Circuit Emojis & Symbols

⣀⣀⠀⠑⢄⠀⠀⠠⠤⠤⠤⠀⠠⠤⠤⡀⠑⢄⣀⡀⠀⠀⡠⢊⠤⠀⠤⠤⠀⠄⠤⠓⠤⠀⠤⠤⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀
⠀⠀⠙⠦⡀⡑⠒⠒⠒⠒⠂⠒⠐⠒⠒⢌⠢⡀⠀⠀⠀⡎⢰⠁⡎⠉⠉⣉⢉⠉⠑⢈⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠤⢀
⣈⣀⣀⣀⡀⣀⣀⣀⣀⣀⢀⣀⣀⣀⠈⠢⡑⠌⠂⠀⠀⡇⢸⠀⡇⠀⠘⠀⠀⠀⡀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⣀
⠀⠀⢀⡤⠂⠂⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⡇⠀⠀⠀⠊⠀⠘⢄⠈⠢⠄⠀⠈⠢⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠊⡠⠄⡀
⢀⡴⠋⠐⠈⠉⠉⠉⠉⠁⠀⠀⠀⠈⢢⠀⢆⠀⠈⠢⠤⠠⠤⠄⡑⢄⠀⠀⡇⠀⠈⠉⠁⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⠈⠢⡀⠀⣉⣉⣉⠔⠉
⠉⠀⠀⠀⠒⠒⠒⠒⠒⠒⢄⠀⠀⠄⠘⢄⠀⠑⠒⠒⠒⠄⠀⠀⠈⠂⠱⠀⢧⡀⠉⠂⠀⠐⠒⠒⠀⠐⢲⠀⡇⠀⠀⠀⠀⡈⢢⡀⠀⠀⠀⠀⠂
⠀⢀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠁⠀⠀⠀⢀⣁⣉⣉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣉⡉⣉⡉⠉⣉⠍⠕⠀⢸⠀⡇⡀⠀⠀⠀⠑⢄⠙⢄⠀⠀⠀⠇
⢄⠀⠑⠒⠒⠒⠒⠒⠒⠢⡀⠀⣠⠂⢰⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⠀⠀⠀⠒⠒⠒⢂⠔⠐⠀⠀⠘⢄⠣⡡⡀⢢⠀⠀⠀⠑⡄⠑⢄
⠀⠈⡉⠉⠀⠉⠀⠀⡀⠙⠌⠉⠀⠀⢸⠀⠀⡈⠁⠉⠁⠠⠊⠁⠀⠀⢰⠀⠀⠈⠁⠉⠉⠡⢊⠉⠁⠀⠀⠀⠅⠁⠨⠀⠇⠸⠀⠀⠇⠀⠀⠈⡇
⠀⡘⠁⡠⠀⠀⠀⠀⠈⠂⠂⠀⠀⠒⠁⡠⢊⠄⠀⠀⠀⠀⠐⠁⠀⠂⢸⠀⠀⠀⠀⠀⠀⢂⠅⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢡⠁⠀⡇
⠀⠅⠸⠀⠀⠈⠉⠉⠉⠉⢉⠉⢉⣉⣩⠔⠁⠠⢀⠉⢉⣉⠉⠢⡀⠀⢸⠀⠀⠠⠊⠉⠉⠁⠀⡄⠠⠀⠈⠉⠉⠁⠁⢀⠐⢄⠀⠣⠀⠰⠀⠀⠇
⠀⠁⢸⠠⠋⠀⠀⠀⡁⡔⢁⠀⠀⠀⠀⠒⠀⠀⠀⠂⢄⠀⠀⠀⠇⠀⢸⠀⠀⡆⠀⠀⠀⢠⠀⡇⠈⠀⠀⠀⠀⣀⣀⡀⠑⢄⠑⢄⠀⢐⠀⠀⡆
⠀⡇⢸⢀⠀⠀⡠⠒⡠⠃⠈⠀⠢⡀⠀⠀⠀⠀⠐⠦⡀⡁⠀⡀⡀⢄⠈⠢⡀⢅⠠⡀⠀⢸⠀⡇⢈⠀⡆⠀⠀⠀⠀⠉⢢⡀⠁⠀⠀⢀⠀⠀⡇
⠀⡂⢨⠀⠀⠀⠀⠀⠠⡄⠀⠀⠐⢌⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⡢⡄⠈⡌⢣⠈⡆⢸⠀⡇⠀⠀⡇⠀⡇⢰⠀⡂⠀⡇⠀⠀⠀⢘⠀⠀⡇
⠀⢁⠈⠃⠤⠤⠤⠀⠐⠡⠀⠀⢀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣀⡀⠀⢸⠈⠀⠀⢸⠀⠃⠜⠄⠃⠀⠀⡇⠀⡇⠀⠀⠀⠀⠇⠀⠀⠀⢸
⢄⠀⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠄⠈⠆⢸⠀⠀⡄⢸⠀⠀⠤⠄⠀⠀⠀⠑⠤⠬⠄⠁⠀⠀⠀⠀⠀⠀⢠
⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠐⢀⠈⡂⠀⠈⠀⡀⠁⢸⠀⠀⠂⠂⠀⠀⠀⠀⠐⠀⠢⡀⠀⠂⠀⠀⠀⠀⢸
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡀⠙⡄⢸⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⣁⡁⠀⠀⠀⠀⠥⠬⡑⠌⠉⡁
⠢⡀⠀⢀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢄⠈⠀⡇⢸⠀⡀⠀⢘⠀⠠⠁⢸⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⠒⠀⠀⠁⠐⠀⠀⢨⠀⠀⡂
⠀⡄⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠘⠀⠀⠇⠸⠀⠁⠀⢸⠀⠀⠀⡸⢀⠄⠀⠀⠊⢀⠤⠤⠤⠤⠤⠤⠤⠔⠀⢀⠀⠀⠰⠀⠀⡇
⠀⡅⠀⠀⠀⢸⠀⢸⠀⠆⠀⠀⠀⠀⢸⠀⠂⠀⢘⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠁⡀⠀⢀⠔⡡⠂⠁⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⢀⠀⠀⠇
⠀⡃⠀⠀⠀⠜⢀⠼⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⢰⠀⠀⡄⢠⠀⠄⠀⠈⠀⠀⡁⠀⠈⢀⡔⡡⠊⢀⢀⠀⠉⠉⠉⡁⡉⢉⠉⠉⠉⠀⡨
⠀⠇⢀⣀⣀⡰⠋⡀⠀⠀⠀⠄⠀⠀⢸⠀⡀⠀⢸⠀⠀⡇⢸⠀⢢⡀⠀⠀⡠⠃⢀⠔⡣⠊⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠊⠀⠀⠀⡆
⠀⡇⠀⠀⠀⠤⠜⠀⠀⡀⠀⠀⠀⠀⢸⠀⡃⢠⠨⡀⢄⠈⠢⡢⠤⠭⠭⠭⠤⠔⡡⠊⠀⠔⠁⠀⠀⠀⠀⠀⠄⠀⠀⣠⠔⢁⠔⠋⠁⠀⠀⠀⡃
⠀⡅⠠⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⠀⠂⠀⠑⢌⠢⡑⢄⣉⣒⣒⣒⣒⣒⣊⣀⣀⣂⡐⣐⣀⣒⣐⣀⣒⣂⣒⣊⣀⡀⠀⠀⠔⡡⠂⢀
⠀⠈⠢⡈⠢⡉⠒⠒⠀⠀⠀⠑⢄⠀⠈⠢⠤⠤⢀⠀⠑⢌⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⢊⠔⢉
⠀⠀⢀⠈⢢⡈⠂⠒⠂⠀⢀⠀⠀⠀⠀⢄⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠔⠁⠈
⡿⢿⠿⠿⠿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⣻⣻⣻⣿⣿⣟⣿⣿

⠀⡇⠀⢀⠀⢀⢀⠀⠀⣀⣀⣀⣀⣀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣠⠞⠁⢀⡔⠁⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠱⣦⠀⠀⠀⠀⡇
⠀⡇⠀⠈⡆⠀⠁⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⡂⠀⠀⠠⠲⠀⠀⠀⠔⠊⠀⠀⠀⠉⠁⢀⡴⠋⠀⠀⡠⠂⠀⠀⠤⡀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⡇
⠀⢧⠀⠀⡇⠀⠈⠉⠉⠉⠑⠁⠀⠀⠩⠀⠀⠀⡃⠀⠀⢀⡀⢀⠀⠀⣀⢠⠀⠀⠀⢀⡴⠁⠀⠀⡠⠊⠀⠐⠠⡀⠀⠉⢦⡀⠀⠀⠑⡄⠀⠀⠀⠀⡇
⠀⠀⠀⢀⠇⠀⢀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠔⠁⢀⠔⠁⠀⠀⠀⠀⠀⠁⠀⢀⡴⠋⠀⠀⡠⠊⠀⢄⠀⠀⠀⠈⠢⡀⠀⠉⠔⠀⠀⡇⠀⠀⠀⠀⡇
⠀⢀⡴⠋⢀⠔⠁⠀⠀⠀⢀⠔⠁⠀⢀⠔⠁⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡠⠊⠀⠀⠀⠉⠢⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠳⡀⠀⠀⠈⠈
⠴⠋⢀⠔⠁⢀⠀⠀⠀⡔⠁⠀⢀⠔⠁⢀⠔⠁⠀⠀⢀⠔⠉⠉⠉⠉⠉⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠈⠕⠀⠀⠀⠈⠥
⠀⠀⢸⠀⠀⢱⠁⠀⠀⢦⠀⠀⠂⠀⠀⢸⠀⠀⢀⠔⠁⢀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡴
⠀⠀⢼⠀⠀⢸⠀⠀⠀⠁⠀⠀⠂⠀⠀⢸⠀⠀⠂⠀⠀⠑⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠑⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠐⠀⠀⡇
⠀⠀⠉⠀⠀⢸⠀⠀⠀⡔⠀⠀⠀⠀⠀⢸⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇
⠀⠀⠀⢀⠔⠁⠀⠀⡠⠃⠀⢀⠅⠀⠀⡸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠈⠈
⠀⢀⠔⠁⠀⠀⡠⠊⠀⢀⠔⠁⠀⡠⠊⠀⠀⠀⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠂⠀⠀⠈⠚⢄⠀⠀⠀⠈⠈⠉⠉⠉⠉⠘⠀⣀⣀⣀⣀⣀⣀⡇
⠔⠁⠀⠀⡠⠊⠀⢀⠔⠁⠀⠰⠜⠀⠀⠀⠀⠀⡢⠤⠤⠤⠤⢄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁
⠀⠀⡠⠊⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠑⣆⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠔
⠐⠉⠀⠀⣤⠀⠀⠀⡠⠠⠤⠤⠤⠤⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢀
⠀⠀⡠⠊⠀⠀⡠⠊⠀⠀⠀⣀⣀⣀⠀⠈⠒⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡠⠊⠀⢀⠔⠉⠉⠉⠉⠁⠁
⠒⠊⠀⠀⢔⠊⠀⠀⠀⡠⠊⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢸⠀⡠⡔⠁⢀⡤⠒⠒⠐⠂⠒
⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⢀⢀⠀⠀⠀⠀⢸⠀⠈⠀⡴⠋
⠒⠒⢄⡀⠀⠀⠀⠁⠀⠀⠠⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⠀⠀⠀⠀⢐⡂⠀⠀⢘⠄⠀⠀⠀⡠⠊⠁⠀⠀⠀⡠⠊⠀⠀⠀⡇⠀⠀⠀⠀⠋⠒⠒
⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡇
⠀⠀⡀⠀⠀⠉⠈⠁⠉⠁⠉⠂⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠉⠢⡀⠀⠀⠐⡀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣔
⠀⠈⠑⢄⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡠⠀⠈⡆⠀⠀⡇⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⡠⠊⠀⠀⠀⡀
⠀⢀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠨⠀⠠⠢⠀⠀⠠⠂⠀⠀⠀⠉⠀⠀⠀⡠⠊⠀⠀⠀⠀⠐⠁
⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠒⠒⠒⠒⠒⠚⠀⠀⠀⡇⠀⠀⠣⡀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠈⠢⡀⠀⠙⠦⠤⠤⠤⢀⠀⠀⠀⢔⠊⠀⠀⠀⠀⠀⢸
⠀⢀⠀⠀⠀⡤⠀⠀⡠⠊⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⢄
⠀⠀⠑⢄⠀⠐⠀⠀⠀⠀⠀⠀⠈⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠱⠀⠀⠀⢸⠀⠀⠀⠀⠑
⠑⡄⠀⠀⠑⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠁⠀⠀⠁⠁⠀⠠⠀⠀⠀⠀⠑⢄
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢸⠀⠀⢀⡤⠂⠀⠀⠨⢆⠀⠀⠑⢄⠀⠀⠀⠀⠑⢄

⠀⠀⡀⠀⠀⠀⠀⠠⠭⠁⠀⢠⠄⠀⠀⠭⠹⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠂⣄⢀
⠀⠀⡤⠀⠀⠀⠔⠀⠂⡄⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠤⡤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢤
⠀⠀⣀⠀⢀⡀⠘⠀⢀⡀⠀⠰⠀⠀⠀⣀⠀⠀⢀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⣀⢀⠂⡀⣀⠀⠀⠀⠀⡀⣀⢀⣀⡀
⠀⠀⣭⠀⠈⠁⠀⠀⠨⢄⣀⣀⣀⡠⠄⡭⠠⠀⢬⠀⠀⠭⠀⠀⠀⠈⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⡀
⠀⠀⣤⣀⣀⣀⣀⠀⣀⡠⠄⠄⠤⢀⣀⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢀⠀⠀⢀
⢈⡀⣀⢀⡀⠀⠀⢤⠄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠤⠀⠀⠥⠀⠀⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠄⠀⠀⠨⠅⠠⢨⠵⠉
⠀⠁⠙⠂⠒⠂⠀⢸⠀⠀⢀⣀⣀⣀⣀⡖⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠒⠒⢇⢀⡯
⠀⠀⠀⠀⠀⠿⠃⢘⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠚⠀⠈⠀⠒⠅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡀⠀⣀⣠
⢑⣒⠀⠀⠀⠂⠀⠉⠁⠀⢩⠀⠀⣶⠄⠀⠀⠁⢀⠀⠈⠀⠦⡇⠄⠀⠀⠀⠀⠀⠴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠂⠠⡏⠀⢆
⠁⡀⠀⠀⠀⠀⠀⠠⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠙
⠀⠀⣀⠀⠀⠀⠀⠐⠀⠀⠗⠀⠀⠀⠐⠀⠀⠀⠺⠀⠀⠒⢻⣄⡠⠐⠀⠀⠀⠀⠀⠠⠀⢰⠰⠀⠀⡄⠀⠈⠶⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠇⠉⠠⠀⠀⠀⠀⠀⠀⠈⠀⠉⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⡈
⠀⡆⠀⠈⠃⠀⠐⢱⡀⠀⠋⣀⠀⠈⢀⡀⠀⠠⠄⠀⠀⠠⠀⠀⠄⠀⡄⠀⠠⠄⠀⠴⠤⡆⠀⠰⠠⠧⠀⠤⠆⢀⠀⠀⠀⠠⠟⠀⠳
⡀⣀⣀⣠⠆⠀⠠⠜⠀⠀⠤⠉⠀⢠⠄⠀⠀⣠⢠⡀⠀⢠⡄⠠⡅⠀⡇⢀⢄⢉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⠂⣀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⠀⠀⠀⢗⠀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠁⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠨⠀⠀⠀⠀⠀⣀⠀⠀⣠⠐⠂⠄⠀⢰⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀
⠀⢰⠈⣙⢀⠀⣈⡋⠉⠀⠈⠁⠀⢘⠁⠉⠀⠀⠈⠉⠀⠀⠙⠳⠄⠀⠀⢸⠀⠀⠀⠀⠴⠀⠀⠀⠠⠀⢄⠀⠀⠀⠀⡇
⠀⠀⠆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠂⣘⠀⠂⠐⠀⢚⠒⠀⠚⠀⠛⠀⠟⠐⠂⠐⡃⠂
⠀⠀⠆⠀⠀⠀⠀⠰⠀⠀⢠⠀⠀⠀⠀⠀⠅⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢘⠀⠀⠀⠀⠭⢀⠀⠀⠀⠀⠃⠀⠀⠆
⠀⠀⠀⢀⠀⠀⠀⠘⠀⠀⢰⠀⠀⠀⠀⠀⡁⠂⠁⠐⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠒⠂⠺⡀⠿⢀⠷⠀⠂⠀⠀⠀⠀⠂
⠀⠀⠀⢐⡖⠀⠀⣰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠁⠀⠀⢰
⠶⢰⠀⠰⠆⠀⠀⠈⠄⠀⠄⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣇⢀⡆⠀⠀⠀⢸⠂⠀⠰⠆⠢
⠀⢠⠄⠀⡀⠀⠀⠈⠁⠀⠀⢨⠅⠀⠀⠄⠀⠀⠀⠀⠀⠀⠺⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⡄⡄
⠀⠸⠀⠀⠀⠀⠀⢂⢦⣤⠀⠀⠀⠀⠀⠆⠀⠀⠰⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠈⡏⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠡
⣀⣀⣀⣂⣀⣀⣀⣀⣀⣘⣉⣈⣈⣀⣁⣒⣘⣀⣈⣀⣀⣀⣀⣀⣀⣒⣀⣀⣀⣀⣋⣓⣛⣃⣀⣓⣐⣇⣀⣀⣀⣈⣀⣀⣂⣐⣃⣀⣀
⣿⣿⡋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡔⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⢉⣀⣉⡉⣀⠀⣀⢀⣀⣀⣀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠔⠁⠄⠀⢀⠀⠐⢄⡀⢀⣀⠀⣀⣀⠈⠢⠔⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠐⠂⠀⠘⠀⠀⠀⠀⠆⢀⠄⠂⠀⠀⠀⠤⠄⠀⠀⡆⠁⠀⠈⠁
⠀⠀⣀⠀⢀⡐⢄⠒⠒⡠⠂⠀⣀⣀⠈⠁⠀⡎⠉⡉⠈⠍⠁⡀⠀⣸⣀⠀⣀⣀⠉⠈⠉⠁⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠄⠃⠀⠠⠀⠑⠠⠀⠀⠀⠀⠀⠤⠀⠠⠀⠀⠀⠐⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀
⠤⠤⠤⠤⠤⢄⠀⢸⠀⠀⠀⢩⠁⠀⠀⠑⠤⠤⠤⠤⠤⠄⢀⠄⠀⠂⠀⠀⠐⠤⠀⠠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢇⠀⠉⣁⣀⠈⢉⣀⠀⠁⠀⠀⠀⠀⠀⠀⠁⡄⠐⢄⡀⢀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠰⠤⠤⠤⢄⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢱⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁


⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡤⣄⠀⠀⠀⢠⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣠⠞⠁⠀⠀⠀⢀⡴⠒⠒⠒⠒⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠙⠳⠤⠤⠤⠼⠉⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⣠⠟⠁⠀⢠⣤⡞⠁⣴⣶⠀⠠⣖⡏⢠⢤⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢦⠀⠀⠀⣿
⠉⠑⢆⡀⠀⠀⠀⠀⠀⣠⣄⠀⢠⠞⠉⣡⡄⠀⣠⠞⠁⠀⢀⡀⠈⠛⢁⠔⠉⠁⠀⠀⠀⠀⠈⠛⠣⠤⠤⣄⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⣿⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠋⠀⡸⠂⡠⠓⣣⠞⠁⡠⠊⠉⠻⠞⢀⡴⠉⣠⠔⠒⠒⠒⠲⠯⠇⠀⠀⠀⠀⠀⠯⠗⣠⣄⠀⢀⠜⠀⠀⠺⣥⡿
⣀⣀⣀⣀⣀⡠⠒⠒⠒⠒⠒⠊⡠⠊⣠⠞⠁⡠⠊⠀⢰⣳⠀⠀⡏⠀⡞⠁⣠⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢦⡀⠙⠋⠉⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠉⣠⠞⠁⡠⠊⠀⢀⣀⠀⠀⠀⠀⡇⠀⡇⡜⠁⠀⠀⠀⠀⠀⢰⣖⡆⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠻⠞
⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣠⡞⠁⡠⠊⠀⠀⠀⠻⠞⠀⠀⣠⠞⠀⡰⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢰⣲⠦⠤⠤⠤⠤⠤⢤⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⠾⠉⣠⠞⠀⠀⢀⠔⠒⠒⠒⠒⠚⠁⡠⠊⢀⠔⠁⠀⣠⠦⠤⢾⡷⠀⠀⠀⠀⠙⠦⠤⠤⠤⠤⠤⣄⠀⠀⠀⠀⠀⠙⠶
⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁⠀⢀⠔⠁⢀⣀⠔⠒⠒⠒⠊⣴⣶⠁⠀⣠⠞⢁⡠⠠⠀⠀⠀⠀⢠⣶⡄⠀⢰⣳⠦⠤⡀⠈⠳⣄
⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠁⠀⠀⠯⠟⠀⠀⠀⠀⠀⠈⠁⣠⠞⢁⡴⠋⠀⠀⠀⠀⠀⠀⣠⠭⠤⠤⠤⠤⠤⣀⠈⠢⡀⠈⠳⣄
⠒⠒⠒⠒⠒⢒⠒⠒⠒⠒⠒⢦⣀⣀⣀⣀⣀⣀⣀⡴⠒⣒⠒⠒⠚⢁⡴⠋⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠈⣟⡆⠈⠢⡀⠈⠳⠤⠤
⠀⠀⠀⠀⠐⠯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⢀⡴⣋⠀⠀⢀⡴⠒⠒⠒⠊⠀⣠⠖⠒⠒⠒⠒⠒⠒⠒⠚⢶⣄⠀⠈⠦⠤⠤⠤
⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⡸⠿⠃⣯⠟⠀⠀⢀⣀⣠⠞⠁⠀⢀⣀⣀⣀⣀⣀⣀⣴⢦⠙⢷⣄
⠒⠒⠒⠒⠒⠂⠒⠒⠂⢒⣤⡒⢚⣏⠛⠛⠙⠓⠛⢛⣧⡟⠙⠋⠀⠀⠸⢿⣇⠀⠀⢿⣽⠇⠀⢀⠔⠁⠀⣠⠤⣞⡧⠀⠈⢉⣀⠀⠙⠷⠶⠶⠶
⠀⠀⠀⠀⠀⡠⠊⠉⠉⠉⠓⠃⠘⠚⠁⠀⠀⠀⠀⠈⣻⠁⠀⠀⠀⠀⠀⠀⠈⠓⠚⠒⠛⠒⠚⠁⠀⣠⢞⠝⠊⠛⣁⡐⠐⠛⢞⡵⠋⣩⡉⠉⠉
⠭⢭⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⢉⡉⠉⢀⣀⡴⠋⠉⣉⡩⠯⠭⠭⠿⠯⠽⣭⠭⠯⠖⠁⠀⠀⠀⠳⠛⠉⠉⠉⠀⡠⠓⠋
⠀⠀⠙⠲⠒⠒⠒⠒⠒⢾⡽⢀⣀⡔⠁⢰⣲⠄⠻⠾⣄⠹⠞⠀⠀⠸⠽⠁⠀⣠⠖⠒⠒⠚⠿⠃⢀⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠊⢀⣄
⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⠀⠯⠟⠀⠀⠀⢯⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠊⠁⣠⠶⠶⠶⠶⠖⣡⠴⠶⠶⠶⠶⠶⠶⠶⠦⣄⠀⠘⠚
⠒⠒⠒⠒⠒⠒⠒⠒⠒⣄⠱⠤⠤⠤⠤⠤⠤⠭⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠥⠤⠤⠤⠤⠜⣡⠞⠉⠉⠉⠉⢢⣤⡀⠀⠈⠳⠶⠶⠶⠶⠶
⠒⠒⠒⠒⠒⢢⡀⠀⠀⠈⣓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢁⡠⠀⠀⠀⠀⠬⠛⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡴⠋⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠳⡄
⣒⣛⣛⣚⣛⣛⣛⣛⣛⣛⣓⣒⣛⣛⣛⣛⡛⠛⠢⡀⠀⠀⢾⡳⠀⠀⠀⡠⠊⣀⠤⠤⠤⠤⢄⡀⠀⠀⠈⠳⠋⠉⠉⠉⠉⠳⣄⠀⡇⢠⣄
⠴⠶⠶⠶⠶⠶⠤⠴⠦⣄⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠓⠒⠓⠓⠒⠒⠋⢀⠞⠁⠀⢾⡷⠀⠀⠙⢄⣀⣀⣀⣀⣀⠀⢰⣲⡄⢸⠀⣇⠙⠚
⣀⣀⣀⣀⣀⣀⣠⣄⡀⠈⠳⣄⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠈⠳⣄⣉⣠⠞⠀⠈⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠈⠳⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠤⠥⠤⠭⠭⠥⠥⠬⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⠀⠀⠲⣄⣈⠻⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠭⠭⠿⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠽⠯⠭⠭⠭⠭⠭⠭⠭⠭⠯⠉⠙⠆⠈⠳⠀⠀⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢴⠀⢀⡀⠀⠀⡇⠀⠀⣀⠀⢰⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠸⣄⠨⣇⠀⠀⡇⠀⠀⡽⠀⣸⠇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠈⢣⠈⢳⠀⡇⢰⠋⠀⡞⠁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸⠀⡇⢸⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸⠀⡇⢸⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠰⠶⠤⠦⠦⣄⠀⠀⠀⠀⠀⠀⣴⣴⣷⣤⣾⣴⣼⣦⣷⣾⣦⣴⣧⣤⣷⣴⡄⠀⠀⠀⠀⠀⠀⡠⠦⠦⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢤⡈⠒⠒⠒⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠒⠒⠒⠒⠒⠚⢀⡤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠙⠦⠤⠤⠤⠤⠤⣿⣿⡏⣶⢶⡶⣾⡦⣶⢰⡆⣶⢰⡆⣿⣿⡧⠤⠤⠤⠤⠤⠔⢋⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⣀⣀⣿⣿⡇⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⣿⣿⣇⣀⣀⣀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠉⠉⠉⠉⠉⣿⣿⣧⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣿⣿⡏⠉⠉⠉⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⡠⠒⠒⠒⠒⠒⠒⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⡗⠒⠒⠒⠒⠒⢢⡈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠋⢀⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠤⠤⠤⠤⠤⢄⠙⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠰⠖⠖⠒⠒⠋⠀⠀⠀⠀⠀⠀⠻⠻⡿⠛⢿⠛⢻⠿⡿⢿⠟⠻⡟⠛⡿⠻⠃⠀⠀⠀⠀⠀⠀⠑⠖⠒⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸⠀⡇⢸⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸⠀⡇⢸⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣀⠜⢀⡼⠀⡇⠸⢄⠀⢇⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢸⠁⢐⡏⠀⠀⡇⠀⠀⣳⠀⢳⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠺⠀⠈⠁⠀⠀⡇⠀⠀⠉⠀⠸⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠀⠠⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡆⠀⢨⠀⠀⡇⠀⠰⡰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢎⠆⢈⡁⠰⡱⠀⠃⠀⠉⠂⠀⠣⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⡀⠀⠀⠀⠀⠃⠀⠃⠀⡀⠀⢸
⠠⠀⠀⠀⠀⠀⠀⢀⠀⢘⠀⠀⠀⠀⡇⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⡀⠀⠃⠀⠀⡀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠰⠁⠀⠀⠀⠀⠀⠠⠊⠀⡠⠊
⠁⢀⠄⠀⠀⡂⠀⠲⠁⢸⠀⠀⠀⠀⡇⠀⠄⠀⠀⠀⢀⠀⠀⠀⠘⠜⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⢨⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⡀
⠔⠁⠀⠠⠊⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠐⠭⠒⠒⠂⠀⡇⠀⠀⠀⠈⠀⠀⠀⣀⠀⠀⠀⠈⠀⠀⠈⠉⠉⣉⠉⠉⠉⠁⠀⢠⠊
⠀⠄⠊⠀⠐⡬⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠤⡀⢀⢄⠀⠇⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢀⠐
⠁⠀⠠⡱⠀⠂⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠒⠀⠈⠊⠀⣀⠀⢀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠐⠀⡀⠀⠊⣀
⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠓⠃⠐⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠄⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠌
⠀⠀⠀⡀⠀⠑⠒⠒⠒⠒⠒⠌⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠒⠒⠀⠀⠀⠀⠨⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⠔
⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠰⠩⠀⠠⠱⠰⠩⠀⠍⠆⠎⠆⠈⠁⠰⠱⠰⠉⠄⠀⠀⠀⠀⠀⠌⠆⠀⠀⠀⠀⠀⠀⠀⠨⠱
⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣠⠠⠀⢠⢠⣀⡀⣀⣀⣀⣀⢀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠔⡄⠀⡔⣄⣀⣀⣀⣀⣀⣀⣀
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠠⢅⠀⠀⠀⠀⠀⠁⠀⠉⠀⢨⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠠⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⢀⡄⠀⢸⠀⠈⠁⠀⠀⢀⡱⠀⠀⠀⠒⠀⡐⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢑
⠀⠂⠋⠈⠉⠉⠉⠉⠉⠉⠉⠑⠄⠀⢀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢸⠀⠐⠊⠀⠈⠁⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠘⠘⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠃⠀⠀⠀⠀⠀⠒⠀⠀⠀⠂
⠲⠝⠀⠀⠀⢐⣢⠤⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠮⠆⠀⠀⠀⠇⠀⢸⠀⠨⠕⠀⠂⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠫⠃⠀⠀⠀⠀⠀⠀⠀⠐
⠒⠒⠲⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠐⠒⠒⠒⠒⠒⠒⠒⠒⢆⠆⠀⠀⠀⡆⠀⢸⠀⢰⡰⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠢⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⢠
⠀⠤⠤⠤⠤⢂⠄⠀⠀⠀⠣⠤⠤⠤⠤⠤⠤⠤⠤⢔⡠⠀⠀⠀⡰⠀⠀⡃⠀⢸⠀⠀⡀⠀⠀⠀⠘⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠊⠅⠈⠑⠐⠩⠀⠀⠀⢸
⠀⠀⢀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣔⡄⠀⠀⠀⡠⠊⠀⡠⠊⠀⢀⠌⠀⠘⠜⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠤⣒⠀⠀⠀⢸
⠀⠀⠐⡊⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⢠⠊⠀⡠⠊⠀⢀⠔⠁⠄⡀⠀⡄⢠⠀⠀⠀⠀⠀⣤⢠⠀⢠⢢⠀⠒⠀⠜⡄⠀⠀⠀⡀⢤⠠⠀⠠⠄⠬
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣈⣀⣀⡀⠀⢸⠀⠀⠀⠀⠔⠁⡀⠀⠹⠀⠀⡀⢄⣀⣀⣀⣀⣀⣠⢡⠀⢠⢥⣀⣀⣀⣈⣀⣀⠀⠀⠀⠀⡁⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠘⡔⠀⢰⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⢍⠀⢀⢅⡀⠀⠀⠀⠀⠀⠀⢀⠄⠘⡔⠀⠠⠂⢣
⠀⠀⠀⠂⠀⠄⠀⢀⢈⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⠀⡇⠀⢀⢀⠀⠀⠀⠈⠈⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⣤⣤⣤⣤⣤⣤⣬⣦⣥⣤⣤⣤⣬⣧⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣧⣤⣬⣮⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⢠⠀⠀⡄⢠⠀⠀⡄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⢁⡐⠂⣀⢀⠀⠂⡐⠀⠀⠐⠂⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⢈⠀⠀⠀⠀⡄⠈⠀⠀⠁⠘⠀⠲⠀⠉⠀⠀⠈⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣃⠈⠀⠈⠀⠃⠀⠀⠀⠀⡄⢀⠀⠂⡀⠀⠀⠘⠂⠠⠄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡡⠊⠀⠥⢈⡰⠂⠀⠠⠀⠀⠀⠀⠠⠈⠠⠆⠀⠀⠀⠀⠈⠁⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠘⠀⠆⠀⢠⠀⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠠⠄⠀⠀⠈⠢⠈⠀⠀⠄
⠀⠀⠀⠀⠀⢉⡌⠀⠃⠀⠀⡒⠀⡀⢀⠀⠀⡄⢩⠀⠀⠈⠁⠀⠠⠄⠀⠀⠒⠀⠀⢀⠄⠉⠀⠀⠤
⠀⠀⠀⢀⠀⠀⠀⠐⠀⢀⠔⠁⠀⠀⠀⠀⢀⠀⡀⠀⡀⠂⣤⠀⠰⠂⠀⠀⠀⠀⠐⠁⠐⠂⠀⠀⠂⠃⡀
⠀⠀⠀⢨⠀⠐⠂⠰⢈⡁⠀⢒⠔⢀⢀⡀⠀⠀⠰⠊⠀⠀⠠⢈⡀⠂⠀⠀⠀⠀⡄⠀⠰⢂⠘⠀⠀⠈⠱
⠀⠀⠀⠀⠤⠀⡀⠀⠀⠀⠆⠁⠀⠀⠀⠀⢠⠀⠂⠀⡀⠀⠀⠀⡀⠋⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⢠⠀⠀⠆⠈⠁⢀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣁⠄⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠈⠁⡄⠉⠀⠀⠀⠀⠘⢠⠀⠀⠀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠃⡄⠀⢀⠀⠀⠀⠀⢀⠠⠄⠀⠀⡄⠀⠀⠀⠀⠐⠄⠀⠀⠤⠪⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⣀⠀⠀⠂⢀⠀⠂⠀⢄⠀⠆⠀⠀⠠⡄⠀⠀⠀⠒⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡀⠀⠀⢁⡀⠁⢀⡉⠀⡀⠀⠀⠑⠄⠀⡌⠁⠀⠀⠊⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠰⠀⠀⠆⠀⠀⠴⠀⠂⠀⠀⠀⠀⠤⠀⠀⠀⠈⠡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠋⠀⠚⠀⠘⠀⠃

⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⠀⠀⠂⠀⠀⢀⠄⠀⢠⢄⠀⢀⢀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠀⠀⠐⠁⡠⡄⠀⡠⡔⠀⠀⠀⠈⠀⠀⠁⢀⠄⠀⠈⠂⡀⠈⠀⠈⠈⠀⠈⠀⠀⠀⠀⠁⠁⠀⠀⠀⠐⠀⡠⠀⠀⠀⡠⡔⠁⠀⠀⠁
⠀⠀⠀⡀⡄⠀⠀⠈⠀⠀⠉⠀⠀⠀⢀⠐⠁⠀⠀⠀⠈⠀⠀⠀⠈⡆⠀⠀⠀⠀⠰⡅⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉
⠀⠀⠀⠈⠁⠀⢀⠄⠒⠒⠒⠒⠐⠂⠁⠀⠀⢉⠀⡀⠀⢀⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠎⠂⠀⠀⠀⠀⡀
⠀⡀⠀⠀⠀⠊⠃⠀⢀⠄⠂⠀⠀⠀⠂⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠔⠁⢀⠔⠀⠀⠀⠀⠀⠈⠉
⠄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠝⠀⠀⠡⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡰⠀⠀⡐⡀⠀⠀⢀
⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠂⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠔⠉⠀⠀⠀⠀⠈⠀⠀⢈⠄⠁⠀⠀⠉
⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠫⠀⠀⠝⠀⠀⠀⠀⠀⠤⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⡄⠁⠀⠀⠀⠀⠀⠀⠫
⢀⠀⠀⠨⠁⠀⠀⠂⠃⠀⠃⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠀⠐⠘⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠃⠃
⠈⠀⠀⠀⠀⠀⠀⠄⡄⠀⠀⡄⠄⠀⠠⠊⠀⡀⠀⡀⠀⠀⠀⠰⢠⠀⢠⠀⠀⠀⢠⡂⠀⠀⠀⠀⠀⠀⠃⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠐
⠢⡀⠀⢔⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡠⠂⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡄⠀⠀⠀⠣⠁⠀⠀⠑⣆
⠀⠈⠢⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡠⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⢀⠘⠊⠀⠁⠀⢀⠠⠀⠀⠀⣀⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠑⠀⠀⡐⠀⠀⡀⠁⠀⠈⠒⠁⡀⠀⠀⡀⠀⡄⠀⠀⠀⠀⢀⡀⠐⠀⠀⠈⠀⡀⠀⠀⠁⠀⠀⠀⠆⠄⠀⠊
⠀⡀⠀⠀⠄⠠⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠠⢢⠈⠊⠁⠈⠂⠀⠂⠀⠀⠀⠀⠐⠂⠀⠀⢔⠂⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀
⠈⠒⠀⢔⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠒⠁
⠀⠀⡀⠊⠀⠀⠑⠀⠀⠀⡖⠄⠀⠀⠀⠐⠂⠀⠒⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠜⠆⠀⠖⠄⠀⠀⠀⠀⠀⠀⢒
⢀⠁⠀⠀⢰⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠰⠲⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠰⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠉⠉⠉⠉⠓⠁⠀⠀⠈⠢⣀
⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠈⠠⡀⠈⠁⠀⡀⠀⠠⠈⠀⢨⣀⠉⠉⠁⠀⠀⠀⠈⠓⠒⠒⠒⠲⢿⠆⠀⠀⣠⢄⣈
⠀⠈⠁⠀⠀⠉⠀⠀⠐⠄⠈⠁⠀⠁⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠊⠁⠀⠈⠅⠀⡀⢀⠀⠀⠀⡠⡀⠀⡠⡀⠀⠀⠈⠁
⡀⠀⠀⠀⡀⡀⠀⡀⡀⠀⠋⠀⠀⠀⠐⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢅⠀⠀⠝⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠈⠁⠀⢀⢀
⠈⠁⠀⠀⠈⠁⠀⠈⠁⠄⠀⠀⠀⠀⠀⢥⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠬⠂⠀⠀⢨⠂⠀⠀⠈⠁
⠀⠀⠀⠀⢀⠀⢪⠀⠀⠀⠀⠀⣄⠀⠀⢀⠀⠐⠭⠀⠀⠕⠀⠐⠕⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⣀⠄⠀⠀⠀⠩⠂⠀⠰⠅
⠀⠀⠀⠔⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠂
⠀⠢⠁⢄⠄⠀⡜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠀⠀⠀⠘⠍⠀⠀⠀⠀⠀⠀⠀⠀⠪⠁⠀⠀⠀⡤⡔
⠀⠀⡀⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠴⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢆⠄⠀⣀⠑⠤
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⣤⡀⠀⢀⣠⣀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢯⣀⣹⣀⣾⣀⣸⠂⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠉⢹⠉⣿⠉⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢸⠀⣿⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⣿⠀⠀⠀⡇
⠀⢀⡤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⡷⠶⠶⠾⠶⠿⠶⠶⠶⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢤⣀
⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⣶⡶⠶⠶⣶⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⣤⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⣀⣠⠾⠁
⣶⣶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⢶⣄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠑⢄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠈⢢
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⢣
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣟⡄
⢀⣤⣤⣄⣀⣾⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⡇
⢸⣿⠿⠛⠁⠙⠛⠛⠉⠀⠠⣂⡠⠙⢿⣿⣿⣿⣿⡿⠿⠛⠛⠟⠹⠿⠃
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠘⠿⠟⠁⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠
⠰⣤⣀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢀⣀⡀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⣶⣖⣶⣾⡿⠛⡀
⢀⣤⣈⠓⢦⣀⠀⠀⣀⣀⡘⠉⢦⠙⠇⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢄⠐⣴⠢⡀⣿⣿⣿⣿⣥⣄⣇⣻⢿⣧⣶⣿⡿⠓⡀
⠠⣙⢿⣁⡀⠙⠑⢆⣼⣯⡇⠀⠘⣿⣄⠀⢰⠄⠠⠄⢀⡤⠀⠆⠂⠀⠀⠀⠁⠀⠀⠀⣹⣯⣽⣿⠀⠉⣿⣿⣾⣿⣟⢿⠀⠐⡟
⠀⣘⣷⣯⣽⣿⣾⣍⠉⠛⠿⣦⣄⠘⣿⣆⣿⡖⠁⢀⠞⣤⣴⣦⢀⣠⠀⠀⠀⠀⠀⠀⠘⡏⠛⠙⢦⣀⣿⡿⢛⣿⣿⡢⠶⠖
⠀⠀⠀⠙⠛⠛⠿⢿⣄⠉⠻⣶⠙⠷⠸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠤⠂⠀⠀⢀⠜⠋⢹⠓⠄⡀
⠀⠀⠀⠀⠀⠀⠀⡈⠙⠻⣦⣉⣠⣷⣤⣛⢿⡉⠛⠋⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢂⠂
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣆⣻⣇⣴⣶⣦⡀
⠀⠀⣾⠛⢿⣟⠛⣿⠛⣿⠛⣿⠛⡿⢛⡿⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠛⠛⠛⠛⠛⣷
⠀⠀⡟⢷⡄⠻⣆⠘⣇⢸⡆⡏⢸⠃⡞⠁⠀⠀⠀⠀⠀⠀⣴⠶⢸⠱⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣷⣄⠙⢦⠘⢆⠘⠄⠃⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢺⠓⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣧⣌⡑⠦⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣻⣛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡷⠤⢍⡑⠀⠀⠀⢀⠴⣚⣉⣉⡓⠦⣄⠀⠀⠀⠀⠀⠀⢻⣽⣭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡗⠒⠒⠂⠀⠀⢠⠃⣾⣿⣿⣿⣿⣷⣾⣷⡀⣀⡴⠀⠀⠸⣻⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡟⠚⠉⢡⣶⣀⠞⣸⣿⡏⢹⣿⠿⣿⣿⣧⡝⣿⣷⠶⠂⠀⢸⠉⠀⠰⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠈⢻⣷⣾⣿⣿⠃⠀⠀⠠⣤⣉⡝⣡⣿⣟⠒⠂⠀⠀⠀⠀⠀⠘⣿⣿⢷⣄⣀⡀⠀⠀⠀⣿
⠀⠀⡇⠀⢀⡾⣏⠀⠉⠉⢡⡀⠀⠀⣀⣰⣿⣿⣿⣿⡛⠛⠂⠀⠀⠀⠀⠀⠀⠘⡷⠋⠁⢙⣿⠀⠀⠀⣿
⠀⠀⣷⠞⠉⠀⠈⠓⠦⢴⡟⠙⠒⢿⣿⡿⣿⡟⢿⡀⠉⠂⠀⠀⠀⠀⠀⠀⡤⡪⠕⠒⢺⡿⡇⠀⠀⠀⣿
⠀⡞⠁⠀⠀⠀⠐⢦⡀⢸⠁⠀⠀⠈⣿⠃⠘⠇⠀⠁⠀⠀⠀⠀⠀⠀⢀⢎⡾⢠⣴⣒⣙⣃⣁⣩⠗⠀⣿⢀
⣸⢁⠀⠀⠀⠀⠀⠀⢷⡈⣄⠀⠸⡀⢸⡀⠀⠀⠀⣠⣴⣶⣿⡶⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣆⡀⢀⡤⠛⣉⢽
⡿⢸⢸⡀⠀⠀⠀⠀⠘⣧⠘⣆⠀⢷⠀⢧⠀⠀⡰⠱⠈⢻⣿⠗⠀⠀⣸⣸⣿⣿⣿⣿⣿⣿⡿⠟⢉⡴⠊⢑⣿
⣇⠘⡏⣷⣄⠀⠀⠀⠀⢹⡆⢹⠀⢘⣷⣌⣽⣞⠁⢀⠴⠃⣿⣠⠤⠼⠿⠿⠟⠛⠛⣿⠟⢉⡤⠚⠁⠀⠀⢨⡻
⢻⡀⢻⡼⣿⣿⡆⠀⠀⠈⠻⣯⡤⠞⠋⠁⠈⣯⢗⡁⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⢐⡏⢻⠉⠀⢠⣶⡋⠙⣎
⠘⣇⠈⣷⣿⣿⣿⣄⡀⣠⢀⡄⠀⠀⣀⣤⣾⣿⠎⣹⠗⢊⣉⠓⢤⣀⠀⠀⠀⠀⠀⡇⢸⣶⢀⣠⠙⣷⠀⢸
⠀⢹⡀⠘⣷⣻⣿⣿⣿⣿⣿⣶⣾⣿⡿⣿⣿⡿⠒⢬⡳⣤⡤⠶⣻⠟⠁⠀⠀⠀⢀⡇⢸⣿⣷⣝⠀⣿⠀⢸⡤
⠀⠘⡇⠀⣺⣷⣝⠿⠿⢿⣛⣭⣷⠞⢋⡁⠀⢀⡀⠀⠈⠉⠉⠉⠀⠀⢀⡤⣤⣤⣴⡇⢸⣿⣿⣷⡄⣿⠀⢸
⠀⠀⡇⠀⢐⠽⣿⣿⣿⣿⠿⡏⣹⠛⡏⣹⠘⡏⣿⠀⠀⠀⠀⢀⡠⢞⡅⣂⣽⣿⣿⡇⢸⢿⣿⠝⠋⣿⠀⢸
⠀⠀⡇⠀⠐⡫⢞⡽⠋⠁⠀⡇⣿⠀⡇⣿⠀⡇⣿⠀⠀⣠⢔⡫⣊⢕⡩⣺⣿⣿⢟⡷⠺⠊⠁⠀⠀⣿⠀⢸

⣿⣷⣠⣿⡇⣾⣟⣛⢰⣿⠛⢷⡄⣿⢠⡾⠛⠳⠀⣰⡿⣆⠀⣿⠀⠀⠀⠘⢻⡟⠃⣿⣟⣛⢰⣟⣛⠂⠛⣿⠛⢸⡇⢸⣷⡄⣿⢠⡾⠛⣳
⣿⠹⡿⢹⡇⣿⣯⣭⢸⣿⣤⡾⠃⣿⠘⢷⣤⣴⢠⣿⠶⢿⡄⣿⣤⡄⠀⠀⢸⡇⠀⣿⣯⣭⢠⣭⣽⡗⠀⣿⠀⢸⡇⢸⡇⠻⣿⠘⢷⣬⣿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⣠⣤⣤⣄⠀⢀⣤⣤⣤⡀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣼⡷⠸⣯⣤⣤⣿⠀⣿⣥⣤⣼⡇⢸⣯⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠀⣿⣧⣤⣿⠇⢸⣷⣤⣼⡏⠀⣿⣤⣤⣿⠀⢸⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⠀⣿⣧⣤⣿⡃⢸⣯⣤⣼⡇⠀⣿⣥⣤⣿⠀⢸⣧⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⢸⣟⠀⣻⡇⠀⣿⠀⢸⡏⠀⢸⡇⠀⣿⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢸⣿⠛⢿⡇⠀⣿⠀⢸⡇⠀⢸⣿⠶⣿⠀⠀⣿⣀⣸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣿⠶⠨⣿⢸⣿⠀⢻⠃⠀⣿⣦⣼⡇⠀⢸⡇⠀⣿⠀⠀⣿⠉⢹⡇⢸⡇⠰⢾⣧⣠⣄
⠀⠀⠀⠀⠀⠀⣸⡷⠟⠉⠁⠀⠀⢀⣿⢸⣿⠀⣿⡇⠀⣿⠅⢸⡇⠀⢸⡇⠀⣿⠀⠀⣿⠀⢸⡇⢸⡇⠀⠀⠀⠉⠛⠷⣧⡀
⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⢈⣿⠸⣯⠀⣻⠇⠀⢿⡄⢸⡇⠀⢸⣇⠀⣿⠀⠀⣿⡀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠙⠛⠋⠀⠀⠈⠛⠋⠁⠀⠀⠙⠛⠋⠀⠀⠈⠛⠛⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⠂
⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⠛⠛⠃⠀⠃⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⢼⣷⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⣸⣯
⠀⠀⢀⣤⣤⣬⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣉⣀⣀⣀
⠀⠀⣿⡃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⣹⣷⡄⠀⠀⠀⠀⠀⠀⠀⣿⠉⣉⣉⣻⣇⣀⣀⣀⣀⣀⣀⣸⣿⣉⣉⠉⣿⡇
⠀⠀⢹⡿⠛⢻⡿⠛⢻⡟⠀⠀⠀⠀⠀⠀⠸⣇⠈⠻⡾⠋⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠉⠉⠉⠉⠉⢉⣉⣉⡉⠉⡉⠉⣿⠄⣿⡇
⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠶⠶⠶⠀⢈⣉⣉⣁⣀⡁⠀⣿⠀⣿⡇
⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⢈⡉⢉⣉⣉⡁⠀⣿⠀⣿⡇
⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⢀⣾⠋⠀⠀⣨⣤⣤⣤⣤⣬⣥⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠶⠶⠶⠀⢈⣁⣈⣉⡉⠁⠀⣿⠀⣿⡇
⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠛⠁⠀⠀⠀⣨⠉⣩⣭⣭⣭⣭⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⢈⣉⣉⣉⣁⡀⠀⣿⠀⣿⡇
⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⣤⣬⣥⣍⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠶⠶⠶⠀⢈⣉⣉⣉⠉⠁⠀⣿⠀⣿⡇
⠀⠀⢸⡇⠀⢸⡇⠀⢀⡁⠀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣁⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⢈⣉⣉⣉⣀⡀⠀⣿⠀⣿⡇
⠀⠀⢸⡇⠀⣼⢿⡄⢀⡅⠀⣿⣉⣿⣉⣉⣉⣿⣉⣉⣉⣿⣉⣉⣉⣿⠀⠀⠀⠀⠀⣿⠀⣿⠀⠶⠶⠶⠀⢈⣉⣉⣉⠉⠁⠀⣿⠀⣿⡇
⠀⠀⢸⡇⠸⣯⣸⡇⢸⡇⠀⣿⠉⣿⠉⠉⠉⣿⠉⠉⠉⣿⠉⠉⠉⣿⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⣿⡀⠛⠁
⠀⠀⠘⣷⡀⠈⠉⠀⣸⡇⠀⣿⡀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠙
⣤⣤⣤⣬⣿⣿⣿⣿⣯⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠛⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⠄⠘⠿⠿⢿⣿⣄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠸⣿⣿⣦⣤⣤⣤⣄⡻⣿⣷⡄⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠱⡄⠀⠀⠁⠀⢀⣀⣤⡴⣶⣦⡀⣿⣿⣟⣁⡀⠀⠀⠈⠉⠉⠉⠙⣿⣿⣧⠘⠻⠋⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢢⡘⢆⠀⠀⢾⣿⣿⣿⣿⣿⡿⠁⠘⣿⣿⣿⠃⣀⣀⠀⠀⣠⣤⣴⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠱⡙⣌⠳⡀⠈⠛⠟⠻⣿⣿⣧⠀⠀⠘⢿⣿⣿⣽⡿⠂⠈⠻⠿⠿⠛⠁⠀⠀⠀⠠⠖⣀⠀⢠⡾⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⣝⢎⢣⡙⢆⠀⠀⠀⠘⣿⣿⣧⠀⠀⠈⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠵⠚⢁⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⢎⠣⡙⣌⠳⡀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠋⣀⡠⢐⣫⢤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢧⡙⢌⢳⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⠉⡠⠴⣒⡫⠵⣚⠩⠒⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡈⢣⡙⣌⠳⣄⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⣋⡡⣤⣶⠶⠷⢚⣩⠵⠒⠋⣠⢴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠹⡄⠱⣌⠃⠘⢦⣤⡤⠖⠚⣉⡥⢔⣒⡭⠖⠚⣉⠤⠒⠋⠁⣀⡠⠴⠚⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠘⢆⠈⠢⣀⡞⠀⠀⣈⡩⠔⠊⢉⡀⠤⠒⠁⠀⣀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⢹⠃⠀⠘⣁⠤⠐⠊⠁⠀⣀⡤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⡾⠀⠀⠩⠁⣀⣠⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢷⣤⠴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⣰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡷⠀⣿⣿⣳⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⣿⣿⣆⠀⠘⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣥⣄⣿⣿⠛⠃⠈⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣿⣿⣿⣿⠅⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣶⣟⣿⣝⣿⣿⠀⠀⠀⣀⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣯⣿⣻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣽⣿⣿⣿⠟⠛⣻⣾⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠟⠛⠁⣠⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠿⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠾⠿⠟⠀⠙⠿⠂



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢀⣈⣧⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣻⢿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⣿⣾⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡯⣫⣟⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣹⣿⢿⣿⣿⣷⣝⣯⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣯⣥⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠟⠛⠛⠉⠀⠙⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⡀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡠⠖⡛⠉⠅⠀⠐⢀⠱⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⠢⠀⠀⠀⢑⠀⢪⡂
⠀⠀⠀⠀⢰⠈⠀⠔⠂⠀⠰⠃⠁⠀⠀⠀⠀⠀⢈⡀⣀⡀⠀⠀⠀⢀⠠⠈⠀⠀⠀⠀⠂⠁⠑⠣⣑⣠
⠀⠀⠀⠀⠘⡆⠆⣄⣀⠀⠠⠖⣚⣤⠢⡀⠀⠀⠈⠁⠐⠄⡀⠀⠀⠀⣀⡀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠳⡀⢀⢲⡐⣟⢯⡳⡽⢷⣮⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣒⠒⠀⠀⠀⠄⠀⣀
⠀⠀⠀⠀⠀⠈⠪⡪⠫⡹⣮⠣⢁⠢⠀⢄⠀⠀⢑⡄⠀⡠⡐⣬⡵⠖⡘⠀⠀⠈⠀⡀⡉⠀⠀⠐⠂⠠⠃
⠀⠀⠀⠀⠱⠵⠀⠐⢔⢄⠀⠀⠀⢐⣀⣥⡶⠌⠂⠀⠀⠉⠚⠡⠱⠈⠁⠀⠀⣆⠀⠉⠀⠉⠁⠐⠃⢠
⠀⠀⠀⠀⠡⡘⠀⠀⠀⠑⠍⠩⠀⠂⠉⠀⡀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠠⡀⠀⠀⠀⠁⠀⠒⠀⠂
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠂⢄⠀⠀⠀⠀⡈⢀⠤⡘⠀⠄⠀⠀⠈⡐⢨⠭⣙⢅⠁⠀⠀⢠⠐⠀⠀⣂
⠀⠀⠀⠀⠀⢰⠋⢅⡀⢀⡜⠃⠀⠀⠠⠀⠨⠀⠁⠀⠊⠀⠀⠀⢀⠀⠂⠒⠈⠐⠀⠀⠸⡬⡀⡂⠘
⠀⠀⠀⠀⠀⠀⣤⡁⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢤⢠⠄⡀⠀⠗⠂⠔⠚⠉⠀⠀⠘⣏⠈⠔⠆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣀⡀⠀⠀⠀⠚⠙⠀⠚⠀⠀⠀⠀⠔⠂⠐⢄⠀⢹⡆⢨⡖⣘
⠀⠀⠀⢠⡴⠖⠒⠃⠀⠀⠀⠈⠁⠀⢀⡀⠈⡄⠀⠀⣀⡤⠤⢄⠀⠀⠀⠸⠀⠭⠏⢀⠀⠈⠛⠋⠉⠁
⠀⠀⠀⠘⠣⠀⠀⠀⠒⠀⢈⢁⢀⣀⣀⡀⠀⠀⠀⠀⠕⠤⠀⡊⠄⠀⠀⠀⠰⠒⠂⠁⠀⠀⠠⢄
⠀⠀⠀⠀⠐⠀⠂⠉⠘⡀⣐⠈⠥⠀⠀⠀⡀⢀⠀⠀⠰⡸⠌⣀⠄⠀⠀⠀⢀⢄⠀⠀⡀⢠⠃⡐⠄
⠀⠀⠀⠀⠆⡀⢀⠀⠀⠀⡅⠀⣀⠀⠀⠤⠓⠚⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠻⡔⣣⠀⠀⠘⠀⠡⠂
⠀⠀⠀⠀⠰⠡⠘⠒⠂⠉⠈⠀⠖⣀⡀⠠⠀⠊⠀⠌⠠⢄⠀⠀⠀⢰⢂⠀⡀⠹⠤⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⣎⡉⠀⠀⢦⠀⠀⠀⠀⠂⠀⠉⡈⠀⠀⠀⠉⠙⠷⠀⠀⠀⠀⠀⠄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠎⠡⠀⠒⠀⠁

⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⢠⡏⠉⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠈⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣦⡄⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⣶⢠
⠀⠀⣷⣶⣾⣴⠚⠓⠛⠛⠛⣶⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⣸⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⠉⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⢸
⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⣸⣿⠶⣦⣴⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⢸
⠀⠀⣿⣿⣿⠻⠦⠤⠤⠤⠤⠟⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣿⠼⠀⠀⠀⠀⠀⠀⢠⣿⣯⣟⡻⡿⠀⠀⠀⠀⠀⠀⠀⣿⣦⣶⣶⣴⣿⢸
⠀⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠯⠿⠟⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣾⠃⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣾⡟⠛⠛⠛⠛⠛⠛⢿⡇⠀⠀⠀⠀⢀⣀⣀⣉⣉⣂⣤⣄⡀⠀⠀⢠⣿⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢿⡟⡻⢦
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣉⣉⣉⣉⣹⣏⢶⣿⠀⠀⠀⠙⠛⠛⢿⣷⢤⡀⠀⠀⠀⠀⠀⢠⣴⡞⠋⠁⠀⠈⢦⣭⣿
⠀⠀⣿⡇⠀⣀⢀⡀⡀⠀⣸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣄⠀⠀⠀⣿⠉⢛⡄⠀⣀⡤⠞⠋⠁
⠀⣀⣝⣛⡛⠛⠟⠟⠻⣛⣛⣣⡀⠀⠀⠀⢿⣿⣿⣭⣽⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠈⠻⠾⠿⠛⠉
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⢀⡀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡄
⠀⢰⣋⣉⣀⣶⣿⢿⣿⣏⣦⠀⠀⠀⠀⠀⠀⣀⣈⢻⡿⠛⠀⠀⠀⢰⣾⠿⠛⠛⠛⠻⢿⣿⣿⣶⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡧
⠀⢸⣿⣿⣿⡏⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⣾⣷⣷⡄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣠⣾⣷⣶⣶⣶⣶⣶⣿⣆
⠀⢸⣿⣿⣿⣿⣦⣤⣾⣿⡟⠀⠀⠀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣯⣿⠀⠀⠀⠀⢿⣿⣿⣏⣿⣿⣅⣀⣀⣽⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠘⠿⠿⣾⣶⣶⣾⣿⣿⠿⠟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⣶⣾⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⢤⣀⣀
⠀⠀⢸⣟⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡳⢀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠶⣛⣛⠷⣶⣦⠀⠀⠀⠀⠀⠀⢰⣇⣀⠤⢠⡈⡿
⠀⢠⣾⡟⢻⣷⡶⠶⠶⣶⡄⠀⠀⠀⠀⠀⣸⠀⢸⣿⡿⣿⡿⠀⠀⠀⠀⠀⢀⣿⣱⣿⣿⣿⣿⣎⢿⡄⠀⠀⠀⠀⠀⡾⣾⣫⡿⣷⣺⠇
⠀⢸⣿⢽⢿⢿⣧⡤⢤⣿⡇⠀⠀⠀⠐⠚⠁⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⣧⢻⣿⣿⣿⣿⡿⣸⡇⠀⠀⠀⠀⢸⢧⣞⡾⣭⠗⡟
⠀⢸⣿⣿⣿⣿⣿⣷⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢹⣦⣛⠿⠿⣛⣵⣟⠀⠀⠀⠀⠀⠻⢼⣱⣗⡏⢸⠃
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⢠⡟⠿⢻
⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⢸⡇⠀⢸⡀
⠀⠀⠀⠀⢸⡇⣶⠿⠯⠭⠭⠽⠿⣆
⠀⠀⠀⠀⢸⡇⢻⣤⣤⣤⣤⣤⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⣠⣿⣿⠟⠻⠛⠛⠻⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⡴⠒⢺⢛⣡⢄⠀⠈⠀⠀⠀⠈⠁⠀⣠⠈⡛⣗⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠉⠙⠻⣿⣿⠟⠿⢿⣿⣿⡄
⢧⡀⢻⠀⠉⠉⠀⠀⠢⠄⠠⠔⠁⠀⠈⠉⠁⣽⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣇⠀⠰⠀⠀⠀⠰⠆⠀⠘⠷⢀⣀⣿⠿⣿⠃
⠀⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻⡁⠀⠐⠤⠔⠀⠀⠘⠑⠀⠀⠁⠀⠀⢸⠆
⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣤⣾⣄
⠀⠀⠀⠀⠐⡿⢻⠛⡖⠒⠒⣲⠛⡟⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣛⡖⠒⠒⠒⢛⡏⠛⣿⣿⡿⠟⠛
⠀⠀⠀⠀⢰⠃⡞⠀⠘⠒⠒⠁⠀⢻⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠈⢳⡙⣟⠿⢿⠋⠙⠦⡀⢸⡿⠿⠄
⠀⠀⠀⠀⡏⢸⠁⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⠀⢘⡇⢸⠀⠀⠀⠀⠀⠸⡆⢧
⠀⠀⠀⣸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢀⡼⢁⠞⠀⠀⠀⠀⠀⠀⣇⠘⡄
⠀⠀⣠⠃⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢶⠿⡖⠉⠀⠀⠀⠀⠀⠀⠀⢹⠀⠳⡀
⠀⠀⠙⠒⢺⡗⡶⢶⡦⠤⠴⢶⡶⢶⡞⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⠀⣷⡶⣶⠤⠤⠴⢶⡶⢶⠛⠒⠚⠁
⠀⠀⠀⠀⢸⡇⢧⠀⠈⡗⢺⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇⠀⢹⡒⢺⡅⠀⢸
⠀⠀⠀⠀⢸⡇⢸⡄⠀⡇⢸⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢻⠀⢸⡇⠀⡇⠀⢸
⠀⠀⠀⠀⢸⡇⢈⣇⠀⣇⣸⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠘⡆⠀⡇⠀⣧⠀⣸
⠀⠀⠀⣴⠚⠉⠉⣽⣀⡿⢹⣀⣺⠉⠉⠛⣢⠀⠀⠀⠀⠐⠒⠀⠒⠘⠛⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⡗⠛⣇⣀⡏⠉⣿⣀⡸⠓⠒⠢⣄
⠀⠀⠀⡏⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡗⠒⠒⠲⠶⠶⠶⠒⠒⠒⠚⠉⣿

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠐⢄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠠⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣾⢻⣦⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⡅⠈⠩⠉⠉⠁⠀⠈⠉⠉⠉⠠⡀⠀⣴⣾⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⡀⢀⠌⣈⣤⡐⢠
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠔⠇⠀⣀⣀⡀⢀⠀⡀⠀⠀⡖⠉⠉⠹⣿⣿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⡀⢨⠂⢿⣿⡿⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣠⣠⣤⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⡇⠀⠈⠢⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⠋⢋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⣿⣿⠀⠀⠉⢉⣉⣉⣉⡉⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠃⠀⠈⠚⠀⠀⠀⠀⠀⠀⡠⡄⢶⡀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⡇⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⢐⠈⠀⡀⠀⠀⠀⠘⢰⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠀⣀⣀⣀⣀⣀⣀⡀⢾⣿⣿⣿⣿⡿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠒⠈⠀⠐⠈⠀⡀⠀⣰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣽⣿⠀⠈⠭⠭⠭⠭⠉⠁⣾⣿⣿⣿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠶⠂⠀⠙⠃⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡟⠛⠛⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⡂⠀⠀⠀⠀⠀⠀⠀⡀⣤⠐⣆⠀⢀⣀⠀⡄⢰⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⣦⠒⠀⠉⠙⠀⠀⣐⠽⠀⠈⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣯⣽⣿⠀⠛⢛⣛⣛⣛⠛⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡠⠋⠉⡅⠀⢀⣀⡠⠤⠔⠊⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠙⠉⠛⢋⣵⡾⠿⠿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠀⠤⠒⠊⠁⠀⠀⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠠⠀⠤⠤⢀⣀⣀⣀⢀⣿⠃⠀⠀⠀⡆⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣤⣤⣶⣶⣶⣶⣶⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠸⣿⡉⠉⠀⠈⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢰⣿⡏⣹⣿⡏⢻⣿⡋⢻⣷⠀⠀⠀⠙⢷⣤⣀⣠⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠙⠃⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⡏⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⠁⠀⢻⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⣇⠂⠀⢀⠤⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⣿⣿⣿⡟⠀⠀⠀⢻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⢉⠀⠀⣀⠄⠀⢄⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠣⡀⠠⡄⠁⣤⠃⠀⠀⠀⠀⡔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣤⡇⠉⠙⠏⡀⠀⠀⠀⠀⠀⠉⠀⠈⠓⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡌⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠂⠐⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠘⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⠤⠶⠶⠶⠶⠶⠦⢤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⣚⣻⡍⠉⠁⠀⠲⠖⠒⠛⠓⠀⠀⠀⠀⠀⠲⠶⠦⣭⡉⠙⠓⠦⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠐⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠻⢯⣍⠉⠉⠉⠛⠛⠶⠶⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣤⣤⣤⡤⠴⠶⠶⠖⠛⠛⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣧⡆⠀⠰⠖⠛⠛⢻⡆⠀⠀⠀⠀⠀⢀⡏⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣻⣏⣀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⠀⠀⠀⠀⣸⠃⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠐⣶⠀⢀⣄⣀⣤⣤⣼⡇⠀⠀⠀⠀⢀⡟⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣌⣽⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠐⠛⣻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣼⠁⠀⠀⠀⠀⣼⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡟⠀⠘⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠐⣿⠄⢠⣤⣤⣤⡤⣿⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠶⣿⠒⠀⠀⠀⠀⠀⠾⠧⠄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢘⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⣿⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⣁⡀⠀⠀⠀⠀⠀⣼⡇⠀⠴⠶⠶⠚⠋⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣀⣤⣴⡶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠖⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣘⣛⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢩⣭⢹⣿⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣬⣭⣼⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣗⠸⠿⢸⣿⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢩⣭⢹⣿⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣈⣉⣸⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢸⣿⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠶⠶⠆⠶⠰⠄⠔⠶⠐⠲⠤⠆⠒⠶⠶⠶

⠀⠀⠀⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⢀⡤⠦⠴⣤⡀⠀⠀⠀⢰⠶⠶⠶⠶⠶⠶⠶⣆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⡏⠀⠀⠀⠀⣿⠀⠀⠀⢠⠋⠀⠀⠀⠀⠳⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣾⠿⣿⣿⡿⡻⣦⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡏⠀⠀⠀⠀⣿⠀⠀⠀⢸⣟⠀⠀⠀⠈⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣾⡿⠿⣿⣾⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣷⣤⣤⣤⣤⣿⠀⠀⠀⠀⠙⠀⢀⡀⠈⠋⠀⠀⠀⠈⠉⠉⢩⣿⡏⠉⠉⠁⠀⠀⠘⠿⠟⠁⠀⠘⠿⠟⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠦



⠀⠀⠀⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⢰⣂⣀⡆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢰⡞⠒⠒⠒⠒⠒⠒⠒⣶⠀⠀⠀⠀⣶⣶⣶⣶⣦
⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⣶⣿⣾⢟⣻⢿⣿⣶⠀⠀⠀⠀⠀⣶⣶⣶⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿⡯
⠀⠀⠀⣿⣿⣽⣺⣿⡇⠀⠀⠀⢹⣿⣇⢿⣿⣺⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣇⣀⣀⣀⢀⣀⣀⣀⣿⠀⠀⠀⠀⡟⣿⣿⡿⠃
⠀⠀⠀⠻⠿⠿⠿⠿⠇⠀⠀⠀⠙⠛⠛⠓⠚⠛⠛⠛⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠀⠀⠉⠉⠉⣩⣿⡉⠉⠉⠉⠀⠀⠀⠀⠧⠿⠿⠧⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⡟⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⢀⣾⣿⣇⠀⠀⠀⠀⠀⠀⣼⠛⠛⠛⣿⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⣦⣄⡀⠀⠀⠀⡾⠛⠛⠛⠛⠛⣷
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠈⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⣶⠲⣶⣰⣿⣿⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿
⠀⢧⣤⣤⣤⣤⣤⣤⣤⣤⡿⠀⠀⠀⠸⣤⣤⣤⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠈⠀⢹⣿⣿⠏⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠐⠒⠚⠓⠒⠂⠀⠀⠀⠀⠀⠀⠻⠿⠇⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠟⠀⠀⠀⠀⠈⠉⠉⠉⠻⠿⠏⠀⠀⠀⠀⠀⠻⠟⠛⠛⠛⠿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠤⠶⠴⠤⠶⠴⠴⠶⡄
⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⡇⠀⠀⠀⢠⡤⠤⠤⠤⠤⠼⢥⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⣄
⠀⠀⡟⠛⠛⢻⡆⠀⠀⠀⠀⠀⣠⣇⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠛⠛⠛⣳⣀⠀⠀⢀⣰⡟⠛⠛⢻⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⢸⣇⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠷⠶⠶⠾⠃⠤⠼⠧⠄⠀⠷⠶⠆⠀⠀⠼⠿⠿⠿⠿⠿⠿⠿⠤⠀⠰⠿⠧⠀⠸⠶⠶⠶⠶⠶⠀⠀⠰⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⢿⣟⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠙⠳⠆⠀⠈⠀⠀⢉⣻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠷⠄⠀⠀⠀⣀⣴⠿⠛⣋⣉⣉⠛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⠀⠀⠀⢀⣼⠟⣡⡾⠟⢛⣙⠛⠿⣶⡌⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠙⠀⠀⢀⣾⠃⣾⠏⣴⡾⠛⠛⢿⣦⠘⣿⡀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠲⠀⠀⢸⡏⢸⡏⢸⡏⠀⠀⠀⠀⢻⡇⣹⡇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⡶⠶⢶⣶⠶⠶⣶⣶⣤⣤⣄⣀⡀⠀⠀⠀⠙⢷⣦⡀⢸⡇⢸⣇⠸⣷⣄⠀⠀⢀⣾⢃⣿⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠛⢿⡄⠀⠉⠳⠀⠀⠙⠃⠀⠈⠙⠂⠈⠻⠟⠛⢿⠶⣶⣤⣤⣭⣿⣿⣿⡄⠻⣷⣬⣉⣠⣴⠟⣡⡾⢃⡾⠃
⠀⠀⠀⠀⠀⣀⣤⣶⡟⠋⠉⠙⠷⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⡙⠊⠉⠻⠍⠙⠿⣶⣬⣉⣉⣩⣥⡾⢋⣴⠟⠁
⠀⠀⠀⠀⠺⣿⡁⠘⠿⠆⠀⠀⠀⠀⣀⣤⣴⡶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢷⣶⣤⣤⣄⣀⣉⣉⣉⣩⣥⡶⠟⠁
⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⢀⣠⣶⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣌⣛⣿⣷⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⢻⣿⣿⣿⣿⡿⠛⢉⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠋⣠⠾⠋⣁⣠⣄⣀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣿⣿⣿⡟⢀⡾⠃⣰⠞⢋⣉⡉⠙⢻⣄⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣿⣿⣿⠁⣼⠃⣼⠃⣰⣿⣿⣿⣷⠀⢻⡄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⣿⠀⣿⠀⢻⣿⣿⣿⣿⠃⣼⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠐⢶⣶⣦⡐⣶⣶⣄⠠⣤⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢻⡄⠘⢷⣄⣸⣿⡿⠋⣰⠏⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠲⣿⣿⣦⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣷⣾⣿⣦⣶⣦⣀⢤⣀⠀⣀⣀⠈⠻⣦⣄⣉⣉⣁⣤⠞⢁⡴⠋
⠀⠀⠀⠀⢀⣠⡀⠘⢿⣿⣿⣶⣿⣿⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢿⣾⣿⣷⣦⣀⣉⣉⣉⣉⣤⡶⠛⠁
⠀⠀⠀⠀⠈⢻⣿⣶⣾⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠙⣿⡿⠛⠉

⢴⠓⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠖⠀⠀⠀⢨⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡆⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠂⠀⠀⠀⠑⠀⠒⠀⠀⠇⠿⠸⠀⠇⠿⠸⠐⠇⠀⠀⠠⠀⡄⠈⠀⠀⠠⠀⠁⠀⠐⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠂
⠐⠀⡀⠀⠀⠉⢈⡀⡀⠀⠀⠐⠄⠀⠈⠤⠈⠀⠀⠇⠛⠸⠀⠇⠛⠸⠐⠇⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠒⠀⠒
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠤⠀⡀⠀⠀⠠⠀⠀⢀⠔⠁
⠀⠀⡀⠀⠀⢠⡀⠀⠠⠠⠀⠀⠠⠀⢠⠀⠀⠀⠘⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠀⠔⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠄⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠄⢠⡀⡀⠀⠀⠀⠀⠰⠇⠂⠰⢂⠠⠀⠀⠀⠀⠀⠄⠀⠀⠁
⠀⠀⡀⠀⠀⠀⡀⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⢂⢐⠂⠀⠂⠘⠃⠀⠘⡁⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐
⠀⠀⠄⠀⠀⠠⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⠄⠀⠀⢠⡄
⠀⠀⠀⠄⠘⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢺⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⢀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠈⠀⠀⠀⠀⠀⠸⠀⠀⠀⠸⠅
⠐⠀⠈⠀⠀⡀⠀⡀⠀⠉⠉⠉⠉⠉⠋⢀⡀⠠⠀⠀⠰⠀⠀⠈⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⢀⠀⠀⠈⠀⠀⠃⠈⠀⠀⠄⠤⠀⠀⠀⠄
⠀⠀⠀⠀⠊⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⢀⣈⠀⠐⢈⡀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣤
⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⢼⡅⠀⠀⡈⠀⠀⠠⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠄⠐⠀⠀⠂⠐⠀⠀⠀⠀⠈⠁⠀⠉
⠀⠀⡁⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠒⠂⡠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠈⠀⠐⠂⠠⠀⠀⠀⢀⢈⠀⡁⠈⢈⠀⡀⠀⠀⠀⠀⢀⢀⠀⢀⠀⡀
⠀⠀⠁⠀⠰⠆⠰⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠉⠁⠈⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⢈⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡁⠐⠇⠗⠀⢈⡁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠐
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⠂⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⡉⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢠⡀
⠓⠀⡀⠁⠈⠀⢠⢀⠙⠍⠃⠀⠀⠀⠀⢈⡀⠀⠂⠈⠀⡁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠈⠀⠀⠀⢀⠀⡀⠀⠃⡁⠀⠁⠀⢀⠉⠈⠁⠀⠀⠂
⡈⠀⢂⠀⠀⠀⠰⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⡀⢀⠀⢀⠀⠀⠀⠀⠒⠀⠖⠀⠒⠀⠠⠀⠀⠰⠀⠀⠁⠀⠀⠀⠡⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣍⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⠾⠋⠉⠉⠉⠉⠀⣀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⣠⡶⠀⢹⡆⠀⠀⠀⠀⠀⣿⣤⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⡾⠛⠶⣦⠀⣿⡀⣤⡀⣠⡾⠋⠀⠀⢀⡟⠀⠀⠀⠀⠀⣿⠁⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⢸⡇⠘⣧⡀⠙⠋⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⣿⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠶⠄⠀⢀⡿⠀⠀⠈⠛⠶⣤⣤⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢰⡏⠀⠀⠀⣠⠞⠋⠉⠀⠉⠉⠀⣀⠀⠀⠀⢠⡾⠋⠉⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠻⣇⠀⠀⣸⠏⠀⠀⠀⠀⣠⡶⠀⢹⡇⠀⠀⢸⣇⠀⢀⣀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢹⡆⠀⣿⡀⢤⣄⣠⡾⠋⠀⠀⢠⡟⠀⠀⢀⣽⠟⠛⠉⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠘⣧⡀⠙⠋⠀⠀⠀⢀⣾⠃⠀⠀⢸⡇⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠛⠶⣦⣤⡴⠶⠛⠁⠀⠀⠀⠈⣻⡶⠞⠛⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠆⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠷⣤⣴⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠠⣶⣦⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⠶⠶⠖⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣀⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣼⣭⣵⣾⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣸⣾⣭⣿⣟⣫⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠸⣿⣉⣉⣉⣩⣭⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⣾⣧⡄
⠀⠀⠀⣠⣾⠿⢟⠛⡟⠿⣝⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠽⢗⣒⣽⣆
⠀⠀⣴⡟⠁⠀⢸⢀⣇⡀⠈⢷⡇⠀⠀⠀⠀⠀⠀⠀⠄⠐⣵⡾⠿⣷⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠹⢯⠥⢶⠒⠡⠤⡀
⠀⠀⣿⠀⠀⢠⢻⢿⡇⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⣸⠀⢸⡀⠀⠀⠘⡄⠀⠉
⠀⠀⣿⠀⠀⠣⠚⠎⠑⠃⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠘⣦⡀⠀⠄⠀⠀⠀⠀⠀⠀⠃⠀⡘⣾⣫⡏⢏⡆⠀⠀⢸⠀⠀⢀⡀
⠀⠀⠘⣷⣄⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠄⠀⢀⡿⠀⣠⡤⡜⠒⡈⠻⣦⠀⠀⠀⠀⠐⠂⠀⡆⠀⠈⠷⠙⠃⠈⠁⠀⠀⡜
⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁⠀⡀⠑⠀⠀⠀⠀⠀⠀⢸⡇⠀⠁⠃⠃⠀⠀⠠⢸⠀⠀⠀⠀⠀⠀⡀⠑⣄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠐⠒
⠀⠈⠀⠀⠄⠀⢀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡱⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠒⠒⠒⠈⠁⠀⠀⠐⠤
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⡀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠜⡇⢀⣼⠁⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠸⣄⠰⠀⡧⠃⣇⠔⢈⣔⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡒⠤⠠⠄⠈⠁⠀⠁⠀⠀⠀⠈⠀⢐⣓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢⠄⠀⠀⠀⢀⠤⣤⣀⣠⠴⠚⠙⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠰⠂⡠⠤⣤⡅⠀⠈⣿⠀⠀⠀⠀⠐⣯⣤⣤⡤⡄⠀⠀⠀⠀⢸⢈⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠚⡀⠀⢘⡟⠂⠒⠁⠀⠀⠀⠀⠀⢹⣿⠋⡐⠀⠀⠠⢤⢀⠊⠉⢣⡇⠐⠈⢹⠠⢄⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠡⠀⣀⠤⠀⠀⠀⠀⠀⢸⡏⢠⠃⠀⢠⢺⢴⠘⣦⣀⣜⣧⠔⣦⢆⡴⢿⣁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡚⠻⠧⣀⠀⠀⠀⠀⠀⢸⣇⣘⣻⣿⠝⣿⣾⣵⢯⣹⣿⣟⣶⡏⠀⠑⢬⣭⣬⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⡄⠀⠀⠀⠁⢀⡄⢄⠈⣺⡙⠏⠻⣷⣋⣵⠌⠙⣿⣿⠻⡿⠀⣀⡤⠘⡀
⠀⣠⠖⠉⡁⠀⢀⠀⠀⠹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡤⢦⡉⠉⠓⢴⡈⠀⢻⣕⠁⠀⢘⣻⣾⡿⢿⣿⠁⠉⡗⡇
⠀⠑⢡⡈⣿⡆⢀⠟⠒⢾⣧⣤⣤⣤⡤⠤⠤⢤⡄⠀⠈⢻⡀⠀⠳⠒⠒⡒⢵⡀⣸⢿⡟⣿⠿⣿⡿⡍⢹⣿⠀⠀⡇⡇
⠀⠀⠘⢣⣼⡇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⢷⠀⠀⠀⠀⢈⢄⡇⠀⠀⢻⣷⣶⣧⡊⠀⠀⢹⠀⠀⠣⠇
⠀⠀⠀⠈⡆⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣇⣀⣀⣀⣀⣡⣇⣀⠀⠀⣻⣿⣿⣾⣦⣤⣿⠀⣀
⠀⠀⠀⠀⢃⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠒⠒⠒⠒⡆⠸⣯⢀⡰⠈⠓⠒⠁⠀⠀⠙⣯⣯⢹⣾⡭⣿⡟⠓⠓⡀
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢀⡇⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⣺⣭⣿⡗⠤⠚
⠀⠀⠀⠀⠣⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠑⠒⠒⠚⢡⣤⣤⡿⢠⡤⠤⢤⡆⠀⠀⢸⡏⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠀⠀⠀⢸⡇⠀⠀⣧⠀⠀⠘⠧⠄⡀
⠀⠀⣐⣿⣷⢶⣶⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣈⣁⣀⣀⣀⡀⠼⠇⠀⠀⠻⠤⠤⠤⠤⠤⠤⠵⠦
⠀⠀⠈⠁⠀⠀⠀⠉
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⣀⠂
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠂⡠⠀⠀⠀⠄⠁⠀⠀⣀⠀⠀⠀⠈⠂⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⡠⠀⠀⠀⠀⠀⢀⣴⠇⠀⠀⠁⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⢈
⠀⠀⡠⠙⠦⠀⠀⠀⠀⢀⢀⡴⠢⠀⠀⠀⠀⢰⣶⣿⠟⠀⠠⢢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠀⡠⠂⠆⠀⠄⠀⠀⠀⠀⢀⠀⠀⢀⣴⣿⠟⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐
⠀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠄⠁⠄⠸⠛⠡⠖⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠉⠀⠀⠀⠁⠆⠀⠄⠁⠔⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠁
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⢀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠄
⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⠐
⠀⠀⠀⠀⠠⠈⠤⣁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠄⠀⠀⠀⠀⠀⠈⠡⡄⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢠⣾
⠡⠀⠠⠀⠠⡀⠀⠀⠀⠙⠒⠀⢀⠀⡀⠀⠀⠀⠅⠀⡠⠈⠳⠀⠀⠠⠀⠀⠀⠴⠃⠀⠀⠀⠀⢀⣶⣿⠟
⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣴⣿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⣿⠟⠋⠉⠉⠙⠻⣿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠟⣋⣥⠴⡆⠀⠀⠀⣴⣶⣶⠀⠀⠀⠀⣳⠶⢬⣙⠻⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⣫⡔⢟⣭⣶⠿⠟⠀⡄⠀⢸⣿⣿⡄⠀⡀⠀⠻⠿⣷⣯⣻⢦⣝⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⣡⡾⣫⣾⠟⠋⠁⠀⠀⠀⠑⠀⣿⡿⢻⡇⠀⠁⠀⠀⠀⠀⠙⠻⣷⣌⡷⣌⠻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⢋⣾⣯⣾⠟⠁⠀⠀⠀⠀⠀⢀⣰⣧⠙⢷⡎⢁⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣽⢷⡜⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⢡⣟⣵⡿⠁⠀⠀⠀⠀⠀⠀⣠⣸⣿⣿⣷⣼⣷⡟⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡹⣎⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⢁⠟⣾⡟⠀⠀⠀⠀⠰⣤⣼⣿⣿⣧⢻⣿⣿⣿⣿⢅⡟⣾⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⢻⣷⡙⡄⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠎⣼⡿⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡇⢿⣿⣿⢯⣾⣷⣿⢠⡇⣼⢀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠹⡘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠸⣄⣿⠇⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣟⠇⠠⡻⢿⣿⣿⣾⣿⣼⣇⡀⠀⠀⠀⠀⠀⠘⣿⣴⠇⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢠⣿⣿⣀⠃⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣇⣠⡀⢀⢤⣼⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣀⣿⡏⣶⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⠈⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣇⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⡧⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢸⠟⣿⡆⡀⠀⠀⢀⣨⣶⣾⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣤⠂⠀⠀⠀⢀⢀⣿⢿⡀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡌⣟⣿⣧⠀⠀⠀⠹⢾⡿⢟⣯⣷⠞⣝⠣⣭⣿⣿⣿⣷⠆⣽⣿⣿⣿⡷⣯⣙⠃⠀⠀⠀⠀⣼⡿⢠⢃⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠘⡌⢿⣧⠀⠉⢀⣭⠾⣻⣿⣫⠀⣿⣧⠈⣿⣿⣿⠏⣴⣿⣿⣿⢯⡻⢮⣿⡆⠀⠈⠀⣰⣿⢣⠏⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡸⣮⢻⣧⣼⣿⣶⣿⡿⣵⣿⢰⣿⣿⣷⣾⣿⣿⣿⣿⣿⡿⣿⣯⡳⣶⣦⠥⠔⣀⠴⡿⣳⢏⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡷⣉⣼⣿⣿⣿⣿⣯⣿⣿⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡻⣿⣿⣮⣿⣿⣶⣦⣿⠊⠋⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⡿⢁⠄⢻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢿⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣍⠻⣿⣿⣿⣿⣿⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣭⣵⣾⣿⣿⣿⠿⣫⣾⣿⣿⣿⣿⣿⣿⣮⡻⢿⣿⣿⣿⣿⣿⣶⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⣿⣿⡿⠛⠁⠈⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⢋⣼⣿⣻⣿⣿⣿⡟⣿⠻⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠉⠉

⠈⣈⣈⣁⣄⢀⢠⠀⠀⡀⠀⠀⠀⠄⡀⠠⠤⠀⢀⠦⠀⠆⣴⠂⠄⠠⠀⡀⠀⠀⠠⠰⠂⠆⠠⠀⠀⠀⠀⠆⠢⠀⢰⡀⢀⠄⠆⠀⠀⠀⠰⠀⠲⠐⠆
⠀⣿⣯⣿⣿⠀⢸⠀⠀⠅⠀⠀⠑⠤⠂⠘⢁⢊⠂⠀⠀⡁⠂⠀⠀⠀⠀⠓⣒⢀⣒⡀⠀⠀⢀⠀⠀⠄⠄⠀⠐⠒⢂⠀⢂⡀⢀⠀⠄⡄⣿⣿⣿⣿⣷
⠀⠀⠀⡄⠀⠀⠠⠆⠀⠁⠂⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡉⠀⠒⠙⠁⠂⠄⠄⡁⠀⠀⠀⠁⠁⠀⠘⠀⡜⠁⣿⣾⣿⣿⣿
⠈⠀⠀⡀⠒⠒⢢⠰⠅⠤⠠⠄⠤⠠⠀⡤⢠⠢⠤⠡⠠⠀⠀⠤⢢⠔⠍⠩⠍⠁⠀⠀⠠⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⢀⣶⣶⣮⠸⢹⠁⠙⠛⠛⠛⠋⠀⣀
⠀⡂⠀⠀⠀⠀⢸⠀⠂⠐⠀⠀⠇⢸⡀⠇⠰⡁⠇⠒⠐⠀⠀⠐⢸⠀⡇⢸⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⢾⣿⠀⠀⠀⠀⠀⠀⠀⠄⠐⠄
⠀⡅⠀⠂⠂⠒⢸⠠⠀⠀⠀⠀⠁⠘⠀⠁⠘⠁⠃⠀⠀⠀⠀⠀⠘⠀⡃⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠀⠀⠈⠑⠊⣭⣀⠸
⠀⡇⠀⠀⠉⠭⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢈⠀⣠⣮⣾⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⠿⠉⠉⠉⠀⠀⡀⣀⠀⢀⡀
⠀⡇⠀⠄⠐⠐⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢐⠠⠀⠈⠿⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠈⠉⠀⢁⣤⣤⣤⡈
⠀⡇⠀⠅⠸⠀⣟⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢼⠰⠀⠀⢠⡰⠆⠖⣀⠠⠤⠀⠐⠀⢐⣐⣂⣒⣐⣂⡀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⡇⠀⠀⠀⠐⢻⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡅⢸⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡇⢸⡄⠀⠀⠁⡋⠉⠁⡇⢀
⠀⡇⠀⠀⠀⠀⣴⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠇⠙⠊⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⠄⠀⠀⠈⠂⠈⠈⠀⠀⠘⠁⠁⠥⠠⠠⠇⣀⢸
⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠰⠉⠀⠈⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠂⠀⢀⠈⢀⢸
⠀⡇⠀⡀⠀⠀⠙⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠈
⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠠⠀⡆⠠⠁⢌⠀⠀⠀⠀⠀⠐⠀⠆⢰⢀⠀⠀⠀⠀⠀⠀⠐⠒⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⠀⢀
⠀⡇⠀⠀⠀⠀⠰⠲⠂⠐⠀⠀⠂⠸⠀⡃⠘⠀⠂⠂⠐⠀⠀⠒⠐⠀⠃⣘⠈⢀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡄
⠀⡇⠀⠀⠔⠶⢂⠀⠀⠀⠏⢀⣀⠀⣀⠀⠉⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠃⠀⠀⠈⠈⠀⠈⠀⠀⠀⠀⠀⠀⢸⠀⠠⠁
⠀⠃⠀⠓⠂⠀⢸⠀⠀⠀⠁⠀⠤⠤⠤⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐
⠀⠄⠀⠀⢀⠀⠘⠀⠀⠀⠀⠐⡆⠀⠀⠖⠳⠆⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⡄⢱⣶⣶⠀⠀⠀⠀⠠⠤⠀⠘⠃⠀⠀⠀⢀⣀⣀⣀⡀⣶⡀⠀⢸⠀⠚⢀⠰⢆⠀⠁⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠇⢸⣿⣟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⢻⠇⠈⠀⠁⠁⢸⠀⠤⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⠀⠀⠀⢀
⠀⠈⠉⠉⠀⢀⠠⡀⢠⠀⠃⡠⢤⡀⠘⠁⠹⠿⠾⠿⠇⠹⠁⠐⠀⠃⠃⠘⠠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⢠⠀⡄⣤⡌⠀⠀⢰⣿⣇⠀⣟⠀⣿⠐
⠀⠀⠀⡀⠋⠥⠀⠂⠘⠀⠀⠙⠛⠃⢒⡁⢤⡀⠀⠀⡶⠖⠀⠀⠀⠰⠀⠀⠠⠄⠒⠀⠀⠀⠀⠀⠈⠂⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠉⠓⠤⠁⠀⠀⠤
⣤⣤⣤⣤⣤⣤⣤⣽⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣼⣭⣿⣽⣯⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣧⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠦⠾⣽⣷⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠃⠠⠄⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠶⣤⠁⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⣺⣿⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⠒⠋⢋⠉⠄⠰⠿⣻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⣉⢠⡤⠀⠆⣒⣂⣩⡤⠴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡫⣶⢀⣃⡴⠴⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣾⣟⣷⣶⣶⣶⣶⡶⡶⡖⣖⣒⣚⣛⣟⣋⣭⡩⡴⠦⠶⠒⡛⢋⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⠿⠟⠛⢛⡛⢉⡉⢉⡉⢩⡅⢡⡅⠠⣄⠀⡆⠰⠇⠘⣃⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢩⡄⠰⠆⠘⣁⣘⣃⣀⣥⣬⣥⣤⠥⠬⠥⠶⠶⠖⠒⠚⠋⠉⠉⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣶⣾⣿⣿⣿⣯⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠤⠖⡒⠋⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠿⠿⠛⠛⢛⡋⠉⡭⠉⢩⠉⠁⠄⠀⢰⠀⠀⠇⠀⣓⣀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢩⠄⠀⡆⠀⠩⢀⢀⡃⣀⣁⣀⣨⣡⣤⣤⠤⠴⠴⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⣂⣠⠤⠶⠒⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣷⣤⣤⣤⣤⣤⣤⡤⡤⢤⠤⠤⠤⠴⢶⢶⣶⣶⡶⢶⣶⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣷⡿⣶⣺⢹⣦⣞⣼⣣⣷⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠓⠛⠛⠛⠚⠓⠓⠚⠛⠛⠛⣛⣛⣉⣩⡭⠿⠿⣛⠻⢴⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠶⢖⠒⢋⢉⣩⠉⠤⡀⡶⠀⠿⢀⣚⡤⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢐⠹⢂⣚⣡⣭⠴⠴⠴⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⢻⡛⡟⣻⠟⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣶⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣴⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣾⣿⣿⣿⣿⠀⠀⠀⠀⠠⢀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠉⢀⣤⣤⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠂⡙⠩⠊
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠉⣉⣉⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡁⠀⠀⠀⠀⠐⠂⢲⠀⠀⠀⠀⠀⢈⣉⣉⠁⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⡇⠛⣿⣿⡟⠃⢸⣿⣿⣿⣴⣿
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠘⠪⡇⠐⠖⠀⠀⠀⠀⠀⠀⢸⣿⣿⠂⠶⠶⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠰⠀⡂⢀⡀⠀⠤⠐⠀⠀⠀⠀⠙⠛⠛⠁⠀⠛⠛⢳⡴⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠁⠒⠒⠈⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠤⢠⣿⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣴⡖⢶⡶⢶⡶⢶⡄⠀⠀⢷⣄⣀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠴⠿⢿⣿⠿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠉⠉⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⠿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⠀⠹⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣿⣿⡇⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣧⡍⠙⠛⡀⠀⠀⠀⠘⠋⠙⠤⠀⠀⠀⠀⠀⠀⠈⠀⡝⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠠⡐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⣇⠀⠀⠀⠀⣠⢇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠇⠀⢠⠖⢉⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡇⠀⠀⡏⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠃⢠⡀⣇⢸⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣼⢷⡈⠘⡄⠀⠀⡤⢀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⢗⡩⢛⣆⠀⠀⠀⢰⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⢠⣛⠅⣊⠥⢚⠳⡀⠀⠸⠸⡀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⣰⡫⢔⡊⠁⠈⠰⢂⡹⢆⠀⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣼⠥⣚⡅⠀⡿⣄⠈⠩⠔⣊⣧⠈⠀⠀⠤⠄
⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⡾⢔⡫⠕⠀⢰⠃⠘⣆⠐⢋⡡⠚⣆
⣟⠷⣬⣍⡛⠓⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣒⠭⡒⠉⣠⠃⠀⠀⠘⣆⠠⠖⣋⢽⠀⠀⠀⠒
⠳⢯⣖⡥⣋⠝⣲⢦⣤⣉⡙⠒⠦⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢸⠫⢒⡉⠀⡴⠁⠀⠀⠀⠀⢸⡄⣋⡤⢾
⠀⠀⠀⠈⠛⠻⠶⣭⣞⠕⣉⠝⡲⢦⣄⣉⡑⠒⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⣚⠁⠀⡜⢀⠀⠀⠀⠀⠀⢸⠃⠡⢒⡿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠾⣴⣫⢖⡡⢛⡵⣲⠦⣄⣉⠓⠲⠤⣀⡀⠀⠀⠀⠉⠁⠀⣯⠴⠂⢸⠀⠢⠀⠀⠀⠀⢠⠏⠠⢚⣭⠇
⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠷⢮⣥⢊⡥⣋⠝⣲⠦⣄⣉⠙⠲⠤⣄⣀⠀⢹⡫⠀⢺⠀⠩⠄⠀⠀⢠⠇⠰⢚⣥⡞⠀⠀⠀⠐⠂
⠀⠀⠀⠈⠱⠒⡭⠴⣂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢮⣴⣊⠴⢋⡽⣓⠦⢤⣉⠙⣲⠦⠤⢼⡄⠉⠄⠀⠀⢸⠀⠨⢵⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠭⢒⡣⠴⣂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢾⣥⡖⡡⣊⡿⢕⢴⢦⡀⡀⠀⠀⠀⠀⠈⠀⢰⠏⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⢚⡭⠖⣂⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⢧⡱⠣⢪⣾⢛⢔⢄⡀⠀⠑⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠥⠒⣊⡥⠄⣀⠄⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢭⣉⣋⣘⣐⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠴⢚⡥⠔⣀⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠽⢒⡮⢗⡢⢔⡠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠥⠒⣫⠵⣚⠤⣂⡠⢆⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⢿⣷⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠘⣧⢸⣿⣿⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠛⣻⣿⢷⡀⠀⠹⣆⠁⢻⡄⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡘⢿⠟⢯⢻⡄⠀⠹⣆⠀⢿⡄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⢷⡹⣆⠀⠹⣆⠈⢿⡀⠀⠈⢳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠈⢷⡽⣆⢀⣹⣧⡈⣧⠀⠀⠀⢻⣝⡳⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠻⣽⣿⠟⠋⠉⠙⢷⡀⠀⠈⣿⠉⠛⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠹⣽⣆⠀⠀⠀⠈⠻⠶⠞⠋⢀⣠⣾⡿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢦⣤⣤⣾⠟⠀⠀⠀⡀⣠⣤⣶⣾⡿⠟⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣄⣀⡀⠀⣀⣀⣴⣶⡿⠟⠛⠉⠉⠀⠀⠀⠀⣠⣶⠟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠾⠿⠿⠿⠟⠛⠉⠁⠀⠀⢀⣀⣀⣤⣤⡶⠟⠋⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠶⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⢦⣤⣤⣤⣤⣶⠾⠛⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡤⠖⠋⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⣸⣇⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡿⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⠀⢸⠇⣀⣤⣤⠤⠤⢤⣤⣤⣀⡀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢛⡟⠀⠀⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣸⡇⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⢸⡆
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣴⠟⣧⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣿
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣴⠏⠀⢿⡀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣼⠃⠀⠀⠘⣧⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⣼⠃⠀⠀⠀⠀⠘⢷⡀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢰⡏
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠙⠳⣦⣄⣀⠀⠀⠀⠀⣀⣤⠾⠋⠀⠀⢀⡿
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⣠⡟⠁
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠒⠒⠚⠛⠉⠁

⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠀⠀⠓⠈⠅⠀⠀⢰⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠍⣾⠈⠙⠒⠒⠒⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠠⡀⠀⠀⠈⠁⠀⠀⡀⠀⢀⡄⠀⡥⠀⠀⠀⣰⠊⠁⠈⠻⣆
⠀⠀⠀⠀⠤⠤⠤⣤⡤⠴⣦⣤⡤⠤⢆⠄⡀⠀⠰⣄⣀⣀⡈⡟⠉⣍⠂⡺⢀⡀⠀⠀⢠⣴⠇⠀⠀⠀⠀⠈⠉⠳⢦
⠀⠀⠀⣀⣀⢀⣌⠁⠀⠘⢻⣿⡿⠆⠈⠃⣄⣀⡤⠻⣿⣿⡐⡟⠀⡈⣠⠇⠀⢉⡆⠀⠈⢿⣆⠀⠀⡀
⠀⠀⠀⠀⠀⢺⣤⡀⣶⣿⣿⠆⠠⢀⠀⠀⠸⠀⠀⠇⢸⢸⣽⣷⣰⡷⢺⠀⠀⢸⢁⠀⠀⠀⠉⠳⠮⠤⠖⢂⣢⡀
⠀⠀⠀⠀⠀⠘⢿⣿⠟⠸⠉⠀⠀⡀⠀⠀⠘⠢⣂⡀⠀⢘⢻⣇⣻⡷⠞⠅⠀⠈⠈⠑⠠⡀⢀⣀⣣⣼⡿⠿⣯⣙⠓
⠀⠀⠀⠀⠀⠀⠘⢯⠀⠀⢀⡀⠀⡠⠀⠀⠀⠀⠀⠉⠓⠺⠞⠋⠘⣏⠔⠁⠀⢂⡀⠀⠀⠀⢸⡄⠁⠀⠀⠀⠈⠛⠕
⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠚⣮⡉⠉⠁⠀⠀⢸⠁
⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠘⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡹⣧⠀⠀⠶⠀⢺⣦⠤⢤⠤⠶⠶⠛⠛
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄⠀⠉⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣝⢧⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢰⠀⠀⢀⠀⡅⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣆⢡⠀⠀⠀⠀⠀⠠⣠⡴⠞
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡙⠳⣥⣀⣀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢷⡙⢏⠛⠛⠯⣉⠁
⠀⠀⠀⠀⠀⠀⢁⠀⡀⠀⠀⠁⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣽⣌⣳⣄⢀⠄⠉
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢤⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠟⣯⠁⠠
⠀⠀⠀⠀⠀⠀⢀⡯⡁⠀⠈⠈⠉⠉⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡻⡇⣿⡀⠁
⠀⠀⠀⠀⠀⠀⣟⠁⠚⡇⠀⠀⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡕⢄⠨⠫⢸⠄
⠀⠀⠀⠀⠀⠰⣧⠀⠨⠇⠀⠀⢈⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⡄⠈
⣶⣶⠶⠶⠶⣶⣿⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣾⣶⣷⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⡖⠛⠛⠉⠉⠉⠉⠉⠙⠛⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣥⣶⣶⣶⣶⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣮⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢁⣿⣿⣷⡒⠀⠀⢰⡇⠀⠀⢠⣼⣆⡀⠀⠀⢀⣰⠀⠸⣿⣿⣿⣿⣿⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣿⢻⠸⠁⠀⠃⠸⡇⢸⣷⠋⢂⣽⢩⣁⣠⣷⣾⠀⠀⠉⢸⣿⣿⡟⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣸⠂⣘⠀⠀⠂⣼⡇⣼⡇⠀⠀⠹⠂⠙⠉⠉⢹⠀⠀⡀⠁⠀⠈⢇⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⢸⡇⣀⣠⣴⣷⣄⠛⠀⡦⠦⠸⣧⡟⠛⢀⠀⢸⠀⠀⠀⠀⠀⠉⢉⠇⠒⠂⠉⠉⠑⢄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢩⣤⡀⠀⠀⣿⣿⡷⢾⡆⠉⢃⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⠀⠡⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⢻⣿⣿⣆⠀⣾⣶⠀⠸⡇⠀⠀⢣⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣸⣿⣿⣿⠀⣯⣭⣄⣂⣡⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡛⠉⢿⣿⣿⣆⡷⣤⠄⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠀⠹⣿⣿⣿⣟⢰⣶⣶⠈⢆⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣘⣿⣿⣿⣿⣿⣿⣶⣾⣷⣶⣿⣿⡇⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠐⠂⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠄⠤⠤⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⡇
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠀⣀⣀⣀⡒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠶⡆⠀⠀⠀⠀⢺⡇
⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠄⠤⠤⠤⠄⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠁⠀⠀⠃⠀⠀⢀⠀⠀⠀⠀⠀⢾⣇
⠀⢸⣿⡆⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠋⠉⢻⣶⠀⠀⠀⠀⠀⠀⢀⡶⠂⠀⠀⠀⠀⠀⠀⠚⠋⠀⢀⠀⠀⠀⠀⣿
⠀⠸⠋⢁⠀⠀⠁⠀⡀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢸⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢬⣑⢱⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠙⠦⣄⣋⣢⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠔⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠑⢿⢾⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⡀⠀⠤⠤⢄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠘⠃⠀⠀⠙⢵⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢠⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⢀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡤⠼⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠁⠉⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠈⠨⠀⠀⠀⢸⡄⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠗⠀⠀⣿
⠀⠀⠀⠀⠠⢨⠀⠀⠀⠸⠃⢀⢀⣀⣀⣀⣉⣯⣨⣄⣔⣀⢀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣿⣇⣀⣀⠒⠀⢀⣿
⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠭⣭⣤⣤⡀⠹
⠂⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉
⣿⣿⢿⠿⢛⠿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠄⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⠄⠀⠀⠀⠀⠀⢄⠀⠤⡀⠀⢄⠀⠀⠀⡠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠁⠀⠄⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠀⠀⠀⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠄
⠀⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠄⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠
⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠁⠀⠀⠀⠀⠀⠄⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠐⠅⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠠⡂⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠔
⠀⢀⠀⠁⠀⠈⠃⠀⠀⡀⠀⠀⠀⠂⠀⠐⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⡂⠀⠀⠀⠀⠀⠀⢀
⢀⡁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠢⠀⠀⠠
⠂⠀⠀⠐⡄⠀⠀⠄⠀⠀⠀⠀⠀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠂⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⢀⠠
⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣂⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⢄⠈⠢⠀⠀⠀⣦⠀⠀⢈⠀⠀⠀⠀⠀⢤⠄⠀⢀⣤⠀⠀⡔⠄⠀⠀⠀⠀⠀⠈⠢⡈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⡂⠀⠀⠀⢀⡀⠸⠂⢠⡀⠀⠀⠢⣀⣀⣀⡤⠈⠢⣀⠀⢀⠀⢸⠀⠠⠊⠀⡠⠊⠀⡀⠒⢀⠀⠀⠀⠀⠀⠈⠢⢀⢀⢀⠐⠀⠀⠠⡀⠠⡀⠂⠀⠀⠇
⠀⠀⠙⠀⡃⠀⠀⠃⠀⠀⠘⠃⠀⠀⠀⠀⠀⡀⡀⠀⠀⠐⠁⠐⠁⢸⠀⠀⠐⠛⠀⣠⠊⢀⠔⠁⢠⠆⢠⠖⣠⠆⠰⠆⠀⣠⠑⢄⠁⢀⠈⠢⡘⠢⡀
⠁⠄⠀⠈⠋⠀⠀⡠⠀⠀⠀⢁⣀⣀⣀⣠⡄⠀⠀⢀⠄⠈⠉⠉⠢⡑⠁⠢⡀⠀⠀⣁⠔⠁⠀⠔⢁⠔⡡⠊⡠⠊⠀⠀⠧⡀⠑⢄⠑⢄⠀⠀⠈⠢⡈⠂
⢄⠀⠒⠄⠀⡠⠊⢀⠀⢀⠖⡁⠀⠀⠀⣈⡀⢠⣶⠁⢴⠄⠀⠀⠤⠈⠲⡄⠈⠲⣄⠉⣀⡄⣠⠜⣡⠌⣠⠂⢠⣤⠀⠀⢀⠈⠢⡀⠑⢄⠑⢄⠀⢀⠈⠂
⠀⠱⡄⠀⠘⠀⠄⢀⠔⠡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠢⡀⠑⢄⠑⢄⡀⢄
⠁⠀⠀⠰⡦⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠈⡆⠀⠁⠈⠈
⠀⠀⠄⠀⡇⠀⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣇⠀⢠⠐⠷⡀
⠀⠀⡇⢀⣇⠀⠀⣨⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢉⠀⠈⡀⠀⠈⠂
⠔⠀⡃⢀⠁⢠⡀⠈⢀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠉⠢⡈⠋⠉⠉⠑
⡠⠊⡠⠒⠁⠈⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠑⢄⠈⢶⠄
⠠⠖⢀⠄⠀⠒⠁⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠑⠀⠀⠒⠆
⢀⠔⠁⠪⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠹⠄⠀⠰⠠
⠁⠈⠒⠒⠐⠒⠀⠂⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠢⡀⠀⠀⠲⣄
⠈⠁⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠢⡈⠉⠁⠢⡈⠓⠤
⠀⠀⠀⠀⠀⠀⠐⢄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠀⠈⠢⡈⠢
⠀⠀⠀⠀⠐⠀⠢⡀⠰⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠉⣀⠢⡈⠢⡈
⠀⠀⢄⠈⠁⠑⢄⡈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡈⠢⡈
⠀⠀⡀⡑⢄⠀⠀⠛⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⡖⠈⠢
⢠⣄⠉⠁⣀⠑⢄⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠴⢀⢄⣀
⠈⡇⢰⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢇⡔⢁⠖⠀⠁
⢠⡇⢘⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠐⢃⠔⢁⠤⢀⠔⠁
⠀⡠⠊⠀⡀⢀⠆⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠔⣁⠔⣁⠔⠉
⠊⠀⡠⠊⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⠃⠀⠂⠐⠃
⠀⠊⠀⠀⠊⠀⠀⠻⠀⠀⠀⠈⠀⠀⠐⠓⠀⠀⢀⠐⠦⡀⠚⠂⠀⠐⠛⢀⡰⠂⢀⠶⠂⠀⠒⠂⠀⠀⠐⠐⠀⠘⠓⡀⡀⠀⠊⠀⠀⠀⠀⠀⠛⠃⠀⠐⠠
⠑⢄⠀⢁⠀⠀⠀⢘⠀⠀⠀⠘⠦⡈⠣⡘⠆⡀⠉⠣⢀⠈⠀⠀⠀⠀⠀⠉⢀⠔⠁⠀⠀⠀⠀⠀⠀⠶⠄⠠⢄⠀⠀⠉⠁⣀⡀⠀⠀⠀⠀⠀⠑⢄⠀⠷⡀
⠑⢄⠑⢌⠢⡀⢰⡄⠀⠈⠛⢄⠀⠈⡆⠈⠂⠈⡆⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠠⡀⠙⠢⡀⠀⠀⠀⠶⠆⠀⢄⠑⢄⠀⠀⠉⠀⠀⠀⠀⠀⡄⠑⢄⠑⢄⠈
⠑⢄⠑⢄⠑⠌⠢⡈⠢⡈⠢⢀⠑⢄⠣⡀⠠⡀⠣⡘⠢⡀⠮⠃⠀⠶⡀⠐⢆⠀⠈⠢⡀⠈⢦⡄⠀⣤⠀⣀⠀⠑⢄⠑⢄⠀⠀⠉⠀⡠⠊⠁⢠⠄⠑⢄⠁
⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠘⠃⠀⠱⠈⠢⡈⠢⡈⠢⡈⠂⠀⠀⠀⠈⠢⡀⠑⢄⠀⠉⠁⠀⠀⠀⠁⠀⠋⠀⠀⠀⠑⠀⠁
⣶⣾⣶⣾⣶⡾⣶⣷⣶⣷⣶⣶⣶⣶⣾⣶⣶⣿⣶⣿⣶⣿⣶⣷⣶⣶⣶⣶⣾⣶⣾⣷⣶⣶⣶⣶⣶⣿⣶⣿⣶⣶⣶⣾⣷⣾⡶⣿⠾⠷⠶⢶⣾⡶⠶⠾⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠟⠛⠛⠛⠛⠿⠶⢶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣿⣿⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣿⣍⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣩⣷⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣀⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣸⣯⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⡾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢶⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠟⢻⣿⠟⠟⠛⢿⡿⠛⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⢿⡿⠛⠛⠻⣿⠟⠻⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣼⠇⠀⠀⠀⢸⡇⢰⣿⠀⢸⣷⠀⣾⣧⠀⣾⡇⠀⣿⡆⢸⡇⠀⠀⠀⠸⣇⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣼⠏⠀⠀⠀⠀⢸⡇⠸⡿⠀⠸⡟⠀⠻⠏⠀⠿⠇⠀⠿⠇⢸⡇⠀⠀⠀⠀⠹⣇⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣴⠟⠀⠀⢀⣠⡶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⢦⣄⠀⠀⠀⢻⣆⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⣰⡏⠀⠀⢠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⢻⣆⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣰⡟⠀⠀⠀⢿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠀⠀⠀⢻⣆⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢠⡟⠀⢠⡟⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⢿⡄⠀⢿⡄
⠀⠀⠀⠀⠀⢀⣾⠁⢠⡿⠁⠀⠀⢀⣾⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠘⣿⡀⠘⣷⡀
⠀⠀⠀⠀⠀⣼⠇⠀⣼⠃⠀⠀⠀⣾⠃⠈⠙⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠸⣧⠀⠸⣇
⠀⠀⠀⠀⢠⡿⠀⢰⡟⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠶⠶⠾⢿⣇⠀⠀⠀⢿⡆⠀⣿⡀
⠀⠀⠀⠀⣾⡇⠀⣿⠁⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⡀⠀⠀⢀⣠⣴⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣿⠀⢸⣧
⠀⠀⠀⢀⣿⠀⢰⡿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣶⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢿⡄⠀⣿
⠀⠀⠀⢸⡏⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⢿⡆
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⡄⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⢸⡇⠀⣸⡇
⠀⠀⠀⠈⣿⠀⠸⣧⠀⠀⠀⠀⠘⣷⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣾⠃⠀⣿
⠀⠀⠀⠀⢿⡆⠀⣿⠀⠀⠀⠀⠀⢻⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⣰⡏⠀⠀⠀⠀⢠⣿⠀⢸⡟
⠀⠀⠀⠀⠈⣿⠀⠘⣷⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⢰⡟⠀⠀⠀⠀⠀⡼⠁⠀⡿⠁
⠀⠀⠀⠀⠀⠹⣆⠀⠹⣧⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡟⠀⠀⠀⠀⢀⣼⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠹⣧⠀⠙⢷⣄⠀⠀⠀⣠⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣄⠀⠀⠀⣠⡾⠃⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠙⠷⣦⣤⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣤⣴⠾⠋⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⡀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠀⠀⢀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠶⠶⠶⠶⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣦⣶⣶⣤⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣤⣴⣶⣤⣤⣶⣦⣦⣴⣿⣦⣤⣄
⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⣿⣇⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣸⣷
⠀⠀⠀⠀⣿⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢈⣿
⠀⠀⠀⢀⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡀
⠀⣴⠾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢷⣄
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠛⠉⠟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣠⣤⣤⣤⣤⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢨⡀⢠⠀⢸⠀⠀⠑⢄⠈⢯⡰⠖⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⣴⠃⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⣀⢄⠀⠀⠘⠂⢀⠾⠂⡇⠀⠀⡠⠊⠀⠀⡀⠀⢃
⠀⠉⠁⠈⠁⢸⡀⠀⠀⢰⠀⠀⠙⠆⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⣀⠔⠁⡠⠊⠀⡠⢊⡀⠀⢀⠜⠋⠀⠀⠑⢌
⢤⡀⠀⠀⠀⠈⠁⠀⠀⣤⠀⠀⠀⣤⠀⠀⢄⣀⣉⡀⣀⣀⡀⣀⣀⣀⣀⣀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠁⠀⢹⠁⠈⢈⡠⠊⠀⡠⠊⡠⠊⢁⠔⢁⠜⠋⠀⣄⠀⢀
⢈⠁⠀⠀⠀⣀⠀⢀⡀⠉⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠠⠶⠄⠀⠀⠴⡆⠀⠀⠸⠇⠸⠀⢀⠋⠀⠀⠊⠀⠊⠀⠶⢃⠔⢁⠜⠛⠀⠈⠢⡘
⡀⠡⢤⠀⠀⠉⠢⡈⠑⢄⠀⠀⠆⠆⠀⠀⠀⠶⠂⠀⠀⡔⠆⠈⠀⠀⠀⠶⠂⠀⠀⠰⠶⠀⣀⡀⠀⢀⠖⠁⡠⠛⢡⠄⢀⠀⢸⠀⠀⡶⢃⠔⢁⠜⠋⠐⠿⡀⠈
⠈⠢⡈⢴⡔⠳⠀⠈⢶⡀⠱⣤⠀⠀⠀⢀⡀⠀⠀⡠⠊⠀⢀⠄⠀⠀⠐⡖⠀⠀⠀⠀⠀⠀⠛⣀⠖⠁⡠⠊⢀⠔⢁⡰⠀⡠⠊⠀⠀⠶⢃⠔⢁⢤⠀⢀⣤⠀⠢
⠢⡀⡇⢈⠀⠀⠀⠠⢀⣀⣀⣀⣀⣀⡠⠚⠀⠘⠙⠀⢀⠔⠁⠀⠀⡶⠀⠃⢰⠆⠀⠀⡀⠀⠘⠃⡠⠊⢀⠔⠁⡀⠈⡠⠊⠀⠀⠀⠀⠶⠁⠀⠀⡍⠀⢠⠈⠑⢄
⠀⠄⡇⢐⠀⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀⠀⢠⠔⠁⠀⠐⡿⠀⠀⠀⠀⢰⠀⠀⠐⠚⠁⠀⠉⢀⠔⠁⡠⠈⡠⠊⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠈
⠀⡃⡆⠘⢁⠔⠁⡠⠄⠀⠀⠀⠀⠀⠢⡀⠘⠦⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡄⠀⠐⠖⠒⠒⠒⠁⠰⠂⡠⠊⢀⢀⠉⠉⠀⠀⠀⠈⠈⠢⡀⠐⠙⠀⢀
⡠⠂⠆⠘⠁⠀⠘⠁⡠⠟⠂⠀⠀⢠⡠⠘⠃⠀⠀⠀⠀⠀⠀⠇⢀⠄⢀⠄⠈⢁⠔⠒⠒⠒⠛⠘⠙⡠⠊⠀⠀⠈⡉⠀⠀⠀⠀⠀⠉⠢⡀⠈⢶⡄⠀⠘⠀⡠⠊
⢀⡴⠃⠀⠀⠀⢠⠊⠀⠺⠂⠀⡴⠁⠀⠀⢰⡤⠤⠤⠤⠄⢀⠔⢁⡔⠃⢀⡔⠁⠀⠀⠀⠀⠀⡠⠊⠀⡲⠀⠀⠈⠊⠉⠉⠉⠉⠉⠑⢀⠈⢶⠄⠀⠀⠐⠟⢀⡀
⠋⠀⡠⠀⠀⠂⠀⠀⠀⢸⠀⠊⠀⠀⠀⠀⢰⡤⠤⠤⠤⠔⢁⠔⠁⠠⡴⠁⠀⠀⢀⠦⠀⠰⠎⢀⠀⠀⠀⠀⣶⠀⠀⠀⠀⡠⠗⢀⡀⠀⠁⣆⡈⠏⠀⠏⠀⠁⢁
⡠⠎⣠⠀⢀⡀⠀⠀⡀⣴⡀⢄⠀⠀⠤⠤⠀⠀⠀⠀⠀⠄⠁⠀⡴⠂⡤⠆⢀⠔⠁⠀⠀⠀⡠⠓⠁⠀⠀⠀⠀⠀⢀⡠⠊⠀⡠⠊⠁⠀⠀⠀⠀⡇⠀⠀⠀⠰⠋
⡀⠌⠁⡠⠋⢠⡒⠀⠈⠢⡀⠀⠑⢄⠀⠀⠀⠀⡠⠄⠤⠤⠤⠊⡠⠊⢀⠔⠁⡠⠏⢀⡠⠊⠀⡶⠆⠀⠀⠀⠀⠀⠘⠀⡠⠊⠀⠀⠀⠀⠀⠀⡠⠃⣠⠂⠀⠄
⠀⠰⠋⢀⠔⠁⠀⠀⠀⠀⠘⠗⠀⠈⠛⢀⡤⠊⡠⠤⠤⠤⠤⠊⢀⠔⢁⡠⠊⠀⠀⠀⠁⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠰⠆⠀⠀⡌⠀⡞⠁⠀⠠⠚⢁
⠀⢀⠔⠁⠀⠀⠀⡠⠠⠄⠠⠴⠖⠀⠀⢀⡥⠊⠀⠾⠀⢦⠂⠀⡃⠀⠈⠀⠀⡀⠀⠀⠊⢀⠶⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡤⠀⠀⠀⠀⡇⠀⡇⠀⠀⠇⠀⠈
⠀⠀⠀⠀⠀⡠⠊⠀⡠⠒⠀⠂⠀⠚⠃⡀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⡠⢊⡀⠀⢀⠔⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⢉⡐⠈⢀⠔⠀⠀⠀⠃⣠⠃⣠⠀⡆
⠀⢘⠠⡶⠊⠀⡠⠊⠀⣀⠄⠁⠀⠀⠈⡁⠀⠀⠀⡄⠀⡀⠀⡠⠥⠤⠊⡠⠋⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⢀⠀⠀⠀⡠⠊⡠⠎⣁⠊⣡⠚⠀⠀⠐
⠀⠀⢄⠀⠰⠎⠀⡠⠊⠀⠀⠀⠀⣠⠈⡎⠀⠠⠊⠠⠊⠠⠊⡠⠂⠈⠉⢀⠔⠁⠀⣤⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⡌⠀⠜⠀⡌⢠⠎⠀⠀⢀⠄
⠒⠂⢄⠱⡄⠀⠫⠁⠀⠿⠀⠀⠀⠛⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⣒⠀⠈⠠⡢⠈⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⡠⠃⡀⠆⡠⠃⠀⠀⢀⠔⢁⠐⠒
⠀⡀⠀⠑⢄⠀⠀⠀⠀⠠⠀⠾⠀⠀⠠⡣⠀⡄⠀⡄⠀⡀⠀⠀⠀⠀⢠⢄⠀⠀⠀⣀⠀⡇⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⢔⠎⢐⠆⢔⠞⠁⠀⠠⡲⠁⡲
⠀⠙⢦⡀⠀⠱⡠⠀⠀⡀⠀⡄⠀⠻⠀⡠⠊⡠⠊⡠⠊⡠⢃⣀⠀⠀⠈⡁⠀⢀⠀⠛⠀⡇⠠⠀⠀⢀⠀⠀⠀⠀⠀⠐⢆⠀⠒⢆⠀⠀⠲⡀⠶⡀⠲⡄⠲⡀⠲
⠢⡀⠀⠙⠦⡄⠀⡠⠊⡠⠊⠀⠀⠠⠊⠠⠊⠠⢊⡤⠊⠀⠀⠁⢀⣄⠀⡈⠐⡀⠱⣄⠈⠊⠀⠑⢄⡀⠯⠆⡀⠀⠠⠀⠀⠑⢄⠀⠑⢄⠀⠈⠢⡈⠢⡈⠢⡈⠢
⠢⢌⢢⢄⠈⠠⠊⠠⠊⢦⠂⠀⠀⡀⠀⡘⠀⡄⠀⣁⠀⠀⠀⠀⣈⠀⠀⠈⠂⡄⠓⠈⠱⢄⣀⣠⠄⠋⡠⠊⡠⠊⢀⡐⠶⡀⠈⠛⠀⡈⠛⠀⠀⠈⠢⢌⢢⢌⠢
⠀⠁⠀⠁⠀⡄⠀⡄⠀⡐⢠⡤⢊⡤⢊⡤⠊⢀⡀⠁⠉⢉⠉⠉⠉⠀⣀⠀⠀⠈⠁⡈⠳⠄⠀⢈⣡⠊⡠⠊⠀⠀⠘⠓⠄⠈⡆⢀⠔⠊⠀⠀⢀⣀⣀⡁⣀⣁⣀
⠀⠀⠀⢠⠊⠠⠊⠠⢊⢀⠀⠀⠀⣀⠀⠀⣤⠈⠙⢄⠀⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⣀⠀⣠⠀⢈⣡⠊⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢀⣤⣀⣉⣀⣀⣀⣈⣉
⡀⠢⡀⢸⠀⠀⡀⢠⠈⠁⢀⣄⣀⣀⢁⣀⠍⠳⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⠌⠀⠋⣀⠁⠀⠈⠁⠀⠀⠀⠀⠀⣀⣄⢸⠀⡇⠀⠀⠚⠆⠁⢀⢤
⡈⠢⡁⠢⠙⢢⠐⢄⠑⢤⡈⠉⠀⠀⠀⠈⠁⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠠⠊⠀⡠⠤⠄⠤⠴⠂⠀⠀⠀⠀⠀⢸⠀⡇⢀⠀⠀⠀⠑⢄⡀⠀⣤⡄
⠈⢂⠈⠢⠈⢀⠘⠆⠑⠄⠑⠄⠠⡢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠊⠀⠠⠶⠀⠀⠀⠰⠆⠀⢠⡖⠁⢀⠂⡀⠑⢄⠀⠀⠈⠋⠀⢀⣀⠀⠠
⠀⢸⠀⠀⠀⢸⠀⢀⠀⢠⠀⠤⠀⠠⠶⠀⠂⠀⠲⠆⢀⠜⠃⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠠⠴⠄⠀⠀⠀⠀⢀⡀⠎⠁⠈⡎⠀⠀⠻⠂⡴⠀⠀⠀⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠶⢾⡷⠲⠶⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⢛⣩⡥⠶⠚⠛⣋⣻⣟⣙⠛⠓⠶⢤⣍⡛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡴⠞⢉⣤⣴⢖⣛⠉⠉⣽⣏⠉⠉⣛⡳⢦⣤⣉⠳⢦⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣩⡾⢋⣤⠞⠋⠀⠀⠀⠀⠙⡂⠉⠉⢐⠋⠀⠀⠀⠀⠹⠳⣤⡙⢶⣌⢷⣄
⠀⠀⠀⠀⠀⠀⢀⣀⠀⣼⢋⡴⠋⣴⠟⠁⢸⠀⠀⠀⠀⠀⢸⣷⣶⣶⣾⡇⠀⠀⠀⠀⠀⡇⠈⠻⣦⠙⢦⡙⣧⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠛⣾⢃⡞⢡⡞⠁⠀⢀⠀⢧⠀⠀⠀⢠⣼⣿⣿⣿⣿⣧⡄⠀⠀⠀⡰⠀⡀⠀⠈⢳⡌⢳⡘⣷⡛
⠀⠀⠀⠀⠀⠀⠀⣼⠇⡾⢀⡿⠁⠀⠀⠈⠉⠉⠳⡤⣀⠀⠛⣿⣿⣿⣿⠛⠀⡀⢤⠞⠉⠉⠁⠀⠁⠈⢻⡄⢷⠸⣧
⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⣼⢣⠀⠀⠀⠀⠀⠀⠀⠙⠂⠁⢀⣽⣿⣿⣯⡀⠈⠰⠋⠀⠀⠀⠀⠀⠀⠀⡜⣷⠘⣇⢻⡄
⠀⠀⠀⠀⠀⠀⢸⡇⣿⢠⡇⠀⠣⡀⠀⠀⠀⠀⠀⢤⠶⣦⣄⠻⣷⣾⣟⣠⣴⠶⡤⠀⠀⠀⠀⠀⢀⠜⠁⢸⡄⢿⢸⡇
⠀⠀⠀⠀⠀⠀⢺⡿⡿⢾⡿⠾⠿⢿⣿⠿⠏⣠⡘⣦⢷⣼⣿⣿⣝⣫⣿⣿⣧⡾⣰⢣⣄⠹⠿⣿⡿⠿⠷⢿⡷⢿⢿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠘⡇⠀⡠⠊⠀⠀⠀⣿⣿⣜⣿⣿⣯⠻⣿⣿⠟⣹⣿⣿⣣⣿⣿⠀⠀⠀⠑⢄⠀⢸⠃⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣧⢸⡄⢻⡀⠁⠀⠀⠀⢠⡿⢸⣿⣿⣿⣿⣶⡌⢡⣶⣿⣿⣿⣿⡏⢿⡄⠀⠀⠀⠈⢂⡟⢠⡏⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢻⡆⢷⡈⢿⣄⣀⠀⠀⣸⡏⠧⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⡼⢻⣷⠀⠀⣀⣠⣿⠁⡾⢰⡟
⠀⠀⠀⠀⠀⠀⢀⣤⢿⡌⢷⡈⢷⡀⠈⠀⣿⣿⣶⣦⣿⠿⢿⣿⣿⣼⣿⡿⠿⣿⣶⣶⣿⣿⠀⠁⢀⡾⢁⡾⢡⡿⣴⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠻⣆⠳⣄⠻⣦⡀⠹⣿⣿⣿⣷⣶⣢⣌⣱⣎⣡⣜⣶⣾⣿⣿⣿⠏⢀⣴⠟⣠⠞⣱⡟⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠳⣌⠛⢦⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡴⠛⣡⠾⢡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠳⢦⣌⡙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⣁⡴⠞⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣬⣙⡓⠶⢤⣤⣍⣿⣿⣩⣤⡤⠶⠚⣋⣥⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⠦⠾⠷⠴⠶⠶⠞⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣼⣤⡀⠀⠀⣀⣠⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢿⡿⠿⠀⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⠦⠠⠀⠘⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⠛⠒⠲⠶⠦⠤⢤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⣠⠜⠋⠉⠉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢠⣤⠀⠀⠀⣾⠁⠀⠀⣠⣤⣄⠀⠁⠘⢷⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⢸⡇⠀⠀⠉⠀⠀⠉⠱⠄⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠉⠀⠀⠀⠀⢠⠀⠀⢰⠇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⣠⣶⡾⠻⣄⠀⠀⣼⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘⣿⠋⠉⠀⠀⠈⠛⠦⢤⣤⠴⠶⠚⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠈⠰⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠶⠶⠶⢦⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠷⠶⠶⠶⣦⣴⠶⠛⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⡽⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣡⡾⢗⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢗⡽⣋⣴⢟⡼⣛⣿⢿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠻⠟⢃⡿⣹⣿⣿⣿⢾⣯⣵⣿⣟⣻⢟⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠠⠋⠀⠉⠉⠉⠰⡟⠩⣠⢟⣽⡫⢞⠽⣻⢟⡿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠎⠀⡰⠿⡾⢷⡾⢿⣿⣶⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠖⠩⣶⡽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣫⣶⢄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⢿⣿⣯⢂⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣯⢿⣿⣷⣻⣿⡿⣣⠄⠀⠀⠀⠀⠀⠀⠠⣺⢟⣻⡿⢛⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠷⠾⠿⡷⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠔⠁⠊⠐⠽⣋⣴⣡⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡿⣀⣨⢅⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣠⣶⣿⣿⡾⣷⡴⢖⡠⠀⠀⠀⠀⢠⠿⣿⣿⣿⣿⣽⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣽⣿⣽⣿⣿⣷⠁⠀⠀⠀⠀⢐⣵⣾⣿⣽⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠛⠛⠛⠛⠛⠉⠙⢻⡟⠁⠀⠀⠀⠀⠀⠀⢉⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⣠⣾⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋

⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⡿⢏⠐⢌⣿⡿⠋
⣿⣿⣿⡿⢛⠐⢂⣷⡿⠋
⣿⣧⡘⠀⣉⣼⡿⠋
⣿⣿⣽⣶⡿⠋
⠱⢿⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣴⣶⣶⡄⠀⣴⣶⣶⣦⠀⠀⠀⢰⣶⣶⣶⣶⣶⣦
⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠿⢿⡿⠿⠿⠿⠿⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⣧⣤⣤⣤⣤⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠗⠀⣶⣶⣴⣶⣶⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣀⣴⣆⣶⣰⣆⡀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣿⠀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠠⣴⣿⠋⠉⠉⠉⢻⣿⣦⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠶⣿⠀⠀⠀⠀⢸⣿⠶⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣴⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠐⠻⣿⣤⣤⣤⣤⣸⣿⠟⠂⠀⠀⠀⣿⣿⣿⣿⣿⠀⠈⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠾⠛⠉⠀⠀⠉⠻⠏⠿⠹⠏⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢾⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠙⠛⠛⠀⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⢀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢰⣶⣶⣶⠀⣶⣶⣶⣦⠀⣶⣶⣶⡆⠀⣿⡁⢈⣿⢸⣿⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⠛⠛⠛⠀⠛⠛⠛⠋⠀⠛⠛⠛⠃⠀⠘⠛⠛⠋⠈⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠈⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⣀⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠄⠀⠀⠀⠂
⠀⠀⠀⢀⠀⠀⠀⢐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠤⠤⠤⠤⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂⠂⠂⠂⠐⠐⠐
⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⢀⠤⠀⠠⠠⠠⠀⠄⠠⠀⠄⠀⠄⠄⠠⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣐⣂⣂⣐⣐⣐⣀⣀⣂⣀⣒⣒⣐⣐⣀⣒⣐
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣪⣵⡾⠷⢶⡭⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣹⡟⠁⠀⠀⠀⠙⡌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣏⣓⢦⡀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣪⡷⠿⣻⣿⣸⠛⣇⠀⢀⣟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣽⠞⠉⠀⠀⢈⠝⣡⣾⣿⡤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣪⠖⠋⠀⠀⢀⡤⣚⡵⠞⣽⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⣭⠞⠁⠀⢀⡠⢖⣽⠞⠉⢀⢾⠏⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠵⠋⠀⠀⣠⣖⣭⠞⣻⠏⠀⢀⣾⡟⠀⣸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⣠⣔⡿⠞⠉⠀⣸⠏⠀⢀⣮⡟⡇⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠀⣠⣴⡿⠞⠉⠀⠀⠀⣴⠏⠀⢀⢮⠟⠘⡇⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣾⠟⠁⣠⣴⡿⠟⠉⠀⠀⠀⠀⠀⣸⠏⠀⢠⢫⠏⠀⡆⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⣀⣴⣾⡯⠵⠒⠲⣼⡿⣅⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⣰⢣⠏⠀⠀⢱⡇⠀⣧⡇
⠀⠀⠀⠀⣠⣞⡵⠛⠁⠀⠀⠀⠀⡸⢠⢟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣰⣣⠏⠀⠀⠀⢸⡇⢠⢻⠃
⠀⠀⢀⢞⡵⠋⠀⠀⠀⠀⠀⠀⢰⠃⡞⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣰⣳⠏⠀⠀⠀⠀⢸⡇⢸⢸
⠀⢀⣏⡞⠀⠀⠀⠀⠀⠀⠀⣰⢃⡜⢁⣿⡟⠀⠀⠀⠀⠀⠀⢀⣠⠷⢷⣴⣿⠏⠀⠀⠀⠀⠀⢸⡇⢸⢸
⠀⢰⡿⠀⠀⠀⠀⠀⠀⢀⠔⡡⠋⡠⣿⡟⠀⠀⠀⠀⠀⢀⡴⣫⠴⠚⢯⢹⣿⣆⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠈⢿⣄⠀⠀⣀⡠⢖⣵⣪⠴⣫⡾⠋⠀⠀⠀⠀⠀⢀⣞⡼⠁⠀⠀⢸⢀⡇⢻⡆⠀⠀⠀⠀⢸⡇⠸⣿
⠀⠀⠀⠙⠛⠿⠿⠿⠭⠥⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⡼⣸⠁⠘⡇⠀⠀⠀⢠⣾⠽⢿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⣰⣷⠇⠀⣸⠃⠀⠀⢠⣿⠃⠀⠈⣿⡜⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡆⠀⢀⡴⡿⠋⢀⣴⠃⠀⠀⠀⢸⣿⠀⠀⠀⢹⡇⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣽⠮⢴⣺⠟⠁⠀⠀⠀⠀⢸⣿⡀⠀⠀⣸⠃⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢧⣣⣀⣴⣟⣴⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣉⣁⣀⣀⣀⣀⣀
⣿⣿⠋⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⣤⣄⣀⣀⣠⣤⣶⡶⠷⠦
⠀⠀⠘⣠⣾⣿⣿⠏⠀⠈⠉⠛⠛⠛⠛⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⣼⣷⣿⣿⣿⢶⡆⢰⣶⠀⣲⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣷⣷⣾⣶⣿⣿⣿⣷⣶⣤⡀
⠀⠀⢸⣿⣿⣿⣿⡝⠃⠈⠉⠤⠉⠉⣽⠛⠛⠛⠛⠛⠙⠙⠋⠋⠋⠛⠛⠛⠋⠋⠋⠉⠉⠙⠻⠿⠿⠿⠿⠛⠁
⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⣀⣠⣀⡀⠙⢤⣀⡀
⠀⠀⠀⠘⠻⢿⣥⠤⠖⠊⠁⠀⠀⠈⠉⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⡴⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡼⠜⣵⣞⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⠲⡎⠀⡠⡀⠉⢥⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⡿⠐⣤⠀⠐⠄⢀⣺⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣜⢰⠄⠸⠀⠈⠋⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠃⡂⠀⠀⢀⠀⠀⢘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⢀⠎⢀⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢡⠀⠇⡀⠎⠀⣴⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⠀⠀⠀⠀⠀⠀⠀⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⠤⢐⡏⢸⠀⠉⠉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⢈⢶⡏⠳⠏⠀⠀⠀⡀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠉⠓⢌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⢀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡉⠈⠃⠀⠀⠀⠀⠀⢸⠓⢤⡀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠓
⠲⢖⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡌⠘⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠙⠦⡀⠀⠀⡼⠂⣨⠄⡄⠀⠠⠒⡄
⠀⠀⠸⠶⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⢀⠔⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠘⠤⡞⠀⠀⡐⣊⡱⠀⠀⠀⠇
⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠘⡄⠖⠤⠬⠃⠀⣆⢰
⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠈⠱⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⡠⠄⣀⠀⠀⠀⣀⠠⠀⠹⡀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢫⣦⠔⠡⠊⠀⡀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡔⠶⠀⠀⠀⡲⣜⢂⡿⠀⠀⠦⣄⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⢀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠈⠉⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⢀⠙⠋⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠥⠒⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡎⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠆⠀⠀⠀⡰⠁⠀⠀⠀⡀⡠⠂⡌⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⡜⠀⠀⠀⠀⠐⠟⠀⡜⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢀⡠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣥⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⠎⠀⠰⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⣀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⠎⠀⠀⡇⠀⠀⠀⠀⢠⠁⠀⡠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡜⠁⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⡜⠀⠀⢰⠀⠀⠀⠀⠀⡞⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⡶⠠⢄⣀⡰⠁⠀⠀⡆⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠉⠛⠦⡀⣇⠀⠀⠀⠀⢣⡄⢀⣀⡀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠈⠻⠄⠊⠀⠀⣞⣷⢦⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠂⠐⠁⠀⠀⠀⣏⠷⡱⢎⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣚⠙⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠉⠒⠤⡀
⠀⠀⠀⠀⣴⣿⣷⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠃⠀⠀⠀⠀⠀⢴⣄⣢⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣸⢽⣷⡿⠛⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⢸⢦⡄⠀⠀⠀⠘⠋⢀⡀⢈⣷⣀⡄⠀⠀⠀⠀⠈⣰⡿⢀⣤⡀⠀⣀
⠀⠀⠀⠀⠀⠹⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⢠⡴⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠉⠀⠾⠏⠠⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠤⠤⠴⠶⠒⠒⠚⠛⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀
⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⢀⣀⠈⣭⣾⣿⡷⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⢸⣿⠀⠹⢻⣽⣄⠔⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣲⣭⣽⣷⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⢿⠀⠜⠉⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢹⣿⠛⠛⠉⠉⠁⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⣹⣿⣯⣍⠉⠁⠀⠀⠀⠀⠀⠀⡜⠉⠙⣷⠀⠀⠀⠘⠋⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠶⠾⢋⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣀⡰⠁
⠀⠀⠀⡠⠊⠀⠂⢀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡀⠀⠀⠀⠛⠓⠒⠒⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⣹⡿⠁
⠀⠀⡎⠀⠀⠀⠠⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠦⠞⢀
⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠙⠋⠉
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢦⣤⡶⠤⣤⡀
⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⢰⣶⣄⣤⣴⣖⣲⣶⡖⡶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡄⠀⣤⣤⠀⠀⠀⣴⣶⣦⠙⣿⣾⡇
⠀⠀⠀⠚⠋⠉⠁⠀⠀⠀⠰⠶⠶⠶⠶⠿⠷⠷⠷⠷⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠓⠒⠚⠛⠛⠛⠀⠙⠛⠀⠀⠀⣏⠈⣹⠗⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣿⣿⣿⠿⣿⠿⢿⣿⣿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣷⣮⣶⣶⣴⣷⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠠⠖⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⠴⠚⠤⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠙⠋⠲⠶⣦⣄⣀
⠀⡠⠠⣭⡀⠀⠀⠀⠀⠄⠠⣀⠀⠈⠀⠛⠁⠠⠠⣦⣤⡀⠀⠉⣡⣶⡀
⠀⢀⣤⣴⣶⣶⣶⣤⣤⣤⣀⣲⡤⢌⣉⡐⠒⠢⢄⠀⡌⣫⡔⠛⠋⠹⣿⡄
⠀⢸⣿⣿⠟⠛⠻⠿⠿⠿⢿⡿⠿⣿⣷⣾⣭⣓⢶⣴⡜⠇⠀⠀⠆⠀⠹⣿
⠂⣾⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⠿⣿⣧⡹⣿⣎⠀⠀⠀⢀⣀⠀⠠
⡀⡇⠀⠀⠀⢀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⢹⡟⠀⠀⣴⢻⡿⠀⠀⠄
⡀⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢿⣦⡀⢹⣆⣧⢀⢠⡄
⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⡿⠇⢀⡿⠈⠰⡌⡇
⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢁⢻⠀⢸⡗⡀⠀⢹⡟
⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠘⠛⡄⡜⡇⠇⠀⣼⠁
⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⣠⠙⢱⡇⣆⠔⠁
⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⡅⠈⣸⣿⣿⣆
⠀⠠⢌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣠⡶⠟⢝⡯
⠀⠀⠀⠀⠉⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣦⠔⠁⠀⠈⠁⠒⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⢙⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⢄⣀⠀⠀⠸⣿⣿⠗⠲⠤⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠋⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀⠶⠀⠀⠀⢁⣒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⢻⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢋⣠⠖⠩⠝⠿⠟⠻⠟⠛⣷⣄⣹⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠃⠀⢠⡄⣠⠀⡦⠀⠀⠘⠉⢻⣟⢦⣉⡓⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡀⠀⠀⠘⠇⠀⠀⠀⠀⢀⣴⣿⣿⣆⠙⠯⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠦⠔⠀⠀⣠⣞⠛⠻⠛⠋⠳⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠑⢶⠠⠤⢒⡏⠁⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠈⠒⠒⠋⠀⠀⠀⠀⠀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⠔⠳⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣋⠉⡇⠀⣢⣼⣷⠷⠾⠿⠿⢿⣶⠶⠾⠵⣦⣄⠀⢀⡴⠿⠴⠦⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡞⢳⡞⠀⠈⢲⡽⢮⡇⡀⠀⠀⠀⢿⢫⢢⢀⠄⢈⡿⠚⠉⠀⠀⠀⢀⡴⣫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠈⠁⠀⠀⣠⠋⠤⠄⠀⠒⠂⠐⠀⠈⡻⠚⠳⠊⠁⠀⠀⠀⠀⢀⡴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣟⣒⠒⠒⠒⠒⠒⠶⠤⠭⠭⠭⣭⣍⣙⣛⣒⣒⣚⣁⣀⡀⠀⠀⠀⠀⠀⡰⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠉⠛⠒⠒⠒⠒⠒⠶⠶⠤⠤⠤⠤⠭⢭⣭⣍⣩⣹⡟⠁

⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣠⣿⡿⠋⠁⠈⠉⡹⠿⣷⣶⡶⠿⢿⣿⣿⠿⠿⠟
⠀⠀⠀⢠⣿⣏⡀⠀⠀⠀⡐⢉⡁⠉⠉⠉⠟⠋⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠐⠆⡄⣀
⠀⠀⠀⢸⣿⣋⠁⠿⠽⠿⠿⡿⠿⠉⠀⠉⠹⣯⡽⠭⠭⠭⠭⠭⠯⠭⠭⠽⠭⠭⠭⠯⠯⠯⠽⠭⠯⠿⠿⠅⠏⠐⠰⠊
⠀⠀⠀⢸⣿⣾⣆⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠸⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠸⣿⣿⣫⣤⠀⠤⢈⡠⠖⠋⠉⠉⠒⠢⠤⠄⠃
⠀⠀⠀⠀⠈⠻⠿⠿⠦⠒⠉


⣿⣿⢟⠿⢛⠿⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⣶⡂⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣠⣀⡀⡀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣠⡤⠶⣤⡀
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣾⢻⣿⣿⡄⣬⣻⣯⡻⣧⠀⠀⠀⠐⠿⣿⣼⣿⣽⣿⡦⠀⠀⠀⠀⠻⢿⡀⠊⣸⡿⠇
⠀⠀⠀⠠⡃⢀⠄⠀⠀⠀⠀⠙⠓⠟⠛⠛⠛⠛⠿⠛⠊⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⡞⣹⠂⠀⠀⠀⠠⠋⠈⢧⠀⠀⠀⠀⣀⢀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⢠⠤⠤⢤⡄
⠀⠀⠀⢠⣞⢻⠁⠀⠀⠀⠀⠈⢆⢄⠎⠀⠀⠀⠀⣿⢽⣿⣿⣯⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠀⠀⠀⠀⠀⣷⣶⣀⢿
⠀⠀⠀⣾⢿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⢿⢿⣷⡿⣤⣤⡄⠀⠀⠸⣿⡿⠃⠀⠀⠀⠀⣤⣶⢉⣙⣛⡾
⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⠒⠒⠒⠒⠒⡄⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡀⠀⠀⠀⠀⢐⢐⠄⠀⠀⠀⠀⠀⠀⣴⣤⣿⡿
⠀⠀⢸⠀⠀⠀⠀⠀⡆⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠿⠀⠀⠘⣿⠇⠀⠀⠀⣴⠺⢏⣤⠀⠀⠀⠀⠀⢤⣰⣈⡿⠁
⠀⠀⠈⠒⣒⣶⣒⠒⠁⠀⢀⠀⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠟⠻⠿⠻⠀⠀⠀⠀⠘⠳⠦⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠔⠉⢢⣠⣤⣤⡄⠀⠀⢀⡀⣀⣤⡄⠀⠀⠀⠀⣀⣀⣶⣶⢀⠀⠀⠀⠀⣾⠁⠙⡆⠀⠀⠀⠗⢋⣄
⠀⠀⠀⠾⢀⡴⢿⠿⠿⠛⠁⠀⠀⠸⣿⡷⣿⣶⠀⠀⠀⠀⣿⣿⣿⣋⣿⡇⠀⠀⠀⢻⣠⣴⣳⡆⠀⠀⠀⢫⢀⠇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠉⠁⠉⠉⠈⠀⠀⠀⠀⠈⠓⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⠀⠀⠀⡠⠐⠒⢤⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⢀⣴⣾⣿⣶⣆⠀⠀⠀⠀⠀⣴⡉⢿⣳⡄⠀⠀⠀⠀⡸⢁⣴⠃⠀⠀⠀⢸⠀⠀⠀⠀⠡⠀⠀⠀⢘⠠⡀
⠀⠀⠀⢯⣿⣏⣿⣿⣿⡇⠀⠀⠀⠀⢿⣷⠐⣹⠃⠀⠀⠀⡐⢻⠿⡕⠀⠀⠀⠀⢸⣿⠀⠀⣸⡷⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠘⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠃⠀⠀⠀⠀⠀⠀⢌⠁⠀⠀⢸⠀⠀⠦⠀⠀⠀⠀⠀⠐⠂⠀⠃⠀⠀⠀⠀⢠⠀⠀⡀
⠈⠀⠀⠈⠠⠀⠀⠀⢀⠀⠀⠆⡀⢸⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠃⠠⠂⠀⠁⠀⠠⢆⠐⠀⠀⠰
⠀⠰⠷⠀⠀⢀⡄⠀⠈⠀⠀⠂⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⢰⠀⡄⠀⠀⠀⠀⠀⠐⠀⠀⡆⠀⠀⠄⠀⢀⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠂⠇⠀⢸⠀⠀⢀⠢⡀⠑⢌⠊⡃⠑⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⡠⠊
⠉⠐⠂⠀⠀⠀⠆⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇⠀⠀⠀⠀⠈⠀⠁
⠀⠠⠀⠈⠢⡁⠀⠈⠂⠀⠀⠃⠠⡀⠰⠀⠌⢀⠐⠂⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⠀⠀⠆⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠐⠄⠈⠀⠀⢀⠀⠢⠀⠀⢀⠘⠀⡄⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠑⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠀⠀⡃⠀⠀⠀⠀⠀⠀⠐
⠀⢀⢀⡈⠉⢀⠀⠈⠀⣁⠀⠈⢁⠀⡀⢁⡀⠁⠀⠀⡀⠀⠀⠀⠀⠤⠀⠠⠆⡠⠊⢀⠀⡆⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢔⠀⠀⠘⠀⠀⠀⠀⠀⠀⠊⢀⠔⠁⠀⡁⠠⣀
⠀⠈⢁⡍⠁⠉⠁⠈⠈⠈⠀⠀⠀⠈⢨⠐⠌⠉⠪⠀⠆⠠⠌⠀⠤⠀⠘⠀⠀⠀⠀⠀⠀⠁⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠠⠀⠀⠙⠀⠂⠡⠀⠄⠤⡀⠀⠀⠃⣄⠀⢀⡀⠀⠀⠀⠀⠀⢀⠄
⠄⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠙⠀⠀⠈⠀⠀⠁⠀⠀⢀⠀⢀⠄⠀⠀⠂⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠔⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠎⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠤⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢⡀⠑⠀⠆⠀⠀⡀⠈⢂⠘⠁⠃⠀⠀⠀⡀
⢀⠀⠀⠀⠻⠃⠀⠀⢘⡀⠀⠂⠀⠀⠀⠀⠀⠐⠰⠂⠀⠁⠈⠀⠀⠀⠀⠈⠁⠀⢘⠀⠀⠂⠰⠀⠀⠁
⠤⠤⠭⠭⠭⠭⠭⠥⠄⠤⠈⠀⠀⠀⠀⠄⠤⠤⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡄⠁⠀⠀⠀⠔⢡⢀⠀⢀⠄⡀⠐
⣀⡀⡀⠀⣀⣀⢀⡀⢀⠀⠀⡀⢀⠀⣀⡀⠐⢄⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡂⠀⠈⠀⠁⠈⠀⠠
⠀⠀⠀⠀⠀⠀⢄⠈⠈⠑⠀⢤⠠⡀⢄⠈⠢⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠘⠂⠄⠀⠀⡀⢀⠉⠈
⠀⣀⠀⠁⠤⠠⠀⠁⠈⠀⠀⠆⠀⢸⠀⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡐⢌⠂⠀⡀⠀⠀⠁⡁⠀⠂⠐
⠀⠀⠀⠀⠂⠀⠘⠂⢐⠀⠀⡀⢀⠀⠀⢬⠂⡘⠀⠀⣀⢀⢀⠀⡀⠀⢄⢂⠀⡁⢸⠀⠀⡁⠀⡀⣄⠀⡀⢀
⣿⡯⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠖⠀⠀⠒⠀⠀⠖⠀⠀⠒⠀⠂⠀⠀⠒⠒⠒⠢⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣁⠐⠁⣠⠘⠀⣠⠂⡀⠈⠁⣈⠁⠉⠉⠈⠁⢘⠉⠉⠢⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢡⠆⠉⡠⠊⠀⡸⠀⣀⠀⡇⠀⠀⠁⠀⠃⠀⠦⠀⠈⠢⡀⠀⠈⠢⡈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠆⠚⠀⡠⠊⠠⠆⠀⠀⡇⠀⠀⡀⠀⠀⢠⠄⠀⠀⠀⠈⠲⠀⠀⡀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⡁⠊⡐⠂⠠⠎⡀⠈⠀⠀⡀⠀⠇⢀⠀⠉⠀⠇⢀⠘⠃⠀⠀⠀⠀⠠⠆⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠎⡠⢊⠔⠃⠠⠤⠥⠤⠤⠊⠀⢀⠔⠁⡠⠊⠁⠁⠉⡠⠊⢉⠉⠉⠉⠉⠉⡉⠑⣄⠑⢄
⠀⠀⠀⠀⠀⠀⢀⡀⣁⠐⢋⠔⢁⠶⠀⠰⠆⠀⠀⠀⢀⠔⢁⡠⠊⠀⠀⡀⠰⠎⠀⠖⠁⠶⠀⠀⠀⠀⠈⠖⠀⢀⠀⠑
⠀⠀⠀⠀⠀⠀⠈⣁⠉⠈⢃⠔⠁⠐⠒⠀⡀⠀⡂⠐⠁⣀⠈⡁⠀⡀⠚⠁⠀⠀⡰⠄⠀⠀⠀⠃⠀⡠⠊⢀⠤⠭⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠊⢁⠄⠐⠁⠀⢀⠜⠁⠀⠁⠀⠁⠠⠀⡠⠊⢀⠎⢀⠶⢀⡠⠊⢠⡄⠀⠀⠀⠀⠊⢀⠔⠁⠠⠤⠤⠤⠤⠄⡖
⠀⠀⠀⠀⠚⢀⠔⣁⠀⣀⠠⠖⠁⢀⡄⢀⠀⠀⠀⠠⠊⠠⠊⣀⠔⠁⠀⠈⢁⠔⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⡆⢀⠂⠠⢓⡀
⠀⠀⠀⠀⠚⠁⢀⠄⠀⠉⠠⠆⠀⠠⡔⠁⠰⠂⠲⠀⠀⠀⢀⠉⠀⠠⠀⠐⢁⠴⠀⣀⠀⠀⠀⠀⣄⠛⠀⠠⠊⡰⠄⡇⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠚⠁⡠⠂⠀⡀⠀⠛⣀⡀⠀⢸⠀⢸⠀⢰⠄⠉⡠⢡⡄⢀⠐⠁⠠⠄⠉⠀⠀⠀⠀⠁⡠⢊⡠⠊⡠⠊⠀⠄⢀⠆
⠀⠀⠀⠀⠀⠀⠈⠁⣤⠊⠀⠖⠀⠈⠀⢀⠜⢀⠌⡠⠒⠐⠁⠔⠁⠔⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠊⢀⠄⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢸⠁⠀⠁⢰⠁⠸⠰⠁⠘⠀⠰⠀⢀⠀⡏⢀⠀⡏⠀⠀⠀⠀⠀⠰⠀⡈⠁⠀⠂⠐⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠦⢸⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠇⠈⠀⡇⠀⠀⠐⢀⠠⡀⠈⠁⠀⠔⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠘⡢⢊⠰⢃⠔⠁⢂⡀⠘⠂⠉⠀⠀⠀⠃⢰⠀⡇⠀⡀⠀⠀⠀⠈⠀⢈⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠊⠰⠁⠔⣡⠀⢀⡈⠁⢀⡀⢀⠑⢌⠢⡀⠈⠁⠀⠠⡈⠲⠀⡀⠀⠀⠈⠁⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⠄⡠⠀⢀⣀⡀⣀⢀⡄⠀⢠⠑⢄⠑⢌⠢⠤⠤⡄⠉⡠⠀⡠⠊⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠤⡀⠀⢠⠄⠀⠀⢠⡄⠐⠂⠐⡄⠑⡀⠀⣠⠊⡠⠊⢀⠔⢉⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠢⠀⠀⠀⠀⠀⠀⠠⡄⠀⠧⠠⠆⠀⠤⠊⠀⠶⠁⠤⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣄⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⡟⠁⠀⠀⢀⣠⣾⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡏⠀⠀⣠⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣾⣿⣿⠟⢻⣷⣴⣾⣿⡿⠋⠁
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠿⠟⠋⠁⢠⣾⣿⡿⠟⠁
⠀⠀⠀⢰⣿⣿⣿⣿⣿⡏⠀⠀⠐⠄⠀⠀⡙⠻⣷⣄⣀⣀
⠀⠀⠀⢸⣿⣿⣿⣿⡿⠳⡀⠀⠀⠀⠂⠀⢀⣠⣬⣿⡿⠟
⠀⠀⠀⠸⣿⣿⣿⣿⣷⡀⠙⣄⢠⡄⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣾⣿⠐⠆⣸⣿⡟
⠀⠀⠀⠀⠀⠈⠻⣿⣿⡉⣹⣿⣙⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠋



⣿⡏⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⢒⡛⠭⠉⢂⣀⠒⠠⠤⠶⣤⡖⣂⡀⠉⠭⠝⣒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢋⠩⢔⡂⠭⣐⣂⣭⣑⣦⣶⣶⣗⣚⣛⣛⣓⣚⡾⢿⣷⣥⣶⢬⣕⣪⢙⠲⣄
⠀⠀⠀⠀⠀⠀⢀⡴⢋⠕⡪⢔⡨⣐⡬⣟⣿⠿⠛⡋⠋⠙⠉⠙⡏⠛⠻⠛⠿⣯⣭⣛⠲⢭⣛⠦⣍⠳⣝⠢⡙⢦
⠀⠀⠀⠀⠀⡴⢋⠚⡡⡪⢊⢵⠟⣡⠞⡏⠀⠀⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⠋⡹⢶⠈⠳⣌⢧⡈⢷⡌⢎⢧
⠀⠀⠀⠀⣼⢡⠃⡔⡱⠱⢱⠁⣼⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⣟⠀⠀⠸⣾⠡⠈⠘⠀⡊⡇
⠀⠀⠀⠀⣿⢀⠘⠀⡇⠀⢸⢠⡏⠀⠀⠀⠀⣀⣠⠤⠤⠴⠶⠶⠦⠤⣤⣀⡀⠀⠀⠇⣼⠉⠳⡄⣰⢃⠎⡜⢠⢀⣵⡇
⠀⠀⠀⠀⣿⡜⡀⢄⢇⠰⢰⠈⣧⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⣤⠇⠀⣠⣿⠗⡡⢊⠔⡡⣫⠏⡇
⠀⠀⠀⠀⡇⣿⡐⡌⢎⢢⠑⡕⢌⠳⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡖⢟⡫⢔⡨⢒⡥⣪⠞⠁⠀⡇
⠀⠀⠀⠀⣧⠉⠻⣌⠢⡱⢕⢬⡑⠬⢚⣽⣓⣶⢤⣤⣤⣤⣤⣤⣴⠶⠶⠞⢛⡋⠭⢐⠪⠅⡒⠥⣢⣵⣯⡥⠴⢶⠖⠟⠒⠶⠒⠖⠲⠶⠒⠲⢦⢤
⠀⠀⠀⠀⣿⠀⠸⠋⢻⢮⣑⠩⣙⠲⠦⣤⣉⣉⣀⣀⣈⣉⣉⠀⠀⠴⠆⠑⣂⠠⣉⣂⣬⣵⠞⢻⠉⠉⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠇⣼
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢹⠛⠶⢬⣍⣙⡛⠒⠖⠒⠒⠒⠀⢒⣈⣃⣽⣾⠿⠻⠋⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢠⠇
⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠈⠀⠀⠀⠁⢹⠉⠙⠛⠛⠛⡏⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⡞
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠒⠛⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⠖⠚⠋⠉
⠀⠁
⠀⠁

⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⡀
⠛⠛⠻⠿⠿⠿⢶⣶⣶⣶⣦⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠉⢩⠉⠛⢷⡀
⠄⠀⢀⣀⠐⠂⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠻⠿⠶⠾⠟⠀⠀⠀⠀⢀⠀⠐⠀⠻⣄
⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁⢀⠙⣦⡀⠀⠀⢀
⡤⠀⠀⠚⠭⠉⢠⠤⠄⠀⠀⠀⣄⠀⠀⣀⣀⣀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣛⣛⣋
⠀⠠⣲⠉⠀⠉⠁⠀⠀⠀⠳⠆⠀⣠⠞⠉⠉⠉⠉⠉⠉⠉⠙⠃⠀⠀⠀⠀⠀⠄⠈⠀⠀⠂⠰⠇⠐
⠒⠒⠲⠲⠶⠶⠶⠶⠶⠶⠶⠶⠞⠁⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠈⠀⡀⡰⡩⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠠⡎⠀⠀⠠⠀⠁⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⡤⠃⠀⠂⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⠴⠁⢀⡕⠀⡀⠀⠀⠁⠈⢪⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⣀⣀⡠⠤⠤⠴⠷⠀⠀⠀⠀⠄⠀⣸
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠤⠤⠤⠶⠒⠒⠒⠛⠉⠉⠀⢀⡜⠁⠀⠀⠀⢀⣀⠠⣤⠤⢒⣒⠒⠨⠜⠁
⠒⠚⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠤⠴⠖⠒⢋⣀⣀⡠⠤⣤⠊⠀⢀⣈⡉⠔⢁⠤⠄⠚⢅
⠀⣀⣀⣀⣤⠤⠤⠶⠒⠒⠚⢋⣉⣉⣥⠤⠤⠖⣒⣚⣭⣭⣵⠶⢦⣙⠦⡈⠉⣉⣀⠤⠤⣒⣁⣤⡤⠶⠖
⢉⣉⣀⣠⠤⠤⠶⠒⠚⣋⣉⣩⣴⡶⢀⠴⢲⡾⠋⠉⠁⠆⠀⠀⠀⠙⢷⣬⣭⠶⠶⠛⠋⠉⠁
⣭⣵⣶⠶⢷⣦⣴⠋⣉⣡⡶⣶⠀⡴⠃⣴⠟⠀⢈⠁⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢵⣖⠒⢛⠲⣌⡛⠛⣛⣉⣻⠯⠞⢠⡾⠋⣀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⢠⡤⠤⠒⠊⠉⠑⠤⠤⠒⠈⠁
⣿⣽⣧⣙⠑⠢⠍⠉⠉⢀⣀⣤⣴⠟⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈
⣈⡀⠉⠛⠿⣶⡶⠾⠟⣛⣩⠭⣄⣀⡀⠀⡄⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠉⠓⠦⠤⠴⠒⠊⠉⠀⠀⠀⠀⠛⠃⢠⠄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢡⠋⡁⠭⠤⠬⠭⠭⠭⠭⠭⠭⠭⠭⡭⢈⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡇⢀⡀⣀⡀⠀⣀⡀⠤⠠⠄⠀⠠⠄⠤⢀⡀⢀
⠀⠀⠀⢀⠄⠒⡌⠤⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠉⡉⠬⠥⠶⢤⡀
⠀⠀⠀⡆⠀⡜⡐⢉⡸⢒⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⡠⠒⡁⠄⠒⠒⠒⢤⡈⠲⡀
⠀⠀⠀⠇⠀⢢⡕⠁⣠⠊⠀⡅⢀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⠊⡠⢊⠀⠀⡔⠉⠑⡄⠙⣆⠘⡄
⠀⠀⠀⢰⠀⠸⠤⠊⢹⠀⠀⡇⠘⣦⠀⠀⠀⠀⠀⠀⠰⠀⢠⢃⢮⠊⠀⠑⢈⠀⠀⢀⠇⠀⢘⡆⢠
⠀⠀⠀⠀⠑⢄⡀⠀⡜⠀⠀⡇⠀⢸⡀⠀⠀⠀⣔⠠⢅⣀⠃⠎⢨⠀⠀⣠⣁⣑⠐⡡⠊⠁⢮⠁⠈⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠢⡇⠣⠴⠅⠃⢸⠁⠀⠀⢰⠀⠉⠑⠊⠭⡒⡄⢁⠈⠏⠀⠀⣧⡁⠀⠀⣼⠈⢸⠇⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⢀⠟⠀⠀⠀⠈⠀⠀⠀⠀⠀⢱⠀⠀⠱⣄⠀⣠⡃⠱⡀⠊⡘⡄⣸⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢖⠉⠀⠀⠀⠀⠰⣀⣀⠀⣀⣠⣼⠀⣠⡧⠤⡉⡌⠀⠀⠱⠠⡱⢠⠓⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠈⡄⢂⠸⡀⠁⠎⠀⠀⠸⢆⠀⣀⡞⡱⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⢀⡘⢌⢆⠑⢄⠣⣀⡠⠃⢀⠠⡪⠊⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⣀⠑⠒⠒⣚⡃⣁⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢼⣀⡀⣀⠅

⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠀⠀⠈⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠄⠄
⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄
⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠂⠀⠘⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠂
⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀
⠈⠀⠀⡀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠐⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠠⠂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠃⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠂
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡂
⠈⠀⠀⡀⠀⡀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠄⠀⠀⠀⠈⠀⠀⠐⠁⠀⠀⠀⠉⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠐⠂⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠡⠀⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠃⠀⠀⠐⠁⠀⠀⠘⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠈⠈⠀⠀⠈⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠅⠀⢀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠨⠂⠀⡀⠈
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠨⠂⠀⠄⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠂
⠀⡀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠁⢀⠄⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠈⠀⠐⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⠉⣿⠉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣠⡶⠛⠋⣿⠙⠛⠶⣄⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡾⠃⣠⠖⠛⠻⠙⠳⣦⠈⢳⡄⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣸⡃⣸⡃⠀⠀⣠⡀⠀⠈⣧⠀⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣤⣴⢟⣓⣛⠳⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⠉⢻⡉⢹⡉⠀⠈⠻⠉⠀⠈⣿⠉⣿⠉⣽⠁
⠀⠀⠀⠀⠀⢀⣴⢇⡶⠋⠉⠉⠛⣦⠻⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⣿⢀⣠⡾⠁⣰⠇⢠⡏
⠀⠀⠀⠀⠀⠈⣷⢸⡁⠀⠀⠀⠀⢸⡧⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇⣿⢉⣁⣤⠞⠁⣰⠟
⠀⠀⠀⠀⠀⠈⠳⡎⢷⣄⡀⢀⣠⠞⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠉⣉⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠛⣶⣏⣭⣍⣵⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡇⢈⣿⣯⠁⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠀⣸⣿⢹⣴⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⢰⣷⣶⡿⣿⢸⡟⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠘⠃⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣱⣿⠓⣛⣌⣓⠞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⡿⠟⣾⠁⠀⠙⣧⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣸⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣸⣄⣀⣀⣙⣷⠹⢦⣤⡴⠣⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠉⠉⠉⠉⠉⠉⠉⠉⣩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⠙⢿⡿⠋⣿⠉⠳⠛⣷⣰⡞⠳⠟
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⡟⠛⣵⠟⠉⠉⠻⣎⠛⢻⣆⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠘⢯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⢰⡇⠀⠀⠀⠀⢸⠆⣯⣭⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣄⠳⣤⣀⣀⣤⠟⣠⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⣷⢀⣭⣭⡀⣾⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠛⠉


⡿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣻⣿⣿⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠤⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠜⠉⠀⠀⠈⠀⠀⠈⠀⠐⢠⡀⠑⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡜⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠹⣆⢱⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡦⠀⢠⢡⠀⠀⢻⣰⠀⡇
⣤⣄⡪⠎⠉⠑⢒⠒⠒⠠⠤⠄⢀⣀⡀⠀⠂⠀⠀⠀⠀⠀⠛⠁⠀⠁⡼⠀⠀⣽⢸⢀⡇
⠘⠋⠉⠋⠙⠓⠒⠶⠦⠤⢬⣄⣐⠀⠀⠉⠡⡀⡀⠀⠀⠙⠒⠒⠒⠊⢠⠔⠾⠗⠯⠴⠦⠀⠀⡀⠀⠀⢀⢄⣤⢄⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠆⠈⡢⣄⣀⡀⠀⢀⣀⣬⡴⠒⣀⣞⣛⢳⡶⠴⡖⠀⠠⠮⠾⣫⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠓⠀⠀⠀⠰⠃⠘⠛⡏⣟⠐⠛⠖⣺⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢞⣡⠗⠖⠖⠚⠁

⠀⣿⣿⣿⣯⣭⡇⠀⠀⠀⠤⠀⠀⠀⠠⠄⠀⠀⠀⠀⣶⡶⢶⣶⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⣿⠟⠉⢙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣴⣶⣾⣶⣶⣾⣶⣶⠀⠀⡟⠛⠛⠛⠛⣿
⠀⣿⣦⣀⣨⣿⡇⠀⠀⣠⣷⣶⠶⢶⣶⣦⡀⠀⠀⠀⣿⣯⣽⣿⠀⠀⠀⠻⠷⠿⠇⠤⠿⠶⠿⠀⠀⡇⠀⠀⠀⢀⡿
⠀⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀
⣶⠶⠲⠶⠶⣶⠖⠆⠀⣤⣴⣤⣤⣤⣼⣤⡄⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠐⣿⣿⣿⣿⢆⠀⠀⠀⠀⠀⢠⠃⣀⠜
⣿⣠⣶⣶⣆⣿⣦⡆⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⣿⣿⠋⠀⠀⠀⠀⠀⢹⣿⣿⣿⡞⠀⠀⠀⠀⢠⣿⣭⣤⣤
⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠀⠀⠀⠀⠤⠇⠻⠿⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣒⣒⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⣀
⢰⠤⣤⣶⣶⣤⣶⡄⠀⠀⢸⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠃⠀⠀⣿⣿⣿⠋⠉⢹
⠙⠋⠉⠙⠉⠉⠛⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠐⠒⠒⢲⡖⠒⠒⠂⠀⢸⣿⢻⣟⡟⡟⣻⡷⠀⠀⣿⣿⣿⣦⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⠀⠀⠀⠀⠀⠘⠚⠃⠃
⠀⢀⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄
⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⣶⣶⣤⡀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⣧⠀⠀⠀⢸⡇⠀⠀⠈⢉⡿⠿⠛⢷⡀⠀⠀⠀⠀⢻⡟⠉⠉⠉⠀⠀⠀⠠⣾⠿⡿⣿⡆⠀⠀⠀⠀⠘⣿⣿⡟
⠀⣿⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠈⠛⠛⠋
⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⢤⡀⠀⠀⠀⣠⣶⣶⣦⡀
⠀⠀⢰⣿⡿⠀⠀⠀⠀⢰⣲⡶⠶⠶⠶⣶⠀⠀⠀⠀⢀⣀⣀⣤⡄⠀⠀⠀⡇⢠⣶⣶⣄⢸⡇⠀⠀⠺⢿⣿⣿⡿⠟
⠀⠀⢈⣿⡇⠀⠀⠀⠀⢸⣿⣇⣀⣀⣀⣉⡀⠀⢠⣾⣿⣴⣶⣾⠇⠀⠀⠀⡇⠙⠾⠿⠏⢸⡇⠀⠀⠀⠀⠀⡇
⣀⣀⣀⣻⣃⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣳⣶⣶⣶⣶⣞⣃⣀⣀⣀⣠⣴⣷⣤⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠉⠉⠉⠉⠛⠛⠻⠭⣛⡳⢶⣤⡀
⠀⠀⠀⠀⠀⣠⠾⠛⠓⠉⠉⠉⠉⠉⠒⠲⠤⣄⡀⠀⠀⠉⠳⣌⠛⢶⣄
⠀⠀⠀⢠⡞⠁⠀⠀⣀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠉⠲⣄⡀⠀⣸⠇⠀⠹⣧⡀
⠀⠀⢠⠏⠀⣠⡶⠛⠉⠉⠀⠈⠉⠉⠛⠷⣦⣄⠀⠀⠀⠉⠛⠁⠀⠀⢀⡼⣷⡀
⠀⢀⡟⢀⡼⠋⢀⣤⠶⠖⠒⠶⠦⣤⣀⠀⠀⠙⢷⣄⠀⠀⠀⢀⣠⠞⣡⠴⠜⣷
⠀⢸⠇⢸⠁⢰⠋⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⠀⠀⠙⢷⡀⠠⡏⢠⠊⠀⠀⠘⣼⡇
⠀⢸⡀⢿⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠈⢿⡀⠹⡌⡄⠀⠀⠀⣐⣷
⠀⠘⡇⠸⣀⡈⠉⠓⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⣷⠀⢹⡐⣀⡴⠚⠉⢿
⠀⠀⡵⠦⢤⣉⠙⠲⢤⣀⠉⠓⢤⡀⠀⠀⠀⠀⠀⢱⠀⠀⢸⡀⠀⣇⡅⠀⠀⠀⠘⡄
⠀⢀⡇⠀⠀⠈⠙⠓⠤⣈⠛⢦⣀⠈⠲⣄⠀⠀⠀⢸⠀⠀⢸⡇⠀⣿⡀⠑⠲⠴⡾⠁
⠀⠈⣷⠀⠀⠀⠀⠀⠀⠈⠑⠢⡈⠳⢤⡀⠑⢄⣠⠏⠀⠀⡾⠀⢠⡟⣁⣀⢤⣾⡁
⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⢦⡀⠁⠀⢀⠞⠁⠀⢼⣠⠤⠚⠉⠀⢹
⠀⠀⠀⠈⠛⠷⠶⠶⠶⣶⣤⣤⣀⡀⠀⠀⠀⠙⠶⠚⠁⠀⠀⠀⠀⠀⠀⣀⠴⠛⠻⢤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠲⠶⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠸⠿⠶⣲⡴⢤⣄⣀⣀⠀⠀⠀⢀⡖⠒⡶⠤⢤⣤⣈⣉⡭⠿⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠦⠶⠚⠁⠀⠀⠀⠀⠉⠉⠛⠓⠾⠿⣿⣶⣤⣤⣂⡀⠠⡠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⢸⡇⠚⠀⢀⡤⠦⣄⠀⠐⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠦⢾⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡀⠀⣸⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣶⡞⠃⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠉⠋⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⡀⢀⣠⣤⣀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⢸⡇⠉⣠⣤⣌⠹⣆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⣀⣤⠤⠦⢤⣄⠀⠀⠀⢸⡇⢸⡇⢸⡅⠀⣸⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠘⠩⠖⣛⣛⡳⠮⠉⠀⠀⢸⡇⢸⡇⣄⣉⢉⣁⡴⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠈⢡⣄⠉⠀⠀⠀⠀⢸⡇⢸⡇⠀⠉⠉⠁⠀⠐⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣧⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣸⡇⢸⡇⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠄⠃⠑⠁⠊⠤⠃⠄⠁⠁⠺⠈⠈⠀⠩⠀⠐⠃⠓⠤⠁⠁⠡⠅⠁⠘⠂⠄⠱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠻⠟⢻⣿⢻⣿⣿⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣻⣿⣿⣻⣿⣛⣛⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣿⣿⣿⣟⡛⠛⢻⣿⣻⣿⣿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⢛⣛⣛⠛⠀⠀⠀⠀⣿⣿⢟⣭⣶⠶⢶⣮⣝⠛⣛⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢛⣩⣴⣿⣿⣿⡏⠀⠀⠀⠀⣿⢣⣾⣿⣏⣾⣷⢹⣿⣧⠙⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣿⢸⣿⣿⣿⢋⣵⣿⣿⣿⢈⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⡌⣿⣿⣿⣿⣿⠀⠀⠀⠀⣽⣎⢻⣿⣿⣺⣿⣿⣿⢟⣼⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣛⣃⣿⣿⣿⡿⠃⠀⠀⠀⠀⣿⣿⣷⣭⣛⡛⢛⣋⣥⣘⣉⣹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠃⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣙⡛⢿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣦⡙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣀⣠⣴⡘⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣮⡛⢿⣿⣿⣷⣄⣠⣴⣶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣝⠻⣿⣿⣿⡿⠟⠋⠁⠀⣴⣶⣶⣶⢂⣴⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⠁⠀⠀⠀⠀⠀⠿⣿⣿⣿⡘⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣾⣿⣿⣿⠀⠀⠀⠀⠀⢠⣶⣶⣶⡖⣴⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⢿⠁⠀⠀⠀⠀⠹⠿⠿⠿⡇⠻⠿⠿⠿⠿⠿⠿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠀⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢨⠁⢯⠗⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⣠⣌⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣹⠏⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⠀⠀⢸
⠀⠀⠀⠀⠀⣠⠞⣡⠞⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⢸
⠀⠀⠀⠀⢸⠁⡞⠁⣰⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠂⡸⠀⠀⣸
⠀⠀⠀⠀⢸⠀⡇⣾⢠⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⢸⠡⣺⡆⣸
⠀⠀⠀⠀⢸⠀⡇⢫⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣤⠈⡇⡉⠐⠃
⠀⠀⠀⠀⢸⠀⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠳⣌⠻⠀⡇
⠀⠀⠀⠀⢸⠀⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡇⠘⠐⠀⡇⡇
⠀⠀⠀⠀⢸⠀⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⠀⡃⠴⢰⠀⡇⡇
⠀⠀⠀⠀⢸⠀⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⢸⠀⡇⠰⢘⠀⡇⡇
⠀⠀⠀⠀⢸⠀⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢸⠀⡇⢰⢸⠀⡇⡇
⠀⠀⠀⠀⢸⠀⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡇⢸⢸⠀⡇⡇
⠀⠀⠀⠀⢸⠀⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡇⠸⠘⠀⡇⠃
⠀⠀⠀⠀⢸⠀⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡇⠀⢈⠀⡇
⠀⠀⠀⠀⢸⠀⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡼⢀⡇⢰⢨⠀⡇
⠀⠀⠀⠀⢸⠀⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣰⢋⡴⢋⡴⢋⡼⠀⡇⢈
⠀⠀⠀⠀⢸⠀⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⢰⠋⡴⠋⠀⠀⡇⠈
⠀⠀⠀⠀⡜⢠⡇⠻⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⢸⠀⡇⢸⡿⠀⡇⠁
⠀⠀⠀⠀⡇⠌⢠⣶⡌⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⢸⠀⡇⠀⡇⠀⡇⠁
⠀⠀⢀⡴⠃⠘⡆⣿⠀⣹⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⢸⠀⡇⠀⡇⠀⡇
⠀⠀⡎⢰⣢⠀⡇⣾⠀⠘⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⢸⠀⡇⠀⡇⠀⡇
⠀⣀⠙⣆⢹⠀⡇⣽⠀⣸⠘⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⢸⠀⡇⠀⡇⠀⡇
⠘⢿⡆⢹⢸⠀⡇⣿⢰⠯⢂⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⢸⠀⡇⠀⡇⠀⡇
⠀⠀⡆⢸⢸⠀⡇⣿⢸⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⢸⠀⡇⠀⡇⠀⡇
⠀⠀⡇⢸⢸⠀⡇⣿⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⢸⠀⡇⠀⡇⠀⡇
⠀⠀⡇⢸⢸⠀⡇⣿⢸⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⢸⠀⡇⠀⡇⠀⡇⠀⢀⡀
⠀⠀⡇⢸⢸⠀⢧⠙⢄⠳⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⢸⡀⢧⠀⢧⠀⢧⠙⢆⠙
⠀⠀⡇⢸⠀⠳⡌⠱⢄⠱⣌⠱⣌⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧⡘⢦⡙⢦⡑⢦⠙⢦⡙⢄⡑
⠀⣠⠇⢸⠀⠀⡇⠀⢸⠀⠀⠑⢄⠱⣌⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢻⠀⠙⢦⠙⢦⠑⢦⡙⢆⠹
⠞⠁⢀⣾⡀⠀⡇⢀⣸⣄⠰⣻⠄⣱⣄⠑⣄⠙⢛⣙⡛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⣀⠀⢸⠀⣀⣀⡙⢆⠙⡄⡓⢨
⠀⠀⣏⠀⣹⠀⡇⣾⠁⢈⡇⡏⢸⡁⢈⡇⢸⠰⣏⠀⣹⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⢈⡇⢸⢰⡏⠀⣷⢨⠀⡇⠄
⠀⠀⠉⠛⠉⠀⡇⠈⠙⢿⣀⠙⢦⡙⠋⠁⢸⠀⠉⠛⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠙⠋⠀⣸⠀⣹⣿⡁⢸⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠦⠀⠙⡄⠀⠸⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⢠⠞⠁⠀⠛⠛⠁⢸⠀⡇⠄

⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⠂⠘⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠒⠀⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄

⠀⠀⠀⠀⠀⣀⡀⣄⡀⠀⠀⠀⣠⣤⣤⢤⣄⠀⠀⢠⣤⡀⠀⠀⣠⡀⠀⠀⣴⣤⣄⣤⠀⠰⠶⢶⣶⣶⣶⡦⠀⠀⡴⠀⠀⢀⣤⣶⣢⣄
⠀⠀⠀⣠⣾⠟⠋⠙⠃⠀⠀⢠⣿⠉⠉⠉⠉⠀⠀⣿⢿⣷⠀⠀⣧⡇⠀⢸⣿⠀⠈⠁⠀⠀⠀⢀⣿⠁⠀⠀⠀⢸⣿⠀⢠⣿⠏⠀⠈⠉
⠀⠀⢰⣿⠁⣀⣀⡀⠀⠀⠀⢸⣿⣤⣤⣤⠀⠀⢰⣿⠀⢻⣧⢰⣿⠀⠀⣼⣷⣶⡶⠆⠀⠀⠀⢸⣿⠀⠀⠀⠀⣾⡏⠀⢸⡟
⠀⠀⢸⣧⠀⠙⠛⠛⣿⡆⠀⣿⡏⠉⠉⠀⠀⠀⢸⣿⠀⠀⢿⣿⣿⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢀⣿⠃⠀⢸⣇⠀⠀⢀⣤⡄
⠀⠀⠈⠻⠷⠶⠾⠿⠛⠁⠀⠿⠿⠿⠿⠟⠀⠀⠾⠇⠀⠀⠈⠻⠏⠀⠘⠿⠿⠷⠿⠂⠀⠀⠀⠿⠃⠀⠀⠀⠘⠟⠀⠀⠈⠻⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠠⣤⣶⣶⣶⣖⣤⠄⠀⣴⣶⣶⣦⠀⠀⣰⡾⠽⠗⠀⠾⠿⢿⡿⠿⠿⠀⠀⣴⡆⠀⢀⣶⣶⡀⠀⢰⡦⠀⠀⣴⣶⠿⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⢰⣿⣠⣄⠀⠀⠀⠿⣷⣤⠀⠀⠀⠀⣾⡇⠀⠀⠀⢀⣿⠃⠀⢸⡿⠙⣿⡄⣷⡇⠀⣾⠟⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⢀⣿⠏⠉⠉⠀⠀⣀⡀⠀⣻⡇⠀⠀⢰⣿⠀⠀⠀⠀⣸⡿⠀⠀⣿⡇⠀⠸⣿⣿⠀⠸⣿⡀⠉⠛⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠸⠿⠿⠿⠿⠃⠀⠙⠿⣶⠿⠃⠀⠀⠸⠟⠀⠀⠀⠀⢿⠇⠀⠀⠿⠁⠀⠀⠛⠋⠀⠀⠙⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⠀⠀⠀⠀⢠⣤⣶⠾⠿⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠷⢶⣶⣦⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣰⣤⣀⣀⣀⣄⣄⣄
⢰⠶⠉⢉⡵⠛⠉⠑⠢⡀⠀⠀⠀⠀⣼⡃⣹⠀⠀⣏⣸⣷⠀⠀⠀⢠⣋⠚⡛⠛⣛⣛⣻⡿⠎⣆⠀⠀⠀⡴⠒⠒⠒⠒⠒⠒⠶
⢸⡇⠄⣿⠀⢐⣾⡃⠀⡣⠀⠀⠀⠀⢿⣏⠁⠀⠀⠀⣹⡿⠀⠀⠀⢸⡏⠉⠉⠉⠉⡉⠉⠉⢹⡇⠀⠀⠀⡇⢘⣿⣸⣏⣻⣿⡁
⠀⢷⢘⣉⣢⣀⣀⣀⡴⠃⠀⠀⠀⠀⠈⠛⠄⡀⠀⠠⠋⠀⠀⠀⠀⢸⣗⠀⠀⠂⠀⠂⠀⠀⣸⡇⠀⠀⠀⣷⣜⢠⣤⣤⣤⣄⣧
⠀⠈⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠸⡿⢀⠀⡀⣀⣀⣀⣀⣾⡇⠀⠀⠀⠙⠛⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⣀⢠⠀⠀⠀⠀⠀⠀⠀⣸⣛⣛⣻⣛⣛⣿⣷⣿⣗⠀⠀⠀⠀⣿⠁⡼⢿⣧⠈⡇⠀⠀⠀⠀⠀⠀⡞⡛⠒⠒⠒⣒⡆
⣠⠖⠚⡋⣭⣭⠉⠙⠒⢄⠀⠀⢸⡉⠩⠭⠍⠅⡬⢤⠤⠤⣼⠀⠀⠀⠀⢿⠀⣹⠿⢥⡀⡇⠀⠀⠀⠀⠀⠀⡇⠌⠉⠉⠃⢼⡇
⠣⣄⡀⠑⠛⠋⠊⠀⣀⠜⠀⠀⢸⡅⢇⣀⡁⠀⣾⣿⣿⣿⢸⠀⠀⠀⠀⢸⣼⠡⣂⠆⣷⡇⠀⠀⠀⠀⠀⠀⡇⠠⠂⠉⠠⠀⡇
⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⠀⢸⠟⠘⠉⠃⠇⣿⣿⣿⣿⢿⠀⠀⠀⠀⢸⡈⠓⠶⠚⢁⡇⠀⠀⠀⠀⠀⠀⣗⢁⠀⠀⡨⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⢠⠤⠤⠤⠤⠤⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀⡴⠒⠒⠓⠓⠓⠒⠖⣆⠀⠀⠀⠀⡖⠒⢲⣶⢒⠒⡆
⢸⠰⠤⠆⠻⠚⠓⠛⠰⢼⠀⠀⠀⠀⠀⠀⢠⠒⠋⠁⡇⠀⠀⠀⠀⠀⢹⡇⡎⣟⣽⢻⣷⡆⠀⣸⠀⠀⠀⠀⡇⠁⢀⠠⡄⢹⡇
⢸⢀⠀⡄⢴⠰⠆⢦⡴⢸⠀⠀⠀⠀⠀⠀⢸⠀⢀⣠⡇⠀⠀⠀⠀⠀⢸⣇⣉⣙⣛⣿⣉⣀⣤⣿⠀⠀⠀⠀⡗⢀⡈⠠⠇⢸⡇
⢸⣈⣉⣉⣿⣿⣇⣙⣋⡼⠀⠀⠀⠀⠠⣞⠟⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠚⠃⠀⠀⠀⠀⡟⠂⢀⣠⡀⠿⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠚⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣮⣝⢛⣹⡆⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡶⢦⡀
⠀⠀⠀⢀⡴⠋⣻⠿⠟⠀⠀⠀⠀⠀⢀⣴⣷⡶⢸⣿⣷⡄⠀⠀⠀⠀⠀⣠⠟⠋⠉⠉⠙⢳⡀⠀⠀⠀⠀⣯⠛⠛⠋⣉⢙⠛⠋
⠀⢀⡴⢟⡣⠚⠁⠀⠀⠀⠀⠀⠀⠀⠸⡿⡟⣿⣸⡛⣿⡇⠀⠀⠀⠀⢠⡷⢢⠀⠀⠀⢴⠲⣇⠀⠀⠀⠀⡿⠀⢰⣶⣿⣻⣇⣀
⢀⡿⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡉⢛⢛⠋⣼⠃⠀⠀⠀⠀⣯⡇⣾⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⡿⢿⠿⠿⡿⢽⣿⡿
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡽⣽⡍⠁⠀⠀⠀⠀⠀⠀⠑⠚⠀⠀⠀⠙⠚⢡⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⡆⠠⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⢁⠜⠡⠀⠀⠀⠀⢀⡀
⣀⠀⠀⠐⠂⠀⢀⠀⠀⣀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢰⠀⠒⠀⠀⠀⠄⠰⢂⠔⠠⠄
⠌⠀⠀⢠⡀⣠⠈⠀⠀⠉⠀⠉⠀⠀⠀⠀⠤⠀⠀⠀⠶⠀⠀⠰⠂⠀⠒⠠⠀⡈⠀⠀⠀⠀⠰⠡⢊⢔⡀⠉⠠
⠢⡑⢀⢠⡀⢀⡑⢀⠈⠁⠀⠀⠁⢀⠔⠁⠀⠀⠀⡉⠀⠀⠉⠠⠆⡀⠈⡠⠊⠐⠂⠀⠐⠀⢈⠔⠁⠈⠀⠉
⠀⠀⠈⠀⠀⠀⡁⣀⣁⣀⠐⠃⠐⠁⠠⠀⠀⠰⠀⠀⡆⠀⢀⠀⠊⡠⠊⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠃⢠⡍⠠
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠊⠀⠀⠃⠀⠀⠀⡀⠀⣈⡀⢀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠐⠂⠐⠀⠠⠀⠀⢀⡐⠆⠁⠀⠀⠀⠄⠀⠀⠀⠃⠀⠀⠠⠄⠖⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢠⡄⠐⠂⡠
⡠⠂⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠂⠀⠉⠉⠉⣉⠉⠀⠐⠂⡀⢀⠋⡠
⠀⢔⠀⠀⠀⠀⢠⠀⠀⠀⣀⠈⠁⠀⠀⠀⠀⡄⢩⠀⠀⠈⠁⠈⠠⠀⠀⠀⠃⠀⠀⠄⠁⠒⠀⠈⠃⠁⠈⠀⠠
⠀⢈⠔⠁⠄⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠂⠠⠊⡠⠂⠀⠀⠀⠈⢁⠄⠁⠀⠀⠀⠀⢂⠄⠀⢀
⠀⢁⠀⠀⠀⢀⠀⠈⣀⠀⠁⢂⠀⢀⠀⡀⠀⠀⠠⠊⠀⠀⠠⢊⡀⠀⠀⠀⠀⠀⠀⠀⣀⠘
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠒⢀⠁⠀⠀⠀⠀⢠⠀⢀⠄⡀⠀⠀⠈⠀⠃⠀⠀⠀⠀⢃⠄⠁⠀⠀⠐
⠀⠃⠊⣀⠔⠁⠀⠀⠀⠀⢠⠀⢀⠄⠀⠀⢀⠀⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⡀⠊⠀⠀⡀
⠀⢈⢢⠀⠰⠀⢠⠀⠀⠛⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⢠⠈⠁⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁
⠠⡀⠁⠀⠀⠀⠀⠀⠃⣀⠘⠀⢀⠔⠀⠀⠀⠀⠠⠀⠀⢀⡀⠀⠀⠀⠀⠑⠄⠀⢀⠒⠐⠂⠂⠀⠀⠒⠐⠀⢀
⠀⠈⠢⡀⠁⡠⢂⠔⠀⠀⠀⠄⠡⣊⠀⠀⠂⢀⠀⠂⠀⢄⠀⠃⠀⠀⠠⠄⡀⠀⠈⠠⡈⠀⠀⠉⠈⠀⠑⢌⠈
⠢⠀⡄⠀⠂⡔⠱⠀⢀⠄⠠⠀⠀⢁⡀⠀⢀⡈⠀⠀⠀⠀⠑⠄⠠⡄⠁⠀⠀⢀⡐⠆⠈⠃⣈⠃⠀⠀⢤⠠⡕
⠀⠀⢀⠔⠀⠁⠀⠐⠁⠊⠒⠁⠰⠀⠀⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠖⠀⠀⠰
⠀⠀⠀⠀⠀⢇⠉⢀⡀⠈⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠃⠈⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠐⠎⠁⣄
⢄⠢⡀⠀⠈⠠⠑⢄⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢀⡀⠀⡀⠂⠀⠀⠀⠀⠐⠄⠰⠂
⠀⠁⠀⠀⠀⠇⠀⠀⡀⠡⠤⠀⠠⠄⠀⠖⠀⠀⡀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠈⡁⠀⠀⠆⠈⠦⢠⠀⠀⠓

⠀⠀⠀⠀⡀⠀⣀⣀⣀⢀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡀⠤⠤⢤⠤⠤⠄
⠀⠀⠀⠀⠑⢄⣈⣈⣈⣀⠀⠀⠀⠀⠠⠈⠢⡀⠡⡎⠀⠀⠀⢀⡴⠂⠈⠀⡠⠀⠉⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠒⠀⠁⠘⠀⠈⠉⠑⠀⠀⠀⠀⣀⡄⢰⡇⠀⠀⠰⠋⢀⠄⠀⠊⠀⡠⠒⠒⠒⠒⠡
⠀⠀⠀⠠⠤⢄⠀⠀⠐⠀⡀⠀⠀⠁⠀⠀⢀⢅⠜⠁⠀⠀⠀⠀⠁⠀⠀⡠⡪⠊⠙⠏⠉⠢⣀⣀
⠀⠀⠀⠀⢽⠀⠑⠄⠀⠰⠃⠀⠀⠀⣔⣔⡕⠁⢀⠔⠁⠀⠀⠀⠀⡀⠠⠇⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠃⢀⡄⠀⠀⠀⠀⠀⠑⠀⠄⠀⠀⠀⠈⠀⠀⠆⠒
⠀⠀⠀⢀⠀⡀⠀⢸⠀⠀⠀⠀⠀⢱⣶⡒⠒⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡇⠀⡔⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⢠⠂⠒⠢⡀⠐⣗
⠀⠀⠀⠀⡇⠀⡧⠄⠀⠀⡀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠈⠂
⠀⠀⠀⠀⠈⠢⡢⡀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠐⠀⠀⠀⠀⠈⠢⣀⠀⠠⣄⣀⠐⠀⠄⠀⠀⡤
⠀⠀⠀⠘⢦⠀⠈⡮⠾⠆⠀⠀⠀⠂⠀⡘⠀⠙⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⡉⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⣠⣤⣤⣤⡀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠓⠒⢺⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠖⠛⠛⠛⠛⠛⠛⠒⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠒⠛⠋⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⣠⠶⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠓⠶⣦⡀
⠀⠀⠀⠀⠀⣼⠃⢀⡴⠶⠶⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠲⢦⡀⢸⡇⠀⢠⢞⡽⣷⣆⠈⣷
⠀⠀⠀⠀⢀⡏⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⢸⡇⠀⠸⣝⣲⣾⠏⠀⢹
⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⡇⠀⠀⣐⣛⡂⠀⠀⢸⡆
⠀⠀⠀⠀⠈⣇⠀⣿⠀⠀⠀⠀⠀⠀⣼⠛⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠗⠀⠀⠀⠀⠀⢸⡇⢸⠃⠀⡞⢣⠴⣝⣆⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠣⣜⣲⣫⠏⠀⢸⠂
⠀⠀⠀⠀⠀⣿⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⢒⣛⠂⠀⠀⣾
⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⠀⠀⠀⢧⡼⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⡜⢯⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⣀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠽⠏⠀⠀⢀⡇
⠀⠀⠀⠀⠀⣿⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣿⠀⡞⡏⢹⠉⡟⢆⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠸⣇⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠇⠀⣿⢸⠛⡗⢺⢺⡗⡟⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠈⢉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿⢸⠓⡗⡿⢸⡧⡇⣾
⠀⠀⠀⠀⠀⠹⣦⣀⠀⢸⡟⠻⡿⠭⠭⣿⠽⢿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣟⣙⣏⣉⣻⣟⣛⣧⠀⠀⡿⠸⣻⣏⣟⣹⣽⢃⡿
⠀⠀⠀⠀⠀⠀⠈⢹⣏⣉⣛⣛⣛⣛⣛⣛⠒⠾⠷⠶⠶⠶⠦⠤⠤⠤⢴⣷⣤⣧⣤⣴⣧⣤⣿⣤⣤⣧⣤⣤⣬⡥⢴⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠧⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣍⣐⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⡇⠀⣿⣶⣶⣶⣶⡶⠶⣶⣶⣶⣶⣾⡇⠀⢀⣾⣿⡟⢻⣿⣿⣿⣿⣿⣿⡋⣹⣿⣷⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠘⠿⠿⠿⠿⠷⠶⠿⠿⠿⠿⠟⠁⠀⢸⣿⣍⡁⢈⣉⣿⣿⣿⣏⣹⣿⣿⣉⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇
⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣼⣿⡿⠋⠙⢿⣿⣅⣼⣿⠟⠀⠀⠀⠀⠹⢿⣿⣿⡿⠋⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀
⡭⣭⠿⠿⠿⠿⠷⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠶⡶⢿⠿⢿⠿⣯⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⡤⣤⢀⡤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠳⡄⠀⠀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣗⣒⣒⣿⠀⠀⠀⢠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢹⡀⠀⢀⣈⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡇⠀⠸⠿⠿⠿⠿⠿⠇
⣿⣿⣿⡏⢹⣶⣄⠀⠀⠀⠀⢸⣶⣶⣶⣶⣶⣶⣶⣶⣾⢸⣿⣟⣿⣿⠀⠀⠘⠋⢉⣉⣉⠘⠛⠀⠀⢀⣤⣶⡿⣷⣶⣤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣷⣾⣿⣿⡇⠀⠀⠀⠀⠉⠉⢁⣽⣿⣅⠉⠉⠁⠸⣿⣿⣿⡿⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢠⣿⣿⣿⣇⣾⣿⣿⣷⠀⠀⢳⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠉⠉⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠈⢣⣀⠀⠀⠀⠀⣿⠉⠉
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣇⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣠⠔⠛⠉⠉⠛⠲⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠉⠀⠀⠀⣿
⣤⣤⣤⣤⣤⣤⣼⡇⠀⠀⠀⠀⣠⣄⣀⣹⢿⣿⣿⣿⡀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠘⣦⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣾⡉⠉⠙⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠻⣿⣿⣿⣿⣿⣿⠏
⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣇⢷⠄⠀⠀⠀⠀⠀⠰⣷⡆⠀⠀⠀⠀⠀⠀⢰⣾⠆⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⢀⣀⣄⣀
⠀⠀⢸⠀⢠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠟⠙⠻⣶
⠀⠀⢸⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⠝⠉⠀⠀⠀⣤⣸⣷⣿⣷⣷⣠⣙⣿⣀⠀⣠⣿⠁
⠗⠀⢸⠀⡠⠃⠀⠀⠀⡞⢛⠛⠛⡚⡛⠛⢛⠚⡛⡛⠛⢳⡆⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⢶⣿⡿⠛⠉⠙⠻⣿⣾⠞⠛⠿⠛⠁⠀⠀⠀⠀⠰⣿
⠀⠀⢸⠊⠀⠀⠀⠀⠀⡇⠐⠀⠀⠄⠤⠀⠘⠀⣦⡄⢠⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠀⠀⠀⠀⠀⣹⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠑⢄⣸⠀⠀⠀⠀⠀⠀⣯⣨⣈⣅⣁⣉⣈⣀⣨⣍⣭⣨⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣦⣀⠀⢀⣠⣿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⡀⢀⠀⠀⠀⠀⠀⠁⠿⢻⣿⣿⣿⣿⠿⠄
⠀⣀⣼⣄⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⢀⣠⣥⣧⣧⣧⡼⣀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠁⠉
⠐⣿⣿⣿⠆⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⡇⠀⠀⢀⣶⣸⢰⣿⣿⣿⣿⣿⣾⣇⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⣷
⠀⠈⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣭⢽⢸⣿⣿⣿⣿⣿⣿⡧⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠭⢽⢸⣿⣿⣿⣿⣿⣿⡯⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠓⡟⡞⠿⡞⢻⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠃⠃⠃⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒⠀⠀⠙⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠲⠦⠤⣀⣀⡀⠀⠀⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⡇⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣦⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇

⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣿
⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠑⡀⠀⠀⠁⠀⠀⠀⠠
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠰⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⡀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠘⢻⣿⣶
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣆⠙⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠀⠀⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠋⣴⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣾⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⣾⣦⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢹⡟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⡌⢿⢲⡄⠀⣰⣶⣇⢀⣴⣀⣤⡾⢋⣴⠟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⡆⢀⣿⠺⡋⢷⣦⣈⢻⡿⠐⠹⠟⣹⣷⣿⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⣨⡙⣁⣶⠸⢺⣿⣾⡇⠀⠠⠴⢿⣿⡟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣸⣿⣷⣤⣿⣿⣿⣿⠟⠀⠀⠄⠀⢸⡿⢿⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠺⠀⢈⡇
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣴⣷⠀⠀⠀⠀⠀⠀⢠⣿⠹⣿⠶⠂⠀⠀⠠⢼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠈⠙⠋⠁⠀⠀⠈⠁⢶⣧⣀⡀⣠⣤⠀⡾⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣦⠀⣠⡄⠀⢤⣴⣿⡿⠛⠻⣏⠰⠁⡇⢡⠀⠠⠀⣀⠀⡀
⠀⠄⠀⠀⠀⠀⠄⠂⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠿⠳⣯⠙⡂⢸⡅⣨⣷⡶⠆⣿⠀⠀⣿⠈⠄
⠀⠀⠀⠂⠀⠈⠀⠁⠀⠀⠀⠐⠀⠀⠀⠉⠀⠀⢀⣼⣿⡄⣽⡀⠹⡄⣰⣀⣤⠿⠀⠀⠛
⠀⠀⠀⠀⠂⠉⠒⠦⡌⠢⠐⠀⠀⠀⠀⠀⠀⠐⢉⣭⣿⠇⠨⢁⣀⠈⣙⣉⣡⣤⡥⠀⠀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⣀⣀⡀⣀⠀⡀⠘⠿⢛⡽⢠⡄⠀⠀⢉⣙⣻⣿⣿⣿⠀⡊⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠁⠀⠈⠋⢉⣭⣽⣿⣿⣿⡿⠟
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢠⣎⣼⣷⣾⡿⠿⠿⣿⣶⣦⡀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢰⡿⢋⡾⠃⠀⠀⠀⠀⠈⢿⣿⣿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠄⢀⣂⣰⣶⠶⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠄⠒⣂⣉⣭⡤⠤⠶⠦⠘⠂⠐⠒⠐⠒⠐⠒⣦⣴⡿
⠀⠀⠀⠀⠀⠀⠀⢠⠶⣲⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠒⠒⠒⠒⠒⢒⣒⣒⡒⢒⠲⣽⡿⠋
⠀⠀⠀⠀⡀⠤⠤⢼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣿
⠀⢀⠔⡡⠐⠂⠀⠺⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠒⡺⡇
⠀⡎⠎⢀⢔⣮⠿⣶⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣒⣂⣀⣀⢠⣷⠇
⢠⢸⠀⢸⡎⠇⠀⠙⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⠀⠘⢸
⠈⡎⡄⢸⢧⠇⠀⠀⢹⠼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⡞⡇
⠀⢱⢣⠈⡾⡸⡀⠀⢸⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⢤⣧⡇
⠀⠀⢫⢆⠸⣷⣣⠀⠀⡇⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹
⠀⠀⠀⢫⢦⠘⢷⣷⡴⣿⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⡾⡞
⠀⠀⠀⠀⠳⣕⢄⠈⠉⠁⣦⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢠⣥⠇
⠀⠀⠀⠀⠀⠈⠑⠯⣒⣦⡇⡎⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠺⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠐⠚⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠠⠴⠤⢼⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⢧⡿⢭⣀⣒⣒⣒⣒⣒⡒⢒⠒⠒⢒⣊⣉⣽⣴⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣒⣊⣒⣒⣒⣂⣀⣀⣨⣍⣭⣉⣉⣙⣐⣒⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿