⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠒⠓⠀⠀⠀⠀⠀⠁⠐⠒⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⢌⠑⡄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠈⠀⣤⠈⡉⡆⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠵⠀⠠⣟⠠⢾⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⡀⠀⠀⠀⡤⠤⠔⢀⠤⡀⠓⠓⢲⢢⠀⠀⠀⣸⠀⠀⠸⢆⠀⠀⠀⠀⠀⡆⠀⠀⠀⢜⢆⠰⠂⠒⠋⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠤⠊⠀⠀⠀⠀⡇⠺⠀⢻⣀⠆⢈⣉⣸⢸⠀⠀⢰⠘⡀⠀⠘⠈⣆⠀⡄⠀⠀⢷⠀⠀⡄⠈⠦⠬⢅⠱⡘⣂⢹⠂⢄⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡀⠤⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⢁⡉⠙⡆⣠⢈⠇⢰⠒⡞⠀⠀⣿⠀⢣⠀⠀⡇⡘⡄⢹⡄⠀⢸⠀⢸⠡⡀⠀⠀⡄⠣⢌⠒⠁⠀⠀⠀⠉⠒⢂⣤⡤
⠀⠒⠛⠭⢷⣤⠤⠤⠒⠉⠀⠀⠀⠀⡀⢤⡾⠋⠀⠀⠧⣭⣼⠠⠼⡜⣧⢣⢠⢿⢀⠏⢃⠀⢇⣇⠸⡜⣱⢸⢸⠀⣸⡀⢱⠀⠀⣷⠀⠀⠣⡐⠤⠍⠈⠛⠫⠥⣄⣀
⠀⢀⡠⠒⠉⠀⠀⠀⠀⢀⡀⣄⣒⠭⢴⠎⠀⠀⡀⠀⢠⠀⠀⠀⡸⠀⡼⡆⠈⢸⡼⠀⠈⢆⠘⢾⢆⠰⢀⠃⣾⢤⣇⣹⣸⠀⠀⡇⡄⠀⠀⡈⠲⢤⡤⠤⠔⠒⠉⠉
⠛⠛⠂⠒⠒⣶⡖⢒⡋⣉⡭⠀⣠⠖⠁⠀⠀⡘⠀⠀⢸⢣⠀⠀⡇⢰⠃⠹⡀⢀⣇⡁⠤⠚⢆⠈⡎⢣⡂⠀⣏⡜⠀⠀⢻⠀⡔⡇⡇⠀⢰⡧⠄⣀⣉⡒⢤⡀⠀⠀
⠀⠀⠀⢀⠔⠡⠘⢤⣓⡜⡡⠞⠁⡀⠀⠀⠜⠀⠀⢀⡇⡄⢣⠀⡇⢸⠐⠂⠹⡍⢹⠀⠀⠀⠈⢢⠱⠀⠱⣠⠟⠀⠀⢠⢃⢼⠀⣇⠇⢀⣆⠈⢆⠀⠀⠀⠀⠁⠀⠀
⠀⠀⠰⠷⠚⠓⠛⡻⠟⠋⠀⢀⠜⠀⢠⠊⠀⣀⣤⢾⢸⡄⠀⠣⡸⡇⠀⠀⠀⠈⠢⣇⠀⠀⠀⠀⠉⠇⠀⠹⠀⡠⠀⣋⣅⢸⣾⠏⣠⠞⠰⢷⢂⠑⠦⣀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣶⣯⠤⠐⣒⣭⠏⠀⠐⡳⠊⢩⣶⢻⢼⡞⡘⢦⡐⠀⢃⠀⠀⠀⠀⢈⣀⣂⣠⠄⠀⠀⠀⠀⠀⢨⡴⠛⠉⠉⢸⣧⠚⣿⣤⠀⢻⠆⠈⠉⠐⠃⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠋⣠⣷⡿⠁⠀⠤⣊⣴⣟⠻⣌⣷⡞⡇⡇⢄⠙⢢⡈⢆⠀⠾⠛⠋⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠀⡇⡟⢺⡆⡼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⡣⠚⡩⠋⢀⣠⢾⡿⣯⡷⣿⡟⡧⡝⡜⡇⠈⠀⠑⡄⢸⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣧⣧⠀⠃⡏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠀⠾⠒⠚⠉⣶⠿⠛⠉⠁⡹⠿⣷⣽⣼⣽⣀⡀⠀⠱⠀⢧⠀⠀⠀⠀⠀⠀⢀⣴⡤⣄⣀⡀⠀⠀⠀⠀⠀⡜⢀⠆⡇⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠈⠉⠙⠚⠽⣶⣦⣌⢆⠀⠀⠀⠀⠀⢞⢩⠉⠙⣻⠇⠀⠀⠀⡠⠊⢠⡞⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠫⣿⣆⣀⠀⠀⠀⠈⠢⠬⠗⠃⢀⡠⠖⠁⡏⠀⡞⠀⢸⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠈⠻⣧⣉⠒⠒⣤⠤⣤⣤⡒⠭⡄⠀⠀⡇⣸⡝⠀⢸⠀⠈⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠤⠀⡀⢀⢀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠈⢿⣍⡒⠓⢂⣨⠿⣿⣦⡇⠀⠀⣧⡳⢀⠀⠇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⢀⠀⠀⠀⠹⣿⡵⢾⡓⠀⡟⢯⡗⠀⡰⣣⣴⠁⡜⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠂⠀⠀⠈⢿⣄⡉⠂⠀⢀⡯⠜⢋⢿⣻⠜⣑⢀⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠀⡜⠀⠀⣤⠖⠛⠶⣦⠀⠀⠀⠀⠀⠀⡃⠉⠂⢅⡂⠀⠹⣯⣑⣆⣇⡗⠠⠃⡸⠉⠀⠄⡇⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡇⠀⢸⡃⣾⢿⣷⣹⡇⠀⠀⠱⡀⠀⡇⠀⠀⠀⠈⢷⢢⣝⣿⣿⣿⠇⡇⠀⠿⠀⠀⠀⠁⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡇⠀⠘⢧⣙⠛⠋⣿⠃⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⠀⣣⠈⡿⣋⡉⣩⠁⠀⠀⠀⠀⢀⠆⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⢻⡀⠀⣿⠀⣀⣀⣀⣀⣱⡒⠤⡀⠐⢀⡰⠃⠀⢣⠃⠁⠢⢄⡀⠀⠀⢀⠊⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣸⡇⠀⢸⣿⢧⡟⣾⡹⣏⣷⣰⠙⠃⠉⠀⠀⡠⠃⠀⠀⡔⠈⢽⣄⣴⣳⢺⣙⣯⡹⣒⠾⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣠⣴⣿⣿⣇⠀⠀⣿⡏⠉⠁⠉⠈⠈⠹⡦⢄⣀⠤⠚⠁⠀⡄⣀⣸⠢⡘⣞⡷⠮⠷⠛⠒⠓⠚⠮⢿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿⣿⠟⠁⢹⡀⠀⢸⣷⠀⣀⡠⢤⡴⣖⡟⢸⡀⠀⡔⢦⢄⣱⠀⠀⠀⠈⠻⡇⠀⠀⢀⡀⢀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠀⠀⠀⣸⣧⢀⡀⠟⠋⠈⢷⣫⠖⠉⣀⠔⣇⠎⠀⠀⠢⠁⢰⠀⠀⠀⠀⠈⠻⣉⠀⠀⠈⡏⢻⣷⢾⡆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣀⣴⣾⠁⢈⡡⠔⠈⠀⠀⢘⠥⠔⠊⠀⠀⠀⠀⠀⠀⠀⣱⡈⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⢱⡘⣿⠟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡿⠛⠛⠛⢇⠈⠀⠀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⣨⢸⣄⠀⠀⠀⠀⠀⠀⠀⠈⢂⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡃⠀⠀⠀⠈⠆⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⢁⣿⡌⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⣤⣀⣀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⣀⠤⣾⣿⣿⣿⠃⣸⣿⡇⢻⣿⡌⠢⡀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠒⠀⠀⠀⠀⠀⠀⠐⠒⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠐⠄⢢⣀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠁⠰⠀⡇⠃⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠖⠃⠀⢺⡁⠎⣇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢀⠄⠀⠀⢠⠒⡒⢁⡄⢢⠈⣉⠈⠷⠀⠀⠀⢼⠀⠀⢈⢧⠀⠀⠀⠀⠘⡆⠀⠀⠀⠏⢄⣋⠈⡉⡀⢲⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠇⠊⠀⠀⠀⠀⢸⣈⣃⡘⠦⠞⡠⠤⠤⢋⡆⠀⢸⠈⡆⠀⠸⡀⢆⠀⡆⠀⠈⢧⠀⢰⠀⠘⠒⠒⢄⠳⠩⢴⠏⠒⠤⢀⡀⠀⠀⠀⠀
⠀⣀⣀⡀⡤⠔⠒⠉⢁⡀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢰⢰⡆⡇⠰⠉⣽⡀⠀⢾⠀⢸⡀⠀⡇⡘⢆⢹⡄⢸⢸⠀⡸⠱⡀⠀⢰⡈⠒⢍⠀⠀⠀⠀⣀⣈⠩⠿⠛⠀
⠀⠉⠉⢙⡹⠒⠒⠊⠁⠀⠀⠀⣀⡤⢐⡽⠊⠀⠀⠘⠒⠚⠓⠚⡟⢸⡘⠄⢻⢠⠇⢱⠀⢱⣧⠘⠜⣱⣌⣾⠀⠷⡄⠃⠀⢸⣃⠀⠈⠳⣉⠐⠀⠈⠉⢒⣒⡦⠄
⣠⣤⡒⢁⣀⣀⣀⣀⣀⣤⠴⠬⠒⣊⠍⠀⠀⡐⠀⠀⣦⠀⠀⡜⠀⡜⢣⠀⢈⡞⠀⠀⣣⠀⢸⡣⡘⠀⢠⡏⡺⠒⢱⡘⠀⣸⣹⠀⠀⣰⠀⠙⠯⣙⠉⠉⠀⠀⠀
⠈⠀⠀⠀⢈⡝⢩⡐⡄⣢⠃⡠⠞⠁⠀⠀⡐⠁⠀⢰⠈⢆⠀⢃⢠⠃⣀⣧⠤⢷⠒⠈⠀⠣⡀⡇⠹⡄⢰⡷⠁⠀⢀⠇⡰⢹⢸⠀⢀⠛⣃⠒⠂⠀⠡⠦⠀⠀⠀
⠀⠀⢀⢔⣡⣌⣤⣴⣷⠥⠊⠀⡰⠀⠀⠜⠀⠀⢀⡾⡁⠌⢆⠸⢸⠈⠀⠀⠣⡘⡄⠀⠀⠀⠑⢼⡀⠘⣼⠈⠀⢀⡾⠊⣷⣸⠌⢠⡞⣄⡈⠢⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠉⢀⣠⡝⠋⢁⣀⢤⡔⠁⣠⣊⠤⠔⣲⡟⡇⡿⣄⠀⠣⢿⠀⠀⠀⠀⠈⠚⡀⡀⠀⠀⠀⠁⠀⠈⠰⣀⡴⠾⠇⣏⢏⣰⣏⠀⠉⡗⠂⠬⢴⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠈⢻⠟⡉⣤⣶⠟⠀⢀⠔⣡⡴⣞⠟⣼⡿⣰⠈⠣⣄⠘⡄⠀⣠⣤⠶⠶⠶⢾⠅⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⡟⢹⢸⠹⡆⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⢎⡴⢟⡷⠁⢀⣤⣵⡾⣯⣷⢮⡹⣎⢷⠘⠌⠢⡈⢱⠼⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⠀⢹⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠚⠁⣤⣊⠤⠖⣻⣿⠷⠛⠛⢿⣧⣝⣜⢺⡀⠀⠀⢱⡀⢃⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⢸⠛⠀⠈⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡰⠉⠉⠉⠿⠿⣷⣶⣆⣀⠁⠸⡆⠀⠀⠀⠀⠀⢰⡿⢿⣿⣷⡆⠀⠀⠀⠀⡸⠁⡎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠿⣷⣜⡀⠀⠀⠀⠀⢹⡲⡠⢀⡽⠁⠀⢀⢴⠊⠀⡞⠁⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠈⠺⣿⡆⠤⣀⡀⠀⠉⠉⠁⣀⠤⠊⠁⠈⡇⢸⡁⠀⡆⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠈⠻⣷⡤⣀⣏⡉⢺⣿⣦⡉⡄⠀⠈⣧⢯⠁⢀⠇⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠉⣰⠀⠂⠂⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠘⢷⣦⢄⣲⡞⠉⣿⡿⡇⠀⢸⠇⡡⠊⡜⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠠⣀⠀⠀⠻⣯⡙⠢⠈⠃⠙⡇⡠⢗⣿⣇⡼⡀⢠⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⡎⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠑⠠⢁⠠⠀⠘⢷⡖⠤⡀⡖⡏⢀⠎⡹⠏⢰⠓⠂⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠀⠀⡾⢩⣴⣦⣹⣧⠀⠀⠠⠀⠀⠀⠀⠀⠉⠒⡤⣀⢻⣷⣿⣿⠁⠆⠀⡇⠀⠈⠘⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⠂⠀⢷⡸⢿⡾⢻⡿⠀⠀⠀⠡⠀⡂⠀⠀⠀⠀⠈⢆⠛⣿⠟⠛⢺⠀⠀⠁⠀⠀⢀⠃⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⠻⡆⠀⢸⡇⠀⠀⠀⠀⢡⡧⣀⠀⠀⠀⢀⠎⠀⣄⠞⠫⢏⡀⠀⠀⠀⢀⠊⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣧⠀⠘⣷⣶⣛⣿⣛⢯⣷⢀⠳⠦⠒⠁⠀⢀⠎⠀⠀⡄⠋⣦⡀⣰⢥⡤⣖⠖⡦⢬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣴⣾⣿⠀⠀⢻⡷⠛⠒⠉⠋⠘⠻⣆⡀⢀⣀⠴⠁⠀⡀⠀⢣⢄⢳⣻⣝⣮⡼⠭⠽⠼⢦⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣾⣿⡿⠟⠹⡇⠀⠘⣿⡀⠀⢀⣀⣤⢤⡵⢸⠁⠀⡤⣄⡀⢇⠉⠉⠀⠑⢿⠁⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠀⠀⠀⣿⡀⢀⣹⠟⠉⢷⣚⡥⠛⢀⡠⣇⠜⠀⠈⢎⠉⠀⠀⠀⠀⠀⠑⢏⠉⠉⠉⢹⠳⣶⢤⣧⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⣠⣴⡏⠉⢉⠡⠀⠀⠀⠈⣏⠤⠒⠁⠀⠀⠀⠀⠀⠀⣢⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠘⡄⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣴⠿⠿⠟⠻⠀⠁⠀⢀⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⡆⢱⡀⠀⠀⠀⠀⠀⠀⠀⠱⡀⡃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⠀⠀⠀⠀⢡⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⢁⣿⠸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣤⣄⣀⡀⠀⢢⠀⠀⠀⠀⠀⠀⠀⢀⠠⣺⣿⣿⣿⠃⣼⣿⡇⢿⣿⡐⢄⡀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠒⠓⠀⠀⠀⠀⠀⠁⠐⠒⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⢌⠑⡄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠈⠀⣤⠈⡉⡆⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠵⠀⠠⣟⠠⢾⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⡀⠀⠀⠀⡤⠤⠔⢀⠤⡀⠓⠓⢲⢢⠀⠀⠀⣸⠀⠀⠸⢆⠀⠀⠀⠀⠀⡆⠀⠀⠀⢜⢆⠰⠂⠒⠋⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠤⠊⠀⠀⠀⠀⡇⠺⠀⢻⣀⠆⢈⣉⣸⢸⠀⠀⢰⠘⡀⠀⠘⠈⣆⠀⡄⠀⠀⢷⠀⠀⡄⠈⠦⠬⢅⠱⡘⣂⢹⠂⢄⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡀⠤⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⢁⡉⠙⡆⣠⢈⠇⢰⠒⡞⠀⠀⣿⠀⢣⠀⠀⡇⡘⡄⢹⡄⠀⢸⠀⢸⠡⡀⠀⠀⡄⠣⢌⠒⠁⠀⠀⠀⠉⠒⢂⣤⡤
⠀⠒⠛⠭⢷⣤⠤⠤⠒⠉⠀⠀⠀⠀⡀⢤⡾⠋⠀⠀⠧⣭⣼⠠⠼⡜⣧⢣⢠⢿⢀⠏⢃⠀⢇⣇⠸⡜⣱⢸⢸⠀⣸⡀⢱⠀⠀⣷⠀⠀⠣⡐⠤⠍⠈⠛⠫⠥⣄⣀
⠀⢀⡠⠒⠉⠀⠀⠀⠀⢀⡀⣄⣒⠭⢴⠎⠀⠀⡀⠀⢠⠀⠀⠀⡸⠀⡼⡆⠈⢸⡼⠀⠈⢆⠘⢾⢆⠰⢀⠃⣾⢤⣇⣹⣸⠀⠀⡇⡄⠀⠀⡈⠲⢤⡤⠤⠔⠒⠉⠉
⠛⠛⠂⠒⠒⣶⡖⢒⡋⣉⡭⠀⣠⠖⠁⠀⠀⡘⠀⠀⢸⢣⠀⠀⡇⢰⠃⠹⡀⢀⣇⡁⠤⠚⢆⠈⡎⢣⡂⠀⣏⡜⠀⠀⢻⠀⡔⡇⡇⠀⢰⡧⠄⣀⣉⡒⢤⡀⠀⠀
⠀⠀⠀⢀⠔⠡⠘⢤⣓⡜⡡⠞⠁⡀⠀⠀⠜⠀⠀⢀⡇⡄⢣⠀⡇⢸⠐⠂⠹⡍⢹⠀⠀⠀⠈⢢⠱⠀⠱⣠⠟⠀⠀⢠⢃⢼⠀⣇⠇⢀⣆⠈⢆⠀⠀⠀⠀⠁⠀⠀
⠀⠀⠰⠷⠚⠓⠛⡻⠟⠋⠀⢀⠜⠀⢠⠊⠀⣀⣤⢾⢸⡄⠀⠣⡸⡇⠀⠀⠀⠈⠢⣇⠀⠀⠀⠀⠉⠇⠀⠹⠀⡠⠀⣋⣅⢸⣾⠏⣠⠞⠰⢷⢂⠑⠦⣀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⣶⣯⠤⠐⣒⣭⠏⠀⠐⡳⠊⢩⣶⢻⢼⡞⡘⢦⡐⠀⢃⠀⠀⠀⠀⢈⣀⣂⣠⠄⠀⠀⠀⠀⠀⢨⡴⠛⠉⠉⢸⣧⠚⣿⣤⠀⢻⠆⠈⠉⠐⠃⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠋⣠⣷⡿⠁⠀⠤⣊⣴⣟⠻⣌⣷⡞⡇⡇⢄⠙⢢⡈⢆⠀⠾⠛⠋⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠀⡇⡟⢺⡆⡼⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⡣⠚⡩⠋⢀⣠⢾⡿⣯⡷⣿⡟⡧⡝⡜⡇⠈⠀⠑⡄⢸⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣧⣧⠀⠃⡏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠀⠾⠒⠚⠉⣶⠿⠛⠉⠁⡹⠿⣷⣽⣼⣽⣀⡀⠀⠱⠀⢧⠀⠀⠀⠀⠀⠀⢀⣴⡤⣄⣀⡀⠀⠀⠀⠀⠀⡜⢀⠆⡇⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠈⠉⠙⠚⠽⣶⣦⣌⢆⠀⠀⠀⠀⠀⢞⢩⠉⠙⣻⠇⠀⠀⠀⡠⠊⢠⡞⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠫⣿⣆⣀⠀⠀⠀⠈⠢⠬⠗⠃⢀⡠⠖⠁⡏⠀⡞⠀⢸⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠈⠻⣧⣉⠒⠒⣤⠤⣤⣤⡒⠭⡄⠀⠀⡇⣸⡝⠀⢸⠀⠈⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠤⠀⡀⢀⢀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠈⢿⣍⡒⠓⢂⣨⠿⣿⣦⡇⠀⠀⣧⡳⢀⠀⠇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⢀⠀⠀⠀⠹⣿⡵⢾⡓⠀⡟⢯⡗⠀⡰⣣⣴⠁⡜⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠂⠀⠀⠈⢿⣄⡉⠂⠀⢀⡯⠜⢋⢿⣻⠜⣑⢀⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠀⡜⠀⠀⣤⠖⠛⠶⣦⠀⠀⠀⠀⠀⠀⡃⠉⠂⢅⡂⠀⠹⣯⣑⣆⣇⡗⠠⠃⡸⠉⠀⠄⡇⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡇⠀⢸⡃⣾⢿⣷⣹⡇⠀⠀⠱⡀⠀⡇⠀⠀⠀⠈⢷⢢⣝⣿⣿⣿⠇⡇⠀⠿⠀⠀⠀⠁⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡇⠀⠘⢧⣙⠛⠋⣿⠃⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⠀⣣⠈⡿⣋⡉⣩⠁⠀⠀⠀⠀⢀⠆⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⢻⡀⠀⣿⠀⣀⣀⣀⣀⣱⡒⠤⡀⠐⢀⡰⠃⠀⢣⠃⠁⠢⢄⡀⠀⠀⢀⠊⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣸⡇⠀⢸⣿⢧⡟⣾⡹⣏⣷⣰⠙⠃⠉⠀⠀⡠⠃⠀⠀⡔⠈⢽⣄⣴⣳⢺⣙⣯⡹⣒⠾⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣠⣴⣿⣿⣇⠀⠀⣿⡏⠉⠁⠉⠈⠈⠹⡦⢄⣀⠤⠚⠁⠀⡄⣀⣸⠢⡘⣞⡷⠮⠷⠛⠒⠓⠚⠮⢿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿⣿⠟⠁⢹⡀⠀⢸⣷⠀⣀⡠⢤⡴⣖⡟⢸⡀⠀⡔⢦⢄⣱⠀⠀⠀⠈⠻⡇⠀⠀⢀⡀⢀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠀⠀⠀⣸⣧⢀⡀⠟⠋⠈⢷⣫⠖⠉⣀⠔⣇⠎⠀⠀⠢⠁⢰⠀⠀⠀⠀⠈⠻⣉⠀⠀⠈⡏⢻⣷⢾⡆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣀⣴⣾⠁⢈⡡⠔⠈⠀⠀⢘⠥⠔⠊⠀⠀⠀⠀⠀⠀⠀⣱⡈⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⢱⡘⣿⠟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡿⠛⠛⠛⢇⠈⠀⠀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⣨⢸⣄⠀⠀⠀⠀⠀⠀⠀⠈⢂⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡃⠀⠀⠀⠈⠆⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⢁⣿⡌⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⣤⣀⣀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⣀⠤⣾⣿⣿⣿⠃⣸⣿⡇⢻⣿⡌⠢⡀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀