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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⢀⣀⡐⢄⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡜⠁⠀⣿⡌⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣸⣷⣤⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠊⣼⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡜⣼⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠉⠲⣄⣀⣼⡇⠀⠀⠀⠀⠀⠀⠻⠿⣿⣟⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠁⠀⡏⠑⠌⠓⢬⣧⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⡀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠇⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠈⠉⠓⠒⠲⠤⢤⣀⠀⠂⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀ ⠀⠀⠀⠀⢀⣠⠤⠖⠒⠒⠒⠦⢤⡀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠒⠋⢉⠟⠀⠀⠀⠀ ⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢠⡞⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀ ⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢠⡟⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠑⠢⢤⣀⣀⠀⠀⠀⠀⢀⣀⡤⠖⠯⣀⠀⠀⠀⠀⠀⠀ ⢀⡟⠀⠀⠀⠀⠠⠴⠤⣀⠀⠀⠀⠀⠀⢸⣠⡟⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢻⠀⠀⠀⠀⠀⠉⠢⣄⣀⡀⠀ ⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢈⣿⡇⠀⠀⠀⠀⢸⠉⢢⣀⡀⢀⣀⣴⠟⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀ ⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⣷⠀⠀⠀⢀⡞⠀⢰⠏⠉⠉⠁⢸⡀⠀⠀⠀⠈⠓⠶⠤⣤⣄⣀⣠⡤⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁ ⠀⠹⣆⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢸⠁⠸⡆⠀⣠⠞⠀⢀⡞⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢰⣧⣀⣀⡀⠀⢀⣀⣠⠴⠃⠀ ⠀⠀⠹⡓⠦⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹⡴⠁⠀⢠⠞⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣸⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀ ⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⢁⡠⠴⢧⡀⠀⠀⠀⠀⣀⠔⠳⣄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣠⡿⠋⠀⠀⠀⠙⢦⣀⡠⠞⠁⠀⠀⠈⠙⠶⣤⣀⡀⣰⠃⠀⠀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠋⠉⠉⣹⠏⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣄⣀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⣤⠞⠁⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠶⠤⠤⠤⠤⠤⢤⣞⡥⠖⠋⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠒⠀⠀⠒⠒⠺⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢄⣈⠆⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⡤⠖⢄⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠒⠂⠠⡀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⡠⢠⠂⠀⠀⠀⠡⡀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⣷⣾⠀⠀⠀⠀⠀⡇⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⢨⠢⠔⡀⠀⠠⠘⠛⠛⠀⠀⠀⠀⢸⡇⠀ ⠀⠀⠀⢀⣀⣀⠀⠀⠀⠰⠀⠀⠀⠀⠡⡀⠀⠈⠀⠒⠂⠄⡀⢀⠀⡀⠀ ⠀⡴⠊⠀⠀⠀⠉⢆⠀⡔⢣⠀⠀⠀⠀⠐⡤⣀⠀⠀⠀⠀⠀⣀⠄⠀⠀ ⢸⠀⠀⠀⢠⠀⠀⠈⣼⠀⠀⠣⠀⠀⠀⡰⡀⠀⠉⠀⠀⠰⠉⠀⠁⠠⢄ ⢰⠀⠀⠀⠀⠇⠀⢀⢿⠀⢀⠇⡐⠀⠈⠀⠈⠐⠠⠤⠤⠤⠀⠀⠀⠀⢨ ⠀⢓⠤⠤⠊⠀⠀⢸⠀⠣⠀⡰⠁⠀⠀⡀⠀⠀⠀⠸⠀⢰⠁⠐⠂⠈⠁ ⠀⠀⠑⢀⠀⠀⠀⠈⣄⠖⠉⠑⢄⠠⠊⠀⠢⢄⣠⣃⣀⡆⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠑⠠⢀⣀⠎⠀⠀⠀⠈⡄⠀⠀⠀⢠⢃⠠⠃⠐⡀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⠯⠉⠤⢴⡃⠁⠀⠀⠀⡇⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠀⠠⠂⠀⠀⠀⠀⠑⢄⠀⠀⢀⠲⠁⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠘⠒⠑⠔⠁⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣴⡎⠠⠊⠀⠠⠚⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢠⡍⠀⠀⠀⠀⣾⣤⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠈⠙⠡⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢷⣤⣮⣤⣴⣶⣶⣶⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣯⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢀⠔⠊⠁⣰⠙⠻⢿⣿⣿⠟⠁⠻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀ ⢈⣶⣂⣀⠀⡐⠁⠈⠁⡒⠛⠋⢀⠀⢀⣿⡟⣿⠃⠀⠀⠀⠀⠀⠀⠀ ⠈⠉⠉⠛⠛⡷⡀⠀⠀⡇⢠⣤⣿⣐⣿⣿⣧⣿⢀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣣⠏⠐⠒⠦⠤⠬⢽⣹⣿⣿⣿⣿⣟⠀⡠⠐⠒⠒⣦⡀ ⠀⠀⠀⠀⡔⠘⢄⠀⢠⡄⠀⢀⣼⠟⠉⠉⠳⣿⢇⣼⣤⣤⣴⣾⣿⣿ ⠀⠀⠀⢰⣀⣴⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿ ⠀⠀⠀⣼⣿⣿⣿⣿⠋⠙⠛⠻⠷⣄⡀⢀⣰⣿⣿⣿⣿⣿⣿⡿⠛⠀ ⠀⠀⠀⠉⠹⠛⠋⠁⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠄⣀⡀⡀⠤⠐⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⢤⠔⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏⠀⣀⣀⡀⠀⠀⠀⠀⢀⠀⡔⢻⣦⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠸⠀⠰⠛⣇⣹⡜⡄⠀⠀⠸⢠⣿⣿⣀⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢀⠔⡉⠤⠐⠒⠒⠒⠂⠠⠬⣁⠒⠠⢄⡀⠀⢠⠀⠐⠠⠿⢿⡇⠀⠀⠀⠀⠈⠛⠉⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⡐⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠄⡁⠒⠱⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⣒⣀⣴ ⠰⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠌⣉⣶⣶⣦⣄⣀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⢁⣠⣴⣾⣿⣿⡟⠁ ⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠉⠛⠿⠿⠛⠆⠤⠄⠀⣀⣀⣀⣀⣀⡀⠔⢁⣤⣾⣿⣿⣿⡿⠟⠁⠀⠀ ⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠀⠀⠄⠀⡀⠀⠁⠐⠒⠂⠠⠤⢤⣤⣤⣶⣾⣿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀ ⢇⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠘⡈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠉⢂⠀⢱⡀⡰⢠⡃⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠐⢄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠑⢄⡁⠂⠤⢀⣀⠀⠀⠀⠀⣀⣀⣼⣿⠀⠀⠀⠀⣮⣤⣶⣶⣦⠋⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠈⠑⠂⠤⠄⠀⠀⠠⠾⠿⠿⢻⣿⠀⠀⢀⣴⣿⣿⣿⡿⠁⡀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⢃⠤⠐⠁⠀⢰⣾⠟⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠆⠀⠀⠀⠀⢸⡟⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢰⠀⠀⠀⠀⠀⠀⣇⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠸⠀⠀⠀⠀⠀⠀⢹⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⡀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⣆⠁⠀⠀⠀⠀⠀⠀⠀⣿⣷⢠⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

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⬜⬜⬜⬛⬛⬛⬛⬜⬜⬜⬜⬜⬜⬜⬜⬜⬛⬛⬛⬛⬜⬜ ⬜⬜⬛🟦🟦🟦🟦⬛⬛⬜⬜⬜⬜⬜⬜⬛🟦🟦🟦🟦⬛⬜ ⬜⬛🟦🟦🟦🟦🟦🟦🟦⬛⬛⬜⬜⬜⬛⬛🟦🟦🟦🟦🟦⬛ ⬜⬛🟦🟦🟦🟦🟦🟦🟦⬛🏿⬛⬛⬜⬛🟦🟦🟦⬛🟦🟦⬛ ⬛🟫🟦🟦🟦🟦🟦🟦🟦🟦🏿🟫🏿⬛🟦🟦🟦⬛🟦🟦🟦⬛ ⬛🟦🟦🟦🟦⬜⬛🟦🟦🟦🏽🟫🟫🏿⬛🟦🟦⬛🟦🟦⬛⬜ ⬛🟦🟦🟦🟦⬛🟫🟦🟦🟦⬜🟫🟫🟫⬛🟦⬛⬛⬛⬛⬜⬜ ⬜⬛🟦🟦🟦⬛🟫🟦🟦🟦⬛⬜⬜🟫🟫⬛⬛⬜⬜⬜⬜⬜ ⬜⬜⬛⬛🟦🟦🟦🟦⬛⬛🟦🟦⬜🟫🟫⬛⬜⬜⬜⬜⬜⬜ ⬜⬜⬛🟦⬛⬛⬛⬛🟦🟦🟦🟦⬜🟫🟫⬛⬜⬜⬜⬜⬜⬜ ⬜⬜⬜⬛⬛🟨🟨⬛🟦🟦🟦⬛⬜🟫🟫⬛⬜⬜⬜⬜⬜⬜ ⬜⬜⬜⬜⬜⬛🟨🟨⬛⬛⬛⬛🏽🟫🟫⬛⬜⬜⬜⬜⬜⬜ ⬜⬜⬜⬜⬛🟦⬛🟫🟨🟨🟨🟫⬛🏽⬛⬜⬜⬜⬜⬜⬜⬜ ⬜⬜⬜⬜⬜⬛⬛⬛⬛🟫🟫🟦⬛🏽⬛⬜⬜⬜⬜⬜⬜⬜ ⬜⬜⬜⬜⬜⬜⬜⬜⬛⬛⬛🟦⬛⬛⬜⬜⬜⬜⬜⬜⬜⬜ ⬜⬜⬜⬜⬜⬜⬜⬜⬜⬛🟦🟦🟦⬛⬜⬜⬜⬜⬜⬜⬜⬜ ⬜⬜⬜⬜⬜⬜⬜⬜⬜⬜⬛⬛⬛⬜⬜⬜⬜⬜⬜⬜⬜⬜

⣦⡾⢃⣤⡿⠛⣒⡛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡐⢊⠉⢱⠖⣲⢤⣀⠀⠀⠀⠀⢠⡾⡗⡛⠒⠈⣤⣤⡀⠀⠀⠀ ⣻⣿⠿⢛⣣⠜⠸⣟⣿⡇⠀⠀⠀⡀⠀⠀⠀⣀⣤⣤⣶⣶⣿⠿⠿⠿⠿⠿⠿⣿⣶⣦⣍⡛⢻⣦⣄⠀⢼⡿⡇⠥⠙⢘⣹⣿⣟⠀⠀⠀ ⣿⣭⠘⣁⠀⠀⠀⢫⣩⡽⣦⠞⠉⣠⣴⣾⡿⠟⠛⠉⠉⢀⠀⠖⡌⠀⠀⠀⠁⠀⠈⠙⡻⢿⣶⣄⠙⢷⣞⣛⣧⣇⡤⣈⣿⣿⣿⠀⠀⠀ ⣿⡟⢸⣷⠀⣰⣾⣫⣭⡿⢛⣶⣿⠟⢋⠡⡐⢌⠱⣀⠠⢄⠢⢑⡈⢆⠑⣂⠒⡰⢄⠀⠀⠈⡙⢿⣿⣟⣿⡿⢻⣿⣿⣷⡿⠟⢟⠀⠀⠠ ⢸⣸⣹⣿⣼⣿⣿⣿⣷⣿⣿⣏⣰⡈⠢⣁⠃⠎⡰⢀⡃⠦⡘⠤⡘⢄⠣⢄⢣⠐⡌⢒⡀⢀⠜⣤⣿⠿⠉⠀⠀⢻⣿⣿⡅⠀⠈⡆⠀⣶ ⠈⢇⡦⣿⣿⣿⠏⠁⠩⢿⣿⠛⢿⣿⣿⣶⣬⣴⣁⣢⠘⢠⠑⣂⠱⣈⠒⡌⢢⠑⡌⠂⣌⣶⡿⢿⣿⣄⠀⠀⠀⠈⢿⣿⣿⣶⡍⠀⣀⠛ ⠀⠘⣿⠸⣿⣿⣦⣀⣔⣿⣿⣧⠀⢿⣿⡌⠙⠛⠻⠿⣿⣶⣮⣤⣒⡠⠑⢌⢂⠡⣨⣾⣿⣿⣇⠘⣿⣿⡄⠀⠀⠀⠘⣿⣿⣿⡧⣼⡇⠀ ⠀⠀⠸⣧⠹⣿⣿⣿⣿⣿⣿⣿⣇⠈⣿⣿⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣶⣮⣾⣿⣿⣿⣿⣿⡔⠸⣿⣷⡀⠀⠀⠀⠸⣿⣿⢹⡟⠀⠀ ⠀⠀⠀⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⣿⣿⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡠⠹⣿⣷⠀⠀⠄⠀⣿⣟⣿⡇⠀⠀ ⠀⠀⠰⢙⣹⣿⠇⡠⠿⢿⣿⣿⣿⣿⡀⠹⣿⣧⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⣿⣇⠀⠀⢸⣿⡏⠹⣇⠀⠀ ⠀⠀⡈⠋⣿⡿⠐⡄⠣⢼⣿⣿⣿⣿⣿⡀⢹⣿⣧⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⣆⢠⣿⠟⣿⣆⠙⣇⠀ ⢀⠒⢁⣾⡿⢃⠱⣈⠱⢠⢸⣿⣿⣿⣿⣷⠀⢿⣿⣆⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⣾⡟⢠⠹⣿⡆⣿⢷ ⣾⣤⣿⠏⡁⠦⡑⢄⠃⢆⣹⣿⣿⣿⣿⣿⣧⠈⢿⣿⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣶⣾⣿⠏⡐⢂⠆⣿⣿⣹⣛ ⠿⣿⢋⡐⢌⠱⡈⠆⡹⢼⣿⣿⠣⡝⢿⣿⣿⣆⠘⣿⣿⡀⠀⢀⣤⣼⣿⠿⣋⠷⡘⢆⠣⢒⠰⡘⡐⢣⠍⠭⣉⠡⢂⠱⡈⣴⣿⡇⢿⣿ ⠀⣿⠠⢐⠊⡔⢡⠊⡵⢫⣿⣧⢫⠜⣣⣮⡻⢿⣷⣾⣿⣷⣾⡿⢟⡹⢢⠝⡨⠑⢌⠂⠓⡌⢲⡄⣉⠒⡌⢒⣠⣥⣮⣶⣿⡿⢋⣰⠾⠡ ⠙⣿⢀⠃⣍⠰⡁⢎⡵⢋⣿⣟⠠⢉⣾⣿⠋⠶⣩⢛⠿⢫⠕⡊⠇⡉⢆⡘⠤⡉⠜⣿⠔⡠⢛⠡⣄⣶⣾⣿⠿⠟⢛⣿⡿⠲⣾⠋⠀⠐ ⠀⢹⠀⠎⡄⢣⠘⣬⢚⡥⣿⣿⠀⡙⣿⣷⣌⡐⠠⠉⠌⢡⢘⣀⣣⣬⣤⣶⣴⣧⣮⣤⣶⣤⣷⣾⡿⠟⠋⣄⠢⣩⣾⣟⣱⡾⠃⠀⠀⠀ ⠀⢈⠘⡰⠘⡄⠳⣌⢳⡘⣿⣿⢀⠡⠌⠻⠿⣿⣿⣿⣿⣿⡿⠿⠿⢛⠛⠛⠛⠛⠛⠛⡛⠿⢛⠩⠐⢌⢒⣠⣾⣿⣯⣍⡉⠅⠂⡀⠀⠀ ⠀⠀⠐⡡⢃⢌⡳⣌⢧⡙⣿⣯⢀⠃⢎⡑⢢⠐⡄⢢⠐⠤⡐⢢⠑⡌⠬⠡⢍⠊⢅⠣⢐⠂⣡⣊⣵⣾⣿⠟⠻⣿⣏⠻⢿⣷⣤⡙⠲⣤ ⠀⠀⠘⡄⢃⢬⢳⡘⢦⡙⣿⣿⣿⣮⣄⣘⡠⢃⠘⠤⠉⠆⠱⡈⢆⠱⣈⣑⣈⣬⣴⣶⣿⣿⡿⢟⠛⡩⡔⠎⠁⠘⣿⣷⠠⠌⠻⣷⡘⣿ ⠀⠀⢈⠔⣩⠞⢦⡙⢦⣙⣿⣯⠘⡛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⠿⢛⠻⣿⣯⠰⡐⢎⡱⠁⠌⠠⢁⠠⠘⣿⣧⠈⡅⢚⣿⣯ ⠀⠀⡐⢮⡱⢎⢧⡙⢦⢹⣿⡇⡱⢘⢆⠳⢄⠢⡔⡰⠢⢔⠢⡔⡒⡔⢪⡑⠎⡥⢊⠽⣿⣇⠉⡀⠄⡐⢈⠂⠂⢀⣤⣿⣿⡆⡘⠌⣿⡧ ⠀⠀⡘⣇⢞⡩⢖⡹⣌⢻⣿⢃⡌⢣⠜⣌⢊⡕⠬⣑⠩⢆⡓⢬⡑⠬⠑⠌⢃⢈⠁⡀⢿⣿⡀⠄⠂⠀⣄⣠⣶⣿⡿⠛⣿⣇⠰⠁⣿⡗ ⠀⠀⠸⡜⢪⡕⡫⣔⢣⣿⡿⠐⠬⠡⠎⠰⠁⠎⠒⠁⡉⠂⠌⠀⠄⠂⠡⠈⠄⠂⠄⢀⣸⣿⣧⣴⣶⣿⡿⠿⠛⠁⢀⠀⣿⣯⠐⡁⣿⣟ ⠀⠀⠈⢉⣃⣞⣑⣬⣚⠛⠁⠀⠠⠁⠄⠂⣁⢀⡁⢌⠠⠅⠌⠰⠈⠤⠡⣒⣒⣛⣾⣛⣟⣿⣛⣯⣭⣥⣄⣠⣀⣤⡶⣴⣟⡃⠸⠴⡷⠋

⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⢀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢀⢀⠀⠀⠀⢸⡿⠛⠛⠛⠛⠛⠉⠛⢿⣿⠀⠀⠀⠀⠀⠀⠀ ⠀⠠⣿⣿⣿⣄⠀⣼⠀⠀⠀⢂⣀⣀⡀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀ ⠀⢸⣿⣿⣿⣿⡷⠋⠈⠀⠀⠀⠀⠀⠀⠀⠈⠘⠣⡀⠀⠀⠀⠀⠀ ⠀⠈⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣦⡀⠀⠀ ⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣦⠀ ⠀⠀⣸⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇ ⠀⣤⡟⠛⠋⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠈⠋⠈⢿⣿⡿ ⢀⡉⠀⠀⣀⣤⣄⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣤⣤⣄⠀⠀⣴⡄ ⠘⢇⠀⠰⣿⣿⢟⢼⣿⣿⣿⣿⣿⣿⣿⣿⡿⢜⠿⠿⠿⠀⡀⠀⠀ ⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⣸⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡞⣿⣷⣮⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠸⣸⣻⣏⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡿⡀⠀⠀⠀⠀⠀⣾⡞⡝⣿⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣾⣿⣶⢦⣤⣀⠸⠻⢭⣥⡻⣧⠀⡙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⢠⣴⣾⣿⣿⣿⣏⣶⣾⡽⣿⣷⣟⣿⣿⣿⣻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣀⣀⣀⠀⠀⠀⠸⣿⡿⠘⠻⢿⣿⣿⠟⠛⠿⠿⠃⢍⣿⣿⢸⣿⣿⣿⡽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣰⣟⠛⠛⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣜⢿⣿⡿⡷⡿⣼⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢰⣿⠃⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣯⣾⣿⡀⠀⠙⠻⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀ ⢸⣿⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⡀⠀⠀⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀ ⢸⣿⡀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣽⣿⣿⢟⣛⣳⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀ ⠀⣿⣇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⢻⣾⣿⣿⣷⡽⣄⠀⠀⢀⣾⣿⣷⣄⠀⠀ ⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣄⡀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢹⣦⠀⢸⣇⠀⠹⣏⢧⡀ ⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⢸⣿⡄⠈⠛⠀⣶⠟⠼⠇ ⠀⠀⠀⠹⣿⣿⣷⣤⡀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⡿⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠙⣿⣿⣿⣿⣶⣄⠀⠀⠈⠻⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⢟⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣧⡀⠀⠀⠈⠻⢿⣿⢸⣿⣿⣿⡿⢟⣫⣾⣿⣿⠿⣛⣵⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢈⣾⣿⡟⠙⠚⠛⠛⠋⠉⠀⠘⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠘⠻⣿⣿⣷⣶⣒⣒⢢⡄⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⣏⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠈⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠿⠿⠿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢠⠁⢰⣴⠀⠀⠀⠀⣿⡇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢀⣸⠀⠈⠙⠃⣀⣀⡀⣤⡄⢸⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢰⠃⠉⠁⠂⢦⣀⢀⡀⠤⠤⣥⣌⡀⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠘⠦⡀⠀⠀⠀⠈⡄⠂⠉⠀⣀⠵⢵⡒⠠⢴⡋⠉⢑⠖⠦⢄⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠈⠲⢤⣤⣴⡥⢄⡨⠋⠀⠒⢄⡑⣄⠀⠈⠄⠀⠀⠀⠈⡝⢀⣀⠀⠀⠀⠀ ⠀⠀⠀⠀⠈⢿⣿⣿⣆⡇⠀⠀⠊⠉⠉⡆⠈⠢⣶⣀⡀⢀⡰⠋⠀⠀⠉⠑⡄⠀ ⠀⠀⠀⠀⠀⠀⢿⣿⣿⣔⠀⠠⡀⠀⢀⡇⠀⠀⠠⣇⠈⢹⠀⠀⡀⠀⠀⠀⠘⡄ ⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣶⣶⢆⠀⠀⠈⠉⠈⢄⠀⠀⠀⠂⠀⢀⠇ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠑⠦⣄⠀⠀⠀⠁⠀⠈⠀⢀⠎⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠲⠖⠒⠊⠁⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠈⡖⡤⠄⠀ ⠀⠀⢀⡀⠀⠀⠀⡐⠁⠀⠀⠠⠐⠂⠀⠁⠀⠀⠀⠀ ⠀⠰⡁⠐⢉⣉⣭⡍⠁⠂⠉⠘⡀⠀⠀⠀⠀⠂⠡⠀ ⢀⣊⠀⡄⠻⠿⠋⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⢀ ⡎⣾⠀⠁⣴⡆⠀⠡⢺⣿⣆⠀⢠⢱⣄⠀⠀⠀⠀⠈ ⡑⠟⠀⠀⠀⠀⠀⢀⣸⡿⠟⠀⠀⠈⢿⣿⡦⡀⠀⡰ ⠙⠔⠦⣤⣥⣤⣤⣤⡤⠆⠀⠀⠀⠀⢀⢀⠀⠈⠎⠀ ⠀⠀⠈⣰⡋⢉⠉⠁⠒⠂⢇⢠⡆⠀⠸⢴⣿⠀⠘⠀ ⠀⠀⠘⡿⠃⠀⠨⠒⢆⣸⣿⠁⠀⡠⡇⠈⠋⠀⠰⠀ ⠀⠀⠀⠛⠒⠒⠁⠀⠈⠷⡤⠤⠐⠀⠘⠒⠒⠖⠁⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢧⠀⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡞⡆⠀⠈⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢡⢻⡄⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢥⢊⢷⡈⠔⠠⠐⢆⣩⢢⡀⢀⣿⣦⡀⠀⣤⠀⠀⠀⢀⣀⣤⢴⡶⡻⣍⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢎⡱⢊⠗⣌⠣⡜⣨⠑⣎⠹⣲⠿⣯⠻⡾⣟⣧⣴⡞⣯⢻⡜⣧⢻⣵⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣏⠲⣄⣀⠀⠀⠀⠀⠀⢸⡘⢆⡫⠜⣤⠳⣘⠤⣋⠴⠃⠇⠙⠄⠓⠑⠢⠒⠦⡙⢬⠫⡝⢾⣣⣯⣤⢤⣤⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠸⣿⡻⢟⠧⣠⠘⠻⢄⡀⠀⠀⠀⡿⡘⢤⡛⣄⡇⠃⢀⠀⠀⡀⠀⠀⠄⡀⢀⠃⠘⠀⠣⡘⢇⠻⡄⢇⡛⠿⣼⣼⠃⣀⣀⠀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀ ⠀⢈⣳⢩⠒⡭⢲⡀⠀⠈⠓⢤⣀⡷⣩⡶⡛⣥⠀⠀⠀⠀⢀⠀⠀⠁⢂⠐⠠⠀⠀⠌⠐⠈⡌⠓⡜⢣⢍⠳⣌⠯⡱⣄⠈⠉⠁⠉⠁⢈⠀⠁⢀⠀⢀⠀⣄⣤⠶⣻⠟⠁ ⠀⠛⠲⢧⡹⣐⢣⠹⢦⡀⢦⡘⣼⡿⣋⠴⢛⡁⠀⠠⠁⢀⠠⠀⠌⠀⠀⠠⠁⠀⠂⢀⠁⠂⢌⠱⣈⠇⣎⠳⣌⠳⡱⡌⠖⡄⠂⠌⡐⠠⢀⣂⣤⠖⡞⣹⢣⡾⠋⠀⠀⠀ ⠀⠀⠀⠈⢳⡱⣊⢕⢪⡙⢮⠳⣉⠖⣡⡾⠋⠀⢰⡀⠀⠠⠀⠀⠀⢀⠀⠁⠠⠐⠀⢂⠈⡐⡈⠔⣌⠚⣤⠛⣌⢳⠱⡸⢱⢈⡡⠤⢖⡻⢍⠳⣌⠳⣜⠗⠋⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠈⢳⡱⣊⢦⡙⣌⢣⠕⣪⠑⢦⡑⢲⡬⣇⠠⠐⠀⠁⢀⠀⡀⠌⠐⡀⢁⠂⢄⢠⡑⢎⡔⣫⢔⡫⣜⡡⢇⡳⡱⢪⢕⡋⢦⢣⡙⢖⣬⠟⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢻⡜⣶⣋⢦⠓⡎⢥⠛⡤⢭⠋⠀⢸⡄⠀⠌⡐⢀⠂⠄⡈⠄⡐⠠⢌⢢⠦⣙⢦⡹⢔⠮⡱⢦⠹⡜⢲⠍⣇⠞⣌⢣⢲⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⣹⠳⡜⢦⠛⣜⢢⡛⢬⡏⡀⠀⠀⢿⣆⠰⣈⢦⣐⠠⣐⠢⣜⡱⢎⡣⢞⡡⢖⡍⣎⣳⡽⣌⠳⣉⢧⢚⣼⡚⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢠⢇⡻⢌⠧⡛⣼⡧⢜⡡⣇⠁⢀⠀⢾⣏⢧⡙⢦⣃⠯⣔⢫⡔⠣⣜⠱⣣⣙⡦⠟⠉⣷⡱⣌⢣⢣⢎⡣⢞⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⡼⢠⠙⣎⢳⡹⢴⠻⣎⠵⣙⠦⣅⡓⠀⠸⣶⡙⢦⢃⡞⢬⠲⣌⠓⣬⡷⠛⠉⠀⠠⢀⡷⡳⣌⠇⡞⣰⠹⣌⣯⢛⡳⣒⢖⡲⣒⠖⣦⠤⠤⢀⡀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⡏⠤⡙⡌⢧⡹⢾⠀⠈⠷⣎⡹⡜⣭⢛⠿⡽⡜⣣⢣⠞⣡⢳⡼⠛⠹⠃⠀⠀⠠⢡⡼⢳⡱⢪⡙⣖⣡⠓⣼⢧⢫⠴⣩⠎⡵⣸⢘⡴⢋⡽⠻⠤⣀⠀⠀ ⠀⠀⠀⠀⠀⠀⣟⠰⢠⡙⢮⢵⣻⡄⠀⢠⠋⠳⢽⣰⢋⢮⡱⢎⡵⢪⡙⣖⢫⠗⣤⣀⣀⣀⣄⣶⠫⡝⣣⡕⣣⠝⡴⣊⡝⢦⣋⢎⢧⡑⢮⠱⢎⡱⢎⡵⢊⡟⠛⠉⠀⠀ ⠀⠀⠀⠀⠀⠀⣏⠲⣁⢾⣩⣞⠶⣷⡀⡎⠀⠀⠀⠉⡛⠲⠽⣎⣜⢧⣙⢦⣋⠾⡰⢭⡹⣙⢞⡰⢫⣴⣟⡱⢣⢏⠶⡱⢎⡳⡜⣎⠶⣙⢦⢛⣬⡳⠗⠛⠳⠇⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣯⢳⡰⢊⡷⣯⢿⣹⣟⣧⠀⠀⠀⢠⠃⠀⠀⠀⠈⠉⠙⠲⠛⠚⠽⠷⠓⡟⠚⠋⣩⡟⣴⢫⡝⣮⢳⣝⣻⡵⣹⣬⣳⠽⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣴⣿⡧⡽⣭⣻⣽⣯⢷⣻⣞⣷⣦⡀⡼⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢰⠁⢀⣼⢧⣻⣜⣧⢿⣼⣳⢾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣸⢿⣿⣷⡝⣾⣷⣿⣿⣿⣿⣽⣾⣟⣿⣷⣦⣤⣀⡀⠀⠀⢸⠀⠀⠀⠀⢀⣮⣶⣿⣽⣯⣷⢿⣞⣿⢾⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣿⡓⣞⡿⣿⣳⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⡿⣟⣾⣯⣷⣿⣿⣯⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣷⠙⣦⢻⣽⣿⢻⣽⣿⣿⡟⣿⣯⣿⣿⣾⣿⣿⢻⣽⣿⣿⣷⣿⣯⣿⣿⣽⣿⣿⢻⣽⣿⣽⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠸⣷⣊⠶⣭⢻⣿⣿⣯⣷⣿⣿⣿⡿⣿⣿⢿⣻⣿⣿⣿⣿⣿⣻⣿⢿⡽⢏⡳⢬⢣⣓⢮⡻⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠙⢿⣾⣼⣿⣳⣿⣿⣯⡷⣿⡷⣿⣿⢿⣿⣿⣿⢿⣿⣻⣿⣿⢯⡛⣜⢣⡝⣎⢳⢬⠳⣝⢯⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣿⣷⣿⣿⠛⢷⣻⡽⣯⢿⣿⡾⣿⡿⣟⣿⡿⣞⣯⢳⣌⢳⡜⡬⢓⢮⣹⡞⡿⣜⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⢿⡏⠙⠀⠀⠈⠙⠿⢿⣾⣟⣿⣽⣿⣻⣽⡻⣜⢯⣲⢯⡛⣴⢻⣋⠷⣩⢓⡎⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⢿⣿⠏⠉⠙⢯⢎⡳⢎⣵⣏⣶⣯⢒⠥⣋⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠏⠀⠀⠀⠈⢧⡝⣾⣿⣿⣿⣿⣾⣿⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣳⡿⣞⡿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⣴⠿⢟⣛⣩⣤⣶⣶⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢀⣴⣿⠿⠸⣿⣿⣿⣿⣿⣿⡿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢠⠞⠉⠀⠀⠀⣿⠋⠻⣿⣿⣿⠀⣦⣿⠏⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀ ⢠⠏⠀⠀⠀⠀⠀⠻⣤⣷⣿⣿⣿⣶⢟⣁⣒⣒⡋⠉⠉⠁⠀⠀⠀⠈⠉⡧ ⢻⡀⠀⠀⠀⠀⠀⣀⡤⠌⢙⣛⣛⣵⣿⣿⡛⠛⠿⠃⠀⠀⠀⠀⠀⢀⡜⠁ ⠀⠉⠙⠒⠒⠛⠉⠁⠀⠸⠛⠉⠉⣿⣿⣿⣿⣦⣄⠀⠀⠀⢀⣠⠞⠁⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⣿⣿⣷⡄⠞⠋⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣑⡙⠻⠿⠿⠈⠙⣿⣧⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡿⣿⣿⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠑⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⠀⠀⠑⢄⣰⠉⠢⣀⠐⡀⢀⣀⠤⠐⠂⠉⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⣼⠁⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢴⣶⣾⣿⡟⠲⢤⣀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠀ ⠀⣨⣿⣿⣷⣄⠀⠈⠑⠢⠄⠛⠛⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠞⠁ ⠈⠉⠻⣿⣿⣿⣷⣄⠀⣤⣤⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡷⠂⠀⠀⠀⠀⢀⣀⣤⣶⣿⡿⠟⠁⠀⠀ ⠀⠀⠀⠈⢿⣿⣿⣿⣧⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⠞⠁⣀⣤⣴⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠆⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢆⠀⠀⢀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢈⣶⣶⣾⣿⣦⣤⠄⠀⠀⢠⣿⣿⣿⡿⠟⢻⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⣿⣿⣷⣄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣁⣴⣿⡿⠟⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠒⠒⠒⠤⠤⠤⠀⢀⣀⡀⠀⠀⠀ ⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠈⢻⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⡿⠛⠙⠃⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣀⠀⠀ ⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⡸⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣥⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡌⠁⠀⠀ ⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⡇⠀⠀⠀⡸⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠁⠀⣀⣠⡤⠖⠚⠉⠉⠁⠀⠀⠀ ⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢀⠃⠀⠀⠀⠀⠀⢩⠉⠉⠉⠉⢻⠉⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⡷⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣜⡀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⣼⣠⣾⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣇⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠏⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⢒⡾⠛⠛⠐⠲⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠉⠀⢀⣠⠞⠀⠀⠀⣠⡶⠿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢹⠛⠓⠚⠉⠀⠀⠀⠀⠘⣡⠤⢤⡈⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣧⣀⣠⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢽⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⣿⠟⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠋⠰⣹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠓⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⣤⣴⣯⣭⣥⣖⣚⣓⣒⣶⣶⠶⢿⣟⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣱⣏⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⡵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢀⣠⠶⠋⠉⠀⠀⢀⡾⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠹⣿⣿⣿⣿⣟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢠⢄⡀⠒⠋⠀⠀⠀⠀⠀⢠⣾⠿⠦⣤⣀⣀⠈⢹⣿⣿⣿⡿⠁⠀⠀⠀⠀⠃⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠈⣿⣿⡏⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠈⠉⢩⡿⠛⠛⢻⠃⠀⠀⠀⠀⠀⢀⣿⣿⣿⢿⣿⡿⠟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢀⣾⣿⣿⣿⣿⣷⣶⣦⣤⣼⡇⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡜⢀⢔⣿⡄⠀⣠⣾⣿⣿⣿⢸⣿⣷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠉⠉⠉⠉⠉⠛⠛⠛⠋⣿⢷⡀⠀⠀⠀⠀⠀⡿⠀⠀⣔⣡⣤⣿⣿⣀⣚⣻⣿⣿⣿⣿⢸⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢳⣄⣀⠀⠀⠀⡇⠀⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣾⣿⣿⣸⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡟⠀⠉⠉⠛⢺⠷⠶⠦⠤⠤⠤⠶⡏⢀⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⢹⠀⠀⠀⡠⠖⠊⠉⠉⠉⠙⣶⣦⣀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣿⠁⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⢿⣿⣷⢸⣿⣿⣿⡇⠀⡠⠊⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣦⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠘⢧⡀⠀⠀⠀⣸⣧⠀⠀⠀⠀⣀⣾⣿⡿⠋⠀⠀⠀⠀⠈⢣⣿⣿⡿⢠⣾⣷⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⡆ ⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⣼⣿⣶⣦⣶⣿⣿⣷⣦⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⣵⣿⣿⣿⣿⣿⣿⣟⣭⣿⣿⣿⣿⣿⣿⣿⠇ ⠀⠀⠀⠀⠀⠀⠀⣼⡀⢀⣠⣾⣿⣿⣿⣿⣭⣿⣿⣯⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀ ⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀ ⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠉⠉⠻⠿⠶⢿⣦⣀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀ ⠀⠀⠀⠀⠀⠈⣹⣿⠿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣾⣿⣿⣿⣿⡇⠈⠉⠉⠙⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠉⠣⠼⠟⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⣠⣤⣀⣀⣀⡀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣄⠀ ⠀⢿⣿⣿⠿⢿⣿⣿⣷⠆⢰⣿⣿⣿⣿⣿⣿⣿⣿⡄⠰⣾⣿⣿⡿⠿⣿⣿⡿⠀ ⠀⢸⣿⡇⠀⠀⠀⣈⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⡁⠀⠀⠀⢸⣿⡇⠀ ⠀⠸⣿⣷⠀⢀⣾⣿⣿⡇⠸⣿⣿⣿⣿⣿⣿⣿⠟⠁⣼⣿⣿⣦⡀⠀⣿⣿⠇⠀ ⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⡄⠹⣿⣿⣿⢁⣤⡤⠀⣾⣿⣿⣿⣿⣿⣶⣿⣿⠀⠀ ⠀⠀⠸⣿⣿⣿⡿⠛⠉⠉⠙⠢⡈⠻⠿⠿⠟⢁⠔⠋⠉⠉⠛⢿⣿⣿⣿⠇⠀⠀ ⠀⠀⠀⢻⣿⣿⠀⢰⣾⡆⠀⠀⢸⣶⣶⣶⣶⡏⠀⣴⣷⠀⠀⠀⣿⣿⡟⠀⠀⠀ ⠀⠀⠀⢸⣿⣿⠀⠀⠉⠁⠀⠀⣸⣿⣿⣿⣿⣇⠀⠈⠉⠀⠀⠀⣿⣿⡇⠀⠀⠀ ⠀⠀⠀⠘⣿⣿⣷⣤⣀⣀⣠⣴⣿⠿⠿⠿⠿⣿⣦⣄⣀⣀⣤⣾⣿⣿⠃⠀⠀⠀ ⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀ ⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀ ⠀⠈⠉⠉⠉⠉⠉⣹⣿⣿⣿⣿⣿⣷⣄⣠⣾⣿⣿⣿⣿⣿⣏⠉⠉⠉⠉⠉⠁⠀ ⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀ ⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡄⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⠀⢀⣠⣴⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⣶⠂⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀ ⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠟⠁⢸⣿⡿⠛⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠛⠉⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢸⣿⣿⣿⣿⣿⣿⡇⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢹⣿⣿⣿⡆⠉⠻⡄⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠘⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇⠀⠀⠀⠀⠀⢀⡀⠤⠤⠴⠶⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢹⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠤⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠈⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⣤⡀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⣸⣿⣷⠀⠀⢀⡤⠔⠬⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⢀⡞⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣀⣀⡠⠔⠊⢩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠁⠀⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠷⠒⠉⠉⠉⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡰⠋⡆⠀⠀⠀⠸⡀⠀⢀⠏⠀⠀⠀⠤⠛⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠖⠛⠉⠁⠀⠀⠀⣀⡠⠞⠁⠀⢸⠀⠀⠀⠀⠙⠲⡎⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠺⣿⡉⠀⠀⠀⠀⠀⡆⠀⠀⠀⣠⣾⠀⣀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⣀⣀⠀⠀⢀⢤⠀⠀⠈⠑⢤⡀⠀⠀⣰⠀⠀⣠⣾⣿⣿⠻⠉⠋⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠈⠑⡞⠈⠈⢆⠀⠀⠀⠀⠉⠉⢁⣠⣾⣿⣿⣿⣿⠀⠁⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣤⠃⠀⠀⠘⡄⢀⣀⣠⣤⣶⣿⡿⣿⣿⠉⠻⣿⠀⠀⠀⠀⢸⡄⠀⠀⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠩⠀⠀⢹⠿⣿⣿⠃⠀⠉⠀⠀⠘⡄⠀⠀⠀⢸⠛⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⡜⠀⠹⡏⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⡸⠀⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠁⠀⣼⡟⠁⠀⠀⠈⠉⠉⠛⠛⠿⠿⢿⡇ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠼⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⣄⢀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⣀⣀⣀⡠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⡤ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⣿⡿⠟⣻⣿⣿⣿⣿⣿⣥⣄⣀⣀⣀⡀ ⠀⢀⣴⣶⣶⣦⣤⣄⠀⢷⣼⣿⣿⡿⠻⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀ ⢀⣿⡿⢻⣿⣿⠿⠟⠀⠀⠟⠛⢿⠿⠿⠷⡶⠚⢻⣿⣿⣿⣿⣿⣋⣁⠀⠀⠀⠀ ⠹⡟⠀⠸⡿⠁⠀⠀⠀⠀⠈⢄⠈⠀⠀⠀⠁⢀⣾⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⣤⣤⣶⣟⣿⣻⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡾⣿⣿⡣⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢻⣿⣿⢿⣿⣿⡝⠻⢷⣄⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⡿⠁⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⣠⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀ ⢸⣿⣯⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀ ⢼⣿⣿⣆⠀⢀⣀⣀⣴⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀ ⢸⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢻⣿⠋⠙⢿⣿⣿⡀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢸⠿⢆⣀⣼⣿⣿⣿⣿⡏⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⡀⣨⡙⠟⣩⣙⣡⣬⣴⣤⠏⠀⠀⠀⠀⠀⠀⣀⡀ ⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣀⣤⣾⣿⣿⡇ ⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣇⠀⢸⣿⣿⠿⠿⠛⠃ ⠀⠀⠀⠀⢠⣿⣿⢹⣿⢹⣿⣿⣿⢰⣿⠿⠃⠀⠀⠀⠀ ⠀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡛⠀⠀⠀⠀⠀⠀ ⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠓⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠈⣿⣿⣿⣿⠀⣿⣿⣇⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣷⣶⣿⣿⣿⣿⣶⣿⣿⣿⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣶⣤⡈⠙⢿⣿⠉⠀⠀⠙⣿⣿⣿⣿⠀⠀⠀⠀⠀ ⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢿⣷⣶⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀ ⠀⠀⢠⣿⣿⣿⡇⠈⣿⣿⣿⡏⠀⣿⣿⡆⠈⠋⢉⣀⣀⣀⣈⠉⠉⠀⠀⠀⠀⠀ ⠀⠀⢸⣿⣿⣿⣷⣶⣿⣿⣿⣿⣶⣿⡟⠁⣠⣾⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀ ⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⢹⣿⡟⢀⣾⣿⣿⡿⢿⣿⣿⣿⡿⢿⣿⣧⠀⠀⠀ ⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⠁⢸⣿⣿⣿⡇⢸⣿⣿⣿⡇⢀⣿⣿⡆⠀⠀ ⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⡇⠀⠀ ⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣧⡀⠀⢀⣼⣿⣿⣿⡇⠀⠀ ⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀ ⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄⠀ ⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⣤⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢀⣠⣤⠀⢸⠁⠙⣷⣄⠀⢰⡟⠃⠀⠀⠀ ⠀⠀⠀⣠⣾⠿⠿⢿⣧⠘⢀⠄⠼⣿⣆⣸⡇⠀⢠⣦⠀ ⠀⢀⣾⠟⠁⠀⠀⢠⣿⣧⢫⢀⣴⣿⣿⣿⣧⢸⣿⢿⡇ ⢀⣾⠏⠀⠀⠀⣴⠋⠀⠈⢯⣿⣿⣿⠋⠉⣽⣿⡟⠀⡇ ⣸⡏⠀⠀⠀⡼⠁⣀⣤⢶⣾⣿⣿⣿⣶⣼⡿⣿⠃⠀⡇ ⡿⠀⠀⢀⣜⣠⠾⠋⠁⣾⣿⣿⣿⣿⣿⡟⢀⡟⠀⠀⡇ ⣧⡶⠿⠿⠟⠁⠀⠀⠀⣿⣿⣿⣿⣿⡿⠉⣾⠃⠀⢸⠁ ⠁⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⣹⠟⠁⠀⠀⠛⢻⣤⠏⠀ ⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠋⠀⠀ ⠀⠀⠀⠀⠐⠁⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⡇⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠙⢦⡀⠀⣀⣠⡤⠤⢤⣀⣀⠸⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠈⠉⠙⢦⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡀⣷⢠⡄⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢀⣀⣀⣀⡀⠀⠹⣟⣾⡧⣾⣧⠴⠚⠁⠀⠀⠀⠀⠀⠀⢠⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠈⠉⠙⠓⠺⠿⢥⣭⣁⡥⠖⠂⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣠⡤⢤⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠶⢶⣠⠄⠀⠠⢸⡇⠀⠀⠀⣿⣷⣾⠃⠀⠀⠀⠀⠀⢸⣶⣿⡧⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠘⡇⢀⡴⠶⢭⠛⠁⠀⠀⠀⠀⠂⠀⠀⠛⠉⠀⠀⣷⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢹⡀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠉⢹⠂⢀⡾⠁⠀⠀⠀⠀⢣⢸⡀⠀⢈⡇⠀⢳⣤⣤⣤⣤⣤⡴⠀⠀⢰⠟⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣇⠀⢀⣀⣀⣀⣠⣤⣤⠤⡤⠀⡘⠀⡼⠁⠀⠀⠀⠀⠀⣸⠀⠓⠦⠞⠁⠀⠀⢧⠀⠀⢀⡟⠀⠀⠀⣯⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠁⠈⠋⠙⠁⠀⠰⠀⠀⢠⢠⡇⠀⡇⢰⠃⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢤⡀⠈⠓⠒⠋⠀⢀⣠⠞⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠈⢣⣦⡀⠀⠈⢹⠁⢸⠀⠺⢄⣀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠛⠛⠲⠶⣟⣹⠏⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣆⠀⣨⠉⠉⠁⢀⠻⢤⣈⡁⠂⠤⣌⡙⢢⠀⠀⢸⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⡀⠀⠀⣸⠛⢦⢀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠃⣳⡾⠟⠁⢀⡀⠀⠀⠀⠀⠈⠙⣷⢠⣁⡟⠀⠀⢸⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⣠⠇⠀⢨⡏⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢶⡶⠀⠀⠀ ⠀⠀⠀⠟⠇⣠⣿⠿⠧⠀⠀⠀⠀⠀⠀⢸⣎⡋⠙⢓⢶⡌⣧⠀⠀⠀⠀⠀⠙⠒⠤⠤⠀⣠⠤⠤⢾⡉⠁⣀⣄⣼⣱⡇⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠚⠡⠆⠀⠀ ⠀⠀⠀⠀⢠⡟⠉⠄⠀⠀⠄⢀⠀⠀⠀⠈⠉⠉⠉⠹⣆⣇⣹⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⢿⣿⣿⠿⠆⠀⠀⠀⠀⣆⡄⠈⣿⠀⠀⣴⣇⣤⡄⠀⠀⠀ ⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠃⠀⠀⠀⠀⠀⡴⠿⢡⡞⢻⣿⣿⣏⡙⠋⢀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣝⡒⠤⢤⡀⠀⠀⠀⢀⣀⢘⣟⣽⡿⠁⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⢘⠀⠐⠦⠉⠀⠰⢣⠞⠓

⠀⠀⠀⠀⠀⠀⠄⢄⠀⠀⠀⠀⠀⢀⡀⠀⠀ ⠀⠀⠀⣠⣀⡈⠀⠀⠑⢾⡷⠐⠈⠀⠁⠀⠀ ⢆⠀⠀⠛⠻⢁⠄⣀⡐⠂⠀⠀⠀⠘⠀⠀⠀ ⠀⠑⡀⠀⠀⠀⠀⠹⡃⢀⢀⢴⣤⡐⢂⠀⠀ ⠤⠠⣚⠖⢩⡑⠐⠉⠉⠁⠈⢄⠀⢁⠀⠀⠀ ⠈⢀⡀⠸⠈⠁⢠⣤⣤⣤⣀⠀⠈⢁⠀⣀⠀ ⠀⠘⠀⣾⣶⡄⠀⠻⣆⣠⡿⠁⡐⠀⠠⠀⡄ ⠀⠀⠘⢏⠙⢿⣆⠀⠈⠛⣠⣴⠀⣤⣤⡠⠀ ⠀⠀⠀⠀⢉⠟⠭⠀⣀⣀⠹⠿⢄⡛⠝⠁⠀ ⠀⠀⠀⠀⠩⠤⠤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⢿⣿⣶⣤⡀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⠻⢿⣿⣮⣻⣿⣿⡄⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡅⠀⣉⣿⡇⣿⣻⠇⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⠆⢴⣧⣗⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⡇⢀⣿⣛⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠈⢻⠾⣿⣟⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠋⠀⠀⠀⠀⡀⢱⠘⣦⡀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⡀⠀⠀⠅⠈⢿⣦⡀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡈⠌⢻⣷⣄ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⣿⡌⠀⠈⠀ ⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠀⠀⠀⢒⣒⠈⠁⠄⠀⠀⠀⠀⠀⠀⠀⣿⡧⡇⠀⠀ ⠀⠀⡠⠒⠉⠀⠀⠀⠀⠄⠂⠉⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⣰⣿⡇⠃⠀⠀ ⣠⠀⠄⠀⠒⠀⠀⠈⠉⠀⠉⠢⠀⠀⠀⠀⡀⠄⠈⠀⠀⠀⢀⣼⣿⡟⡈⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠆⡀⠀⠀⠀⠀⢀⣴⣿⡿⢋⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠐⠈⠉⠻⠫⠐⠁⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠙⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠹⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⣀⣠⣤⣴⣶⣶⣾⣟⠟⠀⠀⠠⠃⢠⠎⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠑⣟⢿⣷⣶⣶⣶⣤⣤⣀⠀ ⠀⠸⣿⣿⣿⣿⣿⡿⢻⠋⠀⠀⠀⠁⡰⠃⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⡀⠀⠀⠀⠈⢺⠛⣿⣿⣿⣿⣿⣿⠃ ⠀⠀⢿⣿⣿⣿⡿⢳⠃⢀⡀⠀⠀⡼⠁⠠⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠑⠄⠀⠳⡛⢿⣿⣿⣿⡟⠀ ⠀⠀⢸⣿⣿⣿⣤⡇⠃⠀⠀⠀⢰⠃⠀⢡⡎⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡄⠀⠀⠀⠀⢱⢸⣿⣿⣿⠃⠀ ⠀⠀⠀⣿⣿⢫⡟⠸⠀⠀⠀⠀⡏⠀⠀⡎⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⣄⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⢦⠀⠀⢫⡹⣿⡟⠀⠀ ⠀⠀⠀⢸⣿⣿⠁⣏⠀⠀⠀⢸⠁⠀⡸⠀⠀⣼⠟⡇⠀⠀⠀⠀⠀⠀⠘⣼⡆⠀⠀⠈⣧⠀⠀⠀⠀⠀⠘⡆⠀⠀⢧⠀⠀⢻⣿⠃⠀⠀ ⠀⠀⠀⠀⣿⠇⠀⠮⠀⠀⠀⡏⠀⠀⡇⠀⡰⠧⠤⠼⡀⢰⠀⠀⠀⠀⠀⠸⣿⣦⠀⠀⠘⣧⡀⠀⠀⠀⠀⠹⡀⠀⠸⡄⠀⠈⢿⣄⠀⠀ ⠀⠀⠀⢠⡟⡄⠀⡳⠀⠀⢰⠇⠀⣼⠁⡰⢁⣀⣀⠀⠹⡄⢣⠀⠀⠀⠀⠀⠙⣌⠛⠦⡀⠱⡙⢦⡀⠀⢣⠀⢣⠀⠀⢇⠀⠈⢏⣏⠛⢖ ⠀⠀⢠⢳⠃⡇⠀⢑⠀⠀⢸⠀⡜⢉⣦⣯⡵⠶⢦⣌⠐⠜⢄⠳⣄⡀⠀⠀⠀⠈⢲⣈⣭⣵⣿⣑⠳⣄⠈⡇⠘⡆⠀⢸⠀⠀⢸⠘⡆⠀ ⠀⣠⠃⡞⠀⢱⠀⢸⠀⠀⢸⡜⢠⣿⠟⢁⠐⢿⣷⣌⢳⡀⠀⠑⠸⠯⠉⠑⠒⢢⡟⠁⠘⣿⣶⡍⠻⣬⡷⣧⠀⢳⠀⠨⡄⠀⢈⠆⢹⡀ ⡜⠁⣼⠇⠀⠈⡆⠈⡆⠀⢸⠐⣾⠃⠀⣿⡖⣶⡽⣿⠀⣧⠀⠀⠀⠀⠀⠀⠀⣾⠀⣷⣶⣷⣹⣿⠀⢹⡾⢾⢀⣾⡀⠀⡇⠀⢸⠀⠀⢣ ⣠⡾⢼⡁⠀⠀⠹⡄⢱⠀⢸⠐⣻⠀⠀⣿⣏⣿⣧⢿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣿⢿⣿⢿⣻⠀⠀⡷⠞⠉⣯⡇⢀⡇⠀⣸⠀⠀⠘ ⣿⡝⣾⠀⠀⠀⠀⢳⠈⢇⠸⡀⠹⡄⠀⠘⢇⣐⡨⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣔⣂⡸⠃⠀⢸⠓⠂⢰⢿⡇⢰⠀⢀⡇⠀⠀⠀ ⣷⣛⣼⠀⠀⠀⠀⠀⠣⡈⢆⢇⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠋⠁⢀⡷⢫⣗⡸⠀⡼⠀⠀⠀⠀ ⢷⣝⣾⡄⠀⠀⠀⠀⠀⠑⣌⢾⡀⠀⠀⠀⠀⠀⠀⣠⡖⠒⠒⠒⠒⠒⠒⠒⠒⠦⠤⣄⠀⠀⠁⠁⠀⠀⠀⡜⠛⣫⣷⠃⡼⠁⠀⠀⠀⠀ ⠸⣿⣸⣇⠀⠀⠀⠀⠀⠀⠀⢣⣇⠀⠀⠀⠀⠀⢸⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⡇⠀⠀⠀⠀⠀⢸⣧⢿⢿⡟⡸⠃⠀⠀⠀⠀⢀ ⣧⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⢠⡿⣼⡹⣾⡟⠁⠀⠀⠀⠀⠀⢸ ⠈⠁⠈⠻⡽⡄⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠓⢦⡀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⢀⣴⣟⡳⢧⣻⡽⠀⠀⠀⠀⠀⠀⢀⡯ ⣿⣶⣤⣄⣹⡽⣆⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠚⠉⠀⠀⠀⢀⣠⣴⢯⡳⣞⣿⣷⣹⡇⠀⠀⠀⠀⠀⢀⣾⢳ ⣿⣿⣿⣿⣿⣿⣯⠷⣄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣒⣦⢤⣤⣠⣀⣄⣠⣤⣤⣴⣶⢻⡽⣻⣮⣗⣻⢼⣻⡟⢾⡇⠀⠀⠀⠀⢀⡾⣭⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣽⣳⢦⣀⠀⠀⠀⠀⠀⠹⡏⠁⠉⠓⠦⣉⡉⠛⣙⣩⠵⢚⠩⠳⣽⣽⣼⣿⣾⣷⣷⣿⣿⠆⠀⠀⠀⣠⣾⣻⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠑⢄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡠⢤⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠑⢄⣴⣿⣿⣦⣀⠤⠒⠉⠀⠀⡜⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⣀⢸⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠋⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀ ⢀⣀⡀⠠⠤⠤⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⡤⣄⠉⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀ ⠀⠈⢦⠀⠀⠀⠀⠀⠀⠈⠉⠁⢰⠙⠆⣠⣤⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀ ⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⡇⠀⠀⠀⠀⠀⣀⡀⠀⠘⣤⢄⠀⠀⠀⠀ ⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⣼⠃⠀⣤⣀⡤⠾⠿⣿⣷⣴⠃⠀⢣⠀⠀⠀ ⠀⠀⠀⠀⠀⡤⣵⣀⡠⠚⠙⠲⠤⣀⡴⠋⠳⠊⠙⠚⠀⠈⠳⡀⠀⠀⠈⠙⢢⡀⠀⢇⠀⠀ ⠀⠸⡉⠑⠚⠦⢌⡉⠀⠀⢸⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⠀⠀⣰⠃⠀⠸⠀⠀ ⠀⠀⠑⡄⠀⠀⠀⠈⡇⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠓⠚⠁⠀⠀⠀⡇⠀ ⠀⠀⠀⠈⣦⣄⠀⢀⠇⠀⠀⠀⠀⠀⣿⣿⡿⠿⢿⣿⣶⣶⣤⡀⠀⠀⠀⢀⡤⠒⠲⡦⡇⠀ ⠀⠀⠀⠀⢸⠈⠉⣡⣶⣦⣄⠀⠀⠀⠈⢿⣷⡄⠀⠀⠉⣿⣿⠇⠀⠀⡰⠃⠀⠀⢰⠁⠘⡄ ⠀⠀⠀⠀⠈⡆⢠⣿⣿⣿⣿⣷⡄⠀⠀⠈⠻⣿⣦⣄⣾⣿⠏⠀⠀⡜⠀⠀⠀⠀⠘⠀⠀⡇ ⠀⠀⠀⠀⠀⠘⢾⣿⠋⠙⢿⣿⣿⣆⠀⠀⠀⠈⠻⢿⡿⠃⠀⢀⣸⠁⠀⠀⠀⠀⠀⠀⢠⠃ ⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠙⣿⣿⣧⠀⠀⠀⠀⠀⢀⣴⣾⣿⡏⠀⢰⣿⣷⣶⣶⣆⠎⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⣿⡿⠿⠋⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⠀⠘⠿⠿⠿⡫⠋⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠑⠒⠒⠤⠤⠤⠤⢤⠔⠛⠋⠉⠓⠤⠤⠤⠊⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⠊⠀⠀⠀⠀⠀⢀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣴⢶⣾⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢠⡀⠀⠀⣀⠀⢉⡶⠠⣤⣀⡀⠀⠉⠉⠛⠙⠀⠀⢀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣰⡇⠀⠘⢁⣴⣿⠀⠀⣠⣿⣿⣽⣦⣤⡀⠀⠀⠀⠀⠘⢦⣄⠀⠀⠀ ⠀⠀⢀⠞⠉⠙⢀⣤⣾⣿⣿⠀⣼⣿⡿⠛⠛⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀ ⠀⣠⠃⠀⠀⠀⠙⢟⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢳⡀ ⠀⠇⠀⠀⠀⠀⠀⠀⠈⠺⣷⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⣧ ⣸⠀⠀⡠⠀⠀⣰⢟⡛⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿ ⡇⠀⠊⠀⠸⢿⠿⠆⠀⠌⣿⣿⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣿ ⣧⠁⠀⠀⠀⠀⢰⠀⠀⠀⣿⣿⣿⣷⣄⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⢀⠔⠀⣼⠇ ⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣧⣀⠚⠿⢯⣀⣀⠀⠤⠔⠉⢀⣾⠏⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣯⣿⢿⣿⣿⢄⠀⠀⠀⠀⢀⣠⣴⠿⠃⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠿⣷⣶⣶⣶⣶⠾⠟⠋⠁⠀⠀⠀⠀

⠠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠛⠉⠀⣀⣀⣴⣊⣀⠄ ⠀⠀⠀⠀⠀⣇⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⣯⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⡟⣽⣶⣿⣿⣿⡿⠿⣿⣟⣛⡿⠯⠭⠭⢤⣤⠾⢶⣶⣖⣟⠛⠛⠛⠛⢻⣷⡚⢲⠞⠖⠚⠿⠭⢭⣟⣛⣛⣛⣋⣉⣁⠤⠤⠚⠋⠀⠈⢻⡒⠳⣄⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢰⣿⣟⣿⡭⠽⣶⠞⠛⠉⠉⠀⠀⠀⠀⣰⣾⣥⣤⣈⠻⣿⣞⣦⠀⠀⠀⠀⢻⣷⠟⠀⠀⣤⣤⣶⠶⢶⣮⡝⢿⣍⠫⣄⠀⠀⠀⠀⠀⠀⠀⢻⡷⣽⡆⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣴⢾⡿⠋⣧⣤⣿⡷⡝⣿⣯⡳⣄⠀⠀⠀⢻⡄⠀⠐⠁⣾⣾⣦⣾⡏⠻⣦⠉⢷⣮⡳⣄⠀⡀⠀⠀⠳⡄⢳⣀⡇⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢸⣇⠀⠀⢀⣿⡟⠀⠀⠀⠀⣰⣾⠁⢸⡇⠀⣿⣿⠘⣇⠀⠈⠻⣷⣌⡶⣶⡀⠀⠻⡄⠀⣄⣿⣻⡟⠋⡇⠀⠟⠁⣸⡏⡟⠷⣝⢮⡳⡄⠀⠹⣎⣿⠁⠀⠀⠀⠀⠐⠀ ⠀⠀⠀⠀⢯⠙⠷⣄⣸⡟⠀⡀⢀⡴⠞⠁⣿⡆⠀⠈⠛⠛⠛⠛⠋⠀⠀⠀⣼⢹⡿⠶⣝⡶⣄⠽⣆⠈⠉⠉⠛⠛⠓⠚⠀⠀⣿⡇⡗⠀⠈⠛⣿⣮⣅⠀⠘⣿⡇⠀⠀⠀⠀⠀⢠ ⠀⠀⠀⠀⠀⠛⠶⣖⡿⢀⣞⡷⠛⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠙⠛⠦⢬⣃⡀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠃⠀⠀⢀⣿⠇⢉⡷⢦⣌⣷⠀⠠⣄⡀⠀⠈ ⠀⠀⠀⠀⠀⠀⠀⢨⣧⠾⠯⠤⢤⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⣿⢻⠀⠀⠀⢸⣃⡴⠋⠀⠀⠈⠙⠃⠀⠈⠉⠑⠚ ⠀⠀⠀⠀⠀⠀⠀⠘⡃⠀⣠⠶⠁⢸⣴⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠀⠀⠀⣸⠋⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⣠⡶ ⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠶⠶⠤⣤⣿⡀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡿⠀⠀⠀⢹⣏⡳⣶⣖⠒⠋⠉⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠸⣿⡍⠳⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣿⠃⠀⠀⣸⠇⠀⠀⠀⠀⠀⠙⠯⢭⣉⣁⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⣿⡇⠀⠙⣦⠀⠀⠀⠀⠀⠠⡖⠒⠒⠒⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢿⠀⠀⢠⡿⢤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢸⡇⠀⢠⠏⠳⣄⠀⠀⠀⠀⠉⠉⠓⠒⠛⠉⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⢸⠀⢠⣿⠁⠰⢬⡙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⢸⣇⣴⣋⡴⠟⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣻⠏⠀⠀⠀⠀⣸⠀⢸⡏⠀⠀⠀⡇⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣸⡧⠞⠉⠀⠀⠀⠀⠈⢳⣦⡀⠀⠀⠀⠀⢀⣠⡴⠟⠁⣴⢃⣀⣀⣤⣴⣾⣿⠀⣿⡇⠀⢀⣾⠃⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣷⣠⠞⠁⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡼⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢺⣿⣿⡀⠀⠀⠀⠀⠀⣀⣤⡾⠿⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠿⢿⣿⣿⡓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⣸⠁⠘⣇⢀⣠⠶⠚⠻⠗⠶⠶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣉⡙⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣠⠏⣠⡴⠛⠉⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠃⠀⠀⠀⣠⣴⣿⣿⣿⡟⠉⠁⠀⠀⠀⠉⠉⠛⠓⠦⠄⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀ ⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠛⠁⠀⠀⢀⣴⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠛⢦⣄⡀⢀⡼⠃⠀⠀

⠀⠀⠀⠲⣦⣤⣀⣀⠀⠀⠀⣀⣀⣠⣤⣀⣀⠀⢀⣀⣠⣤⣶⣶⠟⠀⠀⠀ ⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠈⢿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⡿⠃⠀⠀⠀⠀⠀ ⠀⠀⠀⠲⣶⣶⣾⣿⣿⠀⢨⠙⢿⣿⣿⠏⣅⠀⢸⣿⣿⣷⣾⠟⠁⠀⠀⠀ ⠀⠀⠀⠀⠈⠻⢿⣿⣿⢷⣶⣶⣾⣿⣿⣶⣶⣾⠟⣿⣿⣿⣋⠀⠀⠀⠀⠀ ⠀⠀⢀⣀⣀⠐⢶⣿⣿⣧⠁⠀⠋⠁⠈⠋⠀⢀⣾⣿⣿⡿⣷⣶⠀⠀⠀⠀ ⠀⠀⣼⣿⣿⣷⣤⣙⣿⣿⣷⣶⣶⣴⣴⣴⣶⣿⣿⣿⠟⣡⣿⣿⣧⣄⣀⡀ ⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡿⠿⣿⣿⣿⣿ ⣿⡿⠛⠿⠟⠉⠉⠉⠸⠋⠀⠻⡿⣿⣿⣿⣿⠻⠇⠀⠀⠈⠀⠀⠈⠉⢸⠃ ⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡀⠀⠀⠀⠀⢸⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠢⡀⠀⠀⢀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡔⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⡹⠀⠀⠘⢄⠀⠈⠲⢖⠈⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⢠⠞⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠉⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠇⠀⡤⡀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣾⣿⣷⣶⣤⣄⣉⠑⣄⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢀⠞⢁⣴⣾⣿⣿⡆⢇⠀⠀⠀⠀⠀⠸⡀⠀⠂⠿⢦⡰⠀⠀⠋⡄⠀⠀⠀⠀⠀⠀⠀⢰⠁⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢆⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⡴⢁⣴⣿⣿⣿⣿⣿⣿⡘⡄⠀⠀⠀⠀⠀⠱⣔⠤⡀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⡜⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢣⠀⠀⠀⠀⠀ ⠀⠀⠀⡼⢠⣾⣿⣿⣿⣿⣿⣿⣿⣧⡘⢆⠀⠀⠀⠀⠀⢃⠑⢌⣦⠀⠩⠉⠀⡜⠀⠀⠀⠀⠀⠀⢠⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣣⡀⠀⠀⠀ ⠀⠀⢰⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠱⡀⠀⠀⠀⢸⠀⠀⠓⠭⡭⠙⠋⠀⠀⠀⠀⠀⠀⠀⡜⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡱⡄⠀⠀ ⠀⠀⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢃⠀⠀⠀⢸⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣆⠀ ⠀⢸⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⣆⠀⠀⡆⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡠⠖⣡⣾⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⠀ ⠀⡏⣾⣿⣿⣿⣿⡿⡛⢟⢿⣿⣿⣿⣿⣿⣿⣧⡈⢦⣠⠃⠀⠀⠀⠀⠀⢱⣀⠤⠒⢉⣾⡉⠻⠋⠈⢘⢿⣿⣿⣿⣿⠿⣿⣿⠏⠉⠻⢿⣿⣿⣿⣿⡘⡆ ⢰⡇⣿⣿⠟⠁⢸⣠⠂⡄⣃⠜⣿⣿⠿⠿⣿⣿⡿⠦⡎⠀⠀⠀⠀⠀⠒⠉⠉⠑⣴⣿⣿⣎⠁⠠⠂⠮⢔⣿⡿⠉⠁⠀⠹⡛⢀⣀⡠⠀⠙⢿⣿⣿⡇⡇ ⠘⡇⠏⠀⠀⠀⡾⠤⡀⠑⠒⠈⠣⣀⣀⡀⠤⠋⢀⡜⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠉⡃⠈⢀⠴⣿⣿⣀⡀⠀⠀⠀⠈⡈⠊⠀⠀⠀⠀⠙⢿⡇⡇ ⠀⠿⠀⠀⠀⠀⠈⠀⠉⠙⠓⢤⣀⠀⠁⣀⡠⢔⡿⠊⠀⠀⠀⠀⠙⢦⡀⠀⠐⠢⢄⡀⠁⡲⠃⠀⡜⠀⠹⠟⠻⣿⣰⡐⣄⠎⠀⠀⠀⠀⠀⠀⠀⠀⢣⡇ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠙⢦⣀⢀⡴⠁⠀⠀⠀⠀⠉⠁⢱⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⢏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⡴⠁⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⠤⠖⠋⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⠈⠁⠀⠀⠀⡔⠛⠲⣤⣀⣀⣀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⡠⢔⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠤⠒⣀⠀⠀⠀⠀⣀⠟⠀⠀⠀⠑⠢⢄⡀⠀⠀⠈⡗⠂⠀⠀⠀⠙⢦⠤⠒⢊⡡⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⣒⡁⠬⠦⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢺⢠⠤⡀⢀⠤⡀⠠⠷⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡀⡱⠧⡀⢰⠓⠤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣄⣾⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⣽⢿⣵⣿⡟⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣀⣀⣽⣷⣖⡤⣵⣿⣧⣭⠁⠀⠀⠀⠀⠀⠀ ⢀⡤⢦⣄⡈⠛⠈⠛⠋⠉⠋⠻⣿⣿⡿⣧⣀⠀⠀⠀⠀⠀ ⣾⠁⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠉⠻⣦⠀⠀⠀ ⢹⡄⠀⠀⠘⣿⣦⡀⠀⠀⠀⣠⣾⣿⢯⣯⣤⡀⠈⣷⣄⠀ ⠈⢷⡄⠀⠀⠸⣿⣿⣦⣤⣼⣿⣿⣿⢾⣿⣿⣷⠈⠇⡹⡦ ⠀⠈⢿⣷⣄⡀⠘⢿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀ ⠀⠀⠀⠹⣿⣿⡄⠀⠙⢣⣿⠿⠿⠟⠛⣿⠟⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠉⠀⠀⣠⡿⠃⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⠀⠀⠀⠀⠶⠻⣷⣄⣀⢀⠀⠀ ⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⢰⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⡆ ⢸⣿⣿⡆⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⣿⣿⣿⣿⡇ ⢸⣿⣿⣇⠀⠀⢀⡀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⢠⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⡄⠀⠘⣿⣶⣤⣀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⢀⣠⣴⣾⠃⠀⢀⣾⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⡄⠀⠘⢿⣿⣿⣷⣦⣀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⢀⣤⣾⣿⣿⣿⠃⠀⢀⣾⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣄⠀⠈⠻⣿⣿⣿⣿⣷⣤⡀⠈⢿⣿⣿⣿⡿⠟⠛⣿⠿⠛⡿⠿⣿⣿⣿⣿⣿⣿⠟⠀⣀⣴⣿⣿⣿⣿⡿⠃⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠈⠻⣿⣿⣿⣿⣿⣦⡀⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⢻⡿⠁⢀⣾⣿⣿⣿⣿⣿⠟⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠈⠛⢿⣿⣿⣿⡿⠆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠴⣿⣿⣿⣿⣿⠟⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠉⠻⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⠟⠋⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡤⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣴⣶⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣼⠋⠙⡆⠀⠀⠀⠀⠀⠀⠀⣰⠋⠙⣆⠀⠀⣿⣿⣿⣿⡇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣦⣴⡧⠀⠀⠀⠀⠀⠀⠀⣿⣦⣴⣿⠀⢰⣿⣿⣿⠏⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢿⣿⣿⠇⠀⠀⠀⣀⠀⠀⠀⢹⣿⣿⡟⠀⠸⠿⠛⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠛⠋⠀⠀⠀⠈⠛⠁⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠈⠑⠒⠋⠉⠑⠒⠋⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡻⢿⣧⣴⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢉⣴⣿⣿⢋⣶⣤⣄⣀⣀⣀⠀⢀⣀⣀⣤⣶⣷⣌⢿⣿⣷⠭⠛⣿⣴⣧⠀⣠⣆⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⣠⣾⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣷⡘⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡀⠸⠻⣿⡇⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣿⣿⣿⢿⢸⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡀⠻⣿⡀⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠀⢠⣿⠛⠇⣶⣾⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠙⠂⠸⠋⢿⣿⣿⣿⣿⣿⣿⡿⠁⠁⢀⡾⠁⣾⣾⣿⣿⣿⣿⣿⠟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⢻⡿⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⡿⠃⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠛⢋⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠤⠤⠾⠿⠷⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉⠀⠀⠀⠀⠀⠀⠈⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠶⣄⡈⠂⠔⠀⠀⠈⠀⠀⠈⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡘⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠒⣉⣉⠛⢿⣿⣿⣄⠀⢀⣠⣤⣶⡾⠛⠋⠙⠁⣀⠔⠁⣠⣬⡀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠐⠁⢞⣯⠎⣸⠿⠛⠉⠀⠘⢿⣟⢭⣭⡅⠀⠀⠀⠀⡠⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⠀⠀⢀⣠⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠜⠻⠃⠀⠀⠀⠰⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⣀⣀⣤⣼⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣠⣤⣤⣤⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣶⣿⣿⣿⡀⠀⠀⠀⠀⠀ ⠠⠔⠒⠉⠉⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣼⠟⠛⠛⠛⠛⠻⢿⣷⣤⣄⣀⣠⣤⡶⠦⠤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠛⠉⠁⠒ ⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁⠀⠀⠀⠈⢳⣆⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠀⠐⣶⣿⣿⠿⣿⠖⢦⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠸⣿⣿⣇⣿⣀⡈⠀⠀⠋⠉⠙⠛⠛⢲⣦⠤⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣶⣿⡿⠃⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠻⣿⣄⡉⠙⠛⠛⠛⠛⢻⡿⠋⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⠶⠶⠒⠉⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠲⣤⣤⣤⣤⣴⣾⣿⡿⠟⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠢⠤⠤⠤⠤⠔⠊⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀ ⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀ ⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀ ⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀ ⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

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

⠸⣷⣦⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠀⠀⠀ ⠀⠙⣿⡄⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠉⣿⡿⠁⠀⠀⠀ ⠀⠀⠈⠣⡀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⣰⠟⠀⠀⠀⣀⣀ ⠀⠀⠀⠀⠈⠢⣄⠀⡈⠒⠊⠉⠁⠀⠈⠉⠑⠚⠀⠀⣀⠔⢊⣠⠤⠒⠊⠉⠀⡜ ⠀⠀⠀⠀⠀⠀⠀⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡔⠊⠁⠀⠀⠀⠀⠀⠀⠇ ⠀⠀⠀⠀⠀⠀⠀⡇⢠⡤⢄⠀⠀⠀⠀⠀⡠⢤⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀ ⠀⠀⠀⠀⠀⠀⢀⠇⠹⠿⠟⠀⠀⠤⠀⠀⠻⠿⠟⠀⣇⠀⠀⡀⠠⠄⠒⠊⠁⠀ ⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠰⠤⠖⠦⠴⠀⢀⣶⣿⣿⠀⠙⢄⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠿⡿⠛⢄⠀⠀⠱⣄⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢸⠈⠓⠦⠀⣀⣀⣀⠀⡠⠴⠊⠹⡞⣁⠤⠒⠉⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⡌⠉⠉⡤⠀⠀⠀⠀⢻⠿⠆⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠰⠁⡀⠀⠀⠀⠀⢸⠀⢰⠃⠀⠀⠀⢠⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢶⣗⠧⡀⢳⠀⠀⠀⠀⢸⣀⣸⠀⠀⠀⢀⡜⠀⣸⢤⣶⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠈⠻⣿⣦⣈⣧⡀⠀⠀⢸⣿⣿⠀⠀⢀⣼⡀⣨⣿⡿⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠈⠻⠿⠿⠓⠄⠤⠘⠉⠙⠤⢀⠾⠿⣿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠊⠉⠉⠀⠀⣿⣿⣿⠿⠋⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡠⠼⠴⠒⠒⠒⠒⠦⠤⠤⣄⣀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠋⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣇⠔⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢀⡟⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⠤⢴ ⠀⠀⠀⠀⠀⠀⣸⠁⣾⣿⣀⣽⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⠉⢿⣦⠀⠀⠀⢸⡀⠀⠀⢀⣠⠤⠔⠒⠋⠉⠉⠀⠀⠀⠀⢀⡞ ⠀⠀⠀⠀⠀⢀⡏⠀⠹⠿⠿⠟⠁⠀⠰⠦⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⢘⡧⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀ ⠀⠀⠀⠀⠀⣼⠦⣄⠀⠀⢠⣀⣀⣴⠟⠶⣄⡀⠀⠀⡀⠀⠉⠁⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀ ⠀⠀⠀⠀⢰⡇⠀⠈⡇⠀⠀⠸⡾⠁⠀⠀⠀⠉⠉⡏⠀⠀⠀⣠⠖⠉⠓⢤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀ ⠀⠀⠀⠀⠀⢧⣀⡼⠃⠀⠀⠀⢧⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣧⠀⠀⠀⣸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀ ⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢠⠏⠀⠀⠀⠀⠈⠳⠤⠖⠃⡟⠀⠀⠀⢾⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠈⠦⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠙⢦⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢠⡇⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠸⡇⠈⢳⡀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡼⣀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣷⠴⠚⠁⠀⣀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⡴⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣼⢷⡆⠀⣠⡴⠧⣄⣇⠀⠀⠀⠀⠀⠀⠀⢲⠀⡟⠀⠀⠀⠀⠀⠀⠀⢀⡇⣠⣽⢦⣄⢀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⡿⣼⣽⡞⠁⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠈⣷⠃⠀⠀⠀⠀⠀⠀⠀⣼⠉⠁⠀⠀⢠⢟⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣷⠉⠁⢳⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠏⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠹⡆⠀⠈⡇⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢳⡀⠀⠙⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢀⡄⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢳⡀⣰⣀⣀⣀⠀⠀⠀⠘⣦⣀⠀⠀⠀⡇⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⢸⡇⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠉⠉⠀⠀⠈⠉⠉⠉⠙⠻⠿⠾⠾⠻⠓⢦⠦⡶⡶⠿⠛⠛⠓⠒⠒⠚⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠖⠲⢤⡖⠒⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⣀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⣄⠀⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⠀⠀ ⠀⡞⠉⠻⠁⢹⠀⠀⡏⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠋⠀⠀⠀⠀⣀⡤⠴⠒⠊⠉⠉⠀⠀⣿⣿⣿⠿⠋⠀⠀ ⠀⠳⢤⡀⠀⡞⠁⠀⡇⠀⠀⢀⡠⠼⠴⠒⠒⠒⠒⠦⠤⠤⣄⣀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠋⠁⠀⠀⠀⠀ ⠀⠀⠀⠈⠷⡏⠀⠀⣇⠔⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢀⡟⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⠤⢴ ⠀⠀⠀⠀⠀⠀⣸⠁⣾⣿⣀⣽⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⠉⢿⣦⠀⠀⠀⢸⡀⠀⠀⢀⣠⠤⠔⠒⠋⠉⠉⠀⠀⠀⠀⢀⡞ ⠀⠀⠀⠀⠀⢀⡏⠀⠹⠿⠿⠟⠁⠀⠰⠦⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⢘⡧⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀ ⠀⠀⠀⠀⠀⣼⠦⣄⠀⠀⢠⣀⣀⣴⠟⠶⣄⡀⠀⠀⡀⠀⠉⠁⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀ ⠀⠀⠀⠀⢰⡇⠀⠈⡇⠀⠀⠸⡾⠁⠀⠀⠀⠉⠉⡏⠀⠀⠀⣠⠖⠉⠓⢤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀ ⠀⠀⠀⠀⠀⢧⣀⡼⠃⠀⠀⠀⢧⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣧⠀⠀⠀⣸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀ ⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢠⠏⠀⠀⠀⠀⠈⠳⠤⠖⠃⡟⠀⠀⠀⢾⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠈⠦⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠙⢦⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢠⡇⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠸⡇⠈⢳⡀⠀⢹⡀⠀⠀⠀⢀⡞⠁⠉⣇⣀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡼⣀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣷⠴⠚⠁⠀⣀⣷⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⣳ ⠀⠀⠀⠀⠀⠀⡴⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡴⠚⠉⠉⠀⠀⠀⠀⢸⠃⣀⣠⠤⠤⠖⠋ ⣼⢷⡆⠀⣠⡴⠧⣄⣇⠀⠀⠀⠀⡴⠚⠙⠲⠞⠛⠙⡆⠀⠀⠀⠀⠀⢀⡇⣠⣽⢦⣄⢀⣴⣶⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀ ⡿⣼⣽⡞⠁⠀⠀⠀⢹⡀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⣼⠉⠁⠀⠀⢠⢟⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣷⠉⠁⢳⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠙⢦⠀⠀⠀⡠⠁⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠏⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠹⡆⠀⠈⡇⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠹⣧⠞⠁⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢳⡀⠀⠙⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢀⡄⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢳⡀⣰⣀⣀⣀⠀⠀⠀⠘⣦⣀⠀⠀⠀⡇⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⢸⡇⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠉⠉⠀⠀⠈⠉⠉⠉⠙⠻⠿⠾⠾⠻⠓⢦⠦⡶⡶⠿⠛⠛⠓⠒⠒⠚⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣶⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣶⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢀⡀⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠀⢀⣀⠀⠀⠈⠉⠀⠀⣀⣀⠀⠀⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣀⣶⣿⣿⣿⣾⣇⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣼⡏⠀⠀⠀⣀⣀⣉⠉⠩⠭⠭⠭⠥⠤⢀⣀⣀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣿⠷⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠈⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀ ⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣤⣄⣠⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀ ⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀ ⠀⠀⣀⠀⢸⡿⠿⣿⡿⠋⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠻⠿⠟⠉⢙⣿⣿⣿⣿⣿⣿⡇ ⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⡿⣿⠳⠀ ⠀⠀⡞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡀⠀⠀ ⢀⣸⣀⡀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣰⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⢿⣿⠿⠃⠀ ⠘⢿⡿⠃⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢻⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⡸⠁⠀⠀⠀ ⠀⠀⠳⣄⠀⠀⠀⠹⣿⣿⣿⡿⠛⣠⠾⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠳⣄⠙⠛⠿⠿⠛⠉⠀⠀⣀⠜⠁⠀⠀⠀⠀ ⠀⠀⠀⠈⠑⠢⠤⠤⠬⠭⠥⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⠤⠤⠒⠊⠁⠀⠀⠀⠀⠀⠀

⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣠⣤⣶⣶ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣾⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⡏⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿ ⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⣿ ⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠙⠿⠿⠿⠻⠿⠿⠟⠿⠛⠉⠀⠀⠀⠀⠀⣸⣿ ⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢰⣹⡆⠀⠀⠀⠀⠀⠀⣭⣷⠀⠀⠀⠸⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠉⠀⠀⠤⠄⠀⠀⠀⠉⠁⠀⠀⠀⠀⢿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⣷⠀⠀⠀⠀⡠⠤⢄⠀⠀⠀⠠⣿⣿⣷⠀⢸⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⡀⠉⠀⠀⠀⠀⠀⢄⠀⢀⠀⠀⠀⠀⠉⠉⠁⠀⠀⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿

⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢻⣿⡗⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄ ⠀⢻⣇⠀⠈⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⣹⣿⡿ ⠀⠀⠹⣆⠀⠀⠀⠀⠙⢷⣄⣀⣀⣀⣤⣤⣤⣄⣀⣴⠞⠋⠉⠀⠀⠀⢀⣿⡟⠁ ⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀ ⠀⠀⠀⠀⠈⠻⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡾⠋⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣼⠃⠀⢠⠒⣆⠀⠀⠀⠀⠀⠀⢠⢲⣄⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢰⡏⠀⠀⠈⠛⠋⠀⢀⣀⡀⠀⠀⠘⠛⠃⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀ ⠀⠀⠀⠀⣾⡟⠛⢳⠀⠀⠀⠀⠀⣉⣀⠀⠀⠀⠀⣰⢛⠙⣶⠀⢹⣇⠀⠀⠀⠀ ⠀⠀⠀⠀⢿⡗⠛⠋⠀⠀⠀⠀⣾⠋⠀⢱⠀⠀⠀⠘⠲⠗⠋⠀⠈⣿⠀⠀⠀⠀ ⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀ ⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀ ⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢳⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⢰⠎⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢆⣤⡞⠃⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⣀⣾⢳⠀⠀⠀⠀⢸⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣀⡤⠴⠊⠉⠀⠀⠈⠳⡀⠀⠀⠘⢎⠢⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀ ⠳⣄⠀⠀⡠⡤⡀⠀⠘⣇⡀⠀⠀⠀⠉⠓⠒⠺⠭⢵⣦⡀⠀⠀⠀ ⠀⢹⡆⠀⢷⡇⠁⠀⠀⣸⠇⠀⠀⠀⠀⠀⢠⢤⠀⠀⠘⢷⣆⡀⠀ ⠀⠀⠘⠒⢤⡄⠖⢾⣭⣤⣄⠀⡔⢢⠀⡀⠎⣸⠀⠀⠀⠀⠹⣿⡀ ⠀⠀⢀⡤⠜⠃⠀⠀⠘⠛⣿⢸⠀⡼⢠⠃⣤⡟⠀⠀⠀⠀⠀⣿⡇ ⠀⠀⠸⠶⠖⢏⠀⠀⢀⡤⠤⠇⣴⠏⡾⢱⡏⠁⠀⠀⠀⠀⢠⣿⠃ ⠀⠀⠀⠀⠀⠈⣇⡀⠿⠀⠀⠀⡽⣰⢶⡼⠇⠀⠀⠀⠀⣠⣿⠟⠀ ⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡶⠤⣷⣅⡀⠀⠀⠀⣀⡠⢔⠕⠁⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠫⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⣀⣠⣤⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣧ ⠀⠀⣀⣤⣶⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⣼ ⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⢷⣆⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣿ ⣾⣿⣿⣟⢛⣛⣛⣛⣋⠭⠥⠿⣿⣿⣷⣤⠀⠀⠀⢀⣀⣀⣠⣀⡀⢿⡇⠀⣸⡇ ⣿⣿⣿⠻⢧⠙⢯⡀⠈⠉⠙⠛⠳⢦⣝⢿⣷⢠⣾⣿⣿⣿⣿⣿⣯⣬⡥⢰⡟⠀ ⢹⣯⣛⠯⢿⣾⣷⣍⡳⣤⣀⠀⠀⠀⠉⠳⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀ ⠈⠹⣿⣿⣾⣿⣿⣿⣿⣷⣭⣛⠷⢦⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀ ⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⢋⣵⡾⣣⣤⣦⢻⣿⣿⣿⣻⠻⣿⣿⣿⡏⠖⢻⠀⠀ ⠀⠀⠀⠀⠀⠀⠈⠉⠉⣱⣿⡟⠼⣻⣿⣿⢸⣿⣿⡇⡛⠀⣿⣿⣿⣧⡠⣸⡇⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⢹⣿⣿⣿⣏⢿⣿⣷⣕⣧⣿⣿⣿⢿⣿⡿⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣠⢿⣿⡟⣿⣷⣭⣻⠿⢿⠿⠷⢞⣫⣵⠿⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⣎⢿⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠁⠻⣷⡝⣩⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⣿⣿⣿⠉⠙⢻⢟⣿⡇⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢿⡧⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀

⣴⣡⡞⠉⢹⡆⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⡀⢢⣄⠀⠀⣴⡃⠀⢠⣤⠀⠀ ⢿⠁⠀⠀⢈⣵⡤⢀⣠⠴⠚⠋⠉⠁⠀⠀⠀⠈⠉⠛⢦⡉⢢⣉⣤⡀⣸⣿⠀⠀ ⠚⢸⣇⣼⣿⣭⣾⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠋⠙⣿⡏⠁⠀⠀ ⠀⠄⢿⡏⠀⢙⣯⠙⣿⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⣠⡞⢻⣇⠀⠀⠹⣿⣶⢀⠈ ⠀⠹⡜⣿⣶⣿⣿⣆⠹⣧⠀⠀⠹⣿⣿⣶⣦⣴⣿⣿⣷⡀⢿⡆⠀⠀⢻⣷⡏⠀ ⠀⠀⢁⡿⠛⢿⣿⣿⡄⢹⣆⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧⠈⣿⡄⠀⣼⡟⡇⠀ ⠀⠀⣸⠇⠀⠀⢻⣿⣿⡀⢻⡄⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣷⣰⠟⢷⡈⠀ ⣄⡼⠋⠀⠀⠀⣸⡿⣿⣷⠈⢿⡀⠀⠈⢿⣿⡿⠿⠿⠿⠿⠿⠦⣼⠏⠀⢘⣇⢘ ⢹⠁⠀⠀⠀⠀⣿⠀⠈⡻⢷⣼⣷⣤⠶⠛⠉⠀⠀⠀⢀⠀⠀⠀⢀⣀⣠⡾⢃⠟ ⠀⠀⠀⠀⠀⠀⣿⠀⢸⡏⠀⠉⠉⠀⠀⠀⠀⣀⠘⠀⢈⣠⡴⠞⠛⢹⡟⡰⠁⠀ ⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛⠷⠶⠶⠶⠟⠛⠛⠛⠛⠛⠛⠉⠀⣠⣴⣯⣉⠁⠀⠀ ⠀⠀⠀⠀⠀⠀⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠶⠛⠉⠈⢷⡈⠓⢆⢵ ⠀⠀⠀⠀⠀⠀⣿⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠹⣧⠀⠀⠀⠀⠀⠈⣷⠀⠈⠚ ⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢀⣠⣴⠞⢻⡀⠀⣠ ⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⣟⣛⠉⢀⣀⣸⠃⠠⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣾⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣠⣴⣶⣶⣿⠁⠀⠙⠿⣿⣿⣿⡟⠶⠀⠀⠀⠀⠀⢲⣄⠀⠀⠀⠀⠀ ⠀⢀⣶⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⡀⠀⠀ ⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⢸⣿⣿⡄⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣦⠀ ⣿⡟⠉⠀⠀⠈⢻⣿⣿⠿⢿⣿⣿⣷⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⡿⢿⣿⣿⣇ ⢻⠁⠀⠀⠀⠀⠀⠉⣠⣶⡟⠛⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⣿⣿⠋⠀⣾⣟⢻⣿ ⠀⠀⠀⠀⠀⠀⠀⠈⠻⠙⢇⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠟⠻⠟⠀⠀⣾⣿⠈⡟ ⠀⠀⠀⠀⠀⣀⣤⣴⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢸⣿⡿⠇⠀ ⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⣀⡤⠊⠀⠀⠀ ⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠻⠿⠿⢿⣿⣿⣿⠛⠋⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠉⠉⠁⣽⣿⣿⡿⠋⠀⠀⠀⠀⣠⠖⠁⠀⠀⠈⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠇⢀⣀⣀⣀⣀⣀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡄⠀ ⠀⠀⠀⠀⠀⠀⠀⢸⣀⠴⠋⠉⠁⠀⠀⠀⠀⠀⠉⠙⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠛⠁⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀ ⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⣾⢙⣶⡄⠀⠀⠰⢤⣠⡤⠤⠔⠒⠂⠉⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀ ⠀⠀⠀⠀⠀⠀⣮⣞⣹⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠃ ⠀⠀⠀⠀⠀⢰⠛⠿⠁⣈⣀⣀⣀⣤⠀⠀⠀⢠⠖⠒⠲⡄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢰⠧⠤⠔⠂⠐⠈⠈⠀⠀⠀⣠⠔⠊⠁⠀⠀ ⠀⠀⠀⠀⢠⡟⣇⠀⠉⢿⣿⣿⣿⣿⠀⠀⠀⢯⡐⠲⣠⡇⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠸⣦⡟⠀⠀⠈⢿⠟⠛⢻⠀⠀⠀⠀⠙⠚⠋⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠳⣄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⣀⠬⠷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠃⠢⢄⠀⠈⢣⡀⠀⠀⠀⠀⠀⢀⡽⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣤⠔⠊⠁⠀⠀⠀⠈⠳⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠈⠀⠀⠘⡿⢆⠀⠀⣠⠔⠉⠀⠀⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠐⡏⠸⠀⠀⠀⠀⠀⠀⠀⢢⠀⠈⠳⢄⣀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⣀⡀⠀⠀⠀⠱⡈⠣⡀⠀⢠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠻⠦⢤⣀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠤⠼⠛⠁⠀⠀⠀⠀⠘⣆⠙⢶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠒⠒⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠳⣾⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⣀⡤⠔⠲⣶⣆⣀⡀⠀⠐⠤⠤⠔⠒⠉⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣥⠤⢴⠚⠉⠀⠀⠀⠈⠉⠒⠂⠤⠤⢤⡤⠞⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀ ⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀ ⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀ ⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀ ⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢠⣶⣶⣦⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡆ ⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⣿⠀⠀⠸⣿⣿⡿⠁⠀⢀⣿⠈⠉⠙⠛⠿⢿⣿⣷ ⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠈⢿⣦⡀⠀⠈⠉⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⢸⣿ ⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣤⣴⡾⠟⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿ ⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇ ⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀ ⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀ ⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀ ⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⣶⣶⣶⣶⣶⣶⠾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⢸⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢸⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠙⢤⡷⣤⣦⣀⠤⠖⠚⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣠⡿⠢⢄⡀⠀⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⠷⣶⠂⠀⠀⠀⣀⣀⠀⠀⠀ ⢸⣃⠀⠀⠉⠳⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⡭⠋ ⠀⠘⣆⠀⠀⠀⠁⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀ ⠀⠀⠘⣦⠆⠀⠀⢀⡎⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⣠⠔⠋⠀⠀⠀⠀ ⠀⠀⠀⡏⠀⠀⣆⠘⣄⠸⢧⠀⠀⠀⠀⢀⣠⠖⢻⠀⠀⠀⣿⢥⣄⣀⣀⣀⠀⠀ ⠀⠀⢸⠁⠀⠀⡏⢣⣌⠙⠚⠀⠀⠠⣖⡛⠀⣠⠏⠀⠀⠀⠇⠀⠀⠀⠀⢙⣣⠄ ⠀⠀⢸⡀⠀⠀⠳⡞⠈⢻⠶⠤⣄⣀⣈⣉⣉⣡⡔⠀⠀⢀⠀⠀⣀⡤⠖⠚⠀⠀ ⠀⠀⡼⣇⠀⠀⠀⠙⠦⣞⡀⠀⢀⡏⠀⢸⣣⠞⠀⠀⠀⡼⠚⠋⠁⠀⠀⠀⠀⠀ ⠀⢰⡇⠙⠀⠀⠀⠀⠀⠀⠉⠙⠚⠒⠚⠉⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢧⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠙⣶⣶⣿⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⢿⣳⠞⠳⡄⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠹⣄⣀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣰⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀ ⠀⠀⠀⠀⢀⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠒⣶⣶⠆⠀⠀⢀⠔⠁⢠⠀ ⠀⠀⠀⠀⠘⢀⠼⠤⠀⠀⠀⠄⡠⠐⠈⠀⠀⠀⠀⡰⠟⠁⠀⢀⠔⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢜⡁⠀⠀⠀⣖⣦⠀⠀⠤⣤⠄⠐⠂⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⡸⠷⢁⡀⢀⡀⠈⢉⠤⢄⠀⠈⡀⠀⠀⠀⠀⣔⡁⠄⠄⠀⠀⢀⠠⠂⠁ ⠀⠀⢠⢻⠀⠹⣿⠿⡇⠀⠡⠔⠜⠀⠀⢁⠀⠀⠀⠀⢡⠀⠀⢀⠄⠊⠁⠀⠀⠀ ⠀⠀⠀⠫⡀⠀⠐⠤⠃⠀⠀⠀⠀⢀⠀⠀⢂⠀⠀⠀⠀⢃⠀⠸⠀⠀⠀⠀⠀⠀ ⢀⠠⠐⠂⠉⠢⠀⡀⠀⠀⠀⠀⠖⠉⠉⠀⠀⢧⡀⠀⡠⠒⠀⡠⠀⠀⠀⠀⠀⠀ ⢸⡘⠀⠀⠀⠀⢢⠈⠂⠀⠀⠀⠘⢤⣄⣤⠄⠀⠈⢊⠢⣠⣎⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠉⠀⠒⠒⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⣾⠿⠆⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠄⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⣀⣀⠀⠀⢀⣀⡀⠤⢊⠆⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢤⣥⠒⠉⠉⠑⠂⠠⠤⡤⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⠀⠀⠀⠀⠀⠀⠀⠐⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀

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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣄⣀⣀⠀⢀⣀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⡿⠋ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣼⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀ ⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣶⣶⣦⣤⣤⡀⠀⠀⢿⣷⣄⣹⣿⣿⣿⣿⣿⡿⠟⡿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀ ⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣤⣤⣅⣀⣀⣀⡀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀ ⠀⡀⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⢻⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣾⣧⢿⣿⠏⠀⢸⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠿⠁⠀⠀⠀⠀⠈⠿⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀ ⠈⠻⡎⠟⠀⠀⠈⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣶⣶⣶⣿⣿⡿⢟⣫⣭⣟⠿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣵⣮⣭⣷⣾⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢹⣿⣿⣿⣿⡎⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠀⣴⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡇⠈⠙⠻⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢰⣿⣿⣿⠟⠁⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⢸⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⢀⣴⣇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢸⣿⣿⣄⠀⣀⣴⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣷⣄⣿⣿⣿⣿⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣤⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢸⣿⣿⣿⣿⣌⠻⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⢀⡄⠀⠀⣠⡴⠶⠲⢶⣄⠀⠀ ⠀⣿⣿⣿⢻⣿⣿⣿⣿⣿⡿⠛⣁⣤⠀⠀⢀⣠⣿⣷⣠⡾⠋⠀⠀⠀⠀⣿⠄⠀ ⠀⠙⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣾⣿⣿⣿⣿⣶⣶⠞⠁⠀⢰⡟⠀⠀ ⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡄⠀⣾⡁⠀⠀ ⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣟⠋⠁⠀⢤⣼⣷⣾⠃ ⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡅⠀⠸⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⢸⡿⠿⠧ ⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⠏⠙⠻⢿⣿⣷⠀⠀⠈⠀⠀⠀ ⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⡟⠀⢀⣴⣿⣿⡏⠀⠀⢀⣸⣿⣿⡆⠀⠀⠀⠀⠀ ⠀⠀⠐⢿⣿⣿⡿⠟⠛⠛⠋⠀⠀⠈⠉⠉⠉⠀⠀⠀⠘⠿⠿⠿⠃⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⡀⠀⠀⣠⣤⠶⣶⡿⠿⠿⣷⣶⣤⣀⠀⠀⠀⠀⣀⣀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⣿⣿⡉⠉⠉⠉⠙⠿⣷⣛⣉⣠⠼⠧⠤⠤⣄⣀⣈⠉⠻⣷⣶⠾⠿⠛⠛⠉⠛⢛⣻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⣿⣿⣷⡄⠀⢀⣤⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⢉⡗⠶⣄⠀⠀⠀⠀⣠⠶⠋⢩⡅⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⢹⣧⢫⣻⡶⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠀⠀⠈⠓⢤⠠⢾⣅⠀⢀⡞⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠀⠹⢯⡿⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠈⠳⡄⠉⢿⣟⢠⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⣿⣀⣀⡤⣤⠶⠶⠶⠤⠤⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠈⢷⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠀⣼⡇⢀⡴⠛⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠙⠶⣤⣀⣀⣀⠀⠀⠀⠹⠀⠘⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠀⣿⠛⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠙⣿⣷⡀⠀⠀⠀⠀⢸⣇⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠀⣿⠀⢷⠀⠀⠀⣀⡤⠿⠶⠦⢤⣀⡀⠀⠀⠀⣾⠁⢀⣰⣿⣿⡇⠀⢠⡇⠀⢸⣿⡀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠀⢿⣷⣾⣦⡶⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⢶⣴⣿⣿⣿⣿⣿⡿⠁⠀⣾⠀⠀⢸⡿⢧⡀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠈⣿⠿⢿⣿⠟⢳⢀⣼⠏⠀⢠⣿⠇⠀⠙⢦⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⢀⣠⣾⢿⡇⠈⠳⣄⡀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠙⠛⠉⠀⣠⣾⣿⡋⠀⣠⡿⠋⠀⠀⠀⠈⢳⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⣠⡿⠋⠀⠈⢿⣦⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⢁⡿⠣⠼⠛⠳⣄⡀⠀⠀⠀⠀⣷⠇⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⣰⣿⣥⡴⠶⠒⠶⠿⢷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⠀⠀⣰⠋⠀⠀⠀⠀⠀⠈⠙⣦⡀⠀⣠⠏⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⣿⠋⢠⡇⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⣠⡾⠋⠀⠀⢀⣠⣾⡧⠤⠄⠀⠀⠀⠀⠀⠀⣼⣷⠴⣯⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⢰⡇ ⢨⠀⠀⣿⠀⢸⡇⣀⡤⠤⠤⣤⣀⣿⠈⠛⠷⠶⠟⠋⠀⠀⣠⣾⣿⣥⣀⠀⠀⠀⣠⠾⢦⡀⢀⡼⠃⢸⣦⣿⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⣿⡄⢸⡇ ⢸⠀⠀⣿⣆⣠⠏⠉⠀⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⠀⢠⣟⠁⠀⠀⠈⠱⡴⠞⠁⠀⠀⢳⡏⠀⠀⢸⡿⠁⡇⠀⣠⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣋⡀⠀⠀⣿⠁⢸⡇ ⢸⠀⠀⠹⣿⡋⠀⠀⠀⠀⠀⢰⠖⢿⡇⠀⠀⠀⠀⣠⠾⠋⣿⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠘⡇⠀⣰⠟⠁⢀⣧⠼⠃⢸⣏⠀⠀⣀⣀⣠⣴⡾⠿⠋⠉⠉⠉⠙⢶⣄⣿⠀⠀⡇ ⠀⠀⠀⠀⠻⣿⣄⠀⠀⢠⡾⠛⣧⣼⡗⠂⠀⠀⠈⠛⢻⣾⣯⣤⣤⠀⠀⠀⠀⢸⡄⠀⠀⣸⣧⣾⠷⠶⠲⠿⣧⣤⣴⣿⡿⠿⣿⠛⠛⠋⠉⠛⠲⢤⡀⠀⠀⠀⢠⣿⡟⠀⢠⡇ ⠀⠀⠀⠀⠀⠙⢿⣧⣤⣘⣷⣤⡤⠟⠀⠀⠀⠀⠀⠸⢿⣤⠤⠴⢿⡇⠀⢀⣰⣟⣤⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠈⠳⣄⠀⢀⣠⡤⠼⠻⡆⢀⣤⢯⣿⠁⠀⠐⡇ ⠀⠀⠀⠀⠀⣠⣿⠏⠉⠉⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠛⠉⠛⠛⠛⠉⠉⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠈⢯⠉⠉⠀⠀⠀⢹⡞⢡⣿⠃⠀⠀⠀⡇ ⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⣿⡄⠀⠘⡆⠀⠀⠀⠀⢸⣷⡿⠋⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⣼⣿⣄⣀⣠⣤⣤⡴⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⣤⣀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠞⢧⠀⠀⢻⣷⠀⠀⡇⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⢹⣿⡀⡇⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⠉⢻⡟⠛⠛⠋⠉⠀⠀⠀⠘⡇⠀⢸⣿⣤⢶⠇⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠈⣿⣇⢹⡄⠀⠀⠀⠀⠀⠈⣻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⣿⣀⣿⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠈⢿⣷⣷⠶⠶⠶⠶⠶⠚⠉⠉⠙⠿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢠⣧⠞⣿⣿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⣠⣤⠿⠻⢿⣦⣤⡀⠀⠀⠀⠀⣠⣤⣾⠟⠛⠻⠿⠿⣶⣶⡾⠿⣿⣿⣿⣯⡛⠲⠾⠷⠤⠤⠤⠴⠒⠛⠉⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⢸⠀⠀⣠⡿⠛⣉⡿⠛⠳⣦⡀⠀⠀⠀⠘⢉⣡⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣼⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⢸⠀⣰⢿⣴⠟⠁⠀⠀⢀⣸⣧⣤⡶⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⠶⠦⠤⠀⠀⠀⠈⠉⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⢸⠀⠙⢿⣷⣶⣶⠶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⡤⠶⠶⣤⠶⠖⠲⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠸⡄⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⣷⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷⠿⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣄⠀⠀⠀⢀⣠⠤⠀⡀⣤⣀⠀⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇ ⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠘⠛⠛⠛⠛⠛⠘⠛⠚⠙⠛⠃⠋⠟⠘⠘⠘⠃⠛⠓⠃⠃⠀⠀⠘⠃⠀⠈⠉⠁ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⡀⢀⢀⡀⢀⢀⡀⡀⢀⡀⡀⣀⢀⠀⡀⢀⢀⡀⡀⢀⡀⣀⣀⢀⠀⠀

⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⡀⠀⠀⠀⡀⠠⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀ ⢀⣰⡦⢄⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⣀⣀⣀⣀⡀⠀⠀⠀ ⠀⠾⣿⣦⡈⣀⣉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⣀⣠⡴⠊ ⠀⠀⠘⣿⣷⣿⣯⣄⢀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣯⣤⣶⣿⠿⠋⠀⠀ ⠀⠀⠀⢸⣿⣿⣿⡏⠀⢢⣠⣄⡀⠀⣺⣿⡿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀ ⠀⠀⠀⣾⣿⣿⠙⢿⣦⣬⣿⣿⣿⡶⠟⠉⠀⣿⣿⣿⣿⠁⠀⠐⠒⠠⡀ ⠀⠀⠀⣿⣿⣿⣄⠆⠙⡻⢿⣿⣿⣷⣶⣶⣾⣿⣿⣿⡶⠂⢀⡀⠤⠌⠀ ⠀⠀⢠⣿⣿⣿⣿⣦⣤⠀⠀⠀⡈⠉⢹⣩⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀ ⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠘⠛⠃⠉⠻⢿⣿⡿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿ ⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⠀⠀⠀⠀⠀⠹⣿⣿ ⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠘⢿⣿⡟⠀⣠⣤⣤⣀⠀⢹⣿ ⣿⡏⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣦⣀⣀⣠⣴⣿⣿⣿⠟⠀⣸⣿ ⣿⡇⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⣠⣴⣿⣿ ⣿⡇⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁⣀⣤⣶⣿⣿⣿⣿⣿ ⣿⣷⡀⠈⠛⠛⠻⠟⠛⠛⠛⠛⠉⠉⠁⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿ ⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿ ⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿ ⣿⡇⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿ ⣿⣿⡄⠘⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿ ⣿⣿⣷⡀⠘⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿ ⣿⣿⣿⣿⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣠⣿⠟⠁⣠⣿⣿⣿ ⣿⣿⣿⣿⣿⣷⣄⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡿⠛⠁⣠⣾⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⠉⠙⠛⠛⠛⠛⠛⠛⠉⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⣟⢿⣛⣟⡻⣟⢿⣻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣠⠴⠛⠛⠛⢷⣫⢾⣹⢮⡽⣞⢯⣞⡽⢮⡽⣽⣳⣄⠀⠀⠀⠀⠀ ⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⣿⣣⢟⣮⢗⣯⣞⢮⣽⢻⣜⣧⣛⢾⣳⣄⠀⠀⠀ ⠀⠀⣾⣟⠀⠀⠀⠀⠀⠀⣼⣳⢏⡿⣜⢯⢶⣫⡟⣼⡳⣽⢲⢯⣳⢽⣺⢧⡀⠀ ⠀⣼⣏⢿⢦⣀⣀⣀⣴⠾⣏⡷⣻⢼⣫⣾⣟⣿⣽⣿⣳⣯⣯⢳⣏⡾⣣⡟⣧⠀ ⣼⢟⣮⢯⢯⡝⣯⣛⣮⣟⣵⣯⣷⣿⣟⣷⢏⠷⠒⠲⡝⢷⡿⣏⡾⣵⢻⣼⢫⣧ ⣿⢞⣵⣫⣾⣟⡿⣯⢿⣽⣳⣟⣾⣳⢯⡇⡏⠀⠀⠀⢘⣞⣿⣻⣟⣾⢷⣭⡷⣻ ⣿⣿⢯⣟⡷⣯⢿⣽⣻⡾⠷⠿⠾⠿⣿⣷⡙⢦⣀⡤⢞⣼⣟⣷⣻⣞⣯⢿⡽⣿ ⣿⣿⠿⢞⡛⣏⠉⠀⠀⠀⠀⠀⠀⠀⠙⢿⣻⣷⣿⣻⣟⠿⠋⠈⠉⠙⠚⠿⢽⣻ ⣿⡤⣛⢬⡱⡹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿ ⠀⢳⡣⢎⡵⣱⢚⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀ ⠀⠀⠻⡵⣚⠴⣫⠜⡳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀ ⠀⠀⠀⠙⢮⡚⡴⢫⡕⣣⢛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢔⡽⠋⠀⠀⠀ ⠀⠀⠀⠀⠀⠙⠺⣕⡼⢡⢏⡞⡴⢪⡝⣱⢲⡒⢶⠲⣩⠝⣢⠽⠋⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⣤⣽⣱⣘⣥⣣⣽⢦⣟⣒⣋⠁⠀⠀⠀⠀⠀⠀⠀

⢰⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀ ⠀⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿ ⠀⠘⢿⣿⣿⣿⣿⣦⣀⣀⣀⣄⣀⣀⣠⣀⣤⣶⣿⣿⣿⣿⣿⠇ ⠀⠀⠈⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀ ⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀⠀⠀ ⠀⠀⠀⢠⣿⣿⡏⠆⢹⣿⣿⣿⣿⣿⣿⠒⠈⣿⣿⣿⣇⠀⠀⠀ ⠀⠀⠀⣼⣿⣿⣷⣶⣿⣿⣛⣻⣿⣿⣿⣶⣾⣿⣿⣿⣿⡀⠀⠀ ⠀⠀⠀⡁⠀⠈⣿⣿⣿⣿⢟⣛⡻⣿⣿⣿⣟⠀⠀⠈⣿⡇⠀⠀ ⠀⠀⠀⢿⣶⣿⣿⣿⣿⣿⡻⣿⡿⣿⣿⣿⣿⣶⣶⣾⣿⣿⠀⠀ ⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀ ⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀

⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⢀⣀⣤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣀⠀⠀⠀⠀⠀⠐⣤⡀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⣀⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣠⣾⡿⠁⢀⣴⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠘⣿⣷⡀⠀⠀⠀⠀ ⠀⠀⠀⣰⣿⡿⠁⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⠀⠀⠸⣿⣿⡄⠀⠀⠀ ⠀⠀⣸⣿⣿⠃⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢹⣿⣿⡄⠀⠀ ⠀⢰⣿⣿⡏⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀ ⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀ ⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⢠⣾⣿⣿⠃⠀⠀⠹⣿⣿⣷⠀⢀⣴⣿⣿⣿⣶⣶⣶⣦⣴⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀ ⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇ ⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣠⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣄⡀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧ ⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿ ⣿⣿⣿⣿⣤⣤⣤⣄⣀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⢿⣿⣿⣷⣄⡀⠀⠀⠀⣀⣠⣤⣤⣼⣿⣿⣿⡏ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⠀⠀⠉⠉⠙⠛⠻⠿⢿⣿⣿⡿⠟⢁⣠⣾⣿⣿⣿⠿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⢿⣿⣿⣿⣦⣄⠙⠻⣿⣿⣿⡿⠿⠛⠛⠉⠉⠁⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣴⣿⣿⣿⠿⢋⣵⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⣭⡛⢿⣿⣿⣷⣄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⢁⣶⣿⣿⡿⠋⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠻⣿⣿⣿⣦⠉⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠋⠀⠀⢸⣿⣿⣿⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠸⣿⣿⣿⡄⠀⠈⠛⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⠁⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠙⢿⣿⣿⣶⣄⣀⠀⠀⠀⠀ ⠀⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡇⠀ ⠀⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡇⠀ ⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀ ⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀ ⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀ ⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀ ⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀ ⠀⠘⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀ ⠀⠀⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀ ⠀⠀⠸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀ ⠀⠀⠀⢻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇⠀⠀⠀ ⠀⠀⠀⠀⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠋⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡄⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡄⠀⠀⠀⠀⠀⠀⣰⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⣠⣄⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⡸⢛⣷⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⢰⣷⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⢏⡏⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠘⣦⡀⠐⠂⠠⠴⠀⠀⠀⠀⠀⠀⠀⣀⡴⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⠀⠀⠀ ⠀⢰⠦⣄⠈⢿⣶⡶⠤⠤⠤⠔⠒⠒⣶⠒⡫⢋⡄⠀⢸⠀⠀⠀⠀⢀⣀⠤⣤⠀⠀⠀⠀⠀⢀⡞⠉⠛⢆⠀⠀ ⢰⡟⢷⠂⠉⠚⠿⣿⣄⣀⣀⣀⣠⣴⣛⣯⠔⠋⠀⠠⠼⠦⠔⠒⠊⠁⠠⡶⢧⡄⠀⠀⠀⠀⠸⣧⠀⡀⠈⣿⠀ ⠀⠙⢮⡀⠀⠀⠀⠈⠉⡖⠛⠋⠉⠉⠁⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠞⠁⠀⠀⠀⠀⠀⡏⠿⢧⢀⠈⡆ ⠀⠀⠀⠙⢄⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣇⠀⠸⣯⠀⡿ ⠀⠀⠀⠀⠀⠓⢄⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠐⠒⢖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠉⣶⡇ ⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣴⢦⠞⠁ ⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠸⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⢀⡇⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡠⠚⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠤⠤⢤⣸⠓⠢⠄⢀⠀⠤⠔⠊⠁⠀⠀⡼⠀⠀⠀ ⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⢀⡜⠉⠉⠛⢧⠀⠀⠀⠀⠀⠀⣸⠥⣄⣀⣀⣀⠠⠖⠉⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢀⣀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢸⠇⠀⠀⠀⢀⣴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢿⣿⣷⣦⣀⣀⣀⡀⠤⠤⠤⠚⠁⠀⠀⠀⢸⡤⢤⡤⢤⡴⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⡷⠞⠿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠻⠿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠙⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣷⣶⣄⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⠿⢷⣶⣦⣤⣤⣀⠀⠀⢀⣠⣾⠿⠛⠉⠀⠀⠀⠀⠈⢿⣧⠀⠀ ⠀⠀⠀⠀⠀⠀⢀⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣷⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀ ⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀ ⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀ ⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀ ⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀ ⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀ ⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄ ⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇ ⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿ ⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿ ⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿ ⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟ ⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⡀⠀⠛⠉⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇ ⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⣴⡿⠟⠉⠉⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀ ⠀⠀⠀⠀⠀⠙⠿⣷⣦⣀⢀⣾⡟⠁⠀⠀⠀⢼⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡟⠀⠀⠀⠀⠀⠀⠉⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⠷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣴⣶⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⣀⣀⣀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⡿⠉⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢿⠀⠀⠀⠀⢀⣀⣀⣀⣘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢸⡀⣠⣴⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠴⢖⣿⣿⣯⣉⠉⠉⠉⠉⠙⠛⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠖⠒⠒⠛⠉⠙⢿⣿⠛⢻⣿⣿ ⠀⠘⣿⣿⣿⣿⣿⡿⠟⠋⣠⡴⢋⣡⣄⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⣠⡴⠚⠋⠉⠁⠈⠙⢶⡄⠀⠀⠀⠀⠀⠙⣧⡀⡛⣿ ⠀⠀⠘⢿⣿⡿⠋⠀⢀⡼⠋⣴⠋⠁⡾⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠘⣷⣶⣿ ⠀⠀⠀⣰⣿⠇⠀⠀⡿⠁⡼⠁⢀⡾⠁⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⢹⣿⣿ ⠀⠀⢰⡟⠀⣀⠀⢸⡇⠀⢷⠴⠋⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢸⣿⣿ ⠀⠀⡿⢠⣾⣿⡆⠀⢿⣄⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢀⣾⡿⠟ ⠀⢰⣷⣿⣿⣼⣿⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣏⣠⠤⠶⠒⠋⠁⠀⠀ ⠀⢸⣿⠽⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣶⣿⣿⣧⠀⠀⠀⠀⠙⣿⠒⠒⠒⠒⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣸⣿⣤⣤⣉⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣿⡿⢿⣷⣿⠇⠀⣄⠀⠀⠀⠀⠀⠀⢸⣿⢀⣈⣙⡿⠛⢻⣩⡴⣾⡿⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣿⡋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣮⣉⣉⣁⣒⣨⣿⡿⠛⠀⠀⣀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠘⢿⡀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠙⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠷⠦⢤⣄⣀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣰⡿⡡⠐⠀⠀⠀⠙⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣮⣉⠉⠉⠉⣽⢋⠀⠀⠀⠀⠀⢀⣼⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣷⣄⣼⠋⣉⣀⠀⠀⢀⣴⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣶⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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⣿⡅⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠈⣿ ⣿⡇⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⢸⣿ ⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿ ⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿ ⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿ ⣿⣿⡆⠀⠀⠀⠀⠀⢀⣤⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣤⡀⠀⠀⠀⠀⠀⢰⣿⣿ ⣿⡿⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⢿⣿ ⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢸⣿ ⣿⡇⠀⠀⢀⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⡀⠀⠀⢀⣿ ⣿⡇⠀⠀⢸⣿⣶⣶⣶⣤⣤⣀⠉⣻⣿⣿⣟⠋⣁⣠⣤⣤⣤⣤⣿⡇⠀⠀⢸⣿ ⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣾⣿ ⣿⣿⣧⡀⠻⣿⣿⣿⣿⣿⠟⠉⢿⣿⣿⣿⣿⡿⠉⠻⣿⣿⣿⣿⣿⠟⢀⣼⣿⣿ ⣿⣿⣿⣷⣄⠙⢿⣿⣧⣄⣀⠀⠘⠛⠛⠛⠛⠃⠀⣀⣠⣼⣿⡿⠋⣠⣾⣿⣿⣿ ⣿⣿⣿⣿⣿⣶⣄⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⣠⣶⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣈⠉⠉⠉⠉⠉⠉⣁⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠋⠀⢹⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣶⣿⣿⣿⣿⣇⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏ ⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀ ⠀⢀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀ ⣴⡟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢁⣀⠀⠀⠀ ⣿⡇⢰⣿⣷⣤⡈⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠉⠀⠀⠀⣴⣾⣿⣿⠀⠀⠀ ⠸⣷⡀⠹⣿⣿⣿⣦⡀⠹⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠀⠀⠀ ⠀⠘⢿⣄⠈⠻⣿⣿⣿⣄⠘⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁⠀⠀⠀ ⠀⠀⠀⠙⢷⣤⡈⠙⠿⠿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠉⠻⠷⣶⣶⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠄⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠸⣷⣮⣿⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠈⠁⠲⢖⠒⡀⠀⠀ ⠀⠀⠀⡠⠴⣏⠀⢀⡀⠀⢀⡀⠀⠀⠀⡀⠀⠀⡀⠱⡈⢄⠀ ⠀⠀⢠⠁⠀⢸⠐⠁⠀⠄⠀⢸⠀⠀⢎⠀⠂⠀⠈⡄⢡⠀⢣ ⠀⢀⠂⠀⠀⢸⠈⠢⠤⠤⠐⢁⠄⠒⠢⢁⣂⡐⠊⠀⡄⠀⠸ ⠀⡘⠀⠀⠀⢸⠀⢠⠐⠒⠈⠀⠀⠀⠀⠀⠀⠈⢆⠜⠀⠀⢸ ⠀⡇⠀⠀⠀⠀⡗⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢀⠎ ⠀⢃⠀⠀⠀⢀⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷⡃⠀ ⠀⠈⠢⣤⠀⠈⠀⠀⠑⠠⠤⣀⣀⣀⣀⣀⡀⠤⠒⠁⠀⢡⠀ ⡀⣀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀ ⠑⢄⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀ ⠀⠀⠑⠢⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀ ⠀⠀⠀⠀⢀⠠⠓⠢⠤⣀⣀⡀⠀⠀⣀⣀⡀⠤⠒⠑⢄⠀⠀ ⠀⠀⠀⠰⠥⠤⢄⢀⡠⠄⡈⡀⠀⠀⣇⣀⠠⢄⠀⠒⠤⠣⠀ ⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀

⠀⠀⠀⠀⠀⢀⣤⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣽⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢰⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢸⣿⣿⣿⣿⣿⣿⣷⡀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠹⣿⣿⣿⡟⠉⠉⠙⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⣀⣠⣿⡿⠉⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢸⣿⠿⣧⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢸⠁⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⠿⠋⡀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢸⡀⠀⢀⣄⡈⠙⠻⠟⢁⣠⣈⠙⢁⣠⣶⣤⣾⣿⣦⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠈⢷⣤⣾⣿⣿⣷⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣷⠀ ⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡇ ⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠉⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣧⠀⢀⣴⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠀⣿⣿⡇⠈⣿⣿⣿⣿⣿⣿⣿⡀⢾⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣤⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠾⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⠿⠯⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠀⠀⠙⠋⠁⠈⠛⠋⠈⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⢠⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⣰⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢰⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⣤⣤⣶⣾⣿⣿⣿⡷ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀ ⣿⣿⣿⡇⠀⡾⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀ ⣿⣿⣿⣧⡀⠁⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⢹⠉⠙⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⣀⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢃⠈⠢⡁⠒⠄⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⠟⠁⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀ ⠸⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⠇ ⠀⢻⣿⣿⠿⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠻⢿⣿⡟⠀ ⠀⠘⣟⠁⠀⢀⣠⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣶⣤⡀⠀⣹⠃⠀ ⠀⠀⠈⠳⣴⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠞⠁⠀⠀ ⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡀⣀⣤⣤⣤⣤⣤⣀⠀⣾⣿⣿⡿⠛⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⡿⣛⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢰⣿⡇⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⢿⣿⣮⣛⣡⣿⣿⠛⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠿⢻⣿⣿⣿⡿⠹⠁⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣀⣤⣾⡇ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢀⡤⠐⠻⣿⣿⡿⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣷⣤⣄⠀⠈⢻⠇⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⠋⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡞⣿⣿⣿⣿⣿⣿⢼⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣧⠙⣿⣧⣿⣿⣿⣿⣿⣿⣾⣿⢋⡇⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢹⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⠃⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⣿⣿⣿⣿⠋⢿⣿⣿⣿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⢺⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣷⠀ ⠀⢸⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⡇⠀ ⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀ ⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀ ⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀ ⠀⠀⢹⣿⣿⣿⣿⣿⡿⠛⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋⠛⢿⣿⣿⣿⣿⣿⡏⠀⠀ ⠀⢀⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⡀⠀ ⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀ ⠀⢸⣿⣿⡿⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⢿⣿⣿⡿⠀ ⠀⢸⣿⣿⡇⠀⠉⠉⠉⠛⠛⠿⣶⠄⠀⠀⠠⣴⠾⠟⠛⠛⠛⠛⠀⢸⣿⣿⡇⠀ ⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀ ⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⣠⣶⡀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀ ⠀⠀⠀⠈⠻⣦⡀⠀⠘⠻⠿⣿⣧⣤⣤⣤⣤⣼⣿⠿⠟⠃⠀⢀⣴⠟⠁⠀⠀⠀ ⠀⠀⠀⠀⠀⠉⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠉⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⣶⣶⣶⣶⣶⣶⠾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠰⠊⠐⠄⡀⠀⠀⠀⢀⣀⣀⣀⣀⠀⣀⠠⠂⢉⠇⠀⠀ ⠀⠀⠀⠀⠈⢂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠎⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠈⠂⠤⠀⡀ ⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠤⡈⠆⠀⠀⠀⠀⡇ ⢀⠔⠒⠤⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠐⠁⣾⣧⡡⢸⠆⠀⠀⡖⠀ ⠇⠀⠀⠀⡇⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⡄⢉⡁⠔⡊⠀⠀⠸⠀⠀ ⢆⡀⠀⣰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⠐⠁⠀⠀ ⠀⠀⢫⠘⡄⠀⢸⡀⠀⠀⠀⠀⢀⣦⡀⠀⠀⡀⠁⠐⢄⠀⠀⠀⠀ ⠀⠀⠀⡄⢣⠖⠁⠱⡀⠀⠀⠀⠘⢃⢼⠑⠒⠀⠀⠀⠀⠑⢄⠀⠀ ⠀⠀⠀⣇⡠⢃⠴⡎⠈⠢⠤⠤⠒⠁⠀⠡⡀⠀⠀⠀⢄⠀⠀⢱⠀ ⠀⠀⠀⠀⠹⡉⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠤⠒⢮⢤⢠⠀ ⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠁⠀⠀⠀⡆⠁⠀ ⠀⠀⠀⠀⠀⠘⡄⢃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢠⠃⠀⠀ ⠀⠀⠀⠀⠀⠀⠐⠤⣓⣄⡀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢀⠆⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠞⠋⠂⠈⠙⠉⠉⠉⠉⠑⣦⣶⣒⠖⠀⠀⠀⠀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀ ⣿⣧⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⢸⣿⣿ ⣿⣿⡄⠘⣷⣤⡉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢉⣥⣶⠇⢠⣿⣿⣿ ⣿⣿⣿⣦⠈⠻⣿⣷⣄⠻⡿⠋⠙⠉⠋⠛⢿⡟⢡⣶⣿⡿⠃⣰⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣦⣈⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠐⢿⠿⢋⣤⣾⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⡤⠄⠀⠀⠀⢀⠤⡀⢰⣿⡿⠛⢿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣷⡾⠀⠀⠀⢸⣶⡇⢸⠟⠁⠀⠈⢿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠁⠤⠬⠤⠄⠉⠀⢀⠀⠀⠀⠀⠸⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⡿⢟⣵⢄⣀⡀⠀⠀⣀⣠⣰⣿⡶⣆⡄⣀⠀⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣟⣾⡏⣿⣿⣿⣿⣿⣿⣿⡷⣿⣯⣿⣿⣿⣇⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣤⡈⢷⢹⣿⣿⣿⣿⣿⣿⠁⡿⢟⣿⣿⣿⣹⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁⠉⠻⣿⣿⠟⠀⠘⢸⣿⣿⡿⣡⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠘⠀⠀⠀⢠⣭⣭⣵⣾⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿ ⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠛⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⣠⣴⣾⣶⣿⣿⣶⣶⣶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣠⣼⣿⣿⣿⣋⣠⣿⣿⣿⣿⣿⡖⢶⣶⣤⣤⣀⣤⣶⣿⣷⡤⠶⢦⡀⠀ ⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠘⢿⣿⣿⣿⡿⢁⣾⡀⠀⠀⣷⠀ ⠀⠀⠻⣿⠟⠛⠛⠻⠟⠛⠋⢹⣿⣿⣿⢣⡆⠀⠈⠛⠛⠋⠀⢻⣿⣿⣶⠟⣻⣦ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢇⣾⠃⠀⣠⣤⣴⣤⣤⣀⠉⠙⢁⣴⡿⠁ ⠀⠀⠀⠀⠀⢠⠀⠀⠀⢰⣤⣿⣿⢋⣬⡄⢀⣾⣿⣿⣿⣿⣿⣿⣧⠀⢿⣿⠀⠀ ⠀⠀⠀⠀⠀⣠⠻⠿⠿⠿⠿⣛⣵⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣄⣼⣿⡇⠀ ⠀⠀⠀⠀⣠⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠆⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀ ⢀⣀⣴⣾⣿⣿⡇⣬⣭⣭⣭⣭⣭⣶⣶⣿⣷⡄⢈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀ ⠰⢾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢐⣛⡻⣿⣿⣿⣿⣿⣿⠻⣿⣿⠀ ⠀⠁⠀⣠⣶⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⡵⠿⢿⣿⣿⣿⢟⣄⢹⡏⠀ ⠀⠀⣰⣿⣿⣿⣿⣆⢲⣶⣶⣶⣶⣶⣶⣶⣿⢇⣷⣾⣿⡏⣟⣯⣶⣿⣿⡾⠀⠀ ⠀⠀⣿⣿⣿⣿⣿⣿⣦⠹⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠋ ⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣬⠉⠿⣛⣻⣿⣯⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀ ⠀⣠⣶⣿⣿⣿⣿⣿⣿⠿⠿⠦⠄⠀⠀⠉⠉⠉⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⠿⣿⠟⠛⠛⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠿⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡈⠻⢿⣿⠿⣿⣿⣿⠇⣴⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣶⡆⠀⠀⠀⠀⠉⠀⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢛⣉⣍⡛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⣉⠛⣛⡉⠁⠀⠀⣀⣤⣶⣦⣤⣀⣶⠄⣿⣿⣿⣿⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠇⠀⠀⣸⠛⠉⠉⠙⣿⣿⣿⡆⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠟⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⣿⣿⣮⡻⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢋⣩⣭⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣶⡾⣿⣟⠛⣋⠁⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⣩⣤⣶⣤⣄⡀⠀⢀⣼⣿⠟⠁⣩⣿⣿⢿⣿⡿⣽⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢧⣾⣿⣿⣿⣿⣿⣿⡄⠻⠿⠁⢠⣾⣿⠟⣡⣾⣿⠛⢮⡛⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢳⣾⣶⡄⠀⠈⢿⢸⣿⣿⣿⡟⣩⣶⣿⣿⣿⣷⣄⢿⡿⢫⣾⣿⣿⠏⣠⣼⡿⠾⣻⣽⣤⣶⡶⣦⣄⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⡿⠃⠈⠛⢿⣦⣀⣈⠈⢿⣿⡏⣼⣿⣿⣿⣿⣿⠿⣛⣂⣀⡛⢿⠟⣡⣾⣿⡿⢁⣾⣿⣿⣿⣿⡷⢹⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⣾⣼⡿⠋⠉⠀⠀⠀⠀⠀⠹⢿⣼⢧⡈⠻⠀⣿⣿⣿⣿⢋⣴⣿⣿⣿⣿⣿⣷⣄⢿⡟⠋⠀⣾⣿⣿⡿⠉⠩⠷⠿⣿⣿⣿⣿⡿⣫⣶⣷⣄⠀⠀⠀⠀ ⠀⠀⠀⠀⣼⣿⠋⠀⢠⣶⣶⣶⣾⣿⣿⣿⣿⣷⡄⣶⣶⣆⣿⣿⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠁⣀⣤⣶⣜⢿⠁⢸⣷⡄⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀ ⠀⠀⠀⠀⣿⡇⠀⢰⣯⣿⠉⠉⢩⣍⣉⣉⣭⣭⣥⣭⣍⣻⣥⣭⣭⣜⠿⣿⣿⣿⣿⣿⣿⣿⠿⢓⠸⢿⣿⣿⡟⠀⠀⠀⢻⣧⣀⡀⠀⠈⠐⣿⣿⣿⣿⣿⣿⣿⣿⡄ ⠀⠀⢀⣼⠿⠁⠀⢸⣿⡇⠀⢸⣿⠿⠟⠿⠿⠿⠿⢿⣿⣟⠛⠛⣻⣥⣴⣶⣄⠉⠛⠿⠟⢱⣿⣿⣿⡄⣀⠈⠀⠀⠀⠀⠈⠉⠛⠁⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⡇ ⣠⡴⠟⠛⠁⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⢠⣼⣻⣿⣿⣿⣿⣿⣿⠾⣿⠿⠛⠁⣀⣴⣿⣷⠈⣛⠻⠟⠁⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃ ⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢿⣿⡇⢀⣾⣿⡃⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⣟⡛⠻⣿⡁⠀⣿⣇⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠟⠿⠁⠀⢤⣤⣶⠆ ⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⡇⠸⣿⣿⡇⠀⠀⠀⣤⠾⠻⣿⣿⣶⣿⣿⣿⣦⠻⠃⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⢠⣼⣿⡿⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⢸⣶⠁⢐⡿⠏⠀⠀⠀⢠⣶⣿⣷⠘⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠋⠀⠀⠀⠀⣀⣼⣿⡿⠁⠀ ⠀⠀⠀⠀⠀⠀⠀⠜⠉⠀⠀⠀⢻⣧⠘⣿⣦⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣿⣿⠟⠁⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠀⠘⣿⣧⠀⣾⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠀⠀⠀⣼⣿⠆⣿⣿⣿⣿⣿⢿⣿⣥⣴⣶⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⢹⣿⣿⡟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⣤⣤⣤⣤⣶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠎⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⡉⠉⠛⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⠟⠉⢼⣿⣿⣦⣆⢀⣠⣴⣦⣤⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣶⣶⣦⣄⡀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣦⣤⣤⣄⣀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠉⠉⠙⠛⠿⢿⣶⣦⣀⠀⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣏⠀⣠⣴⣾⡿⠟⠛⠋⠉⠁⠀⠀⠀⠀ ⠀⢀⣠⣤⣤⣤⣤⣤⣤⣀⣈⠙⠻⣷⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠟⠋⣁⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀ ⠈⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⢿⣶⣄⣿⣿⣿⣿⣿⡇⠀⠀⠀⣾⡆⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣧⠀⠀⠀⣸⣿⣿⣿⣿⣿⣤⣶⡿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠁ ⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢿⠃⠀⢀⣼⣿⣿⣿⣿⣧⡀⠀⠸⡏⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣤⣄⠀⠀⠀ ⠀⠀⠊⠉⠁⠀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣶⣿⣿⣿⠿⠿⣿⣿⣿⣶⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠈⠉⠑⠀⠀ ⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⢿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⠿⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⢀⣴⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⡿⠓⠶⣤⡀⠀⠀ ⠈⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⣴⣶⣶⣶⣤⣤⣼⣿⣿⣿⣿⣿⡌⠻⣦⢻⡀⠀ ⠀⠸⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⢤⣴⣾⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣄ ⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⠰⣄⠀⠀⠉⠛⠛⠿⠧⠀⠘⠛⠛⠋⠉⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿ ⠀⠀⠈⢿⣿⣿⡿⠏⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀ ⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⣼⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠲⢶⣶⣿⣿⣿⣿⣿⣿⣧⣴⣶⣶⣦⣤⣄⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠋⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⣿⠇⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠃⣼⡇⣸⠏⠀⠀⢻⣿⣿⣿⣿⡏⣸⡆⡟⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⢋⠔⠁⠀⠀⠀⢸⣿⣿⡿⣿⡇⠛⢠⠁⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠏⢰⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣶⡇⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⢿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣄⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣤⣄⠀⢤⡀⣀⢄⣼⣇⢀⢀⣄⣤⠎⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠘⣷⣿⡻⣿⣿⣿⠇⢂⣴⠏⣀⣤⠔⠀ ⠠⣄⣀⠐⣬⣿⢹⣿⣿⠀⠀⠈⢻⡹⠞⠋⠉⢠⣿⡿⠟⠟⠁⠀⣀ ⠀⠈⢿⣿⣾⣯⡸⡟⠃⠀⠀⠀⠀⠱⡀⠀⠀⠈⠉⣠⣦⣶⡾⠛⠁ ⠀⠀⠐⠻⣿⣯⡀⢀⠀⠀⠀⠀⠀⠀⢡⢀⣀⣬⣽⡿⠟⠛⠉⠁⠀ ⠀⠀⠀⠀⠀⠙⠻⣿⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⣠⠶⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⡀⠀⠀⠀⠀ ⢠⡞⠁⣀⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⣸⣿⣦⠀⠀⠀ ⠘⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠛⠟⠀⠀⠀ ⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇⠀⠀⠀⠀⠀ ⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀ ⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀ ⠀⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡽⣦⣤⣀ ⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⣷⢛⠛⠀ ⢀⣤⣽⣿⠿⠟⠛⠋⠉⠉⠉⠉⠛⠛⠻⠿⣷⣤⣀⣀⣼⡟⠀⠀⠀ ⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣧⡀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠗⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠖⢉⣌⢆⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠀⠈⠉⠲⣿⣿⡜⡀⠀⠀⠀⠀ ⡔⢉⣙⣓⣒⡲⠮⡇⠀⠀⠀⠀⠀⠀⠘⡿⡇⡇⠀⠀⠀⠀ ⡇⠘⣿⣿⣿⠏⠀⠀⠠⣀⡀⠀⠀⠀⠀⡇⠈⠳⡄⠀⠀⠀ ⢹⠀⢻⣿⠇⠀⠀⣀⣀⠀⡍⠃⠀⠀⣠⣷⡟⢳⡜⡄⠀⠀ ⠈⣆⠀⠋⢀⢔⣵⣿⠋⠹⣿⠒⠒⠚⠁⣿⣿⣾⣷⢸⠤⡄ ⠀⡇⠀⠀⢸⢸⣿⣿⣶⣾⡏⡇⠀⠀⢀⡘⣝⠿⡻⢸⡰⠁ ⠀⢳⠀⠀⠈⢆⠻⢿⡿⠟⡱⠁⠰⠛⢿⡇⠀⠉⠀⡸⠁⠀ ⠀⠈⢆⠀⠀⠀⠉⠒⠒⣉⡀⠀⠀⢇⠀⡇⠀⠀⢠⠃⠀⠀ ⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⢉⡱⠀⠀⠉⠀⢀⡴⠁⠀⠀⠀ ⠀⠀⠀⠀⠈⠓⠦⣀⣉⡉⠁⢀⣀⣠⠤⠒⠥⣄⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠰⣉⣀⣀⡠⠭⠛⠀⠀⠑⠒⠤⠤⠷⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠞⠉⠀⠀⠀⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣦⠀⣠⣤⣄⣀⣠⠤⠞⠋⠛⠛⠛⠉⠙⠛⢶⣄⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢀⣤⣄⡀⢀⣶⢤⣀⣉⣯⣍⡁⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢿⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀ ⠀⠀⠀⠀⣠⠞⠋⠉⠉⠛⣾⣇⡀⠙⠋⠁⠀⠈⠁⠀⠈⠙⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀ ⠀⠀⠀⢰⠏⠀⠀⠀⠀⢀⣾⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠚⠛⠛⠛⠛⠛⠓⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀ ⠀⠀⠀⢸⣄⠀⢠⡴⠶⣫⡷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀ ⠀⠀⠀⠀⠻⠷⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⡟⣧⠀⠀⠀⠀⢾⡗⠛⠋⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣀⣀⡀⠀⠀⠀⣠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠙⣆⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠈⣿⠀⠀⢰⡏⠈⠙⠶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢹⡄⠀⠀⠀⠀⣷⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠙⠓⠒⠋⠀⢠⣿⡇⠀⠀⠀⠀⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠸⣇⠀⠀⢀⡼⠏⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⣾⢸⡇⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡀⠀⠀⠀⠀⢹⠀⠀⣿⡀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⣰⠏⢸⡇⠀⠀⠀⠀⢸⡇⠀⠈⠻⣇⠀⠀⠀⠀ ⣠⡶⠶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⢶⣄⠀⠀⠀⢼⣟⠀⠈⠉⠀⠀⠀⢀⣾⠀⠀⠀⠀⠙⠳⢶⣄⠀ ⢿⡇⠀⠀⠀⣠⠞⠛⠻⣦⠀⠀⠀⠀⠘⣷⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⠏⠁⠀⠀⠀⢻⣄⠀⠀⠀⠀⣤⣾⡇⠀⠀⠀⠀⠀⠀⣠⡿⠀ ⠈⠛⠒⠶⢿⣥⣄⣀⡀⣿⠀⠀⠀⠀⠀⢹⡄⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠙⠳⠶⢾⣟⡇⣸⠃⢠⡾⣆⠀⠀⢀⣿⠁⠀ ⠀⠀⠀⣼⠉⠀⠀⠉⣧⠙⣦⡀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠈⠉⠓⠲⠶⠶⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⣾⣟⡇⣰⠏⠀⣼⣀⠿⣄⠀⣰⡟⠀⠀ ⠀⠀⠀⢻⣄⠀⢀⣼⠃⠀⢸⠇⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⢾⣄⡀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠶⠚⠋⠁⣤⠴⠋⢹⣿⠟⠀⠀⠈⠙⣻⡎⠙⠃⠀⠀⠀ ⠀⠀⠀⠀⠈⢛⣉⡀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠛⢿⡉⣿⣋⡉⠉⠁⣀⣀⣀⡤⠶⠚⠋⠀⠀⢀⡾⠋⠀⠀⢠⣶⡿⠛⠁⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣿⠉⠙⠓⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠈⠿⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣠⡴⠋⠀⣴⣦⣀⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠛⠁⠀⠀⠀⠻⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⣀⡤⠴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠻⣆⣀⣠⡴⠶⠦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢶⣦⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣵⠛⠛⠳⡆⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡾⠁⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡟⡴⠛⠋⠀⠀⠀⢀⣤⠞⠛⠛⠛⠛⠦⣤⡀⢠⡟⣧⠀⢀⣤⠼⠞⠋⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⣠⠾⠀⣧⣀⡤⠖⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠐⢲⣶⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⢰⡶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠂⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⠀⠀⢸⢻⠞⠀⠀⡀⢘⠀⠲⠀⠀⠀⣰⣧⣷⡄⠘⡶⠉⠁⠐⡆⠻⠋⡖⠒⠀⠎⣳⠀⡄⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀ ⠀⠀⡄⠀⠀⡀⠀⠀⣹⣿⠤⠔⢁⠼⠀⠀⠡⠥⠠⠤⣯⡄⢀⡴⣃⠀⠀⠸⠖⠃⢰⡇⠠⠇⠐⠀⡇⠠⠖⠫⠿⠄⢿⡗⡦⠚⠀⢾⠟⠉⣷⡴⠄⠀⠦⣤

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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠋⠁⠀⠀⠀⠀⠉⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⠂⠀⠀⠀⠀⠀⠀⠀⢀⡴⣆⡀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⣿⣆⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢠⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣼⣿⣿⡆⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢀⡴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡛⠋⣉⣼⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠘⣧⡀⠀⠐⡦⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀ ⠀⠀⣤⡀⠀⠘⢿⣶⣄⣀⣀⠈⠁⠀⢀⣀⣀⣀⣀⣀⡤⠴⣾⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢿⣷⠀⠀⠀⠀ ⠀⣠⣿⡉⠓⢦⣈⢯⡻⣷⠈⠉⠉⠉⠉⠀⠀⠀⣼⣇⡤⠊⣁⠼⠀⠀⠘⡇⠀⠀⢀⣀⡠⠴⠚⠋⣹⠟⠀⠀⠀⠀⠀⠀⢀⢸⠁⠀⠀⠈⢇⣤⠀⠀ ⠸⣿⠋⢏⠉⠀⠈⠉⠛⢯⡳⠦⢤⣤⣤⡤⠴⢒⣋⣭⣖⠊⠁⠀⠀⠘⠒⠛⠒⠋⠉⠉⠀⠀⠒⣾⠻⣶⡆⠀⠀⠀⠀⠀⠀⢺⣧⡀⣤⠀⠸⣿⠀⠀ ⠀⠈⠻⣎⠀⠀⠀⠀⠀⠀⠉⢹⡶⠒⠛⠉⠉⠉⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡴⠟⠀⠀⠀⠀⠀⠀⠀⢘⡇⣿⢿⠀⢀⠈⣧⠀ ⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠈⣧⡏⠀⣻⡆ ⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⣀⣀⣀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠿⢷⡀⣿⠀ ⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠉⠉⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⣿⡇⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢴⠖⡶⠟⠁⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡏⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡇⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣼⠁⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣈⣤⡤⢤⣀⢸⠓⠶⢤⣀⣀⣀⣀⡤⠶⠚⠉⠀⠀⠀⢠⠏⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠙⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣤⣀⣀⣀⣀⣠⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣀⡴⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣁⠀⠀⠀⠀⠀⣹⡆⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣴⢿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⢘⡉⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠻⠿⠿⠟⠷⠶⠶⠶⠶⠶⠒⠒⠒⠉⠉⠀⠀⠀⠀⠀⠀⢿⡟⠙⣶⠏⢳⣞⢙⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣟⡷⠟⣻⠛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠉⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿ ⣿⣿⣿⠟⠉⠈⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿ ⣿⡟⠁⠀⠀⠀⠈⣿⡿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿ ⠏⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣿⣿ ⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠙⢿ ⠀⠀⠀⢀⠴⠁⠀⠀⠀⠀⢀⠎⠀⠀⢀⡤⡀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠘⢆⢀⡜ ⠀⢀⠔⠁⠀⠣⡀⠀⠀⡠⠁⠀⠀⠀⢸⣉⣱⠀⠀⠀⠀⢠⠀⠀⠙⢡⠂⢹⣧⣄ ⣔⠁⠀⠀⠀⢀⡨⠗⠁⠐⡄⠀⡠⠀⠤⡛⠛⠀⠀⠀⠀⠀⡇⣠⡀⢀⣴⣿⣿⣿ ⣿⣍⠉⠉⠉⠀⠀⠀⠀⠀⢈⠦⢄⡀⠀⡸⠀⠀⠀⠀⠀⠀⡎⠀⠈⢿⣿⣿⣿⣿ ⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⠆⠀⢠⣾⡏⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣄⣀⢠⣿⡓⠀⠀⠀⡀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣏⠈⠛⠑⠀⠀⠀⠈⠢⢀⣠⠤⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣶⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀⣠⣠⣤⣤⣤⣤⣤⣤⣤⣦⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⢀⠃⠀⠜⠀⠀⠠⠊⠀⠀⡠⠂⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⠀⡘⠀⡌⠀⠀⡐⠁⠀⠠⠊⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢢⠃⢠⠀⠀⠰⢀⠠⠊⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠈⠁⠈⠄⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⡒⠀⠉⠀⠀⠀ ⠀⠀⠀⡀⠀⡔⠁⠀⡀⡀⠀⡠⠤⠤⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀ ⠀⢀⡀⡇⡍⠀⠀⣾⢤⡇⠰⠀⠀⠀⠀⠀⠀⠈⠱⠀⢀⠤⡀⠀⠀⠈⠆⠀⠀⠀ ⠇⠀⠀⠱⠀⠀⠀⠷⠞⠁⠧⠀⡀⢀⣀⠀⠀⢠⠃⢠⡷⢤⣾⠀⠀⠀⢘⠄⠀⢠ ⠈⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠉⠀⠀⠉⠁⠀⠘⢦⡤⠏⠀⠀⠈⠀⠀⠠⠁ ⠀⠀⠑⢼⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⡀⠄⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠙⣧⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢀⠄⠊⠙⢿⣿⡀⢐⣢⣤⣤⣤⣤⡤⡄⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⡠⠁⠀⠀⡐⠀⠀⠂⠩⠩⠽⠭⠭⠁⠂⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠌⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠈⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⠛⠛⠋⠉⠉⠉⠙⠛⠻⣿⣶⣦⣀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣦⡀⠀ ⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⡄ ⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣴⠶⣶⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠾⢿⣷⣶⣶⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⢀⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⣀⠀⠀⣿⣿⣿⣿⣿ ⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁ ⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀ ⠻⠿⠿⠿⠛⠛⠉⠉⠁⠀⠈⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠀⢀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠋⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢀⠀⠀⠀⡆⠀⠀⢀⡤⢿⡀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣼⠀⠀⠆⡇⢀⣿⣿⡇⠐⡇⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠙⠯⠿⠟⠗⠚⠁⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣷⢦⡄⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠐⠃⢀⣠⣤⣤⣤⣠⡤⠞⠃⠀⡰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡆⢹⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⢾⡉⠀⠀⣀⣠⣤⡤⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢧⡟⢸⠀⢶⣄⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⠷⢾⠋⠀⠀⠀⠀⠰⢿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠸⠶⠾⢸⢧⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⠴⠊⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢠⣾⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⣠⠟⠉⠉⠓⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢄⡀⠀⠀⠀⠀⣠⠠⣳⠀⠀⠀⠀⢿⣿⡿⣄⠀ ⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⣀⣠⠤⣾⠋⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠠⣤⣀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢚⡟⠯⠟⠀⠀⠀⠀⠀⠉⠈⢿⡀ ⠀⠀⢀⣠⠞⠁⠀⠀⠀⣠⠔⠚⠉⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠘⣏⠓⢦⡄⠀⠀⠀⠘⣦⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃ ⢀⣴⠟⠁⠀⣀⡀⢀⡴⠋⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠹⡆⠀⢻⡆⠀⠀⠀⠸⡆⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⡼⠁⠀ ⣾⣿⠑⣄⡎⠀⣿⠞⠁⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠹⣄⡾⠃⠀⠀⠀⠀⢹⡀⠀⠀⠈⣇⠀⣞⢳⠀⢰⠇⠀⠀ ⢻⣿⡀⣸⡿⠋⠁⠀⠀⠀⠀⠀⠈⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢽⣇⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠉⢹⠉⢹⠟⠋⠀⠀⠀ ⠀⠉⠳⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠸⡟⠳⡀⡴⠻⣦⡶⣿⠀⠀⠀⣠⠏⠀⢸⡇⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠀⠀⠀⠀⠀⢳⣼⡟⣿⣆⣾⡶⠋⠀⢀⡴⠋⠀⣀⡾⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠿⠿⠥⠤⠤⠔⠋⠀⢀⡠⡿⠃⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⠴⢋⡼⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⡏⣀⡀⠀⠀⠀⠀⠀⠀⠙⠢⣤⡀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⡠⠤⠒⠋⣁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠈⠹⣄⠁⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠑⠒⠒⢺⣄⠀⠀⢀⡀⠀⠀⠀⢸⣇⡀⣀⣠⡤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠹⡄⠀⠀⠈⡇⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣀⣴⡿⣄⣀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠶⠆⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠠⣷⣏⡴⢋⣸⠦⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣧⡴⠋⢳⡀⡞⠹⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠁⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛⢷⣀⡼⠛⠻⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀ ⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣦⠀⠀⠀ ⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡀⠀ ⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢠⣾⣿⣿⣦⠀⠀⢸⣿⣿⣷⠀ ⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⡄ ⠀⠈⠙⠿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠉⠛⠋⠁⠀⢀⣾⣿⣿⣿⡇ ⠀⣤⡀⠀⠀⠉⢛⡋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⠃ ⠀⢿⣿⣷⣦⡴⣿⣿⣿⣦⡀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀ ⠀⠈⢿⣿⣿⣇⠀⠈⠛⠟⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀ ⠀⠀⠀⠻⣿⣿⣦⣄⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀ ⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀

⠀⠀⢀⣴⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢠⣿⣿⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢀⡾⠋⠀⣰⣿⣿⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠘⠀⠀⢠⣿⣿⠃⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢸⣿⡇⠀⠀⠀⣼⣉⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀ ⠀⠀⠀⢹⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣿⠉⠁ ⠀⠀⠀⠸⣿⣿⣄⠀⠀⠘⢿⣿⡵⠋⠙⢿⣦⡀⠀⠀⣤⣠⠀⣠⣿⡅⠀⣿⠀⠀ ⠀⠀⠀⠀⠈⠻⢿⣿⣶⣤⣄⣀⠀⠀⠀⠈⠻⣷⣄⣠⣿⣿⡼⠋⠛⣡⡼⠋⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠷⠶⠶⠾⣿⡿⠋⠻⣟⠉⠁⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⣿⡿⠷⣶⣤⣄⡀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⢀⣠⣤⣶⠾⢿⣿ ⣿⡇⠀⠀⣨⣿⣿⣷⠀⢠⣿⣿⠿⠋⢉⡉⠙⠿⣿⣿⣄⣤⣾⣿⣿⣅⠀⠀⢸⣿ ⢸⣧⠀⣰⣿⣿⣿⡏⢀⣿⣿⡏⢠⣾⣿⣿⣷⡄⢹⣿⣿⣿⣿⣿⣿⣿⣆⠀⣼⡇ ⠈⣿⣴⣿⣿⣿⣿⡇⠈⣿⣿⣧⣾⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⠁ ⠀⢹⣿⣿⣿⣿⣿⣿⣄⠙⢿⣿⣿⣿⣿⣿⠏⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀ ⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣄⣈⠉⠉⣉⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀ ⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀ ⣴⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣦ ⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⠛⠛⢉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿ ⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁ ⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⢀⠤⠐⠒⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣰⠇⠀⠀⠀⢠⡀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣠⠟⠀⠀⠀⠀⣮⣾⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⡘⠀⠀⠀⠀⠀⠀⠯⠭⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⡀⠱⣄⣀⠐⠀⣀⣀⣀⠠⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀⠀ ⢰⣳⠁⠚⢿⠄⣀⡠⣔⢪⠒⠁⠼⠄⠐⠈⠵⣧⠀⠀⠀⣇⡀⠱⡀ ⠀⠈⢀⠀⠀⡜⠀⠀⠀⠀⠑⡀⠀⠀⠀⢀⠜⠁⠀⠀⠀⡆⠹⡄⡣ ⠀⠀⠀⠑⢠⠀⠀⠀⠀⠀⠀⠐⡀⠀⢊⠁⠀⠀⠀⠀⠀⠸⣀⣹⠇ ⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⡄⠀⠀⠀⠀⠀⠀⡘⠀⠀ ⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⢀⠔⠀⠇⠀ ⠀⠀⠀⢀⠔⠉⢂⠀⠀⠀⠀⠀⡜⠒⠀⢺⠁⠐⠂⠈⠁⠀⠜⠀⠀ ⠀⠀⠀⠸⡀⠀⠀⠁⠢⠄⣀⡔⠀⠀⠀⢸⠀⠀⠀⢀⠠⠊⠀⠀⠀ ⠀⢀⣠⠒⠉⠀⠀⠀⢪⠀⠀⡱⠀⠀⠠⡎⠉⠉⠉⠀⠀⠀⠀⠀⠀ ⠀⠈⠛⠑⠒⠒⠀⠈⠁⠀⠀⠙⢵⡵⡯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀

⣿⡅⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢨⣿ ⣿⣿⡄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢠⣿⣿ ⣿⣿⣿⡄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢠⣿⣿⣿ ⣿⣿⣿⣿⡄⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢠⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣶⣄⡀⢀⣴⡿⢻⡏⠉⠀⠀⠉⢹⡟⢿⣦⡀⢀⣠⣶⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠸⣷⠀⠀⠀⠀⣾⠇⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠙⠷⣦⣴⠾⠋⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⢠⣿⠁⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⠟⠻⣧⠀⠀⢰⣿⣿⡄⠀⠀⢠⣿⣿⡇⠀⠀⣼⠟⠻⣿⣿⣿⣿⣿ ⣿⣿⣿⡿⠁⠀⠀⣿⡄⠀⠀⠙⠛⠛⠀⠀⠛⠛⠋⠀⠀⢠⣿⠀⠀⠈⢿⣿⣿⣿ ⣿⣿⠏⠀⠀⠀⠀⣸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⠀⠀⠀⠀⠹⣿⣿ ⣿⠃⠀⠀⢀⣤⣾⣿⣿⣿⡄⠸⣷⣦⣀⣀⣴⣾⠇⢠⣿⣿⣿⣷⣦⡀⠀⠀⠘⣿ ⣿⡀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠙⠛⠛⠛⠛⠋⣠⣾⣿⣿⣿⣿⣿⣿⣦⡀⢀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⢸⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⠟⢻⣿⠀⠀⠀⠀⣸⡿⠻⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣴⣿⣿⡿⠋⠀⠀⢸⣿⣀⣀⣀⣀⣿⡇⠀⠀⠙⢿⣿⣿⣦⠀⠀⠀⠀ ⠀⠀⠀⣸⣿⣿⡟⠁⠀⠀⣠⣾⡿⠟⠛⠛⠻⢿⣷⣄⠀⠀⠀⢻⣿⣿⣇⠀⠀⠀ ⣿⣿⠿⠿⠿⠿⠿⠶⣶⣾⣿⠋⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⠶⠾⠿⠿⠿⠿⢿⣿ ⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿ ⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿ ⣿⣿⣶⣿⣿⣿⣿⠿⠿⠿⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠿⣿⣿⣿⣿⣶⣾⣿ ⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠈⢻⣿⣷⣶⣶⣾⣿⣿⠋⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀ ⠀⠀⠀⠀⠘⢿⣿⣿⣦⣀⠀⣿⡇⠈⠉⠉⠁⢸⣿⠀⢀⣴⣿⣿⡿⠋⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡇⠀⠀⠀⠀⢸⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⣷⠀⠀⠀⠀⣿⠀⠀⠸⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠀⠀⠀⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⡿⠟⠉⠁⠀⠀⠀⠉⠛⢷⣦⡀⠀⣶⣿⣿⣦⠀⠀⠀⠀⣿⣿⣿⣿⣿ ⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠈⠉⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿ ⣿⣿⡟⠀⠀⠀⢸⣷⠀⠀⠀⢰⣿⠀⠀⢹⣷⣴⡶⠿⠿⠿⠷⣶⣶⣿⣿⣿⣿⣿ ⣿⣿⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿ ⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⡆⠀⢠⡿⠁⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠘⣿⣿⣿ ⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣾⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡿⠀⠀⢹⣿⣿ ⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢸⣿⣿ ⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⢿⣿⡿⠃⠀⠀⠀⢸⣿⣿ ⣿⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣿ ⣿⠟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⣿ ⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⡏⢹⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀ ⣾⡟⠻⠷⣶⣤⣀⣀⣠⣶⣿⣿⣿⣿⠇⠸⣿⣿⣿⣿⣶⣄⣀⣀⣤⣶⠾⠟⢻⣷ ⠙⠿⣶⣄⡀⠉⠛⠻⢿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⡿⠟⠛⠉⢀⣠⣶⠿⠋ ⠀⠀⠈⠙⢻⣷⣦⣀⠀⠀⠉⠛⠻⠟⠀⠀⠻⠟⠛⠉⠀⠀⣀⣴⣾⡟⠋⠁⠀⠀ ⠀⢀⣀⣤⣿⣿⡏⠛⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⢹⣿⣿⣤⣀⡀⠀ ⣾⣿⣿⣿⣿⣿⡇⠀⠐⠟⠙⢻⣷⣦⣤⣤⣴⣾⡟⠋⠻⠀⠀⢸⣿⣿⣿⣿⣿⣷ ⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿ ⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⡟⢿⣿⠿⠿⣿⡿⢻⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀ ⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⠸⡟⠀⠀⢻⠇⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀ ⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀ ⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡷⢶⣶⣦⣴⣶⠶⢾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⣧⠀⠀⣼⡆⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣾⣿⣶⣶⣿⣷⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠉⡷⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠈⠑⠦⠤⠤⠤⠤⠤⠀⠒⠒⠒⠒⠒⠒⠒⠢⠤⠴⠚⠉⠀⠀⠀⡼⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠉⢹⡀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡶⠒⣆⠈⢣⠀⠀⠀⠀⠀⠀⠀⣸⠂ ⠀⠀⢀⣠⢤⣀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢸⣿⣆⠘⡇⠀⣧⣀⣀⣀⣀⣀⣠⠇⠀ ⢀⡴⠉⠀⠀⠀⠓⡄⠀⠀⠠⣯⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⣿⣿⠤⠞⢀⡏⠀⠀⠀⠀⣸⠃⠀⠀ ⡼⠁⠀⠀⠀⠀⠀⣷⠀⠀⠀⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⠁⢀⣤⠴⠋⠀⠀⠀⠀⢀⠏⠀⠀⠀ ⡇⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠉⠉⠀⠀⠳⡤⣄⣀⣀⡜⠀⠀⠀⠀ ⠙⢦⣀⡀⠀⣀⣾⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠈⠙⣏⠁⢸⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⢰⠒⠊⠁⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠸⡄⠀⣇⠀⠀⣠⠞⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⢦⣀⠀⠀⢸⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢷⠀⢸⡴⠚⠁⠀⠘⢦⡀⠀⠀⠀⠀⠀⠈⠛⠋⣰⢴⠉⠙⠒⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀ ⠀⠀⠀⠀⠀⢸⡆⠀⠑⣆⠀⢀⣠⠞⠙⢤⠀⠀⠀⠀⢀⡤⠞⠁⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀ ⠀⠀⠀⠀⠀⠹⣢⡴⠋⣡⡖⠉⡟⠀⠀⠀⠉⠒⠒⠋⠁⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠙⣦⣄⠀⠀⠀⠘⡆⠀ ⠀⠀⠀⠀⠀⠀⠀⠹⡍⠀⣧⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⣀⡤⠼⣦⣏⣀⣀⠀⡇⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡼⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⠞⠉⠀⠀⠈⠹⡙⠧⡼⠃⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠿⢗⠛⠒⠦⠤⣤⣤⣤⠤⠤⠤⠴⢇⡀⠀⠀⣀⣀⣀⣴⠃⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠊⠓⠚⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉⣷⣟⡻⣭⡩⠟⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⣀⣤⡆⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⡖⠀⠀⠀ ⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢛⣿⢿⣿⡟⠋⠀⠀⠀⠀ ⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⡿⠋⠀⠈⠀⢸⣿⡇⠀⠀⠀⠀⠀ ⢻⣿⣿⣟⠻⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠻⠉⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀ ⢸⣿⣟⠿⠀⠀⠉⠛⢿⣿⡇⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀ ⠀⣿⣿⡄⠀⠀⠀⠀⠀⠈⡇⢀⠠⠀⠀⠀⠁⠀⠀⡀⠀⠠⠟⠛⠀⠀⠀⠀⠀⠀ ⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⢀⢀⡀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⣿⠀⠔⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠻⠿⠇⠒⠈⡄⠀⠀⠀⠀⠀⠀⠈⡉⠁⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠴⠄⠀⢀⠔⠀⠀⠀⠄⠌⠀⠀⡁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⡀⠀⡀⠙⠄⠀⢀⠀⣠⣾⠄⣤⠊⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡑⠁⠂⠀⣈⣥⣾⡿⢿⠀⠀⢄⠀⢀⣠⣤⣤⣴⣦ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠀⠋⠿⠀⠀⠀⠄⠀⠌⣴⣿⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠉⠀⠏⠀⠀⠈⠉⠙⠋ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠂⠄⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢀⣀⡰⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀

⢀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⡀ ⢸⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⡇ ⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇ ⢸⣿⣿⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⢀⣿⡟⠀⢀⣴⣄⠀⠀⠀⠀⠀⣿⣿⡇ ⢸⣿⣿⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⢸⣿⠇⠀⠀⠙⢿⣷⣄⠀⠀⠀⣿⣿⡇ ⢸⣿⣿⠀⠀⠘⢿⣿⣄⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⣠⣿⡿⠃⠀⠀⣿⣿⡇ ⢸⣿⣿⠀⠀⠀⠀⠙⢿⣷⠄⠀⢸⣿⠇⠀⠀⠠⣾⡿⠋⠀⠀⠀⠀⣿⣿⡇ ⢸⣿⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠿⠟⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⡇ ⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇ ⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇ ⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⣿⣿⣿⣿⣧⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿ ⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿ ⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿ ⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠻⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢸⠀⠀⠈⠻⣿ ⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣿⣧⠀⠀⢸⡀⠀⠀⠀⣼ ⠀⠀⠀⠀⠀⢀⡴⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⣰⣶⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠈⢳⡀⢀⡴⠃ ⠀⠀⠀⢀⡴⠋⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⣿⡉⠛⣷⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠛⠟⠃⡠⠂⠉⣿⣏⠀⠀ ⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠦⣄⡀⢀⡠⢾⠁⠀⠀⠀⠀⠀⠀⠀⢻⣧⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢰⠁⠀⣰⣿⣿⣿⣷ ⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠟⠋⠀⠈⢧⡀⠀⠀⣀⠤⠀⠤⢄⠙⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⢀⣠⣀⠀⠈⣠⣾⣿⣿⣿⣿⣿ ⣿⡷⠦⠤⠤⠤⠤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠻⣤⡐⠁⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡤⠊⠀⠈⢦⣾⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⠢⢄⡀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⣠⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠘⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⢠⣿⣿⡍⠙⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠘⠻⢿⠛⠆⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠚⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⣿⣿⣿⣿⣿ ⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⣀⣆⣘⣦⣽⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣄⠀⠀⠀⢀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⠀⡜⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡉⠛⠲⠿⣿⣿⣿⣿⣿⣿⣿⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⠻⣿⣿⣿⣿⣷⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣦⡀⠀⢰⡆⠈⠙⢿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⣤⣭⣬⣥⣀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣆⢮⣫⣦⠈⠉⠛⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⠀⠀⢸⣿⣿⣿⣿⣯⠛⢮⣻⡽⠀⠀⣠⣾⣿⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⢸⣿⣿⣿⣿⣿⠀⠀⣨⣭⣥⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡈⠙⠿⡿⢸⣿⣿⣿⣿⣿⣇⣤⣈⠛⠛⠉⢀⣤⣾⣿⠀⠀⠀⠀⠀⠘⣿⣿⢆⡇⣀⣠⣤⡄⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠗⣠⣾⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠻⠿⠋⣁⣤⠀⠀⠀⠀⠀⢻⠟⣾⠇⣿⡿⠋⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⠻⠛⠀⠀⠀⠀⠀⢀⣾⣿⠘⠋⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⢿⣿⣿⣿⣿⣮⣁⠀⠀⠀⣠⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠋⢸⣿⢿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄⡺⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠁⠀⠸⠇⠀⠈⠛⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣦⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡈⢻⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⢟⣡⢻⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀ ⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣶⡆⠲⢶⣶⣿⣿⣶⣶⢺⣿⣿⣧⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀ ⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠈⠉⠉⠀⠘⢿⣟⠟⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀ ⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀ ⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢢⠀ ⠀⡄⣿⢻⣿⣿⡟⣿⣿⣿⣿⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠇⣿⣿⣿⣿⡇⣿⣿⣿⡏⣿⣼⡆ ⢸⡇⣿⢸⣿⣿⣿⠸⣿⣿⣿⡇⠘⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⠋⠀⢸⣿⣿⣿⡇⢿⣿⣿⡇⣿⡇⠀ ⠈⠃⣿⡼⣿⣿⣿⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠃⢸⣿⣿⠇⠿⠁⠀ ⠀⠀⠹⠇⢿⣿⣿⡆⠘⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠸⠿⠛⠀⠀⠀⠀ ⠀⠀⠀⠀⠈⠛⠋⠁⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀