Drunk Emojis & Symbols

⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⢄
⠀⠀⠀⠒⠉⠘⠀⠀⠀⠰⢄⣀⠈⠀⠠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠
⠃⠀⠁⠐⠤⢀⠁⠀⠀⢀⠠⠏⢀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢠⠀⠀⡄⠀⢠⠀⠀⠡⠜⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠃⢂⣴⡀⢄⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠰⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⢂⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠀⠈⠉⠉⠀⠒⠠⣤⢀⠀⠀⠐⡶⠣⣤⡄⢫⠀⠀⠀⡠
⠀⠀⠀⠀⠄⠀⠀⠄⠀⠠⠀⠀⠀⠈⠁⠑⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠃⡶⠋⠈⠙⢷⠀⢈⠶⠂⠤⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠳⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠒⠈⠁⠁⠶⡀⣸⠑⠁⠀⠀⠀⢆⣁⠀⢑
⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠂⠤⠎⣅⠈⠁⠈⠢⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡖⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⣀⡀⠀⠀⠀⢀⣠⡖⠊⠀⠀⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⣿⡿⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡠⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠓⡁⢀⡉⠂⢄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠒⢸⠀⠀⠀⠀⠀⠀⡀⠀⠄⣀⢀⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⢀⠈⠀⡸⠂⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠐⠉⡉⠹⠿⠿⠋⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⢊⢀⡀⠀⠊⢃⡀⠔⠈⠆⠘⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢤⠀⠀⠠⣄⠀⠀⠀⠀⠀⢠⡀⠀⠀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⣀⠈⢀⠀⠀⢧⠀⡁⠊⠳⡀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂⠘⠄⠀⠀⠘⡄⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠜⣛⠀⠀⠀⠈⣓⡤⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⢙⠁⠀⠀⠈⢒⣀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⢄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠎⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⣀⢀⣊⣀⠀⢅⠀⣸⠀⡀⠀⡀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠆⢲⣤⣾⠃⣔⣰⠘⠱⣄⡀⠈⠣⡀⠄⠀⢀⠀⡌⡠⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⣏⠎⡞⠏⠀⠀⡷⠎⠀⠘⢿⠆⠀⠀⠂⠙⠀⠀⠊⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡤⠥⠛⣖⠂⠠⢀⠀⠈⠁⠇⠀⠀⠠⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⡆⠀⡖⠁⠀⠀⡨⠃⠀⠀⠄⠀⢰⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠃⣰⠇⣀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠑⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠀⡀⠈⡕⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠘⣄⣧⠖⠂⠀⠀⠀⠀⠈⢁⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⣠⣤⡼⠀⠀⠀⠀⠀⠀⠀⡄⢈⡄⠥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠚⠟⠠⠀⠉⠛⠀⠀⠀⠶⠶⠆⠀⠁⠀⠀⡀⠠⠄⠂⠉⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠚⠓⠒⠂⠀⠀⠀⠀⠀⠐⠀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠂⣄⡀⠀⠀⠀⠀⠀⡆⠀⡶⠖⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣀⣶⠏⣀⢆⡁⠠⡀⠀⠀⠀⠀⠘⠁⠀⢇⠀⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⡄⢷⡶⠊⢁⠤⠀⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡖
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡎⢀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢻⣏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠜⠂⠲⠿⠓⠶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⠔⠛⣈⡗⠐⢢⠒⡚⢦
⠀⠀⠀⠀⠀⡀⢒⣿⢤⡅⡀⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠐⢆⠉⠣⡤⢄⠁⢸⠋
⠀⠀⠀⠀⠀⠀⡎⠀⣸⡁⢀⣄⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠈⢿⣀⠀⢡⠀⣁⠸⢂⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠶⠚⠓⠀⠀⠉⠉⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⢁⣀⠄⠀⠀⠀⠀⠀⠀⣛⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣾⣿⣶⣾⣿⣿⣿⠶⠀⠀⠈⡘⣻⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣬⣍⣻⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣿⣿⠃⠙⠈⠳⣍⠉⠙⠛⠛⢛⣽⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⣹⣤⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⡛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣯⣽⣿⣿⣯⣿⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣀⣭⣿⣿⣿⣷⣬⡆⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣠⣴⣿⣶⣶⣶⣤⡄⠀⠀⠀⠀⠈⠉⠁⠀⠙⡿⠿⣿⡇⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣸⣤⣴⣶⣶⣾⣷⣶⣤⣹⣿⡿⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢸⣯⢸⣿⠏⢿⣛⣛⣻⣿⣿⣿⡆⠺⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠈⠛⣾⠛⣿⢿⡟⢹⣿⣿⠁⠀⢸⣿⣏⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠈⠐⠈⢁⡿⠛⡏⠀⠀⣿⣿⣏⡙⠛⠋⣸⣿⣿⣿⣿⡷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣻⣿⣿⣦⠉⠛⠻⢿⣛⣿⣿⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⣏⡀⠁⠀⠀⠀⠀⣹⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠄⡀⠀⠀⢠⡞⠁⠀⠀⠀⠀⣸⣿⡟⣷⡄⠀⢀⣤⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⠘⠀⠀⣿⠷⣶⣤⣀⣠⣴⣿⣿⣴⣿⡇⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⣿⡶⠄⠈⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠀⠀⠀⠸⢀⠀⣼⣏⠠⠤⠤⠤⠤⣤⣤⣄⣈⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡼⢿⣭⡟⠁⠀⠂⢤⣤⣤⣤⣄⣠⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡄⢀⣄⠁⠀⠀⠀⠀⢐⠃⠀⠙⠳⣤⢀⣀⠀⠀⣼⣍⢿⣿⣿⣿⣿⣿⠁⢻⣇
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠉⠛⠀⢤⣤⣴⣶⣾⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⣾⠈⠄⠀⠀⠉⣺⣿⣿⣿⣿⠇⠀⢈⡏⢷⣄
⠀⠀⠀⠀⠀⠙⠛⠳⠶⠾⢷⣿⣄⣈⣿⣿⢿⣿⠇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⠇⠀⠀⠰⠇⠀⠙⠦⢤⡀
⠀⠀⠀⠀⠲⢦⣤⣄⣀⣠⣇⡀⣈⡉⢹⡇⣸⣿⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠁⠀⠀⢰⡟⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⡀⠀⠉⢉⡉⣉⣙⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡀⠀⠚⣿⡿⠛⠁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠙⠛⢷⡄
⠀⠀⠀⠀⣈⠉⠛⠿⣿⣿⣛⠋⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⣸⡏⢀⣦⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂
⠀⠀⠀⠀⠈⠛⠲⠾⠿⠿⢿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⢠⠿⢧⣶⣤⡤⢐⣿⠏⠁⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠿⠟⣠⡾⢁⡄⠀⣠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⣿⡟⣱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣴⣾⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⣩⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣾⡿⠟⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣩⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⣡⡾⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠿⠿⠿⠿⠻⢿⠟⠙⠿⢿⠿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣛⣛⣻⣟⣿⣿⣿⣟⣛⣛⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢠⣶⣿⣿⣿⣶⣿⣿⣿⠟⠻⠃
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣻⣟⠋⠏⠇
⠀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠶⣤⠄
⠀⠀⣿⣿⣷⣶⣶⣾⣿⣿⠋⠀⠀⠀⠀⠀⠉⠀⢀⡀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠘⣿⣿⣿⡏⠫⡍⠛⠀⠀⠀⠀⠀⠀⠀⠀⣉⡀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⢃⢀⣎⡁⡹⡤
⠀⠀⠀⠈⢿⣿⣧⣤⠳⠄⠀⠀⠀⠀⠀⠀⠀⠈⡙⠚⠁⠐⠢⢼⠿⠀⣤⣤⣤⡠⠈⣠⣤⣏⣿⣇
⠀⠀⠀⠀⠀⠙⣿⡟⠀⠐⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⢔⠧⠤⠥⠞⠁
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠙⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢰⠇⡀⠀⠀⠀⠀⠀⢀⡿⡆⠱⠒⠒⠒⠉⠉⠂
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀
⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠢⣀⠀⠀⠀⢡⠘⡷⣀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⠑⠢⣠⣌⢆⠻⣮⣓⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠈⠛⠶⣌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠓⠦⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⡤⢤⣀⣀
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠀⠀⠁⠉⠳⡄
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠲⡄
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀
⠀⠀⠀⣏⢀⣤⣠⣤⣤⣠⣤⣤⣆⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⠀⠀⡸⠁⠀⠀⠀⡀
⠀⠀⠠⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠙⠶⠤⠼⠋⠧⣄⣀⣤⠞⠁⠈⢹⠁⠀⢀⡇
⠀⠀⠀⢙⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣸⠀⢠⠋
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢣⣤⣤⣄⣀
⠀⠀⠀⢠⣿⢠⠓⢦⠠⢀⠀⠒⣆⣀⣠⠐⠂⢲⣄⠀⢀⣀⠤⢤⡤⠔⠢⠐⠆⢸⡇⠄⢸⠁⠀⠈⠉⠳⣦
⠀⠀⠀⢸⣿⢸⠀⢈⠀⠀⠀⠀⠘⠿⠋⢰⡄⠘⠿⠿⠋⠀⠀⠀⡆⠀⠀⠀⠀⢸⡇⢀⡿⠒⠲⢦⡀⠀⠈⠳⣄
⠀⠀⠀⢸⣿⢸⠀⠘⢄⡀⠀⠀⠀⢀⡴⠃⠳⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⢸⡇⣸⠁⠀⠀⠀⠙⢦⡀⠀⠘⢧⡀
⠀⠀⠀⡚⣿⠀⠀⠀⠀⠈⠁⠁⠉⠁⠀⠀⠀⠈⠓⠀⠀⠔⠚⠁⠀⢲⣦⠀⢸⢸⡇⢿⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠳⡄
⠀⠀⡠⠋⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⢸⠈⣧⠈⠳⡄⠀⠀⠀⠀⠀⠹⣆⠀⠀⠹⣆
⠀⣸⠁⠀⡇⠀⢠⡾⠋⠙⢲⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠁⠀⠀⠀⠈⠀⣿⠀⠀⢧⡄⠀⠀⠀⠀⠀⠘⣦⠀⠀⠙⣆
⠀⢻⡀⠀⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡆⣿⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢹⡄
⠀⠘⢦⢰⡇⡆⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠃⢿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⣇
⠀⠀⠀⢰⡇⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠿⠛⠛⠁⠀⠀⠈⠑⠢⣄⢠⢸⡇⠀⣀⣴⠃⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡿
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣸⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢸⡇
⠀⠀⠀⣸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠀⠀⠂⠱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⢀⣤⠶⠶⣤⣤⣤⣤⡴⠞⠁⠀⢀⣾⠁
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢄⠀⠀⠀⠀⠀⢀⡜⢡⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⢀⡴⠛⠁⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣤⣶⠃⠀⠑⠲⠤⠤⠖⠋⣀⡴⠈⠻⢶⣄⠀⠀⠀⠀⠀⠀⣀⣴⠟
⣼⠁⠀⠀⠀⠀⠈⠑⠂⠤⠄⢀⡀⢀⣀⣀⣀⣀⣉⠉⠁⣀⠀⠀⠤⠄⠒⠊⠉⠀⠀⠀⠀⠀⢹⡆⠀⢀⣠⣴⠾⠋
⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠓⠛⠛⠉
⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠉⠻⣷⡶⢦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⢦⣬⣭⣭⣍⣭⣍⣭⣭⣭⣭⣤⠾⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⡇⠘⠛⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠙⠒⠚⢃⡼⠇⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠻⣤⣼⠀⠀⠀⠀⢠⡶⢆⠀⠀⠀⢀⡴⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠲⣄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡤⠶⠶⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⠙⠋⠀⠀⠀⠛⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣀⡠⠶⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣦⣀⠀⠀⠀⠀⢾⡁⠀⣳⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠙⠚⠁⠀⠀⠸⣇⡀⣹⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢸⣭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⢾⣭⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠸⣇⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠄⠀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢀⣤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢿⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⠀⠲⢤⣄⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠘⠶⣤⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⣿⠟⣶⣤⣀⣿⠀⣰⡾⠛⠛⠛⠳⢦⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠈⠛⠛⠛⢉⣽⠇⣸⠏⠠⢤⣀⠀⠀⠀⢱⡄⢸⡇⠀⣴⠞⢛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠤⠤⠀⠤⠐⠛⠁⠀⣿⠀⠀⢸⣿⡟⣷⣤⣼⠇⣾⠀⣾⠁⡼⠋⣍⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠁⠀⠀⠀⠐⠒⠒⠒⠒⠲⣤⡀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠘⣇⠀⠀⠙⠛⠋⣠⡟⢠⠟⠀⠻⣄⠻⠆⠜⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠉⠓⠲⠒⠉⠉⠉⠰⠶⠶⠚⠉⢠⡟⠀⠀⠀⠈⠑⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠉⠉⠛⢦⡀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣰⠟
⠀⢀⠀⠀⢀⡄⠀⠀⠀⢠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣠⠚⠁
⢠⡎⠀⢰⠏⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⠋⠁
⠀⠁⠀⡇⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢻⡌⠛⠳⠶⠦⠀⣤⣤⠄⢤⡶⠶⠖⠛⠋
⠀⠀⠀⡇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣶⠀⣽
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠈⣧⠀⢀⠟⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠈⠳⣾⡀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡍⠉⠉⣷⠄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⣠⠾⠛⣻⡷⠶⠾⣧⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠋⠀⠛⠁⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⠋
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡸⠋⠓⠒⢦⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠄
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣷⡀⠀⠀⠀⠀⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⠶⣤⣀⣀⣀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡄⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⡀⠉⠉⠉⠀⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⡆⠀⠰⣆⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢰⡟⠀⠀⠀⡟⠉⠒⠒⠒⠢⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠸⡇⠀⠀⠙⣆⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⣼⠁⠀⠀⣸⠃⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠸⣇⠀⠀⠀⠀⠈⢷⡀⠀⢸⡇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠈⢷⠀⣼⠃⠀⠀⢸⣇⣀⠀⠀⠀⠀⠀⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠉⠉⠙⠛⠒⠤⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⡄⠙⠻⠶⣦⣄⡀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠻⠷⠶⠶⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⡀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣩⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠜⠀⠀⢀⣠⠤⠴⠖⠲⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠘⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡁⢀⣔⣲⠄⠀⠀⢠⣖⣢⡀⣈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡄⢸⠈⠁⣀⠀⠀⠀⠀⣀⠈⠁⡇⣠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⠸⣄⣀⣁⠐⠂⠐⠀⣈⣀⣠⠃⣟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⣈⠛⠒⠒⠛⣁⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠀⠀⠀⢏⠝⠉⠉⠉⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠡⠃⠳⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠙⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⡀⠀⠀⠀⠉⠓⠲⠶⠶⠖⢚⣉⠀⠀⣠⣄⠀⠈⠳⢤
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣠⢾⠁⠀⠀⠀⠀⠀⠔⠤⠀⠀⠘⡇⠀⠀⠀⢻⡶⠒⢒⡄⡧
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡇⡾⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⡇⠀⠀⠀⢸⣯⠭⢴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣀⡇⣇⠀⠀⠀⠀⠀⠀⢔⢰⠄⠀⠀⡇⠀⠀⠀⣸⡿⠢⠜
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⣱⠸⣄⠀⠀⠀⠀⠀⠄⢸⠀⠀⠀⠉⠉⠉⠉⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠼⠀⢳⠉⠓⠒⠶⠶⠤⠼⠦⠤⠶⠶⠒⠚⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢲⡖⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⣀⡀⠀⢸⡇⠀⢀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠈⠉⠉⢹⡏⠉⠉⠉⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠋
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠠⠋⠓⠑⠛⣆⡀⠐⠀⠄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⠀⠀⠀⠂⠈⢳⡀⠀⠐⠯⡘⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⢐⠃⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠁⠀⠀⠈⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡄⠀⠈⠀⠀⠉⠽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠜⢁⠀⠀⠀⠀⠈⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠋⠀⠀⠀⠀⠀⠈⠀⠀⠐⢇⠀⠀⠀⠀⠀⢠⠴⡀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠂⠁⡰⠃⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢤⠁⢐⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⡨⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠘⠅⠤⠒⠁⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠐⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⣀⣀⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⡇⠀⠈⠢⡀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⢀⠀⠀⠀⠀⢱⠀⢀⡠⠦⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⢀⠎⠀⠀⠀⠀⠈⠏⠁⠀⣀⣀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠉⠦⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠤⢠⠆


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡺⠿⠟⠛⠛⠿⠿⣶⣢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⡾⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⠃⠌⠩⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⢀⡠⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⡴⡵⠃⠀⡤⠜⠀⠀⠀⠀⠒⠻⣦⠀⠀⠀⢠⣾⣿⣦⣹⣷
⠀⠀⢠⣮⣾⣿⣿⣿⣆⠉⠉⠘⠛⠛⠻⣖⢄⣼⡿⠁⠀⡘⠋⠉⠀⠀⠀⠀⢀⡀⠘⠀⠀⠀⣾⣯⣾⢿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣷⢾⣁⣀⡀⢇⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⢿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣄⣇⠏⣌⣄⡑⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡩⢂⠀⡂⡷⡇
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⠟⠉⠉⠉⢉⡩⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣪⣾⠟
⠀⠀⠀⠙⠿⠿⢿⡿⣛⣿⣿⣿⣿⣿⣿⠁⢠⡤⣶⢎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡯⠟⠊⠁
⠀⠀⠀⠀⢀⣔⣽⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⣿⠉⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣾⠿⠛⠁
⠀⠀⠀⠀⠘⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣦⣴⣫⣷⣄⠀⠀⠀⠀⠠⣾⣽⠋
⠀⠀⠀⠀⠀⠈⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⠐⢀⠤⠲⢌⣯⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⢄⠺⣄⠀⢸⢹⣿⣮⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠿⢿⣻⣿⣿⣿⣿⣿⡇⠀⠀⠘⢷⡿⠈⣿⣿⣿⣝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣇⠀⠸⣿⣿⣿⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⠀⠀⣿⣿⣿⣿⣷⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⢹⣿⣿⣿⡫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠈⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢝⣿⢿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣟⡯⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠛⠛⠓⠒⠒⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⡾⠃⢀⠤⠒⠠⢤⡀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⡴⠥⠘⢧⠔⠁⠀⠀⢀⡀⣷⣶⢯⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⠶⠟⠉⠁⠀⠀⠀⢀⠀⠀⠈⠀⠁⠀⠘⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢸⣾⠁⠀⠀⢀⠀⠀⢠⣾⡁⠆⢰⠁⠀⠀⠀⠐⡮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠨⠉⠉⢀⠀⢰⣿⢦⡀⡾⠉⢙⡸⠀⡀⠀⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣞⣱⠀⠸⡀⣿⠁⣈⠻⢧⡀⠀⠀⢡⠘⢾⣇⡤⠖⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡠⠄⡇⢧⠀⠀⠁⡀⠁⠀⢀⣠⣴⣈⠫⠧⠐⠦⢸⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣨⣿⣦⣸⣮⣗⠀⢀⣰⣄⠙⠁⠛⠉⠁⠀⠀⣄⣴⠃⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠛⡿⣿⢎⠙⡟⠋⠀⢰⠁⠀⠀⠀⠀⣧⠀⠉⠢⡀⠀⢸⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠃⠀⠋⢹⡜⣧⠤⠖⠁⠀⠀⠀⠀⠀⡏⠀⠀⠀⣨⢆⢸⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠏⡘⠀⡎⢰⠎⢀⠇⠀⠀⠀⠀⠄⢼⡇⠛⣰⣟⢆⠀⣞⠃⣄⢿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠠⡔⠀⠐⠀⡐⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡿⢸⠂⠀⠑⠀⢸⡇⠀⠐⠀⠰⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡶⣀⠠⣀⠃⠀⠈⠓⢦⣥⢄⡀⠀⠀⢀⡼⠋⡖⢹⡄⠀⠄⡀⢸⡇⢸⠀⢀⢄⠌⢁⡶⠂⠀⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠘⠢⢴⡈⢦⠀⠀⠀⠀⢹⠲⡍⠛⠉⠁⠀⠀⣠⠞⠙⣆⠀⠈⣺⡇⠚⠀⢠⠋⡰⠋⠀⠈⡁⠠⠀⢀⠉⢢⡀
⠀⠀⠀⠀⠀⣀⠤⠒⠚⠧⣀⠀⠀⣀⡿⠀⠀⠀⠀⢀⡼⢀⡱⣆⠦⠖⠚⠉⠀⠀⠀⢸⡄⣸⠃⠁⣹⠂⡟⡴⠁⢀⠔⠁⠀⠀⠀⠀⠈⢀⡇
⠀⠀⠀⠀⡸⢁⠀⠀⠀⠀⠀⢙⡾⠏⠀⠀⠀⢠⡔⡃⢠⡞⢳⠈⠳⢦⣀⠀⠀⠀⠀⢸⣃⣟⣊⡁⠉⢦⣼⠇⢠⠁⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⢀⠗⠁⣠⠴⠒⠂⠒⠏⠀⠀⠀⠀⣰⣯⠀⣉⡟⠀⠈⠀⠀⣀⣈⣉⣒⢴⠈⠁⠀⠀⠀⠀⢹⡆⣙⣶⣇⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⢠⠃⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⢋⠔⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⢈⡀⠀⠀⠀⠀⣀⡼⠓⠁⠀⠈⠃⠀⠀⠀⠀⠀⢀⡞
⠀⠀⢸⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⠟⡠⠖⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⢈⠀⠀⣠⡈⠉⠁⠆⠀⠀⠀⢀⡴⠋
⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⡷⠚⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢴⣢⢀⠤⠐⠀⠀⠀⠀⠀⠆⠀⣴⠾⠿⠤⠴⠶⠶⠶
⣀⣀⣀⠙⣷⠀⠀⣀⣔⣩⡴⠾⠯⠤⠚⠓⠛⠛⠛⠋⠉⠉⠉⠛⠛⠂⠀⠉⠉⠉⠉⠀⠹⡆⠀⠀⠀⠂⠈⢀⠀⢀⡞⠁
⠀⠀⠀⠀⠈⠓⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠳⠶⠴⠖⠋⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⣤⠌⠀⢂⠀⠤⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⡐⠁⠀⠀⠀⠀⠄⠖⠈⠀⢠⠀⠀⠀⡜⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡆⠉⠀⠀⡀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠼⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠊⠀⠒⠈⠡⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠂⠀⠈⠉⠉⠉⠉⠉⠉
⠂

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠤⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣀⠤⠤⠔⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠃⠄⣀
⠀⠀⠀⠀⢀⠞⡀⠀⠀⠀⠀⠀⡑⠤⡍⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠳⠽⢢⣀
⠀⠀⣠⠤⠞⠉⠀⠙⢆⡠⠲⠅⢂⠄⠀⡀⢠⠄⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡈⡃⠀⠑⠐⠉⠢⡀
⠀⠀⢇⣀⠄⠀⠀⠀⢊⢀⡔⠀⡀⠹⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⠀⠀⠀⡀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠧⠱⣌⡁⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠚⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⢤⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠸⣀⠟⠉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠁
⢀⡰⢄⡀⠀⠀⠀⠀⠄⠀⠀⠙⣦⣀⠀⣀⣀⣠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⢙⣄⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠹⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠠⠂⠘⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠻⢿⠟⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠃⠐⢂⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠠⠃⠀⠄⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠈⠁⠁⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠢⠖⠆⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⡼⠃⢀⠤⠐⠠⣄⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠰⠦⠀⠧⠜⠁⠀⠀⢠⢸⣷⡼⢧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠞⠛⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠩⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣀⣷⠃⠀⠀⢀⡄⠀⢠⣮⣅⢂⠘⠀⠀⠀⠀⠱⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⣂⠀⠀⡄⢰⣿⠷⣄⣾⢀⠨⠜⠀⢄⢰⡃⠀⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣽⡆⠀⠃⢸⠁⠤⠙⠻⡄⠀⠀⠠⠈⢻⣧⠒⠋⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡂⢹⡸⣄⠀⠀⢀⠀⣀⡠⣴⣾⠬⠛⠓⠘⢃⠗⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠛⣷⣦⣿⣿⣀⣠⣼⣦⠀⠠⡁⠀⠀⠀⢸⠖⠫⡀⠀⠀⠸⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣇⠿⢹⢵⠤⡇⠀⢀⠌⠀⠀⠀⠀⢸⡇⠀⠈⠣⡄⢰⡇⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⡇⢀⠀⡁⠜⣇⠛⠋⠁⠀⠀⢠⢀⣤⢸⣇⠀⠀⡴⠛⢸⡇⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠐⠀⡀⠀⠁⣰⠇⠀⠀⠀⠀⠁⠙⠃⢠⠿⡎⣧⠘⠢⡘⠊⡇⠀⠀⡘⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢉⡀⠀⠀⠸⠀⠹⢦⣀⠄⠀⠀⠀⠀⠀⢸⠴⡡⡇⠀⠀⠀⠀⣧⠀⠀⠀⡈⠙⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠘⣝⣂⡑⢆⠀⠀⠀⠈⢿⣑⠢⢤⠄⠖⠁⠘⢀⡾⢄⠀⠀⢤⡇⢸⠆⠀⡱⢁⡴⠉⠠⠀⠘⠒⠀⢤⡀
⠀⠀⠀⠀⠀⠀⣀⡀⣼⡁⠀⠀⠈⢻⠄⠀⠀⠀⠀⣸⠈⢧⣄⣀⡤⠴⠖⠋⠀⠈⣆⠀⡶⠓⠐⡄⣼⢠⠏⠀⡠⠐⠂⠉⠀⠀⠀⢹⡄
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠉⠓⢄⡴⠋⠀⠀⠀⣠⡞⠁⣼⠆⠳⢦⣀⠀⠀⠀⠀⠀⡿⢸⠥⠒⠘⢄⣡⡟⠀⠌⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⢀⡥⠋⢀⣠⠤⠤⠴⠋⠀⠀⠀⢀⡼⠁⠲⡼⠁⠐⠀⠀⠉⠑⠢⢄⣠⠒⠉⠉⠉⠉⢐⡄⠻⣅⡎⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⢠⠊⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠠⠉⠁⠀⠀⠖⠒⠋⠉⠉⠁⠃⠀⠀⠀⠀⠀⣀⠷⠛⠉⠙⠢⠀⠀⠀⠀⠀⢀⠎
⠀⠀⢯⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⢂⠤⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⣿⢺⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢸⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⣠⣴⣯⠀⠤⠤⠤⠤
⢀⣀⡈⠛⡆⠀⠀⣠⣐⣡⡴⠾⠥⠤⠟⠒⠚⠓⠒⠒⠒⠚⠟⠓⠒⠂⠀⠈⠉⠃⠘⢧⠠⠄⠀⠒⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠙⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠶⠊⠁⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⣤⡤⠁⠀⠄⠠⠤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠈⠀⠀⠀⠀⠃⠰⡂⠀⢀⠀⠀⠀⡘⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡆⠀⠀⢀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠼⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠉⠉⠈⠢⣢⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣹⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠀⠈⠉⠉⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠋⠛⠀⠀⠭⠷⠀⣾⠯⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⢠⣄⣠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠈⠃⠐⡂⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠈⠂⢈⡂⠠⢄⡀⣠⣴⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠈⠁⠀⠀⠀⠀⠀⠀⠙⠂⢀⣩⡛⣿⣿⣿⠋⠁⠄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⣿⠙⠘⣇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⡠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⡀⠀⣿⣧⡀⠸⡄⢻⣦⣄⢹⡄
⠀⠀⠀⠀⠀⠀⢻⣸⠃⠀⠀⠀⠀⠀⠀⢧⢡⡀⠀⢣⡀⣿⣿⣷⣴⡇⢸⣿⣿⣿⣷⠠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣟⢃⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⢸⡟⣿⣿⣿⣿⡃⢸⡿⠿⣟⡉⠀⢸⠂⠖⠘⠆⢘
⠀⠀⠀⠀⠀⠀⢀⣿⣌⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⠿⠿⠿⠟⠛⠛⠳⠀⠀⠀⣍⠉⣴⢷⣶⣶⣴⡶⡚
⠀⠀⠀⠀⠀⠀⣼⠋⢻⣦⠀⣠⢂⣴⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠸⠀⣎⠘⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⡿⠀⠀⢹⣷⣕⠟⡿⣿⣿⢂⠀⠀⠀⡀⠀⢀⣀⠀⣦⣰⣶⣶⣶⣆⠉⠛⠹⢿⡿⣠⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠿⣷⣭⣸⣉⣬⣈⣥⣤⣶⣶⣶⣾⣦⡬⢿⡿⢋⠀⠈⠁⢠
⠀⠀⠀⠀⠀⠀⣷⡰⠃⠀⠀⠒⠺⠿⠿⢿⣿⣿⢿⣻⣿⠟⠀⠀⠠⠄⠀⠑⢋⡀⣂⢆⣄⡅
⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠛⠁⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⠴⠶⠛⠻⠷⣶⢦⣰⣄⣨⣻⣷⣦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠛⢛⣿⣿⡿⢿⠟⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠋⠉⠉⠛⠿⠿⢿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⠀⣰⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇⠀⠀⠀⠀⠘⣇⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡗⠶⢶⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⢸⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢄⣠⠱⠀⠀⠀⠀⢻⡿⣿⣦⣀⠀⠀⢸⠀⠘⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⡇⠈⠻⢿⣷⣦⣬⣄⣥⣴⣿⣷⣄⣀⣀⣠⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠉⠙⠛⠛⠛⠉⠈⠙⠛⠟⠛⠋⣿⣿⠁⠀⣤⠤⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣧⣀⠀⢻⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣆⣠⠞⠉⠀⠀⠀⠀⠉⢻⡿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠋⠀⢀⣿⣇⣀⣀⠀⢠⡶⢶⡀⠀⠀⠀⡏⠀⠘⣿⡅⣴⡾⢛⣻⣷⠶⠿⢶⣿⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢶⣶⣶⡾⠛⠉⠁⠀⠉⠙⢾⣿⡧⠙⢦⡀⢰⡇⠀⠀⠹⣷⣏⠀⢿⠀⠀⠀⠀⢸⣿⣆⣀⣸⡓⠂
⠀⠀⠲⡶⠺⡄⠀⠀⠀⠀⠀⢹⣿⡠⢀⣤⣴⣦⣄⡀⠀⠙⣯⢂⠈⠳⣼⠀⠀⡀⠀⢻⣟⠦⣼⡷⠟⠁⠀⣼⠹⡏⠀⠈⣷⣤⣤⣀⡀
⠀⠀⠸⠓⢴⠆⠀⠀⠀⠀⠀⣾⣿⣏⣾⠋⠀⠀⠉⢷⠀⠀⢸⡷⣄⠀⠈⠀⢠⣧⠀⠀⢿⣄⠀⠀⣀⣤⡾⣣⠾⠷⠶⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣷⡀⠀⠀⠀⣸⠃⠀⢸⣯⣿⣷⡄⠀⢸⠙⣧⣠⣬⡿⠻⠛⠛⠉⠰⡏⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⣀⡤⠶⠾⠿⠿⠿⢿⣿⣿⣯⠻⠦⠤⠞⠁⠀⢀⣾⣿⣿⡈⠻⡶⠟⠀⠉⠉⠁⠀⢄⣑⣄⣀⣀⣹⣄⠀⠀⠀⠠⢎⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⡹⢿⣿⣧⣀⡀⢀⣀⣴⣿⣟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠳⡌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣠⠞⠁⠀⢈⠿⠛⠛⠛⠋⠁⠈⠻⣧⠀⠀⠀⢀⡴⠞⠛⠓⠲⣄⡂⠀⠙⢦⠈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣰⠋⠀⣠⠴⠶⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠔⢃⡀⠀⠀⠀⠀⠀⣿⣰⠃⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣃⡀⢰⠊⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠛⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠦⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⡦⢄⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡀⠀⠀⠀⢀⠔⠈⠀⠠⠈⠀⢀⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⠸⠀⢆⠀⠄⠀⡀⠐⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠘⠻⠒⡽⠃⠈⠂
⠀⠀⠀⡠⠂⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠤⠤⠤⢄⣀⠈⠁⠀⠀⠀⠄⠦⠤⠤⠤⠀⠀⡀
⠀⠀⢀⡁⠀⠀⠀⠂⢩⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠐⠠⡀⠀⠈⠑⠢⣄
⠀⠀⢼⠀⠀⠀⠂⠐⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠈⢦
⠀⠀⠘⡀⠀⠀⠀⠈⢼⠁⠀⠀⠀⠀⠀⠀⠁⠈⠀⡈⠍⠛⠀⠀⠈⠀⠁⠈⠉⡉⠉⢉⠉⠉⠉⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠂
⠀⠀⠀⠳⠀⠀⠀⣀⠙⠀⠀⠀⠀⠀⠀⠄⠈⠢⠠⠤⠂⠡⠔⠠⠀⠠⠀⠀⢄⡀⠐⠢⠤⠀⡀⡀⢤⣶⠀⠀⠀⣀⠤⠄⠒⠈⢀
⠀⠀⠀⠀⠉⠂⠀⢀⠀⠀⠀⠀⠀⡃⠀⢀⢬⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠀⠀⠈⠁⠈
⠄⠀⠤⠁⠀⠐⠂⣁⣀⡀⠀⠀⠀⠁⠊⠁⠀⠉⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠠⠄⠈⠍⠉⢀⡀⢀⣐⣒⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⡠⠘⠣⣀⢀⣈⡍⠁⠈⠀⠀⠀⠀⠀⢰⠂⠐⠒⠒⠊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢷⠀⡀⠀⡔⠋⡀⢀⠀⡬⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠔⠂⠈⠒⠑⠊⠀⠀⠣⠘⠒⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡨⠄
⠀⠀⢀⠂⠤⠀⣀⡀⠀⠘⠀⡠⠔⣊⠤⠊⠀⠀⠀⠀⡀⠀⡀⠀⠐⠉⠉⠉⢨⡂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠊⡠⠊
⠀⠀⠀⠈⠑⠒⠦⠬⣉⣹⣁⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⠐⣐⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⠊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠀⠜⠀⠰⠤⢀⣉⠐⠒⠠⠤⠤⣸⡄⠊⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠢⠤⠆⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⣤⠇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡸⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢰⠃⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡏⠀⢀⡠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠓⠒⠤⠤⣤⣤⣄⣤⡮⠤⡼⠖⠚⠉⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠮⢟⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣷⡶⣯⣭⣿⣿⣿⣯⣭⣽⣿⢿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣏⠀⣈⣿⠾⢯⣿⣉⣿⣿⡀⢀⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡽⢿⣿⢻⣤⣼⡟⠉⢹⣿⡟⢻⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣦⣼⣿⣿⠁⠀⢹⡿⣿⣿⣹⣿⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠭⣛⣛⣛⣛⣛⣛⡸⠽⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡈⢣⢸⢇⠎⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣢⣿⣿⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣵⣿⣯⣽⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢧⣟⡿⡾⢾⠶⣟⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠙⡾⢽⣭⣟⣹⡏⢹⣅⡼⡆⠀⠀⣠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣏⣹⠓⡿⢤⣯⡼⣯⡽⢧⡤⢷⢀⡴⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢧⠀⠀⣸⠦⣯⣹⣿⣾⡟⠚⣟⠶⢻⠒⠛⡏⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣄⣟⢺⠧⢼⡇⠙⣏⢛⣏⠓⣻⡓⢚⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢹⣍⣽⠓⢺⡇⠀⢹⣏⣿⣙⣹⣏⣉⣼⢰⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢦⣻⡀⢼⠢⣿⣝⣽⠇⠀⠈⡯⢾⣍⣉⣿⣭⢼⣾⡏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠻⡇⣾⣉⡿⢤⡟⠀⠀⠀⡷⢺⡯⡭⢿⠦⠾⡿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹⡷⢭⣽⣾⠁⠀⠀⢰⣟⣹⡗⠶⣿⡓⣾⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡖⢿⡏⠀⠀⠀⢸⣧⣤⣏⣋⣹⣿⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠈⢿⣹⠇⠀⠀⠀⢸⠶⠾⣧⣉⡽⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⢿⠀⠀⠀⠀⢸⣓⢚⣷⠏⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠸⡇⠀⠀⠀⠸⣯⡟⠁⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⢻⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⢳⣄⠀⠀⠘⡇⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣍⣷⣤⣼⡧⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠄⠀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⢟⣫⣄⣴⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠕⠿⠿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡄⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢻⣿⠁⢴⣿⣿⣿⣿⣷⠀⠐⣾⣿⣿⣟⡒⢘⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣶⠀⠉⠋⠀⠀⠉⡇⠀⠙⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣹⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁⠀⠀⠀⠀⠧⢤⣤⠶⠂⠀⠀⠜⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠇⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠒⠳⠶⠤⢤⡆⠠⡾⠀⠀⠀⠀⢀⠄⡤⢛⡻⢥⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣠⠀⠀⠀⠀⠀⣇⠀⣿⠀⠀⠀⠀⠈⢀⣃⣤⣼⠈⠀⠀⠈⠙⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠻⣿⣷⣄⠀⠀⠀⠘⣦⠙⣄⠀⠀⠀⢀⡄⣡⣿⡉⠙⠛⠛⠁⠰⣦⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿⣯⠘⢷⣙⣿⣿⠿⡿⠿⠟⠈⢷⢠⠀⠀⠀⠰⠇⠹⢿⣧⡤⠆⠐⠀⢰⡇⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠋⠀⠻⣿⡀⠀⠙⢮⣗⠀⠀⠀⢀⠄⠺⠤⠧⠤⠤⠖⠀⠀⠈⡿⣷⠖⠒⠀⠀⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠉⣳⣦⣶⡏⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⠛⠳⣶⡶⠀⠁⠈⠀⠀⢀⠷⡀
⠀⠀⢀⣤⠶⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣦⡀⢰⣿⣿⣿⣿⣆⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢲⡖⠂⡠⠊⠀⠈
⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠻⣿⣿⣿⡅⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠛
⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢀⣂⡀⢀⢀⡀⠤⢄⡀⠀⠀⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠠⣊⠥⠒⠉⠀⠀⠀⠀⠀⠀⠉⠐⠨⣐⠄⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡰⡡⠊⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠳⢌⢂⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠌⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢀⠀⠆⠀⠀⠀⠣⠡⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠎⡜⠀⠀⠀⠀⠀⠀⠀⣨⠗⠒⠒⠣⣄⢰⠀⠀⠀⠐⠀⠀⢡⠱⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⠀⠀⠠⠀⠀⠉⠟⠀⠀⠀⠀⠀⠀⠳⠀⠠⠂⠠⠀⠀⠀⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠄⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠐⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠀⡀⠀⠀⠀⠀⠀⣣⡀⠂⠀⠀⠐⢀⡼⠀⠀⠀⠀⠀⠀⠀⢇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⢣⠀⠀⠄⠀⠀⠐⡃⠈⢲⠶⠖⠲⠁⠀⠀⠀⠀⠀⠀⠁⡜⡐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢢⠣⡀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠄⢀⠜⡔⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⢝⠦⡀⠀⠀⡄⠀⠀⠄⠀⠀⠀⠀⠀⠠⢀⠰⡫⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⠬⣒⠤⢀⢀⣀⠀⠀⠀⡀⡀⠤⣒⠥⠊⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠀⠀⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⣛⣛⢋⠉⣉⠙⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⣶⣶⣴⣦⢸⣿⠃⢹⡇⠘⢾⡟⢠⡟⠉⠉⠉⢛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁⢸⡄⣟⠁⠛⠹⠆⠉⠓⢀⣁⣑⣛⣁⣸⣇⣀⣀⣀⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠛⣻⠂⢀⣸⣧⣤⠤⠶⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠉⠉⢁⣀⣴⠊⠀⢀⠴
⠀⠀⠀⠀⠀⠀⢰⢸⡇⢴⠀⢰⣧⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⢺⡇⠀⠀⠀⣿⡷⢲⣒⡶⣶⣻⠛⢆⣿⠀⣀⢸⣯
⠀⠀⠀⠀⢀⣀⣸⣸⣇⣿⠀⠈⠁⠀⠀⡶⣶⠲⡖⢲⢺⣿⣿⡇⠸⣮⢿⠀⠀⠀⠛⢀⡘⠟⠀⠈⠛⠀⠀⠁⠀⣿⢸⣿⢀⣶⣆
⠀⢀⡞⠩⠁⣀⡈⣉⣁⠉⠍⠙⢦⡀⠀⠯⡏⠐⡷⢾⠀⠿⠛⠇⠀⠁⣀⠀⣀⠀⢀⣼⡅⠀⢾⡆⠀⠰⣿⡆⠀⣿⡜⠋⠀⠈⠀⣀⡀
⠀⠀⡇⣤⣤⣴⢶⢠⡶⣄⣶⣦⢈⡇⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⢷⢴⠻⠔⣇⠀⣼⠽⡇⠀⣟⢳⡀⢀⣯⡿⣄⠙⢁⢠⣿⣆⡆⠘⠁
⠀⠀⢻⣇⠻⠿⢾⣼⠧⠿⠃⣿⣼⠃⠀⠀⢀⣸⣿⣿⣽⣃⠀⠀⠘⠄⠑⣒⠋⠘⠳⠛⠃⢸⣯⢈⣧⠸⠷⠿⠛⠀⠈⢳⣻⠯⠂
⠀⢠⠼⢯⣔⣆⣈⣩⣄⣀⣺⠴⠛⠑⣆⢏⣉⣩⣿⣧⠤⢼⡇⠈⣏⣧⣼⣹⠀⠀⢠⢒⠤⢠⡬⠉⡁⠀⣀⣀⠀⢀⡤⠠⡁⠀⢴⣤⠄
⠀⢸⠈⣀⠀⣠⡀⣰⣦⠀⣠⡀⢠⣄⢹⠈⡇⠀⣿⡇⠀⣸⠀⢺⡿⣿⣿⠿⡿⠀⠜⠬⠗⠠⠽⠖⠁⢸⡁⠈⠳⠏⠀⢀⡇⣀⠈⠁
⠀⠘⡟⢹⣾⠇⣿⣿⢸⣸⡏⢧⣾⠘⡾⠃⣇⠀⣿⣷⣀⡏⠀⠘⣷⠀⢐⢿⡇⠰⠀⠀⣠⡴⠲⣄⠀⠘⠧⣄⠀⠀⢀⡼⠁⣿⣿⠟
⠀⠀⣧⣠⢉⠠⡙⠇⡀⣙⣂⢬⡗⢔⡿⠀⢈⡅⢠⡀⢀⣀⡀⠀⠻⠼⠿⠾⠃⠀⢀⣠⡟⠀⠀⢸⣇⡤⢄⠀⠉⢷⡟⠁⠀⠈⣁⡀
⠀⠀⠛⠬⣏⣙⣙⣋⣹⡥⠽⠖⠚⠉⠀⠀⠘⢦⣾⣷⣯⡴⠃⢀⣤⢀⣀⠀⢀⣰⣋⠸⡇⠀⠀⢸⡏⠀⢸⡇⠀⠸⣅⠀⢠⠞⠁⠀⠉
⣀⠞⢦⡄⣀⠀⣀⣤⡤⡄⠀⡀⠀⢿⡷⠀⡏⠉⣿⠀⣿⠁⢹⣾⣿⣿⠟⣰⠏⠀⠉⣷⢳⣀⣠⣾⡞⠋⠉⢳⡄⠀⢸⡆⡏
⢨⡶⠼⠆⢹⡽⠛⠛⠉⠃⣸⣽⢠⣏⣷⠀⢻⠀⢿⢀⡯⠀⣼⢸⣿⢹⠇⣿⠀⠀⠀⣿⡏⠉⣷⣼⠁⠀⠀⢸⠇⢀⡾⠁⢹⣄⣀⣠⠼
⢠⣄⡄⠀⠀⠀⠴⣿⡆⠀⠧⠿⢀⠌⡄⠀⠘⠲⠺⠞⠓⠒⠋⠘⠛⠚⠀⠘⠦⣤⠞⣿⠀⠀⣿⡿⢄⣀⣠⠟⠀⠀⠀⠀⡾⠉⢠⣤⣤
⠚⢶⠛⠠⠴⣱⠴⢋⡀⣰⣆⢻⡿⠀⡗⣠⡀⠀⠀⠀⣀⢤⡎⣳⣀⠀⠀⠀⠀⢩⠀⡻⣄⣠⠏⡀⠘⢋⡴⠊⠉⠑⢦⠀⢇⠀⠮⣿⠿
⠴⠆⢀⣀⡉⠋⠀⠈⠙⠾⠋⠁⠀⢀⣔⣛⣓⣄⠀⢴⡙⠦⠟⠗⣚⣳⠀⠤⠀⠀⠄⢃⠘⠉⠠⠁⠌⢸⠀⠀⠀⠀⠈⡇⠈⡆⢀⣁
⠀⠀⣿⠁⡭⠉⠉⣉⣷⡆⠀⠀⢠⡗⠢⠗⡒⢚⡇⢸⣸⣿⢻⢫⣿⠇⠀⡿⡄⠀⠀⠱⣄⢀⢀⠊⠀⠸⣄⠀⠀⣀⡾⠁⣠⠇⠺⠿⠂
⠀⠀⣿⣀⣠⣴⠞⠛⣩⡿⡵⠀⢪⣤⣑⣭⣤⡭⣼⠀⠛⠛⠛⠛⠋⡤⠀⠛⠳⠤⡀⠀⠸⡶⠁⠀⠀⣠⠟⠛⠉⠁⣤⡀⠁⣤⣤⡄
⠀⠀⠈⠙⣯⣄⡴⢞⡥⠚⣁⢀⠈⢿⣿⣿⣶⣧⠏⢀⢀⣴⡄⠀⢺⡁⠀⠀⠀⠀⣈⠇⢰⣧⠀⠀⠀⡇⠀⣴⣶⠀⠉⢁⡄⠘⠋
⠀⠀⠀⠸⣯⠵⠚⠉⠀⠠⣻⣟⠄⠀⠀⠀⠀⠀⠀⢹⠾⡿⣿⠃⠀⠹⡟⡟⣿⣿⣿⠀⠸⠁⠀⠀⠀⢸⠀⠈⠈⢀⣽⡁⡷⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣧⣿⣟⣛⣛⣀⣀⣀⣀⣀⣴⣃⣀⣀⣀⣈⣉⣉⣃⣀⣀⣀⣀
⣿⣿⠛⡿⢛⠻⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⢤⠴⠶⣶⡖⠒⣒⣒⠒⢲⣶⠶⠦⡤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠈⢧⠘⠯⠝⠀⢣⡼⠀⠫⠽⠃⡸⠁⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠱⡀⠀⠀⠀⠈⠒⠚⢋⣉⣉⣉⣉⡉⠓⠒⠃⠀⠀⠀⢀⡎⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢳⠀⠀⠀⣠⠖⢋⣩⣤⣤⣤⣤⣍⣙⠲⣄⠀⠀⠀⡞⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢣⣠⠊⡠⠾⠭⠤⠤⠤⠤⠤⠤⠬⠷⢄⠑⢄⡼⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡸⢁⣞⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣳⡈⢇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠃⣼⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣷⠘⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⡯⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢽⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⡆⢻⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⡟⢠⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠹⡈⢧⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡼⢁⠎⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠙⣦⡙⢯⣉⣩⣭⣭⣭⣭⣍⣉⡽⢋⣰⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢫⠙⠲⢬⣉⣓⠒⠒⣂⣉⡥⠖⠋⡝⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣀⣀⣀⣨⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣇⣀⣀⣀⣀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡞⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⢹⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠫⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠽⠻⠝



⣿⣿⠿⠿⡿⢻⠿⡿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠿⠛⠛⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢏⠑⠂⠀⠠⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⡄⠈⠉⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣤⣰⣤⢤⣾⣒⣚⣿⣿⣿⣶⣯⣼⡿⣭⠀⠀⠀⠀⠀⠠⠋⣴⠒⣽⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⢻⣿⣿⠿⠋⠉⠀⢨⠃⠀⠀⠀⠀⠀⢠⣤⣤⡾⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⣡⠞⠀⠀⠀⠀⠀⠀⠈⢻⣷⣠⡠⠀⠐⠚⠉⠀⠀⢙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠁⠀⠀⠀⠀⠀⠀⣀⣀⣈⣿⣟⣀⣀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⢿⡯⠁⠀⠀⠀⠀⠀⠀⠀⠱⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠚⠁⠀⠀⣰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡼⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠘⡄
⠀⠀⢰⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠄⠀⠀⣀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠁⠀⠀⠸⢄⣀⣁⠀⠀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢷⢀⣤⣤⣄⠀⣀⣀⣠⡶⠿⢛⡏⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠈⠙⠛⠋⠉⠉⠉⠁⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢰⡄⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⣴⠃⢀⡴⢖⣶
⠀⠀⠀⠀⠀⠀⣿⡀⢸⡅⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤
⠀⠀⠀⠀⢀⣠⣬⣿⣦⣝⣶⣦⣴⣶⣾⣿⣿⣶⣾⣿⣿⣿⣯⣤⣤⣤⣀⣀
⢀⠀⢀⡴⢋⣴⣶⣿⣿⣿⠿⠛⠉⠁⢠⡿⠋⠀⠀⠀⠈⠻⣧⠀⠈⠉⠉⠛⠛⠷⢶⣦⣤⣀⡀
⠙⠶⠋⣰⡿⣻⣿⣿⠟⠁⠀⠀⠀⠀⣿⡇⣾⡆⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣄⡀
⣀⣀⣴⠟⣱⢿⣿⠋⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⡀
⠀⠀⣠⠼⠁⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣶⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⡄
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢀⣤⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⢰⣿⠏⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠰⣶⣦⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠃
⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠉⠉⠉⢻⡿⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉
⠀⠀⠻⣿⣶⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠠⠤⠶⠾⠛⠋⠻⣦⣀⠀⠀⠀⣠⡆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠉⠉⠉⠛⢿⣦⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠋⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⡄⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⢈⣿⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠉⠻⣷
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⡶⠾⠿⠛⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⡠⢫⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⡐⡱⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⡀⠀⠀⠀⠀⠀⢴⣦⣈⠏⡴⣤⣤⣤⣤⣤⡶⠶⠞⠛⠉⠁
⠀⠀⠀⢰⣿⠁⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠹⢿⣦⡀⠀⠀⠀⣠⣿⠋⠊
⠀⠀⠀⣸⡟⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠿⣷⣶⣿⡿⣷⣷⣤⡄
⠀⠀⠀⣿⠇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢿⣟⣾⣷⣯⣭⡶
⠀⠀⢠⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢠⢊⠟⠿⣾⠏⠉
⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠁
⠀⠀⢸⣿⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⠿
⠀⣀⣨⡍⠉⠉⣿⠋⠉⠉⠉⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠁
⠀⠻⣿⣯⣤⣤⡿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠉⢉⣡⣤⣤⡤⠤⠤⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣧⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⢰⡏⠁⠀⡄⠀⢀⣀⡄⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⣤⣀⠀⠀⢰⠀⠈⣻⡆
⣀⣀⣀⣈⣻⣶⣶⣷⣚⣋⣉⣷⣶⣛⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣶⣏⣙⣛⣷⣾⣶⣾⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⠟⣋⣹⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠛⠛⠻⠛⠛⠿⠿⠿⢻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣥⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⢀⣤⠶⠖⠂⠀⠈⠻⣿⣿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠞⠛⣳⠀⠀⠰⠋⠙⠃⠀⠀⠘⠟⢡⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠳⠖⠂⠀⠠⢄⡀⠀⠀⠀⢰⢲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣦⡶⣶⡄⠖⠂⠀⠀⠀⠀⡼⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠶⠶⠆⠺⣋⣵⠞⠽⠖⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⢀⠀⠀⠒⢆⠙⢦⡴⠞⢁⣠⣼⠋⠙⡀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠟⣧
⠀⠀⠀⢂⡤⠤⠝⣳⡦⠌⣈⣤⣶⣿⣿⠟⠀⠀⠈⡿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀
⠀⣠⢒⣥⣤⡀⢞⠁⣠⣾⣿⣿⣿⠁⠀⠀⠀⢀⠞⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣸⠢⣄⡀
⠀⡇⣟⣽⣛⢿⡆⢿⣿⣿⣿⠟⠃⠀⠀⣀⡤⣾⠀⠈⡏⠁⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⢠⠃⠀⠀⠉⠉⠓⠲⠆
⠀⠹⡝⠛⢫⣼⢇⣿⠟⠋⠀⠀⢀⠐⠏⠀⠀⠸⠀⠀⣡⣄⠀⠀⠀⠀⣀⡀⠀⠀⠀⣰⠃
⠀⠀⠘⠻⠷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠙⢦⠀⠀⠁⢷⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠃⠀⠀⠈⠳⠤⠈⠁

⣿⣿⠛⢛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢿⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢹⡏⠉⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⢸⣧⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⡿⠃⠀⠀⠀⢙⣧⣀⣤⣶⣶⠶⠶⠟⠛⠛⠛⠛⠿⠶⠶⣶⣶⣿⣁⣀⠀⠀⠀⢹⣧⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠶⠷⣾⣿⡿⠃⢀⣤⣴⠶⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⢶⣤⡀⢹⣿⣿⣶⡶⢶⣤⣄
⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⣰⡟⠛⠛⣷⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣏⠀⣸⣷⣤⣈⣙⢷⣦⡀
⠀⠀⠀⣠⣿⣥⣤⠶⠾⣿⣿⢿⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⠻⢿⣭⣙⠳⠿⣿⡄
⠀⠀⢀⣿⢿⣯⣥⠶⠛⠋⠁⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠉⠙⠛⢻⡿⣿⡀
⠀⠀⢸⡇⠘⢿⡄⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⢀⣴⠟⠀⢸⡇
⠀⠀⢸⡇⠀⠈⠻⣧⣀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣴⠿⠁⠀⠀⢸⡇
⠀⠀⠸⣧⠀⠀⠀⠈⠙⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⢶⣄⡀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⠀⠀⠀⣿⠃
⠀⠀⠀⣿⡀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠉⠛⠻⠿⠶⠶⠟⠋⠛⠿⢦⣤⣤⣤⣤⣿⠟⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡇⢠⣿
⠀⠀⠀⢻⣇⢠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⢀⣴⠶⠆⠀⠀⠀⠰⠶⣦⡀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⣀⣀⣀⣀⣠⣶⠟⠁⣼⠏
⠀⠀⠀⠀⣿⡀⠙⣷⣦⣤⣤⣤⣤⡶⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⡄⠀⠀⠀⢀⣤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⢿⡍⢻⡆⢠⡿
⠀⠀⠀⠀⢸⣧⣰⡟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢻⣿⠇
⠀⠀⠀⠀⠀⣿⣿⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠛⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⣿⡿
⠀⠀⠀⠀⠀⢸⡏⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢸⡇
⠀⠀⠀⠀⢀⣿⠃⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠀⠀⠀⠀⠀⣰⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣷
⣦⣤⣀⣴⡾⠋⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣶⣶⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠙⢷⣤⣀⣤
⠈⢿⣏⠁⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⢈⣹⡿
⠀⠀⠙⢷⣦⡀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠈⠛⠷⣦⣤⣤⣀⣤⣽⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠻⠶⠾⠿⠿⠷⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣅⣀⣠⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠈⠛⢷⣤⣀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣄⠀⠀⠀⠀⠀⢀⣠⣴⡾⠋⠁⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⣼⣿⣄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣿⣄⣠⣤⣴⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡍⢻⣟⣉⣽⣿⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠁⢠⡿⢋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⠁⢹⣿⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⣼⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⣇⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⢰⡟⠀⠈⣿⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⠀⠀⢸⣿⡄⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⣾⣇⢀⣤⣿⠾⠟⠋⢻⣆⠀⠀⠀⠀⠀⣠⡿⠛⠷⣦⣤⣀⠀⣿⣧⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠈⢹⡿⠃⠀⠀⠀⠀⠈⠻⠶⠶⠶⢶⣴⠿⠁⠀⠀⠀⠈⠙⢿⡟⠁⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣧⡀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣈⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷⣦⣄⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⡁⠀⠀⠀⠀⠀⠀⢀⣿⠁⣀⣤⡶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠛⠻⠷⣶⣤⣄⣀⠀⠀⠀⠰⣶⣦⣴⠟⠁⠀⠀⢀⣀⣠⣤⣾⠿⠛⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠉⠙⠿⢶⣤⣄⣠⣿⡁⢀⣤⣴⠶⠟⠋⠉⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⠟⠻⢿⠿⣋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠷⠶⠶⠶⠶⠶⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠙⠛⠿⣦⣤⣤⣄⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⠶⠿⠛⠛⠉⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⣿⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢰⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣈⠉⠉⠉⠉⠀⠀⠀⠀⢀⣠⡾⣿⠿⣦⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⣡⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣮⡙⠻⠿⢶⣶⣶⡶⠾⠟⢛⣡⣴⠟⢷⣌⡛⠻⢷⣦⣤⣤⣤⣶⠿⢛⣫⣼⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⣤⣤⠀⣤⣤⢀⣤⣄⠀⠀⢠⣤⢠⣤⠀⣤⢠⡄⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣾⡃⣿⣤⠠⣿⣄⢸⣧⣿⠁⠀⢸⣿⣾⣿⠀⣿⢸⡇⢼⣧⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡾⠃⢿⣦⠀⢿⡤⠀⡟⢿⠄⠀⠸⠟⠿⠿⠀⠻⡾⠇⠸⢷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢲⠞⠉⢉⣉⠙⠻⠿⠓⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠁⠀⠀⠀⠙⠃⠀⠀⠈⠡⣄⠀⠉⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣀⠀⠀⠀⠀⠀⠚⠉⠁⠘⢓⣒⣄⡀⠀⢈⣷⣞⣉⣹⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⢈⠉⠉⠁⠁⢠⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠀⠉⠉⠉⠼⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠄⠀⠀⠁⠀⠀⠀⣼⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⢀⣀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠃⠀⠀⠀⠛⠀⠀⠀⣼⠐⣿⠛⠃⠁⣀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡿⠖⠚⠛⠛⠋⠛⢷⡄⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡆⠀⠀⠀⠀⠀⠀⠈⣷⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⡇⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⠀⠀⠀⠀⠀⠀⢀⣴⡅⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢀⣤⡾⡋⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⡇⢻⡷⠶⣶⡾⠳⠙⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⡿⣷⡶⠤⠀⠀⠀⠀⣀⡤⠤⠤⠤⠤⢤⣾⣿⣿⣿⣿⡇⠟⣿⡿⠕⠈⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠘⠻⠋⠛⠿⠷⣤⡄⠀⣀⣠⣤⣤⡀⢐⣂⡈⣽⣿⠿⠿⠛⢠⡇⣫⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡌⠀⠀⠀⠀⠀⠀⠈⠉⠉⢭⣭⣭⣉⣉⡙⠃⠭⠄⠂⢀⠀⠀⣸⣧⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠃⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠒⢛⣻⣿⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⢉⣼⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣌⡑⠢⠤⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣴⡶⠿⢛⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢶⣶⣦⣤⣤⣤⣤⣭⣬⣭⣭⣥⣬⣉⣭⣤⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠛⠘⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡛⠃⠀⠈⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⠀⠉⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠂⠀⠀⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⡠⠐⠈⠳⡌⠂⠀⡀⢴⠧⠗⢄⡀⠀⠀⠀⠀⠀⣶⣶⣦⣨⣧⣤⡀⠀⠀⠀⠀⠀⠈⢡⠇
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠀⠀⠀⠈⠀⢸⡄⠈⢆⠀⠀⠀⠀⢿⠟⠑⠀⠉⠉⠀⠀⠀⠀⠀⢀⠔⡝⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠃⠀⠀⠀⢀⣄⢸⡄⠀⠀⠀⠀⣧⠀⢸⠀⠀⠀⠁⠀⡼⠀⠀⣤⣤⠀⠀⠀⠀⣠⢃⡴⠃⠀⠀⠀⠀⠀⠀⡈
⠀⠀⢀⠀⠀⠀⠀⣴⣿⡗⠈⣧⠀⠀⠀⠀⢹⡀⠘⡄⠀⡀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢾⡁⠀⠀⠀⢀⠀⠀⡰⠁
⠀⢀⠚⢀⠀⠀⠜⢿⣏⢈⠂⢸⡄⠀⠀⠀⠸⣷⡆⡷⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆⠀⢀⡼⠁⠀⠀⡇
⠀⠈⢒⣡⣀⡀⡇⢸⣿⠀⠀⠘⡇⠀⠀⠀⠀⣯⠎⠀⡀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠾⢶⣵⣴⣇⣰⠟⠁⠀⠀⠀⠇
⠀⢻⡿⡅⠀⠈⠃⢸⠣⠀⠀⠀⡇⠀⠀⠀⠀⢿⣶⣋⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠙⠻⠛⠁⠀⠀⠀⠀⣠
⠀⠀⠈⠘⠄⠀⠀⣸⣦⡀⠀⠀⠗⠀⢀⣀⣴⣧⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣾
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⣤⣴⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⡀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠉⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⣿⣿⠛⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡋⡸⢶⡇
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⣡⣈
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠰⠄⠀⣵⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠘⢀⠀⡤⣤⣤⣤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢼⣿⣇⠀⠀⣰⠻⠟⠻⡇⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⢠⣿⡿⣻⢟⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠻⠆⠒⠉⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠐⣸⡿⢟⠁⠙⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠠⣀⢿⣠⡀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⢿⣿⣷⣭⣬⣵⣿⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣁⣻⣇⠀⠀⠀⠀⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⢀⡟
⠀⠀⠀⠀⠀⢾⡂⠈⠉⠉⠉⠩⠅⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣧⣀⡀⢀⣀⣧⣴⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⣿
⠀⠀⠀⠀⠀⠸⣦⣤⣤⣤⣤⣤⣤⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣉⠀⠈⠉⠉⠁⠀⢒⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠑⡠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡏⠀⠹⣿⣿⣿⠏⠉⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⡐⠿⠻⣄⣀⠐⠒⠚⢀⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠉⠁⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠛⠛⠁⠐⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠓⠒⠀⣘⡃⠀⠀⠀⠀⠀⠀⣨⠀⠀⢀⣀⣤⣤⣀⣀⣀⣀⣀⣡
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡀⣚⠀⠀⠀⠀⠀⠀⢰⣿⣄⣀⣀⣠⣀⣀⣀⣀⣤⣤⣭⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠿⣿⣣⠄⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⣿⠃⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣷⠰⠀⠱⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⣿⣁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⠀⠰⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⠀⠰⢿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠴⠾⣖⠛⠰⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠰⢿⢼⣧⣤⣠⡈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠂⠈⠀⠐⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠿⠿⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢑⣰⠇
⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠂⡀⢀⠀⠐
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠙⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠠⡀⠈⢧⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⡞⠑⠠⣀
⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⣰⡁⠀⡉⡆⠀⢘⠀⠁⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴⡀⣀⠀⣢⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡠⠒⠈⠉⠁⠢⠔⠉⡀⠈⠀⠂⢄
⠀⠀⠀⠀⡘⠀⣊⠴⠀⠰⠁⡆⠤⠑⢄⠀⠀⢀⡐⢄
⠀⠀⠀⠀⠇⢨⠵⡒⠈⠀⢳⠂⠖⠀⠀⠑⠤⣉⠚⠈⡀⣀⣀⣀⡀⠀⠀⠀⠤⠤⠤⠤⠄⠤⠤⠄⡀
⠀⠀⠀⠀⢰⡀⠈⠁⠰⠀⠈⠢⡀⠀⠀⠀⠀⠀⠉⡰⢅⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⡆
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠑⢄⠀⠈⠂⠄⡀⠀⢀⠼⠁⢘⡠⢀⠀⠈⠁⠉⢀⠠⠩⠭⠀⡀⠂⢌⠆⢸⢝⣉⣱
⠀⠀⠀⠀⠀⠀⠑⣤⡀⠀⠠⣈⠒⠠⡤⠋⠨⣍⠒⠄⠀⠈⠀⠈⡆⠀⠀⢧⣌⡅⢀⣀⣘⠀⢀⢠⡐⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⡄⡏⠒⢌⡁⠂⡟⠀⠊⠀⢮⠱⠢⣈⠐⠀⢀⠇⠀⠀⡸⠀⠀⠀⠭⠁⠀⠀⢸⠇⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⡇⢀⢀⠈⠙⡁⠀⠐⢁⠀⠘⢆⠈⠉⢒⠁⠀⢀⠔⠁⠀⠀⠀⠢⢆⠀⠀⣾⡘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡠⣅⠢⡕⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠃⠈⠈⡁⠀⠈⠐⠐⠋⣑⡾⠤⣆⠀⠉⠀⢠⠏⡀⠀⠀⠀⠀⠘⠂⢆⢠⠁⠇⠀⠀⠘⡀⠀⠀⢀⠀⠤⠒⠀⠈⠱⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠁⢠⢸⠀⡅⠀⠀⡤⠖⠩⠚⢋⡩⣄⠓⢄⠀⢸⠀⠀⠉⠁⠀⠀⣂⠠⢻⠃⠀⠈⠉⠉⠀⡱⠲⠉⠀⠒⠀⠀⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⢘⠀⡄⢠⠀⠀⡑⠒⠊⠉⠀⡄⠐⠀⠀⢃⠸⡄⠀⠀⠈⡇⠁⠀⠀⡜⣀⠊⠀⠀⡱⠊⠀⡀⠰⡄⠀⠀⢀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠈⡀⡅⠀⠈⡄⠀⢃⡁⠀⡀⠐⠰⠐⡤⢤⠈⢦⢃⡀⠈⠉⠀⠠⠄⢼⠹⢠⠘⡠⠊⠀⠀⠀⠀⠀⠡⠈⠂⢉⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢃⢱⠀⠀⠈⠐⢚⡁⠊⠀⢜⠃⠓⡠⣀⠀⠈⣤⠀⡈⠉⠁⠀⠀⣉⡧⠜⠊⢛⡤⠒⠸⠤⢀⠀⠱⡕⢄⢠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠳⠀⠴⢀⠹⠆⢔⠒⠈⢀⠌⠰⠀⢁⠎⢀⣁⠠⣠⠜⠉⠓⠸⡤⠒⠣⣔⠀⢡⡀⠀⠘⠀⣦⠈⣰⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⠢⠅⡚⠤⣠⡤⡃⠁⠈⢦⡀⠁⠙⠤⡉⠠⣀⠈⢦⣠⡀⠬⠤⠄⠀⠀⠑⠚⣂⣉⠉⠭⠅⡠⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⢲⢀⡀⠀⠑⠂⠭⣚⠶⠂⠜⢂⣋⣀⣒⡉⠈⠭⠥⠀⠀⠀⠀⠐⠒⠀⠉⠩⠄⠀⢀⣀⠀⢬⠅⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⢈⠁⢲⢤⣀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⡠⠤⡤⠀⠀⠂⡖⠂⡎⠉⠁⠀⠀⠀⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⠀⢸⢸⠀⠈⡇⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⢆⡀⣗⢀⣈⠡⠥⠔⠁⠀⠀⠀⠀⠀⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠸⡤⡀⢸⢸⠀⠀⡇⠐⠂⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢢⢌⡑⠛⢸⠄⠀⡇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠦⠼⠓⠚⠃⠀⡇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣀⡠⠄⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡈⠂⢀⠀⠀⠀⠀⠀⡇⠀⠉⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠄⠔⠓⠊⢉⡀⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣈⠀⠀⠀⢀⣇⠀⣀⣀⣀⡀⠀⠤⠌⠦⠀⠀⠒⠒⠀⠉⣁⡀⠠⠄⠀⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠇⠀⠀⠀⠀⠠⠤⠀⠀⠀⠐⠒⠂⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣶⣿⣭⣿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠉⠉⠉⠋⠉⠁⠉⠻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠖⠒⠂⠀⢀⡖⠛⠳⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⠀⠚⠛⠉⠀⢾⠈⠋⠁⠀⢩⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠠⠀⠀⠨⠑⠀⠀⠀⢈⠁
⠀⢀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢤⡀⠀⡴⠁⢀
⠀⢸⡏⠀⠈⠉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠀⠀⠀⠀⡠⣠⣀
⠀⠈⡇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢉⠲⠤⢤⣾⠟⣡⣿⣗
⠀⠀⣇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠈⢦⠀⠈⣠⣾⣿⣿⣿⣧⡀⢀
⠐⣉⣥⣄⡀⣪⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢲⢖⡄⢀⠝⣱⣿⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣅
⠀⡰⣏⡀⠀⠬⣭⡆⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢩⣰⣿⢏⣾⣿⣾⣿⠇⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠂
⠀⠳⠺⠆⠀⠐⡒⠁⠀⠀⠀⠀⠀⠠⣒⣭⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠐⡄
⠀⠀⠳⣜⣋⡟⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡄⡇
⡰⠁⠂⠠⣭⠓⠳⡄⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠃
⡇⠀⠀⠀⠙⠂⠀⢳⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡐
⠳⡀⠀⠀⠀⠀⠀⣜⣀⣤⣴⣿⣿⣿⣿⣿⠿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠈⠢⡀⠀⣠⠾⠿⠟⠛⠋⠉⠙⠿⠋⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⡿⠀⠀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⠀⣼⣿⣿⣿⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡄⢸⣿⣿⣿⣿⣿⡇⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⡏⢴⣾⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⢻⡇
⠀⠀⠀⠀⠀⠀⣰⣿⣿⢃⣽⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠪⠻⣿⣿⡿⠛⠁⠀⠀⠤⣄
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠤⠀⠈⠉⠀⠀⠀⠀⢀⣀⡊⠁⠊
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡿⠿⠿⠟⠻⠛⢿⣿⣿⣷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠋⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⣀⣀⡋⠁⣀⠤⠀⠀⠀⢰⣹⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠷⢄⡀⠀⠀⠀⠖⠚⠙⠉⠁⠀⠀⠀⠀⠘⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠐⠀⠀⠐⢀⣀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⠷⠶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠰⡓⠐⢶⡄⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠟⢀⣼⣿⠄⠀⢰⣿⣿⡆⠀⠀⠠⣠⣶⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠉⢀⣀⡀⠉⠉⠀⠀⠀⠀⠐⠌⠁⠐⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⢿⣿⡷⠀⢀⣤⣤⡀⠀⣴⣤⠀⣠⡀⠠⡀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⢟⣠⣄⡀⠀⠀⠀⠈⠻⢿⠗⠀⢿⡿⠀⢿⣷⠀⠈⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⡿⠿⠃⠐⠀⠀⠐⠃⢠⠄⠀⠀⣠⣦⡀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⢧⣠⡞⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⢿⡿⠃⢰⣿⠆⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣾⡇⠀⠀⠀⠀⠀⠀⣿⡿⠏⠀⣤⣶⡄⠈⠛⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⠇⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠻⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣽⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⢠⣶⠂⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⣶⡄⠀⢀⡀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠿⠁⢰⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⠶⡶⢶⣶⡶⢶⠶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠈⣤⡄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡄⠀⠀⠀⠀⠀⣀
⠖⠠⠀⠀⠀⠀⠀⠀⣿⠁⠀⡀⠄⠀⣤⢠⡀⠀⠀⠀⠉⠁⠀⢀⣤⣤⣾⠟⠃⠄⡀⠀⠀⡀⢰⠏⠛⠉⠀⠀⠀⣼⣾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠟⠛⠇⠀⠉⠀⢠⠃⠟⠁⠀⠀⠆⠀⠀⢰⡇⠀⠀⠙⣧⠀⠀⠉⠓⠘⠏⠀⠀⠀⠀⠀⠀⢰⣏⣠⣶⣿
⢺⡏⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⢸⣶⣦⡀⢀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣍⠀⠀⠀⠀⡹⠀⣤⠶⠀⠀⠀⠀⣿⢻⡟
⠘⠁⠀⠀⠀⠀⣠⢾⡿⠛⠋⠁⢙⣿⣭⣥⣿⡿⠿⢷⡀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⡿⠇⠀⠀⠀⠀⠀⠻⠘⠃
⠀⠀⠀⠀⢀⣀⡛⠚⠃⠀⠀⠀⠈⠉⢨⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠘⠟⣀⣤⣶⣶⣦⣴⣄⡀⣦
⢀⡄⠀⠀⢀⣇⣿⠛⠂⠀⠀⠀⠀⣿⣿⢻⠀⠀⠀⠀⢸⣠⣤⣼⣷⣷⣶⡄⠀⠀⠀⠀⠀⣇⠝⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠖⣿⢀⡶⣴⠀⡴⢺⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡀⢀⣙⠂⠙⣚⣃⠘⣊⠓⠈⣓⣉⢀⣠⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣁⢰⣽⣿⣵⣾⣽⣯⣷⣿⠀⢠⣿⣓⢸⣷⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠿⠋⠻⠿⠋⠘⠻⠟⠃⠿⠿⠟⠻⠿⠻⠟⠻⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⢻⡿⠛⣼⣆⣿⠀⢰⣻⡄⠀⡿⣦⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠺⠗⠀⣸⣷⢀⣿⣏⣿⣆⣿⣿⣷⣸⣷⣹⣿⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⡄⣩⢉⡠⡉⣩⠉⡍⠉⣍⣩⣉⡭⡈⢉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠧⣿⢺⣧⠔⢿⡸⣇⡰⣿⠙⠛⣧⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣶⠟⠀⠉⠀⠈⢁⣈⣀⣈⣀⣈⣉⣠⣤⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⣠⠄⠀⠀⠁
⠀⠀⠀⡀⣀⠀⠀⠰⠁⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠍⣶⠐⠆⠀⠀⠘⡄
⡔⠈⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡈⠁⠀⠀⠀⠀⡅
⠑⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⢿⠆⠂⠀⢴⠀⠃
⠀⠀⢆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⢈⠣⠀⢀⣀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠈⠤⡀⠀⠀⠀⡌⠀⠄
⠀⠀⠀⠄⠀⡀⠘⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠰⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⡄⠀⠀⠀⠢⠤⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡀⠀⠀⠀⠀⠀⢀⡠⠞
⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠉⠑⠒⠒⠀⠋⠭⠀⡁⢀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣔⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⡃⢄⠀⠸⣧⠯⠃⠀⠀⢀⣀⣐⠀⠉⠁⠀⠀⠐⠀⠠⢀⠀⠀⠐⠻⡟⠂⠀⠀⠀⠀⠐
⠀⠐⢤⠃⠀⢀⠔⡅⠀⠀⠐⣋⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⣀⣂⣀⠀⣴⠆⠾⠇
⠀⠀⠀⢀⠔⢁⠢⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠽⠒⠅
⠀⢀⡔⠁⢌⢢⢳⡇⠀⢹⡃⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠤⠑⡄⠀⠀⠀⠀⣤⡞⠀⠹⠆⠬
⠀⠫⣉⠉⠚⢦⣻⠁⠀⡈⠳⠎⢸⣀⠄⡟⢃⡠⢧⣀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡠⠒⠀⠋⠀⠑⢲⠊
⠀⠀⠈⠉⠓⠒⠛⠀⠀⡷⠴⢶⣺⢅⡜⠀⠀⠹⢪⠋⣹⠒⠶⠤⢤⠴⠶⡊⠟⣄⠴⠄⠈⠁⠲⠉⠀⢠⠄
⠀⠀⢀⣀⣀⡀⢀⣀⠀⠫⠻⣷⣮⡟⢠⢒⡒⢄⠀⠉⠀⠃⠰⠢⠌⠦⠆⠈⠃⠈⠀⠀⠀⢀⡈⠀⠀⠈⠂
⠀⠀⠙⢄⠀⢠⡋⠶⠥⡀⠀⠀⣀⠀⢧⡇⠈⢪⡄⣰⠞⠉⢢⠀⠀⢤⠀⠀⢰⠠⠂⠀⡠⢊⢁⠝⠑⠒
⠀⠀⣀⠀⠑⢼⢢⣔⠀⠉⠳⣄⣈⣯⠚⠿⣄⠌⣷⡟⠀⣀⠎⠀⠒⠈⣁⣠⠤⠒⠂⠀⠀⠁⠃⠔
⠀⡰⢤⡂⠐⣢⠉⠈⢳⡉⣉⠽⡏⠉⡑⠲⢬⡟⢙⣶⣚⣉⠙⢧⠀⣸⠑⠀⡗⠀⢀⠀⠀⡠⠀⠀⠀⠠⡀
⠀⠰⠔⠄⢀⡽⡄⠀⠀⠻⠴⡞⠁⠀⢦⠀⢠⠀⢀⡀⢀⡰⠀⠀⠀⢇⢀⡔⠱⢄⡜⠀⢠⠁⠀⠀⠀⠀⠄
⠀⠀⢀⣀⣸⢀⠇⢀⠀⠀⠀⢹⠀⠀⡆⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡨⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢘⢒⠢⡿⠤⠒⡃⠀⠀⢸⠀⠀⡇⠀⢸⠀⠃⠀⡆⠀⠀⠀⠴⡇⠀⠀⠀⢠⡼⠤⢀⡆⠄⠤⠒⠁
⠀⢀⣸⠂⠈⠑⠎⢸⠀⠀⢿⡛⣏⠀⠉⠉⠉⠘⠒⠒⠃⠀⢠⡤⠐⠀⠤⠶⠂⠀⠀⠀⠙⠂
⠀⠘⠤⣑⣦⣤⣔⡃⠴⠀⡰⠉⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇⠃⡰⡟⣭⣕⡄⢯⡟⠀⠀⠀⢀⣸⠙⣳⠆
⠀⠀⠀⠈⣒⣒⡉⠀⠀⠀⠀⠀⠴⡀⠀⠀⠢⡁⢀⡀⢀⣀⢧⣆⠘⠷⢝⠿⠃⢀⠀⠂⠀⡠⠢⡀⢙
⠀⠀⢠⠊⡉⢳⣹⣀⠠⠰⣷⠄⠣⡁⠠⠀⢠⣌⠀⠲⠟⠉⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⢀⡫⠒⣠⠏
⠀⠀⠈⠦⣑⣊⣁⣀⠤⠒⠉⠀⠀⠈⠢⠘⠋⠈⠀⠈⠀⠈⠀⢠⡀⢢⠘⢑⡀⠀⠀⠀⢏⣋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠁⠊⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡽⠃⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡄⠈⠁⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⢻⡀⠀⡇⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢁⣀⠈⠳⡀⢣⠈⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⠁⠀⠀⠀⠀⠀⠀⠉⠙⠢⣄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠁⣠⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠳⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢉⣿⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠛⠒⢦⡄⠀⢀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠈⢻⡙⣎⣧
⠀⠀⠀⠀⠀⠀⢠⢏⡄⣤⣀⡀⢸⣿⠉⠉⠁⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠈⣧⢹⣎⢧
⠀⠀⠀⠀⠀⠀⡾⡼⢸⣿⣿⡟⡼⢸⢠⡠⣶⣶⣦⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢿⣮⠳⣄
⠀⠀⠀⠀⠀⢸⢳⢃⣿⣿⣿⢱⡇⠸⡇⡇⢿⣿⣿⡜⣇⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⣿⣷⣌⠳⣄
⠀⠀⠀⠀⠀⣿⢸⢸⣿⣿⡟⡾⠀⠀⢷⢹⢸⣿⣿⣧⢻⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⣿⣿⣿⣷⡌⢷⡀
⠀⠀⠀⠀⠀⠘⢮⣛⢟⣋⡼⠁⠀⠀⠸⡜⣆⢿⣿⡿⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣿⣿⣿⣿⣆⢻⡄
⠀⠀⠀⠀⠀⠀⢨⡿⠉⠁⠀⠀⠀⠀⠀⠙⠮⣽⣭⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠘⣿⣿⣿⣿⡿⠶⢿⡀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡤⠾⠛⠉⠁⠀⠀⠈⢷
⠀⠀⠀⠐⠶⠾⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠷⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⣀⣤⣴⣾⣿⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠚⠉⢿⣿⣿⣿⣿⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⠾⠟⠋⠁

⣿⣿⠻⢿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣶⣾⣿⣿⠿⠿⣿⣿⣷⣶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠁⣀⣀⣀⣀⣀⣀⡀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣽⣿⣿⣿⠿⠋⠁⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⡟⠁⣠⣾⣷⣶⣤⣀⠈⠙⢿⣿⣿⣿⣯⣀⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁⢀⣤⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠁⢀⣾⣿⣿⡄⠈⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⣴⣿⣿⣿⣿⣷⡀⠘⢿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⣼⣿⣿⣿⣿⣿⣿⣿⣄⠈⠻⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡿⠋⠁⣀⣤⣶⣶⣿⣧⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠻⣿⣿⣿⠀⢸⣿⣿⣿⠟⠁⢀⣴⣾⣿⣿⣿⣿⣿⣿⣧⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄⠀⠁⠀⠀⠀⠉⠛⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢰⣦⣄⠀⠉⠛⠛⠻⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢸⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⡇⠀⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀⠉⠻⠇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢠⣦⣀⠀⠀⠀⢀⡀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⢻⣿⣿⣿⣿⣿⣿⡿⠟⠁⣀⣴⣿⣿⣿⡇⠀⣿⣿⣿⣦⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠻⡿⠿⠟⠋⠉⢀⣠⣾⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣦⡀⠙⣿⣿⣿⣿⣿⣿⣿⡟⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⠀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣷⡄⠈⢿⣿⣿⣿⣿⠟⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡄⠘⣿⣿⡿⠃⢀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⡀⠉⠻⣿⣿⣿⣿⣿⣿⣿⠇⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠛⠉⢀⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⣿⣿⣿⣿⣶⣄⡀⠉⠛⠿⣿⡿⠃⢀⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⢀⣠⣶⣿⣿⣿⣿⠉⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⡀⠀⠈⠉⠉⠉⠉⠉⠉⢀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠿⠿⢿⣿⣶⣶⣶⣾⣿⡿⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠆⣀⣤⡀⣠⣄⠰⡆⣠⣄⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠆⠻⠴⠃⠳⠼⠋⠇⠻⠿⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⡄⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠐⠀⢀⠀⢄⠀⠀⠀⠀⠀⢀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠨⢀⠌⠃⠀⠀⠀⠁⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⡄⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠂⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⡀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢄⠁⠀⠠⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠐⠀⢀⠔⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠈⠀⠀⠀⠀⠉⠙⠀⠀⠀⠁⠀⠀⠂⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠢⠋⠀⡠⠊⠈⢐⡄⠀⢄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡀⠰⡄⠀⠀⠀⠀⠀⠀⡀⠀⢸⠄⠈⠜⠀⠀⠀⠀⠀⠀⠉⠐⠠⠄⠀⠀⠠⠄⠀⠀⠀⠄⠿⢀⣆⢠⠎⠀⠀⠀⡜⠀⠀⠀⡅⡀
⠀⠐⠀⠀⢃⠀⠈⢆⠀⠀⢀⠀⠀⡟⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⡀⠀⢀⡌⠀⠀⢀⡜⠀⣸⠂⠐⠐⠒⠂
⠀⠀⠀⠀⠀⠐⠤⠸⠦⣀⠜⠁⠈⠀⠀⠀⠀⠀⠀⢀⠀⠄⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠉⠉⠁⠑⠤⠤⠋⠔⠚⢁⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣷⣶⣶⣾⣿⣼⣿⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⢄⠈⠀⠀⢀⠀⠀⠅⠑⢶⣦⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣁⣀⡴⠖⢾⡟⠀⢀⠀⠀⢀⢀⡄
⠀⠀⠀⠃⠦⠈⢱⠚⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠶⠈⠡⠈⢲⣶⠄⣠⠠⣤⡀⢠⡤⣀⣀⣀⢀⠄⣀⣤⠀⠄⠰⡶⠒⠄⠉⠏⠁⢀⠀⠀⠀⠈⠀⠐⠲⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠀⠛⠀⠀⠀⠈⠟⠁⠁⠀⠙⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⠻⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡸⠟⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠈⠓⠖⠒⡀⠤⢤⠀⠀⢀⣀⠀⠀⢌⡀⢀⠀⠀⣠⠶⠤⡤⢀⠀⠰⠛⠀⠸⠁⠉⠉⠀⠅⠀⠀⠀⠈⠁⠀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠠⡖⢆⠀⠰⣴⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣀⠈⠀⠀⠈⠀⠀⠀⢱⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠽⠁⠀⠀⠀⠁⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠘⢶⡄⠀⠀⢀⣀⠀⢀⡈⠰⠆⠀⠀⠀⠁
⠀⠀⢤⣴⠾⠶⠶⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠘⣛⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠂⠄⠈⠁⡀⠈⠈⠒⢀⣀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠈⠁⠀⠀⠆⠁⠀⠐⠦⡈⠉⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⡰⠒⠇⠀⠰⠂⠀⡀⠀⢀⠀⠈⡧⢀⠀⠀⣀⠈⠉⠁⠀⠀⠀⣃⡦⠀⢀⠇⠐⠀⠀⠀⠺⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠋⠀⠀⠀⠀⣀⣷⣄⢈⡧⠀⣧⠈⡷⡀⢨⣦⣀⠀⠀⠀⣠⡝⠀⠀⠀⠀⢀⠤⠈⠀⢠⡦⠀⢀⠀⠘⠛⠅
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠣⡾⢀⡨⠉⡍⢛⡍⣯⠉⡍⡈⢺⠀⠀⠁⠛⠀⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⣇⡘⡄⢀⢁⠉⡍⣀⢠⢀⡁⣸⠀⠀⠀⣀⡀⠀⢀⠟⠀⠀⠀⠰⠆⠘⠇⡀⢀⠀⡀⠌
⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣘⣋⡶⣀⠏⡇⠀⢿⡜⠙⠇⠘⢿⣟⠃⢙⠛⡆⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⣁⠘⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡟⠘⠃⢀⠻⠃⢀⠓⠀⠘⠀⠐⠎⠁⠆⠉⠰⡧⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢆⠂⠀⠀⡀⠀⠘⠋⠀⢀⠘⠁⠀⠘⠁⠀⠀⠖⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢢⣶⠱⠀⢿⠎⠀⣀⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⢥⠀⡛⠩⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣣⣾⠃⠀⠀⠀⠸⢛⠄⠀⠀⠶⠃⠚⠛⢠⣤⣿⠂⠁
⠀⠀⠀⠀⡀⠀⠄⠀⢀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠘⣞⠀⣟⠢⠤⡉⣀⠒⠀⠀⠀⣀⠤⠤⠤⠐⠒⠀⠀⣈⠔⠀⡀⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠀⠘⠛⠁⠀⠀⠀⡀⠀⠉⠀⢨⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠠⠄⠒⠋⣀⠄⠨⢁⣀⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⢠⣊⣅⡰⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠑⠢⠤⠄⢈⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⠀⠀⠉⠀⡀⠀⠀⠀⠄⠀⡄⠠
⠀⠀⠀⠘⠉⠞⠀⠙⢍⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠄⠀⣰⣆⠀⡀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠶⢎⡛⣠⣀⢠⡄⠀⠀⢈⠀⠈⠱⠀⠀⠠⡄⢸⠀⡇⢀⠆
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠌⡆⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠘⠳⠦⠴⠁⣁⣀⣴⣄⠀⠐⠀⠀⢬⣻⢨⣻⣯⢹⣶⢃⡢⠀⠠⠀⠁⢨⠀⠀⠀⡇⢸⠀⣇⢸
⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠾⠿⡧⣥⢿⣼⡇⠀⠐⠐⠒⠐⠀⠀⠀⠁⠈⠀⠁⠈
⠀⠀⠀⠀⠠⠤⠀⠛⠠⠤⠤⠀⠀⠈⠑⣄⡀⠐⠀⠉⠁⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁⢻⠶⠜⠃⠀⠰⣤⠖⡩⡀⠀⢴⣤⠀⠀⠠⠈⠓
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢑⠑⡇⠀⠀⠁⣀⠀⢀⡤
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠾⢗⠀⠀⢤⡀⠀⠀⠀⠀⠄⡀⢸⣦⠀⠀⠈⠀⠀⡅⠀⠀⢀⠀⠑⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠉⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠀⠀⠀⡐⠿⠁⠈⠋⠀⠀⠠⠀⠰⠀⠀⠀⠈⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠠⠠⠄⠤⠀⠄⠀⠠⠀⠴⠤⠀⠐⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⢂⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢿⣿⣿⡏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠤⢦⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⡄⠀⠀⠠⠦⠴⡄⠀⠀⠀⠀⠀⢀⡤⠒⠀⡈⠳⢮⣧⣶⣾⢾⠇⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠠⠒⠃⠀⠀⠀⠀⠀⢀⢈⣴⣾⣿⣿⣿⠂⠘⠟⠻⠿⠒⠶⡌⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⣭⡏⠻⠀⠀⠀⠀⠀⠀⣆⢘⣁⣠⠤⠟⠛⠫⠿⣹⠿⠟⠋⠑⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⣑⠷⠄⠀⠀⠀⠀⠀⠀⠈⠃⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢻⠛⠁⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⠶⣦⡀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋⠀⠈⠻⠒⠒⠶⢲⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠙⣄⠀⠀⠀⠀⠀⠰⡇⠀⠐⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⢄⠀⠀⠑⠢⠤⠤⣀⣀⠁⣿⣿⠁⠀⠀⠀⠙⡉⠂⠄⠠⠤⠤⠀⠀⠀⢀⡔⢸⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠑⢄⡀⠀⠀⠀⢠⣞⢷⣽⣿⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⡈⠣⡀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠼⠙⢷⣍⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⠲⢤⣠⡤⡇
⠀⠀⢀⠊⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢄⠢⣑⠐⠢⠤⠔⠁
⠀⠠⠃⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣧⡀⠀⠀⠀⣀⣴⣭⠴⠄⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⡿⠛⠛⠒⠒⠒⠒⠊⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣷⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣨⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⣶⣄⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋⠀⠀⠀⠀⠀⠈⠻⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢠⣄⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠄⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠛⠳⠀⢶⠆⢠⠐⢀⢋⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⢶⣵⡤⣑⢲⣖⣡⣾⣿⡾⠿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁⠘⢧⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢐⣀⣤⣭⣿⠋⠉⠉⠉⠙⢻⣿⣷⠂⢀⡘⠆⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠋⠉⣻⣿⣿⠀⠀⠀⠀⠀⠀⣿⡟⢯⡤⠄⢸⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠠⢀⠀⠀⢀⠒⠤⠄⠀⠀⠠⠞⣑⡷⢸⣧⣠⡞⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠠⠀⠀⠐⢀⣬⡿⠀⠀⢹⣄⣒⣛⢸⡿⠋⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠟⠳⣾⡀⢸⡟⣿⠗⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡽⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⢀⣠⣘⡦⠏⢈⠀⠆⠀⠀⠀⠀⠀⠘⣡⣤⣘⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣄⣀⢀⡀⠶⡾⠾⠍⠯⠀⠐⠉⠀⠀⢰⣤⡆⠀⠈⡟⢿⡿⢿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣟⣉⠤⠄⣶⠿⠀⠀⠀⣠⠀⠀⠀⠀⠈⠻⣵⣄⢀⣿⣶⡇⢸⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⢛⠁⢟⡿⢛⡋⠲⠀⠀⠀⣠⡆⠀⢀⠀⠀⠀⠀⠘⢦⡙⠟⢿⣤⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⣵⡀⠉⠀⠋⢀⠀⠀⢀⡀⠀⠀⠐⠀⠀⠀⣰⣀⣀⠀⠀⠨⠿⠀⠈⠻⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡆⠋⠀⢰⠁⠀⠈⡄⠀⠟⠿⠀⠀⠀⠀⠀⠼⠛⠋⠉⠀⢳⣾⠘⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠻⠿⠞⠀⡀⢀⡟⠀⠀⠨⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣶⣶⣦⠀⠀⣠⢸⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣾⠜⢁⠼⠀⠀⠀⠀⢀⢸⠀⠲⡎⣽⠟⠃⠀⠀⠀⢀⡄⡞⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⠾⠛⠁⢠⠂⡀⠀⡀⠊⠠⠆⠀⠸⠃⠀⠀⠀⠀⠀⠘⠀⠁⢻⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠿⠟⠛⠻⡿⠂⢿⠀⢸⠀⠀⠘⠀⢠⠞⢰⣤⠼⠷⣦⣤⣶⣶⠟⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⡆⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⢠⡆⠀⡄⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠣⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⣶⡯⠇⢼⡇⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⣿⣿⣷⣾⣷⣿⡇⡐⢿⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠋⠈⠁⠉⠈⠀⠙⠢⣄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠰⣶⡄⠀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⣴⣦⡤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⣠⡤⠤⠤⠂
⠀⠀⠀⠀⢸⣿⣿⣽⣧⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠞⣡⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠘⢛⣭⣷⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣾⠛⡩⠭⠭⣙⠻⣷⡄
⠀⠀⠀⠀⠀⠈⠙⢻⡏⠀⠀⠀⠀⠀⠀⠐⢷⣦⣀⠀⢿⣿⣧⣎⡤⠒⠢⣌⠳⣌⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣀⣤⣤⣤⣶⣶⣶⣾⣿⣿⣾⣿⣿⣷⣬⣁⣉⢎⠣⠈⢦⡙⢿⣦⡀
⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠉⠉⠀⠀⠀⢠⡾⠋⠁⠀⣈⡙⢿⡉⠙⠛⠿⣷⣵⣄⠙⢦⠙⣿⣄
⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠿⠍⣼⠇⠀⠀⠀⠈⠙⣷⣗⡄⢳⢸⣿
⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠈⢿⣾⣸⢸⣿⡄⢀
⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡸⣧⠙⠛
⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠴⣦⣤⣀⠀⠀⠀⠀⢻⣿⣷⣽⣷⡄
⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣷⠉⠀⠀⠀⡾⠟⠻⣿⣿⠿⠊
⠀⠀⠀⠈⠙⠛⠿⠶⠶⣶⣶⣶⣶⣶⣶⡶⠶⠿⠆⠀⠀⠀⢀⣼⠋⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢀⣀⣀⣠⣤⠶⠛⠁⠀⢀⣼⡇⠀⠀⠀⢿⣦⣶⡿⠋⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠹⣯⠀⠀⠀⣠⣴⣿⣿⠃⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⣿⣿⡿⠛⠁⣰⠟⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢠⣶⣦⣤⣾⣧⣾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠈⠛⠶⠶⠞⠋⢀⣾⠀⠀⠀⢀⣿⠁⠀⠀⠀⠈⠉⢉⣿⣿⣿⡿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠁⠀⠀⠀⣸⡟⠀⠀⠀⠀⣀⣴⡿⠋⠉⠉⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣍⣉⣛⡓⠒⠒⠶⣆⠀⣠⠤⠦⠴⠶⠶⠶⠲⣿⡃⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠒⠒⠒⠶⠶⠶⢾⣿⠛⠋⠉
⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣸⡇⠀⣹⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⣿⠀⢀⡟⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⠿⠆⣼⠁⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠿⠴⠞⠛⠛⠿⢶⣦⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠷⠶⠶⢶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣄⣀⣀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⠈⢿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣷⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠠⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢩⡍⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡿⠺⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣼⣧⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠛⠀⠻⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡁⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣤⣯⣼⡴⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠛⠿⢿⣿⣟⣋⣽⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⣄⡀⠉⠋⣤⣿⢷⣄⡀⠀⢦⡀⠀⢎⠃⠀⡀⣀⣘⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⢰⡾⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⣠⣄⣬⡄⠘⣒⣹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠈⢩⡟⠛⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠚⠉⠉⠀⠀⠒⠲⠤⣀⣀⠤⠀⠀⠀⢀⠀⠀⠠⠋⠀⢠⢿⡏
⠀⠀⠀⠀⠀⠰⣡⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⢀⠀⠙⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠚⠳⢤⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡁⠐⠚⠉⠉⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⢤⣥⣸⣧⠐⠀⡂⠀⠠⠂⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠑⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣴⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠉⠁⠀⠈⡽⠳⢄⡀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⡀⡀⡞⠁⠀⠀⠈⠓⠦⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠆⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠂⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠢⠴⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⡀⠀⠀⠀⠀⠀⠀⣠⣠⠴⠶⠖⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣭⣥⣶⣖⣶⣶⣬⣵⠀⠀⢀⣀⣴⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⢿⣿⣿⣿⣿⣿⠿⠟⠿⠿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁⣀⣉⣉⣉⡉⠉⢻⣿⣧⣀⣴⣿⡿⠿⠿⠿⠿⠿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⣠⡾⠿⠛⠛⠻⠿⢧⠀⠹⣿⣿⡿⢛⣤⣶⣶⣶⣦⣄⠀⢻⣿⣇⢀⣠⣾⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠛⠉⠉⠀⠀⠉⠉⠁⠈⢿⣿⣿⣿⠟⠉⠈⠙⠻⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⣰⣶⣶⣶⣄⠀⠈⠙⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠐⠉⠀⠈⠙⠻⢦⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣴⣿⡿⢿⣿⣷⣶⣤⣀⣀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢟⣴⣶⣮⠉⠛⠿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⣼⣿⢻⡇⠀⠀⠀⣿⣿⠉⠛⠿⣿⣿⣷⣦⣤⡀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⣿⠇⠘⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠙⢻⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⣿⣿
⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⣿⣿⣀⡀⠀⠀⠀⠀⣸⣿⣇⠀⠀⢰⣿⡟⢀⣤⣤⣤⣤⣤⡄⠀⣸⣿⡏⠀⠀⣿⣿
⠀⠀⠀⣴⣿⡿⠃⠀⠀⠀⠀⠀⣴⣶⣄⠀⠀⠀⠘⠻⠿⠿⣿⣿⣶⣿⡿⠟⠿⠂⠀⣼⣿⡇⠈⠛⠛⠛⠛⠛⠃⢀⣿⣿⠁⠀⢠⣿⣿
⠀⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣟⣁⣀⣀⣀⣀⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣾⣿⠏⠀⠀⢸⣿⣿
⢠⣿⣿⠏⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣿⣿⣯⣿⡿⠿⠿⠿⡿⠿⠿⠿⠿⠏⠛⢛⣛⣿⣿⣿⣿⣿⣿⣵⣦⣄⠀⢸⣿⡏
⣿⣿⠃⠀⣼⡿⠛⠀⠀⠀⠀⣴⣾⡿⠛⠉⠁⠀⠀⠈⠉⠙⠻⣿⣶⣄⠀⠴⠶⠶⠶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⣻⣿⠀⣸⣿⡇
⢻⣿⣧⠘⢿⡇⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢰⣿⣿⠁
⠀⠻⣿⣷⡄⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢡⣿⣿⠃
⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⣾⣿⠃
⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣾⣿⠏
⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣦⣤⣄⣀⣀⣠⣤⣴⣶⣿⠿⢋⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠘⠻⠿⠿⣿⣶⣶⣤⣤⣈⣉⠉⠉⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⢀⣴⣿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⢿⣿⡆⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣴⠀⢀⣤⢼⢣⣦⠀⠀⢰⣿⣿⠇⠀⠠⠄⠀⠀⣤⣤⡄⠀⣿⡇⢸⠀⠀⣤⣤⣿⡄
⠀⠀⠻⠟⣱⠀⠁⠀⠀⢻⡁⡀⠀⢸⣿⣿⠁⠀⢀⡄⠀⠀⢹⣿⠀⠀⣿⡇⠸⠀⠀⣿⣿⣸⡇⢘⡀
⠀⠐⠙⢗⣿⣤⡬⢥⣤⡿⣓⢀⠀⠈⣿⣿⠇⠀⣠⠀⠀⢀⣸⣿⠀⠀⢿⠇⡄⠀⠀⣿⣿⡿⠟⠳⠖⠤
⠀⠈⠛⣿⠣⣤⡼⢧⣬⡶⠽⡍⠁⢀⢈⡉⡉⠉⠀⠀⠀⠈⠉⠉⠀⠀⣾⠁⡆⠀⢉⣉⣉⡁
⠀⠀⠀⢀⡰⢏⠀⠀⢩⠷⣄⠀⣰⡿⠄⠠⢿⣆⢀⣿⣶⣾⡄⠸⣤⠀⢉⠉⠛⣩⣽⢯⣹⣩⢾⣿⣦⠀⣀
⠀⠀⠀⠿⢀⣴⠩⠉⠋⠆⠀⠀⢻⡁⠀⠀⠈⡟⠘⡷⣺⣿⠇⠀⣿⠀⢨⠇⣴⠃⠈⠉⡁⠀⣈⣩⠻⣫⠟
⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⣼⡵⠸⠅⠠⣯⠀⠈⠉⠁⢠⡾⠃⠀⠙⠺⠃⠀⣀⣀⡝⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣄⠀⣠⡄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣘⣦⣾⣷⣾⣿⣿⣿⣿⣿⣿⣧⣾⣃⣼⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢆⣸⣖⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢬⣲⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣽⣶⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢻⣿⣿⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣾⣿⣿⡿⠁⠀⣀⣤⣤⣤⣄⣀⠀⠀⣀⣤⠖⠛⠉⠉⠉⠙⠻⣶⣄⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠁⣠⣾⣿⠟⠋⠁⠀⠀⠉⠉⠉⠀⠀⣀⣀⣀⣀⠀⠀⢹⣿⣧⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢴⣿⣿⣿⠁⣼⣿⡿⠃⠀⣠⣴⠾⠛⠳⢶⣄⣠⠞⠉⠁⠀⠉⠻⣆⠀⠙⠛⠀⠀⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⡇⠀⠛⠋⠀⢠⡾⠋⠀⢀⣀⠀⠀⢻⡏⠀⠀⢠⣤⡀⠀⢻⡆⠀⠀⠀⠀⠀⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠿⠃⠀⠈⣿⡀⠀⠈⠉⠀⠀⢸⡷⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢀⣼⠛⢿⣦⣄⡀⢀⣠⣾⠃⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⣴⣾⣿⠁⠀⠀⠹⣿⣿⣿⣏⣁⡀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠨⢾⡿⠋⣏⠀⠀⠀⠀⠹⣿⡙⠳⠦⠀⠀⠀⠀⠀⠀⠆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⢀⣠⣿⠀⠀⠀⠀⠀⢻⣏⠳⣄⠀⠀⠀⠀⢀⠈⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⡀⠀⠀⠀⠀⣼⡿⠀⣿⠀⠀⢀⠂⠀⠐⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠾⢷⣤⣀⣤⣾⡿⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠠⠀⠀⠨⢙⣟⢻⠁⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣨⣷⣾⣧⠀⠀⠀⠐⠀⠂⠀⢀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠠⣿⣿⣿⣿⣧⠀⠀⠠⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⠏⠂⠀⠀⠀⠀⠀⢀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⡄⠀⢀⠀⠂⠀⠀⠉⠀⠈⠙⠛⠳⣦⠀⠀⠀⠤⠄⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣥⣀⠁⠄⡀⠀⠀⠀⠀⠄⠀⠘⠄⠸⣷⢀⣠⣤⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠻⠿⠷⠶⠶⠾⠶⠿⠿⠟⠛⠋⠁


⠀⠀⡀
⠀⠁⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣿⠇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⢀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⣿⣹⣇
⠀⠀⠀⠀⠀⠀⣴⢀⡀⠀⠀⢰⣟⣿⠁⠀⣼⣷⠖⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣬⣿⣿⣥⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠲⣾⣧⡀⠈⣿⣻⡇⠀⠀⢀⡀⣦
⠀⠀⠀⠀⠀⢠⡿⣿⠁⠀⢠⡟⣿⡇⠀⣼⣷⠟⠀⠀⠀⠀⢀⣠⣶⣿⡿⠯⣷⠛⠛⠟⡉⢻⠛⠛⠿⣦⢤⡀⠀⠀⠀⠀⠹⣾⣧⡀⠘⣿⢻⡆⠀⠈⣿⢿⡆
⠀⠀⠀⠀⠀⣼⢿⡏⠀⠀⣼⢿⡟⠀⣰⣷⡟⠀⠀⠀⢀⣴⣿⡿⢽⡏⠀⠀⠉⠀⠉⠀⠁⢈⡲⠀⣤⠿⢷⣿⣦⡀⠀⠀⠀⢻⣾⣇⠀⢹⡿⣷⠀⠀⢸⡿⣧
⠀⠀⠀⠀⠀⣿⣾⡃⠀⢀⣿⣾⠃⢀⣿⣾⠃⠀⠀⣠⣿⣿⢏⠀⠀⠀⠀⠀⢀⡄⠀⠐⠂⠀⠁⠘⣧⡀⢰⡿⣿⣿⣆⠀⠀⠈⢷⣿⡀⠘⣷⣿⡀⠀⠈⣷⣿
⠀⠀⠀⠀⠀⣿⣾⡁⠀⢸⣧⣿⠀⢸⣯⡟⠀⠀⣰⣿⣿⡇⣶⠀⠀⠚⠀⠀⠀⠁⡠⠀⠀⡄⠀⠀⠀⢶⡃⣿⣻⣿⣿⣧⠀⠀⢻⣽⡇⠀⣿⣼⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢿⣾⡇⠀⢸⣟⣿⠀⢸⣽⡇⠀⢰⣿⣿⣿⣷⣿⣍⠘⠃⠸⣅⠀⣠⡝⢀⡀⠓⠄⠀⣶⡀⠀⢶⢹⣟⣿⣿⣇⠀⢸⣏⡿⠀⣿⣻⡇⠀⢈⣷⣿
⠀⠀⠀⠀⠀⢹⣽⡇⠀⢸⡿⣿⠀⢾⣻⡇⠀⣿⣿⣿⣿⣷⡟⣶⣿⡟⠛⢻⣿⣿⣷⣿⣤⣴⣦⣶⣄⣤⡄⣿⠁⣽⡿⣿⣿⡀⢰⣟⣿⠀⣾⢿⡇⠀⢸⣯⡟
⠀⠀⠀⠀⠀⠘⣯⣿⡀⠘⣿⣿⡀⢸⣿⡇⢰⣿⣿⣿⣿⣿⣿⣯⣿⣇⡀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠻⣾⣿⣉⣛⡻⣿⣿⣿⡇⢰⡟⡧⠀⣿⣿⡇⢀⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠸⣏⣷⠀⢿⣼⡇⢸⡿⣷⢸⣿⣿⣿⣿⣿⣿⣧⢸⡏⠁⢸⣿⠿⣿⣷⢸⣿⢼⡇⠀⣨⣭⣭⣍⠙⢮⣿⣿⡇⣼⢿⡇⢸⣷⣾⠃⣼⣻⡟
⠀⠀⠀⠀⠀⠀⠀⢹⣟⣷⢸⣏⣿⠀⣻⣿⣼⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⢸⣿⡅⣿⣿⢸⣿⣿⡇⠀⣷⣿⣿⣯⡇⠀⣿⣿⣧⡿⣿⠀⣿⣩⡗⣼⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⣿⣿⣛⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠘⣿⡇⢹⣿⢸⣿⣿⡇⢀⢸⣿⣿⣿⡇⢀⣿⣿⣿⣿⠋⣸⣟⣿⣿⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣽⣿⡿⣧⠹⣾⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⢸⠈⣛⢛⡛⠁⢸⣿⣿⣿⡟⣼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣧⠹⣾⣿⣿⣿⣿⣿⡇⢸⣿⠀⠀⣿⡇⢸⣿⢸⣿⣿⡇⣹⣈⣻⡛⣻⡵⠛⠛⠻⣯⣙⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠒⠉⠛⢿⣷⣼⣿⡿⣿⣿⣿⡇⢸⣿⠀⠀⣿⡇⢈⢛⣘⡛⢿⡇⢨⢻⢋⠞⠁⠀⣀⣁⣀⡈⠙⣯⡛⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠈⠀⠈⠛⠿⣿⣿⣿⣿⡇⣿⣿⠀⠀⢟⡕⠁⠀⠙⠻⠷⠦⠬⠠⣬⣄⠀⢐⣫⣿⠿⢿⣦⠘⠛⠉⠙⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣛⠳⠤⠀⠀⠀⠀⠁⠂⠌⠉⠛⠑⠛⠛⠃⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⢉⣭⡶⠾⢛⣁⣠⡔⠒⠦⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠛⠒⠲⠦⠤⣄⣀⠀⠀⠀⠀⢸⣿⣿⢳⡄⠒⠛⠋⠙⠛⣷⠀⠀⡵⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⠉⠈⠈⠱⠶⠤⢤⣄⣀⡀⠀⠉⠑⠂⠤⣾⡟⠁⣼⡷⠀⠀⠀⠀⠀⠸⣇⣴⠃⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣤⣀⣀⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠦⣄⣀⠀⠀⣼⣿⠃⠿⣶⣶⣶⣶⠦⣉⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠿⠿⠿⠮⠙⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠛⠛⠦⣄⠀⠀⠁⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠤⣤⣤⣀⣀⣀⣀⠀⠀⢀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⢰⡟⠁⣠⡾⠋⠁⢀⣀⣤⣴⣶⣿⡿⢷⣤⣀⣀⣉⣻⣿⣿⡑⢶⣿⣿⣿⠁
⠀⠀⠀⠀⢹⡇⠐⣿⡅⠠⢺⣿⣿⣿⣿⣿⠏⠀⠀⠙⠻⣿⣿⣿⣿⣭⡿⠋⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠙⢦⡙⢿⣦⡈⠻⣿⣿⣿⡏⠀⣀⡀⠀⠀⠀⠀⠉⢉⣁⡀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⠛⠷⣮⣿⣿⣿⣀⠀⠉⠳⠀⠀⠀⠘⠉⠁⢀⠀⣸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⢶⠙⣿⣿⣿⠓⠖⠂⠀⠀⠀⠀⠒⠞⠋⠀⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣾⣿⣿⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣏⠺⢎⣧⡀⡶⣶⢈⣁⣁⡀⠀⢀⣴⡱⠖⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠋⡰⠛⠛⠛⠛⠛⠛⠛⢻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⢠⣾⣷⣄⣀⠀⠀⠀⠀⡄⣾⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠠⠤⠤⢤⣌⣉⣛⢶⢞⡼⠋⠗⢦⡤⣀⡀
⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⣿⡿⠁⠀⠀⠀⠐⠒⠲⢤⣙⣶⡛⠛⠋⠀⣸⣹⣿⣑⣿⣿⣿⢶⣦⣤⣀
⠀⠀⠀⠀⠀⢠⡿⣿⣧⣀⣽⠁⠀⠀⢀⡴⢺⣯⠽⡛⠋⠉⠁⠀⢀⠔⣥⡿⢦⣿⣿⣿⣧⣠⣿⡿⢿⡀
⠀⠀⠀⠀⠀⢸⠃⠈⢿⣿⡟⠀⠀⢲⢿⣿⣎⣹⢷⣏⣔⢒⡔⢲⣉⡶⢏⣱⣾⣿⢿⣿⣿⣿⡿⠁⠸⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠈⢿⣿⠀⠀⢰⣠⣿⣿⣷⣶⣍⣟⢛⡟⢻⣉⣷⣾⣿⣿⣧⣠⣿⣿⡿⠁⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣿⠀⠀⠘⣿⣿⣯⣁⠀⢀⣀⣀⣀⣀⣀⣀⠀⢻⣯⣤⡉⠛⠧⣄⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣴⣒⣒⣲⣶⠚⠛⠛⠛⠋⡉⠀⠀⠀⠱⣄⡿
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿⠀⠀⠀⢹⣿⡋⢙⣿⣿⠿⣿⣿⣿⠟⢿⣿⣯⣭⣤⡄⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⠀⠀⠀⢸⣿⣷⣿⣿⣷⣤⣾⣿⣿⣦⣾⣿⣿⣿⣦⣬⣽⣿⣿⠃⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢸⣇⢠⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡇⠀⠀⠀⠀⢿⡁⣙⣿⣿⣿⣟⠁⣻⣿⣿⣿⡋⢙⣿⣿⣿⡏⣧⠀⠀⠈⡇
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣟⣟⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣽⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣂⣀⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⡟⠋
⣀⢠⡄⠀⠀⠀⠀⢀⠐⣛⠛⠉⠙⠉⠉⠈⠉⠛⠗⠠⣤⣀⣀⠀⠀⠀⠀⠒⠲⠬⠛⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⣿⣿⣷⣶⣾⣿⣿⣿⣿⡍⣀⣀⣄⠀⠀⡀⠀⠀⠀⢀⣙⠏⣟⡿⢶⣿⠀⠀⠀⠀⠀⠀⠹⣶⣿⠿⣿⢿⣿⣿⣿⣿⣿⡿⠋⣥⡝⢻⣃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣧⣴⣧⣄⡀⣤⣤⣿⣿⣿⣟⠛⣿⠋⠀⣀⣀⠀⠂⠀⠀⠀⢀⣿⣾⣿⣭⣽⡟⢩⠁⠀⠋⠉⠉⠉⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⠿⢷⡄⠀⠖⠀⠀⢨⠿⠿⠹⣧⠀⠀⠀⠀⠀⢀⢠⠤⠤⠶⠾⢷⣦⡀
⢿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣀⠀⠀⢀⣀⡘⠀⢀⣧⣤⢀⣤⣶⣤⣶⣶⣶⠆⠀⠀⠀⠀⠻
⢸⣿⣿⠛⠉⠈⠛⠛⠛⠛⠛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⢛⣿⣿⣿⣿⣩⣍⣉⠋⠽⠀⢀⣉⣛⣂⢹⠿⠛⠛⠛⢿⣿⣿⡿⢇⣠⣶⣾⣷⣴⡄
⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠀⠀⣠⠨⠞⣿⡉⣿⣾⣿⡿⠃⢀⣾⣿⣿⣿⣿⣇⡀⣠⣄⣀⣦⣶⣾⣷⣾⣿⣿⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠻⠏⠀⠀⠀⠀⢠⠎⡁⡀⠀⣿⣗⣿⣯⣁⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣥⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠒⢠⠀⠀⠀⡀⠀⠀⣀⡄⢀⠚⢻⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠿⠛⠻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⣠⣧⣴⣾⣻⣿⣷⣦⣾⠹⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣿⣿⠿⠿⠿⠟⠛⠛⠛⠃⠿⢥⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣈⣉⣭⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⡖⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⠿⠟⠷⠟⠻⠛⠿⡆
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠐⠛
⠀⠀⠀⣀⣠⡀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣏⡐⠋
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀
⠉⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠻⠿⠻⠛⠋⠿⣿⣿⢤⣀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⢰⡇⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠈⠉⠁⠈⠰⠀⠀⠀⠀⠀⠀⠀⢡⠙⠲⢿⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠁⠉⠁⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢸⣿⠀⠀⢘⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⣽⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⠀⠀⢨⣀⣀⣀⣀⣀⣀⣀⣤
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣌⣋⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠋⠉⠉⠉⠉⠉⠑⠒⠲⠤⣄⡀
⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣠⠔⠒⠂⢄⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣹⡇⠰⢻⡟⢲⣷⡖⢘⣃⠒⣄
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⢂⣉⣭⠭⠿⠄⡀⢀⠩⢾⣇⣻⡉⠃⢸⡇
⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⢀⣤⣴⡟⠋⣩⣶⣾⠟⠿⠋⠉⠉⠀⣈⣙⣛⣤⣾⠃
⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢲⣾⠟⠉⠀⢤⣾⣿⡌⠁⠀⠀⠠⣤⣉⣛⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠘⠛⠃⠐⠛⠻⢶⣝⠋⠙⠂⠀⠀⠀⠀⢲⣶⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠉⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⠀⠀⠀⠀⠀⠈⠙⢩⣿⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠖⢦⠀⠀⡤⠤⠤⠤⠴⠒⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢲⠾⠀⠀⠀⠐⠻⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠘⡇⠀⠀⠀⠀⢀⡷⣄⠀⠀⠀⠀⠀⣀⣤⠶⠶⠀⠀⠀⠀⠸⠆⠀⠀⠀⣀⣴⣛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢽⣦⡀⠀⠀⠀⢅⢈⡃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡉⢭⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⢻⡏⠙⠷⠀⠀⠀⠀⠀⠙⠛⠿⠂⠀⠀⠀⠈⢿⡟⠋⠉⠁⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⢓⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⢻⡙⠛⠉⢉⣱⠖⠀⠀⠀⠀⠀⢀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠻⣿⣿⠿⠋⠀⠀⠀⠀⠀⣠⡞⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⣁⡤⠤⠤⠠⠄⠼⠄⠀⢠⢤⡄⠀⠀⢀⣴⣾⠟⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣴⣾⡿⠿⠋⠀⠀⠀⠀⠀⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⠀⠀⠀⢀⡀⠀⠳⣦⣠⡀⠀⠀⠀⠀⠀⠀⢠⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡁⣄⠀⠀⠀⠀⣀⠀⠁⠀⠠⠬⠉⠓⠢⣄⠀⠀⢠⣴⠛⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⢸⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⣴⡟⠉⠀⠀⠀⠀⠀⣀⣠⣷⣶⣶⡖⢘⣧⡄⢰⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⢀⣾⣿⣿⡀⠀⠀⠀⡀⣀⣤⣶⣾⣿⣆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⣿⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣇⣾⣿⣿⣿⣷⣄⣠⣴⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣠⣚⢢⠀⠀⠀⠀⠀⣷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⠀⠀⠀⠀⢀⠼⣁⣪⠉⠀⠀⠀⠀⠶⠿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡛⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠃⠀⠀⠀⠀⠀⠀⣠⣜⠣⠙⠋⠀⠀⠠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⡅⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠐⠄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣱⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⠄⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⣀⡀⠀⠀⠠⢄⠚⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠈⠒⡤⠂⢄⠀⠀⠀⠀⠀⠀⣶⢑⣈⡄⠛⠁⠀⠀⠀⠀⣥⠐⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠉⠢⡀⠀⢄⣀⣴⠀⠀⠀⢴⡙⣄⣨⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⢙⣄⠀⠀⠀⠑⢶⡑⣤⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣾⠉⠨⠿⠁⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣰⣿⣿⣷⣄⠀⢠⣶⡷⡉⠀⠀⠀⠀⡊⠀⢐⠀⠀⠀⠀⠤⣾⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣾⣿⣿⣿⠿⠋⠉⠢⡀⠀⠈⢆⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣴⣿⣿⡿⠟⠁⠀⠀⣀⣤⣾⣷⣦⣤⠇
⠀⠀⠀⠀⠀⣸⡟⠀⣴⣿⣿⠟⠋⠀⢀⣠⣴⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⢰⠟⣠⣾⡿⠛⢁⣠⣴⣾⡿⠟⠛⠉
⠀⠀⠀⢠⢏⣼⠟⣋⣤⡶⠟⠛⠉
⠀⠀⢀⣿⡿⠷⠛⠉
⠀⠀⠈⠁

⡿⢿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⢿⣿⣿⡿⣿⣿⢿

⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠘⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣖⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢻⣵⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠈⢻⣿⣭⣍⣻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢃⠀⠙⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣆⠀⠈⠻⣿⣿⣿⣿⣿⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡄⠀⠈⠻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡝⣦⡀⠀⠈⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡳⣄⠀⠀⠹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⣀⡈⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠚⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡄⠀⠀⠀⠀⠀⢠⡄⠸⠷⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⡜⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡻⠋⠀⠀⠀⢀⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡈⢹⣟⣀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠀⢀⡀⠀⠀⠞⠀⠂⠻⠂⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣁⠀⠈⠛⠃⠀⣴⣾⣿⣿⣿⣦⡀⣠⣤⣤⣬⣴⣿⡌⠀⢒⡀⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⠜⢻⣿⣿⣿⣿⣿⣿⣿⣧⣤⡾⢟⣤⣆⡀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⣀⣀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣔⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣭⣭⣭⣿⣭⣭⣽⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣾⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣏⡟⠉⠑⢄
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠟⠛⠛⠛⠋⠙⣿⣿⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣾⣿⣃⣀⣀⣀⣠⣤⣀⣰⡄⢸⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⠁⠀⢠⣴⣶⣖⡤⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠉⠚⠻⣧⣀⠀⢀⠀⡠⠊⡼⠀⠳⡤⡀
⠀⠀⠀⠀⠀⠈⣿⣿⡇⢿⠳⣌⠻⣿⣿⡁⢸⢷⡀⠀⠀⠀⠀⢀⢀⡴⠒⡜⢁⣼⡷⡠⠀⠀⠘⡷⣤⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣮⡑⣿⡁⠹⢿⠏⠀⠈⠀⠀⠀⠀⠀⠀⢁⠴⣶⠴⠛⠛⠛⠓⠀⢄⠀⠘⠎⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠉⠛⠢⣄⡘⠃⠉⢰⣾⠂⠁⣤⣄⡈⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⢃⠀⠡⢽⣼⣶⣬⡛⣻⣷⣶⢟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣿⢻⣿⠟⢷⣤⣄⣀⣀⠀⢀⣠⣤⠤⠐⠃⠀⠀⠀⠀⠙⢶⣄⠀⠉⠙⠻⡈⠿⣷⡿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠀⠉⠈⣿⡆⠀⠀⠉⠛⠛⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⠀⠀⠘⠶⠋⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠘⢷⣠⠀⠸⣿⠀⠀⠀⠀⠀⡼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣵⡆⠀⠀⢇
⠀⠀⠀⠀⠀⢀⣠⠟⢆⠀⠀⠙⢷⣴⣿⣦⠀⠀⠀⣼⡁⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣷⣟⣠⠴⠛⡆
⠀⠀⠀⠠⠚⠉⠀⠀⠈⢣⡀⠀⠀⠈⠉⠛⠧⣄⣰⠃⠉⠆⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢰⠋⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⡀⣀⠀⣀⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛⢻⣟⠛⠛⠛⠛⢻⡿⠛⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣤⣤⣶⠟⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⡿⠋⠛⣷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⢿⣦⣴⠟⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠸⣭⠗⠀⠀⠀⠻⣽⠇⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣄⠀⠀⠀⠘⠾⠋⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣬⣽⣿⣶⣤⣤⣤⣤⣤⣶⣿⣯⣥⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢺⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⣼⡇⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⣿⡋⠉⠉⠉⠉⢹⡇⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⢿⡇⠀⠀⠀⠀⣾⡇⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⣿⠃⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⠷⠶⠶⠾⠿⠶⠶⠶⠶⠿⠶⠶⠾⠏⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡟⠛⠛⢿⡟⠛⠛⠛⠛⠛⠛⠛⠻⣿⠛⠛⢻⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣸⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠐⣿⣀⣌⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠤⠦⠄⠶⠶⢶⣶⣷⠀⢰⣤⠤⠴⠦⠤⠶⢶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⠁⠀⠀⠀⠀⠀⠀⠀⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠓⠒⠀⠀⠀⠀⢾⠀⠀⢀⠀⠓⠒⠒⠀⣀⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⢁⢄⢈⣀⢉⠁⢸⠀⠡⠀⠀⠁⡄⡉⣍⣭⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⢸⣿⣇⠀⠐⢻⠀⠀⠄⠀⠀⠘⣿⣧⣤⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢉⠁⠰⠆⠀⢾⡰⠀⠀⠀⠀⠈⡁⢰⠞⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣾⣷⣿⣦⢰⣿⣄⠙⠀⠀⠀⣼⣿⣿⣷⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⠀⠠⣿⣿⣿⣿⠋⣹⣁⡀⠀⣬⣶⣿⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⡙⠀⠠⢟⠆⣀⡽⢿⣿⠁⠴⣊⠃⣿⣿⣻⢤⣼⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⣀⠀⢁⠀⣾⠀⠀⠉⠁⠉⠠⣦⣶⢯⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠖⠹⠀⠀⠀⠀⠀⠒⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣠⠴⡶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢁⣴⠋⠀⡎⠀⠇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⠁⢀⣼⣠⡴⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡰⠀⠀⠀⠀⠐⠄⠐⠀⠒⢭⡀⠐⣿⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⢈⠛⠄⢄⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡿⠀⠀⡶⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠈⠒⢠⣀⠀⠀⠀⢀⡞⠀⠘⢱⠿⠌⢀⠚⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⢀⡖⠋⠀⠀⠉⠃⠘⠊⠙⠁⠀⠈⠂⠀⠉⣡⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⢟⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠄⠀⠀⠘⠀⢹⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣜⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠻⡄⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢰⣴⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⢈⣄⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣠⣶⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⢀⣾⣿⣿⣦⣀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣏⢸⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣜⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡄⢨⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣇⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣼⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢀⢀⡴⠀⢀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡏⠀⠐⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠈⠀⠀⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣄⣀⣀⣀⣤⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠉⠉⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣷⣶⣦⣤⣄⣀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⠟⠉⠛⠙⢿⣿⡿⠛⠉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣾⣿⣿⡿⡿⠛⠿⠻⠏⠀⠀⣀⣀⡀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⡇
⠀⣼⣿⣿⡟⠀⠀⠀⠠⣴⣤⣶⡾⠛⡿⠻⣷⣄⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿
⢠⡿⠋⠈⠀⠀⠀⣀⠀⢀⣅⡈⣠⣄⣰⣶⣾⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣸⡇⠀⢀⣰⣶⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⣿⣄⣤⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⢤⡄⠀⠀⠚⢹⡆
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⣾⣧⠶⠛⠙⠶⣄
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠠⠞⠻⠿⠁⠀⠀⠀⣨⠟⠁⠀⢀⠔⢲⡌⠳⣄
⠀⠀⢻⣿⣿⣿⡿⠉⣉⣉⣉⠙⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⢀⡴⠁⢠⠞⠁⠀⣈⡓⢦⡀
⠀⠀⠀⠹⣿⣿⡇⠸⠋⠈⠙⣷⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢀⠎⠀⣠⠆⠂⠉⠀⣀⡹⠆⢙⠻⣟⠒⠦⣄⡀
⠀⠀⠀⠀⠙⣿⣧⠀⠀⠐⠻⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡆⠀⡴⠃⠀⢠⠔⠋⠀⠀⣠⠿⡄⠈⠳⣄⡀⠙⠲⣄
⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⠁⠀⠀⠀⠐⠃⠀⠠⠖⠋⠀⣠⠃⠀⠀⠈⠙⢦⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⣤⣤⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠙⢦⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⠿⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠛⢮⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣟⡀⠀⠀⠈⠉⠛⠿⢷⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠙⠛⠶⣦⣄⡀⠀⠀⠀⠉⠙⢛⣿⣶⣶⣶⡶⠟⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⣄⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣰⠁⠘⣦
⠀⠀⠀⠀⠀⣴⠟⣁⣤⣤⣤⣀⠀⠀⠀⠉⠻⢶⣄⡀⢀⣴⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣦⣀⠀⢰⠏⠀⠀⠀⠀⢀⡴⠁⠀⠀⠘⣧
⠀⠀⠀⠀⣼⡿⠛⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⣬⡿⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⣿⣿⠇⠸⣿⣿⣿⣿⣷⣤⣀⠀⠀⣀⣴⠋⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⢰⡟⢀⣠⠴⠀⠀⠀⠀⢀⡠⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠙⠃⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⣿⣶⠟⠁⠀⣀⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣧
⠀⠀⣸⡟⠁⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡇⠈⢿⣷⡀
⠀⣴⠟⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣧⣤⣀⣙⣿⣦⣤⣀⣤⣼
⣼⡏⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣭⣽⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠉⠁
⢿⡇⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⠉⠉⠉⠉
⠘⢿⣄⢸⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠈⠻⣾⣿⣿⣿⣿⣷⣦⣤⣄⣠⣴⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⣿⠿⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢈⡟⠀⠀⠀⠙⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣉⡉⠉⠙⠓⠒⠒⠶⠦⠤⢤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⠉⠉⠉⠙⠛⠒⠒⠲⠶⠤⠤⢤⣤⣀⣉⣉⣉⠙⠛⠒⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⣠⠖⠛⠛⠒⠒⠶⠦⠤⣤⣀⣀⡀⠀⠈⠉⠉⠉⠛⠓⢦⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠶⠤⢤⣄⣸⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡟⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⡀⠉⠉⠉⠉⠛⠓⠒⠒⠲⠦⠤⠤⣤⣤⣀⣀⣀⡀⣀⡤⠞⢹⡆⣇⣸⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠈⠉⠉⠛⠓⢂⠀⠤⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⡏⠀⢀⣼⡇⣏⡤⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠙⠛⠒⠒⠶⠯⠥⢭⢉⣓⣆⠀⠉⠉⠃⠐⠒⠠⠄⠤⡧⠚⠉⣼⢧⢟⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢆⣀⣀⠀⣀⠍⠁⠉⠀⠒⠒⠦⠤⠄⢀⡇⠴⠋⠙⠒⠚⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠦⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠶⠤⠤⣤⣀⣀⣧⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠛⢰⠋⢳⢰⢦⣴⡆⣟⡷⢠⣳⡀⡷⣼⡆⢦⠞⠀⣆⡼⢠⣞⡄⣶⠀⢸⠀⡇⣾⠿⠰⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠙⠉⠈⠀⠈⠁⠁⠀⠉⠉⠁⠁⠈⠁⠈⠀⠀⠈⠁⠉⠉⠉⠉⠉⠈⠋⠁⠉⠉⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⡀⢀⣀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⢠⣤⣥⣄⠀⠰⣷⣤⡀⠀⠀⠀⠀⠁⠀⠀⠠⣶⣖⣾⡦⠀⢧⡈⠛⣠⠇⠀⣀⣉⣀⡀⠀⠀⠀⡔⠸⠀⠀⠸⣿⠆⠀⠀⠀⣄⣩⣤⠀⠀⠀⢸
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠛⠛⠛⠂⠀⠈⠁⠀⠉⠀⠀⠀⠛⠃⠀⠀⠀⠛⠘⠋⠀⠀⠈⠙⠛⠃⠀⠀⠀⠂⠁⠀⠀⠀⠋⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⢀⡀⠀⢀⣄⠀⠀⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠇⢠⣿⣇⢀⣾⡿⠀⣸⣿⠿⣿⣶⣄
⠀⠀⠀⠀⠀⢀⣶⣿⡀⠀⣤⡄⠀⢀⣀⠀⢴⣶⠀⢠⣿⣿⣿⣀⣾⣿⣿⣿⣿⠃⢰⣿⡏⠀⠀⣸⡿
⠀⠀⠀⣴⣿⠿⠛⣿⣧⠀⢿⣿⣄⣽⣿⡆⠘⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡿⠀⠀⢠⣿⠃
⠀⠀⠀⢿⣇⠀⠀⠸⣿⡆⠘⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣾⡏
⠀⠀⠀⠘⣿⡄⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣼⡿
⠀⠀⠀⠀⢹⣿⡀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠉⠙⠛⠿⢿⣿⡟⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢻⣷⣶⡿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠷⢶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⠀⠀⠉⠛⠿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⢀⣠⣶⠿⠟⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠿⢿⣿⣶⣤⡀⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⡄⠀⠀⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣦⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢠⣿⠏⠀⠀⠀⠀⢀⣠⣶⡶⠿⠛⠛⠷⠆⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⡏⠀⠀⠀⠀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⣿⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠇⠀⣾⡇⠀⠀⠀⠀⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⢠⣿⠀⢠⣿⠁⠀⠀⠀⠀⣾⠿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⣸⣿⣿⡿⠿⠛⢻⣿⠀⠀⠉⠛⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⢸⣿⡇⠀⠀⠀⢀⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⣾⣿⡟⠁⠀⠀⢠⣿⡏⠀⠀⠀⢸⡿⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠸⣿⠿⠿⠷⣶⣾⣿⡟⠀⠀⢠⣿
⠀⠀⠀⠀⠀⣾⣿⣏⣿⠃⠀⠀⠀⠘⢿⣷⣤⣤⣀⣼⡇⠀⠀⠀⣿⣷⠀⠀⠀⠀⢀⣿⣿⣀⣤⣤⣶⣿⡄⠀⠀⢀⣿⣿⢇⣄⠀⢸⣿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣄⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⣸⣿⡏⣼⡏⠀⣸⣿
⠀⠀⠀⠀⠀⠈⣿⡘⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢀⣿⡟⠀⠉⠀⠀⣿⣟
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠹⣿⣿⣿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣀⣀⣤⣤⣶⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⣿⠾⠿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣆⠀⠀⠀⠀⠀⣾⣏⠀⢙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠛⠛⣿⡿⠿⣿⣿⣿⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣇⣀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣤⣀⣀⠉⠉⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣤⡾⠟⠛⠛⠛⠛⠓⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⢠⡾⠋⠀⠀⣀⣴⣾⠿⠟⣴⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢠⣿⠃⠀⢀⣾⡟⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣼⣿⣀⣰⡟⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠸⠿⠿⠇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣾⡇⢠⣶⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⢸⣿⠁⣾⡟⠀⣿⠀⢿⣿⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⢻⣿⠁
⣿⡏⢰⣿⠇⢰⣿⠀⠀⠈⠙⠛⠿⢿⣿⣿⣶⣦⣤⣄⣀⣀⠀⠀⠀⢀⡀⣀⣀⣠⣤⣤⣴⣶⣶⣿⣿⠿⠟⠋⠁⠀⠀⠈⣿⡇
⠛⠁⢸⠟⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⢀⡀⢀⠈⣇⣀⣀⣀⣴⣛⠛⢷⡞⠛⣳⣆⣀⣀⣀⡏⢀⠀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠘⠃⠙⠀⡏⠉⢩⡟⣩⣿⠀⠘⠇⢸⣿⣍⢿⡍⠉⡇⠘⠁⠛⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠸⠇⠺⠀⡇⠀⠈⠛⢃⣼⠀⠀⠀⠸⣇⠙⠛⠁⠀⡇⠸⠂⠿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠰⡆⢶⠀⡇⠀⣠⠞⣋⡵⣶⡶⣶⡶⣍⡛⢦⡀⠀⡇⢰⠄⡶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢠⡄⣴⠀⡇⠀⠘⠛⠉⠀⢿⣀⣸⠇⠈⠙⠛⠀⠀⡇⢰⠀⣦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢠⡄⣠⠀⡇⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⢠⡀⣤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢈⡁⢈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠁⣉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠘⠃⠙⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠁⠛⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠸⠇⠺⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠂⠿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠰⡆⢴⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠆⡶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢠⡆⢴⠀⡇⠰⠞⠳⠶⠓⠶⠚⠲⠖⠳⠶⠛⠶⠀⡇⢰⡄⣦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢀⢀⣀⣀⣇⣘⣋⣛⣛⣙⣛⣋⣛⣛⡙⠛⣋⣛⣀⣇⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⡩⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢀⠞⠁⢠⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⠐⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠄⠀⠠⠴⣊⠀⢶⡔⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⢘⠓⠀⡄⠀⠀⠀⠀⠀⠀⡸⠀⣸⠃⠀⡤⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠂⠘⠢⢤⡀⠀⠀⣠⠇⠀⡡⠟⣁⡀⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⢰⠋⠀⠀⠀⠁⠈⠁⠉⠀⠀⠁⠀⠀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡷⡀⠘⠓⠒⠀⠀⠀⠀⠀⠀⠐⠒⠚⠃⢠⣰⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⠀⡃⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⣰⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⢱⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣲⣿⣿⣿⣷⣤⣄⣀⣠⣴⣾⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⣸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⢀⠁⠀⢸⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣯⣟⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡁⠀⣀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠊⠀⠀⠈⠀⠀⠀⠉⠀⠀⠈⣧⠠⠤⠔⠒⢒⠶⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠠⣉⣡⣀⣁⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠠⠋⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠠⠤⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠃⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⡇⢠⠃⠀⠀⠀⠀⠀⡰⠃
⠀⠀⢀⠀⠐⠈⠑⢀⣴⡁⠀⠀⠠⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠂⠀⠀⠀⠀⠀⡜⠁
⠀⢀⠔⠒⠶⢄⠀⡜⠠⢸⠀⠀⠄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡼
⠀⡌⠀⠀⠀⢠⠁⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⡤⠆⠀⠠⠋⠀⠐⠊⠈⠉⠉⠑⢤⡀⠀⠀⢰⠁
⢠⠁⠀⠀⠀⡆⠀⢘⠂⡀⠀⡀⠀⠃⠀⠀⠀⢠⠃⠈⠁⠀⠀⡇⠀⠉⠉⠉⠉⠉⠀⢹⠀⢀⠇
⡌⠀⠀⠀⢠⠃⠀⢨⣀⡡⠂⠄⠀⣂⡀⠀⠀⣸⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠊⠁⠈⠉⠉⢹⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇
⠣⢄⣀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠇
⠀⠀⠀⠉⠀⠉⠒⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢰⣃⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠤⠤⠤⣇⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⡤⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣩⣯⣻⣿⡟⠈⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡟⠁⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡅⢀⠙⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⢠⡀⠀⠀⠀⠀⠐⠳⠻⠿⠟⣺⣿⡛⣫⣧⣬⣥⡎⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠙⣦⣤⣀⣀⣀⡀⠀⠀⠀⠈⠙⠙⠻⠿⠿⢿⠳⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠳⠀⠀⠀⠀⠀⠛⠷⣄⠀⠀⠀⠈⢣⣄
⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠓⠦⡀⢰⠙⢺⠈⠀⠀⠀⠀⠀⠀⠁⠒⠤⣀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⡄
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⢿⣷⠄⠀⢀⣀⡀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠒⠘⠁⠉⠉⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⣏⣀⣤⠤⠀⠀⢀⣀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡗⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡿⠿⣿⣿⡏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⡿⠀⠀⠔⠒⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⢿⡏⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡰⠄⠀⠀⠀⠀⠀⠀⢀⡈⣠⣤⣀⣀⣤⣶⣶⣤⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠖⢦⡀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠸⣿⣦⣰⣶⣷⣶⣶⡆⡶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠹⠶⣶⠤⠤⠤⠀⠀⠀⠀⠀⠀⠁⠈⠛⠛⠛⠻⠙⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠇⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠳⠀⠀⠀⠀⠀⠀⠃⡷⡄⠀⠀⠀⠀⠱⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠒⠦⣀⠀⡜⠳⣇⠉⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣤⣀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⠓⠮⢅⡀⢺⠒⠶⠤⠄⠀⠀⠀⠊
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠸⠤⠤⠴⠒⠂⠀⢠⡤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇
⣠⣄⣠⣤⣠⣸⣤⣤⣄⣄⣄⣼⣄⣤⣠⣤⣤⣠⣤⣤⣤⣤⣤⣀⣤⣅⣼⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣄⣤⣄⣤
⣿⣿⣉⣻⣉⣏⣿⣋⣿⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠚⠛⠋⠉⠲⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣡⡶⠈⢁⠸⠿⣶⣶⣀⣤⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⣿⠟⣩⠀⡀⠖⠒⠚⣛⣉⣬⣿⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡁⠾⠃⢈⣡⡾⠟⠛⠓⠛⠻⠛⠉⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⡠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢁⡽⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⡎⣁⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⣿⡇⢠⡇⠀⠀⣠⣴⠶⠛⠋⠀⠀⠀⠿⠢⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣏⠹⠇⣸⡇⠀⠈⠁⢠⣤⡤⠀⠀⠀⢶⣶⠦⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⣧⣰⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠉⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠈⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⣢⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⢿⣦⡶⢧⠀⠀⠀⠀⠀⠀⠀⢀⣦⣠⣄⡀⠰⠀⠀⠀⣠⡖
⠀⠀⠀⠀⠀⢀⡴⠿⡉⠀⣰⠟⣧⠀⠘⣆⠀⠀⠀⠀⠀⠀⠈⠹⣯⣽⠁⠀⢀⡴⠟⠧⠶⠀⠀⢀
⠀⠀⠀⢀⡴⠋⠀⠀⢠⡼⠃⠀⢿⣧⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⡐⠋⠀⠀⠀⠀⠀⢀⡠
⠀⠀⣠⠟⠀⠀⠀⠀⠈⠁⠀⠀⢸⡿⣆⠈⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠞⠛⠙⠁
⠀⣴⠏⠀⠁⠀⠀⠀⠀⣤⠀⠀⠸⣇⠹⣧⡀⠈⠛⠿⣷⣶⣤⡀⠀⠀⠀⠀⠀⡉⠷⠶⠶⣤⡀⢄⣀⡀
⠈⠁⠀⠀⠀⣠⣴⠾⠋⠙⡆⠀⠀⢿⡄⠈⠻⣶⣤⡀⠀⠈⠙⣷⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠙⢦⣤⡄⢀⠔⠈⠉⠲
⠠⠤⢤⣤⡾⠋⠁⠀⠀⠀⢠⠀⠀⠈⢿⡀⠀⠈⠙⠿⣦⡀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢏⠙⣿⠁
⠀⢀⡀⠈⠻⣦⡀⠀⠀⠀⠘⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣦⡴⠎⢸⠇⠈⠁
⠀⠻⣿⣦⠀⠘⣷⡄⠀⠀⠀⠙⡀⠀⠀⢻⣆⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠻⣷⠀⠸⣿⠀⠀⠀⠀⢣⡀⠀⠀⣿⣧⡀⠀⣠⣿⡇⠁⠀⠀⠀⠀⠸⠃⠀⠀⠀⢸⡇⠀⣽⡇⠀⠀⠀⢀⠈
⠀⠀⠀⠀⢹⣇⠀⣿⡇⠀⠀⠀⠘⡇⠀⠀⢸⡟⢿⣶⣿⣿⠁⠀⠀⠀⠀⠀⣤⠀⠀⠀⢀⡞⠀⠀⣿⡏⠒⠒⠂
⠀⠀⠀⠀⠀⣿⡀⢸⣿⠀⠀⠀⠀⠇⠀⠀⠈⣿⠈⣻⣿⠁⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣼⠁⣠⠾⠈⠇
⠀⠀⠀⠀⠀⢸⣧⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢻⣾⡿⢉⢷⡄⠀⠀⠀⣠⠎⠑⠀⠀⠸⣇⠘⠁⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠟⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠹⣦⠙⢆⠀⢰⡅⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⢀⠀⠀⠀⢾⣿⠀⠀⡇⠀⠀⣤⣼⡟⠀⠀⠀⠹⣧⡈⠳⣹⣇⠀⠀⣴⡾⠟⣿⠀⠀⠀⠀⠀⠸⣧
⠀⠀⣀⣀⣀⣀⣀⣀⣸⣿⡆⠀⠀⠀⢀⣿⠋⠀⠀⠀⠀⠀⠘⢿⣤⣿⢿⣆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠚⠛⠋⠉⠛⣉⣿⡿⢻⣇⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⣇
⠀⠀⠀⣠⣾⣿⣏⣥⣶⠆⢻⡄⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⢰⣿⣿⡿⠛⣩⠋⠀⣸⣷⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠄⢀⣾⡿⠁⠀⠀⣾⡇⠀⣀⣀⣀⣀⠀⠀⢻⡄
⠀⠀⣼⠿⡟⠁⡘⠁⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⣶⢆⣾⡟⠀⠀⠀⢠⣿⣤⠾⠛⠛⠛⠛⠛⠆⢸⡇
⠀⠀⡟⠀⣤⡄⠃⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡟⠲⢁⣾⡟⠀⠀⢀⣤⣾⠿⠷⠆⠀⠀⠀⠀⠀⠠⡀⢿
⠀⠐⠀⢰⣿⠛⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⢁⣾⡟⢀⣠⣴⠟⠉⠀⠠⣤⣀⣀⣀⠀⠀⠀⠀⠁⣿
⠀⠀⠀⣾⡿⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠿⠛⠃⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢠⣿
⡀⠀⡀⣿⠃⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠀⠀⠀⠀⣸⣿
⠁⠀⠁⣿⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠛⠶⢤⣤⠀⠀⣠⣿⠇
⠀⠸⡄⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠛⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⢠⡀⠀⠀⠀⢀⣼⡿⠋
⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠋⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⢀⠈⢦⠀⠀⠀⠀⢀⣀⣀⣀⣤⣾⣿⠿⠋
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⢸⠀⠀⢃⡀⣠⣾⠟⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⢹⡇⢁⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠀⠀⠀⠀⢀⣿⣦⣴⣯⣿⠏⠁
⠀⠀⠀⠀⠈⢿⣄⠣⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⣀⣴⠟⠉⠀⠈⠉
⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⣀⣴⣿⢛⣁⣤⣤⣴⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⡿⠀⠀⡆⠀⠀⠀⠤⠤⠤⠄⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣠⣤⠴⠒⠋⠉⢁⣀⠤⠄⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣄⣀⠀⠀⢀⣉⠉⠙⠳⢦⣄
⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⣶⡋⠁⠀⠀⠀⠀⠀⠀⢴⡄⠀⠠⠤⠀⠀⠀⠀⠀⠀⠈⠙⣶⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⠶⠶⠦⣤⣤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠁⢀⣀⣀⣤⣴⠞⠋⡀
⠀⢠⣭⠿⣿⠷⠦⠤⢤⣤⣤⣤⣄⣠⣤⣄⠀⢀⣀⣀⣀⣀⣠⣀⣤⠤⠤⠤⠴⠖⠒⠚⠋⠉⠻⠋⠁
⠀⠀⢀⡆⠉⠙⠓⠒⠒⠠⠴⠶⠶⡤⠄⠀⠀⠴⠲⠐⠀⠀⠀⠀⠤⠖⠠⠀⠀⠀⠉⠀⠀⣀⣀⣤⡤⠴⡖⠉
⠀⠀⠈⠇⠐⠒⠶⠤⣤⣤⣤⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡴⠶⠶⠞⠛⠋⠉⠉⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠃⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀⠀⢸⠇
⠀⠀⠀⠀⢿⡔⠀⠁⠀⠀⢀⠤⠤⠤⠄⠀⠀⠀⢀⣀⣤⡤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠁
⠀⠀⠀⠀⢹⡟⠦⢤⠀⢀⡎⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠉⠐⠈⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠉⠀⠀⠀⣐⠶⠖⠷⠦⣤⢄⣀⣀⣀⣠⣤⣤⣤⣤⣤⡔⠚⠒⠉⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠠⠒⠉⠀⠀⠀⠀⡀⠈⠀⢀⡴⠋⠁⠀⠀⠤⡄⠀⠉⠉⠲⢄⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⢻⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠊⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⣧⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠰⠁⠀⢸⡀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠈⠂⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⢀⣀⡀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠤⠐⠂⠉⣁⣤⠌⠒⠒⠲⠖⠛⠁⠀⠀⠀⠐⠀⠀⠄⡁⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣄⡀⠀⠘⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⣠⠟⠁
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⣤⣤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠛⠁
⠀⠀⠀⠐⢧⣄⣀⣐⠂⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠳⠶⠶⠶⠖⠒⠒⠊⠋⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣛⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠙⠛⠛⠛⠻⠿⠟⣳⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠲⠄⠀⠠⠴⠶⠶⣄⠀⠈⢮⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⢴⣤⠀⠀⠠⣤⣄⠀⠀⠀⠀⢻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢠⠏⠀⠀⠀⠉⠀⠀⠀⠀⠀⣾⡿⢻⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⢤⡤⢤⣄⡀⠀⠀⠀⣿⠀⠫⠀⠤⠰⠄⠀⠀⠀⠀⠀⠀⠉⠁⣳⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡖⠋⠉⠀⠀⠀⡠⠤⠈⣽⡆⠀⠀⢻⡀⣴⠦⠤⠤⠄⢰⡀⠀⠀⠀⠀⢠⣄⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⢻⡏⠛⢛⣧⣀⠀⠀⠀⠸⣿⡇⠀⠀⠸⣧⠈⠓⠲⠒⠁⠘⠀⠀⠀⠀⠀⣾⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠋⠉⠉⢹⣷⣅⢸⣿⣿⣿⣧⠀⠀⣿⣿⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠂⠉⠉⠙⣿⣿⢸⣿⣿⣿⣿⡇⠀⣿⣿⣀⣠⣤⣾⣿⣷⣶⣶⡶⠒⠀⠀⠀⠀⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠆⠐⠒⠚⣿⣿⣸⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⡇⢸⣿⡏⠁⠀⠀⣀⣂⣴⠞⢋⣿⡟⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠐⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣷⠈⣿⢀⣠⡤⠾⠋⠉⠀⣠⡾⠋⠀⠀⣾⠟⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢤⣴⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡏⣿⣤⣾⣿⡁⠀⠀⣀⣴⣿⡟⠁⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢄⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢻⣿⠃⠉⠙⣿⡿⠻⣤⠾⢋⣼⠋⠀⠘⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣉⡙⠛⠛⠛⣉⣉⣠⣴⣿⡟⠀⠀⣾⣿⠇⠀⠀⢠⡾⠁⠀⠀⢠⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⠿⠛⠁⠀⣸⣿⣿⠀⠀⢠⠟⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⡟⠀⠀⠋⠀⠀⠀⠜⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⡛⢿⢛⡛⠻⡟⢛⣻⡿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠠⠾⠇⠀⠀⠠⢦⡀⠀⠀⠀⠀⠀⣰⠶⠀⠀⠀⠚⠦⠀⠀⠀⠀⠀⣀⣠⣴⢆⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡄⠀⠀⠀⠀⠉⠉⠀⢀⠈⠆⢀⠀⠉⠁
⠀⢀⠀⠀⠀⠚⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠊⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⢁⣌⠉⠀⠀⠄
⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠈
⠀⠀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠃⠀⠀⠀⠶⣆
⠀⠀⠀⠒⠒⢦⠀⠰⠦⡄⠀⠀⠀⠀⢀⣄⣀⠀⠀⠀⠀⠀⣀⣤⡤⠀⠀⠀⠀⠀⠀⣀⣄⣀⡄⣀⣄⡠
⠀⠀⠀⠀⠀⠀⠆⣀⠀⠀⡀⠀⠀⠀⠀⡈⠉⠛⠛⡾⠚⠛⠉⢀⠄⠀⠀⠀⠀⠀⢀⣹⣷⡅⡀⣹⣿⡆
⢀⡀⠀⠀⠀⠀⠀⠉⠀⠠⠀⠀⠀⠹⡀⠈⠀⠀⠈⠀⠈⠀⠀⠀⠀⠄⠀⠀⢉⠀⠀⠈⠉⠀⠉⠀⠀⠈⢀
⠀⠣⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⡀⠀⠄⠈⠀⠀⠀⠀⠠⡄⣀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⢠⠁⠐⠀⠐⠀⢡⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢡⠨
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠅⠘⠙⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢀⣀⣀⡀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠘⠳⠀⠀⠀⠀⠀⠀⠞⠛⠀⠀⠀⢠⣤⡄
⠀⢀⠉⠛⠛⠺⠗⠻⠋⠁⡀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠒⠀⡆⠐⠒⠀⠀⠂
⠀⠀⠁⠀⠂⠁⠀⠁⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠉⠠⠌⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡠⠤
⠀⠀⠲⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⢀⠀⠔⠁⠀⠀⠀⠀⠀⠘⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠠⠀⠀⠀⠤⠤⠀⠀⠀⠤⠠⢧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⡇⠤⠄⠤⠄⠤⠄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢠⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⡗⠒⠒⠒⠒⠒⠒⠚⡄
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠸⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠈⠱⡥⠤⠤⠤⠤⠤⠤⢤⠆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⡤⡄⠀⡠⢄⠀⣠⢄⠀⢠⢄⠀⢀⠤⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⢸⠀⡇⢸⠀⠐⢸⠀⢸⠀⠂⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⢸⠀⡇⢸⠀⠀⢸⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⢸⠀⡇⢸⠀⠀⢸⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⢸⠀⡇⢸⠀⠐⢸⠀⢸⠀⡀⠀⢸
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠲⠃⠀⠳⠂⠀⠳⠎⠀⠑⠞⠀⠐⠞⠀⠘⠖⠁⠀⣸
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣛⣿⣿⣻⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⡁⢰⠀⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠇⠘⠁⢸⠘⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠠⡄⠠⠀⠀⠀⢠⡄⠀⢠⡄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢹⡀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⠀⠰⣃⠀⠀⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠳⠀⠀⠀⠸⠇⠀⠸⠇⠀⠄⠀⠧⠀⠄⠀⠀⠀⠀⠸⠀⠀⠞⠀⠀⠇⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⠃⡇⠈⠇⡔⠀⠁⡰⢀⡆⠈⢀⡃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠳⠀⠃⠀⠀⠃⠀⠀⠃⠈⠁⠀⠘⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢄⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⠄⣠⠀⠄⠀⢀⠀⠀⢠⠄⠀⢠⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢼⡃⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⢸⠀⠀⢸⠀⠀⢸⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⣼⠀⠀⢸⠀⠀⢸⡄⠀⡀

⢀⣤⡤⢤⣄⡀⠀⠀⢀⣀⠤⢄⣀⠀⠀⢀⣀⣀⡀⠀⠀⢀⣀⣀⠀⣀⡀⠀⠀⢀⣀⣀⠀⠀⣀⣀⠀⢀⣀⣀⣀⣀⠀⠀⢀⡠⢤⣄⠀⠀⣀⠠⢤
⠀⣿⡇⢀⣽⡿⠀⣰⣿⠁⠀⠀⢻⣷⡀⠀⣿⡇⠀⠀⠀⠀⣿⣏⠀⠀⢹⣷⡀⠀⡟⢻⣷⣄⠀⡇⠀⠀⣿⡇⠀⡈⠁⠠⣿⣤⣀⠉⠀⢸⣿⣄⣀
⠀⣿⡏⠉⢻⣶⡀⢻⣿⠀⠀⠀⢸⣿⠃⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠃⠀⡇⠀⠙⢿⣶⡇⠀⠀⣿⡏⠈⠃⠀⠀⠈⠛⠻⣿⡄⢀⠉⠛⢿
⠠⠿⠧⠤⠼⠟⠀⠀⠙⠷⠤⠠⠞⠋⠀⠠⠿⠷⠤⠴⠇⠠⠿⠷⠤⠴⠟⠁⠀⠠⠷⠄⠀⠈⠻⠇⠀⠴⠿⠧⠀⠴⠇⠘⠧⠤⠴⠛⠀⠸⠤⠤⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠀⢠⠀⢀⠀⠀⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠘⡆⠀⢸⠀⠄⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠄⠀⠀⣿⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠼⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣧⠀⠀⢼⠀⠀⠄⠀⣿⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢀⠀⡀⢀⠀⢰⠂⢸⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⠰⠃⠸⠀⡎⠀⡟⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠤⠤⠶⠶⠖⠚⠋⠉⠉⠉⢉⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⢤⣀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠖⠛⠉⠁⢀⣀⣀⡴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠉⠓⠋⠀⠀⢀⣠⣤⠤⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⢠⣚⣩⠤⠶⠛⠉
⣀⠤⠤⠶⠶⠲⠦⢤⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠶⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠴⠒⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠚⠉⠉⠉⠋⢳
⠀⣀⣠⡤⠤⠶⠖⠛⠉⠉⠁⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢷⠿⡗⠚⢛⣛⣛⣛⣛⣲⡄
⠉⠁⠀⠀⠀⠀⣀⡤⠶⠒⠒⠚⠋⣉⣀⡤⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢺⠃⠈⢹⡜⣍⠉⠉⢻⡌⣧
⡤⠴⠖⠚⠋⠉⢁⣀⡤⠴⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣹⠦⡄⠀⢳⡌⠻⡇⠘⡇⢹
⣀⣀⡤⠶⠒⠋⠉⠀⠀⠀⠀⢀⡠⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡼⠁⡤⠟⠀⠀⠈⠳⢿⡟⢻⣼⣤
⠉⠀⠀⠀⠀⠀⠀⡴⠒⣋⣭⠵⢿⢸⣉⠉⣴⣄⣀⠀⠀⠀⠀⠀⠀⠀⡾⠘⠀⡼⠁⠀⠀⠀⣀⡤⠈⣧⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣻⡧⠐⣆⣩⣀⢼⣆⣇⢸⡎⣹⠀⠀⠀⠀⠀⢰⡇⠀⣸⠃⢀⡤⠖⠋⠁⠀⠀⡿⣆⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣠⡖⢋⣹⠄⠸⣧⢌⡙⠿⣟⢳⡄⠀⠀⠀⠘⣇⣰⢿⣤⣤⣤⣤⣤⣤⣤⣤⡟⠛⢦⡀⠙⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⢁⡔⢹⣿⠀⠀⠹⣌⣛⣦⣌⠀⢙⡄⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠻⡇⠀⠀⠈⠠⣄⢱⡄
⠀⠀⠀⠀⠀⠀⠀⢧⣴⣻⡿⠞⠋⠀⣀⡤⠀⠁⢹⠈⠙⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣴⣦⠀⠀⠀⡇⠀⠀⠀⠐⢺⠾⠇
⠀⠀⢀⣠⠖⠓⠲⢄⠈⢻⠇⠀⠖⠋⣁⣀⣤⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣆⠀⣿⢹⠀⠀⠇⡇⠀⠀⠀⠀⢸
⣠⡶⠋⡀⠀⠀⠤⠴⠀⡸⠶⠞⠛⠋⠉⠉⠁⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠙⠀⣿⠘⡇⢸⠀⣿⠀⠀⠀⠀⡞
⠁⠀⠀⠉⠒⠊⠀⠀⡀⣠⠀⢀⡔⠚⡣⠀⠀⡀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿⠀⣧⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⢤⡴⠄⠀⠀⡼⢡⡏⢀⡞⠀⠀⡇⠀⢠⡇⢸⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣿⠀⢻⣤⣄⣸⠀⠀⠀⢸⠁
⠊⢡⡄⠠⠦⠀⡘⠁⠸⠁⡞⠀⠀⠀⡇⠀⠾⠀⢸⡇⠈⠓⠆⣀⠀⠀⠀⠀⠀⣿⣿⡿⣿⠀⠸⡿⠿⢿⡇⠀⠀⢸
⠀⠈⠰⠠⡀⠀⡇⠀⠀⢠⡇⠀⠀⠀⣇⠀⠀⠀⠈⢧⠀⠀⠀⠈⠁⠀⠀⢀⢴⡷⠀⠀⣿⢠⡄⢷⡀⢶⣿⡄⠀⡇
⠀⣀⡀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠘⢧⠀⢀⣴⣿⢷⡀⠀⠀⠀⠀⠀⣏⠋⣁⣠⢤⠏⠀⠀⠀⠛⢶⣍⣹⠀⠱⠶⠶⠶⡶⠶
⠘⠁⠰⠖⠲⠀⣇⣀⣀⣸⡇⠰⠒⢄⣀⠈⢳⡿⠛⠁⠀⣧⡀⡄⢀⡴⠚⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠓⢦⡀⣀⣀⣀
⣀⣀⣀⣀⡀⠀⣿⠿⠿⠻⡇⠀⠀⠈⠉⠁⠀⠹⣤⡄⣠⡏⠄⣰⠿⢤⣄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⠀⠀⠀⠳⡙⠛⠿
⠉⠉⠉⠉⢀⣤⡋⠀⢀⣠⡇⠉⠙⠛⠛⠓⠂⠰⢭⠴⣧⠶⠋⠀⠀⠀⠈⠛⠉⠉⠉⠓⠦⠤⣾⣀⣀⡀⢀⣀⠀⠃⠀⠀⠀⠀⠛
⠿⠿⢟⣆⠈⠛⠛⠛⠉⠙⠀⠘⠿⠿⠂⣀⣀⣀⣠⠞⢁⠀⠖⢾⣭⠭⠝⠓⠶⢤⣀⣤⠤⠤⠤⢤⣤⣀⠀⠉⠤⠶⠤⡄
⠀⠀⢤⣿⣿⣿⣿⣿⠿⠿⠗⠒⠒⠛⠉⠓⠙⠛⠛⠚⠫⠙⠂⠀⠐⠲⠲⠦⠀⠀⠉⣷⣄⣀⣀⣤⣤⣬⣽⣤⣤⣶⡾⠋⠉⠙⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢀⣀⡀⠀⠀⠀⠀⣄⣠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣤⡴⢶⣶⣶⣤⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣠⣴⠶⠛⠛⠛⠂⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠈⠙⢿⣧⡀⠀⠀⠹⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⢀⡾⠁⠀⠀⢀⣤⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⣀⣴⠾⠟⠛⠻⠷⣦⣄⠀⠀⠀⠘⢷⡀⠀⠀⣸⠁⠀⢀⣴⠟⠋⠁⠀⠀⠀⠈⠉⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠘⣇⠀⠁⠇⠀⢠⠞⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠳⡀⠀⠸⡀⠀⠀⠀⠊⠀⢀⣴⡾⠿⠿⠿⢷⣦⣀⠀⠀⠀⠀⠀⠀⣀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⠷⠶⣤⡀⠀⠘⠄⠀⠃⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠐⠂⠀⠀⠉⠲⠄⠀⠀⠀⣼⣾⣀⠞⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⣰⣶⡀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠴⠚⠉⠙⠛⢿⣦⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠃⠀⠀⠀⡼⠁⠀⠀⠀⠀⣠⣿⠟⠉⠀⠀⠉⠀⠀⠀⠸⠧⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠹⣧⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠸⠁⢠⣤⠀⠀⢰⡿⠁⠀⠀⠀⢠⣶⠶⠆⠀⠀⠀⠀⠈⠁⠈⠙⣷⡀⠀⠀⠀⠀⢹⡆⠀⢠⣿⣿⡄⠀⠀⠆
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⠁⠀⠀⠀⢠⡟⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠁⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠼⡿⠆⠀⠀⠸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠀⠀⢀⡀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠁⠀⠀⠀⠀⠁⠀⠀⣠⡄⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣴⣶⣾⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠛⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⣠⠒⢆⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢟⣭⡻⣿⣿⣿⣿⣿
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣾⣤⣤⣤⣷⡄⠀⠀⠀⢸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠛⠛⠛⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠙⢿⣿⣿⠟⠀⠀⠀⢸⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠀⢀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣠⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣳⡿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠿⠟⣋⣽⣿⡿⣿⡄
⠀⠀⠀⠀⠀⠞⠋⢁⣡⣴⣴⣿⣿⣿⣿⣷⠘⣷⡀
⠀⠀⠀⣠⣴⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⠆⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⢀⣴⠂⠀⣴⣀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⠃⠀⠠⣈⢛⣿⠂⠁⠀⢀⣀⠲⣾⣿⡟⣿⣶⠊⠉⢀⡀
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠁⠀⠀⠘⠛⠉⠙⠂⠀⡃⠀⠀⢀⣈⣯⡾⣿⣿⣿⠠⡾⠃
⠀⠀⠀⢹⣿⣿⣿⣿⠟⣛⠻⢿⣿⣷⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠠⡋⣡⣤⢀⣠⣶⡿⢿⣿⡿⠟⢠⣷⠁
⠀⠀⠀⠘⣿⣿⣿⣿⠘⠉⠓⠀⠙⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣑⡟⠛⠋⠂⠀⠳⠈⠁⠀⠀⠙⢃⠀⠠
⠀⠀⠀⠀⢻⣿⣿⣿⣧⡀⠚⠻⠀⠀⠀⠀⠀⠁⠀⠁⠄⠀⠀⠀⠀⠀⠉⣥⠄⠀⠀⠠⠄⠀⣀⣀⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣶⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠳⣶⡄⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⠉⠻⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣟⠀⠀⠀⠙⠓⠶⣶⣶⣤⠔⣀⡀⠰⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠀⠈⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠐⢿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠓⠒⠢⠤⠒⠋⠉⠀⠢⣤⣄⡀⠀⠀⠙⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⠲⠤⢤⣄⣀⣀⣀⣀⣀⣀⣀⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀
⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠉⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠁⠀⠀⠠⠐⠀⠀⠂⠐⠀⠀⠈⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠉⠉⠉⢉⣭⠍⠛⠛⠩⣭⠉⠉⠉⠉⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⣠⢴⢠⡄⠀⠀⠀⠀⢠⠄⡦⣄⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢠⠀⢠⢠⡄⠀⠀⠀⠀⢠⡀⡄⠀⡄⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡬⠬⢭⢠⡭⠭⠭⠭⠭⢭⠀⡭⠩⢥⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢨⠭⢭⢠⡭⠭⠭⠭⠭⢭⡄⡭⠭⡅⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢣⠀⢩⢭⢠⠭⠭⠭⠭⠭⠭⠀⣭⡍⠀⡞⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⢣⡀⠉⠀⢭⣭⡭⢭⣭⡭⠀⠉⢀⡜⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠢⣄⡀⠀⠀⠁⠀⢀⣠⠔⠋⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠖⢲⠒⠒⠚⠓⠒⠒⢲⠒⡖⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠈⠒⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⣤⡧⠀⡇⣼⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠁⠀⠀⣤⢸⢸⠁⠀⡃
⣿⡏⢸⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢁⣹⡄⠀⡆⠀⡇⠸
⣽⣁⣸⡏⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢙⠉⢣⠀⡁⠀⡇
⠛⢿⣿⡧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠸⠀⠈⠀⠁⠀⡇
⠀⢠⣿⣧⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⡆⠀⠀⠀⠄⠀⣧
⠀⢸⣿⡿⢀⠸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢁⠀⠀⠀⠀⠀⡇
⠀⠸⡟⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇⠘
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⢰⢸⡏⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠈⣿⣇⢸⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷⢽⣏⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⣇⡀⠀⠀⠀⠀⣷⣄
⠀⠀⢹⣿⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣛⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⡇⠻⠀⠀⠀⠀⣿⣿
⠀⠀⠘⣿⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⡇⡇⠀⠀⠀⠀⣿⠛
⠀⡖⠀⣇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⠁⠉⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡇⠀⠀⠃⠁⠀⠀⠀⠀⡇
⠀⠀⠀⣿⢸⠻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠚⣘⣿⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⡆⠀⣀⣀⣀⣀⡇
⠀⠛⠀⠃⢸⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠾⠉⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⡇⠀⢁⣽⢹⡍⡇
⠀⠐⠀⡇⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣠⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⡇⠀⢸⣯⣿⡟⡇
⠀⠀⠀⡔⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⡇⠀⢸⣿⣿⡧⡇
⠀⠀⠀⣿⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⡇⠀⢹⣿⣿⣧⡇
⠀⢀⣀⣉⣀⡀⠀⠀⠠⠤⠤⠄⠐⠤⠤⠤⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣉⣉⣀⣀⣁⣀⣉⣉⣉⣉⣁⣀
⠀⠀⠀⠁⠀⢨⠀⠐⠂⠀⠠⠄⠀⠀⠐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⡇⠀⠀⡇⡀⠁⣿⣿⡆⡇
⠀⠀⠠⡟⠀⠟⠀⠀⢀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⡇⠀⢠⣼⣿⡇⣇⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠛⠛⠛⠛⠛⠛⠳⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢓⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⡭⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⣉⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠻⠥⣄⡀⣠⠔⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣇⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠿⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠠⠖⢒⣠⣄⣀⡀⢸⠁⣀⡀⠀⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣴⠯⡤⠀⠀⠀⣰⠃⠀⠀⠀⠉⠛⢮⣽⣿⠀⠀⠀⢠⠇⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⣲⣾⡏⠁⢀⣠⠴⢃⡠⠴⠒⠒⢦⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠉⢉⡒⢄⠀⠈⢧⡀⠀⠀⠀⠉⠁⡿⠚⠉⠀⠀⠈⠀⠀⢀⣠⠞⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠙⠀⠀⠀⢙⣦⠤⠶⠒⠾⠁⠀⠀⠀⠀⠐⠒⠋⠉⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⢠⡏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢸⠴⢻⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠟⢦⡀⢀⡠⠖⠛⣉⣉⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢻⡀⠐⠙⠓⠒⠉⢀⣠⠤⢤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣠⡀⢻⡌⢦⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠋⢩⠇⠀⠀⠀⠀⠈⠉⠉⠉⠉⡿⣍⠉⠉⠉⠁⠀⠉⠉⠙⣆⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⡇⠈⠳⣄⡴⣄⠂⠤⢄⡴⠃⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠠⠷⡄⠀⠀⠀⢀⣜⡗⣧⠴⢾⡋⡟⠢⢝⡲⢼⣧⡞⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠘⢦⢀⢼⣅⡀⠉⠙⠓⠲⠛⠧⠤⣀⠈⠙⠚⠻⠿⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⢤⣤⣀⣀⣀⣈⠛⠁⠀⠉⢙⣶⠦⢤⠀⠀⠒⠾⡁⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠈⢹⢹⠉⠉⢯⠀⠀⢈⡦⠤⠴⠊⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠐⠲⠚⠾⠖⠂⣸⠀⠀⠀⠙⠲⠤⣄⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣀⣤⢤⡞⠢⣤⢿⠒⢦⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣗⡽⢮⣄⣀⣙⣉⣼⣴⣻⣴⣷⣺⣿⣶⣶⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣶⡶⠤⣤⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣴⠲⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁⡔⠢⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡧⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⢴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠇⠀⠀⠀⠀⢸⣿⣿⣯⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣭⣭⣽⣭⣽⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⠀⠀⠀⠀⢸⣿⣿⠏⣤⠤⠙⣿⣿⣿⣿⣿⣿⣿⣿⠋⡤⢤⠙⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠉⠀⠀⠀⠀⢸⣿⣿⣆⠉⠀⣠⣿⣟⣿⣽⣿⣿⣟⣿⣄⠁⠀⣰⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠒⠛⠢⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣾⣿⣿⡶⠖⠻⢿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡯⠑⠈⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠼⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣟⣃⣀⣀⣸⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡜⢃⠀⠀⠀⠀⠘⣿⣿⡟⣩⢍⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠛⠀⠀⠀⠀⠀⢻⣿⣆⠛⠀⢀⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⣼⡿⠓⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠓⠚⠛⠛⠚⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠐⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢠⣄⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁
⠀⠀⠀⠀⠀⡆⠀⠐⠀⡂⠋⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠂⡖⠀⠶
⠀⠀⢠⠀⠀⡇⠀⠀⠀⠘⠂⢠⡄⠀⠄⣄⠀⠈⣿⠛⠛⠙⠋⠁⠀⡏⠀⠈⢨⠁⢸⡭⠨⠁⠉⠉⠉⢹⠙⠛⢫⡄⠁⠀⡯⠀⠀⠘⠂
⠀⠄⠀⠴⣼⣧⣀⠤⠄⠤⠠⣰⣄⣄⣠⣼⣀⣄⡿⠠⠀⢤⠰⠠⢀⡅⠀⠀⠠⢴⣼⣇⠠⠼⠀⠄⢤⣼⣀⣄⣤⣤⣠⣤⡧⠀⠀⠀⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⠀⠀⠀⠰⢄⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠲⠂⠀⠀⠀⠀⠀⣠⠏⡐⠶⠃⠀⠀⠀⢀⡐⠂⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⢀⣞⠉⠳⠞⠀⠀⠀⠀⠀⠀⠀⠡⡌⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⡂⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢦⡵⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⠸⠃⠀⠀⠀⠀⠀⠀⠀⣾⠤⣌⠆⠀⠀⠀⢰⠁⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢥⠊⠀⠀⢔⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣤⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⡔⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⢠⡏⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢢⣀⡀⠀⠀⠀⢤⡐⠀⠁⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⡀⠀⠈⠢⡤⠀⢄⠀⠀⠀⠀⠀⠀⠠⣎⢂⢈⡰⠂⠁⠀⠀⠀⠀⢈⡤⠂⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡐⠉⢆⠀⠀⢄⡀⣠⠀⠀⠀⠀⡖⢢⠀⣩⠜⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡐⠀⠀⠀⠑⡄⠀⠀⠀⠑⢤⡈⢆⢙⡸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡠⠒⠣⣰⣁⠁⠑⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡐⠁⠀⠀⡠⠊⠈⠢⡀⠀⠀⡠⣅⡜⠉⠀⠀⠀⢀⠊⠈⠲⠀⠀⠀⠀⢀⣠⢉⠀⠐⠗⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠃⡜⠀⠀⢀⠔⠁⠀⠀⢀⠜⠓⢄⠑⠉⠈⢢⠀⠀⠀⠀⠢⠄⠘⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠃⡜⠀⠀⡰⠃⠀⠀⡠⠖⠁⠀⠀⠀⣁⠦⢀⡀⠡
⠀⠀⠀⠀⠀⠀⡌⠌⠀⡠⠊⠀⢀⠤⠊⠀⠀⢀⠠⠒⠉⠀⣀⠠⠘⠋
⠀⠀⠀⠀⠀⡐⠌⢀⠔⠁⡠⠐⠁⢀⡠⠐⠈⣀⠠⠐⠊⠉
⠀⠀⠀⠀⢰⠏⡠⢃⠴⠊⡀⠄⢂⠡⠔⠒⠉
⠀⠀⠀⢠⣏⣨⢔⣡⠔⠍⠒⠉
⠀⠀⢠⡿⠟⠛⠉


⣿⢿⠿⡿⡿⠛⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣟⣛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⠒⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠈⠀⠠⠇⠘⠙⢲⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡄⠀⠀⠀⢀⠔⠋⠉⠋⠀⠀⠀⠉⠁⠉⢦
⠀⠀⠀⠀⣠⠞⠃⠀⠠⡄⠀⠀⠀⠀⠀⠀⠺⡄⠀⠀⣠⡤⠼⠷⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠗⠀⠀⠀⢸⠀⠀⠀⣀⣀⣀⣤⣀⣀⡀⢈⡇
⠀⠀⠀⠀⢻⢀⣤⣶⣶⣶⣶⣦⠀⠀⠀⠰⢠⠇⠀⢰⠃⡔⠲⢤⣿⣒⡓⠦⣀⠀⢀⣾⣺⡇⠀⠀⠀⠀⢸⠇⢀⣿⣿⣿⣿⣿⣿⣿⣟⢉⣀⣴⣿⣧
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⢀⡼⠀⠀⢸⠰⡃⠀⢸⠀⠉⠙⠉⠛⠛⣾⣫⢹⡇⠀⠀⠀⠀⠘⢦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣻⡇
⠀⠀⠀⢰⣿⠉⣿⣿⣿⣿⣿⣿⣿⣶⣦⡴⠃⠀⠀⢸⠀⡇⠀⢸⢠⠀⠘⠀⣾⠀⣧⠿⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡋⠁⠀⢿⣷
⠀⠀⠀⠘⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⡄⠙⠒⠻⡾⠀⡇⠀⣿⠀⢨⠀⡎⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀⠈⣿⣧
⠀⠀⠀⠀⠹⠿⣿⡿⣿⡿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠒⠲⡾⡇⣸⠁⠀⡟⠀⢸⠀⣇⣽⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢦⣛⡍⣳⣔⣛⣢⣞⢝⣁⡼⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⠁
⠀⠀⠀⠀⣀⣠⠔⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠲⣄
⠀⠀⠀⡞⠁⠀⠀⠀⠀⠉⠙⠚⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡤⠖⠁⠀⠀⠀⠓⢲⡄⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⠀⠉⠉⣦
⠀⠀⠀⢳⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠄⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⠖⠙⠋⠀⠀⠀⠀⠀⠈⠀⠀⠻⡄
⠀⠀⠀⠘⢢⡴⣺⣍⣶⣳⣤⡴⣤⠀⠈⣣⠀⠀⠀⠀⠀⠀⢸⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠘⣆⠀⠠⠀⢠⣶⣄⠁⢀⣤⡴⠞⠁
⠀⠀⠀⠀⢸⣇⣿⣿⣿⣿⣿⣿⣿⣗⣾⡿⠿⣶⡀⠀⠀⡴⢀⣙⣿⠀⠈⣀⣤⣀⡾⠓⠚⡋⠀⠀⠀⠀⠈⢦⡀⢀⣼⣿⣿⣿⣶⣮⣿⣿⣶⣤⣀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢸⡇⠀⢸⢰⠟⠙⣿⣶⣾⠉⠀⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⣿⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⢸⢹⠀⢸⡇⢹⣿⠀⣀⠀⡆⢠⢸⡇⠀⠀⠀⠀⠀⠀⠳⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⠸⡼⣄⣸⡇⢸⣿⠀⣿⠀⡇⠸⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⣼⡏
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣟⣿⣷⠶⠿⠃⠀⠀⠙⠛⢻⣇⢸⣿⠀⠛⠀⠃⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠶⠤⠤⠴⠚⣉⠇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣯⣿
⠀⠀⠀⠀⠈⠙⠻⠿⠽⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠓⠚⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠐⠐⠒⠈⠁⠀⠀⠀⠱⣀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠾⠁⠈⠉⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⢀⢤⡎⠀⠀⠀⠀⣀⣸⠖⢤⡀
⠀⠀⠀⠀⠀⢀⡖⠙⠁⠀⠀⠀⠀⠄⠀⠀⠀⢈⡆⠀⠀⠀⢀⠤⠔⠀⠀⠀⠀⠀⠀⠀⠄⠀⠅⠀⠀⠀⠀⠀⢀⠴⠃⢀⡤⠀⠀⠀⠀⠀⠤⣀⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠳⠦⣄⣀⣀⣀⣀⠀⠀⠀⢱⠋⠀⠀⠀⠀⢨⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⣨⠀⠀⠀⠀⠀⢸⡀⠰⠉⠀⠀⠀⠀⠀⠐⠄⠈⠙⡶⠃
⠀⠀⠀⡔⢉⣭⣅⢠⡄⣡⡉⣩⣝⣿⠄⣾⠋⠀⠀⠀⠀⠀⢰⠀⠀⠦⣤⢞⣶⣾⣻⣿⡛⠛⠧⠖⠓⢆⠀⠀⠀⢹⡟⢤⣴⣄⣀⡀⠀⠀⢠⣶⠞⣉⢳
⠀⠀⣸⢰⠏⠀⠸⣿⣇⣿⣿⣿⣿⣿⣄⡿⠀⠀⠀⠀⠀⠀⠈⡆⠀⣿⣷⣾⣿⡟⢻⣿⡇⣷⠊⠈⢱⢸⠀⠀⠀⠀⢣⢸⣿⣏⢹⣷⣤⣤⣾⡟⡎⠈⡇⡇
⠀⠀⡏⢸⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠳⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⠈⡇⠀⠀⠀⠘⣾⣿⣿⢸⣿⣿⣸⣿⡇⠀⠀⢃⠇
⠀⠀⢿⡈⠆⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⣧⠀⠀⠀⠀⠀⠀⠀⢠⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⠀⠀⢸⢰⠃⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⢸⣿⡇⠢⠎⡼
⠀⠀⠀⠙⠶⣤⡐⣿⣿⣿⣿⣿⣿⢸⣿⢻⡅⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⠛⢦⡴⠃⣸⠀⠀⠀⠀⠀⡏⣿⣿⢸⣿⣿⢸⣿⡟⡚⠉
⠀⠀⠀⠀⠀⠀⣿⣽⣛⡻⠿⠿⠿⠘⢛⣈⣹⠀⠀⠀⠀⠀⠀⠸⣠⣤⣿⣟⣻⣿⣿⣯⣥⣰⠋⠛⠛⠁⠀⠀⠀⠀⠀⣇⢽⣯⣤⣻⣏⣤⣿⣣
⠀⠀⠀⠀⠀⠀⠹⠽⣟⣉⠉⢁⣀⣤⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠆⠀⠢⠄⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠛⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀⠀⠀⣠⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡄⢰⠟⣡⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠁⢀⣾⠏⠀⠀⠀⠀⢀⣴⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠐⣶⡄⠁⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⣠⡿⠁⠀⠀⣾⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢃⣼⠏⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⠟⢁⡄⠀⠀⠀⡀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢷⣦⡀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢈⠛⢷⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣰⡿⠁⢸⡇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⠟⠀⠀⣿⠁⢿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⢰⣿⡁⠀⠀⠰⠿⠶⠶⠾⠃⠀⠀⣤⣤⠀⣤⣤⡄⠀⣤⣤⠀⣤⣤⡄⠠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠷⢶⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⢠⣤⣤⠀⣠⣤⡀⢠⣤⣤⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⣤⣤⡄⢀⣤⡄⠀⣤⣤⡄⠀⠀⠀⠀⠀⣤⣤⠀⣠⣤⡄⠀⣤⣄⠀⣤⣤⡄⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠀⠀⠉⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠈⠁⠀⠈⠁⠀⠈⠉⠀⠀⠈




⣿⢿⡿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣛⣿⣿⣿⣿⣿⢿⣿⣿