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⠛⢿⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣼⣛⣉⣉⣀⣀⣀⣀⣰⡇⠀⠀⠀⠀⠀⢀⣶⠿⠲⠤⢤⣄⣀⣀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣶⣶⣾⣿⣿⣿⢿⣿⣿⣿⣿⣟⣻⣿⣿⠿⠿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠉⠛⢿⣿⡟⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣤⡀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠉⠙⣳⣤⣤⣶⡦⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⣙⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣼⣿⣧⣆⡀⠀⠀⢀⣤⡴⠾⠿⠋⠉⠉⠁⠀⠀⠉⣹⠟⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⠛⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠟⠛⠙⢿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣶⣮⣍⡀⠀⠀⠀⠀⠀⠀⠀⠲⢶⡏⠀⠀⠀⠄⠈⠉⡟⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⢛⡅⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠀⠀⠀⠀⠙⠙⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⣠⡿⠐⠀⠀⠐⠂⠀⣼⠁⠀⠀⠈⠻⡶⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠙⠛⠓⠶⠶⠶⣶⣶⣾⡶⡶⢤⣤⣀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣀⡀⠀⢀⣼⠃⠀⠀⡄⠀⠀⠷⡇⠀⢹⡿⢿⢳⣄⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⡿⠋⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣶⠟⢹⠋⢙⠀⠷⣼⣿⠛⣿⣦⡀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣧⣴⣇⢠⡄⠀⣠⣀⣹⠀⢸⣷⠀⠀⢻⣄⠈⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⣀⡴⠞⠉⠀⠀⢠⡀⣤⣀⣀⣀⣀⣴⠟⠁⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⠈⣷⣤⣤⣶⣿⠏⢭⣙⣿⣷⠀⠀⠀ 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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣤⣿⡏⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣠⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠘⠁⠀⢴⣿⣽⠇⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣼⠉⠙⢠⡟⠶⠾⢇⣰⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣾⠏⠀⢀⣾⣿⡿⠛⢉⣹⡿⠿⠿⢿⣄⠈⠙⣷ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣻⣿⣿⡿⠷⢶⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⡇⠀⠀⠈⢿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣠⠂⠀⠀⠁⢻⣿⣿⣄⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⢰⣇⣀⣀⣿⠿⠿⠖⣼⡿⢻⡟⢡⣿⣿⣿⣿⣿⣿⡟⠁⠀⣴⣿⠟⠁⠀⣰⡟⠁⠀⣀⣠⣤⣾⠳⣆⢸ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣸⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⡇⢺⡇⠀⠀⠀⠀⢀⡟⣿⣿⣦⡀⠀⠀⠀⠘⠀⠀⠀⠀⠘⢿⣿⣉⣛⣹⣷⣶⣿⣷⣂⣧⣾⣿⣿⣿⣿⣿⣿⠃⠀⠘⠋⠁⢀⠤⠾⢿⡟⢷⣆⠈⢿⡋⠁⠀⠘⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⠃⠀⠀⠘⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣷⣾⡇⠀⠀⠀⠀⠈⠿⠿⢿⣿⣿⣦⣄⠀⠀⠀⣷⣄⠀⠀⠈⢻⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠃⠀⠀⠸⣧⠀⠻⣷⣾⣿⣿⢿⣆⢸ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣤⣤⣾⣶⣾⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⢠⣿⣿⣿⣿⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡙⣛⣿⠓⠻⢿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣤⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣤⣤⣤⣄⣀⠀⠀⠶⠶⣾⣿⣄⠈⠁⠀⣹⣆⢻⡎ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⡿⢼⣷⠀⠛⠀⣿⣟⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⣿⣿⣦⣿⠟⠛⠈⢛ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣧⣄⣀⠙⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⣁⠀⢀⣄⠀⠲⠶⠖⠒⠂⠈⠛⠛⠉⠂⡹⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠙⣟⣉⣿⣿⣻⣿⠿⠟⠋⠉⠁⠀⠠⣬⠹⣧⣦⣤⣬⣉⠻⠿⢿⣯⡝⢷⣶⣶⣿ ⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠲⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣇⣀⣈⡍⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠿⢻⣿⢿⡀⢀⠀⢦⡄⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣧⡙⣿⣿ ⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠱⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣼⡟⠙⠁⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡇⠀⢰⣿⣾⣷⡀⠀⠘⢿⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣩⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢠⣴⡆⠀⠀⢰⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣹⣇⣀⣀⣀⣀⣀⡟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣿⢹⡏⣿⡇⠉⣾⣿⣿⣿⣿⣦⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿ ⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠺⠟⠀⠀⠀⢸⣿⣿⣿⠇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⡿⠋⠉⠉⠉⠉⠛⢀⡀⢀⣤⣤⣤⣤⠶⠄⢀⣤⢠⣤⣤⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿