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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣉⠿⣿⣿⣿⣫⣼⣱⢎⡿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣌⢳⣻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⣮⣽⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡆⢧⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣜⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠿⠟⣫⡄⣾⣿⣿⣿⡀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣷⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣀⣸⣿⣵⣿⣿⡿⢿⣥⠨⡝⣰⣯⣿⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢋⠉⣿⣿⣿⣿⠀⠈⢻⣆⠡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⡘⢌⣿⣿⡇⣠⣴⣤⣟⡼⡿⠟⣹⣯⣿⣿⣿⣿⣿⣿⣿⠫⢍⠹⡘⠿⢿⠿⣛⢛⡟⣻⢛⡟⡻⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡐⢎⣽⣿⣿⣿⣿⢟⠩⣀⣰⣾⣿⡿⣿⣿⣿⣿⣿⣿⡏⠖⡬⡑⢌⡃⢎⡘⡌⠶⡸⣥⣏⣶⡙⣎⠾⣍⠿⢿⣟⠻⢿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣐⣷⣾⣿⣿⣿⡿⢊⣴⣾⡿⣛⣦⣹⣿⣿⣿⡟⣏⠱⢌⠓⡌⠔⢣⡙⢆⠹⣜⡳⠱⣍⠞⡭⢻⢎⡳⣿⣛⡿⣜⢢⠃⡽⣿⣮⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠜⣛⣿⣿⣿⣿⣦⣧⣿⣟⡰⢍⣾⣿⣿⣿⡟⣉⢛⡱⢋⠴⡘⢌⠢⡉⢆⠱⣌⡖⡩⢆⡹⢰⡉⢦⡑⣿⣿⡛⡜⣦⣝⠰⡌⡝⣿⣿⣞⣿⣿⣿⣿⡗⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠎⡄⢻⣿⣿⣟⣿⣗⠛⡡⢚⣼⣿⣿⣿⣿⡓⢬⣛⠴⣉⠖⡩⢦⠳⣌⢂⠒⡄⠻⣅⠳⡼⡡⢜⢢⡙⣿⣿⡵⡹⢬⣍⢳⣛⡛⣖⢻⣿⣿⣟⣻⣿⠃⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠄⣃⢿⣿⣿⣿⣇⠈⡔⣻⣿⣿⣿⣿⣿⡙⢞⣛⠳⡜⢮⡱⢎⣱⢢⠌⢒⠨⡱⢌⠳⣰⡙⢮⢥⣻⣿⣿⣷⣙⠶⣌⠳⣌⣷⣎⡳⢎⡿⣿⣿⣿⣆⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠐⣌⢚⣿⣿⡋⠔⡨⣽⣿⣿⣿⣿⣿⡇⠜⡌⣎⠳⣍⡳⣝⢮⡜⣧⠊⠄⢣⡗⡌⢳⠰⣉⠞⣾⣿⣿⣿⣿⣿⡜⣬⢳⣌⣿⣿⣱⢋⡶⣽⣿⣿⣭⡄⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠒⡌⡌⣿⣿⡱⢰⣿⣿⣿⣿⣿⣿⣿⠆⠉⡔⢌⠳⣌⢳⡭⣷⣾⡱⢈⡘⠤⠓⡌⡡⠓⡌⢸⢸⣿⣿⣿⣿⣿⣷⡮⣝⣾⣿⡿⣌⠧⣝⣾⣿⠟⢿⣦⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢣⠜⡰⢹⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡂⠐⢠⠊⡕⣎⡳⣽⡻⢿⡑⢂⠰⣁⢣⠐⡡⠑⠨⠖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣬⢛⣿⣿⡟⠀⠀⠉⠁ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠚⡔⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡁⢆⠱⣘⢦⢻⣽⡿⣭⡓⢌⠢⡍⢂⠡⢀⠁⠒⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣆⡻⣿⣿⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣌⠳⣌⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⡳⣘⢎⡷⣿⣟⢶⡙⢆⢣⡑⢌⡐⠂⡌⢼⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⢷⣣⠽⣿⡟⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡷⣎⡟⡤⢋⠴⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡺⣼⣽⣿⣮⢳⡹⢌⡲⣷⣦⢘⡰⣘⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣛⢾⣿⡇⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣜⣿⣇⢫⠌⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣋⠧⡓⡌⠜⣿⣦⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣶⢫⣏⣿⡇⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣼⣹⣿⢸⡸⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢲⢨⠱⣌⢲⠡⡜⢨⠔⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⡗⣮⣿⡇⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢧⡳⣽⡶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⢯⡓⡬⣑⠎⡵⣚⢧⡗⡘⠦⣙⠴⣨⢟⡻⢿⡿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣯⣿⡞⣥⣿⡇⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣗⣣⢟⣿⣿⣿⣳⣿⣿⣿⣿⣿⡳⣽⢮⣳⡜⣱⡹⢞⡵⣊⠖⣌⢳⡳⡌⢧⢣⢏⡖⣣⢞⡹⢶⣙⢮⣿⡟⣾⣿⣿⣿⣾⣿⡝⣶⣿⡇⠀⠀⠀⠀⠀ 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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣼⡛⢧⣏⢞⣿⣿⣧⣛⣮⠷⣽⢾⣽⣫⢷⣏⡷⣽⣚⡟⣾⣱⢯⣳⣿⣿⡹⣞⡵⣿⡜⣎⣿⣿⣟⠼⣣⢾⣿⣿⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡽⢶⣹⢳⣎⢯⡖⣿⣿⣷⣞⣻⣽⣻⢾⣽⣻⢾⣽⣳⢯⣞⣷⣿⣿⢿⡹⢶⡝⣮⡽⣿⡞⣥⢫⣿⣿⣣⢓⢎⣿⣿⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣹⢳⣯⣳⢮⡳⡽⢮⡽⣻⣯⢷⣳⢯⣟⡾⣽⣻⣞⣿⡿⣟⢯⡻⣝⢮⡝⣧⣟⠶⣽⣻⣿⢒⣯⣿⣿⣇⢏⠦⣻⣿⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⢫⡟⣼⢣⡿⣽⡝⣧⡻⣵⢫⣿⣿⣿⡿⣿⣷⣿⣾⢿⣹⢞⡭⣷⢫⣞⡝⣧⢻⣟⢶⡹⢷⣯⣿⣿⣿⣿⢎⡳⢿⣿⡆⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣏⢷⡹⢎⡷⣹⡷⣿⣷⣏⢷⣻⣿⣿⣿⣿⣿⣿⣿⣿⡳⣝⣮⣽⣟⡳⢮⡝⣎⢷⣻⢧⡻⢥⣻⣿⠏⣿⣿⡧⢃⢞⣿⡇⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡝⣮⠹⣭⢚⡵⣻⢿⣿⣾⣯⣷⣻⣿⣿⣿⣿⣿⣿⣯⢷⣿⣿⡿⣭⢫⣝⡻⣼⢣⡟⣎⢷⣩⣿⣿⠀⠘⣿⡷⡉⢾⣿⡇⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡝⡶⡹⣜⢫⢼⡳⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⠼⣱⡓⢮⡳⣭⢳⡝⣮⣓⢧⣿⡇⠀⠀⣿⣷⢁⣻⣿⠃⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣏⠷⣱⣮⠓⣎⢷⡹⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣷⣣⢿⡱⣯⢷⡹⣎⢷⣛⢶⡹⢾⣿⠃⠀⣸⣿⢋⠴⣿⣿⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣎⡟⡴⣣⠛⡬⣏⢷⣯⢿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣽⣾⡿⣋⠶⣩⢗⣏⡷⣹⢮⣝⡮⣝⣿⡿⠀⠀⢹⣿⢌⠢⡽⢿⡇⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⢛⡴⢧⡛⢴⣫⠾⣽⣯⣿⣿⣿⣿⣯⠀⣸⣿⣿⣿⢏⡳⣍⢾⣹⢮⡝⣾⡱⣏⢾⡱⣿⣿⡇⠀⠀⠘⣿⣎⡱⢎⣿⣧⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣎⠷⣚⣧⡙⢶⣭⣛⡷⣯⣿⣿⣿⣿⣿⣴⣿⣿⣿⡟⢮⡱⢞⣭⢳⡏⣾⡱⣟⣮⢳⣿⣿⡟⠀⠀⠀⠀⢻⣷⣽⣿⡿⣿⣧⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣙⢧⢣⡝⣦⡳⣭⣿⡽⣿⣿⣿⣿⣿⣿⣿⡿⢯⣙⢮⡵⣫⣜⣧⣛⢶⡻⣝⢶⣫⣿⡿⠁⠀⠀⠀⠀⢸⣿⣿⣿⣇⣿⣿⠂⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡹⣎⢧⡹⣎⠷⣏⣷⢿⣿⣿⣿⣿⣿⣟⡯⣝⢧⣹⢦⠽⣱⢫⢶⡹⣎⠷⡭⢶⣿⡿⠁⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠋⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡝⣮⢏⡷⣹⡟⣽⢯⣿⣟⣿⣿⣿⣿⡿⡼⣍⢮⢳⣏⡻⡵⢯⡳⣝⣮⠻⣵⣻⣿⠁⠀⠀⢠⣾⡿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡞⣵⢻⣜⡳⣿⣹⡿⣿⣾⣿⣿⣿⣿⢳⡽⣘⣏⢶⣡⠿⣝⢧⡻⣝⢾⣹⣾⣿⠃⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀ ⠀⠀⠈⠂⠈⠂⠛⠃⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣽⣚⠿⣮⢽⣳⢯⣿⢿⣽⣿⣿⣿⢧⡻⢴⢫⡜⢯⡞⣟⢮⡳⣝⢾⣯⣿⣿⠃⠀⠀⠀⡀⢀⣽⣿⣿⣿⣯⡅⢀⡀⠀⠀⠀⠀

⠀⠀⠀⠀⡀⢀⡅⣯⣿⣿⣿⣽⢀⡀⠀⠀⠀⠃⣿⣿⣯⢾⣝⡳⢮⣟⡞⢯⡜⢫⢴⡻⢧⣿⣿⣿⣽⢿⣿⢯⣳⢽⣮⠿⣚⣽⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠒⠃⠛⠂⠈⠂⠈⠀⠀ ⠀⠀⠀⠀⠁⠟⣿⣿⣿⣿⣿⢿⠾⠀⠀⠀⠃⣿⣾⣹⢾⣝⡻⢧⣝⠿⣡⢶⣏⣘⡽⢳⣿⣿⣿⣿⣾⣿⡿⣹⣿⡳⣜⢻⣵⡞⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠃⣿⣿⣿⣿⣿⣿⡿⣾⢠⠀⠀⠁⣿⣻⣵⠻⣮⣝⡳⢯⡵⡻⣏⢳⢮⣍⡼⡿⣿⣿⣿⣿⣟⣿⢯⣽⡟⣹⡷⢏⣮⡝⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠋⣿⣿⣿⣿⣿⣾⣠⠀⠀⠀⠀⠁⡿⣿⢶⡭⠷⣎⡹⢶⢫⣱⠽⢦⣹⢧⣝⡯⣟⣿⣿⣿⣿⣿⢿⣷⣏⠷⣎⡹⢧⣎⡹⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠂⣿⣿⣇⣿⣿⣿⢸⠀⠀⠀⠀⠁⡿⣿⣫⢶⣝⡻⢶⣛⣧⣜⣫⡵⢮⣙⢯⡿⣿⣿⣿⣿⣿⣿⣿⡽⣿⣭⡳⣦⡝⢣⢧⣙⣾⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⣧⣿⡿⣿⣽⣷⢻⠀⠀⠀⠀⡟⣿⣿⢳⣮⣟⡱⣾⡏⢳⣭⢞⡱⢮⡟⣿⣿⣿⣴⣿⣿⣿⣿⣿⣯⡷⣛⣭⢶⡙⣧⣚⠷⣎⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣧⣿⢎⡱⣎⣿⠘⠀⠀⡇⣿⣿⡱⢾⣏⡱⣾⡝⢮⣹⢾⣍⡳⢏⣿⣿⣿⣸⠀⣯⣿⣿⣿⣿⣯⣽⠾⣫⢴⡛⢧⡴⢛⣧⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⡇⢿⡽⠢⢌⣿⢹⠀⠀⡿⣿⣝⡮⣝⢮⣹⡷⣏⢗⣩⠶⣋⡿⣾⣽⣿⣿⢿⠿⣿⣿⣿⣿⣿⢿⣯⢷⣏⡬⠛⣣⡴⡟⣎⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣿⣿⠴⢋⣿⣸⠀⠃⣿⢾⡹⢶⣛⢷⣎⡹⢷⣯⡱⢿⣣⣷⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⡹⢷⣎⠓⣮⣱⠷⣏⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠃⣿⣻⢁⣷⣿⠀⠀⡇⣿⢧⣓⣮⡝⢳⣭⡳⢮⡓⣱⠼⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⡳⢼⢫⣜⡹⡶⡝⣿⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⡇⣿⢾⡉⡷⣿⠘⠀⣿⣿⣩⢷⣎⡟⢣⣼⡻⣝⢫⣭⡿⣿⣿⢷⣯⣿⣿⣿⣿⣿⣿⣻⣷⣯⣾⣿⢿⣻⡵⢚⣭⠹⣮⡝⣿⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⡇⣿⢞⢃⡧⣿⣿⠏⣿⣻⢥⡻⢧⣻⢷⣎⡝⢮⡳⣟⣽⣮⣝⡳⣿⣿⣿⣿⣿⣿⣿⣿⣻⢷⣏⣷⣿⡷⣹⡷⢎⡹⢷⣏⣿⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⡆⣿⢿⡳⢎⣿⣿⣿⣿⣯⢷⡹⢶⣟⢻⣧⡝⣞⢫⣷⡭⢞⣹⢿⣾⣿⣷⣿⡿⣿⣿⣿⢫⣵⡻⣧⡝⣽⡿⢣⣼⡟⢫⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣿⣻⠦⢏⣇⣿⣿⣯⢒⣿⣻⣽⠶⣟⣧⡝⢮⣝⡻⢯⣟⡿⣿⣞⣻⣽⡾⣟⢯⣳⢷⣯⣻⡽⢮⡽⡳⢮⣳⣯⢳⣹⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣿⣿⢎⢓⣣⣿⣿⢫⣥⡞⣿⡽⣮⡝⢶⡹⢿⣿⣿⣷⣞⢯⣳⣽⢾⣻⣽⢾⣻⣽⣻⣞⣷⣿⣿⡖⢯⣎⢳⣹⢶⡽⣿⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣿⣿⢾⣣⠼⣟⣿⣿⣎⡜⣿⡵⣞⡹⣿⣿⣳⢯⣱⣾⡟⣚⣽⡷⣏⢷⣫⣽⢾⣽⠷⣮⣛⣧⣿⣿⢞⣏⢧⡛⣼⡟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣿⣿⢏⢣⣜⡞⣷⣿⣿⡞⢎⣲⡷⣛⡶⢟⣭⡹⣶⡝⢳⣧⣛⣭⡶⣛⢭⣻⡵⣞⡯⢗⣫⣵⡞⣯⣷⢿⡽⣞⣣⣟⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣿⣿⢲⢳⣎⡝⣧⣿⣿⣿⣾⣶⣷⣻⣼⡟⣎⡹⣯⡝⣫⢧⢧⣛⣭⡞⣏⡜⣫⣝⢫⣱⢻⣦⡝⣚⣹⣷⣿⣿⢿⡿⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣯⣿⣹⢦⡝⡣⣗⣿⣿⣿⣿⠒⡵⠳⣍⠾⢭⡽⢮⡹⢽⣫⢭⡻⢦⡝⢎⡵⢎⡱⢳⣭⡝⣧⡜⠿⣎⡙⣧⣿⣿⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⣧⣿⡱⢦⡹⣜⡹⣷⣿⣿⢒⣸⡜⡹⡵⢟⢣⣹⢶⣞⡝⢣⣵⠞⣣⠗⣏⡛⣜⡹⠶⣍⡳⢧⣎⠷⢏⡱⢮⡗⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⡇⣿⡹⢦⠳⣎⡓⣷⣿⣿⣿⠿⣎⡔⡻⡶⣏⡱⣾⣏⡱⣼⡿⡌⢶⡿⣿⣿⣾⡓⢮⣑⡸⢷⣌⢻⣏⡹⠶⣏⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⡄⣿⢻⣥⠳⣌⡽⡳⣯⣿⣽⣿⣿⣵⡚⣵⢺⣽⡞⢫⣱⡷⢎⣋⢵⡎⣽⢳⣜⠳⣌⡹⣜⡻⣌⠷⣋⡱⢲⣟⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣿⣿⣙⠶⣏⡳⢮⣟⣿⣿⡆⣿⣾⡝⣣⡞⣜⢿⣎⡳⡼⣟⢎⡳⢎⡱⠾⣏⡥⡞⡏⣌⠻⣜⢻⢣⣹⡷⣿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⡇⣿⣿⡹⢼⣫⢷⣯⣿⣿⠐⣿⣿⡍⢲⡼⣟⢮⣝⢳⣼⡟⣭⢳⣭⣳⡼⢟⣣⢷⢮⣜⠿⣎⣹⢶⡹⣿⣿⢸⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⡄⣿⣿⣾⡎⣟⡾⣟⣿⣿⠠⣿⣿⣹⡼⣜⠾⣏⣱⡾⣏⠷⣬⡻⣜⡻⣏⡱⣼⡺⣟⢧⢳⣮⣟⣧⡝⣿⣿⠰⠀⠋⡿⣿⣿⣿⢿⠻⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠁⣿⣿⣳⢭⣾⣽⣿⣿⣿⠁⣿⣿⣳⣎⡝⢧⣛⣭⡳⣜⡿⣔⠿⣎⠷⣎⡻⣜⢞⣫⣵⡻⢧⣗⢮⡽⣿⣿⠰⡿⣿⣿⣿⡽⡞⣯⣷⣿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⡇⣿⣼⢻⣧⣞⣿⣿⣿⣿⡇⣿⣸⢾⣋⡵⢎⡥⣼⡾⣏⠳⣭⠳⣧⡙⢯⠹⣼⡮⢛⣥⡿⣳⢯⣜⡻⣷⣿⣿⣿⣿⣿⣿⣿⢿⣛⣮⣿⣿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⡇⣿⣶⡝⣿⣾⣿⣿⣿⣾⡟⣿⢮⣙⢶⡹⢞⣣⡖⢏⢣⢧⡌⡳⢳⣌⠖⣊⡵⢞⡹⣱⡜⣳⢮⣽⡳⣿⣿⣿⣿⣿⣳⣿⣿⣿⢟⣣⣗⣿⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⡇⣿⣥⡞⣿⣯⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⢿⡻⢟⣨⠴⣙⠦⡘⡗⢧⣚⡵⠎⣑⡬⡓⢯⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡶⣽⡳⢧⣿⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⡇⣿⣮⡗⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠔⢨⡜⠡⢲⣌⠱⢨⢲⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⡸⢸⣿⣹⣼⡿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⡇⣿⣏⢫⣶⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⣦⣿⠜⡌⡓⠧⣋⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⠌⢫⣇⣿⣜⡿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⡇⣿⢾⣛⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣘⡰⢘⣦⣷⡲⢌⡹⢳⣮⣿⣽⣼⡺⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠴⢋⡤⡟⣎⡷⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⡟⣿⠽⣣⢷⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⢼⡌⠂⡐⢌⡑⢣⢆⡙⢶⣟⣿⡷⢎⣘⡳⣂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠒⣌⠳⣌⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠁⣿⣿⡻⣆⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠒⠁⢀⠡⢂⡍⠢⢌⡓⣭⡿⣽⢻⢦⣘⠱⢆⡁⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⡔⠚⣇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠁⠉⠀⠀⡟⣿⣿⢛⣬⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠨⠑⡡⠐⢣⣁⠰⢂⡑⢿⡻⣽⡳⣎⡕⠊⢠⠐⡂⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⢹⡰⠜⢣⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣦⢿⠟⣿⣾⣝⠧⣌⡿⣿⣾⣝⡮⣷⣿⣿⣿⣿⣿⢸⢸⡌⠓⡡⡌⠓⠤⡘⢈⡱⣾⣷⡭⢳⣌⠳⢌⡔⠉⠆⣿⣿⣿⣿⣿⣿⣿⢰⡱⣿⣿⡌⡌⠒⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⡄⣭⣿⣿⣽⡶⢋⣱⣿⣿⣌⢳⣬⡜⣿⣿⣿⣿⣿⣾⠞⣉⠰⢳⡌⡗⢣⠄⠊⣧⡜⢮⣝⡳⣍⠳⣎⡌⠜⡇⣿⣿⣿⣿⣿⣽⡨⠔⡋⣿⣿⢚⣌⠐⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⣆⣿⣿⣿⡿⢎⡳⣎⣷⣌⠳⣌⠶⣙⣷⣿⣿⣻⢥⢮⡙⣰⠳⢌⡱⠨⢒⠌⢢⣱⢎⡱⢮⡜⠳⣛⢞⡙⣿⣿⣿⣿⣿⣻⡔⠈⣇⣿⣿⣿⢿⣃⠄⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠃⣿⣻⣟⣿⣿⢻⣖⡛⣛⢳⣍⢬⡹⡵⣿⣿⡙⢢⢜⡡⡼⠳⣅⠻⡄⠒⢂⣌⠳⢦⡩⠖⣉⠴⣛⢬⡓⣿⣿⣿⣿⣼⢚⡡⠛⣗⣿⣟⣿⣿⢻⡄⠎⣿⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⡗⣿⣿⣿⣿⣞⣿⣿⡝⡌⠰⣝⣦⡜⡛⣿⣿⡑⢦⡉⢰⡹⢆⡩⡖⣌⠱⢆⡉⠢⢌⡘⠴⢋⡱⢛⣉⡟⣿⣿⣿⣾⢍⡰⣟⣿⣧⣦⣿⣿⣿⣿⣛⠜⣷⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⡇⣿⣿⣿⣿⣿⡿⣿⣮⣿⡽⠃⢢⣜⡿⣛⣿⡳⢎⢻⡭⠞⣍⠱⡳⣜⠹⢆⡙⢣⠔⡌⠓⢌⠱⣏⡟⣿⣿⣿⣹⣦⣛⡿⣾⣴⢊⡿⣿⣿⣿⣾⣷⣐⡿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣆⣷⣿⣿⣿⣿⣿⣿⣿⡿⢿⠻⣟⢿⠿⣍⠾⣎⡙⣶⣏⣥⡸⠶⡌⡘⢎⡃⢌⡑⡬⠖⡏⣿⣿⣿⣿⣿⣿⡿⣿⣾⣰⣀⠩⢟⣿⣿⣿⣿⣽⢎⡐⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣄⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡻⡟⢛⣻⡟⢛⣛⠿⢿⠿⡘⠹⢍⠫⣿⣿⣿⣿⣿⣿⣿⣯⣹⠟⡿⡼⣟⣤⣴⣠⡇⣿⣿⢌⡘⡆⣿⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⣄⣷⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠡⣆⢻⠈⠀⣿⣿⣿⣿⠉⢋⣿⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⣿⣹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣯⣰⡝⠨⣥⢿⡿⣿⣿⣵⣿⣸⣀⣇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣻⣷⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣷⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣘⡀⣿⣿⣿⣾⡄⣫⠟⠿⡿⣶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⢿⣜⣷⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣾⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⠿⣿⣿⣿⣴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⠻⢧⡆⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣽⣮⣟⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣶⣶⣤⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣻⢳⣌⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⢎⣱⣼⣫⣿⣿⣿⠿⣉⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⣿⡿⢟⡤⣿⣽⣿⣿⣿⣿⣿⣾⡽⢟⣢⠜⣿⣿⣿⣀⣽⠳⣡⣿⣿⣿⣿⣿⣿⣿⣼⣯⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣏⣰⢥⡉⡌⣿⢾⠀⣟⠻⡻⡟⣿⣿⣿⣿⣿⣿⣿⣾⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⡿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣽⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣈⣱⣄⠂⡄⡘⣬⡟⣿⣿⣿⣯⣿⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⢪⡑⡈⠐⢢⠅⢋⡿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠻⠛⠟⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣦⣶⣶⣿⣿⣿⣿⣿⣶⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⢂⠘⠢⢐⣚⠙