⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠐⠒⠂⠀⠈⠉⠉⠉⠉⠉⠐⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠊⠁⠠⠤⠴⠒⠚⠍⠉⠩⠍⠍⠛⠛⢀⡀⠀⠤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠀⠀⠀⠐⠒⠐⠒⠒⠒⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣤⣴⣶⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⣿⣿⣿⣿⣿⣿⢄⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣼⣿⣿⣿⣿⣹⣿⣿⣟⠶⡄⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠾⣧⣵⡾⣿⣷⡿⢿⣿⣧⣏⢻⣄⠀⠀⠀⣄⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⣦⣟⡾⣷⡿⠏⣴⠾⣿⣷⣾⣧⡛⢷⡄⠀⠈⢷⡿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⡴⣿⣾⣟⣿⣋⣿⣷⣯⣿⠯⢸⡿⣏⣯⠀⢻⣦⣀⠀⠀⠈⣵⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⣾⣟⣿⣿⣿⠟⣀⠀⣿⢿⢻⣿⡿⣿⡿⣿⢟⡳⢛⢛⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⢽⣿⡿⢿⣿⢻⢷⢾⣿⣾⣿⠃⠀⢿⡼⠀⢁⠂⡇⢌⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣿⣿⡟⣺⣿⠈⠚⢻⣿⠟⠁⠀⠀⠀⠀⠀⢸⢸⡇⠰⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⣿⣿⣿⢮⣿⠀⠀⢸⠏⠀⠁⠀⠀⠀⠀⠀⡄⡸⢿⢰⡁⡇⠑⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢢⢿⣿⢿⡞⢿⠀⠀⠸⡀⣶⣶⠄⠀⠀⠀⠀⣄⡧⣽⣀⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣉⢻⣿⣿⣿⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⢃⣿⡳⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⡠⢀⢠⣿⣧⣿⣏⣯⣽⢶⣄⣀⠀⠀⠀⠀⠀⢤⣾⣿⣇⣿⡏⠁⠀⠈⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡀⢙⡠⢻⣹⣯⣿⣟⣿⣿⣯⣳⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⢿⣻⣗⢅⢀⣀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠙⠀⡆⡇⠞⣡⣿⢽⣿⣿⣾⡿⣿⡛⢻⠀⠀⠀⠀⠀⠀⢸⡿⢻⠈⠛⢽⣷⣿⣟⠶⢀⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢰⣩⣷⣶⡾⠿⢿⢿⣿⣿⡿⢿⠁⣿⠀⠘⠠⢀⠀⠀⢀⠔⠙⠒⠢⣀⠀⠀⢹⣿⣍⠓⠸⢄⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⣄⣼⡟⠋⠀⠀⠀⠀⢙⣿⣯⣀⣆⣤⡿⠀⠀⠀⣀⣥⠮⡁⠀⠀⠀⠘⡀⠉⠒⠞⠉⠉⠀⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⣿⣿⣿⣿⡄⢿⣱⣄⡶⣁⠀⠈⠻⡽⢧⠃⠀⡔⢺⣿⣿⡟⢻⢦⡀⠀⣐⣧⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣯⣮⣯⣷⣿⣾⡁⠀⠀⡳⢄⠢⡘⠃⢾⣿⣿⣿⡞⠋⠓⠺⠶⠚⣾⠀⠀⠀⠀⢘⡘⡀⠀⠀⢀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻⡿⢿⣤⡄⣠⠧⣈⠑⠠⣀⣀⣿⣿⣿⣏⣛⠒⠢⠤⠤⢿⣷⠀⠀⠀⢨⣿⠃⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢛⡏⠽⠛⠙⠋⠉⠉⢰⡀⠈⢢⣌⠪⢭⣿⣿⠏⡏⢁⢀⣀⠀⡀⣨⣟⡃⠀⠀⢸⣿⠀⠀⢀⣿⣙⠁⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣠⠃⠀⠀⢀⠀⢸⠨⡄⡃⢸⠀⢰⠈⠂⢾⣿⡿⠀⢸⢌⡁⠒⠬⠉⠀⢿⡇⡀⠀⣿⡿⡄⠀⢸⣦⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠰⣇⣀⢀⡴⠚⠄⣏⡀⢠⣝⢯⡄⠘⠠⡀⢸⣿⡿⠀⠨⣀⠈⠒⠠⢀⣀⣸⣿⣤⣸⡿⣹⣳⢢⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⠽⡈⠉⣿⣶⣶⣿⣿⠭⠫⠎⠂⠐⠈⠂⠹⠿⠃⠀⠰⠀⠉⠒⠀⠀⠀⠙⠷⣿⣿⣛⣦⡟⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣀⠤⢒
⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠒⠐⠂⠒⠐⠂⠂⠀⠀⠀⠀⠀⠀⠈⠉⣽⠟⠛⠀⣩⣩⡿⠉⠀⢰⣿⣿
⠤⠠⠴⠤⠭⠍⠛⠛⠻⠿⠿⠿⠿⠟⠟⣁⣼⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣾⣯⡢⣠⣾⣿⠋⣀⣴⡤⣾⣿⣿
⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢾⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣠⣿⣿⣿⣿⣿⣿⢯⡿⢛⣵⣂⡏⢩⡙