⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠈⠻⣆⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⢀⣀⠀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢱⡄⠀⠀⠀⠀⠘⣷⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢠⣭⣓
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢰⡆⠀⠀⣿⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⢻⡄⣀⣀⣀⣀⣹⣧⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡆⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡶⠾⠛⢿⡟⠉⠉⠉⠉⢿⡍⠛⠛⠓⠻⠆⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢿⣿⣦⠀⠀⠘⣿⣆⠀⠀⠀⠸⣧⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀
⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣷⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⢻⣿⣆⠀⠀⠀⣿⣆⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⣿⣿⡶⠶⣿⣿⡿⢿⣿⣿⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢿⣿⣿⣧⠀⢸⣿⣿⡄⠀⠀⢹⣿⡄⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⣿⠀⣠⡶⠛⣿⣿⣿⡄⠐⣿⠿⣇⠸⣿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⢸⡇⢹⣿⣇⠈⣿⣿⣿⠀⠀⢸⡟⣿⡄⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢹⡟⠉⠀⠀⣿⣿⠹⣧⠀⣿⠏⢿⡀⣿⡧⢻⡆⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⢿⡀⢸⣇⠀⠙⣿⡀⣿⢽⣿⣇⠀⢸⣿⣿⣷⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⣇⠀⠀⢸⣿⣿⠀⢻⡄⣿⠀⠘⣷⣿⡇⠀⢿⡄⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⣿⠘⣷⠘⣿⠀⠀⠙⣧⣿⠀⢹⣿⡀⣸⠋⢿⣿⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⡄⠀⢘⣿⡏⠀⠈⣷⣿⡄⠀⠸⣿⣧⠀⠈⢿⡄⠀⠀⢸⣧⠀⠀⠀⠀⠀⣿⡄⢹⣧⣿⣠⡴⠶⣿⡿⠀⠀⢹⣇⡿⠀⠀⢿⣿⡀⣼⡇⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⡇⠀⠀⠘⣿⣇⠀⠀⠈⢿⡄⠀⠀⢿⡄⠀⠀⣿⣧⠀⠀⠀⠀⣿⠁⠀⢿⡟⠁⣀⣀⣾⣧⣤⣴⣶⣿⣷⣶⣾⣿⣿⣧⣿⣧⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⣿⡇⠀⠀⠀⠈⢻⣟⠛⠋⠉⠋⠀⠀⠀⢻⣆⠀⢿⡘⣧⠀⠀⠀⣿⠀⠀⢸⢷⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠙⣿⣿⣿⣶⠶⠀⠀⠀⣸⣟
⠀⠀⠀⠀⠀⠀⠘⣿⠘⣧⣸⣇⣀⣀⣀⣠⣤⣬⣤⣤⣶⣶⡴⠶⠀⠀⢻⣆⢸⡇⠘⣧⠀⢰⡟⠀⠀⢸⠿⠋⠀⢰⣿⣿⣿⣿⣿⣿⣏⡀⣀⠀⢠⡿⣿⣿⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠃⣿⣴⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠷⠆⠀⠀⢻⣾⡇⠀⠹⣧⣸⠇⠀⠀⠀⠀⠀⠀⠸⣿⡛⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⡷⠞
⣇⠀⠀⠀⠀⠶⢾⣿⣿⡟⠁⠈⢳⣿⣿⣿⣿⣿⣿⣧⣴⣶⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠿⠿⠷⠿⠀⠀⠀⢠⡟⠀⠀⢀⣾⣿⢃⣤
⢿⡄⠀⠀⠀⠀⠀⣿⡏⠁⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣠⡾⢃⣿⣾⡍⣿⡇
⣌⢿⡄⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⢻⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢀⣴⠟⠀⢸⣿⣿⣷⣿⡇
⣿⣿⣿⣦⠀⠀⢀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢻⡇⠀⠀⣼⣿⣿⣿⣇⠀
⣿⣿⣿⣿⣷⣄⠀⠻⣦⣘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣼⡃⠀⠀⣿⣿⣿⣿⡿⠀
⣿⣿⣿⣿⣿⣿⢷⣄⠹⣿⠛⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⡟⠁⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣿⠉⢉⣴⣿⣿
⣿⣿⠻⣿⣿⣿⣿⣿⣿⡾⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣴⣿⣿⣿⣿
⣿⣿⠀⠸⣿⣿⣿⣿⢿⣧⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠆⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⣿⣿⣿⣿
⣿⣿⡇⠀⠈⣿⣿⣿⣿⣿⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⢹⡇⠀⡇⠈⣿⣿⣿⣿⡄⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿
⡇⣿⡇⢠⡇⠀⠈⣿⣿⣿⡇⠀⠈⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿
⢡⣿⡇⢸⡇⠀⠀⢸⡿⣿⡇⠀⠀⣿⣾⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣽⡇⠀⠀⣿⣿⣿⣿⣿
⣸⣿⡇⣼⠃⠀⠀⣿⠀⢺⡇⠀⠀⣿⣿⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿
⣿⣿⢣⡟⠀⠀⣰⠇⠀⢸⡇⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢹⣿⣿⣿
⣿⡿⣼⠃⠀⣰⠏⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⣿
⣿⣿⡏⢀⣼⠋⠀⠀⠀⢸⡇⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⣀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀