⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠶⠒⠒⠒⠒⠒⠶⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⣋⣤⡤⠶⠚⠛⠛⠛⠓⠶⢤⣤⣀⠈⠙⠷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠶⠛⠉⠀⣀⣀⡠⠤⠤⠤⢄⣀⡀⠀⠉⠛⠶⣤⣶⣌⡛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣿⠿⠋⠀⢀⡤⠖⠋⠉⢱⣆⠀⠀⣴⡀⠈⠉⠓⢦⣀⠀⠀⠙⢿⣝⠛⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣡⠞⠁⠀⣀⡴⠋⠀⣴⣄⣀⡿⠻⣦⣼⡇⢻⣄⠠⢷⠀⠈⠳⢄⠀⠀⠙⢷⣄⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⡾⠃⠀⣠⠞⢁⣄⠀⣸⠏⠚⣿⠁⠀⠈⠉⠀⠈⢻⠒⠸⡇⠀⣰⡀⠹⢆⠀⠀⠙⢷⡀⢈⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣱⠛⠀⠀⡴⠃⠀⣸⡏⢻⡏⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⡀⠀⢻⡆⢨⣇⠀⠈⠳⡄⠀⠈⢻⡄⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣹⠋⠀⢠⡞⢱⣄⣰⡟⠀⠈⠁⢠⢸⣿⠀⠀⠀⠀⢀⠀⠀⢹⠀⠈⠉⠀⣿⡄⣠⣆⠙⣆⠀⠀⠹⣆⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⣸⠃⠀⢀⡎⠀⣾⣾⣿⠃⠀⣰⢀⡞⣼⢻⣀⠀⠀⠀⢸⠀⠀⠸⣆⠀⠀⠀⢸⣧⣹⣿⡀⠘⣆⠀⠀⢹⣆⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣰⠇⠀⢀⣾⠃⢸⡇⡿⡏⠀⢠⠃⣾⠇⡇⠈⣿⠀⠀⠀⢸⡀⠀⡆⣿⣆⠀⠀⢾⠻⠁⢹⣧⠈⠹⡄⠀⠀⢻⡀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣠⡿⠋⠀⠀⢀⡟⠀⠀⣼⣿⠶⠟⢰⠃⠀⢰⡏⡼⢹⣾⣿⠀⢹⡆⢸⡀⣾⡇⢀⣷⠘⣿⡆⠀⢸⡇⢳⠀⣿⡶⣦⣿⡀⠀⠈⣿⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀
⠀⠀⠀⣴⠏⠀⠀⠀⠀⣾⠀⠀⢰⣿⠁⠀⠀⢸⡄⢠⡏⢃⠃⠺⣿⠿⡷⠾⠷⠞⠧⠃⢧⡼⠈⠆⠸⡿⣄⠀⣿⡜⡆⠀⠀⠘⣯⣇⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀
⠀⠀⣾⠃⠀⠀⠀⠀⣸⡇⠀⠀⣾⠇⠀⠀⢠⣾⡇⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠃⠻⣷⠀⠀⠀⠹⣿⡄⠀⠀⢿⠀⢀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀
⠀⣾⠃⠀⠀⠀⠀⢸⣿⠀⠀⢰⡏⠀⠀⠀⢸⣏⣶⣶⠶⣶⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⡶⠶⣶⣤⣻⠀⠀⠀⠀⢹⡇⠀⠀⢸⡇⢸⣆⠀⠀⠀⠀⠀⠈⢻⡄⠀
⢸⠇⠀⠀⠀⠀⢠⣿⡟⠀⠀⢸⡇⠀⠀⠀⠀⢧⣿⠀⢹⣿⣿⣿⡟⠇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡇⠰⣏⡟⠀⠀⠀⠀⢀⣿⠀⠀⠀⡇⠈⣿⡄⠀⠀⠀⠀⠀⠀⢻⡄
⢾⡧⠴⠶⠶⠿⠛⣿⡇⠀⠀⢸⡇⠀⠀⠀⣆⠘⠙⠆⠸⠿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠁⠚⢹⠀⡜⠀⠀⠀⢸⠸⠀⠀⠀⡇⠀⠿⣷⠿⠶⠶⠶⠦⠤⣼
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⠘⣆⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡾⠁⠀⠀⠀⢸⠈⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⡄⠀⠐⡇⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⡜⠀⠀⢰⡘⡇⠀⠀⠀⡇⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢿⣧⣴⣶⣬⣽⡇⠀⠀⡇⠰⠘⡎⠁⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠉⢱⡇⣼⠀⡇⠀⠀⠀⣇⢳⡀⠀⢸⡇⠀⢽⡏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢻⣿⠁⠀⠀⢹⡀⣇⢻⣄⠀⠀⠀⠀⠀⠘⢿⣛⣛⣻⠟⠁⠀⠀⠀⠀⠀⠀⣼⢻⡇⣸⠁⠀⠀⢰⠃⠰⣷⣄⣿⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣼⣿⣿⣿⡾⠛⢁⡾⡇⠀⠀⠀⢸⡇⢻⡌⢿⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⢣⣾⢀⣿⠀⠀⠀⡿⠀⠀⠈⢻⣟⣳⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣠⣶⣿⣿⣿⣿⡟⢁⣴⠟⣠⣷⣴⠀⠀⠘⣿⡈⢳⡜⢷⣄⠉⠛⠶⡤⣀⡀⠀⣀⣀⡴⠶⢺⣿⣿⡿⣡⣾⠃⣼⠏⠀⠀⣸⠁⠀⠀⢀⡿⠉⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀
⢸⣿⣿⣿⣯⣟⣿⣿⡟⠁⢀⣿⣿⣿⣿⣦⡄⢹⣧⠈⢿⡚⠋⣧⣰⠞⣷⠀⣹⠿⣭⢀⣴⠲⣿⡿⠋⠛⣿⠁⣸⡟⠀⣰⣸⠇⠀⠀⢠⣾⢁⡾⠃⠀⠀⠀⠀⠀⠀⠀
⢸⣿⣿⣿⠀⣿⣿⣿⡇⢴⣾⣿⣿⣿⣿⣷⠝⢦⡙⢶⡌⠉⠒⠺⠧⠤⠬⠿⠅⢀⣈⡿⠥⠤⠽⠷⠒⠊⣩⡾⠋⠁⣰⣿⠃⠀⠀⣠⣿⡿⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠸⣿⣿⣿⣿⣿⣾⣿⠁⣼⣿⣿⣿⣿⡟⠉⠀⠀⣿⠈⢿⣆⠀⠀⠀⠀⠀⠀⢀⣀⠀⠁⡄⠀⠀⠀⢀⣴⡿⠁⠀⣴⠟⠁⢀⣴⣿⡿⠋⠀⠹⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢹⣿⣿⡛⢿⢿⢃⣼⠁⠀⠚⠿⣿⠇⠀⢠⣼⣇⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠛⢀⣰⠇⠀⠀⠀⣬⠟⠁⢀⣺⣥⣶⣿⠟⠛⠁⠀⠈⠙⢦⠹⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣿⣿⣿⣶⣿⣾⣷⣿⣶⣴⣾⢿⣀⠀⣸⠁⠉⠿⡆⠀⠙⠻⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠵⠶⠒⠛⣩⠿⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⢦⠈⠙⠀⠀⠀⠀⠀⠀⠀
⠀⢿⣿⡿⠿⣿⣿⣿⠟⠀⠀⠀⠀⠒⣿⠛⠃⠀⠀⠀⠈⠳⣄⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⣠⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢿⡄⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⡀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⢷⠢⠤⠤⠤⠤⢰⣞⠀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠋⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠁⠀⣿⠙⠳⢶⣄⡀⠄⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠞⠉⠁⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⢿⣿⣿⣿⣧⣤⣀⣠⣤⡴⣾⠇⠀⠀⠀⠈⠙⣻⢶⣦⣄⡀⠀⠀⠘⢧⡀⣤⠏⠀⠀⠀⢀⣤⣴⣾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀