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