⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠈⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠓⠒⠲⠦⠬⣭⣉⠁⠀⠀⠀⠈⠻⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀⠀⠀⣸⣿⣿⣷⠀⠈⠛⢿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠚⠁⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣿⣿⣿⣿⡄⠀⠀⠀⠙⠛⠻⣿⣿
⣿⣿⣿⠋⠀⠀⠀⠀⢠⣶⣾⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⡨⠝⣻⠿⣿⣿⣿⣿⣿⣿⣿⡟⠐⠒⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⡿⣿⡷⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⠁⠀⠀⠀⠀⠀⠘⠻⢿⡿⡣⠂⠀⠀⠀⠀⠀⢀⡠⢚⡵⠛⠉⠀⠀⠈⢻⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡏⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⡰⠋⡠⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡑⢦⡀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⡇⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⢠⠎⠀⡴⠁⠀⠀⢀⣀⠤⠔⠒⠊⠉⠉⠁⠀⠀⠉⠉⠉⠒⠒⠤⢄⡀⠀⠀⢱⡀⠘⢦⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢠⣶⡀
⣿⠁⠀⠀⠀⣠⣴⡿⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⡼⢁⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣄⡱⡀⠀⠱⡄⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⣿⡿⠁
⠃⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⡰⠁⠀⣰⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠘⢆⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠈⠉⠀⠀
⠀⠀⠀⠀⢀⣾⡇⠀⠀⢀⠖⠀⠀⠰⠁⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⢆⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣴⣿⡟⠀⠀⢀⠎⠀⠀⢠⠃⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠘⠀⠀⠀⠀⠀⠘⡇⠀⢀⣠⣤⣄⠀
⠀⣠⣾⣿⣿⠃⠀⠀⡜⠀⠀⠀⡌⠀⠀⢀⠇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢰⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠀
⠀⢿⣿⣿⣿⠀⠀⢰⠁⠀⠀⠀⠁⠀⠀⢸⠀⣿⠀⣷⠀⠀⠀⠀⠀⣠⢶⠇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡠⣇⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠀⢸⡆⠀⠀⠀⠀⢸⣿⠟⠁⠀⠀⠀
⠀⠀⠙⣿⡯⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡜⠀⣿⠀⣿⠀⠀⢀⠴⠋⠁⢸⠀⠀⠀⠀⠀⠀⠀⡸⠀⢀⡜⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⢰⢻⣆⢸⢧⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀
⠀⠀⠀⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⡆⢸⣧⠖⠁⣴⡒⠲⢼⠀⠀⠀⠀⠀⠀⢠⠃⢀⠎⠀⡠⠔⠒⢒⡄⠈⠲⣄⠀⠀⠀⠈⡇⢹⣿⠸⡀⠀⠀⠀⠀⣿⣿⣶⡆
⠀⠀⠀⣿⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠈⣷⠞⠁⠀⠀⠈⠙⠢⣼⠀⠀⠀⠀⠀⠀⡜⠀⡞⠀⣎⣀⠤⠚⠉⠀⠀⠀⠈⠢⡀⠀⠀⢇⣸⠿⣇⡇⠀⠀⠀⠀⢿⣿⣿⣧⣀
⠀⠀⠀⢻⣧⢀⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⢸⡿⠀⢹⣾⠀⠀⠀⠀⢼⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⡆⠀⠀⢸⣿⠃⠀⢣⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠈⢆⢸⢷⠀⠀⢻⡄⢄⠀⠀⢨⢻⣿⣿⡟⠁
⠀⠀⠀⡼⠀⣿⠀⠀⠀⠀⠀⢳⠀⠀⢸⣿⣶⣶⣿⣿⣿⣯⣵⣒⢤⡀⠈⣧⠀⠀⠀⠀⠀⣶⠁⠀⠀⠀⠀⠀⡠⢔⣶⣾⣿⣿⣷⣶⣴⣟⢸⢸⠀⠀⠈⣇⣀⣑⣤⣀⡀⠹⡇
⠀⠀⡼⠁⠀⢻⠀⠀⠀⠀⠀⠸⡀⠀⠈⡟⠉⠁⠀⢀⡈⢉⡛⠿⣷⣮⣦⡇⣆⠀⠀⠀⠀⣿⠀⠀⠀⠀⡴⣫⣾⡿⠟⢋⣉⣉⠉⠉⠙⠻⡞⢸⠀⠀⠀⢱⠀⣰⠃⣿⣿⡽⢦
⢀⠞⠁⠀⠀⢘⠀⠀⠀⠀⠀⠀⢷⠀⠀⣇⠀⠀⢠⠃⢀⡄⠘⠀⠈⢻⡷⠸⣸⡄⠀⠀⠀⢸⡄⠀⠀⠀⢞⡿⠋⠀⣼⠁⣀⠈⢃⠀⠀⢀⡇⡇⠀⠀⢰⢸⣰⠃⠀⠸⡎⠻⠿
⠁⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⢹⣧⠀⢸⢄⠀⠈⢧⣀⣀⡴⠀⠀⠀⠑⠀⢣⢿⡄⠀⠀⠈⡟⡄⠀⠀⠋⠀⠀⠀⠹⣄⠀⣀⠎⠀⣀⡜⣼⠃⠀⠀⣾⢸⠃⠀⠀⠀⢸⠀⠀⠀
⠤⠤⠒⠚⢉⢿⠀⠀⠀⠀⠀⠀⢸⢫⢆⠘⡟⠳⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣞⣆⠀⠀⢸⡜⢆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⣿⡝⣹⠀⠀⡸⢙⠈⠄⠀⠀⠀⠀⣇⠀
⠀⠀⠀⠀⠸⢸⠀⠀⠀⠀⠀⠀⠈⡏⡎⢦⢱⠀⠀⠈⠉⠙⠛⠛⠇⠀⠀⠀⠀⠀⠹⣎⠳⡀⠀⢫⠳⢕⣄⡀⠀⠀⠉⠉⠉⠉⢠⡾⠋⢠⡇⠀⣰⣏⡧⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⡗⠈⣿⡀⠀⠀⠀⠀⠀⢣⢸⡸⣯⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠢⡈⢧⠀⠈⠉⠒⠂⠀⢀⡠⠖⠉⠀⠀⣜⠀⣰⣯⠎⢱⢹⠀⠀⠀⢲⡄⠀⣇⠀
⠀⠀⠀⢠⠃⠀⢻⣵⡄⠀⠀⠀⠀⠘⡄⢇⢣⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢁⣴⣹⡏⠀⢸⢹⠀⠀⢀⡟⠙⣆⢸⠀⠀
⠀⠀⠀⢸⠀⠀⠘⣇⠻⣆⠀⠀⠀⠀⢣⠘⡌⣆⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡹⣡⣾⠞⠁⣿⡀⢸⡌⠀⠀⢸⠀⠀⠈⢦⡇⠀
⠀⠀⠀⡼⢣⡾⢷⣹⡄⠙⢷⣄⠀⠀⠈⢳⡘⣼⣾⠿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡛⣷⣿⡛⠁⠀⠀⢸⣣⢸⠃⠀⠀⢸⠀⠀⢿⠀
⠀⠀⠀⣇⡞⠀⠀⢳⢳⡀⠀⠙⠳⣄⠀⠀⢻⡏⡼⡞⢻⠻⣿⣿⣿⣷⣶⣤⣄⣀⠀⠉⠉⠉⠁⠀⠀⣀⣤⣶⣿⣿⣿⡿⣿⡾⠏⠈⠻⣦⡀⠀⠀⢇⢳⡀⠀⠀⢸⠀⠀⠀
⠀⠀⠀⡿⠀⠀⠀⠸⡇⠁⠀⠀⠀⠘⣿⢤⡀⠱⣿⡣⣾⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣶⣶⣿⣿⣿⣿⡿⠿⠕⠛⠉⢃⠎⠀⠀⠈⢷⡀⠀⠘⡆⠱⡀⠀⢸⠀⠀⠀
⠀⠀⢰⠇⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢻⣿⠙⢶⣼⢄⢻⠀⠀⠀⠀⠈⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⡞⠇⠀⠀⠀⠸⣷⡀⠐⠼⣠⢱⡀⢸⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡟⠀⠀⠈⢿⣼⣇⠀⠀⠀⠀⠀⠙⠳⣭⣛⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡜⢨⠂⠀⠀⠀⠀⣿⠁⠀⠀⢱⠱⣣⠘⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢧⢻⢿⣆⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⡏⠉⠻⢿⣿⣿⣿⠀⠀⣠⡞⣱⠃⡤⠄⠀⠀⠀⢻⡆⠀⠀⠀⢧⠻⠀⣇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡿⡇⠀⠀⠀⠀⠹⣦⣙⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠉⠉⠁⣠⡾⠋⡞⡿⣾⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⡾⠀⢸⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣰⠃⣷⡄⠀⠀⠀⠀⣽⡉⢮⣕⠦⣄⠀⠀⠀⠀⢠⡿⠉⢿⣄⠀⢀⣠⡾⢋⡴⢻⣿⡀⢻⣧⠀⠀⠀⠀⣼⣀⠀⠀⠀⠀⡇⠀⠀⣇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠞⢹⡄⢹⣟⣶⣄⡀⢰⡿⠃⣾⢯⡉⣯⡕⠢⢤⣠⣿⠧⢤⣠⠽⣗⢉⡥⠋⠉⠀⠀⢹⣧⠀⢻⣆⡤⡠⠖⢛⠏⠀⠀⠀⠀⢐⠀⠀⠸⠀
⣄⠀⠀⠀⠀⠀⠀⠀⣾⠔⠚⠒⠊⡅⠀⠈⣿⣿⣿⣿⡿⠀⣼⡇⠀⠙⠒⠉⠉⠑⠬⢿⡶⢾⡶⠆⠉⠉⠀⠀⠀⠀⠀⠀⢟⡆⠘⣿⣤⣶⣾⠏⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢠
⣿⣦⡀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣁⠀⠀⢸⡆⢉⣹⡇⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⡽⢻⣿⡿⠀⠀⠀⠀⠀⢀⡟⡄⠀⠀⠈⣇⣿
⣿⣿⣷⣄⠀⠀⠀⡼⠀⠀⠀⠀⢀⢿⠀⠀⠀⢻⡼⡜⠣⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢿⠉⠀⢸⣿⠇⠀⠀⠀⠀⠀⡼⠀⢹⠀⠀⢀⣾⣿
⣿⣿⣿⣿⣷⣦⣰⠃⠀⠀⠀⠀⣸⠀⢧⠀⠀⠀⢳⣽⡀⠙⠿⠶⣖⣒⣤⣄⠀⠀⠀⠀⣸⣏⣦⡀⠀⠀⣀⡠⠤⠞⠊⠁⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⡜⠀⠀⠀⢇⣴⣿⣿