⠀⠀⠚⢿⣿⡻⠿⢿⣿⡻⠿⠿⠷⠶⠶⣶⣦⣤⣤⣤⣄⣀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⢀⣀⣤⣤⣤⣴⣶⠶⠶⠶⠿⠿⠿⣿⣿⠿⠿⣻⣿⠗
⠀⠀⠀⠀⠈⠻⢦⣀⠀⠉⠛⢦⣄⠀⠀⠀⢀⣠⣴⣿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⢿⣧⣤⣀⠀⠀⠀⢀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠉⠳⣄⣶⠏⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⢷⣆⡼⠋⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⣰⡿⡣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣻⣦⡀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⢠⡾⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡓⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⢀⣤⠶⣾⡿⢻⣿⣿⣿⣷⣶⣶⣤⣄⡀⠀⠉⠀⠀⠀⠈⠀⠀⣀⣤⣶⣶⣶⣿⣿⣿⡿⠻⣿⡶⢦⣄⠈⣷⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠈⠀⠀⠀⠙⢧⡀⠙⠿⣿⡿⠟⣣⡾⠫⠛⠓⠄⠀⠀⠀⠀⠔⠛⠫⢻⣆⡻⠿⣿⠿⠟⠁⣰⠟⠁⠀⠀⠈⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠢⢙⠳⠦⠤⠶⠞⡋⠄⠄⠀⠀⡠⣤⣤⣤⣤⡀⠀⠀⠄⠌⡛⠶⠦⠤⠶⢛⠁⠂⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠂⠨⠭⠭⠍⠓⢄⣊⣠⢀⡒⠏⠒⢼⠔⠊⠕⣀⢀⣈⣂⠔⠉⠭⠭⠭⠀⠂⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⡆⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⢋⣩⠕⢊⣤⠶⠞⠛⠛⠛⠻⠶⢦⣌⠒⢭⣉⠓⠢⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡏⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⣦⡢⡀⠀⠀⠀⠁⠀⠀⡠⠖⠉⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠈⠓⠦⡀⠀⠈⠁⠀⠀⠀⡠⣠⣾⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⠮⣷⣤⣀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠊⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⢀⣠⣴⡿⠞⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠒⠈⠀⠀⠀⠀⠀⣀⣀⣤⡴⠞⠋⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠈⠉⠙⠛⠓⢒⣲⣤⣤⣄⣀⣀⣤⣤⣔⡒⠒⠛⠛⠉⠉⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣾⠟⠉⠁⠀⠀⠀⢨⣿⠇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢿⣏⠀⠀⠀⠀⠉⠛⢿⣦⠀⠀⠀
⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠸⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡼⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⡄⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣦⡾⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢨⣇⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀
⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀
⠀⠀⠀⠀⣠⡾⠿⣿⣄⠀⠀⠀⠀⠀⠀⠹⡧⡀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⡰⡿⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀
⠀⠀⣰⣿⣋⡠⢀⣨⣿⡿⠿⠗⢃⣠⠀⠀⠈⠊⠄⠀⠀⢀⣠⣴⣿⣧⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣠⣿⣧⣤⣀⠀⠀⠀⠜⠋⠀⠀⢠⣌⠛⠾⢿⣿⣏⡀⠀⠀
⢀⣾⠿⣻⠏⣴⡿⣻⠋⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⣴⡟⠋⣴⠋⠉⢙⠷⢄⠀⠀⠀⢰⣿⣇⠀⠀⠀⢀⠴⢛⠉⠙⢶⡌⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠈⢻⡻⣷⡄
⠉⠀⢰⡏⠸⣿⡀⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣼⠃⠀⣰⠃⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠈⢣⡀⠀⢻⡄⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠈⣇⣸⡿⠀
⠀⠀⣿⢡⠀⠈⠛⠿⠶⣦⣼⣤⣤⣄⣀⣀⣀⣀⣀⣻⣇⢰⡟⠀⢠⣿⠀⠀⠀⠀⠀⠀⣸⡏⣿⡀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡷⢀⣿⣀⣀⣀⣀⣀⣀⣤⣤⣼⣤⡶⠾⠟⠋
⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣭⣿⣿⣽⣿⠿⣷⣶⣿⣶⣶⣶⣶⣶⣾⣿⣅⣙⡛⠶⠶⠶⠶⠶⠾⠷⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢻⠛⣧⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠐⠚⠛⠛⠻⠿⠿⠶⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣯⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣼⢿⡟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠻⡞⢷⣄⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣹⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠘⢿⡦⣤⣀⡀⠀⠲⣤⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⣶⠾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡖⣖⣖⢒⣖⣖⢲⠒⡶⡖⣶⣲⡲⠂
⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠿⣶⣍⡙⠛⠲⠶⠿⢿⡿⠿⠭⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠑⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠁