⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⣿⣿⡿⠟⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡴⢚⣵⣾⣿⣿⣷⣶⣶⣦⣤⡤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⣴⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣉⠓⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⠀⠀⢀⡴⣣⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⢜⣙⠓⠦⠤⠄⠀⠀⠠⠤⣶⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠀⢀⣿⣿⣆⡴⣫⣾⣿⣷⢹⣿⣿⣿⣿⢟⣭⣿⠿⣛⣭⣽⣿⣿⣿⣿⣿⣿⣿⣾⣮⣴⣤⣶⣶⣶⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢿⣄⠀⠀⠀⣿⣿⣻⣿⣿⣿⣿⣿⣦⡙⢿⣿⣱⣿⣿⣿⣿⣿⣿⡷⡶⠶⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠒⠒⠲⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⢿⣷⣦⣤⣻⣿⣟⣿⣿⣿⣿⠟⠻⣿⣶⣽⣿⣿⣿⣿⣿⢟⣋⣠⣀⣠⣬⣤⣽⣿⣿⣿⡿⠿⠿⠿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣾⡿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠙⠿⣿⣿⣿⣿⣮⣿⣿⣿⣶⣶⣾⠟⠁⣾⠄⠀⡛⣻⣯⣽⣿⣿⣿⣭⣉⣁⡉⠉⠉⠀⠀⠀⠀⠤⠲⠾⠀⠁⠙⡀⣁⣽⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠉⠙⢻⡟⣻⣿⣴⡶⠟⢁⡠⠞⢁⣶⣾⣿⡿⠿⠛⠋⠉⠉⠉⠉⠛⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⡟⣠⣿⣿⣿⣾⣾⣭⠄⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣤⣤⣖⡊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣶⣯⡾⢽⣿⣿⠁⠀⠀⣿⣼⠟⠁⠀⠀⢿⠛⢻⡝⠛⠓⠶⢤⣀⠀⡏⡄⠀⢇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⢻⣿⣽⣿⡏⠀⠀⡜⠙⠯⠀⠀⠀⢰⠇⠀⡿⠀⠀⠀⠀⠀⠈⠉⠳⠃⠀⢸⣸⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡇⢀⣞⡀⠀⠀⠀⠀⠀⢻⠀⠀⢇⣠⣶⠿⠿⢿⠷⢦⣀⠀⠀⣷⡇⠀⠀⠀⠀⠀⠶⢿⢷⣤⣀⣀⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣾⣿⢛⣷⣆⠀⠀⠀⠈⠑⢫⣿⡟⢫⣶⠒⡆⠀⠀⢻⡆⢰⡿⠁⠀⠀⠀⠀⠀⢠⡾⠛⠉⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⢻⣇⠘⣇⢹⣷⣄⠀⠀⠀⣿⡟⣾⠊⠹⣤⠃⠀⢀⡾⠁⡟⠁⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠙⣦⠈⠛⢸⣿⠀⠀⠀⠘⢷⣭⡀⠀⣀⣠⣶⡿⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠹⣿⣶⣿⠃⠀⠀⠀⠀⠀⠈⠙⠛⠛⣿⣉⣴⡾⠃⢠⡏⢹⠄⠀⠀⠀⢀⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣟⡙⢷⡀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣤⣀⣠⣿⠴⠋⠀⠀⠀⠀⠀⠉⢛⣶⡿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣿⠯⠭⠤⠴⠆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣓⡶⠯⠟⠷⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩⡽⢦⡀⠀⣦⠀⠀⠀⢀⣠⠴⠾⠿⣏⠛⠻⣷⡶⡆⠀⠀⢀⣠⣴⣖⣲⣶⢄⣠⣤⣤⣤⣭⣭⣽⣾⣒⣒⣟⣟⣛⣶⡶⠴⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡄⠀⠙⢷⣿⣷⣤⠿⢋⡁⠀⠀⠀⠀⢳⣠⣼⡇⣿⣠⡾⠟⠛⠹⠿⠛⣿⣋⣁⣈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠹⡀⠀⣠⢬⡟⣠⡞⠑⣽⣦⠀⠀⢠⣿⣩⣿⡇⠙⠋⠀⠀⠀⠀⠀⠘⠟⠋⠙⠻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢡⣾⢣⣿⢱⠋⡷⣺⡅⢸⡇⠀⣼⡟⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⢠⡼⢷⠋⠀⠉⡡⡄⠈⣷⡀⢹⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⣿⣰⠋⠀⠀⠀⠈⠁⠀⢸⣷⡾⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠘⣿⡟⢢⡀⠀⠀⠀⠀⠀⣸⠯⡇⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⢛⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣈⠛⠛⠛⠿⠷⠶⠶⠺⠋⣰⢁⣾⠏⠀⠀⠀⠀⠀⠰⣦⣴⣶⡶⡶⠶⠟⠋⠈⢻⣿⡿⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣛⣻⣿⣿⡿⠛⢛⣿⠛⡛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⡀⠀⠀⠀⣼⡏⠰⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀