⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⢿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣶⣶⣶⣶⣦⣶⣿⣿⣿⢿⣻⣞⡿⣾⣽⣻⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⣟⡿⣿⢿⡿⣿⢿⣿⢿⣻⡽⣯⢷⣯⣟⡷⣯⢷⣯⣟⡿⣿⢿⡿⣿⢿⣿⣻⣟⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣻⣽⣻⡽⣯⢿⣽⣻⣞⣯⡷⣿⣽⣻⢾⣽⣻⣽⣻⢾⣽⣻⡽⣯⢿⣽⣻⣞⣷⣻⣞⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣳⣟⣾⣳⢿⣽⣻⣞⡷⣯⢷⣻⢷⣯⣟⡿⣞⡷⣯⣟⡿⣞⣷⣻⣽⣻⣞⣷⣻⣞⣷⣻⣞⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⢷⣻⣞⡷⣯⢿⣞⡷⣯⢿⣽⣻⣽⣻⢾⣽⣻⣽⣻⢷⣯⢿⡽⣾⡽⣞⣷⣻⣞⣷⣻⣞⣷⣻⣞⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⢿⣻⡽⣯⢷⣯⢿⣽⣻⢾⣽⢯⣟⣾⣳⢯⣟⡿⣞⡷⣯⣟⡿⣞⣯⣟⣷⣻⣟⣾⣳⣟⣾⣳⣟⣾⣳⣟⣾⣻⢿⡿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀ ⠀⠀⠀⠀⣿⣿⣿⣿⡿⣟⡿⣼⢿⣻⢿⣻⢿⡼⣿⣼⣻⢿⡼⣟⡿⣼⣻⢿⣼⣻⣟⡿⣧⣟⣿⣻⣼⣻⡼⣧⢿⡼⣧⢿⡼⣧⢿⡼⣧⢿⣼⣻⣟⣿⣻⣟⣿⣿⣿⣿⣿⠀⠀⠀⠀ ⠀⠀⠀⠀⣿⣿⣿⣳⣟⣯⢿⡽⣯⣟⣯⣟⣯⣟⡷⣯⣟⣯⣟⣯⢿⡽⣯⣟⣾⡽⣞⣿⣳⣟⣾⣳⢯⡷⣿⡽⣯⢿⡽⣯⢿⡽⣯⢿⣽⣻⣞⣷⣻⣞⣷⣻⣞⣷⣻⣿⣿⠀⠀⠀⠀ ⠀⠀⠀⠀⢹⣿⣿⣷⣻⢾⣯⣟⣷⣻⢾⡽⣾⣽⣻⢷⣻⢾⡽⣾⢯⣟⣷⣻⢾⣽⣻⣞⣷⣻⢾⣽⣻⡽⣷⣻⡽⣯⢿⣽⣿⣽⢯⣟⣾⣳⣟⣾⣳⣟⣾⣳⣟⡾⣿⣿⡇⠀⠀⠀⠀ ⠀⠀⠀⠀⢸⣿⣿⡷⣯⣟⡾⣽⢾⡽⣯⣟⡷⣯⣟⣯⣟⣯⢿⣽⣻⣞⡷⣯⢿⣞⣷⣻⢾⣽⣻⣞⡷⣟⡷⣯⢿⣽⣿⣿⣿⣿⣿⣻⣞⣷⣻⣞⣷⣻⣞⡷⣯⢿⣿⣿⡇⠀⠀⠀⠀ ⠀⠀⠀⠀⢸⣿⣿⣟⡷⣯⣟⣯⢿⣽⣳⣯⣟⣷⣻⢾⡽⣾⣻⣞⡷⣯⢿⣽⣻⣞⡷⣯⢿⣞⡷⣯⢿⣽⣻⣽⣿⣿⠟⠁⣨⣿⣿⣿⣟⣾⣳⣟⣾⣳⢯⡿⣽⣻⣿⣿⡁⠀⠀⠀⠀ ⠀⠀⠀⢠⣾⣿⣿⢿⣽⣳⣟⡾⣟⣾⣳⣟⣾⣳⢯⡿⣽⣳⣟⣾⡽⣯⣟⣾⣳⢯⡿⣽⣻⢾⣽⢯⣿⣾⣿⣿⠟⠁⣠⣾⣿⣿⣿⣻⣞⣷⣻⣞⣷⣻⢯⣟⣷⣻⣿⣿⣷⣄⠀⠀⠀ ⠀⠀⣴⣿⣿⡿⣯⣟⣾⣳⢯⡿⣽⣞⣷⣻⢾⣽⣻⡽⣯⢷⣻⢾⡽⣷⢯⣷⣻⢯⣟⣷⣻⣟⣾⣿⣿⣿⠟⠁⣠⣾⣿⣿⣿⣻⣞⣷⣻⣞⣷⣻⢾⡽⣯⣟⣾⣳⢯⣿⣿⣿⣦⡀⠀ ⣠⣾⣿⣿⡿⣽⣳⣟⡾⣽⣻⡽⣷⣻⢾⣽⣻⣞⡷⣿⣽⣻⡽⣯⢿⣽⣻⢾⡽⣯⣟⣾⣳⣿⣿⣿⠟⠁⣠⣾⣿⣿⣿⣻⣞⣷⣻⣞⣷⣻⢾⡽⣯⣟⣷⣻⢾⣽⣻⢾⡽⣿⣿⣷⡄ ⣿⣿⣿⣯⣟⣷⣻⣞⡿⣽⣳⢿⣳⣟⡿⣞⡷⣯⣟⣷⣿⣷⣻⣽⣻⢾⡽⣯⣟⣷⣻⣾⣿⣿⠟⠁⣠⣾⣿⣿⣿⣻⣞⣷⣻⣞⣷⣻⢾⡽⣯⣟⣷⣻⢾⣽⣻⢾⡽⣯⢿⡽⣿⣿⣿ ⠙⢿⣿⣷⡿⣞⡷⣯⢿⡽⣯⣟⡷⣯⢿⣽⣻⣷⣿⣿⣿⣿⣿⣷⣯⢿⣽⣳⢿⣾⣿⣿⠟⠁⣠⣾⣿⣿⣿⣻⣞⣷⣻⣞⣷⣻⢾⡽⣯⣟⣷⣻⢾⣽⣻⢾⡽⣯⢿⡽⣯⣿⣿⡿⠃ ⠀⠀⠻⣿⣿⣿⣽⢯⡿⣽⣳⣯⢿⣽⣻⣞⣿⣿⣿⣿⡀⠙⢿⣿⣿⣟⣾⣽⣿⣿⠟⠁⣠⣾⣿⣿⣿⣻⣞⣷⣻⣞⣷⣻⢾⡽⣯⣟⣷⣻⢾⣽⣻⢾⡽⣯⢿⡽⣯⣿⣿⣿⠟⠁⠀ ⠀⠀⠀⠘⢿⣿⣿⣯⣟⡷⣟⣾⣻⣞⡷⣯⢿⡽⣿⣿⣿⣦⡀⠙⢿⣿⣿⣿⠟⠁⣠⣾⣿⣿⣿⣻⣞⣷⣻⣞⣷⣻⢾⡽⣯⣟⣷⣻⢾⣽⣻⢾⡽⣯⢿⡽⣯⢿⣽⣿⡿⠋⠀⠀⠀ ⠀⠀⠀⠀⢸⣿⣿⡷⣯⢿⣽⣞⡷⣯⢿⡽⣯⢿⣽⣻⣿⣿⣿⣦⡀⠙⠛⠁⣠⣾⣿⣿⣿⣻⣞⣷⣻⣞⣷⣻⢾⡽⣯⣟⣷⣻⢾⣽⣻⢾⡽⣯⢿⡽⣯⢿⣽⣻⣿⣿⡅⠀⠀⠀⠀ ⠀⠀⠀⠀⢸⣿⣿⡿⣽⣻⢾⣽⣻⡽⣯⢿⣽⣻⣞⡷⣯⢿⣿⣿⣿⣦⣠⣾⣿⣿⣿⣻⣞⣷⣻⣞⣷⣻⢾⡽⣯⣟⣷⣻⢾⣽⣻⢾⡽⣯⢿⡽⣯⢿⣽⣻⢾⣽⣿⣿⡇⠀⠀⠀⠀ ⠀⠀⠀⠀⣸⣿⣿⣟⣷⣻⣟⣾⣳⢿⣽⣻⣞⡷⣯⢿⣽⣻⣞⡿⣿⣿⣿⣿⣿⣻⣞⣷⣻⣞⣷⣻⢾⡽⣯⣟⣷⣻⢾⣽⣻⢾⡽⣯⢿⡽⣯⢿⣽⣻⢾⣽⣻⣞⣿⣿⣆⠀⠀⠀⠀ ⠀⠀⠀⠀⣿⣿⣿⣞⣷⣻⣞⡷⣯⢿⣞⡷⣯⢿⣽⣻⣞⡷⣯⢿⣽⣻⣟⡿⣞⣷⣻⣞⣷⣻⢾⡽⣯⣟⣷⣻⢾⣽⣻⢾⡽⣯⢿⡽⣯⢿⣽⣻⢾⣽⣻⣞⡷⣯⣿⣿⣿⠀⠀⠀⠀ ⠀⠀⠀⠀⣿⣿⣿⣿⣾⣷⣯⣿⣽⢻⣾⣽⣯⡟⣾⢳⣯⣿⣽⢻⡞⣷⣯⣿⣽⡞⣷⢻⣾⣽⣯⡟⣷⢻⣾⣽⢻⣾⣽⣯⣿⣽⣯⣿⣽⢻⣾⣽⢻⡞⣷⣯⣿⣿⣿⣿⣿⠀⠀⠀⠀ ⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣾⣯⣿⢾⡽⣞⡿⣽⣻⢾⣳⣯⢿⣽⣳⣟⣾⣳⢿⣽⣻⣞⡷⣯⢿⣽⣻⢾⣽⣻⣞⣷⣻⣞⣷⣻⢾⣽⣻⢾⣽⣯⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣻⣽⣻⡽⣯⣟⡷⣯⢿⡾⣽⣞⡷⣯⢿⣞⡷⣯⢿⣽⣻⢾⣽⣻⣞⣷⣻⣞⣷⣻⢾⣽⣻⢾⡽⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣞⡷⣟⡷⣯⢿⡽⣯⣟⡷⣯⢿⣽⣻⢾⣽⢯⣟⣾⣽⣻⣞⣷⣻⣞⣷⣻⢾⣽⣻⢾⡽⣯⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⡿⣽⣻⡽⣯⣟⡷⣯⢿⣽⣻⢾⣽⣻⣞⣯⣟⣾⣳⣟⣾⣳⣟⣾⣳⣯⢿⡾⣽⢯⣟⣷⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣟⣷⣯⣿⣷⣯⣿⣽⣿⣾⣽⣻⣞⡷⣯⢷⣻⣞⣷⣻⣞⣷⣿⣾⣷⣯⣿⣽⣯⣿⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠿⠿⠿⠿⠛⠿⣿⣿⣷⣯⣿⣽⣻⢷⣻⣾⣷⣿⡿⡿⠛⠿⠿⠿⠿⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣾⣽⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀