⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣚⣯⣭⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢾⡿⣟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣶⣭⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢫⣿⣽⣿⣿⣿⣿⣿⣿⡿⡟⠹⣿⣿⡿⣿⣿⣿⣿⣷⣾⣷⣏⣟⣧⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢷⣿⣿⣿⢿⣿⡿⢹⣿⠟⢡⠇⠀⢻⣿⡇⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠃⣿⣿⡁⡾⠫⠀⡼⠀⠀⠀⢻⣗⠀⠘⣿⡏⢿⣿⣿⣿⣿⣿⣿⣷⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⣿⣿⡏⠀⣿⣤⠽⣷⡦⡼⠁⠀⠀⣤⠤⢿⣾⣭⣈⣣⠈⢿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣸⠏⠀⠀⠀⢹⡄⠀⠀⠀⠀⢠⡾⠉⠀⠈⠙⣦⠘⣿⣿⣿⣿⣿⣿⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣿⣿⣿⡃⠹⣦⠀⠀ ⣠⡾⠀⠀⠀⠀ ⠸⣧⡀⠀⠀⢀⡿⠀⣿⣿⣿⣿⣿⣿⣿⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⣿⡇⠀⠙⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠙⠳⠤⠴⠚⠁⠀⣿⣿⣿⣿⣿⣿⣿⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⣿⣿⠆⠀⠀⠀⠀⢀⣤⣤⣠⣀⡤⠤⠤⢤⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⣿⡃⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⣿⣿⣿⣧⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣿⣿⣿⣿⣿⣿⣶⣤⣀⠻⢀⣀⣀⣀⣀⣀⡀⠀⠀⣰⣇⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡤⠤⠭⠟⠛⠛⢹⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⡟⣿⣿⣿⣏⣿⠿⠋⣽⡇⠀⠀⠀⠀⠀⣸⠇⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡿⣿⣿⣇⠹⣿⡿⠟⠁⠀⢹⡯⣀⠀⠀⢀⡀⣰⠟⠒⠉⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣹⣿⣿⣿⠴⠋⠀⠀⠀⠀⢸⠁⠈⠉⠉⠉⣰⠋⠀⠀⠀⢀⣿⣿⠟⢹⣿⣿⣿⢳⢸⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⢻⡉⢿⡆⠀⠀⠀⠀⠀⢸⠒⠒⠒⢒⡾⠁⠀⠀⠀⠀⣸⣿⡿⠖⣿⣥⠼⣿⣿⡶⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣥⣾⢷⡌⠻⠶⢶⣶⠀⠀⢸⡆⠀⢠⡞⢡⣄⡀⠀⣀⣴⠟⣅⡴⠋⠁⠀⠀⠈⢻⡵⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣽⠁⢠⠉⠛⠺⣽⢻⣄⡀⢸⡇⢠⠟⣀⣼⣷⢭⣛⣩⡾⠛⠁⠀⠀⡀⣠⡞⠓⠛⠻⣞⣆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⣷⠀⢸⡀⠀⠀⠙⠲⣬⣉⠛⣷⣟⣋⣉⣥⠿⠀⠈⠁⡀⡀⠀⠐⢠⣼⡏⠀⠀⠀⠀⠹⣜⡆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⠇⠘⣧⣸⠀⠀⠀⠠⢀⠀⠘⣿⢹⡟⠁⠀⠀⢀⠐⠀⠐⠀⣿⢀⢨⣾⡟⠀⠀⡀⠠⠀⠀⠘⣾⢆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⠠⠀⠙⣿⡇⠀⢈⠐⠠⢀⠠⣿⣿⠀⠌⢀⠈⠀⡀⠂⠂⢈⣿⣶⣟⢯⣿⡄⠀⢀⠐⡀⠐⠀⠉⢷⡳⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⢀⠀⠀⢂⣽⠁⠂⠈⠄⠀⠌⢐⣿⣿⠀⠄⠈⠠⠀⠁⠠⠄⠸⣿⣿⣿⣎⠿⣷⡄⠈⠀⠁⠄⠀⢀⠈⠳⣝⢆⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠟⠀⢈⠠⠀⢄⣿⣿⠀⡃⢈⠐⢀⠂⢸⣿⣿⠀⠌⠐⢀⠑⢀⣦⣴⣄⣹⣿⢿⣿⣻⡿⠛⢶⣄⣂⣠⣶⠦⢦⣄⣈⢧⡳⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡖⣲⣤⣶⡻⠷⠲⢶⣤⡐⢠⣿⣿⣯⠀⡐⢀⠐⡀⠂⣾⡟⣿⠀⠜⠀⢄⣢⣀⣻⣦⣄⡈⠙⢛⡿⣿⣤⣄⡀⢈⠙⠣⠀⠰⣀⢈⠍⠃⢻⣽
⠀⠀⠀⡠⠖⡟⠉⣿⡝⠳⢶⢦⡿⣍⢀⠒⡆⠤⠙⢿⣿⣟⣼⡇⡐⠄⡈⠰⠠⢁⣿⡹⣿⠀⠃⠌⡐⠿⣭⣍⣩⠽⠛⠋⠉⠀⠘⣎⢷⢻⡄⠌⡐⠈⠄⡀⠆⡐⠠⡈
⠀⠠⢊⠄⠈⠀⣸⣽⣣⡄⠈⢯⢷⠘⣧⣰⣴⣶⣼⣦⢿⣿⣽⡇⡐⠄⡂⠅⠌⠤⣿⢘⣿⠀⡩⢐⠠⢂⠄⡈⢳⣶⢒⣻⠟⠃⠀⠸⡞⡇⢿⠀⢤⠉⡐⠠⢒⠠⢁⢐
⠴⠁⠀⢨⡀⢠⣧⠏⠁⠀⠀⣸⣿⣇⣻⣿⣭⣷⣻⣞⣯⣞⣿⠇⡐⢡⠂⣉⠚⣰⡟⢸⡿⢀⡑⠢⢑⢂⠰⢐⢂⣿⠋⣁⣠⡶⠂⠀⣿⡇⣾⠈⣄⣢⣵⣾⣿⣿⡿⣿
⣤⢤⣤⣭⢭⣭⣥⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠿⣷⢾⡷⣿⢾⡷⣼⢦⡶⣴⢶⣾⡷⢾⡷⢶⡬⣵⡈⢢⠁⢎⠤⡉⣿⡿⠋⢀⣠⣼⣿⣽⣷⣾⣾⣿⣿⣿⠟⢯⠞⠁
⠮⠾⠴⣮⠞⣴⢎⠶⡵⢮⠶⡵⢮⠶⢶⢶⡼⢮⡵⣮⢶⣵⢮⡶⡵⣮⡵⣮⠷⣦⡽⣧⣾⢧⡼⠾⢃⠥⡉⢆⠢⡑⣼⣿⣿⣿⣿⣿⣿⣽⣾⣽⣿⣟⣭⣿⣽⣟⠀⠀⠀⠀