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⠀⢸⠁⠛⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⡟⠀⠀⠐⢤⡈⠑⠢⢄⡀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢀⡇⠀⠀⠀⠀⠈⠳⢦⣄⡈⠑⠢⣤⠔⠁⣀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢸⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⡲⢤⣀⠀⠈⠈⠳⣄⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠀⠉⠲⠤⠄⠈⠑⢄⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠈⢆⠀⠀⣀⡴⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢳⣀⣨⡻⠯⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠁⠀⠀⠄⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠘⣄⠸⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠾⠃⠘⠣⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⢀⣠⠤⠖⠒⠒⠛⠒⠒⠲⢤⣀⢣⠀⠀⠀⠀⠀⠀⠈⠛⠉⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣇⠀⠀⠀⠀⠂⠈⢹⠀⠀⡀⠈⣳⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢀⠀⠀⠀⣰⡟⠀⣴⡏⢀⡿⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠘⡆⠀⡀⣠⠞⢉⡇⣸⣼⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠘⡆⠀⡀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠙⢦⠀⠀⠀⠀⢳⠀⣴⡃⠀⡼⠀⣿⡇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢱⡄⢱⡀⠀⡾⠀⡴⠒⠈⠒⣆⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣸⡄⠈⠉⠛⠃⢸⣼⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢧⠀⢣⠀⡇⠀⠀⠀⠀⠀⣽⠀⠀⠀⠘⠦⣀⡀⠀⠀⣠⠇⢀⡤⠞⢁⣤⠞⢛⣷⣶⣾⠋⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢸⣤⠀⠀⡇⢤⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠉⠙⣻⡵⠚⢁⡤⠾⣿⣇⣴⣿⡏⠙⣿⣦⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⢧⣈⡳⠶⠚⠁⠀⠀⠀⠀⠀⢀⣠⠴⠋⣁⡤⠚⠉⠀⠀⣻⣿⡛⣿⣿⣿⣿⣿⠿⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⣀⡀⠉⠉⠙⠓⠒⠒⠲⠖⠚⢉⣠⠴⠚⠁⠀⢀⣠⣴⡿⠛⣿⣇⠙⢿⣿⣿⣿⣧⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣍⣉⠉⠈⠙⠓⠒⠒⠒⠛⢁⣀⣠⣤⣶⣾⠿⠛⠁⠀⠀⡼⠛⠛⠛⠛⠛⢿⡿⠁⣰⡇⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢠⡏⠀⢸⣿⠙⠛⠛⠻⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⢰⠃⡐⠀⠀⠑⠢⣄⠁⢺⣿⠇⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⠁⠀⠀⡼⠀⢨⠇⣸⣽⣿⣶⣶⣶⣶⣶⣦⡀ ⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡌⠀⠀⢠⠃⠀⡞⢰⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢻⣷⣦⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠇⢠⠁⠀⠀⡟⠀⡼⢠⣾⣏⡽⠻⣿⣟⣫⣿⣿⣷⠋ ⠀⠀⠀⠀⠀⠀⠠⡁⠀⡀⠀⠀⠻⣿⣟⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⢻⠀⠈⠀⠀⣸⠁⣰⠁⣸⣿⣏⠀⢠⢿⣿⠏⠁⢻⡇⠀ ⠀⠀⠀⠀⠀⠀⠀⢳⡀⠁⠀⠀⢀⠈⢻⣿⣦⣀⣀⣴⣶⣄⠀⠀⠀⠀⢀⣤⣤⣼⡆⠀⠀⢀⣇⡴⠃⣼⣿⡟⣻⠀⠎⢹⣷⠿⣷⠘⡇⠀ ⠀⠀⠀⠀⠀⠀⠀⢀⣹⣄⡀⠠⠏⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣇⠀⢀⡼⢋⣴⣿⣿⣿⣿⣧⣄⣼⣿⣿⣾⣏⣼⣷⠀ ⠀⠀⠀⠀⠀⠀⡜⣡⣶⣾⣷⣤⣴⣿⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⠀⢈⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣽⣿⣿⣿⠀ ⠀⠀⠀⠀⠀⠀⡇⣿⠞⠋⢹⣿⣿⠉⠉⠉⠉⠉⠙⣿⣿⡏⠉⠉⠉⢉⣿⣿⠋⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⣿⣿⡏⠈⢿⣿⠀ ⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⢸⢸⡿⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⠀⢸⣿⠀ ⠀⠀⠀⠀⠀⠀⢸⣼⡆⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣼⡇⠀ ⠀⠀⠀⠀⠀⠀⣤⣿⠇⠀⠸⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⠇⠀⢻⣿⡆

⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⢀⣤⣶⣪⣨⣈⡀⠐⠲⠋⠛⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠸⣿⣿⣿ ⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⣶⣪⣿⣿⣥⣿⣿⣿⡚⠁⠐⢓⡺⣷⣶⣽⣮⣢⡀⠀⢀⣀⠀⢠⡤⢸⣧⡀⠀⠀⠀⠀⠀⠈⢿⣿⣿ ⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡽⢛⡿⠟⣻⡟⣻⣿⣿⣿⠟⠛⠛⡛⢷⣬⣿⣿⡻⡧⠀⠀⠀⠀⠈⠉⢸⣿⣷⡄⠀⠀⠀⠀⠀⠸⣿⣿ ⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣴⣻⣿⣿⣿⡏⣹⡿⠿⠉⡧⢠⠄⢠⣥⣿⣿⣯⣻⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀⠀⠀⠀⠀⢹⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠉⠀⠈⠙⠻⢷⢔⣩⣙⣻⣷⣾⣿⣿⣗⡱⠻⡹⡽⣿⣦⠀⢸⣿⣿⣿⣿⣿⣷⣶⣦⣴⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠈⠙⠺⢿⣿⣿⣿⣿⣿⣿⡔⠙⠾⠞⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⡹⣣⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⡓⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⢅⠀⡈⣯⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠈⠫⠿⠷⠏⠀⠀⠀⠶⣶⣶⣶⠦⠦⠌⢻⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠑⠹⠟⠹⠆⠀⣼⣿⣿⣿⣿⣿⣿⣿⡟⢓⢹⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⢿⣿⢿⡷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⢄⠀⠀⠀⠀⢠⠍⠀⠀⠀⠀⣠⣿⡿⠿⣿⣿⡿⣻⣿⣮⡗⢯⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠋⠃⠧⠀⠠⠤⠄⠀⠀⣴⣿⠿⢶⣎⡎⢏⢾⡩⣿⣮⣿⢿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣄⠀⠀⠀⠀⠀⡀⢀⡤⣿⣿⣃⡛⢜⣟⣯⣞⣥⣞⠪⢾⣾⣼⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⣏⢆⠀⠀⠀⠀⠁⠉⢰⣿⣟⢇⣆⠀⢿⣶⣷⡿⣺⢄⠌⢯⠿⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡔⢿⠁⠀⠀⠀⠀⠀⠀⠻⣾⠹⡇⠀⢞⣛⣾⣟⢓⢬⣙⠝⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢣⠟⠀⠀⠀⠀⠀⠀⠀⢠⣟⠓⣿⢰⢋⠻⣷⡝⠟⣧⡤⢈⠠⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⢺⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡷⣿⠐⣸⡻⣿⡞⠋⣹⣿⣆⢟⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠸⡁⠁⢀⠃⠀⠀⠀⠀⠀⠀⠀⢀⣔⣿⡓⣧⣀⣭⣿⣽⣿⡐⣺⣬⢿⣯⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢹⡇⡠⠃⠀⠀⠀⠀⠀⣀⠔⢊⢡⡿⡯⣽⡌⢾⣹⣿⢹⣿⣿⣿⣿⠟⣛⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⢸⡯⠀⠀⠀⠀⣀⠔⢊⣤⡾⣿⣿⣟⣮⣿⣿⣿⢯⣿⣷⣿⣿⣷⡿⣷⣬⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⣤⡠⠄⢊⢤⣾⣿⣿⣿⣿⡿⣿⣯⣭⣗⢙⢿⣿⣿⣾⣯⣿⣿⣿⣽⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⢁⢄⣾⣷⣿⣿⣽⡿⣟⣼⣿⣧⣽⣿⣿⣿⡷⡯⠿⣿⡿⢿⡾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣾⣿⡿⣟⣿⣟⣧⣾⣿⢿⣿⣿⡿⡿⠿⢵⢾⣾⣿⣭⣻⣿⣯⣶⢿⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢟⣭⣾⡿⣿⢛⣿⡟⡫⠙⠩⠂⠈⠀⢩⣷⣿⡝⣮⣝⢿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⣤⣤⣼⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣧⣤⣤⠀⠀ ⠀⠀⠿⠿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⠿⠀⠀ ⠀⠀⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⠀⠀ ⠀⠀⠛⠛⢿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⡿⠛⠛⠀⠀ ⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀ ⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄ ⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⠘⠛⣿⣿⡟⢻⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⡟⢻⣿⣿⠛⠃ ⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀ ⠀⠀⣿⣿⡇⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢸⣿⣿⠀⠀ ⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀