⠰⣤⠤⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡠⠃⢀⡀⠀⡰⠁⠀⢀⠎⠀⠀⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⠀⣠⠞⠀⡜⠀⠀⠐⠁⣠⠇⠀⠀⠄⠀⠈⠉⠀⡌⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡈⡔⡁⠀⠀⡠⡁⠀⠀⢚⠃⠆⠀⠀⢦⠀⠀⠀⢸⠀⠀⠀⠀⠘⠀⠀⠀⠂⠀⠐⢜⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⠴⠛⠰⠀⢠⡼⠋⡅⠀⢀⠆⠘⠀⠂⠀⠀⢀⠀⠀⡿⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢨⠈⢉⣀⡀⣀⣤⣤⣄⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⡇⠀⠀⠀⠀⠀⠇⠀⡘⠀⠃⢠⠀⠀⡰⠁⣀⡀⠀⡀⠠⡀⠣⠀⠀⢠⠎⠀⠀⢀⠘⢻⣿⣿⣿⣿⣿⡻⢿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠱⡆⠀⠂⠀⠀⠀⠀⢠⢀⡆⠈⠀⣰⣷⠟⠉⢈⢴⠁⠰⠀⠀⠀⠀⠀⡀⠀⠀⠘⢠⣿⣿⣿⣿⣿⣿⡟⢄⠙⣏⠙⠿⠂⠐⣬⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢠⢷⡀⠀⠀⢠⠀⠀⡰⠁⢃⢀⡖⠁⡄⠀⠀⠁⡾⣀⠀⢠⢀⡠⠘⢱⣧⠂⢀⣳⠋⠀⢼⢏⣿⣿⣭⣙⠈⢻⣮⣳⣤⣦⣬⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⣣⢠⣀⣌⡔⠀⠀⡐⢻⠏⠀⠀⠸⠀⠀⢞⡻⢋⡴⡇⠉⣷⣶⣿⠇⣠⣿⣿⣦⣷⢀⣼⣿⣿⣏⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⠟⠓⠉⠈⠽⠃⠹⣦⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠘⠀⠀⠀⣀⣀⣿⣿⣿⡿⣿⣿⣿⣿⣥⣤⣀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠌⠂⠀⠀⠀⢀⡀⠤⠌⠁⢻⣿⣿⢿⣿⠁⠀⠀⠋⠿⠿⠿⠻⠂⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠙⢦⡀⠀⢀⠎⠀⠀⠀⠀⠈⢻⣿⠲⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠘⠛⣧⡊⠊⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀
⠉⠉⠐⠂⠀⠀⠀⠀⡤⠤⠤⠤⠮⠤⢀⣀⠀⠉⠒⠊⠀⠀⠀⠀⠀⠀⠀⢿⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⡇⢁⣿⣿⡟⠛⠛⠉⣡⣿⠟⠃⠈⠂⠀⠀⠀
⠠⠌⠙⠋⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀♡ ⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⢛⡽⡀⠔⠊⠀⠀⠀⢀⣄⠊⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀♡ ⠀⢣⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀♡ ⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀♡ ⠀⢿⣤⣤⣤⣴⣶⣶⣦⡀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⡠⠀⠒⠒⠰⠦⢏⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀♡ ♡ ⠘⠿⣿⣿⢿⡿⣿⡭⡷⢄⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠐⠈⠛⢦⣄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠊⠀⡇⠀⢳⡀⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠉⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡇⠀⠸⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⢠⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠸⡴⠁⠀⠀⠀⠀⠀⢠⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⢆⠀⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⢸⠎⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⢠⠻⡄⠀⠀⡴⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣳⠜⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠉⠁⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠃⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠈