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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣀⠀⠙⣿⣾⣿⠄⠀⠀⠀⠀⣾⣿⠟⠉⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠌⠀⠠⠀⠀⠀⠙⠀⠉⠉⠂⠀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⣀⣠⠀⠰⢄⠀⠂ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⠊⡜⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠈⢿⣷⣄⠈⠻⣿⡆⠀⠀⠀⢠⣿⡏⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠁⢀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⢠⢣⡄⠀⠂⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠀⠀⠌⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠸⣿⣿⠷⠀⠈⠂⠀⠀⠘⢼⣿⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣷⠞⠀⠀⠀⠀⠀⢀⡤⢀⠀⢀⠀⠀⠀⡈⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠈⠉⢀⠐⠀⣀⢠⠒⠀ ⣴⣶⣶⣄⣴⣶⠀⢠⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣴⣶⠀⠀⠀⠈⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠗⠁⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠈⠀⠀⠀⠀⠀⠶⠶⠀⠈⠐⠀⠀⠐⠋⢀⣀⣀⡀⠀⠀ ⠀⠀⣿⣿⡏⠁⠀⢸⣿⡀⢴⣶⣤⠀⢠⣤⠀⣶⣤⡀⠀⣿⣿⡷⠂⠀⠀⢠⣶⣖⠀⣠⣄⠀⢴⣶⣄⠀⣠⣤⣶⣶⣤⠀⠠⣿⣿⠀⢰⣶⠀⠀⠀⣴⣶⣶⣄⠀⠀⣀⠒⣶⣤⡀⠘⠉⠀⢠⣴⡆⢀⣶⡄⠀⢤⣶⡄⠀⢠⣔⠂⣶⣤⠀⠀⠁⠀⢠⣶⡆⠀⣤⣄⠀⣴⣦⡄⠀⣠⡄⠐⣶⣄⠀⢰⣶⣶⠀⢀⣤⣶⣶⣦⡀⢺⣿⣿⡷⠀⠀ ⠀⢀⣿⣿⡇⠀⠀⢸⣿⡇⢨⣿⣿⠀⠿⠿⠀⣸⣿⣇⠀⣸⣿⡇⠀⠀⠀⢸⣿⣿⠀⣿⣿⠀⠸⣿⣿⠀⡿⠿⠀⢘⣿⡅⠀⣿⣿⣶⣿⡃⠀⠀⣿⣯⣀⣽⣿⡆⠸⣿⣇⠈⠛⠋⠀⠀⠀⢸⣿⡇⠀⣽⣿⠀⢸⣿⡇⢰⣿⣿⣤⣼⣿⡇⠀⠀⠀⢸⣿⣷⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿⡅⠀⣿⣿⠀⣻⣿⣇⠈⠛⠃⠀⣿⣿⡇⠀⠀ ⠀⠠⣿⣿⡇⠀⠀⢹⣿⡇⢸⣿⣿⠀⣠⣤⠐⣿⣿⣧⠀⢻⣿⡇⠀⠀⠀⢸⣿⣿⠀⣿⣿⠁⠀⣿⣯⠀⣠⣴⠖⢻⣿⡆⠀⣿⣿⠿⣿⡄⠀⠀⣿⣦⠀⣀⣄⡀⠀⠉⠿⢿⣶⣄⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿⡇⠠⣿⣿⡁⢁⣠⡀⠀⠀⠀⢸⣿⣿⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿⡇⠀⣿⣿⠀⠈⠛⢿⣿⣶⡄⠀⣿⣿⡇⠀⠀ ⠀⠐⣿⣿⡇⠀⠀⢸⣿⡇⢨⣿⣿⠀⣿⣿⣀⣽⣿⡷⠀⢹⣿⣇⠀⠀⠀⢸⣿⣿⠀⣿⣿⡁⠀⣿⣿⠀⣿⣿⣀⣸⣿⣇⢀⣿⣿⠀⢻⣿⠀⠀⣿⣿⠀⣿⣿⠇⠀⣶⣆⢈⣿⡿⠀⠀⠀⢸⣿⡇⢈⣿⣿⠀⢸⣿⡇⠀⣿⣿⠀⣸⣿⡇⠀⠀⠀⢸⣿⣿⠀⣿⣿⡀⢸⣿⣧⠀⣿⣿⣀⣸⣿⡇⢀⣿⣿⠀⢰⣶⣆⣸⣿⡇⠀⣿⣿⣇⠀⠀ ⠀⠈⠛⠛⠃⠀⠀⠘⠛⠃⠈⠛⠛⠀⠘⠛⠉⠛⠛⠓⠀⠈⠛⠛⠃⠀⠀⠘⠛⠛⠀⠛⠛⠀⠀⠛⠛⠀⠙⠛⠋⠛⠛⠋⠘⠛⠛⠂⠘⠛⠃⣀⡈⠙⠛⠛⠉⠀⠀⠙⠛⠛⠛⠁⠀⠀⠀⠘⠛⠃⠀⠛⠛⠃⠘⠛⠃⠀⠈⠙⠛⠛⠉⠀⠀⠤⠀⠘⠛⠋⠀⠛⠛⠀⠘⠛⠛⠀⠈⠉⠛⠛⠀⠀⠈⠛⠛⠀⠈⠙⠛⠛⠛⠁⠀⠙⠛⠛⠃⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠀⢀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠃⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠰⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡰⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠀⠀⠀⠀⠀⠁ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⣠⠈⠙⠂⠀⡅⠀⠃⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢠⠂⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠶⠀⠘⠀⠂⠃⠀⠀⠀⠠⠄⠀⠂⠐⠂⠀⢁⡀⡀⠤⠃⠀⡀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠁⢀⡴⠦⠉⠄⠀⠀⠀⠀⠂⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠘⡋⠴⢤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠈⠐⠌⣠⠁⠈⡤⠴⠀⢠⣶⠀⠀⠀⠀⠉⠀⠲⠀⠐⠂⠀⢀⠠⠁⠑⠠⠀⠰⠜⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢠⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡛⢯⣍⡙⠫⢽⣲⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣝⢿⡀⢀⠈⠀⠐⠦⢉⡛⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠻⣿⠟⠁⠀⠈⡀⡄⠨⠳⣄⠀⠀⢀⠙⢦⡙⢿⣿⣟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⢠⡄⠀⠀⢀⡀⠣⠖⣁⢱⡟⠓⣦⡈⠳⣄⠙⢆⠙⢿⣷⣊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣠⣾⡿⠃⣠⣴⣿⠀⠻⣇⢹⣦⠻⣆⠁⢙⡶⢤⡀⠈⠳⡀⠘⣿⡙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⣩⣿⡿⠫⢤⡈⢰⢇⠀⣿⢹⡏⠇⢸⠁⠀⢙⢄⠹⣦⣀⠙⣆⠈⢷⡛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢠⡷⢾⣿⣿⠁⠒⠋⠀⢸⡿⡆⢿⡄⢷⡄⠹⣄⢰⠈⠘⣧⡈⢻⣷⣌⢳⡀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢠⠞⢡⣯⠖⠋⢸⣿⠀⠀⠀⠠⢤⢷⡟⠈⢿⡆⢻⣄⠙⢧⣳⣄⠈⢿⣶⡉⠻⣷⡙⣆⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡟⠀⡟⢡⣀⣿⣿⠏⠀⠀⣠⣴⣿⣦⣷⣀⠀⣿⣦⠹⣆⠈⠿⣿⣇⠀⠹⢿⣦⡙⢧⡹⣧⠹⠀⡆⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢰⣧⠒⣿⣾⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣿⣿⣷⣿⣿⣷⣶⣶⣽⣿⣴⣿⣿⡷⠀⡇⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢸⡟⢠⠏⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣌⠉⠹⣿⣿⣿⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀ ⠰⠒⠒⠊⠉⠉⢹⡇⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣷⣴⣿⣿⣿⡈⠀⠸⠀⠀⠀⠀⠀⠀⠀ ⡀⠀⣠⣀⣀⡄⢸⢿⡏⠀⣿⣿⡿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠻⣿⣿⣿⣿⣿⣿⣶⣶⣾⢿⡿⣿⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀ ⠓⠀⠛⠛⠀⠀⠈⠻⣆⢰⣿⣿⡟⠛⠻⠾⣿⣿⣿⣽⣿⣿⣿⠋⠀⢶⡄⢘⡻⣦⣿⣿⣿⣿⣿⣴⠿⠁⣿⣿⡇⠀⠀⡃⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣿⣿⡄⠀⠀⣴⠟⣿⠋⢳⣿⡿⠃⢰⣶⣦⠵⠀⠻⣶⡎⠉⠉⠋⠉⠘⡀⠀⣿⣿⡇⠀⠰⠁⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⢠⠁⢠⣿⣼⣿⣿⣶⣦⡶⢦⣝⣥⣶⣶⣿⣷⣀⡀⠀⠀⣿⠀⢸⣿⢹⠇⠀⡠⢧⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⢻⣇⠀⢸⠀⣨⣴⡿⠟⠟⠋⠀⠃⠴⠊⠛⠙⠀⠻⢿⣿⣿⣆⣾⣿⡀⠀⡟⣸⠀⣠⣧⣀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣦⢸⡄⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣻⣟⠈⣷⡃⣴⣿⣿⡿⣶⣷⣆⣀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⣧⠹⣿⣿⡿⠿⣿⣷⣶⣶⣿⡿⠷⠶⠚⣿⣿⣿⠛⢻⢸⠟⢸⣿⣿⣿⡿⠿⣿⣿⣁⣹⠋⣇⣀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⢸⡠⣿⣿⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⣿⣿⠀⠈⣿⠀⣿⣷⢿⡟⢲⣿⢏⣀⡿⢶⣀⠻⠿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣉⣿⡿⠆⠀⠀⠒⠋⠙⠉⠀⠀⠀⠀⠀⠈⣿⡇⣀⡟⣸⠟⢡⣾⠃⣴⡿⠼⢿⣅⠸⢿⡇⠂ ⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣺⣿⠏⡈⣿⣷⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣄⠉⢷⣼⣻⣯⣞⡻⢯⣶⡟⣻⡦⢴⣿⡇⢰⣙⡆ ⠭⠓⣶⡞⣓⠤⣤⠶⢿⡿⢹⣿⣴⣿⣿⣷⣿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣀⡀⢀⣤⣾⣿⣿⠟⢹⣿⣻⣷⠉⠲⠯⡉⡙⠶⣤⠾⢀⣤⣿⡀ ⠀⠀⠉⠛⢻⡄⢬⣠⡾⠀⣼⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠁⢀⠌⢸⣿⡿⠀⠀⠀⠘⠿⠿⣭⣡⠤⠟⣻⡇ ⠀⠀⠀⢸⠃⣧⢞⣿⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⠇⣿⢰⣾⣷⣿⣿⣅⠀⠠⠀⠀⠀⠀⢀⣠⡄⠀⣿⠀ ⠀⠀⠀⢸⠞⠀⣼⡇⠀⠸⠏⢹⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣟⣽⣿⣿⣿⣧⡟⢰⡀⠀⣷⠀⠀⠀⠉⠁⠀⣿⡀ ⠀⠀⠀⣼⠀⢰⣧⡇⠀⠀⠀⠘⠿⣷⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⣿⣿⣿⠟⠀⢸⡇⠀⢿⣷⠀⠀⠀⠀⠀⣿⡇ ⠀⠀⠀⣿⠀⢾⡿⠀⠀⠀⠀⠀⠀⠘⠟⢧⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠀⠀⣀⣼⣿⣿⡟⠀⠀⠀⣿⡇⠀⠀⠈⣷⣦⣀⡀⠀⢸⡇ ⠀⠀⠀⡇⠀⠟⠁⠀⠀⠀⠀⠀⢀⣄⡀⢸⡏⢻⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢀⣠⣾⣿⣿⠏⠀⠀⠀⢰⣿⣧⠀⠀⡄⠀⠉⠉⠀⠀⢸⡇ ⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⠀⠙⣿⡉⠉⠙⠻⠿⣿⠿⢿⠟⣿⣿⣿⡋⠀⢀⣠⠇⢸⣿⣿⠀⠀⠳⠀⠀⠀⠀⠀⢸⠀ ⠀⠀⢰⡇⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠘⢿⡀⠀⠀⠀⠀⠈⢀⣱⣿⠟⠉⠁⠀⢏⣿⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄ ⠀⠀⣼⠁⢰⣧⠆⠀⠀⠀⠀⠀⠀⠀⣾⠀⠸⣿⡦⠀⠳⣿⣧⣄⣀⠀⠀⣮⠟⠁⠀⠀⠀⠈⣿⡿⠀⢸⣿⣿⠐⣀⠀⠀⠀⠀⠀⠀⠘⣷ ⠀⠀⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢿⣷⠀⠀⠈⠻⣿⣿⣶⡞⢃⣀⣀⠀⢀⡆⢸⣿⡇⠀⠘⣿⣿⡄⠿⠀⠀⠀⠀⠀⠀⠀⣿ ⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⢉⣠⠈⠛⠛⣰⠟⠀⢸⣿⠀⠀⠀⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿ ⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠈⠀⠀⠀⠀⠀⠤⠚⠗⠁⠀⠀⣿⠋⠀⠀⠀⣿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿