⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⣶⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⣿⣇⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⠸⣿⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⣿⣿⣿⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢻⣿⣟⠉⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠸⣿⣿⣿⡄⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⣿⣿⣿⡀⡀⡀⡀⡀⣠⣾⣿⣿⣶⣄⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢀⣤⣤⣀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⣿⣿⣿⡀⡀⡀⡀⣼⣿⠋⡀⠈⠙⢿⣿⣄⡀⣀⣀⣤⣤⣄⣠⣾⣿⠿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⡄⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⣿⣿⣿⣿⡀⡀⢰⣿⠇⡀⡀⡀⡀⡀⠙⢿⣿⡿⠿⠛⠻⣿⣿⡟⡀⡀⡀⠙⣿⣿⠉⡀⡀⡀⡀⣿⣿⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢸⣿⡀⡀⣿⡆⡀⣿⡿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⣿⡀⡀⣴⡷⣶⣿⣿⣦⡀⡀⡀⡀⣿⣿⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠸⣿⡀⡀⢹⣷⡀⣿⡇⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢸⣿⡿⡀⡀⣿⣰⡟⠉⠈⠻⣿⣶⣶⣶⣿⣿⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⡀⡀⡀⣿⡀⣿⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠸⣿⣷⡀⡀⡀⣻⡀⡀⡀⡀⡿⠋⠁⠈⢻⣿⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⡀⡀⡀⢿⣷⣿⠏⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠙⢿⣿⡿⠿⣿⣿⣤⣤⣾⠋⣿⣆⡀⢸⣿⠇⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⡀⡀⡀⠘⣿⣿⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠛⠻⣿⣿⣶⣿⠋⡀⣾⡿⡀⡀⡀⡀⡀
⡀⡀⡀⢀⣾⣿⣄⡀⡀⡀⡀⡀⣿⣶⣶⣿⣿⣿⠁⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠻⣿⣄⡀⣀⣾⣿⡄⡀⡀⡀⡀⡀
⡀⡀⣴⣿⠛⠈⠿⣿⣶⣀⡀⡀⡀⣿⣿⣿⣿⣿⠛⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠈⠙⠛⠉⡀⣿⣷⡀⡀⡀⡀⡀
⡀⡀⠻⣿⣄⡀⡀⡀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⡀⡀⡀⡀⡀⣾⣿⣷⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⣿⡀⡀⡀⡀⡀
⡀⡀⡀⠈⠿⣿⣦⣀⡀⡀⡀⡀⡀⠈⣿⣿⣿⣿⡟⡀⡀⡀⡀⠻⣿⠟⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣴⣶⡄⡀⡀⡀⠛⣿⣿⠿⣶⣤⡀⡀
⡀⡀⡀⡀⡀⡀⠉⠻⣿⣿⣿⣶⣦⣤⣿⣿⣿⣿⣷⣤⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣴⢶⣦⡀⡀⡀⡀⡀⡀⣿⣿⡏⡀⡀⡀⢀⣿⡟⡀⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢩⣿⠿⢿⣿⣿⣿⣿⣧⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠘⣧⣀⣼⠇⡀⡀⡀⡀⡀⠈⠉⡀⡀⡀⠉⣿⣿⣿⣤⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⢸⣿⣤⢸⣿⣿⣿⣿⣿⣿⣄⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⠉⠁⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣶⣶⣿⠋⡀⠉⡀⡀⡀⡀
⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣿⣛⣿⣿⣿⣿⣿⣏⠙⣿⣿⣤⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⡀⣀⣾⡿⠛⣶⡀⡀⡀⡀⡀