⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠘⠹⣶⣿⠷⢃⡆⠻⢈⣿⣿
⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⢉⣴⣿⣷⢱⢠⡘⣿⣿
⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⣀⣴⣿⠿⣫⣾⢈⣸⣷⠹⣿
⠄⠄⠄⠄⠄⠄⠄⠄⠄⣀⣤⣶⣾⣿⣿⣿⣷⣶⣶⣬⡩⣵⣿⣿⣿⡘⢹⣿⢠⣄
⠄⠄⠄⠄⠄⠄⠄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢻⣿⣿⣞⡄⢿⣜⣿
⠄⠄⠄⠄⠄⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣋⠄⠙⠉⠛
⠄⠄⠄⠄⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⠇⠄⠄⠄⠄
⠄⠄⠄⠄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠄⣿⡄⠄⠄⠄⠄
⡀⠄⠄⢠⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠄⠄⠘⠇⠄⠄⠄⠄
⣶⣾⣿⣷⡹⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣛⣵⣿⣷⣶⣤⡀⠄⠄⠄⠄
⣿⣿⣿⣿⣿⣮⣿⡿⠿⣛⣢⢩⣭⣭⣭⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠄⠄⠄
⣿⣿⣿⠿⣫⣾⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄⠄⠄⠄
⠿⢟⣭⣾⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠄⠄⠄⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣁⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠴⣶⠟⠀⠀⠀⠀⠀⢀⡶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠈⣧⣀⠀⠀⠉⠑⠲⢄⡀⠀⠀⠈⠑⢶⡖⠂⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠤⠖⠒⠋⠉⠉⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⣀⣀⣀⣀⣘⡆⠀⢀⡇⠈⠙⢦⡀⠀⠀⠀⠙⢆⡀⠀⠀⠈⠧⣄⣀⣀⣠⡤⠶⠋⠉⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠖⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡇⠀⣼⠓⠲⠶⠶⠬⣥⣀⣀⠀⠈⠱⣄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢁⡾⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⢦⣌⡳⣄⢀⣀⣀⡀⠠⠤⠤⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⣷⣄⣀⠀⠀⢀⣀⣠⠤⠶⠿⠭⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠩⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣍⠉⠉⢅⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣺⠇⠀⠀⠀⣸⠁⠀⠀⠀⠀⢀⣴⣾⠟⡉⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡈⠓⠦⢤⣀⠠⠄⠀⠀⣹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢸⢈⠀⠀⣰⡇⠀⠀⠀⣠⢶⡿⠋⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡤⠶⠞⣻⡿⠟⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡄⠀⠀⣿⢁⠀⠒⢉⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⣠⣇⠀⠀⢸⠀⠀⣴⠋⠀⠀⡈⠀⣠⠖⠛⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠝⣦⠀⠀⠀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣧⠐⠒⠉⠁⠀⢻⡀⠀⢸⣤⡞⠁⠀⠀⢀⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢠⠏⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠀⠀⠀⣀⣠⡼⣧⠀⢸⡏⠀⠀⠀⡤⢸⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡟⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠋⠉⠀⠀⣸⣆⠈⣷⠀⢠⡞⠁⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡏⠀⣸⠃⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣾⡿⠙⣦⠸⣶⠋⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠠⠎⠀⢀⡟⠀⠀⡏⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠑⣄⢈⣽⠷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣰⢿⡿⠁⠀⠈⢻⢿⣀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢰⡇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢸⡄⠀⠘⣿⠁⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡞⢡⣿⠁⠀⠀⣰⠟⠀⠙⠂⠀⠀⠀⠀⠈⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⢸⠄⠀⠀⢦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢠⠀⠉⢧⡀⠀⢻⡄⠀⠀⠀⠼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠐⣼⠃⡀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⢠⡏⠀⠀⠀⣟⠀⠀⠀⣾⡆⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢧⣀⡀⣿⣽⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⠀⠀⠀⠉⠀⠀⢿⡄⠀⠀⠀⣿⡳⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣇⢸⡏⠐⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡿⠀⠀⠀⠀⢠⠟⠀⠀⢠⣿⣧⠀⠀⠀⠁⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢹⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⣶⣾⡿⣷⣍⠎⡝⣛⠻⡛⢟⢳⢲⢛⡖⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⠠⠁⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢳⠃⠀⠀⠀⢠⡏⠀⠀⢀⡞⣿⣓⠀⠀⠀⠀⣿⡀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣧⡙⢦⠱⣿⣿⣟⠴⣡⠓⣍⠎⣆⠣⢎⡜⡱⠜⣌⠻⣷⡀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠤⠤⠴⢤⢦⣬⣍⣟⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢿⡏⠄⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡌⠀⠀⠀⢀⡏⠀⠀⢠⣟⡿⢻⡇⡃⠀⠀⣾⡇⢷⡀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⡆⠀⣰⣿⣧⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣧⡙⢆⡏⡼⣻⣿⣷⣎⡝⡤⢫⠴⣙⢬⠲⣍⠳⣌⠣⢽⡇⠀⣼⠁⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡄⢀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢸⡆⡀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⠇⠀⠀⠀⣼⠃⠀⢠⣿⠏⠀⠸⡇⠹⠀⠀⣿⣧⠀⣿⣄⢿⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣸⠁⢠⣿⠻⣿⡀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡗⣜⢣⡜⣡⠽⣿⣿⣿⣟⣷⣧⣞⣌⠶⣉⠶⣉⠶⣉⢞⣷⠀⣯⠀⠈⠑⠂⢄⣀⣠⡶⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⠀⠁⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡧⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⠀⠀⢀⣾⡇⠀⢠⣿⠋⠀⠀⠀⢻⡀⠀⣸⣿⡟⠀⠘⢯⠾⣷⡄⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣰⡿⠀⢾⡏⡐⢻⣁⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⡟⢬⡓⡜⢦⡹⣙⣿⣿⣽⣶⣝⣎⢟⡻⢟⡶⡿⣶⢯⣾⡏⠙⠛⠛⠶⠞⠛⠛⠉⠁⠀⠀⠀⠀⢀⣠⢞⣁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⠞⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⡀⠀⣠⡞⣹⣇⣠⣿⣁⣀⣀⣀⣀⣼⣷⡿⣿⠛⣧⡃⠀⢸⠓⢌⠻⣄⠀⠀⠀⣰⠏⡀⠀⠀⠀⣠⣿⠁⢸⡟⣴⠈⣿⡆⡄⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⡏⡞⣥⡙⡆⢧⢣⡛⣷⡻⣮⣟⢻⣾⣷⣯⣼⣵⣭⣾⣿⠀⠉⠓⠒⠶⠶⠶⠦⣤⢤⡤⡤⡴⠾⠟⠛⠉⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠞⠋⠡⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣴⣿⡿⠋⢹⡿⣹⡏⠉⠉⠀⠀⠀⠀⢻⣥⣿⠀⢻⡄⠀⠸⡆⠀⠀⠈⠃⠀⣰⠟⠀⠙⢦⣀⣀⣿⣧⣸⣿⡿⠃⢹⣸⣷⠹⣆⠀⠀⠀⠀⣯⡇⠀⠀⠀⠀⠀⠀⠀⢠⠧⣿⡟⢼⢢⠳⣍⢎⠧⣙⢞⣷⡭⢿⣶⡜⣚⢟⡿⣭⣫⢝⡛⢷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠉⠁⠀⠀⠀⡡⣷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣧⣿⡿⠁⠀⠀⢸⣷⣿⠃⠀⠀⠀⠀⠀⠀⠈⢷⣿⣆⢸⣷⡀⡀⣇⠀⠀⠀⢀⡼⠏⢀⡠⠄⡈⣹⣿⣿⣷⣿⣟⠁⠀⠈⢷⣿⡆⠸⣆⠀⠀⣸⢿⠃⠀⠀⠀⠀⠀⠀⠀⡞⢁⣿⢣⢟⡥⣛⠴⣋⠼⡱⢎⡞⣱⢣⢏⣿⣜⠮⣜⡹⢷⣮⡙⠶⣌⢻⢿⣴⣠⣂⣄⠠⠀⠄⡀⠀⠄⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠖⠊⠉⠁⠀⠀⠀⠀⠀⠀⣀⣼⣟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠑⢦⣄⣀⣸⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⡈⢻⣷⡀⠹⣆⠀⣠⠞⠁⡀⢁⠠⣰⣾⣿⣿⣹⣿⠏⠉⠙⠓⠦⣈⣸⣷⠀⠘⡆⠀⣘⡟⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸⣿⣍⢾⠲⣍⢞⡡⢏⡵⢣⠞⣥⢋⣞⢴⣻⣿⢆⡻⡜⡽⣯⡳⣍⢮⠹⣽⡎⠉⠉⠛⠛⠶⢦⣿⣟⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⣽⣶⠟⡡⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣾⣿⡘⢙⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣈⠳⢄⠀⠀⠀⠉⠀⠀⢻⣷⡄⠙⠺⠁⠄⡐⢠⣠⣾⡯⣻⣿⢟⡽⠃⠀⠀⠀⠀⠀⠈⠙⣿⣧⠂⠠⢘⣸⠇⠀⠀⠀⠀⠀⠀⠀⡾⠀⣟⡿⣘⢮⠳⡜⢮⡱⢫⡜⣣⠻⣔⢯⣾⣟⢿⣿⣯⣵⣿⣛⠿⣷⢩⣎⣷⣿⠡⢀⣀⠀⠀⠀⠀⠀⠉⠙⠶⢤⣀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠶⠚⠉⠁⣰⡾⢋⢅⡊⠴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠔⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠛⠉⠉⠹⠏⠙⠛⠿⣿⣿⣷⣌⡓⠀⠀⠀⠀⠀⠀⠻⣟⣧⡀⢁⣢⣴⡿⠟⠋⣰⣟⣩⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠁⢢⡟⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⣿⡿⣡⢏⣳⡙⣦⢝⡳⡜⣥⢟⣼⣿⢻⡿⣾⣿⣿⣭⣛⢿⣗⣎⣳⣾⢻⠛⡷⣶⣬⣁⠒⡦⣄⠀⠀⠀⠀⠀⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⠀⣀⣀⣠⣠⡤⠤⠴⠶⠒⠛⠋⠉⠁⠀⠀⠀⢀⣴⠟⢉⠰⡁⠦⢌⣿⢿⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣸⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢟⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⣤⣤⣜⡛⠿⣿⣷⡶⠀⠀⠀⠀⠀⠀⠈⠻⡷⠟⠉⠀⠀⠀⣾⣟⣯⣽⣷⣶⣶⣶⣶⣶⣶⣦⣄⠤⠤⠿⡿⣎⣽⠁⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⣿⣗⡣⣏⠶⣙⢦⣏⣶⣽⡾⣟⢯⡜⣧⣝⣧⢫⡽⣷⣎⢯⡽⣻⡿⢛⠷⣭⣄⠀⠉⢿⡷⣦⣈⠛⢦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡞⢉⣁⣤⠤⠶⠖⠚⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠠⣌⠱⡘⣂⣿⠫⢼⣇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠠⡿⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡞⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣧⣿⣿⣷⡬⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢿⣟⠻⠉⠉⠉⠙⠻⣿⣿⣷⣴⠿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠿⣾⣷⠾⣿⢿⣟⠿⣝⢧⢻⡜⣮⣵⡟⣏⢖⣫⡜⢯⢿⣮⣽⡟⠱⡈⢆⠡⠋⠛⠶⣬⡙⠾⢻⣆⠄⡙⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣞⣯⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⢄⠠⢈⡼⢡⢒⣵⠟⢁⠣⠾⣿⠈⣇⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⣶⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠹⣆⠀⠀⠀⣦⣀⣀⣠⣾⣿⣿⣿⣻⣿⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠋⠁⠀⠸⣾⣾⣶⣤⡀⠀⠀⠈⢿⣿⣿⣏⣶⣿⠴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣻⡇⢏⠴⣻⣯⣛⢮⢏⣧⣽⡟⢧⡛⣜⢮⡱⡞⣭⠞⡽⣿⣌⠱⠈⠂⠀⠀⠀⠀⠀⠉⠳⢦⣽⣧⠐⡈⢣⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉⠀⠂⠌⡐⠠⣠⠿⣬⡵⠟⠡⠈⢄⢏⣵⣿⡄⢻⠀⠀⠀⠀⠀⠀⠀⡿⠀⢠⠂⠀⠀⣼⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠠⡀⠈⠑⠦⣀⣿⡉⠉⠉⠿⣿⠟⢍⣉⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⢀⠀⠀⠀⣽⣿⣿⣿⣿⡀⠀⠀⠸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡜⢪⠜⣽⣷⡹⣎⣿⡿⣙⢮⠳⣝⢎⡶⡹⣜⠮⣝⡳⣭⢻⣧⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠉⠛⢦⣍⡧⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠋⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠙⡄⢈⠐⣀⣷⡿⠟⡉⢐⡈⠄⡁⢎⣾⣟⣻⡇⠈⡇⠀⠀⠀⠀⠀⢀⡇⠀⣼⠀⠀⢰⣿⠀⠀⠀⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢰⠀⢣⠀⠀⢠⠀⠉⠛⠶⣤⣀⠘⠂⣰⡿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣿⣤⣤⣾⣿⣿⣿⠿⢿⡇⠀⠀⠘⣿⢟⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣌⠣⡝⣼⣿⣳⣿⢫⡼⣙⣎⠟⣜⠮⣕⣏⢎⡟⣬⢳⢎⣏⣿⡇⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠈⠁⠀⠀⠀⠀⠀⠀⢰⣠⡶⠏⢁⠠⢀⠠⢀⠠⢀⠠⣼⣿⣏⣿⣷⠀⢷⠀⠀⠀⠀⠀⢸⠇⠀⣿⠀⠀⢸⣿⠀⠀⠰⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⡆⠸⡆⠀⠸⡄⠀⠀⢆⠀⠉⠰⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢰⠀⠀⠰⡀⠀⠈⡄⣿⡿⡉⠹⢿⣿⠈⢤⣼⠇⠀⠀⠰⠏⣈⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠤⣈⠦⣁⣿⣿⢿⡸⣆⢧⣉⡌⡶⡸⣆⡸⡌⣎⡴⣡⣎⣼⠿⠁⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⢡⡌⠐⠠⠂⠄⠂⠄⢂⠰⣼⣿⣷⡟⢯⢿⡄⠘⡆⠀⠀⠀⠀⢸⠂⠀⣿⠀⠀⣼⡏⠀⠀⢬⣿⠘⡆⠀⠀⢰⠀⠀⠀⠀⠀⣿⡇⠀⠸⡄⢳⠀⠀⢳⠀⠀⢸⡄⠀⢰⡀⠀⠀⠀⣇⠀⠀⢰⡴⠋⣷⠀⠀⡇⠀⠀⣧⠀⠀⠹⠙⢧⣤⣤⠤⠶⠾⠟⠏⠀⡄⠀⠀⢰⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⢎⡱⢎⣵⡯⣿⢧⡻⣜⠾⣘⢮⡳⡽⣜⡳⣝⣾⣼⣿⡋⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⣷⠀⠀⠈⠀⠀⠀⠀⣀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⠋⠀⠀⠀⠘⣧⠈⠄⠡⢈⠐⡈⢔⣢⣿⣟⢣⠹⡘⣾⡇⠀⢿⠀⠀⢀⣴⢿⡇⠀⣿⠀⠀⡾⡇⠀⠀⣽⣿⢀⢻⠀⠀⢸⠀⠀⠀⠀⠀⣿⣧⠀⠀⠁⠘⡇⠀⠘⡇⠀⠀⢇⠀⠀⢧⠀⠀⠀⢸⡄⠀⠸⡄⠀⣾⠀⠀⢻⠀⠀⠸⡄⠀⠠⡀⠀⠀⡄⠀⠀⣇⠀⢸⠀⢡⠀⢐⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣏⠶⣉⢾⡏⣷⣿⡷⣹⠮⣽⢹⡎⣗⣳⣭⣷⣟⡻⢍⡞⣳⣦⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠘⢷⡀⠀⣀⣠⠴⠚⠉⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢈⠐⢂⠡⣘⣴⡟⠳⢀⠆⢣⢍⣿⣿⠀⠈⡇⢀⡾⡁⣾⡇⠀⣿⠀⠀⢹⡇⠀⠀⣿⣿⡂⢸⡇⠀⢸⡆⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠸⠄⠀⢸⡄⠀⢸⡄⠀⠸⡆⠀⠀⠀⢧⠀⠀⢹⠆⠛⡆⠀⠀⣇⠀⠀⢷⠀⠀⢧⠀⠀⢿⠀⠀⢹⠀⠀⣇⠈⠂⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣾⢿⣏⠶⢡⡿⢱⣏⣿⡗⣧⢻⣬⢳⣽⡿⢿⢫⠟⣿⣜⡣⡜⣥⢺⢳⣆⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⣀⣤⠿⠚⠉⠀⠀⠀⠀⢀⣠⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡌⢢⣵⠟⠃⡌⠐⠡⣈⠺⢶⣼⣿⡇⠀⣹⡟⠛⠐⢾⣿⡀⣿⡅⠀⣸⣿⠀⢘⡯⢽⣇⢂⢻⠀⢸⣷⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⢣⠀⠀⢳⠀⠀⠀⠸⡆⠀⠘⡆⠀⢻⠀⠀⢹⡀⠀⠘⡆⠀⠘⡆⠀⠘⡆⠀⠈⣇⠀⢸⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⡟⣾⡏⢆⣿⠇⣥⡿⢾⣟⡼⣳⣾⢿⣻⡟⣌⠳⢊⠔⡙⢷⣽⣢⢇⡫⡜⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡽⠷⠛⠉⠀⠀⠀⣀⣤⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠾⢛⠡⡈⠥⠘⡠⢃⢤⣳⠿⣣⢛⣧⣰⠟⠀⠀⠀⠸⣿⣧⣿⡄⢀⠻⣿⠀⢨⣗⣫⣿⠄⡚⣇⠈⣿⡄⠀⠀⠀⢻⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠘⣇⠀⠀⠀⢳⠀⠀⢻⠀⠈⡆⠀⠀⢷⠀⠀⢹⠀⠀⢷⠀⣸⢳⠀⠀⢹⡀⠀⠧⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣹⡿⣘⣿⢃⢦⡟⡨⢸⣟⡽⣿⡞⢯⡱⢎⡼⢿⣦⠑⡌⢢⠑⠢⢌⠛⡻⢷⣽⣔⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠁⠀⠀⣀⣤⡴⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⡁⠎⢠⠃⠌⣄⣣⣴⢟⡋⠃⠹⡔⢻⣿⡟⠀⠀⠀⠀⠀⠉⣿⣼⡗⢨⢸⣿⣇⠸⣿⢾⣿⣧⢂⠹⠀⢭⣷⠀⠀⠀⢹⣿⢫⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠙⠦⣄⠈⠇⠀⠀⣧⠀⠹⡀⠀⣈⣦⡀⠀⠃⠀⠈⣿⣇⣸⡇⠀⠀⡇⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠠⣼⡿⡱⠋⣽⠌⣾⠃⡵⢸⣏⠶⣉⢮⠱⡜⢂⠦⣉⢿⣾⡄⠣⢌⠱⡈⢆⣱⣽⠾⠛⠛⠃⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠁⠀⢀⣠⡴⠾⠛⠉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⢀⣤⣭⡴⠾⠞⠋⠉⠈⠀⠀⠀⠀⣿⡽⣿⠁⠀⠀⠀⠀⠀⠀⠨⢿⣿⣾⣿⠿⣿⣸⣧⢇⣻⣿⡆⡡⠃⢄⢺⣇⠀⠀⢸⣿⣣⢚⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⡤⢬⡤⠴⠖⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣴⣿⡿⢁⠃⣿⣸⣇⠣⠄⢸⣏⢮⠱⣾⡡⠘⡄⠒⠤⢊⠿⣿⣵⣌⣦⢿⣛⡭⠴⠖⠒⠋⠉⠉⠉⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠂⢀⣠⣴⠞⠋⡡⠔⠊⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣻⣯⡥⠤⠶⠶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢏⣽⣧⠀⠀⠀⠀⠀⠀⢀⣤⠾⠟⠉⠀⠀⢹⣷⣿⢎⢖⣻⣿⡤⢉⠄⢂⢻⡄⠀⢸⣿⣧⣯⣾⢿⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠈⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⣰⣿⣿⠱⣈⠆⣿⣾⢃⠒⡀⢸⡯⢎⡳⢜⡻⣥⠌⠱⡈⣆⣦⠷⠟⠙⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⣴⡟⠉⠈⠀⠸⣄⣀⣀⡀⠀⠀⢨⠆⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠞⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢏⠱⣨⣿⠸⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⢠⣿⣿⣏⢮⣼⣿⣿⡄⢊⠔⡈⢿⡀⢈⣿⣿⢿⠿⡿⣛⠟⡷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣴⡿⠔⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣴⡿⣽⡇⠒⣬⠐⣿⡟⣈⠒⠄⢸⣿⡱⢎⣣⡳⣌⡾⠷⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠀⠀⠀⠀
⠀⠀⣠⡾⠃⠀⠀⢀⣼⢟⡟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠉⠀⢀⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠟⢩⠘⡄⡃⣿⣇⠱⡃⠀⠀⠀⠙⢷⣄⠀⠀⠀⢀⣠⣔⣦⣯⣿⣿⣟⢏⠳⣿⣾⣅⠊⡔⢉⢷⠀⣻⢿⣯⡚⡵⣩⠞⡱⣌⣿⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⢠⣾⣟⡳⣿⢨⠑⡰⡌⣿⡗⡨⡘⡄⣨⣷⣿⣼⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣴⠟⠁⠀⠀⣠⡟⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢾⠋⠁⠈⠀⠀⠀⣴⠟⢙⣧⢊⡇⠀⠀⢀⣤⠶⠋⠙⠟⠻⠿⠿⣟⣿⠿⠧⠟⠴⠚⠃⡽⣿⣮⣗⠈⠆⡌⠡⢌⢿⣿⣷⢱⢣⣾⡿⢻⡍⢾⣿⡍⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣟⢻⣿⣹⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⢀⣶⣿⢿⡿⣡⣿⣧⠚⡐⠿⢿⣧⣥⣿⣟⡽⠋⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣼⠋⠄⠀⠀⣼⠏⡐⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠙⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣾⣟⡯⠓⠈⠀⠀⠀⠀⠀⠀⣾⠃⠀⢸⣿⡄⣃⣤⡾⠋⠁⠀⠀⠀⠀⣙⣸⣯⡿⠋⠀⢀⡴⠒⣄⠀⢹⡿⣿⣧⢫⣖⢌⠱⡈⢎⣿⣿⣧⣿⣧⣝⣣⣞⣻⣿⠏⡗⢢⠜⡙⠶⢦⣀⡀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⡿⣿⣿⡳⢬⣻⠷⣿⡟⠈⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⣴⣯⣿⣹⣾⣗⢸⣇⢻⣧⡑⠌⣂⣩⣿⠿⠋⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪
⠏⡌⠀⠀⣼⢇⠊⣴⡇⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⡧⠴⠶⠖⠚⠛⠻⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⡶⡟⢿⡹⣽⡿⣛⣧⠟⠃⠀⠀⠀⠀⠀⠀⠀⢠⣼⠇⠀⠀⠈⠝⣷⡟⠁⠀⠀⠀⠀⠀⣠⣾⣴⠿⡫⠀⢀⡴⠋⠀⠀⠘⡆⣸⠇⢸⣿⣧⣌⢳⣦⠑⡬⢸⣿⣧⠉⠉⠉⠉⠉⠉⠻⠀⠙⠳⣎⡥⢙⢢⠩⢍⠳⣛⠫⢍⠭⣩⠙⡜⢢⡓⢼⣽⢧⣋⢷⣿⣿⠏⠀⠀⠀⠀⠀⢀⡴⠟⢋⠠⠜⣻⣶⣿⢿⢿⣿⣏⣼⡏⠉⠉⡏⠛⠋⠉⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡛
⠐⡈⠀⢰⡟⡠⢃⣾⡇⠀⠀⢠⢠⣤⡴⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢤⣤⣠⣴⢾⡛⣭⢚⡥⣛⣶⠟⣽⡧⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⠀⠀⠀⠀⣾⠋⠀⠀⠀⣀⣤⣴⣾⡟⡍⢎⡑⢀⡴⠋⠀⠀⠀⠀⠀⣽⣿⠀⠘⡄⢽⣮⢇⢮⢳⣔⠩⡝⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢮⣆⠹⡨⡑⢦⡉⢎⡒⢥⡙⣌⠣⣜⣿⣯⣶⣹⣾⣿⣫⠁⠀⠀⠀⢀⡴⢋⠒⡉⠤⢃⣾⣿⣹⣩⠶⠋⠀⢹⣼⣧⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢟⡏⢶⢩
⠈⡔⠀⣿⢡⠑⣂⢻⡇⢀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢷⣽⢆⣏⣶⠟⢥⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⠀⣸⣧⠷⠶⡿⢛⣿⣿⢣⠽⣖⣩⣦⣤⠭⠤⠖⠒⠒⠒⠒⠒⠶⠿⠤⣤⣉⣀⠹⣿⣮⡱⢎⡕⡢⢟⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣵⣈⠖⣑⠊⡜⡰⣌⡴⠓⣾⣃⠎⠻⢹⣿⠓⠆⠀⠀⣠⠞⣋⠢⡁⢎⣑⣪⡿⢫⡶⠛⠁⠀⠀⠀⢾⣿⣿⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⣏⠞⣜⢪⠱
⢁⠒⠤⣿⠢⣉⢤⣭⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣩⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣠⡟⠁⠀⠁⢈⣥⣿⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣽⣚⢬⡳⢬⣼⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠮⠷⠞⠋⠉⠀⣼⠃⠀⠀⣲⡟⢡⠊⢀⡠⠞⢩⠐⣢⣱⢼⣾⣿⡫⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣾⣿⢻⡹⡜⣯⣿⠩⠜⣠⠃
⢨⠘⣂⢿⣷⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢻⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢋⡙⣿⣇⣰⠏⠀⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢯⣶⢼⣽⣿⣥⡀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⡼⢏⠰⣁⠒⣄⢢⣭⢲⣿⡯⣽⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣷⣷⠀⠃⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⢿⡻⢿⣮⢣⢧⡹⣷⡟⢌⠒⡄⢃
⠢⡑⣬⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣶⡆⠀⠀⠀⠀⠀⢀⣤⠞⡋⢅⠢⡘⠤⣶⠏⣿⣧⣤⠞⠋⠀⠀⠀⠀⠀⢀⣠⠤⠤⣤⣤⡤⠤⠤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣞⣿⣿⣟⢧⣄⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⠱⣈⢒⡤⢷⣚⣷⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠈⣿⣹⡇⢰⠀⠀⠀⠀⠀⢰⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⣻⣯⢖⡹⢎⡿⣿⣢⣷⠟⢌⠢⡑⢈⠆
⣱⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣧⡤⠀⢀⣤⠞⣋⠡⡘⠰⡈⢆⡡⣿⣿⣦⡾⠋⠁⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⣤⠀⠈⣷⠀⠀⠀⢈⣿⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣭⣟⢷⡍⠳⣄⡀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⡡⢎⠶⣌⠳⣜⣣⣾⣿⡷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢹⡜⢻⡾⡆⠀⠀⠀⢠⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡿⢿⡹⣌⢳⣹⣿⢎⡱⢫⡜⣽⡿⠓⠈⠂⠁⠈⠀⠀
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⣷⣢⢌⣄⠣⡄⠣⢌⠱⣈⣦⣿⣯⡿⠉⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⣰⠾⠁⠀⡀⣹⠆⠀⠘⣏⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠙⢾⣻⣟⣧⡈⠳⣆⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢣⣋⡜⢮⣜⣼⠟⣫⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠈⢷⠀⠻⣿⡄⢀⣤⡟⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⣋⢧⡙⢦⢓⡬⣓⣾⠏⣎⢑⣧⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⡟⠋⠛⠛⠷⠾⠛⠋⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⣰⡟⣇⠀⠠⠟⠁⠀⠐⡀⣰⠏⠀⡀⠀⠉⠳⠤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠈⢻⣿⣬⡻⢦⡈⠻⢧⡀⠀⠀⠀⠀⢀⣴⡿⢫⢜⣱⣶⠾⠛⠉⠠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠸⣇⡀⠹⣿⣿⠟⢡⣿⠃⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⢟⡹⣌⢳⢩⠦⡙⢎⠎⢴⣽⠫⣜⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣧⣄⣀⣀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⣰⠛⣧⠘⢧⣀⣀⣀⣀⣤⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠐⡄⠀⢧⡀⠙⠛⠿⢷⣤⠙⠒⠒⢂⣴⣿⣯⡼⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣇⠀⠀⠀⠙⣦⠂⢄⣁⣤⠟⠁⠀⠀⠀⣀⡤⣴⣚⣽⡿⢏⡳⢜⡣⡕⢎⠣⡉⢆⡉⢆⢺⣾⣽⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡜⢛⢾⣿⣼⢿⠂⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⠳⣤⣭⣩⣭⡽⠚⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠔⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠒⠦⢤⣄⡀⢸⣿⣿⡄⠀⠀⠀⠈⠛⠛⠋⠁⣀⡤⠴⠚⢩⣠⣾⠿⣋⠷⣘⠧⢚⡡⠱⡈⠆⠥⣑⣢⣭⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠉⡆⣾⢧⡏⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣟⢧⣤⡤⠖⠒⠛⠋⠉⠉⠀⣀⣴⣾⠻⣍⠧⣓⠭⣊⢅⠪⠔⣌⣱⣬⠷⠗⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⣟⠳⡜⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡘⣼⡏⢺⠁⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠈⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⢀⡀⣀⢠⣤⣶⢟⡻⣱⢊⡵⢊⡞⢌⡒⣡⣮⠶⠛⠋⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣾⣿⣟⠷⢊⠱⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡟⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣍⢆⣿⡟⡜⢮⡑⢦⣋⠴⢋⣔⡿⠞⠉⠂⠀⠀⠀⠀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣾⣽⣿⡿⣿⣞⣉⠦⠅⠂⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡹⡜⡣⣝⠲⢌⣾⠞⠋⠀⠀⠀⠀⠀⠀⢀⣾⠏⠃⠀⡀⣀⠠⡄⣤⣖⣿⣿⢾⡻⠿⣚⠙⡲⢥⣻⣦⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⣤⣀⡀⠀⠀⠙⢷⣱⠳⣌⣳⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡣⢱⣘⣴⣦⣿⣾⡟⡯⢓⠬⠍⠐⠁⠀⠀⠑⢦⠹⣿⣷⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⠉⡓⣦⠀⠀⠻⣷⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣷⣿⢿⡿⢓⡎⢼⡛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⠘⣿⣧⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⢠⡾⠛⠉⠉⠳⣄⠀⠈⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⣧⣤⣀⣈⣷⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⣷⠈⢿⣧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠀⠀⠀⢠⡟⠀⠀⢢⡄⠀⠘⣧⠀⠈⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣙⣿⣿⣯⣽⣿⣟⣶⣤⡀⠀⠀⠀⠸⢈⠜⣿⡄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠁⠀⢿⡀⠀⢹⡄⠀⠀⠽⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⢶⣿⠚⠋⠉⠉⠽⣆⠀⠀⠀⠉⠻⣿⡄⠀⠀⠀⠣⢌⣻⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡹⣄⠀⠘⠧⠀⢸⡇⠀⠀⠀⢹⣎⠙⡛⢛⠛⡛⢛⡻⢽⣿⠋⠐⠘⣿⠀⠀⠀⠀⠀⠹⣶⡀⠀⠀⠀⠀⠙⠀⠀⠀⢃⠂⡷⣏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠛⠶⠶⢛⡽⣿⠀⠀⠀⠠⢻⡎⠐⠛⠙⠈⠁⠀⠀⠹⣄⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⢌⠒⣹⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠖⢋⣽⣽⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠈⠙⠒⠚⠉⢸⡿⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠈⣷⣀⠀⠀⠀⠀⠀⠀⠉⠿⣶⡤⣀⠀⠀⠠⢌⣸⡟⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣠⣾⣫⣴⠦⠽⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⡃⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⢀⣿⠃⠀⠀⠀⠀⠨⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣿⢆⡚⣴⡿⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⣾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⢘⢳⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠸⢹⢷⣤⣀⠀⠀⠀⠀⠀⡀⢬⢹⣿⡾⠟⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠹⣇⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⢸⡛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⢳⡝⢯⣵⣦⣤⣴⣢⣷⠾⠛⠋⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣃⢿⡆⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠘⢧⡻⢽⢿⣟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣎⢻⡄⠀⠀⠀⠀⠀⠘⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢼⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢈⣿⣬⡳⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡹⣿⣜⣻⣆⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢹⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠻⢿⣿⣤⠀⠹⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡡⢏⡿⣶⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠋⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠠⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⢿⣞⢷⣄⠀⠝⢷⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣫⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠱⣏⣽⣿⣷⣟⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⠦⠤⠤⠀⠀⠐⠒⠒⠉⠉⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢀⣿⡭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⠀⠀⠀⢹⣿⢌⠿⣦⠈⠃⢿⣦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣵⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣾⠟⣭⢓⡟⣷⣯⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣼⡟⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⢾⣯⡚⠽⣷⡀⠀⠈⢿⣄⠀⣀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⢬⠛⣔⢫⡜⣱⢯⣿⣯⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢀⣱⣿⡹⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠁⣿⡜⠳⡜⣿⡀⠀⠀⠹⣶⠞⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⢎⡝⣬⢓⣼⣿⢟⣹⣿⢟⡻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢢⣿⢧⡓⣥⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⣧⠙⡼⣘⣿⡀⠀⠀⠘⣷⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡼⣘⢦⣻⣾⢣⣛⣾⠳⣎⠵⣣⢏⣿⡿⢷⣶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣠⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠛⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⣀⣿⣏⢧⠹⣰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠐⢿⡘⣤⣷⣿⣷⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⣫⣵⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡴⣩⣶⡿⣡⠗⣾⣏⠳⣌⢳⡱⢮⣿⡳⢌⠦⡩⢜⢩⣿⡿⠿⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⣠⡿⢃⣯⢲⡙⡴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⠭⣐⠲⣿⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⣤⣾⣟⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣳⣯⠳⣥⢻⣟⡬⢳⡌⢧⣙⢎⣿⣅⣈⣁⡀⡀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⣀⣶⡟⣉⠦⣹⢦⠹⣜⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⢹⡇⠀⠀⠀⠀⠀⣼⠉⢻⠸⣿⠐⡢⣙⣿⣿⠀⠀⠀⠀
⠀⠀⠀⠀⠠⠔⠋⠁⣀⣴⢿⣯⠟⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣲⣿⣱⢫⣖⣿⣷⣼⠷⠾⠛⠛⠋⠋⠉⠉⠉⠉⠉⠛⠛⠛⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⠀⠀⠀⠀⠀⠙⠓⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠚⠉⠁⠀⠀⠀⠀⣠⣶⡿⡣⠱⢄⠒⣹⡧⣛⣼⠃⠀⠀⠀⠀⠀⢀⡞⠉⠒⣸⠁⠀⠀⢀⣠⣤⣿⠀⢸⠀⣿⣬⣵⠛⣿⣹⡀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣠⣶⡛⣭⠞⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⢻⣻⠽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠋⢸⡃⡉⠦⡉⡔⡷⣩⡟⠀⠀⠀⠀⠀⠀⢾⠀⠀⢰⡟⠀⠀⠀⢸⡁⢠⣿⠀⠀⡜⣼⡋⢄⡋⢾⣿⡇⠀⠀⠀
⠀⠀⢀⣠⡶⠟⡍⢦⡽⠃⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⣧⡾⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡻⣾⣛⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣔⣲⣯⣷⢿⡇⠀⠀⢾⠠⡑⢢⠑⡰⣟⣼⠃⠀⠀⠀⠀⠀⠀⠈⠉⠀⣼⠁⠀⠀⠀⠈⠓⠞⠻⣆⢠⠷⢹⡇⢢⠡⣻⣿⡇⠀⠀⠀
⣀⣴⡟⢣⡙⠜⣼⠏⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣗⣻⢳⣮⣟⡓⠶⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡤⢔⡒⣞⣭⣷⡶⣟⠻⣜⡟⠀⠀⠀⡯⢐⠡⢊⠜⡰⣯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢼⡇⡅⢣⣿⣾⠀⠀⠀⠀
⡛⢥⢊⡕⢌⠊⠁⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡸⢢⡙⢟⡻⣶⣭⣦⣹⡙⢖⡖⣲⡐⣆⠲⡡⣍⣣⣍⣦⣱⣵⣾⣿⢻⡙⢮⡱⢎⣳⡿⠁⠀⠀⢸⡇⢡⢊⠴⣈⢾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢈⣽⣿⠃⠀⠀⠀⠀
⡘⢆⠣⡘⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⣖⠦⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠙⢦⣽⣲⣿⡻⣝⠻⡟⢿⣿⡟⠛⠛⠛⢙⣻⢯⡙⢧⡍⣿⠟⣦⡙⢧⠱⣎⡞⠁⠀⠀⠀⣿⡁⠆⡌⢆⡸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⣭⡿⠁⠀⠀⠀⠀⠀
⡘⢢⡑⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠬⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢒⣼⡿⣍⠳⣌⢳⣩⢶⡗⠁⠀⠀⠀⣾⠟⣦⡙⢦⣿⢫⡝⢤⢫⠜⣱⡟⠁⠀⠀⠀⢰⡟⣀⠣⡘⢄⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⢃⣸⠟⠀⠀⠀⠀⠀⠀⣼
⡘⠆⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⣾⢯⠱⢎⡵⢊⢇⣽⠋⠀⠀⠀⢠⣿⢯⡙⢦⣽⣟⢣⢇⡺⣑⢮⣹⠏⠀⠀⠀⠀⢀⡿⢐⠠⡑⢬⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟⠁⠀⠀⠀⠀⠀⣀⣾⢫
⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢟⡱⢪⡙⠶⡈⢥⡾⠁⠀⠀⢀⣴⣿⣛⠦⣝⣾⢏⡜⢮⡘⢖⣡⡾⠃⠀⠀⠀⠀⠀⣼⠓⡈⢆⠱⣼⠏⠀⠀⠀⠀⠠⣄⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠀⠀⣠⡶⡟⣬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢤⣀⠀⠀⣠⡾⢣⣋⢼⡥⢋⢔⡹⠋⠀⠀⢀⣴⣿⡿⣏⡜⣳⢿⡱⢎⡼⡑⣎⣧⠟⠁⠀⠀⠀⠀⠀⣰⡇⠱⡈⠆⣽⠏⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣶⠶⡤⣤⣤⣤⣤⡤⡤⠶⠶⠛⠉⠀⠀⠀⠀⠀⢀⢠⣴⡿⢫⢵⡾⠃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⣿⡕⣣⠾⢣⠐⣡⠚⠁⢀⣠⡶⣟⣷⢿⡱⣮⢿⣋⠶⡱⢎⢲⣹⡾⠃⠀⠀⠀⠀⠀⠀⢠⡟⢠⢃⠜⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠞⢻⣧⢹⠙⠉⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡳⣡⠚⣄⣣⣤⡴⢾⡛⢧⣳⣾⡟⣧⡿⣋⠶⣉⢮⠱⣍⡾⠋⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⢃⣬⡾⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠒⠦⠶⣶⣋⣟⣛⣟⢻⡳⣟⡻⣛⣿⠉⠁⠀⠀⢸⠏⠁⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⢟⠻⡍⢣⠜⡤⢛⣼⣿⣽⠿⣙⠖⣭⠪⡕⣪⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡐⣣⣾⡿⠑⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠁⢉⣼⠃⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣊⠵⣈⠧⣈⣴⣿⡿⢏⢧⠛⣬⠹⢤⣳⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣋⣤⠾⣫⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢠⠎⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣸⣶⢿⠿⡿⡹⢜⡩⢖⡙⣦⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡵⠟⠁⣴⠏⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣠⠏⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡜⣎⠳⡱⢍⢎⣱⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠟⠋⠀⢀⡾⠁⠀⠀⠀⠀⠀⣠⡴⠛⠁⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⣴⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀