Equipment Emojis & Symbols

⠀⠀⠀⠀⠀⣀⠤⠒⠀⠈⠉⠉⠁⠐⢄
⠀⠀⠀⢠⠊⡠⠐⠈⠁⢀⡀⠀⠈⠑⢄⠑⡄
⠀⠀⢀⠁⠎⠀⡀⠈⠠⠀⠀⢄⠐⡀⠀⢀⠘⡀
⠀⠀⠈⠀⠀⢰⠀⠈⠀⠀⠀⠀⠆⢰
⠀⠀⠸⠀⢂⠀⢆⠐⠄⠀⡀⠄⢀⠂
⠀⠀⠀⠑⠀⡀⠀⢢⠈⢃⠠⠒⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠉⠈⡆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠑⠤⠴⠉⠉⠉⠧⠀⠄⠤⠏⠉⠉⢱⠤⠠⠤⠞⠉⠉⠹⠆
⠀⠀⡀⠀⠀⠀⠀⠀⠇⠤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠢⡤⢤⣀⠦⠀⠀⠠⢴⠤⠄⡼⠄⠀⠀⠴⠤⠄⣰⠄
⠀⠀⠀⡄⠀⠀⢀⠠⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡄
⠀⢀⠀⡀⠀⠀⠈⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⢱
⠠⠀⠀⢐⠀⢀⢰⠂⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⠈⠃⠀⠀⠀⠀⢡
⠈⠁⠐⠜⠃⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡠⠊⠀⠈⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡘⠀⢀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⣄⠀⠀⠂⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢨⠀⠀⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠁⠂⠒⠒⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠄⠤⠤⠤⡤⠀⠠⠄⠤⢤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⡇⠀⢰⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡀⡄⠀⠀⠀⠀⣆⢀⣰⠀⠀⠀⠀⢀⢀⣀
⠀⠀⠀⠀⠀⢀⢤⢔⡁⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡇⠀⠀⠀⠀⡃⠀⣼⠀⠀⠀⠀⢸⠀⢠
⠀⠀⠀⠀⡰⣩⠊⠁⢑⠀⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠉
⡀⠠⠀⠤⠿⠧⠀⡔⠀⠠⠈⢢⠀⠀⠀⠀⠀⠀⢀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢

⠀⠀⠀⠀⠀⣿⢼⠀⢠⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠉⠈⠈⢈⠈⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠃
⠀⠀⠀⠰⣄⠀⠠⠰⢌⡀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠐⠈⠈⠂⠀⠘⠁⠈⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢀⢀⡀⠀⠀⠀⠆⠲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠁⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⠟⠂⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣧
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠗⠠⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡞⠀⢠⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡟
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠠⠃⠀⠀⠀⠠⠆⠀⠀⠀⠀⠈⠀⠀⠈⠧⠀⢠⠇⠀⠀⠀⠀⢀⠀⠀⢔⡫⠞
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠴⡐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡟⠀⠀⢠⡀⠀⢀⠴⠒⠀⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠊⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢆⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⡜⠁⠀⠘⡆⠀⠸⡆⠀⠀⠀⠀⠉⠈⠂⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⡀⠀⢠⠇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⡿⠀⠀⠀⠀⢀⠀⢤
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠷⠀⠀⢀⡀⠀⠀⠁⠀⠀⠀⠀⢠⠾⠛⠀⠀⠀⠀⠀⢠⠂⠓⢳⣝⢄⠀⠀⣀⠀⠀⢀⠔⠁⠁⠠
⠀⠀⠀⠀⠀⠀⠄⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠘⠧⡀⠀⢻⠁⠀⠠⢀⠂⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⣀⢀⠀⠀⠀⠀⠀⠾⠓⠀⠘⠀⠃⠀⠀⡄⠀⠀⠈⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠊⠀⢀⡀⠌⠀⠀⠀⠈⠀⠀⠀⡀⠀⢀⠀⠀⠀⠘⢦⠀⢀⣀⡐⠒⠊⠐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⡨⠎⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠡⡀⠀⣒⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠇⠀⠘⠰⠀⠀⠀⠀⠀⢀⠈⠀⡀⠀⢰⣾⡄⠀⠀⠀⠀⠀⢀⡰⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠠⠀⠀⠡⡀⠁⠀⠀⠻⡇⠀⠀⠀⠀⠐⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣬⠘⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢢⡀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠘⠁⠀⠠⡄⠈⠀⠀⡀⠁⠀⠀⠈⠀⠀⠒⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠈⠀⢀⡤⠒⠈⠉⠀⠉⠛⠒⠠⢄⠀⠀⠀⠀⠈⠀⠄⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⡴⠃⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⣤⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⢠
⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸⠀⢀⡿⠁⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⡀⠑
⠀⠸⣄⣀⣤⣤⠀⡴⠆⠀⠀⠀⠰⠁⠀⠀⠀⣐⡠⠔⠀⠀⠀⡇⠀⠈⠀⠀⠙⠀⠀⠘⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⢀⡔⠁⠀⠀⡐⠀⠁⠀⠄⠀⠀⢸⠁⠀⠀⠘⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠈⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⡟⠀⠀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠐
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⣴⡦⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⣀
⠀⠀⠁⠀⠀⠀⠉⠓⠿⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⡠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠈⠙⠁
⠀⠀⠀⠁⠀⠑⠀⠠⠀⠀⠀⠐⠀⣠⡿⠃⠀⢠⡐⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⢀⠈⠓⠪⣍⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠙⠢⢄⣠⠋⠀⠀⠘⠁⠙⢲⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⣾
⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣾⣿⣿⣿⣿⠀⢸⠀⢰⠐⠞⠻⠖⠀⣤⠀⣿⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⡾⠀⠘⠛⠛⠛⠛⠃⢠⣆
⣿⠟⠛⠛⠛⠂⢸⠀⢸⠀⠀⠁⠀⠀⡇⠀⣿⡄⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⢸⣷
⢯⡀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠰⠂⠀⠀⠀⠀⠀⢸⡇
⠀⠙⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢠⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⡇
⠀⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⢸⣿⠀⠀⠀⡟⠀⠀⠀⢻⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠎⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠟⠀⠄⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⣿⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣒⡒⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣸⣿⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⢻⡏⣤⣶⣟⢧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⡇⣸⡧⢿⣿⣿⣴⠇
⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣉⣉⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠂⠄⠀⢸⢹⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡆⢰⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⣴
⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠀⢸⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⢢⠀⣜⣇⠄⡇⡄
⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⠋⠉⠑⠲⠁
⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇
⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢰⠋⢙⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢸⠘⢃⠃⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⢠⠞⠀⠀⢠⠄⠀⠀⠀⠀⠘⠒⠋⠀⠀⡠⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⢀⡠⠔⠊⠁⠀⠀⠀⠐⠀⠀⠀⡠⠚⢁⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⡀⠀⠀⠀⠀⠓⠋⠁⠀⠀⠀⠀⠀⣠⠔⢁⡐⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢸⣿⣿⠀⠀⠤⠤⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠠⠠⡀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢸⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠘⠛⠋⡅⠀⠀⠀⠘⡀⠠⡀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠷⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⢱⠀⠀⠀⢀⣃⠀⠀⡨⠉⠉⡆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣁⡀⠤⠊⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣾⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⠉⠉⠉⠉⢙⣿⡇⠘⡦⣉⠁⠒⠢⡀
⠛⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⡁⠀⢱⠀⠀⠃
⠀⠀⠉⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠙⠛⠛⠛⠋⠀⡰⢄⡑⡀⣘⣤
⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠉⡻⣾⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⡡⢋⠜⢂⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⠃⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠛⡵⠃⢀⠌⠈⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡅⠀⣀⣀⣀⣀⣀⣀⡀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⡰⢁⡀⠀⠔⡡⣅⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⡈⢆⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢠⠊⠠⢊⣼⡣⠗⠁⠀⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠧⠠⣤⡥⠤⠦⠈⣄⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⡐⠁⠀⠙⠊⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⢾⣦⣄⡀⠀⢸⡂⠀⣀⣤⣾⡆⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⡝⠿⢿⣧⣸⣥⣾⠿⠟⠉⠈⡄⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⢄⠀⣿⣿⣿⠇⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⢀⠁⠀⠈⠻⣿⡿⠟⠀⠀⠀⠀⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣾⣍⠁⠀⠀⠀⢐⠠⠄⠐⠒⠒⠒⠐⠚⠀⠢⢄⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠁⠀⢀⠔⠁⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠁⢾⡇⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠏⢠⠀⠀⡌⠀⠀⠠⠊⠁⠀⠀⢀⠇⠀⠈⠲⡀⠀⠈⠃⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠐⠚⠐⠂⠀⡄⡇⠀⠀⡄⠀⠀⠀⠀⡜⠀⠀⠀⠀⠁⠀⠀⠀⠰⠒⠒⠒⠒⠛⠛⠛⠒⠒⠒⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣇⣀⣀⡧⠤⠤⠤⣴⣁⡤⠤⠄⠔⢦⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⡇⠀⠀⠀⠓⠐⠃⠀⠀⠀⢰⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠛⠛⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡇⠀⠀⠀⣶⣶⣶⠀⠀⠀⢘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠋⣛⠛⢛⠛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠒⠒⠂⠀⢰⠂⠀⠀⠀⠒⠀⠀⠚⠁⠀⠀⢸⠀⠀⠐⠃⠀⠀⠉⠃⠀⠀⠛⠈⠉⠉⠉⠉⠉⠉⠋⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⠄⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡼⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡎⠩⠩⠍⢁⠒⠠⢬
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠜⠁⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠖⠀⣁⠤⠐⠀⠀⠀⠀⠀⠬⡀⢅⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠃⠈⠐⠲⡄⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⢠⠔⠁⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠁⡠⠊⠉⠑⢄⠈⣤⠜⠀⠀⠀⠀⠱⠄⠠⠂⠀⠀⠀⠀⠢⠀⠔⠁⠀⠀⢀⠊⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢃⣶⣶⣶⡆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠉⠉⠉⠀⠀⡌⡉⡍⡉⡉⢉⡉⣷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣺⣀⣨⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⡀⠀⠀⠀⣸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠒⠉⠀⠁⠉⠉⠉⣉⠙⠛⠋⠭⠝⠋⠉⠉⠉⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣥⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣪⡿⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁⠀⠀⠃
⠀⠀⠀⢀⡀⠊⠉⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠒⠿⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢤⡦⣀
⠀⠀⠀⠸⠀⠀⠀⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠤⠤⠒⢻
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡇⠀⠀⠀⠃⢱⠀⠇⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢄⠀⠀⠀⢀⡜
⠀⠀⠀⠀⢐⢲⣧⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⣀⠔⠋⠑⠊⠁⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠁⠀⠚⡀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠒⠊⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠠⠦⠄⠀⠀⢄⢀⡀
⠀⠀⠀⠓⠀⠀⠠⠐⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⣑⠀⡇⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⠀⠆
⠀⠀⠀⠀⠀⠄⢤⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣄⣄⠀⣀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠀⠈⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠓⠄⠄⠀⠤⠋⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠒⠒⠂⠀⠑⠢⠼
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⡠⠤⠤⠤⠤⠤⠠⠤⠤⠄⠢⠀⠀⠀⠀⠀⠠⡆⠂⠀⠒⠒⠒⠀⠀⠄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⢄⠀⠘⡀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⠦⠔⣦⠀⠀⢰⠔⠒⠾⣄⠀⡊⠀⠀⡽⠐⠢⢀
⠀⠀⢀⠠⠐⠁⠡⠀⠀⠄⠀⠄⠰⢐⢸⠀⠀⠀⠸⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⡠⠚⠀⠀⠻⣄⠀⠀⠀⠘⠀⠀⠂⠀⠀⠄⠀⢠⠈⠢⡼⠁⠀⢀⡞⠉⠢⡀
⠀⠀⢣⠀⠀⠀⠀⢃⠀⠀⠀⠈⠡⣘⠾⡁⠋⠀⠡⢉⠀⢡⡤⠤⠤⠤⣤⣤⣶⣖⣾⢣⡈⠉⠋⠁⣸⠁⠀⠀⠉⠀⠀⠉⠀⠀⠃⠀⢸⠀⠠⠉⠐⡤⢞⡀⠀⠠⠁
⠀⠀⠀⠡⠀⠀⠀⠈⢆⠀⠀⠐⢒⠯⢟⡹⢖⠒⠒⡞⠒⢺⠿⠿⠿⣿⣟⡟⢛⣛⣛⠒⠙⢦⠀⠐⡗⠒⠒⠒⠒⠀⠀⠒⠒⠒⠃⠀⠸⠠⠁⠀⡰⠁⠀⠀⠉
⠀⠀⠀⠀⠑⡀⠀⠀⠘⠃⠀⠀⠈⠀⢠⠃⠨⠀⠀⢠⡀⠘⡋⠉⠉⠉⠉⠉⠉⢉⠝⠀⠀⠸⡆⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡐
⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⡘⠀⡈⢡⠀⢇⠀⢀⣣⡀⠀⠀⠉⠐⠀⠊⠀⠀⠀⠀⠀⠀⢧⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡐
⠀⠀⠀⠀⠀⠀⠈⢢⣰⡄⠀⢀⠀⢠⠀⠀⠆⠈⠆⢸⡥⣷⠀⠀⠀⠰⠶⢶⠀⠀⠀⠀⠀⠀⠸⡄⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢘⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⠀⠈⠢⡄⠀⠀⠈⠄⠠⠀⠁⠘⡄⠀⠀⠰⠒⠚⠀⠀⠀⠀⠀⠀⠀⣧⠀⠸⡆⠀⠀⠀⠀⠀⠀⡔⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠄⠀⡆⡀⠘⠀⠀⠀⢘⠀⠀⠇⠀⠀⠀⠀⠀⠀⠸⡀⠀⢷⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠊⠒⠐⠒⠀⠃⠃⠐⠒⠒⠀⠚⠒⠃⠂⠀⠀⠀⠀⠀⠀⠀⠓⠂⠘⠂⠒⠒⠀⠚⠀⠒⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠊⠋⠀⠀⠀⠈⠋⠂⠌⠅⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠁⠀⠉⠉⠉⠉⠉⠉⠉⡩⠋⠁⠀⠀⠀⠇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠲⠤⠤⠤⠤⣄⣀⣀⣀⣀⣤⠤⠤⠤⠴⠒⠒⠐⠚⠉⠉

⡀⣀⣀⢀⡀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⢿⢟⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣟⣿⣿⣛⣿⣟⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠸⠷⠰⢦⣬⣿⣯⣦⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡄⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠥⢀⠀⢰⡿⠻⠺⣿⠛⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣶⣶⣿⢡⣤⡄⣬⣟⢧⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⣷⡶⣶⣶⡄⠛⣟⣛⣿⣿⣿⣿⡟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⣿⣿⣿⢸⣿⡇⣿⡏⠀⠱⣜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⢸⡇⣿⡆⣿⠀⣫⡉⠛⠛⢛⡛⠓⠃⠀⠀⠀⠀⠀⢸⣶⡖⢀⣠⡤⠴⣸⠟⠛⠛⠓⠘⠇⠀⠀⠈⣖⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠚⠐⠹⢿⠇⠈⠛⠁⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠰⠿⣿⣇⠜⠠⣴⡟⢹⣇⡛⣧⠾⠛⠻⡶⢶⣾⡋
⠀⠀⠀⠀⠀⠀⣀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠙⢧⣶⣾⡿⢿⣻⣯⣥⣭⡆⠌⣷⢰⡀⢹⣽⠄
⠀⠀⠀⠀⠀⠀⣿⡄⠉⣉⣒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⠛⠁⠀⠋⠉⢿⣧⣼⠏⠀⠀⠙⠷⠌⠡⠂
⠀⠀⠀⠀⠀⠀⣿⣇⡸⢿⡉⠙⠒⠛⠛⠛⠓⢄
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠘⠋⠛⠓⠲⠦⣤⣸⣇⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣆
⠀⠀⠀⠀⠀⣸⣿⡗⠁⠉⠀⠀⠀⠀⠀⠈⢿⣷⡆⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣿⣽⠛⠛⠛⢧⡀⠀⠒⢄
⠀⠀⠀⠀⠀⣿⢹⣧⣀⠀⠀⠀⠀⠀⠀⠀⠈⣎⠻⣶⣿⣿⣿⣷⣶⣦⡆⠀⠀⠀⠀⣤⣭⡍⣈⣉⣀⣬⣯⡄⠀⢴
⠀⠀⠀⠀⠀⠚⣛⡿⢠⡀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣿⣟⣀⣐⣉⣁⠀⠀⣆⣽⡿⠿⢿⣿⢋⡍⢻⠿⠃
⠀⠀⠀⠀⢀⣿⠉⣰⠏⡀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⢿⡿⡿⢦⠀⠀⠲⠾⠤⣄⣉⣿⣿⣯⡌⢰⣶⣶⣶⣿⣿⣿⣿⡿⠆
⠀⠀⠀⢀⣙⣹⡀⠀⠀⠠⠰⠀⠀⠸⣿⣿⣿⣿⠿⢿⣿⣯⣧⣧⣴⠷⠞⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠘⠛⠛⠉⠉⠁
⠀⠀⠀⠸⢿⣿⣿⡄⡀⠀⣁⡀⠴⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣿⣽⣯⣯⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠟⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡴⠀⠀⠀⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠎⣡⠾⡇⠀⠀⠀⠀⠀⡀⠀⠛⠓⢠⣴⠲⡄⣠⡤⣤⡤⣀
⠘⠿⠷⢶⡃⠀⠀⠀⠀⠀⠀⣀⡀⠐⠒⠒⢚⡧⣽⠉⢩⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬
⠀⠀⠀⠀⠙⣄⣠⢀⠠⠴⠶⣿⠀⠀⠀⠀⣽⣿⡟⠀⢸⠀⠐⠀⠀⠀⠀⠀⡆⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌
⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⢻⠀⠀⠀⠤⣬⣭⠅⠀⢸⡤⡀⠀⠀⠀⠀⠀⡇⠀⠼⠀⢸⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⢛⠀⠀⠀⠀⢼⡏⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠠⠞⠀⠀⠀⠀⠀⠒⠀⠠⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡼⠀⠀⠀⠀⢽⡅⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡧⣠⣀⢠⡀⠨⣇⣀⣄⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⢧⠶⠂⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠁
⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠘⢻⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⢀⣌⠀⣥⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠸
⠀⠀⠀⡜⠀⠀⠀⠀⠧⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⣤⠄
⠀⠀⣰⠆⠀⠀⠀⠀⢠⡄⠀⠀⠀⠈⠈⠀⠙⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⣾⠀⠀⠀⠀⠀⡂⢧⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠐⣸⠇⠀⠀⠀⠀⠀⠘⢺⡄⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤
⢇⡼⠀⠀⠀⠀⠀⠀⠀⠇⠵⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠰⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠨⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣬⣇⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐
⠀⢄⠁⠀⢸⠇⠀⠀⠀⢠⣀⠤⠄⠀⠓
⠀⠈⠳⣄⡀⠢⣌⡐⠂⣿⡇⠀⣀⣀⢈⠁⠀⠀⠀⠈⢀⡀⢀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠉⠑⠒⠮⠭⠭⠿⣋⣁⡉⠉⠀⠀⠀⠀⠀⢃⣀⠈⣻⣶⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁⠉⠓⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠄⠀⠀⠀⠰⠄⠀⠀⠀⡐⡀⣖⡉⡀⠀⠀⠀⠀⢀⠄⠂⡭⠉⠀⡄⠀⠀⢢
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⡀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠥⠀⠉⠙⠂⠀⠀⠃⠀⠀⣅⠀⠀⠛⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠤⠄⠤⠀⠀⠀⠀⠀⠀⡈⠁⠀⢀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⡄⠈⠁⠂⠀⠀⡂
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠤⠤⠠⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣔⠁⠀⡐⠉⠢⠠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡄⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢙⡀⠀⠅⢀
⠀⠀⠀⠀⠀⠄⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠶⠮⣤⣉⣸⣧⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠸⠥⠙⠻⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⡀⢲⡔⠀⢄⣀⢴⣀⠉⠉⣭⠥⠬⠭⠥⠤⡆⠀⠒⠀⠀⠀⠘⡇
⠀⠀⢘⡁⢀⢀⣀⣀⣄⠼⠆⠀⠐⠀⠀⠒⠆⠀⠀⠂⠐⠒⠂⠀⠒⠀⠀⠷⠖⠀⡀
⠀⠀⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠂⠀⠀⠉⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⣀⡀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠀⠈⠂⡄⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠒⠁⠀⠑⠒⠋
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢢⡤⢲⠀⠀⠀⣠⠀⡴⠐⢤
⠀⠀⢀⠀⠀⣀⡀⡀⢀⢀⣀⡀⢠⡄⠀⠀⠀⠀⠀⢀⠻⡆⠀⠀⠀⣠⠠⢥⡴⠤⡄⠀⠀⠀⠈⡀⠀⠀⠀⢻⠀⠀⠓⠎⣁⣀⣀⠁⠶⠃⠀⢸⠇
⠀⣀⢨⠂⠊⠔⠌⠂⠑⢐⠂⠀⠀⠠⠄⠀⠀⠀⠶⠦⠠⠄⠀⠀⠀⠸⠀⢐⠂⠀⠧⠀⠀⠀⠸⠇⠀⠀⠀⠈⠳⠖⠂⠀⠹⠷⠞⠀⠀⠒⠶⠋
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠘⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠀⠂⢁⢀⠀⠠⠀⠀⠀⠀⠄⡕⠀⠀⠀⠀⡐⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⢀⠀⠄⠈⠀⠀⠀⠀⠀⡠⠀⠈⡦⠄⠀⠀⠀⠀⠀⠀⠀⡠⠂⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠐⠊⠀⠁⠀⠂⠁⠀⣀⠄⠁⡀⠀⠀⠐⠈⠀⠀⠀⠠⡆⠠⡆⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠈⠉⠁⢠⡠⠐⠆
⠀⠀⠀⠀⠀⠀⠊⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠈⠁⠀⠀⠀⠠⠀⠉
⠀⠀⠀⠀⢀⡄⠂⠀⠀⡄⠀⠂⠁⠀⠀⢠⠆⠀⠐⠖⠈⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠒⢃⠀⠀⠀⠘⠀⠀⠀⠀⠐⠃⠀⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡆⠀⠙⡂⠀⠠⠀⠀⠀⠀⠴⡄⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠴⠀⠀⠀⡴⠄
⠀⠀⠀⠀⠀⠢⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠁⠀⠈⠀⠀⠀⠀⠀⠠⣀⠀⠤⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁⡀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⠉⠀⠨⠭⠍⠁⠉⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠣⡦⠀⠀⠀⠀⠈⠙⠀⠀⢀⡀⠀⠀⠀⠀⠀⠨⠀⠀⠘⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⢀⡀⡀⠀⠀⡅⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⠀⠀⠀⠀⠆⢀⠈⠁⢀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠒⠀⣀⣐⠀⠸⠄⠀⠂⠁⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠰⠄⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠰⠄⠈⠑⠀⠀⠰⠀⠀⠀⢐⠂⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⠱⢄⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠄⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠁⠈⠂⠠⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠖⠀⠀⢀⡔⠀⠀⠀⠠⠈⠇
⠀⠀⠀⠀⠀⠙⠀⠀⠈⠃⠄⣀⠀⡀⠀⠀⠀⠊⣀⠀⠠⡀⠘⠁⠀⢁⠰⢦⠐⠝⠶⠂⠀⠀⢀⡠⢎⠁⠀⠀⠀⠀⠐⠀⠐⠂⠂⠁⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢄⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠉⠁⠺⣄⠀⠀⠀⠀⠀⠕⠀⠀⢄⠨⠁⠀⠀⠠⢢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠂⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠔⠈⠅
⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⡀⠀⠀⠀⠐⠃⠀⠀⢀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⠊⢀⠀⠀⠀⠀⠐⠠⠂⠐⠈⠢⠀⠀⠀⠀⡀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⠀⠄⠀⣀⢀⣀⣠⠀⠄⠈⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠄⡀⡀⠀⠀⠙⢀⠇⠁⢀⠀⠂⠁⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠈⠀⠀⠀⠀⣇⡀⠂⢐
⠀⠀⠀⠀⠀⢰⠁⠐⠠⠁⠀⠒⠀⠀⠠⠐⠃⠀⠈⠃⢀⣀⠨⠀⠀⠀⡀⠀⠀⠀⠀⠀⡘⠀⠀⡠⠆⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠙⢠⠒⣆⠉⠄⠀⠀⠀⠀⠀⠈⠁⢀⠐⠒⠂⠀⢀⠄⠀⠁⠀⠀⠉⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠘⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠄⠀⠀⠐⠄⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⢠⣎⣠⠀⠀⠀⠀⠀⠀⢰⣆⣠⠀⠀⠀⠀⠀⠀⠀⣰⠗⠀⠀⠀⠀⠀⢀⣰⣟⣠⠀⠀⠀⠀⠀⠀⣠⣎⡄⠀⠀⠀⠀⠀⠀⣠⣆
⠀⠀⠀⣠⡾⠋⠉⠀⠀⠀⠀⢀⣴⠟⠉⠁⠀⠀⠀⠀⢠⡠⠎⠀⠀⠀⠀⠀⣠⠖⠋⣹⠏⠁⠀⠀⣠⠶⠚⠉⣽⠁⠀⠀⠀⢀⣠⠤⢲⠟⠁
⣠⣤⡾⠋⠀⠀⠀⠀⠀⡀⣴⠟⠁⠀⠀⠀⠀⠀⣠⣶⡟⠉⠀⠀⠀⠀⠀⠊⠀⠀⡴⠋⠀⠀⠀⠀⠁⠀⠀⣠⠏⠀⠀⠀⠰⠋⠀⢀⡟
⠉⠸⠃⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠰⠛
⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢀⡴⠒⢦⠀⠀⠀⠀⠀⠀⣠⣶⡦⠀⠀⠀⠀⠀⠤⢤⣄⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⡎⠉⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠀⠀⠀⠀⠀⠀⣠⡜⠛⣧⠀⠀⠀⠀⠀⢈⡿⣿⡷⠀⠀⠀⠀⢀⣾⠿⣷
⠀⢀⣄⠀⠀⠈⢢⠀⠀⠀⠀⣨⠖⠀⠀⠀⠀⠀⣠⡞⢿⣻⠏⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠠⠈⠉⠣⣀⡾⠏⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠘⢷⡜⠉⠀⠀⠀⠀⠠⡾⠋⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⢰⡟⠁
⠀⢀⣀⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠨⡏⠉⠉⡅⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡻⠟⠀⠀⠀⠀⠀⠀⣼⣯⠖⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⣀⣴⣿
⠀⠘⡟⠛⠛⡇⠀⠀⠀⣨⡷⢙⣦⠀⢠⣤⡀⠀⠀⢠⡾⢁⡼⠃⠀⠀⠀⠀⠀⢀⣴⠟⠛⠃⠀⠀⠀⠀⢀⠀⠋⠁⠀⠀⠀⢀⣤⣾⣿⠿⠋
⠀⢰⡇⠀⠀⡇⠀⠀⠀⠿⠦⠶⠿⠿⠶⠿⠀⠠⠞⠉⡴⠋⠀⠀⠀⠀⠀⢠⣴⡟⠁⠀⠀⠀⠀⠀⣠⣾⣿⣷⡦⠀⠀⠀⢠⣾⠛⠋⠁
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠓⠋⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠀⠀⠀⠀⠀⠈
⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⢀⣤⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣶⣶⣶⣶⡄
⠀⠀⠙⠳⣦⣰⣷⠀⠀⠀⣾⣦⣤⣼⣿⡇⠀⠀⢸⣿⣿⣿⢿⠛⠂⠀⠀⠛⢻⣿⠛⠋⠉⠀⠀⠀⠀⠀⢀⣴⣿⡟⠃⠀⠀⠀⠉⠉⠉⡩⠜
⣰⡖⠒⠛⣿⣿⣿⠀⠀⠰⠿⠭⢭⣿⣿⡇⠀⠀⣿⠃⠀⠀⢸⠀⠀⠀⠀⠀⣸⣏⡀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⢰⡆
⠙⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠉⠀⠀⠀⠘⠀⠀⠀⠀⠀⠛⠻⠃⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣤⣴⣤⡀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠁⠀⠀⢠⣤⣤⡀⣾⠛⣷⠀⠀⠀⠀⡀⠀⣿⣻⡧⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⢼⣿⣛⣿⡷⠀⠀⠀⠽⠒⠒⠒⠒⠒⠂⠀⠀⢻⣧⣼⠃⣙⡛⠁⠀⠀⢰⣿⣿⣿⡏⠋⠁⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠛⠛⡟⠛⠃
⠴⠾⠿⠿⠿⠷⠶⠀⠀⢰⣶⣦⣶⣤⣤⡄⠀⠀⠀⠀⠀⢸⣏⣹⠆⠀⠀⠽⣿⣶⣿⠏⠀⠀⠀⠀⠀⠻⡷⠦⠄⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⣀⡀
⢼⡟⠛⢿⡆⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠀⢀⣤⣤⣄⣀⣠⣤⣤⡀⠀⣴⣶⣿⣶⣷⣶⣦⠀⢀⣀⣤⣶⣾⣿⠎⠀⠀⠀⢠⣾⣷⣿⣿⣷
⢸⣧⣥⣼⡇⣀⣀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠚⠛⠛⠛⠓⠚⠁⠀⣿⣥⣿⣿⣿⣼⣏⠀⠀⢉⣽⡿⠯⣉⠄⠀⠀⢠⣾⣿⣿⡿⠿⠛

⠀⠀⠀⠀⢀⠴⠂⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠀⠐⢄
⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠒⠒
⠀⠀⠀⠀⠉⣉⣽⣿⠿⡏⠁⠀⢠⠊⠉⢲⣀⡾⠋⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣽⣿⡿⠏⠁
⠀⠀⠀⠀⠀⡿⠋⣁⣤⡇⠀⠀⢸⠀⠀⣰⠋⢧⡀⡀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢁⣠⡆
⠀⠀⠀⠀⠀⣶⣾⠿⠛⡇⠀⠀⠀⠠⡀⠂⠀⠀⠁⡐⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠟⣀⣤⣶⡇⠀⠀⠀⠀⠈⢔⢄⠀⠀⠂⠀⠀⠀⠠⡈⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⢀⠇⠀⠆⠀⠐⠀⢠⡤⠴⡆⠀⠀⠀⡟⢁⣤⣶⡇
⠀⠀⠀⠀⠀⣿⠿⠛⢁⡇⠀⠀⠀⠀⠀⠈⡀⢱⠆⠀⠀⠀⠀⠀⠙⢀⣀⣀⡐⠁⠀⠀⠀⠀⠀⡜⠀⡘⠀⠀⠀⠀⢐⠒⠿⠅⠀⠀⠀⣿⡿⠛⢉⣀
⠀⠀⠀⠀⠀⣃⣴⣾⡿⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠹⠀⠀⠀⠀⠀⢰⠃⢀⠃⠀⠀⠀⠀⢒⣛⣭⡅⠀⠀⠀⣏⣤⣶⡿⠇
⠀⠀⠀⠀⠀⡿⠋⣁⣤⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢇⠀⠄⠀⠀⠀⠀⠀⠠⣬⣬⡇⠀⠀⠀⡿⠛⢉⣠⣦
⠀⠀⠀⠀⠀⣶⣾⠿⠛⡇⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠜⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠩⠥⢖⡂⠀⠀⠀⣷⣾⡿⠟⠁
⠀⠀⠀⠀⡀⠋⣁⣤⣾⢇⡀⠀⠀⠀⠀⠀⠈⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣃⠀⠀⠀⠀⠀⠀⠀⣘⢒⣲⠇⠀⠀⣀⠛⢁⣤⣶⠗⡀
⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠠⠠⢀⣀⣀⣀⣀⠉⠉⣉⣙⣉⣁⣈⠭⠂⠀⠀⠀⠀⠀⠂⠉⠭⠅⠀⠀⠠⠠⠤⠤⠤⠤⠂
⠀⠀⠀⠀⠈⠢⠄⠀⠠⠊⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⣀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣀⠀⢠⠀⠀⢠⠀⣀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣦⡯⠐⠁⠀⠀⠀⠒⢸⣧⡆⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠠⣁⡟⣤⠀⠀⠀⠀⠀⠀⢠⠙⣅⡔⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠓⠷⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⣀⡱⠃⠀⡀⠀⠀⠀⠀⠀⠀⣀⢘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢀⡀⠀⠀⠀⠀⠀⠀⡰⠁⢠⣥⣤⡦⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠨⠗⠂⠀⠀⠀⠘⠬⣿⡟⠦⡀⠀⠀⠀⠴⢪⣿⡧⠊⠀⠀⠀⠀⠀⠀⢀⡧⠄⢺⡇⠀⣹⠢⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⣦⢀⠀⠀⣌⣠⡚⣉⠀⠀⠀⠀⠀⢀⣤⠊⠍⢹⣆⡌⠛⠛⠁⠀⣶⠋⠁⢦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡆⠀⡿⠏⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠈⠓⠒⠒⠒⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⠒⠶⠶⠒⠒⠦⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠠⠈⠟⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠶⠱⠀⠐⠈⠁⠀⠀⠀⠉⠁⠀⢀⠀⠀⠀⠀⡀⠀⠀⠹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠓⠂⠸⡀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⢄⡠⠐⡄⠠⠆⠀⠂⠐⠈⠀⠀⠀⠁⠈⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠒⠆⠀⡇⠀⠀⠀⢀⠊⠃⠁⠁⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⣷⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠰⠀⢀⣀⡀⢀⣤⣄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣒⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⢀⡀⢀⡄⢠⡶⠤⠰⢖⣶⠘⣟⢸⡇⣿⠉⣿⢸⡄⢻⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠍⠀⠸⠀⠀⠀⠀⢰⣛⣦⢾⠉⣻⠈⣇⢘⡧⠹⣆⡴⠸⠯⠿⠇⠛⠂⠃⠈⠉⠁⠈⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣤⡄⢠⡄⠀⠀⠀⠘⠷⠗⠈⠛⠛⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠠⠄⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠠⠆⠠⠤⠀⠔⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⣧⠀⠀⠈⡄⠀⠀⠜⠈⠈⠁⠁⠈⠀⠀⠀⢀⢀⢀⠀⠀⢀⠀⡄⠀⠀⠀⠐⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣍⡁⢿⠀⠀⠀⠈⢆⠀⢠⡀⠀⡄⠀⠐⠰⠘⠁⠀⠀⠀⠀⠀⠀⡈⠀⠀⢀⠀⢀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⠂⢸⠀⠀⠀⠀⠀⠱⢄⠀⣀⠀⣀⢀⢀⠀⠀⡄⠠⠐⠀⠀⠀⠃⠀⠀⠈⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠆⢸⡇⠀⠀⠀⠀⠀⠀⠙⠢⣈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⣄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⠤⣤⣤⣤⣤⣤⣴⣶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢒⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷
⣤⣤⣤⣤⣤⣤⣤⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣶⣦⣤⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣤⣤⣴⣴⣴⣶⣴⣦
⣿⣿⣇⣀⣀⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⡌⢠⣤⢠⣥⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃
⠀⢠⠀⠈⠐⠀⠀⠀⢐⠀⢀⢰⠘⠃⣠⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⡀⠐⠀⠀⠀⠀⠈⠈⠀⠁⠀⠉⠀⠀⠀⠀⠀⠐⠃⠸⠄⡀⠠⠀⠀⠀⠀⠀⠐⠃⠀⠇⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠘⠇⠀⠐⠀⠀⠀⠀⠰⠀⠀⢤⡤⠀⡄⢀⠀⠀⢀⡆⢰⡂⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⡄⠀⠀⠦
⠐⠄⡀⠀⢀⠀⠈⠀⠀⠢⠀⠁⠀⠀⠌⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠴⠀⠆⢴⢤⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⡠⡄⠀⠰⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠐
⠀⠀⠀⣀⠀⠀⠤⠀⠷⠀⠀⢀⡀⠀⠀⡇⠀⠀⢀⡀⡀⠀⠀⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⢰⠀⠀⠀⠓⠤⠤⠀⠀⠀⢀
⠀⠀⠈⠁⠀⠀⢰⡆⠀⠀⢀⡌⠀⠀⢠⢣⠤⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠄⡀⠉⢁⡀⠄⠀⠠⠀⡀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠈⠃⠀⠀⠀⠀⡰
⠠⠀⠀⠀⡀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠸⣄⣤⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠋⠁⡀⠀⠙⠄⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠐⠀⠀⣄⠀⠠⢠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂
⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠀⠀⠀⠀⠀⠘⢿⠂⠀⣼⠀⠂⢠⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠋⡀⠆⠑⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠃
⠤⠤⠀⠀⠠⠆⠀⠀⣤⠐⢀⠀⠘⣿⣤⠀⠀⠀⠀⠀⠀⠶⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠄⠀⢠⠄⢄⣀⡀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⡎⠂⠀⠀⠀⠀⠀⠠⠠⠐⠒⠒⠂⠈⠁⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⡼⠁⠀⠀⠀⠀⠨⠆⠸⠠⠅⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⣾
⠠⠤⠖⠀⠀⠈⠀⠀⠰⢤⣠⣄⠤⠚⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠀⠀⠆⠂⠀⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣄⠀⠀⠀⠀⢰⡀⠀⡀⠀⠀⠀⠁⡀⠀⠀⡁⠃⢀⠠⣭⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢁⠐⠀⠂⠀⠀⠀⠄⠀⠰
⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠦⠸⠄⠚⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠀⠀⠀⡆⢀⠀⠀⠔⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠸⡦⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠱⠠⠤⠶⠇⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠆⠀⠠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠚⠀⠀⠘⠒⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠰⠶⠃⠀⠀⠀⠀⠀⠀⠢⡈⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂
⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠓⠒⠦⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠐⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠈⡷⢸⠀⠀⠀⠀⠀⢠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠐⠄⠀⠀⠘⠀⠀⡄⢀⠀⢠⠀⠀⢠⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⢀
⠅⡁⠀⠂⠀⠀⡀⠀⠘⠄⠀⠀⠀⠀⠀⠁⠀⠐⠦⠀⠀⠀⠀⠂⠀⠀⠇⠈⠠⠆⠀⠉⠠⠅⠀⠀⠀⠀⠀⠁
⢐⡦⠀⠀⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⢠⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂
⠀⠛⠀⠁⠓⠀⢀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⢀⡁⠒⠀⠀⠀⠀⠜⠈⠀⠀⠀⠀⠀⡀
⠀⡄⠀⠤⠀⠀⠈⠄⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠖⠀⠀⠀⠐⠦⠀⠀⠀⠠⠀⠀⠀⢸⠔⡇⠀⡇⠁⠀⠤⠤⠤⠤⠤⠤⠤⠤⠴⠴⠭⠤⠤
⠀⠁⠀⠀⠀⠀⠀⡀⠀⠠⠂⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣶⠴⠤⠀⠠⠤⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⢟⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡶⣦⣤⣤⣀⡀⡀⠀⠀⡀⢀⠀⢀⣠⣶⠿⡻⣵⠫⣺⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡻⠊⠁⠚⠉⠩⠛⠛⠻⠦⢵⣽⣾⢿⡯⣖⢌⣨⢗⡫⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⢿⣬⣀⠀⠀⢀⣀⡀⠀⠀⠀⡜⡱⣣⢟⡺⣷⣭⣶⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡀⠉⠛⠛⠣⠶⢦⣬⣀⣀⡀⣗⣭⣾⡿⣟⡛⠀⠑⠉⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⢻⣦⠠⠀⠀⠀⠀⣀⣈⣉⣉⣛⣋⣓⣀⣀⣀⣀⣀⣤⣤⣤⣬⡓⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠂⠉⠙⣕⡄⠉⠻⣿⢛⠟⣻⣯⢿⠋⠫⡫⡫⡫⡫⡮⡪⡺⡽⠹⡻⡑⡕⢽⢸⣀⢀
⠀⠀⠀⠀⣠⣄⡚⠅⠀⠀⠀⠀⠀⠈⠻⣶⡆⣟⢵⢟⢕⢕⢅⠅⡨⡪⡪⣪⣪⡪⣪⣪⣪⣪⠪⠮⢾⢞⣼⣿⡟
⠀⠀⠀⠀⠘⡦⠦⠥⣔⣐⣠⠄⣀⡀⠀⣉⡻⠽⠷⣯⡭⠭⡭⠭⠽⠶⠶⠶⠄⠖⡲⠟⠒⠂⠿⠛⠋⣹⣿⣻
⠀⠀⠀⠀⡴⢧⣀⣀⠀⠠⠸⠓⠒⠾⠃⢢⢻⡀⡹⣯⣶⣿⣯⣿⣭⣥⣔⡂⢒⣀⡀⠠⢄⣀⣀⣠⣾⣯⣻⣹⠇
⠀⠀⠀⠀⠉⠑⠒⠛⢹⣷⣖⡶⠤⡤⣤⡤⣸⣵⡷⣿⠒⠀⠈⠄⠀⠉⠉⠉⠋⠁⠀⠈⠉⣩⣏⣭⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⡠⠃⠈⠉⠉⠉⠛⠓⠚⠛⠛⠻⠽⣷⣲⣤⠄⣀⣒⡚⠓⠲⠄⣀⡀⠀⣟⢞⣿⣿⢿⠛⠙⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣱⣀⣀⠀⠤⢤⣤⣤⣄⣄⠀⠤⠤⣄⣈⡘⢛⣳⣶⣶⣿⣧⣶⣶⣒⣠⣽⣮⠾⠻⠿⠿⢿⠿⡂
⠀⠀⠀⠀⡤⣴⢫⡟⡉⡻⠭⠭⠍⠉⠀⠀⠈⠉⠉⠉⠉⢉⢉⠉⠁⠀⠀⠀⠀⠀⠀⠠⢈⡉⠳⠒⠈⠀⠀⢸⡪⢐
⠀⠀⠀⠀⠈⢺⢸⣆⠫⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢂⣀⣀⣀⣀⡀⡀⠙⢀⣀⣀⣀⣀⣘⣦⣕⠅⡀
⠀⠀⠀⠀⠀⠀⠑⠽⣷⣶⡖⣲⢒⡲⣶⣶⣶⣶⣆⣉⣩⢿⡝⣿⢻⡖⡶⠶⡶⡦⢦⢶⠿⡶⢪⢖⠧⠞⠟⠟⠻⠃⠕⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⠁⠈⠈⠁⠉⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡏⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣤⣿⠂
⠀⠀⢢⠀⠀⠾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣅⡀⠀⠀⠀⠀⡌⠀⠀⠀⠀⣿⠟⠁⡆
⠀⠀⢀⠱⡤⠀⠰⢖⡥⢐⠀⠠⠀⠀⠀⠀⣉⣭⣉⣭⣉⠀⠀⠀⠀⠑⠀⠀⠀⣄⠀⠀⠐⠁⠀⠀⢀⣷⣶⣤⣄⣦⣠
⠀⠈⢂⡩⠝⠒⠪⠥⠊⠁⠀⠀⠀⠀⠀⣸⡿⠛⣿⠛⢿⣇⠀⠀⠀⠀⠀⠀⠴⠿⠦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⠿⣡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣹⣭⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⣿⣿⡟⢿⡿⣹⠖⣺⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⣁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣿⣿⣿⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣼⠏⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⣿⣷⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣿⣷⠶⠄
⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣛⣿⣟⣛⣿⣟⣻⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠿⠀⢃
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣩⣀⣏⣉⣇⣸⣁⣹⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣶⢶⣶⡶⠿⣻
⠀⠄⠄⢤⣿⡧⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠣⡼⢦⢧⢤⢧⠾⡤⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠾⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠙⠛⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤
⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⠀⠀⠀⠀⠀⠀⣴⠑⡆⠀⣧⢎⠀⡴⠂
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠊⠘⠀⠹⠜⠀⠙⠤
⠀⠀⠀⠀⠀⢰⢻⠛⠛⠛⠛⠛⠛⠛⣿⠲⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⠀⠀⠒⠒⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠀⡏⣿⠉⠉⠉⠛⠋⢹⡏⡆⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⣈⣉⣉⠁⠸⡇⠀⠀⠀⠀⠘⣿⠛⠛⢛⣛⠛⠛⠛⢻⡟⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⣤⠴⠦⠿⠤⠤⠴⠶⠶⠾⠷⠒⢶⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣿⠀⠀⠼⡯⠀⠀⠀⢸⡅⠀⠀⠀⢠⠏⢹
⠀⠀⠀⠀⠸⣿⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠼⠀⠀⠀⠀⠀⠀⠸⠾⠶⠶⠶⠒⠒⠒⠛⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣾⡄⠀⠀⠀⢸⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡇
⠀⠀⠀⠀⠀⣼⠛⡆⠀⠀⠀⠀⠀⢰⣿⣿⢿⣷⣦⡀⠀⢀⣴⣿⣿⢿⣷⡄⠀⠀⢴⣾⣿⡿⣷⣆⠀⠀⣴⣿⡇⠀⣠⣴⣶⡆⠀⠀⠀⠀⠀⠙⠯⠏
⠀⠀⠀⠀⢰⣇⣀⣹⠀⠀⠀⠀⠀⢸⣿⣿⣀⣨⣿⡇⠀⢸⣿⣿⠃⠀⢹⣿⡄⡀⣾⣿⡟⠀⠘⣿⣇⢠⢸⣿⣷⣿⣿⠟⠋
⠀⠀⠀⠀⢸⠋⢹⢸⡇⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⡄⠀⢸⣿⣿⠀⠀⢸⣿⡇⡇⣿⣿⡇⠀⠀⣿⣿⢸⣸⣿⣿⣿⣄
⠀⠀⠀⠀⣼⠀⠈⢸⡇⠀⠀⠀⠀⣸⣿⡟⠀⠀⣹⣿⠀⢸⣿⣿⠀⢀⣼⡿⠁⠃⣿⣿⡇⠀⣠⣿⠇⠀⣿⣿⡟⢿⣿⣷⡀⠀⠀⢀⡴⠖⠚⠛⣳⡄
⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⢿⣿⣷⣶⣾⡿⠋⠀⠈⠿⣿⣶⣾⠟⠁⠀⠀⢹⣿⣷⣶⡿⠋⠀⠀⣿⡿⠁⠈⠻⣿⣷⠀⠀⠈⣿⢒⢲⣛⢹⣿⠇
⠀⠀⠀⠀⠘⠷⠶⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠳⣬⣤⠟⠁

⠀⠀⠀⠀⠀⠀⢀⣴⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠃⣿⠀⠀⠀⠈⠉⠙⠒⠒⠦⣤⠴⠒⠛⠒⠒⠒⠒⠒⠒⠒⡗⠀⠀⠀⠀⠀⠀⢀⡴⠖⠲⢦⡀⠀⠀⠀⢀⣤⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⣏⡀⠀⠀⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠉⠀⠐⠒⠒⠂⣺⠙⠀⠀⠀⠀⠠⢶⣏⡞⠁⠀⠀⢻⣀⣤⣀⡞⣵⠓⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⡈⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠠⠈⢀⣒⠒⢰⡏⠀⠀⠀⠀⠀⠀⠘⣏⠁⠀⠀⢀⡾⠁⠀⢹⡇⠃⠀⠀⢀⡾⠛⠃
⠀⠀⠀⠀⠀⠀⠁⠒⠒⠪⠭⢙⡒⠤⣄⡀⢀⣿⠀⣀⣀⣠⠤⠤⠤⣄⣸⢡⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠖⠋⠀⠀⠀⠀⠓⠦⢤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠛⠉⠛⠉⠒⠒⠀⠉⠉⠀⠀⠁

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⠳⣄⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⢤⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡠⠚⠛⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠉⠉⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⢿⡿⠃⠀⠀⢀⣀⣼⣄⣀⣀⣧⣀⣀⣸⣇⣀⣀⣼⣀⡀⠀⠀⠀⠈⠻⡿⠃⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⡇⠀⠐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡇⠁⠉⠉⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣅⣀⣀⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡧⠤⠤⠤⠤⠤⢴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⠿⠟⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⣤⣶⣄⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣠⠞⠀⠀⣤⣷⣄⠀⢸⡇⠀⠀⠈⠉⠉⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠒⠚⠁⠀⠀⠀⠿⣿⠟⠀⠘⢧⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠠⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠋
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠄⠀⠉⡄⠀⢂⡠⠀⠀⠀⠀⡠⠀⠐⠐⠂⠀⠀⠀⠀⡄⠀⣀⠀⠈⠉⠁⣀⠠⢀⣄⠀⠀⠀⠀⠀⠀⠐
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠆⠀⠀⠀⠀⠶⢾⡀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠠⢤⠀⠀⠐⠄⠀⠀⠀⠀⠀⣉⠠⢀⢸⡀⠀⠀⠀⠀⡔⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠘
⠀⢠⡂⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠆⠀⠀⠀⠀⣄⠄
⠀⠀⠓⠒⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠱⣀⡠⠄⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⡦⡄⠀⠀⢀⡠⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂
⠀⢠⠀⢀⠀⠀⡄⠀⠀⠈⠙⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⣰⠋⠀⠀⠐⠂⠁⠀⠀⣠⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠐
⠁⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⡄⠸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⡴
⠠⠂⠀⠀⠒⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠃
⠀⣀⠈⠂⠉⠀⠠⠂⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡑⠀⠀⠀⠀⠀⠰⠖⠂
⢀⠁⢀⠀⠀⠊⠀⠒⠒⠒⠂⠂⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠓⠆⠀⠰⠞
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠁⠀⡀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠈⠀⠀⠀⣄⡀⣀⡄⠀⠀⠀⠀⠉⠋⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠣⠤⠂⠀⠀⠀⡠⠀⠀⢀⠀⠬⠀⠀⠀⠀⠀⠪⡁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠀⠀⠈⠀⠀⠀⣂⠀⠀⢐⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠾⡲⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀
⠀⠀⠀⠁⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠁⠀⠀⠂⠀⠀⠀⠀⢀⠐⠂⠀⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⡀⠓⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⠀⠀⠀⠀⠠⠀⠁⣤⠀⠀⠠⠀⠀⠂⠘⠑⣄⠀⠂⠐⠀⠐⠀⠁⠍⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡄⠘⠸⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠴⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠅⠀⠀⠀⠐⠒⠃⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠦⠀⢀⠙⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀
⣀⡀⣀⡀⢀⢀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⠀⠀⢁⠀⠀⣀⡁⠈⠀⠀⠀⠀⣈⢂⣀⢀⣀⣀⡀⠀⢀⣈⠀⢀⡀⣀⢀⡀⢀⣀⡀⠀⣲⣀⣀⠀⣀⣀⣀
⣿⣿⡿⠻⡿⠛⠿⢿⡿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠸⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛⠛⠛⢿⡆⠀⠀⠀⠀⢠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠶⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣤⣀⣼⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⣶⣶⣶⣆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣍⣹⣿⣿⣿⡆⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠋⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠤⠤⠄⠀⠿⠿⠿⠿⠿⠿⠿⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠆⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⣿⣿⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

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



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


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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡆⠀⠀⢸⣿⢸⣿⠀⠀⠀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣴⡆⣿⣿⢸⣿⣿⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⣄⣴⣿⣿⣸⣿⣿⡷⠶⠾⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣏⣽⣿⣯⠿⠟⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣰⣿⣼⡟⣿⣩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠿⠿⣺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣹⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣯⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⣷⡖⣻⣿⡇⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡇⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⡇⣿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡖⣶⣼⡏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⣿⣶⣯⣿⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⢸⣏⣻⣿⣟⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠛⠸⣿⣿⢹⣿⣿⡷⠶⢶⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠻⠇⣿⣿⢸⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠸⠃⠀⠀⢸⣿⢸⣿⠀⠀⠈⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢠⠦⣑⡠⢄⠀⠀⠀⠀⢀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠡⠜⣀⣱⡮⠴⢆⠄⠀⡸⣿⠂⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠺⢙⡟⠁⢄⡈⠀⠀⠀⠊⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⠳⡄⠈⠳⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢀⡇⠀⢰⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢦⠹⣤⠼⢷⡁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠹⡄⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠧⠀⡀⠀⠀⠀⢸⠃⠸
⠀⠀⢀⠀⢀⠀⣀⡀⠀⠀⢀⡔⠀⠀⠠⠀⠀⢀⡔⠄⠀⠠⠀⠼⠀⠀⠀⠀⢀⠀⠈⠁⡀⠀⠀⣼
⠀⣶⣤⢢⠘⢦⣟⢍⠉⠉⠉⢣⢀⡤⠀⠐⠴⠋⠙⠀⠀⠀⢠⠀⠨⡀⠠⠀⢹⠀⠀⠈⢄⠀⣠⣿⣠⡄
⠀⣿⣟⠀⠃⠇⠈⠳⣽⡄⠀⠘⣯⡂⠀⠀⠘⣸⣿⣿⣿⡄⢸⡄⠀⠰⣠⡄⠀⣿⠤⠀⢰⢸⣿⣇⠀⠑
⠀⣿⠿⡉⣻⠁⠀⠀⠁⠀⠀⠀⢯⣷⠀⡰⠀⠘⢿⣿⠟⢁⠸⡆⠀⠄⢑⡈⠀⣩⠢⠄⢀⠈⠻⠿⢧⣰
⠀⠸⢤⣭⠿⡂⠀⠀⠀⠀⠀⠀⣭⠛⢿⣽⣇⣞⠀⠀⠉⠙⡄⠉⠀⢸⡀⠀⠀⠛⠀⠀⠈⢰
⠀⠀⠀⠸⠄⡇⠀⠀⠀⠀⠀⢨⠿⠿⢿⡪⣿⡟⠀⠀⠈⠤⠇⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠻⠆⠄
⠀⣐⠀⠀⢀⠉⠳⠶⠶⠶⠊⠉⣾⡀⣿⣿⣿⢿⠤⠤⠄⡄⣄⠀⢀⡄⠁⠐⠐⠠⠶⠴⠚⠡⠆⠀⠒⠚
⠀⠈⢹⠉⡇⢀⣤⣷⢠⣸⠀⠀⠿⡿⠿⠿⠋⠀⠀⢤⠀⢍⠉⠉⣉⠇
⠀⠀⠻⠷⠻⠎⠰⢿⣾⡿⠷⠦⠀⣿⣾⣅⠀⠀⡅⢀⠀⢤⣨⠀⢥⠐⡻⣷⣼⣂⡬⠿⠓⠿⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠍⠛⠿⠀⠓⠚⠖⠋⠀⠀⠈
⠀⠐⠁
⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣴⣿⣿⡃⠀⠀⣠⣴⣶⣦⣄⠀⢀⣀⣄⣀⡄⠀⢀⠘⢳⣴⠆⠀⠀⠀⣤⣤⠀⢀⣴⣾⢿⣷⣦⡀⠀⣠⣂⡀
⠀⠀⢀⣀⣀⠛⡟⣩⣤⡄⢸⣿⣿⣿⣿⣿⣇⠘⠘⣋⣤⣤⣀⠘⣷⡟⣿⡄⠀⠀⠘⠿⠿⠃⢾⣿⣿⣿⣿⣿⣷⠀⠁⠻⠿⢄
⠀⣴⡏⣿⣽⣵⡄⢿⣿⠟⠸⣏⣙⣛⣛⣸⠏⢀⣾⣿⣡⣬⣿⣷⡈⣉⡉⠁⣠⣶⣶⣶⣦⣄⠘⢿⣧⣤⣿⡿⢃⠸⡥⢦⣅
⠀⢿⣿⣿⣼⣿⢇⣤⣴⣦⣄⠈⠛⠛⣛⠁⡀⠸⣿⠟⠻⠟⠻⣿⠃⠚⠁⣾⡟⠋⠛⠋⠛⢻⣧⠀⠉⣩⣭⣤⣄⢘⡱⠚⠂⢡⣶
⠀⠀⣙⡛⠛⢡⣿⡉⠉⠉⢹⣷⠀⣄⣨⣿⠃⠀⠙⠶⢶⠶⠶⠋⣤⣿⡀⣿⡇⠐⠂⠐⠀⢸⣿⠀⢸⣟⠉⠉⢻⡇⠀⣀⣤⣬⡉
⣴⡟⣋⡻⣷⡌⢿⣇⣶⣶⣿⡟⠀⠙⠧⠾⠇⠀⠀⣠⣴⣶⣶⣦⣝⠛⠀⠙⢷⣤⣤⣤⣴⡾⢃⣤⣬⣿⣶⣤⠾⢁⢰⡟⡇⠀⢸
⢿⡇⣀⡀⣿⠇⢀⠙⠛⠛⠉⣠⣶⣶⣶⣄⠀⠀⣾⣿⢿⠜⢁⣿⣿⣧⠀⠀⠀⣉⡉⠉⠁⢰⣏⣥⣬⣹⡆⢠⣾⣷⠀⠛⠷⠶⠊
⠀⠙⠛⠛⠁⢰⣿⣿⣦⡀⢸⣿⡟⠋⢱⣿⡇⠸⣿⣿⣄⡓⢈⣻⣿⣿⠀⣰⣿⣿⣿⣿⣆⠘⢏⣉⣉⣹⠇⠚⠛⠯
⠀⠀⠀⠀⠀⠀⠙⠿⠏⠀⠘⣿⣮⣲⣿⣿⢇⡀⠻⣿⡟⠃⠛⣿⣿⠟⠀⣿⢟⣿⣿⡻⣿⠀⣠⣭⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⡯⠁⠺⠟⠀⠈⠙⠛⠛⢫⢅⡠⡀⠙⠶⣾⣿⠶⠃⠸⣿⠉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢀⣾⠿⠿⠿⣷⡄⠐⢸⡋⣷⢢⢀⣶⣶⡄⢠⣼⡄⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⠇⠀⠀⠀⠸⡷⠀⠐⠋⠈⢀⣈⡛⠛⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣶⣿⠿⠃⠚⠂⠈⣼⣿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣇⠀⠙⢿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠙⣻⣿⢿⡿⣿⣛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠄⡿⠿⣿⣿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣒⡶⢶⣒⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡉⠀⠈⢉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠈⠓⠚⢉⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣾⣷⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇

⠀⠀⠈⣳⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠂⠇⠀⠀⡀⢠⠁
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠑⢀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡀
⠀⡀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠁⠆⠀⠀⠀⠀⠒⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠄⠀⢀
⡋⠖⠁⣀⠰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⡁⠀⠀⠂⡀⠐⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠄⠨⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠁⢈⠀⠀⠀⠀⢀⡴⢻⡆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⢀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⠐⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢀⠔⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠦⢄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡄⠁⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠠⠀⠐⠠⠄⠋⠀⠐⠀⠀⢀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠾⠟⠋⡀
⠀⣀⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⡀⠀⠀⢔⠀⠀⠁⠀⠀⠀⠀⠀⢀
⠉⠁⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠄⠀⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠈⠳⠤⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠃⡀⠄⠀⠀⢀⣄⠀⠀⠐⠂⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠂⠤⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠙⢷⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠸⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠠⠄⢠⡀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠈⠀⠀⠀⠀⠙⡆⠀⠐⠂⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠛⠿⢶⣤⣀⡐⡹⠁⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠈⠐⡀⠠⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣀⠀⠀⣀⣀⣀⠀⠀⠂
⠀⠀⠤⠀⠀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠁
⠄⠠⠁⠐⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠂
⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠄⠀⠀⠀⠀⠐⠀⠠⡀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢀⠈⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡄⠐
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠐⠀⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⢤⠂⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠜⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠂⠀⠀⠀⠀⠀⠑⠀⠀⠀⢀⣠⠊⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠠⠀⠀⠐⠂⠊⠀⠀⠀⠠⠀⠀⠀⠈⠀⠐⠈
⠄⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⡀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠂⢀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠁⡀⢠⠀⡀⢀⡄⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡠⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠰⠀⠈⠀⠀⣀⠀⠀⠀⠈⠈⠀⠀⠔⠀⠀⠀⠰⠀⠀⠈⢢⡀⠀⠀⠀⢠⡞⠠⠐⠠⠁⠀⠐⠚⠷⠀⠛⠁⡀⠀⠀⢀⠁⣸⠊⠑⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠂⠶⠀⠀⠀⠀⠘⠐⠂⠈⠁⠀⠀⠃⠀⠀⠁⠀⠨⢠⡞⠀⠀⠀⠀⠀⠀⠠⠶⠀⢾⠷⠈⠛⠁⠀⢘⣦⠰⡄⡠⢯
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⠀⠀⠀⢰⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠐⠁⠶⠄⠘⠷⠀⠀⠀⠀⠙⠛⠐⠙⠃
⠀⠀⢀⠀⠀⠀⠀⠀⠈⣵⠀⠘⠀⡀⣰⡄⠀⠀⠣⡀⠀⠈⠀⠐⠀⢀⠄⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⠀⠀⠀⠀⠈⠐⠂⠐⠒⠊⠁⠀⠀⠀⠀⢀⡀⢀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠁⠐⠀⠢⠐
⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠈⠳⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠑⢯⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠈⠀⠀⠘
⠀⠀⠀⠀⠀⣀⠀⠀⢈⠁⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠃⠠⠀⠀⠀⠐⠀⣾⠒⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣀⠀⠅⠀⣆⠔⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⣼⠒⠀⠀⠀⠐⠀⠂⠀⣰⠃⠀⠀⠀⡀⣀⠀⠀⠀⠈⡄
⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠀⠀⠌⠀⠀⠀⠐⠃⡼⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠰⡴⠁⠀⠀⠀⡃⠤⠀⠑⡀⠀⠀⠁
⠀⠀⠈⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠋⠀⠄⠰⠄⢀⠀⠀⠀⡨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⡇⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠔⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⢦⠀⠀⡐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⡀⠀⠘⢎⡀⢀⣄⣦⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡠⠊⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠄⠚⢋⡁⠊⠒⢄⠀⠀⠀⠀⠀⠤⠈⠀⠀⠐⠉⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⡋⠌⠀⡠⠃⠀⠀⠀⣀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠁⠙⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⢢⠀⠀⠀⠀⠀⢀⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠰⠁⠠⠤⢱⠀⠀⠀⠀⠀⠀⠀⡀⢄⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⡄⠀⠀⢀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠑⠤⠁⣀⡄⠃⠀⠀⠀⢊⠀⠀⠀⠀⠀⠀⠷⢀⣀⠁⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠈⠁⠉⠀⡄
⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⡀⠀⣠⡀⠙⠾⡀⠀⠸⠆⢡⠔⢀⠲⠀⠀⠀⠉⠁⠀⣠⡿⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠖⡈⠉⠓⠢⡀⢸⠀⠙⠊⠀⠀⠀⠀⠀⠀⠘⠻⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢾⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠠⡄⢀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⡰⠂⠉⠀⠢⡘⠀⠈⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⠀⠀⠈⠁
⠀⠀⠀⣀⠓⠀⠈⣶⠀⢀⣤⠃⠀⠀⠀⠀⠀⠀⢇⠺⢃⠀⠀⢰⠀⠴⠃⢤⠀⠀⠀⠀⠀⠀⠐⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠉⠛⢷⣯⣭⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠈⢆⠇⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⢠⡀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⡠⠤⢄⡀⠀⠀⠀⠀⡀⠀⠀⡀⢀⠀⠀⠀⠀⢄⠀⡀⠀⢀⠀⠀⢀
⠀⠀⣷⡀⠀⠀⠀⠀⡨⠛⢓⠦⣶⣶⠀⡎⠀⠀⠀⢱⠀⠀⠀⡠⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠢⠀⡀⠀⠑⠀⠀⠂
⠀⠀⣿⠹⣆⡀⠀⢸⠀⠄⠄⠀⢸⠀⠀⠱⢄⣀⡠⣮⡀⠀⠀⣾⣿⣿⣏⣀⣀⡀⡀⢀⠀⠀⠀⠐⡔⠄⢀⢀⠀⠀⢂
⠀⠀⢻⠀⠘⠃⠀⠸⠣⣄⣀⠤⠃⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠈⠛⠛⢾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⢂⠁⠀⠀⠀⠄
⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠐⢀⡿⠀⠀⠀⠈⢁⣀⣀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠡⠒⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⡄⠀⠀⠀⢁⣶⣶⣿⣿⡆⠀⣴⣿⣟⣿⡿⠏⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁
⠀⢲⣶⣦⣤⣄⡀⠸⣦⣤⣧⣤⣀⠀⠀⠻⣿⣿⡿⠃⠈⢉⣿⣿⣿⣿⡄⢰⠀⣤⡀⠈⢡⡄⠀⠀⢀⠀⠀⠀⠐
⠀⠀⠙⡿⠟⠓⠁⠀⠀⢰⡏⠀⢘⡧⠀⠀⠀⢠⣄⡀⠀⠈⠋⠉⠉⠁⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⢀⣴⡇⠀⢀⣤⣤⡀⠀⠛⠗⠛⠁⢠⣦⣅⡀⠀⠈⢢⠀⠀⢀⠀⡴⠠⠀⣀⡀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠠⠠⠀⡄
⠀⠈⠉⠉⠀⢸⣧⣤⠟⡀⠀⠀⠀⠀⢹⣿⣿⡿⢀⣰⠈⡇⠀⠢⣀⣷⣾⣅⠀⠁⠀⢠⠀⠀⠀⠀⠈⠒⠠⠄⠄⠊
⠀⢤⣄⠀⠀⠀⠀⠡⠛⠃⠀⠀⠀⠀⠡⡈⠿⣧⠀⣽⡳⠁⠀⠚⠀⠟⠻⢥⡁⡄⠀⠀⠀⠸⠀⠀⠀⠀⢠⣦⣀
⠀⠈⠻⠇⠀⠀⠐⣾⣿⡇⠀⠀⠀⠀⠀⠀⠱⠶⡞⠊⠁⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⡀⠀⠙⠻⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⠀⠀⠛⠀⠀⠀⠀⠀⢐⠂⠀⠀⢀⠂⠈⠀⠘⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠶⠀⠀⣞⡿⠃⡧⣄⠀⠀⠀⠀⠀⠶⠀⣤⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣀⣀⣀⣀⣉⣀⣀⡀
⠀⠀⡖⠀⠀⠀⠀⡀⠀⠀⠀⠀⠓⠋⠈⠁⠀⠀⠀⠀⠈⠻⠇⠘⠂⠀⠀⠀⢸⣿⡷⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠐⡄⠀⠰⠆⠘⠃⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⢧⣤⣤⣤⣤⣤⣤⣤⡼
⠀⠛⠒⠶⠤⠤⣄⣀⣀⠀⢸⠃⠀⢀⠤⠀⢤⣀⠀⢀⠠⠆⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠲⠶⠿⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⣄⣴⣦⡄⠘⢿⢧⠉⠀⠀⣠⡾⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⠿⠇⢀⡴⠒⣄
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠦⠀⠀⡅⠙⠛⠀⣀⣄⠀⠀⢠⡶⠉⠀⠀⠀⠈⣥⠀⠀⢰⣶⣦⠀⠀⠀⢰⡋⣠⢞⣡⣦⡀
⠀⠀⠀⠀⢀⡄⠀⠋⠀⠁⠐⠀⠘⢷⡄⠐⢿⡟⠋⠀⠀⠀⠀⢠⡄⣤⠺⠂⠀⠀⠈⠙⠉⠀⠀⠀⠀⠉⠱⣿⣿⡿⠋
⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⢸⠋⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠙⠋⢻
⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣄⡀⠀⠀⠀⠀⠀⣠⣀⣴⡶⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⢀⠜⢻⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⡠⠊⢀⡌⠈⠢⡄⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⢀⣔⡁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠘⡄
⠀⣜⣀⣀⠈⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⡼⢄⣀⣠⣄⡀⢘⣆
⠀⢿⣿⣏⣹⣶⣦⣶⣾⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⡄⠘⠂⢠⣬⡁⠘
⠀⠈⠻⣿⣯⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣈⣁⣀⣀⣈⣛⣁⣀
⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⢸⣿⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠬⠅⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠘⠿⠃⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠭⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣦⣤⡇
⠀⠀⠤⣤⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣍⣿⡇
⠀⢀⣤⣿⣯⣥⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡶⡇
⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠔⢢⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠥⢤⡀
⠣⡀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣹⣢⣖⣏⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⢪⠀⢰⡗
⠀⠈⢢⠀⠖⢱⣶⣶⡶⠀⠀⠀⠀⠀⠀⢔⠉⢱⣏⠠⠄⣳⡏⠉⡶⠀⠀⠀⠀⠀⠀⡈⣆⢀⡟⠉⢉⡏⠁
⠀⠀⠘⣸⠀⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⢛⡲⢞⡗⠙⡏⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⢸⠆⠘⠁
⠀⠀⠀⠛⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠉⢆⡰⠉⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣫⠄⠀⠀⠀⠀⠀⠄⠀⠀⢄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⢛⡀⠀⠀⠀⢀⠀⣑⠠⠵⠕⠃⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠁⠁⠀⠀⠀⠀⠘⠘⢀⡀⠈⠁⠀⡄⠀⠈⠈⠀⠀⢸⠀⠀⠀⢠⢫
⠀⠀⠀⠀⠀⠀⠀⣰⣻⠀⡀⠀⡀⠀⠂⠀⠀⠀⠀⠈⠿⠢⠀⠀⡀⠀⠀⠀⣀⢼⣆⠉⠁⢸⡑
⠀⠀⠀⠀⠀⠀⣰⠿⢹⠉⣃⡁⠅⢀⡀⠈⠠⢠⣤⠄⠈⠈⠀⡴⠇⠀⠰⠊⠀⢸⠀⠀⢒⡞
⠀⠀⠀⠀⠀⣰⡝⠃⠀⠀⠀⠁⠀⠀⠉⢣⠀⠀⠀⠈⠂⠀⠀⢀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⡴⠿⣶⣷⣴⣤⣤⡄⡆
⠀⠀⠀⣀⣀⡀⢀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠂⣖⠀⠐⠞⠁⠆⠀⠀⠀⠰⠀⠈⠃⠣⡀
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠈⠁⠉⠀⠀⠘⠁⠉⠉⠉⢻⠆
⠀⠀⠠⠤⠜⠗⢠⣄⢠⣐⣛⠻⠅⠀⠀⢠⠀⠀⠀⠀⠠⠷⠆
⠀⠀⠀⠀⠈⠉⠛⠙⠻⠉⣹⣿⣯⢭⣿⠩⢥⡦⠄⢦⠽⠿⠶⠶⠮⠷⠖⠒⠺⠥⠟⠿⠷⠮⠥⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣎⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠎⣱⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠆⡀⡀⠂⣠⡾⠣⠄⣀⠀⠀⠀⠀⠀⠐⣆⠀⠃⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠈⢀⣤⣀⠀⠀⠀⣎⣀⣀⠀⠈⢡⢰⣶⡆⠀⠀⠀⠉⠁⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⢹⠶⡏⠀⠈⡷⠀⣼⣿⣿⣿⣿⣶⠟⠈⠋⠀⠀⠉⠉⠀⠀⠀⠀⠙⠷⠘⠿⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠓⠖⠋⠀⠙⠲⠚⠷⠀⠸⣿⣿⣿⣿⡏⠀⠀⡀⠀⢀⡙⠳⣤⡀⢰⠇⠴⠰⠆⠄⠨⠀⠀⡠⠾⠷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡰⠶⢤⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣔⣅⡈⠛⢁⣸⠇⠈⠻⣆⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢀⣀⣛⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢾⡄⠀⠀⣠⡄⠀⠀⢶⣲⡀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠹⣆⠀⠀⠀⠀⠀⢀⣤⡶⠛⡋⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠰⠤⠄⠹⠀⣀⣈⣁⣀⣀⣸⣿⣧⠀⠀⠀⠀⠀⣸⣿⣿⡆⠀⠀⢻⡄⢺⡗⢀⣴⠟⠁⢰⣿⡇⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⣦⡆⠀⢘⡷⠶⣤⠀⠘⠿⣿⣿⣿⣿⣇⣿⣿⡆⠀⣀⡀⠀⠘⠿⠿⠃⠀⠀⢸⡇⠀⣰⡟⠁⠀⠀⠈⠉⠁⠛⣩⣀⣈⣀⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠾⠡⠤⠌⠧⠀⠺⠿⠿⠿⠿⠿⠸⠿⠷⠀⠛⠃⠀⢀⠐⠃⠐⠃⡀⢸⣯⢠⣿⠃⢸⣾⣾⠀⠀⢀⠀⠉⠀⠀⠈⠉⢄
⠀⠀⣀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠲⠶⢦⣄⡀⠘⠁⠀⠀⡀⠁⣼⡇⣼⡟⠀⢸⣿⣿⠀⣀⠻⠀⠀⠀⠀⠀⠀⠸
⠀⣴⣧⣄⣀⠘⠛⠀⡄⠰⠀⡔⢀⠀⠀⠘⣧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠙⣦⡈⠃⠈⠁⢠⣿⢡⣿⠃⠀⠀⣉⣣⣄⣿⡆⠀⣤⡀⠀⢀⡄
⠀⠀⠀⣼⠋⢀⣈⠃⠀⠀⠀⠀⢋⡀⠀⠀⣿⠀⣴⣤⣴⡀⠀⠀⠀⡂⠀⠀⠈⣿⣄⠀⢀⣾⡏⣼⡿⠀⠀⠀⢽⣿⣿⡾⡷⠀⠉⠁⠀⠈⠁
⠀⠀⠘⠁⠀⠀⡤⠀⠀⠀⠀⠀⠡⠤⠀⠀⣿⠸⠿⠿⠿⠗⠀⠀⠀⠁⠂⠀⠀⢸⣿⣤⣾⡿⣱⣿⠇⣼⣇⡤⡄⠀⠀⣠⣤⣶⣿⣿⣿⣿⠶⠒
⠀⠀⣠⣤⠀⠀⠀⠀⢀⣤⣄⠀⠰⢛⡆⢰⣿⠀⣀⣤⣴⣶⣶⣦⣄⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠏⠈⠋⠉⡱⢣⠀⠀⠡⠀⠉⠉⠉⠉⡀⠂⠁
⠀⠸⠟⠁⠀⣷⣾⡄⠀⠻⠋⠀⠀⠈⠀⢸⣿⠀⠉⢳⣿⣿⣿⣿⠙⠉⠀⠀⢠⣿⣿⣿⣿⠟⠃⠀⠘⢲⣞⠁⠿⢦⡄
⠀⠁⣤⣦⣀⠈⡿⠀⠀⠀⠀⢰⣟⣠⢄⠈⣿⣆⠀⠀⠉⠙⢉⣵⡂⠀⠀⢠⣿⣿⡿⠟⢁⣠⠀⠀⠀⠉⠉⠁⠀⠎⠀⢀⡀⠐⠀⠀⠀⢀⡞
⠀⠀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠛⠉⢹⠎⠀⠘⢿⣷⣤⣀⡀⠻⡿⠃⠀⣠⣿⣿⠋⠀⠈⠉⠀⠀⠀⠀⣴⣾⠀⠀⠀⠀⠀⠄⠚⠀⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠲⣶⠋⠀⠀⠀⡀⠉⠻⣿⣿⣷⣦⣀⣴⣿⡿⠁⠐⠀⠠⠆⠀⡜⠀⡀⠙⠛⠀⣀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠉⠋⣀⡀⠀⠀⣀⠀⠀⠙⢿⣿⣿⣿⣿⡇⠮⠉⢭⡩⠨⠏⠀⠀⣷⠀⠀⠀⣍⠀⠀⠀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⣰⣶⣶⡄⠀⠀⠈⡆⠀⢀⣼⣿⣿⣿⣿⣷⣄⠀⠼⠧⠀⠀⠀⣸⡟⢠⡄⠈⠉⠑⢦⠈⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠃⠀⠀⠀⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣴⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢀⣾⣿⣿⣿⣿⣿⣿⡟⠁⠀⣀⣠⠤⠤⠖⠒⠶⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣀⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣰⠏⣀⣤⡤⠶⠖⠒⠒⠒⠒⠒⠒⠒⠶⠶⢤⣤⣀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠴⠶⠦⠤⠤⣤⣤⣀⣀⡀⠘⢧
⠀⠀⠀⠀⣼⣿⣋⣡⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⡉⠛⠶⣼⣀⣤⣖⣚⣋⣩⣤⣤⣤⣤⣤⡤⠤⠤⠤⢤⣤⣤⣭⣭⣿⣾⣧
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⢀⣼⣼⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣷⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣀
⠀⠘⠛⠟⠻⠛⠻⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⠿⠛⠻⠿⠿⠿⠟⠻⠿⠿⠿⠟⠛⠛⠛⠻⠛⠟⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣛⣛⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠻⠿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣿⣟⡛⣿⣿⣿⠿⠻⢻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠉
⠀⠀⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⣤⣀⡀⠀⠀⢀⡀⠀⠀⠀⡁⠀⠀⠀⠀⠀⡀
⠀⢀⡀⠀⠀⢁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠈⠙⠉⠑⠲⡀⠀⠀⠠⠀⠄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠐⢻⡁⠀⢀⠎⠀⠀⠀⠀⠀⠀⠠⡆⠀⠠⡄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠅⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅⡦⣀⠄⠀⠘⠛⡀⠀⠀⡤⠤⠶⠶⠶⢷⡤⠴⡏
⠀⠈⠀⠀⠢⡀⠀⠀⠀⠀⠀⠈⠃⠱⠠⠀⠐⠃⠨⠃⠀⠀⠀⣀⣈⣉⣀⡀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢸⠡⠋
⠀⠀⠀⠀⣰⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢘⢠⢁⣀⡄⠉⠩⠋⠊⠑⠀⠀⠀⠃⠀⣸⠀⠀⠀⠀⠀⣿
⠀⡤⠄⢠⣃⠀⠒⠂⠀⠀⠴⠶⠄⠀⠀⠀⠠⠤⠌⣀⠀⣀⡀⢀⠀⠀⢀⣆⠤⠬⠤⢬⠂⠉⠒⣐⣀⣀⣰⡿⠤⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢿⡿⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢻⠳⢤⣖⠉⠀⠠⡇⠀⠉⣲⠤⠚⡟⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⢸⠀⠀⠈⢱⠀⠈⡇⠀⡏⠀⠀⠀⡇⠀⣤⣀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⣿⣿⣿⣦⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⣇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠢⣄⡀⢸⠀⢀⡇⠀⡇⠀⣀⠼⠿⠧⡈⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠔⠊⠀⠀⠀⠀⠀⢉⠿⠀⣨⣏⠰⣷⣉⠀⠀⠀⠀⠀⠑⠠⢄⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢁⠀⠀⠀⠀⢀⡠⠔⠊⠀⣠⠔⠻⠟⠢⣄⠈⠑⠢⣄⡀⠀⠀⠀⠀⡨⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠤⠒⠁⢀⡠⠖⠉⠀⣀⢴⣦⣀⠀⠉⠲⢤⣀⠉⠲⠤⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄⠊⠁⢀⡠⠔⠈⠀⠐⡇⠈⠙⠢⢄⡀⠈⠑⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡉⠀⠀⢠⠂⠁⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠉⠲⡄⠀⠀⢙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢼⠀⠀⠀⡠⠔⠊⠉⠉⠙⠢⢄⡀⠀⠀⡧⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⢀⠠⠐⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⢰⢶⠷⢶⡶⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡿⠟⠹⠀⠀⠀⠀⠀⣸⡀⠀⠀⢠⠀⠀⠀⠸⠀⢀⣷⡀⠀⢠⣿⣬⣠⣀⡴⡟
⠀⠀⠀⠀⢀⡄⠀⠀⢰⣷⣍⠁⢀⠘⠀⠀⠀⠀⢀⡟⢃⠀⠀⠈⠀⠀⠠⠌⢅⣼⡿⠷⠆⠘⢏⣈⣉⣉⣡⡗
⠀⠓⠃⠀⠀⠁⠀⠀⢸⡟⠀⠿⣿⣆⠀⠀⠀⠀⡘⢠⣞⠞⠀⣀⡀⠀⢀⠀⠀⠙⠁⠀⠀⠀⠀⠉⠙⠃⠈
⠀⣤⡆⠀⠀⠆⠀⠀⢸⡇⢀⠀⠈⠉⣁⠀⠀⣠⣤⠞⠉⡷⠞⢣⡙⠈⠁⠑⠰⠾⠁
⠀⢠⣤⣤⡤⣦⣤⢶⡾⣷⡈⢻⣷⣾⡎⠳⠞⠿⠋⢀⡀⢱⣶⡞⣷⢰⡇⣤⡀⠛⠂⠀⠀⡀
⠀⠘⠋⠹⠇⠀⠹⠀⠈⠿⢁⠀⣿⣻⡧⠐⠃⠘⠘⠃⢠⢨⣿⠇⢸⣿⠀⡆⠿⠳⢦⣄⡘⠃
⠀⠀⠀⢀⣀⣀⣀⣀⠀⣰⠫⢏⡿⣿⣧⣤⣤⣤⣴⠀⠀⠰⠀⠀⢠⡟⢂⡀⠖⠀⠀⠈⠹⠓⢤⣀⠙⠉
⠀⠀⠀⠀⠠⠀⢾⠋⠀⣿⠀⠈⠃⣿⣷⣿⣟⣿⣿⠀⡀⠀⠀⠀⠀⠖⠛⠛⠀⠐⠓⠀⠄⢀⠀⠀⠗
⠀⠀⠀⠀⢀⡌⠆⠀⠀⣿⠂⠀⠀⣿⣿⣜⡟⠋⢸⢠⣼⣄⠀⠄⠀⠀⠀⠀⠀⢆⣤⡀⠀⠈⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⠀⠀⡇⢀⣿⣷⠒⣿⣼⡟⡧⠀⠀⠀⠀⠀⠀⠀⢤⡉⠁⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⠇⣤⠛⠩⠋⡃⢸⣟⠃⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠸⢄⣧⠀⠀⠘⠁
⠀⠀⠠⠠⠄⠀⠄⠚⠀⠿⠄⠀⢹⢠⠇⠀⠒⠒⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡼⠀⠀⠀⣀⡀⠘⠛⠁⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣿⣾⠟⣻⣏⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠻⠛⠓⠲⠷⡶⠂⠀⢸⠦⢿⠿⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢁⣁⢀⠀⠀⠁⣁⠀⣤⠶⠀⠀⠀⠄⠠⠐⠒
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⡖⠳⡀⠈⣿⡇⠀⠀⡐⠿⠀⠀⠀⣀⠰⠷⢽⡄⠐⢀⢈⣔⣁
⠀⠷⠖⠂⣀⡈⡅⠤⠀⠀⡿⠂⣄⠠⣟⡂⠀⠚⠋⠏⢹⢯⠀⢈⣾⠃⠒⠂⢀⡀⠀⠋⠁⠀⠀⠀⠀⠂
⠀⣤⣠⡸⣰⡇⠀⢠⢠⡤⢷⣁⡋⠀⣿⡗⠃⠀⠀⠁⠘⢆⢙⡎⣻⠇⠀⠀⣶⠀⠀⠀⡤⠀⡄
⠀⠛⠇⠙⠁⠘⣇⣈⣥⣴⢼⡏⡜⠀⠁⠉⠈⠈⡄⠑⣖⣀⡄⣠⣿⠂⠀⠀⣼⠄⠀⠀⡘⠆⠀⠘⠅⠴⠲⠆⢠
⠀⠂⢰⣶⣾⣿⡟⡟⠉⠻⠉⠁⠀⠘⠀⠀⠀⠀⠓⢀⠉⡜⡿⠽⠿⠧⠀⠀⠘⠁⠀⡀⠉⠈⠙⠂⠐⠂⠐⠲⠅
⠀⣿⠸⠟⠋⠛⠃⠁⠀⢸⠛⠋⠛⠛⠋⠉⠉⠉⠉⠹⡏⠐⠉⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⡀⠀⠀⣿⣶⣄
⠀⠁⠀⠀⠀⠀⠀⠒⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠂⡇⠀⠀⠛⠙⠋
⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⢲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠰⡆⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢏⠀⡗⠀⠀⡇⠀⢸⠂⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠳⡈⢢⠀⡇⡴⠋⡔⠋⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠉⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⡔⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣟⡍⣭⢹⣭⡍⣭⣩⣩⡝⣿⣿⣀⣀⣀⣀⣀⡤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⣀⣀⣀⣀⣿⣿⣯⣥⣭⣬⣭⣥⣭⣭⣬⡅⣿⣿⣀⣀⣀⣀⣀⠔⠉⠉
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠉⠉⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠉⠉⠉⠉⠉⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠔⢁⠜⠀⡇⠳⣄⠣⡀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣇⠀⡧⠀⠀⡇⠀⢸⠄⢹⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠁⠀⠀⠀⠰⠇⠀⠀⠀⠈⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠠⠤⠤⠤⠄⠀⠀⠀⠠⠄⠤⠄⣤⠤⠀⡘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣾⡗⠒⠒⠒⠲⠒⠒⠒⠒⠲⠶⠒⠒⠖⠿⠖⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⢸⣾⠇⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠼⠧⠀⠀⠀⠀⠤⠶⠒⢢⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠷⠶⣴⠤⠤⠤⠤⠴⠤⠦⠤⠤⠤⠤⠶⠤⠤⠤⠴⢦⡼⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣀⣀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣶⣤⣤⣤⣴⣶⣶⣶⣦⣤⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⣤⣤⣤⣦⣤⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠛⠃⠛⠛⠛⠛⠛⠛⠛⠛⡻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠾⠿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠽⠶⠶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⠤⢀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣧⣀⣠⣤⣤⣤⣤⣀⣠⣤⣄⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣐
⠀⠀⠀⠀⠀⠀⢁⠤⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠒⠀⠒⠒⠂⠀⠀⠀⠂⠒⠀⠀⠀⢲⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⠤⠤⠤⠶⠤⠦⠤⠶⠶⠦⠶⠶⠶⠶⠶⠶⠧

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢤⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⡀⠤⠀⡀⠀⠀⠀⠀⠀⠁⠀⠂⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠠⠠⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⣤⠀⠀⠀⠀⠤⠁⠀⠀⠀⠀⠀⡀⠤⠤⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠐⠀⢠⠠⠐⠈⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠤⠚⠉⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠤⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠋⠁⣀⡤⢞⣕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⡂⠒⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡰⠒⠁⢀⣠⢴⣪⣯⠮⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠘⢄⠢⠀⠀⠈⢦⡀⡀⠀⠀⠀⠀⠀⠀⢀⡀⡠⠔⠀⠀⢀⡤⣺⡧⡳⠛⠁⢀⣠⢴⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⣠⠦⠀⠀⠀⠀⣀⠠⢐⠄⠀⠀⠡⡀⠀⠀⠀⠈⠉⠀⠀⠈⠀⡀⢐⣥⠗⠋⠁⠁⢀⡤⣲⢫⡣⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⢀⠞⣻⡟⠻⠒⠈⠁⠀⠀⠀⠀⡀⢠⣄⣬⣆⣀⣀⣀⣀⡀⠀⣠⡼⠞⠙⠉⠀⡠⣴⣺⡵⠗⠓⠁⢀⣔⠦
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⡀⢑⠊⠁⠀⠀⠀⠀⠀⢠⣀⡦⠐⠂⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠋⠀⡀⠀⣔⣽⠾⠛⠉⠀⣠⡴⠺⠋
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠈⠁⠀⠀⢀⢀⣄⠴⠶⢿⣯⣡⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣨⣾⠟⠉⠈⡀⡠⡖⠝⠉
⠀⠀⠀⠀⢀⠈⠀⠀⠀⣀⢠⡄⠶⠀⠓⠸⡉⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠉⠉⠙⠛⠛⠓⠒⠚⠋⠈⠀⢀⣔⠞⠙⠁
⠀⠀⠀⠐⢠⡀⡦⠸⠒⠉⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠤⠤⠀⢀⣂⡀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⢀⢄⠢⠪⠊⠁
⠀⠀⠖⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠐⠀⠀⠀⠐⠉





⡿⠿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣻⣛⣻⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠉⠀⠀⠀⣈⣙⡦⠀⠀⠀⠀⠀⠀⠀⡔⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⡏⠉⢢⡐⢋⡥⣄⠀⠈⠂⠘⠀⠀⠀⠀⠀⢐⠌⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠙⠒⠷⠓⠒⠊⠀⠀⠀⠀⠀⠀⠀⠦⠀⠈⠉⠐⠒
⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⢜⡠⠀⠀⠀⠀⠈⣄⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠠⠀⠀⠀⢠⠀⠀⠀⠀⡇⠀⠀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠈⠊
⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⣀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⢸⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠓⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡔⠈⠹⠄⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠃⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠘⠢⠤⠤⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠀⠀⠃⠀⠀⠀⣈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐
⠀⠰⠀⠲⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⢀
⠀⠀⠀⠀⠀⠈⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠤⠀⠀⢠⡎⠁⠀⠀⠀⢑⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠒⠶⠶⠚⠋⠀⠀⠀⠀⠁⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣝⣁⣀⣀⡀
⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠀⠈⢁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⠧⢤⡤⠀⠀⠀⠀⠀⠁

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⠶⣄⠀⠀⠀⠀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣍⡀⢈⡆⠀⣠⠾⣻⠟⠋⢀⣀⣀⣀⣀⣀⡤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠙⡦⠾⣁⠈⣻⠛⠻⣟⣡⢾⡗⠋⠸⠛⢚⣿⣶⠶⣾⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠛⣾⠁⠀⢹⣟⣻⣦⣤⡿⠛⠿⣿⣿⡛⢿⣯⠛⠉⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢷⣤⣤⣾⡟⠋⠛⣿⠋⠁⢀⡷⣤⣤⡼⢿⣗⡤⢽⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣴⠚⠿⠛⣄⣒⣴⡿⠿⠿⢿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢹⣷⣤⣾⣿⠿⢿⡉⠳⣤⡠⠈⠙⠶⠶⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢬⣿⠟⣱⢻⡧⢀⣸⡇⠀⠀⢹⡶⣆⠒⠂⠀⠀⠘⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡾⠟⢰⢋⣽⣫⣾⡇⠜⠓⢶⡿⣷⡀⡆⠀⠃⢈⡳⣞⠃⠀⠀⠈⠙⢓⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡯⣔⠀⣿⡿⠋⠁⡇⠙⣦⠀⠁⠂⢸⡻⣧⠀⠀⠈⣇⠈⢻⣌⠉⠀⠀⠀⠉⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⡘⣡⣾⢿⡇⠀⠀⣇⠀⣯⡀⠀⣠⠧⢋⢻⣦⡠⠀⠀⠀⠀⣏⠿⣟⠓⠒⠀⠀⠘⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⡿⠃⢸⡁⠀⡆⣽⡀⠀⣹⠆⣠⠂⠘⢾⡇⠙⢦⡀⠀⠀⠙⢸⠀⠳⣖⠒⠀⠀⠐⠺⣆
⠀⠀⠀⠀⠀⠀⢀⡤⠾⣿⠙⣇⠀⠀⣇⠀⠰⢻⣏⠘⠧⡜⠁⠀⠐⠈⣝⣆⠀⢳⣄⣀⣤⡀⠀⠀⠩⠳⣦⡰⠂⠀⣘⣆
⠀⠀⠀⠀⠀⢰⢏⡄⠠⣿⠀⠘⢦⠀⠈⣷⡀⠀⠸⣷⠄⣿⣤⣐⡿⠀⣈⢻⡀⠀⠈⠉⠉⠉⠳⣄⠈⠀⠈⠙⠦⣄⠉⠹⡆
⠀⠀⠀⠀⠀⢸⢸⠃⢰⣿⠀⠐⠘⣆⠀⠀⠹⣴⡀⠟⣷⣌⢧⡘⢦⣤⣇⣆⠹⡄⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠒⠚⠿⣦⣿⡄
⠀⠀⠀⠀⠀⠸⣆⠀⠘⠉⣦⠀⢀⠘⢦⡀⠀⠀⠛⠦⠼⣽⣎⣧⠀⠈⠉⠙⢿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆
⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⣿⣧⡀⠀⠂⠹⡄⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢳⡴⠀⠙⠉⢳⡄⠈⣄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⣴⠀⣿⡄⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⢷⡸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢿⣿⡄⠀⠀⠀⣰⠞⠛⢿⣤⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠾⠟⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡴⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⠿⠋⢀⡈⢠⢨⡈⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⣤⣿⣿⣿⡟⡤⠀⡼⢀⡞⢠⣷⡘⣿
⠀⠀⠀⠀⠀⠀⠀⣰⠏⣴⢃⣾⣿⣿⣿⡿⢃⡼⣡⠞⣠⣻⣿⠿⠁
⠀⠀⠀⠀⠀⠀⢀⡏⣰⢇⣿⡿⣿⣿⣿⣷⡿⠾⢷⠟⠃⣛⡃
⠀⠀⠀⠀⠀⠀⢸⠁⡏⣼⢿⢷⣿⣿⣿⡇⠀⠀⢸⠛⠀⠹⡁
⠀⠀⠀⠀⠀⠀⣾⢸⢠⡏⣿⢸⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⡏⢸⢸⡇⣿⢸⡟⣷⣀⠀⠀⠀⠀⠀⠀⠒⠛⡏⠀⠀⠀⠀⠀⠀⠀⡤⣄⡀
⠀⠀⠀⠀⠀⠠⡇⢾⠸⡇⢛⠀⢣⠻⣿⡀⠀⢠⣤⣤⣤⣤⠀⡇⠀⠀⠀⠀⠀⠀⠌⠀⠀⠉⠓⠤⣀
⠀⠀⠀⠀⠀⠀⣇⠘⣄⢗⠈⣦⠈⠃⠈⢻⣷⠘⣿⣿⡇⠀⡼⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠹⣄⡘⠀⢆⠘⣧⣴⣦⡾⠋⠀⠈⠉⣠⠞⠁⠀⠀⠀⠀⢀⡎⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠚⢻⡿⠛⠋⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⢀⡞⠀⠐⠑⣵⠂⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣀⡈⠙⠺⠁⢠⠀⣀⠀⠀⠀⠀⠀⡼⢹⣧
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁⠑⢤⣀⠀⠀⠀⢸⣦⣄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠻⠳⣒⣆⢤⠈⠘⠙⠗⣴⠆⢀⡾⣱⣿⠏
⠀⠀⠀⠀⣀⣤⠾⣋⢌⣿⣦⣀⠀⠁⠒⠒⠊⢀⡟⣻⣶⣦⡀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠟⠇⠀⠀⠈⢀⡾⣱⣿⠃
⠀⠀⣠⠞⠛⡿⢿⣾⣿⣿⣿⠻⣏⠂⠀⠀⠖⠊⢻⣸⢾⣾⣿⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣰⣿⠏
⠀⢸⡏⢰⢇⣄⢢⣿⣿⣿⣿⢣⠹⣆⠀⠀⠀⠀⠘⣧⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣼⣱⠏
⠀⣿⢠⡇⣸⣽⣷⡆⣿⣿⣿⣷⡇⢹⡄⠀⠀⠀⠀⢿⡘⢿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢯⢻⠏
⠀⣿⣸⡅⣿⣽⣯⢰⢻⣿⣿⣿⡟⠀⣧⠀⠀⠀⠀⢸⡇⣻⣏⡩⣶⢤⣄⠀⠀⠀⠀⠀⣠⣤⠖⢴⡓⣄⣼⠃⢡⠏
⠀⢿⣿⡗⢉⣽⢛⢙⢸⣿⡿⣿⣷⣶⣽⡄⠀⠀⠀⢸⠟⠉⠁⣤⣾⣣⡭⠟⣶⣤⣀⡊⢣⠈⢇⠈⡇⠸⡧⢂⠏
⠀⢸⡿⣿⣦⡿⠍⠟⣸⣿⣷⣿⣿⣿⡿⣇⠀⠀⢀⠎⠀⠀⠀⢉⣀⣤⠔⢊⣷⣌⡻⢿⡌⠆⠘⠀⠀⠀⣷⠌
⠀⣼⢧⣿⣿⣷⠀⠶⢿⣿⣿⣟⣑⣫⣴⡿⣎⠉⢫⠀⠀⠀⠀⠀⡠⠔⢚⣵⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⡿
⠀⢿⡈⡿⠿⢿⢄⣼⣿⣿⡿⠛⠉⠉⠀⠀⠈⢳⠀⢣⠀⠀⠀⠀⠒⣊⣽⠿⣽⠷⠂⡿⢄⡀⠀⠀⣀⡼⠁
⠀⠈⣷⣷⣆⢴⣿⣿⣿⠟⠀⠀⠀⠀⠀⣀⡀⠈⣇⢀⡶⠒⠛⢻⣿⣿⣿⡆⢖⢪⡠⡹⢤⣈⣉⠟⠁
⠀⠀⢸⣿⣿⣦⣾⣿⣅⣀⣠⣤⣶⣿⣿⣿⣿⡇⣿⣿⡇⠀⠸⠎⣿⡿⣾⣥⣤⣤⣵⠾⠋⠁
⠀⠀⠈⢧⠙⡿⣿⣿⣿⣿⡏⢿⣿⣿⣿⣯⣥⣾⠟⢸⡇⠂⠒⠀⣿⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠳⠋⠛⠋⠉⠉⠁⠘⠀⠘⠁⠀⠀⠀⢻⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⢀⠀⠀⠀⠀⢈⡷⢤⣀⠀⠀⠀⠀⣴⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡜⣁⡤⠴⠾⠷⣤⣤⣴⢋⡴⣿⣫⣳⣄⠀⠀⣿⢸⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠶⢦⣤⣀⡀⠰⠿⢷⣓⣦⣠⢤⣴⣿⣿⣣⡆⠀⠀⠀⠀⠙⢯⠈⡇⠐⣼⡿⢍⡑⣄⣽⠾⠁⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠍⠀⠀⠀⠠⣀⣈⣿⣿⣦⠀⠀⠀⠁⠀⠀⢸⣿⡇⠛⠿⠃⠀⠀⠀⠀⠈⣇⣿⠐⣼⣿⡆⠘⣏⠀⣠⠴⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠙⠓⠀⠀⠀⠀⠈⠀⠀⢹⣿⣿⣧⡤⠚⠋⠉⠉⢹⡏⠉⢤⡀⠀⠀⠀⠀⠀⠀⣿⣌⢁⣿⣿⠃⠀⢿⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣄⡀⢀⣠⠀⠀⠀⠀⠀⣹⣿⠏⠀⠀⠀⠀⣀⡼⠃⠀⣀⣹⣿⣶⣦⣤⣀⣀⣈⠻⠿⢛⣁⣀⣤⣟⣈⣿⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠋⠉⠛⠻⣿⣦⣀⠀⠀⣿⠏⠀⠀⠀⣴⠟⠉⠷⣾⡋⠉⠁⠀⡽⢻⡟⠛⠻⣿⣿⡛⠛⠛⠿⠍⠉⠉⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢿⣷⣼⠏⠀⠀⢀⣾⠃⠀⠀⠀⢸⣟⠂⢀⣰⠀⢸⡇⠀⠀⠙⢟⢮⠃⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠻⣶⡀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⣼⡟⠀⠀⠀⠀⣼⠉⠛⣿⣟⠀⣾⠁⠀⠀⠀⠸⡆⠀⠀⠀⢠⣷⡾⠃
⠀⠀⠀⢀⡄⠀⢸⡇⠀⠀⠀⠈⠻⣤⠀⠤⠊⢹⠃⠀⠀⠀⠀⣽⠁⠀⠀⠀⠀⠇⠀⢀⣿⣿⢰⠇⠀⠀⠀⠀⢰⣧⣤⡶⠿⠛⠉
⠀⠀⠀⢸⣿⣦⣿⠁⠀⠀⠀⠀⠀⢫⠀⠀⠀⣸⣧⣤⣆⣀⣼⡟⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⡟⠀⠀⠀⠀⢀⡟⠉
⠀⠀⡠⣾⠋⠀⠙⠤⣼⠿⠷⠾⣦⣼⣇⠀⠀⣉⠻⣿⣿⣿⠏⠀⢀⡀⡀⡄⠀⣆⢸⡿⠈⠟⠀⠀⠀⢀⡤⠋
⠀⢰⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡻⢿⣺⡃⠀⠋⠀⠀⣿⣿⣿⣿⣿⣾⣧⣧⣀⣀⣑⣒⣿⠟
⠀⡇⠀⢰⠀⠀⠀⠰⡀⠀⠀⠀⣸⠀⠀⢻⡇⠀⠙⠇⠀⠀⠀⣼⡿⠟⠻⢿⣥⣀⠀⠉⠙⠛⠉⡙⠁
⠀⡇⠀⢸⡇⢰⡀⠀⠹⣦⡀⢠⡟⠀⠀⠈⠁⠀⠀⢸⡂⠈⣿⠏⠀⠀⠀⠀⠈⠉⠛⠛⠛⠒⠋⠁
⠀⢻⢆⢸⡇⠈⣿⡶⢦⣼⣿⣾⣇⠀⠀⠀⠀⣀⣴⠯⠄⠊⠁
⠀⡀⢠⣿⡇⠀⡟⠀⠀⠀⠀⠉⠀⠉⠉⠉⠁
⠀⠙⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠊⠉⡆⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠵⡒⠁⠀⢇⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢦⣀⡼⠀⠀⠸⡯⠀⠁⠀⠀⢀⠀⠀⢠⣿⣄⠀⠀⠀⠀⠓⠀⠀⠠⠄⠀⠈⠛⢇⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⣼⠁⠀⠁⠀⠀⢸⠀⠀⠘⡏⡇⠀⢀⣤⡈⠀⠑⠀⠀⠀⡀⠀⠀⠀⢱⠀⣸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠘⠻⢤⡀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⠀⠀⢸⠇⠀⢀⠊⠈⠁⠀⠀⠃⠀⠀⠙⠿⠿⠿⠋⠀⠛⠉
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡖⠀⠀⡀⠀⠹⡆⠀⠀⡇⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠠⠇⠀⠀⠀⠀⡄⠀⠀⠀⢠
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢣⠀⠀⠅⠀⣠⠇⠀⠀⡇⠀⠀⠀⠀⠀⣸⠘⣦⠀⠀⠀⠀⡄⠀⠈⢩⠉⠉⡇⠀⠀⢠⡮⡀⠀⠀⢘
⠀⠀⠀⠀⡆⠀⢠⠀⠀⠀⠈⠟⠛⠟⠛⠏⠀⠀⠀⡇⠀⠀⠀⠀⠉⢻⠀⠉⠻⠉⠉⠋⠁⠀⠀⢸⠀⣗⡇⠀⡖⠉⠀⠀⢀
⠀⠀⣠⣤⣿⣀⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣶⣶⣶⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣤⣷⣧⣤⣷⣤⣤⣤⣼⣤⣬⣤⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⣄⣀⣀⠀⢀⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡿⣯⡶⠿⢾⣝⢿⠠⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠰⣷⠀⣿⢾⣷⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡿⢦⣤⡼⣿⣾⠁⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠻⣿⠛⠉⣋⡼⣿⡽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⡃⢶⠄⣄⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⣷⣶⣾⡿⠂⢄⠀⠀⠙⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠁⠄⠀⠘⢦⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⠄⠐⢃⣀⠭⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠔⠚⡈⠁⠀⠐⠢⣶⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠄⠁⠘⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠂⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠴⠁⠀⠀⢀⣁⡀⠀⠀⣤⣔⠀⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠿⠿⠀⠀⠙⠛⠀⠁⠀⠀⠘⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⡆⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢸⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠠⠇⠀⠒⡂⠀⠀⠀⠂⠫⠣⠀⣀⠀⠀⢸⢤⢿⡀⠀⠀⠀⢸⡇⡇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠸⡂⢈⣀⠀⠀⠀⠐⠀⠀⣰⣏⣆⠀⠈⠐⠊⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣣⣵⣼⠋⠍⠁⡀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⡖⠉⠉⠀⢸⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡘⠢⣄⠍⠐⠂⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⠆⢸⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠈⠀⠄⠀⠁⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠣⠤⢦⠬⣤⡤⠤⠗⠚⠡⠤⠤⠤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣼
⠀⠠⠀⠸⠀⡄⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠇⠐⣄⠀⠀⠰⠀⢢⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡀⠀⠀⠠⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⡀
⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡁⢀⠀⣁⣠⢀⡅⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠤⠀⠤⠤⠤⠤⠜⡀⡀
⠀⠌⠀⡄⢠⠆⢀⡀⢠⠀⠀⢹⠁⠀⠉⠈⠀⠀⠀⠀⠈⡇⠀⠆⠰⠀⠠⠤⠄⠂⠁⡐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁
⠀⠉⠀⠁⠈⡏⢈⠀⠠⡀⠀⣸⠀⣀⠀⠀⡀⠀⠀⠤⠀⠇⠀⠀⠀⢀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⢀⡀⠠⣈⠃⠀⠀⡐⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠀⠀⢀⠀⠀⠠⠀⠂⣀⠐
⠀⠀⠀⠀⠠⠆⠀⠀⠀⢀⡀⠀⢁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠒⠂⡑⡄⡅⠀⠀⠀⠐⠠⡤⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⣁⡒⠀⠠⠀⠒⠈⣀⠀⠐⠀⠀⠀⠀⠈⠀⠐⠄⠀⠀⠠⠄⠀⠆⡄⢀⠀⡆⠀⠑⠁⠀⠀⠀⢎⠀⠀⠀⣀⠘⠀⢀⡂⠶⡅
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠐⡱⠦⠤⠔⠊⠀⠀⠀⣇⢀⠙⣠⠀⢀⠠⠦⠀⢰⠀⠭⠃⢠⠠⡄⠀⠁⠀⠀⠀⢀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠔⠁⠀⠀⠐⠀⠀⠀⠀⠘⢦⡀⠈⠁⠉⠁⢂⡠⠬⠀⠀⠉⠀⠋⠀⠠⡀⠀⠀⠐⠈⠐⠉⠐
⠀⠂⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠂⠰⠀⠀⠀⠀⠀⠀⠉⡁⠒⠒⠊⠉⠀⢠⠀⠀⢰⠀⠑⠀⠠⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠔⠀⠈⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⡊⠐⠀⠄⠀⠘⠀⠑⠁⠦⠀⠀⢀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠂⠂⠀⠀⠀⠀⠀⣀⠀⠀⢀⣠⣤⣄⡀⢀⣀⣀⡀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⠈⠉⠉⠁⠁⠀⠀⠀⠈⠁⠀⠀⠀⡀⢸⣿⠀⠀⢸⠟⠉⢻⡇⣿⡇⢹⡇⠀⠀⠰⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠍⠀⠁
⠀⠀⠀⠀⢨⠈⡅⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠃⢸⣿⠀⠀⢸⣆⣀⣸⡇⢸⣧⣾⠇⠀⠀⢀⣀⣀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢉⠆⠺⠀⠆⡀⠠⡄⠀⠀⠀⠀⠱⡀⢀⠔⠀⢸⣿⠀⠀⢸⡇⠀⢸⡇⢸⡏⣹⡇⠀⠀⢸⠉⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠘⠃⠀⠘⠁⠘⠛⠛⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠀⠂
⢀⠀⠐⠉⠀⠀⠀⡆⠀⠂⠀⠀⠀⢠⢀⠀⠀⣀⠀⠀⢸⡛⡟⣻⠻⢿⠛⢟⣿⢻⡇⠀⠀⠀⡄⠀⠆⠐⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠈
⠀⠉⠀⠀⠒⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣷⣷⢾⡾⣾⡷⣷⡿⡾⡇⠀⠀⠀⠁⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠑⠀⠀⢸⠀⠠⠀⠀⠀⣀⠀⠀⠠⠀⠀⠐⢀⠀⠌⠐⢆⡃⠂⠀⠡⠠⢀⡐⠀⠀⠀⠀⠀⠠⡀⠠⠂
⠀⢀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠆⠀⠀⠀⠰⠀⢀⣀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠵⠀⠀⠀⠀⠊⢔⠀⠀⠀⠀⠀⠙⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠂⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⠀⠀⠁⠀⠂⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠀⠀⠂
⠀⠀⢰⡆⢰⡆⠀⠀⢠⡾⢷⡄⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠐⠁⠊⠁⡐⠀⠀⠀⠉
⠀⠀⢸⣧⣼⡇⡶⣦⠸⣧⢸⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠈⠀⠀⠀⠀⠀⠈
⠀⠀⢸⣿⢸⡇⢀⡿⠀⣿⣸⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⡄⣡⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠈⠛⠘⠁⠙⠛⠀⠙⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⡹⠟⣧⣦⣤⡺⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣤⣦⣜⣛⣠⡘⢳⡄⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⠾⢿⣿⣿⣿⣿⡟⠉⢠⡜⠃⠀⢿⠋⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⣿⡿⢧⠀⠀⠀⠀⠀⠀⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣼⠟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠻⠿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⡤⠀⠀⣀⠀⠠⠘⠃⡄⠐⠀⠄⢀⣀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⣀⡩⠥⠄⠒⠒⠀⠀⠀⠀⠀⠒⠂⠤⠬⢉⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⡉⠥⠀⠀⠀⠀⠀⠀⠠⠍⣉⡐⠂⠤⠠⠁
⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠭⠤⠔⠒⢀⠀⠀⠀⠀⠀⡐⠒⠠⠆⡍⢈⡁
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣈⠩⠥⠄⠀⠀⠀⠀⠀⠠⠬⣷⣷⣷⣶⣦⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠠⠥⠄⠒⢂⣉⠀⠀⠀⠀⢈⣉⡐⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠒⢂⣉⠭⠠⠄⠀⠀⠀⠀⠀⠠⠠⢽⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣠⣄⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣿⣿⣿⣿⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣿⣿⣻⣛⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠀⠠⠀⣀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣡⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣥⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢀⣾⣧⣄⣀⣀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⣠⠞⠙⡽⡫⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠤⠀⠀⠀⠀⠘
⠀⠀⣾⣿⣿⣉⠉⠉⠙⠓⠲⠦⠥⣴⣦⣵⡶⠊⠁⡀⠀⠊⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠹⡆⠀⠀⠀⠀⠀⠑⠢⣄⠀⠀⠀⠀⣰
⠀⠀⠘⠟⢻⠋⠀⠀⠐⣶⡶⠀⠀⠀⣼⠃⠆⠀⠀⠊⠀⠀⠀⠀⠀⠀⣤⢛⣦⣀⣀⠈⠀⠀⠀⢸⣤⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠏⠙⠂⢄⠀⣰⠟⣛⠀⠀⠤⣿⡶⠃⠀⠀⠀⠀⠀⢠⡾⠛⢻⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⢸⡁⢀⡾⣟⡿⣟⣿⣿⣿⠀⠀⠀⠀⠀⣿⠧⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣭⠛⢿⡭⢁⣻⣿⣿⠟⠀⠀⠀⠀⠀⢸⡄⠀⠤⠀⣀⡉⠁⣀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⡵⠿⠟⠁⠀⠀⠀⠀⠀⠀⠈⣇⠀⠐⠛⠋⠙⠋⠁⠀⠈⠁⠀⣠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣠⢸⡌⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠶⠇
⠀⠀⡄⢰⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹⠀⠀⠀⠀⣴⡤⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠈⠁⠀⠛⣋⣀⡀⣀⠀⢀⣾⣷⣤⡴⠆⠀⠀⢀⡴⠃
⠀⠀⣿⢶⣦⣤⠤⠈⠤⠀⠉⠐⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⢈⡀⣀⠾⡑⠄⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠘⠋⠛⠛⠉⠀⠛⠛⣛⡋⢀⣤⠴⠞⠋⠁
⠀⠀⣾⠦⣀⡀⠀⠁⠉⠁⠙⠓⠄⠸⣇⡀⠁⠀⠀⠀⠀⠀⠀⠙⢮⡢⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⡟⠀⠓⠒⠀⠀⠉⠉
⠀⠀⠻⣦⣄⣈⣈⠙⠂⠒⠒⢒⠘⢣⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡂⠀⠀⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠁⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣷⣇⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷⠟⠁⠀⠀⠀⠀⢠⣶⣾⠏⠠⢀⠠⡀
⠀⠀⣤⡶⣶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⢿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠐⠛⠛⠓⠒⠂⠸⣷⡦⣤⣤⡀
⠀⠐⣿⠀⠙⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⢊⡥⠀⠀⠀⠀⠀⣿⢰⠏⠈⣿⣀
⠀⠀⢸⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢡⣤⢘⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠐⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⡆⠀⠀⠀⠀⠀⠀⡅⣿⠀⠀⢨⣹⡦⡀
⠀⠀⠘⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⢁⣬⡍⠙⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⡰⢋⠁⠀⠠⡁⠀⠀⠤⢀⠂⢀⡇⢹⠀⠀⠀⣿⢧⡈⠑⠄
⠀⠀⠀⡇⠀⢸⣿⣿⠙⠻⣿⣿⣿⡷⠈⢛⣡⣾⣿⡟⢋⣍⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⡃⠐⡀⠀⠀⠀⠁⠈⠀⠀⠀⡸⠁⡾⠀⠀⠐⢹⠀⠛⢦⡀⢡⡀
⠀⠀⠀⡇⠀⠘⣿⣿⠘⣂⡀⣻⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⡏⡇⠀⠀⠀⠀⠀⠀⣹⡄⠑⠄⡀⠚⠂⣴⠀⢀⡴⢁⡼⢃⣀⠀⠀⠸⣧⠀⠀⠘⢻⡇
⠀⠀⠀⢃⠀⠀⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣛⣻⣿⣿⣿⡅⠀⠆⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠉⡯⠥⠴⣿⠁⠰⡇⣠⢀⠀⢹⡆⠀⣠⣦⠃
⠀⠀⠰⢜⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⢷⣦⣦⠀⠀⠀⠈⢿⠀⡤⠀⠀⢀⠚⠁⠀⠀⠀⣿⠀⠠⡄⠘⠁⠀⢸⡟⠉⢻⠇
⠀⠀⠀⣾⣤⣾⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⡿⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠛⠂⠀⢣⠀⠻⠀⠀⣀⠟⠛⠁
⠀⠀⠀⠙⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢖⣀⣀⣰⣧⡴⠾⠄⠳⠦⠐⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠲⡤⠤⠄⠀⠰⠮⣁⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠁⠀⠀⠘⡀⠀⠀⠀⠀⠀⠉⠢⣄⠐⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠔⠃⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⢸⠦⣀⣰⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣸⠀⠀⠀⠀⢣⠈⠑⠒⠓⠒⠠⠤⠤⠖⡐⠒⠖⠈
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢋⠀⠀⠀⠀⠘⡄⠀⢀⣐⣂⣀⣀⣠⠴⠒⠒⣷⠆
⠀⠀⠀⠀⠀⠀⠘⠶⡀⡰⢣⡄⢆⡴⠤⣄⢀⠶⠺⡏⠀⣠⠀⡏⠹⡀⠘⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠣⣗⠀⠉⠀⠀⠑⠤⠤⠚⠠⡤⠈⠉⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⣀⣀⠀⢀⣀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠒⠽⡅⢠⣏⣹⢨⠀⠀⡔⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠈⠳⢅⢀⡯⢵⡤⢚⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⢄⠀⠀⠀⠈⠁⠀⠀⡄⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠙⠒⠤⣀⣀⣀⣀⡿⠋⠁⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠄⡆⠀⠀⠀⠀⠘⡄⠀⠀⠀⡮⡇
⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⡆⠀⠀⠀⠀⡜⠀⠀⠀⠂⠇⠀⠀⠀⠀⠀⠱⠤⡀⠴⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⡇⠀⠀⠀⠰⢁⠀⠀⠀⢀⣒⣀⣀⠠⡤⠄⠚⠁⠀⢢⠄⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠃⠀⠀⠀⠀⢀⠜⢫⡉⠁⢀⣌⠀⢂⡇⠀⠀⠀⠀⢸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⡀⠀⢠⠚⠑⢢⣀⡉⢂⡺⠀⣈⠡⠶⣄⠀⠀⠀⡸⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⢇⠉⢉⢇⠀⠐⠂⠒⠈⢣⠀⠂⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠆⢸⡀⠀⠀⠀⠉⠁⠈⡆⠀⠀⠀⣀⠎⠀⡄⠀⠀⠀⠀⡸⠀⡄⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠁⠀⠑⠤⣀⣀⣀⣀⠔⠉⠉⠉⠉⠀⠀⠀⠈⠂⠤⠤⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⠀⠟⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⡶⠶⣄⠘⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⡿⡄⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠉⠋⠀⣘⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣸⠆⠀⠀⡴⠉⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢶⣦⢰⣶⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠉⡙⠻⣤⣠⣤⣤⣠⣼⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢲⣶⠚⣿⠉⠙⠋⣿⡇⠈⠋⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⢠⠀⣉⠀⠋⡤⢠⡞⠁⣘⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⢻⣦⣀⣀⠜⢠⣾⣿⣟⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣤⡀⠀⠙⢷⣦⣿⠛⠟⣉⠀⠀⣤⡜⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣌⠻⠂⠘⠃⢾⡆⠀⠀⠘⢿⣦⣭⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢁⣼⠟⠛⢷⣶⠀⠀⠀⠀⣶⣉⡉⢹⡄⢰⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⢸⡃⠀⠀⢀⡽⢇⠀⠀⣿⠀⠙⣁⢸⣧⠈⠁⠿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣸⢷⣄⠀⡼⠃⣾⠀⣾⠀⠀⣤⠟⠋⠀⠙⢻⣦⠙⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠈⠉⠀⠀⠸⢆⣿⠀⣼⠋⠀⢰⡞⠁⠀⣿⡇⠛⠁⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠀⣿⠀⠀⠸⢷⣴⠾⣏⡀⠶⠀⠀⠀⠀⢾⡟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⢻⡶⠶⣤⡹⣿⣿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢁⡸⠇⢠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡜⢻⡖⢻⣿⠿⣿⡀⠀⡀⠀⠀⠀⢀⣠⣤⡀⠀⠀⣀⠐⠛⠛⢷⣤⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣧⠘⢻⣾⠿⠛⣷⣾⣿⡍⢉⠛⣤⣤⣀⠀⠀⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢈⣏⣠⡞⢣⣤⣿⣷⣞⠛⣷⠶⢻⡟⠀⠀⢶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⠻⢿⡿⠇⡸⠃⠠⠼⣿⣀⣿⣄⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠱⡀⣠⠗⠂⠀⠈⠿⠧⣱⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⠀⢰⣿⣦⣄⣼⠃⠀⣠⠐⣤⠀⢀⡅⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣿⡟⠛⠙⠛⢿⣿⣿⣶⣤⠔⠋⠁⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣶⢠⣿⠘⢧⡀⠃⠚⣁⣼⠿⣷⣤⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠗⢉⣀⣉⡀⠹⣷⣄⠹⠶⠚⠛⢉⣴⣿⠈⠉⠁⡀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠁⣴⡞⢋⡉⠻⢶⣄⠉⢳⣶⠶⠿⠟⠉⣿⢀⣴⣿⣻⣶⠛⢁⡀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡶⣏⠀⠈⣉⠀⠀⢹⣷⠀⠀⣠⣤⠀⠀⣿⠈⠛⠛⢋⡀⠠⣿⠃⠀⠛⠁⠀⢲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⠻⠟⠃⠀⠀⢈⣿⢠⡼⢃⣽⠃⠀⣿⠀⠀⢀⠸⠷⡀⠛⠂⠚⠃⣤⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠶⣢⣤⡀⠀⢸⡟⢸⣿⢻⣿⠀⣶⣅⣠⣼⠟⢀⣤⠟⢣⣴⣶⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⢀⣴⢿⠟⢷⡄⠀⣸⡟⠙⣿⡀⢸⡇⠸⣿⡟⣣⡿⠉⠛⠋⢉⣠⡾⠻⠤⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⣼⣿⢈⣁⣨⡿⠀⠘⠿⢿⣿⣧⡈⠇⣀⣠⡾⠋⣀⣀⠿⣦⠙⣿⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⣿⣎⠉⠉⠀⠀⠀⠀⠀⠀⣉⣡⣄⣽⣯⣴⠄⠛⢋⣤⡌⠛⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠠⠀⠈⢹⡇⠀⢀⣾⣦⣤⣤⡾⠟⣉⠙⠻⣯⡁⠻⠷⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⠀⠁⢀⡁⣠⡿⠋⣩⠉⣩⠿⢶⣄⠐⢀⣼⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⣿⣧⠼⣀⣠⣤⣶⣦⣜⠳⣾⣿⠉⣤⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⢀⡄⢻⡏⠀⣼⡿⠛⠿⠉⠙⢻⡇⢸⡿⣦⡀⠙⠣⣄⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⢨⣇⣸⣇⠀⢿⣿⠛⠶⠃⠀⠀⠀⠾⠃⠈⣳⡀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⢉⡹⢿⣷⣀⠙⠷⣤⣄⠀⠑⣾⡆⠀⢰⣿⣠⣾⠇⠀⢠⣾⠁⠀⠀⢠⡘⠿⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⢸⡇⢸⡇⠀⠀⠀⠉⠉⠻⠿⡈⠀⢠⣾⣿⣿⣥⣴⣾⡿⠁⠀⠀⠀⠈⠁⠀⠲⣮⠋⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⢸⡇⠈⢿⣧⡐⣤⠀⠀⠀⡰⠟⢰⠟⢻⡈⣿⡏⠉⠁⢀⣤⣶⣿⣿⣿⣷⣦⣄⠀⢺⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣦⣄⡈⠻⣦⣄⠙⣿⣿⣿⡷⠿⢣⡄⠀⣠⠎⢠⣿⠇⢀⣴⠟⠋⠁⠀⠀⠀⠉⠻⣿⣷⡀⠘⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠛⢿⣷⣦⡀⠈⠉⣥⣿⠙⢷⣦⣀⣠⡾⠃⣠⣿⡿⣶⡟⠁⠀⠀⠀⢀⣀⡀⠀⠀⠙⣿⣿⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⢷⣄⠉⠻⠿⣿⠿⠟⢿⣦⣄⡈⠙⠋⣠⣾⣿⠟⣿⠏⠀⠀⠀⠀⢰⡞⠉⠁⠀⠀⠀⣿⣿⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⡹⢷⡄⠀⠀⠀⠀⠀⠛⣿⣿⠿⢿⣿⡟⠁⣾⡇⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⠀⣰⣿⡟⠀⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡟⠓⣆⣱⠂⠀⠀⠀⣴⠟⠉⣿⠀⠀⠈⠁⣴⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣿⡿⠋⠀⠰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢴⣿⣿⡆⠀⠀⢰⠁⢸⡿⢩⡇⠀⠀⣾⡟⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡾⠿⠟⠁⠀⠀⠀⢠⣶⡇⠸⠇⠀⢠⣿⠃⠀⢿⡏⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠈⠀⠀⢿⣿⡿⠀⠀⠀⢸⡟⠀⠀⠀⢀⣼⠟⠀⢴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠙⣿⡿⣿⣷⣶⡄⠀⠈⠛⣀⠀⠀⢀⣾⣇⠰⠶⠶⠿⣧⣀⠳⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣌⡛⢿⣿⣿⡿⠇⠀⠚⢿⣆⠀⣸⡟⠁⠀⠀⠀⣀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠀⠀⠀⠀⠈⣧⣰⡿⠀⠀⠀⠀⢾⡀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡟⠛⠿⠟⠛⠿⠟⠛⠿⠟⠛⣼⠟⠛⠻⠿⠟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣏⣙⣿⣏⣉⣉⣉⣉⣉⣩⣏⣿⣿⣿⣟⣛⡛⠛⠛⠛⠛⢛⣛⣻⣿
⣿⣿⣶⣿⣷⣶⣾⣶⣾⣿⣿⣿⣿⣿⣶⣶⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢹⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⡿⠒⠉⠉⠲⣄⢠⠖⠉⠉⠒⢤⣤⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡴⠀⠀⠱⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⠼⡧⡀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⣠⠊⠀⠑⠦⢰⠀⠀⠀⠀⠀⢸⡇⣀⠀⠀⠀⠀⡤⠤⠊⠀⠑⢄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠇⠀⢠⠀⠀⠈⠂⠀⠀⠀⠋⢰⡇⠈⠀⠀⠀⠐⠁⠀⠀⡀⠀⠘⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⡆⠀⠀⡁⢰⠀⠀⡤⠀⠀⠀⢨⡇⠀⠀⠀⠦⡀⠀⡁⢈⠁⠀⢠⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠙⢤⠊⠀⠈⠀⠀⠀⠀⠖⢀⢼⡧⣀⠲⡀⠀⠀⠀⠃⠀⠑⣤⠊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠘⢆⠀⡀⠀⠀⠀⠀⠀⢀⡜⢣⡀⠀⠀⠀⠀⠀⢀⠀⣠⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠓⠬⣉⣀⣈⣂⠴⠊⠀⠀⠙⠦⢐⡀⠀⣈⡡⠞⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⢋⣉⣉⣉⣉⡃⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠉⠉⠉⠉⠉⠉⠉⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣤⣖⠭⠔⠒⠉⠉⠉⠀⠀⠀⠀⠉⠛⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⣤⣤⣤⣤⣤⣤⣤⣤⣧⣶⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢧⠀⠀⠀⢀⡜⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉

⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⣶⢦⡀
⠀⠀⠀⢀⠀⣄⠀⠀⠀⠀⢆⡆⠀⠀⠀⠀⣹⣄⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⣠⠞⢵⠀⠈⠢⠈⠓⠤⣄
⠀⠀⢀⠀⠀⢘⣆⠀⠀⣘⣙⣻⡄⠀⠀⠀⡇⢘⣿⠀⠀⠀⠀⢸⣏⠀⠀⠀⠈⢰⠞⠁⠀⠀⠀⠐⢆⠀⠈
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⣈⠉⠁⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠹⡑⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⣀⢀⡀
⠀⠀⠀⢰⡶⣒⢿⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣟⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢀⡇⠀⠀⠀⠀⠀⣀⠃
⠀⠀⠀⢸⣿⢿⠿⣿⠀⠀⠉⠉⠘⡇⠀⠀⠀⢐⠶⠖⠀⠀⢠⠄⠀⢤⠀⠀⠀⡇⢸⠈⡇⠀⢀⠔⠋
⠀⠀⠀⠚⠋⠉⠉⠛⠀⠀⠀⢐⠊⠀⠀⠐⠒⠒⠒⠒⠀⠀⠙⠒⠒⠚⠃⠀⠀⠁⠀⠚⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⢜⡆⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⡀⢀⣨⡇⠀⠀⣀⠠⢚⡃⠀⠀⠲⠲⢔⠶⡲⢖⠶⡲⢲⠆⠀⠀⠀⢈⠀⠀⣹⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠉⠉⠉⠀⠀⠈⠀⠁⠈⠀⠁⠉⠈⠀⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠉
⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⡤⢤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⡄⠀⠠⡄
⠀⠀⠀⢜⠓⠂⠀⢠⡄⠀⠃⢸⠀⢱⠀⠀⠸⠛⠛⢩⣧⠀⠀⠖⠀⠀⢸⠄⠀⠀⢠⡾⠛⠁⠀⣴⡇⠀⢸⡇
⠀⠀⠀⠘⠦⠀⠠⠞⠁⠀⠀⠸⠀⠀⠄⠐⠶⠒⠒⠚⠲⠆⠀⠰⠤⠤⠞⠀⠠⠊⠈⠀⠀⠀⠈⠀⠁⠀⠼⠃

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣘⣳⡄
⠀⠀⠀⡠⠀⠀⠀⣠⣶⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⣰⣿⣿⡟⡇
⠀⠀⠀⢳⡀⠀⠐⠿⠟⠁⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣦⣄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠻⠟⠋⠀⣧
⠀⠀⠈⢄⠱⡤⠴⠶⠴⠖⣉⠤⠒⣚⡀⠆⠀⠀⠀⠀⠀⢉⣩⣉⣉⣍⣉⠁⠀⠀⠀⠀⠀⠑⠂⠠⠀⢀⣄⠀⠠⠀⠒⠋⠀⠀⠀⠀⣿⣷⣶⣦⣤⣠⡆⣀
⠀⠀⠱⡈⢖⣚⣩⣍⣉⣛⡡⠚⠉⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠾⣟⡆⢀⡄
⠀⠀⠀⠙⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠁⠐⠲⠷⠒⠈⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢺⢤⡖⣿⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡛⣿⠾⠾⡗⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⡙⢛⣿⠇⣷⠟⢚⡝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡆⠉⠉⠑⠊⠉⠉⢰⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡇⢠⣴⣿⣿⣶⡄⢸⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⡄⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠷⢴⡇⣿⣿⢸⡧⠾⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠸⣋⣠⠤⠤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠼⠷⠿⠿⠾⠧⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣾⣷⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⢠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣤⣶⣶⣦⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣤⣄⡀
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⡍⢹
⢸⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣭⣭⣶⣮⣋⣵⣾⣭⣽⣏⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢀⡘⠿⠿⠿⠟⠁⣘⡀
⠘⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣘⣇⣀⣇⣀⣸⣀⣸⣁⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣶⣶⣦⣶⣾⠿⠿⡇
⠸⠀⠀⢀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣈⣣⣈⣇⣀⣹⣀⣽⣁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣿⣿⣶⣾⣿⡇
⠈⠉⠀⠀⢉⣿⣿⡏⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⡫⣹⣉⣏⣝⣩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⢿⣿⡿⣿⣿⣿⠿⠟⠃

⠀⠀⠀⠀⣠⣤⣤⣄⠀⢀⡀⣀⣠⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣿⣟⣿⣷⣿⡇⠘⠓⠛⠛⠚⠐⢠⣺⣷⡄⡀⠀⠀⠀⠰⢆⣂⣂⠈⠀⢀⣀⣠⡶⠋⠀⠀⠠⣀⠀⠀⠀⠀⠳⢎⡸⠀⠀⠀⠀⠄
⠀⠀⠈⠛⠿⠟⢿⠋⠀⠀⠀⣠⠶⢄⠀⠀⠉⠉⠀⠀⠀⠲⢣⣞⠁⠈⢻⣛⣆⠛⠋⠁⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠸⠟⠻⠇⠀⣠⡄
⠀⠀⠀⣠⣴⣤⣌⡀⠀⣀⡤⣷⣤⡾⠃⠀⡀⠀⢠⠀⠀⠀⢾⣿⡄⠀⠈⣿⡏⡇⣰⠆⣠⡦⠀⡴⡄⠀⠀⢠⡶⣦⢠⣶⣦⡸⣧⣸⣿⡀
⠀⠀⢰⠻⣿⣿⡿⠟⠋⠑⠀⡼⠁⡀⠐⡤⡾⣿⡯⠑⢏⠀⠘⢏⡁⣠⣤⣍⡠⠃⠋⠊⠁⠀⠐⣃⡀⢠⣦⠘⢷⡿⢻⣧⣴⣷⠘⠋⠹⣧
⠀⠀⡏⠀⠘⣯⣠⣄⣀⣀⣾⣇⠀⣧⠈⠳⢤⣮⣦⠴⠋⠀⠀⠀⠉⠙⠛⡉⣠⡄⢠⣠⣦⢠⣿⠛⣿⡌⣿⣤⣴⡆⠀⠻⢿⠿⠀⣀⡀⢀⡄
⠀⠀⢿⠀⢤⠁⠉⠻⢮⣿⣿⣿⡇⠈⠷⣶⣆⠀⠀⠀⢤⣰⠚⢣⣤⣄⣺⣧⣸⣷⢻⣿⣿⣎⣿⣦⣿⠇⠻⠿⠛⠋⠀⠀⢰⣾⣿⢿⡇⠈⠇
⠀⠀⠈⢷⣤⡀⠀⠀⠀⣈⡽⠟⠁⠀⠀⠉⠀⣀⣤⣴⡄⠈⣀⣿⡏⠙⠛⣿⡏⢿⡞⡿⠙⠟⠈⡉⠁⠀⠀⠀⣧⠀⠀⠀⣸⣷⣤⣄⣧⣀⡀
⠀⣤⣶⣾⣍⣛⣻⣿⡛⠉⢀⢤⡒⠄⣇⡀⠹⠛⣿⡏⢺⣿⣽⡜⢿⣶⣿⠻⠗⠈⠀⠀⠀⡀⠛⠁⠲⠄⠀⠤⠞⠀⠀⠘⣿⣷⣾⣿⣿⣿⠃
⠀⣿⣟⠛⠛⠛⠛⠛⢻⠃⡜⠀⠀⠀⠀⠂⢀⠀⢹⣿⡘⣿⣷⠶⠀⠉⢁⠀⠀⠀⢹⡀⠀⠸⡄⢠⠤⠒⠀⣀⣠⣤⣤⣀⠀⠙⢿⣿⣿⠃
⠀⣿⠿⡀⠀⠀⠀⢀⡏⠀⢷⠀⠀⠀⠀⢈⡆⠀⠀⠉⠀⣀⡀⠀⠀⢰⣬⣟⣲⠤⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣟⣿⣿⡟⠀⢠⡀⠙⠁
⠀⠻⣧⣇⣀⣀⣀⡼⠁⡀⠀⠑⠂⠔⠐⠀⢀⣠⠤⠒⠉⠙⣁⣀⣤⣼⣿⣿⡿⢿⣿⣶⡆⠀⡁⠀⠀⠈⠙⢿⣿⣿⣿⠀⠰⡏⠘⠳⠆⣀⡀
⠀⠀⠀⠉⠉⠉⠁⠀⢄⠀⠠⣀⡀⠰⠞⠛⢉⣀⣤⣶⣺⣿⣿⣿⣻⣿⣿⣾⣿⣿⡿⠃⠈⡥⠁⠀⠀⠀⢀⡀⠈⣵⡆⠀⣼⣿⣦⣦⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣆⡔⠀⠸⣃⣤⣶⣾⣿⣽⣿⠿⠿⠟⠛⠛⠛⠉⠹⣿⡿⠋⠀⠂⡠⠔⠉⡁⠀⢤⣿⣷⡄⢸⡆⠀⠻⠿⠛⠁⠐⣿⠂
⠀⠀⠀⠃⠁⡔⠀⠀⠉⠉⠀⠈⠛⣛⠋⠉⠉⠀⠀⠀⠀⠀⢈⣶⠃⠀⠀⠉⠈⠆⡆⠀⠀⣀⣀⢿⡿⠄⢉⣩⠥⠔⠧⢄⡀⠀⠀⠀⠬⡁
⠀⠀⠀⠂⠀⠀⠀⠀⠂⣤⡞⢉⣻⣅⣈⠑⣤⠢⠐⠀⠻⢷⣾⠋⠀⠀⠑⠈⠢⣠⠖⣆⠀⡇⠈⡎⠁⣰⠋⣠⣶⣶⣶⣦⣙⡄⠀⠠⣤⣄
⠀⠀⠀⠀⡀⠰⡆⠀⣾⣤⣾⣿⣽⠶⠶⠟⠾⠷⣉⡴⠠⠐⠀⠀⠀⣤⣦⣤⣤⡘⣦⣈⠛⣷⣤⠇⢰⡇⣬⣿⣿⣿⣿⣿⣿⣇⠀⢨⣷⣾
⠀⠀⣠⣾⣷⡀⠉⠘⡿⢋⠉⠀⢻⡀⠀⠀⢤⣤⢸⣷⡀⠐⡅⢰⣟⣛⣿⠻⡟⣿⣎⠙⢻⣿⣿⣆⠈⣧⠸⣿⣿⣿⣿⣿⠏⢹⡀⠘⠿⠟⠂
⠀⠀⠛⣿⡟⠓⠀⢸⣿⡾⠆⢠⣾⣿⣷⣦⣄⣀⠞⠙⣿⡀⢇⢸⣿⣛⣿⣠⣇⣿⣿⠀⠘⣏⡕⡻⣷⣿⡦⠄⣉⣉⣉⡤⠒⠉⢿⡵⣦⡀
⠀⠀⠀⣿⡇⠀⢠⣿⠉⠓⢦⣾⣿⣿⢿⣿⠟⢅⠀⠀⢸⣧⢪⢸⣿⢙⠏⡿⡏⣿⣿⠀⠀⠙⣾⣱⣹⣿⢴⢒⣍⣩⣭⡍⡭⣽⠸⣿⣒⣳
⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠘⢯⣛⣛⠛⣋⡠⠆⠀⠀⣴⣛⢦⡀⣿⡞⡿⣷⣗⢿⣿⡆⢰⠄⠈⠛⠿⣿⡸⣙⠞⠙⢿⠇⢉⡼⠀⣿⣿⣾⡆
⠀⠀⢀⣿⡇⣴⠋⢿⣳⡄⠀⠀⢨⠏⠛⠂⢲⠀⠀⢸⣿⡦⡌⢷⢹⣷⣤⢿⡼⠼⣿⣧⠘⡄⠀⠀⠀⢸⣦⣄⣉⣘⣛⣋⣥⣴⣾⣿⡿⠙
⠀⠀⢸⣿⡇⢹⡀⡏⢿⡷⠀⠀⣸⣀⠀⠀⡇⠀⠀⠈⠛⠓⠘⠉⣿⣿⡆⣾⡿⠾⣿⣿⡙⡄⠀⠀⠀⠈⣿⠿⠿⢿⣿⣿⡿⢟⣹⡞⢷⣰
⠀⠀⣾⣹⣷⠀⠉⢡⣴⣶⠆⠀⠀⢸⠁⠀⢉⠇⢀⣀⣾⣀⡀⠀⣾⡿⣻⢻⡟⢿⢻⣿⡧⡇⠀⠀⠀⢀⣿⣭⣭⣭⣧⢸⣿⣿⣶⡷⣶⣄
⠀⢰⣿⣹⣿⣧⡀⢀⡏⠀⣾⠆⠠⠚⠚⣧⠎⠀⠀⢹⢿⡏⠁⠀⢻⣷⠹⢿⡇⣆⢘⣿⠇⡸⠀⠀⢰⣯⣛⣿⣿⣿⣿⠸⣿⣿⡿⠿⠿⢿⡆
⠀⠈⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⠁⠀⠀⠈⠛⠿⠿⠿⠿⠿⠋⠀⠁⠀⠀⠘⠋⠋⠛⠛⠛⠉⠀⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⡀⠀⡠⠀⠀⠀⠀⠄⠩⣀⠆⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⣁⡁⠀⠤⠌⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⠁⢀⡴⠥⠘⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠛⠫⠴⣪⠶⠀⠀⢙⠡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⡤⣄⠁⠀⠀⠀⠀⠀⠀⡂⢰⠀⠀⠀⠘⣤⠀⠀⢀⢀⠀⡄⡆⢠⡴⠁⠛⠫⡱⠀⠀⠀⠀⢿⢫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢁⠒⠁⠀⠀⠀⠀⠀⠀⠐⠵⠘⠀⡀⠀⢸⠈⠀⠀⢨⠀⠀⠀⠩⠊⠜⠀⠀⠀⡇⠀⠀⠀⠀⢠⠪⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠆⠀⠤⠀⠀⠀⠀⡠⢄⠀⠀⠀⠠⠀⠉⠢⣟⣦⡀⠀⠈⠀⠀⡀⡠⠃⠀⠀⠀⢀⣧⡤⠐⠀⠀⢆⡞⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣾⡥⣴⣦⡄⢰⠀⠀⠁⠀⢤⢀⠊⠀⠀⠀⠀⢸⠀⢩⠀⠀⡀⡌⠀⠁⠀⢀⡤⠖⠋⡟⠓⠀⠀⠀⢉⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣛⡂⢽⠀⢈⠠⡄⠰⠀⡤⢠⡀⠀⠰⠆⢠⣤⣴⠀⢸⠀⡂⠀⢸⠾⠋⠀⠀⠘⠉⠀⠀⠀⡇⠀⠀⠨⡙⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣞⢿⠕⠀⢀⡄⠀⠉⠍⠀⠀⢉⠛⠓⢖⡀⠀⠀⠈⠀⢈⢜⠠⠆⠀⠀⢠⠀⠀⠀⣼⡄⠀⠀⠀⠃⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⡚⠛⠚⠀⠀⢸⣀⠐⠀⣘⣠⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠀⠊⠀⡄⢀⠼⣇⣠⣀⠀⡇⠀⠀⢂⠠⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠁⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⠐⡀⠁⠀⠀⣀⣇⡙⠀⣍⡙⣍⣤⣳⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠒⣺⠂⠐⡲⠶⠆⠀⡇⢀⠀⠀⠈⠀⠉⣾⠀⠀⠸
⠀⠀⠀⠀⠐⢒⠂⠁⢀⡰⠃⠀⠀⠀⠸⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠂⠐⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠇⠀⣴⡀⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠀⢀⣰⠋⠀⠀⠀⠀⠀⢰⢶⣀⢀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⢻⡟⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡇⠠⠀⢀⡀⠠⠋⢱⠃
⠀⠀⠀⠀⡤⠤⠼⠿⠦⠄⠦⠀⢠⡈⠻⢸⠾⠮⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠴⡧⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠠⠤⠤⠴⢾⣿⢾⠿⠛⢾⣶⣖⡒⠶⢶⡶⠒⠶⢶⡒⠤⢀⡀⠠⢶⣗⣠⣤⣤⠤⢧⣤⣀⣀⣰⣶⠒⣓⣶⣶⣶⣤⣤⣤⣤⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠶⠿⠿⠿⠁⠾⠓⠚⠛⠛⠃⠁⠈⠛⠛⠛⠉⠉⠀⠈⠀⠛⠛⠓⠒⠒⠒⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⣶⠆⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀
⠀⠀⢰⣿⡆⠀⠚⣀⣐⠃⠀⠀⠀⠀⠰⢏⣉⣉⣉⣉⡹⠆⠀⠀⠀⠀⠀⢴⢿⠿⠿⡆⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣷⣶⡄⠀⠀⠀⠀⢸⣿⣶⣿⣿⠷
⠀⢀⣿⣍⣿⡄⢸⣯⣽⡇⠀⠀⠀⠀⠀⠟⣿⠉⡉⡿⢷⠀⠀⠀⢰⣻⢻⣽⣼⣾⣿⣏⡟⣿⡂⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣶⣦
⠀⠺⠋⠀⠙⠗⠻⠷⠞⠗⠀⠀⠀⠀⠐⣿⡟⠛⠛⢻⡿⠂⠀⠀⢸⣽⣽⡿⢹⠻⠟⣿⣯⣯⡇⠀⠀⠀⠘⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢀⡞⢿⡟⠛⠛⠃
⠀⠈⠁⠈⠉⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢠⠖⠦⢄⣀⡠⠴⠲⡄⠀⠀⠀⠀⢰⣿⣷⣶⣶⣶⣜⡄⠀⠀⠀⠀⢸⠭⠉⠉⢉⣼⠷⠀⠀⠀⢰⣯⣴⣿⣮⣴⣾⣮⣤⣾⡅⠀⠀⢸⡍⠉⠉⠉⣱⠟
⢰⠃⠀⠀⠀⣿⠀⠀⠀⠙⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠠⠀⢀⡾⢁⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢸⠂⠀⢀⣼⢋⡄
⠙⠛⠷⣮⣶⣿⣖⣵⠾⠛⠋⠀⠀⠀⢸⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⢸⠤⠀⠚⠁⢸⡇⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢸⡅⠀⠄⠁⢸⡇
⠀⠀⠀⠈⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠈⠋⠋⠈⠋⠋⠁⠀⠀⠀⠈⠁⠉⠉⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣶⣻⣿⣿⣿⣿⣟⣦⠀⠀⠀⠀⠌⢡⠔⠒⠢⡌⡤⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠖⠝
⣾⣿⢿⡟⣻⣿⠛⢻⣏⡍⢳⠀⠀⠀⣿⣿⣿⣦⡙⣿⣿⣿⠀⠀⠀⠐⣂⢣⠀⠀⢀⡏⡒⠀⠀⠀⠀⢀⡀⢠⡙⣧⡠⠤⡀⠀⠀⠀⠀⠀⣾⠁⠛⢳⡄
⣿⣿⢀⡐⣟⣹⡧⢸⠟⠁⣳⠀⠀⠀⠻⣽⣽⣿⣿⣿⣯⠟⠀⠀⠀⠀⠀⠂⢣⣀⡎⠲⠀⠀⠀⠀⠀⡏⠶⠮⣉⣈⢧⡐⠧⢄⠀⠀⠀⠀⠙⠲⡶⠞⠕
⠙⠿⠼⠓⠚⠈⠉⠈⠉⠉⠁⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠙⠳⠶⠿⠭⠴⠶⠞⠋⠀⠀⠀⠀⠘⠓⠕


⠀⠀⣠⣾⠿⠿⣿⣷⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⣿⣷⡾⢃⣼⣿⡆⠀⠀⠀⠀⠀⣿⡷⢘⣛⣻⣿⣿⡇⠀⠀⠀⠀⢙⢙⣟⣿⣯⣿⣆⠀⠀⠀⠀⢤⠤⠄⢀⣤⡀⠠⠄⢤⠀⢀⢤⣶⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠻⣿⣟⣻⣿⣿⡟⠀⠀⠀⠀⠀⢻⠀⠾⠿⠿⠿⠿⠇⠀⠀⠀⠀⠾⠸⣿⣿⣿⣿⣿⠄⠀⠀⠸⣈⣓⡂⢻⣿⡟⣐⡒⡚⠀⠀⠙⠛⡾⠿⠿⠿⢿⡟⢛
⠀⠀⠀⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⠸⢠⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠠⡿⠉⢿⠄⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠚⠁⣾
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⡂⠀⠀⠀⣀⣠⡄⠀⠀⢠⣄⣀⠀⠀⠀⢐⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠑⠒⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠒⠊⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠐⢄⣠⣄⡠⠄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠀⠀⠀⠀⠤⠀⠈⠀⠀⠀⠀⢀⠀⠀⠁⠛⠛⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠈⠰⢶⣉⡉⠊⠈⣉⣱⠶⠏⠉⠀⠈⠀⠀⠀⠀⢀⡤⠞⠛⠳⠄⠂⠱⠁⠂⠴⠓⠂⠠

⠀⠀⠀⠀⠀⠆⣠⠤⠤⢤⡜⣠⠤⠤⢤⣐⣦⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠂⢄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡠⠀⠀⠀⠈⠀⠀⠀⠠⣀⠀⠀⠀⢀
⠀⠀⠀⠉⠉⠛⠉⠀⠀⠀⠀⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠓⠒⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢝⡓⠒⠋

⠀⠀⠀⠀⢀⣠⠤⠴⡂⠀⠀⠀⠐⠲⠦⢤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠶⣦⣄⣠⡔⠀⠒
⠀⠑⠒⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⠦⠠⠒⠋⠡⠛⠓⠲⠆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⡐⠲⢄⡀⠸⣆⠠⠀⠀⡀⠠⠀⣀⠤⠒⣁⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⠀⠐⠀⢀⣀⠀⠂⣀
⠀⠂⠀⠁⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠈⠑⠊⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

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

⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣒⠒⡖⣢⠨⣡⣶⡔⠦⠃⢀⠀⠂⠀⡤⠀⠒⠒⣂⣀⡐⠒⠒⠂
⠀⠀⠀⠀⣀⠤⣺⠁⢠⣤⣤⣤⣶⣶⣶⣾⠀⠀⣾⣦⣴⣿⣿⡧⠀⠨⢎⠌⠜⡼⠠⣖⡻⠷⣔⠃⡎⠀⠀⡀⢰⣷⠀⠁⢻⣿⡇⠀⠿⠿⠿⠿⠗⠐
⠀⠀⠀⠸⢐⠥⠚⡁⢸⣿⣿⣛⣟⣻⣿⣿⠀⠀⠙⢿⣿⣿⡟⠁⠀⣐⣠⣂⣊⣾⡀⠢⠄⠉⣹⡶⣄⠀⢀⠁⠸⠿⠀⢠⠘⠛⣁⣤⣤⣦⣤⣄⠀⠀⡀
⠀⠀⠀⡋⠤⢂⠥⡇⠈⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠄⠙⠉⠀⠀⣸⡿⠿⠿⠿⢿⣧⣀⠀⢀⡟⠃⠙⠆⠘⠀⢠⣶⣦⠸⠀⣼⣿⢟⣩⣍⡝⠻⣿⣄⠁
⠀⠀⠀⡇⠊⡡⠔⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢀⣼⡟⠁⠀⢀⡀⠀⡉⢻⣧⠈⠁⠆⠀⠀⠀⠀⠈⠻⠋⠀⢸⣿⡇⣾⣿⣿⣿⣇⢹⣿
⠀⠀⠀⡇⣈⠔⣊⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠹⣿⣇⠀⠀⢘⡁⠉⠁⣸⡟⠀⠀⡃⠀⠀⠀⠀⠀⠀⢠⠀⠘⣿⣧⡹⠿⣿⠿⣣⣿⡟⡀
⠀⠀⠀⣋⡠⢊⠅⠘⠼⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣼⠀⠀⠙⢿⣷⣤⣨⣁⣠⣴⠟⠁⠀⠀⠃⠀⠀⢠⠀⠀⠀⠈⠀⠀⠈⠻⢿⣷⣶⣿⠿⠋⠀⠂
⠀⠀⠀⠧⢈⣤⢊⣠⡀⠀⠀⠀⠀⠀⢀⢼⠀⠀⡒⠄⡁⠠⠐⠀⠀⠈⢻⡻⠻⠟⣿⠀⠀⠀⠀⡆⠀⠀⠀⢄⣀⣀⣀⣠⡄⣠⡠⠠⠤⠤⠤⠤⢠⠲
⠀⠀⠀⠀⡀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠐⠊⠀⠀⠈⠢⣿⠤⠚⠀⠀⠀⠘⠒⠀⠒⠞⢀⡴⣤⠀⠘⠀⠠⡶⣦⠈⠓⠓⠚⠓⠒⠒⠂⠋⠂⠈⠈⠁
⠀⠀⠀⠋⠀⠀⠀⣀⠄⣆⣂⣖⡠⡀⠀⠀⠀⡌⠁⡀⡀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠚⠿⠂
⠀⠀⠀⠀⠀⠠⡎⢰⠊⠉⠙⢿⣿⣮⣢⠀⠈⠀⠁⠠⣿⡴⠂⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣥⣥⣤⣤⡀⢱⡀⠀⡊⠁⠰⠈⠀⣧⢀
⠀⠀⠀⠀⣐⢃⡡⠁⠀⠀⠀⠀⠹⣿⣿⣇⠀⠀⣠⣤⣿⢀⡀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠉⠉⣉⣉⡉⠉⠙⠛⢻⣿⡇⢸⡇⠀⠀⡀⠀⠀⣰⠏
⠀⠀⠀⠀⡉⢀⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⢸⣿⣿⣿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⢷⣆⠀⢸⣿⡇⢸⡇⠀⢀⠀⢠⡴⠃
⠀⠀⠀⢠⡁⣈⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⠀⠈⠛⠛⠁⠀⠀⢀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⠀⠸⣿⡇⢸⡇⢀⠀⠀⣸⣿⠀⠻⠃
⠀⠀⠀⠀⡁⢌⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⠀⠀⢠⣄⣰⡆⢀⠀⣿⣿⠀⠀⠀⠀⠐⠘⢿⣷⣿⣽⣿⡛⠋⠀⠀⣿⡇⢨⡇⢸⠀⠀⣿⣿⡆
⠀⠀⠀⠀⢅⠈⣭⣉⠛⢶⣄⠠⠐⠂⢩⣙⢻⣿⡄⠀⠈⢿⡿⠃⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣿⡇⢠⡇⠸⠂⠀⣿⣼⠇
⠀⠀⠀⠘⠂⠁⠈⢿⣷⣄⠻⣧⠔⢠⣿⣿⡆⣿⣷⠀⠀⠀⠀⠀⠀⢄⠻⠿⠿⠿⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣾⠿⠿⠿⢃⣸⠇⠀⠀⠄⠀⡀
⠀⠀⠀⠀⢰⣾⣿⣆⠻⣿⡄⢻⣇⣿⣿⣿⡇⣿⣿⠀⠁⠀⠀⠀⠄⠂⠀⠀⠈⠉⠁⠁⠈⠻⢿⣿⣿⣷⣾⣟⣃⣉⣀⣀⣉⣑⣒⡄⠀⠀⠸⠀⠰⠁
⠀⠀⠀⠀⣿⣿⣿⣿⡆⢹⣷⢸⣿⣿⡿⠏⣰⣿⡟⠀⠀⠠⠀⠀⢷⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠠⠀⠐⠀⠈
⠀⠀⠀⠀⠻⣿⣿⣿⡏⢨⣟⣿⣧⣤⣴⣾⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⣲⠄⣀⡲⠄⢠⠰⠤⠀⣀⣀⡀⠀⠀⠀⢀⠠⠐⣒⣒⣒⡀⠄⠁
⠀⠀⠀⠀⠄⠈⠛⠛⠴⠛⠋⠀⢀⠉⠁⠀⠈⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠘⠛⠀⠀⢹⣿⣷⠀⠁⠀⠓⣿⣿⠿⣿⣿⠿⢷⣾⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢠⠂⠚⠀⠠⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⡆⠀⠀⠀⣿⡇⠀⣿⡏⠀⣾⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠨⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠡⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠘⣿⠀⢹⣷⠀⢸⣇⣼⠏
⠀⠀⠀⠀⠀⠀⠒⣴⣶⣶⢽⣿⢿⡍⠃⣉⣶⣶⣿⡙⠻⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠀⢀⣿⠀⣸⣿⠀⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣼⣿⣽⡧⠧⣿⣿⣿⣿⡷⠀⠀⢐⢂⠀⠀⠀⠀⠀⠉⠉⠈⣈⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣴⣾⣧⣀⣿⣧⣸⣿⣷⣦⡀
⠀⠀⠀⠀⢀⠀⠀⠙⠛⠛⡉⢩⣉⣠⡂⠙⠻⠿⠛⠁⢠⠀⠈⠀⠀⠀⡦⠠⠤⠀⠤⠤⠤⠀⢀⠀⠀⠀⢸⣿⣿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⠒⠐⠂⠂⠒⠒⠒⠓⠚⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠠⠤⠤⠤⠤⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠒⠒⠒⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠐⡆⠀⠲⡆⠀⢰⡀⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠘⢓⠀⠛⠇⠈⠙⡇⠛⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠴⠀⢀⣀⠀⡀⣀⣓⡇⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⡇⠀⠀⠋⠀⠀⠨⢽⠛⠛⠩⠉⠍⠃⠒
⠀⠀⠀⠀⠀⠀⠀⢤⠈⢭⠀⣨⢽⣈⡳⢗⣐⠆⣎⡋⣧⡇⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⢫⣈⣉⠉⠈⠉⠉⠸⠭⢆⡋⣱⣬⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠈⠁⠈⠀⠁⠈⠉⠀⠀⠀⠉⠄⢀⣃⠀⠀⠀⠀⠀⠄⠀⠐⠀⠸⢄⠉⠉⠉⠉⢩⣀⣀⣀⣀⣨⣀⡀⠀⣀⣀⢠⣄⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣠⣤⣤⣶⣒⣒⡒⠒⠒⢀⣐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠸⣇⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠛⠿⠿⠻⠛⠟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠃⠋⠉⠉⠉⠉⠉⠉⣉⣉⡉⣁⠤⣧⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣥⡀
⠀⠀⠀⠀⠘⠀⠀⠄⠤⠠⠤⠤⠇⠰⠒⠛⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠢⢄⣄⣀⢀⡀⢀⢀⠀⠀⠀⠀⠀⢀⣠⣠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠉⠊⠐⠉⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⡯⠈⠀⣾⣧⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⠇⣨⡝⠂⣸⣿⡏⠀⠀⠀⢋⠁⢤⠤⠒⢂⣷
⠀⠀⠀⢀⡀⠀⠀⠐⠀⢠⠥⠄⢹⣿⣿⣿⣿⣿⡿⠀⠀⠒⣞⠀⠈⡀⢠⣿⣿⠁⠤⢤⣄⡐⢠⢯⢐⣦⣍⡿
⠀⠀⠘⡄⠀⠀⠀⢀⣂⣀⣭⣵⣀⣿⡿⠟⠋⠁⣀⣀⣀⣀⠈⠉⠒⠋⠚⠛⠁⠀⠀⠀⠀⠫⠉⠭⠜⠛⠉
⠀⠀⠀⠙⠛⡛⢛⡻⠛⠉⠀⣹⠛⠛⢋⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠀⠀⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⢠⠉⣐⢎⠐⠁⠀⠀⠀⠀⡇⢠⣿⣿⣷⣶⣶⣶⣶⣦⣤⣬⣤⣤⡀⠀⢿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⢸⢰
⠀⡼⠣⢍⡁⢱⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡿⠟⠻⣧⣐⣲⣆⡼⣸
⡘⢠⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣨⠛⣙⣤⣶⣿⣿⣿⢟⡠⠋⠔⠁
⠇⢸⡀⠀⠀⠀⠀⢀⣀⡀⣸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⢾⣟⠁⣿⣿⣿⣿⡟⠉
⠈⠀⢱⢤⡒⣈⣭⣤⠄⠀⢀⠗⠘⠛⢛⡿⠿⠿⢿⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣶⣿⣿⣿⢤⣤
⠀⡔⠁⠀⠚⠛⠋⣁⡤⠖⠉⠛⢶⣤⣤⣀⣈⣉⠉⠙⢌⠻⣿⣿⣿⣥⠖⣺⣿⣿⣿⢿⣫⣷⣿⣿⠀⣀⡀
⠀⠈⠒⠒⠒⠛⠛⠳⠤⠄⠀⣶⠻⠿⣒⣚⠛⠛⠛⠓⣚⣓⣍⠻⣿⣿⣷⠞⠛⣫⣵⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣐⡒⠫⠉⠩⠉⣙⣓⠒⠓⠾⠛⠿⣶⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠒⠠⠬⠥⢉⣑⣒⠒⠂⠨⠥⠄⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⡠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡰⠠⢍⣉⠑⠒⠒⠤⠬⣭⣉⣉⣉⠀⢻⣿⣿⣿⠟⠁⣀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣈⣉⠛⠓⠶⠤⣤⣄⣀⣀⠀⠀⠉⠁⠀⢸⣿⡿⢃⡴⠊⠑⣢⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⡂⠭⢉⡓⠒⠦⠤⢤⣉⣉⡙⠛⠻⠶⠶⠾⠋⠠⡏⠑⢢⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣚⠁⠀⠀⠈⢉⠚⠂⠀⠨⠉⠐⠂⠀⠀⠉⣍⣩⢿⠒⠒⢂⡄⠈⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣤⣀⣀⡉⠒⠒⠀⠀⠀⣠⢮⢟⣰⢃⢞⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣦⣤⣽⣷⣶⡖⣡⡾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⢀⡀⠀⠀⠀⣠⣤⣤⣤⣤⣄⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣤⣤⣤
⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡿⠛⠉⠉⠻⣿⠄⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⢠⣶⣤⠀⠙⠀⠀⠀⠀⠀⣿⣈⣿
⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠰⡇⠀⠰⠀⠀⣹⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠐⠃⠈⠛⠋⠀⣤⡆⠀⠀⣴⠶⣿⣿⣿⢶⣦
⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣱⢤⣀⣀⢴⡇⠀⠀⠀⢻⣻⣟⣻⣟⡟⠀⠀⠀⠀⢴⣦⡄⠀⠛⠁⠀⠀⣿⣤⣿⣿⣯⣿⡿

⠀⣠⠴⣶⢲⡶⣄⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⢲⠀⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⢀⣶⣶⣆⠰⡆⠶
⢰⣛⣶⣿⣿⣷⢛⡇⡇⠀⠀⠀⠀⡟⢏⠀⠀⠀⠀⠀⡧⢤⡤⢤⡬⣼⠀⠀⠀⠀⡎⠙⠧⠀⠀⠀⠀⠀⢰⣿⣟⣿⡀⢠⠂
⠈⠻⣿⠛⣿⣏⡿⢰⠃⠀⠀⠀⣸⠁⠸⡄⠀⠀⠀⠀⡷⣿⡷⢾⡷⢾⠀⠀⠀⠀⠘⠢⡤⠀⠀⠀⠀⠀⣺⣟⣖⣿⡆⢸
⠀⠀⠙⠶⠛⠋⠀⠘⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠳⠿⠷⠾⠷⠞⠀⠀⠀⠀⠲⠾⠿⠷⠀⠀⠀⠀⠛⠓⠒⠚⠓⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠰⡄⠠⠀⠀⠀⠀⠀⠀⢷⡀⣠⠏⠀⠀⠀⠀⡥⠀⣤⠀⠀⢀⡀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⢀⠃⠙⠆⠀⠀⠀⠀⠀⠀⠀⣻⣇⠀⠀⠀⠀⡜⡥⡔⠓⢃⡀⣌⡅⠀⠀⠀⠀⠿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⡇⣏⢱⠀⠀⠀⠀⡐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⣠⡤⢤⡀⠀⠀⠀⠀⠀⠀⡀⠴⢦⡄⠀⠀⠀⢀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⣤⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡃⣴⡄⢹⠀⠀⠀⠀⢠⣾⣿⢿⣦⡁⠀⠀⠀⢸⢻⡿⡄⠀⠀⠀⠀⠀⢀⢈⣱⣿⡿⠸⡄
⠀⠀⠀⣼⣿⣷⡀⠀⠀⠀⠀⠈⣳⣦⣴⡋⠀⠀⠀⠀⠘⢿⣯⣿⣿⠃⠀⠀⠀⢸⢸⡇⠹⣆⠀⠀⠀⠀⠘⣿⡙⢻⣷⣦⠁
⠀⠀⠼⠿⠿⠿⠷⠀⠀⠀⠀⠀⠛⠏⠻⠓⠀⠀⠀⠀⠘⠶⠬⠉⠁⠀⠀⠀⠀⠸⠽⠷⠤⠼⠧⠀⠀⠀⠀⠀⠱⠟⠋⠁

⠀⢀⣤⣄⢰⣿⣿⣆⠀⠀⠖⠂⠀⠒⠒⠒⠒⠂⠀⠀⠀⠀⢰⠀⠁⠀⠀⠀⢠⣤⣤⣦⣤⣦⣤⣤⠀⢀⣠⣴⣾⣿⣶⣦⣄
⠀⣾⣿⣿⣿⣿⣿⡿⠀⠀⣋⣉⠉⢁⣀⣉⣉⡁⠀⠀⠀⢀⡇⢸⠃⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⣿⠟⠛⠻⡟⠁
⠀⠹⣿⣿⣿⣿⡿⠁⠀⠀⠅⠀⠤⠤⠤⠤⠤⡅⠀⠀⠀⠘⠣⠾⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠈⠙⠛⠋⠁⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠛⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠸⠃⡇⣨⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠳⡇⢨⡷
⠘⠉⠀⠀⠀⠀⢠⡄⢠⠀⠀⠀⠀⠀⠀⠸⠑⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢰⠀⢠⠀⠀⠀⠀⠀⠀⠘⠑⠚
⠀⠀⠀⠀⠀⠀⠘⠓⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⢠⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠄⠀⠀⠀⠀⠀⢀⠀⣀⣀
⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⡤⡀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⡤⡀⠀⠀⠀⠀⠀⣄⠀⣀
⠟⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠄⠀⠀⠈⠓⠈⠓⠃⠀⠀⠀⠀⠀⡟⣤⢻⠰⣏⠟⠀⠀⠀⠀⠀⠐⠂⠀⠀⠠⠀⠀⠀⠀⠈⠓⠈⠓⠃⠀⠀⠀⠀⠀⡿⣴⢿⢰
⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠸⣟⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠸
⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠤⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⣀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⢰⣖⡦⡦⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠱⠀⠀⠀⠀⠀⠀⢰⣖⡦⡦⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠌⠱
⠈⠁⠀⠓⠀⠀⠀⠀⠀⠰⡇⠀⠐⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠓⠀⠀⠀⠀⠀⠸⡇⠀⠐⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠸⠳⡇⢨⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠳⡇⢨⡷
⠘⠉⠀⠀⠀⠀⢰⠆⢠⠀⠀⠀⠀⠀⠀⠸⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠀⠀⠀⠸⠑⠚
⠀⠀⠀⠀⠀⠀⠘⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠘
⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠄⠀⠀⠀⠀⠀⢠⠀⣀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⡤⡀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⡤⡄⠀⠀⠀⠀⠀⣄⠀⣀
⠟⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠄⠀⠀⠈⠓⠀⠓⠃⠀⠀⠀⠀⠀⡿⣤⢻⠰⣏⠟⠀⠀⠀⠀⠀⠐⠂⠀⠀⠠⠀⠀⠀⠀⠈⠓⠈⠓⠃⠀⠀⠀⠀⠀⡟⣤⢿⠰
⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠸⣏⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠸
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠠⢠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠤⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⠀⠀⢀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢠⣒⡦⡦⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠱⠀⠀⠀⠀⠀⠀⢰⣖⡦⡦⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠌⠑
⠈⠁⠀⠓⠀⠀⠀⠀⠀⠰⡇⠀⠐⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠓⠀⠀⠀⠀⠀⠰⡇⠀⠐⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡈⠁⠁⠀⠈

⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣟⡟⣻⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠻⢿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⡀⢈⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣶⣿⣼⣷⣾⣿⣯⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣏⣼⣿
⠀⠀⠀⠀⠘⠛⠛⠛⠋⢉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⣿⣿⡿⠃⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡍⠀⡀⠤⠀⠀⠿⠿⠟⠀⠀⢀⢨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠳⠽
⣴⣶⠊
⣿⡟
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡀
⠯⠄⠢⠀⠀⠀⠀⠀⠀⠀⠰⡆⢰⣤⡴⠟⠁⠀⣶⣶⣶⣦
⠀⠀⢶⡖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉
⠀⠀⢸⡧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡶⠶⠂
⠒⠂⠀⠉⠑⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠤⠄⠀⠒⠂⣠⣻⡄
⠀⠀⢀⡀⠤⠄⠐⠒⠀⠉⠀⢀⣀⠀⠠⠤⠄⠀⠀⠒⢾⠈⠪⡇
⣠⣔⠡⠤⠀⠂⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⡢⢺
⠁⠀⡠⡐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢀⢔⣧⢤⠄⣀⣀⡀
⢠⠀⢆⠣⢵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠕⠁⣻⣛⡿⢥⠛⣓⡩⠯⠟⣀⣰⠆⡤⢠⢄⣀
⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⣠⢤⠤⡤⡤⣀⠀⠀⠀⠀⠀⠘⡅⠪⣺⣽⣠⢿⡚⣹⢧⠎⡞⡰⠃⠊⢁⣁⠔⡸⢉⣼⣵⣲⣦⡀
⠀⢀⠀⠀⠀⠀⠀⠀⡴⡛⠑⣠⠪⣪⡨⡨⢓⠄⠀⠀⠀⠀⣧⡪⠪⣿⢯⡞⢙⣃⡾⢾⡤⢤⠎⡝⢁⠞⠐⢁⣁⡼⢷⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡫⡢⠎⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⢸⣪⡆⣿⣴⠶⢮⡭⡽⢙⣣⡶⡚⢀⠅⡰⢁⣋⡘⢸⣳⠃
⠀⠀⠀⠀⠀⠀⠀⢨⡎⢜⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡣⠊⢸⣧⠟⡟⢀⡧⠟⡙⡿⠋⡎⢀⠧⠎⢉⣠⣷⠃
⠀⠀⢠⠀⠀⠀⠀⠀⣗⢕⢧⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢻⢔⢝⣍⡀⡟⢻⣜⣽⠋⣇⢠⡓⣮⡠⢤⣃⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣕⢝⢆⠀⠀⢀⣼⢿⢽⠀⠀⠀⠀⠀⢸⡕⠡⣯⢫⢹⣬⡇⢸⣸⡍⣛⣵⠛⡧⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣵⡠⠋⢪⡫⡢⣱⠃⠀⠀⠀⠀⠀⠀⡧⡪⣺⣦⢾⠀⣦⣼⣩⣬⣋⣷⣜⣷⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠚⠚⠁⠀⠀⠀⠀⠀⠀⠀⢿⡪⢪⣿⢨⣧⢟⣧⣬⢠⣬⢿⣭⡬⣽⡟
⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢄⣀⠀⣀⢀⡤⠄⠀⠀⠀⢸⡊⡊⣿⠶⢫⣼⠇⣶⡶⣶⣶⠀⣶⣶⣶⣤⢄
⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠐⣛⠘⠁⠸⠤⠉⠒⠀⠀⠀⠀⠀⠀⠈⣋⠀⢿⢴⢻⠠⣤⡷⣴⠇⣅⣠⣿⣾⣿⣽⣽⣿⣿⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⡀⢠⠀⠄⠀⠀⠀⠀⠀⢿⢐⢽⡹⣸⡻⣪⣼⡀⣶⠿⣭⣿⣿⡿⣻⣿⣿⢷⡾⡏
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠇⢩⡁⠀⠣⠜⠀⠀⠀⠀⠀⠀⠀⠀⢸⡰⣻⣷⣏⣠⠾⡀⡿⢇⣷⢷⣾⣷⣾⣿⠷⣻⣿⠾⠁⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠼⠾⠅⠭⠭⠙⣛⣛⣿⠿⠽⠾⢿⣿⣻⣿⣿⡷⠉⣩⣀⡤⠤⠀⠀⠁
⠀⠀⠀⠀⠀⠸⣀⣀⣀⣀⣠⣤⣄⣐⣲⡒⠒⠒⠨⠭⠍⠩⠭⠉⠉⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠠⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⢀
⠀⠖⠲⠀⢸⠀⠀⠀⠀⠀⢀⢀⣀⣄⣨⣀⣄⣸⣀⡇⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⢀⠤⠀⠀⠀⠀⠀⢛⡀⡀⠀⠀⡀⢀
⠀⣍⢈⡄⡜⣄⣸⠉⡄⠀⠈⠉⠉⠉⢉⠉⠉⢩⠀⠁⡇⠀⠖⠰⠠⠄⠐⠒⠈⠀⠀⠤⠀⠀⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠈
⠀⠁⢀⠀⠸⠁⡜⠀⠣⠀⠠⠄⠠⠀⠈⠠⠀⠈⠤⠀⠇⠀⢀⠀⠠⠄⠀⠀⢀⠎⠀⠀⠀⠀⠀⠠⠠⠤⣏⡁⠀⠀⠌⠁⠐⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠈⠁⠀⢀⠈⠀⠀⠀⠀⠀⢀⡀⠂⢀⠀⠀⠈⠁⠸⠀⠀⡨⠀⠀⡆⠈⠣⠀⠁⠰
⠀⠀⠀⡀⠰⠈⠀⠂⠀⡤⠀⠠⠄⠀⠀⠀⠠⡀⠀⠀⠀⠂⠀⢁⠑⠘⠦⠀⠀⠀⠀⣠⠲⠆⠦⠀⡂⠀⠀⠀⠀⠀⠀⠊⢢
⠀⠀⢀⡉⡁⠒⠒⠂⠁⠠⠄⠈⠀⠀⠀⠀⠀⠀⠈⠡⠀⡀⠀⠂⠀⢠⠆⠀⢠⠐⢈⡅⠀⠀⠀⢘⠂⠀⡄⣤⣈⠀⠀⠗⠳⡆⠀⡀
⠀⠀⠀⠀⡆⢀⠤⡀⠀⠀⠀⠀⢠⠄⢐⠀⠀⠀⠂⠀⢈⡞⠓⠒⠈⠀⠀⠀⠘⡆⠈⠱⣦⢠⣄⠛⠁⢰⠃⠒⢇⡸⣐⠂⠀⠀⠀⠀⠀⢠⠰
⠀⠀⠀⠀⠀⢁⠀⠁⠀⠀⠀⠀⢀⣀⡒⠀⠀⠀⠀⠠⠋⠀⠀⡀⠀⠐⠀⠀⠀⠈⠢⣄⣐⠀⣀⣡⠔⠚⠀⠀⠀⣐⢁⠀⠐⡤⠀⠂⡀⠀⠁⠀⡁
⠀⠁⠀⠀⠀⠈⡆⠀⠂⢐⠀⠀⠠⠠⠀⠀⠀⠀⠀⠔⠈⠉⡉⠀⠜⠠⠀⠀⠀⠀⠀⠢⠉⠉⠁⠀⠀⠠⠀⠀⠘⠀⠈⠀⠐⢠⡀
⠀⠀⠀⠀⠀⠔⠀⠀⡀⠈⠠⠀⠠⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠔⠀⢀⢀⠀⠀⡀⢦⠀⢄⠔⡙⠀⠄⠘
⠀⠀⠀⠀⠀⠀⠔⡈⠁⠐⠀⠀⢀⡀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂⠄⠀⠀⠀⠀⠀⠘⠀⠀⠉⠉⠀⠙⠁⠀⢠⠀⠀⠀⠢⠒
⠀⠀⠂⠀⠀⣀⣀⣀⣈⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⣰⣶⠶⣶⡆⣶⠶⣶⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠠⠁⠠
⠀⠀⡀⠀⠄⠳⠄⠀⢸⠀⠀⠀⠀⢐⠀⣀⣀⡢⢸⣿⠀⠀⢿⡅⠀⢻⡇⣿⡀⣼⠇⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡁
⠀⠀⢧⠀⡝⢻⠀⠀⢸⠀⠀⠀⠀⢠⡀⠀⢠⠅⢸⣿⠀⠀⢸⡗⠒⢻⡇⣿⠿⣿⡄⠀⠀⡶⠶⠒⠒⢶
⠀⠀⠚⠁⠍⠈⠀⠂⢘⠃⠒⠀⠀⢀⠑⠐⠁⠀⠸⣿⣶⡆⣽⠇⠀⠸⠇⠿⣴⠿⠁⠀⠀⠃⠀⠀⠀⢸⠀⠀⠰⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⢀
⠀⠀⢠⠀⠠⠀⠀⠃⠩⠁⠉⠀⠀⡂⠀⠀⢀⠀⠀⢠⡤⣤⣤⣶⣶⣶⣶⣶⣶⡄⠀⠀⢀⠀⢀⣐⡀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠀⠑
⠘⠒⠤⠠⠤⠔⠂⠁⠀⡀⠀⠀⠘⠉⠑⠒⠚⠀⠀⢸⣧⣧⣿⣦⣿⣤⣿⣧⣾⡇⠀⠀⠸⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠉⠉⠉⢉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠁
⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠐⠀⠘⢘⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⢀⠃⠄⠀⠀⠀⢀⢂⠀⣠⠀⠀⠀⠀⠀⣁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠁⠖⠀⠀⠀⠀⠀⠁⡌⠀⠀⣑⡄⠂⠀⠐⠜⡡⠀⠀⠀⠀⠀⠂⠁⠀⠐⠁
⠀⠈⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⡐⠀⠀⢀⠈⠀⠈⠀⠀⠀⠀⠂⠑⠢⠁⡀⠀⠀⠀⠜⠁⠁⠀⠈⠀⠀⠄
⠀⠈⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠀⠀⠃⠠⠄⠀⠀⠀⠀⣀⠀⠀⠀⠂⠀⠀⠀⠄⠠⠀⠠⠀⠀⠀⠘⠀⠀⠀⠀⠆⠀⠀⠡
⠀⠀⢠⡄⣴⠀⠀⠀⢠⡶⣦⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⡀⠀⠀⠀⠈⠁⣀⠀⠀⠀⠀⠀⠀⠆⠰⠀⢀⠠⢀⠒⠂⠐⠀⠀⠀⠉
⠀⠀⢸⣇⣿⢠⣶⡄⢸⡆⣿⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢂⠀⠱⠚⠀⠀⠁⠀⠀⡌⠀⠀⠘⠀⡀
⠀⠀⢸⡇⢹⠀⢰⡇⢸⣇⣿⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠂⠁⢁⢈⡀⡄⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢠⠀⠂⠀⠐
⠀⠀⠘⠃⠘⠀⠛⠓⠈⠛⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠈
⣿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣛⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢻⣦⣤⣀⠀⠀⠀⠀⢠⡴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠞⠁⠀⠈⠳⣄⠀⠀⢻⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⢹⡷⢦⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⣠⠞⠉⠉⠉⢷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⣄⠀⠈⠙⠋⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠸⣧⣤⡟⠀⠀⠀⠀⢀⣸⣀⣀⣀⣸⣀⣀⣀⣸⣇⣀⣀⣸⣃⡀⠀⠀⠀⠀⠘⢷⣤⡿⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠉⣻⠀⠀⠀⡇⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦⠀⠀⠀⠀⠀⡇⠀⠀⢼⡋⢙⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠋⠀⠀⠀⠓⠒⠒⠒⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠃⠀⠀⠈⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣅⣨⡷⠒⠒⠒⠒⠒⠒⠒⠲⣿⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⡘⠃⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠒⣿⣀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣻⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣀⣀⠀⠀⠀⠀⢿⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⡀⠀⠀⢸⡇⠀⠀⠸⠧⠤⢴⡏⢉⣷⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⣦⣀⣀⣠⠞⠁⠀⠀⣾⠋⢹⡆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠟⠀⠀⠁⠀⠀⠀⠀⠘⠛⠛⠁⠀⠈⠳⣄⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠺⣇⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



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

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

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

⠀⠀⠈⠀⠀⡅⠀⠀⠀⠀⢻⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠙⠛⠛⢁⠀⠀⠀⠀⠄⠀⠀⠀⠤⢤⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢘⣀⣘
⠀⠀⠀⠀⠀⡃⠀⠀⢀⠔⠉⠀⠀⠀⠈⠒⣄⠀⠀⠂⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆
⠀⠀⠀⣤⣤⡅⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⣿⣷⣿⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡄
⠀⠀⠀⣿⣿⡇⠀⠘⣷⣶⣲⣖⣶⣶⣶⣶⣶⡇⠀⣿⣿⣿⡇⣿⡇⠀⠀⣀⣀⣀⣀⣀⡀⠀⢰⣿⣿⣿⣿⣿⣷
⠀⠀⢀⣿⣿⣇⠀⠀⠹⣿⣿⣿⣷⣿⣿⣿⡿⠁⠀⣿⣿⣿⣹⣿⡇⠀⠀⠻⣿⣿⣿⡿⠁⠀⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⠀⡤⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⣠⣤⣄⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠘⠛⡃⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣼⣤⣧⣤⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⢰⣤⡆⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠈⡏⢹⣭⡽⠉⣏⣭⡏⢹⠀⠀⠀⠀⠀⠀⠸⠟⣿⣿⠟⢿⣿⡿⠻⣿⣿⠟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣟⠀⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⠀⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠸⣿⠇⠀⢿⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

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

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

⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠈⠐⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣍⠛
⠀⠀⠰⠀⠀⠀⢀⠀⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⣬⠊⠚
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⠐⢘
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡀⣀
⠀⠀⠀⢰⠀⠣⠤⠤⠤⠖⠀⡆⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠖⠲⣆⢀⡅⢸⢫⡟
⠀⠀⠀⢸⢠⣀⣀⣀⡀⠀⡀⡇⠀⠀⠀⠀⠀⢹⠈⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⡇⠈⡆⠈⠎⣼
⠀⠀⠀⢸⠸⠿⠿⠿⠟⠙⠃⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠐⠂⡇⠀⡅⠀⠸⢶⡄
⠀⠀⠀⢸⠀⣩⣭⣭⣍⣁⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡇⠀⡇⠀⠐⠻
⠀⠀⠀⢸⠀⠛⠛⠛⠋⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡷⠾⣇⢀⡇⠀⠀⡿⢿
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠁⠁⠀⠁
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣿⣿⣟⣿⣻

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