⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠈⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣇⠀⢀⣠⠇⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠈⠀⠒⠀⠠⠤⠄⣿⣟⡉⣿⠹⢿⡟⠉⠀⠀⠈⣿⡿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣭⣥⣀⠀⢀⣀⣀⣾⣿⣧⣤⣈⣂⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣋⠀⠐⢢⠀⡀⠀⠀⡀⠀⠂⠁⠈⠉⠛⣷⣿⣾⣿⣽⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠄⠈⡀⠀⠀⠀⠈⢡⣼⣆⣰⣷⢰⠀⠢⠀⠀⣼⡿⣯⡿⠛⠛⠛⠜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡠⠀⡠⠊⠀⠀⠀⠀⢀⣾⣿⣿⡟⠁⣮⡄⠀⠀⡀⣤⣀⣀⣸⣶⣷⣷⢆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀
⠀⣰⣠⠚⠀⠀⠀⠀⣀⠔⣹⠋⢨⠏⠀⠘⠻⠛⠛⠉⣹⣝⢿⣿⣿⣿⢿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀
⣰⢋⡀⠈⠐⠀⠄⠘⠠⠞⡁⠠⠊⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣮⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠹⣿⢿⡿⡿⢿⡶⢶⣶⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⠧⢽⣞⣷⣼⣳⣮⣿⣤⣄⡀⢀⠀⢀⠀⣀⣈⣡⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠈⠉⠉⠁⠉⠀⠀⠉⠉⠁⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠋⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣤⣶⣄⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣤⣧⣶⢶⣶⣶⢶⣶⣤⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢄⣤⣾⣟⣷⣻⢾⣟⣾⡽⣟⣾⣽⣻⢿⠂⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⣾⣻⢾⣽⣾⣻⣟⣾⢯⣟⡿⣽⣾⣻⠃⠀⠀⠀⡡⡀⠀⠀⠀⣠⠖⠒⢶⡗⣾⣠⠖⠛⠲⣌⣻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢱⡿⣯⣟⣯⣷⢯⣷⢿⣽⣻⢯⣿⣻⣞⠋⠀⠀⠀⠀⢰⡠⠀⠀⠀⢧⡀⠀⣠⡗⣹⢿⣋⠋⢛⡿⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡄⣿⣟⣷⣻⣽⡾⣿⣽⣻⣾⣻⢯⣷⠿⠁⠀⠀⠀⠀⠀⣼⡇⠃⠀⠀⠀⠉⠉⠉⠀⠉⠀⠉⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⢻⣽⡾⣯⣟⣾⣽⢷⣯⡷⣿⣽⣻⠝⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⣠⣴⣶⣶⣦⣤⠀⠀⠀⣠⣴⣶⢶⣦⣄⠀⠀⠀⣠⣴⣶⣶⣦⣄⠀⠀⣠⣴⠶⣶⡤⠀⣼⡆
⠏⣷⡿⣽⣻⡾⣽⣟⣾⣽⢷⠋⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⡄⠀⠀⣼⡟⠁⠀⠀⠀⠹⣿⠀⣾⣟⣁⣀⣀⣀⢹⣷⡀⣾⠏⠀⠀⠀⠀⢻⣷⠀⣿⣧⣀⡀⠀⠀⣿⡇
⠀⠹⣿⡽⣷⣟⡿⣞⠟⠊⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⡿⠃⠁⠀⠀⣿⡇⠀⠀⠀⠀⢠⣿⠀⢿⣟⠋⠉⠛⠛⢛⣛⠡⣿⡆⠀⠀⠀⠀⣰⣿⠀⠀⠉⠛⢿⣶⠀⣿⡇
⠀⠁⠻⡽⠓⢛⣁⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⡿⣟⡿⣃⠂⠀⠀⠀⣿⡿⢷⣶⣶⡶⠟⠁⠀⠈⠻⢷⣶⣶⡶⠟⠃⠐⣿⠻⢶⣶⣶⡾⠟⠁⠠⢶⣶⣴⡾⠟⠀⢿⡇
⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣯⣷⡿⢟⠑⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠛⠻⢿⣾⣿⣷⣿⣿⣿⠿⢛⡉⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠑⠓⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀