⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠤⠤⠤⠤⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⢒⣫⣭⣴⢶⡶⣶⣻⣞⡿⣞⣿⣳⢶⣶⣻⢿⣖⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣡⣴⡾⣿⡽⣾⡽⣯⣟⣷⣻⢾⣽⣻⢾⣽⣻⢾⣽⣻⢾⡽⣷⡿⣖⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣁⣴⣾⢿⣽⣻⢷⣻⢷⣻⢷⣻⢾⣽⣻⢾⣽⣻⢾⣽⣻⢾⡽⣯⣟⣷⣻⣟⣷⣜⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣁⣴⣿⣻⣞⡿⣾⢽⣯⣟⣯⣟⣯⣟⣟⡾⣽⣻⢾⣽⣻⢾⡽⣯⣟⣷⣻⣞⣷⣻⢾⡽⣦⣱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣶⡿⣽⣞⡷⣯⢿⣽⣻⢶⣻⢾⡽⣾⡽⣾⣽⣳⢯⣟⡾⣽⢯⣟⣷⣻⣞⣷⣻⢾⡽⣯⢿⣽⣻⢮⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⣾⢯⣟⡷⣯⢿⣽⣻⢾⡽⣯⣟⣯⡿⣷⠿⢳⣯⣟⡿⣾⡽⣯⣟⣾⣳⣟⣾⣳⢯⡿⣽⢯⣟⡾⣽⣻⢷⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⣯⣟⡾⣽⢯⣟⡾⣽⢯⣟⣷⣻⡾⠀⠀⠀⠀⣿⢾⣽⣳⣟⣷⣻⣞⣷⣻⢾⡽⣯⢿⣽⣻⢾⣽⣳⢯⡿⣷⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡋⣼⢿⡾⣽⣻⣽⣻⢾⡽⣯⣟⣾⣳⢿⡗⠀⠀⠀⠀⣿⢿⣞⣷⣻⣞⡷⣟⡾⣽⢯⡿⣽⣻⢾⣽⣻⢾⡽⣯⢿⣽⡜⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣿⣟⡿⣧⢿⣼⣻⣟⡿⣧⢿⡼⡟⢸⡇⠀⠀⠀⠀⣿⠘⡿⣼⢧⣟⡧⢿⣿⣻⣟⡿⣧⣿⢿⣼⣻⣟⣿⣻⢿⡼⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣾⣻⢾⡽⣯⣟⣾⣳⣯⢿⣽⣯⣿⡇⠸⡇⠀⠀⠀⠀⣿⠀⢿⣽⣿⣾⡷⢹⣿⣳⣯⣿⣿⠚⣿⣞⣷⣻⢾⡽⣯⢿⣽⢹⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⠃⣷⢯⡿⣽⣳⣟⡾⡏⣿⣿⣥⣬⣧⣥⣀⣁⠀⠀⠀⠀⠁⠈⠈⠁⠀⠉⠁⣈⣙⣥⣼⣭⣧⣀⣿⢾⣳⢯⡿⣽⢯⣟⡾⡜⠄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⢈⣿⢯⣟⣷⣻⣞⣿⡅⠈⢀⣠⠤⠤⢄⡠⣍⠩⡚⠂⠀⠀⠀⠀⠀⠀⠘⣻⡭⢉⡡⠤⠤⣀⡄⠁⠸⣿⢯⡿⣽⣻⢾⡽⣏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⢸⣯⣟⣾⣳⣟⢾⣿⢠⡾⠁⣠⢾⣭⣶⢌⠓⠓⠃⠀⠀⠀⠀⠀⠀⠈⠋⠊⣴⣶⣶⡦⡀⠹⡄⠀⠈⣿⣽⣳⢯⢯⡿⣽⡣⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠲⣟⣾⣳⣟⣾⠀⠛⠈⠇⠀⢻⣿⣿⡿⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠷⡿⣿⣿⡿⠀⠜⠀ ⣿⣳⢯⡿⣽⣳⢳⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⣷⣻⢾⣽⠀⠀⠀ ⠐⠒⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠈⠉⠙⠛⠋⠴⠂ ⣿⣏⠻⣯⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣱⠸⣷⢯⣟⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⣷⡙⢋⣿⢯⡏⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⡟⡻⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢘⡿⢏⣡⢹⣿⢯⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⡇⡇⠘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠸⣴⡿⡿⢺⣿⣿⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠉⠻⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣿⣿⣿⣿⣿⡿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡆⢿⣶⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡼⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⢦⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡗⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢿⣿⣿⣿⣿⣿⡗⠀⠀⠈⠉⠒⠒⠒⠒⠊⠁⠀⠀⠀⠀⣿⠿⢿⣿⣿⣿⣿⣟⣷⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠙⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠉⠀⢸⡿⣽⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⢨⣿⣽⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⠋⠈⠒⠠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣀⠤⠄⠐⠉⠀⡇⢸⣿⡅⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠢⢄⠀⠀⠀⠀⠀⣀⠤⠒⠒⠒⠚⠉⠉⠁⢰⣿⣯⠁⠀⠀⠀⣿⠠⠏⠂⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀