⠀⠀⠀⢀⠢⡐⢢⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠢⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠐⡈⢄⠣⡘⢄⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠉⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠂⢡⠘⡄⢣⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡⠈⠄⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠘⠤⣉⠆⡡⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠡⠈⡄⠌⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢀⠳⣀⠣⡐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠁⡐⢌⡱⢢⡜⢤⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠠⢄⢣⠸⡀⠄⡀⠀⠀⠀⠀⠀⠀⠀⣄⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⢼⢧⣿⣧⣻⡸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠢⠱⡱⢨⠐⡀⠀⠀⠀⠀⠐⡄⣿⣿⣿⣶⣠⢖⣶⣶⣿⣿⣿⡀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠑⢎⡿⣿⣿⣿⣷⣧⢊⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠱⡐⣃⠓⣌⠐⡀⠀⠀⢀⠓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣥⠂⠀⠀⠀⠀⠀⠀⠀⠈⠈⠞⣿⣿⣿⣿⣿⣯⣖⡡⢀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢈⠱⠌⢆⡡⢂⠀⠈⣶⣏⠉⠉⠙⠻⢿⠻⠿⠿⢿⣿⣯⡌⠙⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⢻⣿⣿⣿⣿⣿⣧⣇⠢⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠊⡒⢄⠣⢌⠀⠘⣿⣷⣤⣤⣾⣧⡀⠠⣤⣩⣿⣿⠫⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢊⠿⣿⣿⣿⣿⣿⣗⡣⢄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡈⠖⡌⠜⠀⠛⡋⠏⣼⣿⣿⡟⢻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢫⢿⣿⣿⣿⣿⣿⣎⣒⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⢸⣆⠀⢀⡈⠛⠁⠈⠁⠸⠃⠠⠀⠒⠭⣥⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡽⢿⣿⣿⣿⣿⣯⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢒⡡⠞⣁⠀⠘⣿⡄⠀⠑⠂⠐⠎⠀⠀⣰⡇⠀⢙⡓⠒⠒⠒⢬⣵⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⢟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⢋⡴⠋⣠⠎⣡⣄⠀⠻⣷⡀⠀⠀⠈⠀⣀⣴⣿⢡⣄⣾⣿⣿⣿⣿⣷⣤⣬⣝⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡹⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡪⠞⣡⠖⣰⣿⣿⣿⣿⣿⣿⣿⣶⡝⡳⣖⠾⠻⢿⡟⢁⣾⣿⣿⣿⢳⣿⣽⣽⢿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡵⢋⡔⢫⣴⣾⣿⣿⣿⣶⣾⣶⣤⣤⠌⠀⣿⣮⡛⣀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⡑⢢⢀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣴⣁⣼⣿⣿⣿⣿⡿⠛⠛⠛⠛⠃⠙⢿⣿⠿⡿⢿⡛⡆⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣻⣿⢼⣿⣿⡟⠅⠀⠀⠀⠀⠠⡤⣄⡀⠀⠀⠀⠀
⢈⠅⢎⠰⢀⠀⠀⠀⠀⠀⠉⣉⠉⢹⡻⢿⣿⣿⢟⣣⡏⡰⢠⢂⠈⢂⠀⢘⣻⣻⣻⡪⡄⣿⣿⣿⢿⣿⣿⣿⣿⠟⡿⣷⢻⣟⣿⣿⡟⠐⠀⡄⢐⡞⣒⠀⠘⠻⢶⡀⠀
⠀⠘⢌⠢⢅⠊⠄⠀⠀⠠⣘⠊⠁⡘⡟⢠⠉⢊⣿⣿⣧⢰⣈⠳⣤⠁⡀⢃⣸⣽⣻⠽⠜⢉⠙⠛⡾⣿⣿⣿⣏⣄⣲⣽⢾⣿⠙⠩⣔⡁⢀⡠⡆⢚⣏⢟⠛⡷⡁⠀
⠀⠈⠀⠊⡔⢩⡘⢄⠂⢶⣟⣙⣠⡿⣰⠁⢠⣸⣿⣿⣿⡇⡞⢿⡵⢽⢡⣏⡼⡇⣾⣟⣚⣿⣿⣿⣷⣿⣿⣿⣿⣯⣿⣯⣿⣿⠆⠄⢰⡈⠀⡐⠡⣑⠮⡾⡳⠻⣝⠀
⠀⠀⠀⠀⠈⢂⠜⣢⠵⠛⣻⡵⣾⠱⢃⠠⢨⢻⡇⣿⣿⣇⢿⣄⣷⣱⡼⢼⣇⣽⣿⣿⡼⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⠙⣃⠶⣠⠄⠀⢁⠀⡻⢿⣿⡿⢷⣠
⠀⠀⠀⠀⠀⠀⢈⡀⣠⣞⡽⡷⢑⠨⣒⢒⠑⣷⠀⢿⣿⣿⣰⣛⡵⣟⣼⣿⡿⣽⣿⡑⢀⣿⡇⠀⣶⣾⠐⣿⣿⣿⣿⣿⣿⣏⠀⠁⠂⠈⣀⢴⠀⢐⣨⣤⣶⠿⠟⠃⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣽⡸⣟⢰⠮⠣⢐⣋⠄⣿⡇⢸⣿⣿⡟⡣⢼⣿⢻⡟⣿⡫⣳⣦⠀⣿⡇⠀⢻⡏⡇⣼⣿⣿⣿⣿⣿⠃⡀⠐⢲⣽⢟⣧⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡜⢋⠆⠛⡲⠆⣜⠠⢤⠀⣿⣷⠈⣿⣿⣧⡇⣪⢻⡸⣿⣿⣷⡧⣿⠀⣿⣿⠀⢻⣷⡇⢸⣿⣿⣿⣿⣽⡃⡔⣋⣧⣿⣟⣯⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢈⠢⠂⢀⡈⠄⢒⠑⣄⠢⢸⣿⡆⢹⣿⡏⡿⢰⣿⣿⡿⣿⣿⣱⣿⠄⠹⢿⡀⠘⠋⠁⣸⣿⣿⣟⣯⠟⢔⣼⣿⣿⣿⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⡔⠀⢰⠃⠘⣤⢣⡍⣿⣷⡜⣾⡇⣷⣿⣿⣿⣾⣷⣿⣼⣿⣤⣴⣾⣿⣶⣶⣿⣿⣿⣿⣯⢫⣾⣿⣿⣿⣿⣿⣿⣿⣽⣵⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠐⠁⠀⠐⠠⢊⡄⣿⣿⡇⢨⣿⣺⢟⡝⣿⣿⢇⡿⣿⣿⣿⣿⡟⣿⣿⣿⢸⣿⣿⢿⣯⣯⣿⣿⣿⣿⣿⣿⡿⣵⣿⣿⣮⡱⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠌⢿⣿⡇⢸⣿⣟⣷⣿⣿⣷⣿⣷⣿⣿⣿⣞⡟⡜⠿⢿⣸⣯⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠘⢽⡻⣿⣷⣿⡲⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡯⣍⢿⣹⡿⠿⡿⠭⡹⣿⢯⣿⣷⡲⠟⠻⢉⣱⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⠀⡙⢳⢿⣻⡿⣶⡩⢄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣏⠿⣿⡸⣩⡷⣖⠡⠚⠹⡟⣿⣷⣌⠀⠀⢀⣼⡿⣋⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠁⠎⡽⢻⣿⣽⣎⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣼⣧⣼⣽⣾⣿⣿⣶⣷⣷⣶⣷⣶⣶⣿⣭⣭⣤⣤⣭⣿⣱⠿⣫⣶⢟⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣭⢿⣾⣷