀⠀⣠⣴⣾⣿⣿⣿⣦⣄⠀⡁⢹⢦⣄⡀⠀⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠀⠐⠊⠀⠀⣤⠚⠉⠁⠁⠀⠀⣠⣖⢾⣽⣿⣿⣿⠟⢛⡿⣿⡿⠋⠀⠀⠀⠚⠉⠓⠶⠦⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠴⠛⠉⠐⠚⣻⣿⣿⡇⢀⠀⠀⠛⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⠿⠀⠀⠀⠀⠀⠈⠓⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⢦⣄⡀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⣉⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⡄⠀⠀⠀⠀⠀⠀⢀⡀⡐⠀⠀⠀⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⠀⣝⠇⡀⠀⠀⠂⠀⠀⠀⢀⡄⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⡈⣿⣆⡆⢀⡀⢀⠁⠀⠠⣤⠮⠙⠡⠀⠞⠃⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⣠⣾⣷⣿⣿⡠⡜⠀⠀⠀⡀⠀⠛⠈⠀⠄⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⡄⠈⡇⠀⠀⢠⣿⣋⣮⣽⣷⣭⣽⡳⢤⡤⢿⠀⡠⠘⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡶⢚⠠⠶⢾⣿⡷⡄⠶⣿⠟⣫⣿⡾⢋⠟⣣⠁⢈⠂⢻⡛⠚⠄⢀⣀⡌⣄⠀⠀⠃⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠠⠤⠞⠒⠐⢯⡾⣯⡟⣰⣿⣾⣿⣶⣌⠑⠌⡀⠂⠀⡁⠀⠀⠈⠃⠐⠑⠠⢀⠀⢛⡠⠼⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠈⢻⣿⣿⣷⣿⣿⣿⣿⣈⡷⠆⡶⠀⠀⠀⠀⠀⢀⣀⡴⢀⠀⢀⠄⠊⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠠⣄⠄⠸⠳⣈⡀⠀⠀⠀⠀⠉⢷⣈⣻⣾⣿⣿⣿⡇⢁⠀⠀⠀⠀⠀⠀⠈⠋⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢀⠀⠄⡀⠢⠬⢆⠀⠀⠈⠂⣈⣦⠗⢿⣿⣿⣿⣿⡦⠄⠀⠄⠉⢢⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣒⡛⠀⠀⠅⡰⢀⡀⢠⣆⠀⣀⡹⡯⣀⣿⠟⠋⠉⠀⠀⠀⡀⠀⠀⠁⠚⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⣠⠟⣿⠟⠀⠀⠀⠁⠈⠉⠉⠀⠉⠁⠀⣲⣾⣧⣦⣀⠀⠀⠀⠀⠀⢠⠐⠃⢤⠠⠗⠀⠠⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡇⠀⠀⠂⠀⠒⣠⡧⣾⢥⡤⡴⢋⢉⡷⣢⣤⣄⣤⠶⣓⣺⣽⣿⡻⢻⣿⡆⠀⠀⠀⣴⠦⠠⠄⡀⠀⠀⠀⠀⠀⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠁⠐⠂⠠⠜⠂⣀⠀⠀⠘⣽⠁⠋⠆⠰⠋⠟⣟⠁⡋⠉⣙⢿⠼⣟⢿⣽⣧⣀⡀⠰⠇⣂⡶⢴⠦⠀⡀⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠠⡉⢈⣿⠓⡞⠧⡙⠟⠣⠺⠁⢀⠂⡕⠈⣡⣴⣿⣧⣶⡉⣷⢷⣘⢾⡓⠰⣺⢿⡷⢤⣄⡲⣭⣀⠠⠀⠉⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⢸
⠀⠀⢠⠀⠒⢓⠂⠉⠉⠤⣼⢷⡇⠀⠀⠀⠀⠀⠀⠋⠀⢸⣟⢸⣽⣿⣿⣿⣷⣽⣿⠋⢉⠋⠘⠓⠻⠄⠐⠀⠁⠈⠉⠁⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸
⠀⢀⢈⡀⠀⠀⠀⡔⠀⠒⢿⣸⡇⡿⠀⠀⡴⠀⠀⢠⠀⣾⣿⣿⣿⣿⣽⣿⣿⡏⠀⠀⢈⣆⣀⠀⠹⣤⠓⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⡀⡀⠄⠂⠛⠠⢎⣹⡿⣡⠦⠞⠳⠗⠒⠂⣤⢝⣿⣿⣿⣿⣿⣿⡿⣀⣸⠀⠀⠈⠄⠙⣈⡀⣻⡏⠀⠂⠀⠤⣀⠤⣀⠀⠄⠀⠀⠀⠀⠀⡀⢀⠁
⠀⠀⠀⠘⠀⢡⡄⠀⣀⠁⣸⣿⡿⠏⠀⠁⠘⡀⠀⠀⠀⠀⠿⢿⣿⣿⣯⣭⣽⣶⣶⣦⣦⣡⣠⡰⠃⣀⠈⠁⢀⡀⠂⠀⠨⡌⠓⠀⠨⠣⠆⣀⡀⠀⠀⡏
⠀⠀⠀⠀⠀⠈⠀⠀⠀⣾⠋⠛⠀⠀⢀⢤⠋⠥⠄⡲⠒⣀⡀⠘⣿⣿⣿⣿⣿⣿⠿⣇⢌⠙⠻⢿⡒⠻⣧⢤⡰⢄⡆⡰⠔⠉⠉⠁⢔⡀⠀⠀⠉⠂⡞
⠀⠀⠀⠀⢦⣤⠄⠀⠸⠀⠉⠁⢀⠀⠚⠓⠃⢀⠀⢀⣾⢡⣍⣼⣿⣻⣿⣿⣿⣧⠀⢈⠣⠘⢢⡀⠙⢦⠉⠂⢉⠲⡄⠤⠀⠙⡆⠱⢄⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠄⠀⡴⢇⢀⠄⠿⠀⠠⠈⣡⡬⠍⣸⠚⢿⣿⡧⡽⣏⢷⡄⠃⠈⢆⠱⠀⠈⠘⢆⠈⢣⠸⡁⠀⠀⠐⠀⠈⠲⠌⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠁⠀⠀⠀⡀⠠⢄⠀⠉⡟⠀⠁⢚⣹⣿⣱⠸⡬⣤⢳⡀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠃⠁⠐⠦⠀⠠⣄⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠁⠠⠀⠀⠀⠓⠀⠁⠃⠔⠀⢞⣽⣹⠁⢳⠹⡄⠁⠀⠀⠸⠀⠀⠁⠀⠘⡈⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⣤⡤⢠⣀⠠⠄⠡⢾⠏⠀⢸⠀⣝⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠉⠀⠁⠀⠀⠀⣸⠋⢠⠀⠘⠀⢸⠄⠆⠆⢀⢰⠀⠀⢸⣆⡟⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠦⠀⣀⠀⡘⣀⠀⠀⡀⢠⠘⠀⠀⠀⠈⣦⣣⣠⡈⣾⡟⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠿⠀⡆⠁⠘⡄⠀⠀⡄⠀⠀⠙⢿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⢸⡇⠀⠀⣧⠀⡄⠈⠀⠀⠀⠀⠈⠙⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡟⢤⡆⡏⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢳⠀⡓⣄⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡄⠈⠓

⠀⠀⣄⠀⣀
⠀⠀⠂⠈⠀⠐⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⡀
⠀⠀⠌⠀⠠⢂⢀⠀⠀⣠⣴⣶⡄⠀⠀⠀⠀⠀⠓⠀⠀⢐⣆⠀⠀⠀⠀⠀⠀⡀⠐⠠⠐⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢾⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠷⠤⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠐⠤⢈
⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛
⠀⠀⠀⠀⠀⢠⣔⣲⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠣
⠀⠀⠀⠀⠀⣉⡿⣁⡀⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣌⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠛⠛⠛⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠐⢇⣻⠐⢠⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠀⢁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢰⣶⣿⣷⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⠻⠁⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣟⣿⣋⣓⣶⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡍⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠑⠛⡇⠀⠀⠀⢀⣀⠠⠀⠀⠀⠀⠀⠐⠶⣦⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠀⠈⣿⡤⠔⠀⣈⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠠⢶⣾⣯⠙⠿⠿⠒⠢⢤⣤⣽⣿⣿⣾⣛⣿⡦⢀⣼⣿⡆⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣶⣂⠈⢀⣩⣿⣿⣿⣿⣿⡍⠙⢿⣿⣿⣿⣇⡔⠀⢀⠠⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⡟⢻⣿⣿⣿⠏⠀⢀⣾⣿⣿⣿⣿⣿⡟⢛⠤⣀⣄⣈⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣅⣈⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⠻⠿⠟⠀⠀⠘⢿⣿⣿⣿⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠪⠻⣧⠠⠀⠀⠀⠠⢆⣈⣿⣿⣿⣷⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡹⠄⠀⡜⢢⠀⠀⠀⠙⠻⢩⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠄⠀⠘⠁⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⡃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠠⠾⣿⣿⣿⣿⠀⠁⠘⣷⣶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⣀⣠⣿⣿⡿⠇⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⣿⣿⣿⠿⠿⣿⣿⣿⣿⠁⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠙⡄⠀⠿⣷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⡏⡂⠀⠀⠈⠻⢇⡀⠘⢀⡿⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⡏⠢⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠑⢤⣼⠇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⡳⣄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢊⣿⣿⣿⠷⢶⣦⣀⠀⠀⠀⠀⠀⠹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢱⡇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣇⣾⠟⠋⠁⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⡙⢷⣄⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠙⢷⡄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠻⢾⡿⠿⢿⣿⣟⢿⣿⣿⣿⡟⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠳⢍⠙⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⡿⢿⣿⣿⡟⠿⣿⡿⢿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠞⠁⠀⠀⠙⠻⣿⣷⣦⣍⣙⡒⠾⢭⣤⣈⡙⠲⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣝⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⠋⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣬⣍⣙⠓⠲⢮⣭⣳⢮⣷⣵⡝⣿⣿⣿⣿⣿⣿⣿⣿⣦⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠭⢷⣒⣶⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠛⠓⠂⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣶⠈⠀⠹⣿⣿⣿⣿⣿⢧⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠰⠿⠋⢁⠀⣀⣀⣀⡀⢀⡠⠤⠒⠉⠁⠙⣿⣿⣿⣿⣧⡐⢀⣿⣿⣿⣿⣿⣷⡁⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢰⡶⠿⠗⠀⣠⠊⠉⠀⠉⠻⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠸⡟⠘⣿⡞⣿⣾⣿⣿⣿⣿⣿⣿⣷⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣷⠬⢄⣀⢠⣀⡇⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣀⡼⡟⢺⢋⠳⡀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠀⠀⢃⠀⡇⠀⠙⠢⢄⣀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣆⠀⠈⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⡇⡿⣸⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⢠⠉⠉⠑⡄⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⡇⠁⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣏⠄⠀⠀⠈⢆⢈⡤⠴⠒⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⡟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠘⡟⣦⠀⠀⠀⠀⠡⡄⠐⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡏⢿⣿⣿⣿⣿⡟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢱⠘⣷⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⣠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠇⢸⠻⣿⣿⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡇⠈⢿⣄⠀⠀⠀⠀⠈⠂⠀⠒⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠈⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣀⠔⣠⡿⠃⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⢋⣼⡿⣡⣾⠏⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣼⣿⠋⠀⠀⠀⣸⣿⣷⣶⡄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣸⣿⣿⡿⠁⠀⠀⠀⢀⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⣿⣿⣯⠆⡀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠜⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣛⣋⣿⣿⠟⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠟⠷⣤⣄⠈⢿⣿⣏⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣤⣀⡀⢀⣙⠻⣶⠛⠛⠻⣿⡟⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⣟⠩⢉⣿⣷⡍⠀⠀⠀⠀⠞⣩⣿⠛⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠙⠻⠿⠿⠋⠉⠀⠀⠀⠀⢰⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣶⣶⣦⡄⠀⠀⠀⠀⠈⠟⢾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣭⣿⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⢯⡝⣿⠻⢿⡷⠀⠀⠀⣠⡄⢀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⢘⢳⡶⠾⠻⠷⠀⢀⣀⣹⣢⡈⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⣿⣮⡘⡇⠀⠈⢰⣾⣿⣿⣿⡟⢷⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢶⣝⣻⡿⢿⠧⣈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣧⡀⢷⡀⡙⠛⠻⣦⡄⣠⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⢳⣿⣦⢸⡀⢾⡏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠀⠀⠀⠩⢻⣾⡇⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠄⠀⠺⣏⠀⠀⢿⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢂⣶⠀⠀⢹⣧⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠂⠀⣯⡟⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡁⠀⢸⣿⡥⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡼⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⣀⣤⣖⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣷⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣌⣿⣯⣟⣹⣬⣿⣷⣿⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣿⣿⣿⠿⠛⠻⠻⠿⠿⠿⠿⣿⣿⣿⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡧⢶⣶⣦⣀⣀⣠⣶⣶⣖⣀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣈⣻⣿⣿⡏⣒⡀⣿⣿⣿⣿⣿⣿⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡹⠟⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠦⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⡇⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⣅⠀⡌⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣍⣁⠲⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣆⠀⠀⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢰⣿⣿⣦⣀⠀⣰⣟⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣸⣿⡃⣼⠃⠀⠰⠆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣻⠏⠀⡀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠘⠋⠰⠏⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠠⣶⠆⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢫⡿⣁⣤⣴⣶⣶⣦⣀⠀⠀⠀⣰⣿⣠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢸⠃⣿⠉⠈⠻⠟⠈⠛⠇⠀⢰⣿⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⢈⠀⠈⠁⠀⠀⠀⢠⠔⠃⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠿⠋⢽⣷⡀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣇⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⢿⠇⢹⣿⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣀⠀⠀⠀⠀⠙⠿⣷⣤⣀⣀⣠⠴⠋⠀⢸⣿⡌⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣼⡟⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠠⠞⠋
⠀⠀⠀⠀⣠⣶⣾⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠻⣿⣟⡻⠿⠖⠂
⠀⠀⠀⣾⣿⣿⣿⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣝⠿⢿⣶⣶⣤⣄⡀
⠀⠀⠸⣿⣿⣟⡁⠄⠉⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠁⠀⠀⠉⠙⠛⢿⣦⡀
⠀⠀⠀⢻⣿⣿⣷⣶⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠈⢦⡀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠈⠛⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⠿⠖⠿⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⠀⠀⠀⠈⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠨⠗⠀⠐⠒⠂⠀⠀⠠⣿⠯⠛⠀⠘⣧⡄⠀⠀⠀⠀⠀⠀⠤⣀⠐⠐⠀⠁⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣼⠀⠀⡀⠀⢠⡌⠉⠀⠀⠀⠀⠈⠠⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⡤⣴⠏⠀⠀⠀⠀⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡀⣤⡄⠀⠀⠀⠀⣸⣟⣉⣽⡿⣶⣀⣀⠀⠀⠀⢼⣻⣿⣷⠀⠀⠀⠀⠀⠠⠀⢠⣤⣾⣿⣿⣿⣿⣿⣟⣉⣳⣦⣤⡀
⠀⠀⠀⢀⣀⡄⠀⣽⣿⣷⡄⠀⠀⠈⠩⣭⡅⠀⠀⢩⣭⡽⠄⠀⠀⢼⣽⣯⡿⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⣾⠿⢿⡆⢿⠄⡀⠃⠀⣴⣴⣿⣧⡌⡷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤
⠀⠀⠀⠷⠀⠰⠇⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣶⣤⣶⣶⣶⣦⣤⡃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋
⠀⠀⠀⠀⠀⠂⠀⠀⢐⠀⠀⠀⠀⠀⣴⣷⣿⣾⣵⣿⣷⣦⠀⢘⣛⣛⡛⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠈⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠋⠋⠛⠉⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣷⡿⢋⣤⣴⣶⣶⣶⠖⣠⣀⠀⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⢀⣀⡈⠁⠟⠁⠁⠀⣠⣿⡟⠉⢿⣿⣿⣇⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣾⠡⠉⠀⠀⠀⠀⣰⣿⡏⠈⢀⣾⡏⠉⣠⢤⡙⢉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢠⣿⠋⣽⣿⣿⡿⠣⠊⢠⣾⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⢸⡟⢹⣿⣿⣿⣤⣀⠀⠘⠛⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⢀⣤⣿⡿⣿⣿⣟⠀⠘⠻⢷⣦⡀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢸⡟⢡⡖⣍⢻⣿⣶⣤⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⢰⣿⡷⢶⣉⢶⢸⣦⣌⡈⣩⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣾⣿⣧⣘⢙⣂⣾⢿⣯⣭⣁⣰⣿⠀⠀⢀⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⢿⣿⣿⣿⣁⣾⠀⠀⣿⠟⠁⠀⠀⠀⠀⣿⠃⠀⣲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡃⡾⠻⢻⣿⢿⡀⣠⠀⠀⢠⡾⡆⠀⠀⢀⠀⠀⠀⣠⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⡟⠟⠀⠀⠀⡏⢘⣧⣝⣵⣿⡟⠀⠀⡀⢀⣾⣷⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣬⣿⣟⡀⠀⠀⠀⢰⣿⣟⡿⣿⣿⣷⣾⣿⣿⣏⠉⣉⣶⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⢉⣩⢴⡿⣿⢛⣿⣷⣄⠀⠀⣻⣯⣿⣾⣿⣿⡙⣿⣿⡿⠓⣿⣿⣿⠟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣟⣼⣿⣿⣿⠦⠿⣧⡀⢿⣿⣿⠟⣿⣿⣿⠏⠉⢀⡀⠈⠙⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠉⠀⢨⣷⡄⠀⠼⣷⣾⣿⢟⠀⣉⣏⠡⠀⢴⡋⠉⣧⠀⡙⢿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⢸⣿⣿⣄⣶⣟⡿⠁⠀⣺⣿⣿⣿⠀⠈⠓⠚⠃⢀⣴⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣿⣿⣿⠃⣠⡼⠛⣿⣋⣽⣿⣷⡐⣠⣶⣾⡻⣿⡿⠛⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠈⠻⠈⠻⠋⢀⣠⢬⡟⠻⢿⠿⠩⠿⣿⣿⡷⠟⠁
⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠸⣇⢀⣽⠀⠀⠀⠀⠀⣸⣷⣤⣄⡀
⠀⠀⠀⣾⣿⣿⣿⡿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⣤⣄⠀⠉⠀⢠⣿⠃⢀⣾⣿⣿⣷⠈⢿⡄
⠀⠀⠀⣿⣿⣿⠃⠀⠉⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠋⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡏⠀⠈⠁
⠀⠀⠀⢻⣿⣿⣷⣶⣶⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣹⡿⢿⠁
⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⢀⡘⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠠⠐⠒⠊⡓⡀⢄⡀⠀⠀⠀⠀⠊⠐⣌⠉⠁⠉⠉⠁⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡠⠋⠀⠈⢂⡉⠢⣀⣀⡄⠸⠊⠀⠈⠐⠢⢄⡀⠀⠀⠀⢹⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠤⠊⠀⠀⣀⣴⠆⠉⠠⡘⡜⠀⣇⣀⣀⣤⡶⠔⠃⠈⢆⠀⠀⠘⡀⠀⠈⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠋⠐⠐⠒⠋⠉⠁⠀⡀⠀⢠⠃⢸⠀⠈⠀⠁⠀⠀⠀⠀⠘⡀⠀⡦⡗⠀⠀⠀⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠔⣧⠀⡈⡀⠀⠈⠀⠀⠀⠀⢀⣀⡀⡎⢀⢿⠀⣀⣀⠀⠀⠀⣠⣤⠀⢷⠈⠁⢽⡄⠀⣰⠁⠀⠘⡀
⠀⠀⠀⠀⠀⣠⢞⣁⠔⠩⠙⠀⡅⠀⠚⠻⠇⠀⠀⠈⠋⠁⡇⠘⢎⡆⠙⠛⠀⠀⠀⠘⠉⠀⣸⠆⠀⢁⣆⢀⡍⠀⠀⠀⢠
⠀⠀⠀⠀⠀⡇⠀⠳⠌⠀⠀⣠⡇⠀⠀⠀⠀⡠⢤⠀⠀⠀⡔⣸⡀⡁⠀⠀⠠⠤⡄⠀⠀⠀⡹⡄⠉⠀⠱⣤⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⢱⡀⠀⠀⠑⠓⠴⣙⡀⣀⣀⡀⣀⠀⣤⡤⠌⠴⠛⠋⣿⠢⡄⠀⣀⢠⡤⣄⠔⣵⡀⠁⢀⡼⠋⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠙⠣⣤⠳⢺⠦⡄⠀⣰⡑⠀⠂⢼⢂⠦⠠⠓⠚⢄⠄⠀⣁⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠈⠛⢤⡀⠀⢰⠙⠁⠀⠀⠀⡼⠀⠀⠀⠀⣠⡶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣐⠀⠀⠀⠀⠂⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⢀⠀⠀⠀⢀⠤⠄⠀⠀⠀⡈⠑⢄⠀⢀⣤⡒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠄⠘⠐⠆⡄⠀⢀⡆⠀⠀⠀⠀⠌⡙⠉⢙⣉⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠉⠲⠄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠈⠱⠄⠋⠀⠀⠈⣇⠐⢂⡀⣥⡽⢳⡤⢿⡶⠤⠄⠠⠀⢀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠋⠐⠂⠈⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠌⠋⠀⠀⠘⠁⢨⠓⢀⠀⠀⠀⠒⠶⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠙⠢⡤⠀⠀⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⢠⠸⣇⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠐⠀⠴⠀⠄⠰⡤⠒⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⢦⠀⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠠⡀⠀⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢣⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠡⠋⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⢄
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠤⠤⠤⢤⣀⣀⣱⣄⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠋⠐⠒⠒⠲⠖⠠⠤⠤⠤⠤⠤⠤⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⠛⠛⠛⠛⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⡾⠁⠀⢠⣂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⣸⠇⠀⢈⠏⢹⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⣿⠀⠀⠺⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣾⠀⠀⠀⠀⣴⠞⠋⠉⠁⠀⠀⠈⠉⠉⠛⠶⣤⣀
⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣧⡴⠇⠀⣀⣀⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠳⢼⡇⠀⠀⠺⠤⠜⠘⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠹⣆
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠳⣼⠁⢀⣤⣤⣄⣀⣀⣹⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣦⢄⢻⡄
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠋⣿⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠢⠸⡇
⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⢀⡏⠓⢾⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢸⠷⢤⡾⠀⠀⢀⡔⣶⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⡾⠀⠀⢀⠀⠀⣸⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠇⠀⠐⠭⠇⠀⣿⠃⠀⠀⠀⠀⣀⣠⣤⣤⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⣯⠤⠶⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣌⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉



⣶⣶⢶⠶⠶⡶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⣶⡶⣶⣶⣶⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣡⣦⣄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢰⡦⠀⢀⣠⡾⠛⠛⠒⠒⠒⠚⡛⣦⣄⠲⢄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛⠒⠿⢶⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣿⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⢠⠚⠁⡄⠸⣇⠀⡀⠀⠀⠀⠀⠀⡠⠀⠐⠋⠀⠘⠁⠚⠿⢿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡄
⠀⠀⠀⠀⠀⠈⠄⡄⠰⠀⢛⣼⠀⠀⠀⠤⠂⠉⠀⢀⡀⢀⣀⣐⣄⣠⠀⠀⠈⠉⠻⠷⠀⠀⠀⢀⣀⣀⣀⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠈⠾⣦⡳⡀⡟⠀⠀⠀⠀⢀⢴⠀⡀⠙⠿⢿⣿⣿⣽⣝⣢⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠙⠿
⠀⠀⠀⠀⠀⠀⢀⠠⠟⠿⣿⣿⠀⣴⠋⠀⡂⠀⠈⠉⠲⣶⣤⡝⠋⠃⠀⠹⠗⠒⠢⡀⠀⣀⣤⡶⢦⣤⣞
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠹⣇⢸⠇⣴⡹⠁⠄⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣇⣀⣉⡿⠛⠐⣼⣿⠛⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣠⠎⠉⠉⢿⠀⠀⠀⠉⣤⣷⠀⠈
⠀⠀⠀⠀⠀⠰⡀⢆⠀⠀⢀⠀⡋⢸⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣀⡴⠞⠡⠄⠀⠀⠠⠀⠀⠀⠀⣠⢏
⠀⠀⠀⠀⠀⠀⠙⢦⣥⣄⣈⢳⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠞⠁⢸⠂
⠀⠀⠀⠀⠀⠰⠁⠈⠀⠈⠉⢿⡏⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢿⡆⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠁⠶⠐⠆⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣿⡟⠀⢑⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠑⢆⡀⠀⠀⡀⣿⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢴⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡦⣤⣰⡿⠀⠀⠀⡀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⡀⢁⠬⠁⠌⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡹⣁⢰⣧⡞⠂⢤⠀⠈⠀⠀⠀⠀⠐⠖⠆⢸⡷⠋⣠⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢯⡀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢲⣄⡈⠳⣏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢷⠀⠀⢰⡇⠀⠀⠀⠀⠀⣾⢉⢻⣶⠿⠀⣀⣷
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠲⢿⡇⡀⠘⡟⠀⠀⠀⠀⠀⢹⣧⣾⠁⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢋⡒⣾⣿⣀⣶⡇⠀⠀⠀⠀⠀⠀⡉⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠻⠿⠋⡕⠀⠀⠀⠀⢀⠐⠉⡒⠖⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⢲⣆⠁⠑⣄⠀⠀⣾⡠⢠⣇⠨⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠋⠆⠀⢻⡆⠀⣿⢃⣾⠃⢸⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠰⡸⡄⠈⠟⠀⢹⣾⡏⢠⡏⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣀⢷⡁⠀⢸⠀⠈⣽⡁⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠸⠏⠿⡀⢸⠀⠀⠙⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣙⣛⣛⣹⣯⣩⣉⣿⣋⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣭⣽⣭⣽⣿⣯⣭⣭⣭⣭⣿⣭⣭⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣶⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⢛⡻⢿⣿⠇⠀⠀⠀⠘⣿⣿⠿⠿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣿⣿⣿⡿⠀⠀⠀⠀⠀⢿⣷⣿⣿⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠛⠀⠀⣄⣠⡄⠀⠈⠻⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⣉⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠉⠉⠐⠢⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣄⡠⠊⠁


⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⠶⠶⠶⠶⢶⣶⠶⠶⣶⠶⠶⠶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠳⢬⡁⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⢌⢣⡉⢀⡉⠒⠀⠐⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡈⠀⠀⠑⠀⠡⠄⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠤⠤⠄⠂⠋⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠂⠀⠈⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠈⠑⠄⠀⣉⠀⠀⠀⠀⠀⠀⢀⡃⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⡂
⠀⠀⠀⠀⢐⠁⠀⢠⠢⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠈⣊⠘⠃⠀⠀⠀⠀⠀⠓⢀⠎⠀⠀⠀⠀⠀⢀⠂⠀⢀⠀⠀⠀⠀⠀⡀⠠⣲⡀
⠀⠀⠀⠀⠙⢤⠞⠑⢄⡊⠀⠀⠀⡀⠃⠀⢠⡀⠘⠆⠀⢨⠐⢀⠀⠸⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠘⡀⠇⠀⠸⠀⠀⠀⡀⠀⠀⠀⠀⢖⡈⠔⠑⠄⠸⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⡇⠀⡠⠈⠤⠄⠁⠤⠀⠠⠀⠀⠚⠁⠈⢰⠀⡃⢀⠀⡀⠀⠀⠠⠀⡇⡏⢀⣈⠙⠀⠑⠦⠈⠂⠰⠒⠚⢀⡀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠃⢀⣤⣤⡀⠀⠀⠀⠀⠀⠞⣯⡳⣦⡔⠀⠀⡄⠀⠀⢠⠁⠀⠰⣼⢋⣹⣷⠀⠀⠀⠀⠀⢤⡶⣄⡈⠁⠀⠈
⠀⠀⠀⠀⠀⢄⡀⠀⠀⠐⠈⠂⠛⢠⣿⣷⠀⠀⠀⠀⠀⢿⠿⠃⠁⠁⠀⡀⠀⢘⡇⠀⢠⠀⠈⠀⠈⠙⠛⠀⠀⠀⠀⢰⣾⣿⠈⠙⠁⠀⠀⠂⠀⢠⠔
⠀⠀⠀⠀⢤⠀⠁⠀⡀⢀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡈⡔⠀⠒⠁⠀⠄⠀⢠⠙⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⡀⠈⡁⠖
⠀⠀⠀⠀⠀⠈⠁⠋⢠⠁⢣⣜⡄⠀⠀⠀⠀⠀⠓⠃⠀⠀⢀⠠⠂⣀⠁⢠⠀⠀⠀⢠⡀⠀⣁⠈⠢⡄⠀⠀⠙⠋⠀⠀⠀⠀⡀⠼⠰⠂⠈⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠁⣸⢄⠀⢄⡀⠀⡡⠴⠶⡖⠒⠂⣾⠛⠚⠣⡀⠈⠁⠀⠀⠀⠀⠈⠀⠈⠁⠀⠚⠓⠛⢿⠉⠀⣰⢲⠳⠤⡀⠠⣄⡠⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⡠⠀⠍⠂⠀⠁⠀⣆⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⠀⡔⠀⠁⠄⠠⠀⡀⢠⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⢀⠃⠀⠀⠀⠀⣾⣙⠙⠂⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣐⠁⠀⠀⠀⠀⣎⣹⡟⣀⢀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠀⢸⡀⠈⢀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠏⠀⠀⠀⠀⠀⠀⠁⠀⢹⢄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡀⠔⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢃⠀⠀⠀⠀⡌⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠑⠄⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠄⠊⠀⠔⠈⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢆⠀⢆⠀⠀⠐⠀⠐⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠈⠑⡀⠈⢀⠄
⠀⠀⠀⠀⠀⠶⢿⠄⠊⠀⢀⡔⠁⠀⠉⠐⠀⠒⠒⠒⠀⠄⠂⠈⠀⠈⣲⡀⠀⢱⡆⠀⢀⣏⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⣱⡄⠀⠈⠠⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢚⠍⠑⠓⠋⠙⠛⠁⠙⢑⠦⣀⣀⠀⠀⠀⠀⠀⠀⣀⣠⠴⡻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠉⠙⠒⠒⣒⡒⠒⠚⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠈⠩⠉⢩⡍⠉⠍⠁⢒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠆⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⡸⠀⠰⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠐⠆⠀⡄⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠠⢠⠇⠀⠸⡦⠄⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠀⢼⠀⠀⢱⡠⠄⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡛⠦⣼⠀⠀⠀⠭⢝⣛⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⣓⠦⠽⠀⠀⢸⠥⣟⣛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠠⢈⠑⠺⠀⠀⢘⡒⠋⡀⠀⠻⢄⠀⠀⠀⠀⠀⠀⠀⢀⠴⠧⠀⡈⠓⢺⠀⠀⠘⠒⢁⠠⠈⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣀⣳⡋⠀⠀⠨⢁⡲⢁⡄⠀⠀⠵⠀⠀⠀⠀⠀⢠⠣⠀⠀⠤⠷⢊⠡⠀⠀⠀⢳⣆⡤⠀⠀⠨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠐⠊⠀⠀⠀⠀⠀⠀⠁⠒⠒⠂⠈⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡹⠉⠈⠋⠁⠀⠁⠈⠛⠀⠒⠐⠒⢠⠁⠀⠂⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣼⣷⡄⠀⢈⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠞⢿⠉⢡⠅⢄⡈⣷⡧⠤⢠⣤⣤⣀⣀⡀⠠⢐⣠⠤⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⡂⠒⠠⢸⡇⢭⣥⣌⢈⣙⣚⣛⡃⠿⠏⣇⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⣥⣴⡶⢈⣰⡶⣶⣶⣤⠤⠤⣭⣅⣈⣀⣀⣹⡃
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠚⠙⠵⠤⠦⢠⣬⣭⢁⣉⣁⣀⠒⠛⠛⠛⠛⠟⠟⠛
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣄⠀⢸⠀⠀⠀⠀⠀⣿⠉⠙⠛⠛⡇⣟
⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣯⠉⠀⠀⠸⠀⠁⠀⠆⠀⢹⠂⠀⠀⠀⣿⠿⠈⠙⠀⠁⠁
⠀⠀⠀⠀⠀⠀⢈⡿⣿⡿⣿⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⢸⣷⡩⠀⣷⠙⠆⠀⠀⢰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠎⡡⠵⠀⡙⠀⢂⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠂⡗⠠⠈⠠⠀⢸⣦⠀⠠⠴⢾⣧⠀⠀⠃⠀⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠘⠃⠀⠀⠀⢾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣴⡖⠂⠂⢀⣀⣈⣡⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠃⠀⠀⠲⠲⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣠⠆⠀⠀⠀⠀⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠘⠁⠰⠀⠰⠄⢀⡀⠸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⢰⣆⢀⣶⠀⠀⠰⣦⣀⡿⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⣾⠃⠀⠀⠀⠘⣿⡇⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣠⡞⢻⣇⠀⠀⢀⡼⠛⣷⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡉⠀⠈⠁⠀⠀⠈⠁⠀⠈⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣶⣦⣤⣤⣤⣤⣶⣶⣿⣿⢯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣼⠟⠋⣻⠟⠛⠉⠉⡿⠁⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⣾⠏⠛⣻⡷⣦⠀⠀⠈⣷⡀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣼⡟⠀⠀⣿⡇⠀⠀⠀⠀⠸⣷⡄⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⢀⣠⠶⠖⠛⠒⠶⢦⣄⠀⠀⢀⡎⠀⠀⢀⣿⡇⠘⠛⣿⠻⠂⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠘⣿⡄⠀⠀⣠⡴⠶⠒⠒⠲⠶⣤⡀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢰⡶⠈⢻⡄⢸⠃⠀⠀⢸⣿⡇⠀⠀⣿⣀⡀⠀⠀⠀⠀⠀⢻⣿⣧⡀⠀⠀⠀⣿⡇⢀⡾⠃⢴⡆⠀⠀⠀⠀⠀⠙⢷⣄
⠀⢀⣿⠁⠀⠀⠿⠿⢷⣶⣿⣁⠀⢀⣷⣼⣇⠀⠀⣸⣿⣧⡐⠟⢿⠉⠋⠀⠀⣀⣀⣀⣿⣿⡿⢿⣦⣤⣾⣿⣧⣼⡇⠀⢀⣹⣶⡶⠿⠿⠃⠀⠀⣿⡄
⠀⠸⣿⡄⠀⠀⠀⣀⡾⠉⠉⠙⠛⣼⡿⠁⣹⠿⠿⠿⡿⠸⣷⡀⠈⠀⠀⣰⡟⠉⠀⠈⢻⣿⡁⠀⠉⠙⠛⣧⡀⢻⣧⡚⠛⠉⠈⢿⣄⠀⠀⠀⢀⣿⡇
⠀⠀⠻⣿⣦⣄⣀⣙⣃⣠⣤⣶⣿⣿⣥⡞⠁⠀⠀⢸⡇⠀⠙⢿⣷⣶⣶⣿⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠈⠳⣤⣿⣿⣦⣤⣄⣈⣛⣀⣀⣴⣿⠟
⠀⠀⠀⠀⠉⢉⡿⠿⣿⡋⠀⢨⣿⡽⣇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠉⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢨⣯⣽⣅⠀⠉⣿⠿⢿⡉⠉
⠀⠀⠀⠀⢠⡟⠀⢀⣿⡇⠀⠠⣽⠤⣿⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⣿⢿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡤⣿⡄⠀⠀⣿⡄⠀⢹⡆
⠀⠀⠀⠀⠸⣧⣤⣿⠿⣧⡀⠀⠟⣰⡏⠀⠀⠀⠀⢿⡆⠀⠀⠀⢀⣿⡟⢸⡇⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⢸⣇⠹⠆⢀⣴⠿⣿⣦⣤⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡿⣶⡾⠋⣿⠀⠀⠀⠀⠘⢿⣤⣤⣤⣿⡿⠁⠈⢿⣤⣀⣀⣀⣼⣿⠇⠀⠀⠀⠀⠀⣸⠉⢻⣶⢾⡏⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠸⣧⣠⡞⠀⠸⣧⣤⡟⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠹⣦⣴⡟⠀⠻⣤⣴⠏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⡻⡿⢛⠿⢿⢿⢻⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⡠⠐⢄⠀⠀⠸⣧
⢀⣶⠟⠛⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢀⠊⠀⠀⠀⠆⠀⠀⣿
⣾⠃⠀⠀⡀⢀⠈⠻⢷⡄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠶⠶⢶⠶⠶⠿⢤⣤⣁⠀⠀⠀⠀⢠⠀⠀⣿
⣿⠀⠀⢰⠀⠀⠑⢄⠈⠻⣦⡀⣀⣤⡶⠟⠋⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠘⠀⠀⣿
⣿⡀⠀⢸⠀⠀⠀⠀⠑⢀⣸⠟⠋⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⢰⡟
⢸⡇⠀⠈⡀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⠻⠿⠦⠀⠀⠘⣦⣾⠃
⠈⣿⡀⠀⢡⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢠⠀⡀⠀⡀⠀⠀⠈⢿⡀
⠀⠸⣧⠀⢀⡾⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠔⠁⠀⠀⠀⣠⣖⡊⠙⠓⢴⠁⠀⠀⠀⠘⣷⡀
⠀⠀⢻⡆⢸⠁⠀⢠⡿⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣠⡆⠀⢇⠀⠀⠀⠀⠸⣧
⠀⠀⠀⢿⣾⠀⠀⠈⠀⠀⢀⣱⢄⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⢿⣿⠿⠃⢀⡎⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠈⣿⠀⠀⠀⠐⠢⡞⣿⣿⣄⣠⡄⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠠⠴⠊⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⡇⢿⣿⣿⡿⠃⢀⠇⠀⠀⠀⣀⡠⣤⣄⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠘⢦⣈⣁⣀⡠⠊⠀⠀⣀⠾⣷⣶⣿⠏⠀⠀⢠⠀⠀⠉⠑⠢⡀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠈⣧⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠘⢤⡀⢀⡴⠃⠈⠉⠉⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠈⠁⠀⠄⠀⠠⠂⠀⠀⣀⡤⠞⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣤⣄⣀⣁⡀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠶⠚⠋⠙⡄⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⡏⡙⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠸⣧⣀⡀⢀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡆⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⡏⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣴⠟⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣇⣴⠟⠁⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⢷⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡟⠋⠁⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠿⠛⠒⠒⣾⣤⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢰⣃⣀⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⡀⠀⠀⣰⢋⠁⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣠⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠶⣿⣦⣤⣅⣀⣠⣤⣇⠀⠀⡄⠀⠠⠀⢀⣇⣀⣀⣤⣴⠾⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣛⣿⣶⣷⣶⣿⣾⣟⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠁⠐⢠⠤⠐⠈⠀⠊⠀⠁⠀⠂⠂⣈⠃⠈⢄⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢄⠀⠄⠉⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠐⠆⡠⠆⠀⠀⠀⠀⠀⠀⠀⢀
⢀⠀⣀⠀⠀⠀⠀⠘⡀⠀⠆⠈⠛⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⠀⠀⠀⠈⠋⡀⢢⠀⠀⢆⠀⠀⢳⢉⠖⠀⠀⡄
⢬⡋⡶⠀⠠⡦⠀⠀⠐⠀⢂⠁⠀⠀⠀⢠⣄⠀⠀⠁⠀⠁⠒⠀⠐⠂⡂⡀⠀⠀⢅⡧⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠈⠁⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢳⠀⠀⢈⠀⠀⠔⠁⠀⠢⠜⠀⠀⠀⠀⢠
⠀⣀⠀⠀⢀⡀⠀⢰⠀⢀⠀⠀⠀⡤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⠂⠀⠘⠀⡐⠄
⠀⠁⠀⠰⠉⠼⠀⠀⠑⠀⠂⢡⠀⠘⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡠⠁⡀⡀⠈⡆⠀⠀⠀⠀⠀⠀⢠⠔
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⢀⠀⠁⠶⠆⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⡅⠀⠐⠑⠈⠀⠀⠀⠀⠘⠀⠀⠋⠰
⠀⠀⢀⡴⠢⡀⠀⠈⠤⡈⡣⠂⠀⠛⠀⢀⠁⠀⠀⠀⠀⠄⠲⣀⡀⠈⠁⠀⠀⠀⠖
⣀⡀⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⢐⣷⠸⡡⠀⠠⠒⣇⠀⣤⣜⠃⢡⡀⢄⡀⠀⠀⠀⠀⢰⡅⠹⠁⠄⠀⠀⠀⠀⠠
⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠂⠁⠀⣇⠘⢀⡀⠀⠀⢄⠂⠁⠀⣅⠀⠀⠀⠀⠠⠔⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠁⢁⠂⠀⠀⠀⠀⠀⠀⡄⢀⠈⠁⠀⠀⠈⠀⢠⡲⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠸⠈⢣⡀
⠀⠀⠀⡠⠄⠂⠀⠀⠘⠈⠐⠀⠀⠀⠀⡀⠄⠀⠛⡘⠇⢁⣀⣀⣀⠥⠚⠀⢀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠁⠑⠃
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠐⣀⠂⣤⡁⡴⢞⠂⡀⠀⠑⢄⠀⠀⠀⠀⠠⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⡀⠪⠘⠐⠁⠈⠀⠑⠊⠢⡀⠀⠀⠀⠀⠀⠀⠑⠁
⠀⠀⠀⠀⠀⢀⠄⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠂⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠘⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠦⣀⡠⠴⠢⠔⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡘⠀⠀⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⡀⠠⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠤⠘⠁⢄⠀⠀⠀⠀⠀⣠⠄⠘⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣢⣤⣴⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠈⣿⣿⣦⠀⠀⣠⡾⣿⡇⠀⠀⠀⢠⣦⣤⣄⡀⠀⠀⢀⣤⣤⣶⠀⠀⠀⣿⢫⠤⡀⠀⠀⠀⠀⡰⣬⢻⠀⠀⠀⣿⣷⣤⡀⠀⠀⠀⣠⣴⣿⡗
⠀⠀⠀⠀⠀⠛⡇⠹⣷⣷⡿⢉⡿⠁⠀⠀⠀⠀⠿⢅⣀⡙⣶⣶⣋⣀⠨⠟⠀⠀⠀⠸⣾⣦⣿⣢⣤⣴⣺⣴⣿⠞⠀⠀⠀⢻⣿⣿⣿⣦⣀⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⣾⠇⣤⠚⠟⢢⡄⣿⡆⠀⠀⠀⠀⢠⣿⠾⢻⡟⢿⣿⠾⣷⠄⠀⠀⠀⠀⣠⣞⣯⢋⣿⢉⣽⣷⣦⠀⠀⠀⠀⠀⣾⣿⡌⠿⠿⠟⣸⣿⣦
⠀⠀⠀⠀⠀⣻⣷⠄⠀⠀⠀⢴⣿⡁⠀⠀⠀⠀⠘⠳⠾⠋⠀⠀⠛⠶⠞⠁⠀⠀⠀⠀⠸⢿⡞⠃⠀⠈⢳⣿⠟⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠸⢿⡿⠃


⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⢄⡀⠀⠀⠀⢀⡠⢤⣦
⠀⠀⠀⠀⢹⣷⢿⣱⣄⣄⣴⣹⢷⣿⠁⠀⠀⠀⠇⡀⡀⠀⢀⡀⠀⡀⠀⠁⠀⠀⠀⢸⣇⠈⠁⢂⣠⣠⠊⠉⢸⡿⠀⠀⠀⢀⠄⠲⢶⢄⣤⣠⡤⠄⢀
⠀⠀⠀⠀⠀⣹⣿⣏⣿⣿⣏⣿⣻⡁⠀⠀⠀⠀⠀⢀⡀⠀⢿⡧⠀⢀⠀⠀⠀⠀⠀⠀⣩⡶⠂⣾⣿⣿⠂⠲⣌⠁⠀⠀⠀⠷⣤⣤⣠⣾⣿⣦⣀⣀⣤⠗
⠀⠀⠀⠀⠀⢿⣿⣿⠀⠁⢨⣿⣿⠇⠀⠀⠀⠀⠀⠋⠀⡀⠸⠃⠀⠀⠇⠀⠀⠀⠀⠀⠻⣦⣸⡿⠹⠻⣗⣠⡟⠁⠀⠀⠀⠀⠈⠏⣉⠝⠻⣍⡉⠝
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠉⠙⠁


⠀⠀⠀⠀⣴⡶⢤⣀⠀⠀⣀⡤⢠⢶⠀⠀⠀⣠⠤⠖⠲⣤⣀⣀⣤⠖⠦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⣢⣄⠀⡀⢀⣤⣆⠐⡆
⠀⠀⠀⠀⠘⣦⠈⠙⣿⣿⡛⠉⣠⡟⠀⠀⠀⠈⠑⢤⡀⠈⢹⡏⠁⣀⡔⠋⠁⠀⠀⠀⠀⠀⠀⠘⢶⠊⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣟⣻⣿⣟⣿⣿⠆
⠀⠀⠀⠀⠀⢸⡙⡇⠛⠛⢲⢋⡿⠀⠀⠀⠀⠀⠀⠀⢻⡾⠃⠹⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⢰⣲⠀⠀⠀⠀⠀⠀⠀⠘⣿⣬⣭⢿⣭⣴⡿
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠈⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢲⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⠤⢤⠞⠁⠀⠀⣸⡀⠀⠀⠙⢦⠤⢤⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⢸⡦⠴⠖⠋⠁⠉⠓⠶⠤⣾⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⣀⢸⡄⠀⠀⠂⠀⠀⠂⠀⠀⣼⠀⣠⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡀⠦⣄⣀⣀⡤⠆⣰⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⡿⠲⠤⣤⡤⠴⠺⡷⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⡼⠁⠀⠀⣖⡆⠀⠀⠹⡄⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡻⠁⠀⠀⠀⣈⡀⠀⠀⠀⠹⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠳⠃⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡤⠤⢤⡤⣤⠤⠤⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⠇⢿⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠀⠀⠚⣭⣶⣿⣷⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⣴⣾⣿⣷⡀⠈⣿⣿⣿⡇⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⣛⢉⣩⣭⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⢛⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣰⣿⣤⠀⢀⣠⣿⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⠛⢛⣉⣉⣭⣬⣽⣿⣦⣀⣀⣀⣴⣿⣿⡿⣣⣾⣿⡿⢋⣿⣟⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣻⣟⣵⣿⣿⡟⢠⣿⣿⣿⡏⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣬⡙⢿⣿⣿⣿⣿⠿⠛⢛⣉⣉⣉⣩⣭⣥⣤⣶⣿⣿⣿⣿⣿⠟⢁⣿⣿⣿⣿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣦⡙⢿⣿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⢛⣉⣥⣴⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⢀⣶⣦⣬⡙⠿⣿⣿⣿⣿⣦⡹⣿⠿⠛⠻⠛⠛⠛⢛⣋⣥⣶⣿⣿⣿⣿⣿⣿⣷⣤⣬⣉⠁⣁⣴⣿⣿⣿⡆
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣌⠻⢿⣿⣿⣷⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⣛⣉⠽⣟⣫⣴⣾⣿⠋⣾⣿⣿⡇⣴⣄
⠀⠀⢀⣴⣦⣄⣈⡙⠻⢿⣿⣿⣿⣦⡙⢿⣿⣟⠻⠛⠻⠿⠿⠿⢛⣫⣴⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠟⢸⣿⣿⡟⢠⣿⣿
⠀⢠⣾⣿⣿⣿⣿⣿⣶⣤⡙⠿⣿⡿⢻⣾⣷⡾⢿⣿⣿⣿⣿⡿⢿⣿⣿⡿⠟⣛⣯⣥⣤⣴⣷⣿⣟⣋⣠⣴⣿⣿⣿⢁⣿⣿⡿
⢀⣙⣙⠛⠛⠛⢿⣿⣿⣿⢿⣷⣾⡆⣷⣿⣿⣴⣷⣿⠛⣿⣿⣿⣎⢻⣿⣿⣦⠀⠉⠉⠉⠉⠛⠛⠻⣿⣿⣿⣿⠟⢡⣾⣿⣿⠇
⣿⣿⡿⠿⠛⢛⣛⠟⣽⢃⣿⣿⣿⡇⢹⣿⣿⣷⣹⣿⡇⢹⣿⣿⣿⣆⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣙⡻⢧⣴⣿⣿⡿⠋
⠋⣡⣴⣾⣿⣿⢋⣾⡇⢸⣿⣿⣿⡇⢸⣿⡟⢿⣿⣿⣿⡈⢿⣿⣿⣿⣇⢹⣿⣿⡇⠀⠀⠀⠀⠠⡄⠘⠛⠁
⣿⣿⣿⣿⠟⣵⣿⣿⡇⢸⣿⣿⣿⠇⣾⣿⣇⠸⣿⣿⣿⣷⠈⣿⣿⠿⠟⠀⠉
⣿⣿⠟⢡⣾⣿⣿⡿⢀⣿⣿⡿⢣⣾⣿⣿⣿⠀⠿⠿⠟⠛⠁
⠟⢁⣴⣿⣿⡿⠏⣴⣿⣿⣿⠇⠘⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⢀⣴⣾⣿⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣶⣤⠀⢰⣿⣿⣿⣿⣿⣿⣿⣦⣜⠿⣿⣿⣿⡇⣠⣴⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣵⣿⣟⣩⣥⣤⣤⣤⣉⡙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⣶⣿⣿⣿⣿⣿⣿⣿⠿⠋⠈⣿⣿⣷⠀⣸⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣦⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠉⠉⠉⠀⠀⢀⣠⡄⢹⣿⣿⡄⣿⣿⡿⠀⠀⠉⠉⠛⠛⠛⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠰⠛⠂⠀⢀⠀⠀⠀⢸⣿⣿⠀⢿⣿⡇⠈⠉⠃⠀⠀⠀⠶⡆⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢄⡀⠀⠀⠀⠀⢀⠤⢺⣿⠏⠀⣾⣿⣷⣄⡀⠀⠈⠁⠀⠀⣀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠄⣩⡛⢆⠈⡩⢳⡄⠾⠋⠀⢀⣿⣿⣿⡿⣩⠳⡀⠀⡘⢊⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠣⡀⠉⣗⠔⢁⠜⠆⠀⠀⣼⣿⣿⠏⡜⢄⠉⢛⢎⠔⠠⠾⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠙⡅⠀⠀⠀⢠⢁⣜⠿⠟⠁⢠⠀⠀⠁⢫⡔⠈⠀⠀⡂⠻⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⠈⠁⠀⠁⠀⢸⠀⠀⠀⢰⡇⠀⠀⠀⡇⠀⠀⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠌⠉⠣⡀⠀⢸⠀⠀⠀⠀⠀⢸⡀⠀⡠⠋⠙⢄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣭⡈⠁⠀⠀⠀⠈⢀⣫⣿⣷⣶⣶⣿⣿⡃⡈⠀⠀⠀⠀⠁⢘⣽⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⣷⣶⣿⣶⣿⡿⠿⠿⠛⠛⠿⠿⣿⣿⣷⣾⣶⣷⣾⡿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⢀⡐⣦⣦⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⣀⢈⣩⣍⠉⠡⡷⣐⠂⡡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⣦⣿⣵⣿⣿⣿⣿⣿⣿⣤⡈⠀⠂⡅⢀⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠎⠉⠉⢽⡟⢼⡿⣫⣿⣿⠛⡀⢐⠊⠄⠁⡳
⠀⠠⢴⣶⠶⠞⠉⠀⠀⠀⣤⣀⣀⡀⢈⠠⡄⢈⣡⣾⠏⠏⠟⠑⠀⠁⠘⠢⠄
⠀⠀⠀⠀⠀⠀⠠⣼⣤⣤⠶⠍⠻⠿⡟⢻⡈⡟⠟⡇⠈⡀
⠀⠀⠀⢠⣴⣦⣤⢮⣿⣟⣃⢀⣱⣦⣄⣽⡅⠀⠀⢳⠀⡀
⠀⠀⣀⣙⡯⢿⠽⣟⣿⠻⠳⠘⣧⠻⣎⢇⠱⡄⠀⠈⢧⠰⡀
⠀⠀⢛⣿⠇⣌⡻⠙⣝⠣⣥⡀⠈⢳⡜⠓⠃⠁⢀⣠⡄⠑⠥⡄
⠀⠀⠨⠻⠘⠏⠳⡄⠈⠀⣀⡙⣂⣥⣠⣾⣟⣤⣈⣯⣶⣶⣾⣶⢦⣀
⠀⠐⣄⣻⡾⠦⣺⡾⣷⣶⠃⠤⢄⠉⠻⣿⣿⢿⣿⣿⣷⣾⣾⣿⣷⡦⢅⡲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠻⣀
⠀⠊⠈⢸⡗⣾⠻⣿⠛⠇⡁⢐⣲⣿⣶⠿⣯⠀⠻⣿⣿⡷⣽⣿⣾⣭⣛⠛⠛⠒⠾⠽⢷⣶⡶⠦⠶⠖⣚⣫⠿⠋⠁
⠀⠀⠀⠘⡄⠹⣄⡆⠀⣾⠵⡊⠿⠳⡹⡄⠘⢧⡀⠙⠻⣶⣤⣈⠙⠛⠿⠿⢷⣶⣶⣶⣶⣶⣶⠿⠟⠛⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠐⠧⠐⠻⣿⣷⣿⣥⣅⢰⡀⠙⢿⣄⡀⠙⢶⣤⣀⠉⠙⢻⣿⣶⣶⣶⣶⣾⣿⣭⣤⠄⠠⠀⣀⣀⡠⠄⠀⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣁⣀⣀⣈⣛⣶⣦⣽⣛⣛⣁⣀⣀⣈⣉⣉⣉⣁⣀⣀⣀⣒⣛⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣂⣊⣴⣿⣿⣷⣿⣿⣢⣤⣄⠚⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⠈⣿⣿⣿⣿⣿⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⣿⣿⡿⣛⣿⣿⣿⣿⣿⣿⣿⡿⢻⡟⢻⡇⠘⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⠏⠀⠀⣿⡀⠸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣿⣿⣿⣿⡿⢿⡟⢉⣰⠏⠀⣴⠀⢿⡇⠀⠀⠈⣿⣿⣿⡿⣋⣿⣟⣉⣻⣤⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣷⢿⣿⣿⡟⣹⣿⣿⠇⢸⠇⠀⣼⡟⠀⠈⠁⠀⠀⡼⣿⣧⣿⠳⣼⡟⠛⠻⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠠⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣸⠀⠀⣿⡇⠀⠀⠀⠀⠞⢠⣿⣿⣷⣮⣿⣗⣤⣴⢼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢀⠙⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⣿⣄⠀⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⣿⡁⠀⡠⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣼⣿⣷⣮⡀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⣼⣿⣿⣿⣿⠆⠀⠀⠀⠀⢻⣿⣿⣿⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣿⣿⠉⠛⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠸⢿⣼⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⢠⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⡿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⣿⡟⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡆⣈⣉⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡦⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⠄⢀⣠⠎⠉⢷⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡰⡞⠈⠀⠀⠀⢀⠈⠵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢧⢱⠀⠒⠂⡀⠈⠀⢀⠉⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⡀⠈⠓⠂⠀⠀⠀⡀⠀⠉⠀⠙⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡀⠐⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⣀⣹⠀⠀⠀⠀⠀⡀⠤⠀⡂⠱⠌⡰
⠀⠀⠀⠀⢀⠔⠊⠒⠒⠀⢄⡀⠀⠰⠈⠈⠉⠀⠀⠀⠁⠀⠈⠈⠀⠉⠁⠠⠀⠀⠀⠀⠘⠰⠈
⠀⠀⠀⠔⠃⠈⠀⠀⠀⢀⠀⠀⠁⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡄⣆
⠀⠀⡎⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠡⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡔⠈⠠⠀⠀⠀⠑⢢
⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠨⠀⢀⡴⠒⠒⠒⠲⢄⡀⢠⠀⢸⠀⠀⠈⠀⠀⠀⠂⠀⡇
⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢸⡧⠀⠀⣿⠆⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠘⡀⠀⣤⠀⠀⠀⠀⠹⠃⠀⡇⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠙⡒⠒⠠⠤⠤⠤⠴⢤⠂
⠀⠀⠀⢱⠀⠉⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⠈⠲⠤⠤⠤⠴⠚⢡⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣇⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠤⠄⠒⠒⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⢤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⢛⠻⢿⣏⣿⡿⣿⣶⣝⢿⣿⣿⣿⣷⡔⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣥⣶⣤⣤⣀⠁⠈⠙⢿⣿⣛⣿⣮⢻⣿⣿⣿⣿⣮⡳⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠙⣿⣙⣿⣇⣿⣿⣿⣿⣿⣷⡹⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣦⠀⠘⢿⣯⣿⣾⣿⠏⢻⢿⣿⣷⢹⣝⡾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⢿⣟⣿⡏⢾⡷⠞⢛⣿⡏⣯⣷⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢧⡼⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠘⣿⣽⣷⣎⠙⣉⠾⣿⣷⢻⣿⣇⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣨⡀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠸⣿⡿⠬⠿⣿⡺⠿⣿⣟⣿⣿⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡌⠁⢠⣤⣶⣶⠒⠀⠉⠛⠻⣿⣿⣿⣿⣿⣧⠀⢻⣶⣶⣤⡁⢰⣦⣿⣿⣿⢧⣻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⡄⠀⣿⠿⣿⣿⡿⠓⠀⠀⠘⢿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣇⢸⣿⣿⣿⣿⣿⣿⣿⢻⣽⣿⡿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢋⣾⣻⣼⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⠠⣿⣿⣿⣿⡿⠋⠀⠀⢫⣾⣿⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⢲⡶⣘⣿⡉⠁⠀⠀⠀⠀⣾⣿⢏⢼⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠁⠀⠀⠀⠀⠀⠀⣀⣴⣾⡟⣳⣦⣥⣶⡟⣷⠀⠀⠀⠀⢠⣿⡵⣫⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⣀⡀⠠⢤⣶⣿⣿⠟⠀⢻⣎⣻⣿⣛⣾⠃⠀⠀⠀⠀⠀⣿⣾⢟⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠁⠀⠀⠀⠙⢿⣾⠿⠛⡄⠀⠀⠀⠀⠀⣟⡵⣫⣸⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣾⣦⡀⠀⠀⠀⢯⣾⢣⣿⡿⣿⣿⣿⠇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⢴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⡤⢞⡯⠋⢀⣿⣦⡀⠀⡿⣡⡿⢿⣾⣿⣿⡟⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⢾⣿⠋⢽⢧⡀⠀⣀⣀⠤⢖⣪⠵⠚⠉⢀⣴⣿⣿⢻⣿⣦⠘⣫⣶⣿⣿⣿⢟⡾⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠹⣿⡿⠁⠀⢈⣳⡬⡥⠴⠒⠊⠉⠀⠀⣀⣴⣿⣟⣻⣿⣿⣏⠿⣿⣿⣿⣿⣿⡷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣽⢻⡿⠇⠀⢀⣾⣿⣟⠀⠀⠀⠀⠀⣠⣾⣿⢛⣿⠟⣿⠿⢋⡤⠚⣡⣾⡿⣫⣷⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣟⣿⣿⡆⠀⠀⠛⠟⠃⠀⠀⣠⣴⠿⣿⣬⣿⣿⡿⢓⡡⠞⣡⣴⣿⣿⣿⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⡿⣿⣿⣦⣤⣄⣠⣴⣾⣽⣿⢿⣿⣿⡽⠟⣩⠴⢋⣴⣾⣿⡿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠙⠧⣁⠛⠼⠯⢹⣯⡩⠯⠹⠟⢒⣉⣭⠴⢚⣋⣤⣶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠒⠒⠛⠛⠙⠒⠛⠋⠉⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⠿⠻⡿⠛⠿⢿⠿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣴⣶⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⣾⣷⣤⣤⣤
⣠⡄⠈⠃⠉⠘⠁⣦⡄⠀⠀⠀⠀⠀⠀⠀⡸⠏⠀⠻⠙⠉⢧⠀⠀⠀⠀⠀⠀⠀⠸⢉⣤⡀⢀⣤⡙⠃⠀⠀⠀⠀⠀⠀⢻⡿⠋⠁⠀⠀⠈⠻⢿⣤
⢸⣏⠁⢐⣀⡀⢈⣹⠁⠀⠀⠀⠀⠀⠀⠠⣯⣁⣤⣤⣤⣄⣽⡆⠀⠀⠀⠀⠀⠀⠇⣾⠟⠃⠈⠻⣿⠸⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⣀⣀⠀⠀⢸⡟
⡀⠲⠂⠀⠉⠁⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣿⣿⣿⣾⠟⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠙⠛⠀⡠⠃⠀⠀⠀⠀⠀⠰⣿⣷⣀⠀⠈⠉⠀⣠⣾⠿
⢛⠀⣘⠀⠐⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⡿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣷⣶⣶⣿⠛⠃
⠘⣴⡇⢠⠄⡤⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⠭⢭⢭⠽⡄⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⣶⢰⡆⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⠠⠃
⠈⠛⠆⣸⡀⣇⠰⠛⠃⠀⠀⠀⠀⠀⠀⠀⣾⢉⣻⣛⣼⡉⠷⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⢿⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⢿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢶⣆⢀⣶⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠟⠈⠁⠀⠉⠙⢿⠟⠀⠀⠀⠀⠀⢀⣠⣄⠀⣀⡀
⠀⢀⠀⠐⠒⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣠⣤⣤⣤⣀⣈⠀⠀⠀⠀⠀⢠⠋⣸⣾⠿⠿⠿⣷⣄
⠣⠁⠀⠀⠀⠀⠀⠡⠇⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢰⣿⢻⡟⠁⠠⠀⠀⠀⢻⡛⣷
⠠⠀⠀⢿⣿⠿⠀⢀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⣿⢩⠉⠀⠀⠀⠀⠀⠀⠀⠹⣏⠳⢄⠀⠀⠀⡀⠈⢛⡯
⠀⠑⠤⠀⣀⠠⡔⠁⠀⠀⠀⠀⠀⠸⢿⣿⣷⣄⠀⠀⠀⣠⣾⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠐⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡏⠉⠹⣷⣄⢿⣞
⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠈⣬⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⢀⣿⣜⡞
⢠⠂⠀⠀⠀⠀⠀⡓⡄⠀⠀⠀⠀⠀⠀⠀⢸⡿⢃⣴⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢘⠀⡄⣤⢚⠎⠀⠀⠀⠀⠀⠀⠐⢿⣿⣷⣤⣾⣿⣿⣿
⠈⠉⠺⠭⠶⠭⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠖⠲⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠓⠻⠆⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⡐⠐⢶⠄⠦⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⠈⠡⡗⢤⢳⠄⣤⣱⠉⢨⠕⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠃⢹⣲⠇⠊⠀⠀⣼⣷⣦⣚⣙⡁⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⡖⣸⣤⡾⠎⠀⠀⠀⠙⢯⣟⡟⣷⣿⣜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠝⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠀⠨⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠒⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢺⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠈⠉⠁⣶⠆⠀⠀⣶⡆⠁⢸⣼⣏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠠⢀⠄⠛⠳⣤⠴⡋⠗⢀⣼⡿⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹⠗⡄⠇⠀⠀⠘⠀⠀⠁⠀⠘⢿⣷⣜⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠁⡄⢸⠀⠀⠀⠀⠀⢰⠀⡄⣎⣿⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣏⠀⢄⠀⣸⠀⠀⠀⠀⠀⠸⡁⠀⠏⣿⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠎⢈⠁⠀⠄⠀⠀⢀⠀⠉⠀⢲⣿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡔⠛⠃⠀⠂⠀⠀⠐⠀⠄⠀⠀⠘⣯⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠣⢀⣀⣀⣀⠀⠠⠤⠔⠓⠒⠒⠐⠒⠂⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠤⢤
⠀⠤⠒⠀⡁⠐⣀⣠⠄
⠀⠀⠀⠀⠀⠐⠀⠣⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⡀
⠀⣀⣀⣤⣄⣀⢀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⣿⣷⣄⡀⠀⡀
⠀⠀⠙⠟⠻⠟⣛⡀⠘⠛⠉⠉⠁⠀⠀⣿⣿⣿⣭⣱⡆⠀⠀⠀⠀⠀⠀⢀⠟⠁⠀⠙⡋⠃⠹⡀⠀⠛
⠀⠀⠰⠶⢰⣿⢋⣹⢀⠂⠀⠀⠀⠀⠀⠘⠿⢿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠠⠞⠉⣿⠉⣿⣿⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠇⠘⣚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣹⣂⣴⣀⣠⣠⣠⡠⠄
⠀⠀⠀⠀⠒⢸⠏⠉⢿⣿⠀⠒⠒⠒⠒⠒⠛⠛⠛⠒⠒⠛⠛⠛⣿⠃⠙⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠉⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠉⠈⠙⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡠⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⠀⠀⡀⠐⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⠤⠀⠀⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠬⣉⠐⠁⠀⠂⠐⠠⢂⠄⡀⡀⢀⠀⠀⠀⠊⠊⠄⠀⢁⠁⠈⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠒⠀⠀⠐⠢⠁⠬⠀⠡⠒⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠠⠄⠄⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠁⠀⣈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢱⠁⠈⢢⢀⠀⢱⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢸⠀⠈⠀⠀⠐⡇⢁⠈⠀⠀⠀⠁⢠⣀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⢘⡠⠀⠀⠀⠀⠀⠁⠙⢢⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⢴⡀⠀⠆⠀⠀⠀⢘⡠⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠣⡄⡎⠀⠀⠀⡬⡀⠄⡀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⢱⠘⠀⠀⠑⠀⠀⠀⡜⠩⢐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢤⠀⠰⠧⠱⠘⠆⠀⠀⡠⠾⠶⠭⢖⣞⡮⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣟⢀⣤⠀⠌⠠⠶⠚⠄⠀⠀⠀⠀⠈⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⡂⠑⢖⣲⠀⠀⠄⡂⠐⠠⢂⠔⡠⢂⠖⠔⠕⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⢐⠁⠀⠘⠀⠀⠤⠂⠰⡶⠤⠬⣄⣡⡀⢤⢔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠅⡪⠀⠀⠀⠀⡨⢐⠀⠇⢨⠀⡀⠀⠐⠄⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠄⠨⠂⠀⡄⠂⠀⠀⠐⠀⠘⡈⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣀⡀⣀⡀⢀⣜⡽⠀⠀⠁⠠⠀⠃⠃⠀⠀⠠⠪⠂⠀⠐⡆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠲⣁⡀⡀⢀⣠⡤⠚⠝⠒⠒⠂⠚⠉⠛⠑⠋⠀⠀⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⢀⠲⡄⠀⠀⡠⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠢⡜⢱⠀⠣⠤⠧⣤⠎⠙⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⣧⠈⢧⠀⠀⡴⠙⢄⣀⡎⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢫⢣⡀⠑⢌⡲⢤⣀⡼⠀⠀⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠳⣝⠢⢄⣉⠑⠚⢒⣚⡩⠥⢺⡤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣀⣠⣀⠀⠀⠀⠀⠉⠒⠂⠬⠝⠓⠚⠉⠉⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡠⢤⣀⠀⠀⠀⠀⡇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠼⠶⠠⠤⡀⠀⠀⠀⠑⠐⠷⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⡇⠀⡘
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠈⠳⠀⠀⣀⠄⠒⠒⠢⢄⠀⠀⠀⠀⠀⠀⡼⠀⢠⠃
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⡠⢾⠀⡰⠃
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡷⡖⡏⠀⢸⠀⡧⣂⠢⣄
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⣇⠀⢸⠀⠳⠽⠃⠈⡆
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠉⠒⠚⠀⠀⠘⡆⠈⢧⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠵⢖⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠲⠤⠄⠀⠀⠀⢸⠤⢄⣀⣋⣻⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⣠⡃⠀⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡦⠤⠤⠖⠚⡗⠒⢶⠒⠋⠀⠐⢲⡏⠣⣌⡉⠓⠚⠉⢉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡸⠁⠀⢸⡀⠀⠀⢀⡜⠙⢦⣀⠈⠉⠉⣉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠙⠲⠖⠋⠀⠀⠀⠈⠉⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢲⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠔⠀⠀⠰⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡦⠛⠘⠣⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡃⠀⠀⠀⣻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣧⣤⣤⣤⣤⣤⣤⣤⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠗⠀⠄⠐⠿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢠⣿⣿⠟⠉⠉⠉⠉⠉⠉⠙⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣸⠿⡟⠀⠀⡀⠀⠀⠀⡀⠀⠈⡿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠛⠏⠀⠀⣄⠀⠀⢀⡄⠀⠈⠟⢻⣿⣿
⠀⠀⠀⠀⠀⢹⣄⠀⠄⠄⠉⠀⠀⠀⠉⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⠀⠒⠂⠀⠀⠀⠀⠀⠐⢂⢠⣾⣿⡿
⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠂⠀⡼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡳⠶⠦⣤⣤⡤⠶⠲⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠲⠤⠤⠴⡒⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣑⡨⡄⠠⠠⠄⠠⠈⡉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣌⠀⠀⠀⠀⢈⡄⠀⠀⡀
⠀⠀⠀⠀⡀⠚⠁⠀⠀⠰⠂⠀⠀⠀⠐⠈⠀⠀⠙⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢄⣀⠰⣿
⠀⡀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣸⠃⠀⠄⠀⢻⠀⠀⠀⠀⠀⠈
⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠉⠉⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠘⣧
⠀⠂⠀⠀⠀⢀⣀⠀⠂⠇⠀⠀⠀⠀⠀⠇⠀⢀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠢⠀⠀⠀⠀⣀⡴⠿⠀⠀⠀⠀⠀⠀⠸⠷⣄⣀
⣶⣶⣶⣶⣶⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶

⠀⠀⣇⠀⠁⠀⠀⢴⣄⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣷⣟⡙⠂⣠⣤⣀⡆⠀⡍⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡄
⡿⠿⠻⠿⠋⠀⠀⠘⠀⠘⣿⣤⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠏⢹⣿⢠⡄⠀⠀⠀⠀⠐⢦⣿⣿⡟
⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠿⠒⡱⡄⡀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠿⠈⡇⠀⠀⠀⣠⣄⠀⠉⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠚⠁⠈⢶⡄⠀⢐⡇
⠜⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣖⠃⣤⡀
⠑⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠶⣄⣀⣴⣿⣦⣀⡀⠀⠀⠀⠀⠀⡼⠟⣿⠟⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⠛⠋⠉⠒⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠲⢤⣠⣄⣤⣀⣄⣀⣠⣿⣀⣘⣿⣠⣄⣤⣀⣠
⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⡏⠁⠈⣿⠉⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣤⡀⠀⢠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠛⠻⢿⣿⡿⠛⠉⠙⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠁⢠⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢀⣿⣿⢁⡀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⣤⡄⢀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠼⠿⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣟⣀⣉⢩⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠰⠿⠶⡾⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣧⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⡼⠁⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⠄⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠼⠛⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⢿⣿⣧⠀⠀⠀⠀⢀⡀⢴⡆⠰⠀⢂⡀⠀⢠⡄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⣷⣿⣘⣿⢆⠠⢀⠀⠁⠈⠁⢚⣐⠃⡀⣀⠀⠀⠄⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⢁⣥⣿⡗⣤⣃⢦⠘⠳⣦⠠⠖⡀⠰⠊⠁⠀⠀⠀⠀⣠⡀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣸⣿⣿⣧⡞⣼⣫⠏⠳⠠⠾⠨⠁⠚⠰⠗⠀⠊⠀⠄⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢀⣠⣼⣿⣾⣿⡛⣿⣽⡘⠧⠥⢁⣶⡘⠁⢈⡀⢀⢉⡄⠈⠉⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡦⣻⣿⣼⣿⣿⣿⡿⡏⣿⣿⣫⡿⢊⢓⣈⠁⠄⠙⠡⠠⠀⠄⠀⠐⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣿⣧⣶⣬⣉⠻⠛⠉⠀⢀⣹⣏⣻⢾⡤⠜⠀⣈⢃⡚⠂⠠⠄⠀⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠛⠻⢿⣿⣿⣶⣦⣄⡆⠺⣿⣯⣻⣓⡒⡐⠏⠁⣅⠘⢒⠀⡀⠈⠁⠀⠐⠃
⠀⠀⠀⠀⠀⠳⡄⠀⠀⡀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣦⣴⣴⣶⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠃⠁⢀⠈⠲⠀⠀⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⢷⠶⠛⠻⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠿⠿⠿⢿⠻⢿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣠⣤⣤⣦⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢔⣠⣾⣿⣿⣿⣿⣿⣧⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣁⣥⠖⠛⠛⠛⠿⣿⣿⣿⣿⣿⣷⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡗⣒⣂⡀⠀⠀⠉⠙⠛⢿⣿⣿⣿⣧⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢋⣹⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⣀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄⠀⠀⠀⠀⣀⡀⠀⠀⠸⣿⣿⣿⣿⣆⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⢰⣯⣥⣾⠃⠀⢀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡿⣿⣿⣿⣿⣧⡀⠀⠈⠈⠉⠀⢠⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠹⣿⣿⣿⣿⣿⣷⡶⠆⠀⠐⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣹⡄⣈⣛⣛⣛⣛⠻⢫⡄⠀⠼⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠟⠉⠙⣿⣿⣿⣿⣿⡄⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣆⠘⢿⣿⣿⣿⣿⣿⣧⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠘⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡽⣿⣿⣿⣿⣿⣆⠈⣻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⣾⠟⠛⠟⠚⠛⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠊⠁⠀⠀⠀⠀⠺⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠫⠛⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠉⢿⡿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠈⠻
⣿⣶⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠛⠋⠁⠀⠸⠀⢀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠙⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢤⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⣿⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⣀⣿⣿⣿⣿⣿⣇⣾⣷
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⠛⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣾⣿⣿⣶⣾⣷⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣠⠟⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠏⠀⠀⣾⣿⣿⣯⣿⣯⣿⣿⣿⣏⣿⣿⣿⣿⠆⠀⠀⠀⠈
⣤⣴⣿⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⠛⠋⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠼⠛⠻⡄⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣯⣿⣽⡇
⠀⠀⠀⠀⠈⣿⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⢀⠀⠀⢰⡟⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿
⠀⢸⠀⠀⣿⠁⢠⠀⠀⠀⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠃⣿⡄⠀⠹⣧⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣄⣿⣧⣀⣤⣀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣀⡀⠀⠀⠀⣿⣿⣿⣿⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣄⣀⡀
⣭⣭⠽⠿⠿⣿⡿⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣭⣭⠙⠛⠷⢼⣿⣿⣿⣿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠮
⡽⠿⣿⣿⣭⣭⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠛⠛⣦⣼⣿⣿⣿⣿⡟⢧⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⠷
⣶⣖⣖⠒⠚⠻⠗⠲⠆⠀⠀⠀⠐⠛⠓⠶⠂⠀⠀⠀⠛⠳⠶⢶⣀⣈⠉⠙⠻⠿⠶⢦⣀⣀⠉⠛⠿⠿⠿⣯⣟⣛⣛⠻⠿⠿⣶⣶⣞⣛⡛⠛⠶
⣉⣻⣿⣿⣭⣭⣅⠀⠉⠉⠙⣯⣤⣀⣀⡀⠉⠉⠉⢫⣤⣤⣀⣈⠉⢙⣛⢻⣬⣭⣅⣈⡉⣙⣛⡻⣿⣭⣭⣍⣉⣙⣛⣻⣿⣯⣭⣭⣍⣉⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠜⠁⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠔⠢⡀⠀⢀⠔⠉⠀⠀⠀⠉⠢⠤⠊⠉⠒⠨⠕⠢⡀
⠀⠀⠀⠀⠀⠘⢄⠀⠀⠇⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠉⠘⣀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢠⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⡆⡇
⠀⠀⠀⠀⠀⠀⠸⡸⡀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠑⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢎⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠔⠊⠁⠀⠆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠋⠑⠒⡄⠈⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡒⠤⠤⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⠜⠀⠀⣆⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠂⠉⠀⠈⠑⠂⠤⠐⠂⠁⠀⠀⠀⠈




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠍⠩⠉⠉⠛⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠛⠫⣝⠛⢻⣟⣻⣿⠽⠷⢿⣛⠿⢯⠿⠿⠽⠟⠛⠛⠛⠛⠛⠛⠋⠉⠋⠉⠁⠈⢿⣄⠀⠀⠀⠀⠀⠀⠨⢍⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠁⠀⠘⣳⢊⠁⠌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢬⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿
⠀⠀⠀⠀⠀⢤⣦⣤⣤⣠⣝⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢾
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣾⣿⣿⣿⣷⣧⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⣀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠉⢛⣻⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠐⠛⠁
⠀⠀⠉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢿⡎⠉⠻⣿⣿⣿⣿⣆
⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣄⢙⣶⠋⠈⠉⠛⠿⣿⣦⡀
⠈⠉⢉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢉⣿⠏⠀⠀⣀⣀⣀⠀⠀⠹⠇
⣀⣴⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢿⣿⣿⢰⡟⠁⢠⣶⣿⣿⣿⣿⣿⡷
⢠⠿⢻⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡑⠢⡀⠀⠘⣿⣿⠀⠁⠀⣼⣿⠟⠉⠉⠙⠁
⠁⢀⡾⢋⣿⣿⣿⠃⣹⣿⡿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⣸⠂⢠⡸⣿⣷⣡⣠⣿⡿
⠀⠈⠀⢸⣿⣿⡟⢰⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣾⣿⣷⣿⣿⡇
⠀⠀⠀⡯⠉⡿⠁⡼⠋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢐⠆
⠀⠀⠀⠁⠀⠀⠘⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⣶⡄⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⡿⠀⣸⣿⣿⣿⣿⠟⣿⠋⢛⠊⣿⠁⣿⠿⠻⠀⠛⣿⢻⡷⣿⣷⠊⠈⢹⠇⠈⠉⠀⠀⠀⠐⠋⠹⡧
⠀⠀⠀⠀⠀⠁⢀⣽⡟⢰⡿⠋⠔⠁⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⣷⣾⣏⠀⠀⢻⠀⠀⠀⢰⡖⠀⠀⠀⣰⡷
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠗⠀⢻⡄⠄⣼⢹⣷⣠⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣶⢀⣴⣿⡆⠁⠈⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⣿⡇⠀⣬⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣏⣼⣿⠀⣴⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠙⠳⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣎⠙⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⣶⣿⣽⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠓⠀⠚⠛⢹⣿⣿⣟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠋⠻⢿⣻⣛⣟⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣠⣶⢂⣂⢀⢏⡟⢉⣿⣿⣿⣿⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⣉⣀⡀⣉⣉⠁⣀⣋⣉⣻⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠂⠛⠹⠇⠿⠿⠺⠿⠿⠿⡿⣿⣿⣩⡝⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡦⣄⡠⠂⠀⠄⠴⢀⠠⣤⣤⣤⢤⣴⣦⣿⣿⣿⣿⣿⣷⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠃⡁⠀⠰⠖⠶⠲⠦⠇⠀⠉⠻⠿⣯⣻⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠐⢂⡤⢠⣅⣄⣀⣀⣀⣀⣀⢀⡀⠀⢀⣀⡀⢀⣊⣙⣽⣿⣿⣿⣿⣯⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠉⠹⣻⢿⣿⣿⢻⣿⣙⡿⡟⠿⡛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣺⣿⠇⠘⡏⠉⠈⠁⠀⠁⠁⠉⠹⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠐⠂⠀⠤⠠⠒⠰⢆⠛⠛⠓⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⢿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢘⣿⣿⣿⣿⣿⡿⠟⠻⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠑⠀⠆⠂⠦⠀⠀⠀⢀⡠⣿⠟⠋⠉⠁⠄⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⣀⣀⣀⡀⠀⣀⣠⣔⢂⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣻⣿⣿⣿⣿⣿⣿⢿⣿⡤⢐⡀⣀⠀⠀⠀⠄⠈⢷⡀⠈⢮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⣼⣿⢸⣿⠶⠟⠛⠋⠀⠀⠀⠀⠀⠀⠙⢷⣾⣧⣠⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣚⣿⡿⣻⣿⣧⡈⢷⣿⣷⣔⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣽⣸⡿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣿⠿⠉⠛⢿⣿⣾⢿⠉⠋⠛⠿⠿⠤⠀⠀⠀⠀⠀⠑⢹⣿⣿⣿⣿⣿⡿⠛⠛⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢛⣹⣿⣿⣧⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⣠⣾⣷⡿⢟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢄⠀⠀⠀⠀⠀⠀⠀⠈⣉⣽⠿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⡀⣀⣀⠀⠀⠈⠀⣈⡴⣞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⢿⡏⠛⠑⣲⠚⠻⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢈⡘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰
⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡄
⠀⠀⠁⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⣿⣼⡇⢸⣿⣧⠙⣿⠃⢸⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⢹⡇⢸⣯⣿⠀⣿⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠛⠘⠃⠙⠁⠛⠁⠋⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⣤⡴⠆⣶⢸⡆⢿⣟⠀⣿⢹⡇⢹⣿⡏⢸⡏⣿⢸⣷⣾⢰⡶⠆⠀⢀⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣇⠠⠈⣿⠀⢹⣿⡇⢸⡏⠀⣿⢿⡇⠀⣿⠁⢸⣇⣿⢸⣿⣿⣸⠋⠀⠀⠛⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠨⡇⠀⠙⠶⠀⠻⠃⠘⠛⠃⠛⠈⠋⠀⠙⠀⠈⠙⠋⠘⠋⠃⠛⠃⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠦⠒⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠀⣖⠀⠀⠀⠀⣀⠀⠀⡀⢐⠀⠥⠒⠉⠀⠀⢀⡀⢠⠜⢲⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠓⡀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠔⠈⠑⠂⠁⠀⠈⠂⠀⢀
⠀⠀⠀⣄⠠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠐⠈⠂⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⠦⠀⠀⠈⣐⠁
⠀⠀⠀⣏⠀⠟⠀⢠⢨⡁⠀⠀⠀⠈⠒⠁⠀⠀⠈⡈⠀⠀⠘⡐⢂⠀⠀⠐⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢎
⠀⠀⠀⠀⠓⠀⠀⠀⠓⠉⠢⡀⠀⠀⠈⠀⠀⠀⠠⠦⠄⢀⣨⠠⠴⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⡆⠀⠀⠀⠀⠸⠆⣄⠀⠀⡀⠀⢀⠀⠀⠤⠀⠉⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠠⣶⡄⢀⣒⣼⡓
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢙⡁⠀⠀⠰⠿⠀⢈⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣠⡁⠄⠀⠀⠀⠀⢅⠁⠢⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⣄⠉⠡⠋⠘⠌
⠀⠀⠀⠀⠉⠂⠂⠚⠀⠀⠀⠀⠊⠀⠰⠈⡄⠀⠀⠀⠀⠉⠃⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠔⠀⠂⠈⠀⠀⠀⠀⠈⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠠⢀⠁⠀⠀⠀⠀⠀⡅⠀⠠⣶⠆⠀⠢⠤⠀⠀⠀⠀⡐⠁⠀⠀⠐⠂⠠⠀⢀⠁⠀⢀⡀⠀⠀⠐⠄⠐⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠂⢀⠂⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠂⠀⠀⠠⠚⡔⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠀⠊⠁⠀⠀⡀⠀⠀⠀⠀⠇⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⢆⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⡐⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⠐⠢⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠢⡈⠢⡡⠀⠀⠀⠠⣹⠓⠂⠀⠃⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠱⠄⠀⠀⠀⠀⡅⠁⠁⠐⠀⠀⠀⠀⠀⢀⠀⢠⠀⠠⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⢖⣈⠶⢌⠀⠀⠀⠈⠂
⠀⠀⠀⣂⡠⠐⠀⠃⢀⠚⠀⠀⠀⠀⠀⢈⠂⠒⠁⠀⠀⠀⢈⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⣀
⠀⠀⠈⠚⠢⢀⣀⠠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠄⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⠀⣴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⡄⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣬⣹⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⠉⠉⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣶⠄⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣏⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣼⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣣⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠉⠉⠉⣿⠋⠉⠉⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠈⣹⠋⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠐⢛⠇⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠔⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡉⠐⠀⠀⠈⡉⠉⠂⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠔⢀⠞⠒⠬⣀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⡴⠋⡀⠀⣤⠀⠉⡷⠀⢰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠛⣆⠈⠁⠀⠀⠀⢀⠛⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠘⣦⣀⣈⣁⡠⣞⣢⣤⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡹⠥⣑⣤⣶⠭⠁⢣⡉⠢⡀⠱⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠵⠄⢰⡿⠀⠀⡉⠀⠴⠚⣉⢹⢦⡀⢀⡇⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢰⠀⠀⢘⠋⣉⣠⠦⠞⠚⠁⠁⠸⡄⠨⠙⠒⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢈⢦⠀⠾⠉⠀⠀⢀⠀⠀⢄⢂⠤⠾⠦⡴⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠴⠁⠈⠉⢀⡗⣄⡩⢄⡔⠒⠃⠈⠀⠀⢣⡀⡀⢉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡜⠲⢺⠈⠀⠀⠀⠀⣀⠠⠄⣒⢨⡴⠿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⡄⠈⡗⣀⣡⣄⡰⠤⠒⠒⠁⢁⡤⢒⡻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠁⠀⢀⡠⠄⢐⣈⡥⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠵⢤⠗⠠⠚⢰⠛⡇⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠘⡄⡇⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢃⠃⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⣠⣼⠘⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡿⠋⠀⣿⣿⡿⠟⠁⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⡈⠛⠋⠀⠀⠀⠈⢉⡀⠀⠀⠙⠛⠁⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠛⠋⢀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠀⠀⢚⡫⠥⢤⡀⠀⠀⠀⠀⠀⠀⡠⠖⠈⠹⣍⠉⠀⠀⠀⠈⢿⣿⡿⠿⠿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⣬⣷⠀⣞⣠⠖⠋⠉⠆⠀⠀⠀⠀⠜⠉⠒⠲⢤⣘⡇⠀⠀⠀⠀⠀⠉⢠⣴⣶⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⣯⣹⡇⡏⠀⣔⣐⣦⠀⠀⠀⠀⠐⠀⡰⢒⣦⠀⠈⡇⠀⠀⠀⠀⠀⠀⢿⣍⡹⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⢻⡇⠜⠀⠌⠿⠋⡖⠉⠚⠀⠀⠠⡝⣛⣋⢠⠔⠀⠀⠀⠀⠀⢀⣀⣘⣋⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⠓⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣰⠟⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡴⠋⠁⣠⣤⣈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠶⢤⣤⣀⣀⣀⣀⣁⣈⣠⣤⣴⠞⠋⠁⠀⢀⠋⠀⠀⠙⣦⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣠⣤⣭⣭⣍⡉⠙⣛⣶⣿⣥⣤⣄⡀⢀⡞⠀⠀⠀⠀⢸⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⣴⡟⢉⣠⣾⣿⣿⣻⣾⡋⢁⣠⣾⣿⣿⣿⠟⠀⠀⠀⠀⢀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢠⣾⣿⣿⣿⠿⠾⣷⠌⠙⢻⡿⠿⠿⠿⠛⠉⠁⠀⠀⠀⠀⢀⣼⠃⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⡀⠀⢀⣼⠟⣿⣿⣿⣟⠓⠆⠀⠀⠀⢸⠁⡆⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠉⠈⠉⠙⢧⣹⡏⢸⣿⣿⣿⣿⣭⡍⠀⠀⠀⢸⣄⣧⣤⣤⣤⣶⣶⣾⣿⣿⣿⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⠄⠀⠀⠀⠀⢸⡟⠁⠐⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⣾⣿⡦⢠⡿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠛⠛⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣾⣿⣯⣟⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⢹⣿⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⢿⣿⣛⣛⣛⣻⣿⣿⡿⠟⠛⠁

⠀⠀⠀⠀⢀⣠⣶⣶⣶⣶⣦⣤⣄
⢀⣤⠀⢴⣿⡿⣿⣥⣴⣦⣭⣟⡻⢿⣦⣄
⢸⣿⣷⣤⣤⠘⠛⢋⠁⢨⠟⠙⠿⣶⣈⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦⣤⣤
⢸⣿⣿⣿⣿⣧⣶⣿⡗⢰⣥⠤⠀⢙⣛⠿⣷⣤⣀⠀⠀⠀⠀⠤⠔⢚⣿⣷⣒⡛⡛⠛⣻⣻⣿⣲⣦⣤⠄⢠⣴⣶⣾⡿⢛⣿⣽⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣥⡀⣿⠀⠀⠀⡈⠁⠠⠶⠍⠁⠀⠀⠀⠀⠀⢠⠾⠿⠿⠛⢻⣿⣉⣉⣽⣿⣿⣿⣿⣷⣦⡀⢀⣠⣶⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣶⡤⣶⠀⠀⠀⣀⣀⠀⠀⠀⠀⡀⠀⠀⣠⣶⣾⠿⠿⢿⣿⣿⠿⠟⠉⠉⣾⣶⣾⣿⣿⠿⠿⠿⣿⣿⣿
⠈⠻⠟⢻⣿⠿⠛⣻⣿⣿⣿⣦⡬⠿⢡⣗⣰⣿⣿⣦⣤⠄⠀⠀⠐⠈⠁⠀⠀⠀⠀⣠⡀⠀⠀⠀⠂⣚⣯⣭⣭⣀⠀⠀⠀⠀⡈⠉
⠀⠀⠀⠀⢀⣀⣿⡝⠋⣿⠉⠉⣤⣁⣼⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣡⣶⣄⠀⠀⣍⡉⣀⣄⣀⡰⡷⢆⣿⣿⣀
⢠⡀⠀⠀⢸⣷⣄⣄⠀⣿⠀⠿⠏⠙⠛⠿⣿⣿⣿⣿⣿⣿⣷⣀⣘⣠⡀⠀⠀⠀⠀⠀⢉⣭⢿⣅⢙⡉⠩⠀⣼⣿⣿⣶⣋⣻⠿⢧
⢸⣿⡀⢶⣿⡿⠿⠇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣿⣿⣿⣿⣿⣿⣀⣦⠀⠀⢐⣿⣿⣇⣀⣾⣿⣤⣠⣿⣿⣿⣟⡝⠛⠛⠓
⢸⣿⣷⣶⣶⣶⣶⣦⣠⡏⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⡿⠿⢿⠻⠇⠈⠃⠀⠰⢿⠟⠿⢟⣻⣿⣇⠾⡶⠶⠾⠇⣠⣦⣤⣴
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣉⣀⣀⡐⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠀⠾⣿⣿⣿⡷⢴⣯⣼⠀⣠⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⡙⠷⣅⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠛⣏⣼⣿⣿⣿⣦⣼⣿⣿⣴⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡘⠂⠀⠀⠀⠀⠀⠀⠀⠀⣤⠸⠄⠀⢰⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣯⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⢻⣷⣸⢙⠢⣄
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢹⣌⢿⣧⣾⠀⣿⠛⢦
⠀⠀⠀⠀⠀⠀⠈⢿⣶⣤⣄⣀⣠⣤⣼⠿⣿⣿⣿⣷⣿⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡆⠀⢻⣿⣿⣿⣷⣄⡇⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠂⠈⠉⠀⠉⠁⠂⠀⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⡀⠒⣿⠟⠁⠈⠉⠙⠒⠲⠤⢤⡀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠟⢁⠔⠀⠴⢶⣶⣿⣓⠦⣤⣧⣾⣷⣦⣀⣤⡶⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠃⣰⣿⣵⣿⣶⠾⣿⣿⡿⢿⣿⣿⣿⡿⢛⣽⣿⣤⡀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⠠⠀⣿⣿⣿⣿⣿⣷⣌⢻⣾⣿⣷⣿⣧⣴⣿⣷⣿⣿⣷⡀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠉⠙⠛⠻⢯⣟⣿⣷⣝⡻⢿⣿⣿⣿⣻⣿⣿⣿⣿⢿⡶⠃⣴⣥⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⢤⠾⢿⣿⣶⣿⣿⣿⣿⣿⡿⠟⠛⠉⣰⣿⠟⠁⠀⠞⢠⣾⣿⡿⠟⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣤⣴⣶⣾⣷⣿⣧⠈⣻⡿⠛⠋⠉⠉⠀⢀⣀⣠⣼⣿⠏⠀⠀⣸⣿⣿⣿⣿⠃⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠋⠉⠙⢿⣿⣿⣿⣷⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⡿⠃⠀⠀⢠⣿⣿⣿⣿⠇⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠛⠛⠁⠀⢀⠁⢠⣿⣿⣿⣿⡿⠀⠀⠀⢀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⣀⣂⣴⣿⣿⣿⣿⠟⠁⠀⠀⢠⡟⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣦⡀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⢻⣿⣿⠟⠋⠉⠁⠀⠀⠀⣠⣷⣶⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⡏⣿⣿⡄⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠋⠉⠙⢻⣟⣿⣿⣿⣿⣿⣦⣄⡀⠀⢀⡴⠊⠁⠀⢸⡇⣿⣿⣿⣶⣶⣿⡟⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⠃⠀⠀⠀⠀⣧⢨⣿⣿⣿⣿⣿⡃⢠⡟⠻⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⣿⣿⠛⢻⣿⠟⢈⣹⠟⠁⠀⠀⠀⠀⠀⠘⢶⣽⠙⣿⣿⣿⡳⣼⠃⠀⠀⠀⠈⠙⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⣠⡾⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⡏⠙⣿⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣼⠟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠸⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣥⣬⣧⣴⣤⣽⣮⣼⣯⣽⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣠⣤⡤⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣻⣿⣿⠋⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠛⠃⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⢠⣿⣿⣿⣿⣿⣿⣶⣤⣤⡄⢀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣧⣿⣿⣿⣿⣿⣟⠛⠋⠉⠹⠿⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣦⠶⠶⠞⠛⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣥⠀⠀⠀⣴⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣿⣿⣶⡾⠛⠄⡈⢻⣿⠛⠛⠿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢻⣿⣿⣿⣿⣶⣄⠙⢺⣿⣀⣀⣀⣠⣿⣿⣿⣦⡀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⣿⣿⣿⣷⣻⣟⡛⣿⣿⣿⣿⠋⢹⣿⣿⣦⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣅⡢⠀⢿⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣷⣤⣹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠏⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠠⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⢻⣿⣿⣿⣿⣿⣿⡁⠀⠀⠈⠛⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⡀⠀⢐⣚⠂⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⣿⣿⣿⣿⣿⡿⠿⠻⡇⠀⠛⠿⠃⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⣿⣿⣿⣿⣿⣿⣄⣰⡧⠀⠀⠀⢀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢐⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣓⠂⠀⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢐⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠠⠀⠀⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⢀⣨⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢐⠀⠀⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢐⠀⠀⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⢿⡾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⡃⡼⠃⠀⠀⠀⠀⣴⣶⣶⣼⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢤⠀⠀⠀⠀⠈⢃⣀⣀⡸⢱⣀⠤⠙⠛⠛⠛⠿⣿⣿⡿⠃⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⢠⣄⣀⣠⠊⠁⠀⠐⢁⠜⢣⠀⢸⣏⣶⣤⡀⠘⣿⣿⣷⡄⠐⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠈⠒⢤⠤⠀⠠⠀⠘⠀⢀⠀⠆⢠⣉⡙⠻⠛⢹⡏⠉⠉⠁⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⢏⠤⠒⠚⢄⠀⠢⠴⢶⡊⠁⠀⢰⠀⡏⢙⡀⠀⠠⡀⠀⠀⣴⠟⣻⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠱⣁⣀⢶⣾⠂⢣⡀⠀⠸⣨⠗⠈⠙⠚⠀⠛⠳⡦⡁⠇⠀⠠⠵⣾⣿⣝⢿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠀⠐⡇⠛⠿⣤⣽⣿⣿⣇⣀⣰⠁⠀⣀⣀⠀⠠⢆⡀⠀⠙⣄⠀⠀⠀⠈⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⢠⣀⡇⠀⠒⢃⠜⠋⢡⣂⡄⠀⣉⠲⠄⠋⢀⣠⣴⠀⣀⠤⡀⠉⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡡⠨⢳⡀⠀⡏⠀⠀⢨⡽⢗⠻⠀⠀⡎⠁⡪⠽⠋⠰⠥⡀⡨⢰⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠰⡅⡀⡇⠙⠂⠖⠂⠘⠁⢀⣎⠀⣀⠔⠋⠉⠑⠢⣾⠼⣀⣈⢣⠸⣌⠆⡀⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠂⠳⢌⣘⡦⡄⣠⡀⠀⣤⠈⠈⢶⠁⠠⡄⣠⣶⣴⠈⡞⠀⡏⡆⣠⠃⠊⠐⠂⠐⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠕⠠⢊⠼⢿⠀⠉⣀⣀⣄⠀⠀⠸⠀⠀⢀⠁⢀⡀⠀⢰⠊⠀⠀⢹⠍⠉⡓⣎⠔⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠲⠡⡐⠆⠆⠀⠻⠽⠟⠀⠀⠀⡆⠧⡸⠀⠿⠃⠀⠸⠀⠀⠇⢸⠈⢑⠈⢘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⣿⠖⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⣶⣶⣶⣿⡖⠒⠒⠚⠧⠁⣂⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠋⠀⠀⠀⠀⠀⣴⣾⣷⣦⡀⠀⠀⠈⠛⠟⠛⠁⠀⠀⣠⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣶⣤⣀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⠀⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢀⣀⡀⠀⠀⠀⣰⣶⣶⣦⡀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢤
⠀⢨⣿⣿⢟⢿⣿⣧⠀⠀⠀⣰⣞⡛⣳⣦⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠶⢤⠁⠆⠰⠆
⠀⠸⣿⣯⠈⠠⣿⡗⠀⠀⢠⡟⠛⠁⠙⠛⡇⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢿⣤⣬⣇⠁⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⢶⢆⣶⣶⣇⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄
⠀⠀⠀⠤⠤⠤⠆⠀⠀⠂⠀⠉⠉⠈⠉⠈⠀⠀⠀⠠⠄⠀⠤⠀⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠻⡿⠿⠃
⠀⠀⠀⠀⢀⢀⠀⠀⣈⠥⠀⠀⠰⣠⡆⣴⢰⣴⢰⠀⣤⣤⢀⡆⣦⢰⠀⣆⣤⡄⠀⠀⠀⠀⠀⠀⠀⠓⠂
⠀⠀⠀⠐⠻⠳⠀⠀⠉⠁⠀⠀⠈⠁⠀⠈⠈⠈⡈⡀⣁⡈⡀⡅⠀⠈⠀⠉⠁⠁⠀⠀⠀⠀⠀⢠⡀⠀⠀⣠⡄
⠀⠀⢠⡀⣀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣤⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣿⡶⣾⣏
⠀⠀⠀⠁⠈⠁⠁⠀⠀⣶⡄⠀⠀⢠⣶⠀⠀⣀⣻⠟⢉⠙⢿⣃⡀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⣿⠙⢹⣿
⠀⠀⠀⠏⠀⢒⣛⠀⠀⠘⡇⠀⠀⢸⠇⠀⠀⣭⣿⠀⠀⠀⢈⣯⡅⠀⠀⠀⠈⢹⣿⠀⠀⠀⠀⠀⣿⠀⢸⡟
⠀⠀⠀⣿⣿⣿⡗⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⢹⡷⣶⣶⣿⡍⠁⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⢸⣶⣾
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣍⠀⠀⠀⠀⠶⠖⠀⣬⣿⡄⠘⠿⠗⠀⢸⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣾⣿⣿⣿⣿⣷⣦⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠈⠻⣾⣿⣿⢿⣿⠛⠛⢛⣛⣻⢾⣟⣿⣿⣿⣆⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠏⣼⣛⣙⣳⣭⣵⣶⣿⣿⣿⣷⣤⣤⣤⣄⡀⠀⣀⡀
⠀⠀⠀⢀⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣀⠀⠀⠀⠀⠀⠀⠀⠁⢼⡛⢷⠸⣄⠀⠙⠻⠿⠛⠋⠛⠛⠻⢿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠈⠻⣿⣿⡿⠛⠟⠻⢻⣿⡟⠛⢛⣻⣿⣿⣿⣟⣡⣴⣶⣦⣄⣀⣀⠀⠈⠙⢺⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠒⣋⣀⣀⠤⠤⠶⢺⣻⣿⡿⠣⣤⣜⠻⠿⣿⣿⡿⠿⢿⡿⣿⠿⠃⠀⠀⣾⠀⡟⠉⣇⣤⣤⣾⣶⣶⣦⣀⣀⣀⡀
⢀⣤⣤⣴⣿⣿⣿⣿⣷⣶⣶⢀⡿⣝⠷⢤⣀⣀⣠⠞⣉⡉⠱⡄⠈⠀⠈⠀⠀⢀⡞⠁⡸⠁⠈⠙⠛⠛⠛⠛⠛⠿⠿⣿⠿⠃
⠘⠻⢿⣿⠿⠟⠻⠿⣿⡿⠭⠋⠀⠈⠛⠲⠾⠿⠶⣞⡉⡟⢠⡗⠒⢶⣦⣀⢀⣿⡇⣤⡇
⠬⠿⡿⣦⣶⣶⣶⣄⠀⢀⠀⠀⠀⠀⠀⢀⣠⣶⣾⠉⠀⣷⡘⠷⢦⣤⣄⣩⣿⣿⣷⣾⠇⢠⣤⠞⢯⢽⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⣤⡤⠤⠬⢷⣬⡒⠶⠶⣶⣾⣿⣿⣗⣀⣠⣤⣼⣿⣿⣾⣿⣿⣿⣽⣿⣿⣏⣿⣾⠿⠖⣁⠤⠤⠀⣠⣶⣶⣿⣟⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠚⠳⠂⢒⣶⡼⠋⣀⣠⣤⠶⠛⠛⠛⠩⠁⠀⠈⠭⠿⠟⠛⠉⠉⠀⠴⠞⠛⠛⠛⠶⠀⠀⠀⠀⠘⠻⣿⣿⣷⠶⠶⠶⠒⠂

⠀⠀⠀⠈⠆⠀⠀⠃⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠈⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣦⣤⣤⣴⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⡀⣢⣉⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢹⣿⣿⣿⣿⢿⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⢠⣆⢢⣲⣶⣠⣀⡀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠘⠋⠀⠉
⠀⠀⠀⠀⠀⠀⣜⣶⣿⢿⠎⠑⡛⢽⣿⣟⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⡔⠃
⡀⠀⠀⠀⠀⢠⣟⡫⠇⠈⡀⠀⡇⠈⠈⡿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠳⡀
⢿⠆⠀⠀⠀⣞⣛⣒⠀⠂⡆⠀⡇⠀⢈⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⢰⣿⠷⠤⠤⠤
⣾⡆⠀⠀⠀⠴⣖⣚⠀⠀⠹⡿⠇⠀⠈⣙⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠈⢂⢀⡀⠀⠀⠀⠀⠀⠀⣾⣤⣦⣄⠀⢠
⣿⡇⠀⠀⠀⠈⠟⣶⣧⠀⡀⣐⣀⣀⢠⣻⡦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠒⠻
⡻⠃⠀⠀⠀⠀⠘⠻⣻⣿⣿⣿⣴⣷⣿⠏⠛⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢻⠀⠒⠒⠐⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣤⣤⣶⣦⠠⣶⣶⣶⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⢻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣄⣄⡀⡀⡄
⠀⠀⠰⠖⠂⠀⠀⠀⠒⠚⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣀⣀⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠼⠍⠉⢼⢗⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⣀⡿⣿⣿⡂
⠀⠀⠀⠀⠈⠡⠻⠓⡈⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠀⠀⠀⠤⣯⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣤⣤⣤⣼⣿⢚⣛⡁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⢤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⢲⡞⠂
⠀⠀⠀⠘⡀⢸⠀⠐⠀⡰⠁⠈⠀⠀⠀⠀⠀⡴⢧⡖⠀⠀⠐⠊⠀⠈⠹⣮⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢦⠠⠀⢠⣥⡀⠄⢰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⣿⡷⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣦⣿⣿⣤⣾⣆
⠀⠀⠀⢀⣀⡀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⡿⢿⠿⡿⢿⣿⣧⣶⣦
⠀⠀⠀⠈⠁⠙⣿⣿⡅⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⡇⠀⡆⠀⢸⣿⣿⡿⠃
⠀⠀⠀⢀⠠⠀⠛⣿⠃⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢻⣀⣿⠀⠀⡀⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⡇⠀⠃⠆⠸⣿⣿⣿⠗
⠀⠀⠀⢸⠛⠰⣶⣶⡆⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠂⠈⠟⠛⠃⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⢷⠇⠀⣿⣿⣿⡶

⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠦⢦⣤⡀⠀⢀⣀⣶⢿⣷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠈⢷⢦⣸⣿⣷⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⡉⢉⣻⢦⡀⢻⣽⣿⣿⡿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠘⡗⡿⣄⣿⣿⣿⡇⠰⠃⢰⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⠀⠀⠘⢛⣿⣹⣿⣿⡄⠀⠠⢄⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡛⡦⠀⢸⣿⣿⣿⠿⠁⢰⣴⡿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣿⣶⣿⠿⠙⢿⡈⢩⣏⢻⡭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠈⠅⠫⠉⢸⠆⠀⣼⡇⠘⠏⢸⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣠⣾⡏⠀⠀⠀⠀⠀⢠⣀⢡⢸⣷⠶⠿⠛⢷⡢⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⢣⠀⠀⠀⢀⠸⠛⠁⠀⠀⠀⠀⠀⠀⣸⡦⠠⠜⣇⠀⠀⠀⠀⠹⣶⡀⠀⠀⠀⣠⠜⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⢈⡥⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠘⡇⠂⢆⣼⠀⠀⠀⠀⠀⠀⠉⠳⣤⣤⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⢰⠀⠈⢿⣿⣷⡀⠀⢀⣤⡤⢄⠻⣟⣶⣽⠷⡇⠀⠉⠀⠀⠀⠀⠀⠀⢰⣟⣋⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⠷⠂⠀⠀⠻⣿⣯⣭⣷⠏⠈⢻⡇⢙⣿⡀⠀⠑⠄⠆⠀⠀⠀⠀⢀⣠⣿⠝⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠃⠀⠀⠀⠙⠻⢾⣇⠀⠀⠀⠀⣠⠊⠻⣾⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⣾⡿⠛⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢆⣠⡄⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠿⠟⠛⠻⠷⣶⣤⡀⠀⠀⠀⠀⣠⣴⠾⠟⠻⠿⢿⣽⣿⣾⣷⣀⣠⡀
⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⣠⠛⠉⠀⠀⠀⠀⠀⠀⠉⠁⠙⢿⣿⣿⣿⣄
⠀⠀⠀⠀⣄⣾⣿⣿⠉⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡄
⠀⠀⠀⠰⢻⣿⣿⠇⠀⠀⠀⠀⢰⣀⠴⠈⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃
⠀⠀⠀⢠⡟⣿⣿⡖⠀⠀⠀⠀⠈⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦
⠀⠀⠀⠀⣩⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⣀⡠⠀⠴⠒⢐⡩⠋⠀⠀⠀⠀⢠⣿⣿⠫⡉
⠀⠀⠀⠀⠈⣻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠔⠒⠊⠉⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⣰⣿⣿⠾⠃
⠀⠀⠀⠀⠀⠿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠎
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⢀⡀⢀⡀⠀⣶⣿⣿⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣁⡀⠤⠄⠒⠂⠈⠉⠀⡔⢀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿⣿⣶⡄⠀⠀⠀⠀⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣌⣴⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠜⠻⡾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡶⡀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⣠⡼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉



⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⡀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⠰⣿⢿⣶⣤⣤⣴⣯⡶⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠠⣶⣾⠀⠀⠀⠀⠁⠀⣼⠃⠈⠀⡼⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⢹⣦⡀⣦⠈⠁⠀⢀⣀
⠀⠀⠃⠀⢸⣿⠋⣿⣿⣿⡿⢿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣷⣦⠈⠁⠀⠀⠀⠀⢀⣼⣿⠀⠀⣰⣇⣴⠏⠀⠄⠀⠀⢀⡤⠂⣰⣿⣿⠁⠄⠀⠠⠀⣻⣿⡗
⠀⠀⠀⢀⣿⣿⣷⣌⢻⣭⡴⠞⣯⣿⣷⠀⠀⠀⠀⠀⣠⣸⣿⡏⠀⠀⠀⠀⠂⢀⣾⣿⣿⡄⢠⣿⣿⡃⠀⠀⠀⣠⣶⡟⠱⠀⠈⠙⠡⠀⢴⡛⣷⠀⠄
⠀⠂⢨⣽⣿⣿⣿⣟⣵⣶⣶⣾⣿⠯⠥⠄⢀⠀⠀⢀⣿⣿⣿⣧⣄⠀⠐⠂⠀⠈⠻⠿⢿⣿⣿⣟⣾⣧⣶⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⡌⠉⢁⠀⠶⣤
⠀⠀⡉⠁⠈⠙⠿⣿⣿⡿⢿⣿⡁⠉⠂⠀⠀⠀⠀⢰⣿⣿⣿⡇⣿⡄⠀⠄⠀⠀⠀⣼⡏⣻⣟⡛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⣀⠀⢿⣦⣤⣶⣤⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠘⠋⠀⠠⠀⠘⠃⠀⠀⠀⣛⣿⣿⣿⢿⡇⠀⠀⠀⢀⡀⠈⠛⠛⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠛⠋⠻
⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⢹⣾⣟⣿⣿⠇⠀⠆⠀⣿⣿⣷⣶⣾⣷⣿⣶⣾⣃⣤⣄⠀⠀⡀⠀⠈⠀⠀⠀⠁⠀⣠⠘⠀⠀⠈⠁
⠀⡀⠀⠀⠃⠀⠀⣿⣆⠀⠀⢸⠀⠀⣠⠊⠀⠀⠀⠀⠘⣿⣿⣿⡏⠀⠀⠀⠠⠀⢼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣟⡀⠀⠁⠀⠀⠘⢄⢰⡾⠛⢿⠴⢦⡀
⠀⠹⣦⣄⡀⠀⠀⣿⣿⣧⠀⢸⣷⣿⠃⠀⠀⠃⠀⠀⣰⠛⡟⢟⢳⡄⠀⠁⠀⠐⢿⣿⣿⣿⣯⣿⣿⣿⣿⣭⣍⠁⠀⠀⠠⢤⡤⣴⠚⡀⣩⣤⠸⢂⡿
⠀⠀⠹⣿⣿⣶⣌⠻⣿⣿⣧⣾⣿⢃⠀⠀⠀⠖⠀⠸⠶⢦⣤⠶⠶⠶⠂⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⡟⢿⣿⡿⠂⠈⠀⡀⠈⠛⠋⠈⠁⠈⣋⢤⠉⡄⠒⠆
⢠⡄⠀⠙⠛⠛⣿⣷⡜⣿⣿⡿⡃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠰⠀⠈⠉⠸⣿⠿⠉⡟⢻⡁⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⣤⣶⠀⠀⠈⣲⣾⡁
⠀⠀⠀⠀⢠⣿⣟⢿⡇⣿⡟⢰⠇⠀⣄⣶⣆⡄⠀⡀⠀⠀⢠⡀⢀⣨⡅⠈⠁⣀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠙⠳⠈
⠀⠠⠄⠀⠀⠙⠿⣷⣤⣭⠵⠋⠀⠀⡺⢿⣿⠟⠀⠊⠀⠀⠀⠋⠈⠙⠀⣀⠈⠁⡄⠀⠰⠀⠀⠀⠀⠀⡃⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢁⠀⠉⠀⠈⠀⣄⠀⠠⠦⠀⢾⢛⣫⠂⡐⠀⢀⠀⠀⠀⠀⠐⡩⠂⠄⡚⠀⠀⠀⠠⠄⠀⠀⠀⠈⣀⣠⣤⡀⠀⠀⠁
⠀⠀⠐⠂⠀⠀⠀⠁⠀⠰⢾⣏⣯⣡⣴⣶⣄⠀⠀⠀⠠⠆⠀⠐⠆⠀⠀⢁⡈⠀⠀⠈⠀⠀⠀⠀⠈⢀⠐⠈⠙⠀⠀⠄⠀⠀⠀⢴⣦⢨⣾⣮⣀⡟
⠀⠰⠀⠀⠀⣠⣴⣶⣄⠻⣾⣿⣷⣿⣯⣿⣿⣤⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠐⠂⢀⣤⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢐⠁⢠⡻⡟⠛⠙⠻⠇
⠀⠀⠀⠰⣄⣿⣿⣿⡿⠖⠉⠉⠁⠙⠿⣍⠉⠠⠄⠀⠘⠀⠀⣦⡖⠀⢀⠀⠀⠀⠀⢲⣦⣬⠿⠇⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣼⣿⠇
⠀⠀⠠⡄⠈⣩⣽⡏⢀⠀⣼⣿⣆⠀⠀⡈⡇⠀⠀⠀⣠⠖⠞⠩⠙⠾⡝⠁⠈⠀⢀⣰⡟⢃⡴⠶⡄⠀⠰⣀⡇⠀⠀⠀⡀⠀⠀⠓⠀⠈⣿⡏⠀⠒
⠀⠰⠗⣴⣿⣿⣾⣷⡟⠸⣏⣿⡟⢇⠙⣿⡇⠀⢤⣤⡏⠀⣰⠚⣦⠐⢰⠀⠀⠿⠇⣹⣦⡙⢧⣤⢗⠀⠀⠀⠂⠀⠀⢰⣷⠀⠀⢀⡐⢠⣿⠃⠀⠀⠀⠀⠘
⠀⠀⠁⠙⣿⣿⣿⣿⡇⢓⢦⣭⣵⣋⠁⣿⠀⠀⠂⠛⢯⡀⠌⠛⠡⢀⡮⠴⠆⠀⠀⠿⣾⣷⣬⠀⣞⠻⢛⡿⢳⠀⠀⠾⢿⡁⠐⠋⠁⣾⣿⠀⢐⠀⢀⣠⡆
⠀⠀⠀⠀⣬⠍⢩⣝⣿⣬⣋⣀⣀⣩⣾⡟⠀⠀⠀⠀⠁⠉⠒⠓⠚⠋⠀⠀⣠⠀⠀⠀⠚⠉⣙⢠⣿⣿⣿⣿⣿⡇⠀⠀⠈⠻⢷⣶⣾⢿⣿⣤⣂⣀⣿⣿⠃
⠀⠀⠀⠀⠁⠀⠛⠋⠀⠙⠛⠿⠟⠛⠉⠈⠃⠀⠠⠀⠀⠀⠐⠀⠀⠀⠐⠆⠉⠠⠀⠀⠀⠀⠋⠈⠉⠀⠙⠉⠀⠉⠀⠈⠀⠀⠐⠱⠰⡶⠉⠛⠛⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠝⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢟⠻⡏⠸⣿⣿⡿⣿⣿⠟⠻⠿⡿⠟⢋⣿⣿⣿⠿⠛⠁
⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⠀⠀⠠⠐⢩⠥⠀⠈⣀⠀⠜⠛⠋⠈⠁⢀⣀⣤⣤⣄⣚⣉⡉⠁
⠀⠀⠐⠒⠯⣭⣉⣈⣈⡁⠀⣀⣀⣀⠀⠀⠈⠁⠘⠛⠛⠛⠃⠀⠀⠈⠃⠀⠀⠀⠀⠀⠒⠒⠠⠶⠞⠛⢿⣿⢦⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⣻⠆⠀⠀⠀⠀⠀⢈⣶⡀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢸⣌⠁⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣄⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣆⠀⠀⣄⣠⣀⡀⠐⠶⠂⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣶⣤⣤⣐⣚⣿⣿⣿⣿⡇⠀⣋⣠⠼⠛⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⡆⣠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣾⠂⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣿⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣰⣿⣿⡄⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠟⠿⠿⠛⠛⠛⠛⠋⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠏