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⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⣀⣠⣀⣉⢻⣷⣾⣯⣰⣶⣿⣿⣿⣷⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠐⣿⣯⡍⣿⣏⠉⠙⣿⣯⠉⠉⣭⣿⣶⣦⣀⢀⣶⣶⣤⣴⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⣼⣿⣿⡿⠛⣿⠿⣿⣿⣿⣿⡟⡈⠙⠿⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠠⣿⣿⣿⣿⣿⢀⣠⣿⣿⣷⣤⣿⣿⣟⣛⣻⡍⠈⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢉⠡⠀⠘⣽⣦⣀⢠⣽⣿⣧⡌⠂⢵⣮⣿⣿⣹⣿⣿⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣻⣿⣿⠄⢠⣼⣿⣿⣧⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣾⣿⣿⣧⠊⠀⣤⣿⣿⣿⠿⣿⣿⣿⣿⡀⠌⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠈⠻⠟⠘⠻⠟⠻⠟⠋⠉⠻⠟⠙⠻⠿⠛⠉⠀⠻⠿⠛⠻⠿⠿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣧⢂⢲⣿⡿⣯⡴⠶⠞⣯⣹⣿⣿⣞⣾⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⣧⡿⣿⣿⠿⠀⠔⠒⠛⣿⣿⢻⣿⣿⣿⣿⣿⣿⡿⢙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣟⣉⣉⣉⣙⣉⣉⣹⣉⠉⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢿⣿⣿⣿⣷⠹⢿⣶⣤⣴⣶⠿⢿⣿⣸⣿⣿⣿⣿⣿⣿⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢤⣿⡟⢉⣻⡛⣿⣛⡟⢿⣇⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢻⣿⣿⣿⡤⢈⠁⠉⠁⠀⠀⣲⣿⣿⣿⣿⣿⡿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ 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⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣟⣿⡿⣙⣲⣽⣫⣿⣿⣿⣿⣿⣿⣧⢀⣀⣠⣶⠟⠁⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⣻⣻⣿⢿⣷⣬⣝⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠴⡄⠀⠀⢰⣶⣦⣿⣷⣿⣿⣿⣿⣿⣿⣿⣛⣹⣾⣧⣄⣀⣠⣴⣿⣿⣾⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀⠀⠙⠛⠻⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⣾⣿⠟⠛⠻⠟⡿⠉⠀⠀⠀⠀⢀⣹⣿⣧⣀⠙⠻⠛⠛⠿⣿⣟⠛⢿⣿⣿⣷⠂⠀⠀⠀⣤⣿⣿⡿⢟⣁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⣿⠿⣿⢿⡿⣟⡿⣛⠿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠐⠲⣶⣿⣿⡿⢿⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⢰⣧⠙⠿⢿⣿⣷⣤⡀⠀⠴⣦⣄⠀⠛⠿⣿⣟⠀⠀⣴⣾⣿⣿⠟⣩⣾⣿⠿⠛⢉⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣿⣶⣿⣷⣿⣾⣷⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠉⢙⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⣀⣀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠂⠀⠀⠈⢷⣮⣽⡖⠀⠀⠙⢲⣤⡀⡘⣻⣶⣎⣿⣿⠏⣠⣶⡿⠋⢀⣠⣶⣿⠿⠛⠁⠀⠼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠿⠟⠛⡛⠋⢻⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⣀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠙⠻⢿⣯⣍⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠀⠀⠀⠶⣄⣀⡈⠙⠻⢷⣤⣀⢸⡿⠟⠃⣻⣿⣿⡟⣡⣾⡟⠋⣀⣶⣿⣻⣭⣤⣴⣤⣤⣦⣼⡿⠟⠻⠋⠉⠉⠥⠄⠠⠀⠀⠐⠚⠛⠛⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡀ ⣀⣴⣿⠿⢁⣀⣤⣤⣿⣿⡿⣽⣿⣿⢯⣽⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠈⠙⠛⠿⢷⣤⣄⣀⠀⠀⠃⠠⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⠦⠼⠿⠿⣷⣄⣀⣿⣿⣿⣤⠾⠿⠿⠿⠟⠛⠛⢉⡉⠁⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢉⣴⣿⣥ ⣿⢟⣯⣴⣿⣿⣿⣿⣿⡟⢱⣾⣿⣿⣿⣿⢈⡱⠈⢾⣿⢏⡛⢿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⡄⠀⠀⠂⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠞⢋⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠄⢩⠉⢹ ⣿⣿⣿⣿⣿⣵⣿⣿⣯⢀⣾⣿⣿⣿⣿⣇⠆⣂⠡⣿⣿⠂⡀⠈⣽⣿⣿⣿⣿⣿⣦⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠝⠿⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢐⠊⠁⠚⠃⠰⠿ ⣿⣿⣿⣿⣿⡿⢋⣠⣬⣿⣿⣿⣿⣿⣿⡷⣘⠄⠲⣿⣧⠐⡄⢂⣿⣿⡇⠀⠹⣿⣿⡄⠙⠷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠚⠓⢦ ⠀⢹⣿⣿⠏⣀⣾⣿⣿⠿⣿⣿⣿⣿⣿⣟⠨⢌⡱⣿⣿⠐⡌⢦⣾⣿⡇⠀⠀⢿⣿⡿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⣄⣁⠶⣿ ⢀⣼⣿⡿⠀⣴⣿⡿⠃⢜⣻⣿⣿⣿⣿⣧⢿⣦⣽⣿⣿⠀⢲⣾⣿⣿⣷⡀⠀⠹⣿⣧⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀ ⣿⣿⠟⠃⠰⣿⡿⠇⠁⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢹⣿⠀⠸⣿⣷⣀⠐⣻⣿⡄⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠎⠁⠀⠀⠀⠀⠀⠂⠀⠒⠒⠀⠀⠴⠂⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠲⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⢠⠈⠀ ⠟⠋⠀⠀⠀⠼⢁⡰⣰⣾⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡗⠋⠙⣿⣿⠀⣹⣿⣧⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢀⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⢀⣀⡸ ⠀⠀⠀⠀⢠⣶⡷⣘⣿⣿⡟⢠⣿⠏⠈⣿⡷⣿⣿⣿⡟⣿⣿⣿⠁⠀⠈⣿⠁⠀⢹⣿⣿⣄⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠈⠀⠀⠀⠀⠀⠀ ⠃⠀⠀⠀⢸⡟⠉⠻⣿⡿⠀⣼⡟⠀⠀⢾⣿⣿⣿⣿⠸⣿⣿⡏⠀⠀⢀⣿⣾⠏⠀⢹⣿⣿⡄⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⣠⢀⠀⣀⡀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⣿⡇⣠⣠⣼⣿⣿⣿⣿⣰⣿⡿⠂⠀⠀⢾⠟⣿⣆⣀⣼⣿⡿⣿⡄⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠈⠀⠁⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠ 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