⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⢻⣿⣟⣯⡟⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⣝⠢⣝⣻⡾⣝⣣⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣟⡿⡡⠒⢌⡷⣻⢳⡜⣾⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣻⢾⡱⠁⠐⢌⡾⣽⣣⢞⡷⣿⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣟⣿⠾⡝⠦⠁⠀⠀⠸⣼⣳⣟⣾⣿⣻⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⢿⡯⣏⠟⡁⠂⠀⠀⠀⢱⡸⣷⣻⣞⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡔⠉⠐⠀⠀⠀⠀⢐⢢⡳⣽⣳⡿⣞⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⡀⢆⢰⢰⡈⡎⢷⣹⢷⣏⣿⣿⡾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⡳⢮⣵⣻⢯⣟⡿⣾⣽⡿⣿⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣠⠤⡄⢲⣖⣶⣦⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⡽⣾⣽⣻⢾⣽⣿⣾⣿⣿⣿⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢶⢾⣹⣏⡿⣽⣏⢿⡘⢆⡾⣽⣯⠀⠀ ⣀⣀⡀⣀⣀⣤⣄⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣠⣈⡹⣿⣷⣯⣟⣯⣟⣾⣿⣿⣿⣾⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣻⢾⡵⣯⣟⣧⡿⣽⣳⣞⡣⢜⢪⡵⣻⣾⡀⠀ ⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⢿⣟⡿⣻⠿⠿⢿⢿⣷⣿⣾⣽⣻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣠⣾⣯⢷⣻⢿⡽⣳⠞⡧⡙⠤⠁⠂⠑⡈⢆⡙⠷⣿⠁⠀ ⠀⠀⠉⠻⣿⣿⣿⢿⣿⣻⣿⣟⣯⣿⣯⣷⡿⣟⣷⣟⡿⣞⡷⣯⣞⡷⣶⢦⣌⡁⠛⢷⣿⣽⣿⡋⠙⢳⢦⣀⣀⣠⣾⢿⡝⡯⣏⡽⣪⣕⣣⡝⡰⢠⠀⠀⠀⠀⠐⢤⡚⣼⣳⣷⣤ ⠀⠀⠀⠀⠈⢿⣿⣿⢿⣿⣽⢿⣻⣽⢯⣷⢿⣻⡽⣾⣻⣽⣻⠷⣯⢿⡽⣯⣟⡿⣷⣾⣾⣽⣿⣿⣷⣦⡌⢉⣿⣿⣯⣟⣾⣷⣻⣞⣷⣾⣳⣽⣣⠇⡌⢀⠀⠀⠀⠢⣝⡷⣿⣿⡟ ⠀⠀⠀⠀⠀⣾⣿⡿⣿⣻⢾⣻⡽⣞⡟⣮⢏⣧⣙⠶⡑⢎⠁⡉⠈⠀⠁⢀⢈⣈⣁⣹⣜⡯⠿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣯⣞⡴⢂⠀⠀⠀⠠⢹⡾⣿⣿⠁ ⠀⠀⠀⢠⣾⣿⣿⣿⣟⣯⣿⣳⢿⣽⣻⢮⡟⣶⡹⣎⡽⣘⡞⣴⣋⢾⣩⠏⡞⠴⠙⠋⠈⠀⠀⠀⣹⣿⣳⢿⡿⢹⣞⣧⢿⣹⢯⡿⣽⣳⢯⣟⡷⢯⣞⡧⢎⠠⠀⠀⠱⣿⣿⣟⠁ ⠀⠀⣰⣿⣿⣿⣿⣽⣿⣻⣾⣽⡻⠞⠷⠏⠻⠖⠋⠓⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣽⣻⡜⡀⠸⣿⣟⣾⣳⣟⣷⣯⡿⣾⡽⣯⢿⣹⣏⢦⠁⠀⠸⣽⣿⡏⠀ ⠀⠾⠋⠛⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡽⣷⢯⡗⠀⠀⠹⣿⣟⠿⢿⣿⣾⣿⣷⣿⣿⣯⣷⣮⣳⣎⣤⣐⡿⣿⡁⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣽⢯⠇⠀⠀⣜⣷⣷⡆⠀⠁⠙⠳⠿⡾⣍⡻⡝⡿⣿⢿⣿⣿⣿⣿⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢯⡛⠀⠀⢰⣞⣿⣿⣿⡀⠀⠀⠀⠀⠀⠉⠉⠛⠙⠋⠛⠋⠛⠉⠁⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠁⠀⠠⣟⣾⣿⡿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⡀⣼⢿⣿⣻⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡰⣹⢾⣿⣿⣿⣿⢿⣯⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣻⣿⡿⣿⣾⣿⣿⣿⣿⣽⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣏⣿⣿⣿⣿⣿⣿⢿⣏⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢿⣻⣿⣷⣿⣿⣿⣿⣻⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣿⣿⣾⣿⣿⢿⣿⣻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣿⣿⣻⣯⣷⣿⡿⣿⣟⣯⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣽⣟⣯⠛⠑⠛⠻⢻⠿⣿⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣽⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀