⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢀⣀⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣾⠿⠛⢉⣠⣤⡟⣀⣀⣤⣤⣶⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣞⠛⣉⣿⣿⡏⢹⣿⣿⣟⠛⠋⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠉⠉⠉⠉⠉⠙⠿⣿⠛⠛⣿⢿⣿⣿⡿⠟⠛⠉⢉⠛⢷⣿⣿⣯⣿⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⢙⣿⣿⣿⣿⢷⣯⣽⣟⣛⡻⣿⣿⢷⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⡶⣤⣀⠀⠀⠀⠀⡼⠁⠠⡶⠛⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣌⢻⣷⠶⠤⡇⠀⠀⣰⡾⢿⣿⡆⠀⠈⣿⣿⣿⣿⣿⣿⡿⠋⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣄⣷⣤⣼⡏⠀⢾⣿⣷⣀⣤⣿⣿⣽⣿⣯⣿⣶⣤⠨⣭⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣭⣿⣿⢏⣹⣶⣾⡿⢿⣿⠿⠛⠻⡌⠙⢻⣉⡽⠋⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣵⣞⠁⠘⣧⠘⠋⠉⠉⠀⠀⠉⢳⣿⣿⣿⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣥⣈⣳⣦⣀⠀⠀⢠⣤⣄⣼⡟⣣⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣉⣿⣿⣿⣷⣾⣿⣭⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠒⢲⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡼⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠉⠙⠻⢦⣈⢿⣿⡿⠿⢿⣿⣽⡟⠁⠀⠀⠀⢸⠇⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⣇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣔⠒⠶⠤⣄⣀⠉⠛⣿⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⣧⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠙⣆⣀⠀⠀⠉⠙⢶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⣄⣀⣠⣞⡋⠉⠓⢲⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢯⡀⠉⠓⠲⡾⣾⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⣫⢽⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣹⠆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠷⣄⠀⠉⠓⢦⣼⣻⠇⠘⠛⠥⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣡⠼⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢧⠀⠈⠳⢤⣀⡼⢋⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⢀⡤⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢧⡀⠀⣠⠞⢹⣿⡷⠲⢤⡤⠤⠤⠤⠤⠤⠖⠒⠒⠚⢛⣿⣏⡥⠞⢋⣠⣽⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠹⠾⣧⣴⠟⠀⣧⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⣸⢹⡼⣇⣴⠿⠛⠁⠈⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠈⠛⠳⠶⠞⠋⠀⢀⣰⠏⠀⢰⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⢸⣷⠋⠀⠀⣠⠴⠛⠁⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢻⡀⠀⠀⠀⠀⠀⠀⣠⣾⠙⠳⠦⢯⣀⣀⡀⠀⠀⠀⠀⣾⡷⠀⣸⣼⡜⣧⡴⠋⠁⢀⡤⠞⠉⠘⣇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠋⠀⠀⠀⠀⠀⢀⣼⡿⠙⠳⢦⣀⣀⠀⠉⠉⠙⠛⠛⠛⠛⠋⠉⢉⢀⣿⠈⠻⣦⠞⠉⠀⢀⡤⠖⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⣠⠉⠉⠛⠒⠒⠒⠒⠒⠀⠚⠛⠉⣄⠘⣧⠀⠘⢧⣠⠾⠋⢀⣠⠞⠉⢷⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣿⡄⠸⣆⠀⠀⠹⣦⡴⠋⠀⣀⠴⢰⡷⣄⠀⣀⣠⣤⣤⣀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢸⣷⠀⢹⡆⠀⠀⠈⠳⣤⡞⠁⢠⠏⣠⢿⡏⠙⣿⣧⣾⣿⣷⣄⠀
⠀⠀⠀⠀⢀⡏⠘⣆⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⢸⡾⡇⠀⢳⠀⠀⠀⠀⠈⢷⣶⠋⢀⡟⠀⣇⠀⢻⢹⣤⣄⠙⢿⣷
⠀⠀⠀⠀⠘⢧⢀⡟⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⡟⣿⠃⠀⠀⠀⠀⠀⠀⢸⢻⠀⠀⠀⠀⢸⣧⢇⠀⠘⣧⠀⠀⠀⠀⠀⠛⢧⣾⣁⣴⠿⢿⣾⣿⣟⠿⣦⣼⡿
⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⢶⢤⣀⠀⠀⠀⣀⡷⣿⣄⡀⠀⠀⠀⠀⠀⣸⣸⡀⠀⠀⠀⠸⣿⣾⡀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣤⣤⣼⡿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣄⣀⣀⣀⣀⣠⣿⣧⣀⠀⠀⠀⠀⠀⢈⣱⣼⣭⣷⠀⠀⠉⠉⣿⣗⣂⣀⣹⠄⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢉⣉⣭⠴⠾⠚⠛⣿⣯⠭⠉⠛⣿⠛⠛⠛⠒⠒⠶⠶⠾⣯⣉⣹⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⡾⣿⡄⢰⠀⠀⠀⢀⣸⡿⣶⡄⠈⣿⣭⠵⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢿⣧⡿⢻⣾⠶⠶⠴⢾⠸⢇⣿⠄⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠛⣻⢿⣾⢰⣤⣦⡞⢿⣿⡤⠆⠀⠈⣷⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⢿⠇⢸⢹⣸⣿⣽⣷⠈⢟⠃⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢷⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡤⠤⣾⠉⠀⢸⠀⠀⢀⡴⢋⣷⣀⡀⠀⠀⠀⠀⠀⠀⢠⣄⣿⠀⠀⠀⠀⠀⠀⠀⢉⣉⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠈⣧⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⣆⠀⢸⡇⠀⣇⠀⠀⣏⣤⠖⠋⠁⢨⡿⣧⣙⠲⢤⣀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⣀⣴⠟⠉⠉⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡞⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣤⣸⣶⣿⣤⣀⡽⠶⠛⠉⠀⠀⠀⣠⣞⠁⠀⠉⠙⠓⣏⠓⠄⠀⠀⢼⢉⣇⠀⠀⠀⠋⠩⠏⠀⢀⡤⠖⣮⢷⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠉⠉⠉⠉⣩⡽⠛⠉⠀⠀⠀⠀⠀⠀⣴⡿⠋⠛⠲⠤⡄⠀⠙⢧⠀⠀⢠⡏⢸⢿⣀⣀⣀⠀⢀⡴⠂⠀⠀⢀⡸⢿⡇⠀⠀⠀⢀⣤⠟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀