⢸⣀⠜⢄⡀⠀⠰⣶⣤⣀⠀⠀⣠⠔⠚⢛⣛⣦⡀⠀⠀⠀⠀⠀⠀⡐⠉⠉⢶⠄
⠈⢅⠀⡈⠱⣄⠀⠈⠛⠻⢷⡎⡡⣶⣿⣿⣿⢿⣿⡀⠀⠀⠀⠀⣴⣯⣝⣲⠊⠀
⠀⠀⠠⢅⢀⡎⣷⣄⠀⠀⠀⢿⡰⠊⠁⠀⠸⡼⡏⣇⠀⠀⢀⣾⣿⣿⣿⡏⠀⠀
⠀⠀⠀⠈⢻⣾⣿⣿⣷⣤⣀⣼⣧⡀⠀⣠⠴⣗⣳⣼⣄⣴⣿⣿⣿⣿⠟⠀
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⠏⢹⠁⠀⡟⢹⢿⣿⣿⣿⡟⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠦⢅⣀⣨⣤⠴⡞⣿⣿⡟⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⠈⠲⢦⡾⣛⣾⣁⣸⠋⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡍⡿⠟⢻⡔⠚⠚⣷⣼⣾⠉⢹⠃⠀⣀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣷⠀⠀⢧⠀⠸⠋⠸⠌⠃⢸⠀⢰⣿⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⣸⣤⣤⣶⣶⣶⣶⣾⣤⣿⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⠀⠀⠀⠀⠀⠀⠉⣿⠿⠏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢹⠀⢸⠀⠀⠀⠂⠁⠀⠀⢿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣤⣼⣀⣄⣀⡀⡀⢀⣀⡸⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠙⡿⠋⢹⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⣀⣀⣤⡷⠶⡿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠃⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣿⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢰⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⢯⡉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠬⠏⠉⠀

⢸⣀⠜⢄⡀⠀⠰⣶⣤⣀⠀⠀⣠⠔⠚⢛⣛⣦⡀⠀⠀⠀⠀⠀⠀⡐⠉⠉⢶⠄ ⠈⢅⠀⡈⠱⣄⠀⠈⠛⠻⢷⡎⡡⣶⣿⣿⣿⢿⣿⡀⠀⠀⠀⠀⣴⣯⣝⣲⠊⠀ ⠀⠀⠠⢅⢀⡎⣷⣄⠀⠀⠀⢿⡰⠊⠁⠀⠸⡼⡏⣇⠀⠀⢀⣾⣿⣿⣿⡏⠀⠀ ⠀⠀⠀⠈⢻⣾⣿⣿⣷⣤⣀⣼⣧⡀⠀⣠⠴⣗⣳⣼⣄⣴⣿⣿⣿⣿⠟⠀ ⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⠏⢹⠁⠀⡟⢹⢿⣿⣿⣿⡟⠀⠀ ⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠦⢅⣀⣨⣤⠴⡞⣿⣿⡟⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⠈⠲⢦⡾⣛⣾⣁⣸⠋⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡍⡿⠟⢻⡔⠚⠚⣷⣼⣾⠉⢹⠃⠀⣀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣷⠀⠀⢧⠀⠸⠋⠸⠌⠃⢸⠀⢰⣿⠃⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⣸⣤⣤⣶⣶⣶⣶⣾⣤⣿⠃⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⠀⠀⠀⠀⠀⠀⠉⣿⠿⠏⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢹⠀⢸⠀⠀⠀⠂⠁⠀⠀⢿⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣤⣼⣀⣄⣀⡀⡀⢀⣀⡸⠄⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠙⡿⠋⢹⠇⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⣀⣀⣤⡷⠶⡿⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠃⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣿⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢰⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⢯⡉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠬⠏⠉⠀