⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡌⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⣀⣀⣀⠀⠀⢀⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⣀⡀⠀⠀⣀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣋⠀⠀⠀⠀⠀⠀⢹⣿⠛⠛⠁⣴⣿⠛⠙⢻⣷⡄⢸⣿⠀⠀⠀⣿⡇⠀⠀⢰⣿⠟⠛⠻⣿⡆⢹⣿⠀⢀⣿⣿⡀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⠀⣿⡇⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀ ⣿⡇⠀⠀ ⣾⣿⠀⠀⠀⣿⣿⠀⣿⡇⣸⡟⣿⣇⢸⡿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⢿⣷⣄⣀⣴⡿⠁⢸⣿⣄⣀⡀⣿⣧⣀⣀⠸⣿⣄⣀⣠⣿⠇⠀⢸⣿⣿⠃⢸⣿⣿⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⢻⣿⠀⠀⠰⡀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⠈⠉⠉⠉⠁⠉⠉⠉⠉⠀⠈⠉⠛⠉⠁⠀⠀⠀⠉⠉⠀⠀⠉⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⢀⣿⡧⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⠀⠭⣿⣿⡄⠈⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠁⠐⠉⠀⠀⣿⣿⡤⣄⡀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⣤⡆⢀⣠⣤⣤⣄⡀⠀⣠⣤⠀⠀⢠⣤⠀⢠⣤⣤⣤⣀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⡷⠀⠀⠀⠀⠀⢠⣸⣿⣳⣿⡻⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⣼⡟⢠⣿⡟⠁⠉⢻⣷⠀⣿⡇⠀⠀⢸⣿⠀⢸⣿⠉⢹⣿⠀
⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠊⣻⣶⣤⣿⣿⡿⠿⠯⢛⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠁⢸⣿⡀⠀⠀⢸⣿⠂⣿⣇⠀⠀⢸⣿⠀⢸⣿⠿⣿⡏⠀
⠀⠀⠀⠀⣼⣿⣿⣖⣀⣤⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠘⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⢿⣷⣤⣴⣿⠏⠀⠸⣿⣦⣴⣿⠏⠀⢸⣿⠀⠹⣿⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣭⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠀⠀⠀⠉⠀
⠀⠀⠀⢠⣿⣿⣿⣿⡿⠋⢀⣴⣿⣿⣿⣭⡹⣿⣿⣿⣬⣹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⣿⣿⠀⣠⣾⣋⣹⠁⠙⢛⡛⠻⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢻⣿⣿⣽⣿⣿⣋⣡⣤⣤⡀⠀⣵⣤⣴⣦⣘⣻⣿⢿⣇⠀⠀⠀⣶⡆⠀⠀⢰⣶⣶⣶⣶⠀⠀⣴⣶⡆⠀⠀⢴⣶⣶⣶⣤⡀⠀⣶⣶⣶⣶⠆⢰⣶⣶⣶⣤⠀
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠿⠿⡛⠻⢿⣿⠄⠀⠀⣿⡇⠀⠀⢸⣿⣦⣀⡀⠀⣰⣿⢻⣿⡀⠀⢸⣿⠀⠈⢻⣿⡄⣿⣧⣀⣀⠀⢸⣿⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠟⣉⡥⠄⠀⠉⠉⠓⠚⠛⠓⠦⣌⡃⠀⠀⠀⣿⡇⠀⠀⢸⣿⠉⠉⠁⢀⣿⣧⣤⣿⣷⠀⢸⣿⠀⠀⣸⣿⠁⣿⡟⠉⠉⠀⢸⣿⠛⢿⣧⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡴⠚⢁⣠⣤⣀⡀⠀⣀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠿⠷⠶⠶⠸⠿⠷⠶⠶⠼⠿⠀⠀⠘⠿⠇⠸⠿⠶⠾⠟⠃⠀⠿⠿⠶⠶⠆⠸⠿⠀⠘⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠀⣠⣾⠟⠁⠀⠁⠀⠙⠲⠭⣶⢤⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣰⠟⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠻⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣷⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣄⠉⠀⠀⠀⠀⢀⡀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣧⣼⣿⠀⠀⣴⣄⣸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣴⣿⣿⣝⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣽⣿⠟⢧⣜⣛⣷⡂⠉⣻⢿⣿⣿⣿⣿⣿⣿⡿⠿⠻⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣧⣤⣿⣿⣿⣿⣿⡏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⠛⠻⢿⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⢹⣿⣧⣽⣿⣫⡽⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣟⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⣼⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣮⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣛⣶⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀