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