⠀⣰⣶⣿⣺⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⡿⣧⠈⠉⠿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⡇⢹⣶⡀⠙⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⡟⠃⢸⣿⣿⠀⠛⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⠇⠀⢸⡏⣽⣿⡄⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣆⡀⢸⣿⠉⢿⣷⡀⠈⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠏⣿⣇⣸⣿⣀⠘⣿⡧⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠈⢹⡏⢸⣿⣥⣿⠀⢹⣿⣿⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⢀⣶⣶⣾⠛⠛⢻⣶⣶⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠈⣿⣿⠙⣿⡿⣷⣿⣿⣯⣉⠉⠉⠉⠻⠿⢿⣷⣦⣴⣾⡿⠿⠏⠉⠉⣀⣀⡈⠉⠉⠉⠉⠿⠿⠿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⢀⣽⣿⣶⡛⢷⠉⠉⠉⢻⣿⠘⣶⣆⠀⠀⠈⢉⣹⣿⠏⠁⡀⣰⣶⣶⣿⣿⣷⣶⣶⣶⣦⣀⣀⠀⠘⢉⣹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⢰⣿⣿⣿⠀⣿⡆⠀⠀⠀⢸⣿⠟⠛⠻⣧⣤⡼⠛⠀⠀⣤⣿⣿⣿⠛⠛⠀⠀⠛⣿⡟⠛⠛⡉⠉⣭⣥⣾⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⢸⣿⡏⢸⣿⣅⡀⠀⢠⣼⣿⠁⠀⠀⠀⠹⢿⣇⢀⣠⠿⠋⠀⠀⠹⠿⠿⠿⠿⠋⠉⢡⣤⣠⣿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠈⠛⣿⣿⣻⣿⣷⣶⣿⣿⣉⣀⣀⣀⣀⣶⣾⣿⡾⠉⠷⠶⣷⣶⣦⣀⣀⣀⣀⣿⣶⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⢹⣿⣿⠛⣿⣿⣧⡀⡉⣉⣝⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⡏⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠃⠀⠀⡾⢷⣤⠀⠀⠀⠙⠛⠛⠿⣶⣦⣤⠀⢻⣿⣧⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡆⠀⠀⢻⡟⠉⣀⡀⠀⠀⢀⣸⣾⣉⣿⣿⡆⣼⡏⢻⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠘⢿⣦⡛⠁⠀⠀⠈⠙⠛⣿⣿⠛⣳⣿⡗⢸⣿⠛⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⣿⣿⣿
⡇⠀⠀⠀⠀⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠜⠁⢹⡇⢸⣿⠚⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣾⡟⠟⠛⠀⢸⣿
⡇⠀⠀⠀⣤⣿⠿⠏⣿⡿⠿⠷⣶⣤⠀⠀⠀⢻⣧⠸⢿⣄⠈⠿⢿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠉⠉⠀⠀⠀⠀⢸
⡇⠀⠀⠀⣿⣿⠀⠀⣿⣧⡄⠀⣾⣿⢤⣤⣶⡞⠉⠀⢸⣿⡀⠀⠈⠉⣽⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⡇⠀⠀⣾⣿⠛⠀⠀⠀⣼⣧⣿⣿⠛⠀⠀⢸⡇⠀⠀⠈⢻⣷⠄⠀⠀⠘⣿⡿⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⡇⠀⣠⣿⣿⠀⣤⣿⣥⡿⢿⣿⣿⠀⠀⠀⢸⡇⢰⣼⣿⣤⣿⡆⠀⠀⠀⠀⠷⣦⠉⠻⠿⢿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼
⡃⢸⣿⣏⣹⠿⠃⠀⢻⣇⣸⣿⣿⠀⠀⣿⡏⢿⡿⠃⠉⢿⣍⣿⡆⠀⠀⠀⠀⠹⠷⠤⠶⠮⠭⠉⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⡅⢸⣿⣏⠉⠀⠀⠀⠘⣿⣿⣿⣿⢤⡀⣿⣷⣾⠁⠀⠀⠀⠉⠛⢷⣦⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠛⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⣿⣀⠀⠀⢀⣤⣤⣤⣴⣟⣻
⠀⠘⢿⣷⣄⠀⠀⡀⣤⣿⠏⠛⣿⣤⣿⡀⣹⣏⠀⠀⠀⠀⢀⣶⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣷⡄⠀⠀⠀⠀⠀⣠⣾⠛⠀⠀⠛⠛⠓⠒⠛⠛⠀⣤⣤⣿⡿
⡀⠀⠀⠙⠿⣿⣾⣿⠛⠁⠀⠀⠙⢿⣍⣷⡋⢹⣇⠀⠀⠀⣿⡉⠁⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠃⢸⣿⡇⠀⣀⣠⡶⠾⠿⠉⠀⠀⠀⣤⣤⣤⣼⣿⣿⠿⠛⠛⠀
⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣿⠉⢳⡎⢹⣷⣤⣾⣯⡁⢰⣾⠉⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢸⣿⣿⣿⠉⠉⣳⣄⣀⢰⣶⣶⣶⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣾⣧⡤⢸⣿⣦⠀⣿⡿⠇⠀⠀⠀⠀⣤⠤⠾⠉⠉⠀⠀⠀⠀⣸⣷⣦⣤⣴⣦⡿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡗⠈⢻⣿⣿⣦⣿⣧⣽⣟⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡟⠃⢀⣸⣿⠿⣿⣿⡟⢛⣛⠻⣷⣄⡀⠀⢀⣀⣀⣀⣶⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⣶⡾⡏⠁⠀⠘⠻⣷⣿⣿⣿⣾⣿⣷⣶⣾⣿⡿⡿⣿⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀