Public holiday Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡟⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣀⠀⠀⠀⠀⣶⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠙⠀⠀⢹⡄⠀⣇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠠⡄⣀⠀⠀⢀⣀⠀⠀⢳⠀⢹⠀⠘⠀⠀⠀⠀⠀⣰⠃⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣸⣿⠀⣀⣤⣤⣜⠀⣠⣤⣥⣤⡀⢠⣥⣤⣤⡀⢠⣄⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⢻⣿⠐⣹⣥⠾⣿⠀⣿⡏⠀⢻⡇⢸⣿⠁⢸⣿⠈⣿⣄⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⡘⠿⠀⠀⠸⠿⠀⠻⠷⠾⠿⠆⣿⡿⠶⠿⠃⢸⣿⠶⠾⠏⠈⠘⣿⡏⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠘⠟⠠⠤⠒⠰⠿⠛⠀⠀⠤⠴⠆⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣨⣉⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣱⣿⠏⠀⣼⣧⢸⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⢱⣿⠏⢀⣐⠸⠷⠸⠟⠿⠀⠀⠀⠀⠂⠀⠀⠀⢠⣐⠢⠤⣄⣈⠉⠉⠛⠒
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⢠⣿⡟⣀⢸⣿⠀⠀⢀⠀⣶⡆⢰⡆⠀⠀⣶⡄⣴⡆⠈⠱⣶⡀⢉⣶⠆
⠀⠀⠀⠀⠰⠿⠀⠀⠀⢠⣿⣿⣬⣥⣼⣿⣤⠰⠃⠀⣿⡇⢸⡇⢰⠀⣿⡇⣿⡇⡀⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡭⠉⢻⣿⠋⠸⣷⣤⣿⠃⠸⣿⣤⣴⡿⠃⣿⣧⣭⣤⡄⢸⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⡖⠈⠉⠁⠀⠀⢨⠉⢭⠀⢢⠈⠉⠉⠉⠀⠈⠁⠙⠂⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⣿⠀⢸⠄⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⠛⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠷⠈⠓⣆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠀⡐⠈⠙⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠁⠀⠐⠂⢄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠌⠁⠀⠀⢤⡄⠠⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣐⡥⠜⠃⠠⠘⠀⠀⢨⣷⠃⠀⠢
⠀⠀⠀⠀⠀⢀⢠⠠⠋⠀⠀⠀⠀⣀⠉⢂⡦⠰⠎⢣⣄⡈⠁⠠⣄⡀⡀⢠⡄⠶⠈⠉⠂⠀⠁⠀⠀⢫⠘⡃⠠⡔⣈⡼⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠁⠀⣠⡞⠋⠉⠑⡇⠀⠈⠃⠀⠀⠉⠈⠋⠁⠀⠀⠄⢀⠀⠀⠀⠀⠀⠈⠂⢡⠐⠺⢁⠀⠀⠀⣠
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢀⡴⡏⠁⡤⣤⣀⣣⠀⠀⠀⠈⢀⠀⠀⠁⡀⠀⠀⡀⠓⠄⠃⠈⠀⠀⠀⠜⠀⠘⢄⡀⢒⠓⠈
⠀⢀⡀⣀⠒⡌⠙⠀⠠⢂⡁⣠⠄⠶⠀⠈⠙⠎⢆⠀⠀⠀⠀⠀⠈⠄⢀⠴⠘⡁⣀⠂⠀⠃⠉⠐⠀⠈⠀⢀⠤⠞⠁
⠀⠊⡉⠁⠈⠙⠀⠀⢔⠉⠋⠉⠀⠰⠟⠀⠾⢿⠐⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠁⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠
⠠⠀⣒⠀⠠⢀⠋⠐⠀⠀⠨⠒⠀⢀⠀⢀⠀⠀⠀⠀⠀⠍⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠃⠑⢄⣶⣁⡠⣀⣥⣦⡀⠄⢦⡀
⠀⠀⠌⠈⠉⠐⠀⠀⠀⠀⠀⠀⠐⠾⢤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠘⠛⠃⠸⣿⣟⣛⣫⡥⡗
⠀⠁⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣠⠀⠀⠀⠓⠒⠲⠤⡄⠠⠀⡀⠀⠀⢀⠤⣈⠀⢀⣀⠤⠞⠑⠀⠀⠀⠀⠀⠀⠛⠅⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠦⠐⠉⠽⠅⠠⠀⠀⠀⠀⠀⠀⠹⢶⣶⣞⣤⠀⢅⠀⠙⠶⠂⠀⠢⠴⡂⠈⠂⠀⠀⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠉⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢧⡃⠄⠀⠀⠀⠊⠱⠉⠁⠘⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⡀⠐⠀⡈⢠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣌⢁⠈⠁⢰⡏⠈⣥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠅⠀⠅⠨⠀⠅⠀⠄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡂⢀⡔⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡰⠆⠴⠀⢶⢰⢆⠴⢤⠶⢰⠆⠰⠄⠖⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠈⠉⠁⠉⠈⠉⠉⠈⠁⠁⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣆⣀⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠑⠒⠀⠠⠤⠤⠀⠤⠤⠤⠒⠚⠂⠉⠁⠀⠀⢀⠠⠀⡄⠢⠀⢰⡄⢰⡀⠀⠐⠀⠀⡄⡀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢨⠀⠀⡗⠎⠀⠈⠇⢸⠠⡄⠀⠀⠀⡆⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⡰⠀⠀⠆⠰⢸⡀⠀⠀⠀⡇⠀⠀⠚⡆⠀⠀⠀⠀⠀⠇⠘⠀⠃⠐⠘⠀⠃⠐⠄⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠋⠀⠄⢰⡆⢨⠀⠀⠀⠀⡃⠀⠀⠀⠐⠀⠀⢀⣀⠠⠄⠒⠒⠒⢀⠊⠀⠐⠒⠢⠤⣀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⢀⡀⠀⠀⠈⠃⠀⠀⢀⣉⠀⠤⠐⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠉⠢⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠒⠶⠖⠶⠲⠶⠲⠶⠒⠖⠲⠒⠲⠒⠶⠒⠶⠒⠶⠒⠶⠲⠶⠲⠖⠒⠖⠲⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠂⠀⠀⠀⠂⠐⠀⠐⠀⠐⠀⠂⠀⠂⠀⠂⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⡀⠈⠀⠘⠀⠈⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠆⠀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠄⠀⠀⠀⠀⠈⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠂⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠸⠀⠀⠐⠒⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⣿⠿⠿⠿⡿⣿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢸⣆⡤⣤⠀⢀⣤⡀⠀⢀⡀⠀⠀⠀⢀⡀⡠⣄⠀⣴⠀⠀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠛⠀⡟⢀⣏⢀⡗⠀⢸⢇⠞⢹⡦⢾⠏⢀⣼⠻⡇⠀⢸⡇⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠓⠊⠈⠀⠧⠄⢹⡏⣴⡽⠃⣿⠀⠧⠏⠀⣉⡀⢸⠧⠒⠉
⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⡽⠀⠀⠀⠀⢀⡴⡄⠀⢸⠇⠀⠀⠀⠛⠀⠀⡴⠋⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢀⡿⠀⠉⠀⣀⣀⠀⢀⣞⠜⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠳⠄⠤⠞⠁
⠀⠀⠀⠀⠀⡔⢹⡇⣠⣄⠘⣧⣌⣷⣼⠁⠀⠶⠀⣠⢤⡾⠀⢀⠴⢄⠀⢠⡄⠀⠀⢰⡋⠷⠀⣸⠂
⠀⠀⠀⠀⠀⠛⠀⠷⢀⡟⠀⣏⢀⠏⣿⠀⣰⠃⢰⣇⣰⡇⠀⣏⣠⡏⠀⣿⠀⣠⠏⡈⢷⡐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠄⠉⠁⠀⠈⠉⠘⠂⠊⠉⠘⢧⡤⠁⠀⠓⠀⠈⣹⠏⢾⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⢤⣤⣀⡀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⠙⠛⢶⣤⣯⣒⣉⣉⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⠤⠤⠤⠖⠒⠈⢠⠋⣤⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠆⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣀⠱⡀⠂⠁⢀⠁⠂⠀⣰⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣠⠤⢤⠄⠀⢀⣀⣀⠀⠐⠐⢄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢈⠐⠦⠷⠶⠶⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠤⠒⠈⠀⠀⠀⠀⡀⠀⠀⠄⠁⠀⠀⠀⠀⢀⠌⠀⠀⠀⣓⡠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢘⠢⢀⠀⠀⠀⡠⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠤⠾⣁⠼⠀⠈⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠰⠂⢠⡊⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⠁⡠⠀⡊⠈⠂⠀⠀⢠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠠⠁⡀⠀⠀⠉⠒⠀⠠⠤⠀⢀⣀⣀⡠⠊⠀⠄⠅⠰⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃⡴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠘⠀⠂⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠃⡜⠀⠀⠀⠀⠒⠠⠤⠤⠀⠀⣀⠊⠀⡔⠀⢀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢄⣀⠀⠀⠀⡔⠁⢀⠎⠀⠀⠀⠀⠀⠀⣀⡀⠠⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠒⠐⠃⠀⠀⠀⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠑⠀⣠⠀⠀⠀⣩⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠈⠀⠀⠄⢠⠒
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠂⠒⣤⠘⢇⠉⠉⠁⠀⠐⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⣆⠀⠀⣄⣀⣤⣰⣀⣰⣠⢄⡠⠄⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⠃⢁⠀⠄⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⡀⠀⢀⡀⠳⢤⠔⢊⠴⠚⠀⠀⠉⠓⣮⡗⡶⠊⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠆⠀⠀⢧⡀⠠⠄⠄⠀⠀⠀⠀⠀⠀⠈⠇⡇⠀⡸
⠀⠀⠀⠁⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⠀⠀⠉⠀⢠⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠤⠤⠤⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣨
⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐

⠀⠀⠀⠀⠀⠦⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠀⠈⢃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄
⠀⢰⢀⠀⠀⣠⠀⠑⡄⠀⡠⠒⠁⠀⠀⠀⠀⠄⢀⣇⠐⣹⠀⠔⡡⠂⠢⡑⢄⠀⠀⠀⠀⠀⡴⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠨⢀⠱⡊⠀⣠⠂⢁⡜⠀⡠⠤⠭⣄⠀⠀⠰⠁⠈⢁⠔⡡⠊⠠⠀⠀⠈⠂⡑⢄⠀⢠⠞⠄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐
⠉⠉⠣⡈⡔⠈⢀⡳⠺⣠⠊⠉⠉⠉⠙⢂⠀⠋⢉⠝⡁⠊⠀⠂⠀⠀⠀⠀⠀⠈⠢⡙⢇⡀⠀⠀⣠⡏⠉⠉⠉⠉⠉⠉⠉⠉⣙⠌⠉⠉⠉
⠀⠀⠀⠙⠮⡐⠉⠠⠇⠁⠀⠀⠀⠀⠀⠀⠂⠜⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠄⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⢌⠊
⠀⢀⠀⠀⠀⠈⢲⠐⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣥⡅⠀⠀⠀⠀⠀⢀⡢⠁⠀⠀⡀
⠀⢈⡃⣀⠀⠀⢘⡀⠈⣄⠀⠀⣀⡠⢊⠔⠁⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⢌⠹⣦⡀⣀⣀⡠⣑⢁⣀⣀⠚⡈⠂⢄
⠀⠀⢫⡹⡀⠀⠈⢱⠀⢻⠀⡠⢉⠔⠁⠀⠃⠈⠀⠃⠈⠓⠀⠀⡏⠦⢄⠀⡜⠠⠠⡀⠈⠋⠳⠃⠀⠑⢌⠻⣈⠛⣵⡅⠀⡨⢃⠌⠀⢀
⠀⠀⠀⢻⢷⡀⠀⠘⡅⡠⠋⠔⠀⠀⠀⠀⠈⠅⠀⠀⠀⠭⠀⠀⡇⠐⠈⠈⠃⠐⠊⠁⠁⠀⠀⠀⠀⠀⠀⠐⠈⠻⡇⠀⡐⢠⠂⢠⡶⠂
⠀⠀⠀⠀⢳⢱⡀⡠⠋⠄⠀⠀⠀⠀⠀⠊⠃⠀⠅⠿⠄⠉⠢⠴⠄⠄⠀⠀⠠⠠⠀⠀⠐⠈⠂⠀⠀⠀⠀⠀⠀⠐⢈⠪⡐⠁⠀⡐
⠀⠀⠀⠀⠘⠳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠜
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠠⢈⠀⠀⠀⠦⠠⠀⠀⡍⠤⢠⠑⠀⠀⠀⠹⠗⠈⠂⠈⠁⡠⠤⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⢰⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠠⢘⠀⠀⠀⢠⢆⠀⠀⠁⠘⡃⠘⢄⠈⠀⠐⠀⣃⠀⣷⠀⠒⠂⢁⡰⠀⠀⠀⠀⠀⠀⠀⠀⠄⢐
⠀⠀⠀⠑⢌⠢⡀⠀⠀⠀⠀⠀⢘⠀⠘⠒⢂⡄⢈⣈⠀⢂⢰⠸⠀⠼⠀⠑⡄⠈⠒⠁⠀⢋⠀⠰⢄⠈⡄⠀⠀⠀⠀⠀⠀⡠⢊⠔⠁
⠀⠀⠀⠀⠀⠑⢌⠢⡀⠀⠀⠀⠀⠀⠈⠀⠘⠑⠁⠀⠙⠊⠈⠁⠀⠓⠋⠀⠀⠀⠀⠀⠀⠈⠁⠁⠈⠐⠋⠀⠀⠀⠀⡠⢊⠔⠁
⠀⠀⠈⢠⠈⠈⠈⠀⢈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠼⠉⠁⠐⠢⡀⠀⡾⡀⠀⠀⠢⡀⢆⠒⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠉⠁⠉⠀⠉⠉
⠀⢠⠀⠈⠃⠀⠀⠀⠀⢠⡈⠂⠀⠀⠀⠀⠀⠀⢠⠀⠇⠐⡀⣱⠠⠀⠄⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠋⠢⢀⢀⠌⢤⣌⡑⠄
⠀⠀⠉⢀⣠⠤⣀⣀⠔⠁⡨⠢⡀⠀⠀⠀⠀⠀⢸⠀⠒⠊⠀⡜⠀⠰⠼⠀⢆⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠈⠂⠤⡌⠰⢤⣁⠁⠂
⠀⠠⠖⠉⢄⠢⡀⠀⠠⠚⠀⠀⠘⠢⡑⢄⠀⠀⠘⠀⠈⠀⠉⠀⠃⠊⠁⠑⠘⠚⠀⠃⠀⠠⠄⢀⠔⡠⠚⠙⠒⢦⠄⠀⠠⠁⠄⠀⠀⠸⠒
⠐⠀⠀⠤⠀⠢⡈⠀⡈⡄⠀⢀⠔⠉⠈⠢⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⡊⠀⠀⠀⠀⠀⠙⣀⡂⠆⠀⠀⠀⠆
⠀⠈⠢⡄⠘⠂⠈⢓⠎⠀⣰⢄⡘⠃⡈⠐⢄⢂⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⠡⠘⢄⠙⡆⠀⠀⠀⠀⠀⠈⠩⠄⠂⡄⡜
⠀⠐⠀⣌⠢⠀⠘⠡⢐⠠⡐⠀⠀⠑⢄⡀⠒⠀⣉⢤⠑⠀⠀⠀⠀⠀⠀⣀⠀⠀⠔⣁⠀⠀⠀⠈⠆⢹⠀⠀⠀⠀⠀⢠⠒⠠⠀⡔⠁
⠐⠙⠑⡋⠖⣂⠤⠊⠑⠀⠀⠀⢀⣀⠤⣄⠬⠼⠃⠂⠛⢄⠑⠀⠀⠀⠀⠀⠀⢀⠶⢅⠑⠒⠢⢀⠘⣀⣗⡀⠀⠀⠀⣇⠀⠀⠼⠃
⠀⠔⡠⠀⠈⠀⠀⠀⢀⠀⣀⠬⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠒⠀⠤⡁⢀⣈⣦⡀⠀⠣⣀⠴⠃
⠃⠁⠀⠀⢀⡠⠤⠙⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢏⡉⡑⢿⡄⣸⠅

⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⡇⠀⠀⣠⣤⣤⠀⠀⠀⣤⣤⠀⠀⠀⢀⡄⠀⠀⠀⣶⣶⣶⣦⡄⠀⢰⣶⣶⣶⣄⠠⣤⣄⠀⠀⢠⣶⡆
⡇⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⠀⠀⢀⣾⣿⡄⠀⠀⣿⣿⠀⣹⣿⠀⢸⣿⡇⢸⣿⠆⠙⢿⣧⢠⣿⠟
⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⡿⢹⣿⡄⠀⣿⣿⠿⠿⠟⠀⢸⣿⡿⠿⠋⠀⠀⠈⢻⣿⡏⠀⡀
⡇⠀⠀⣿⣿⡏⠉⠉⢹⣿⣿⠀⣸⣿⣷⣾⣿⣷⡀⣿⣿⠀⣾⢳⡄⠈⣿⡇⠀⢀⢰⡀⠀⢸⣿⡇⠰⣿⠂
⡇⠀⠀⣿⣿⡇⠀⠀⠸⣿⣿⢰⣿⡟⠀⠀⢻⣿⣧⢹⣿⠀⠘⠋⠀⠀⣿⡇⠀⢸⣿⢟⠂⢸⣿⡇⢘⣿⠃
⡇⠀⢠⣴⣆⣴⣦⣆⣤⣦⣠⣠⣀⣠⣠⣠⣠⣠⡀⠘⠋⠀⠀⠸⡷⠀⠉⠁⠀⠀⠈⠀⠀⠀⠉⠁⠚⣿⠃
⡇⠀⠈⠙⠻⠙⠻⠋⠻⠙⠻⡟⠋⠉⠝⠛⠙⠛⢡⣄⢸⣿⣧⠀⠀⠀⣿⠀⢶⣦⢠⣶⠆⣠⣤⣤⣄⠉
⡇⠀⠀⣿⡇⢸⣿⠀⠘⠏⣓⣟⡹⠃⠀⣿⡇⠀⢨⣭⠀⣙⣟⡊⠀⢠⣿⡇⠈⢿⣿⠏⢰⣿⠏⠙⣿⡇
⡇⠀⠀⣿⣷⣼⣿⢠⣾⣿⣿⣿⣿⣷⠀⣿⣇⠀⢸⣿⠀⣿⡟⣿⡄⢸⣿⣷⠀⠀⣿⠀⠈⣿⣇⡀⠙⠃
⡇⠀⠀⣿⡏⢹⣿⠸⣿⣿⣿⣿⣿⣿⠆⣿⣿⠀⢸⣿⠀⣿⡇⣿⡇⣿⣿⣿⡆⠀⣿⠀⣀⠈⠛⢿⣷⡄
⡇⠀⠀⣿⡇⢸⣿⠀⠙⠻⢿⡿⠿⠋⠀⣿⣿⣤⣤⠀⠀⣿⣇⣿⠇⣿⣿⣿⣷⠀⣿⠀⣿⣇⠀⣠⣿⡇
⡇⠀⠀⠿⠇⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠀⠀⠘⠛⠋⠀⠀⠻⠇⠀⠀⠛⠀⠈⠛⠛⠛⠋
⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⣀⣀⣠⠖⠲⡀⠀⢀⣀⠀⢀⡤⠤⣄⡔⠒⣒⠲⣎⠱⣄⠤⣄
⠀⠀⠀⠀⡇⠀⠹⠆⠀⠙⡆⡏⢉⠹⣼⠀⠒⠀⣷⠀⢁⡠⠋⡆⣠⠖⠋
⠀⠀⠀⠀⢱⠀⠀⣠⠀⠀⢿⠃⣨⡄⢹⣀⣰⣋⡘⠦⣞⣀⣈⡲⣇⠀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠈⣇⠀⣻⠧⣤⣼⡉⢁⡼⠛⠚⡏⠀⣍⢳⠇⢀⠹⣧⡈⠋⣁⢽⡁⠀⢼⠀⠀⠀⡞⠉⢳
⢀⣀⣠⠖⢲⡈⡩⠓⠒⢼⠀⢸⠀⢹⡆⣾⡇⠀⠛⣸⠀⡤⡄⢱⣇⣸⠁⢯⡈⠁⣀⢧⠖⠲⡇⠀⢸⡧⠦⡄
⢯⠀⠘⠇⠀⢹⡇⠀⣿⠈⡇⠈⣉⡯⠤⠼⠛⠚⠉⡭⣍⠁⠉⣡⢬⡁⠀⠀⢉⣉⣁⣸⣆⢀⠠⠀⠬⠄⠴⡧⢤
⠘⣆⠀⢤⡄⠀⢿⠦⠤⠞⠉⠉⠁⠀⠀⠀⣀⣀⣸⣃⣈⡓⠚⣁⡀⣧⣀⡀⢸⠁⠀⠀⠀⢨
⠀⠹⣄⡸⠓⠒⢫⢤⣰⢦⣠⢤⠀⠀⠀⠈⡇⡀⡀⠀⠀⠀⠀⠀⠀⠠⠄⢿⢸⡇
⠀⠀⠀⠀⢰⡋⢻⡀⡇⣼⢁⡾⣤⠀⠀⠈⣗⠒⠒⠒⢚⠀⠇⠨⠉⠉⠁⣽⠘⡇⠀⠀⠀⠀⠄⠀⠀⠂
⠀⠀⠀⣀⣺⣝⣦⡵⠇⠓⠿⠞⠓⠒⠛⡇⢿⠀⠀⠀⠘⠀⠀⠲⠇⠀⠀⣿⠀⡧⠀⠀⠀⠀⡁⠀⠀⡄
⠀⠀⢸⡍⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣧⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠈⡇⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠐⠀⠀⠀⢈⠀⠀⠁
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠐
⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠀⠘⠛⠚⠓⠓⠒⠓⠚⠒⠚⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠛⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⣄⠀⠀⠈⢧⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠘⡀⠀⠘⣆⢰⠀⠈⠉⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠱⡀⠀⠘⡄⡷⠖⠚⠉⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠶⠤⣄⠀⠀⢹⠀⣄⠀⠱⡄⠀⡿⠃⠉⠉⠉⠉⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢄⡀⠀⠐⡄⠀⠙⢦⠈⣇⠘⢆⠀⢹⡾⠔⠒⠒⠒⠒⠂⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢠⡀⠙⢦⡀⢹⡀⠀⡈⢳⣹⡀⠈⢧⣞⡤⠤⠖⠂⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠙⢦⠀⠳⡀⢧⠀⡇⢀⠹⣧⠀⠌⢀⣠⢴⣒⣯⠥⠴⠶⠶⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠳⣤⠴⡶⠗⡺⢾⡧⠀⠀⠚⠛⠋⢉⣠⠴⠒⠋⣀⣀⣀⣀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⢀⡜⠁⡞⠀⡜⢡⠏⣴⢟⡆⠈⢲⠶⣉⡀⠤⠒⠉⠁⢀⣀⡀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡠⠊⠀⡼⠀⡜⠀⢠⠞⡽⡼⡇⠀⢸⠀⢸⠓⢦⡔⠒⢋⣩⠤⠄⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⠀⠀⠊⠀⠀⣠⠴⠋⢰⣿⠃⢻⠀⣼⠀⢸⠀⢸⠀⠀⢠⡤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠤⠤⠴⠖⠋⠁⠀⢀⣿⠃⠀⠈⢧⠹⠀⣾⠀⢸⠀⡆⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠈⢦⠀⢻⠀⠸⡀⠳⡀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠳⢤⣀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣭⣻⣿⣾⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⡀⠛⢩⣿⣿⣿⣿⣅⣀⣶⣤⢶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡼⣿⢿⣿⠿⣾⣯⣿⠿⣷⣾⢟⣉⣯⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠫⣼⣛⠈⣭⣽⣷⣴⣿⣿⢛⡭⣡⡴⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⢘⠀⠰⣷⢏⣀⡴⡿⣦⣽⣿⣿⣿⣹⣿⣿⣿⣿⣟⣩⣭⠝⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣄⠀⠀⠀⠲⣿⠀⣰⣷⠉⣩⡄⠉⡇⣿⣿⣿⣿⣿⣿⣯⣟⡻⠟⢹⡏⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡧⠀⠀⢰⣿⣄⣿⡟⠉⠉⠿⣤⣷⣿⣿⣿⣿⣟⡙⠿⠋⠁⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣴⣤⣶⣾⣿⣿⡿⠃⢻⣏⣹⡿⠁⣽⣿⣿⢿⣿⣿⣶⢶⣾⣧⣴⣾⣿⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣻⢹⣿⣿⣿⣿⣿⣿⣿⠟⠉⢀⣄⣬⣁⣀⡶⢹⠉⣺⣿⣿⣿⣿⣟⣾⣵⢞⣛⠛⠛⠲⡷
⠀⠀⣀⣴⣦⡀⠀⠀⠁⠘⠊⣿⣿⣿⣿⣿⡇⠀⣰⣿⣄⠚⠹⠙⢷⣾⣴⣿⣿⣿⣿⣿⣯⣤⣶⣴⣶⣦⣤⣤⣤
⠀⣺⣿⣿⣿⣿⣦⣠⣤⠀⠀⠘⣿⣿⣿⣿⣷⠀⢥⣴⣯⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⡿⠿⢿⣉⠉⠀⠉
⠀⠹⣿⣿⣿⣿⣿⡉⢻⣄⣾⠿⣸⣿⣿⣿⣿⣯⣹⠤⣭⣅⠻⠇⠛⣻⣿⢿⣿⣿⣿⣀⢸⡀⢀⣓⠀⣴⣿⣿⣿⣦⢀⣴⣶⣤⣄⣀
⠀⠀⠈⠙⢻⣿⣿⣷⣾⣿⣿⢠⣿⣿⣿⣿⣿⠫⡔⠘⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠇⢻⣿⣿⣿⣿⣿⡄⣀⡴⣾⢿⣷⣼⣿⣿⡟⠁⢸⠉⠉⢻⣿⣿⣟⡥⠌⠿⠿⠿⣿⡿⠏⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣏⣨⣵⣿⣯⣾⣻⣿⣧⣤⣼⣤⣤⣾⣿⢿⡷⣚⢫⠄⠀⣠⣤⣤⣶⣿⣿⡟⠛⠻⣿⣿⣿⣿⣿⠶
⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣽⢃⡻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠭⣽⠓⠒⠀⠚⢀⡈⢈⣭⣿⡿⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠐⣿⣿⣿⣿⠁⠀⣾⣿⣿⣿⣿⣿⣿⡏⢙⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣽⠀⠀⢂⠁⠚⠛⠛⢉⡉⠀⠀⠀⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠙⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⢿⠛⣿⠻⣤⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣼⣷⣆⣸⣿⣯⣤⣴⡿⣷⠀⠀⢰⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⣿⣿⠻⣿⣿⢠⠰⢾⣾⣿⠿⢟⡿⠛⢛⣿⣿⢿⠿⢿⣯⣿⣿⣿⡿⠛⠉⢩⠟⠛⠙⠀⢭⢤⠀⢸⣿⣿⢿⣿⣇
⠀⠀⠀⠀⠀⣿⡟⣿⣿⠀⠀⣿⡏⠀⢻⣧⣼⡞⢉⠊⠀⠀⣯⠀⠀⢨⠟⠇⠸⢀⡜⠉⠙⣟⡿⠇⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⣿⣷⢸⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣇⣿⡇⠀⠺⣿⣇⠀⢿⣿⡈⠁⢠⠀⠀⠀⣿⠐⠂⢹⠖⠒⠂⢸⡄⠀⠀⠸⠇⠀⠀⠀⣸⠀⠀⢀⠄⠉⠀⠀⠚⣿⡟⠘⠿⣿⡗
⠀⠀⠀⠀⠀⠛⠛⣿⢿⣦⠀⠹⠷⠀⠀⠹⠧⠀⠀⠀⠀⠀⠛⠀⠀⢸⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠟⠃⢠⣾⡿⠟
⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⢸⡅⠀⢠⣄⣤⡄⣤⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠏⡌⢻⣤⣀⡀
⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡑⠶⣴⣿⡤⠖⣩⡟
⠀⠀⠀⢸⠀⠀⠀⢲⠀⢻⠆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠞⠛⢧⢸⡏
⠀⠀⠀⢸⠀⠀⠀⢸⠀⡞⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠴⢾⣿⠶⢼⠇
⠀⠀⠀⢸⣄⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠛⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⣼⣛
⠀⠀⠀⣰⣦⡀⠀⠀⠀⠀⠀⢀⣠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣷⡆⢻⣦⡙⠻
⠀⠀⠀⠈⠙⠿⣿⣶⣶⣶⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡄⢠⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⣠⣿⣧⣿⣿⣷⠀⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡏⠀⠈⢹⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣾⣿⢡⣶⣮⢻⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡄⠀⠀⡤⠀⠀⣤⠶⠂
⠀⠀⠀⠀⠀⢻⣿⣮⣛⣫⣾⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠙⠻⠿⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠛⢉⣏⠈⠃⢤⣄
⠀⠀⠀⠀⠀⢠⣤⣄⣰⠿⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡤⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠙⠉⠉⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⡿⠿⢿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⡇⢀⡤⢠⡄⢰⣦⠄⣄⢰⣧⠄⡄⣤⡄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠸⡇⢸⣇⣾⡇⣸⣏⢀⠏⢸⡇⠴⠃⠻⠷⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠙⠁⠁⠘⠛⠀⣿⠀⠀⢸⡇⠀⠀⢀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠈⢁⡀⠀⠦⠰⠃
⠀⠀⠀⠀⠀⠀⣠⢾⡏⠉⣳⡄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠁⠀⠀⢠⡷⠁⢀⡀⠀⠐⠋⠉⠓⣲⠗⠀⢐⠒⠂
⠀⠀⠀⠀⠀⠀⠁⢸⣿⣠⠟⠸⠯⣴⠄⣴⠏⠗⣾⠁⢀⣿⠀⣠⠖⣿⠀⣴⠏⠜⢰⢀⠞⣽⠀⣿⠀⢰⡏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠼⣿⡅⠀⠀⠀⢿⡠⢿⣄⡰⠉⣷⡄⢿⡴⢿⠔⣿⡴⠿⣀⠔⢻⡏⢰⡏⣰⡟⢠⠎⢳⣔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠈⠀⠈⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠀⠙⠁⢠⡏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⢦⠀⡠⣠⡀⢠⡀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⡸⢸⠁⣿⡇⢸⣧⠚⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠈⠉⠘⠃⠀⢠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡐⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠾⢶⣦⣤⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠻⣦⡀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠉⠀⢀⠄⠀⢉⡙⠻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠄⠳⠛⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢭⠀⠀⢀⠲⠀⠂⠀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⡀⠈⠄⠄⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣇⠀⠄⠀⠁⠉⠰⣼⠀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡎⠀⠀⠀⠘⣤⠀⠀⢠⠀⠀⢀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠀⠀⣋⡤⠴⠋⠈⠎⠉⠀⠀⢠⣼⣟⠁⠈⠰
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠜⠣⠀⠀⠀⠀⠀⢁⠩⠂⣴⠀⢴⠙⣦⡀⢀⠉⠀⢀⣄⠀⠀⠀⣠⠤⣦⠚⠉⠠⠀⠈⠁⠀⠀⢻⡃⠚⠂⢀⡡⢒⣉⠾⠉
⠀⠀⠀⠀⠀⠀⢐⠁⠚⠐⠂⠀⡀⠐⠉⠀⢀⡴⠞⠛⠲⢶⠂⠀⠸⠧⠀⠀⠀⠋⠑⠿⠂⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡘⢃⡀⠃⠠⢰⡋⠁⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠀⠀⠀⢰⠏⠃⠀⠀⠀⢸⠂⠀⠀⠀⠀⠄⠀⠈⠀⠤⡀⠀⠀⠀⠁⣤⠃⣠⠄⠰⠀⠀⠀⠂⢤⠉⢠⡄⠐⢀⣰⡄⡠⠚⠢
⠀⠀⠀⠀⠀⠀⠰⢷⡆⠀⠀⡀⡢⠄⠚⠛⠀⠀⠫⣿⠷⣴⠞⠀⠀⠀⠀⠂⠐⢄⠀⠀⠢⠀⢠⠀⠆⣃⠉⢁⡠⡄⠀⠄⠀⠔⠁⠀⠀⠐⡤⠤⠠
⠀⠀⣄⣠⡤⣉⣆⠀⠀⠀⠐⣁⣀⢠⢞⠘⠋⣀⠀⠀⣙⡘⢄⠀⠀⠀⠀⠀⠀⠀⢒⢠⠒⠃⠈⠄⡤⡁⠄⠈⠀⠀⠈⡀⠀⠀⠀⣀⠒⠛⡁⠀⠀⠡
⠀⠐⣈⠁⠀⠉⠈⡀⣀⠰⡈⠁⠋⠁⠀⠀⠾⠟⠀⠺⢛⣇⠊⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⢠⡁⠀⠠⠀⠀⠀⠀⠀⠀⢠
⢐⡀⠂⠄⠀⠤⠄⠛⠑⠆⠈⠈⢠⠕⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠭⠷⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠘⠲⠀⠰⡋⠀⡄⠂⡡⠤⡀⠠⠀⢆
⢁⠀⢈⠃⠒⢒⡈⠄⠀⠀⠀⠄⠀⠀⠀⣤⣍⣀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠅⠀⢨⡉⡟⠓⠤⠒⠂⡄
⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⠂⠝⠉⣗⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠢⠄⠀⠀⠀⠀⠀⠀⠉⢷⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢲⠦⢦⡶⠇⠀⠀⠀⠉⠁⠀⠉⠑⣮⣉⠁⠂⣄⠀⠀⠤⠊⢱⣆⠀⠰⠒⠋⠙⠁⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠈⠢⠀⠀⠀⠀⠠⣄⡃⠡⠄⠎⠓⠂⠐⠀⠀⠀⠀⠀⠀⠀⡀⠻⠿⠯⠯⠦⢈⠢⠄⠀⠈⠛⠁⠀⠈⠃⠫⡀⠀⠑⠀⡀⢠⡠
⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡐⠀⠈⠧⣜⠠⠄⠀⠀⠀⠙⠈⡶⠉⠁⠈⠁⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠠⠄⢀⡉⢤⠀⠀⠑⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡉⠘⠃⠀⢒⡈⢡⠚⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⣤⡤⢤⡤⢤⡤⢤⣄⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄
⠀⠀⠀⡿⠷⠮⠴⠼⠧⠿⣿⠀⠀⠀⠀⠀⠀⣻⣥⡿⠁⠀⠀⠀⠀⠀⠀⡀⠀⠹⣿⠁⣀⠀⠀⠀⠀⢨⡉⡶⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀⡤⠻⣿⣿⠃
⠀⠀⠀⡇⠀⢀⡴⣷⠀⠀⣷⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠻⢷⡐⠀⡿⢀⡟⠃⠀⠀⠀⠈⢹⣿⣆⢀⢀⣾⣷⡀⠀⠀⠀⠀⣴⣛⠳⢶⡌⠉
⠀⠀⠀⡇⠀⠘⠛⠿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⢻⡆⠀⠀⠀⠀⠀⠀⠀⠒⠙⠄⠣⠊⠔⠛⠀⠀⠀⠀⢸⣿⠘⣿⠊⣿⡿⠁⠀⠀⠀⠀⡝⠏⠟⠃
⠀⠀⠀⠙⠖⠒⠒⠒⠒⠚⠛⠀⠀⠀⠀⠀⠘⠃⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠛⠀⠛⠃⠀⠀⠀⠐⠛

⠀⠀⠀⢠⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⠀⣠⠴⠂⠀⠀⠀⠀⠀⢠⡄⢲⠆⣤⡀⠀⠀⠀⠀⢀⣴⢀⠀⠀⠀⡀⡀⠀⠀⠀⢲⣀⣀⣀⣠⠤⢶⡲⠤
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣀⠀⠔⠊⠙⠓⠤⠀⡀⠀⠀⠀⠾⢦⣾⡟⢿⣧⡾⠇⠀⠀⠀⢈⡹⡉⣌⣡⣈⣏⣃⠀⠀⠀⢸⣿⣿⣿⣟⡯⠭⠭⡅
⠀⠀⠀⠈⠻⣿⡿⢻⣿⣿⠇⠀⠀⠀⢙⢻⢷⠶⠶⠾⠿⡟⠃⠀⠀⠘⣛⣻⣷⣀⣰⣿⣟⡛⠀⠀⠀⠸⣧⣧⢫⣧⢻⣦⡿⠀⠀⠀⢸⠭⠭⠵⣖⣒⣭⣭⣓
⠀⠀⠀⠀⠀⠈⠀⠀⠙⠁⠀⠀⠀⠀⢶⣬⣼⣘⣘⣃⣣⣧⡆⠀⠀⠀⠙⢥⡝⢻⡛⣧⡝⠃⠀⠀⠀⠐⣯⣰⣷⡀⣇⢸⡇⠀⠀⠀⢸⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠙⠁⠉⠀⠀⠀⠀⠀⠈⠉⠈⠀⠁⠉⠉⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠐⠜⠴⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣋⣀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠈⢣
⠀⠀⠀⠀⢲⣶⣶⣶⣶⡖⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠈⣢⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⣄⠀⠀⠀⠀⠈⡍⠉⠉⠉⡍⠀⠀⠀⠀⠀⠀⢇⣤⣄⣄⣨⠂
⠀⠀⠀⠀⠈⠹⠻⠟⡏⠀⠀⠀⠀⠀⢸⣥⣴⣿⣿⣷⣶⣿⠄⠀⠀⢸⣷⣶⣶⣶⣶⣶⣶⣾⠄⠀⠀⠀⠀⠇⠸⠿⢠⡇⠀⠀⠀⠀⠀⠀⠀⢻⠑⣽⠃
⠀⠀⠀⠀⠠⠏⢘⡇⠹⠄⠀⠀⠀⠀⠀⢧⣀⣀⣀⣀⣀⠜⠀⠀⠀⠈⢍⣉⣉⣉⣉⣉⣉⠝⠀⠀⠀⠀⠀⢸⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠃
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⣟⣿⣏⣹⣿⣻⡇⠀⠀⠀⢠⣤⣤⣤⣤⣀⡀⢀⡄⠀⠀⠀⣴⣿⣽⣯⣿⣿⣷⡄⠀⠀⠀⠛⢉⠀⣀⠀⠀⠀⢹⠀⠀⠀⠀⠀⣟⢹⡏⣿⢹⡇
⠀⠀⠀⢸⢩⢩⢽⣉⣽⡉⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢸⢁⠀⡰⠀⡴⣄⢠⢸⠀⠀⠀⠀⢸⠀⡎⣀⣀⣀⢸⠀⠀⠀⠀⠀⢸⣸⡇⣇⣿
⠀⠀⠀⢸⣬⢭⢬⡭⡥⣥⡇⠀⠀⠀⠈⠛⠟⠿⣿⠿⠟⢯⠀⠀⠀⠘⣦⣤⢤⢤⢤⣬⣤⡞⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣘⡀⠀⠀⢀⣄⢸⣿⣧⣧⡿⢀⣄
⠀⠀⠀⠀⠙⠺⠾⡷⠗⠋⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠼⠼⠷⠋⠀⠀⠀⠀⠀⠘⠤⠼⠿⠿⠿⠿⠏⠀⠀⠘⠻⠷⠿⠥⠽⠿⠿⠛

⠀⠀⠀⢀⠀⡀⢀⣠⣀⠀⡄⠀⠀⠀⡀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⣤⢠⣆⢰⢦⣴⠄
⠀⠀⠀⠉⢘⡇⠈⠛⠃⢘⡇⠀⠀⠀⢻⣷⣶⣨⣿⣠⣶⣾⡇⠀⠀⠀⠀⠀⢱⣶⠀⠀⠀⠀⠀⠀⠀⢶⣄⣠⣤⣤⣤⣤⣄⡀⠀⠀⢈⣭⣭⣭⣭⣼⣯⣽⡄
⠀⠀⠀⣀⠠⡇⠀⣹⠀⠀⡇⠀⠀⠀⠈⠙⢛⣿⣿⣿⠛⠋⠀⠀⠀⠀⠀⢀⠈⣿⣿⠀⠀⠀⠀⠀⠀⣻⡯⣥⣤⣤⣤⣤⡬⠇⠀⠀⠸⣿⣤⣼⣧⣧⣤⣬⡇
⠀⠀⠀⠉⢈⡇⠀⢺⠀⢠⣄⠀⠀⠀⠀⠐⠛⣼⣾⣦⠓⠂⠀⠀⠀⠀⣄⣸⣤⣿⣧⣦⣄⡀⠀⠀⠀⠉⠁⠀⠀⠶⠖⠀⠀⠀⠀⠀⢰⡇⢸⠰⣿⡂⣼⣷
⠀⠀⠀⠀⠀⠁⠀⠙⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠐⠚⠑⠋⠙⠂

⠀⠀⢀⡀⠤⠠⠄⠀⣀⠀⢀⠀⣆⢠⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄⡠⢄⠀⢰⣶⡆⣠⣄⠀⠀⠀⠀⢠⢤⣠⡤⡴
⠀⠀⠰⣴⣴⣼⣧⢙⣾⠘⢢⣶⣟⣩⠏⠀⠀⠀⠈⠙⠫⠁⠐⢲⡀⠈⠣⣤⣭⡄⠛⠛⢀⡀⡔⠒⠋⠀⢀⠀⣇
⠀⠀⠐⠷⠶⠮⡻⢿⣿⣟⠲⢿⣿⣷⣾⠏⣰⠟⣡⣶⣶⣄⡀⡀⠉⠢⠀⠀⡟⢁⡏⠈⠉⢿⡁⠀⠀⡸⠃⠀⡈⠂
⠀⠀⠀⠀⠐⠛⠈⣛⣓⠟⣴⠀⣿⣿⡟⣰⢃⣾⡿⢋⡥⠶⠛⠪⣀⠃⡅⠀⠵⠚⢀⠂⠀⢰⠁⠜⠋⠁⢀⣸⡅
⠀⠀⠀⠀⣷⠉⣤⠊⠙⣇⠸⣇⢻⣿⣧⠿⡞⠛⢵⣫⣴⣾⣿⣿⣿⡇⠀⢸⠆⢄⡦⠀⢀⠼⠂⣠⠤⢔⣅⣴⣤⣤⣆
⠐⠿⠿⠶⠙⡃⢻⡆⢸⣿⣧⣘⠷⡽⢡⠁⠑⠄⡀⠙⣛⣩⣭⣉⡉⠳⠰⠛⠀⠸⠁⢠⠃⠀⠉⠲⠶⣿⣿⢿⡟⠓
⠀⠀⠠⠾⠿⣋⣐⢅⢨⠛⢿⣿⣷⡇⠒⠓⡀⠀⢣⠀⢱⣶⣶⣬⣍⡓⠸⣿⡷⠥⣴⣮⣤⣀⠀⠒⢤⠈⠛⠲⠛⠊⠓
⠀⠀⣤⠀⠚⢛⣩⣴⠜⠻⢦⣌⣙⠃⠱⡀⢘⢀⡇⣀⢀⠙⠻⠿⠟⠡⣾⣿⠀⣸⣿⡿⠋⠁⢀⣂⣤⡀⣀⠀⡆
⠀⠀⠀⠈⠀⠐⠈⠅⢐⡷⣦⣌⣉⣹⠤⠓⠈⠀⠀⠀⠀⠀⠈⡂⣰⠀⠘⢿⡀⢿⡿⢀⣶⣿⣿⣿⣿⡍⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣄⣈⠑⠜⢷⣯⣥⣀⡀⠀⠉⢿⣠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠚⣖⡛⠿⡖⠀⠀⠀⠀⢀⠀⠀⠀⡛⠛⠛⠛⢋⣵⡞⣧⠉⠻⣿⣷⣄⢸⠟⠴⠟⠃
⠀⠀⠐⠀⠀⠀⠀⢀⠋⠀⠀⣸⣶⣭⣛⢧⠀⠀⠐⠶⡀⠬⠛⡀⠐⣦⣶⣾⣿⠏⢰⣿⡇⠀⢹⣿⣿⡇⢿⣦⠙⠿⣶
⠀⠀⠀⠠⠒⠂⠀⠎⠀⠀⠀⢻⣿⣿⣿⣿⣆⠀⠀⠀⠈⠻⣿⣥⣔⣺⡿⠋⢀⣴⣿⣿⡇⠀⡸⠛⠉⠃⠈⠙⠃
⠀⡊⠈⣓⣀⠀⡀⠀⠀⠀⢰⣼⣄⣉⠛⠻⣿⡐⠀⠀⠀⠀⠘⢷⣶⣚⣻⣶⡈⠉⠉⠉⠁
⠀⠀⠀⢇⣀⡽⠀⠀⠀⠀⣽⣷⣯⣉⠙⠲⢤⣇⠈⠄⠀⠀⠀⠀⠻⣿⢟⠿⢅⡒
⠀⠀⠀⠤⠤⠆⢀⠄⢄⠀⢿⣿⣿⣿⣿⣷⣦⣉⡀⠈⢂⠀⠀⠀⠀⠙⠞⠐⠤⠉
⢀⣠⠔⠁⢀⡔⠁⢀⠼⣷⡾⠛⢷⡛⠛⠿⢿⣿⣧⠀⠀⠀⠀⢀⠀⠀⢂
⢀⡄⣀⣀⠁⣀⢤⣯⡀⢿⣠⡾⠟⠛⣟⠲⠦⣬⡙⠆⠀⠰⠀⠎⠀⠀⠰
⠘⣉⣁⣸⣿⣷⣾⡙⠛⢲⡿⡙⠲⣶⠎⣷⣶⣤⣉⠻⠀⠀⠃
⠀⠠⠬⠿⢛⠟⣫⠗⣶⣟⢀⣿⣤⠋⠀⢙⠛⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠘⢱⡿⣿⢫⢻⣏⠳⠀⢀⣼⡟⠶⣤⣉⠻⡄
⠀⠀⠀⠀⠀⠀⣋⣼⣿⡀⣿⠑⠂⠀⢉⣡⣸⣷⣶⣍⣻⠇
⠀⠀⠀⠀⠀⠘⢩⣿⠿⡟⠀⠀⠀⠰⠟⠋⠆⢿⣿⠛⠋
⠀⠀⠀⠀⠀⠈⡱⢹⠀⠁⠀⠀⠀⢀⣶⣿⠹⣶⡍⠓
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⣿⡆⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠀⡟⢡⠆⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠃⡼⢡⡟⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠏⠰⠃⠚⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢏⣿⣿⡏⢀⡞⡽⢰⠏⠁⠀⠀⠀⠀⠀⢠⣤⠀⣞⣰⠃⣴⡇⢠⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢃⣾⣿⡟⠀⡾⣰⢃⡟⠁⠀⠀⠀⠀⣠⣤⡄⠀⡼⢩⢃⡾⣼⢀⡿⡼⢣⠏⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠁⣼⣿⡿⠁⠘⠓⠃⠚⢀⣀⣤⠴⠶⡾⠛⠋⠀⠀⠁⠁⠈⠀⠉⠚⠀⢠⡟⠚⢱⡵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⣸⣿⣿⠃⠀⠀⣠⣴⠞⠋⠁⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⢷⠀⠀⠀⠀⡜⠁⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃⠀⢰⣿⣿⠇⠀⢠⣾⠟⠁⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⢃⡏⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⢠⣿⣿⡏⠀⢀⣿⡇⠀⠀⠀⣷⢀⣾⡏⠀⣀⡀⠀⠀⣀⠀⢠⣿⢣⠞⠀⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣶⣶⣿⣿⣿⣶⡶⠈⢿⣇⠀⣀⡴⠃⣼⡟⠀⣼⡿⠁⢠⣿⠋⢀⣿⡷⠋⠀⣼⡿⠁⢀⣾⠏
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⣿⣿⡿⠛⠛⠁⠀⠀⠉⠉⠉⠀⣼⡿⢁⣼⡿⠁⢠⣿⠃⠀⣼⡟⠀⠀⣼⡿⠁⢠⣾⡟⠀⢀⡆
⠀⠀⠀⠀⠀⠀⢀⡤⠒⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠷⠋⣿⢁⡰⢻⣏⣀⠞⣿⡀⢀⠼⣿⢁⣰⢻⣿⠁⣠⠎
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠰⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠉⠉⠀⠀⠉⠁⠀⠈⠉⠁⠀⢉⣉⣠⣾⠷⠚⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⠉⣡⡿⠃
⠀⠀⠀⠀⠀⠘⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠤⣤⡤⠴⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠒⠂⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠦⠤⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠑⠄⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠉⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠁⠀⠄⠀⠀⠊⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠰⢷⠀⠀⠀⠀⠀⠀⠙⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⡂⠀⠡⠂⠀⠄⠀⠀⠀⠀⠠⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠔⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⡁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⡀⠀⠄⠞⠂⡑⢄⠀⠀⠀⠀⠔⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠊⠈⡀⢀⠀⠀⠀⠀⠘⢄⠁⠀⠀⠀⠈⠢⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢴⡀⠀⠀⠀⠀⠀⠀⠀⠄⠰⡄⠀⠄⠈⠕⠀⠀⠀⠀⠀⣬⡤⠴⠓⠀⡐⠀⠀⠠⠑⠀⠀⠠⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠢⡑⢄⠂⠀⠀⢈⠢⡀⠢⡑⣌⠢⣀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣦⠀⠊⠁⢈⣣⡚⠀⠀⢂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢀⠁⠁⢀⠀⠋⠜⠂⠈⠑⠶⠈⢢⡄⠀⡀⠤⣖⡱⠐⠀⠀⢀⠀⠰⡣⠀⠀⠨⠓⠁⠀⢀⢘⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⢌⠢⢃⠂⢄⠈⠀⠀⠒⠲⠦⣤⣤⠖⠋⠁⠰⠂⠁⠘⠛⠛⠃⠁⠀⠀⠁⠘⠁⡤⣌⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢡⣌⠓⡀⠄⠐⣊⣤⠗⠋⡙⠦⢰⢤⠂⡿⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⢀⡆⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⠜⢋⡡⠶⡛⠁⠀⠀⣚⣅⢀⡉⡆⠰⡏⠀⠀⠀⠉⠉⠉⠉⠙⢻⠀⢸⠀⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢁⠡⠂⠀⠀⠀⠢⡙⢯⠀⠈⠁⠀⠀⠀⢠⣘⣇⠀⠀⣠⣤⡀⠀⠀⠀⠌⠀⢸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠢⣑⠄⠀⠀⠀⠰⡏⠀⠀⣹⣿⠁⠀⢙⣷⣤⣤⣤⣤⡞⢠⠍⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠊⠁⠈⠑⠒⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠉⠁⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠦⠀⠄⠂⠀⠀⠀⠁⠀⠔⠃⠀⠀⠀⠀⠀⡤⠂⠤⡄
⠀⠀⠀⢀⣯⣤⣶⣶⣄⣄⠀⠀⠀⠈⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⢠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠆⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠘⠛⠟⠁⠀⠸⡆⠀⢀⡤⢀⣀⣀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠊⣉⠥⠐⠛⠁⠀⠈⠈⠩⠉⠀⠂⠀⢼⣆⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⠁⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⢸⣿⠀⠀⣠⠶⠦
⠀⢀⠈⡠⠂⠀⠀⠀⠀⠀⢸⣿⠀⠀⣴⠀⠀⠀⠄⠀⠀⡀⢀⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠘⠁⠔⠀⠀⠀⠀⠀⣸⠏⠀⠀⢼⣒⠲⠞⢒⣠⣈⡀⣸⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠓⠀⠀⡆⡂
⠀⠀⠀⠰⠖⠀⠒⠀⠀⠞⠁⠀⠀⠀⠀⠉⠛⠛⠛⠻⠿⠶⠞⠁⠀⠀⠚⠓⠚⠓⠀⠐⠐⠛⠓⠛⠂⠀⠀⠀⠀⠀⠄⠀⢀⢯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢡⠋⠔⠁⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⡀⣠⢀⡀⢀⣠⣤⣤⣶⣶⡆⢾⠁
⠀⠀⠠⢀⠀⠀⠀⠀⠀⠄⠀⠠⢁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡟⠋⡽⠋⠹⠈⠈⢣⡼⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⣀⡀⠂⠀⠀⠤⣤⣤⣤⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠦⠄⠀⠀⠀⡀⠠⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡟⢸
⠀⠀⠀⠀⠀⢀⠴⠶⢾⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐⠀⠄⠈⠆
⠀⠀⠀⠀⠠⠀⠀⠀⠘⠄⠐⠓⠀⠂⠐⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠤⠤⠖⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⡅
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠀⠀⠀⠀⠠⠤⡄⣤⣤⣤⣠⣀⣤⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⢐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⡄⠀⢀⠀⠀⠂⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⡀⢢⠁⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠃⠀⡍⠀⠀⠀⠀⠀⠀⡀⠀⣎⢈⡃⠀⠈⠀⠁⠀⠀⠀⠊⠉
⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⢀⡀⠀⠀⠈⠉⠀⢉⡁⠘⠀⠀⠀⠀⠀⠁⡇⢸⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠐⠀⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠁⠀⠀⠀⠀⠓⢚⠳⠂⠖⠀⠀⡀⠀⠓⢶⠸⠖⡀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠃
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣀⠀⢸⠀⠀⠀⠀⠀⠎⠀⠀⠀⢄⠀⠀⠲⠃⠠⠆⢀⠢⠆
⠀⠀⠀⠀⠀⠀⡀⠰⠷⠅⠤⠆⠖⠀⠀⠀⠶⠲⠂⠉⠁⠀⠃⠀⢸⠀⠀⡀⠀⠀⠀⠀⠀⠋⠁⠠⢀⠀⠀⠀⠀⢀⠔⣦⠁
⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⢂⣠⣤⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⢏⡀⠀⠀⠀⠠⠔⠒⠈⠁⠀⠈⠁⠉⠈⠁⠂⠂⠠
⠀⠀⠀⠈⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠘⠴⠇⠀⣸⡀⢀⡀⢀⣠⠃⡀⣀⣀⣀⣆⣆⢀⣓⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠄⠀⠀⠀⠀⠀⠀⢀⠜⢎⡢⠄⣀⡴⠋⠉⠁⠒⠉⢰⡛⠛⠛⠛⠋⠁⠛⠿⠃⠀⠉⠉⠛⠳⢿⣿⣶⡆
⠀⠀⠀⠀⠘⠂⠤⢄⣀⠀⠀⠀⠀⣀⠤⠐⠈⠀⠀⠀⠩⠛⠉⠀⠀⠀⠀⠡⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠄⣀⠈⣁⠒⠒⠒⠀⠒⠂⠁⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠐⠒⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣀⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⡇⠀⠀⢹⡇⠀⠈⣿⠀⠀⠛⡶⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣆⠀⠀⣠⡞⣿⠓⠲⣿⡀⠀⢸⣧⡄⢠⡟⠀⠀⣼⠃⠀⢀⣿⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⡴⠛⣦⡀⠈⠁⢿⠀⠀⠸⠧⠀⢸⡏⠉⠉⠀⠀⢰⡿⣇⣠⠾⠃⠀⣠⠞⠉⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⢷⡄⠀⠈⢷⡄⠀⠈⠷⠀⠀⠀⠈⠋⠀⠀⠀⠀⠲⠛⠀⠀⠀⠀⠀⢸⣇⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣀⣭⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡶⠶⠛⠛⣉⣉⣩⣤⣤⣤⣍⣉⡛⠳⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠶⠟⠛⣉⣉⣡⣤⣤⣤⣬⣉⣉⡛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣽⣿⣿⣿⣿⣷⠶⠟⠛⣉⣩⣥⣤⣤⣤⣤⣬⣍⣙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠻⠿⢿⣿⣿⣿⣿⣿⡿⠶⠞⢛⣉⣥⡤⠶⠶⠶⠶⢦⣬⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠛⠶⠶⢦⣤⣤⣤⣴⠶⠶⠛⣋⣭⡴⠶⠞⠛⠛⠛⠶⢦⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⢶⣤⣤⣤⡶⠶⠾⠛⠉⢀⣀⢀⡀⠀⠄⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⢸⣶⣷⡆⠠⡀⣤⣤⢨⡝⡏⣩⢡⡄⠀⢤⡀⢨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⡇⠸⠾⠟⠗⡁⠁⠂⣿⢸⡇⠂⣾⢸⡇⠄⠈⢷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⢀⡤⠀⠀⠘⠛⣿⡏⠀⠀⠀⠀⠸⢧⣤⠟⠸⢧⣤⡟⢸⣧⣤⡄⢸⡇⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣄⠢⡍⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔⣡⠄⣠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣄⠀⠹⡶⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠞⢡⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠏⡼⣵⠐⣠⢠⠀⡀⣼⠀⢀⢀⡀⡄⡀⠂⢠⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠃⠚⠘⠄⠧⠐⠧⠘⠇⠃⠙⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢰⠖⣶⠃⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠀⡀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣦⣄⣥⣾⣽⣦⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠈⠉⠉⠱⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⢀
⠀⠀⢠⣄⣨⣗⣦⣾⠂⣾⣶⠶⡾⢷⠀⣀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠄
⠀⠀⠈⢙⣿⣿⣯⣥⠀⠋⠰⡶⠀⠛⠿⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⢠
⠀⣻⣲⡶⠟⠁⢀⠀⠀⡎⠁⠀⢰⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⡻
⠈⠉⠉⠀⡄⠀⠀⠀⣼⡀⢦⡀⠘⢇⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠉⠀⠐⡆⢉⡇⢮⣵⡀⡾⠴⠀⠀⠀⣦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢧⣴
⠀⠀⠀⠀⠀⠀⣶⣴⡟⠉⠃⢈⡟⠀⠀⠈⠁⠠⢿⣦⠀⢃⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠄⠐⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣤⠴⠋⠀⠈⣹⣶⣟⠁⢠⣄⠀⠀⣠⣤⣿⡆⠀⢇⠀⠀⠀⠄⠀⣞⠀⠀⠀⠀⠀⠀⠀⢲⡇⠀⠘⠣⠶⣗⣀⢸⠿⠂⠀⠀⠘⠒
⠀⠀⠀⠀⠀⢤⠼⠷⣶⠉⣿⢻⣷⣿⡻⢷⠟⠁⠀⢻⣿⠀⢀⣁⠀⠂⢀⣿⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠛⠷⠀⠀⠈⣵⡒⠒
⠀⠀⠀⠀⠀⣤⠄⠈⠙⣤⣨⣼⣿⣇⠀⠀⠀⠀⠈⠩⢹⣷⢾⣇⢀⣠⠟⠛⠒⠂⠀⠀⣀⠀⠈⠑⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣭⡥⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠛⣿⡝⢫⡔⠀⠀⢠⡀⠀⠛⠀⠂⣿⡿⠁⢶⢃⣀⠀⠈⠉⢻⢠⡇⣰⠖⠛⠉⠁⣡⠄⢺⡇⠀⠀⠀⠀⠻⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠿⣿⣿⣷⣆⠀⠀⠀⠀⠁⠀⠀⠀⢸⠿⣶⢦⡞⠻⠿⣿⠷⣤⣴⣾⣿⠣⠀⠀⠁⣾⣿⠀⢿⡆⠁⢀⣴⠂⠀⠀⠀⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣉⠛⡄⠀⠠⠦⠼⠀⠀⠀⠉⣰⣧⣀⠥⠤⢄⣠⡼⠉⠛⣿⡌⠷⠁⣠⡚⠁⢀⡀⠀⣀⣀⠀⠋⠐⠶⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣷⡀⠀⠀⢠⡇⠀⠀⠀⠀⣿⡏⡄⠐⠐⣎⠹⣿⣷⣄⡀⢐⣘⣁⠉⠈⠛⠃⠀⢸⢇⡈⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡄⣠⣤⣦⢻⣶⣶⣄⠐⠙⠳⠄⠢⡄⠙⢢⣿⣾⠁⡀⠀⠘⢏⢢⣶⣦⣠⡀⠘⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠋⠻⣿⢎⣿⣿⡏⠃⣀⠴⣷⡦⢀⢶⠟⣙⠁⠛⠁⠀⢠⢤⣾⣟⣂⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠘⠛⣠⠀⠀⠈⠻⡀⠈⠃⢠⠆⠀⠀⡨⢟⣿⠛⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠉⠉⠒⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣴⣶⣤⣤⣈⣿⣿⣇⣠⣄⠀⣶⣶⣄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢠⡄⢠⣄⣹⠀⠀⣹⣿⣏⠛⢿⣿⣿⡿⣿⣿⡇⣸⣿⣟⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣷⠀⠈⠁⣤⣼⣿⡤⣔⣸⣿⣿⣿⣿⠿⣿⣶⣦⡾⠱⠟⢾⠉⢍⣀⣦⠴⠞⢓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣾⣿⣿⣋⣀⠀⠀⠼⢿⣷⣔⡙⠛⣩⣿⡏⠀⠈⠻⣿⣦⣄⡀⠀⠓⢌⠛⢇⡠⢂⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣽⡿⢟⣿⢿⣦⣴⡄⠉⠛⡇⠘⢿⣿⠁⠀⠀⠀⠘⣿⢿⡟⠛⠶⠀⡄⠀⣡⣾⠯⣁⠀⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⣄⠉⢠⣶⢾⠟⣿⣿⣧⡊⠀⠉⠀⠈⠛⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⣽⢷⣿⣿⣶⣍⡀⠈⢡⣾⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢉⣿⠀⠙⢀⣿⣄⣝⣽⡏⢫⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢄⠻⠻⢿⡄⢹⣿⣿⣿⣬⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣆⡀⣬⠀⠒⣿⡟⣿⣀⣴⠏⠁⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠼⣧⣰⣾⣧⠈⣹⣿⣿⣍⠁
⠀⠀⠀⠀⠀⠀⣠⡜⠉⠉⠻⠁⠈⣙⣿⣷⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢾⣿⣼⣿⡷⠸⠅⣀
⠀⠀⠀⠀⠀⠀⣩⠀⢠⡴⣤⣄⠀⠠⢻⣁⣀⣈⣇⡀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠏⠀⠹⡿⣿⠃⠐⠀⠈⣈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠫⢿⣄⣄⣽⣿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣼⡤⢄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢫⡇⠈⠻⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⢿⠁⠃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡀⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢿⣿⣷⡆⠀⢰⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⠀⠀⠀⠻⠃⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⡨⢿⠙⣿⡿⠈⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣿⠾⠲⣆⠄⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣴⣤⣾⣤⣴⣶⣶⣿⠿⠆⠸⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢿⢀⣤⡿⢶⣿⡅⠀⢠⣶⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡅⠹⠛⠷⣯⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢱⠾⡿⠁⠀⢶⣻⣄⠛⠋⠶⠄⠀⠀⠀⠀⠀⠀⠒⠃⢾⡉⠹⢿⠀⠀⠤⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠛⠃⡄⠀⣶⣏⡃⠙⠀⠀⠀⠀⠀⠀⣠⡀⠀⢠⡀⠀⠉⣷⠊⠛⠿⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠇⠉⠀⠀⠀⠀⠀⠀⢲⣿⠛⢿⣿⡏⠀⣴⣿⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣐⡄
⠀⠀⠀⢠⣦⡄⠀⠀⠀⠀⠀⠋⠄⠀⠀⠀⠐⠒⠀⠀⠱⠀⢀⠀⠸⣹⣿⢳⡆⠀⣾⣹⣿⡿⢿⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⣤⣦⣤⣦
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢰⣀⣶⠀⠀⠀⢀⣀⣸⣀⣀⠀⠘⠤⠄⢻⠀⠀⠀⠀⠿⠯⠥⠥⠼⠃⠀⠀⢻⡯⣝⠛⠀⠀⠀⠷⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⢠⡶⢶⠀⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠉⣿⣿⣿⣶⡆⠀⠠⣿⢟⣷⡿⠃⠀⠠⠒⠶⠶⠖⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠻⠿⠿⠀⠀⠀⠀⠿⠦⠤⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠙⠟⠁
⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣿⣿⣿⡿⠀⠀⠀⠰⠶⣾⣶⣶⠀⠀⠰⠶⣾⣂⣀⠀⠀⠀⡀⢸⣦⠀⠀⠀⢀⣶⣿⡏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⢺⡿⢷⡦⠀⠀⠀⠰⠴⠿⠿⠿⠀⠀⠀⠀⢹⠉⠁⠀⠀⠀⢙⣍⣙⠃⠀⠀⠠⠤⠤⠤⠿⠧⠀⣰⣷⣾⣧⣤⡀⠀⢾⣦⡎⠱⠿⠆
⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠋
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢾⡥⠙⠀⠀⠀⠀⣴⣶⣶⡄⠀⠀⢰⣦⣀⣠⣞⠀⠀⢠⣀⣂⡄⠀⠀⠀⠀⣠⢀⣀⣀⠀⠀⠀⣬⠀⢀⡃⠀⠀⠀⠀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠈⠁⠈⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠋⠛⠛⠋⠀⠀⠀⠀⠙⠈⠁

⠀⠀⠀⢠⠞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠀⠀⠀⠀⢀⡄⢀⣀⣀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢀⣶⣠⡄⠀⠀⠀⡶⢶⠶⢤⡀
⠀⠀⠤⠀⠀⠙⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠘⠛⠛⠉⠛⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠛⠟⠀⠀⠀⢀⣷⣾⡶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡛⠋⢉⣽⡿⠏⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⢻⣿⣶⣼⣿⣿⣴⣦⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠟⠁⠀⢸⣿⣿⠿⠛⢻⣿⣿⣿⣿⡄⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠔⠒⠋⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⣴⣿⣿⣿⣟⣋⣀⣀⣀⣀⣸⣿⣿⣟⣤⣼⣿⣿⣿⣿⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⢸⣿⣿⣿⣿⠿⠟⠛⣻⣿⣿⣿⡿⠿⠿⣿⣿⣯⠁⢨⣿⣿⣿⣿⣿⡇⠀⠀⠀⠐⠈⢳⠀⠂⠈⠁⠀⠀⠀⠀⠀⡀⠐⠀⠀⡀⠀⠀⡴⠃⣠⠆⠀⠠⠂
⠀⠀⠀⣸⣿⣿⣿⣿⣏⣴⣬⣿⣿⣿⣷⣢⣄⣺⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇⠀⠀⡆⠀⡠⠂⡌⠀⠀⠀⣄⠀⠀⣀⠄⢀⠀⣠⢾⣀⡠⢾⣠⠔⢱⣀⢴⠁
⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⡟⠋⠙⣻⣿⣿⡟⠛⠛⠻⠃⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠋⠀⠀⠉⠀⠀⠀⠀⠈⢁⠆
⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣿⣤⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠛⠓⠒⠦⣄⠀⣠⠎
⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣷⣠⣰⣿⣿⣿⣀⣄⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⣀⠀⠐⢦⣀⣀⣀⣀⣠⡤⠾⠟
⠀⠀⣰⡿⠋⠀⠘⠛⠶⠷⠶⢶⡦⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⡟⣻⣾⢻⣿⣼⣟⣷⡀⠀⠀⠉⠉⠉
⠀⠀⠈⠀⢀⣀⣀⣤⣤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣼⢷⣯⣽⢧⣟⣻⣷⡯⢟⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⣠⠞⢉⣽⣧⡀⠀⠀⢛⣛⣛⣉⣭⡭⠵⠶⣚⣩⠍⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣟⣻⣯⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦⠀⠀⣰⡿⢫⢞⡽⣧⠀⠀⢠⣾⣁⡴⠋⣡⡾⢷⡀⠀⠀⠉⠤⡤⠴⣒⣚⣫⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣹⣿⠿⣧⣼⣯⡀
⠀⠀⠀⠀⠀⠀⢸⡿⣻⢟⡵⣻⡇⠀⢻⣷⣫⢎⢞⡿⠀⠀⢀⣿⣯⡴⠊⢁⠴⠞⠋⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠠⠖⣛⣛⣓⣀⠀⠀⡿⣿⢻⣿⣶⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠾⠛⠁⠀⠀⠈⠙⠛⠋⢀⣤⠶⠋⠁⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣰⣿⠿⠿⠿⠿⠿⠇⠀⣿⢻⡟⢿⣧⣿⠖⠁
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠞⠋⠀⠀⠀⠀⠀⢀⣤⣤⠒⢲⣤⢀⠀⠀⠀⠀⢠⡿⢿⣧⣼⣧⣤⣤⣤⣤⣴⣶⠆⠀⠛⠋⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⣀⣴⣾⡿⠛⠁⠀⠀⠀⠀⠀⢴⣇⠀⣿⣿⡆⢸⣿⢸⠃⠀⠀⠀⠘⢿⣿⣿⣧⣭⣉⣉⣉⣉⣉⡍⠀⠀⠀⢴⣶⣤⣙⠳⣦
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄⢿⡆⠸⣿⣷⠈⣿⣼⠀⠀⠀⠀⠀⠹⡧⣿⣟⣿⡿⢿⡛⣛⠛⠀⠀⠰⣿⣦⡈⠛⢿⣷⣌⠃
⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⣿⣤⢿⡟⢳⡏⢿⠀⠀⠀⠀⠀⠀⠀⠻⣭⣿⣻⣿⠁⠋⠀⠀⠀⣧⡈⠻⣿⣦⣀⠉⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠌⠛⢈⣁⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠙⢿⣦⣌⣙⠿⠋
⠀⠀⠀⠀⢿⣟⢿⣿⣿⢿⣿⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⣴⣾⣿⣿⣿⣿⠿⠋⠀⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠉⠉
⠀⠀⠀⠀⢸⣯⣾⣿⣟⣼⣿⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢈⠉⠉⠉⠁
⠀⠀⠀⠀⠘⠛⠛⠛⠿⠿⠿⠻⠇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣷⣶⣶⣦⣴⣶⣶⠀⠀⠀⠀⠀⣭⣍⡛⠀⠀⠀⠰⠂⠠⠀⠀⠀⠰⠶⠶⠶⠆⠀⢲⣶⣿⣿⠃
⠀⠀⠀⠀⠀⢹⣉⠉⠉⠉⠉⣹⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⡋⠀⠀⠀⢻⢿⡇
⠀⠀⠀⠀⠀⠈⠿⠿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠈⠿


⣶⣶⣶⣾⣶⣷⢾⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣾⢿⡾⠷⢾⡾⢷⣶⠷⠶⠾⠷⠶⠶⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⣰⠟⢷⡀⠀⠀⠐⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⢰⡏⠀⢸⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⠁⠀⢸⠇⠀⠀⠀⠀⠸⠽⠃⠀⢀⡶⢼⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠴⠃⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠉⢳⡄⠀⠀⠀⠀⠀⠀⣿⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⠀⠀⠀⡀⣿⠀⡿⠀⡀⠀⠀⣀⠀⠀⠀⠀⢼⠄⠀⠀⠸⡘⡆
⠀⠀⠀⠀⠀⢠⡜⠒⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢻⠁⠀⢰⡇⢹⣾⠃⢠⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢱
⠀⠀⠀⠀⠀⣲⠙⠀⠀⣼⠁⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⡇⢸⠀⢀⣿⡁⣸⣧⠀⢸⣿⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠈⠀⠀⠀⠀⣟⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⡇⠘⠷⠟⠘⠿⠃⠘⠷⠟⠻⣤⡾⢻⡄⣀⣠⣤⠶⠶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⡟⠉⠀⠀⠀⠀⠀⠀⠀⢮⡵
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢀⠀⠀⠀⣾⠀⠀⠀⢠⢤⣒⡮⠽⠂⢀⡴⠋⠁⢸⡇
⠀⠀⠀⠀⠀⠛⡟⠂⠀⠀⣼⠃⠀⠀⣀⢰⠾⠀⠀⢰⠇⠀⠀⠀⠈⠁⠀⠀⠀⣴⠏⠀⡀⠀⢸⡇⠀⠀⢸⡉⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢄⣯⠉⠃⠀⢠⡏⠀⢀⣄⠀⠀⠀⠀⠀⣸⠇⢀⣸⡟⠀⣼⠃⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠉⠀⠀⣠⠏⠀⠀⠼⠼⠆⠀⠀⠀⠀⢿⡀⠘⠃⠀⣰⠏⠀⠀⢀⣀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠳⠦⠴⠟⠁⠀⠀⢹⡇⠘⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠲⢄⡉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⡈⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⣤⣴⣦⣤⡄⠀⠀⢐⠒⠒⠂⣀⡀⠀⠀⠀⢀⣰⣾⣾⡆⠀⠀⠀⠠⢤⠤⠀⠀⠀⠀⠀⣠⣾⣧⣠⠂
⠀⠘⠿⠿⠿⠿⠇⠀⠀⠘⠀⠀⠣⠀⠀⠀⠀⠻⡿⠛⠻⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⠀⠀⠀⠀⠄⣦⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣾
⠀⠀⢸⣿⣤⣄⠀⠀⠀⠀⠸⠿⠿⠿⠄⠀⠀⠀⠀⠚⠁⠟⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢰⣾⣶⣾⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢠⣾⣿⣄⠀⠀⠀⠀⠠⢨⡂⠀⠀⠀⠀⢸⡇⣶⣿⣿⠀⠀⠀⠀⠀⣠⡺⠟⠁
⠀⠀⢹⣿⣾⡿⠀⠀⠀⠀⢠⢏⠷⡄⠀⠀⠀⠈⣇⣻⣿⡟⠀⠀⠀⣠⣦⠟⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢠⣤⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣀⣤⣶⡦
⠀⠀⣾⣿⣿⣿⡄⠀⠀⠈⠉⠉⠉⠈⠁⠀⠀⠀⠀⡇⢥⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡄⠀⠀⠀⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠀⠀⠀⠀⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠈⠙⢦⠀⡴⠋⠁⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠃⠀⣿⡀⠀⠀⠀⠛⠀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢀⡼⠃⠹⢦⣀⣀⣤⢾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠛⠶⠶⠞⠋⠀⠀⠀⠀⠀⢠⡄⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠡⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⢠⣿⠃
⠀⠀⠀⠀⢀⣤⣤⣤⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠈⠓⠤⢤⣄⡤⠤⠴⠚⠋⠀⠀⣼⡟
⠀⠀⠀⠀⣿⣿⣿⣿⡟⠉⠉⠉⠉⠉⠙⠳⢦⣀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄
⠀⠀⠀⠀⠘⠛⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣠⡼⠿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠲⠦⠤⠤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠸⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠄⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⣀⣤⣴⣾⣿⣿⠟⠋⢿⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣀
⠾⠿⠿⠿⠟⠋⠉⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⡼⠛⠛⠻⠿⣷⣶⣤⣄⣀⣠⣿⠉⠉⠙⠓⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠋⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⣁⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⢿⣿⣿⣿⣷⣦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣿⣿⣿⣿⣿⣿⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠆⠀⠀⠰⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸⣿⣿⣿⣿⣿⣿⣓⠶⢬⣙⡛⠶⠶⠶⢚⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢛⣛⣛⣛⠻⠿⣭⣛⠲⢬⣍⣙⣛⣛⣩⡅
⠀⠈⠉⠉⠉⠉⠉⠉⡇⠀⠀⠀⢸⠁⠀⠸⢖⠀⠺⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣘⣩⣭⣭⣭⣙⡓⠶⣭⣙⠲⠶⠬⡭⠥⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣯⡴⠶⠶⢦⣬⣍⡓⠶⣬⣙⣛⠒⠒⢚⣫
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠁⠀⠀⠀⠀⠀⠉⠛⠳⢶⣬⣍⣉⣉⣩⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠚⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣨⣀
⠀⠘⠒⠒⠐⠒⠒⠒⠒⠒⠒⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⢻⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠋⠉⠁⠙⠟⡟⠃
⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠁
⠀⠀⣰⣶⣶⣾⡯⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢈
⠀⠸⠛⣽⡟⠻⣷⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠀⠀⢀⣾⠃⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠄⢀⣿⡇⣴⢏⣴⣦⡄⣴⢀⡤⣦⡶⢀⢤⣄⣤⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⠤⠐⠒⣾⠏⢠⢏⡟⢱⣟⣼⡣⣋⣼⡿⠰⣡⡞⣿⢡⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠛⠳⠐⢁⣤⡄⣰⠏⠀⠘⠈⠁⠈⢹⡟⠀⠉⣸⠏⠀⠁⢀⢜⡿⠁⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠀⠀⠀⢀⣰⣾⡿⠞⣱⡇⠀⠀⠀⣴⡇⣈⠀⠀⠀⢁⣾⠃⠀⠥⠋⠈⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡠⠖⠉⣾⡟⣀⣼⣟⣴⡦⣄⣸⡟⣠⡛⢀⣤⣤⣾⠋⣤⣶⣤⡄⣤⡄⢀⣤⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣀⡠⠤⣾⡿⠉⢰⢟⣿⠿⢲⢻⡟⣸⡟⣰⡿⠋⣾⠃⣾⠟⣵⠟⣸⡟⢁⣾⠇⣿⠟⣡⠴⠒⠂⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⣰⡿⠁⠀⡟⠈⠧⠔⠁⠾⠈⠻⠊⠻⠔⠁⠿⠚⠗⠂⠿⠊⠻⠄⣿⠏⣸⡟⠋⠀⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣰⡟⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⠏⡔⣽⢣⣀⣠⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣤⣀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣰⠏⠀⠛⠁⠀⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡼⠋


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

⠂⠀⣀⡀⠀⠐⠀⢀⣀⠀⠀⠘⣷⣤⡴⡿⠀⠀⢀⠀⠄⡀⠘⠂⠀⣀⡀⠀⠐⠀⠁⣀⠀⠀⠘⣷⣤⡴⡿⠀⠀⢀⠀⠄⡀
⠀⢸⠉⠀⠘⠀⠀⡴⠪⣥⠆⠀⠀⠉⡁⠀⠀⢰⠦⣴⡟⢡⠂⠀⢸⠋⠀⠘⠀⠀⡴⠪⣅⠆⠄⠀⠈⡁⠀⠀⢰⠦⣴⡞⢡
⠀⠀⠀⣀⠀⠀⡏⠁⠃⠀⠀⠀⠀⠀⠠⠺⠀⡹⢟⣿⢹⠆⠀⠀⠘⠀⢀⠀⠀⡎⠀⠂⠐⠀⠀⠀⠀⠠⠺⠂⡹⢟⣿⢹⠆
⠀⠰⠆⠈⠋⠀⠈⠠⠀⠀⠀⡴⠈⠀⡠⠾⠁⣤⡶⠤⠀⠀⢠⠀⠰⠆⠈⠋⠀⠈⠠⠀⠀⠀⡴⠈⠀⡠⠾⠁⣤⡶⢤⠀⡀
⡶⠂⠀⠀⠀⠺⠃⠀⠀⠂⠀⡀⠀⢄⠀⠀⠈⢟⠓⠚⢀⣿⣿⡶⠂⠀⠀⠀⠺⠃⠀⠀⠂⠀⡀⠀⢄⠀⠀⠈⢟⠓⠚⢀⣿
⠙⠀⠀⠀⠀⠀⡠⠀⣀⡀⠀⠉⠀⠀⡀⠄⠀⠘⠃⠀⣀⡘⠋⠙⠀⠀⠀⠀⠀⠠⠀⣀⡀⠀⠉⠀⠀⠀⠄⠀⠘⠃⠀⣀⡘
⠀⠀⠀⠂⢡⣂⠀⠀⠈⠁⠀⠀⡄⠀⢀⠀⠐⠦⠀⠈⠛⠃⠀⠀⠀⠂⠢⢤⣂⠀⠀⠈⠁⠀⠀⡄⠀⢀⠀⠰⠤⠀⠀⠛⠃
⠈⠀⠐⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠙⡛⠑⣀⠀⠀⠀⠀⠈⠀⠐⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠙⡟⠃⣀
⣢⣀⡀⠀⠀⠀⠀⣅⠀⠀⠀⠀⣈⣀⠱⣶⣇⡀⠉⠀⠀⠀⠰⣢⣀⡀⠀⠀⠀⠀⣍⠀⠀⠀⠀⣘⣀⠱⣴⣆⡀⠁
⢀⠁⠀⢸⣧⠀⠀⢻⣷⡶⠾⠛⠁⠀⠀⠈⠉⠀⣠⣶⣶⣄⠀⢀⠁⠀⢸⣧⠀⠀⢻⣶⡶⠾⠛⠁⠀⠀⠉⠉⠁⣠⣶⣶⣤
⠘⣄⣦⣼⠟⠀⠀⠘⠉⠀⣤⠀⣀⠤⠴⣿⠀⢠⠘⠋⠑⡿⠀⠘⣄⣦⡼⠟⠀⠀⠘⠁⠀⣤⠀⣀⠤⠴⡟⠀⢠⠙⠋⠑⡿
⡀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⢸⣅⠀⢀⣬⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⢸⣄⠀⣀⣼
⠀⢠⡖⠒⢠⠀⠀⢐⡲⠀⡀⠀⠛⠿⠯⠉⠀⢀⡀⢀⣡⠆⡀⠀⢠⡶⠂⢠⠀⠀⢀⡲⠀⡀⠀⠛⠿⠏⠉⠀⢀⡀⢀⣡⠆
⠀⢘⠀⠀⠀⠀⡠⠋⠈⢛⠁⠁⠀⠀⠂⣠⡀⢼⣭⣿⢧⡘⠀⠀⢘⠀⠀⠀⠀⡠⠋⠈⢛⠁⠀⠀⠀⠂⣠⡀⢼⣭⣿⢧⡘
⠀⢀⡀⠰⡤⠀⠣⠀⠀⠀⠐⢀⠀⠀⠉⣨⠄⠊⣑⠛⠘⠁⠀⠀⢀⡀⠰⡤⠀⠡⠀⠀⠀⠐⢀⠠⠀⠈⣨⠄⠊⣑⠛⠘⠁
⣀⡈⠁⠀⠀⣠⡄⠈⠀⡀⠀⠋⠀⠀⠊⠉⠠⣿⣏⣩⠀⣦⣸⣀⡈⠁⠀⠀⣠⡄⠈⠀⡀⠀⠊⠀⠀⠊⠉⠠⣿⣏⣩⠀⣦
⢯⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠦⠀⠐⠈⠀⠀⢱⡄⠀⠐⢻⡿⢯⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠦⠀⠑⠀⠀⠀⢱⡤⠀⠐⢻
⠀⠀⠀⡀⠀⠀⠈⠀⠲⠆⠀⠀⠀⠀⠂⠁⢀⣀⠀⠠⣶⡆⠀⠀⠀⠀⡀⠀⠀⠈⠀⠲⠆⠀⠀⠀⠀⠂⠁⢀⣀⠀⠠⣶⡆
⠠⠀⢀⠄⠉⠒⠀⠀⠀⠀⠀⠀⠃⠀⢰⢤⣤⢍⠀⠀⠀⠀⠀⠠⠀⢀⠄⠉⠒⠀⠀⠀⠀⠀⠀⠃⠀⢰⢤⣤⢁
⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⢆⣀⡆⠀⠦⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⢆⣀⡃⠀⠶
⠚⠖⠂⢠⡄⠀⠀⣷⣀⣀⣤⣤⠖⠒⠈⠿⠷⠂⠀⣀⣀⠀⠈⠚⠒⠂⢠⡄⠀⠀⣷⣄⣀⣤⣤⠖⠒⠈⠿⠿⠂⠀⣀⣀
⢰⠀⡀⢸⡿⠀⠀⢸⠿⠋⠉⠀⠀⠀⢀⣤⠀⠀⢺⡿⢻⣿⠀⢰⠀⡀⢸⡿⠀⠀⢸⠟⠋⠁⠀⠀⠀⢀⣤⠀⠀⢺⡿⠻⣿
⠀⠓⠻⠛⠁⠀⠀⠀⠀⠀⠛⢠⠖⠁⠉⠫⠀⠘⠀⠀⠀⠋⠀⠀⠓⠻⠛⠁⠀⠀⠀⠀⠀⠛⢠⠖⠉⠉⠻⠀⠘⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⢀⡀⡀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄⠀⠀⢸⡇⠀⢀⡆⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⣆⠀⠀⣇⠀⢸⠇⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠈⠻⣦⠀⠘⣆⠀⠻⠀⠈⠀⠾⠁⣀⡴⠈⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢠⣿⠏⠀⢹
⠀⠀⠀⠀⣴⣋⠀⢀⡀⠀⠙⠳⣤⡀⠀⠀⣿⢃⡇⠀⠀⠀⠈⠀⠆⠉⠀⠉⠀⠀⠈⠀⠘⠉⠀⠀⢀⡤⠤⠤⠤⢤⣼⣿⡇⠀⠀⢀⣿⡿⠀⠀⡾
⠀⠀⠀⢰⠏⣏⢳⡏⣿⡀⠀⠀⠀⠙⠳⢾⣿⠟⠀⠀⠀⡴⡄⣶⡀⠰⡶⣶⡄⠀⡶⣶⡄⠀⣶⣶⡄⢐⣄⣶⡄⠀⣿⣿⡏⠙⠒⢾⣿⡧⠴⠊⠁
⠀⠀⠀⢸⡀⠘⡆⢀⣿⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⣀⠀⣿⣧⣿⡅⣿⣧⣿⡇⢸⣧⣿⠇⣸⣧⣿⠇⠿⢿⣿⡇⠀⣿⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠈⢧⣀⣣⡾⠃⢀⣀⠀⢠⠖⢒⣿⡿⠤⠶⠋⠀⠻⢃⣿⠃⠙⠃⠻⠃⠹⠟⠀⠀⠻⠟⠀⣀⣀⠀⠻⠃⢰⣿⣿⠀⠀⠀⢿⣿⡇⠀⢀⣀
⠀⠀⠀⠀⠀⠉⠉⢀⡞⠉⠉⢳⡈⠀⢸⣿⠇⠀⠀⠀⢀⣴⣿⡉⢳⠀⠀⣼⣿⡆⠀⢰⣿⡗⠀⠀⣿⣿⡾⠻⣆⢘⣿⣿⠀⠀⠀⢸⣿⠁⡜⠉⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣸⠃⠀⣿⡟⠀⠀⠀⠀⣾⣿⢿⡀⢸⠀⢀⣿⣿⠁⠀⣾⣿⡇⠀⢠⣿⣿⠀⠀⠈⠉⣿⣿⠀⠀⠀⣸⣿⣾⠁⢠⣿⡿⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠉⢀⣼⡟⠀⠀⠀⠀⠀⢻⣿⠀⢉⡟⠋⠁⠘⢿⣤⡼⢹⣿⡇⣠⠏⢿⣿⠀⠀⠀⠀⠘⢿⣆⣀⡴⠃⠻⠇⠀⢸⣿⡇⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠴⠞⠋⠀⣀⣀⣀⡀⠀⠀⢉⣛⣉⠀⠀⠀⠀⢀⣀⡀⠀⢙⣛⠁⣀⡈⠁⢀⣠⡀⠀⣀⣀⠉⣁⡀⠀⠀⠀⠀⠈⠿⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢱⠋⣿⣿⡆⠀⣥⣿⣀⠀⠀⠀⠀⢨⣼⡇⠀⡏⣸⠘⣿⣿⠀⢀⣸⡇⠀⣇⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⢸⣿⣿⠉⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⣿⣿⠀⣿⣿⡇⠀⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⣿⣿⠇⠈⣿⣿⠀⠀⠀⠀⠀⢸⣿⡷⠀⣿⣿⣀⣿⣿⠀⢻⣿⣇⠀⠈⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠛⠛⠀⠀⠀⠀⠀⠛⠛⠁⠀⠈⠙⠛⠋⠁⠀⠈⠛⠛⠋⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠴⠶⠶⠶⠶⠶⠒⣒⣚⣋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣾⣿⣿⠿⠿⠿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⢠⡇⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢰⡏⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⣼⠃⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⣇⣀⣀⣀⣤⣤⣴⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣿⠟⠛⠛⠛⠋⠙⣿⠈⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⣿⠁⠀⢀⡟⠀⠀⠀⠀⠀⠀⢻⡀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣾⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠶⠾⠿⠿⢿⣿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣧
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣼⣿
⠀⠀⠀⠀⠀⠀⣸⣟⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣶⠾⢿⣿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⢀⣀⡀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡆⠀⠒⡖⠉⢳⡄⠀⢤⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣠⢲⠗⠮⣧⠀⠀⣇⡀⣸⠇⠀⣸⠃⠀⣿⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣦⡤⠞⢳⡀⠈⠘⡆⠀⠸⠆⢀⡏⠉⠁⠀⢠⡿⠦⠴⠋⢀⡴⠋⠁⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠟⢳⡄⠀⠹⠂⠀⠙⠂⠀⠀⠉⠀⠀⠀⠈⠋⠀⠀⠀⠀⠸⣧⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡷⠶⢛⣋⣉⣭⣭⣭⣭⣍⣛⡲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣛⣋⣭⡤⠴⠶⠶⠶⠦⢭⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣉⣭⡴⠶⢚⣛⣛⣛⣛⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⣭⣭⣉⣉⣩⣭⡥⠶⢚⣋⣭⠭⠷⠯⢭⣝⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠦⢤⣤⡤⠶⠶⠛⠉⢁⢀⠀⠀⠀⠀⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢸⣿⣷⡀⠀⠶⣦⣵⠙⢰⢰⡄⠰⡄⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⠈⠉⠉⠁⣄⠀⡟⢿⡀⣸⢸⡇⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠐⠀⣡⡊⠁⠀⠀⠀⠉⠀⠀⠀⠀⠈⠋⠁⠈⠙⠁⠈⠉⠉⠈⠁⠀⠀⡀⢌⡲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠫⢔⡄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⠈⢶⠬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢡⠒⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠑⠓⠄⠔⡐⣠⢸⠀⡐⢠⢀⢠⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⢀⡤⠖⠁⠀⠀⣹⡆⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢀⣽⡗⣰⠟⠀⠀⠀⢀⣨⣭⣄⠻⢦⣄⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠇⣴⡿⠁⣼⠏⠠⡄⠀⠀⢾⣏⠀⠀⣷⡄⠀⠀⠀⠀⢸⣿⠄
⠀⠀⠀⠀⠀⠀⣞⠀⡠⠋⣸⣿⠃⣼⡏⣠⠾⠃⠀⢠⣶⡙⡛⠋⣻⣇⠠⣤⡀⢴⣯⢉⣶⡄
⠀⠀⠀⠀⠀⢀⣈⡅⠀⢠⣿⡟⢰⣿⢉⣴⠶⢠⣾⢺⣿⠃⢸⡇⣿⡇⠀⣼⡇⣾⡏⢸⡿⠀⢳⡄
⠀⠀⠀⢀⣼⠋⣿⣀⡄⣾⣿⠁⣿⡇⣾⡏⠀⣾⡏⢸⣿⢀⣿⠃⣿⡇⣰⠟⠁⠿⠇⢸⡇⣠⠟⠁
⠀⠀⠀⢸⡏⠀⠈⠛⣹⣿⠇⠀⣿⣧⠹⠷⠄⠻⠃⣿⡇⠉⠁⢸⣿⠇⣠⠔⠂⠀⢀⡿⠉⠁
⠀⠀⠀⠈⠻⠄⣀⣴⠿⠃⠀⣠⡭⠁⠀⠀⣰⡿⠁⣿⡗⠀⠀⣬⡯⠀⢿⣦⣄⠠⠊
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⣼⡟⠀⠀⠀⢠⣿⠇⢤⠉⠀⣀⢠⣿⠇⣀⣀⣤⡄⣤⡄⣴⣦⣾⠟
⠀⠀⠀⠀⠀⠀⢠⣿⡇⢰⣿⠁⣰⠖⡀⢸⣿⢀⣾⡆⣾⠋⢸⣿⢰⡿⢸⣿⠁⣿⠃⣿⡏⢿⡆
⠀⠀⠀⠀⢰⡎⢠⣿⡇⣼⡿⢸⡟⠀⠀⣾⡏⢸⣿⠸⣿⠀⢸⣿⢸⡇⢸⣟⢸⣿⢀⣿⡇⠀⢻⡄
⠀⠀⠀⠀⢀⣩⣽⣿⠀⣿⡇⠸⣧⣀⠀⢹⣷⠘⠿⠄⠻⠷⢺⣿⡈⠛⠈⠉⢀⣉⢹⡿⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠠⠞⠉⢧⠸⣿⠀⠈⣵⠂⠀⠀⢀⡴⠀⠀⠀⠘⢛⡥⠀⠀⠀⠘⢿⣾⣇⣀⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠳⠸⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⡸⣦⡀⠀⠀⠀⠈⠛⠷⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠒⠒⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⢠⡄⢀⠊⠀⠐⠦⡀⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣈⠉⠙⠓⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠤⠀⠀⠛⡀⢠⠇⠀⠀⠂⠂⠼⠇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠁⠀⠈⠉⠙⠓⠲⠶⢤⣄⣀⡉⠉⠛⠒⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⠀⠀⠀⠀⠀⠀⠀⠀⢹⠳⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠤⠤
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡠⠁⠀⠐⠂⠀⠀⠀⠀⠒⠀⠀⠀⠀⠱⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⢤
⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢸⡤⣤⢀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⡈⠛⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠋⠔⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠈⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠐⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠠⠀⡀⠀⠀⠓⠐⠁⠀⠀⠀⠀⠀⠒⡌⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣴⠟⢻⣧⢀⣼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠙⠷⠾⣿⡾⠃⢠⡞⢻⡗⣴⠟⠷⣰⠟⣿⠇⣺⠟⢃⡾⢠⡞⢻⡆⣸⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⠟⠁⠀⠘⠛⠛⠓⠙⠓⠒⠛⠛⠛⠒⠛⠓⠘⠓⠚⠛⠛⠑⠋⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠄⠠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡂⠀⠀⣀⠊⠀⠈⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⠁⠀⠁⠀⠀⠇⠓⠠⠄⠠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⢠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠦⠀⠀⠰⠀⠀⠀⠒⠀⢤⡄
⠀⠀⠀⠀⠀⣀⣉⡂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠟⠀⠀⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠈⠇
⠀⠀⠀⠘⠿⠟⠛⠃⠐⠿⡷⠆⠀⠀⠁⠀⠀⡀⠀⠀⠀⢀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⣴⣶⣄⡀⠀⠀⢀⣠⣶⣦⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⢀
⠀⠀⠈⠛⠿⠋⠀⠀⠈⠿⠟⠁⠠⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠸⣿⠟⠃⣠⣈
⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠲⢠⡶⠀⡰⠠⡀⠀⠀⠠⠼⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⡀⢸⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠀⠐⠚⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⠃⠀⠠⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣦⣄⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠆⢠⠀⣀⠈⠙⢷⡀⠀⠀⣀⠉⠛⠁⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣤⣤⣈⠛⠿⣆⠀⢷⢠⠞⣉⣤⠶⢞⡛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠉⠙⠲⠌⠃⠀⠣⠚⠅⠒⠒⠦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠁⠀⢀⠀⠀⠚⠉⠛⢿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡴⡄⠀⢸⠉⡇⠀⡿⢙⡆⠀⢦⡰⠀⠀⠘⠟⠀⣠⡄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣻⠀⢰⣇⣹⠀⢸⠋⠁⠀⡟⠉⠀⠀⡼⠁⢀⣀⡀⢀⣾⠏⠀⠀⣠⣾⠟⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡆⠀⠀⣧⠈⠧⠘⠇⠈⠃⠈⠀⠀⠀⠁⠀⣀⡘⠁⠀⠀⠈⢉⣿⠋⠉⠉⣹⣿⢿⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡿⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⢠⣿⡆⢸⠋⡇⠀⠀⠀⠀⣸⡏⠀⠀⣰⡿⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡇⠀⠀⠀⣾⣿⠛⣆⠀⣾⡇⠀⣸⣿⡇⠈⡟⢛⣿⠇⠀⢀⣿⠀⠀⢰⣿⠗⣡⠴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⣰⣿⠛⠦⣿⠞⢿⡇⢀⡏⣿⡇⡼⠀⣼⡟⠀⢀⢾⡟⠀⢀⣾⢃⠞⠁⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⡍⠀⠙⠷⠶⠃⠀⠈⠛⠋⠀⠹⠟⠁⢰⣿⣠⠔⡁⢸⣇⡰⠃⠛⠃⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡹⣄⠀⠀⠀⠀⠀⠀⢰⡏⢳⠀⣾⠁⠀⠉⢠⡿⢻⠀⠉⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣽⠀⢀⣀⣤⣄⠀⢸⡇⣸⠀⣿⠀⠀⠀⣿⡇⡸⠀⠀⠀⠀⠀⢀⡈⠧⠄⠀⣤⠀⠀⢀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡤⠿⠚⠉⠁⠹⣿⣇⠀⠙⠁⠀⠉⣤⡀⠀⣿⢁⠇⠀⠀⣠⡄⠀⢸⣿⡄⠀⣰⠄⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢿⣿⠀⣠⡄⠀⢸⣿⡇⠀⣿⡟⠀⠀⢸⣿⠀⠀⣸⣿⡇⠀⡏⠘⢁⡤⠆⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⠋⣤⡀⠀⠀⢸⣿⠀⣿⡇⠀⣾⢿⡇⢠⢿⡇⠀⠀⣾⣿⠀⢀⡟⣿⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠘⠛⠛⠲⠦⣈⢳⡀⠀⢸⣿⠀⠹⣧⣠⠇⢿⣧⠞⠈⢿⣤⡼⠀⠙⠶⠋⠀⣿⠃⠀⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢠⣤⣤⡈⠂⠑⠀⢸⣿⠀⠀⠀⠀⠀⠀⠁⣀⣤⣤⣭⣀⣀⠀⠀⠀⣸⠇⢀⡾⣦⣍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠒⠲⢦⣄⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠠⡟⠉⠀⠀⠀⠉⠉⢛⣶⣾⡃⢀⠎⠙⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢻⡆⠀⠀⢀⡿⠀⠀⠀⠀⠐⣄⡀⠙⠒⠶⠶⠖⠒⠛⠉⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⢠⣄⠀⠀⡼⠃⠀⢀⡾⠁⠃⠀⣴⢀⢳⡈⠋⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠁⠒⠈⠀⢀⡴⠋⠀⠀⢀⡼⠃⠘⡆⠙⢶⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠔⠚⠉⠀⠀⠺⠗⠈⠠⡤⠀⠿⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⣀⠀⠀⠀⠀⠀⠀⠀⢻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⢀⡀⠘⠉⠈⠁⢸⡇⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⡄⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⣧⠀⠀⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢿⠀⠸⠇⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡆⠀⠀⢠⡄⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠁⡄⠀⡼⠁⠀⠀⠀⣠⡴⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠡⠀⠀⠉⠀⠀⠜⢀⠄⠀⣠⠞⠉⠀⣠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⠄⣤⡀⠀⣤⡰⠆⣠⡁⢾⣬⡿⢰⣿⡀⣿⣽⢶⣿⡿⠘⣷⠯⠊⣡⡀⠀⣨⡁⣀⣠⡀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠈⠁⠉⠀⠀⠉⠐⠙⠀⠋⠋⠉⠃⠋⠀⠈⠃⠀⠀⠛⠀⠀⣉⣠⠤⠘⠈⠉⠉⠁⠀⢈⣡⣤⠀⢴⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⣄⢠⡄⣤⠤⣄⢠⡄⠤⢠⡤⣤⢠⡄⠤⢤⣄⢠⡄⣤⠦⣄⢰⡮⠥⢠⣄⢠⡄⣴⠦⡄⣤⠤⠄⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢸⠟⠿⠇⣿⣤⠟⢸⣏⡥⢸⡟⠉⢸⡏⡤⢸⠉⠻⠀⣿⣤⠟⢸⣯⡥⢸⠝⠿⠍⢿⣤⠄⣿⣍⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣉⠉⠀⢀⡀⠀⠀⡀⠉⡠⣀⠀⠀⣶⠲⣆⢰⣦⠰⣦⠖⠀⢀⠀⢬⢉⡒⠦⢤⣀⠀⠀⢉⡉⠛
⠀⠀⠀⠀⠀⡀⠀⠀⣀⣤⠛⠀⠀⠙⠃⣀⡬⠛⠉⠀⠛⢀⠂⠻⠾⠋⠟⠛⠇⠻⠀⠀⡙⠁⠄⠘⠛⠆⠀⠛⠈⠛⠙⠃
⠀⠀⠀⠠⣾⣷⡖⠈⠉⠀⠀⠀⢠⡴⠚⠉⠀⠀⠀⢀⡴⠁⠀⢀⡄⠀⠀⠀⠇⠀⠀⢧⡈⢦⡀
⠀⠀⠀⠀⠛⠙⠃⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣰⠋⠀⠀⠀⡾⠀⠃⠀⠀⢸⠀⠀⠈⢳⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠋⠀⣠⡞⠁⢀⡄⠀⠈⠁⠀⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⡞⠀⠀⠀⠀⠀⠀⢸⡇⠀⣧⠀⢹⡄⠀⠀⠀⠀⠉⠀⠀⠀⠈⠋⣀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠁⠀⠈⠹⣆⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡞⠃⠀⠀⢦⠈⢿⣤⣤⣇⠀⠀⢀⣀⣤⣤⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣸⠇⠀⠀⠘⣺⣷⣾⠁⠀⢨⡗⠋⣉⡁⠀⠀⠸⣆⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢈⡇⠀⠀⠀⠘⣁⣴⣇⡀⠀⡾⢿⡿⠭⠀⠀⢠⡾⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠙⠶⠶⠒⠛⣩⠏⡏⠉⢻⣷⡀⠃⠀⠀⢀⣼⠋⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁⢀⡇⠀⠸⡏⢳⡄⠀⠀⢨⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠛⣻⣗⣀⡴⠋⢻⡀⢀⣹⡄⠙⠳⠶⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢸⠳⣾⡏⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⢄⠠⠉⠠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⡄⠀⢠⠀⠄⠐⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⣽⣿⣿⣿⣧⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠧⡀⠈⠛
⠀⠀⠀⠀⠀⡀⠀⠉⠀⢄⠂⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡜⣿⣿⣯⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠐⠿⠃⣤⡄⠀⢰⣦⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣠⠀⠀⠀⠀⠀⠀⠉⣁⡀⠀⢁⡀
⠀⠀⠀⠀⠀⢠⡄⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠃⠹⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠛⠃⠁⠛⠋
⠀⠀⠀⠀⢰⣾⣷⣄⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣰⣠⠏⣿⡇⠻⣿⣿⣿⢦⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏
⠀⠸⣿⡇⠀⣿⡇⠀⢸⣿⠀⠀⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠙⠁⠀⠙⠃⠀⠈⠙⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠍⠉⢉⠒⠒⠢⢄⠀⠀⠀⠀⠀⢰⣶⣶⣶⣿⣿⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⢸⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣄⠀⠀⣠⡾⠂⠀⠀⠀⠀⢸⠁⠀⠈⠁⠉⠁⢸⠀⠀⠀⠀⠀⣸⣿⡿⠛⣿⣿⡟⠻⣿⣷⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣤⣾⣿⠁⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠁⠀⣼⠀⠀⠀⠀⠀⠈⠉⢱⣿⣿⣿⣿⣷⠉⠉⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠥⠁⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠈⠙⠛⠛⢿⣿⣿⣿⣿⣿⠿⣿⡏⠀⠀⠀⠀⠀⠀⢸⠀⠠⣤⣤⣭⣭⣤⣧⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠈⠻⠉⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠳⠤⠿⠿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠰⠶⠿⠿⠿⠿⠿⠿⠶⠶



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠻⡄⠀⠀⣠
⠀⣀⣀⢰⣦⣀⣀⡠⣶⢀⣀⡀⠀⠀⠀⠀⠀⠰⣿⡟⢻⣿⡟⢻⣿⡟⢻⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣁⣀⡰⣿
⠀⣿⣿⣼⣧⣿⣿⣯⣿⣼⣿⡇⡆⠀⠀⠀⠀⠀⣸⢻⡟⢻⡟⢻⡟⢻⡟⢻⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟
⠀⣿⡟⠛⡟⣿⣟⢻⡿⠉⢿⡇⡇⠀⠀⠀⠀⠀⣿⢸⡇⢸⡇⢸⡇⢸⡇⢸⠄⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⡜⠀⡇⠈⡆
⠀⣿⣶⣷⣾⣷⣿⣾⣿⣷⣿⡇⡇⠀⠀⠀⠀⠀⠻⣼⡇⢸⡇⢸⡇⢸⣇⡼⠀⠀⠀⠀⠀⠀⡠⠖⢿⣽⣿⣋⣿⣏⡷⠢⢄⠀⠀⠀⠀⢰⠷⠶⡿⠶⠿⡀
⠀⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣁⡇⠀⠀⠀⠀⠀⠀⠈⠓⠾⢧⣼⠷⠚⠋⠀⠀⠀⠀⠀⠀⠀⠘⠶⢤⣌⣉⣉⣉⣠⡤⠖⠃⠀⠀⠀⢀⠇⠀⠀⠇⠀⠀⢣

⠀⠀⠀⠀⠀⠀⢐⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠶⠶⢤⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣀⣠⣄⣀⣀⡀⠈⠙⣦⠡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡇
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠄⡀⠀⢂⣠⣠⣤⣐⣲⣤⣤⣤⣤⣀⣀⣘⣷⣿⢉⣀⠀⠈⠩⠙⣳⣶⠋⠄⢂⣠⣤⠒⣠⣐⡠⠠⠀⠄⣒⣄⣂⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢐⠪⣢⠶⠶⠶⠿⠛⣫⣩⠟⠛⣲⣶⡿⠟⠉⠉⠭⠿⠿⠍⠩⠭⠭⠿⠿⠿⠿⣿⡿⠟⠓⣲⡮⠽⠟⢛⣛⣛⣛⠛⠫⢉⢳⡄
⠀⠀⠀⠀⠀⠀⢈⠰⣿⡀⡀⢀⡠⠞⠁⠀⠀⢸⡏⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡏⠀⠀⢎⠀⠀⠀⡀⠈⡆⣯⠀⡗⠀⡄
⠀⠀⠀⠀⠀⠀⠠⠁⣿⠀⡴⠁⠀⠀⠀⠇⠀⠘⢣⠀⠀⠀⢈⡄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡇⠀⢀⠀⠙⢆⡒⠋⠀⢷⡇⢰⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠰⡅⢻⠜⠁⠀⠀⠀⠀⠛⠛⠛⡟⠀⠀⠀⢀⠈⠁⠁⡰⠖⠀⢀⠀⠀⠀⠀⠀⠟⠉⠉⢺⡇⠀⠀⠀⠀⠀⠙⢄⣀⡜⣇⠀⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠠⢺⠀⡀⠂⠁⠂⠘⡀⠀⡠⠇⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠘⢄⠀⡠⠛⠑⠀⠀⠀⠀⠠⡐⠦⠀⠇⢻⠀⣿⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠂⣿⠀⠁⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠐⡝⠀⠀⠀⢸⠀⣿⠀⡄
⠀⠀⠀⠀⠀⠀⢠⡀⣿⠀⠀⠁⣤⠆⠀⣼⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠒⠚⠈⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠁⣿⠀⠀⠈⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⡀⠀⠀⣿⠠⠁
⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠄⠠⠤⠄⡀⢨⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢨⡀⠙⠊⠀⠂⠄⡀⡇⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠀⣿⠀⠁⠀⠀⠂⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠅⣁⣀⠌⠙⢺⠃⠃⡆⠀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⣄⠀⠀⠀⠠⡀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠆⠀⠀⠀⠰⠇⠀⠀⠓⠠⠀⠈⠁⠒⢿⠆⢀⠀⡇⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠁⣯⠀⠣⡀⠀⣠⠶⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠠⠀⠀⠀⡀⠀⠀⡀⠀⢡⠀⠀⠀⠀⢀⢸⠀⡇⠈⡇⠀⠁
⠀⠀⠀⠀⠀⠀⢸⡀⣿⢀⣰⠈⢦⡀⠀⠸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣠⡄⢣⡘⠆⠄⠈⡇⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠻⠶⠶⠶⣶⡿⠶⠶⣤⢤⣼⠷⠤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⠀⠀⠀⡠⠂⢐⡦⠀⣠⣿⡶⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠻⠁⠦⠄⠐⠒⠂⠉⠀⠘⠆⠠⠁⠀⠈⠉⠉⠉⠩⠉⠙⠛⠛⠛⠛⠙⠉⠝⠓⠲⠶⠟⠛⠓⠿⠤⣤⣤⣤⡼⠋⠂⠈⠗⠀⡇
⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠤⠀⠀⠠⠬⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠂⠰⠒⠂⠂⠐⠐⠒⠀⠒⠀⠖⠒⠂⠀⠐⠐⠒⠀⠂⠀⠀⠀⢘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠤⠤⠤⠤⠠⢤⠀⠤⠤⠀⠤⠤⠀⠤⠄⠠⠤⠤⠤⠤⠄⠀⠀⠈⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⢤⡀⣠⣠⣀⣤⠀⡄⢠⣠⣀⢠⠀⡄⢠⣄⣀⢀⠀⠀⠀⠀⢸⣿⠀⠠⣀⣧⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⢀⠀⢀⣀⣀⠀⠀⠀⡀⠀⣀⣀⡀⠀⢀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠸⣿⣆⠐⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣿⣻⣄⠹⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠈⠁⠉⠉⠈⠉⠁⠉⠉⠁⠈⠉⠉⠉⠈⠉⠉⠀⠀⠀⠀⢻⣟⣿⣿⣿⣿⣧⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠁⠋⠀⠘⠀⠃⠂⠘⠁⠊⠙⠂⠙⠀⠁⠀⠂⠀⠀⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠒⠒⠒⠒⠒⠒⠀⠒⠒⠂⠒⠒⠂⠒⠒⠂⠒⠒⠒⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡹⣦⣀⣶⣶⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠄⠤⠤⠤⠴⠤⠄⠠⠀⠤⠦⠤⠄⠤⠄⠤⠴⠀⠤⠤⠀⠀⠀⠀⠀⣾⣽⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠠⠄⠠⠠⠤⠠⠀⢠⠠⢀⡠⠤⠀⠄⠤⠀⢠⡤⠠⠠⠠⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣀⣀⣀⣀⢀⣀⡀⢀⠀⢀⣀⢀⡀⢀⣀⣀⣀⣀⡀⣀⣀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢟⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠁⠀⠉⠁⠀⠁⠀⠉⠉⠁⠉⠀⠁⠀⠀⠁⠈⠉⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⠁⠉⠁⠁⠈⠁⠉⠁⠉⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⠀⣴⠄⣿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⣿⣿⠀⠸⠆⠟⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⣿⣿⠀⠀⢠⣄⢀⣄⠀⠀⣠⡀⢠⡄⠀⣤⡀⠀⢀⣤⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣴⣴⣿⣿⣤⠀⢸⣿⢸⣿⠀⠀⣿⡇⢸⡇⠀⠘⢿⣄⣾⠃⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⣿⣿⠉⠀⣸⡿⠸⣿⡀⢀⣿⠇⢸⣇⠀⠀⠈⣿⠃⠀⠀⠀⢴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠛⠛⠁⠀⠉⠛⠛⠉⠀⠘⠛⠛⠃⠀⠛⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠒⠋⠉⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣤⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⢀⠄⡌⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠛⢧⠀⠀⠀⠀⠀⠀⠀⠆⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡋⠁⠠⠠⠌⢷⣀⠀⠀⠀⠀⠓⠀⡾
⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠈⠀⠀⠀⠀⠀⡄⣹⣷⣤⣈⣀⠁⡼⠁
⠀⠀⡰⠬⠃⠈⠈⠀⠒⠂⠀⠀⠀⠀⠀⣠⠞⢡⡀⠀⠀⠀⠀⠀⠒⠺⠞⠛⢛⡽⠋⠉⠛⠁
⠀⢈⡭⠁⠀⠀⠀⢀⡫⠀⠀⠀⠀⣠⠞⠿⠋⠀⠉⠂⠤⠄⣠⣀⣀⣠⣤⠔⠋
⠀⠀⠐⠃⡆⡄⢰⠓⠙⠀⢀⣠⠞⡁⠀⠈⠢⠄⣀⣀⢠⣶⣿⣗⣽⠞⠁
⠀⠀⠀⠀⠁⠀⠀⠀⢀⡴⢻⡡⠀⠀⠠⡀⠀⠀⠀⠉⠉⢍⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡼⢫⡀⠈⠁⢄⡀⠀⢀⢌⣯⣿⣶⡷⠋⠁
⠀⠀⠀⠀⠀⢀⡴⣷⠋⠑⠠⢀⣀⠈⠙⠛⠛⣿⡿⠋
⠀⠀⠀⢀⡴⠋⣠⠿⣦⣀⣀⣴⢨⣛⣛⣿⡿⠁
⠀⠀⠀⠈⡠⠚⢁⡴⠊⣳⢿⣿⣾⣿⡿⠋
⠀⠀⠀⠀⢀⠔⠉⡠⠚⠡⠊⠁⠉⠉
⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠉⠀⢠⠷⠀⠀⠀⠀⢨⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢺⡗⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠠⢀⣀⡠⠀⠀⠀⣡⣿⡍⠀⠀⠀⠀⢠⡀⠀⠀⠀⢽⠅⠀⠠⢿⡟⠀⠀⠀⠀⠀⠀⠈⠁⠠⣼⣷⠄
⠀⠀⠀⠀⠀⠘⠁⠁⠀⠀⠘⣡⣿⣌⠢⠀⠀⢠⣿⣷⡀⠀⠀⢺⡒⠀⢠⣿⣿⡄⠀⠀⢠⣆⠀⠀⠀⠴⣿⣿⠶⠆
⠀⠀⠀⠀⠀⠖⠒⠂⠀⠀⠔⣩⣿⡝⠢⠀⢀⣿⣿⣿⣧⠀⢐⠱⣝⠀⢠⣾⣿⣄⠀⠀⣾⣹⡀⠀⠀⠾⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣱⣿⡝⠢⠀⣼⣿⣿⣿⣿⣇⢁⡼⣌⠁⠉⣽⣯⠉⠀⣸⣷⢺⣇⠠⠴⠾⣿⣿⣟⣉⣀
⠀⠀⠈⠁⠀⠀⡇⠀⠀⠀⠚⠁⢺⠈⠂⠸⠿⠿⢿⠿⠿⠿⠀⢸⠀⠀⣾⣿⡿⠷⢠⣿⣷⣼⣿⡄⣠⡾⢿⡿⣿⣍⠁
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠀⠀⠀⠘⠀⠀⠈⠉⠉⠈⠁⠀⠀⠀⠘⠃

⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⣀⢠⢀⡀⠀⠀⢀⣤⣶⣶⣦⡀⠀⠀⠀⢀
⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⢸⣅⢉⠀⠀⠀⡎⠉⠉⠉⠉⠙⠀⠀⣿⣿⣿⣿⡯⠭⠯⠍⠀⢸⣿⣿⣿⣇⣈⣈⡁
⠀⢰⣛⣻⣷⣿⡄⠀⠀⡊⢲⠀⠁⠘⠀⠁⠀⠀⠱⣠⢰⢠⣄⡶⠀⠀⠹⠿⠿⠿⠭⠭⠭⠁⠀⠈⠿⠿⠿⠷⠶⠶⠆
⠀⠈⠙⠛⠛⠛⠁⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⣤⣤⣦⡄⠀⠀⠀⠸⣷⢀⠀⠀⠀⠀⠀⠀⣰⢏⢁⢅⠉⢷⡀⠀⠀⠀⣼⣿⣏⣱⠀⠀⠀⠀⢀⣨⣿⣿⣇⣀
⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠸⢇⡄⠀⠀⠀⠀⢽⡈⠧⠥⠅⣸⠁⠀⠀⠀⢩⣉⢉⣹⠀⠀⠀⠀⠀⠂⠈⠈⠱
⠀⠀⠠⠤⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣿⠶⠶⣾⡇⠀⠀⠀⠀⠀⠡⡵⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠸⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣦⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⣠⣴⣿⣷⡆⠀⠀⣀⣰⢢⡤⣤⢰⣄⡀⠀⠀⣿⠿⠿⠿⡿⠀⠀⠀⠀⠘⢿⣿⣗⣲⡄⠀⠀⢸⠛⠛⠛⣛⠛⡇
⠀⢸⣿⣿⡿⠋⠀⠀⠸⠟⠋⠉⠓⠛⠉⠙⠻⠀⠀⠘⢆⣀⡰⠃⠀⠀⠀⠂⠀⠾⡟⢛⠛⠀⠀⠀⢸⠀⠀⠰⢿⠄⡇
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠢⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠉⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠛⠉⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠄⢂⡞
⠀⠀⠀⠀⠀⠀⠤⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠀⠀⠀⠀⠀⣾⠋⠉⠁⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡘⡠⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠑⠄⠀⠀⠀⠀⢀⡴⠋⢳⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢏⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣠⣴⣏⠀⠀⠁⠓⠒⠖⠊⠻⣧⣀⠀⠀⠀⢸⠀⠀⡆⠆⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠾⠋⠨⡁⠀⠀⠀⠠⠀⠀⠀⢀⠬⣻⣷⣌⠘⠀⡄⠀⠐⢀⣼⠃
⠀⠀⠀⠀⠀⢠⡀⠀⡴⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⡀⠐⠀⣰⢛⣖⣇⣾⣻⡿⢷⣶⣧⣤⣼⠇
⠀⠀⠀⣘⠓⠢⠍⠂⠀⠀⠀⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⡁⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⡋⡋⠉⠉⣡⠾⠋⠀⠀⠀⠁
⠀⠀⠈⠉⣉⠁⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⡹⠋⠀⠙⠲⢤⣀⠀⠀⠠⠀⠈⠀⠀⠀⢃⣇⣴⠟⠁
⠀⠀⠐⠋⣠⠄⠀⠀⠀⢀⠀⡙⢦⡀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢏⠀⠅⡠⠊⠀⠀⠀⡉⣩⡻⣿⢚⡶⡿⣻⡿⠋
⠀⠀⠀⠀⠋⢀⡇⢸⠀⠸⡇⠁⠀⠁⠀⠀⣠⣴⠟⠁⡀⠀⠀⠀⠙⠢⢄⣅⣀⣠⢊⣾⢟⣗⡿⣏⣮⡿⠋
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⢁⡀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠉⠉⡉⢛⠛⢛⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠈⡾⠍⠀⠀⠀⠀⠉⠒⠠⣤⣄⣀⣀⡀⣁⣸⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⡀⠂⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠑⣀⠀⠀⠀⠐⠤⣀⠀⠁⢠⣠⢅⣴⢰⡭⣏⣯⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡆⡧⠛⠢⢄⡀⠀⠀⠀⠘⠗⢶⢽⣽⢽⢭⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠹⣷⠁⠀⠀⠀⠈⠑⡲⡦⣄⣄⣀⣀⣈⣤⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⡶⠻⣷⡤⣀⢠⢄⡳⡂⠇⣀⡇⣯⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢀⠔⠋⠀⣠⠞⠉⠀⣠⡌⠳⣮⣽⣲⠿⣴⣗⣧⣷⡿⠋
⠀⠀⠀⠀⠀⠴⠊⠁⢀⡴⠞⠉⢀⡴⠛⢉⡿⠿⠶⢾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣀⡴⠋⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠔⠉⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣷⣶⣶⣶⣶⡾⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣷⣷⡾⡿⠾⠷⠶⢶⣶⠾⠿⠶⠶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘

⠀⠀⠀⠀⠒⠲⠴⠖⠒⠖⠲⠴⠒⠂⠂⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡠⠀⠄⣀⡀⣀⣀⠀⠀⠀⡀⢀⡀⡀⢤⡀⠒⠒⠀⠀⠀⠃⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠈⢡⢠⠈⠀⠉⠠⠁⠄⠀⠀⠀⠐⢄⠃⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⣮⣑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡈⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡶⡀⠀⠂⠀⠀⠀⠀⡔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠘⠁⠹⠇⠀⠀⠀⠀⠀⠸⢉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠃⠀⠀⠘⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢀⠀⠀⠀⠀⠠⠛⠀⠀⠀⠈⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡤⠀⠀⠀⠃⠀⠀⠀⢀⠆⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠚⠀⠀⠘⡀⠈⠀⠀⠀⠀⢈⡀⢀⠀⣠⢦⡄⠆⠠⠀⠀⠀⠀⠀⠀⠀⠸⢀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⠀⠀⠀⠀⠛⠉⠀⠈⠉⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⣤
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⡠⠀⠀⠰⠀⠀⠀⠐⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠛⠓⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠘⡄⠀⠀⠀⠀⠞⠛⠶⠶⠴⠶⠤⠤⠠
⠀⠀⠀⠀⠀⠀⠀⢁⠀⡀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠘⢶⡀⠀⠀⠀⠐⡗⠀⠀⠀⠂⠐⠩⠴⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠧⠀⠁⠀⠀⢡⠀⠀⠀⠀⠇⠀⠀⢊⣡⣬⡁⠀⠀⠀⠀⠀⠀⠀⡄⡠⠤⠀⠀⡠⠆
⠀⠀⠀⠀⠀⠐⠂⠀⣈⠀⠐⠁⠀⠀⠀⠘⣶⣴⣃⣦⣼⠃⠀⠀⠀⢠⠀⠀⠀⠈⢀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠟⠻⠟⠋⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠄⠘⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠋⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠉⠀⠀⠁⠀⠘⠀⠀⠁⠐⠀⠀⠐⠂

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⣀⣀⣀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠈⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠉⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠛⠋⠁⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⡞
⠀⠀⠀⠀⠀⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣷⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⢁⠹⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⢀⡴⠋⠁⠛⢿⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⢱⠏⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢀⣴⠟⠂⢄⣀⠀⣀⣹⣦⠀⠀⠀⠀⠀⡀⢀⢰⡄⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⢀⡴⢻⡗⠠⠀⠀⠈⠈⠉⠀⠙⢷⣤⠀⠀⡄⢇⠀⡈⡀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⣠⡶⠋⠀⠈⠀⡀⠀⠁⠀⠀⠀⢀⡎⣡⢝⣿⣦⣅⣤⣆⠣⠀⣾⠃
⠀⠀⠀⢠⣄⡘⢦⠀⠃⠋⢀⡠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⠀⠀⠀⠀⠀⠀⠁⠂⠠⠤⡤⣿⢼⠾⠾⠷⢛⣭⠟⠛⠛⠿⠋
⠀⠀⠰⠶⠦⠌⠀⠁⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢥⡦⠚⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⢘⠀⣠⡼⠋
⠀⠀⢠⠴⠂⡀⠀⠀⠀⠀⠀⢌⡓⠀⠀⠀⠀⠀⢀⣤⠞⠉⠿⢋⠔⠁⡠⠈⠘⠓⠶⠤⣴⣤⣤⣠⣤⣤⣶⠞⠉
⠀⠀⠀⠠⠟⢀⡆⣰⠀⢰⠸⠆⠙⠀⠀⠀⢀⣴⠟⠁⠀⠐⠈⠣⢥⠊⢄⠀⠀⠔⣡⡾⣯⣾⢼⣹⣷⠟⠁
⠀⠀⠀⠀⠀⠸⠃⠙⠀⠈⠀⠀⠀⠀⣠⣶⠟⠁⠀⠢⠀⠀⠀⠀⠀⠉⠙⠓⠺⠶⢿⠾⠷⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⢷⡆⠏⠀⠀⠈⠂⠄⢀⡀⡀⠀⠄⠀⡄⢸⣠⡾⠛⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢡⠀⠀⠈⠠⢀⡀⠄⠀⠀⡠⡢⢹⣋⢟⣟⣗⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡴⡺⠷⣀⠀⠀⠀⠈⠑⠦⢦⣿⣾⣽⣿⣵⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢿⡏⠁⠀⠀⠉⠒⠦⢤⣀⣀⢈⠀⡉⣉⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢀⡴⠟⠁⣠⡼⢿⣦⡀⠀⣀⢰⢀⡸⢨⣩⢹⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢀⠴⠋⠀⣠⠞⠉⠀⣨⠻⢮⣝⣷⣽⣨⣺⣽⣽⣾⠟⠁
⠀⠀⠀⠀⠀⡤⠊⠁⢀⡴⠞⠁⣠⡾⠛⣻⡿⠶⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠞⠉⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠔⠁⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⡠⠔⠊⠉⠉⠉⠁⢒⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⠀⢠⣎⣀⠀⠀⠀⠀⢀⠔⠉⠀⢛⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄⠀⢝⠂⠀⠀⠈⠙⢲⡀⢠⠀⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⡉⠉⠋⠀⠀⠀⠀⠀⠀⢱⣇⣀⡀⠀⠀⠀⡇⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⡀⡰⣖⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⠀⠆⡄⠀⠀⠀⠀⠀⣸⠋⠑⢄⠀⡇⢀⠀⡾⠀⠀⠀⠀⠀⠀⠠⣀⡳⠜⠀⡏⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠈⢀⡀⠀⠀⠀⣰⠃⠀⠀⠈⠷⠋⡠⠈⠴⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣴⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⢤⠠⠒⣓⡄⢠⢀⠤⢡⠒⠢⢴⠃⠀⡠⠀⢀⡠⠊⡱⠷⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠈⠖⢥⠀⠀⢠⠃⠀⠀⠀⠑⡏⣀⠴⢋⠔⠙⠦⠤⠊⢀⡀⠀⠀⠀⢀⠔⢲⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢽⡖⠨⡷⠔⠂⠄⠀⠀⡾⠀⠀⠀⢀⣠⢼⡥⢚⡁⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⡠⠎⢳⣀⣠⠗⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠃⠠⡀⠔⠀⠐⢦⣀⣇⠀⡠⠞⠉⠀⢀⡟⡉⢁⡀⠀⠀⠒⠀⠀⠀⠢⡀
⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⡿⠀⠄⠉⠋⠀⠑⠀⡇⠀⣏⠀⠀⠀⠀⣼⠁⡀⠄⠩⡒⠀⠒⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠒
⠀⢀⡗⡁⠀⠀⠀⣀⣂⣀⣃⣤⣼⢥⣿⣿⣱⠁⣠⣨⣁⡖⡀⣉⣞⣀⣑⣤⡠⠾⠷⠏⠘⠚⣢⠈⠂⠀⢄⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣳⡜⢻⡄
⠀⡌⠁⠁⢠⠀⣀⡤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠙⡖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠉⠀⠉⠂
⠀⡃⠀⠘⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠇
⠀⠿⠲⢶⠃⠀⠀⣠⠔⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡏
⡴⠛⠈⠚⠲⡀⢸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⣠⣇⣀
⡇⠀⠀⠀⠀⡇⠀⠀⢠⠁⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⡆⡴⠁⠀⠀⠈⡷⠀⠀⠀⠀⠀⢈⠀⠀⠃
⠘⠠⠄⣀⠴⣧⠃⠀⠘⢀⡀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⠀⡠⠀⢁⢧⠀⠀⠀⡰⠃⠰⠄⠉⢹⡂⠘⠆⠁⢀
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠙⡤⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠄⠀⠀⠀⠀⠀⠈⡏⠍⠉⣉⢱⠀⡔⠀⠀⠠⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠈⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠱⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⣤⠴⣤⡁⠂⢄⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⡤⢾⣦⡰⡟⠻⡉⠉⠙⢦⢌⠢⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡕⠁⠈⠁⠈⠛⠒⣻⢏⠎⣂⢰⠀⠀⣼⠀⠑⡈⢂⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠒⡄⠀⠀⠀⠀⠎⢀⠇⠯⡀⠘⠔⠁⠀⠀⠀⠈⡄⢂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡔⠀⠀⠀⠀⠀⠉⠀⠖⠺⠀⠀⠀⠀⠀⠸⣮⠿⡀⡀⢠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⢀⠀⠀⢄⠀⠀⢀⣀⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢐⠵⠾⡇⠁
⠀⠀⠀⠀⡀⠨⠰⠉⠉⠉⣷⠨⠀⡀⠈⠓⢄⣇⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡄⡀⢀
⠀⠀⠀⠎⠇⠀⠀⠀⠀⠀⠀⠀⠀⡉⡄⠀⠀⠀⠙⠶⠒⠂⠀⢀⠀⠀⠀⠀⠀⠑⠂⢲⢁⠁⠈
⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⡄⠵⠤⠤⠤⠤⠤⡧⠾⠴⠓⠲⢟⣚⡛⢍⠉⠙⠛⠛⢋⠆
⠀⠀⢰⠀⠘⠠⡷⠀⠀⠀⠰⡧⠈⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡏⠈⢻⡆⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠘⣄⡀⠀⠀⠀⠀⠀⠄⠁⠰⢢⢀⠀⠀⠀⠀⠰⠀⠀⡠⢄⠙⠒⠚⠁⠃⠀⠀⠀⠀⡔
⠀⠀⡄⠀⡠⠐⠒⠒⠒⠒⠒⠒⠠⠀⠀⡷⢤⡁⡐⠂⠀⠒⠀⠒⣘⡜⡠⠀⠡⠄⠀⠬⠅⠀⠀⠈⢆⠀⠀⠠⠀⠀⠀⠀⠀⡠⠂
⣆⠀⠀⠀⠃⠤⠤⠤⠤⠤⠤⠤⠄⡆⠀⠂⢀⠧⠀⢀⡀⠀⠀⡀⠠⠖⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⢀⠦⠀⡖⠐⠀⠂⠈⠀⠠
⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠰⠠⠀⣠⠑⠶⣷⣴⡀⠀⠀⠀⠀⠀⡔
⠣⣘⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⡠⠂⠐⠒⠓⠒⠒⠓⠒⠀⢀⠀⠀⠀⠀⠀⢀⡤⠊⠈⠁⠐⠃⠀⡀
⣀⣈⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣌⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀⣎⣶⣶⣶⣦⣬⣤⣦⣤⣖⣒⣒⣚⣉⣉⣉⣉⣉⣉⣉⣉
⣿⣿⣋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣀⣀⣀⡀⠀⠀⠀⠀⠀⡠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣦⣤⡾⠀⠅⠀⠀⡈⢑⠤⣀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⢱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⡀⢅⠈⠀⠄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠉⠉⠉⠛⠻⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⢨⡇⠀⠀⠐⠀⠐⠄⠠⠀⠀⠉⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⣿⡄⠀⠀⠀⠨⠀⠀⠀⠀⡀⠠⢀⢸⡇⣀⣀⣀⡀⢀⠀⠀⠀⠀⠀⡀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⣾⠃⠀⠀⠀⢹⣷⡀⢠⠀⡋⠉⠈⠉⠑⠢⠤⣐⣘⣫⣵⠇⠀⠉⠒⢤⡀⠀⠀⠀⠂⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣰⣿⠇⠀⠀⠀⠀⠀⣿⣧⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⢀⠀⠹⣟⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⠁⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢂⠀⠹⣄⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⢹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠠⠀⠈⢻⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠐⢄⠀⠙⢿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠣⠀⠹⣿⣿⣦⡀⠀⠘⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠛⠿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠘⠿⣿⣷⣤⡀⠈⠻⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⢀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠂⠀⠈⠙⠿⣿⣿⣿⣿⣶⣶⠶⠊⠁⠀⠀⠀⠈⡍⠙⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢀⠴⢃⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⡘⠀⠀⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠰⠀⠂⠄⠀⠠⠂⣿⠈⠀⠀⠐⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠇⠀⠀⢀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠰⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⡌⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠡⠀⠀⠀⠆⠀⠀⠀⠀⠇⠀⠀⠀⠄⠀⠀⠐⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠆⠲⠦⠤⠠⠀⠤⠶⠒⠶⠒⠊


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⡆⠀⠘⡞⠉⢳⡄⠀⢦⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⢠⡖⡟⠒⢿⠀⠀⣧⣀⡼⠁⠀⡞⠀⢀⡇⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⢘⣦⠒⠙⢧⡀⠀⢧⡀⠘⠃⢠⠇⠀⠀⢀⣸⠛⠒⠋⠀⣰⠋⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡀⠈⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢛⣦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣄⣀⣀⣀⣤⡄⣀⣀⣤⡤⠴⠶⠦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣏⣩⡥⠴⠖⠒⣛⣛⡒⠲⠮⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡴⠶⢚⣛⣩⣭⣭⣭⣍⣛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣛⠛⠿⠿⠿⠿⠿⠖⣛⣩⠥⠶⣒⣒⣲⠶⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢬⣍⣛⣛⣛⣛⣛⣩⡭⠖⠚⠉⠉⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠉⡭⣯⡉⡉⣀⡀⣐⠖⣂⢀⠀⣀⠀⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠓⠛⠃⡀⠀⡷⣿⠀⣿⢸⠀⠘⢷⠃
⠀⠀⠀⠀⠀⠀⠠⠀⣠⠖⠀⠀⠉⠛⠁⠀⠀⠀⠙⠖⠃⠘⠶⠋⠘⠒⠂⠙⠀⠀⠀⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢋⢆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠢⣌⡢⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠎⡄⢀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠘⠆⠰⢠⢠⢰⠁⠀⠠⠄⢀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⢀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⣀⡻⡦⠑⠘⠰⠉⠥⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢒⣋⠁⠀⠀⠀⠀⢒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠰⠀⠐⢀⡀⠈⠁⠀⠉⠥⢂⡀⠀⠀⡘⠦⡉⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠐⠨⠋⠀⠀⠀⠀⠀⠐⡒⠀⡀⠀⠐⠋⠀⢸⡇⠇⠀⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⣄⠀⠃⢸⠇⣴⠀⡴⠟
⠀⠀⠀⠀⠈⠀⠡⠄⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠈⢀⠇⠀⠀⠘⣂⠹⡶⠇⠈⠀⠀⠈⠀⣤
⠀⠀⠀⠀⠀⠀⢀⠔⠋⢠⠀⢀⢀⠀⠀⠁⠀⠀⠀⢸⠀⠀⣠⡤⠭⠀⠁⠀⠀⠀⠀⠀⠀⠰⠶⠖
⠀⠀⠀⠀⠀⠚⠁⠀⠀⠁⠀⡈⠈⠀⠀⠀⠠⠀⠀⡏⠀⠀⠀⠰⠆⡀⠀⠀⠀⠀⠀⠀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⠀⠀⠀⡇⠀⠀⠀⠴⠟⢁⠀⡀⢀⡄⢠⡀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡇⠀⠀⠀⢀⡴⠃⠈⠃⣼⠁⠈⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⡇⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠃

⡷⣞⣷⣿⣶⣿⣾⢷⣾⡶⣿⣞⡷⣾⣷⠿⣶⢿⣾⣷⣾⣶⣿⣾⣷⣾⣷⣿⣾⣷⣾⣷⣿⡖⣟⣞⠓⣺⣷⣿⣾⣷⣾⣷⣿⣶⣿⣾⣷⣾⣷⣿⣾⣷⣾⣷⣿⣶⣿
⣧⣿⣧⣶⣿⡟⣷⣮⣿⣿⣿⣹⣷⣿⣴⣿⣾⣽⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣳⣞⣫⣮⣛⣽⣾⣿⣿⣧⣿⣴⣿⣾⣧⣷⣾⣽⣧⣴⣷⣶⣶⣾⣿⣿
⠉⠈⠁⠈⠁⠱⡄⠁⠈⠁⠉⠁⠉⠈⡱⠋⢡⠟⠁⠁⠈⠁⠉⡀⢩⠈⠃⠉⡡⠉⠈⠅⠸⠟⡉⢤⠙⠈⠁⠙⠁⠉⠈⠃⠈⠁⠉⠀⠁⠈⡁⠉⠀⠉⠈⠁⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⡰⠁⠀⠁⠀⠀⠀⠀⠀⠀⣬⡀⠀⣠⠖⠃⠀⠀⠀⡈⠀⠅⢀⠀⢀⠀⠀⠂⠀⠀⢀⠀⠀⢄⠀⠀⠀⠁⠀⠄⠀⢀
⢀⠀⠀⠄⠒⠂⠤⠀⠀⠀⡀⠀⠀⡇⠀⣠⠆⠀⡠⠤⠒⡒⠄⠰⡀⢀⠀⠀⠓⡠⠀⡄⠁⠬⠤⢨⡆⢂⢀⠁⠀⠁⠀⠨⠓⣄⠀⠀⠀⠀⢀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡑⠀⠀⠄⠘⠁⠀⢀⡀⠀⠀⠑⠬⠁⡁⠖⠀⠠⠉⠀⠴⠁⠖⠀⠐⢁⡀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢀⡸⡖⣃⣋⣖⡧⠀⠀⠀⠒
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠐⠀⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⠀⠀⠰⠗⢩⠐⠰⠀⠐⠁
⠀⠠⠊⠁⠀⢀⡤⠊⠀⠀⣠⠈⠀⠈⠀⠒⠀⠀⠉⠃⢀⠂⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠠⠛⡁⠀⠛⠃
⠀⠀⠀⠀⢀⠈⠀⡠⠂⢀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠢⠀⢀⠠⢈⡉⢄⡀⡀
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⡄⠀⠁⠀⠘⡆⠀⠰⡄⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄
⠀⠀⠀⠀⢀⠼⠀⠤⠀⠀⣸⠀⠀⠅⠀⠀⢡⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠐⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠
⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣶⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠒⠶⠆⠂⠂⠒⠒⠒⢦⠤⡀⠀⠀⠀⢀⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠁⠀⠃⠀⠀⠀⡀⠀⠀⠣⠄⠀⠈⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠒⠀⠠⢀⣀⣉⠀⠔⠠⡀⠀⢀⣠⠀⡀⠄⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠉⠉⠈⠀⠈⠉⠉⠁⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⣀⡠⠌⠀⠍⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠚⠈⠁⠀⠀⠀⠀⠀⡰⠃⠀⠈⡙⠂⠀⠀⠀⠀⠁⠀⠀⠦⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣰⠒⠛⠋⠟⢥⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡇
⠀⠀⠀⠀⠀⢀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡤⠀⠀⠀⢀⠀⠀⠀⠀⠀⢈⡀⠀⠇⠀⠀⠀⠐⠁⠈⠃⡀⠀⠀⠀⠀⠀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠁
⠀⠀⠐⠄⠀⠈⠀⠃⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠤⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⡠⠀⠀⠐⠄⠀⠀⠀⠀⠂⠀⠀⠂
⠀⡤⢊⠄⠀⠀⠀⡀⠠⠀⠀⠀⡡
⠀⠄⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠰⡄⠀⡆⠄
⠀⠠⠀⠀⠀⠀⠀⠈⠘⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢰⣿⣿⣿⣿⣿⡇⡄⠀⢀⣶⣿⣿⣿⣿⣿
⠀⠀⢹⣿⣿⣿⣿⣿⣷⠁⠀⠀⢿⣿⣿⣿⣿⣿⡾
⠀⠀⠀⠙⠿⠿⠿⠛⠁⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡾⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠟⠃⠢⡤⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣷⠆⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠛⠙⠀⢠⡄⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠰⡆⠀⢀⡄⠀⢸⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣷⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⢹⡀⠈⠁⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⠇⠀⠀⢠⠏⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⡀⠀⠀⠇⠀⠀⠾⠀⠂⢠⠏⠀⠀⠀⣠⠴⠋⠀⢀⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⡀⠀⢰⣆⣶⠄⣴⣦⠀⣶⢲⡆⣶⣲⡆⢶⣤⠆⠞⠁⠀⠠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⡄⠀⠈⠙⠶⠸⠏⢿⠠⠟⢻⠆⡿⠉⠁⣿⠉⠁⠀⡿⠀⠀⠀⢀⣀⣤⠴⠚⠃⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠚⠉⠁⠀⠀⠀⠀⠀⠐⠚⠛⠀⠹⠿⠇
⠀⠀⠀⠀⠀⢹⡇⢹⣷⣬⡇⢸⡏⢹⣦⢻⣯⡝⢸⣿⣹⠆⣿⣭⠃⢹⢷⣼⡇⢹⡏⢻⡆⢻⣯⡍⢸⡿⣦⣿⢰⡟⠙⠆⣿⣭⠋
⠀⠀⠀⠀⠀⠺⠇⠸⠂⠙⠃⠸⠷⠞⠃⠺⠷⠖⠘⠿⠀⠀⠿⠶⠆⠺⠂⠙⠃⠺⠷⠞⠃⠺⠧⠞⠸⠇⠙⠛⠘⠻⠖⠃⠿⠦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠶⠖⠒⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢤⣀⡀⠀⠀⠒⠲⠶⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠁⠀⠰⣿⢳⡆⣰⢷⡘⢷⡼⠁⠀⠀⠓⠦⣄⡉⠛⠲⢤⣄⡀
⠀⠀⠀⣠⣄⣀⠠⠶⠚⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⢀⠔⠐⠿⠾⠃⠟⠻⠇⠸⠇⡈⠢⡀⠀⠀⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠙⣿⣿⡋⠀⠀⠀⠀⠀⠘⠋⠁⢀⡤⠀⠀⡴⠋⠀⠀⣰⠃⠀⠀⠀⢠⠀⠀⠹⣆⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⣠⠞⠁⠀⠀⠠⡏⠀⠀⢠⡄⢸⡀⠀⠀⠙⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠼⠃⠀⣰⠃⠀⠀⠀⠀⠀⢸⡇⠘⡇⠀⢠⠀⠀⠀⠀⠘⠷⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠸⠂⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⢀⡤⠞⡖⠒⠒⣁⡤⠴⠒⠒⠒⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⣸⠁⠀⠀⡏⠀⢀⠃⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠖⠁⢠⡧⠈⢑⡄⠈⠉⠀⠀⠀⢄⠈⠱⠀⠀⡔⠈⠳⡄⢀⡼
⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⡿⠀⠀⣼⣀⡴⠋⠀⣄⠀⠀⣧⢀⡀⢤⠀⣤⠀⠈⠂⠤⡿⠋
⠀⠀⠀⠀⠀⢠⠎⢀⠤⢤⡀⡇⢀⣴⡟⢉⠄⣴⠀⣿⢈⡇⣿⢀⡇⢾⣠⡏⢠⠎⠉⡼⠉⢦⠀⣀⡤⠶⠶⠦⣄
⠀⠀⠀⠀⠀⠘⡀⠸⠀⠠⢃⡿⠋⢰⡇⠸⡄⠿⠄⡏⠚⠃⣿⠉⠁⠀⢸⣇⡸⢶⣊⠜⣀⣼⣴⣇⠀⡴⠲⡄⠀⡇
⠀⠀⠀⠀⠀⠀⣈⠀⠀⢠⢿⡇⠀⠘⡇⠀⠰⣄⠀⡇⠀⠀⠿⣠⣔⡈⣹⠀⠀⠀⢀⡞⡱⠃⢠⣈⡇⢇⡈⠀⡠⠃
⠀⠀⠀⠀⣴⠋⠁⢉⡆⠋⣼⠁⠀⠀⠳⣀⡠⠋⠀⡡⠔⠈⠸⣇⢀⣠⠋⠙⢦⡀⢸⠋⠀⠀⠀⣀⠤⠤⡀⠉
⠀⠀⠀⠀⢧⡀⠀⠉⢀⡴⠃⠀⡴⢳⠀⠀⠀⡴⠋⠀⠀⠀⠀⠈⠉⠁⣠⠊⠀⢷⠘⣄⠀⡠⠊⡄⠀⠀⡨⢷⠤⣄
⠀⠀⠀⠀⣀⠉⣛⡋⠉⠀⠀⣸⠵⠋⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠟⠀⠈⢳⠗⠚⠁⠀⠀⠓⣋⡀⠈⣷
⠀⠀⠀⢎⠀⠀⠀⢙⣦⠒⠉⣿⠀⠀⢀⡟⢰⡔⡄⢀⠞⠁⠀⢱⡄⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠌⠁⠿⢀⡟
⠀⠀⠀⠀⠁⢀⠔⠁⡈⡇⠀⣿⠀⠀⣼⠁⠀⡰⠃⣾⠀⠴⣠⠞⠀⡇⠀⠀⠀⢀⡀⣄⢠⠳⠶⠀⠀⠀⠈⠓⠚⠉
⠀⠀⠀⠀⢠⡏⠀⠿⠭⠓⠀⣿⠀⢠⣇⣤⠞⠁⢠⡧⠖⡋⠁⡠⠠⡇⣴⠙⡇⢸⡇⣿⠠⢭⠆⣠⠔⠂⠀⠒⣄
⠀⠀⠀⠀⠘⣇⠀⠀⠀⣠⠀⣿⡤⢾⡏⠀⡴⠐⣤⡇⠀⡇⠀⣇⡠⢧⠙⠒⠛⠀⠉⣿⠀⢀⠞⠁⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠈⠛⠒⢚⠵⠋⣿⠀⢸⡇⠀⠣⠴⠃⠓⣂⣭⠥⠶⠶⠶⠶⠖⠦⠤⢴⠷⣴⠯⠤⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣀⣿⠀⢸⡇⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣠⠏⠗⠀⣷⠀⠀⠀⠀⠙⠦⢤⣀⣀⣤⠤⢔⣚⡥⢤⡾⠥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢤⠀⡠⠞⠋⢀⡴⠋⡴⠂⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⡸⠁⠧⠴⠚⠁⠀⠀⠣⠤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣼⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢵⣷⣊⠀⠀⡖⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⠀⠀⢸⠃⠀⠀⢸⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠎⠀⠀⣾⠀⠀⠀⡏⠀⢀⡤⡄⢀⡠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣲⠄⠀⠀⢀⠠⠀⢇⠀⣀⡠⠧⢀⠀⠳⢌⠋⣐⡡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠁⠀⠀⠀⠹⡎⠀⠀⠀⠱⠀⠀⠓⠬⣁⠭⠗⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠰⡒⢦⠄
⠀⠀⠀⠀⠀⠀⢀⠔⣒⡓⢒⣋⣒⡄⠀⢰⠀⠀⠀⠉⢀⢇⠈⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡎⠀⠀⠀⠀⠓⠁
⠀⠀⠀⠀⠀⠐⠧⠊⠉⠀⢸⠀⠀⠀⠀⣠⠗⠤⠤⠔⠃⠀⠉⢋⣉⡿⠠⣔⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠃⠀⠀⢀⡼⠁⠢⡀⠀⠀⠀⣀⠤⠊⣸⠇⠀⠀⠉⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠢⡤⠋⠔⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⡿⠀⠀⠀⠀⠀⢠⢞⡥⡶⠈⡭⠬⣵⠂⠀⠀⠀⠀⠀⠈⠑⠲⣔⣛⣒⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠈⠋⢰⠐⢃⠇⠀⠀⠀⠀⠀⡔⠢⡀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠓⠉⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢀⣴⠟⡽⠀⠀⠀⠀⠀⠀⣸⢣⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠛⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠐⠛⠓⠒⠃⠀⠀⠀⠀⠀⠀⠛⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠓⠒⠦⣤⠀⠀⠀⠀⠀⡼⠛⠒⠲⠤
⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⡀⠀⠀⠀⠀⣤⠤⢤⡏⠀⠀⠀⢀⡏⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸
⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠇⠀⠀⠀⢸⠇⠀⢰⠃⠀⠀⠀⠸⠥⣤⣤⣄⣀⣸⠀⠀⠀⠀⣾
⣼⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣼⡇⠀⡜⠀⠀⠀⠀⡿⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢠⠇⠀⠀⠀⢸⠇⠀⢀⡇⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⡿⠀⠀⢸⠁⠀⠀⠀⣼⠉⠉⠉⠉⠉⡿⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢠⠃⠀⠀⠀⢸⠃⠀⠀⡟⠀⠀⠀⢀⡏⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢼⣀⣀⡀⠀⡿⠀⠀⠐⢧⣤⣀⣀⣸⠁⠀⠀⠀⠀⠸⣤⣤⣀⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⣄⣀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣻⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⠀⣀⣤⣤⣄⣤⢤⣴⠶⠶⠶⣶⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⣤⣀⣤⠶⠶⠶⣤⡾⢋⣉⠙⢷⣟⡉⠙⢯⣤⣄⠈⣷⣦⠈⣿⢲⡀⢸⣷⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡏⣴⣶⣆⢹⣧⣾⡿⡆⢹⠓⠛⠛⡆⢸⠋⣹⠀⣼⣤⡞⢀⡷⢾⣠⠟⠹⠧⠞⠛⠞⠁⢻
⠀⠀⠀⠀⠀⠀⠀⣼⠿⠛⣛⣿⢠⣧⣤⣴⢃⡾⠶⠶⣾⣡⠞⠛⠻⠾⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢀⡏⣼⣿⠁⠙⠛⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⢀⣿⡸⣿⡇⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⣀⣤⣤⡀⣤⣤⣤⠀⣶⠛⠙⠳⠶⠋⠈⣳⡄⠀⠀⣿
⠀⠀⠀⠀⠀⣸⣯⣧⣿⡇⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⣿⠏⠹⠏⠈⣿⣿⠀⠸⣇⣙⠷⣤⠀⠀⣠⡾⠋⠀⠀⠀⢻
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢻⡀⢺⡟⢹⡆⠀⠀⠀⣸⢿⠀⠀⠉⢉⡇⣽⠀⠀⣿⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢰⣿⣿⣼⣿⡇⠀⠀⠀⠠⣶⠻⠾⠇⠈⡇⠈⢷⣀⣀⣠⠟⠘⠶⠶⠞⠛⠁⢻⣤⣤⠿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣾⢸⣿⢇⣿⡇⠀⠀⠀⠀⠹⣦⣀⣀⣰⠇⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⡏⣿⣿⢸⢸⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⣸⢃⣿⡏⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠰⠀⠀⠀⣷
⠀⠀⠀⣿⣸⣿⡇⡆⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⠶⠟
⠀⠀⠀⠈⠙⢿⣇⣧⠀⣇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣴⠶⠶⠶⠶⠚⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠤⠶⠶⠶⠖⠛⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠤⡤⠒⠒⠒⠒⠂⠐⡖⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠖⠂⢩⠶⠿⠙⠉⠀⠀⠘⠟⠛⠙⠉⠀⠒⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⢂⡀⡄⠈⠀⠤⠤⢤⡤⠤⠤⣤⢤⡀⡆⠈⠀⣉⠁⠐⠂⠳⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢁⡄⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠰⢀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣀⠀⠀⠀⠀⠀⠀⢠⠖⠀⡠⠀⠀⠀⠃⠈⠀⠉⠈⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠊⠀⠁⠀⠀⢀⠠⠓⠖⠁⡀⠉⠀⠀⠀⠀⠀⢀⡀⠤⢀⡤⠶⡶⢿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠊⠀⠀⠀⠀⢀⠄⠈⠀⣠⠞⠁⠀⠂⠀⠀⠀⠙⢦⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠃⠀⠀⠀⠀⡰⠁⠀⠀⡜⠁⠀⠀⠌⠀⠀⠀⠈⠀⠀⢣⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠂⠀⡀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠑⠂⠀⠘⠀⠀⠀⠰⠄⢀⣀⡀⣀⠀⠀⠀⡇⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠰⢷⡴⠻⡿⠿⠦⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠐⠦⠤⠀⣀⣀⠁⠀⠀⠈⠁⠐⠃⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢳⠼⣦⡀⣿⠉⢹⣿⠗⠀⠢⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠏⠘⣡⠆⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣈⠙⠉⠘⣿⣷⣼⡋⠈⠋⢀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀⠇⠀⡾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠰⠀⠙⠶⠶⠶⠛⠿⠿⠷⠦⢀⠀⠁⡀⠀⠀⣤⠀⠑⠀⠞⡈⠀⠀⠀⡀⠀⢘⡄⣪⠃⡜⠁⢙⠃⢆⠅
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢠⣌⣉⡀⢈⠀⠀⠈⠀⠀⠀⠀⠑⡄⢋⠜⢁⡞⠁⡠⢞⡴⠁
⠀⠀⠀⠀⠀⠀⢀⠂⠐⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠁⠀⠀⠀⠀⠐⠶⣀⠀⢞⣐⡭⠞⢟⣤⠞⠁⠘
⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⠂⠀⠀⠀⠀⡀⠠⢼⢝⠳⢯⡐⠦⣞⣪⠦
⠀⠀⠀⠀⠀⠀⠐⢀⡃⠐⠀⢲⡎⠉⠉⣶⡎⠉⠙⣦⣀⣤⣤⣤⣤⣤⠧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡬⠑⠂⠈⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠙⠒⠊⠉⠙⠒⠺⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠀⠀⠀⠀⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⢀⣦⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⠀⠀⠀⢀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⡄⢀⠀⣶⡧⠀⠀⢀⣿⣿⠟⢳⣄
⠀⠀⠀⠀⠀⠀⠐⡾⢁⠀⠀⠀⠀⠀⠀⢰⢋⠉⠀⠀⠀⢰⡀⡀⠀⠀⣤⠏⠓⡂⠀⠀⠀⠠⣗⢚⡄⢉⠃⡀⠐⢩⣿⣷⡖⠁⠛⢃
⠀⠀⠀⠀⠀⠀⢀⡷⢻⠀⠀⠀⠀⠀⣰⠿⠟⠀⠀⠀⠀⠤⠓⠃⡀⠀⢠⡄⠘⠟⢀⡄⠀⠀⣻⢼⡃⠀⠰⠀⡘⣿⡟⠉⠀⣀⡴⣿⡀⢀⡄
⠀⠀⠀⢀⢀⣰⣿⣃⣀⠁⢠⡄⠀⢀⣁⡄⠀⠀⠐⢶⠀⢼⠃⣿⠁⣶⠈⠁⢀⣀⠀⢳⡴⠞⠓⠐⠛⠦⣄⠀⠈⠛⣡⡶⠛⠁⠠⠸⠃⢀⡧⡀
⠀⠀⡰⠀⢼⣿⡿⢅⣠⡀⡨⢦⡀⠸⡿⠃⣀⡀⣿⡿⠀⣾⠶⣿⠀⠉⠠⣦⣾⡟⢲⣿⣤⣤⣤⡤⠤⠶⠞⣛⡀⢞⠙⡾⠀⣠⠶⠁⢀⣼⠄⢷⡀
⠀⠀⢧⠸⣿⣀⢶⡎⠀⢈⠃⠄⡇⡆⠈⠐⠿⠷⠈⠁⠸⠁⠀⠸⣧⠀⠠⠃⠘⠃⣾⣤⣤⣠⣤⣀⣤⢠⡶⣿⡧⠈⢢⡿⠋⢀⠄⠀⢀⣅⠀⢠⣅
⠀⠀⠘⡇⣾⣽⣔⢑⣄⣡⢈⣤⡠⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢻⣄⠀⠀⠀⠀⠹⣿⣼⡷⣦⡿⠴⠌⠿⣾⠃⠀⠀⠁⢴⠉⠀⠀⣸⣄⣀⣀⣸⣆
⠀⠀⠀⠈⠻⢿⣿⠿⠤⠀⠉⠞⠁⠀⠀⢀⣼⣤⣀⣀⡄⠀⠀⠀⢸⡟⢀⠀⠀⢀⠄⠙⠳⣶⠶⠶⠖⣲⡞⠋⠀⠀⠀⠀⠀⠀⣀⣼⣯⣥⣤⣤⣤⣽⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⠉⢡⡀⠀⠀⣼⠁⠀⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠟⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠋⠙⠒⠀⢿⡀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢸⢐⠃⠀⢀⠠⡀⠄⡀⡀⠀⢠⠀⠀⡄⡄⡀⢀⠀⡄⠠⢀⣠⣀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠰⠸⠁⠘⠀⠃⠘⢘⠁⠀⠠⠀⠀⠁⠀⠁⠈⠀⠅⠆⠘⠈⠀⠐⠸⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢫⣾⢢⣤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢲⣀⣀⠠⠯⠭⡦⠤⢤⠀⠀⠀⠀⠰⠿⠥⡿⢡⡿⠁⠀⡿⠻⣶⣶⣦⡄⠠⢤⣤⣤⡶⣶⠒⢻⠟⢿⡋⣱⡀
⠀⠀⠀⠀⠀⠀⠴⠒⡏⠁⠸⠇⠀⠀⣀⣘⣃⠤⢼⡀⠔⠀⠀⠀⠀⠀⠸⡿⠁⠀⢸⣷⣾⣿⣤⣾⠀⠀⢸⣇⣿⡇⣿⠀⠘⣷⣾⠟⠿⣧
⠀⠀⠀⠀⠀⠀⢠⠴⡇⠰⠀⠈⠉⠁⠀⠀⠀⠀⠀⢇⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠈⠻⡟⠀⠀⢸⡷⠟⠧⣿⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⢸⡌⣧⠀⠄⠀⢠⠀⢀⡆⠰⠾⣆⠘⡄⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢸⠠⢹⡀⢄⠢⡈⠲⠐⠂⠀⠀⠈⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡦
⠀⠀⠀⠀⠀⠀⢸⡇⡌⢧⠀⠀⣀⣀⣀⡤⠤⠤⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢿⣕⣝⡿⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡚⠻⣶⣀⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⡉⡻⣿⣆⡿⠋⠀⠀⠀⠀⠀⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠁⠈⠓⠦⡿⠋⠀⠀⠀⠀⠀⢴⣛⣻⣿⠇
⠀⠀⠀⠀⡀⠀⢀⣤⡶⠖⣰⠿⠻⣿⡿⠻⢿⡄⠀⢀⠠⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣉⣻⠟
⠀⠀⠀⠀⠀⢰⣷⣶⣶⡿⣿⣧⣼⡿⣿⣤⣿⣿⡐⢿⣷⣤⣉⡛⢧⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠈⠀⠁⠀⠀⠀⠀⢀⣴⡿⠿⣄⣽⠃
⠀⠀⠀⠀⠀⣿⣤⣤⣤⣴⣿⣿⣿⣄⣰⣿⣿⣿⣷⣦⣉⠛⠿⠿⣿⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⠳⢶⣿⠃
⠀⠀⠀⠀⠀⠸⡟⠿⠛⣋⣽⣆⣠⣿⣷⣀⣾⡏⢎⡛⢿⣿⣷⡶⠃⠀⠀⠀⠀⠀⡜⢀⠀⡠⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠐⠌⠻⠿⣿⡛⠙⢿⣿⣿⡿⠋⠀⠈⢿⡶⠖⠋⠀⠀⠀⠀⠀⠀⠘⠠⠷⠮⠊⠁⠀⠀⠀⠀⠀⠀⢀⡟⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡽⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠜⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡠⠁⠰⠐⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡌⣠⡤⠴⠂⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠠

⠀⠀⠀⣠⡄⠀⠀⣀⡀⠀⠀⣀⣠⢄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠘⡻⡧⠤⠞⠟⠋⠀⡌⠈⣿⡿⣦⢬⡆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⣶⣄⣀⣀⠀⠀⠀⠠⠀⠀⠀⠀⠄⡀
⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠂⡈⠨⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⡉⡉⣱⠀⠉⠉⠁⡑⠦⠀⡀⠄⠁
⠀⠀⠀⠀⡠⠀⠠⠄⠀⠀⠀⠄⠁⠀⡁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢜⡾⠀⠀⠀⠀⠎⠀⠀⠑
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⢰⠁⠄⣸⠀⠀⠈⠑⠢⢄⣆⠀⠀⢀⠀⠀⠀⠉⠁⢀⡀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠐⣆⠀⠀⡰⠃⠀⠀⠀⠀⡇⠀⢠⣇⠀⡆⠀⠂⢀⠀⡏⢉⢉⣀⠀⣀⣀⠀⡀⠈⠁⠀⠀⠀⠲⢴⣤⣿⣶⣤⠶
⠀⣀⠎⠀⢀⠀⠀⠀⢠⡄⠀⠀⠀⠃⠀⠀⡏⠆⠉⡆⠇⢣⠐⠇⠸⠸⠹⠰⠇⠃⠈⣤⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛
⠀⠀⠀⠀⠊⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⠀⢈⠒⠦⣄⣐⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠨⣤⡦⠄
⠀⠀⠔⠁⠀⡠⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⠄⢼⢀⢀⠄⡏⢩⠍⡇⠀⡀⡀⠀⠀⡔⠀⠀⠀⠀⠀⠶⠀⠀⡮⠏⣑⠚⣦
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⠀⠸⠘⠀⠇⠃⠃⠀⠀⣕⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡗⠂⠶⢠⡏
⠀⢀⠀⠐⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠎
⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠊⠉⠀⠶⠀⠀⠀⠘⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⢰
⠀⠀⢱⠀⠀⠴⠀⠀⠀⠀⠭⠄⠀⠀⠀⠀⠀⠀⢠⠀⣄⠀⠀⠀⠀⠀⠀⣢⣤⠖⠛⠃⠰⠀⡠⠔⠐⠂⠉⠀⠀⠀⡄
⠀⠂⠈⢸⠀⠂⠀⠀⠀⣾⠭⢭⣷⣶⠀⠀⠠⣻⢿⣿⡗⢶⣄⠤⠀⠚⡋⠉⠀⠂⠀⠀⡐⠀⠣⠤⠤⠀⣀⣀⣀⣠⠃
⠀⠀⠀⠀⠱⡄⠀⠀⠀⠦⠀⠀⠀⠻⠀⠀⠀⠀⠈⠰⠉⠀⠀⠀⠀⠐⠒⠂⣤⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣠⠋⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢱⠀⠀⠀⢀⡞⠁⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠳⣄⠀⠀⠀⣀⣀⣹⡉⠉⠁⠒⠁⠀⠀⠸⡄⠀⡰⠋⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡴⠛⠛⢻⡏⣭⠍⡄⢠⠀⠱⡄⢀⣀⡀⣀⣀⠀⠈⠙⠳⢤⡀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡸⠀⢀⠞⠀⠀⠀⢸⡇⠸⠀⠟⢹⠀⢠⠇⢺⣀⡇⣿⠂⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠻⣄⡀
⠀⠀⠀⠀⣀⠴⠚⠉⠀⣰⠋⠀⣠⠀⠀⢸⡇⠀⣶⠒⠒⠒⠛⠒⡖⠛⢲⡆⣰⠒⢻⡖⠒⢲⠀⢶⠒⠒⢦⠀⣰⠚⢛⡿⠒⠦⢤⣀⡀
⢀⡤⠒⠉⠀⠀⠀⢀⡼⠁⠀⣰⣿⠀⠀⢸⣇⣀⠛⠲⡆⠀⢰⠖⡇⠀⢸⡧⣼⠀⢸⡇⠀⢸⠀⠈⢣⡀⠈⠻⠁⢀⡞⠀⠀⠀⠀⠀⠉⠛⣲
⠀⠉⠒⠤⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡇⠀⢸⠀⡇⠀⢸⡇⣺⠀⢸⡇⠀⢸⠀⠀⠀⢳⡀⠀⢀⡞⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠉⠓⠾⢖⠒⠒⠒⢲⠀⠀⢰⡖⠛⡤⠤⠇⠀⢸⢧⣧⠀⠘⠧⠞⠀⢸⡇⠀⠸⠶⠶⠶⡄⡇⠀⢸⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠸⠤⠤⠼⠃⠀⢧⣄⣤⠴⠋⠀⠻⡲⢤⣄⣤⠴⠋⣧⣤⣤⣤⣤⣤⠇⣧⣤⣼⢠⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⡰⠋⠉⠉⠑⠒⠒⠒⢠⡤⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣀⣠⡤⠴⠒⠋⠉⠁⠀⠈⠳⢄⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠈⠈⠐⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⢤⡀⠀⢀⠔
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠸⠠⠀⠀⢠⡄⠀⢀⠀⠀⠀⠀⠀⠀⠈⠐⡒⠀⠀⠀⠀⠀⠀⢄⠠⡜⢣⡜⣤⠀⢠⡄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠐⠆⠰⠀⠀⠀⠀⠀⠀⠰⠄⠄⣤⠢⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣄⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⡄⠀⠀⣀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠁⠒⠀⡠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠛⡄
⠀⠀⠀⠀⢀⠔⣁⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⢠⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠈⠢
⠀⠀⠀⠠⡁⠐⠉⠀⡀⠀⠀⠀⠈⢀⠄⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀⢀⡀⢤
⠀⠀⠀⢀⠐⢀⣤⠐⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⡀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠠⡃⠀⠂⠀⠁⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⢈⠶⠀⣄⠈⠫⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⡀⠀⠀⠀⠀⠀⢀⠠⠈
⠀⠀⠀⠁⠀⠈⡅⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠣⣸⣎⠐
⠀⠀⠑⠰⠀⢀⣀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠊⢀⠀⠀⠀⠀⠀⠠⢄⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠀⠀⠀⠆⠀⠀⠀⠀⢐⣀⣠⡊⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠈⠙⠉⠀⠁⠘
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠒⠄⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⣶⣿⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠘⠠⠄⢀⣀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠁⠁⠠⣀⠁⠀⠀⣀⣦⡀⠀⠀⠀⠀⠘⠤⠃⠀⡄⢀
⠀⠀⠀⠈⡄⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⢀⠆⠀⠀⠈⠀⠀⡆⠀⠀⠀⠀⠀⠔⠁⡃⠘⠂⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠻⠩⠦⢀⣀
⠀⠀⠀⠀⠘⠉⠉⠁⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠐⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⡀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡀⠗⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠤⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠁⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠐⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⣀⣀⠃
⠀⠀⠀⠀⠀⠢⣤⠀⠀⢀⡀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⡀⠁