⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠉⠉⠉⠛⠛⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠋⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⣀⣿⣇⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣷⣦⣤⣀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢀⣴⣾⠟⠛⠉⠉⠉⢹⣿⡿⣷⣤⣄⡀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠉⣀⣠⣤⣤⣤⣬⣻⣿⣦⠀⠘⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣾⠟⣴⣾⣿⣿⣿⣿⣿⣿⣶⣦⣌⡙⠻⠷⣶⣶⣶⣶⡿⠛⣁⣴⣶⣿⣿⠿⠿⠿⣿⣿⣿⡿⢻⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠸⣿⠀⣿⣿⣿⣏⡀⠀⠈⠉⠙⠻⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⣁⣀⣠⣴⣾⠟⠛⠛⣼⣿⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣿⣷⣄⠀⠸⣿⣿⣿⣿⣷⣿⢿⣿⣿⣿⡿⣛⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣤⣾⢿⡇⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⣿⡙⣿⣷⣿⣷⣿⣿⣿⣿⣭⣶⡿⠟⣹⣾⢟⣫⣭⣙⠻⣿⣝⠿⢿⣿⣿⣿⣿⣿⠿⠿⢋⣵⣾⣇⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣿⣿⣯⡉⠛⠛⠛⠛⠉⠁⢠⣾⣿⣿⡿⠋⠉⠛⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⣄⢀⣠⣴⣾⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣤⣾⠿⣶⣤⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣠⣾⠿⠋⠀⠀⠀⠙⢿⣦⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠏⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⣀⣀⣤⣶⡾⠿⠛⢿⣿⡇⠙⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⣠⣾⠟⠛⢿⣶⣄⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⣿⠉⠙⠻⠷⣶⣤⣄⣀⣀⡀⠀⠀⠀
⣤⣶⣾⣿⣿⣿⠿⢛⣭⣴⡾⢿⡆⠈⣿⠇⠀⢨⡿⢿⣿⣿⣿⣿⣿⣷⡿⠛⠁⠀⠀⠀⠈⠻⣷⣾⣿⣿⣿⣿⣿⡿⢻⡇⠀⠀⣿⡆⠀⣶⣶⣦⣍⡛⠿⣿⣿⣿⣶⣤
⠉⢰⣿⠛⠁⣠⣾⠿⠋⠁⠀⣾⠇⢰⣿⠀⠀⢸⡇⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⣿⣿⠏⠀⢸⡇⠀⠀⠸⣿⡀⣿⡄⠈⠙⠿⣶⣄⠙⢻⣷⠈
⠀⠈⢿⣦⠀⠛⠁⠀⠀⠀⠸⠟⠀⠿⠇⠀⠀⢸⡇⠀⠀⢻⣿⣿⣶⣿⣷⣤⣀⠀⠀⠀⣀⣤⣾⣯⣾⣿⣿⠇⠀⠀⠈⠁⠀⠀⠀⢻⣧⣽⣷⠀⠀⠀⠈⠻⢷⣾⡟⠀
⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⢀⣬⡅⠀⠀⠀⣿⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⡟⠀⠀⣰⣿⣷⡀⠀⠀⠀⢿⣷⠙⠀⠀⠀⠀⠀⣼⡟⠀⠀
⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⢠⣿⠃⠀⠀⣾⡟⣿⡀⠀⠀⠘⣿⣇⢻⣇⠀⠉⠉⠉⠉⠁⠀⣼⡏⢸⣿⠁⠀⢰⣿⠁⠹⣷⡀⠀⠀⠀⢻⣷⡄⠀⠀⠀⣼⡟⠀⠀⠀
⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⣾⠏⠀⠀⣸⡟⠀⢻⣇⠀⠀⠀⠙⣿⡌⢿⡆⠀⠀⠀⠀⠀⢰⣿⢁⣿⠋⠀⠀⠘⠃⠀⠀⠹⣷⠀⠀⠀⠀⠙⠟⠀⠀⣸⡿⠀⠀⠀⠀
⣷⡀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠿⠁⠀⠈⣿⣇⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠈⠁⣸⡟⠀⠀⠀⠀⢀⠀⠀⠘⠛⠀⠀⠀⠀⠀⢀⣀⣰⣿⠃⠀⠀⠀⠀
⣿⣷⡄⠀⠀⠀⠀⠙⣿⣿⠿⢶⣶⣤⣤⣀⣀⡀⠀⠘⢿⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠘⢀⣀⣀⣠⣤⣶⡶⠾⠿⠛⣻⣿⠃⠀⠀⠀⠀⠀
⢻⣿⣿⣄⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠉⠉⠛⠛⠿⢿⣶⣤⣤⣄⡀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⣀⣤⣴⣶⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⣴
⠀⠹⣿⣿⣆⠀⠀⠀⠀⠈⠻⣷⡀⠶⢶⣶⣤⣤⡀⠀⠀⠈⠉⠙⠻⠿⢿⣦⣤⠀⠀⠀⠀⣠⣴⡾⠿⠛⠉⠁⠀⠀⣠⣤⣶⡶⠾⠿⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⣴⣿