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04/03/2022
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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡜⠋⠁⠁⠁⠁⠁⠁⠁⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⢈⠘⠙⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠢⠀⠀⠄⠄⠀⠐⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠀⡏⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠍⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡂⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡏⠁⠀⠈⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⡀⢣⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⠀⠐⠐⠐⠐⠀⠂⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠔⠊⠁⠃⠀
⠀⠌⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠇⠀⠀⠀⠨⠀⠀⢀⠀⠂⠀⠀⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢂⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠔⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀
⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠠⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⡊⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀
⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠠⠁⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀
⠀⠚⠂⠒⠐⠠⠤⠤⠀⠰⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⣀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⡇⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⢗⡱⢄⠐⢀⡀⠴⠒⠒⣢⣀⣰⡋⠁⠀⠉⠚⠧⣀⣤⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠌⢿⣷⡿⠓⣂⢤⣢⡡⢊⠁⠿⣿⣦⣀⣀⣠⣾⣿⣿⣿⡆⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠘⡟⣽⣿⣿⣿⣿⣷⣇⣶⣶⣿⡛⣿⣏⠉⢳⡿⠛⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣯⣯⣭⣓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠈⠉⠱⠙⢿⡿⣿⣿⣿⣿⣿⣿⣻⣿⣿⠻⠿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣀⠀⠀⠘⠷⢟⣿⣿⣿⣿⣿⡇⣿⣿⣿⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢃⠀⠠⢶⠞⡁⠀⠌⠛⠿⢿⢹⣿⣿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⢁⠀⠀⠗⠂⠈⢈⠃⠀⠀⠠⠂⠘⠿⣿⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠏⠈⣇⠀⣠⠖⠁⠀⠀⣰⡁⠀⠌⠀⠘⡆⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣗⣤⣴⣶⠟⠉⠀⡘⠀⠀⠄⠈⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡿⣩⣿⠀⢂⡰⠁⢀⠎⠀⣼⡁⡌⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢁⢃⣹⠷⠿⣈⡶⠮⣀⠼⣉⣼⣅⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡌⣌⡇⠀⠀⠈⠳⣶⡉⠀⠈⡉⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠘⠠⣸⠁⠀⠀⠀⠀⠋⢇⣂⣀⣰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⡠⠂⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠋⢀⠜⠀⠀⠀⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡠⠊⠊⠀⡠⠀⠀⠀⠠⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣠⠴⠋⠁⠀⠠⠊⠀⠀⠀⢀⢃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣀⣴⠟⡩⠀⠀⠀⠀⢀⠔⠀⠀⢀⢆⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠁⠈⠀⠀⠀⠀⡠⠂⠀⠀⡠⣡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠐⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⡀⠀⠀⠀⠀⠀⣀⡀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⡟⣽⣫⢟⣽⣯⣿⣽⣾⣝⢮⡝⣭⣟⣿⣿⣿⣿⡟⠃⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⠁⠀⠈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⡷⣽⣿⣿⣿⣽⣯⣿⣿⣿⣞⣳⢯⣿⣿⣿⣿⣷⠀⠀⠀⠀⣼⡿⣿⣿⣿⣿⢃⣀⠀⠀⢠⣾⡟⠋⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣿⣿⣯⣿⣽⣯⣟⢾⣽⣻⣿⣿⣿⣿⣏⣗⣢⠔⡸⡟⠙⡓⠲⠾⣽⠊⠑⢲⠃⣸⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣞⡷⣯⢾⡹⣏⢷⣛⡾⣯⢾⣟⣷⡿⢟⡛⢫⡍⠭⢃⠱⣀⠣⢈⡁⠒⡘⠲⣄⣸⣠⣿⢽⠇⠄⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⣳⣏⢷⡹⣎⢷⣹⢯⣿⠿⢣⠜⡠⠜⡇⣯⡘⢆⠣⡐⠄⡃⠄⡃⠤⡑⠰⢿⣿⢯⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣟⣷⣯⣿⣽⣿⡟⣥⢃⠎⡴⢡⣏⢷⡜⣎⡵⢛⢣⣘⡜⢣⠑⡌⢣⢏⠼⣿⣏⠄⢀⠀⣤⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠡⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡓⢮⡑⢦⠡⣿⢪⣼⡱⣲⠥⣊⠍⣐⢢⣅⣛⡛⣎⢼⣽⣿⣮⣦⣹⡿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠈⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣆⣹⣤⣧⣜⡧⢷⣎⡕⢮⡰⣩⢆⡩⢌⣩⢍⣭⡻⣾⠀⠀⣼⡽⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢮⡵⢶⣶⣬⣿⡽⣻⣼⣛⣽⠶⣞⣻⡜⣎⣳⡼⣳⢿⠀⣰⣿⢇⡤⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⠐⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣟⡾⣣⣽⠶⡾⣿⣳⣟⣿⣎⢿⡭⣷⢾⣭⣳⡽⣶⢿⣿⠟⢉⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢏⠐⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡂⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣼⣻⣟⡷⣟⣾⢯⣟⡿⣾⣟⡿⣶⢯⡷⢿⣻⣧⣴⡿⣯⢿⡿⣿⣿⣿⣿⣿⣿⠟⡑⠠⢂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡔⡈⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢯⣟⡾⢿⡷⡿⢟⡙⢦⡙⢮⣿⣿⢯⡽⣿⣿⣗⠛⢿⣿⣿⡿⣋⠆⡡⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠰⡁⠎⠄⠡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠯⣏⠳⠁⠘⡐⢣⠘⢢⠙⣾⣿⣯⣿⣿⣿⣿⡏⡜⢠⠒⣿⠓⡔⡨⢐⠂⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠔
⡡⢓⡌⢌⠡⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠷⡌⠥⠃⡐⢡⠒⣯⢀⢯⣿⣿⣿⣿⣿⡿⣏⠼⡘⢤⣟⢣⡙⠤⡑⢂⠂⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡐⢢⠘
⣧⠣⡜⢢⢡⢉⠰⢀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢂⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣮⢣⡝⢢⠡⢀⣣⠸⢃⡾⠰⣿⠩⠙⠛⠻⣷⣭⡾⣵⡟⡌⢦⡉⢆⠱⢈⠐⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢈⠤⡑⢢⣙
⢿⣷⣌⠣⢎⠤⢃⠌⡰⢀⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⢆⠩⠿⣿⣻⣿⣿⣿⢿⠿⣿⢿⣿⣯⢭⡳⢌⡇⢢⠑⡻⢌⡩⠱⡩⣿⠠⢁⠂⡁⠀⠢⢇⡰⠘⢷⣷⡜⢠⠒⡀⢂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢂⠅⣢⠑⣣⣾
⠈⢿⣿⣾⡌⡚⣌⢒⢡⠂⡔⠂⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡⠌⢢⠑⢢⠝⢣⡙⢌⠣⠙⠤⢫⣿⣟⣶⡙⢦⡙⢆⠩⠔⢢⡘⠥⣱⡿⠐⡄⠂⠄⠀⠐⡠⠡⢉⠆⡩⢐⠂⡐⡐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡡⢚⡰⣘⠤⣿⡿⠡
⠀⠀⠛⣿⣷⣕⡌⢎⢆⢣⡐⢩⠐⠡⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠈⠤⠉⠆⡘⠄⡐⠂⡌⠰⢡⢋⣿⣗⣺⣽⢂⠳⡈⢆⠹⡠⠜⣣⣏⣿⠐⠠⠉⠄⠐⠠⠐⣁⠢⢘⠠⠁⠆⡁⠐⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡔⢰⠃⢦⣹⣿⡟⠡⠁
⠀⠀⠀⢙⣿⣿⣾⣌⢆⠣⠜⢢⠉⡔⠂⠌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢂⠉⡐⠠⠘⢠⠁⢆⡉⠦⡒⠾⣏⡾⣹⣎⢣⠱⠨⣅⢃⠇⢾⢡⣿⠀⠐⡀⠠⠀⠀⠁⠄⢂⠁⢂⠉⠄⠠⠁⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢃⠜⡠⢃⣹⣿⣿⠏⠌⠀⠀
⠀⠀⠀⠀⠑⠹⣿⣿⣎⡱⣉⠦⡙⠤⣉⠐⡠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠠⢁⠘⡀⠎⡰⢈⠃⣄⡘⣯⡳⢧⠎⢦⡉⢇⠦⡩⠜⢢⢣⡏⠀⠀⠀⠀⠀⠡⠀⠀⠀⠂⢀⠈⠠⠁⠂⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡰⢈⠆⡱⢳⣿⣿⢋⠐⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢻⣿⣷⣧⢊⡕⢣⠰⡁⢆⠱⢀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠐⢠⡑⠄⢣⠘⡁⠀⣿⣿⡬⣛⢤⡉⢆⡣⢱⡉⢦⣻⢟⠀⠀⠀⠀⠐⠀⠀⡀⠀⠈⢀⠀⠂⠌⠐⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⡐⠰⣈⠲⣽⣿⡟⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⣎⠥⣃⢍⢢⠑⡌⡐⠠⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠄⡉⠞⠀⢁⠀⣿⡷⣿⢜⣢⠙⢦⡑⣣⠜⣲⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠐⠀⠂⠄⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢢⠐⡨⠑⣤⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⡌⢆⠣⣍⠰⣀⠃⡄⠢⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢂⠁⠂⢁⢢⠀⢻⣿⣿⣎⠶⡙⢦⠱⢎⡾⣣⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠌⡄⢊⠔⣩⣿⡿⢁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣌⠱⣂⠧⣀⠣⢠⠡⠌⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠐⡈⠤⠁⠂⠈⢆⠉⣿⣷⣏⢞⡙⢦⡛⣬⠳⣽⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⢀⠀⠂⠄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠄⡌⠰⢀⠣⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣷⡐⠦⡑⡌⢆⠒⣈⠐⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠡⠐⡀⠈⠀⠡⢨⠀⠿⣯⢻⣮⣝⣦⣹⣲⠿⣹⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡌⢧⠈⠄⠂⠌⠠⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠠⢈⡐⢠⠃⡜⣶⣿⡋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⣿⣴⠱⣘⠢⡑⢄⠒⡠⠁⡌⠠⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠄⠀⠐⠀⠁⠀⠐⠁⡩⣇⠷⣾⣛⡭⢷⣺⢗⣿⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⣿⡈⣇⠈⠤⢁⠂⠌⢀⠈⠄⡐⢀⠀⠄⠀⡀⠠⠀⠂⡀⠐⡈⠐⠠⡁⢢⠐⡡⢊⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠣⢍⢢⠱⣀⠣⠐⡄⡁⢂⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢂⠠⠀⢠⡎⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⢱⣿⢚⠴⣛⢛⢿⡫⢞⡼⡇⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⣬⡓⡌⡐⣈⠂⠆⡈⠰⢀⠂⠌⡐⠠⠐⡀⠡⠐⠠⠁⠤⢉⠰⢠⠁⢎⢰⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣫⢿⣮⡘⣂⠳⢌⡒⡡⠒⡌⠤⢉⠐⠠⢈⠐⡀⠡⢈⠐⠠⢈⠐⠠⢐⢠⠂⢌⡼⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠂⡿⣜⣲⢥⡾⣮⡕⢫⡴⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⢡⢂⡱⢈⡔⢡⢂⠉⡔⠠⡁⠆⡐⠡⡉⠤⢉⠰⡈⠔⣂⢉⣶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠍⣟⣿⣦⠓⡬⠰⣁⠣⡘⠰⡈⠤⢁⠢⢐⠠⡁⠢⢈⠐⡂⢌⠡⣆⠋⣴⠞⡀⠄⠀⠀⠀⠀⠀⠠⠁⠠⢀⠈⣱⠂⣰⠿⡟⠱⡷⠙⢦⣝⠠⢂⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡌⢖⡡⢎⡒⡌⡒⢌⡱⢠⠃⣌⠡⠒⡌⢢⠑⡌⠒⣌⣾⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣻⣿⣔⠣⠜⡰⢡⠃⡜⡠⢃⡜⢠⠃⡔⢡⠊⡔⡡⢎⡵⢡⣾⠋⠄⡐⠠⠐⠀⠀⠀⡀⠀⠀⠡⢀⠂⣴⢠⢯⢣⢱⠯⠎⡥⢰⠂⠁⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣜⡢⠵⣨⠑⡎⡔⠣⡜⢠⢃⡱⢈⠆⡱⣨⣿⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣟⣿⣧⣑⢊⡱⢢⢑⢣⡘⢆⡓⡌⢆⠳⣌⡱⢎⣜⡷⢡⢈⠒⡈⠄⡁⠂⠤⠀⠄⠀⢀⠀⠀⠁⣌⣬⡟⠶⢮⣱⣞⣱⠃⣆⣀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣖⣡⢋⠴⣉⠖⣡⠃⡆⢆⣣⣾⣽⣿⡿⢋⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠤⡈⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⣝⣿⣿⣦⣕⠪⡔⡘⠦⡱⡘⢬⠓⡴⣩⢾⠞⢠⠃⢢⠘⣀⠒⠠⢁⠂⠄⡈⠀⡀⠠⠀⢄⠩⢼⣏⣿⣿⡻⣹⠿⣾⣥⠦⠱⡌⡐⡀⢂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣾⣾⡶⣿⠻⡏⢏⠻⠿⢍⡐⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡕⣨⠐⡄⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢾⣿⡟⡿⣶⣭⣶⣥⡙⢦⡛⣴⡷⢋⠰⠁⡜⠠⠡⢄⡘⢠⠁⡘⢀⠐⠠⠐⠠⢌⡰⢉⢎⣽⠻⣿⣿⣿⣿⣟⠿⣬⣓⡰⣁⠒⡄⠂⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣸⢻⣿⠟⣧⢳⡙⢦⢋⠜⠈⠁⠀⠀⠉⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡜⢤⠣⢌⠱⡈⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠈⠓⠤⣏⡝⡻⣿⣷⣿⢫⡐⢌⠢⡑⠄⡃⢅⠂⠔⢂⠰⠀⢌⡀⢃⠌⡑⢢⠰⡉⢍⣰⡟⣻⣿⣿⡿⢮⡻⣵⢺⣵⢈⠇⡘⠠⢈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡀⢢⠐⣿⣏⠻⣔⣣⠜⠶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢉
⡝⢦⡙⡌⢦⠑⡢⢐⠡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠈⠙⠳⠇⠻⣿⡦⢘⠢⡑⢄⠣⠌⡐⠌⠒⡈⠤⠉⡄⠂⠆⡘⢄⠃⠆⡵⠋⣥⢞⢣⡙⡤⢫⡝⣿⣜⠳⣦⢍⡊⠤⢁⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠡⢀⠡⡐⢠⢻⡏⣦⠹⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠈⠁⠒⠤⢀⣀⠀⢀⠰⡁⢮
⣿⢧⡝⣜⢢⡙⡔⡡⢊⠔⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣻⡧⢑⠌⡆⡘⠤⣁⠊⢡⠐⡀⠃⠠⠉⡐⠐⡈⠌⡜⢀⠣⠆⡍⣦⢑⣣⡟⣬⢹⣞⡷⢨⠙⠤⢃⠆⠄⠁⡀⠀⠀⠀⠄⠀⠀⠠⠀⠀⡀⢁⠂⠆⢡⣿⡿⠋⠐⢆⠑⠒⠂⢲⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⣼⣿
⣿⢯⡿⣽⣦⡳⡜⡰⠩⡜⣠⠑⡄⠀⠀⠀⠀⠀⠀⢀⠠⠔⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⣅⠚⠛⢿⡌⢎⡰⢡⠒⡄⢡⠂⡐⠠⢈⠐⠠⠐⠠⢀⠁⠠⠈⠰⡁⠲⢍⠢⣗⡯⢔⡣⢾⣼⢁⠎⡑⠂⠌⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠈⠄⣙⠫⣿⣒⠀⠁⠀⠀⠀⠈⠉⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⣼⣿
⣿⣟⣿⣳⣟⣿⣽⣲⡃⢖⡡⢣⠌⣅⢂⣀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⠀⠑⠂⢺⣿⡐⠦⡑⡌⠰⢁⠂⡁⠆⠀⡀⠂⢁⠂⠄⠈⢀⠀⠡⠀⠣⢌⠲⣽⣜⡎⡖⢥⠚⣄⠊⡔⠉⠄⠡⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⡀⠈⠄⡑⠰⠠⠌⣹⣯⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⠉⠂⠸⣷⣿
⣿⣻⣯⣿⢾⣟⣿⣷⣿⣢⣑⠣⣞⡴⠚⢡⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣀⡀⠀⠀⠀⠀⣿⡗⣌⠳⣐⠡⢃⠌⢂⠡⢀⠐⠀⡀⠀⠠⠀⠁⡀⢈⠀⡁⠒⡌⣓⣿⡽⣮⡙⢦⡙⠤⢃⡌⢡⠊⠄⠡⢈⠀⠄⠁⠈⢀⠀⠄⡐⠠⢁⠒⡈⡁⢆⠡⠐⢻⣿⡀⠃⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⣴⣾⣿
⣿⣯⣷⢿⣿⣻⣯⣿⣿⣿⣾⠋⠁⠂⠀⢡⢰⣱⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣄⠉⠢⣄⠀⢰⣿⡜⢤⠓⡬⠑⡌⢄⠃⡰⢀⠊⡐⠠⢀⠁⡐⠠⢀⠂⠤⠑⡌⢲⢭⣿⣿⣷⣹⢲⡍⢎⠥⡘⢄⠊⡌⢁⠂⠌⡐⢈⠐⡀⠂⠄⠠⢁⠂⠆⡡⠌⡐⠌⡡⠘⣿⣿⣤⠀⠀⠀⠀⠀⠀⢀⣘⣦⠻⡼⢙⣲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⣀⡤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣧⠀⠈⣇⣿⡳⣜⢣⡙⠴⡩⡐⢌⠂⡅⠂⠆⡐⢁⢂⡐⠠⢁⠂⠌⡄⠣⢌⡣⢞⣿⣿⣿⣼⢣⡛⣬⠒⣍⢢⡑⢌⠢⡁⠎⡐⢂⠰⢀⠡⠌⢂⠌⠄⣃⠐⢢⠁⢎⢰⡇⢿⣿⣿⣯⡄⠀⠠⡄⠀⣿⡿⣭⣿⠉⠛⠯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠚⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠉⠻⣿⣿⡆⢰⣿⣟⣷⣎⢧⡙⣖⢡⠓⡌⠒⠤⢉⠰⠐⡈⠄⢠⠁⠆⡉⠔⡨⡑⢎⡜⣯⣿⣿⣿⣾⡳⣝⢦⢫⡔⢣⡘⠤⢃⡔⢡⠒⡈⠔⣈⠰⡈⢆⡘⠰⢠⢉⠆⡍⢦⢏⣼⣿⣿⣿⣿⣿⡄⣁⠀⠀⠈⣹⡿⠿⡿⠟⠦⠄⠉⠳⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⠤⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣄⠀⠙⣿⡇⣿⣿⡝⡾⣿⣎⠷⣌⢧⡙⢬⠱⡈⢆⠡⢊⠔⡈⢆⡘⠤⡑⡸⡐⣍⠺⡼⣽⣿⣿⣿⣷⡝⡾⢬⠳⣜⢣⡜⣡⠣⡜⠤⣃⠜⢢⢄⠣⡘⠤⡌⡱⢢⢍⡚⣜⣮⣟⣿⣿⣻⣿⣿⣿⣿⣆⠄⠀⠀⢻⣶⠿⠗⠳⣯⡳⢾⣶⣤⡽⣝⠛⢶⣶⣦⠤⠴⠒⠲⣿⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⣀⣀⣠⢾⡿⠿⢿⣿⣿⣿⡇⠀⣿⣰⣿⣿⣿⣗⡿⣞⡿⣼⢲⣙⠦⣣⠑⡎⠴⡁⢎⡐⢆⡘⢤⢃⠵⣑⢎⣳⣽⣿⣿⣿⣿⣿⣽⡹⣭⠳⣌⠳⣬⡑⠧⡜⡱⢌⠎⡥⢊⠖⣡⠓⡬⣑⢣⠞⣼⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠻⢮⣷⣦⣼⣿⣷⠟⠳⠿⠯⣤⠤⢬⣿⡀⠀⠀⢦⣿⣯⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣷⣷⣶⣦⣤⡀⠀⢹⣿⣟⠀⢸⣾⣿⣿⡿⣿⣿⡽⣾⣱⢯⣞⣱⢣⡝⣘⠦⣙⠢⣍⠲⡘⢦⡩⢖⡭⣾⣱⣿⣿⣿⣿⣿⣿⣿⣿⣶⣛⢬⡓⢦⡹⣑⠎⡕⣎⢚⠴⣉⢞⡰⣙⠲⣍⡞⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣰⢿⡷⠋⠈⠙⠢⣄⠀⠈⠉⠒⢄⡈⠑⡄⣨⡏⣿⡹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢤⣀⣤⣴⣶⣿⣍⡈⢉⡉⠉⠉⠟⠿⢿⣿⣷⠀⠀⠀⣻⠀⢩⣿⣿⣟⣿⣿⣻⣽⣳⣟⠾⣜⣣⠷⡸⢥⡛⡴⣋⠴⣋⡝⢦⡙⣎⣖⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣦⡳⡜⡸⡱⡌⡞⣬⠱⣎⣵⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣠⡿⢟⠳⢤⡀⠀⠀⠈⣷⠀⠀⠀⠀⠉⢦⠘⣦⣶⢣⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⡟⣽⣫⢟⣽⣯⣿⣽⣾⣝⢮⡝⣭⣟⣿⣿⣿⣿⡟⠃⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⠁⠀⠈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⡷⣽⣿⣿⣿⣽⣯⣿⣿⣿⣞⣳⢯⣿⣿⣿⣿⣷⠀⠀⠀⠀⣼⡿⣿⣿⣿⣿⢃⣀⠀⠀⢠⣾⡟⠋⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣿⣿⣯⣿⣽⣯⣟⢾⣽⣻⣿⣿⣿⣿⣏⣗⣢⠔⡸⡟⠙⡓⠲⠾⣽⠊⠑⢲⠃⣸⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣞⡷⣯⢾⡹⣏⢷⣛⡾⣯⢾⣟⣷⡿⢟⡛⢫⡍⠭⢃⠱⣀⠣⢈⡁⠒⡘⠲⣄⣸⣠⣿⢽⠇⠄⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⣳⣏⢷⡹⣎⢷⣹⢯⣿⠿⢣⠜⡠⠜⡇⣯⡘⢆⠣⡐⠄⡃⠄⡃⠤⡑⠰⢿⣿⢯⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣟⣷⣯⣿⣽⣿⡟⣥⢃⠎⡴⢡⣏⢷⡜⣎⡵⢛⢣⣘⡜⢣⠑⡌⢣⢏⠼⣿⣏⠄⢀⠀⣤⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠡⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡓⢮⡑⢦⠡⣿⢪⣼⡱⣲⠥⣊⠍⣐⢢⣅⣛⡛⣎⢼⣽⣿⣮⣦⣹⡿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠈⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣆⣹⣤⣧⣜⡧⢷⣎⡕⢮⡰⣩⢆⡩⢌⣩⢍⣭⡻⣾⠀⠀⣼⡽⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢮⡵⢶⣶⣬⣿⡽⣻⣼⣛⣽⠶⣞⣻⡜⣎⣳⡼⣳⢿⠀⣰⣿⢇⡤⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⠐⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣟⡾⣣⣽⠶⡾⣿⣳⣟⣿⣎⢿⡭⣷⢾⣭⣳⡽⣶⢿⣿⠟⢉⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢏⠐⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡂⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣼⣻⣟⡷⣟⣾⢯⣟⡿⣾⣟⡿⣶⢯⡷⢿⣻⣧⣴⡿⣯⢿⡿⣿⣿⣿⣿⣿⣿⠟⡑⠠⢂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡔⡈⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢯⣟⡾⢿⡷⡿⢟⡙⢦⡙⢮⣿⣿⢯⡽⣿⣿⣗⠛⢿⣿⣿⡿⣋⠆⡡⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠰⡁⠎⠄⠡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠯⣏⠳⠁⠘⡐⢣⠘⢢⠙⣾⣿⣯⣿⣿⣿⣿⡏⡜⢠⠒⣿⠓⡔⡨⢐⠂⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠔
⡡⢓⡌⢌⠡⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠷⡌⠥⠃⡐⢡⠒⣯⢀⢯⣿⣿⣿⣿⣿⡿⣏⠼⡘⢤⣟⢣⡙⠤⡑⢂⠂⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡐⢢⠘
⣧⠣⡜⢢⢡⢉⠰⢀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢂⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣮⢣⡝⢢⠡⢀⣣⠸⢃⡾⠰⣿⠩⠙⠛⠻⣷⣭⡾⣵⡟⡌⢦⡉⢆⠱⢈⠐⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢈⠤⡑⢢⣙
⢿⣷⣌⠣⢎⠤⢃⠌⡰⢀⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⢆⠩⠿⣿⣻⣿⣿⣿⢿⠿⣿⢿⣿⣯⢭⡳⢌⡇⢢⠑⡻⢌⡩⠱⡩⣿⠠⢁⠂⡁⠀⠢⢇⡰⠘⢷⣷⡜⢠⠒⡀⢂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢂⠅⣢⠑⣣⣾
⠈⢿⣿⣾⡌⡚⣌⢒⢡⠂⡔⠂⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡⠌⢢⠑⢢⠝⢣⡙⢌⠣⠙⠤⢫⣿⣟⣶⡙⢦⡙⢆⠩⠔⢢⡘⠥⣱⡿⠐⡄⠂⠄⠀⠐⡠⠡⢉⠆⡩⢐⠂⡐⡐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡡⢚⡰⣘⠤⣿⡿⠡
⠀⠀⠛⣿⣷⣕⡌⢎⢆⢣⡐⢩⠐⠡⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠈⠤⠉⠆⡘⠄⡐⠂⡌⠰⢡⢋⣿⣗⣺⣽⢂⠳⡈⢆⠹⡠⠜⣣⣏⣿⠐⠠⠉⠄⠐⠠⠐⣁⠢⢘⠠⠁⠆⡁⠐⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡔⢰⠃⢦⣹⣿⡟⠡⠁
⠀⠀⠀⢙⣿⣿⣾⣌⢆⠣⠜⢢⠉⡔⠂⠌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢂⠉⡐⠠⠘⢠⠁⢆⡉⠦⡒⠾⣏⡾⣹⣎⢣⠱⠨⣅⢃⠇⢾⢡⣿⠀⠐⡀⠠⠀⠀⠁⠄⢂⠁⢂⠉⠄⠠⠁⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢃⠜⡠⢃⣹⣿⣿⠏⠌⠀⠀
⠀⠀⠀⠀⠑⠹⣿⣿⣎⡱⣉⠦⡙⠤⣉⠐⡠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠠⢁⠘⡀⠎⡰⢈⠃⣄⡘⣯⡳⢧⠎⢦⡉⢇⠦⡩⠜⢢⢣⡏⠀⠀⠀⠀⠀⠡⠀⠀⠀⠂⢀⠈⠠⠁⠂⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡰⢈⠆⡱⢳⣿⣿⢋⠐⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢻⣿⣷⣧⢊⡕⢣⠰⡁⢆⠱⢀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠐⢠⡑⠄⢣⠘⡁⠀⣿⣿⡬⣛⢤⡉⢆⡣⢱⡉⢦⣻⢟⠀⠀⠀⠀⠐⠀⠀⡀⠀⠈⢀⠀⠂⠌⠐⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⡐⠰⣈⠲⣽⣿⡟⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⣎⠥⣃⢍⢢⠑⡌⡐⠠⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠄⡉⠞⠀⢁⠀⣿⡷⣿⢜⣢⠙⢦⡑⣣⠜⣲⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠐⠀⠂⠄⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢢⠐⡨⠑⣤⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⡌⢆⠣⣍⠰⣀⠃⡄⠢⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢂⠁⠂⢁⢢⠀⢻⣿⣿⣎⠶⡙⢦⠱⢎⡾⣣⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠌⡄⢊⠔⣩⣿⡿⢁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣌⠱⣂⠧⣀⠣⢠⠡⠌⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠐⡈⠤⠁⠂⠈⢆⠉⣿⣷⣏⢞⡙⢦⡛⣬⠳⣽⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⢀⠀⠂⠄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠄⡌⠰⢀⠣⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣷⡐⠦⡑⡌⢆⠒⣈⠐⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠡⠐⡀⠈⠀⠡⢨⠀⠿⣯⢻⣮⣝⣦⣹⣲⠿⣹⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡌⢧⠈⠄⠂⠌⠠⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠠⢈⡐⢠⠃⡜⣶⣿⡋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⣿⣴⠱⣘⠢⡑⢄⠒⡠⠁⡌⠠⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠄⠀⠐⠀⠁⠀⠐⠁⡩⣇⠷⣾⣛⡭⢷⣺⢗⣿⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⣿⡈⣇⠈⠤⢁⠂⠌⢀⠈⠄⡐⢀⠀⠄⠀⡀⠠⠀⠂⡀⠐⡈⠐⠠⡁⢢⠐⡡⢊⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠣⢍⢢⠱⣀⠣⠐⡄⡁⢂⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢂⠠⠀⢠⡎⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⢱⣿⢚⠴⣛⢛⢿⡫⢞⡼⡇⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⣬⡓⡌⡐⣈⠂⠆⡈⠰⢀⠂⠌⡐⠠⠐⡀⠡⠐⠠⠁⠤⢉⠰⢠⠁⢎⢰⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣫⢿⣮⡘⣂⠳⢌⡒⡡⠒⡌⠤⢉⠐⠠⢈⠐⡀⠡⢈⠐⠠⢈⠐⠠⢐⢠⠂⢌⡼⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠂⡿⣜⣲⢥⡾⣮⡕⢫⡴⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⢡⢂⡱⢈⡔⢡⢂⠉⡔⠠⡁⠆⡐⠡⡉⠤⢉⠰⡈⠔⣂⢉⣶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠍⣟⣿⣦⠓⡬⠰⣁⠣⡘⠰⡈⠤⢁⠢⢐⠠⡁⠢⢈⠐⡂⢌⠡⣆⠋⣴⠞⡀⠄⠀⠀⠀⠀⠀⠠⠁⠠⢀⠈⣱⠂⣰⠿⡟⠱⡷⠙⢦⣝⠠⢂⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡌⢖⡡⢎⡒⡌⡒⢌⡱⢠⠃⣌⠡⠒⡌⢢⠑⡌⠒⣌⣾⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣻⣿⣔⠣⠜⡰⢡⠃⡜⡠⢃⡜⢠⠃⡔⢡⠊⡔⡡⢎⡵⢡⣾⠋⠄⡐⠠⠐⠀⠀⠀⡀⠀⠀⠡⢀⠂⣴⢠⢯⢣⢱⠯⠎⡥⢰⠂⠁⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣜⡢⠵⣨⠑⡎⡔⠣⡜⢠⢃⡱⢈⠆⡱⣨⣿⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣟⣿⣧⣑⢊⡱⢢⢑⢣⡘⢆⡓⡌⢆⠳⣌⡱⢎⣜⡷⢡⢈⠒⡈⠄⡁⠂⠤⠀⠄⠀⢀⠀⠀⠁⣌⣬⡟⠶⢮⣱⣞⣱⠃⣆⣀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣖⣡⢋⠴⣉⠖⣡⠃⡆⢆⣣⣾⣽⣿⡿⢋⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠤⡈⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⣝⣿⣿⣦⣕⠪⡔⡘⠦⡱⡘⢬⠓⡴⣩⢾⠞⢠⠃⢢⠘⣀⠒⠠⢁⠂⠄⡈⠀⡀⠠⠀⢄⠩⢼⣏⣿⣿⡻⣹⠿⣾⣥⠦⠱⡌⡐⡀⢂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣾⣾⡶⣿⠻⡏⢏⠻⠿⢍⡐⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡕⣨⠐⡄⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢾⣿⡟⡿⣶⣭⣶⣥⡙⢦⡛⣴⡷⢋⠰⠁⡜⠠⠡⢄⡘⢠⠁⡘⢀⠐⠠⠐⠠⢌⡰⢉⢎⣽⠻⣿⣿⣿⣿⣟⠿⣬⣓⡰⣁⠒⡄⠂⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣸⢻⣿⠟⣧⢳⡙⢦⢋⠜⠈⠁⠀⠀⠉⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡜⢤⠣⢌⠱⡈⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠈⠓⠤⣏⡝⡻⣿⣷⣿⢫⡐⢌⠢⡑⠄⡃⢅⠂⠔⢂⠰⠀⢌⡀⢃⠌⡑⢢⠰⡉⢍⣰⡟⣻⣿⣿⡿⢮⡻⣵⢺⣵⢈⠇⡘⠠⢈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡀⢢⠐⣿⣏⠻⣔⣣⠜⠶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢉
⡝⢦⡙⡌⢦⠑⡢⢐⠡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠈⠙⠳⠇⠻⣿⡦⢘⠢⡑⢄⠣⠌⡐⠌⠒⡈⠤⠉⡄⠂⠆⡘⢄⠃⠆⡵⠋⣥⢞⢣⡙⡤⢫⡝⣿⣜⠳⣦⢍⡊⠤⢁⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠡⢀⠡⡐⢠⢻⡏⣦⠹⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠈⠁⠒⠤⢀⣀⠀⢀⠰⡁⢮
⣿⢧⡝⣜⢢⡙⡔⡡⢊⠔⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣻⡧⢑⠌⡆⡘⠤⣁⠊⢡⠐⡀⠃⠠⠉⡐⠐⡈⠌⡜⢀⠣⠆⡍⣦⢑⣣⡟⣬⢹⣞⡷⢨⠙⠤⢃⠆⠄⠁⡀⠀⠀⠀⠄⠀⠀⠠⠀⠀⡀⢁⠂⠆⢡⣿⡿⠋⠐⢆⠑⠒⠂⢲⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⣼⣿
⣿⢯⡿⣽⣦⡳⡜⡰⠩⡜⣠⠑⡄⠀⠀⠀⠀⠀⠀⢀⠠⠔⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⣅⠚⠛⢿⡌⢎⡰⢡⠒⡄⢡⠂⡐⠠⢈⠐⠠⠐⠠⢀⠁⠠⠈⠰⡁⠲⢍⠢⣗⡯⢔⡣⢾⣼⢁⠎⡑⠂⠌⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠈⠄⣙⠫⣿⣒⠀⠁⠀⠀⠀⠈⠉⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⣼⣿
⣿⣟⣿⣳⣟⣿⣽⣲⡃⢖⡡⢣⠌⣅⢂⣀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⠀⠑⠂⢺⣿⡐⠦⡑⡌⠰⢁⠂⡁⠆⠀⡀⠂⢁⠂⠄⠈⢀⠀⠡⠀⠣⢌⠲⣽⣜⡎⡖⢥⠚⣄⠊⡔⠉⠄⠡⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⡀⠈⠄⡑⠰⠠⠌⣹⣯⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⠉⠂⠸⣷⣿
⣿⣻⣯⣿⢾⣟⣿⣷⣿⣢⣑⠣⣞⡴⠚⢡⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣀⡀⠀⠀⠀⠀⣿⡗⣌⠳⣐⠡⢃⠌⢂⠡⢀⠐⠀⡀⠀⠠⠀⠁⡀⢈⠀⡁⠒⡌⣓⣿⡽⣮⡙⢦⡙⠤⢃⡌⢡⠊⠄⠡⢈⠀⠄⠁⠈⢀⠀⠄⡐⠠⢁⠒⡈⡁⢆⠡⠐⢻⣿⡀⠃⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⣴⣾⣿
⣿⣯⣷⢿⣿⣻⣯⣿⣿⣿⣾⠋⠁⠂⠀⢡⢰⣱⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣄⠉⠢⣄⠀⢰⣿⡜⢤⠓⡬⠑⡌⢄⠃⡰⢀⠊⡐⠠⢀⠁⡐⠠⢀⠂⠤⠑⡌⢲⢭⣿⣿⣷⣹⢲⡍⢎⠥⡘⢄⠊⡌⢁⠂⠌⡐⢈⠐⡀⠂⠄⠠⢁⠂⠆⡡⠌⡐⠌⡡⠘⣿⣿⣤⠀⠀⠀⠀⠀⠀⢀⣘⣦⠻⡼⢙⣲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⣀⡤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣧⠀⠈⣇⣿⡳⣜⢣⡙⠴⡩⡐⢌⠂⡅⠂⠆⡐⢁⢂⡐⠠⢁⠂⠌⡄⠣⢌⡣⢞⣿⣿⣿⣼⢣⡛⣬⠒⣍⢢⡑⢌⠢⡁⠎⡐⢂⠰⢀⠡⠌⢂⠌⠄⣃⠐⢢⠁⢎⢰⡇⢿⣿⣿⣯⡄⠀⠠⡄⠀⣿⡿⣭⣿⠉⠛⠯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠚⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠉⠻⣿⣿⡆⢰⣿⣟⣷⣎⢧⡙⣖⢡⠓⡌⠒⠤⢉⠰⠐⡈⠄⢠⠁⠆⡉⠔⡨⡑⢎⡜⣯⣿⣿⣿⣾⡳⣝⢦⢫⡔⢣⡘⠤⢃⡔⢡⠒⡈⠔⣈⠰⡈⢆⡘⠰⢠⢉⠆⡍⢦⢏⣼⣿⣿⣿⣿⣿⡄⣁⠀⠀⠈⣹⡿⠿⡿⠟⠦⠄⠉⠳⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⠤⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣄⠀⠙⣿⡇⣿⣿⡝⡾⣿⣎⠷⣌⢧⡙⢬⠱⡈⢆⠡⢊⠔⡈⢆⡘⠤⡑⡸⡐⣍⠺⡼⣽⣿⣿⣿⣷⡝⡾⢬⠳⣜⢣⡜⣡⠣⡜⠤⣃⠜⢢⢄⠣⡘⠤⡌⡱⢢⢍⡚⣜⣮⣟⣿⣿⣻⣿⣿⣿⣿⣆⠄⠀⠀⢻⣶⠿⠗⠳⣯⡳⢾⣶⣤⡽⣝⠛⢶⣶⣦⠤⠴⠒⠲⣿⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⣀⣀⣠⢾⡿⠿⢿⣿⣿⣿⡇⠀⣿⣰⣿⣿⣿⣗⡿⣞⡿⣼⢲⣙⠦⣣⠑⡎⠴⡁⢎⡐⢆⡘⢤⢃⠵⣑⢎⣳⣽⣿⣿⣿⣿⣿⣽⡹⣭⠳⣌⠳⣬⡑⠧⡜⡱⢌⠎⡥⢊⠖⣡⠓⡬⣑⢣⠞⣼⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠻⢮⣷⣦⣼⣿⣷⠟⠳⠿⠯⣤⠤⢬⣿⡀⠀⠀⢦⣿⣯⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣷⣷⣶⣦⣤⡀⠀⢹⣿⣟⠀⢸⣾⣿⣿⡿⣿⣿⡽⣾⣱⢯⣞⣱⢣⡝⣘⠦⣙⠢⣍⠲⡘⢦⡩⢖⡭⣾⣱⣿⣿⣿⣿⣿⣿⣿⣿⣶⣛⢬⡓⢦⡹⣑⠎⡕⣎⢚⠴⣉⢞⡰⣙⠲⣍⡞⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣰⢿⡷⠋⠈⠙⠢⣄⠀⠈⠉⠒⢄⡈⠑⡄⣨⡏⣿⡹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢤⣀⣤⣴⣶⣿⣍⡈⢉⡉⠉⠉⠟⠿⢿⣿⣷⠀⠀⠀⣻⠀⢩⣿⣿⣟⣿⣿⣻⣽⣳⣟⠾⣜⣣⠷⡸⢥⡛⡴⣋⠴⣋⡝⢦⡙⣎⣖⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣦⡳⡜⡸⡱⡌⡞⣬⠱⣎⣵⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣠⡿⢟⠳⢤⡀⠀⠀⠈⣷⠀⠀⠀⠀⠉⢦⠘⣦⣶⢣⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⡟⣽⣫⢟⣽⣯⣿⣽⣾⣝⢮⡝⣭⣟⣿⣿⣿⣿⡟⠃⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⠁⠀⠈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⡷⣽⣿⣿⣿⣽⣯⣿⣿⣿⣞⣳⢯⣿⣿⣿⣿⣷⠀⠀⠀⠀⣼⡿⣿⣿⣿⣿⢃⣀⠀⠀⢠⣾⡟⠋⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣿⣿⣯⣿⣽⣯⣟⢾⣽⣻⣿⣿⣿⣿⣏⣗⣢⠔⡸⡟⠙⡓⠲⠾⣽⠊⠑⢲⠃⣸⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣞⡷⣯⢾⡹⣏⢷⣛⡾⣯⢾⣟⣷⡿⢟⡛⢫⡍⠭⢃⠱⣀⠣⢈⡁⠒⡘⠲⣄⣸⣠⣿⢽⠇⠄⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⣳⣏⢷⡹⣎⢷⣹⢯⣿⠿⢣⠜⡠⠜⡇⣯⡘⢆⠣⡐⠄⡃⠄⡃⠤⡑⠰⢿⣿⢯⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣟⣷⣯⣿⣽⣿⡟⣥⢃⠎⡴⢡⣏⢷⡜⣎⡵⢛⢣⣘⡜⢣⠑⡌⢣⢏⠼⣿⣏⠄⢀⠀⣤⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠡⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡓⢮⡑⢦⠡⣿⢪⣼⡱⣲⠥⣊⠍⣐⢢⣅⣛⡛⣎⢼⣽⣿⣮⣦⣹⡿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠈⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣆⣹⣤⣧⣜⡧⢷⣎⡕⢮⡰⣩⢆⡩⢌⣩⢍⣭⡻⣾⠀⠀⣼⡽⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢮⡵⢶⣶⣬⣿⡽⣻⣼⣛⣽⠶⣞⣻⡜⣎⣳⡼⣳⢿⠀⣰⣿⢇⡤⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⠐⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣟⡾⣣⣽⠶⡾⣿⣳⣟⣿⣎⢿⡭⣷⢾⣭⣳⡽⣶⢿⣿⠟⢉⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢏⠐⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡂⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣼⣻⣟⡷⣟⣾⢯⣟⡿⣾⣟⡿⣶⢯⡷⢿⣻⣧⣴⡿⣯⢿⡿⣿⣿⣿⣿⣿⣿⠟⡑⠠⢂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡔⡈⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢯⣟⡾⢿⡷⡿⢟⡙⢦⡙⢮⣿⣿⢯⡽⣿⣿⣗⠛⢿⣿⣿⡿⣋⠆⡡⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠰⡁⠎⠄⠡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠯⣏⠳⠁⠘⡐⢣⠘⢢⠙⣾⣿⣯⣿⣿⣿⣿⡏⡜⢠⠒⣿⠓⡔⡨⢐⠂⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠔
⡡⢓⡌⢌⠡⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠷⡌⠥⠃⡐⢡⠒⣯⢀⢯⣿⣿⣿⣿⣿⡿⣏⠼⡘⢤⣟⢣⡙⠤⡑⢂⠂⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡐⢢⠘
⣧⠣⡜⢢⢡⢉⠰⢀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢂⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣮⢣⡝⢢⠡⢀⣣⠸⢃⡾⠰⣿⠩⠙⠛⠻⣷⣭⡾⣵⡟⡌⢦⡉⢆⠱⢈⠐⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢈⠤⡑⢢⣙
⢿⣷⣌⠣⢎⠤⢃⠌⡰⢀⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⢆⠩⠿⣿⣻⣿⣿⣿⢿⠿⣿⢿⣿⣯⢭⡳⢌⡇⢢⠑⡻⢌⡩⠱⡩⣿⠠⢁⠂⡁⠀⠢⢇⡰⠘⢷⣷⡜⢠⠒⡀⢂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢂⠅⣢⠑⣣⣾
⠈⢿⣿⣾⡌⡚⣌⢒⢡⠂⡔⠂⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡⠌⢢⠑⢢⠝⢣⡙⢌⠣⠙⠤⢫⣿⣟⣶⡙⢦⡙⢆⠩⠔⢢⡘⠥⣱⡿⠐⡄⠂⠄⠀⠐⡠⠡⢉⠆⡩⢐⠂⡐⡐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡡⢚⡰⣘⠤⣿⡿⠡
⠀⠀⠛⣿⣷⣕⡌⢎⢆⢣⡐⢩⠐⠡⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠈⠤⠉⠆⡘⠄⡐⠂⡌⠰⢡⢋⣿⣗⣺⣽⢂⠳⡈⢆⠹⡠⠜⣣⣏⣿⠐⠠⠉⠄⠐⠠⠐⣁⠢⢘⠠⠁⠆⡁⠐⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡔⢰⠃⢦⣹⣿⡟⠡⠁
⠀⠀⠀⢙⣿⣿⣾⣌⢆⠣⠜⢢⠉⡔⠂⠌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢂⠉⡐⠠⠘⢠⠁⢆⡉⠦⡒⠾⣏⡾⣹⣎⢣⠱⠨⣅⢃⠇⢾⢡⣿⠀⠐⡀⠠⠀⠀⠁⠄⢂⠁⢂⠉⠄⠠⠁⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢃⠜⡠⢃⣹⣿⣿⠏⠌⠀⠀
⠀⠀⠀⠀⠑⠹⣿⣿⣎⡱⣉⠦⡙⠤⣉⠐⡠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠠⢁⠘⡀⠎⡰⢈⠃⣄⡘⣯⡳⢧⠎⢦⡉⢇⠦⡩⠜⢢⢣⡏⠀⠀⠀⠀⠀⠡⠀⠀⠀⠂⢀⠈⠠⠁⠂⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡰⢈⠆⡱⢳⣿⣿⢋⠐⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⢻⣿⣷⣧⢊⡕⢣⠰⡁⢆⠱⢀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠐⢠⡑⠄⢣⠘⡁⠀⣿⣿⡬⣛⢤⡉⢆⡣⢱⡉⢦⣻⢟⠀⠀⠀⠀⠐⠀⠀⡀⠀⠈⢀⠀⠂⠌⠐⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⡐⠰⣈⠲⣽⣿⡟⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⣎⠥⣃⢍⢢⠑⡌⡐⠠⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠄⡉⠞⠀⢁⠀⣿⡷⣿⢜⣢⠙⢦⡑⣣⠜⣲⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠐⠀⠂⠄⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢢⠐⡨⠑⣤⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⡌⢆⠣⣍⠰⣀⠃⡄⠢⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢂⠁⠂⢁⢢⠀⢻⣿⣿⣎⠶⡙⢦⠱⢎⡾⣣⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠌⡄⢊⠔⣩⣿⡿⢁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣌⠱⣂⠧⣀⠣⢠⠡⠌⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠐⡈⠤⠁⠂⠈⢆⠉⣿⣷⣏⢞⡙⢦⡛⣬⠳⣽⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⢀⠀⠂⠄⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠄⡌⠰⢀⠣⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣷⡐⠦⡑⡌⢆⠒⣈⠐⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠡⠐⡀⠈⠀⠡⢨⠀⠿⣯⢻⣮⣝⣦⣹⣲⠿⣹⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡌⢧⠈⠄⠂⠌⠠⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠠⢈⡐⢠⠃⡜⣶⣿⡋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⣿⣴⠱⣘⠢⡑⢄⠒⡠⠁⡌⠠⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠄⠀⠐⠀⠁⠀⠐⠁⡩⣇⠷⣾⣛⡭⢷⣺⢗⣿⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⣿⡈⣇⠈⠤⢁⠂⠌⢀⠈⠄⡐⢀⠀⠄⠀⡀⠠⠀⠂⡀⠐⡈⠐⠠⡁⢢⠐⡡⢊⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠣⢍⢢⠱⣀⠣⠐⡄⡁⢂⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢂⠠⠀⢠⡎⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⢱⣿⢚⠴⣛⢛⢿⡫⢞⡼⡇⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⣬⡓⡌⡐⣈⠂⠆⡈⠰⢀⠂⠌⡐⠠⠐⡀⠡⠐⠠⠁⠤⢉⠰⢠⠁⢎⢰⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣫⢿⣮⡘⣂⠳⢌⡒⡡⠒⡌⠤⢉⠐⠠⢈⠐⡀⠡⢈⠐⠠⢈⠐⠠⢐⢠⠂⢌⡼⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠂⡿⣜⣲⢥⡾⣮⡕⢫⡴⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⢡⢂⡱⢈⡔⢡⢂⠉⡔⠠⡁⠆⡐⠡⡉⠤⢉⠰⡈⠔⣂⢉⣶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠍⣟⣿⣦⠓⡬⠰⣁⠣⡘⠰⡈⠤⢁⠢⢐⠠⡁⠢⢈⠐⡂⢌⠡⣆⠋⣴⠞⡀⠄⠀⠀⠀⠀⠀⠠⠁⠠⢀⠈⣱⠂⣰⠿⡟⠱⡷⠙⢦⣝⠠⢂⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡌⢖⡡⢎⡒⡌⡒⢌⡱⢠⠃⣌⠡⠒⡌⢢⠑⡌⠒⣌⣾⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣻⣿⣔⠣⠜⡰⢡⠃⡜⡠⢃⡜⢠⠃⡔⢡⠊⡔⡡⢎⡵⢡⣾⠋⠄⡐⠠⠐⠀⠀⠀⡀⠀⠀⠡⢀⠂⣴⢠⢯⢣⢱⠯⠎⡥⢰⠂⠁⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣜⡢⠵⣨⠑⡎⡔⠣⡜⢠⢃⡱⢈⠆⡱⣨⣿⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣟⣿⣧⣑⢊⡱⢢⢑⢣⡘⢆⡓⡌⢆⠳⣌⡱⢎⣜⡷⢡⢈⠒⡈⠄⡁⠂⠤⠀⠄⠀⢀⠀⠀⠁⣌⣬⡟⠶⢮⣱⣞⣱⠃⣆⣀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣖⣡⢋⠴⣉⠖⣡⠃⡆⢆⣣⣾⣽⣿⡿⢋⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠤⡈⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⣝⣿⣿⣦⣕⠪⡔⡘⠦⡱⡘⢬⠓⡴⣩⢾⠞⢠⠃⢢⠘⣀⠒⠠⢁⠂⠄⡈⠀⡀⠠⠀⢄⠩⢼⣏⣿⣿⡻⣹⠿⣾⣥⠦⠱⡌⡐⡀⢂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣾⣾⡶⣿⠻⡏⢏⠻⠿⢍⡐⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡕⣨⠐⡄⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢾⣿⡟⡿⣶⣭⣶⣥⡙⢦⡛⣴⡷⢋⠰⠁⡜⠠⠡⢄⡘⢠⠁⡘⢀⠐⠠⠐⠠⢌⡰⢉⢎⣽⠻⣿⣿⣿⣿⣟⠿⣬⣓⡰⣁⠒⡄⠂⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣸⢻⣿⠟⣧⢳⡙⢦⢋⠜⠈⠁⠀⠀⠉⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡜⢤⠣⢌⠱⡈⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠈⠓⠤⣏⡝⡻⣿⣷⣿⢫⡐⢌⠢⡑⠄⡃⢅⠂⠔⢂⠰⠀⢌⡀⢃⠌⡑⢢⠰⡉⢍⣰⡟⣻⣿⣿⡿⢮⡻⣵⢺⣵⢈⠇⡘⠠⢈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡀⢢⠐⣿⣏⠻⣔⣣⠜⠶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢉
⡝⢦⡙⡌⢦⠑⡢⢐⠡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠈⠙⠳⠇⠻⣿⡦⢘⠢⡑⢄⠣⠌⡐⠌⠒⡈⠤⠉⡄⠂⠆⡘⢄⠃⠆⡵⠋⣥⢞⢣⡙⡤⢫⡝⣿⣜⠳⣦⢍⡊⠤⢁⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠡⢀⠡⡐⢠⢻⡏⣦⠹⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠈⠁⠒⠤⢀⣀⠀⢀⠰⡁⢮
⣿⢧⡝⣜⢢⡙⡔⡡⢊⠔⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣻⡧⢑⠌⡆⡘⠤⣁⠊⢡⠐⡀⠃⠠⠉⡐⠐⡈⠌⡜⢀⠣⠆⡍⣦⢑⣣⡟⣬⢹⣞⡷⢨⠙⠤⢃⠆⠄⠁⡀⠀⠀⠀⠄⠀⠀⠠⠀⠀⡀⢁⠂⠆⢡⣿⡿⠋⠐⢆⠑⠒⠂⢲⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⣼⣿
⣿⢯⡿⣽⣦⡳⡜⡰⠩⡜⣠⠑⡄⠀⠀⠀⠀⠀⠀⢀⠠⠔⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⣅⠚⠛⢿⡌⢎⡰⢡⠒⡄⢡⠂⡐⠠⢈⠐⠠⠐⠠⢀⠁⠠⠈⠰⡁⠲⢍⠢⣗⡯⢔⡣⢾⣼⢁⠎⡑⠂⠌⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠈⠄⣙⠫⣿⣒⠀⠁⠀⠀⠀⠈⠉⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⣼⣿
⣿⣟⣿⣳⣟⣿⣽⣲⡃⢖⡡⢣⠌⣅⢂⣀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⠀⠑⠂⢺⣿⡐⠦⡑⡌⠰⢁⠂⡁⠆⠀⡀⠂⢁⠂⠄⠈⢀⠀⠡⠀⠣⢌⠲⣽⣜⡎⡖⢥⠚⣄⠊⡔⠉⠄⠡⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⡀⠈⠄⡑⠰⠠⠌⣹⣯⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⠉⠂⠸⣷⣿
⣿⣻⣯⣿⢾⣟⣿⣷⣿⣢⣑⠣⣞⡴⠚⢡⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣀⡀⠀⠀⠀⠀⣿⡗⣌⠳⣐⠡⢃⠌⢂⠡⢀⠐⠀⡀⠀⠠⠀⠁⡀⢈⠀⡁⠒⡌⣓⣿⡽⣮⡙⢦⡙⠤⢃⡌⢡⠊⠄⠡⢈⠀⠄⠁⠈⢀⠀⠄⡐⠠⢁⠒⡈⡁⢆⠡⠐⢻⣿⡀⠃⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⣴⣾⣿
⣿⣯⣷⢿⣿⣻⣯⣿⣿⣿⣾⠋⠁⠂⠀⢡⢰⣱⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣄⠉⠢⣄⠀⢰⣿⡜⢤⠓⡬⠑⡌⢄⠃⡰⢀⠊⡐⠠⢀⠁⡐⠠⢀⠂⠤⠑⡌⢲⢭⣿⣿⣷⣹⢲⡍⢎⠥⡘⢄⠊⡌⢁⠂⠌⡐⢈⠐⡀⠂⠄⠠⢁⠂⠆⡡⠌⡐⠌⡡⠘⣿⣿⣤⠀⠀⠀⠀⠀⠀⢀⣘⣦⠻⡼⢙⣲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⣀⡤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣧⠀⠈⣇⣿⡳⣜⢣⡙⠴⡩⡐⢌⠂⡅⠂⠆⡐⢁⢂⡐⠠⢁⠂⠌⡄⠣⢌⡣⢞⣿⣿⣿⣼⢣⡛⣬⠒⣍⢢⡑⢌⠢⡁⠎⡐⢂⠰⢀⠡⠌⢂⠌⠄⣃⠐⢢⠁⢎⢰⡇⢿⣿⣿⣯⡄⠀⠠⡄⠀⣿⡿⣭⣿⠉⠛⠯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠚⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠉⠻⣿⣿⡆⢰⣿⣟⣷⣎⢧⡙⣖⢡⠓⡌⠒⠤⢉⠰⠐⡈⠄⢠⠁⠆⡉⠔⡨⡑⢎⡜⣯⣿⣿⣿⣾⡳⣝⢦⢫⡔⢣⡘⠤⢃⡔⢡⠒⡈⠔⣈⠰⡈⢆⡘⠰⢠⢉⠆⡍⢦⢏⣼⣿⣿⣿⣿⣿⡄⣁⠀⠀⠈⣹⡿⠿⡿⠟⠦⠄⠉⠳⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⠤⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣄⠀⠙⣿⡇⣿⣿⡝⡾⣿⣎⠷⣌⢧⡙⢬⠱⡈⢆⠡⢊⠔⡈⢆⡘⠤⡑⡸⡐⣍⠺⡼⣽⣿⣿⣿⣷⡝⡾⢬⠳⣜⢣⡜⣡⠣⡜⠤⣃⠜⢢⢄⠣⡘⠤⡌⡱⢢⢍⡚⣜⣮⣟⣿⣿⣻⣿⣿⣿⣿⣆⠄⠀⠀⢻⣶⠿⠗⠳⣯⡳⢾⣶⣤⡽⣝⠛⢶⣶⣦⠤⠴⠒⠲⣿⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⣀⣀⣠⢾⡿⠿⢿⣿⣿⣿⡇⠀⣿⣰⣿⣿⣿⣗⡿⣞⡿⣼⢲⣙⠦⣣⠑⡎⠴⡁⢎⡐⢆⡘⢤⢃⠵⣑⢎⣳⣽⣿⣿⣿⣿⣿⣽⡹⣭⠳⣌⠳⣬⡑⠧⡜⡱⢌⠎⡥⢊⠖⣡⠓⡬⣑⢣⠞⣼⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠻⢮⣷⣦⣼⣿⣷⠟⠳⠿⠯⣤⠤⢬⣿⡀⠀⠀⢦⣿⣯⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣷⣷⣶⣦⣤⡀⠀⢹⣿⣟⠀⢸⣾⣿⣿⡿⣿⣿⡽⣾⣱⢯⣞⣱⢣⡝⣘⠦⣙⠢⣍⠲⡘⢦⡩⢖⡭⣾⣱⣿⣿⣿⣿⣿⣿⣿⣿⣶⣛⢬⡓⢦⡹⣑⠎⡕⣎⢚⠴⣉⢞⡰⣙⠲⣍⡞⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣰⢿⡷⠋⠈⠙⠢⣄⠀⠈⠉⠒⢄⡈⠑⡄⣨⡏⣿⡹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢤⣀⣤⣴⣶⣿⣍⡈⢉⡉⠉⠉⠟⠿⢿⣿⣷⠀⠀⠀⣻⠀⢩⣿⣿⣟⣿⣿⣻⣽⣳⣟⠾⣜⣣⠷⡸⢥⡛⡴⣋⠴⣋⡝⢦⡙⣎⣖⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣦⡳⡜⡸⡱⡌⡞⣬⠱⣎⣵⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣠⡿⢟⠳⢤⡀⠀⠀⠈⣷⠀⠀⠀⠀⠉⢦⠘⣦⣶⢣⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⡟⣽⣫⢟⣽⣯⣿⣽⣾⣝⢮⡝⣭⣟⣿⣿⣿⣿⡟⠃⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⠁⠀⠈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⡷⣽⣿⣿⣿⣽⣯⣿⣿⣿⣞⣳⢯⣿⣿⣿⣿⣷⠀⠀⠀⠀⣼⡿⣿⣿⣿⣿⢃⣀⠀⠀⢠⣾⡟⠋⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣿⣿⣯⣿⣽⣯⣟⢾⣽⣻⣿⣿⣿⣿⣏⣗⣢⠔⡸⡟⠙⡓⠲⠾⣽⠊⠑⢲⠃⣸⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣞⡷⣯⢾⡹⣏⢷⣛⡾⣯⢾⣟⣷⡿⢟⡛⢫⡍⠭⢃⠱⣀⠣⢈⡁⠒⡘⠲⣄⣸⣠⣿⢽⠇⠄⠀⠀⣸⣿⣿⣿⣿⣿⣿