⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠐⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡾⣏⢶⣿⣿⣿⡋⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⠻⣿⣿⣇⢀⡴⠖⠛⠳⢌⠉⠙⠛⠿⢿⣿⣿⣦⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣀⣀⠀⠀⠈⢳⡀⠀⢀⣀⠤⢽⣿⣷⣤⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣇⠀⠈⠛⠓⢦⣼⠗⢦⡞⠁⠀⠀⠈⢻⣿⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣀⡤⠾⠁⠀⠼⣗⣒⠦⢤⣀⣸⣿⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡿⠋⠙⢷⣤⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣤⣀⠀⠀⠀⣦⡀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠙⣧⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠮⣽⣓⠒⠚⠋⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢾⣇⠀⠀⠀⠀⠀⢨⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠀⠀⠀⠀⠀⢸⡿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⡀⠀⠀⠀⠀⠀⠀⣾⢿⡏⠈⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠳⣆⠀⣠⠟⠛⠻⣦⣀⣀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡠⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⣠⡿⠸⡇⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢘⡟⠁⠀⢀⡴⠋⠀⢈⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠛⠙⢧⣀⠀⠀⣀⡾⠃⠀⢀⣠⣾⣿⣿⡇⢸⡇⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣾⢿⡶⢠⣟⠀⠀⢀⡞⠁⠀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⢠⣾⣿⣿⣿⣿⡈⣰⣾⡇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣼⠀⣼⣯⡿⢓⣿⣿⡟⢠⣿⣄⠀⠀
⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣇⣿⡿⠃⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢰⡇⣰⣿⡟⢀⣾⣿⠏⢁⣿⣿⡟⠀⠀
⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⣀⣤⠾⢷⣿⡏⢠⣾⣿⢣⣴⣿⣿⠏⠀⠀⠀
⣇⠀⠀⠈⠱⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⣿⠇⠀⠀⠀⠀⠀⠀⣀⡤⢶⡶⠶⣶⡷⠚⠉⠁⠀⢾⠏⢀⣾⡿⠃⣟⣾⣿⠃⠀⠀⢀⣴
⠈⢦⡀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣿⣿⡿⠀⠀⠀⣠⠟⠡⠞⢁⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣼⡿⠁⠀⠀⣠⠾⠿
⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⡼⠁⣿⣤⡄⢀⡞⠁⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣾⣇⣀⣀
⠀⠀⠀⠀⠘⠳⣄⡀⠀⠀⣀⣤⣄⡀⠀⠈⠙⠳⠦⣄⡀⠀⠀⠀⣠⡿⢤⡀⠀⠀⣴⠃⠀⠘⣿⣣⡾⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠙⣶⠞⠉⠀⠈⡻⣦⠤⠤⣴⣯⣬⣙⠳⢤⣠⠏⠀⠀⠙⢦⠞⠁⠀⠀⠀⠙⣿⣃⡀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣤⠀⢸⣿⣿⡀⠀⠘⣧⠀⠈⣿⣎⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠘⠳⠦⢿⣿⣧⡀⠀⢸⣆⠀⢿⣿⡄⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣭⣀⣠⣴
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢀⠀⠀⠹⣿⣿⣿⠟⠉⠉⠉⢻⣿⡄⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⢀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡾⠛⠛⠛⠻⠿⢿⣿⣿⣷
⠀⠀⢀⣰⣦⠤⡟⠀⠀⠀⠀⠸⣆⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠻⣿⡀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⣿⣧⣾⡟⣠⣦⣠⣆⣴⣾⣿⠿⠿⠶⣶⣦⣤⣄⣀⣀⣀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣷⠀⠀⢀⡀⠀⠀⢻⡇⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠉⠙⠛⠛⣿⣿⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠏⠀⣼⣿⠁⠛⠛⠛⠉⠉⠳⣄⣀⣃⣀⣸⣧⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣶⣟⣫⣍⣁⣀⣀⣀⣬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠿⠃⠀⠀⠀⠀⣀⣀⡠⠿⠿⠿⢿⣻⠋⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠉⠉⣉⣭⣽⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⣤⣠⡴⠞⠋⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿