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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⣤⡶⣿⡴⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠿⣿⣿⣺⣅⣯⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣲⢀⣇⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⠱⢈⡿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⢳⡏⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣿⣿⢷⢢⡿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠙⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣻⣿⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡿⣼⣾⣶⣿⣸⣷⣿⠘⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣾⣯⣷⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣾⣰⣤⢀⠀⣀⣯⡟⣿⢿⠘ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⢿⡶⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣿⣦⠀⠀⠀⠋⠿⣾⣠⢏⣿⣿⣾ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠏⣿⣿⣿⣾⡹⣱⣦⢿⠻⠛⠛⠙⠉⠉⠉⠉⠉⠉⠉⠉⠁⡿⣾⢐⢂⢁⠋⡿⣿⣿⢿⣿⣶⣴⣠⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠟⣿⣿⣿⣿⣿⢯⣓⡸⢐⣿⣼⣤⣤⣤⣤⣤⣤⣤⣥⣯⣿⣿⣿⣿⣹⠀⢋⡿⣶⣤⣤⣀⢀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠟⣿⣿⣿⣿⣿⣿⣿⣿⢧⡸⡜⠃⣿⠐⠀⠀⠀⠀⠀⠁⠃⡿⣿⣿⣿⣿⡜⠁⡿⣾⢀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⢳⡥⠐⣯⣿⠈⠀⠡⠂⠌⠃⢁⡯⣿⣿⣿⣿⣄⣾⡇⣿⢸⠀⠉⠉⠛⠿⣿⣿⣿⣿⣿⢰ ⠀⠀⠀⠀⠀⠀⠀⠋⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢞⣣⡜⠣⡅⣿⢻⡉⠉⢩⡏⢿⡿⣿⣿⣿⣿⢠⠀⠀⡄⣿⢿⣿⣿⣿⣿⣿⣿⢟⡿⣶⣠⠀ ⠀⠀⠀⠀⠀⠋⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡻⣖⢭⡜⠣⠀⣿⣽⣷⣾⣷⣿⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⣿⣿⣿⣿⣶⣴⣤⣀⢀⠀⠀⠀⠀ ⠀⠀⠀⠁⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡼⣻⢇⣯⠱⠀⣿⣿⡴⢋⢡⣶⣿⣿⣿⣿⣿⢠⠀⠀⠀⡇⣿⣾⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠁⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣟⣳⢟⣱⢾⡱⠌⡆⣿⣻⢆⣿⡙⣯⣿⣿⣿⣰⠀⠀⠀⠀⡅⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠃⡿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣟⣯⡽⡿⣾⢋⠤⠙⣌⣲⢌⠡⡄⠏⣌⣷⣿⢨⠀⠀⠃⢼⠈⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⡿⣿⣿⣿⣿⣿⣿⣿⣷⣞⣿⣟⣿⣿⣿⣿⣾⣿⢯⣳⣟⣿⣿⢻⣰⠱⢂⠀⠀⠀⠀⣤⣷⠿⡽⢻⠐⠀⡀⠤⠈⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠃⣿⣿⣿⣿⣿⣿⣽⣿⣽⣿⣯⡷⣿⣿⣿⣿⣟⢾⣻⣿⣿⣿⣾⣚⡠⠐⠡⢂⣄⠦⠙⠀⠀⣄⣧⣿⠻⡀⠐⢃⠄⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⡇⣿⣿⡿⣿⣿⣿⣿⣿⡷⣻⣽⣿⣿⣿⣻⣿⡷⣟⣿⣿⣿⣧⢄⠠⢀⡀⠀⠠⢀⠀⡀⠤⠀⠀⠀⣆⣿⣿⠻⠌⠠⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⡇⣿⢿⣿⣿⣿⣿⣿⣟⣾⡿⣟⣿⣿⣿⣿⣟⣯⡿⣿⣿⢢⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣿⣼⣦⢤⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⡇⣿⣿⣿⣿⣿⣿⣹⣿⣏⢧⣿⣿⣿⣿⡼⢿⣿⣿⣼⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢿⠈⠀⣇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⡄⣿⣿⣿⣿⣿⣿⣯⣳⣽⣿⣿⣿⣿⣷⣟⣿⣿⣶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢿⠘⡃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣷⣿⣿⣿⣾⡿⣟⣯⡷⣿⣿⣿⣿⣯⣿⣿⣻⣀⠡⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⠷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⡄⣿⣿⣿⣽⢾⣳⣽⣿⣿⣿⣿⢾⣿⣿⣼⠻⣈⠱⠏⠿⣰⠔⠊⠟⠶⡴⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⡄⣿⣿⡷⣏⣽⡿⣿⣿⣿⣿⣟⣿⡷⣝⠻⣿⣿⣿⣶⣜⣶⡄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣇⣿⣳⣞⡿⣿⣿⣿⣽⣻⢿⣿⣿⣿⣾⣽⢦⣈⡑⠄⠈⠀⠁⠀⠃⠀⠛⠟⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⡀⣿⣻⣿⣿⣿⣷⣟⣿⣿⣿⣿⣶⣛⢬⡘⢣⢆⡈⠱⠆⡉⠱⡏⠟⡿⣳⢎⠉⢂⠻⠈⠀⠀⠀⠀⠀⠀⠀⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣇⣿⣿⣿⣿⣿⣿⣿⣿⣶⣛⣸⢽⣲⡜⢛⢣⣜⡛⢧⡿⡿⣿⣻⡽⢻⢾⡍⠢⣌⢷⠉⠀⠀⠀⠀⠀⠀⡏⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣆⣿⣿⣿⣿⣿⣿⣻⣧⠿⣟⡻⢯⣏⡱⢿⣏⡿⣿⣿⣿⣿⣿⣿⣿⢿⢾⠴⠀⠀⡀⢀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣷⣭⡟⢯⣾⣝⣷⢯⡳⣽⡿⣿⣿⢿⣿⣿⣿⣿⣞⣈⠰⢀⠀⠀⠀⠀⠀⠀⠀⠋⠟⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⣄⣿⣿⣹⢶⡹⣽⢟⣭⡾⣛⣿⡿⣿⣿⣿⣿⣿⣳⣼⠺⢇⣘⠴⠀⠀⠀⠀⠀⠃⠟⡿⣿⣿⣿⣿⢿⠻⠛⠛⠙⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⣿⣻⣜⠿⣳⣟⢯⢷⣯⣿⣿⣿⣯⡿⣿⡿⣙⢥⠣⠀⠀⠀⠀⠀⠀⠃⣿⣿⣿⣿⣿⣷⣮⣰⣿⣿⣿⣿⣿⠿⠈⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⣿⣻⣽⡾⣟⣳⣿⣷⣻⢷⣿⣿⣿⢿⡩⡐⠠⠀⠀⠀⠒⠖⠶⣶⣿⣿⣿⣿⣿⡛⠟⣶⣿⣿⣿⣿⣿⣿⢰⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⣿⣿⣻⣿⣿⣽⡳⣯⣷⣜⣿⣿⢿⡙⠀⠀⠀⠀⠀⠀⠀⠀⡀⣧⣿⣿⣼⢦⢁⡿⣿⣿⣿⣿⣾⣠⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⣿⢿⣽⡻⢧⢻⣾⣏⡷⣿⣿⣿⢿⠹⢀⠀⠀⠀⠀⠀⠀⡀⣷⣿⣿⢿⣿⣿⣿⣿⣿⣰⢀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣷⣿⠻⠙⠈⠀⠍⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡶⡟⣿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣾⡏⣿⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⣽⣞⣿⣿⣿⣻⣟⢿⣿⣿⣿⣿⣻⣈⣾⣽⣇⢿⣿⣽⠃⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⣿⣿⣿⣼⡟⣿⡟⣿⣿⣻⣿⢿⣹⡽⣾⣿⢚⢧⡘⡿⣿⣿⣼⡯⣿⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣷⣿⣿⣞⣿⣷⢿⣿⣿⢿⣿⣽⠿⣎⡷⣏⢧⡞⣿⣿⣽⠟⣿⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⣿⣿⣿⣟⣿⣿⣿⣿⣽⠿⣇⢮⣳⢶⣟⣿⣿⣿⣼⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣿⢿⣿⣿⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀