⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁⠀⠈⠹⣆⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⡶⠖⢲⣶⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠹⣇⠀⢀⣴⣾⣿⣿⣿⡻⠟⠋⢹⣆⣴⣿⣽⡟⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠀⠀⠀⠀⠀⠀⢻⣶⣿⣿⣿⣿⣿⣿⣟⡀⢀⣼⣿⡿⠛⠃⠙⠋⢿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠛⠛⣿⣿⣿⣷⣿⡀⠀⠀⠀⠀⣘⣿⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⠀⠀⣿⣿⢿⡿⣿⠿⣿⣶⣶⣶⣿⣿⣾⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣾⣿⣿⡇⣿⠇⠀⠀⠀⠀⠈⠉⣉⣻⣿⢿⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠈⣇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡄⣶⠀⠀⣀⠀⢀⡉⢩⣿⣾⣿⡟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣹⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣶⣾⣿⡄⢸⣧⣼⣿⣿⣿⡇⣿⣿⣀⣀⣀⣤⣤⣤⡴⠶⣤⣤⣤⣄⣀⣀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠶⠶⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⢿⡿⠿⠇⠛⠋⣽⣿⣿⣿⣿⣿⢻⣿⣿⡇⣿⣿⣇⣀⣀⣀⣀⣀⣀⣤⠾⠛⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⢀⣦⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⢷⡀⠉⢉⣭⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡦⠴⠀⠘⠟⠀⠀⠀⠈⠉⢫⣿⣿⣿⣿⢷⣿⣿⢀⣤⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡌⠳⠶⣦⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⡟⢿⣿⣿⣿⣟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⢠⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⠘⣿⣿⣿⣯⠀⠙⠛⣻⡿⠻⠿⠟⠋⠁⠀⠀⠀⢀⠀⣶⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠁⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⣧⠀⣿⣿⣿⡿⢷⣀⠾⢿⠃⠀⠀⠀⠀⠀⠀⣄⣠⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠛⣱⣿⡇⠀⢀⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⣿⠯⣤⣿⢿⣿⠃⠀⠀⠀⠀⠀⣤⣀⣀⠀⠀⠀⠛⣿⣿⣧⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⡴⢛⣭⡾⠋⠀⢠⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀⠀⣸⠃⣤⡿⣃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀⠀⠀⠀⠀
⠀⠀⣤⠟⢉⣴⡿⠋⠀⠀⢀⣾⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣄⠀⠀⠀
⠀⣼⠃⣰⣾⠏⠀⠀⠀⠀⣾⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈ ⢛⡏⠀⠀
⢰⣇⣾⡿⠁⠀⠀⠀⠀⣰⣿⡇⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀ ⠀⠀ ⣠⣤⣄
⠈⣿⡟⠀⠀⠀⠀⠀⢰⣿⡟⢀⣿⡏⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡧⠀⠀⠀⠀⠀ ⠀ ⠘⠛⠛⡏⠀
⢀⡿⢷⣄⡀⠀⠀⠀⣾⣿⠁⢸⣿⣧⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀ ⠀⣼⣷⣄⠀⠀
⠈⣧⠀⠉⠛⠂⠀⣼⣿⣿⠀⢸⣿⣿⣾⣧⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠻⣷⣦⣤⣤⣤⣤⡤⠶⠛⠋⠀⠀⠀⠈⠙⠷⣤⣄⡀⠀⠀⣠⣾⣿⣿⣿
⠀⠙⠷⠀⠀⠀⢰⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣦⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⣿⡇⠈⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⠹⡆⢸⣿⣿⣿⡉⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠉⢻
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣷⣸⣿⣿⣿⣷⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠘⢿⣿⣿⣿⢻⣿⢹⣿⣧⢘⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣾⣧⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⠙⣿⠀⠀⠈⣿⣿⡟⠸⣿⡇⢹⣿⠎⢿⣿⢸⣿⣟⠛⠛⠛⣿⣿⠁⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⣿⣿⡆⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣿⠀⢻⡄⠀⠀⢸⣿⣿⠀⢹⣿⠀⠹⣆⠀⣿⠀⢿⣿⠀⠀⠀⣿⣟⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢹⠀⠀⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠸⣿⣶⣾⣇⠀⠀⠀⢿⣿⣧⡀⣿⣧⣀⠹⣧⣻⡆⠘⡟⣿⠀⢰⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⣿⠀⣾⡆⠀⣼⠉⢿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄⠀⠀⠘⣿⡈⠛⠻⠿⠿⠃⠘⢿⣷⣠⣿⡇⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⡿⢻⣰⣿⡇⣰⡇⠀⠈⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣏⠛⠶⠦⠀⠘⢷⡄⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠃⠸⣟⢸⣿⠟⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣽⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀�