⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠒⠒⠚⠒⢶⣶⣶⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣆⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡛⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠿⣿⣿⣿⣿⡿⠟⠉⢉⣿⣦⣈⣭⣿⣿⠿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⣿⣿⣿⣏⠀⠸⡿⣿⣷⣴⡟⣷⣿⣿⣿⣿⣿⠇⠙⠄⠀⠀⠈⣿⣿⣿⣿⣿⡿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠛⠿⠟⠁⢀⡀⢹⣀⣼⠿⢿⣯⢙⣙⣙⣃⣀⠀⠀⢀⠀⠀⠛⢿⣿⣿⣿⣧⣀⣀⣵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣏⣀⣀⠤⠴⣤⣿⣿⣿⣯⢵⣾⣿⣿⣿⣿⣩⣭⠉⣿⣿⣿⣿⣷⡇⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣿⠟⠋⠁⠀⡢⠀⣿⣿⣿⣿⣷⠘⠟⢛⣿⠟⣿⢿⣿⣶⣻⡟⠻⣯⣹⡄⢿⣿⣿⣿⡿⠛⠛⠛⠛⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⡅⠀⢸⣦⡀⠀⠀⢻⣿⣿⣿⣤⣬⣴⣾⡷⠶⠶⠶⠿⠿⠿⠟⠿⠿⠿⠛⠿⠿⢷⣮⣤⣀⣀⣤⣤⣤⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⢷⠀⢸⣿⣷⣦⣠⣼⠟⠛⠉⠉⠀⢀⣀⣠⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⢶⣦⣤⣤⣄⣙⡉⠙⠻⠿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⣿⣇⣸⣿⠟⠛⢋⣴⣶⣤⣶⣾⣿⠟⠉⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣴⣶⣾⣿⣿⣿⣿⣶⣤⣀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣤⠿⠿⠋⣙⣤⣶⣶⣿⠏⢹⣿⣿⣿⣿⡤⠴⠾⠿⠿⢿⡿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⠋⠙⠺⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⢁⣠⣶⣿⠛⠿⢿⣿⣧⣤⡾⠿⠿⠭⠤⠤⡤⣤⣤⣤⣈⣧⣀⣠⣤⡤⠤⠤⠶⠶⠒⠒⠒⠺⠛⠛⠛⠿⣿⣿⡄⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⣿⠿⠻⣷⣶⡿⠟⣉⣥⡴⠶⠖⠒⠲⠶⠶⠤⢤⣤⣀⣉⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢟⣣⣶⠿⣻⣿⣷⣟⣩⠴⠞⠋⠉⠉⠉⠉⠉⠉⠓⠒⠻⠯⢽⣻⣄⠀⠀⠀⠀⣀⣠⣤⠤⠶⠶⠶⠶⠶⠶⠶⠿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠟⢻⡝⣶⢿⡾⠈⠻⣄⠀⠀⠀⠀⠀⢀⣴⣾⣿⠛⣿⣦⡄⠀⠀⢹⠦⣠⡞⠋⣉⡠⠔⠒⠋⠉⠉⣉⡉⠉⠛⠛⠺⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈⢻⠁⣼⠃⠀⠀⠈⠳⢄⡀⠀⠀⣾⣿⣟⣿⡶⠻⣿⣿⠀⠀⣸⢰⢋⡴⠊⠁⠀⠀⠀⢠⣴⣿⠛⣿⣷⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡄⢿⡆⠀⠀⠀⠀⠀⠉⠓⠲⢽⣿⣿⣿⣷⣴⣿⣿⣁⡔⠁⠘⢏⠀⠀⠀⠀⠀⢀⣿⣿⣻⡟⠉⣿⡇⢀⣰⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡇⠈⣿⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠈⠉⠉⣉⣉⣩⡵⠞⠀⠀⠈⠳⠦⣄⣀⣀⣀⣿⣿⣿⡿⠾⠟⣻⡽⠟⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⣿⠀⢀⡼⠒⠶⣤⡉⠙⠛⠛⠛⠛⠉⢉⣀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠶⠶⣶⡒⣻⣿⠃⠀⠘⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⢠⡟⠀⢾⡇⢰⡀⠈⣧⡀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠚⠛⠋⠉⠉⢙⣿⣿⣿⣆⠀⢹⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⠀⠀⣿⣁⡾⠁⠀⠈⣷⡈⢻⣄⡈⠻⠶⣤⣤⣀⣀⣀⣀⣤⣤⠤⠴⠶⠚⢋⣉⣤⠶⠖⠛⢛⣛⣛⣿⣿⣿⣿⣿⣿⠀⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⣇⠀⠈⠻⠁⠀⠀⠀⠈⠹⢦⣌⠛⠶⢦⣤⣀⣀⡈⠉⢁⣀⣀⣠⣤⡴⠞⠋⠁⢀⣠⠶⢛⣿⣿⣿⣿⣿⠟⣿⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⢤⣭⣍⣉⣉⣉⣉⣉⣁⣤⣤⣤⠶⠞⠋⣁⣴⣿⣿⣿⣿⡟⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣄⣠⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⣤⠾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⢿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠖⠛⠉⠀⠀⠀⣀⣴⢿⣹⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢱⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⡏⢀⣼⣿⣿⣿⣿⡿⠟⣿⣿⠟⠛⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣾⣿⣿⡟⠉⠈⠙⢯⠩⠉⠻⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⢿⣿⠟⠉⠙⣤⣾⣿⣿⣤⣴⣤⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣅⠠⠀⡀⠀⣶⣿⣷⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣯⣿⡇⠀⢀⣴⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⢿⣿⣿⣿⣿⣿⣿⣽⣿⠶⣤⣀⣀⠀⠀⠀⠀⣀⣀⣤⣶⠾⣟⣷⣿⣿⣿⣧⣀⣾⠃⠀⠟⢿⣿⣿⣿⡿⠿⠃⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠉⠀⠙⣿⣿⣿⣿⣿⣿⣿⡿⢷⣉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⡿⠟⢛⣫⣵⣵⣿⣿⣿⣿⣿⣿⣿⣿⡿⣰⣷⣿⣿⣿⠋⠉⠀⠀⠀⠘⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣤⣤⣤⣦⡌⠉⠉⠈⠁⠈⣿⣶⣮⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠈⠉⠙⣿⣿⣿⡆⠀⠀⣀⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⠿⠿⠿⢷⣦⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣾⣿⢿⡟⠀⠰⣿⣿⣿⠿⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢀⣀⣤⣤⣼⣿⡿⣷⣾⣿⠟⠉⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣧⢀⣠⣿⣿⠏⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠯⠛⠿⠿⠿⢿⣿⣾⣿⣿⣦⣤⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠉⠙⠛⠛⠻⠿⢿⣾⣿⣿⣿⡀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⢰⣤⠀⠚⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⣀⡀⢀⣀⣠⣾⡏⢿⣿⣿⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⣠⣤⣼⠿⠯⠤⠤⢤⣉⣿⣀⣉⣉⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠿⠿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⣷⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀