⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠀⣿⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠐⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡄⠀⠀⠀⠀⠀⠸⡇⠀⠐⠈⠀⡀⠁⠠⠈⠀
⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⡧⠀⠂⠀⠄⠀⠐⠀⠄⠀
⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣷⠀⠀⠂⠀⠈⢀⠀⠂⠀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⢸⡇⠀⣿⠂⠀⣀⣀⣀⣀⢰⣄⡀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣿⠄⠀⠐⠈⠀⠀⡀⠐⠀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⢸⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢸⡇⠀⠀⠀⠀⠀⣿⡃⠀⠠⠐⠈⠀⢀⠠⠀
⠤⠄⠠⠀⠄⠠⠤⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⡿⣟⠳⢿⠛⢻⡿⠿⠾⢷⣿⣾⣳⢿⡿⣷⣾⡇⠀⠀⠀⠀⠀⣿⡂⠀⢀⠀⠄⠂⠀⢀⠀
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣼⡏⠀⠉⠀⠀⢻⣤⣀⡴⠚⠉⢙⠻⣿⣿⡽⣿⢿⣷⠀⠀⠀⠀⠀⣿⡇⠀⠀⡀⠠⠀⠈⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠘⢷⣒⠒⣦⡤⠞⠓⢿⡀⠀⠀⠙⠉⢻⠻⣿⣯⣿⣿⣤⡄⠀⠀⠀⣿⡇⠀⢀⠀⠀⠄⠂⠁⠀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡆⣰⠏⠀⠀⠀⠀⠙⠦⣭⡝⠉⠀⠀⠀⠀⣿⠲⣤⡀⣠⠿⣄⠙⣿⣿⣿⣿⡁⠀⠀⠀⢹⡇⠀⠀⠀⠂⠀⠠⠐⠀
⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣼⠇⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠰⡶⠶⠿⠯⠽⣛⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣆⠀⠀⢻⡇⠀⠈⠀⠐⠀⡀⠄⠀
⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣱⠏⠀⠀⠀⠀⠀⠀⠀⠾⡧⠴⠺⣿⣍⡉⠲⣄⠀⠈⠧⠀⠀⠀⠉⠙⠷⣇⣻⡆⠁⠀⢈⢻⡇⠀⠀⠁⠠⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣾⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡇⠀⠀⠀⢸⡇⠀⠈⠀⠀⡄⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠀⠀⠀⠀⠀⠀⠠⣸⡇⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡇⠀⢀⠈⠀⢀⠀⠐⠀
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣜⣀⣀⣀⣀⣀⣈⣴⣟⡁⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⡀⠈⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠈⠉⠉⠉⠉⠉⠉⠀⢿⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠄⠀⡀⠁⢀⠠⠀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⢸⡇⠀⠀⠄⠀⠠⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢸⡇⠀⠂⠀⠐⠀⡀⠁⠀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣧⣷⠀⠀⠀⠀⢀⣀⣴⡏⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⣠⡤⠤⠶⠾⠧⣤⡐⠀⠀⢸⡇⠀⠄⠈⠀⠄⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡟⣿⠀⠀⠀⣴⠞⠋⠉⠉⠙⢦⡀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠀⡷⠋⠁⠀⠀⠀⠀⠀⠈⠙⢦⡀⢸⣧⠀⠀⠐⠀⠠⠀⢀⠀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢿⠀⠀⠀⠛⠛⠒⣦⡄⠀⠀⠈⠓⠒⠲⠾⢿⣿⣿⣿⣿⡿⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⡯⠀⠀⠂⠀⠄⠐⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡾⢹⡄⠈⢧⡀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⠃⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠐⠀⠠⠀⠀⠂⠀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢣⠀⠈⠳⣄⠀⠀⢻⡄⠀⠀⢀⣷⠀⠀⠀⠀⠀⣰⣄⣁⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠐⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠠⠐⢂⡼⠋⠀⠀⠀⠀⠘⢧⠀⠀⠈⠳⣄⠀⠈⠙⠛⠉⠈⠳⢤⣀⣤⠶⠋⠀⢉⡽⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠸⡇⠀⠂⠈⢀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⣽⠀⢀⠀⠐⣰⠟⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⣨⠗⢦⡀⠀⠀⣀⣤⠶⠋⠁⠀⠀⣠⠞⠀⠀⠀⣀⡤⠖⠋⠉⠉⠁⠉⠉⠒⠦⣄⡀⠀⣿⠀⠀⠐⠀⠀⠄⠀
⠀⠀⢀⣤⠤⠤⢾⠀⠂⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡴⠃⠀⠈⣧⡴⠿⢯⡀⠀⠀⠀⣀⡼⠃⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣿⠀⠐⠀⠠⠈⠀⠀
⣀⡞⠋⠀⠀⠀⢼⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⠁⢀⣤⠞⠃⠀⠀⣦⠞⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠑⢤⠙⡄⠀⠐⠀⠀⠂⠁
⠟⠀⠀⠀⠀⠀⢺⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⡿⠀⠀⢠⠿⠒⠉⠀⠀⠀⠀⢰⠇⠀⠀⠀⢀⣠⡶⠚⠉⠉⠁⠈⠉⠉⠓⠢⣄⡀⠹⣯⠀⠐⠈⠀⠠⠀
⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⠃⠀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢻⡄⠀⠐⠈⠀⠀
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⠀⠀⠀⠂⠈⠀
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠁⠀⠂⠀
⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠁⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠁⠀⠁⠀
⣀⣤⣤⣤⣤⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢀⠠⠁⠈⠀
⠉⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢿⡄⠀⡀⠀⢀⠐⠀
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⢻⣆⠀⠐⠀⠀⡀
⠀⠀⢠⣤⣤⣄⡀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⢸⣧⠂⠀⠠⣦⣀⠀⠀⢹⡆⠀⠐⠀⠀
⠀⠀⢾⡃⣧⣼⡇⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠁⠅⠀⢀⡀⠉⠳⠶⣿⠁⠀⠂⠈⠀
⠀⠀⠘⠿⠿⠟⠀⠀⠀⢼⣿⣧⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⠀⠀⣸⣇⠀⠀⠀⣿⠀⠀⠠⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡷⣦⣄⣀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡗⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣔⠀⠀⠀⢠⡟⢻⡄⠀⠠⢿⡆⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⡛⡿⣹⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣩⡗⠒