⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⢠⣆⣆⣀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢈⢀⢀⢂⠀⠀⠀⠀⠀⠀⢠⠃⠈⣆⠉⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⠖⢃⣄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⢀⢀⢀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠋⣄⠀⠀⠈⠋⢢⣄⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣀⢀⢀⣠⢴⣦⣤⣀⣀⠀⠀⠀⢀⣠⣦⠊⠋⠁⠀⠀⠀⣴⠈⢫⢀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢪⢀⢀⠀⠀⠀⠀⠀⠀⢻⣄⣫⠀⠈⢆⠀⠀⠀⢠⠎⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⣆⣎⠀⠀⠀⠀⠈⠉⠻⣦⣄⠁⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⢫⣤⠚⠋⠉⠉⢛⣦⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⢀⣦⠋⢸⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣆⣆⣀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠘⢊⢀⢀⢀⢠⠠⠠⠄⠀⠀⠀⠀⠀⠀⠊⢄⢀⢀⠀⠀⠀⠀⠀⢀⣠⣮⢆⢦⣤⣠⣏⠀⠀⣮⢃⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⢆⠂⠂⠂⠊⠃⠃⠃⠓⠲⢶⣦⣤⣄⣀⠀⠀⠀⠀⢀⢈⠁⠀⠀⠀⠀⠋⣦⠀⠀⢰⣎⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠃⠉⢫⢄⣠⣊⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⢳⣦⣤⣀⠀⠈⢢⣀⠀⠀⠀⢫⡀⠀⢦⠊⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⢠⡆⢢⣎⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⠀⠀⢢⢀⣠⣾⠃⣦⠎⠀⢠⣏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⣇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⢈⢁⢀⢒⢁⢀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣾⣦⠎⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⣀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣦⢋⠉⠉⠉⠉⠋⢻⣏⠁⢣⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⢀⠇⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⢃⣄⢃⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣎⠀⠀⢢⠉⠃⢢⣄⣰⡃⠀⠀⢧⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⣾⠀⢠⡆⠀⠀⠀⢰⠋⠑⢢⢎⠀⠈⠀⢣⢆⢆⢆⠆⠂⠊⠃⠃⠊⠓⠓⠓⢲⠞⠁⢣⢀⠀⠀⠀⢣⠀⠀⠀⢣⠊⠊⠁⢸⣤⣤⣤⣮⢇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣯⠀⢸⡇⠀⣾⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠉⢂⠀⠀⢪⣆⠀⠀⠀⠋⢆⣠⣏⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢰⣇⠀⢸⠂⠀⢻⣀⢀⣤⠟⠀⠀⠈⠃⠂⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⢻⣦⣶⠞⠋⠉⠀⢀⠀⠀⠀⠈⢫⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⡂⠀⠀⢛⣿⠀⠀⢤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣦⠀⣀⠀⢈⣧⠀⠀⢠⣮⠁⠀⢄⠀⢀⣤⣀⢀⣦⢿⣇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⣧⠀⠀⠀⣿⠀⠙⣿⣿⠋⠁⠉⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠈⠉⣿⣏⠁⠀⠀⣯⠀⠀⢻⢆⢦⣄⣀⣠⡟⠁⠉⠉⠀⠀⠙⣧⣀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣯⠀⣇⠋⣧⡀⠀⢿⡖⢀⣿⠁⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⢀⣿⠇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢫⢁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣇⢀⣎⠋⣦⡟⠁⠀⠙⢿⣦⣤⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣶⣿⠋⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⣾⠀⢫⣀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠀⢠⡏⠀⠀⢀⢀⢀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⢠⣠⣾⠇⢾⠒⠀⠀⠀⠀⠀⠋⠀⢀⡇⢀⠀⠀⠀⠀⠀⠀⠀⣧⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⠋⠀⠀⠀⠀⢀⣾⠋⣠⣿⠁⠀⢀⠀⠀⠀⠀⠀⣠⣏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣖⠀⠀⠀⠈⠋⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢪⣯⣄⠀⠀⠀⠀⣠⣦⡟⢣⣾⠋⣆⠀⠀⢀⠀⠀⠀⣠⣾⠋⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⢀⠀⢠⠀⠀⠀⠉⠋⠛⢲⣶⣦⣤⣤⣤⣤⣤⣤⣤⣦⣦⣶⣶⣶⠺⣿⢛⢫⣿⠟⢫⢋⠋⠛⠛⠋⢩⠃⠀⠀⠀⣠⣞⢀⢀⣂⣠⣶⠛⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢷⣦⣎⣦⣇⣠⣤⣴⡃⠀⠈⢉⠋⠋⢩⣁⣀⣯⢧⣤⣀⢀⣧⠀⠀⢋⠁⠀⠀⠈⠋⢧⣦⣤⣄⢈⣧⣤⣾⣟⢡⣴⢏⣿⣏⣁⣀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⢻⣅⣮⡃⠀⠈⢧⣀⠀⢸⠀⣰⠋⣿⢫⣿⣦⣿⣏⣏⠋⣆⠀⣆⠀⠀⠀⢀⣶⠋⠀⠀⢫⣧⣀⠈⢸⡏⢉⢉⢫⣀⠀⠀⠉⢫⣦⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠁⢀⠈⢠⣏⠀⠀⢰⠋⠁⠀⢸⠀⠈⢣⣤⣾⠋⢸⡋⢿⣤⣎⢁⣤⣿⠃⣦⣤⣮⣅⢠⣴⣦⢮⢿⣯⢷⣿⠋⠉⣯⣆⠈⠛⣦⣀⣮⠃⠀⣠⠏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠃⠀⣎⠀⢸⣆⠀⠀⢪⣦⣤⣄⣮⣆⠆⣦⣌⣸⣦⣮⣧⣼⣆⠀⢋⣠⣤⣿⠀⠀⠀⠀⢣⣠⣧⠀⣀⣼⣿⠋⢳⣷⢮⢿⣤⣠⣦⠋⠀⢀⣦⠋⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⣪⠀⠀⣰⠋⢋⠀⢺⠋⠉⠉⠉⠉⠃⢢⣀⢫⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢀⠀⠉⢫⠋⠁⣧⠀⠀⢠⡇⠀⠀⠈⢣⠀⢀⣴⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠀⠀⠀⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠋⠀⠀⠛⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠐⠛