⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⡇⠀⠀⠀⢰⡇⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢧⠀⠀⠀⢘⣿⡄⠀⠀⢸⡇⠀⠀⢠⣿⡃⠀⠀⠀⡼⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠘⣧⠀⠀⠈⣿⣷⠀⠀⢸⡇⠀⠀⣾⣿⠁⠀⠀⣼⠃⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⢻⣷⣄⠀⠀⢹⣧⠀⠀⣿⣿⡆⠀⣸⣇⠀⢰⣿⣿⠀⠀⣼⡏⠀⠀⣠⣾⡟⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⢻⣿⣷⡀⠀⢿⣧⠀⢻⣿⣿⠀⣿⣿⠀⣿⣿⡟⠀⣼⡿⠀⢀⣾⣿⡟⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀⠀⠀⠀⠈⠻⣦⡀⠀⠹⣿⣿⣆⠘⣿⣧⠸⣿⣿⡆⣿⣿⢰⣿⣿⠇⣼⣿⠃⣰⣿⣿⠏⠀⢀⣴⠟⠁⠀⠀⠀⢀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⡀⠀⠙⢿⣦⣄⠙⣿⣿⣧⢹⣿⣧⢻⣿⡇⣿⣿⢸⣿⡟⣼⣿⡏⣼⣿⣿⠋⣠⣴⡿⠋⠀⢀⣠⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠠⢄⡀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⣀⠙⢿⣷⣌⠻⣿⡧⠛⠛⠉⠁⠀⠀⠀⠀⠈⠉⠛⠛⢼⣿⠟⣡⣾⡿⠋⣀⣴⣾⣿⡿⠛⠁⠀⠀⠀⠀⢀⡠⠄⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣀⡀⠀⠙⠻⣿⣿⣷⣬⡻⣿⠗⠉⠀⢀⣠⣴⣶⣶⣿⣿⣶⣶⣦⣄⡀⠀⠉⠺⣿⢟⣵⣾⣿⣿⠟⠋⠀⢀⣀⣤⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣦⣤⣙⡻⢿⡿⠂⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠐⢿⡿⢟⣋⣤⣴⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⠒⠶⢶⣤⣤⣤⣤⣤⣤⣀⣈⡙⠻⢿⣿⡿⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢿⣿⡿⠟⢋⣁⣀⣤⣤⣤⣤⣤⣤⡶⠶⠒⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣷⣾⠁⠀⢰⣿⣿⣿⣿⡟⠛⠻⣿⣿⣿⣿⠟⠛⢻⣿⣿⣿⣿⡆⠀⠈⣷⣾⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣭⣭⣭⣭⠀⠀⣿⣿⣿⣿⣿⣄⡀⣠⣿⣿⣿⣿⣄⢀⣠⣿⣿⣿⣿⣿⠀⠀⣭⣭⣭⣭⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⢛⣛⣻⠀⠀⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⠀⠀⣟⣛⡛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⡿⡀⠀⠸⣿⣿⣿⣿⣷⣄⡈⠙⠛⠛⠋⢁⣠⣾⣿⣿⣿⣿⠇⠀⠀⢿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠤⠴⠶⠿⠿⠿⠿⠛⠛⠛⠋⣉⣥⣶⣿⣷⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⠟⠀⠀⣾⣿⣶⣬⣉⠙⠛⠛⠛⠿⠿⠿⠿⠶⠦⠤⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠛⣋⣭⣶⣷⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣾⣶⣭⣙⠛⠿⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⠛⠉⠁⢀⣴⣾⣿⣿⠟⣡⣾⣦⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⣴⣷⣌⠻⣿⣿⣷⣦⡀⠀⠉⠛⠻⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠒⠉⠁⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⣠⣾⡿⢟⣵⣿⡷⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢾⣿⣮⡻⢿⣷⣄⠙⠻⣿⣿⣷⣄⠀⠀⠀⠀⠈⠉⠒⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠛⠉⠀⢀⣼⡿⠋⢠⣾⣿⡟⢱⣿⡟⣼⣿⡇⣶⣶⢸⣿⣧⢻⣿⡎⢻⣿⣷⡄⠙⢿⣧⡀⠀⠉⠛⠿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠉⠀⠀⠀⢀⣴⠿⠋⠀⣰⣿⣿⠟⢀⣿⡟⢠⣿⣿⠇⣿⣿⠸⣿⣿⡄⢻⣿⡀⠻⣿⣿⣆⠀⠙⠿⣦⡀⠀⠀⠀⠉⠛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⣰⣿⡿⠃⠀⣼⡟⠀⣸⣿⣿⠀⣿⣿⠀⣿⣿⣇⠀⢻⣧⠀⠘⢿⣿⣆⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⣰⣿⠏⠀⠀⣰⡿⠁⠀⣿⣿⡇⠀⢻⡟⠀⢸⣿⣿⠀⠈⢿⣆⠀⠀⠹⣿⣆⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⢠⡿⠁⠀⠀⣿⡿⠀⠀⢸⡇⠀⠀⢿⣿⠀⠀⠈⢿⡄⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⡾⠁⠀⠀⢰⣿⠃⠀⠀⢸⡇⠀⠀⠘⣿⡄⠀⠀⠈⢷⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢸⡏⠀⠀⠀⢸⡇⠀⠀⠀⢹⡇⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀