⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠄⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢈⠀⠄⠐⠀⠂⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠈⠠⠀⡠⠊⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢀⠀⠤⠐⠃⠃⠀⠁⠈⠈⠈⠆⠀⠀⠀⠀⠀⠸⠀⠀⠇⠘⠾⠂⠟⠀⠃⠀⠀⠀⢠⠹⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⢐⠘⢌⣠⡈⣆⣥⣢⢈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⢀⡄⠀⠀⠁⠁⠈⢰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣶⣷⣿⣿⣽⣻⣻⣿⢸⢠⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⣠⣠⠀⠀⡀⠐⠒⣰⢠⠀⠈⠀⠀⠄⠀⠀⠀⠀⣇⠀⢰⢹⢀⠀⢧⠀⠸⠵⠀⡶⡆⠀⠀⠀⠀⠀⠀⠄⢸⠀⣞⢿⡟⣽⣟⡿⣝⣿⣿⣮⡪⡎⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠁⠓⣀⠀⠀⢃⠀⠤⢤⠀⠀⡘⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠂⡱⡷⣽⣿⡿⣻⡝⡽⡟⠘⡨⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡅⠀⠉⠁⠀⣀⢸⡄⠀⠀⠀⠀⠁⠸⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⠢⡈⢈⣣⡏⢁⡠⣈⢡⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠓⠁⠚⠉⠐⠈⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡜⣆⣿⣝⣿⣿⡇⣧⢺⣶⡾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠩⡭⠭⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⡊⠐⠀⠀⢸⡻⣿⣻⣿⠳⡽⠟⠸⣿⣿⣣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⡏⣟⢿⢄⣄⢄⡴⣿⢻⢜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠤⠤⠄⢪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡱⡽⡻⡘⢘⢙⢈⠬⡡⢅⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡑⢱⢣⢀⣅⢬⣄⢌⠄⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⣠⠀⠀⠀⠀⢰⡀⠘⠘⠎⠏⠇⠋⠈⠀⠀⠀⢰⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠄⠀⠀⠀⠀⠸⣼⡆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠠⠀⠐⠀⠀⢀⣸⢞⢧⢣⠀⡠⡀⠀⢀⠠⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠈⠙⠊⠪⠪⠪⠊⠂⠓⠀⠀⠀⠀⡀⠀⠀⢀⠠⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠔⠐⠁⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠄⠠⠀⠰⠀⠀⠂⠀⠂⠀⠂⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠜⠄⠀⠁⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠂⠀⠋⠀⠉⠀⠁⠀⠁⠈⠀⠀⠀⠘⠀⠘⠰⠀⡄⠀⠀⠀⠀⢰⢀⡂⢠⠀⠰⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠠⠄⠀⠀⠀⠂⠐⠀⠀⠂⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⢠⠀⢠⠀⠔⠀⠀⠰⠀⠠⠂⠠⠀⠠⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠂⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⢀⡀⠀⣄⠀⢀⢰⠂⢠⡆⢰⢂⠀⢲⠎⢆⡆⢦⠇⠸⢷⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠆⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠈⠀⠀⠀⠀⢀⠷⠀⠀⠞⠇⠀⠛⠀⠛⠈⠁⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣆⢤⠀⢀⣾⡄⠀⠀⢠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠒⠒⠀⠈⠈⠈⠈⠀⠀⠐⠒⠠⠄⡀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠘⠉⠚⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⡀⣀⣀⡀⢀⣀⡀⠀⣠⣀⠀⣠⡄⢀⡶⡀⣶⡸⡇⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢡⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⡆⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠞⠀⣿⠴⠁⢸⡯⡟⠀⣓⢮⠀⣇⡷⠘⣵⠇⠿⠹⠇⠀⠀⠀⠀⠀⠀⠀⠤⠄⠐⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠋⠲⠂⠘⠃⠛⠀⠙⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⢀⠀⠀⠀⠀⡀⣀⢀⣀⠀⣴⠖⠆⢠⡄⠀⣄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠈⠀⠁⠉⣁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢇⠀⡟⠀⢸⠁⠀⡇⢺⣿⡏⢺⣰⠆⣹⣋⠀⣾⢷⠀⡟⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠓⠀⠘⠂⠀⠉⠀⠀⠀⠀⠁⠀⠉⠉⠀⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡈⠣⠄⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⣤⣖⣀⣀⠈⠁⡄⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢳⠓⠒⠒⠛⠉⠛⠢⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣽⠄⠀⠀⠀⠀⠀⣾⢒⣒⣒⣒⢒⣒⠲⠴⠂⠀⠀⠀⠀⠀⢴⠀⠀⠀⠰⠐⠖⠐⠚⠲⠀⢤⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⡾⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⢀⣀⢀⡀⣆⠀⠀⠀⠀⢉⠀⠀⠀⠀⡍⠐⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡋⠙⠁⡇⠀⠀⠀⠀⠀⢺⠏⠉⠉⠉⠉⠈⠉⠉⠁⠀⠀⠀⠀⠀⢠⢣⣠⠀⠀⠀⠀⠉⠀⠓⠁⡎⠻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠌⠅⠀⠈⠞⠚⠂⠙⠀⠋⠂⠀⠀⠀⠈⡄⠉⢠⠀⢱⠀⢀⠨⡀⠀⠀⠀⠀⠀⠀⡼⡙⠀⢠⢠⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢦⠀⠀⠀⠉⠑⠐⣥⡇⡎⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠘⠀⢸⠀⠰⠀⠀⠀⠀⠀⢈⢡⠇⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⢣⠀⡀⠀⡴⡌⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢁⣠⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⢀⠇⠆⠀⢧⠘⢃⠀⠀⠀⠀⢏⠆⠀⠀⢰⠘⢆⠀⠀⠀⠀⠸⣐⣐⣐⣒⣒⡖⣢⡀⠀⠀⠀⠀⠀⠀⡟⡇⠀⠀⠀⠱⡅⠀⠀⠀⢴⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠄⢀⠀⠀⠀⠀⣀⡠⠊⣰⠀⠁⠈⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⢸⠀⠀⠀⢣⡈⠆⠀⠀⢨⡸⠀⠀⠀⢸⡀⢎⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠃⠀⠀⠀⠀⠈⢆⠀⡀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢳⠧⡀⠀⠀⠀⠈⠈⠉⠈⠃⠉⠉⠙⠒⠉⠉⠁⠀⠀⠅⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠱⠘⡄⢠⢳⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢣⣡⠚⢠⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⢼⠀⡇⠀⠀⠀⠀⢆⠋⠑⠋⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠱⠇⠨⡅⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡄⠀⠀⠠⠀⠀⠀⣠⣶⢆⣀⣘⢒⡤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⣄⡀⠀⠀⠀⣀⡠⠜⠀⠀⠀⠀⠀⠀⢰⣠⡇⠀⠀⠀⠀⠈⣆⣙⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⣀⠀⢠⡄⠀⠀⠀⠀⠨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠠⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠐⠠⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣿⡯⣿⣳⢿⣿⣾⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⢸⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣀⢐⠂⠺⠰⡂⠄⠀⢠⠀⠀⢴⡤⢤⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⢀⠀⠐⠀⣀⣆⡀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⣿⣽⣻⣯⢿⡽⣺⢿⣾⣯⣳⡇⠀⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡉⢃⠊⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠓⠠⠀⠀⠆⠀⠀⢀⣀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡡⠄⢠⠈⠈⡉⢈⠀⠀⠈⠀⡄⠀⣿⢟⢿⢿⣿⢿⣮⣿⡽⡻⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠐⠡⠇⠆⠴⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⢙⠀⠀⠀⡁⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠘⠾⠸⠀⠹⢸⣱⠙⠠⠁⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠕⠀⡀⠀⢀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠐⠐⠐⠗⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡰⣘⡴⣞⣿⡣⣢⡄⣯⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠐⠠⡄⡖⡀⠁⢊⡆⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⢀⣿⣟⡿⣻⣿⣿⣳⣹⢻⣾⢴⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⣸⢿⡹⣮⡷⣏⢕⡉⣼⢪⢪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⢤⢀⡀⠀⣀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠘⢞⡵⡫⣚⡝⡁⢉⡨⡡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠜⡌⠘⠙⠚⠣⠺⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢹⢎⢟⣌⣨⣊⡄⡃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⢇⢝⠺⠋⠙⡁⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠰⡯⣎⢇⠆⢀⠀⠠⡀⠀⠀⢀⡀⠀⠀⠀⠀⠱⡀⣀⣀⠀⠀⠀⠀⢀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠆⠀⡔⢹⢣⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠄⠁⠀⠢⠄⠀⠀⠀⡣⠐⠀⠁⠀⡑⠀⠀⠀⠀⠀⡰⠒⢀⠈⠘⠣⠱⡸⠜⠀⠀⠀⢠⠀⠈⠐⠄⣀⠀⢀⠀⠀⠀⠈⠁⠓⢄⡼⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⠂⢠⢌⠀⠢⠀⢤⠀⡄⠀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠠⠆⠈⠀⢀⡠⠔⠊⠁⠀⠀⠀⣠⠘⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡰⠡⠀⡄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠠⠂⠁⠀⠀⠀⠀⠑⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠡⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠐⠐⠐⠂⠐⠈⠈⠀⠁⠂⠂⠀⠀⠈⠁⠀⠀⡄⠀⠀⢀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠱⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠤⠜⠒⡗⠒⠒⠂⠂⠁⠁⠉⠉⠉⠉⡉⠉⠉⡉⠈⠁⠁⠉⠈⠀⠀⠀⠉⢩⡍⠐⠈⠄⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠘⡗⠠⢚⠒⠂⠉⢍⠉⠉⠉⡯⠈⠉⢁⠁⠈⡍⠉⢪⠁⡁⠈⢱⠋⠁⠋⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠀⠀⡀⢀⡄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣇⠀⠈⣀⢄⢀⣌⡠⡀⡠⠂⠤⠂⠧⠰⠤⠯⠆⠔⠀⠐⠀⠚⠄⠤⡠⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢀⠄⠄⡴⠰⡖⢰⠀⡔⡉⠀⠀⢺⠐⠊⣆⣀⣰⠀⠈⣗⢑⡵⠀⡇⠣⡘⠄⠀⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠉⠁⠁⢀⡀⡀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇⡀⢠⠞⠆⠈⠇⠀⠃⠈⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢄⠃⠐⠕⢘⠂⠈⠎⠈⠁⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠑⠢⣃⢠⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢠⡀⣠⢱⡤⠴⠤⠤⠤⡦⠠⠰⡇⠄⠄⡒⠒⠒⠒⠒⢔⠆⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⡜⣄⠄⠀⠀⠀⠀⠸⣀⠜⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠉⠒⢴⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢀⢀⡆⡁⣠⣐⡀⡄⠄⡄⠤⠀⠠⠀⡧⠰⠂⠒⠀⢸⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⡄⠀⠀⠨⡇⢀⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠈⡀⢀⣀⣀⣀⣀⢤⠀⠴⠠⠤⠔⠂⡗⠒⠒⠌⠂⢸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣤⠤⡰⠒⠀⠐⠖⢰⠃⠓⡍⠂⠈⠛⣇⠁⠘⣗⢄⡸⠠⠂⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠈⡀⣀⢀⡄⠠⠠⠠⠄⠔⠄⠄⠤⢒⡄⠒⠐⠀⡐⣸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡄⠑⠂⢉⠸⢃⠘⠀⠜⠀⠀⠅⡀⠣⡐⠟⠀⠀⠵⡈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⠒⠚⡚⡒⡒⢒⠒⠒⣒⡐⡀⡉⣃⠉⠇⠉⠉⠉⠈⢸⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⢄⡀⠀⢺⡵⠇⠇⠇⠰⠐⠀⠐⢄⠆⠊⠔⠀⠀⡈⢀⠁⠈⠂⡌⠀⠪⠀⠂⢄⠀⠁⠂⡈⠐⠡⢁⠢⠀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠈⠀⠁⠁⠁⠁⠀⠁⠀⠀⠀⠀⠀⡀⡀⢀⠀⠐⣹⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢷⠞⣴⡁⢠⡦⣁⠪⡰⠡⡂⢨⡠⣈⡠⣈⣀⣂⡐⠀⠄⠑⠀⠊⠐⠄⡀⠀⠂⠄⡐⠀⠁⠂⠅⠈⠂⠈⢠
⠀⠀⠀⠀⠀⠑⢸⠄⠠⣀⡀⡀⠀⠀⠀⠀⠂⠁⠂⠒⠓⠒⠂⠂⠀⠈⠈⠀⠁⠁⠁⠀⠀⠁⢀⠈⢸⢀⢀⠇⠄⠀⠄⡀⠀⠀⠀⠀⠀⠳⡘⡄⣿⡇⢇⣯⠉⠈⠐⠩⠈⠈⠈⠉⠈⠉⠙⠡⠉⠁⠁⠄⠀⠀⠈⠂⠀⠀⠈⠐⠄⠀⠀⠐⠀⠢⠈⠂⠁
⠀⠀⠀⠀⠈⠢⠥⠄⢰⢀⠀⠀⠠⡦⠠⠀⠎⢺⠀⡀⣄⣠⣂⠠⠀⠴⠡⠤⠤⠄⡑⠐⠂⠈⠁⡨⠀⠀⠠⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠸⠁⢇⠹⣅⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠐⡼⠀⣆⡀⡀⠀⠀⠀⠀⠇⠀⠀⠀⠒⠠⠠⠠⠀⢠⠀⠀⠀⠀⠂⡈⢉⠁⡀⢀⠄⠆⠔⠡⠆⠐⠌⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⡶⣰⠀⠀
⠀⠀⠀⠀⠁⡀⠈⠂⠀⠁⠂⠀⠐⠞⠑⡒⠁⠒⠒⠑⠒⠂⠕⠃⠃⠂⠁⠉⠈⠀⠀⠀⠀⠀⠀⣄⢀⠆⡂⠕⠀⢠⠀⠠⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡪⣣⡟⡜⠄⠀
⠀⠀⠀⠀⠀⣿⣻⣷⡶⡠⡀⠀⠀⢺⠈⡄⠄⠔⢰⠸⡀⡀⢄⠴⠤⠤⠧⠤⠔⠀⠊⠒⠓⠚⠌⠏⠀⠆⠀⠀⠀⣦⢣⢏⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⡸⣻⢕⡇⠁⠀
⠀⠀⠀⠀⠀⣯⣻⣿⣿⡸⡼⡀⠀⠀⠀⠁⠀⠐⠸⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡎⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱⡏⣾⠘⠀⠀