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⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠈⠍⠂⠀⠀⠀⠀⠀⢀⠤⠂⠉⠉⠉⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠤⠒⠒⠂⠒⠒⠢⢄
⠀⢠⠄⢀⡂⠬⠐⠀⠉⠀⠀⡀⠈⠠⠀⠀⠀⠀⠠⠁⡠⠰⠎⠉⣉⢦⣠⠌⡄⠀⠀⠀⡀⠀⢀⡀⠠⠄⠐⠀⠉⠑⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠂⠁⠀⠀⠀⠀⢉⣿⣾⣥⡄⠄⠆⠀⠀⠀⠀⠀⡀⢀⠜⠀⠱⠃⠋⡁⢰⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠀⠀⠀⠀⠀⠉⠟⢻⡀⠀⠰⠘⡀⠀⠀⠆⡀⠰⣁⠀⠀⠀⡱⡄⠇⠄⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⣀⠀⢤⠚⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠇⠃⠀⠀⢀⡀⠤⠽⠳⠀⠀⢃⢁⠀⠀⢸⣡⢆⠀⠉⠀⠈⠀⢠⠘⠀⠀⠀⠀⠃⢀⡀⠀⠤⠉⠀⠀⠀⠈⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠸⠈⡄⠊⠁⢀⡀⠠⠀⠐⠂⣉⠰⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡇⠑⠒⠉⠀⠀⠠⠔⠂⠉⠁⠀⢀⠤⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠒⠲⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠞
⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⡔⠒⠈⠉⠁⠀⠀⠀⠀⠀⡨⠂⠀⠀⢠⠖⢉⠇⠀⠀⠀⠀⣰⠖⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⣠⡤⠤⠄⠒⠊⠁⠀⠀⢰⠃⠀⠈⠲⡄⠀⠀⠘⣉⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠭⠀⠛⠘⠓⠈⠉⢡⠀⠀⠀⠀⠟⡄⠀⠀⠀⣇⠀⠀⠐⢓⣙⢒⡷⠀⢺⠀⠀⡔⠤⠄⠀⠀⠒⠒⢐
⠀⠀⠀⠀⠀⠀⠀⠔⠉⠈⠑⢄⠀⠰⡁⠀⠀⠀⢀⣀⣠⠤⠰⠀⠀⠀⠀⠂⣹⢾⣭⣭⡁⠀⠀⠘⣇⠀⠀⠀⡠⢿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠈⡀⠀⠀⠁⠈⠉⡋⠀⠄⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⠀⠀⠀⠀⠑⣆⡀⢰⢡⠂⠀⠀⣇⣀⡀⠀⠀⠀⢀⣸⡀
⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢱⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠘⡀⠀⢀⣀⠀⠀⣠⠗⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠘⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠓⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⡠⠖⠛⢉⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⣀⡀
⠀⠀⠀⠀⢡⢀⡤⠶⠂⡡⣀⡴⡊⠁⠀⠐⡀⠀⠀⠀⡠⠶⠀⠐⠃⠄⢀⣰⠀⠀⠀⠀⠀⠀⡠⣀⠤⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⢸⠻⣓⠴⢾⣿⣿⣿⣦⠀⠀⠀⢣⠀⠐⠊⠂⠈⠀⠰⠦⢀⣄⠀⠉⠀⠒⠂⠀⠠⠖⠁⡠⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⢏⠀⠀⠐⣝⣛⡿⠟⢀⡠⠐⠁⠀⠐⠢⠤⣀⠀⠔⠒⠒⠠⡀⠛⠀⠓⠀⠀⠀⣰⡉⡄⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠈⢢⡔⠔⠊⡁⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⠀⠀⢀⠧⠀⠐⠐⠒⠒⠉⠁⠈⠁⠀⠀⠀⠀⠀⠈⠂⠤⠤⠤⠤⠤⠤⠤⠤⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⢐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠤⠤⠤⠬⢤⠄⡁
⠀⠀⠀⠀⠀⠀⠀⣇⠤⠔⣀⠔⠐⠀⢲⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⡀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠠⠤⢀⡀⠀⠀⠑⢄⠀⠀⢀⠔⠋⠊
⠀⠀⢠⠀⠀⡠⠁⠉⠢⣀⠋⠀⠀⠀⠈⠀⠀⠀⣠⡄⠼⠏⠁⠀⠚⠁⠉⠉⠉⢆⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠑⠐⠁
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠴⠦⠵⠶
⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⠀⡀⠔⠁⠉⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠇⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠈⠐⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⠀⠠⠤⠄⠐⠒⠀⠁⠁⠀⠀⠀⠀⠙⠛⠳⠶⣶⣄⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⡀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⡿⠘⠀⠀⢻⣿⣿⣿⣿⣿⠿⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣔⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⣠⡠⠀⠠⢤⠴⠄⠀⠀⢀⣀⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⢄
⡀⠈⠛⠙⠚⣰⢊⡋⡛⠟⠧⠿⠎⠉⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠁⠀⠤⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠈⠁⠐⠒⠒⠚⠃⠀⠉⠈⠀⠰⠷⠿⢕⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢄⠀⠀⠀⠸⣉⡀⠂⠄⢠⣢⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠤⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠂⠀⠀⢸⡹⣆⡄⠄⢀⠀⠀⠀⠀⡀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠀⠳⠈⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠘⡩⡺⢠⠀⠀⠀⠀⡀⠀⠐⢀⣁⠀⠀⠸
⠀⠄⠀⠠⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠐⠒⠀⠂⠀⠁⣀⡀⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠑⠁⠀⠀⠀⡀⠀⠠⠀⠀⠉⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠠⠤⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠁⠈⠀⡖⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠇⠄⠀⠀⠀⠀⠀⢦⡰⣦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠂⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⡀⣀⣀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⣰⣀⣠⠤⠤⠠⠯⠿⠯⠿⠗⠀⠀⠀⡇⠀⠉⠉⠉⠛⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤
⠀⢀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢁⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢤⠀⠀⠀⠀⠉⠉⠀⠀⠀⠐⠒⠀⠄⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⡀⡐⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⣀⡀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢻⡟⠉⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣠⣀⠀⠀⣿⢿⡇⠀⠀⠠⠤⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣼⣯⣤⣿⡄⠀⠀⠀⠀⣾⣿⣿⣇⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢳⣶⣦⣀⠀⠀⣀⣤⡀⠀⣰⡿⠛⡙⠛⣷⡄⢀⣴⣾⣿⣿⣿⡋⣴⣿⠋⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠘⣿⣿⣿⡶⣟⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣿⣿⣿⣿⣅⣶⣿⣿⡇⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⠀⠀⠀⣸⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡿⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡏⠀⠀⢀⡀⡄⣆⢳⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠀⠀⢀⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⢺⡜⡎⣧⢳⢹⡜⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠧⠼⠘⠈⠁⠀⠀⣶⡄
⠀⠀⠀⠀⢀⣤⣤⣴⡖⠒⣶⣶⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣤⣤⡀⠀⠀⠀⣀⠉
⠀⠀⠀⠀⠀⠈⠙⠛⠷⠤⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢽⣽⣧⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣾⣿⣿⠀⠀⢸⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡷⠿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠚⠋⠁⠀⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡄⠀⢱⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⠴⠦⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢳⣀⣀⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠛⠙⠯⣭⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠋⠍⠅⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣿⣿⠿⡀⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⣴⣿⡿⡟⠻⠆⣽⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣼⡿⠋⡀⠀⢠⣼⡿⠃⢸⣿⣿⣿⣿⡏⠉⢻⡟⠀⢹⣿⡄⠁⠉⠻⠿⡉⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠙⠀⣸⡿⠁⠀⢸⣿⣿⣼⣯⡇⠀⠘⡁⠀⢀⣽⣿⠖⠒⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⢸⣿⠏⢹⣿⠇⠀⢠⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠀⠈⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⠂

⠀⠀⠀⡋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣧⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠄⠀⠀⠀⠀⠀⠠⠡⠊⠀⠄⠀⠑⠀⠀⠀⠀⠀⢀⠤⠐⡶⢐⡟⠉⠓⣄
⠀⠀⠀⠁⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠇⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠈⠢⣌⠥⣝⡭⢯⠓⢺
⠀⠀⠅⠌⣉⠫⠀⠀⠀⠀⠀⠀⠀⠀⠠⡩⢀⠘⣦⠈⡱⠀⠀⠀⠀⠀⠈⢌⠠⠀⠄⠠⢈⠔⠀⠀⠀⠀⠀⠀⠀⠓⢥⠂⠋⠁⠁
⠀⠀⠐⠒⠛⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢰⣶⠠⠐⣀⠐⢀⠠⢢⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠢⡄⠀⠀⠀⠀⠀⢀⣤⣄⠁⠀⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡃
⠀⡎⢀⡀⠈⡕⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⢔⡛⣆⠀⠀⠀⠀⠀⠀⠀⠿⠝⠠⡚⠙⢲⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠅⣈⠀⠅
⠐⡆⠢⡙⠚⠟⠡⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣠⠺⣠⣀⠇⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶


⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠠⠐⠤⠖⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠁⠀⠁⠒⠫⡭⠂⠁⠀⠀⠀⠀⠀⠠⢀⠀⠄⠀⢁⣔⡤⠀⠀⠀⠀⠀⡁⠀⡀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⣲⣄⠀⠶⠶⠖⠇
⢀⠀⠀⠀⠀⠀⡏⠀⡀⠀⠀⠀⠀⢐⣂⠀⡄⠀⠰⠍⡀⠀⠀⠀⠀⠀⠀⡈⠀⠓⠀⡀⠈⠀⠀⠀⠀⠀⠀⢀⢀⡁⠀⢀⣀⠀⠂
⠀⠀⠐⠐⠦⠄⠀⠀⠀⠀⠀⠀⠀⠻⠩⠈⠒⠒⠒⠒⠈⠀⠀⠀⠀⠀⠀⠙⠠⠀⠤⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢀
⠀⣠⠄⣀⣐⠠⡌⠀⢢⠀⠀⠀⠀⠀⠀⡀⠠⠀⠐⠶⣶⠀⠀⠀⠀⠀⢀⠴⠶⠲⡊⠨⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⣠⠔⠀⠀⠁⠀⠀⠀⠀⠀⠈⣿⠈⠁⠃⡀⠉⠀⠀⠀⠀⠀⠀⠀⠒⠀⠓⠲⠒⢄⠀⠀⠀⠀⠀⠀⠈⠀⠛⠁⠈⠆⡄
⠁⠀⠀⠐⠁⠚⠁⡀⡀⠀⠀⠀⠀⠀⠸⣟⢲⡶⠦⢬⡽⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠠⢲⣀⠀⢀⡐⠋
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⠛⠟⠲⠺⠟⠅⠀⠀⠀⠀⠀⠀⠀⣄⣣⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠓⠋⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⣰⡖⢥⣃⡂⠄⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣂⡁⢀⣴⠾⣧⣼⢟⠘⢍⠀⠉⢳⢌⠢⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠊⠀⠈⠀⠉⠑⣊⡟⡄⣡⢀⠀⢀⡼⠀⠑⠌⢂⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠂⠡⠀⠀⠀⠀⠘⢄⡌⠙⢂⠀⠒⠁⠀⠀⠀⢈⡆⢃⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠈⡀⠸⠙⠂⠀⠀⠀⠀⠘⡷⢛⡀⡀⢀
⠀⠀⠀⠀⠀⠠⠂⣀⣀⣐⠂⠄⠀⠐⣄⠀⢀⠤⢤⠘⠀⠀⠀⠀⠀⠀⠀⠀⡘⠒⠚⠇⠁
⠀⠀⠀⢀⠤⠤⠤⠤⠤⠬⠧⠦⢤⡀⠈⠑⠴⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠃⠆⢠
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠀⠀⠀⠀⠙⠉⠁⠁⠢⠄⡀⠀⠀⠀⠈⠀⡙⠠⠀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢁⢠⡄⠀⠀⠀⢠⠀⡄⠀⡀⠀⠀⠀⠈⠢⡀⠀⠀⠒⠚⠓⠒⠒⠒⠒⠗⠛⠉⠉⢝⡭⢭⣍⢂⠀⠀⠀⠀⡰
⠀⠀⠸⠀⠀⠸⠇⠀⠀⠀⢹⠋⠀⠀⡄⠀⠀⠀⠀⠀⡇⠀⡠⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⣤⠂⣿⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠄⡀⠀⠀⢃⡈⠀⢀⣘⠂⠄⠀⢀⡀⢸⠄⠄⠊⢂⠈⠉⠁⠰⠀⠀⠀⠀⠠⠂
⡔⠐⡇⠀⠊⠀⠀⠀⠀⠀⠀⠀⠱⠀⠘⠉⢳⠂⠀⠈⠉⠁⠀⠐⠊⠒⠀⠈⠁⠀⠒⡀⠀⠀⠀⠣⠀⠀⡄⠀⢀⡀⠤⠐⡞
⡯⢁⠀⢠⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠍⠁⠀⠒⠀⠀⠒⠀⠉⡀⠀⠀⠀⠀⠀⠛⠀⠀⢀⢔⣑⡀⠃⠀⠀⠀⠀⠀⠐
⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠇⢈⢀⠀⣄⠀⠀⣠⢀⠈⢠⠀⠀⠀⠀⠀⠘⠈⡠⠢⠈⢹⡟⢻⠀⠀⠀⠀⠀⠰
⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠠⠀⠀⠀⠀⠀⠀⠃⡀⣄⣀⣀⣠⠀⠐⣋⢀⣀⣀⣤⠤⠼⠤⠤⠤⠤⠤⠴
⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢬⠉⠙⠒⠀⠀⠐⡆⠀⠐⠊⠩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣼⠀⢀⣀⣀⠀⠀⢇⣀⣀⣀⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠒⠊⠉⢀⠚⠛⠉⠀⠀⠀⡈⠋⠋⠉⠉⠈⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⠀⢄⣀⠀⠀⢠⡉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠨⢆⠈⠀⠀⡸⠐⠒⠒⠒⠒⠒⠲⡀⠄⠀⠀⠄⠀⠉⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⢰⡐⠄⠀⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠒⠒⠈⣀⠠⠘⠋⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠔⠈⠀⠉⡠⠂⠀⠀⠀⠀⠀⠠⠐⠀⡡⠔⠋⠙⠉⠻⠲⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠔⠁⠀⡠⠊⠀⠀⠌⠀⠀⠀⠄⠈⢢⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠀⠀⢰⠁⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣠⠀⣤⣤⣶⣆⣀⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⡲⠄⢀⣀⡀⠉⠁⡒⠒⠒⠂⠤⠀⣈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡿⠧⡌⣯⠁⣹⠽⢿⣷⣧⠠⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠁⠀⠀⣕⠂⡤⢠⠄⠀⠤⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠴⠃⣿⣷⣸⡦⠀⣿⠁⠀⠀⠁⠐⠠⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠀⢀⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⢈⡀⡀⠘⣆⣀⣀⣽⣿⣿⣷⣄⠀⢈⠂⠄⡀⠀⠀⠈⢄⢀⡴⠃⠀⠀⠀⠀⠀⢆⠠⠺⢉⡗⢒⠤⣠⢲
⠀⠀⠀⠀⠀⠀⠀⠈⢠⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠠⠀⠀⠒⠁⠀⠀⠐⡄⠰⢊⡴⢁⡞⠀⣠⢆⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠁⠃⠈⠀⠀⠀⢠⣄⠀⢀⡜⠵⣋⣴⢏⣠⠞⠀⠏
⠀⠀⠀⠀⠀⠀⡘⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠄⠀⠀⠀⠀⠀⢈⣕⡶⢭⡋⢥⣠⠯⣣⡀
⠀⠀⠀⠀⠀⠀⠠⠰⠄⠤⠠⣤⠚⠓⢲⡴⠒⠒⢦⡀⣀⣀⣀⣀⣀⣇⠘⠁⠀⠀⠀⠀⠀⠀⠉⠐⠠⣈⡑⠪⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠙⠷⠔⠚⠛⠢⠴⠟⠉⠛⠃⠛⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠮⠀⠀⠀⠀⠑⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡠⠂⢤⡀⠀⠀⠀⠀⢀⡠⠔⠂⢉⠀⠀⠉⠑⢢⡀⠀⠀⠀⠀⠀⢀⠔⠊⠉⡁⠀⠡⡐⢀⠀⠀⢀⡀⠀⠀⢀⠔⠈⠣⠄⠠⠜⠁⠢⡀
⠀⠀⠣⡀⠀⠈⠐⠂⢀⠍⢠⠀⢁⠚⠥⠈⠀⠁⠈⠁⠀⠕⠀⠀⠀⢀⠃⠀⠀⠀⠁⠀⠀⠀⠑⢆⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⢀⠜⠂⣈⠕⠀⠈⠀⡡⠐⠒⠢⢄⠀⠀⠀⠀⡀⣀⠜⠁⠀⠀⢸⠠⠈⢈⡈⡀⠀⣠⠀⠔⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠘⠒⠫⠤⠄⠄⡤⠊⢀⠀⢀⡠⠚⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⢳⠀⠀⡆⠁⢀⠀⡘⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠮⠤⠴⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣢⡈⢲⡰⠁⠀⢘⠖⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠀⣀⡀⡠⠔⠢⢄⠀⠀⠀⠀⠁⠈⠀⠀⠈⠀⣀⠀⠀⠀⣀⠀⠀⠈⠒⣄⠀⠀⠀⠀⡤⠊
⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠇⢯⡱⢸⠒⠀⠀⠆⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠠⡤⠴⠎⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⢄⡠⠃⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⢀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠛⠁⠀⣠⠄⠒⠒⡤⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⠆⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠠⠁⣠⡀⡄⠀⢡⣧⠀⠀⠀⢨⠉⠉⡇
⠀⠀⠀⡀⠀⠐⠚⠀⠀⠐⠻⣄⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⢰⠃⠀⣙⡷⠃⠀⣿⠇⠀⠀⠀⠈⠳⠞
⠀⠀⢰⠁⠀⠀⠀⠀⠠⡄⢠⠀⠃⠀⠑⠐⠊⠉⠉⠐⠢⠤⠀⠤⠔⠊⠀⠀⢀⠀⠀⠀⠇⠀⠐⠇⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡈⠀⠀⠀⠀⠀⠀⠁⢸⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⢸⠂⣠⣀⡀⠀⠀⣼⠇⠀⠀⠀⠈⠁⠀⣀
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠲⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡈⠈⠁⢐⡏⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⡧⠀⠐⠒⠒⠀⢤⢄⠈⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠂⠚⠉⠢⣄⠀⠀⠀⠀⠀⠙⠶⠶⢥⣤⣤⡾⠁⠀⠀⠀⠀⡚⠓
⠀⠀⣇⠀⠀⠈⠀⠂⠈⣾⠘⠃⡅⠀⠀⠀⠀⠔⠁⠀⠀⠀⢀⣄⣴⠟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⡀⣠⣖⠠⠀⠐⠀⣷
⠀⠀⠉⡄⠀⠀⠀⠀⠀⠙⠐⡀⡇⠀⠀⠀⠊⠀⠀⠀⠀⢰⠃⠙⠋⠀⠀⠀⠀⠉⠀⢀⡀⠀⠀⢠⡄⠀⠀⠀⠀⠄⠀⢀⢸⣿⣿⡇⠀⠀⢸⡅
⠀⠀⠀⠲⠶⢤⠤⠤⡾⠡⡾⠐⠁⠀⠀⠀⠀⠀⠀⣀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠉⠈⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠙⠛⠶⠶⠶⢴⣄⡺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⣉⠀⢀⡂⠀⠀⠀⠁
⠀⠀⡀⠀⣀⠠⠖⠒⠊⠱⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠁⠀⠠⠔⠒⠂⡄⠀⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⣤⣀⣀⣀⡀⢈⠀⠀⠀⠀⠀⠀⠀⢀⢄⠤⠄⠀⠀⠄⠀⠀⠰⠂
⠀⠀⢸⡀⠀⠀⠒⠡⡐⠀⠰⡀⠀⠀⠈⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⡄⠕⠁⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠈⡃⠀⠐⠂⠚⠁⠀⠀⡁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠤⠀⠀⠀⠀⡈⠀⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡀⡀⠀⠀⠀⡧⡀
⠀⠀⠀⢣⠀⣤⠴⠖⠚⠀⠀⡁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣄⢣⣤⠇⠀⠀⠀⠀⠀⡀⠀⠁⠀⠰⡀⠀⠀⠀⠀⠂⠀⠒⠊⠁⠈⠒⢄
⠀⠀⠀⠈⠆⡀⠀⠠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠙⠋⠘⠻⠂⠀⠀⠀⠈⠛⠀⠀⠀⠀⠈⠀⠀⠉⠉⠐⠲⠀⡀⠤⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠂⠈⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠔⠒⠒⠒⠒⠦⣄⠕
⠀⠀⠀⠀⠀⣀⣀⢠⠴⠶⣒⡶⠶⣬⣱⡿⠖⢂⣰⣿⣦⠀⢀⣀⣤⣤⣿⣤⣤⡴⠦⣀⣒⠶⠦⠤⠤⠴⠤⠄⢄
⠀⠀⠀⠀⠀⡁⠀⢀⡠⠎⠉⠀⠘⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣼⠋⠀⢂⠀⠀⢀⠁⠆⡁⢸
⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠐⠀⠐⠃⠀⠀⠈⢄⠉⠁⠀⠀⠀⠀⠀⠀⠀⢔⠀⠀⢛⠀⠀⠀⠑⢤⠚⠁⢰⡇⢸
⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠈⠉⠉⡇⠀⠀⠠⠀⠀⠠⠉⠀⠄⠀⠀⠀⠀⡈⠁⠀⣿⠀⠀⠀⠀⠀⡑⠄⠜⢣⠸
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢠⡀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠑⠤⠊⠀⠀⠀⠀⠠⢰⡌⠁⢰⢸⠀⡇
⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠇⠆⢸⠘⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⡀⠀⠀⠀⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⠙⠂⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⢀⠀⠀⠵⡠⠀⡉⠏⠀⡇⢸
⠀⠀⠀⠀⠀⠧⠀⠀⠀⠠⡀⠀⠀⡇⠀⠀⠀⠀⠒⠂⠀⠀⠀⠐⠠⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠁⠛⠀⠀⠁⢸
⠀⠀⠀⠀⠀⡄⢈⠢⠐⠈⠁⠀⠀⢡⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⢂⢀⠀⠀⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠢⡇⢸
⠀⠀⠀⠀⠀⠧⠬⠤⣰⠤⠤⠄⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠀⠀⠀⡇⠀⢸⠂⠀⠀⠀⠀⠀⠀⢀⣧⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠓⠒⠒⠤⠤⢭⡤⠎⠁

⠀⠀⠀⠀⠀⠀⣤⠄⣀⣀⣀⣀⡀⣀⠀⣠⠄⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠠⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠀⠀⠛⠀⠳⣤⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⢸⠀⣿⢦⡾⣇⠀⣧⢀
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡆⢹⠀⠀⢻⡆⣿⠉
⠀⠀⠀⠀⠀⠐⠛⠗⠀⠀⠉⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠃⠘⠃⠀⠀⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠕⠈⠉⠉⠓⢌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡤⠤⡤⠤⡮⠀⠀⠀⠠⠀⠬⣮⣇⣤⡤⢄⡠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢀⠔⠁⢀⠇⡸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⣆⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⢮⠥⡽⠀⠀⠀⠀⠀⠀⠀⠀⠮⡅⡽⠂⠈⠢⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡾⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢀⠀⠀⠀⢿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠑⣄⢾⣼⡷⠀⠀⠀⠀⠀⠀⠀⠰⣞⣐⡳⠀⠠⠔⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠈⡎⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⣸⠀⡰⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠛⠛⠛⠒⠒⠒⠂⠀⠀⠀⠐⠓⠓⠒⠓⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠉⣿⠙⢱⡎⢹⡆⠀⠀⠀⠀⠐⠒⡶⠒⠂⢶⢲⠀⠀⢀⣶⡀⢰⡀⢰⡇⢺⠉⠀⣴
⠀⠀⠀⠀⠀⠀⢿⠀⠸⣇⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡿⣄⠀⢼⠗⢷⠀⢳⣼⠁⢸⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠘⠃⠈⠀⠛⠀⠈⠀⠀⠋⠀⠘⠋⠁⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠳⣆⢤⣠⠤⢔⢂⣰⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠠⠀⡀⢀⠠⣄⣠⣤⣿⡚⡟⢁⡰⡎⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⡡⠒⠉⢈⠹⠎⠉⠀⠀⠙⢾⣥⣹⡀
⠀⠀⠀⠀⡀⠠⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⢢⣴⠒⠒⠿⠋⠑⠄⠂⠙⣿⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠤⠄⢢⡀⠇⠀⡰⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠉⡆
⠀⠀⠀⡀⢠⠔⠚⠒⠒⠒⠒⠻⠟⠀⠘⠷⠯⠤⢒⠷⢾⣶⣦⠢⣑⡀⠀⠴⠀⠐⠄⠀⠀⠈⣴⡠
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠄⠀⢈⠏⠀⠀⠀⠀⠀⠐⣄⠀⡔⠜⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⢸⡄⠀⠀⠐⠒⠖⠙⠐⡥⠤⣿
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢸⠄⠀⠀⠀⠀⠀⠀⢆⠸⡐⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠸⠀⡀⢀⢀⠀⢀⠠⢸⢐⠌
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⣷⠶⠤⠤⠬⠠⠄⣊⡜⠋
⠀⠀⠀⠿⠦⢤⠤⣄⣀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⢀⢒⠂⢄⣔⣵⣷⣭⠨⠤⠴⠚⠉⠀⣀⣠⡠⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠙⠃⠒⠒⠻⠚⠛⠋⠋⠁⠀⠠⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠈⠈⠉

⠀⠀⠀⠀⠀⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡤⠴⠖⠒⠦⢤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣁⣠⠤⠦⠤⣤⣀⣙⡆⠪⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⢔⢄⣠⣤⣴⡶⡟⣷⠾⣛⣽⡿⠶⢖⣿⣷⠲⠶⢶⣶⣾⣿⣬⡶⠮⣁⣾⣶⠶⠶⠴⠶⡶⢦⡀⢠
⠀⠀⠀⠀⠀⡁⣿⣀⠀⡠⠜⠉⠀⢸⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢰⠏⠁⢂⠀⠈⠉⢐⣐⠀⣷⢀
⠀⠀⠀⠀⠀⠀⣗⢠⠋⠀⠀⢐⣀⣘⡃⠀⠀⠡⣈⠁⠀⠀⠀⠀⠀⠀⠀⣂⣀⣘⡇⠀⠀⠑⢄⠊⠈⣿⠐⡏⢸
⠀⠀⠀⠀⠀⠃⣼⠁⠀⠀⠀⠈⠀⢸⡇⠀⠀⠂⠀⠀⢊⠀⠀⠀⠀⠀⠠⠁⠀⡸⠄⠀⠀⠀⠀⣑⠰⠹⠀⣿⢸
⠀⠀⠀⠀⠀⠄⣿⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢱⠈⠀⠀⠀⠀⠀⠼⠄⠀⠀⠀⣿⢨
⠀⠀⠀⠀⠀⡆⡇⠀⠈⠶⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⡇⣧⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⣿⢨
⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⢑⡀⢀⠀⢤⠂⠀⠀⣿⢸
⠀⠀⠀⠀⠀⡂⢻⠄⠀⠀⠠⠀⠀⠀⡄⠀⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠂⠀⠈⠑⠲⠆⠀⠀⢀⡇⢸
⠀⠀⠀⠀⠀⡃⣧⠀⠢⡀⠐⠃⠀⠀⡇⠀⠀⠀⠀⢀⡀⠀⢀⠀⠄⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⠈⣷⢸
⠀⠀⠀⠀⠀⠃⢿⣐⣡⣈⣦⣤⣄⣀⣧⣀⡀⠀⠀⠀⠀⠀⠚⠃⠈⠁⠀⢸⠀⠀⠆⠀⠀⠀⠀⢀⠀⣹⣤⡟⢸
⠀⠀⠀⠀⠀⠠⠲⠠⠄⠘⠐⠀⠉⠘⠅⠐⠉⠉⠉⠙⠛⠒⠒⠒⠒⠒⠒⠺⠦⠴⠗⠒⠶⣤⣤⣥⡼⠋⠁⠖⢸
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠒⠒⠀⠒⠒

⠀⠀⠀⠀⠀⢠⢂⡐⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢈⣌⡠⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠎⡀⢀⢔⠜⠀⠀⠀⠀⠀⡰⠃⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠃⠠⣀⡡⠊⠀⠀⠀⠀⠀⠠⠀⠡⠊⠠⠂⠀⠀⠀⠀⠈⠢⡡⠠⠀⠀⠀⠀⠀⠀⠀⠢⠤⠘⠀⠤⠂


⠀⠀⠀⠀⠀⢀⡴⠶⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢄⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠧⠀⠆⠠⠅⠀⠀⠀⠀⠀⠰⠌⠉⠉⠉⠁⠀⠀⠀⠀⠈⠂⡈⠐⢡⠂⠀⠀⠀⠀⠀⠀⠃⠤⠤⠃
⠀⠀⠀⠀⢐⡄⢀⡄⠀⡈⠀⠀⠀⠀⠀⢠⣡⣕⣢⡖⠀⠀⠀⠀⠀⠠⢔⠁⠒⡈⠄⠀⠀⠀⠀⠀⠀⣢⠤⢤⡄
⠀⠀⠀⠀⠈⠛⠋⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉

⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠇⣶⢲⡄⠃⠀⠀⠀⠀⠀⣾⡆⠀⠀⣿⠆⠀⠀⠀⠀⠀⢗⢄⠠⢪⢆⠀⠀⠀⠀⠀⠀⠀⠛⠘⢆
⠀⠀⠀⠀⠀⡇⠙⠚⠁⠀⠀⠀⠀⠀⠀⠈⠑⢲⠐⠙⠀⠀⠀⠀⠀⠀⢀⢕⠕⠈⠁⠀⠀⠀⠀⠀⢖⠁⡲⠀⠛⢄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠋⠈⠈⠋⠀⠀⠀⠀⠀⠀⠓⠀⠀⠒⠁⠀⠀⠀⠀⠀⠐⠁⠃⠒⠂⠁


⠀⠀⠀⠀⠀⠀⠒⠘⠢⠀⠀⠀⠀⠀⠀⠒⠀⣀⣠⡀⠀⠀⠀⠀⠀⠠⠽⢄⠀⡸⣀⠀⠀⠀⠀⠀⠠⣄⠓⠚⡦⠄
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠸⠸⠀⢸⠀⠀⠀⠀⠀⠀⠚⡐⠆⠘
⠀⠀⠀⠀⠀⠠⡁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠲⠃⠐⠳⠇⠀⠀⠀⠀⠀⣤⠀⠈⢰⠊⠀⠀⠀⠀⠀⠤⠚⠘⠄

⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⡟⠛⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠉⠑⠄⠀⠀⢀⠀⠀⠁⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡀⠈⢀⡠⠀⢀⡀⠐⠤⠄⠀⡤⠂⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⡇⠀⢀⡀⠀⠀⡀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠉⢣
⠀⠀⢀⠀⠴⠷⠿⠿⠟⠛⠶⠆⢀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⠀⠀⠀⠇
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢱⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⡹⠀⠀⢹⡆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠁
⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠂⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠘⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⢀⢈⠃⠀⠀⠀⠀⠀⠘⠻⠿⠟⠀⠀⠸⡆⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡴⠒⢒⡀⠤⠞⠁⠀⠘⠉⢒⣋⠛⠈⠍⠀⢀⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⢉⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⡇⠀⠀⡞⠙⠛⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠲⠚⠁
⠀⢠⠀⡠⠊⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠠⠏⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆
⠀⢸⠀⣠⠎⠀⡠⠀⠀⠀⠀⠀⣹⡿⠀⠀⢰⡇⠀⠀⢄⠂⠀⠀⡠⠀⣰⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡄⠀⠐
⠀⠈⠀⠁⠀⠊⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠠⣛⡚⠲⠋⢝⣲⣄⣀⢀⣼⡍⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠨⠙⠘⠑⠀⠂⢸⢕⡀
⠀⠀⠈⠀⠶⠆⠀⠀⠀⠀⠲⠎⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠷⠦⠟⠀⠀⠀⠔⠛⢒⠒⠒⠢⠀⠔⠒⠒⠒⠛⠢⠀⠀⠀⠀⠀⠁⡀⠀⠀⡸⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠓⣼⢂⡤⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⡜⢁⠎⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⢠⠤⢭⣤⡤⣴⡶⢶⡖⠖⢴⠏⢺⠆⠀⠛⠛⠛⠛⠛⠛⠷⢼⠇
⠀⠀⠐⠂⠆⠀⠀⠀⠀⠀⠌⠀⠀⡜⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠸⡄⣠⠟⠁⠈⠁⠀⠂⠳⠏
⠀⠀⠠⢤⠈⠀⠀⠀⠒⢶⣶⣶⣞⣠⠃⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⠀⠀⠑⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⠏⣷
⠀⠀⠀⠀⠀⠠⠂⢠⣀⣸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡄⠰⠁⠛⠉⠀⠀⠀⠢⠤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢠⣄⠀⠀⠀⠘⡀⢻
⠀⠀⠀⠀⠀⡀⠒⠁⠉⠉⡇⠈⢈⣆⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⢸
⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠘⠀⠈⠉⠀⠈⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠘⠋⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣇⡴⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠐⠲⠶⠴⠶⠦⣤⠴⠤⠤⠤⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡍⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⠒⠒⠒⠒⠒⠢⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⠳⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢦⠄⠀⠀⠀⠀⠊⢀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⡀⠀⠀⢂⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠑⡄⠀⠀⢖⠋⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⣠⠴⠚⢄⠀⠈⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠐⢁⡤⠒⠉⠀⠀⠀⠀⠁⡀⣜⠖⠢⡀⠸⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢐⠁⢺⠗⠐⠒⠀⠑⠀⡇⢸⠄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢡⠈⠀⠀⠀⠀⠀⢅⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠑⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠄⠒⠒⠒⠒⠒⠒⠒⠒⠶⠖⠒⠒⠒⠒⠂⢄⠀⠀⠀⠀⣇⢀⡀
⠀⠀⠀⠀⠀⠀⢠⢤⣄⣀⣀⠀⢠⠈⠀⠀⠸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠂⠀⢦⠄⠀⢈⣂
⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠉⠉⠉⠙⢲⠒⠚⠒⠒⡲⠾⠒⠶⡖⠦⠤⣄⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠸⠀⣤⣤⣼⠀⠐
⠀⠀⠀⠀⠀⠀⢘⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠰⠟⡁⢷⡌⠆⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡶⢘⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣸⠱⠔⠒⣀⠀⠀⢸⠀⠀⡀⣄⠰⣏⣀⡼⠁⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⠀⠔
⠀⠀⠀⠀⠀⡔⠛⠒⠒⠒⠒⠚⠒⢢⣽⠁⠀⠀⠘⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠉
⠀⠀⠀⠀⠀⡇⠀⣀⣔⡖⢄⡀⠀⠀⢸⡀⠀⠀⠀⠐⡀⠀⢀⠁⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⡇⢰⠉⡅⡇⢩⠈⠀⠸⣬⢀⠀⠀⠀⠐⠃⢀⠟⠀⠀⠀⠀⠀⠈⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢘⠁⠁
⠀⠀⠀⠀⠀⡇⠈⠖⣆⡆⡲⠔⠀⢸⣨⣌⢁⠴⠄⠙⣻⠷⣦⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠂
⠀⠀⠀⠀⠀⡇⠠⠤⠬⠭⠤⠤⠄⢨⣇⠈⠀⠀⠀⠀⠋⠀⢈⠀⠄⠀⠀⠀⠀⠒⠒⠒⠲⠖⠶⠶⣶⠒⠒⠒⠒⠓⠊
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣷⠰⠚⠋⠛⠋⠛⠛⠛⠛⠀⢩⡁⠀⠠⠤⠤⠄⠀⢈⣶⣶⡍
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠤⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⣒⣐⣂⣶⣀⣐⠴⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣀⣀⠀⠀⠠⠀⠀⠀⠂⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⣀⡄⠂⠀⠀⠐⠀⠀⠀⡊⠀⠀⢆⡄⠀⠀⡇⠀⠢⠤
⠀⠀⠀⠀⠀⠄⠈⠀⠁⡠⡀⠀⠋⠡⠠⠂⠀⡀⠂⠈⠈⠁⠐⢀⣀⡝⠥⠄⣠⣇⠀⠀⠠⠆
⠀⠀⠀⠀⠀⢀⠁⠶⠩⠀⣰⠈⠀⠐⢈⡁⠀⠄⠀⠈⠀⠀⠀⡉⣛⠚⢂⠀⠀⠉⠀⠀⠀⡀
⠀⠀⠀⠠⠀⠠⠀⠈⢁⡈⠁⠨⠐⠂⠈⠍⠉⠭⠅⠩⠤⠀⡽⠁⠄⡠⡎⡅⠺⠏⢛⠓⠀⠔⠀⡀⠠⣠⡀
⠀⠀⠀⢠⠀⠀⠄⠈⢁⣠⠤⠐⢀⠑⠀⠁⠀⣠⡀⠀⠀⠀⠀⡄⢰⠅⢙⣋⠊⠉⠉⠫⡀⠈⠀⠉⠀⠉⠋
⠀⠀⢁⠀⢦⡀⠂⠡⠠⠧⠤⠄⠀⠁⠀⢁⡄⠉⠁⠀⠀⠀⠀⠁⢈⠀⠅⠿⠀⣀⠀⠱⡀⠀⢁
⠀⠀⠀⠐⠀⠈⢀⢰⠛⣻⣧⣃⢀⣚⢁⢈⠁⠐⠲⡉⠀⠈⠙⢓⣂⡖⢀⠀⠐⣩⠀⡲⠚⠊⠀⣀⠟
⠀⠀⠀⠁⠀⠀⠀⠀⠠⣀⠀⢀⠀⠀⢨⠀⠜⠨⠄⢣⢠⠁⠈⠈⠁⢆⠀⠠⢰⠅⠀⠘⣤⣧⠃⢡
⠀⠀⠀⡀⠆⠀⠀⠆⠈⠂⠐⠠⡴⠢⠀⢀⡄⠂⠐⠘⠆⢄⠐⠨⠀⠀⠡⣤⠦⠴⠆⠋⠉⣡⣼⣻⡀
⠀⠀⡜⠀⠑⡀⠀⠑⢀⢀⠀⢸⠂⠀⠀⠈⠉⠱⡠⢀⢀⣼⡆⣁⣤⣶⣦⠹⡄⠐⠠⠄⢪⣄⠀⠀⠅
⠀⠀⠇⠀⠀⡏⢰⠌⠆⡚⣎⠠⠀⠀⠀⠀⠀⢀⡀⠐⠈⠊⠑⠘⣷⣙⡼⠀⠹⣄⠒⠀⢀⠉⠵⠖
⠀⠀⠀⠀⠀⠀⠘⡆⠆⢱⡶⢴⠀⣠⡄⣀⠈⠀⡀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠉⢆⢠⣀⣬⣱⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢉⢁⡄⢠⣯⠀⡇⠀⠂⠈⠰⠤⢀⠀⠀⠁⢀⣀⣠⡤⠴⠀⠉⠂⠁⠃⠃
⠀⠀⠀⠀⠄⠠⡞⠉⠁⠀⠻⡀⣸⠇⠂⠀⠀⡰⢀⠀⠀⠈⠀⠐⠀⠭⠁⠀⠀⢀⡀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠁⠀⠈⠀⠎⢘⡈⠉⠠⡤⠂⠀⠀⠄⠀⠀⠀⠀⠓⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⠀⠠⠤⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠖⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠈⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠒⡈⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡠⠁⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⠀⡠⠊⠀⠀⢀⠤⠩⠭⠌⠤⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠤⠍⠤⠤⠤⠤⠼⠩⠭⢍⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⡀⠄⠀⠁⠀⡀⠔⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠫⡀⠀⢀⠀⠀⠀⠀⠁⠀⡀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⡀⠀⢄⠀⠆⠀⠀⠀⠀⠀⠄⠐⠂⠤⣘⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣎⠊⢉⡁⠺⠆⢓⠿⠝⠁⠡⠀⠀⠀⠀⠔⠁⠔⠁⠀⠐⠀⠠⡀⠀⠀⠠⠊⢀⡀⠀⢀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⠟⠛⠊⠉⠁⢀⣀⣠⣀⣈⠤⠒⠂⠐⠆⠀⠠⠎⡠⠂⠀⠀⠀⠀⠈⡢⣐⢀⠐⠃⠉⠀⠄⠠⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⢬⡞⠋⠉⠉⠙⢾⡀⠀⠀⠀⢰⠀⠀⠀⠆⠀⠀⠀⠀⠀⠠⣁⠌⡈⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠀⠀⠐⣿⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⡄⠀⢱⠬⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠈⡹⣆⠀⠀⠀⢀⣼⢯⠄⠠⠀⠐⠣⠀⠀⡀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⢎⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢆⢀⣀⣠⣤⠮⡿⢓⡺⠋⠃⠀⢀⣀⡀⠄⠐⠀⠀⢁⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⣄⠂⠀⢂⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠈⠀⠀⠀⠉⢇⠀⢀⠀⣈⠁⠀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣀⣀⣀⡸⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠉⠉⠓⢦⣄⠀⠀⢀⣤⠖⠉⠉⠉⠉⠉⠙⠲⢦⣐⡂⠀⠀⠀⠀⠀⡂⠄⢐
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⠤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⢁⠀⠀⠀⠀⠀⠙⠀⠋
⠀⠀⠀⠀⠀⠉⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠤⠴⠾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣌⢁⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⡳⣄⡀⠀⠀⠀⣀⡠⠾⠃⠀⠀⠀⠹⠦⣀⡀⠀⠀⠀⣀⡴⣟⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠐⠀⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠘⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠄⠸



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⡀⠎⠀⠀⠀⠘⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⢨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠞⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠴⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣤⡔⠀⠀⢮⠈⢠⡆⠙⠂⠀⠆⠀⠀⠀⠀⠀⠀⠉⠀⠘⠷⠃⠀⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣄⠸⠏⢠⡆⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣠⠜⠉⠀⠀⠀⠀⠄⠐⠒⠲⠄⠂⠀⠀⠘⢿⣷⣾⡟⠁⠀⠀⠀⠂⠾⠇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣦
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠐⠂⠂⠀⠀⠀⠀⠀⠈⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⠓⠄⠀⠈⠁⠀⠀⠀⠀⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠀⠐⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢠⠀⠀⠀⣀⠀⠀⠀⠈⠀⠄⣀⣄⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠸⣙⡓⢶⡤
⠀⠀⠀⠀⠠⣄⢠⢠⡀⠀⠀⠀⠀⡆⣆⡛⢶⠀⠀⠀⠂⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⢸⡜⣞⠃⠀⠀⠀⠀⣧⣌⠳⣾⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣴⡄⡀⡄⠀⠀⠀⠀⠀⠠⠤⠀⠠⠶⠶⠄⢺⣒⠲⠼⠷⠃
⠀⠀⠀⠰⡌⣇⠃⠀⠀⠀⠀⠀⠀⣇⡙⢷⣬⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠙⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠰⠟⡁
⠀⠀⢀⢧⢳⡈⠀⠀⠀⠠⣤⠀⠀⢅⡙⢦⣉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇
⠀⠀⠈⠀⠳⡁⠀⠘⠳⣷⡝⠀⠀⢀⠹⣦⡙⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⠛⠀⠀⠘⢷⣄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠍⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢑⡈⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⢀⡀⠄⠈⠁⠀⢰⠄⡀
⠀⠀⠀⠀⠛⠙⠄⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠈⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⣸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⣿⣷⣄⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⠹⣿⣿⣷⡄⠀⠘⢿⣿⣦⡀⠀⠘⠋⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣷⡀⠀⠀⠀⠘⠻⣿⣿⣦⡀⠀⠙⠛⠃⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣧⡔⢦⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠘⣿⣿⣧⡀⠀⠀⠀⠀⠈⠻⣿⣿⣦⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿⠿⣆⢿⡀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⢰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠀⠀⠀⠀⣰⢿⣿⡟⢻⡏⢠⣼⣿⠠⣶⣿⡌⢇⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣰⡟⢸⣿⣿⠸⡇⢈⣉⣭⠀⠛⢛⣿⢸⠇⣤⣤⣄⡀⠀⠈⣿⣧⠀⣸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀⠀⠀⢿⣶⣄⠀⠀⠀⠀⢀⣿⠀⣾⣿⣿⠀⡇⢸⣿⣿⠀⣿⣿⣿⡎⣶⢹⡟⡍⢳⡄⠀⣿⡿⢰⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠙⠋⠀⠀⠀⠀⣸⡇⠐⠟⠛⠟⠀⣧⠸⣿⣿⡀⠛⣉⣉⣅⣿⢸⣷⣄⣼⡇⠀⠈⣰⣿⣿⠏
⠀⠀⠀⢰⣶⣦⣄⡀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⠿⠟⣛⣹⣿⠀⡎⢿⠿⠋⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠉⠛⢿⣿⣦⡀⠀⠀⠹⣿⣷⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⠿⢻⡟⣿⠉⢿⣯⣶⣶⣿⡿⣿⣿⡆⡇⠀⠀⠀⠀⠠⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⣼⣿⣾⣿⣿⣿⣿⣿⣶⣿⣿⢿⡟⠛⠗⠘⢿⣧⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⣾⣿⡧
⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠻⣿⣿⡟⢡⣾⣿⣿⣶⣦⠈⢿⢸⡄⠀⠀⠀⠀⢀⣴⣶⠀⠀⢀⣼⣿⣿⡿⠃
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⣸⣿⣿⣷⣄⡈⠻⢷⣄⠙⠿⢋⣸⣿⣷⣾⢸⡇⠀⠀⠀⠀⠈⠛⠁⢀⣴⣿⣿⣿⠟⠁
⣰⣿⣿⠟⠉⠁⠈⠉⠛⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⠛⠿⣦⣈⠙⢿⣦⡈⠛⠿⢁⣿⢸⡇⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠁
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟⠻⢿⣷⣄⠈⠙⠳⣤⠈⠻⣷⣤⣾⣿⣼⠇⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁
⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⠀⠀⠀⠀⠀⣿⣿⡟⠀⣶⣦⣄⠉⢿⣶⣄⠀⣤⣀⠀⠀⠙⢿⣿⡇⠀⠀⠀⠀⠀⣰⣿⣿⠋
⠘⣿⣿⣷⣤⣤⣴⣿⡇⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⢀⣿⣿⣿⣦⡈⠻⣿⣧⠀⣿⣿⣷⣄⠙⢿⣷⣶⣾⣿⡇⢀⠀⠀⠀⢠⣿⣿⠃
⠀⠈⠻⠿⣿⣿⠿⠟⠁⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⢸⣿⣿⣯⡙⠿⣦⡈⠉⠀⠙⠛⠛⠛⠷⣄⣿⣿⣿⣿⡇⢸⠀⠀⠀⠘⠟⠁⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⣦⡈⢻⣦⡀⠻⣿⣿⣶⣶⣿⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⣀⣴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣷⣌⠙⢿⣿⣷⣄⠙⠿⣷⣌⢻⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠿⢿⠟⠳⢤⡙⠻⣿⣆⠀⣤⣽⣿⣿⡿⠛⠋⠀⢻⣿⡧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⢸⣷⡀⠰⣶⣄⠙⢦⣈⠛⣀⣿⣿⣿⢧⣤⣴⡆⢀⣼⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠁⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠸⢁⡙⢦⣌⠙⠗⢈⣿⣿⣿⣿⣿⡿⠀⠀⠰⢇⣸⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⣠⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⣴⣿⡿⠦⠘⣿⣶⣿⣿⣿⣿⣿⣿⣷⣦⣀⣶⣾⣿⣿⣿⣷⢸⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⣿⣿⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⠿⣿⣿⠟⢻⣿⢸⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠸⣿⣷⣶⣤⣀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⢠⣿⠟⣛⣻⡟⢻⡟⢻⣉⣉⠙⣿⣧⣤⡀⢻⣦⣈⠋⢀⣿⣿⢸⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠉⠙⠻⣿⣷⣦⠀⠘⣿⠀⠀⠀⠀⠀⢸⣿⠘⠿⣿⡇⠀⠂⢸⣿⣿⡀⣿⠀⣿⡇⢠⣿⣿⠀⣸⣿⣿⢸⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢻⡆⠀⠀⠀⠀⣸⡟⢰⣶⣿⠃⣠⠀⣸⡿⡿⠃⣿⣇⠈⢁⣼⣿⡇⢰⣿⣿⣿⢸⠀⠀⠀⠀⢀⣴⡟⠀⢠⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⣿⣧⣬⣭⣭⣴⣿⣤⣿⣧⣤⣾⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡇⢸⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⠿⠻⢿⣿⣿⣿⣿⡇⠸⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣍⠁⢨⣿⣿⣆⣿⣀⣟⣀⣿⡿⢂⣠⣾⣿⣿⣿⣿⢧⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣽⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠠⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⣩⡙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⣾⡿⠛⠋⠉⠛⠻⣿⠋⢻⡏⠉⠻⣿⣿⣿⠟⢛⡛⠛⠛⣻⠋⡄⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⢿⣷⢸⣿⡆
⠀⠀⠀⠀⢸⣧⡀⠀⠸⣿⣿⡄⠀⠀⠀⠀⣿⠀⣴⣿⣿⣷⠀⠹⠀⣿⠃⣶⡄⠘⣿⣿⠀⣿⣿⣿⣿⡿⢁⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣦⣭⣼⣿⠇
⠀⠀⠀⠀⠈⢿⣿⣄⠀⠹⣿⣿⡄⠀⠀⢀⣿⠀⣿⠿⠟⠁⣠⡆⢠⣿⠀⣿⣿⡀⠸⣿⠀⠻⠿⢿⣿⢣⠉⠀⠀⠀⢀⣄⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀⠈⠛⠋⠀⠀⢸⡏⠀⠀⣤⣶⣾⣿⠇⢸⡇⢰⣿⣿⡇⠀⡿⢰⣶⣶⣾⠁⡎⣴⣿⣿⣿⠟⠋⠀⠙⠻⠿⢿⣿⠿⠟⠋⠀⢀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣄⠀⠀⠀⠀⢸⡇⢠⡄⢻⣿⣿⣿⠀⣼⠇⢸⣿⣿⡇⠀⠁⠸⣿⡿⢣⠛⣼⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠺⢶⣤⠈⠛⠀⠀⠀⠀⢺⡇⣸⣇⠈⣿⣿⣿⠀⣿⠀⣿⣿⡟⠀⣸⣷⣶⢎⡴⠋⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠘⣥⣿⣿⡄⠸⣿⡇⢠⡟⠘⠛⢉⣠⣾⡿⢟⣡⠞⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⡛⢿⣿⡀⢻⣇⣸⣷⣤⣶⠿⠛⣩⡶⢋⣵⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢶⣤⣭⣭⣭⣭⡴⠶⠞⣋⣥⣾⣿⣿⣿⡿⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠟⠛⠋⠀⣠⣾⠟⠁

⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⢀⣀⠀⢀⠀⠀⢀⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⢄⠀⠀⠀⣠⡀
⠀⠀⠀⠀⡇⠀⡆⢸⠃⠀⡇⠀⠃⢸⢀⠂⢸⠘⡇⠀⠀⠁⠀⠃⠀⠀⠀⠀⠠⡒⠊⠁⠀⠛⠸⠋⠀⠀⠀⠀⠀⠁⠀⠻⠏⠧⠀⠀⠩⠄⡠
⠀⠀⠀⠀⣣⠠⠁⡆⢲⠀⡇⠀⠀⢸⠧⡀⢸⢠⡷⡀⠀⠀⣀⡀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠵⢄
⠀⠀⠀⠀⡇⠀⢠⠁⠸⠀⡇⠀⢠⢸⠀⢃⢸⢨⡇⢷⠀⡇⠀⢹⠀⠀⠀⠒⠃⠀⠀⠀⠄⣂⠉⠀⠔⠀⠀⠀⠀⠀⠰⠀⠀⠠⠐⠀⠀⠀⠳⠒
⠀⠀⠀⠀⠃⠀⠸⠀⠀⠇⠳⠤⠎⠘⠀⠸⠘⠘⠇⠘⠀⠳⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢀⡤⢀⡄⠠⡄⠠⠆⠠⠤⠀⣴⠃⠀⠀⡠⢤⢀⡤⠄⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣯⠆
⠀⠀⠀⠀⢀⣁⠔⢹⢰⣡⣁⠰⣡⣆⠔⠍⡠⣎⠴⣁⠜⡽⠁⡇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠰⠀⢂⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⡄⠀⠀⠀⡄⠀⠒⠂⠀⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡈⠀⠀⠀⢈⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠈⢀⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠀⠀⠀⠀⠀⠉⠀⠒⠀⣆
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⢀⠀⠵⠚⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣼⣿⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠗⠺⠱⠐⠶⠀⠀⠀⠀⠀⠀⠀⡲⠁⠀⠀⠀⠀⠀⠙⠿⠀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣈⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⣀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⡠⠀⠀⠀⠀⠠⠔⠂⠀⠀⠀⠂⠀⠀⠀⠈⠀⢪⠔⠚⠛⠓⠲⣇⠀⠀⠄⠈⠀⠃
⠀⠀⠀⠀⠀⢀⡀⠀⠓⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⢠⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠘⠏⡆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡔⠀⠴⣊⠩⠽⠒⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠸⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡀
⠀⠀⠀⠀⠳⢀⡜⠀⠀⠁⠀⡀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⣝⡀⠀⠀⠀⠀⢸⣿⣷⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠃
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠂⡐⠀⠀⠀⡰⣇⠀⣀⣠⠒⠀⠐⢠⣄⠀⢨⡇⠀⠀⠀⠀⢘⢃⣤⣄⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⡀
⠀⠀⠀⢶⠀⠹⢧⠀⠀⠂⠀⠁⠀⠈⣴⢟⣤⠄⠈⡁⢀⢠⠈⠁⡤⠸⢻⡀⠀⠀⣄⡀⠀⠉⢿⣧⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠘⠁
⠀⠀⠀⠈⢡⠀⡌⠁⢀⡀⠀⠀⡀⠀⢀⠾⣿⡀⠀⡇⢰⠨⠀⠀⣀⠀⡸⠱⠆⠀⠀⠘⠀⠀⢺⣿⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠈⠲⠜⠿⠁⠒⠓⢒⠒⠒⠂⠀⠀⠀⠀⢉⢁⢉⡀⠀⠀⠀⠀⠂⠴⠁⠀⢀⠀⠀⠀⢸⣿⡂⠀⡀⠀⠀⢇⠶⣤⣄⣀⠀⠀⠀⠀⠆⠀⠀⠅⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣰⡟⠀⠀⠀⠀⢳⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇⠐⠀⠀⠀⠈⢹⠇⠀⠉⠙⠒⠒⠖⠐⠂⠐⣎⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠸⠅⠀⠂⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠐⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁
⠀⠀⣀⣽⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠹⢿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⣠⡆⠀⢀⣰⣾⣰⢃⡄⢺⡔⠄⠙⠀⠀⠢⠔⢀⣠⠀⢀⡀
⠀⠀⠀⠀⠀⣠⣤⣾⠿⣤⣿⣠⡛⠖⠛⠿⣿⣏⣱⣠⣄⡦⣤⣰⣶⣧⣦⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠁⠠⣶⣶⣶⣿⣯⣹⣿⣙⣿⣿⣾⣿⣿⡟⠉⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠸⡷⠀⢀⣄⠀⠈⠁⢾⣿⣯⣿⣿⡿⢋⣄⡟⣁⡀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠃⠀⠈⠉⢉⣠⣄⣿⣿⣿⣵⣿⣤⣠⣄⠴⡄⢀⣀⠀⠀⣄⠀⠀⠀⠒
⠀⢀⡀⣭⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠙⠛⠛⠻⠿⠿⣿⣾⣿⣶⣼⣖⢠⡤⣀⣀⣀⠀⠀⠀⠀⠀⣀
⠀⠈⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠟⠶⣾⣾⡯⣿⣿⣿⣿⡿⣻⡧⠱⣦⣄⣀⡀⠀⠉⠉⡀
⠀⠀⠀⠙⠁⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣾⠦⠌⣉⣠⣽⣻⣿⣿⣿⡿⣿⣞⡉⠀⢾⣧⡀⠀⠈⠉
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣻⣿⡃⠀⠈⣿⣏⣟⢿⣽⣿⣿⣿⣤⡸⣿⣿⡟⠋⠁
⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠁⢀⣼⣿⣻⡻⣈⠻⣽⣿⣿⣷⣿⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⣿⣿⣿⣿⣧⠈⠻⣇⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠉⠙⠛⢋⣡⣭⣼⣿⠯⢛⣿⣿⣿⣿⣷⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠘⢿⡷⣼⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣿⠿⠿⠿⠟⠃⠉⠉
⠀⠀⠀⠀⠀⢸⣿⣇⣿⡏⢠⠀⠀⠀⠀⠀⠀⢀⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢠⣤⣽⣿⣿⣷⣿⣾⣷⠞⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠠⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠄⠀⠦⠀⠀⠠⠤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠀⠀⠉⠉⠉⠉⣭⠺⣶⣶⡤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⣴⠦⠀⠠⡄⣵⠞⠛⠭⣵⣶⣦⣿⣦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⣀⠀⠀⠀⠀⠐⣿⣶⣴⠰⠂⢂⣬⣣⡐⣄⠀⠀⠉⠛⠿⣿⣷⡌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠯⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⣴⣿⣿⣿⣶⣿⣿⣶⣭⣉⣀⣰⣇⣉⠉⢻⣿⠪⡓⣄⠀⠀⠀⠀⠈⠑⠢⢌⣢⡀⠀⠀⠀⠀⢀⠔⠉⠈⠉⠁⢆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠫⠅⠰⢿⣿⠗⠈⢪⡑⣄⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡠⠊⠁⠀⠀⠀⠀⠀⠸⡞
⠀⠀⠀⠀⠀⠀⠀⠀⡰⣱⡿⠛⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢼⣷⡆⣁⠀⠀⠈⠻⠦⡱⢄⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢰⣾⣿⣿⣿⣆⠀⠀⠀⠀⠈⠪⡓⣄⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠉⠹⠄⠀⠀⠀⠀⠀⠈⠪⣁⠔⠁⠀⠀⠀⠀⠀⠀⢀⠔⢣
⠀⠀⠀⠀⠀⠀⢸⡠⠔⠂⠐⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⢆
⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠘⡄⠘⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠈⢆
⡔⠚⠓⠒⠒⠒⠒⠒⠒⠛⠛⠓⠚⠒⠚⠻⣿⣿⣿⠋⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣠⣾⣧⡳⣄⠀⠀⠀⠀⠀⠈⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡇⠻⠿⣤⣤⡄⠀⠀⠀⠀⠀⠀⢀⡰⠋⠈⠉⢐⠎⢀⣤⠀⠀⢀⣴⣿⣿⣿⣿⣿⡮⡳⣄⠀⠀⠀⠀⠘⡄
⡇⠀⠀⠀⢀⡴⣶⢻⡟⢶⠦⣄⠀⠀⢸⡇⡇⠀⠀⠀⠈⠙⢧⡀⣀⣄⣠⣤⣋⠀⠀⠀⠠⢃⣴⠟⢁⡠⠊⠁⠈⠻⠿⠛⠛⢿⠃⠈⠪⡳⣄⠀⠀⠀⠘⡄
⡇⠀⠀⣴⠛⢶⠗⢺⡷⠚⣷⠚⢧⡀⢸⡇⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣭⣙⠓⣛⠥⠊⠁⢁⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠈⠪⡳⣄⠀⠀⠘⡀
⡇⠀⢰⣇⣀⣿⣀⣸⣇⣀⣼⣄⣨⡇⢸⡇⡟⡄⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠈⠪⡳⣄⠀⠰
⡇⠀⠈⣇⠀⢻⠀⢸⡇⠀⣼⠀⢰⠇⢸⡇⡇⠘⡄⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⡀⡠⠊⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣁⠔
⡇⠀⠀⠘⢿⣙⣏⢹⡏⣹⢏⡽⠋⠀⢸⡇⡟⣀⡼⠆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⠁⠉⠀⠀⠀⠀⠀⠀⣠⠎
⡇⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⢸⡇⡏⡥⠢⡈⡄⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⡇⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⢸⡇⣷⡘⡄⠱⡈⠄⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣷⡘⣄⡱⢘⡀⠀⠀⠀⢿⡿⠀⠀⠀⣀⣀⠤⠒⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡿⢛⡡⠔⠊⠁⠈⠉⠁⠀⠈⠉⠈⠉⠁
⢇⣈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣠⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡒⠒⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠘⠀⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⣠⠀⣀⠀⢤⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢉⣆
⠀⠀⠀⠀⠀⢀⡀⠂⠀⠈⠠⠀⠀⡴⠱⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⡾⢄⠈⢏⡡⠦⠔
⠀⠀⠀⠀⠀⠈⠀⢰⣷⣃⣠⢠⡇⠧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠁⢱⠀⠈⢳⠖⠉⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤
⠀⠀⠀⠀⠀⠳⡄⠠⠍⠐⠀⠠⢿⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⡐⡤⢤⡷⠖⠀⠀⠀⠀⠀⢀⣠⠤⠴⢅⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠠⠎⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠂⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠖⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠐⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠁⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⣀⡠⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀⠀⠀⠘⠢⠄⢀⠀⠠⠞⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠰⠄⠉⠂⠀⠀⠀⠀⠀⠀⠀⠁⠚⡆⠊⢱⠻⢆⣀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢀⡤⠀⡃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠌⠂⠞⣩⠟
⠀⠀⠀⠀⠀⠈⡀⠤⡈⢧⣎⠠⠞⠠⠖⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡘
⠀⡀⠀⠀⣰⠋⠀⡀⡤⠐⠉⠰⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠂⠀⠀⠉⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⠃⠀⢀⠄⠀⠀⡰
⠀⢇⠀⠀⢨⠬⣁⠴⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⢀⠀⠀⢠⢀⠀⢰⠀⠄⠂⠀⠀⠀⡀⠀⢠⠎⠀⠀⡼⠁
⠀⠈⠀⠀⠀⠚⠃⠰⠋⠀⠀⠁⠀⠀⠄⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡎⠀⠀⠸
⠀⠀⠀⠀⠀⠈⠐⠠⢀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠊⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠂⠀⠀⠀⠀⠘⡛⠂⠀⠀⠀⠉⠁⠀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⡆⠀⠀⠀⣶⡀⢀
⠀⠀⠈⠁⠀⢐⠊⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣁⠒⠂⠄⣀⡰⠀⠀⠀⠀⠀⠀⠀⢼⡆⠀⠀⢸⢀⡇⠀⠀⠁⣾
⠀⠀⠀⢰⠀⠈⠁⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⢀⠇⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢠⡌⠳⠀⠂⢸⠐⠑⠀⠀⠀⠋⠗⢢⡄⠄⢀
⠀⠀⠀⠘⠀⡸⠵⠦⢔⠄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢧⠀⠀⠘⠀⡄⠀⠀⠀⢠⡄⠀⠉⠁⢘⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⢀⠠⠲
⠀⠀⠀⠀⠀⠀⠂⠢⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣤⠈⢰⢀⠇⠀⠀⠀⠁⠀⠈⠧⠈⠀⠀⠾⠈⠁⠀⣼⠄⠀⠀⠀⠐⠃⠀⠀⠀⠛⠂
⠀⠘⠦⢔⣢⠄⡲⠒⢁⠀⠀⠀⠀⠀⠀⠀⠐⢿⣥⣐⡀⣤⠠⠌⠂⠀⢙⣛⣯⡀⠀⣰⡠⠚⠀⠀⠀⠀⠢⠐⠐⠐⠀⠀⡀⠤⠁⠀⢄⠀⠤⠄⠀⠄⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠀⠈⠀⠄⠀⠀⠀⠀⠀⠓⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢼⠁⠈⠈⠁⠀⠀⠀⠒⠒⠒⠒⠓⠒⠒⠶⠦⠤⠤⠤⠤⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠊
⠀⠀⠀⠀⠀⠙⠂⠤⠀⠀⠀⠄⠴⠐⠒⠀⠀⠀⠀⠀⢒⣀⣀⣀⣀⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡴

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠒⠒⠲⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢆⠀⠈⢣⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠣⡀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠈⢺⠤⠤⠤⠤⠤⢤⡟⣻⣟⠛⠛⠛⠛⠛⠛⠛⢛⣿⣓⣆⣀⣀⣀⣀⣀⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⣄⡞⠀⠀⠀⠀⠀⠀⠙⢣⣀⣓⣒⣒⡒⠒⠒⠒⣋⣀⡸⠃⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⡀⠈⠻⡲⠶⠦⠤⠤⠤⠤⣤⣤⣬⣭⣭⣭⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣠⣴⡧⠽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠙⣄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠶⠒⠚⠋⠉⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣀⣠⠴⠞⠻⡄⠈⢢⡀⠀⠉⠓⠒⠒⠒⠒⠛⠉⠉⠁⠀⠀⣀⣀⣀⣠⣤⠤⢤⣶⣶⣄⣠⠤⠤⠒⢺
⠀⠀⠀⠀⡾⣽⡄⠀⠀⠀⠙⣆⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⢚⡿⠯⣍⡁⣀⣀⣠⠤⠼⠃⣈⡇⠀⠀⠀⢀⡼
⠀⠀⠀⠀⡗⠊⠉⠓⠶⢤⣀⠘⢦⠀⠈⠒⠒⠒⠒⠒⠂⠉⠉⠙⠺⠳⠤⣤⣭⠥⠴⠶⠖⠛⣋⣁⡤⡤⢴⡖⠋
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠙⠺⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠶⠒⠚⠋⠉⠁⣀⡤⠶⠏⣿
⠀⠀⠀⠀⡏⠑⠢⢄⡀⠀⢀⡀⠀⠀⠙⠶⠤⠤⢴⣖⠒⠚⠋⠉⠉⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⠁⠀⠀⠀⢹
⠀⠀⠀⠀⢻⣆⠀⠀⠈⠑⡯⣭⠛⠶⢦⣄⡀⠀⠀⠈⠉⠓⢶⣮⣉⣑⣦⠴⠒⠉⠁⠀⠀⠀⠀⢀⣠⠤⠒⣻
⠀⠀⠀⠀⠀⠙⠲⠤⣄⡀⢣⡙⠤⣄⡀⠈⣙⣷⣦⠀⠀⠀⠀⠓⠤⠚⠁⠀⠀⠀⠀⣀⡠⠴⠚⠉⠀⠠⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠮⣕⣒⠬⣍⣁⡼⠋⠑⠲⠤⣀⣀⣀⣀⡠⠤⠖⠚⠉⠁⠀⠀⠀⠀⣀⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣯⣅⣈⣹⣤⠤⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣚⡿⣦⣤⠭⢤⡤⣤⣤⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢩⢾⠏⢠⠏⠉⠉⠉⠉⠉⠻⡻⣌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠥⠴⠧⣼⠀⢸⡖⣒⣒⣒⣒⣒⣒⣳⡜⡆⠹⣄⣀
⠀⠀⠀⠀⠀⠀⣠⠾⠭⠤⣴⣞⠋⠉⠉⡇⠀⠘⣇⠀⣨⢭⡍⢉⠩⣭⠉⢼⠀⢹⣭⣽⣓⣦⣄
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠉⠀⠫⠓⡶⢯⡳⠴⣐⠯⠀⠀⠀⠀⠀⠀⠀⠀⣎⣂⣐⠣⠇⠀⠈⠙⣧
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢺
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⢠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠖⠒⠛⠉⠀⢀⢸
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⢀⠇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⢠⢠⠁⠀⠀⠀⢸⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⢈⠀⠀⠀⠀⠘⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⠀⡇⠈⢀⡇
⠀⠀⠀⠀⠀⠸⡈⣄⠀⠀⠀⠀⠀⠃⢠⠘⡀⢰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⣈⡇
⠀⠀⠀⠀⠀⠀⠑⢿⡉⠀⠀⡸⠠⡀⠈⡀⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠤⠤⢤⣀⣐⣶⣦⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠀⠴⠷⠶⠖⠒⠒⠒⠚⠓⠂⠀⠈⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣦⡄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡇⠈⠀⠈⢸⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩⠛⠿⠿⠿⠛⠁⠀⠀⠀⠈⠛⠛⠛⠉⠉⠃⠀⠀⠀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⡀
⠀⠠⠾⠿⡤⢮⡌⠷⠺⠦⠶⣰⢾⡝⠿⠛⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠋⠉⠉⠃⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠂⠀⠀⠀⠐⠚⠡⠋⠀⠉⠁⠁⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠤⠀⢄⣀⢀⠀⠤⠤⠤⠀⠄⠀⣠⣄⣤⢤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⡤⠄⢀⠀⠀⢀⠙⠉⠉⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢽⣯⡳⠀⠀⠀⠀⠂⠒⠀⠉⠀⠛⠃⠀⠈
⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢛⡷⡆⠁⠠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣆⠀⠀⠀⠀⠀⠈⣃⠀⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣢⣍⡰⠀⠀⠀⠀⠠⠀⠀⠀⢤⣤⠀⠀⠘
⠀⠲⠄⠒⠠⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠉⠁⠈⠙⠀⠁⠀⠀⢀⡤⠤⠴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠉⠀⠀⠀⠀⠤⠤⠤⠂⠀⠀⠁⠀⠀⠘
⠀⠀⠀⠀⠈⠀⠀⠠⠤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢦⠀⣀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠒⠲⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠒⠒⠀⠈⢠⡔⠒⣆⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠒⠊⠅⠂⠀⠀⠀⠀⠀⢠⣌⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⣤⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⢀⠀⠀⠄⠠⠠⢤⢠⠀⠀⠀⡀⡒⡤⠀⠀⠀⠐⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠶⠀⠈⠈
⠀⠀⠀⢀⠠⠄⠒⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣀⠀⠴⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒
⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠄⠀⠒⠦⠤⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠰⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠀⠬⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⠄⢀⡄⣠⣄⠀⣰⠏⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠀⢰⡴⢱⡷⢁⡟⢰⡟⠔⣶⠋⣽⢂⣴⢋⡿⢁⣴⡓⣀⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠃⠘⠁⣸⠃⠿⠁⣼⠇⠸⠏⠀⠋⠸⡇⠸⠥⢺⠏⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠔⠊⠉⠈⠀⠀⠙⠀⠀⣀⣠⠤⡤⢠⢀⠤⠄⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⠃⡟⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⢠⠒⢠⡁⠀⠀⠃⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⢈⣀⠧⣤⢨⣎⢡⡟⣰⡶⠀⡜⣠⠆⣀⣴⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣿⠐⢸⣇⠀⠛⢸⠏⠘⠁⠛⢀⡞⠸⠇⠠⡿⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⡿⢀⠀⠁⠀⢀⣀⣠⠤⠖⠒⠚⠃⠀⠀⢀⡠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⠔⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢠⡴⠶⠶⠶⠶⠬⠭⣭⡉⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠸⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢸⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠾⠛⠿⠿⠿⣭⣿⠿⢷⣶⣶⣶⣶⣤⠤⣤⣤⣧⣏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣤⣤⠤⠤⢤⣤⣤⣄⣀⣈⣉⠁⠒⠒⠚⠯⠭⢍⣓⣒⣶⣬⣍⡙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠴⣃⡟⢀⡤⠤⠤⠤⣤⣈⠉⠉⠉⠛⠛⠒⠶⠦⣤⣄⣀⣀⡴⠟⢻⢩⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠋⠉⠀⠈⠉⠛⠛⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢏⣿⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣀⣠⡴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠦⠤⠤⢽⡀⣸⣏⡉⠉⠒⠒⠒⠲⠦⢤⡾⠻⣦⣀⣀⡀⠀⣿⠀⡀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠙⠉⠀⠉⠉⠉⠉⠐⠒⠒⠺⢧⣤⠿⠤⢀⣉⣉⣿⠀⡇⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡧⠠⠤⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⡇⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡗⠒⠂⠰⡇⢀⣿⠀⠀⣀⠁⠀⠐⠒⠀⣴⠶⢶⡄⠀⣀⠀⢸⡇⠀⡇⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠁⢻⣀⣼⠃⠤⠀⠀⢼⡇⢰⢷⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⡇⢸⢸⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠱⠋⡗⠀⠀⠀⢸⡇⢸⢸⠃⢠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢫⣱⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠞⠀⠀⠀⢀⣸⠀⠘⢸⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⢲⠒⣖⠒⡶⠒⠒⠶⠤⢤⣄⣀⡀⢀⡴⢚⡭⣿⠀⡇⢸⠀⣆⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⠀⠀⠀⢧⣸⣍⣹⠟⠓⠦⠿⣤⣅⡼⠶⠯⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠴⡶⠚⠻⠷⢶⡶⡿⣯⣿⡧⠓⠆⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀
⢴⣿⣶⠖⠲⣶⡟⠷⡮⡦⢶⠟⠙⡷⠦⠀⢿⣃⡿⠛⠁⠈⡇⠀⢀⣤⢶⣶⢶⡄⠤⠔⠐⡦⡔⠂⠈⠀⡼⠉⠉⠨⠏⣩⡟⢓⣖⣲⡶
⠈⠛⠉⠉⠙⠯⣁⡀⠄⡐⠙⣦⣸⠛⣢⣤⠀⠉⡇⠀⠀⠀⠰⣦⣈⡹⠜⠋⠀⡠⠀⠠⠾⢋⠐⡡⢂⠈⠀⡡⠊⢱⢾⡉⠉⣵⠏⠁
⠀⠀⠀⠀⠀⠀⢌⠀⢈⡐⠷⢀⢀⡼⠁⠉⠀⢀⣠⠀⠠⡀⠘⠚⠉⠓⠲⣤⣄⡀⠀⠀⠁⠀⠀⡤⠁⢀⠜⢑⣡⠡⠏
⠀⠀⠀⠀⠀⠀⠈⢺⡌⢉⠤⠥⡎⠀⠀⠀⠀⠩⠁⠉⠀⡙⡇⠀⣀⠀⠤⠀⠀⠀⠑⣀⡀⠀⠊⢀⠞⠁⠀⢨⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⢤⡄⠀⠀⠀⠀⠀⠀⠄⠈⠀⢰⡇⠀⠠⠴⠂⠀⠀⠀⠀⠀⠈⠱⣴⠉⠠⣀⣖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣖⢖⣄⡀⠀⠀⠀⠀⠘⣿⠀⠂⠀⣠⠄⠒⠒⠉⠁⠀⠀⠂⠀⠀⠈⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡱⠋⡤⢚⠀⠀⠀⠀⢀⢿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⢲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠞⠀⠈⠅⠀⠀⠀⠀⢼⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⣢⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡠⠂⠁⠀⠀⠀⠀⢺⡇⢀⠀⣇⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢵⣋⣀⠐⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⠀⠀⠀⠀⠀⠀⠀⠸⣥⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠁⠀⠀⠀⠀⠀⠀⠀⢐⣿⠀⠂⣻⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣾⠀⠅⢸⡀⠀⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⡜⢧⣀⢀⣉⣄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠒⠒⠒⠓⠈⠉⠉

⠀⠀⠀⡠⠤⠤⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠈⢀⠄⡩⠋⢰⢠⡀⡥⠀⠀⠀⠀⢀⠔⠉⠁⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⢀⠑⠊⠀⠖⠐⢱⠄⠀⠀⠀⠀⡠⠇⠒⠒⠀⠈⣉⣀⠀⠒⠂⠼⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠁⠀⠀⢸⠢⠀⠠⠠⠤⠤⠤⠤⠀⠠⠄⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠰⠂⠀⠒⠒⠒⠒⠒⠂⠒⠒⠒⠁⠀⠀⠀⢀⠤⠄⠀⠀⢀⠀⠠⢂⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠓⠈⠘⠘⠀⠚⠉
⠀⠀⡇⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠆⠈⡄⠀⠀⠀⡞⢉⡽⢋⡝⠉⡇
⠀⠀⡁⠀⠀⣀⢀⠀⣀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⡷⠊⡠⠋⡠⠊⡇⠀⠀⠀⠀⠠⠄⠀⠀⢀
⠀⠀⠁⠐⠀⠀⠀⠐⠀⠀⠀⠀⢀⠁⠀⠀⢀⢀⡀⠘⠀⠀⠀⠀⠓⠚⠲⡞⠒⠒⠃⠀⠀⠀⠀⠀⠀⠁⢈⢠
⠀⠀⠀⠃⣀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠁⠀⠘⠉⠃⠀⠀⠀⠀⠀⠀⠒⠀⠈⠙⠀⠀⠀⠀⠀⠀⢀⠀⠺⠑⠈
⠀⠀⠈⢀⠤⠤⠤⢄⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠤⠒⠒⠒⠒⠒⠒⠒⢄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡇⠀⠀⡀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⣎⠤⠔⠒⠒⠒⠒⠒⠤⣈⣄⡀
⠀⠀⢠⠇⠀⠀⢤⠄⠋⠤⡄⠀⠘⣄⠀⠀⠀⠀⢀⠘⠀⠀⠘⠀⠘⠊⠉⠉⠉⠉⠉⠉⠉⠋⠀⠱
⠀⠀⠀⠀⠀⠀⠈⡦⠒⠊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡔⠒⠒⠂⠀⠀⠀⠒⠒⠢⡤
⠀⠀⠘⠤⠤⠤⠐⠓⠀⠉⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠒⠢⡄
⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⢸⠁⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢀⣠⠿⢤⡀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⡎⣩⠖⠒⡄⠀⠀⣃⣀⣀⣀⣀⣠⡇
⠀⠀⠸⠡⠀⡲⠒⢲⢦⠨⡆⠀⠀⢸⠀⠐⢸⠐⠀⢸⡇⠀⠀⠀⢸⠱⣇⡠⢴⠁⠀⢸⠀⠀⠀⠀⠀⢸⠃
⠀⠀⡇⠃⡰⠁⡰⠁⢸⡒⠇⠀⠀⢸⠀⠀⢸⠀⠀⢸⠀⠀⢀⠄⠘⠀⢄⠓⠊⠀⠀⢸⠀⠀⠀⠀⠀⡘⡆
⠀⠀⠣⠉⠁⠈⠁⠀⠁⠜⠀⠀⠀⠸⣀⣀⣸⣀⣀⣸⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠒⠆⠄⠒⠲⣴⣎⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠘⠁⠠⠀⠀⠀⠀⠙⠉⠛⢦⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢖⣶⠖⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢷⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⠉⠉⠀⠈⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠶⣺⢸⡿⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣉⣴⣿⡾⢁⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⠟⣡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠄⠀⠀⠀⠀⠺⠧⠤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠵⣖⣶⢶⡾⠷⠿⡿⢋⡠⠊
⠀⠀⢠⣀⠀⣀⣀⣀⣀⠠⣸⡀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⢶⠀⢀⣿⣟
⠀⠀⠉⠘⠛⠛⠛⠛⠚⠿⣿⣷⣯⣶⠭⠥⢀⠀⡀⠀⠀⢀⣴⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠉⠐⠊⠓⠚⠛⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣻⡿⠿⠾⠧⠤⠄

⠀⠀⠀⠀⢹⠁⣯⢢⣷⢰⡔⣸⠁⡆⠀⢦⠆⡏⢰⢉⠘⠋⡎⡧⣯⡆⢸⠱⡋⢰⣙⢖⢢⡐⢡⠀⠺⡄⡏⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠈⠀⠈
⠀⠀⠀⠀⡏⠉⢯⣭⣩⡉⠉⢹⠀⢸⠋⠉⠉⢩⢉⡉⠉⢳⠀⢰⠉⣉⡉⢉⣩⠍⠙⠂⠀⣼⠉⠉⠉⠉⠉⡩⣽⡆
⠀⠀⠀⠀⡇⢖⣏⡉⢩⡻⠦⢸⠀⢸⠀⢀⣀⣼⣒⣋⠀⢸⠀⠈⡔⠿⣏⣹⣏⡆⠰⡇⠀⣸⠀⣴⡻⠄⠚⣱⢹⠇
⠀⠀⠀⠀⡇⣌⢉⡠⢔⣂⢼⢸⠀⢸⠀⣨⣆⣀⡀⠨⣁⢸⠀⢀⠁⠀⢹⢀⡇⡇⠈⡁⠀⣿⡔⣉⡶⠉⣙⣳⢾⠂
⠀⠀⠀⠀⣧⣈⣉⣉⣉⣉⣀⡼⠀⠸⣄⣐⣢⣤⣔⣂⣀⣼⠀⠸⣄⢀⣀⠓⠗⠃⠠⠇⠀⢹⣉⣁⣀⣀⣀⣀⣸⠇
⠀⠀⠀⠀⡀⠀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢸⡁⣀⠔⡉⣈⠢⡌⢹⠀⠈⡄⢀⡔⠉⠲⣧⠀⡇⠀⢸⢤⣀⡼⠤⢤⡴⠋⡇⠀⢸⢁⡴⠦⠤⣤⠤⡈⡇
⠀⠀⠀⠘⡟⡋⡏⠀⠈⡗⠟⣧⠀⠀⣷⣋⡀⣠⡤⢬⡷⡇⠀⢸⠠⡏⠀⠀⠀⢣⣤⡇⠀⢸⠼⢧⣾⣷⡉⠁⣿⡇
⠀⠀⠀⠈⣧⡁⠙⠒⠒⢁⡠⢾⠀⠀⡇⢸⡀⣯⣇⣸⠀⠇⠀⢸⠖⠣⣀⣀⡠⡏⠀⡇⠀⣾⡞⢆⣻⣭⣤⣴⡏⡇
⠀⠀⠀⠀⠧⠿⠂⠒⠖⠻⠤⠞⠀⠀⠧⠌⠉⠉⠉⠛⠒⠂⠀⠘⠤⠤⠾⠤⠤⠼⠷⠇⠀⠻⠭⠭⠤⠿⠯⠿⠶⠇
⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⣄⠀⢀⡤⠤⢤⡤⣤⠤⠤⢤⠀⢠⡄⠠⠤⠤⠤⠤⠤⡄⠀⢠⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⡇⠀⡤⣒⢒⣢⠀⢸⠀⠈⣶⣾⣥⣤⣤⣽⣷⣾⠀⢸⣴⠒⠒⠫⣒⢦⠀⡇⠀⢸⠀⡠⠒⣖⠲⣆⢸⡇
⠀⠀⠀⠀⣇⡔⠁⠉⠈⠉⠳⣼⠀⠀⡄⣿⠀⡙⡇⣸⡇⢹⠀⢸⠈⣷⣏⠀⠘⡏⠀⡇⠀⢸⠰⡗⢲⠿⡔⠉⡾⡇
⠀⠀⠀⠀⡏⢻⡵⠒⠒⠮⠟⢹⠀⠀⡟⣿⣟⣟⣛⣻⡟⢹⠀⢸⠀⠈⠛⠋⠉⢯⠗⡇⠀⢸⠀⠑⠾⣄⣭⠟⢱⡄
⠀⠀⠀⠀⠙⠓⠒⠒⠒⠒⠒⠋⠀⠀⠛⠓⠛⠛⠒⠒⠓⠊⠀⠈⠓⠒⠒⠒⠒⠒⠚⠁⠀⠈⠒⠊⠋⠉⠉⠉⠉
⠀⠀⠀⠀⡏⢩⠍⢭⣩⢯⡉⢱⠀⠰⡗⣍⢉⠁⠈⠉⠉⡇⠀⢸⠃⠉⠉⢉⡉⠉⠙⡆⠀⠸⢉⣽⣿⣯⡟⣫⣽⠆
⠀⠀⠀⠀⡧⡏⣴⠚⡦⠠⣟⢸⠀⢀⡷⣏⠉⣹⠲⣄⠀⡇⠀⢸⠖⠙⣦⡋⠙⣄⠀⡇⠀⢻⠙⣿⣿⡽⣯⡅⠀⡇
⠀⠀⠀⢠⠐⠭⣽⣦⣗⣀⣯⢸⠀⢨⠿⡉⠙⠈⠀⣀⡽⡇⠀⢸⣠⠞⠉⠙⠚⠙⢳⡇⠀⢸⠀⠀⡏⢓⣿⠀⢰⠁
⠀⠀⠀⠀⣆⣀⡛⠁⣈⣃⡀⠼⠀⢸⣄⣙⣚⣙⣛⣁⣀⡷⠀⠸⣇⣀⣀⣀⣀⣀⣠⠇⠀⠿⠤⣤⣿⣭⣼⢇⣼⠂


⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⠋⠉⠉⢇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⡿⣷⡶⣛⣷⣶⠖⣶⠤⣼⣷⣸⣄⡀⠀⢸⡀⢸⣧⣠⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⡴⣫⠟⣻⣾⡿⠿⠿⣗⣖⣐⡶⢖⢥⣴⣞⡛⠉⠉⠛⠛⠛⣷⠚⢉⣹⣷⠒⢩⣽⠛⠋⠉⠙⣻⣶⡶⣶⣶⣤⢴⣶⣄
⠀⠀⠀⠀⠀⡾⢱⠅⡼⠁⠀⠀⠀⠀⠀⠈⡹⠁⢉⡞⠛⠛⠙⠛⠛⠛⠻⡿⠋⢏⡾⠧⣤⡭⠭⠿⠯⠭⡽⢋⡭⢶⢞⣿⠛⢹⡇⠀⢹⡆
⠀⠀⠀⠀⢸⡇⡿⢸⡏⢠⡖⠶⣶⣤⣄⠀⠁⠀⣼⠁⠀⠀⠀⠀⠀⠀⢰⠁⠀⡾⠀⠀⠀⠀⠀⠀⠀⡐⢰⠏⣰⢣⠋⢅⠀⣼⠁⠀⠀⣿
⠀⠀⠀⠀⣸⡇⡇⣸⡇⢸⠏⢡⡟⢆⢸⡇⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡇⡟⢀⣿⡿⠀⠈⠊⠁⠀⠀⠀⣿
⠀⠀⠀⠀⢿⡇⣇⣿⡇⠈⣱⣾⣷⣆⢾⠳⠀⣤⡇⠀⠀⣼⣿⣽⣦⠀⡇⠀⢸⢁⡤⠄⠀⠀⢄⡀⢸⢹⡇⢸⣿⠃⠀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⣯⣽⡇⠀⢉⡛⠲⠶⠊⢸⠀⠀⠇⠀⠘⡟⢀⣿⠟⠀⠃⠳⢸⠀⠄⠉⠈⠀⢀⡟⢸⣾⠇⣸⡇⠀⢠⠀⣆⠀⢀⣠⡀⣿
⠀⠀⠀⠀⢸⡇⣿⢼⡇⡞⣋⣿⣷⡄⠀⢸⠀⠀⡄⠀⠀⠀⠨⠅⠁⠀⠀⠀⢸⠀⢀⡀⠀⠀⢠⡆⢸⣿⡆⣸⡇⠀⣴⣆⡼⠀⠋⠹⡇⣿
⠀⠀⠀⠀⢸⡇⣯⢸⡷⣏⣛⡛⢿⡿⠀⢸⡆⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⣈⡀⢸⣿⡂⢸⡇⠀⠛⠉⠀⠀⢀⢺⡇⣿
⠀⠀⠀⠀⢸⣧⣿⢹⣿⠈⠻⠾⠿⠃⠀⠈⡇⢀⣄⢈⣀⣂⢠⠄⡀⠀⡀⠀⢸⠀⠀⠀⣉⣁⠈⣵⢸⣿⠀⢸⣧⠀⠀⠀⠀⠀⣉⠸⠇⣿
⠀⠀⠀⠀⠈⣿⣿⢸⣿⠀⢠⣶⣶⣶⣶⣦⡇⠈⢹⠈⢇⡠⠀⠒⣿⠀⡇⢠⣼⠀⠀⠀⠈⠓⠘⠁⢸⣿⠀⢸⣿⠀⢰⣾⣦⠀⣧⣅⡀⣿
⠀⠀⠀⠀⠀⣿⢸⠈⣟⡆⢸⣿⣿⣾⣿⣿⡇⠀⢸⠀⠙⠃⠀⠒⠋⠀⡇⠀⢸⠀⠀⢀⡀⣀⣀⣀⢸⢻⠀⢸⣿⠀⣿⣯⢸⡇⢿⠟⠁⣿
⠀⠀⠀⠀⠀⣿⢸⠀⣿⡇⢸⡟⠻⠿⠛⢻⣿⠀⢸⠀⠀⣰⡾⣷⣦⠀⡇⠀⢸⡆⣿⣿⣿⣿⣷⣎⢿⣸⡆⢸⡇⡁⠹⣭⡿⠁⢠⡄⠀⣿
⠀⠀⠀⠀⠀⢿⡸⡇⡟⡇⠀⢿⣒⡒⠳⢿⢿⢀⣸⡆⠀⠘⢇⣡⢶⡇⡇⠀⢸⡇⢿⣿⣿⢿⣿⡟⣾⡾⡇⢸⡇⠀⠀⠀⣴⠀⣞⣷⠀⣿
⠀⠀⠀⠀⠀⢸⣇⢷⡇⢁⠀⠀⠈⠀⠀⠀⢸⠘⢹⡇⠀⠀⠀⠁⠉⠀⡇⢀⣸⡇⠘⠶⠾⠚⠛⠛⠉⡇⣏⢘⡇⠀⠀⠀⢥⠊⠉⣻⠀⣿
⠀⠀⠀⠀⠀⠀⠹⢾⣷⣤⣦⣀⣀⣀⠀⢀⣹⡄⠘⣧⡀⠀⠀⠀⠀⠀⣧⠈⠁⣷⠀⠀⠀⠀⠀⠀⠀⡇⣷⠐⣇⠀⠀⠀⠘⡐⣉⠝⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠻⠿⠷⢤⠼⣷⣤⣤⣄⣀⡀⢹⡄⠀⢻⡄⠀⠀⠀⠀⠀⠀⡇⢹⡋⣿⢠⠀⠀⠀⠉⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠻⠦⠼⠷⢦⣤⣤⣄⣀⡁⣽⡘⣷⢻⡞⣇⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⠿⠿⣿⣮⣷⣷⠟⠁



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

⠀⠀⠀⢀⠤⠤⠄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⠡⢂⠔⡱⠂⢢⢨⠈⣢⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠀⠀⠈⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⢀
⠀⠀⢸⠘⠥⠋⢠⠔⠥⢫⠈⣩⠀⠀⠀⠀⣸⠥⠤⠄⠒⠒⠒⠒⠂⠤⠤⢨⡇⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠢⡈⠄
⠀⠀⠀⠑⠂⠀⠀⠠⠤⠷⠤⠂⠀⠀⠀⣎⣀⠀⠀⠀⠈⠉⠉⠉⢀⣀⣀⣀⢨⠂⠀⠀⠈⠰⠀⠀⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠤⡄⠀⢀⣀⣀⣀⣀⡀⠀⠠⠤⣤⠴⠃⠀⠀⠁⠠⡀⠀⠈⠂⠈⠈⠀⣀⠀⡅
⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠲⢭⣍⢀⣦⣰⣆⣀⣵⠞
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣄⡀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⢀⠉⠀⠀⡇⠀⠀⠀⢸⢀⡔⢉⡴⠃⣠⡇
⠀⠀⣀⢀⠀⠠⠆⠤⠤⠤⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⢸⠋⡠⠋⣠⠞⠁⡇⠀⠀⠀⠀⠀⠒⢐⠠⡒⢲
⠀⠀⡆⠈⠉⠉⠀⠀⢸⠀⡄⠀⠀⢀⠃⣀⠀⠀⣀⣠⠀⠘⠀⠀⠀⠀⠈⠛⠓⣻⣓⡒⠚⠁⠀⠀⠀⠀⢠⠀⠠⣑⡤⢎
⠀⠀⡇⡀⠀⠀⠀⠬⠥⡄⡇⠀⠀⢸⠀⠀⠀⠀⠉⠈⡅⢠⠀⠀⠀⠀⠀⠈⠁⠒⠂⠊⠀⠀⠀⠀⠀⠀⢸⡀⠐⠟⠂⡀
⠀⠀⠉⢁⡠⠤⠤⠤⡀⠈⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡠⠴⠒⠒⠒⠒⠒⠒⠒⢤
⠀⠀⠀⢸⠀⠀⢀⠀⠸⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣸⣁⠤⠴⠒⠒⠒⠒⠧⢤⣀⣇
⠀⠀⠀⡸⠤⠤⢌⡠⠤⠧⢔⡄⠀⠈⡄⠀⠀⠀⠀⠀⠀⡸⠀⠀⢰⠁⠈⠗⠒⠒⠒⠒⠒⠒⠒⠒⠾⠁⠹
⠀⠀⢸⠀⠀⠀⠀⢳⣀⠤⠔⠇⠀⠀⠈⠁⠀⠀⠀⠈⠉⠀⠀⠀⠰⢀⣠⠥⡤⠤⣀⣀⣀⣀⣠⣤⣆⣀⡠
⠀⠀⠸⡀⠀⠀⠀⡸⠤⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⢀⣀⣀⡀
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⢻⠉⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢣⡸⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⢀⡤⢄⣀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⡤⠤⠶⠮⠭⠵⠤⢄⠀⠀⠀⠀⡇⠀⡀⡇⢀⠀⢸⠀⠀⠀⠀⠀⡞⢠⠊⠁⢉⠆⠀⢀⡗⠖⠒⠒⠒⠒⢾⡇
⠀⠀⢸⡠⠊⢩⠋⢉⠟⢱⠚⡇⠀⠀⠀⡇⠀⠁⡇⠀⠀⢸⠀⠀⠀⠀⠀⡇⠳⢴⠒⡧⠀⠀⢸⡇⠀⠀⠀⠀⠀⡏⠃
⠀⠀⡶⡇⣰⠃⣠⠋⠀⣸⣙⠃⠀⠀⢸⡇⠀⠀⡇⠀⠀⢸⠀⠀⣠⠐⠈⠁⠒⣌⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⢻
⠀⠀⠑⠬⠭⠭⠭⠭⠭⠤⠞⠀⠀⠀⠘⠦⠀⣠⢧⣀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢸⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⢿⣀⡼⠓⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠀⠀⠀⢠⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡴⠃
⠀⠀⠀⠀⡼⠿⠦⠤⠀⠀⠀⠀⠀⠀⠰⢍⡲⣄⠀⠀⠀⢨⣭⣧⢷⣤⣤⢤⣀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣳⣤⡀⠶⢶⢺⢺⢻⣿⢼⢬⡅⠀⣀⡤⠾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⣿⡿⠿⠾⠾⣷⣶⣶⡶⠾⠿⣷⣾⠃⠀⠀⠀⠀⣀⠀⠀⠀⠲⠶⢿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⠠⣀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡈⣿⠀⢀⡄⠀⣀⣿⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣓⢤⣨⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢻⣅⣷⣖⢻⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣽⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⣿⣞⣿⠖⠻⠛⠆
⠀⠀⠀⠀⣀⣀⣶⣦⣤⣀⣀⣠⣤⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⣹⡤⠗
⠀⠠⠤⠤⠿⠿⠻⠿⠿⠿⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣒⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⢛⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣈⣿⣿⣿⣿⡦
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⣧⢚⣯⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠿⠽⠿⡿⠃⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⠉⠀⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠷⣌⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣷⢛⣯⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⢿⣉⠳⣌⠓⣦⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠙⠏⣋⢽⣿⣿⡿⢿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⡙⢦⣈⠳⣬⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣾⣿⠟⠁⣿⣿⢉⣿⠋⣿⠋⠿⠟⣋⡥⠙⢦⣈⣿⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠋⠀⢸⣿⡃⢸⣯⢨⣯⠀⠀⡩⠤⢒⡁⠀⠈⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣼⣿⡿⠋⠀⠀⣸⣿⠇⠛⠁⠘⠀⠀⠀⠀⣊⠥⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⣩⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣁⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠉⠃

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⣀⣠⣀⣀⣠⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⣁⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠡⠱⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠄⠀⠈⢁⠀⠠⠰⠀⠄⠀⢀⠈⠀⠀⠈⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⠀⠂⠀⠀⠀⣾⣤⠀⠀⠀⠀⠃
⠀⠀⠰⠀⠀⠀⣀⠂⢠⡄⠀⠀⠀⣶⠀⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠀⠀⠀⠘⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠄⠀⠀⠉⠀⠈⠀⢂⠀⠀⢸⠀⠘⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⡀⠈⠀⢸⣿⣯⡀
⠀⠀⠀⠴⢈⣁⠀⠠⠆⠀⠀⢣⠀⢸⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⠀⠀⡈⠀⠙⢿⣦⣤⣶⠿⠂
⠀⠀⠀⠀⢸⡅⠀⠨⠀⠐⢠⠀⠀⡈⠠⠀⠀⠠⠀⠀⢠⠀⢠⢠⣤⡀⣦⠠⡆⢰⣦⠀⢀⠀⠀⠸⡟⠀⠀⢀⠀⠀⠆⠀⠀⠉⠋
⠀⠀⠀⠀⢸⠀⢀⠀⠂⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣶⣿⢸⠘⡇⣿⠀⡇⢸⣿⡆⢀⣀⠠⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⠀⠀⣿⡇⢸⣸⠇⣿⡆⣇⢸⣿⠃⢀⠄⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⢸⡃⠀⢠⡄⠀⢠⠀⠀⣤⠑⠀⠄⠀⠁⡀⠀⠀⡀⠀⠀⠀⠈⢀⡉⠀⣁⠀⢀⠀⠀⣴⣷⠀⠀⠀⢐⡄⠃⠀⠶⠖⠒⠊⠉⠘
⠀⠀⠀⠀⢸⡇⠀⠘⠃⠀⠈⠀⠀⠉⠀⠀⠀⠀⠐⠒⠀⠀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂⡂⠀⠈⠈⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠈
⠀⠀⠀⠀⠺⠄⠀⠸⠃⠀⠀⠀⢃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠊⠀⠀⠀⢀⠀⢀⡀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢆⠀⡐
⠀⠀⠀⠀⠀⠒⠀⠠⠠⠀⡀⠀⠈⠀⡆⠀⠀⠀⡇⠀⡲⠀⠀⢰⢁⡄⠀⢰⠀⡼⠀⠀⠅⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠪⠈⠙⠉⠉
⠀⢀⣄⠀⠀⠀⠀⠤⠀⢆⠀⠀⠀⠀⡇⠀⠀⠀⡇⠰⡄⠀⠀⡏⠈⢡⠀⠈⡄⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⠀⠀⠃⠀⠑⠀⠈⠁⠀⠈⠁⡀⠁⡀⠀⠀⠘⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠠⣦⠀⠲⣶⡶⢰⣶⡄⣴⢶⠆⣖⣦⢸⡗⢺⡇⡴⢲⡐⣶⢶⠀⠀⠀⠀⠀⠲⠛⠙⠾⡉⠙⡃
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⢉⣀⠀⠀⠀⠀⠙⠀⠘⠛⠃⠙⠋⠀⠛⠛⠀⠓⠘⠃⠈⠋⠐⠛⢘⠃⠀⠀⠀⠀⠈⠒⠋⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠠⢀⡤⠼⠋⠙⠳⠒⢶⠐⠀⠈⢳⠘⠀⠀⠾⡀⢰⡆⡆⠀⠀⠀⢠⡟⠀⢠⣟⠀⠀⠀⠒⠆⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠸⠋⠀⡄⡅⠰⠈⠀⠀⠹⠀⢰⠏⠀⠀⠀⡷⠇⠀⢷⡇⠀⠀⠀⠈⠳⠀⠀⠙⠃⠀⢀⠃⠃⠐⠒⠀⠈⠉⠉
⠀⠀⠀⠀⡀⠀⠀⠀⠠⡇⠀⠀⠠⡇⠀⠀⡀⠘⠀⠀⠦⠀⠃⠸⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠂⠀⠈⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠇⠀⠀⡃⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠤⠄⠂⠀⠚⠋⠁⠀⠀⠀⢀⣴⣷⡌
⠀⠀⠀⠀⠂⠀⠐⠒⠒⠓⠀⠀⢀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⢦⡜⠃⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠌⠆⠀⠀⠀⠀⠐⠀⠐⠒⠀⠂⠀⠠⠴⠂⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠠⢀⢔⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⣀⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣶⡆⢰⣶⣶⣶⣶⡆⠀⠀⣶⣶⣶⣾⡟⠉⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠸⣿⣿⣿⣿⠇⠀⠀⢿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⠠⠤⢤⣄⣤⣄⣀⣀
⠀⠀⠀⠀⠀⣾⣾⡿⣤⣀⣠⣬⠒⠂⠀⠛⠛⠳⠾⢿⣛⠷⢶⣶⣆
⠀⠀⠀⠀⠀⠉⢁⡺⡇⠸⡟⢉⣉⣉⣿⣷⠒⠦⢤⣀⠀⠈⠘⠗⠛⠥⠆⠀⠀⠁⠀⢰⣢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡛⠛⠋⠉⠀⠀⢀⡀⠉⣿⣶⣥⣠⣦⣤⣵⣶⢷⣶⠚⠻⢿⡇
⠀⠀⠀⠀⠀⢠⢰⠿⠇⣾⣷⠀⠀⠀⠀⠀⠀⠘⣿⠠⠀⠙⠙⠿⠃⠻⣿⣿⡏⢸⡏⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠊⠐⠿⠋⢈⣩⠀⠀⠀⠀⢯⡂⠀⠉⠁⠀⣷⡆⠀⠀⠀⡿⡿⠁⠘⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠿⠀⠀⠀⠀⠸⣙⡆⠀⠀⠀⣉⡃⢐⣻⠀⢱⣷⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⢠⣴⣿⠀⢷⠀⣀⠀⠀⠀⣏⠥⣤⠀⠀⣿⠃⠘⠋⠀⢰⣇⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡴⠏⠙⠀⣿⢾⡏⣹⢦⠈⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⣷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣟⡆⠀⠀⢻⠀⠈⠛⠃⠀⢀⣀⢀⣀⠀⠅⠀⣠⡀⠀⠀⡇⠀⠀⣿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠶⢆⡀⠀⠀⠀⢈⢻⣀⡉⠁⠀⣤⣏⢉⠅⠀⣿⡆⠀⡏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢤⣀⡀⠀⠉⠁⠀⡇⠀⠙⠀⠀⣿⠄⠀⡇⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠑⠒⠶⠧⣀⣀⡐⣺⣿⣿⣰⣅⠤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠴⠒⠂⠀⢋⢉⣀⣀⡀⠇⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⢈⠀⠀⠀⠀⡤⠄⠤⢬⠀⠀⢠⠂⣉⢀⠉⡏⠉⠉⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡔⠐⠂⠐⠄⠘⠂⠉⠹⠄⠭⠭⡤⠶⠀⠴⠆⠀⠀⠀⠀⠓⠲⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⣠⡀⡤⠦⠐⠒⢲⠀⠀⠀⠀⡇⠀⠀⠀⠛⠭⠙⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡅⠠⠤⠤⠴⠀⠴⠶⠚⠒⠒⠒⠒⠛⠂⠈⠉⠉⠉⠉⠉⠁⠒⠚⠣⠤⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠐⠂⠀⠀⠔⠄⢀⡤⢀⡀⡔⡩⠭⠤⡤⢔⣒⠒⠒⠒⢠⠀⢀⠠⠐⠂⠉⠉⢉⠝⡉⠉⡉⠑⡀⢘⠉⠉⣉⠙⠛⠒⠦⡙⠢⡀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠉⣀⡀⡟⠰⠡⠂⢉⣄⣀⡁⠑⢢⢠⠃⠠⢃⠔⠉⠀⠀⠙⣥⠉⠀⡀⠸⠈⢡⠈⣞⠉⡩⠓⡉⢁⠒⢌⢆⠘⣆
⠀⠀⠀⢀⠀⠀⠀⠀⠘⠃⣟⠅⠔⠋⠀⢀⢈⡽⠩⡎⠀⡤⢀⠔⠊⠁⠈⠓⢷⠰⠷⠾⢼⠤⠚⡀⢉⡸⡐⠁⢀⠉⠁⡈⠌⡆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠻⢦⣛⡁⢋⡎⠀⣀⡇⠀⣧⠁⢄⠰⠄⠲⢣⠜⠀⠀⡀⣨⠦⠚⡇⢈⠀⠠⠃⠒⠖⡤⣊⠤⢲⡨⠇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⣐⣞⣋⠈⠁⠨⢋⠔⡆⠀⣧⡳⡤⠜⠀⠐⠗⢤⣀⠀⢀⠉⢰⠉⡇⢸⠂⠀⢂⠴⣋⢭⣔⠬⠜⣃⠄⣈⠧⠤⠄⠐⠒⠢⢄
⠀⠀⠀⢀⡠⠤⠐⠒⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠒⠃⠀⠘⣄⡇⠀⠠⠐⠀⢁⣁⠁⡼⠄⠼⠧⠳⠞⠒⠚⠒⠒⠒⠚⠒⠒⠛⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⣰⣅⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠐⠉⢡⠋⠀⣋⠉⠀⠀⠀⢀⠒⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠒⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠇⠀⠤⢄⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⣀⠤⢘⣃⣀⡒⠢⣄⠀⠀⠀⠀⢸
⠀⠀⠘⡄⡎⢹⡆⠆⠂⠂⠐⠀⢖⡭⠑⣼⠀⠀⠀⠀⠀⠀⢀⡠⢶⡶⠿⠿⠿⠶⣖⠤⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠜⡡⠚⠉⠀⠀⠈⠳⡌⢣⣀⣀⠠⠤⢀⡀
⠀⠀⠀⢹⠘⠛⠀⠄⠀⠀⠀⠀⢜⠠⡢⢹⡆⠀⠀⠀⠀⢠⢋⡞⠁⠀⠀⠀⠀⠀⠀⠑⣜⣆⠀⠀⠀⢸⠀⠀⠀⠀⡈⣼⠁⠀⡠⠒⠲⡄⠀⠸⡘⡇⠀⠀⠀⠀⡁
⠀⠀⠀⢀⠦⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⡞⠀⠀⡠⠐⠒⠒⡄⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠸⡁⠀⢀⡇⠀⢠⣇⡇⠀⢀⡠⠞
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠙⠒⠒⢒⠖⠂⠙⠒⠒⠒⠚⠷⡇⠀⠀⢇⠀⠀⡠⠁⠀⠀⢸⢸⠤⠤⠤⠴⠖⠒⠒⠂⠻⡻⡀⠀⠈⠛⠉⠀⠀⣸⠉⠉⠉
⠀⠀⠀⠈⠢⢀⡀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⣀⣇⠀⠀⠈⠉⠉⠀⠀⠀⢠⣯⠏⣀⣀⣀⣀⠠⠤⠤⠤⠄⠓⠙⠢⢤⣀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠚⠦⠤⠤⠤⠔⠒⠊⠁⠈⠳⢤⣀⣀⠀⢀⣀⡴⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠦⠤⠤⠤⢤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡀⠀⣸⠁⠀⠀⣴⠶⠦⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠞⠋⠉⠉⠙⣛⣿⠀⠀⢰⣧⣤⣤⣀⣀⣀⣀⣉⡉⠙⢿⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣁⣀⣀⢀⡤⠖⠋⠁⣏⠀⠀⢸⠁⠀⠀⢩⠭⠭⠭⠭⣿⠛⣿⠀⠀⢸⣇⣀⡀
⠀⠀⠀⠀⠀⣀⣴⠞⠁⠀⠀⠀⣠⠜⠋⠀⣀⣴⡚⠙⠒⠶⠛⠀⠀⠴⣯⠤⠤⠤⠴⠇⠀⣿⠀⠀⣸⠃⠉⢙⣿⠶⠶⠶⣶⣤⣤⣄⣀⡀
⠀⠀⢀⣤⣞⡉⠈⢣⠀⢀⡴⠚⠁⢀⣤⠞⠉⠈⠉⠙⠛⠒⠶⠤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠻⣤⣤⠟⣀⡴⠋⠀⢀⣤⠞⠋⠀⠀⠉⠉⣟⠛⣶⡀
⠀⠀⠀⣯⠉⢙⡿⠛⠛⣿⠶⠶⣶⣯⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⡼⠿⠯⣍⡙⠛⠛⠓⠒⣶⠞⠋⠀⣠⣾⠋⠀⠀⠀⠀⠀⢀⣠⣿⠟⢻⡇
⠀⠀⠀⢹⡠⠋⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠉⠉⠙⠛⠛⠿⠿⠶⢾⣿⣤⣤⣤⣞⣁⣀⠀⡠⠞⠁⠀⣠⠞⠁⠈⠉⠙⠒⠦⣤⣴⠟⠉⠈⠳⣾⡇
⠀⠀⠀⢸⠁⠀⠀⠀⠀⢿⠀⠀⢸⠀⢀⣠⠴⠖⣶⠀⠀⠀⠀⠀⠈⢷⠀⠀⢻⡉⠉⠙⢻⠟⠶⢶⣾⣥⣴⣞⣛⣛⣛⣿⣿⠉⢇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⠀⢸⠀⢸⡀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⣇⠀⠀⣳⠀⠀⢸⠀⠀⠀⡇⠀⢹⡉⠉⠙⣿⢀⡏⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡷⠊⢹⠀⠀⢸⠀⠀⢸⠀⠘⡇⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⠀⠀⢸⠀⠀⠀⡇⠀⠀⢳⡀⠀⣿⣸⠁⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⢸⡆⠀⢸⠀⠀⢸⠀⠀⡇⠀⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠙⢦⣿⡏⠀⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⡇⠀⠈⡇⠀⢸⡇⠀⢸⡄⠀⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⣠⠴⠶⢤⡀⠀⠀⣸⠀⠀⠀⡇⣀⡤⠶⢦⠀⢹⡅⠀⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣧⠀⠀⣿⠀⢸⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⡀⠙⢷⡀⣿⠀⠀⠀⣇⢻⠀⠀⢸⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠉⠉⠁⠀⢸⡇⠀⠀⣧⣠⣤⣀⣀⣀⣠⡤⢤⡄⠨⡇⢈⠀⠀⠘⡆⠈⡇⣿⠀⠀⠀⣿⠘⣇⠀⢸⡄⢸⡇⠀⠀⢸⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢳⡈⠓⠶⠚⢀⣼⠃⣿⠀⠀⠀⣿⠀⠙⠛⠛⠃⢸⡇⠀⠀⠸⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⠀⠀⣿⠓⠛⠛⠛⠛⠛⠒⠛⠀⠀⠀⠉⠓⠒⠚⠋⠁⠀⣿⠀⠀⠀⣿⠉⠓⢦⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⢻⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⢼⣆⠀⡇⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠙⢦⡀⠀⢸⡄⠀⢸⡄⠀⠀⢠⡴⠶⠖⠚⠋⢹⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠘⠉⢠⠇⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢹⣀⠀⠹⣄⠸⡇⠀⠸⡇⠀⠀⠀⣇⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣻⠶⠖⠋⠀⠀⢸⡇⠀⠀⠀⢹⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠈⠉⠛⠓⠾⠶⢷⣤⣤⣧⣀⣀⡀⢹⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⢠⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠶⠶⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⡆⠀⢀⣯⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⣤⣄⣀⣿⠀⠀⠀⢸⡆⠀⠀⣠⠞⠁⣿⠱⣄⠀⣧⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⢤⣼⣇⣠⣞⠁⠀⠀⣏⣠⡼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠟⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠖⠲⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠳⣤⠶⠶⠶⢤⡞⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡇⠀⠀⠀⠀⢹⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⢸⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⢼⣇⡀⠀⠀⠀⢸⠋⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢶⠖⠛⠓⠚⠓⢷⠒⠚⠛⠛⠛⠛⢛⠳⠶⢤⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡌⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠐⠈⡡⠄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡇⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠁⠀⠠⠀⣀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡷⠖⢦⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡩⠴⠒⠀⠈⣧
⠀⠀⠀⠀⠀⠀⡇⠀⣠⠏⠐⠛⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠁⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⡇⢠⠏⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡤⠀⠘⡇
⠀⠀⠀⠀⠀⠀⡇⠈⠛⠒⠦⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠤⠤⠤⠤⠤⣶⣶⣒⣒⣦⡀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣼⠵⢒⣛⣉⣉⣉⣉⣉⣉⣀⣈⠉⢻⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⠏⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡈⡇⠀⠀⠀⢀⣴⡚⠒⠒⣒⡒⣶⠶⣄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠘⡇⠉⣿⠉⠉⠙⡇⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡧⠷⡶⡶⢶⠶⠷⠶⠿⠶⠶⠶⠿⠶⠿⢶⠶⣶⠶⠶⠶⡶⢦⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠦⣄⣁⣠⣬⣭⡭⠭⠭⠭⡾⠃⠀⡜⢰⠃⠆⠉⠀⠀⠘⠀⠀⠘⠁⠀⠁⠈⡆⢸⠀⠀⠀⠹⡆⠹⡆
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡇⠀⠀⠀⢿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡟⠀⣿⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⡛⢳⡆⠀⠀⢸⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠞⠑⣠⠖⡁⠀⠀⠀⠀⠀⠀⠈⡄⠀⡇⠀⠙⡷⣟⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠘⡟⢻⠀⠀⠀⢸⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠠⠊⡠⠊⡡⠂⡠⠀⠀⠀⠀⠀⡇⠀⣧⠀⠀⡇⢿⠀⠀⢰⠞⠋⢁⣹⠄⠀⠀⠀⠀⡇⢀⡇⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣈⣀⣈⣀⣀⣀⣀⣀⣀⣷⠀⢸⡄⠀⣇⢸⡀⠀⠀⠳⠚⠉⠀⠀⠀⠀⠀⠀⡇⢸⠇⠀⠀⣾⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣭⡏⠉⠉⠉⠉⠉⠀⠘⢆⠀⢳⡀⢹⠈⡇⠠⠄⠀⠤⠀⠀⠤⠀⠀⠤⠀⡇⡼⠀⠀⠀⡟⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠑⠦⠷⠼⢤⣽⣶⣶⣶⠒⠒⠛⠛⠛⠛⠛⣾⣿⣿⣶⠚⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣀⣀⣀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠋⠉⠉⠉⠉⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⠛⠛⠛⢻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⠀⣤⣀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⠀⣿⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⡟⣛⣃⣘⣛⣋⣀⣿⣀⣀⣀⣼⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡟⢺⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣸⣿⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠋⠉⠉⠉⠉⠁⠀⠈⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠁

⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⢠⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⢠⣾⠀⠀⢀⠀⡀⠀⠀⠀⢐⣅⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⢠⣴⠄⢀⣤⠀⠀⠀⠀⠀⠀⠉⠭⠁
⠀⡃⠉⠄⠢⠈⠢⠐⠀⠀⠀⠀⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⢀⣹⡀⠀⠀⡀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣟
⠀⠷⡄⠈⢃⠀⡀⠀⠈⡄⠀⡀⢘⠀⠀⠀⠀⠀⠰⣳⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠐⠋⠙⠁
⠀⠂⠀⠀⠀⠃⠈⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⠀⠙
⠠⣄⣤⡀⠀⠀⠹⣿⣆⠀⠀⠀⠀⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⢀⣶⡄
⠀⠛⠀⠈⠋⠀⠀⠈⠛⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⠁⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⢠⠀⢠⠁⠢⡀
⠀⠀⠈⠐⠂⠘⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⣀⡤⠞⢱⠀⠐⠘⠀⢘⠀⣀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠐⠀⠲⣾⣄⠈⠀⠀⠸⠀⠰⢸⡇⠀⠀⠠
⠀⠀⠀⡆⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⣶⣿⣷⠀⡆⠈⡂⢀⡾⠁
⠀⠀⠀⠕⢂⠀⠐⠫⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⡤⠤⠀⠀⠀⠙⢄⠀⣶⣾⣿⣿⠟⠀⢉⣢⡷⠋
⠀⠀⠀⠀⠁⠀⠠⠂⢹⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠩⠭⠤⠒⠒⠉
⠀⠀⠀⠄⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⢟⣀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠁⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⡀⠂⠀⠀⠀⠠⠀⠀⠈⠀⠀⡀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⢹⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠂⢈⣱⠄⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣀⠴⠔⠒⠋⠉⠀⠀⠀⠀⠀⠓⠠⠀⠀⠀⠀⢁⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⠟⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠰⣟⢦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣯⠁⠀⢸⣻⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠈⠳⣄
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢠⡜⠋⢉⡠⠍⣢⡀⠀⠀⠀⠀⠀⢠⠟⢓⠛⣆⠀⢸⢸⠀⠀⠀⠀⠀⢠⣿⣤⡀⠀⠀⠀⢠
⠀⠀⠀⣠⠾⠷⣄⠀⠀⠀⠀⠀⠀⠘⢕⡎⠱⠶⠰⢆⡹⠂⠀⠀⠀⠀⡟⠘⠆⠁⢸⠀⢸⢸⠀⠀⠀⠀⠀⠼⣿⣿⣿⣷⣿⣽⠿
⠀⠀⠶⣥⡀⣀⣈⡟⠀⠀⠀⠀⠀⠀⠀⠙⢦⠴⠚⠁⠀⠀⠀⠀⠀⠀⣧⣄⣫⣀⣼⠀⢸⢿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠁
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢤⣀⣠⠼⣿⣿⣿⣿
⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⡴⠋⡠⠄⣠⠉⠲⠤⠒⡆⠀⠀⠀⠀⠀⢀⣠⠔⢒⣓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣻⠻⡟⢿⡟⣻
⡟⠂⠀⠁⠀⠒⠃⠀⠀⡇⠀⠀⠀⢠⠖⠉⣴⣪⢞⠧⠃⣨⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠠⠄⠠⠀⠤⢸
⡇⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢸⡑⠠⡐⢋⠡⠐⣨⠔⠁⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⠀⠁⠆⠀⠀⣼
⢣⡀⠀⠀⠀⣠⠶⠤⠚⠀⠀⠀⠀⠘⠽⣦⣸⣥⠖⠟⠁⠀⠀⠀⠀⠀⢠⡴⠊⠩⠻⠷⠀⢈⣿⠀⠀⠀⠀⠀⠸⡇⠀⠇⠀⢀⡇
⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢭⣢⣖⡶⠖⠿⠛⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁
⠀⠀⠀⢀
⢠⠾⣏⢻⣻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠙⠢⢤⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣺⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠛⢹⠆
⠀⢳⡙⠆⠁⠁⣻⣇⠀⠀⠀⠀⣠⠶⠊⠀⠀⠀⠀⠀⠀⣋⢳⠀⠀⠀⠀⣾⢣⣾⠽⠒⠉⠙⢢⡀⠀⠀⠀⠀⠀⣠⠜⠛⠻⢄
⠸⣍⠃⠀⠀⠀⠘⡟⠀⠀⠀⠀⣻⠁⠂⠄⡀⠀⡀⠤⠊⢀⣼⠀⠀⢀⡔⠻⣸⠀⠀⣀⣴⣾⣯⠗⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠳
⠀⠀⡏⠢⡀⠀⠀⠘⢆⠀⠀⠀⠻⣦⣄⡀⠀⠸⠀⣀⣺⠟⠁⠀⠀⣸⣱⣷⣤⡴⣟⣫⠜⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈
⠀⢸⣇⣀⢡⣀⣤⠶⠋⠀⠀⠀⠀⠀⠙⠺⢶⣶⡬⠚⠁⠀⠀⠀⠀⠈⠉⠳⠿⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⡀⠀⠀⣀⣀
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠁
⠀⠀⠀⣀⢠⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⢣⣀⣠⢾⠁⠀⠀⠀⠀⡖⠒⣆⣀⡀⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⣠⠴⠒⠒⠒⠦⢄⠀⠀⠀⠀⠀⠀⢘⡟⢻⣵⣟⣲⡄
⠀⠀⠀⢸⠟⢻⠀⠁⠀⠀⠀⠀⠹⣄⡟⠀⠉⠉⠉⠉⠉⣧⢰⡇⠀⠀⠘⣿⠒⠶⠶⠶⠶⣺⠋⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠉⢳
⠀⠀⣴⣁⡀⠈⢢⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⢹⠦⢄⣀⡤⠖⡿⠀⠀⠀⠀⠀⠀⡧⠉⠉⠉⠉⠉⢹
⠀⢀⣿⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠤⠤⠖⠊⠁⠀⠀⠀⠀⠀⠈⠷⣤⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⡗⠤⠤⠄⠀⠤⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⢀⠤⣤⢖⣑⠦⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣴⠄⠠⠖⡶⠶⠶⠶⠦⣴⡏⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⣷⡄
⠀⠀⠀⠛⠒⠁⠀⠁⠀⠀⠀⠀⠀⠀⢰⣿⣅⣀⣮⠇⠀⠀⠸⣇⣀⣀⡀⣀⣿⠇⠀⠀⡴⠋⢉⠓⢦⠔⠊⢹⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠉
⠀⠀⠀⠀⠀⠀⡤⠒⠈⢰⠂⢄⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠙⠿⠿⠤⠟⠀⠀⠘⠖⢺⡇⢺⡠⠀⠈⢸⠀⠀⠀⡀⠐⠠⠆⢀⠤⣠
⠀⠀⠀⠀⠀⠾⣤⣐⡴⠧⢀⡸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⣇⡼⠧⢄⣁⣀⡶⠟⠀⠀⠨⠗⣟⠀⣶⢤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⣈⠗⢶⡎⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⡚⠛⡳⢾⡆⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠠⡀⠙⠲⣼⡇⢀⠊⠀⠀⠀⠤⠤⣄⣀⠀⠀⢀⡞⠁⢰⣶⠈⠺⡅⠀⠀⠀⠀⠀⢠⣤⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠁⠁⢀⣀⠤⠚⡁⠀⠀⠨⠃⠀⡸⡄⠀⠀⢀⠀⢸⡇⠀⠀⢀⠠⢴⡏⡈⢁⣦⠬⡑⠲⣦⡀⠀⠀⠀⠀⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⠒⠁⠁⢄⡐⠁⠀⠀⠃⡷⠀⠠⠀⢸⢚⠀⠀⢠⣅⠴⠋⠐⠀⢘⣧⣀⠝⠈⠋⠀⠀⠀⠀⠘⠇⢃⡟
⠀⠀⠀⡤⠶⣤⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠈⠀⠀⠀⠀⠑⠀⠀⠀⠐⠊⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⡇⠀⡏⠀⠄⠂⠐⠂⠤⡄⠀⠀⠔⠒⠂⢤⡀⠀⠀⠀⢀⠤⡄⠀⠀⠀⢀⣤⡀⠀⡤⢤⠀⠀⢀⠤⠤⢤⠀⠀⠀⠀⡤⢤
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡖⠈⠀⠀⠀⠀⢊⠀⡄⡑⠀⠀⠀⠀⣴⠰⡀⠀⠀⠘⡀⢳⡀⠀⡎⠀⠀⠸⠐⡆⠉⠀⠀⠀⠀⠀⢸⠀⠀⠴⠶⠶⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢈⠀⢰⠁⠀⠀⠀⠀⡙⡀⢃⠀⠀⠀⠐⠀⠃⢸⠃⠀⠀⠀⠈⡉⠵⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡈⠁⡀⢃⠀⠀⢀⠸⠁⠡⣘⠆⠀⠀⠀⢂⢀⡇⠀⠀⠀⠀⠠⠇⢤⠀⠀⠀⠀⡀⠘⠃⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⠽⣆⠈⠁⠀⠈⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢆⠲⢸⠂⠀⠀⠀⠰⡏⠉⣹⣦⣤⡔⠀⠀⠀⠀⠀⠀⠉⢉⣢⡀⠀⠀⠀⠀⠀⢠⠉⠃⠒⠂⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⢋⡁⣀⡘⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢛⡈⣡⣼⠩⠶⠀⠀⠀⠰⣤⣤⣀⡤⠛⢃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡴⢲⡄⠀⠀⠀⠀⠀⠀⢠⠰⢖⡄⠀⠀⠀⠘⠂⠐⠚⠀⠀⠀⠀⠀⠀⠀⠑⠐⠛⠃⠀⠀⠠⠀⠈⠁⠀⠀⠀⠀⠀⠉⠻⠆
⠀⠀⠀⠀⠀⠀⠙⠛⣁⠀⠀⠀⠀⠀⠀⢾⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠯⠷⣾⠄⡀⠀⠀⠀⠀⠀⠀⠀⡐⠆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠤⠀⡀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⢀⡉⠉⠀⠀⠀⠀⠀⣄⣾⣿⠟⢉⣗⠀⠀⠀⠀⠀⠄⠁⠀⠀⠠⡀⠂⡀⠀⠀⠀⠀⠈⡠⢔⠁
⠀⠀⠀⠀⠀⡤⠦⣧⠀⠀⠀⠀⡄⠀⠈⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠛⠃⠀⠀⣀⣀⠀⢀⠀⠀⠀⠈⢱⣄⠘⠂⠀⠀⠀⠈⠁⠀⠃
⠀⠀⠀⠠⢬⡅⠀⡟⠀⠀⠀⠀⣵⡟⠛⠛⢿⡆⠃⠇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢨⣿⣿⠄⣀⡴⣦⡀⠀⠀⠉⠂⠨
⠀⠀⢰⡎⠘⢯⠉⠀⠀⠀⠀⠀⣿⡦⠀⠊⢸⣧⣠⠀⠀⠀⢀⠀⠀⠐⡄⢨⠄⠀⠀⠀⡇⢸⠀⠀⠨⠗⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣰⣀⣠
⠀⠀⠀⠑⠄⠘⠀⠀⠀⠀⠀⠀⠘⠃⠓⠒⠟⠉⠉⠀⠀⠀⠘⡆⠀⠠⠥⠘⠃⠀⠀⠀⢹⣽⡗⢰⣛⣦⠂⠀⠀⠀⠀⠀⣴⣰⣂⣴⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠔⠚⠁⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀⠀⠀⠀⣠⣤⣴⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠇⠀⠀⠘⢿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠧⣀
⠀⡠⣶⣆⣄⡙⡮⣦⣠⣠⣄⠶⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠢⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢
⠂⠀⠀⠀⠘⠐⠂⠈⠁⠉⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⠠⠤⠤⠤⠄⠐⠀⠀⠠⣤⣤⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⢄⡀⠀⠀⢒⠀⠀⠀⠀⣤⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⢤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣹⢗⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠀⠀⠀⠀⠈⢂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢃⣸⢢⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠘
⠀⠂⠀⠂⠄⠠⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠠⠤⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠁⠀⠀⠀⠤⠄⠀⠀⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⠄⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠞⠇⠀⠀⠀⠀⠀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠶⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⠀⠈⠉⠉⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠂⠃

⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⡀⠈⠁⠒⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠀⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⠠⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠦⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⡣⠤⠾⠃⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⡀
⠀⠀⡀⢀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠋⢀⢤⠴⠆⠋⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠃⠈⠂⠓⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⢀⠁⠄⠀⠀⠒⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⢀⠀⢀⣀⡠⠀⠀⠖⠂⠈⢁⠀⠤⠀⠒⠒⠚⠁⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙
⠀⡀⠀⠀⠐⠀⠠⠄⠰⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⡀⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢩⠀⢀⠀⠀⠀⠀⢀⡀⠀⠠⠄⠒⠈⡉⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠔⠉⢀⡂⠀⢘⠀⠀⢨⠁⠀⢀⠀⣀⡠⠄⡄⠤⠈⠡
⠀⠀⠃⣀⡁⠀⠂⠈⠅⠀⠀⡰⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁⢂⠘⠈⠀⠐⢈⠀⠀⠸⠀⠀⠈⣉⠀⠈⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠌⠀⠈⠁⠀⢠⠘⠀⠀⠀⠀⠠⠾⠋⠀⠀⠀⠀⠘⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⠀⢀⠀⠀⠀⠀⡄⢀⣡⠤⠀⠀⠀⠀⠀⢀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⢀⣤⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⡍⠅⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠨⠁⠠⠤⠃⠣⠄⠸⠯
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⠄⢀⣀
⠀⠀⢀⡁⡀⢀⡀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁
⠀⠀⠊⠀⠁⠢⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠚⠀⠀⠰⡄⣈⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠇⠢⠠⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⣄⠀⠨⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢰⡄⠀⡴⠂⠀⡄⡄⠀⢰⠀⠀⠀⠀⠀⠀⠾⠁⢈⠷⠆⠀⣀⠁⠀⠀⠀⠀⠀⠰⠒⠠⠒⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠠⠀⠀⠠⠜⠀⠀⠀⠀⠀⠀⡄⠉⡁⠘⠒⢒⡓⢊⠉⠙⡆⠀⠀⠀⠀⠀⡀⢀⠀⠤⣁⡀⠀⠀⠂⠀⠀⠀⠀⠠⠒⠉⠀⠄⠠⠈⠉⠒⢄
⠀⠀⠀⠀⢀⡂⠐⠀⠀⠀⠀⠀⠀⠀⢱⠌⠥⠢⠳⠈⠁⠭⠄⢤⠄⠀⠀⠀⠀⠀⠁⠃⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠⠤⠤⠀⠤⠠⠔⠆⢸
⠀⢠⠀⠀⢈⠑⠀⠀⠀⠄⠊⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠐⠒⠒⠀⢀⡠⠋
⠀⠈⣀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠇⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠉⠙⠛⠛⠋⠁⣀⡤
⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠁

⠄⠠⠒⠙⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠙⠀⡀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠅⢠⠂⠀⠀⠀⢀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠄⠀⡄⠄⠀⠀⠈⠡
⠀⠀⠈⠉⠁⠀⠀⢠⠄⠐⠀⠐⠐⠧⠀⠀⠉⠢⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⠀⠀⠀⠀⠅⠉⠒⢤⡀⠀⠀⠀⠀⠀⡀⡀⠀⠃⠁⠀⠀⠀⠀⠈⠁⣀⠀⠀⠃
⠀⠀⠀⣤⠀⠀⠀⠀⠑⢄⠀⠀⢠⠠⠤⠄⡝⢂⠈⡆⡠⡀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⠢⡀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠑⠠⣀⠀⢀⡤⢋⡴⠋⣦⣀⠔⠉⠀⠉⠐⠢⠤⢀⢀⡀⠀⠈⠆⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠆⠰⠥⠐⡚⢡⠏⣠⠎⡔⠁⠀⠊⠉⠑⡀⠀⠀⣰⣏⡿⠢⣀⢰⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠂⢹⣁⡏⣰⢃⠎⠀⠀⠀⠢⠠⠰⠁⢠⣾⡽⠋⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠘⡀
⠀⠈⠀⠁⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠃⢀⢔⡶⢢⢄⠀⠀⣰⣧⠝⠁⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⢡⠀⠀⠀⠀⡄⠀⠀⠀⠄
⠀⠀⠀⢀⢀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢠⢃⡞⡰⢃⠊⠀⣴⣭⠋⠀⢀⣠⣴⣲⣶⣶⣢⣤⡀⠀⠈⠢⡀⠀⠀⠀⠈⡅⢀⠒⠂⠀⠈⠀⠍⣈⠂⠑
⠀⠀⠀⠑⠂⠀⠀⠀⠀⠐⠒⠠⣀⠀⠀⠀⠀⠀⠎⠀⡄⡞⣰⢃⠂⠀⣸⣛⠋⢀⣴⡿⠊⣉⠀⠀⠠⢉⠙⢾⣷⡀⠀⠈⠲⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⣀⠠⠐⠂⠀⠁⠐⡒⢄⡛⢹⠃⣦⢴⠀⠀⠴⣇⡏⡈⠀⢰⠛⡏⠀⣾⡎⢠⠊⠀⠀⠀⠀⠀⠀⠀⠹⣾⡄⠀⠀⠘⢄⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠻⣁⠀⠀⠀⠰⢂⠉⠉⠁⠀⠃⡏⢠⠇⡄⠀⠀⠀⠀⠈⠀⠀⡾⡿⠀⢰⣼⠀⡇⠀⠰⠀⠀⠀⠀⠀⠀⠀⣯⡇⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠑⠢⢤⣀⠀⢈⣹⡡⠸⠀⡇⢸⠀⠁⠀⠀⡀⡀⠀⠀⢰⣷⡆⠀⠘⣻⡀⠰⡀⠀⠚⠀⠘⠃⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠤⠚⠧⠧⠘⠺⠀⠀⠊⠀⠈⠀⠀⢸⣬⠁⠀⠀⠹⢷⡀⠀⠀⠀⠀⠀⠂⠀⢉⢾⡟⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠑⠀⠒⠀⠀⢸⣹⠀⠀⠀⠀⠈⠻⢳⣦⣤⢤⣤⣤⣺⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⣀⠀⠀⢸⠿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜
⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠺⣾⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⠒⠈
⢈⠁⡐⠂⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠋⠉⠉⠀⠀⠀⠀⠀⠈⢉⡝⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠁⠀⠀⠀⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡄⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠣⠄⣀⡀⠀⠀⠀⠀⣀⣀⠠⠤⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠂⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠔⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢀⣀⣈⠙⠛⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣍⣉⡉⠙⠳⠶⣤⣄⡉⠙⠛⠷⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢩⡟⠛⠲⢤⣄⣉⠙⠛⠶⢦⣀⣀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠉⠛⠳⣦⣄⣈⣉⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⢠⡏⢹⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⢸⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⢶⣶⠖⠛⠛⠛⠲⠶⣏⠉⠻⣼⣇⠀⠀⠀⠀⠀⠀⣿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠈⠛⢦⣄⡉⠛⢶⣄⡀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⠈⠙⠲⢤⣈⠙⠳⢦⣿⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⡀⠀⠀⠈⠛⠶⣤⣀⠈⣹⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⠀⠀⠀⠉⠛⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⣤⠴⠶⠖⠚⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢤⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠛⠙⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢸⣦⣴⣷⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠾⠿⣦⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠙⠹⣿⠀⠀⠀⢸⡇⠀⠀⠀⢰⡟⢠⡴⢾⣇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣰⡿⢷⣤⡀⠀⣿⠤⠬⠤⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⣇⠸⠶⣶⣼⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⣀⣀⡀⠐⠿⣄⡀⠙⠿⣶⣝⠛⠛⠛⠛⠀⠀⠀⢸⡇⠀⠀⢠⣟⣛⢻⠀⢸⣿⡇⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠸⡏⠉⠉⠉⣯⠀⠀⠀⠑⠲⢦⣬⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⢝⣸⠀⢸⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⣷⢿⣤⣾⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⣄⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠳⣆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⢸⣏⢉⣼⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢮⣿⣶⣄⡀⠀⠀⠀⢷⣂⠃⣸⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣄⣉⣽⠏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠙⠛⠉⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣤⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⡼⠷⠋⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣷⣤⣀⢸⣀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁⣀⣀⡀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠉⠉⠈⠉⠉⠙⢲⡴⠁
⠀⠀⠀⠀⠀⢀⡠⠤⠤⠾⠽⣛⣹⠗⢛⣦⣶⡿⠛⠓⢶⣾⣿⡖⠒⠲⠶⣶⣶⣾⣷⣶⣶⠺⠥⣤⣼⠿⠷⠒⠒⠶⠶⠶⡒⠢⡀
⠀⠀⠀⠀⠀⢸⡀⡀⢀⡠⠞⢉⠀⠀⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣼⠋⠀⠰⡀⠀⠀⣀⠉⡆⢱⠀⡇
⠀⠀⠀⠀⠀⢸⠈⡔⠁⠀⠀⢸⠀⠠⠸⠆⠀⠀⠐⣃⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠻⠆⠀⠀⠈⠢⡐⠜⠀⢰⣟⠠⡇
⠀⠀⠀⠀⠀⢸⠜⠁⠀⠀⠀⠘⠒⠒⢺⡇⠀⠀⢀⡀⠉⠉⡀⠒⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⣿⠀⠀⠀⠀⠀⠈⢦⡀⣸⢻⠀⡇
⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠂⠀⠀⢀⠼⠁⠀⠀⠀⠁⠈⠀⠀⠂⠀⠠⠀⠀⡀⠀⠘⡄⢀⠔⠋⠀⠀⠀⠀⡁⢂⠓⠂⢰⢸⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠑⠫⠃⡀⢸⢸⠀⠋
⠀⠀⠀⠀⠀⡼⠀⠀⠘⠶⠃⠠⡇⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠁⢸⢸⠀⠸
⠀⠀⠀⠀⠀⢶⠀⠆⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⢠⠄⠀⠀⠀⠀⠘⠈⠀⡀
⠀⠀⠀⠀⠀⢸⠀⡂⠐⠂⢢⠀⠁⠀⠀⠀⠐⠂⠠⢁⠁⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢘⠀⠀⠸⢀⢢⣄⠋⠀⣁⠂⣀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢁⠐⡀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⡀⠀⠀⠀⢀⠀⠀⢸⠀⠐⠲⢅⡒⣀⡙⠃⡄⡇⠀⡇
⠀⠀⠀⠀⠀⢸⠦⡀⠀⠀⠑⡀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠲⠀⠀⠀⠀⠐⠠⠀⠀⠀⠈⠀⠀⢈⠉⢂⠀⠀⠈⠐⠇⠀⠀⡁⢠⡁
⠀⠀⠀⠀⠀⢰⠀⠐⢄⠠⠂⠉⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠄⠰⡀⠀⢠⡀⠀⢸⠀⠀⠇⠀⠀⠀⠠⠐⠆⡇⠀⡇
⠀⠀⠀⠀⠀⢸⡀⢘⣀⢑⣄⣀⡃⠀⠀⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠜⠆⠈⠛⠀⠀⠀⡇⠀⢸⠄⠂⠀⠀⠀⠐⠀⠀⠐⣧⣀⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠁⠐⠒⠒⠀⠀⠤⠤⠀⠀⠀⠠⠄⠠⠰⣇⣀⣸⣤⠤⢠⡔⠁⠀⠜⢀⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣀⡎⡰⠥⣀⠉⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠁⣀⡄⠀⣐⡿⣦⣽⠧⠦⢤⡉⠢⢄⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣊⣀⣀⣀⣰⣟⣻⢶⣶⡟⣏⠈⠢⠀⠀⠀⠹⡗⢄⠑⢄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣩⠊⠀⠀⠀⠀⠈⠉⠒⢊⣷⢨⠂⢃⠀⡇⠀⠀⣰⡄⠀⠑⡄⠣⡀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠕⢢⠀⠀⠀⠀⠀⠀⢎⠁⢹⠒⣟⡁⠀⠣⡔⠉⠀⠀⠀⠀⠈⣆⠱⡀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠱⠃⢀⡤⣴⡀⠀⠀⠀⠀⠀⠀⣤⡔⣿⡄⢡⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠤⠂⠃⠈⠁⠀⠀⠀⠀⠀⠀⢈⠟⣉⣵⠀⠆⠘
⠀⠀⠀⠀⠀⠀⠀⡴⠂⣀⣀⣀⣈⠑⢢⠀⠀⠈⢦⡀⠀⢀⠤⠤⣤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠚⠓⢺⠀⡆⠀⡆
⠀⠀⠀⠀⠀⣀⣀⣇⣌⣀⣀⣀⣈⣧⣠⣀⣀⠀⠈⠉⢦⣸⣀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢸⠀⡃⢀⠁
⠀⠀⠀⠀⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢱⠀⠀⠀⠀⠈⠻⢦⠔⠒⠠⠀⡀⠀⠀⠀⠀⠀⠀⠈⠢⠒⢲⡎⢠⠁⠈
⠀⠀⠀⢠⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠈⡀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠈⢒⠠⣤⠀⠀⠀⠀⠀⠀⣸⢀⣆⣠⣍⣠⣤⣤⣴⣶⡲⠶⠶⠶⠶⠶⠶⡖⡄
⠀⠀⠀⠸⠀⠘⡀⢰⡄⠀⠀⠀⠀⢠⡆⠀⡇⠀⡇⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⢩⢘⠉⠉⠉⠉⠉⠉⠉⢛⠉⠉⠉⠀⡰⣩⡶⢶⣶⡍⢦⠀⠀⠀⠀⠀⠂⠃
⠀⠀⠀⠸⠀⠀⠉⢺⡇⠀⠀⠀⠀⢸⡟⠈⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠘⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠁⡽⣠⠊⢸⡿⠀⠀⠀⠀⠀⠀⢸⠸
⠀⠀⠀⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢄⡀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⡆⡦⡀⠀⠀⠀⠀⠀⠀⡆⠀⢀⡠⢧⠈⠛⠒⠛⠁⠸⠀⠀⠀⠀⠀⠀⡌
⠀⢀⢀⠃⠀⢀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀⠀⠀⣧⣌⡉⠒⠠⠤⠀⠰⠼⠠⠤⠐⢣⠇⢀⡉⠉⣠⡐⠂⣀⣛⠉⠁⠀⠀⠣⡀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢃⡇
⡀⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⡅⠀⢙⠉⠁⠒⠒⠒⠒⠒⠂⠈⢹⢸⠉⠀⠀⠀⠀⠀⠀⢩⡀⠀⠀⠀⠀⠱⡀⠀⢠⠅⠀⠠⠤⠤⠔⠂⢉⢠
⡯⠅⡌⠀⠀⠃⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠸⠀⠀⢁⠨⠌⠉⠁⠂⡖⠒⠒⠀⠚⠈⠉⡃⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠔⠙⢄⠚⡀⠀⠀⠀⠀⠀⠀⠈⠈
⡇⠀⡃⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠃⠑⠄⣠⣃⡙⠛⣿⠷⣧⠀⠀⠀⠀⠀⠀⠀⠇
⠣⢀⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣄⠜⢀⠒⠒⠛⠒⠒⠒⠛⠒⢢⢸⢈⠀⠀⠀⠀⠀⠀⠀⢸⡠⠊⠀⠀⠀⠘⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⢣⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠂⠊⠀⡞⣖⣶⣤⣴⣖⣀⣉⣨⣄⣠⡤⠤⠴⠶⠶⠞⠛⠓⠒⠒⠒⠒⠒⠒⠛
⣶⣶⡶⠿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢁⡀⣀⣀⣀⠁⢢
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣠⠀⢠⣤⣄⣀⣿⡁⡀⠀⠀⠈⢹⣏⢀⣀⡀⠀⣀⣀⣀⡀⢀⣀⡀⡀
⠀⠀⠀⠀⠔⠈⠉⢉⡽⠒⠉⢰⠛⠛⠁⠀⠉⠙⠉⠀⠈⠉⠙⠛⠛⠛⠉⣩⠟⠋⡍⠉⠈⠉⠈⢲⠸
⠀⠀⠀⠀⠈⢀⠆⠁⠀⠠⠀⠸⠀⠀⠀⢂⠐⠂⠀⠀⠀⠀⠀⠀⢠⠄⠀⢹⠀⠀⠈⠢⡠⠆⢈⡟⢠
⠀⠀⠀⠀⠱⠊⠀⠀⠀⠘⠀⢸⠁⠀⠀⡈⢁⠠⠐⠀⠀⠀⠀⠀⠈⠉⠉⣿⠀⠀⠀⠀⠘⠦⢸⢻⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠈⠤⠀⠁⠀⠀⠀⠀⣮⠉⢰⢸⠈⡇
⠀⠀⠀⠀⡉⠀⠀⣄⠀⢸⠀⠘⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠘⠘
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢷⠀⠀⠀⠢⢨⠀⠀⠀⠐⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠃⠀⠀⠨⠤⠈⠀⢀⣠⠄⠀⢰⠄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⢄⠈⠑⠪⢠⡌⠀⠀⢰
⠀⠀⠀⠀⣈⠑⠄⠀⠠⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠐⡀⠀⢀⠀⠁⠀⠀⠈⠀⠀⡄⢸
⠀⠀⠀⠀⢽⠠⠀⢢⣀⣀⠀⠀⣇⠀⠀⠀⠀⠈⠀⠀⠠⠆⠐⠓⠀⢱⠀⢰⠀⠀⠀⠀⠀⠀⠀⣧⢸
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠁⠀⠉⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠾⠀⠸⠤⠤⢄⣀⣀⣃⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠠⡠⠐⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⢁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⡇⠀⠀⠀⠸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠈⠀⠀⠂⠈⠉⠉⠐⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢱
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⡆⠄⠈⠀⠔⠒⠀⠀⢱⠒⠒⠒⠂⠈⡈⡆⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⠀⠀⠰⠟⠀⠣⣀⠀⠀⢀⣠⠏⠀⠈⠉⠁⠀⠠⣇⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠁⠓⠲⠬⠭⢭⣭⠥⣤⡤⠤⠤⠤⠀⠐⢰⠀⠐⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠁⠂⠠⣀⣠⣀⣤⣤⣈⣁⣀⣀⣃⣀⣀⣀⣀⣀⡸⠀⠁⠇⠀⠌⠀⠰⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⢲⡄⠀⠀⠉⠉⣁⣀⣀⣀⣾⡇⢰⣷⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣶⡆⢸⠇⢀⡀⢠⡟⠛⠉⢉⢹⣿⠛⣿⡿⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣿⣥⣿⡄⣿⢃⣴⡟⢁⢰⡟⢹⡟⢀⣿⠃⠓⣺⡆⣾⠂⣾⣦⢾⣦
⠀⠀⠀⠀⠀⢠⣄⣀⢀⣀⣀⠹⣏⢸⣧⡻⠴⠛⠧⠿⠜⠧⠼⠧⢺⣿⢀⠀⠿⠄⠻⠤⣿⡗⠾⠟⠒⠶⠤⢴⠶⠟
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠘⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠂⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⢁⣤⠀⢀⠀⠀⠀⢀⡠⠖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣤⡴⠞⣯⠀⠄⠀⢀⡤⠚⠁⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣀⣴⠞⣁⣿⣀⡆⠏⠁⠀⠀⠀⠀⠀⢇⣀⠠⠀⠂⠀⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣩⣴⠟⠋⠓⠛⠃⣷⠴⠖⠂⠀⡁⠀⣸⣀⣀⡤⠤⠴⠖⠂⠡⠤⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⢹⢋⣡⠾⢃⠀⠀⠀⡿⠦⠴⠶⠊⠀⣠⣿⠯⠤⠤⠤⠤⠥⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡥⠚⣡⠀⠀⠀⣧⡀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡴⠟⠉⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠐⢾⣅⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⣂⠀⠀⠀⠀⣀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠴⠠⡖⠀⠀⠀⠀⠀⠀⠀⢱⡑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⣶⢻⠁⡀⣀⠔⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠼⣞⣿⠾⠷⣭⣀⡀⠀⠀⠀⡖⠀⣀⣀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣫⠿⡂⠀⢸⠧⠤⠔⠀⣸⠿⠭⠭⠥⠦⠁⠀⠀⠀⢀⡀⠀⠀⠰⢁⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣾⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣕⣺⠆⠀⡸⠆⠀⠀⠈⠀⠀⠀⠀⠤⢤⣤⣤⣶⣴⡋⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠸⡀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠑⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⢦⣄⡀⠀⢀⣀⠠⠄⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡩⣶⠪⠋⣩⠤⠾⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⠀⠑⠄⠀⠀⠢⠠⠠⠌⣂⡤⢊⣿⣖⠯⠿⠑⠒⢥⡀⡆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠔⠠⠂⠀⡈⠁⠀⢀⣤⠞⠁⢀⠠⠂⢉⣀⣀⣀⣀⠀⠙⢷⣶⣷
⠀⠀⠀⠀⡀⠀⠐⠠⠀⠀⠀⠀⠀⠈⠈⠈⠀⠂⠀⠀⠠⠊⠀⢀⠆⠱⡾⠏⠉⠙⠋⠀⠈⠂⠈⠀⠻⣽
⠀⠀⠀⠀⠀⠀⠂⠒⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣠⠉⠀⠀⠈⠀⠀⠀⠀⠂⠀⠄⠂⠀⠀⠀⣘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⠤⢄⠒⡄⠘⠀⠀⢧⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠩⠄⢀⠘⠩⡆
⠀⠀⠀⠀⡀⣠⠴⠶⠶⠶⠶⠒⠲⠿⠗⠒⠒⣧⣿⣂⡑⣰⣶⣷⣶⣥⡀⡐⢆⠀⠀⣤⠀⠀⢌⠀⠀⠀⠁⡼⣠
⠀⠀⠀⠸⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⡌⡂⠁⠐⢮⠉⠀⠀⠀⠀⠀⠰⢀⠀⢀⡌⡜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⢠⢐⡄⠀⠀⠀⢀⣤⣠⢻⠆⡼⡈⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⢀⠀⠠⢈⡟⠀⠀⡂⢑⠁⠢⠈⣀⡘⣼⢈⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠡⠘⣅⡱⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⢘⠂⠠⠄⠤⢠⠐⠠⠔⠂⣇⡴
⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢤⠀⡀⢸⢟⠒⡒⠒⠖⠐⠂⢫⡴⠋
⠀⠀⠀⠸⠷⠦⠤⠤⣤⡤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⢲⠁⢞⣤⣥⡲⣯⡥⠬⠵⠖⠛⠉⠈⢀⣀⣤⡴⠶⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠑⠛⠒⠒⠺⠿⠖⠛⠋⠋⠁⠀⢀⠀⠀⠀⠒⠈⠀⠈⠁⠀⠀⠀⠂⠀⠀⠉⠊⠁

⠀⠀⠀⠠⠠⠲⠆⠀⠀⠐⠶⠤⠤⠀⠀⠀⠀⠠⠤⠤⠦⠀⠀⠠⠶⢤⢠⢀
⠀⠀⠈⠀⠐⠐⠀⡀⠀⠀⠂⠠⠀⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⣀⢁⣭⠉⠉⢉⣥⢉⡈⡂⠀⠀⠀⠐⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⢒⢒⠠⠀⢂⠀⣒⡒⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⠉⠉⠉⠉⠁⠈⠀⠅⠀⠀⠀⠁⠉⠀⠀⠈⠉⠁
⠀⠀⢀⠀⠈⠀⠀⠠⠀⠠⠈⠀⠀⢀⠀⠀⢸⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢤⠄⠦⡆⠀⠀⠀⢠⠄⢠⡄⠠⠀⠠⠀⠦⠀⠀⠀⢢⠠⠄
⠀⠀⠀⠰⠤⠰⠀⠀⠀⠈⠇⠤⠤⠀⠀⠀⠘⠀⠤⠀⠀⠀⠀⠀⠠⠰⠠⠁⠀⠀⠀⠄⠀⠀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠀⠐
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠜⠀⠀⠀⠀⠀⠀⢈⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠃
⠀⠀⠐⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⢀⡇⠀⠀⠀⠀⠀⠀⢀⢀⡇⠀⢀⣇⠀⡀
⠀⠀⠀⠤⠐⠦⠂⠀⠀⠀⠦⠆⠠⠄⠘⠀⠀⠀⠀⠀⠀⠉⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠁⠉⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢄⠡⠔⠀⠀⠐⠦⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠤⢶⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠠⠀⠈⠆
⠀⠀⠀⠀⢠⠤⠤⡾⠟⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠰⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠘⠈⠂⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⠛⣝⢛⣉⡛⣻⣛⣹⡟⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠤⠤⠤⢤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠒⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣾⠀⠀⠀⣤⢤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠙⠛⠛⢻⡏⠀⠀⢸⣧⣄⣀⡀⠀⠈⠉⠉⠛⠛⢷⡄⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣛⣉⠀⠀⢀⣠⠴⠚⢩⡇⠀⠀⣾⠋⠈⠉⢉⣛⣛⣛⣻⣿⠶⢾⡇⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠉⠉⣩⠗⠋⠀⢀⡤⠞⠷⠤⠤⠿⠀⠀⣰⣋⣀⣀⣀⣠⡟⠀⣼⠇⠀⢠⡏⠙⠛⢶⣶⢶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⢀⡴⠞⠉⢣⠀⠀⢀⡤⠚⠁⢀⣠⠞⠋⠙⠒⠶⠦⠤⣤⣀⣀⠀⠁⠀⠀⠀⠀⠀⠀⢻⣄⣀⣼⠃⣀⡴⠋⠁⢀⣠⠴⠋⠉⠉⠛⠛⡷⢶⣄
⠀⠀⠈⢿⠛⠳⢶⠶⠷⢶⣯⣤⣀⣴⣋⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣿⣛⡒⠶⠶⠶⠦⢭⣭⣡⠞⠋⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⣀⣷⡶⢿⡇
⠀⠀⠀⢸⢀⡴⠃⠀⠀⢸⠀⠈⢹⡏⠉⠉⠛⠛⠻⠷⠶⢶⣦⣤⣤⣴⣞⣁⣀⢈⣹⠇⠀⠀⢀⣴⠞⠁⠀⣠⠞⠙⠓⠲⠦⣄⣀⢀⣠⡾⠋⠙⢦⣼⡇
⠀⠀⠀⢸⡏⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⢀⣠⣤⡄⠀⠀⠀⠀⠈⠙⣏⠉⠛⢿⠛⠿⠶⢶⣾⣤⣤⣤⣞⣁⣤⠤⠤⠴⠶⣶⡾⠻⡇⠀⠀⠀⠈⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⡟⠉⠀⠀⣷⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⣧⠀⠀⢸⡇⠀⠉⣹⠛⢻⡟⠻⠷⢶⡟⢹⡇⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⢸⣇⡠⢶⡄⠀⢸⠀⠀⢸⡇⠀⢷⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢷⡤⠶⠾⠂⠀⢸⡇⠀⠀⣿⠀⠀⢳⡀⠀⢸⡇⣾⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠸⡏⠀⠘⡇⠀⠸⡇⠀⠸⡇⠀⢸⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠹⣄⢸⣷⡇⠀⠀⡇⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⣷⠀⠀⣇⠀⠀⡇⠀⠀⡇⠀⠸⣦⡴⠶⠞⠋⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢸⠀⠀⠀⣷⠀⠀⢀⣀⠈⠹⣿⠀⠀⠀⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⢹⠀⠀⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠙⠢⣄⠀⢸⠀⠀⠀⣿⢰⡟⠉⠈⡇⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢿⡤⣤⠼⠀⠀⣿⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠠⠀⠀⢦⡀⠹⣆⢸⠀⠀⠀⢹⠈⣇⠀⠀⡇⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⢸⡗⠒⠒⠒⠒⠒⠒⠋⢻⠀⠸⡇⢡⡀⠀⣠⠇⢀⡟⢸⠀⠀⠀⢸⠀⢹⣄⣀⣷⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣼⠀⠀⠙⢦⣈⣉⣁⣤⠞⠁⢸⠀⠀⠀⢸⣤⣀⡀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢸⡄⠀⠸⡏⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢸⠀⠀⠀⢸⡆⡀⠙⢦⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⣄⡀⠀⠀⠘⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡟⣷⡆⢸⠀⠀⣿⠀⠀⠀⠸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢹⡀⠙⢦⡀⠀⡇⠀⠀⣷⠀⠀⠀⢺⠛⠛⠉⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⣠⠟⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠸⣧⣀⣀⢳⡀⣿⠀⠀⢻⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡟⠋⠁⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠷⠶⠾⣦⣤⣄⣀⣇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⡇⠀⠀⣰⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠳⠶⠶⣦⣄⣀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡇⠀⠀⠀⠀⣠⢿⣧⡀⠀⢠⡇⠀⣰⣷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⢦⣄⣼⡀⠀⠀⠀⡇⠀⠀⣠⠾⠃⢸⡇⠳⣄⣈⣧⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⢶⣧⣤⣾⣁⡀⠀⢸⣇⣴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡄⠀⠠⡶⠀⠀⢀⠀⠀⠀⠀⣴⠀⠂⠀⠠⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⣠
⣍⠀⠀⠀⡀⠀⠉⠀⠀⡀⠀⠁⢦⡀⣀⠀⠆⠁⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⣴⣄⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⢈⢽⣿
⢧⠀⠀⠀⠙⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⡽
⠀⢣⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠐⠄
⠀⠈⣆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡘⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠛⠋⠀⠀⠀⠒⠠
⢡⠀⠀⠠⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁
⠂⢀⠀⠀⠂⠀⠀⠠⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠂⠀⠈⠠⣺⣉⠃
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⢸⣿⣿⠂⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⣄⣀⡬⠄⠀⠀⠀⠀⠐⠀⠀⠀⠐
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠸⡿⠇⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠠⡀
⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠸⣿⣿⡷⠀⣤⣦⣄⠀⠀⠀⠈⠀⠀⠀⠠⡀
⠀⠀⠀⠀⢀⠤⠰⠛⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠀⠀⠑⠿⠋⠀⠀⠀⠀⠀⠀⠠⠀⡄⢀⡀
⠀⠀⠀⠀⠰⠄⠶⠟⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢀⡂⡠⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠛⠀⠀⠀⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⢘⡋⠈⠤⢀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠛⡟⠃⣿⣻⣡⡞⠛⠃⣿⣤⣼⡆⣿⣄⣿⣰⠟⠻⣆⣿⠀⢰⡟⠛⣦⣴⠛⣛⡈⢷⡼⠃
⠀⠀⠀⠀⠀⠇⠀⠿⠶⠆⠻⠶⠃⠻⠀⠸⠃⠇⠘⠿⠘⠷⠾⠃⠻⠶⠊⠳⠶⠋⠘⠷⠿⠃⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣤⢤⡄⢠⠀⢠⡄⣤⢤⡄⣤⡄⢠⡀⣤⠤⢤⣄⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠸⣇⣀⡿⢸⣄⣸⠇⣿⢾⡇⣿⠻⣾⡇⣟⣛⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠉⠁⠀⠁⠀⠁⠈⠀⠈⠀⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢛⠛⡛⢛⡛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣤⣸⡇⢨⠀⡅⢨⡁⣿⣛⣛⣛⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡏⢩⠉⣉⢹⡇⠠⠀⠄⠠⠄⣿⣭⣭⣭⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣄⠀⠤⢸⡇⠰⠀⠆⠰⠂⣿⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⢸⡇⠀⢸⡇⠀⢸⡇⠘⠀⠃⠘⠂⠻⠀⠀⠀⠀⠟⢠⣤⣄⣀⡀
⠀⠀⠀⠀⣠⣴⠿⠛⠉⠁⠀⠙⠶⠏⠀⠀⠈⠁⠈⠀⠁⢠⠞⠛⠛⠛⠛⣦⠀⠀⠀⠉⠙⠳⢾⣄⡀
⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⢠⡶⠒⠛⠛⠛⠛⠛⠛⠛⠛⣷⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠀⠒⠸⣧⣼⡛⣷⣤⣤⣴⡟⣻⣦⡿⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧
⠀⠀⠀⠽⠆⢀⣠⣤⣴⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⣦⠀⠀⠀⠀⠀⠀⣤⠘⠁
⠀⠀⢀⣤⠞⠋⣽⠋⠀⣻⠀⠈⢿⡍⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣿⡄⠀⠀⠀⢠⡾⣉⢳⡄
⠀⣠⣿⣅⡀⣼⠃⠀⠀⣽⠀⠀⣼⠛⢳⣄⣽⣦⡀⠀⠀⠀⠀⠐⡏⠙⣿⢶⣼⣇⣀⣀⡀⢸⣧⣭⣼⡇
⢰⡏⠀⠈⢹⡟⠲⠶⠶⣾⠶⠶⠾⣦⣾⠉⠀⠈⠷⠀⠀⠀⠀⠈⣧⡴⣿⠋⢹⡏⠉⢉⣡⣿⣤⣄⣤⣿⣄
⣾⠀⠀⠀⢸⡇⠀⠀⠀⢾⠀⠀⢀⡿⣿⠀⠀⠀⠐⠂⠀⠀⠀⠐⡏⠙⣿⢶⣼⡇⠀⣾⢹⡇⠀⠤⠀⢸⡏
⣿⠛⠛⠛⢻⡟⠛⡛⠛⢛⠛⠛⣻⠟⣿⠛⠛⠛⢻⡇⠀⠀⠀⠈⣧⡴⣿⠛⢹⡇⠀⠿⢸⡇⠀⠲⠀⢸⡇
⢻⡄⠀⠀⢸⡇⢨⣽⣿⡟⠀⣰⡏⠀⣿⠀⠀⠀⣼⠃⠀⠀⠀⠀⡏⠙⣿⢶⣼⡇⠀⠶⢸⡇⢸⠀⣷⢸⡇
⢨⡷⢦⣴⠾⣷⠀⠀⠺⢁⣴⠏⠉⢻⠟⠶⣤⣰⠏⠀⠀⠀⠀⠀⣧⡴⣿⠛⢹⡇⠀⣶⢸⡇⠸⠀⠟⢸⡇
⠸⣇⣰⣧⣀⣸⣧⡴⠞⢿⠁⠀⢠⡟⠀⣠⡾⣃⠀⠀⠀⠀⣀⣠⡏⠛⣿⢦⣼⡇⠀⣿⢸⡟⠓⠀⠚⢻⡇
⠀⠈⠋⠉⠙⠷⢼⣷⣄⣿⣠⣴⣿⠴⠞⠉⠀⠉⠟⠳⠶⠂⣿⣉⣧⣤⣿⣛⣹⡇⠀⣿⣻⡟⢻⣛⡿⢻⣟
⣶⣶⡶⠶⠶⠶⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣷⣾⣿⣿⣾⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⡣⠤⠀⠀⢦⢠⠀⠀⠀⠀⢠⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣧⡄⠁⠀⠂⠀⠀⠀⠈⠁⠀⠀⠀⢀⠛⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⢟⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠇⠶⠄⠀⠀⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠛⡛⡵⡄⠀⢀⠠⠦⠦⠋⠡⠟⠉⢽⣛⡿⠃⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠋⡋⣝⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠂⠘⡽⠿⠻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡀⢀⠀⠀⠀⠀⠀⡴⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢳⣣⣱⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣧⠁⡀⣄⣀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠾⠾⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠓⠒⠒⠂⠠⠅

⠀⠀⢘⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠼⢖⣺⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒⠤⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⢀⠀⠀⠀⠀⣔⢈⡟⣸⢧⢬⣱
⠀⠀⢀⠠⡌⠠⠀⠀⠀⠀⠀⠀⠀⡄⢠⡀⠠⡀⠀⠀⠀⠀⠈⢀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠑⢎⠟⠩⠧⠜
⠀⠀⠤⠼⠤⠔⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄
⠺⠧⠀⠠⡄⡄⠈⠄⠀⠀⠀⠀⠋⠉⣉⠉⠐⠀⠀⠀⠀⠀⡴⡖⢀⣄⠠⢤⠀⠀⠀⠀⠀⠀⠀⠄⠀⢨
⢸⠀⠰⣀⣿⠃⢸⠀⠀⠀⠀⠀⠀⢸⣂⡇⠀⠀⠀⠀⠀⠀⠁⢹⠃⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⢘
⠀⠁⠀⠒⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠙⠋⠀⠛⠁⠀⠀⠀⠀⠀⠀⣤⣄⢀⠂


⠀⠤⠀⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠄⠐⠑⢄⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠁⠇⠀⠀⠀⠀⠀⢀⠀⠂⡀⣀⠔⠘⠀⠀⠀⠀⠐⠀⡆⠀⠀⢤⠀⠀⠀⠀⢀⣵⡷⢈⠉⣩⣹
⠐⠀⠀⢄⡀⠂⠐⠀⠀⠀⠀⡶⡂⠦⠤⠤⠀⠂⠀⠀⠀⠀⠘⢄⣓⣐⠀⠂⠀⠀⠀⠀⠀⠛⠻⠦⠾⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄
⠄⠒⠒⠈⢛⠃⠂⠆⠀⠀⠀⠀⢴⠄⠆⠍⠉⠇⠀⠀⠀⠀⠊⠉⠉⣇⣈⡀⠀⠀⠀⠀⠀⠠⠀⠤⠀⢄
⠀⠀⠀⠞⠡⠂⠀⠀⠀⠀⠀⠀⣿⢤⣤⣄⣲⡆⠀⠀⠀⠀⠀⠀⠀⠆⠁⠈⠁⠀⠀⠀⠀⠠⠀⠀⠀⠴⠂
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠸⠾⠿⠿⠶⠿⠞⠀⠀⠀⠀⠀⠀⢄⣣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠟⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠑⣆⠀⡜⢉⠁⠀⡴⠛⢉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢀⡄⠀⠀⠀⢦⠀⠀⠀⢠⠀⢀⡿⣸⠃⡆⢀⢔⠓⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⣾⠀⠀⠀⠀⠈⠓⠒⠤⠞⠀⠘⣧⠇⢀⡱⠋⡱⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⠻⣤⠔⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠎⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠁⢰⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⡘⡰⢫⡵⡄
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠸⣷⣄⡀⣸⠀⡀⠀⡀⠀⠀⠁⠀⡀⠈⠛⠐⢋⠀⠬⠚⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢉⠁⢰⠀⠀⠀⠀⠀⢰⠀⠀⠀⠰⠀⢀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣀⠀⠈⠓⠒⠦⠀⠀⠈⢁⣀⣤⣴⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣶⣶⣶⣶⣿⢿⠟⠛⡏⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣧⠇




⡟⠉⣿⣛⣻⣛⣻⣛⣛⣛⣟⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣿⣟⣿⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣐⠀⠒⢠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠄⣄
⠀⠀⠀⠀⡠⠄⠄⠀⠀⠀⠀⠀⠀⠎⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠂⣉⠀⠀⢘⡃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠘⠂
⠀⠀⢀⠊⠀⢀⠔⠈⡁⠀⢀⡀⠉⠑⠦⡀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠉⠁⠒⢠⠀⠀⠀⠀⠀⠀⠀⢀⡆⠊⠁⠰⣀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠈⠀⢠⠇⠀⠀⢁⣠⣈⠁⠀⠀⠀⢱⠀⠸⠀⠀⠀⠀⠰⠂⠀⠀⠀⠐⠒⠒⠐⠂⠀⠀⠀⠀⠀⠀⢀⠖⠋⢉⠉⠙⠢⢄⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠈⠢⠼⠀⠀⠀⠀⡻⢁⢑⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⢀⠄⠡⠈⣦⡄
⠀⠀⠀⠀⠀⠢⡀⠀⠠⠀⠤⠊⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡁⠀⠉⠁⠀⡀⢀⠛⠃⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⢓⣲⠂⠶⠔⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⡊⠴⠃⠀⠀⠀⠀⠀⠀⠸⠤⠤⡆
⠀⠀⠀⠀⠀⢠⠇⣀⠔⠃⠓⠠⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠭⠍⢉⣀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢀⠔⠁⡰⠀⠀⠀⠀⠀⠈⠀⠱⢄⠀⠀⠀⠀⠀⠀⢀⡰⡄⣀⠀⢰⡄⠤⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠐⠈⣉⢠
⠀⠀⠔⠁⢀⠐⠓⠠⡄⠀⠀⢠⠀⠚⢦⡀⠳⠀⠀⠀⠀⢀⠀⠰⠤⢀⠱⠾⠵⠶⡅⠀⠹⠋⡀⠀⠀⠀⠀⠀⠈⠑⠪⠜
⠀⠀⡁⢀⠀⠀⠀⠀⣸⠀⠀⣣⠀⠀⠀⢱⠠⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡄
⠀⠀⠀⠀⠘⠂⠒⠊⠀⠀⠉⠉⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠁⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠅⠀⡖⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠠⡀
⠀⠀⠀⠀⠀⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠈⡂⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠤⠤⠔⡃⠎⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⢔⢦⠀⣑⣬⠂⣴⠋
⠀⠀⠀⠀⠀⡀⢀⢀⡀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣄⠰⣮⠻⣟⣈⣇⣼⠋⣠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡽⣧⡿⠛⣏⣭⣸⣋⣱⣾⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠉⢠⠉⠉⣁⠀⡉⢨⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠈⡀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢌⢉⢓⠠⠀⣠⠂⠸⢇⡐⠣
⠠⠀⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠣⢬⡖⠙⠥⢡⠶⠊⠀⢀
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠄⡀⠀⠀⢄⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⢂⠔⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⢲⠀⠈⠁⠄⠀⠄⡇⠀⠨⡐⢤⡀⠚⡬⠀⠀⠀⠀⠀⠀⠠⢤⠀⠂⠀⠂⢧⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡼⠀⠀⠄⠀⢠⢀⠃⠀⠀⠀⠐⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠈⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⣆⣈⣀⣟⡀⠘⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡄⠦⡜⠁
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡃⠀⠀⠀⠀⠰⠁⠀⠁⠉⠃⡄⠀⠀⠀⡀⢤⠆⠑⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠚⠁⠄
⠓⠀⠀⠀⠒⠀⠀⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀

⠀⠀⠀⠀⠀⠀⡀⡀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢒⠀⢀⢀⡤⡶⣲⣶⣆⠤⠄⡀⢲⢀⠄
⠀⠀⠀⠀⠀⠀⠁⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡶⢞⠋⢁⠀⢸⣇⣀⣀⣿⣿⠀⠀⢀⡎⣸
⠀⠁⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⢀⣖⠀⠐⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠤⠀⢸⣷⣿⣿⡿⣿⠀⠈⠀⡇⡇⠀⢀⣀⢀
⠀⠀⠀⠀⠀⣀⠄⠈⠁⠒⠠⠤⠈⠙⠛⢃⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠀⠒⠀⠈⠉⠉⠉⠁⠛⠒⠒⠒⠁⠉⠉⠉⠉⠉
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠉⠖⡀⢁⠂⠀⠀⠁⠀⠤⠤⠄⠒⠒⢒⣚⡛⠓⠀⠀⢈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠑⡆⠈⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢉⡦⢤⡖⠀⠀⠋⠔⢈⠄⠀⠀⠀⠀⢠⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠜⠠⡀⢀⠀⠠⠀⠚⡡⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣼⣀⣠⣤⣤⣄⣀⣤⣠⣠⠇⠠⡄⠀⠀⠀⠀⢀
⠀⠀⢠⠀⢠⣄⠀⠀⠀⠈⢀⠉⢀⠌⠔⠀⢀⡁⢀⡈⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣀⡀⠀⠉⠀⠀⠀⢀⠄⠠⠒⡄⠀⡔⢀⢏⠐⠄
⠀⢀⠈⠀⠈⠁⠀⠀⠀⠠⠀⠀⠀⠀⠠⡤⢴⡇⠘⠄⠀⠠⢀⠈⠂⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢁⣴⠀⠈⢀⠄⠈⠁⠀⠀⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⠀⠜⢂⠓⠺⡅⠀⣄⠐⠀⠁⠚⠀⠀⠀⢀⠀⠱⠄⠀⠀⠀⢀⠈⠀⠀⠠⠊⠁⢀⠔⠁⠀⠀⠀⠀⠀⢤
⠀⠀⠀⢀⠀⠀⠀⠈⢁⣀⡀⠈⠀⠀⠉⠀⠈⠛⠂⠀⠀⠠⠀⠀⢀⣀⣀⡀⠀⠤⡌⠒⠀⢶⡋⠀⠀⢀⡦⢈⡐⠁⠀⠀⠀⠀⠀⠀⡐⠁⡄⠀⠐
⠀⠀⠈⠀⠀⠀⢀⡔⠁⠊⠀⠛⡖⠀⠀⠀⠠⠄⠀⠀⡌⠠⠦⢀⠀⠠⠤⠀⠀⠀⠔⠈⠐⠀⣀⣁⡀⡀⠀⠘⠋⠥⣂⠀⠀⡀⢠⠊⠀⠀⢭⡀⠐
⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⢐⠀⠀⢀⣇⠀⠀⠀⠀⠐⠀⠀⣀⣀⡀⠀⠀⠊⠹⠾⠓⢁⡴⠃⠀⠀⠀⠈⠓⠒⠁⠀⠀⢀⡀⠀⢁
⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠚⠃⠀⡀⠘⠂⠀⠀⠀⠀⠐⠈⠉⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠤⠐⠀⠀⠀⠠⠦⠄⢀⠀⢠⠓⠀⠄⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⠙⠛⠲⢦⣄⡀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠉
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠁⠀⠂⠒⠀⠀⢠⢀⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣⠀⢀⡀⢍⠙⠲⠤⠊⠀⡇
⠠⠄⠐⠒⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠶⢦⡄⠀⠀⠦⠄⠀⠀⠀⠀⠒⠒⠂⠲⠖⡖⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠐⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⢀⡀⠤⠔⠒⠈⣉⣄⠀⠀⢀⡀⠀⣐⡠
⠈⣁⠀⠠⠀⠀⠀⠀⠀⠀⠂⠀⠂⠐⢳⠀⠤⠤⠀⣒⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠷⠦⠤⠤⠀⠔⠀⠠⠔⠒⠟⠉⣁⣤⡌⠳⠓⠋⠡⠔⠂
⠠⠤⠀⣘⣀⡀⣀⣀⣀⣀⣀⣀⣈⣐⣒⣓⣀⣂⠠⠤⠀⣐⣂⣀⣀⣀⣀⣀⣀⣀⣒⣒⣲⣗⣒⣂⡀⠀⠊⠃⠶⠒⠚⠋⠀⠠⠐⠀⣊⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠚⠉⠀⢡⣀⠀⠀⡄⠀⠀⠀⠀⢀⣠⠴⠳⣄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣿⢀⣠⣤⡶⢟⡋⢁⡀⠈⠀⠀⢀⣠⠔⠋⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠸⠟⣋⣭⠄⣿⡅⠀⡇⡰⣤⠖⠋⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠄⣐
⠀⠀⠀⠀⠐⠀⠀⠀⠠⠤⠤⠀⣀⣀⣀⣀⡀⠘⣆⣴⠿⠋⣡⡆⣿⢣⣰⠃⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⡄⠤⠐⠂⠀⠉⠀⠀⠀⢱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⣠⣴⡿⢻⠃⠿⠼⠿⠠⢿⣠⣴⠤⠄⠒⠒⠃⠀⠀⣻⠀⠀⢀⣀⣀⣤⣤⣤⡒⣒⣊⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠒⠀⠠⠀⠄⠷⣾⠟⠫⣶⣼⡆⠀⠀⠀⠀⣿⣃⠀⠀⢀⣀⡴⠃⠀⢀⣿⣿⠋⣉⠉⢠⠄⢀⡄⠀⡤⠀⠤⢀⡇
⠀⠀⠀⠀⠀⠠⠄⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠰⣇⣴⡿⠛⣉⡅⠀⠀⠀⠀⣿⠙⠛⠛⠋⠉⠀⠀⣠⠾⠟⠓⠚⠒⠚⠒⠒⠓⠛⠛⠛⠛⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⡴⠚⢁⣄⠀⠀⠀⢀⣿⣄⣀⠀⠀⠤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡁⣠⣼⠟⠉⠀⠀⠀⠼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠐⠛⠋

⣿⢿⠿⠿⡿⠟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣟⣿⣿⣿⣿⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⢀⣠⡤⣀⡀⠀⠀⠄⢡⡇⠀⠀⠀⠀⠀⢀⡀⠄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠘⢷⣄⠀⠀⠀⣸⣿⣿⣷⣄⢀⠀⠛⢳⣷⠚⢷⡄⠐⠰⠸⠀⠀⠀⠀⠀⠘⠿⠦⠺⠘⣾⣿⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣵⣟⠞⠿⣷⣦⣴⣿⣏⠀⠾⣻⣾⣷⡁⣏⢸⡇⡾⠀⠈⠃⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠈⠉⠓⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡃⠀⠀⣨⢛⡷⠿⣛⣢⣭⣭⣴⣶⣶⣶⣬⣭⣍⡛⠤⢤⣴⣤⡀⠀⠠⢀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣟⣷⣤⠶⢋⡠⠴⠊⠉⠙⠋⠉⠚⠛⣫⡯⠛⢿⣿⢿⣷⣶⣬⣙⠧⣀⠖⢋⣠⡼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣓⠃⣈⡷⢭⠟⣥⠞⠉⠀⠀⠀⠀⠀⠐⢿⣿⠟⠁⠘⠒⣿⣿⣷⣼⣿⣿⣿⣷⣌⠢⣨⣏⡀⠚⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠯⠟⢲⣦⡝⢡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢻⣁⣠⠈⠁⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⡀⠠⣬⠟⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣤⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠹⣍⢋⣄⠂⠀⠀⠀⠀⠲⠆⠄
⠀⢀⣀⣀⣀⡀⠀⠀⠉⢀⡀⣁⡎⣼⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠹⣿⣿⡀⠀⠐⠒⠐⠀⠊⠁
⠀⠘⢿⣁⡏⣿⣷⣶⣠⣬⣉⡿⣸⣿⠀⢀⣾⣿⣶⣷⡆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢳⡄⠙⢦⡤⠔⠚
⠀⠀⠀⠉⠁⠛⠿⡏⣿⣿⠿⠇⣿⣿⣄⡙⠛⢙⣿⣭⣝⠳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢸⣯⡄⡄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⢲⣿⣿⣿⣿⣤⡘⣻⣿⣿⣼⣦⣤⣝⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠀⣿⡗⠳⣤⠥⡄
⠀⠀⠀⠀⠀⠀⠈⡄⠀⢸⣽⡸⣿⣿⣿⣿⣿⣧⣻⡿⠿⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠀⢾⠷⠶⢽⠦
⠀⠀⠀⠀⠀⠀⠀⢀⠄⢐⣻⡆⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⢼⣴⣀
⠀⠀⠀⠀⠀⠀⡐⠡⠈⠀⡀⣷⢹⣿⣿⣿⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⠚⢃⢻⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⣿⣿⣿⣿⣿⣿⡿⡰⠉⣹⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣦⣤⣤⣤⣤⣀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡟⢰⣦⣼⣏⣹⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣡⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣴⣿⣿⣿⣿⣿⡿⢋⠜⢀⣽⣿⣿⡍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠉⠪⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣵⠣⢠⣾⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⡙⠢⠀⠈⠒⠭⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⢩⣶⡛⡶⠄⠀⠛⠛⠿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠙⠈⡁⣤⡄⠈⡑⣶⣭⡝⢛⡛⠻⠿⠿⢿⡛⠛⠋⠩⠐⣾⡀⠈⣸⡿⠃⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠛⠀⠀⠋⠁⣾⢀⡟⣞⠋⠉⠁⣄⡀⠀⢀⣿⡗⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡸⠿⣡⣬⠀⠀⢀⡟⠁⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⣴⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⢀⠤⠀⡖⠒⢶⠞⠳⠀⠀⠀⠐⠿⠷⡖⠒⠄⠀⠤⣀⠀⠀⠀⠀⠀⠀⣄⡀⠀⣄⣠⣤⣯⣤⣤⣤⣬⣧⣤⣤⡄⠀⣤⡄
⠀⠀⠀⠀⠀⢁⠀⠀⡇⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠧⠄⢀⠈⡄⠀⠀⠀⣀⠴⠃⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⣏⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣻⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇
⠀⠀⠀⠀⠀⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠁⠀⠀⠀⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣇
⠀⠀⠀⠀⠀⢪⡻⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡿⢟⡝⠀⠀⠀⠀⠀⠈⣯⣤⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⢨⣴⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣿⢿⣷⣆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣴⣶⣶⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠔⠒⠋⠹⠋⠠⠙⠁⠋⠉⠙⠋⠸⠋⠠⠫⡛⠛⠻⣷⠀⠀⠀⠀⠀⣰⠿⠛⠛⢛⢿⡷⠾⠟⠛⠛⠛⠾⠻⣿⡿⠿⠛⠳
⠀⠀⠀⠀⠀⡆⠀⠀⢈⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠡⠄⡿⠀⠀⠀⠀⠀⣿⡀⡄⠀⠀⠿⠇⠁⠀⠀⠀⠀⠀⠀⠿⠇⡀⠐⠀⣤
⠀⠀⠀⠀⠀⠇⠀⠀⢠⣀⡄⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡶⣿⠀⠁⠀⠀⢲⡖⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠄
⠀⠀⠀⠀⠀⡇⠀⠀⠸⠈⠇⠀⠀⠀⠀⠀⠀⠸⠁⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⣿⠀⠄⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢀⣧⣤⣤⣤⠀⣤⣤⣤⣤⣤⣤⣤⡄⠀⣤⣤⣤⣤⠀⡆⠀⠀⠀⠀⠷⣿⡀⠆⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢘⣿⣿⣿⣏⠀⣮⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⣧⠀⠀⠀⠀⠀⢿⡇⡂⠀⠀⠙⠃⡀⠀⠀⠀⠀⠀⠀⠛⠓⠀⠀⣀⣀
⠀⠀⠀⠀⢈⠁⠻⢿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡗⠀⣿⣿⡿⠏⢸⠟⠀⠀⠀⠀⠀⠸⢇⠀⠀⡀⣹⣿⣀⡀⣀⡀⠀⠀⣀⣿⣇⣀⠈⢀⠈
⠀⠀⠀⠀⠀⠁⠀⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠙⠂⠋⠉⠓⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈