⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣾⡟⠻⣾⣯⣓⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⣦⣴⡴⠾⠛⠋⠉⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣾⠏⠁⠄⣈⣻⠯⠓⠷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣻⠌⢿⣿⣿⣶⣤⡊⠉⠒⢤⣈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⠛⠀⡠⠊⠁⠀⠉⠐⠚⠟⠄⠀⠀⠀⠀⠀⠩⣧⡿⣰⠛⣿⣿⡫⢻⣿⣾⣿⣟⣿⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣼⣟⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⢿⣿⡿⣿⣿⡟⢉⣷⣿⢯⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣰⣿⡩⠃⢀⣴⣾⣿⣾⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⣶⠿⣷⡻⣾⣿⠏⡞⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣯⡟⢀⡴⣿⣿⣿⣿⣿⠿⠏⢻⣿⣿⣷⡄⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠿⢿⣿⣟⣿⣿⣁⠀⡇⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⣿⡇⢸⣵⣿⣿⡿⠊⣰⣶⡄⠀⠹⣿⣿⣿⠀⠀⠀⢀⡼⢻⣿⠏⠁⢠⣶⣄⠀⠙⢿⣿⣿⡇⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢰⣿⢳⡞⣿⣿⣿⡄⠀⣿⣿⡇⠀⠀⣿⣿⡟⠀⠀⠀⢸⡇⣿⣿⣄⠀⢸⣿⣿⠀⠀⣸⣿⢻⡇⡇⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⢿⣇⣣⣻⣿⢿⣿⣦⣌⣉⣀⣀⣼⣿⣏⠇⢀⣠⣤⣈⢷⠺⣿⣿⣶⣌⣓⠋⣀⢴⣿⣟⣼⣧⡵⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⣿⣍⢟⠿⢷⣯⣿⣿⣿⠿⠿⠿⠟⠁⠀⠸⠧⠤⠽⠞⢷⣽⠻⢿⣿⣿⣿⣿⣿⡿⢟⣫⣵⣞⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠉⡿⣿⣿⣷⣊⠭⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠽⠟⠲⠮⠽⣶⣧⣴⣶⢿⣿⣇⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣧⣿⡀⣯⣿⠋⣿⢛⣿⡟⡿⠛⣿⡟⢻⡟⣿⡟⢿⡿⢻⣿⠛⣿⢻⣿⢫⣷⣿⡎⣧⣿⣿⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢻⣿⣿⣿⣿⣸⣿⣾⣿⣳⣿⣀⣇⣏⢸⡇⡿⣇⡬⢻⢸⢹⣰⢻⡛⡈⣾⣿⣿⣿⠟⣩⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⣧⡀
⠀⠀⠀⠀⠹⢷⣯⡙⢿⣷⣿⣜⣿⣿⣼⡧⢹⣸⣇⣇⣹⣇⠘⣯⣸⣿⣤⣷⣷⠿⣿⣿⣧⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⡿⠿⠛⢷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠻⣶⠽⣿⡿⣿⠿⢿⣿⠿⠿⠟⠛⢻⣿⣿⡟⢛⣉⣤⣤⣤⡾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠟⠛⠉⠀⠀⢀⣳⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣠⣶⠏⠉⣷⠿⢿⣷⣿⣶⣶⣶⣦⠤⠿⠿⣿⡟⠛⠋⠙⣿⣄⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⢀⣠⢴⣶⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠀
⠀⠀⠀⠀⢀⣾⠟⠁⠀⣼⠇⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠈⢿⣷⡀⠀⠹⣷⣄⠀⠀⢀⣴⣿⣏⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⣀⠰⣿⣿⣿⣿⣿⠇⠀
⠀⠀⠀⢠⡞⠁⠀⠀⠰⣿⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⣿⣷⡀⣀⣼⣿⣶⣶⣿⣿⣿⣿⣷⣾⣿⣾⣶⣶⣿⣷⣶⣶⣿⣷⣿⣿⣿⠟⠁⠀⠀
⠀⠀⢠⣏⠟⢷⣦⡀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠋⠉⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠁⠀⠀⠀⠀
⠀⠀⣾⣿⠀⠀⠙⣻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠹⣿⢟⠉⠀⠈⢻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⡏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⠦⠤⡾⠏⡘⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠛⢿⣷⣶⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠒⠒⠒⠊⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠀⢹⠳⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠾⠛⠛⠛⠛⠛⠛⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣆⢣⠀⠀⠀⠀⠀⠀⠀⢸⣿⡢⢿⣼⡀⠀⠀⠀⠀⠀⠀⢸⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣜⡄⠀⠀⠀⠀⠀⠀⢸⣿⠁⠸⡟⡇⠀⠀⠀⠀⠀⠀⠇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⠀⠀⣼⡟⠀⠀⣿⡇⠀⠀⠀⢀⡴⣺⡴⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣷⣖⣂⣀⣿⡇⠀⠘⣽⣷⣦⣴⣾⠷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀