⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⡟⣯⣽⣻⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣴⣿⢻⣿⣿⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⣴⣿⣿⣏⣿⣿⣿⡈⢻⣽⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣴⣿⣿⣿⣿⢾⣿⣿⣿⣇⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⠿⠿⠿⢿⣿⣿⣿⣿⣿⢻⢻⣄⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⢹⣒⠢⣿⣄⠀⠀⠚⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⡏⣿⣿⡟⣿⣿⡮⠈⠀⠀⠰⣿⣿⠻⣿⣿⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣼⣿⣇⠉⠋⠀⠀⡀⠀⠀⠉⠉⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣄⠀⠀⢀⣉⡀⠀⠀⣠⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡦⣀⣀⣀⢴⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⠿⢿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠙⠛⢿⣿⡿⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠎⠀⠀⣎⣿⣿⡂⠒⠒⢢⣤⣖⣛⣉⣁⣼⣿⡇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⢀⠀⠀⠇⢹⣿⡍⡉⣹⣿⣧⠀⠀⠀⠀⣾⣿⣇⢀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢿⠀⢧⣀⡇⠘⣿⡇⠀⠉⢹⣧⡀⠀⠁⠈⣿⡿⢹⠎⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⠀⠀⢻⢆⠀⢿⡇⠀⠀⢸⡇⠻⣆⠀⢰⣿⠃⣺⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡄⠀⢸⡄⠃⡈⣿⡀⠀⢨⢸⡦⢻⣧⣽⠿⢰⢿⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡅⠀⠸⡛⡄⠈⠹⠁⠀⢸⠈⣧⠀⠹⣏⢠⠇⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢁⠀⠀⡇⠹⣥⠀⢡⠀⢸⠀⢸⡆⠀⢹⡟⠀⢸⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⠀⠀⣿⠀⠈⠄⠀⢡⠀⠀⠀⣧⠀⠀⢻⠀⢸⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⡄⠀⣿⠀⠀⢠⠀⠀⢣⡄⠀⢽⠀⠀⢉⣧⣿⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢇⠀⣿⠀⠀⠀⠄⠀⠀⠃⠂⢸⡇⠠⠃⢛⣿⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⢠⣿⠀⠀⠀⠀⠀⠀⠈⢢⡘⠃⠀⢰⠈⣿⠶⠒⠿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠿⠘⠀⠀⠀⠀⡄⠀⠀⠈⠋⠢⠄⠈⢆⢻⠀⠀⠐⢌⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠁⠈⢻⣦⡭⣍⠢⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠹⣦⣷⣽⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡆⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢣⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⠀⠀⠘⣼⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢃⠀⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⠀⠀⠀⠘⡆⠀⠀⠰⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠈⡆⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡟⠀⠀⠀⠀⢡⠀⠀⠀⢃⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠘⡄⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⡄⠀⠀⠀⢆⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢹⠀⠀⢿⣿⣿⣿⣿⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⢂⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⣇⢸⠘⣿⣿⣿⣿⣇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠀⠀⠀⠀⠀⠘⡆⡆⢹⣿⣿⣿⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⢱⢧⠘⣿⣿⣿⣿⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠈⡆⠀⢹⣿⣿⣿⣿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄⠀⠀⠀⠀⠀⠀⠹⢰⠈⣿⣿⣿⣿⡆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠇⢇⢹⣿⣿⣿⡇⠀⠀
⠀⠀⠀⠀⠣⠀⡀⠀⡀⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣆⣀⣠⣤⣤⣤⣤⣴⣾⣾⣿⣿⣿⡧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠉⠉⠉⠁⠉⡍⠀⠈⡇⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡙⠶⢄⣀⠀⠀⢣⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃⣿⣿⣿⣾⣿⣿⣿⣿⣿⣏⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠉⢹⠀⠀⡇⠒⠂⠐⠒⠉⢸⢹⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡆⠀⡇⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⡞⠈⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣷⠀⢠⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣯⣷⣿⣿⣿⣿⣿⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⢠⢿⠀⢸⢠⠀⠀⠀⡖⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⡟⢻⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠞⠀⡖⠀⡎⣸⠀⠸⠀⠇⠀⢀⠁⢃⠀⠀⠘⣿⣿⣿⣿⣿⣿⡧⢸⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡇⠀⡇⢸⠀⠀⡃⠀⡇⢸⠀⢸⠀⠀⠀⢹⣿⣿⣿⣿⣿⡗⠨⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⡇⠈⠀⠀⡇⠀⠀⠀⢿⣿⣿⣿⣿⡏⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠀⠀⡇⠀⠀⢰⠀⢡⠀⠀⠀⢃⠀⠀⠀⠘⣿⣿⣿⣿⠇⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄⠀⡇⠀⠀⢰⠀⠈⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢃⢠⠃⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⢈⢰⠀⠀⠀⠀⢱⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢀⠀⠀⢸⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⡇⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡇⠀⠀⠀⠀⢀⠇⣀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢳⠀⠀⠀⠀⢸⡇⢁⣀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⢸⠀⠀⠀⠀⢠⣷⠿⠛⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⣤⣤⣄⡇⠀⠀⠀⢠⠏⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠀⠛⡏⠀⠀⠀⣘⣀⡤⡤⠤⢄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀