Cockatiel Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢘⠀⠀⠠⠀⠈⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⠈⠀⠠⠃⠀⠸⡀⠖⠁⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⣅⠄⠀⠀⠀⠜⠁⠀⠀⢀⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⡀⠀⠁⠀⠀⣠⠆⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣠⠜⠀⡠⠂⠁⠀⠀⠀⢠⠞⠁⠀⠀⠰⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⣠⡔⢊⠡⠔⠈⠀⢀⠀⠄⠒⠈⠀⠀⠀⣠⠞⠑⠈⠁⡠
⠀⠀⠀⠀⠀⠈⠉⠤⡁⢌⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠘⠋⠀⠀⠂⡁⠄⠊⠀⡀⠄⠐⠀⠀⠀⠀⣀⠠⠔⠂⠀⠀⢀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠙⢇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠃⠀⠀⢀⠠⠄⠈⠀⠀⠀⠀⠀⣀⣄⡁⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠈⢂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠶⠉⠀⠀⠀⠀⢀⠀⠀⠂⠈⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡄⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⠂⠈⠁⠀⠀⠀⢀⡀⡄⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢘⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠠⠤⠀⠐⠒⠒⠀⠉⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠈⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠴⠖⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢸⡇⢠⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠉⢉⠉⠀⠉⠉⠉⠁⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠇⠈⡇⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⡀⠣⢀⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠂⠠⠤⠀⠀⣀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠞⡠⠆⡙⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠆⢠⣄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⠠⠐⠈⠀⠀⠈⠀⠉⣍⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠙⠀⠒
⠀⠀⠀⠀⠌⠀⠀⢀⣠⡀⠈⠉⠁⠈⠁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠁⠲⠒
⠀⠀⠀⢄⡠⢀⣀⡼⠿⠃⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⢤⠃
⠀⠀⠀⠀⠰⠀⢭⡀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠⣀⣀⣀⠃
⠀⠀⠀⠈⠑⣨⠶⡁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠦⣤⠚
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⣤⠠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠐⠲⡘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠇⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣀⠀⠀⠀⠀⠀⢀⣀⠦⡴⢄⠌⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡛⠂⠀⠀⠰⠟⠀⠀⠀⠀⢀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣴⢋⠀⠀⡇⠚⠋⠉⠁⠀⠀⠈⠁⠒⢤⡈⠙⠲⠖⠤⢤⣤⣄⡀⣀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠡⢊⡉⠀⢠⡟⠀⠀⠐⢤⡑⠤⠀⠀⠀⠀⠉⠐⠠⡀⠀⠀⠀⠀⠀⠈⠉⠁⢈⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢁⡀⠈⠀⠸⠈⠃⡆⠈⢡⠀⠈⠳⢈⠀⠂⠄⣀⠀⠀⠀⠛⠞⡛⠒⠚⠛⠓⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢠⠊⠀⠀⢀⠃⠀⠂⢡⠀⠀⠄⠀⠀⢺⡈⠀⡀⠁⠈⠠⢀⠀⠀⠀⠁⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠆⠀⠀⣠⠎⠀⠀⠀⠸⠀⠀⠘⡇⠀⠀⠙⡈⠳⡀⠀⢂⡀⠁⠂⢀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⣠⠈⠀⠀⠔⠇⠀⠀⡄⠀⠀⣤⠀⠀⠠⢃⠀⠀⠁⠀⠈⠀⠀⠑⠢⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠤⠤⣀⠞⠁⠀⠠⠀⡘⠀⠀⠀⠇⠀⠀⠙⢂⠀⠀⠆⠐⡀⠀⠈⠄⠀⠐⢄⡀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⠔⠁⠀⡀⠁⠀⠠⠁⠂⠀⠀⠀⠈⡀⠀⠰⠀⢡⠀⠀⠈⢀⠀⠀⠀⠠⢀⠀⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⢀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⠆⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⢀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⠀⠀⠰⠀⠈⢆⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠇⠀⠀⠁⡀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠢⠀⠈⢁⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⠀⠀⠀⠀⠀⠀⠑⢄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⡠⢺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠀⣰⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡇⠀⢀⠀⡿⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡇⠀⠈⣇⢡⠟⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠇⠀⠈⠞⠀⠀⢀⡀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠝⢛⠟⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢠⠀⠀⡀⠄⠠⠾⠓⠂⠀⠀⢘⣶⣷⠆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⣴⣬⠁⠔⠒⢄⡀⣠⣴⠶⠆⠛⠛⣀⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⣸⠀⠀⠀⠉⠁⣄⠀⡀⠀⠀⠂⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠟⠀⠀⢀⠄⢸⠀⢠⠂⠀⠀⠈⡆⠁⢀⠐⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠟⠀⠈⣇⠀⠄⠀⠀⠀⠀⠀⠈⠓⠶⠤⠶⠈⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⡄⢀⣴⣄⢠⣿⣷⣶⣤⠀⠀⠀⠀⠀⠀⢀⢀⣀⠤⠖⢹⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠐⠛⠚⢁⣠⠾⠻⣿⣿⡿⠛⠟⠛⠒⠶⠾⠶⠾⠛⠋⠋⠀⠀⠀⠀⢻⣿⣿⡿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠙⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡄⠀⠀⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣴⡿⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢠⣀⢀⢀⣠⠠⠄⠀⠀⠀⠀⠀⠀⡀⠻⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠛⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡆⠀⠘⣤⣝⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⠀⠀⠀⡀⢀⡴⠶⠟⠉⠀⢰⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⢿⡀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡿⣿⡿⢿⡗⣿⡿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠟⠈⠁⠁⠀⠘⠁⠙⠁⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠋⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠂⠀⠀⠀⠀⠉⢱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠖⠛⠘⠃⠛⣀⠀⠀⠀⠀⠐⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⣿⣿⣶⣶⣞⣿⠟⠛⢿⠷⢦⣴⡶⠶⡤⠤⠾⠓⠀⡠⢚⠀⠘⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣭⣿⣿⣿⣷⣦⣴⣤⣄⣀⠀⠀⠀⠀⠀⢀⣠⢞⣾⣹⡾⠟
⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠿⠛⠋⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁
⠀⢀⣠⣴⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡽⠿⠿⠿⠿⠿⠿⠯⠁
⠀⠉⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢖⣲⣿⣿⣿⡷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⡭⠖⣊⣩⠤⢶⣻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠁⠊⠉⢵⣶⣾⣯⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢪⡙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡄⢰⣆⠄⠀⠀⠀⢈⣙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢠⣁⠉⠋⠈⠁⢰⡾⡆⣄⠀⢹⡟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣠⡴⠂⠀⠀⠀⢀⣡⣴⠟⠀⢈⣙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠈⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢣⠀⢤⡀⠀⣀⣀⣀⡀⣀⡀⠀⢃⢺⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠈⠛⠛⠿⠿⠿⠛⠛⠛⠀⠀⠀⠀⠀⡛⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠠⠀⠁⠀⠀⢣⡐⠆⠀⠀⠈⢊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⡀⠲⠀⠀⠀⠀⠀⢠⡀⣐⠄⡁⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠄⡄⠀⠀⠀⠀⠀⠀⠢⠈⣵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠐⠀⢠⠀⠀⠀⠀⢀⠙⢮⡙⠷⢶⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⡀⡾⠀⠀⠁⠀⠀⠀⠀⠵⣜⠓⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡄⢀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⡀⠀⠀⠀⠻⠄⠀⠢⢽⣧⣾⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠲⠐⠁⠂⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠤⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⣴⣾⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠻⡿⡿⢻⡄⣷⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⣉⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⢸⡿⠉⢀⣼⠀⠀⠀⢤⠀⢀⡀⡐⠛⠀⠀⠀⠀⢚⡩⠔⠉
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⡯⢣⠀⠀⣀⠀⠀⠔⠂⠈⣽⠃⢀
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡀⠀⠀⣿⣆⡀⢀⣷⠏⢀⠠⠀⠀⢀⣴⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠀⢾⣦⣶⣶⣶⣉⡿⡁⠊⢁⡀⠀⠀⠀⢈⣾⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠉⠋⠉⢠⣤⣠⣴⡟⠀⡄⠀⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⡎⠿⠿⢁⡿⠁⢀⠀⣰⣿⣿⡿⠁⢀
⠀⠀⢀⠔⣠⡶⠇⠀⠀⠀⠀⠙⠿⣶⠦⣸⠇⠀⠀⢠⣿⣿⣿⡇
⢀⠄⣰⡾⠋⡴⠁⠀⠀⠀⠀⠀⠀⠈⢻⡏⠀⠀⠀⢾⣿⣿⣿⣿⣦⣀⠀⡀
⣱⡿⠋⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⣵⡀
⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⡿⣿⣿⣿⡹⣿⣿⣮⠛⢦⡀
⠊⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⠀⠸⠁⢻⣿⣿⣧⠹⣿⣿⣷⡄⠁
⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢠⡏⠀⠀⠀⠀⠹⣿⣿⣷⣸⣿⣿⡿⢆
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⡾⠀⠀⠘⠀⠀⠀⠘⢿⣿⣿⡽⣿⣿⣎⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡝⣿⣯⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡻⣿⣧⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣤⣤⣤⣤⣤⣤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣼⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣾⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠐⣛⠛⠿⠟⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⣶⣾⠻⠿⣿⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣤⠙⠁⠀⠀⠿⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣴⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⣿⣿⣿⣿⣿⡦⠀⠀⢰⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠊⠻⣿⣿⣿⣿⡇⠀⠀⣼⠇⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢰⣿⣿⣿⠿⠇⠀⢸⡿⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠈⠁⠀⠀⣀⣀⣀⣸⣿⣧⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⣿⣿⣿⠂⠈⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡿⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢏⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⠀⣼⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠀⠀⠀⣯⣀⣿⡇⢰⡥⠛⠲⢽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠀⠀⠀⠙⠿⠟⠃⢫⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢜⣅⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⢂⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠓⢢⣶⡞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢸⣿⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⢀⡾⣱⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⣠⢾⣽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⣀⣠⣴⣾⣿⣷⠏
⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⠿⠿⠛⠛⠛⠒⠛⠛⣿⣟⣋⡁⠿⠿⢧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠀⠀⠀⠉⠁

⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠇⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁
⠀⠀⠀⣋⢴⣦⣤⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠤⣶⢀⣤
⠀⠀⠘⣵⣈⠀⠻⠋⠀⠈⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠈⠁⠀⣠⣘⡏
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠘⠦⠄⢤⣠⠀⣀⡀⠸⢏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠀⠈⠑⠦⡀⢧⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠠⡀⡤⠀⠀⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠁⠀⠤⠖⠊⠁
⠀⠀⣀⢀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠱⠘⢢⣀⡀⠀⠀⣀⢒⠀⠀⠀⠀⠓⠢⢄
⠀⠈⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⢀⣀⠀⠀⠀⠈⢷⣠⣿⠻⠟⠀⠀⠀⠀⢀⠴⠒⠿⠭⠋⠿⡳⠒⠢⠤⢤⣀⡀⡀⠀⠀⠈⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⠤⠒⠈⢳⣜⣧⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠛⠷⠈⠃⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠛⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠠⠉⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⣤⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠈⠓⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠘⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠸⠵⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⠁⠀⠐⠢⣤⣤⣤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠄⠠⣄⢠⣦⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣤⡾⠉⠀⡀⠀⠀⠿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠀⠀⠀⠁⠀⠐⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠄⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢤⠀⠀⠉⢁⠐⢤⠀⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠐⣰⣶⠢⡀⡀⢠⠈⠀⠀⢳⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⢀⠀⢿⣇⠤⠘⠇⠀⠀⠀⠃⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⢀⢳⣻⡻⢀⡄⠀⠀⠄⠀⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⣼⠚⠈⠀⠀⠀⢻⣧⠀⠴⠀⠀⠠⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⡴⠚⠉⠈⠀⠀⠀⠀⢀⣾⣿⠀⠀⢀⠐⠀⠐⡡⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠡⣀⠤⠀⠀⡀⠀⠀⠀⡄⣠⣾⠋⠁⠀⢀⡄⣠⠞⠈⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠴⠊⠁⢈⣵⣾⣀⣤⣶⣿⣿⡿⢗⣧⣶⣿⢟⡻⠅⠀⢀⣴⣿⣇⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⣀⣠⣠⣵⣿⣿⠿⠿⠿⠻⠿⠷⣙⣿⢯⠋⠁⡦⢠⡴⣿⣏⡫⠵⠋⠁⠀⢀⡤
⠀⠀⠀⠀⠀⠠⠒⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠓⠚⠐⠚⠋⠁⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡤⠒⠖⠷⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠀⠊⡄⠀⠔⠀⠄⠤⠄⠀⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⢀⠀⠂⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⠟⠙⠛⠳⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⣤⣤⡀⢠⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣠⣴⣶⣤⠈⠋⠁⣸⠙⢭⠇⠀⠀⣰⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠐⣿⠁⠀⣤⣤⢾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⡤⠘⠃⠘⠂⣠⣤⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⡀⠀⠈⠉⠉⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡇⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢠⠓⠲⡿⠁⢾⣷⢀⣤⣤⣀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠁⠀⠀⠀⣀⣰⣷⠀⠀⠁⣿⣿⣿⣿⡆⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡾⣶⣟⠀⠘⠟⣩⠋⠀⠀⠀⠈⠻⠿⠿⠃⠀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⠰⠀⠀⠀⠀⠀⠀⢀⣤⣀⣟⣿⢷⣦⡿⣦⡀⢀⣿⣆⠀⠀⠀⠀⢀⠀⠀⢀⣴⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠚⢁⣽⣏⣽⢊⣶⣿⣿⡆⢸⣗⢸⣿⣷⣶⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣿⡾⢫⣿⣷⢿⣿⠇⢸⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⡞⠛⣳⣿⣏⡿⣎⣿⠀⣼⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢹⣿⣿⣿⡀
⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⣼⣅⣼⣿⣏⣹⣉⣿⢿⡿⢰⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢹⣿⣿⣧
⠀⠀⠀⠀⠸⠛⡇⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣏⡽⢹⣿⠿⡿⢠⡎⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⣿⡟⢻⣇⣼⣿⠟⣠⣾⣅⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣰⡿⠷⠿⠿⠋⣠⣶⣇⣿⡝⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠹⣿⣿⣧
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢰⣿⣤⣤⣴⡴⢿⣧⠜⢉⡿⣿⣧⣄⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠙⣿⣿
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⢠⡈⠛⢙⡏⣉⡵⠈⢁⣴⡿⣱⡏⠀⠙⠃⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠙⣿⡄
⠀⠀⠀⠀⠈⣿⣿⡄⠀⣠⣿⠟⠾⣟⣀⡙⢻⣿⣿⣫⡴⢿⣷⠦⡉⢀⣼⠟⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣼⡇
⠀⠀⠀⠀⠀⣿⣿⡇⣰⣤⠾⡶⠊⡙⢋⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⠸⢇⡞⠁⠈⡙⠿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⠷⠰⣶⡄
⠀⠀⠀⠀⠀⣿⣿⠇⠀⢀⠎⠀⡌⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠷⠀⠀⠀⠉⠉⠉⠉⢣⠀⡴⠋⠉⢉⡉⠃
⠀⠀⠀⠀⠀⣿⠃⣠⡾⢃⡕⢉⣶⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠂⠘⢳⣤⣾⣌⣣⣄⣀⣨⣤⣬⣷⣾⡇
⠀⠀⠀⠀⣀⡩⠊⣡⠔⣩⣶⣿⣿⡿⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⡘⢉⠄⠊⣵⣾⣿⣿⡿⡝⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣌⣻⣿⣿⠈⣿⣿⣿⡇
⠀⠀⠘⢋⡡⣾⡇⣼⣿⣿⡟⠀⢱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣝⣿⣄⣯⢸⣿⣧
⠀⠐⠓⠉⠀⣾⣇⣿⡿⠋⠰⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⢻⣿⠀⣿⣿⠄
⠀⠀⠀⠀⢀⠋⣹⣿⠃⢀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⢹⡇⢹⡇
⠀⠀⠀⠀⠈⠺⠋⠛⢠⠉⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣇⣼⣷
⠀⠀⠀⠀⠀⠀⢸⣶⡏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄
⠀⠀⠀⠀⢰⣶⠃⣿⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⢸⠁⣼⠇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⣤⣴⡏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⢀⠏⢡⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷
⠀⠀⠀⣨⡰⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⢀⡿⢡⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠘⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⣄⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⣿⠏
⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣍⡛⢤⣤⣴⡶⢿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠛⠭⢷⣢⣙⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⣀⣀⣀⡠⠴⠖⣺⡷⠀⠀⠀⠀⠀⠉⣉⡁⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠻⠧⠀⠀⠀⣲⡁⡘⠛⡗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠆⠀⠀⠀⠀⠀⠸⠿⠷⠁⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⠲⠀⠀⠀⠀⠀⠑⠲⢀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠋⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⠷⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣽⣸⡾⠶⡦⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣿⢿⡃⠠⠼⠗⠀⠀⠀⠀⠀⠀⠠⢴⡆⢱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢯⠘⠛⠘⠀⠀⠀⣠⣤⡤⢦⠔⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣴⠀⠀⠀⠀⣰⣿⢿⣤⣶⣷⣷⣯⣤⣼⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦⣽⣤⡆⠀⣠⣾⣿⣾⣿⣿⣿⣿⣯⢏⣀⣀⣀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢉⡩⠿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⠿⢛⡠⣄⠀⠀⣀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣁⣨⣤⠽⠿⠿⠟⠋⣸⣷⣥⠤⠤⠙⡟⡇⠀⣁⣀⣐⣛⣻⣟⡓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠻⠿⠯⣭⢦⡏⣹⣦⡶⠶⢶⣾⠶⠶⠾⠧⠬⠍⠛⠉⠋

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

⠀⠐⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡾⠁⠀⠀⠀⠀⠀⠀⣻⣿⣿⣏⣩
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡾⠁⠀⠀⠀⠀⠀⠀⠀⣿⡈⠉⠄⠹
⠀⠈⠀⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢼⣁⣠⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠊⣻⣿
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⣾⡷⢷
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⢨⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⢻⣿
⠠⢄⡒⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠡⢤⣿⣿
⠀⢀⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠉⢉⣉⠛⠉⠋⠛⠿⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣿⣿⣿
⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⣀⣰⣿⣿⣿⣶⣤⣶⣴⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠿⢭⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠛⠿⡿⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⠀⠀⠀⠀⠘⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠈⠉⠀⠀⠀⠀⠀⠀⠀⢹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⢷⣖⢄⠀⠀⠀⠀⠀⠀⢹⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠈⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠤⢠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠑⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣕⣀⣭
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠿⠆⠠⠀⠀⠀⠀⢰⡟⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡿⠁⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠚⠃⠒⠀⠀⠀⠀⠀⣾⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⠿⠛⠯⠙⠃⠀⠀⠀⠀⠀⠀⠀⠠⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡷⣾⣧⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢶⣼⣿⣽⣧⣶⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣿⣾⠆⠀⠐⣞⣿⡏⢹⣿⣣⠆⠀⠰⣿⣿⠂⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢀⡀⠀⠀⢲⣶⣿⣶⠇⠀⣨⣿⣿⣿⣿⣇⠀⠘⣷⣿⣦⡖⠀⠀⢀⣀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣇⠀⠀⠀⣼⣿⣿⣶⣾⣡⣿⣧⣼⣷⣌⣿⣶⣿⣿⣿⡀⠀⠀⣸⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣖⣺⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣟⣒⣮⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⣾⣿⣿⡿⣿⣿⣋⣿⣿⢿⡽⢿⣿⣯⣻⣿⡟⣿⣿⣷⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣻⣽⣿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢻⢿⣏⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣹⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸⣙⣵⣶⣾⣿⣷⣷⡔⣰⣾⣶⣾⣿⣶⣮⣿⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡏⢿⡿⣿⣿⣿⡿⣳⣼⠻⣿⣿⣿⠿⣿⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢡⡊⢓⡉⣉⡁⢰⣿⣿⣧⢀⣉⠩⡓⢃⣦⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣶⣤⣶⣶⣹⠇⢿⡿⢿⠿⠹⢟⣴⣶⣤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⣇⣀⣀⡀⡀⣀⣀⣬⣹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣶⡦⢤⠥⠤⢶⣿⣯⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠁⠉⠛⠛⠛⠛⠛⠛⠋⠀⢈⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⢀⣈⣀⣀⣠⣀⢠⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣤⣄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⡠⢀⣀⣀⣀⣀⣀⣀⠀⠀⣾⣿⡿⠛⠛⠿⢿⣿⡿⢿⣿⣯⠉⠀⠀⠀⠀⠀⢐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⢀⢈⣾⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠄⣿⣿⣿⠟⠙⠛⠻⠿⢿⣿⣿⡿⣿⠇⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠏⠋⠙⠁⠈⠁⠀⠈⠛⣿⣿⣿
⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠀⠀⠀⠐⠃⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡟
⠀⠀⠀⠀⢸⢳⠀⠀⠀⠠⡄⠀⠀⠀⣤⠀⠳⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢀⡇⢄⣄⣠⠄⠀⡄⠀⠆⠀⠀⠀⠰⠀⠀⠀⠀⣏⠉
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢄⠀⡂⠀⠀⢘⠈⠀⠀⠀⢀⠀⢀⣀⢠⠔⠁⠀⠿⢉⡩⡀⠘⡀⠀⠀⠢⠀⠐⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢸⣿⣷⣤⣀⣀⠀⠀⠀⠀⣀⡠⠂⠀⠀⠀⢀⡜⠀⢢⣀⠂⠀⢀⠂⠀⠀⠈⠂⢁⠀⢤⡀⢀⡀⠀⠀⠠⠀⠂
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟⠤⢻⣿⣿⣿⠀⠀⠀⠀⠣⡈⡠⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠁⠘⠚⠐⠳⠀⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠄⠀⠀⢻⣿⣿⡇⠀⠀⠂⠀⠀⠁⠨⡒⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠢⣜⠂
⠀⠀⠀⠀⠻⠿⠿⢿⢋⠋⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠼⠀⠀⠀⠀⠀⠀⠠⠂⠀⢠⠆⠀⠀⡠⢤⠡⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠏⠀⠀⠀⠀⠀⢠⢀⠀⠀⠂⠀⠀⣀⡀⢀⢠⠀⠀⠦⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠘⠚⠗⠀⠀⠀⠀⠀⠺⠓⠃⠠⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠡⠐⢋⡉⣂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⡠⠂⠀⠀⠂⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠂⠀⠀⠀⠔⠁⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡂⠐⡂⡖⠀⠁⠚⠲⡆⠀⠀⠀⠀⠀⠑⢈⣱⠐⡆⠀⠀⠀⠀⠀⣀⣠⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡅⠀⢩⢅⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠞⠁⠀⠘⠚⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⢀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⣽⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⡸⣹⡿⠛⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⡠⣜⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠻⣾⣷⠏⠀⠀⠀⣿⠀⠀⢰⠇⠀⠀⠀⠀⠀⢀⣴⣿⡿⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣬⣿⣷⡄⠀⠀⢸⣧⣠⣾⣤⣤⣤⣴⣶⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢮⣭⡙⢻⢿⣶⣤⡾⣿⣿⠛⣫⣭⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣥⣞⣿⡿⠁⠘⠿⡿⢧⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠿⠟⠛⠋⠜⠛⣛⣦⣾⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢷⠔⣛⠉⣁⠀⠈⠉⠈⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⠦⠬⣿⣿⣾⣿⣾⣿⣴⣤⣿⣧⣀⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣇⠀⠉⠲⡂⢤⡤⠄⠀⢀⣀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠀⢠⠀⠀⢠⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠐⠄⢰⡀⠀⠀⢠⣷⠈⣤⣤⣾⡾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣇⠀⠀⢰⡄⠄⠀⠈⠻⠤⣿⣿⣿⣿⡻⠂
⠀⠀⠀⠀⡄⠣⣴⢀⣀⢠⡀⢠⣷⡄⢠⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⠀⠂⠸⡇⠀⠀⠀⠀⠀⠀⣻⣿⣿⡅
⠀⠀⠀⣌⣶⣿⣿⣿⣏⠉⡤⣿⣿⣷⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⢀⡄⢰⠈⢀⠀⠀⠀⠀⠁⠈⣿⣿⡷⠂
⠀⠀⠀⡿⣿⣿⣿⣿⣿⣖⣿⣿⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢀⡤⠀⣿⠀⠀⠀⠀⢸⣆⣿⡿⠇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣏⣿⣿⣷⣶⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣥⢀⣧⡴⢂⢀⣤⣸⣿⣟
⠀⠀⠀⠀⠀⣈⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠛⣻⡟⢙⣿⣿⣿⣿⣿⣿⡷⢠⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⣿⣿⣿⣿⡿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠰⠂⠈⠛⠻⡿⠯⠁⠚⡿⢿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠁⠈⠁⠀⠙⠛⠛⠛⠛⠁⠁⠀⠀⠀⠈⠛⠁

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠆⠀⢠⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⠇⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣟⣼⡇⠀⠀⢠⣄
⣿⣿⣿⣿⣿⡏⣠⣤⣼⣿⣿⠟⢀⣴⣿⣿⡿⢸⣾⣿⣿⠟⢻⣿⡇⠀⠀⠀⠀⠀⠀⡿⠟⠛⡇⠀⢀⣴⣿⠏⠀⠀⢀⣸⠘⠀⠀⣤⡄
⣿⣿⣿⣿⡿⠷⣿⠛⠋⣿⡇⡴⡋⣉⠻⣿⣦⣼⣿⣿⣿⢰⣾⣿⡆⠀⠀⠀⠀⠀⣀⣿⢠⣴⣧⠀⣾⣯⣽⠶⣦⠀⠘⠛⠀⠀⣄⡏⠁
⣿⣿⣿⣿⣧⣤⣿⣷⣾⣿⣧⡹⠿⠿⣠⣿⣿⣿⣿⣿⣷⣼⣿⣿⡇⠀⠀⠀⠀⠀⠛⠋⠀⠉⠁⠀⠹⣯⣈⣇⡼⠃⠀⠀⠀⠀⠈⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⢸⣿⣿⠿⠿⣿⠛⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣶⡯⣯⠀⠀⣠⣄⠀⣴⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⢠⣴⣿⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠉⠛⠁⠀⠀⡟⠋⠀⡏⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⠋⢉⣹⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣶⣶⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⢸⣿⡄⢻⡆⠀⠀⠀⠀⢰⣿⠏⣸⣿⠃⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠘⣿⣷⠈⣿⠀⠀⠀⠀⣿⡿⢠⣿⡿⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⢹⣿⡆⣿⡇⠀⠀⢰⣿⠇⣼⣿⠇⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⣿⣇⢸⣷⡀⢀⣼⣿⢠⣿⣿⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣾⣿⣿⣟⠛⠻⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠋⠉⠉⠉⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⢻⣿⣀⠀⣠⣀⣀⡀⢀⣿⡋⠁⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣾⡿⠛⠀⣸⣿⣿⡿⠿⠛⠛⠻⢿⣿⣷⣄⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣾⣿⠥⠤⣴⠏⠉⠙⣧⠶⠲⠶⢤⠀⠀⠌⠻⣆⣒⣻⣧⣀⣀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠀⠀⠀⠂⢀⠉⠄⠐⣿⣿⣿⣿⠃⠀⠀⠀⣽⣷⣦⣶⡏⠀⠀⠀⠉⢹⣬⣭⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠙⣿⣿⣿⠀⠀⠀⠀⢾⣿⣿⣿⣆⠀⠀⠀⠀⣸⣿⡿⠧⢈⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⡦⢴⠶⠾⣿⣿⣿⠿⠶⢶⢤⣺⣿⣿⣷⣤⣀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣴⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⠀⠈⠉⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡶⠤⠀⠀⣄⣀⠈⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢰⠂⠐⠺⠗⠂⠈⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠸⠓⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠁⠀⠐⠀⠀⠀⣠⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⣤⣤⣶⠟⠁⠀⡟⠛⠛⠛⠛⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠝⠋⠁⠀⠀⠀⢧⣤⣤⣤⣤⣤⣤⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠁⠀⢀⠀⠤⠀⠂⠁⠙⢛⣿⣿⠿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⣴⣾⣧⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠈⠉⠙⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣄⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠘⣿⣿⡿⠃⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⡶⠶⢶⣤⣤⣴⡶⠿⠿⣿⡷⢀⣀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣭⣥⣴⠶⣶⣶⣷⣤⠀⠀⢠⠾⣗⡚⠉⠉⠀⠀⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⢾⣿⣉⣁⣽⣦⣶⠿⢛⡿⠿⠶⠾⠿⣻⠟⠉
⠀⠀⠀⠀⣠⣴⣿⣿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠈⠙⣿⣦⡈⠻⣟⡛⠉⠉⠀⢀⡟⠀⠀⠀⣠⠞⠃⣀⣀⣤⣤⣤⣄⡀
⠀⠀⢀⡿⣿⣿⣿⡿⠿⠿⠟⠻⢛⡛⠃⠀⠀⠀⠀⠀⠀⢈⣻⠿⣦⣽⣿⣄⣀⠀⠉⣀⣀⣠⣴⣿⣾⣏⣭⣥⣤⣶⡾⠟⠁
⠀⣀⣼⣷⣿⠋⠁⠀⠀⠀⠀⠀⣤⣽⣆⠀⠀⢀⣠⣴⡿⠋⠉⠛⠷⠾⠿⠿⣿⣿⣯⣭⣿⠶⠻⣭⣉⠉⠉⠉⠉
⣾⡿⠋⠉⠁⠀⠀⣀⡀⠀⠀⠀⣉⡉⢙⣦⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀
⢿⣴⣿⣄⠀⠀⣼⡟⠁⠀⠀⠀⠏⣡⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡄
⠈⠻⣿⣿⠆⠀⠈⠀⢘⡥⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣄
⠀⢠⡞⣭⣗⡀⠀⠀⠈⢀⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣸⣇
⠀⠈⠳⣄⣈⣀⡆⠀⣴⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣋⡴⢿⡄
⠀⠀⠀⠈⠉⠉⢙⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⣀⣬⣷⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠴⣛⣷
⠀⠀⠀⠀⠀⢀⣾⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⠛⠉⠙⣧⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿
⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠁⠹⡄⠀⠀⣸⣆⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢰⠿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⢀⡾⠡⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⣿⠟⠛⠛⢿⡉⠉⠉⢻⣿
⠀⠀⣼⠃⠀⣬⣭⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡆⠀⠀⠘⣧⢀⣤⡿⠁
⠀⠀⠹⣇⠀⢬⣭⣛⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣤⣀⣷⣤⣤⣶⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠙⢷⣔⣒⣺⠆⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⣏⠉⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠙⣿⣿⣄⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣤⡀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣻⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣹⣯⣽⢙⡟⣻⢿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⣟⣻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢧⡿⢿⣿⣽⣟⣟⣾⣿⠏⠀⠈⠉⠙⢿⣿⣍⡿⣯⡷⣛⣧⡟⣯⣧⡽⣆⠀⢀⡴⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣭⠿⣽⣟⡶⣧⣾⣽⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⣽⢟⣷⡟⣯⡷⠛⣻⣶⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡷⣯⡟⣧⣟⣳⣻⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣱⣿⣿⣿⣾⣿⡇⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⣿⣧⣿⣿⠾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠉⣿⣿⣿⠋⣰⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣌⣷⣄⠀⠀⠀⠀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠘⢿⣿⠋⢙⡇

⠀⠠⠠⠀⡌⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡄⣦⠀⢰⠄⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢸⠀⠎⢹
⡇⠈⠄⠘⠃⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡦⠃⣠⠃
⠰⠀⠀⠀⠀⠒⠲⠖⢀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⠉⠚⡇⠀⠀⠈⠁⣀
⠀⠙⡄⠀⠀⠀⠀⢀⠟⠀⠁⠀⠀⠀⠀⠀⠈⢿⡁⢤⠀⠀⠶⠈⠉⣳⠜⢳⡄⠀⠀⠀⠀⠈⢱⠀⠰⢄⠀⠀⢀⣄⡐
⠀⠀⠈⠳⢦⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠗⠀⠀⠀⠀⠀⠀⠁⠀⠀⠳⠀⣀⠀⠀⠀⠘⣄⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠈⡄⠈⡇⠀⠀⠀⠀⠀⠀⢠⡄⢉⡀⣠⣤⣤⡀⢀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⡨⠃⠰⠖⠋
⠀⠀⠀⠀⠀⢧⠀⢹⡄⠀⠀⠀⠀⣠⠔⠛⠉⠁⠀⠀⠀⠀⠉⠑⠢⡀⠀⠀⠀⠳⣀⠀⠀⢀⠎⠀⢀⠟
⠀⠀⠀⠀⠀⠸⡇⠀⢷⠀⠀⡤⠊⠁⠀⠠⠄⠀⠀⠀⠠⢤⠀⠀⠀⠈⢢⠀⠀⠀⠉⠀⣰⠏⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣴⡀⠴⢧⡞⠁⠀⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⣠⡄⠀⠀⢇⠀⠀⣦⠴⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠘⣇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠴⠚⢡⡆⠀⠀⠘⣇⠀⣿⠁⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠻⡀⠀⡇⠀⠀⠀⣾⣿⠉⠉⠉⠉⠁⢀⣤⣴⣿⠃⠀⠀⠀⠀⣹⠷⣤⡰⠏
⠀⠀⠀⠀⠀⠀⠀⢇⠀⢁⠀⠀⠀⢹⣿⣶⣶⣶⡶⠋⠉⠀⣹⠏⠀⠀⠀⣹⠊⠁⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠘⡀⠀⠀⠀⠹⢿⡟⠉⠁⢉⣡⠜⠁⠀⠀⢀⣴⠃⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠹⢆⠀⠀⠀⠈⠁⠐⠂⠉⠁⠀⠀⢀⣠⠞⠁⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠈⠛⠶⢦⣀⣀⣀⣀⣀⣠⠶⠞⠋⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠘⣇⡠⢤⠤⠤⠤⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⡍⠉⠉⠛⠓⠒⠤⠤⠤⣄⣀⣀⣀⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⠉⢉⣭⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠐⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠱⢄⣀⡠⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡤⠲⠶⠶⠶⠖⠙⣧⣤⣄⣀⣀⠀⣀⠀⢀⣠⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄⠀⠀⠀⠀⢸⠁⠈⠉⡋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⡴⠀⠀⠀⠀⠀⠀⡦⠀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⢻⠀⠀⠀⢀⡠⠊⠀⠀⢰⡀
⠀⢀⡀⠀⠀⢠⡴⠚⠁⠀⠀⠀⡀⢻⣄⠰⠊⠁⠀⠀⠀⡠⠺
⠀⢨⠑⠀⠠⠄⠤⠀⠀⠀⣠⡔⠉⠁⠈⠡⠄⠀⠀⠀⣄⠁⢼
⠀⠀⠉⠐⠒⠒⠒⠒⠒⠀⠓⠌⠀⠂⠀⡀⠺⠀⠒⠂⠉⠐⠂
⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠀⠀⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠉⠉⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣮⣝⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠉⠀⠀⣀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢿⣿⣦⠹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠈⢿⣿⣷⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣅⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠙⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣿⣷⣶⣿⡿⠋
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣧⣀⣼⣿⡿⠋
⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠉⢹⣿⣿⡿⠋
⠀⢰⣿⣿⠋⠉⠀⠀⠉⠙⣿⣿⡇⠀⠀⣠⣴⣾⣿⣿⣿⣶⣤⣾⣿⠟⢿⣿⣶⣿⡿⠋
⠀⠈⢿⣿⣷⣦⣤⣤⣴⣾⣿⡿⠀⢀⣾⣿⠟⠉⠁⠀⣹⣿⣿⣿⣧⣀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡀
⠀⠀⠀⠉⠛⠻⠿⠿⠟⠛⠉⠀⣠⣾⣿⠋⠀⠀⣠⣾⣿⠟⠁⢙⣿⣿⣿⠋⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⠟⠁⣠⣶⣿⣿⣿⣅⢀⣴⣿⡿⢻⣿⣧⠀⠀⣾⣿⣿⡿⠿⠿⠛⠛⠋⠉⠉⢁⣀⣀⣿⣿⡇
⠀⠀⠀⠀⣠⣾⣿⠿⠛⠋⠉⠁⠀⠀⢰⣿⡿⠉⠉⢻⣿⣿⡿⠋⠀⢀⣿⣿⠀⠀⣿⣿⣧⣠⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⣠⣼⣿⡏⠀⠀⢀⣼⣿⠏⠀⠀⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⠉⠁⠀⠀⣿⣿⡇
⠀⢀⣿⣿⠃⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠋⢀⣤⣶⣿⡿⠋⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⢸⣿⣏⠀⠀⠀⠀⠀⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣤⣿⣿⡇
⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⢀⣾⣿⠇⠀⠀⠀⢀⣀⣠⣄⣀⣿⣿⡇⠀⠀⠀⠀⣴⣿⡿⠟⠛⠛⢿⣿⣿⡇
⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⢠⣾⣿⣿⡿⢿⣿⣿⣿⡇⠀⠀⠀⠀⢻⣿⣷⣤⣤⣤⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠀⠀⠀⢻⣿⣯⣀⠀⢀⣀⣿⣿⡇⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣾⣿⣿⠿⠋⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⢠⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⢻⢆⣸⡀⠀⡜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⡾⠀⠁⢡⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣲⠬⠡⠤⣤⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠟⠀⠀⡠⠲⠈⢧⡠⠆⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣶⣶⢤⡄⠀⠀⠀⠀⢾⠄⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣝⣿⣟⣿⣆⠀⠀⠀⢾⠆⠀⠀⠀⠀⠀⣿⡀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣷⣤⣀⣸⠀⠀⠀⠀⢀⡞⠉⠉⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⡠⢣
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠟⠛⠉⠁⠀⡄⠀⣽⡙⠓⠆⠀⣾⣀⠀⠀⠀⢀⣈⣵⡄⠀⠀⠀⠀⠀⢸⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡱⠀⢻⣇⡀⠧⡜⣿⣈⠉⠉⠉⠉⠉⢀⣹⣄⠀⠀⠀⢠⠀⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠿⣷⣾⡇⠘⠗⠫⣅⣒⣚⣛⣛⣉⠽⣍⠓⠒⠦⢤⣀⡇⠑⢄
⠀⠀⠀⠀⠀⠀⢀⣀⠈⢲⣶⣄⠀⠲⢄⡀⠀⠀⠀⠰⠘⢍⠣⠀⠀⠀⠈⠳⢝⡺⢭⣉⣉⠙⣆⠀⠀⢠⡞⢁⣀⣬⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠙⢻⡛⠛⠷⢤⣬⣝⡄⢀⠀⠀⠀⢠⠑⢄⠀⠀⠀⠀⠀⠈⠑⢾⣯⣽⣿⣷⡄⠛⠉⠁⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠠⠤⠤⠤⠿⠏⣛⢁⡀⠤⣀⣍⠙⠌⣧⡀⠀⡀⠀⠀⠑⠦⣀⡀⠀⠀⠀⠀⠙⢦⡀⠀⠙⢆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⢋⣑⡦⠤⠁⢍⣂⡙⠦⠿⣅⠀⢀⣈⣷⠟⠋⠉⢃⠀⢀⣔⣹⠦⡀⠀⠱⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠒⠒⢤⣽⣁⢀⣵⡀⠀⠀⡝⡗⠃⣿⠟⢤⡘⣶⣄⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠷⣖⣨⡟⠉⠹⡀⠀⠘⠀⠀⠙⣿⡄⠙⢾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠙⣝⣟⡈⠧⣀⣀⠀⢰⡀⠈⠃⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆⢌⠉⠉⢣⡀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀⠈⠑⠢⢤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⢱⣾⢍⣱⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠎⠉⠉⠀⠀⠀⠀⠀⠀⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠃⠰⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⢀⡀⠀⠈⠄⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡇⠀⠀⡀⢤⡅⠀⢰⡆⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠈⠀⠀⠟⣣⠀⠀⠀⠘⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠓⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣤⣦⡀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⡄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⢠⠄⠀⣄⢀⣠⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣤⣤⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠉⣹⣷⡀⠠⠶⠚⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢠⠀⠒⠊⣩⡶⢻⣏⣴⠟⣿⠛⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠰⠆⢁⠉⠁⢀⢈⣁⣶⡿⠁⣸⡿⠋⡀⠿⠀⠘⣁⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡉⠀⠀⢀⠔⠁⠀⠀⣾⢉⣿⣟⣄⣤⢿⣧⠀⢿⠀⠄⢈⠐⠃⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⣿⣆⣴⠃⠀⠀⠀⣸⣏⣼⣏⣿⢿⣿⣿⡿⠀⠄⠀⠀⠎⠘⢦⡄⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡋⣽⣳⣿⣿⣿⠛⣿⡟⣾⣿⢿⣿⣿⣿⣼⡿⣿⡿⠀⠀⢀⡇⠀⢰⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣏⠀⣵⣾⣏⢸⡧⠿⢀⡄⢸⣟⡟⣶⣿⣿⠀⣿⠃⢻⡇⠀⠀⠀⠁⠀⠈⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢡⠘⠀⣰⠿⢩⣿⢸⣷⣀⣸⠃⢸⡿⣷⣿⠛⣿⠀⡿⠀⠈⣱⡆⠀⠀⢀⠀⣿⠀⠀⠀⠀⠛⡄
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣿⠀⣾⡏⠀⠸⠛⢠⠟⢹⣷⠶⠟⠿⢿⣿⣿⡍⠀⠗⠀⠀⢹⡇⠀⠀⠸⠀⢽⡄⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠛⠰⡟⠀⠀⠂⠀⠦⠀⣿⡄⠀⢀⡀⢼⣿⣍⣽⡀⢀⠀⠀⠀⢻⣷⠀⠀⢀⠈⢀⠀⠐⣇⡀⢸⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣰⡿⠀⠘⠀⠀⠈⡤⢘⡟⢨⠟⠊⠉⠀⠙⢿⣟⣁⣠⣤⣄⣀⣸⣟⣷⡀⠿⠀⠀⠀⡸⠋⣷⣼⡁
⠀⠀⠀⠀⠀⠀⠀⢻⡖⠀⣿⠀⠀⠀⢀⠀⠈⠃⢸⣧⣈⡀⠀⠒⠀⠀⠀⢿⡏⠙⠛⣿⡿⢻⡏⠉⠹⢷⡄⠀⠀⠁⡀⠛⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⡇⠀⠀⠀⠛⠁⠀⣽⠘⡁⢻⡏⠉⠉⠀⠀⠀⢸⡄⣷⠀⢿⣧⡹⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣦⠀⠀⢠⡀⣿⠀⣿⠀⣿⢿⣅⣤⣄⣀⣼⡇⢻⡇⠰⣿⡇⢠⡿⠀⢰⠀⠠⠀⢀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠏⠀⠀⢸⡇⢸⠁⣿⣾⠇⠀⠀⠀⠀⠀⠹⣧⢸⣟⡁⢘⣡⣾⣧⣤⣼⣷⣾⣷⣿⣿⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⡀⣼⠇⣸⡆⣿⠃⢤⣤⣄⡀⠀⠀⠀⢹⣿⣿⡿⠾⡟⢿⠛⠋⠏⠁⠙⠁⣿⣿⡆⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣸⠀⣱⡈⠀⢸⠃⠉⠀⠀⠀⠈⠁⠀⠀⠀⣿⡅⠙⠿⣦⡌⠀⡄⢠⠀⠀⠀⠀⢹⣸⡏⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⣿⢇⣴⠏⣤⡆⠀⢠⣄⠀⠀⠀⠀⢀⣿⣿⡀⢤⠜⢻⣤⡇⠀⠀⠀⠀⠀⢸⡿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⣿⣀⣿⢀⣙⡠⠟⠛⢿⡦⠾⢿⠉⠈⠁⢀⠀⢠⡘⣿⣄⠀⠀⠀⠀⢸⠃⣿⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢙⠉⢩⣿⠋⠙⣿⣦⠀⢘⠁⣀⠀⠀⠀⢀⡼⠀⣽⠁⣿⣿⣷⠀⢀⣷⠋⠰⣿⣶⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠠⣝⡿⠀⡀⠀⡐⠙⣷⣟⠀⠀⠀⣄⠀⣸⣧⠀⣿⠀⣿⢀⣿⣧⠘⣏⠀⣷⠀⣿⠀⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣸⣷⢿⡇⠀⠹⣧⣰⣷⠘⢻⣤⠃⢐⣩⡏⢹⣣⣤⣿⣶⣿⣾⣿⣿⣧⡏⢸⡇⢸⠏⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡿⠀⡇⠀⠀⡏⢻⣿⡆⢈⣻⣧⣾⣿⣯⡿⠿⢻⠛⠉⠯⡌⠁⣼⢹⡇⠸⠃⢸⠀⣄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡆⠀⠐⠀⢿⣾⣷⣸⣇⠹⠿⠟⠻⠃⣠⡄⠈⣷⠂⠀⠸⡏⠈⢷⠀⠀⠈⢼⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠘⣹⣿⣿⣿⠀⢷⠀⡁⢠⣿⡇⢰⣏⢰⡇⣰⡇⠀⢸⣀⣦⣶⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⣇⠀⠀⠁⠀⢸⡏⢻⡿⠀⠘⠟⢱⣾⣿⠀⢿⡿⢸⡇⣿⡇⠀⠈⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣸⡏⠀⡀⠃⠀⣾⢃⣼⠷⢶⣆⠀⠂⠈⢀⡀⡈⢇⣼⣧⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣿⡟⢸⡿⠂⠀⣼⢿⡟⠀⠈⢻⡇⣿⠀⣿⡏⢹⣰⡿⢹⣮⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡗⠈⠀⣿⠀⣿⣿⡇⠀⠀⠀⠻⣿⡅⢿⡇⢸⣯⡇⣾⣇⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣬⠟⠢⣬⡀⠾⢆⢿⣿⡇⠀⠀⠀⠀⠈⢳⣿⡇⣸⢿⣧⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣶⡆⣻⣁⡆⢀⠰⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⣾⠈⡏⢉⢸⣇⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⢹⣿⠇⠘⢠⣿⠀⠀⠀⠀⠀⠀⢹⣿⢀⡇⢠⣧⠙⣸⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢹⡇⡏⢿⡄⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⢿⡏⠇⡘⢻⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣾⠁⠃⣼⡇⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠁⢸⡄⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⣤⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡇⠈⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢾⣷⣿⣿⣷⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡗⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⢾⣛⠇⠈⣸⣿⣿⣏⣻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣿⣼⡿⣿⣿⡟
⠀⠀⠲⢶⣾⣷⣿⣿⣿⣷⣬⣼⣷⣴⣿⣿⠿⠟⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⡤⢤⣾⣿⣿⣸⡃⣽⢀⣾⠏
⠀⠉⠉⠀⠀⠉⠉⠙⠋⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠳⣦⣤⣽⣿⣿⣿⣷⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠿⠶⢶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣥⣴⣶⣶⣶⣶⣦⣤⣉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠏⠙⠛⠛⠛⠛⠛⠟⠻⠿⣿⠂⠀⠀⠀⠀⠀⣿⣿⠉⠁⠀⠀⠀⠈⠉⠁⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⢇
⠀⠀⠀⠀⠀⠀⠀⢀⠿⢿⡿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠀⢠⡄⠀⠀⠀⠈⡞⢙⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠊⣇⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣀⡄⠀⠀⠀⠀⠀⠠⢬⣤⡄⠀⠀⠀⠀⠀⠀⣴⡆⢰⡤⢧⣦
⠀⠀⠀⣀⣢⡷⠠⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⠛⠤⢀⡠⣦⡶⡀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⢀⣴⣿⠧⠘⠀⠐⠛
⠀⠀⠀⠐⠋⠀⠀⠠⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⠌⣁⠠⡉⠁⢴⣤⣤⣤⡀⠀⣠⣤⣶⣾⠟⠋⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠑⠀⡁⠂⢁⠂⠄⢀⠀⠀⠀⠀⢀⠀⠀⠠⡊⠄⢈⠀⠊⠀⠀⠀⠐⢀⡘⠙⠛⠣⠄⠻⠋⠁⢡⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠂⠀⢀⠤⣀⠨⣀⠄⠆⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠂⠀⠀⠀⠐⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⡀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠁⠀⠃⠀⠖⢂⠢
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⢀⠠⠐⠻⣀⠇
⠀⠀⠀⠀⠀⠀⡔⣀⣡⠉⠑⠐⠀⠀⠀⠀⠐⠒⠒⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠠⠮⡄⢨
⠀⠀⠀⠀⠀⠀⠃⠘⠁⠑⢀⠀⠀⠀⢀⠠⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⢣⡄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠂⡡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⣀⣀⠈⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠘⢽⡾⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢀⣤⣄⠀⠀⣠⣤⡀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠙⠾⠞⠀⠀⠳⠷⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡤⠴⢶⣤⣤⠶⠤⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠖⠺⡇⢸⠗⠶⠶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⣄⡀⠀⣀⡤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⢤⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⠤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠹⡾⠁⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⣲⣤⣄⣈⣉⡉⠉⠉⣉⣉⣀⣠⢤⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣽⣿⣾⣿⣍⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣉⡉⠉⠉⠉⠉⠉⠉⢉⣉⣠⡼⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡋⠉⠁⠀⠀⠸⠿⠀⠀⠀⠉⠉⢛⡆⠀⠀⠀⠀⠀⢀⡯⣑⣤⣌⣉⣉⣉⣉⣉⣉⣉⣀⣀⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⣉⡇⠀⠀⠀⠀⠀⣿⢧⣄⣈⡉⠉⠉⠉⠉⠉⠉⢉⣉⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣈⡉⠒⠒⠒⠒⠒⠒⠒⢒⣊⣩⢿⡆⠀⠀⠀⠀⠀⠻⣤⣀⣈⡉⠉⠉⠉⠉⠉⢉⣁⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣄⣉⡉⠉⠉⠉⠉⠉⠉⢉⣉⣠⡼⠃⠀⠀⠀⠀⠀⠀⠷⣄⣈⡉⠉⠉⠉⠉⠉⠉⢉⣁⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣴⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⣻⣣⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣌⣝⡻⢦⡀
⠀⠀⠀⠀⠀⣠⣯⣵⣾⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠈⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⣧⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⡝⣿⣿⡿
⠀⠀⠀⠀⠀⠘⣿⣿⣳⣿⡿⢟⣭⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣽⡻⢿⣿⣜⣿⡇
⠀⠀⠀⠀⠀⠀⠹⢿⣫⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣟⢿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡻⣿⣿⣿⣿⢸⣿⣿⣿⣿⡿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⢏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡻⣿⣿⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⠿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⠿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⢀⠀⢀⣱⣴⣴⢛⣋⠐⠒⠢
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⣿⠿⣭⡉⠉
⠀⠀⠀⠀⠀⠀⠀⡟⠈⣾⡗⣌⣷⡈⠙⠈⠀⠀⢀⣰⣿⠋⣓⠀⣠⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠶⢻⡇⠈⢻⠙⠠⠀⠀⠀⡾⠁⢻⣷⡌⠸⢳⠏⣦⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢄⠑⢤⡀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠎⠉⠠⢄⠀⠀⠀⠀⠉⠒⠦⢄⡀⠀⠈⠑⠦⡀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠓⠺⡞⠛⠛⠒⠂⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠈⠓⢄⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⢦⣀⠀⠙⢶⠀⠾⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢐⠤⡹⣧⣠⣄⡈⠶⣬⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⡤⠖⠋⠉⣀⠀⠚⢑⠻⣮⡾⠛⠓⠾⣧⡘⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⣹⢀⣴⠡⠜⡉⣤⣦⣾⣷⣿⣿⣿⣶⣄⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠖⠟⠳⠿⠿⠾⠷⠷⠛⠁⠁⠀⠀⠐⠀⠐⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣤⣤⣀⣀⣳⣼⣁⣠⣤⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣼⣿⣿⣿⡇⠉⣩⣿⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣿⣿⢿⢿⣷⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣤⣴⣶⣄⠀⠀⠀⠀⣀⣴⣾⣿⡿⣫⣿⣿⣝⢿⣿⣿⣦⣀⠀⠀⠀⠀⣀⣴⣶⣤⡀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⣴⣶⣿⣿⠿⢛⣡⣾⣿⣿⣿⣿⣷⣬⡛⠿⢿⣿⣷⣦⣼⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠈⠻⣿⣿⢿⣿⣿⣯⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣭⣿⣿⡿⣿⣿⠟⠁
⠀⠀⠀⠀⢸⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢹⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿
⠀⠀⠀⠀⣠⣿⡇⢸⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⠀⢸⣿⣅
⠀⠀⠀⠀⣷⣾⡇⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣾
⠀⠀⠀⠀⢿⣿⣿⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣶⣿⣿⣿
⠀⠀⠀⠀⣾⣯⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⣿⣽⣿
⠀⠀⠀⠀⠙⣿⣿⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣿⣿⠊
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢸⣿⡟⠉⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⡏⠈⠙⠛⠛⠿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠋⠁⠀⢸⠀⠀⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠒⠒⠋⠉

⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⢶⣶⣶⣶⡶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢹⣠⠏⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡏⠀⢸⠇⠀⢠⠇
⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣤⠇⢸⡇⠀⡟⠀⠀⡾
⠀⠸⡇⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣾⠀⢹⡀⠀⠀⠀⠃⠀⢠⠇
⠀⠀⠳⡄⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⠀⠻⡄⠀⠃⠀⠀⢀⣄⣀⣸
⠸⡏⠓⠿⣦⡀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⢆⠀⠡⠀⣠⡴⠒⠚⠛⠢⠉⠳⣄⣀
⠀⠙⢦⡀⠀⠙⠢⢄⠀⠀⠉⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠻⡄⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⢾⠙⠓⠦⢀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⢰⡞⢷⡀⠙⠄⢰⡷⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⢀⣹⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⢀⣷⡀⠙⠆⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠈⣧
⠀⠀⠘⢦⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠘⣆⠙⠢⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⠀⠀⠀⣀⣸⡀
⠀⠀⠀⣴⠞⠛⠓⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡞⠳⣄⠀⠀⢿⠀⠀⠀⣠⠖⠚⠋⠉⠛⠲⢌⠀⠀⢀⠞⠁⠀⠉⢓⣦
⠀⠀⠀⠈⢛⡶⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣁⠀⠸⡄⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢳⠀⢻⠒⠲⢶⣚⠉
⠀⠀⠀⠀⠈⠓⠦⢤⣤⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠿⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣹⠖⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠐⢯⣤⣤⣤⠤⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣀⠀⠀⠀⠀⣀⣴⣣⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠤⢤⡤⠶⠒⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠈⠉⠛⠛⠛⠉⠉⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⣞⣉⣀⡤⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠴⠟⠒⠛⢻⡥⠴⣦⠤⠴⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣤⣤⡤⠴⠒⢋⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠉⠀⢀⡤⠞⠉⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠥⠤⠤⢶⡶⠞⠉⠀⣀⣠⠴⠚⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣋⣤⠴⠖⠋⠉⠀⠀⠀⢠⡖⠛⣙⠲⠦⣤⣤⣀⣀⣀⣠⣤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠛⣁⠀⣰⠃⣴⣉⡤⠂⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣡⠞⠁⠀⠀⠸⠧⠚⠁
⣶⡶⢶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣷⣹⣷⣬⣿⣶⣤⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⡜⠀⠀⠀⢀⡤⠆⠀⠻⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠰⠂⣤⣴⠁⢀⡴⠚⢉⡄⠒⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡤⠀⣰⠟⠀⣀⠁⠠⠀⠉⠁⠀⠀⡄⠚⠀⡖⣒⠖⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠡⠀⠻⢄⠘⠋⠀⠀⠾⠀⡀⣒⢀⣄⡀⠤⠛⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠡⠄⢀⠄⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠐⠂⢃⡀⠀⠀⠀⠀⢠⢦⣤⣄⡀⠀⠀⠀⢼⠈⣯⡫⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀⠀⡌⠁⠀⠀⣤⣤⠃⢀⡀⡨⠃⠀⠀⠀⠈⠳⠦⠚⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠤⢠⢾⢥⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣠⢀⣠⠀⠾⡐⠂⠘⡚⠊
⠀⠀⠀⠀⠀⠀⡀⠀⡴⠓⣀⠃⢄⠘⠋⠀⠀⠉⠀⠀⠁
⠀⠀⠀⢨⠉⡍⠉⠉⢑⡀⠋⠀⠀⠷⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢵⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠘⢧⣅⡑⠀⠀⠀⠀⢠⠶⠚⠛⠃⠀⠀⠀⢄⢀⡀⠀⠠⠄⠀⠀⠆⠀⢀⠞⠐⠂⢀
⠀⠀⠀⠀⠠⠄⠀⢀⣀⡀⠈⠳⠶⠊⠡⡦⠀⠀⠀⣀⡄⠀⣴⠆⠀⠀⠀⠸⣌⡢⡀⠶⠘⣠⣤⠀⠰⠋⣄⠃⠴⢠⡅⠀⠑⣀
⠀⠀⠀⠀⠀⠀⣀⣫⢔⡉⠁⠀⠀⠀⠠⢄⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⢀⡄⠀⠀⠉⠀⢂⠈⠁⠀⠀⠂⠀⠀⠀⠀⠈⠙⠲⠶⠆
⠀⠀⢈⠃⢀⠀⠉⢀⣩⠶⠋⣀⠀⠀⠙⠀⠠⡶⠀⠀⠀⢠⡆⠀⠀⠀⠀⡄⡄⠠⣄⠀⠋⠀⠄⠉
⠀⠀⠘⠀⠼⠀⠰⠋⣀⠘⠂⠁⠀⢀⣤⣤⠀⠂⠀⢰⠖⠀⢢⡀⠀⠘⢁⠴⠀⠒
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡴⠃⠈⢁⡤⠀⠖⠀⢢⡴⠛⢡⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠶⠒⠀⠀⠀⠀⠀⠈⠸

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⡒⠶⠎⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⠡⠆⡆⡴⣤⡙⢁⡴⡾⣭⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣏⡠⢔⣒⣒⣦⢼⡟⢠⢏⣒⠂⡤⢿⡄⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠐⢻⣣⡞⣉⠨⠦⣮⣝⢠⢿⣦⡷⣬⠽⣟⠷⡜⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣳⡘⠺⠗⢸⣱⡋⠙⣙⢷⡱⠆⢒⣽⣛⠟⠟⣓⣒⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣇⣼⡗⠢⡤⠤⣐⣾⢹⣦⣣⡈⠘⠕⢋⠙⠉⠁⠾⣹⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⡜⢿⣷⣄⠈⠲⢬⢝⣛⣳⢍⢿⠏⢏⢸⣭⠑⠺⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣮⡳⣑⠠⢠⡱⣯⣿⡄⡿⣆⣲⢣⢋⡴⠂⠁⡹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣫⣙⢦⣓⠛⣛⣴⠷⢾⠏⣹⣿⣷⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣯⣭⣏⠂⢡⡟⠼⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣚⠞⠠⠷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣍⣲⢋⡆⠸⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡋⡠⠺⢁⠒⡒⣸⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣏⠈⠞⣯⠘⣷⡑⢦⠃⠂⠆⠌⡜⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢻⠐⡈⡜⣶⠊⠻⣷⣧⣉⡹⢍⣉⣒⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠈⢇⢡⠄⢍⠷⣝⢠⣍⢛⣛⠛⠿⣛⣻⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠈⢦⠲⢈⠰⣈⠳⠦⣬⣙⣛⣠⣭⠵⠞⡩⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠑⢞⡀⡄⢉⡈⠢⠤⠼⠤⠔⠚⣍⠀⣦⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠓⠮⣥⣘⣂⡸⠀⠻⠀⠙⢁⣘⡯⠜⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⡆
⠀⠀⠀⢀⣴⡄⠀⠀⠁⠐⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠁⠐⠀⠀⠠⠇⠀⠀⠀⠀⢸⣅⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⠟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠐⣻⡀⠀⠀⠀⣀⢀⠬⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠥⢷⠀⠀⠀⠸⣁⣠⢔⠗⠀⠀⠀⠐⠒⠒⠁⠂⠈⢠⠀⠀⠀⠀⢰⣃⣀⠀⢄⠀⠄⠠⠀⠁
⠀⠀⠀⠀⠀⠸⠽⡇⠀⠀⠀⠈⠃⠤⣂⠀⠀⠀⠀⠀⠀⠀⢰⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⢒⣻⣤⡠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠖⠁
⠀⠀⠀⠀⠀⡀⠺⡜⣏⠀⠉⠢⠐⠀⠀⠀⢀⠀⠀⠀⠀⢠⡁⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠀⠀⠧⢻⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣨⠈⡃⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢁⣀⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢸⡴⠧⣀⠀⢁⣆⠀⠀⠀⠀⠀⠀⠠⡅⣈⠃⠀⠀⠀⡀⠀⢙⠛⡄⠀⠀⠈⠥⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠙⠀⣠⡜⠀⠈⠣⡀⠀⠀⠀⠀⠀⢠⠀⣸⠀⠀⠀⠂⠀⠀⠀⠣⣘⡄⠀⠀⠒⠀⠀⢸⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡆⠤⡌⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⣥⠀⠀⠀⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠡⡤⡄⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢄⠀⣸
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠈⢁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡁⡆⠀⠀⠀⠁⣠
⠀⠀⠀⠀⠀⠘⢤⡀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡤⠀⠀⠀⠀⠀⡀⠤⠂⣷⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠇⠀⠑⢀
⠀⠀⠀⠀⠀⡲⠦⡀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠉⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⣶⡂
⠀⠀⠀⠈⢞⢀⣐⠍⡀⠀⠀⠀⢲⠞⠑⠢⠄⣀⠀⣀⠀⠀⠈⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠉⠁⠘⠒⠀⠀⠀⠀⠀⠀⠈⠀⢒⠈⠭⠉⠭⢙⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠷⢱⠀⠀⠀⠀⠀⠹⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠒⣄⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠒⣸⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡛⣯⢀⡀⠀⠀⠀⠘⠃⣿⣧⠀⢸⣿⡆⢠⢄⢠⣤⣀⣀⣠⣤⡞⢡⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢷⣻⣿⣷⣴⠾⢃⣿⣿⣿⣿⣷⣼⣿⣿⣦⣾⡟⠉⠙⠛⠛⣿⣠⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⢿⡟⠀⠀⢸⣿⠃⣿⣿⣿⣿⣿⠙⠛⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣽⣿⠇⠀⠀⠈⠁
⠀⠀⠀⠐⠀⡀⣀⠀⢀⣴⣺⣿⣿⢻⣾⣇⠀⠀⢸⣿⣴⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣟⣿⣾⣿⡆⠀⡸⠟⣿⣿⣿⣿⣿⣿⣇⢀⢀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣟⢿⣿⣿⣿⣿⠐⢶⠷⣿⠛⠛⣿⠟⠛⣿⢰⣾⡆⠀⢀⢸⠿⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⢸⠢
⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⢳⡟⣾⣿⣿⣿⣿⡇⣐⣲⣿⡆⠀⣿⠀⠀⣿⡏⣿⢹⡀⠀⠀⠀⢸⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠘⢲
⠀⠀⠀⠀⠀⢀⣀⢐⣿⣿⣾⣿⡟⣿⣿⣿⣿⡇⠀⠉⠀⠃⣨⢛⠂⢀⣿⠁⢹⠀⠃⠀⠀⠀⠈⣿⣿⣿⣿⡄⠀⠀⠀⠀⠠⠘
⠀⠀⠀⠀⠀⠈⠛⠀⠙⠿⣿⣽⣿⣟⣿⣿⣿⡇⠀⠀⢀⠄⠉⠀⠐⠁⢿⠀⠸⡆⠀⠀⣤⡤⢂⣿⣿⡏⣿⣿⡀⠀⡈
⠀⠀⠀⠀⠀⢀⠴⢊⠀⣴⣯⣿⣏⣿⣿⣿⣿⠡⠤⣄⠀⣀⣻⡅⠴⠶⠾⡷⢼⣁⡤⠾⠿⠁⢸⣿⡟⠁⢹⣿⡇⠁
⠀⠀⠀⠀⠀⠀⣤⡿⢀⣿⣷⣿⣿⣿⣟⣿⣿⣀⣀⡹⣷⢁⣘⣁⠲⠶⢾⣃⠨⠁⠀⠀⠀⠀⣸⣿⠃⢀⣸⣿⡧⢤⣀⠄
⠀⠀⠀⠀⠀⠐⠋⠃⠘⢿⣿⡿⣿⣿⣿⣿⠿⠉⠉⠀⠈⠋⠙⠃⠀⠀⠈⠁⠀⠀⠀⠐⠐⠛⠛⠛⠀⢉⢼⣿⡗⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠁⡀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠈⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠃⡤⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢠⠷⠀⠀⠀⠀⠀⠀⠀⠙⠋⢰⡀⠈⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣀⠀⡀⢦⣠⣶⣿⡇⢀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⠀⠀⠀⡀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⣄⣀⣁⣀⣀⣀⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠤⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠥⠤⡀⠀⠈⢢⠀⠀⠀⠀⠀⠂⢀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⢀⣀⣀⡀
⢀⠂⠀⢀⠀⡀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠈⡆⠀⠈⡇⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢾⠋⠁⢀⣍⠉⠒⣄
⠘⢄⣀⡠⠄⢄⣀⡠⠃⠀⠀⠠⠀⠀⠀⢀⠇⠀⢀⠇⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠁⠸⣄⣀⡾
⠀⠄⠄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⠲⠖⡁⠠⠔⠀⠀⠀⠀⠘⠀⠀⠁⠐⠁⠀⠚⠀⠀⠀⠀⠠⠀⠀⢀⣠
⠀⠑⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠈⠐
⠀⠀⠀⢢⢀⡤⠁⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠁⠂⠀⠀⠀⠀⠀⠀⣀⠤⠒⠲⢄⠀⠀⠀⠀⠀⠂⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⢇⠀⠀⠀⠀⢄⢐⣦⡈⡠⡆⢠
⠀⣤⣤⣤⣤⣄⣀⣤⣤⣤⣤⡄⠀⠑⢄⠀⠀⠀⠀⡀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⣠⠇⠀⠀⠐⢖⠞⠁⠀⠈⠱⣆⠆
⠀⠸⣿⣿⣿⡟⠹⣿⣿⣿⡟⠀⠀⠀⠀⠑⣄⠀⢠⠁⠀⠀⠀⠀⠀⢹⠉⡉⠈⡽⠀⠀⠀⠀⢃⡼⡀⠀⠀⠀⢀⠧⠐
⠀⠀⠈⠛⠋⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠈⠢⠎⠀⠀⠀⠀⠀⠀⠀⡇⢀⢠⠃⠀⠀⠀⠀⠈⣨⢩⣢⠤⣔⠎⢁⠁
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⠀⠀⠀⠀⠘⣄⡏⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠈⠁
⠀⠀⠠⣼⡄⠀⢡⠉⠂⠀⠀⢀⣴⣶⣶⣶⣶⣶⠄⠀⠐⠋⠀⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⢁⠀⠀⣠⠁⠀⠀⠀⠈⢦⠂⠀⠀⠐⠐⠃⠀⠀⠀⠀⠀⠀⣴⣾⣷⣦⣀⣹⣦⡀⠀⠀⠀⠀⢰
⠀⠀⠀⠌⠀⠀⢻⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠰⠀⠀⠀⠀⢀⠀⣼⡿⠟⠋⠉⢁⣦⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⡀⠀⣀⡼⡀⠀⠀⠀⠀⠸⠀⠁⠈⠀⠂⠀⠀⢐⠨⠊⠀⠉⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⣸⠀⠀⠀⡀
⠀⠀⠀⠈⠘⠓⠈⠀⠀⠀⠀⠀⠀⠁⠐⠐⡈⠅⠀⠔⠁⠀⠀⠀⠀⠀⡀⢀⣾⣿⣿⡷⠀⠀⢠⣤⣶⣿⣤⣤⣴⣿⡄
⠀⠀⠀⠠⢀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠘⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠳⡄⠀⠀⠀⠀⢸⠁⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠂
⠀⠀⠀⠀⠀⠲⣦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣌⠢⡀⠀⠀⡾⢶⣤⡀⠀⢸⠀⠀⠀⠀⠀⡠⠔⠈⢠⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣍⠑⠒⠤⣀⠀⠀⠀⠀⠀⠀⡇⠚⢉⣁⡀⠈⣢⣞⣀⣠⢤⣁⣀⠀⠣⣀⠠⠐⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠉⠙⠒⠢⠤⠤⢃⡜⠉⠀⠀⠉⠳⣴⠋⠀⠀⠀⠈⢻⡙⠒⢤⡀⠀⠀⢠⠃⠀⠀⠀⣀⠤⠐⠒⠈⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠀⠀⠀⢀⡴⢻⠁⠀⠀⠀⠀⠀⠘⡄⢀⣀⠀⠀⠀⢣⠀⠀⠈⠳⣄⠺⠤⠐⠂⠉⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⢠⠎⠀⢸⡀⠀⠀⣖⣆⠀⠀⢳⢰⣿⡆⠀⠀⢸⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⢀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠕⢠⠇⠀⠀⠀⢣⡀⠀⢿⡿⠀⠀⣾⡀⠛⠁⠀⣀⠞⠀⠀⠀⠀⠀⡄⢣⠀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠲⠶⣒⡈⠉⠁⠀⠀⡸⠀⢤⠀⠀⠀⠙⠦⣤⣤⠴⠚⠀⠈⠙⠋⠉⠁⠀⠀⠀⠀⢀⠘⠀⠀⡇⠑⠢⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⡀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢸⠀⠀⠀⢀⣀⠬⠵⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⢡⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⠀⢸⢠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠸⡀⠀⠀⠸⡀⠀⠒⠀⠤⠤⠠⠤⠀⠐⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⢀⠤⠖⠉⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⣀⠔⠊⠁⢀⣀⣀⠤⠤⠤⠀⠒⣦⠈⢧⠀⠀⠀⠀⠀⠁⠀⠦⠀⣀⡀⠴⠔⠊⠁⠀⠀⠀⠀⠀⢀⡴⢁⠤⠀⣀⡀⠀⠀⠀⠀⠑⠢⣀
⠀⠰⠞⠒⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠘⢆⠀⠀⠀⠉⠉⠁⠒⠂⠤⠤⢑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣀⠠⠄⠒⢌⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢁⡤⢄⣀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠗⠊⠉⠀⠀⠀⠀⡆⠀⠀⠈⠉⠓⣒⠲⠲⠶⠖⠒⣋⡉⠀⠀⠀⠀⡁⠀⠀⠉⠒⠤⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⡴⠋⠀⢇⠀⠀⢀⠜⠀⠙⣄⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡴⠋⠀⠀⠀⠘⡀⢀⠎⠀⠀⠀⠈⢢⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡠⠊⠀⠀⠀⠀⠀⠀⣇⠎⠀⠀⠀⠀⠀⠀⠱⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⡿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⣄⡼⣧⣠⠀⠀⠀⠀⠠⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠶⣤⣬⢙⣃⠀⠀⣀⡽⣰⣎⣯⣄⠀⢀⣰⣋⣇⣠⠴⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠠⠊⣻⣗⠭⢛⡋⢭⠞⣫⣏⠛⠿⠿⠿⠿⣿⣿⣧⣤⣤⣴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⠐⠉⣿⣷⣤⣀⣤⠤⢿⢿⣦⣤⣤⣴⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣷⠚⠯⠉⢉⣹⠿⠀⣈⣥⡀⢾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣿⣷⡶⠞⠛⠛⠉⠉⠻⠟⠉⠙⠿⠻⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⣿⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢢⣿⣿⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⡿⠞⠁⠁⠠⡀⠀⡀⢀⠀⠀⠠⣈⡂⠛⢿⣾⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢳⣿⣿⣿⣷⣾⣿⡿⢿⣾⣷⣿⣿⣿⣿⡖⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⢿⣽⣿⡿⢿⠟⣴⣦⡻⠿⢿⣟⣿⡹⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣄⡠⢀⣀⣰⣶⢾⣿⣻⣷⣿⣶⣄⣀⠀⣤⡶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡽⠉⠙⠋⡙⠋⠘⢿⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢿⣿⢮⣭⣱⣆⣧⣽⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠻⣿⠷⠶⠷⠶⣿⣿⣟⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠣⡐⠈⠀⠀⠀⠁⠞⣿⠷⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠉⠉⠋⠉⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠋⠉⠉⠉⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠷⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢤⡄⠀⠀⠀⠀⣤⠄⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠙⠷⠶⠟⠷⠶⠟⠁⠀⣸⡏⢠⣦⣤⣶⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠶⠂⠀⠀⣀⣼⠏⠀⠈⢿⡍⠀⢁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠶⠤⠉⠻⠶⣦⣤⣤⡶⠞⠋⠡⠶⠶⠀⣼⠟⠛⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⡄⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠸⣷⠀⠀⠀⢠⡿⠁⠀⠀⣤⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣤⣤⡄⠀⠀⠀⣸⡇⠀⠀⣿⡀⠀⠀⣿⠁⠀⢠⣾⠛⠻⡆⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⡇⠀⢸⡟⠀⠀⢠⣝⠛⢷⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣄⣴⠟⠁⠀⠈⣿⡀⠀⠀⠙⢿⠛⠁⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⡈⠻⣦⣄⣀⣀⣀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠁




⡿⠿⠿⠿⠿⡻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠠⣶⣤⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⣷⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⣽⣿⣿⡿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢶⣤⣞⡋⠁⢸⠏⢸⡟⢿⡄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣩⣿⡟⣮⠻⡄⠈⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⠀⠹⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⡴⢒⣖⣒⡂⢥⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⣀⣤⣽⡝⢻⠁⢩⣟⣿⣿⣷⣬⣄⡀
⠀⠀⠀⠀⠀⢀⣩⣽⣿⠿⠛⠋⠉⠉⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠟⠻⠿⠟⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠈⠛⠟⠥⣤⠀⠒⢲⣾⣿⣿⣒⣒⡒⠲⠶⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠶⢿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠂

⣿⣿⣿⠟⠛⠛⠻⠿⠿⠟⠛⠛⠛⠿⠿⠟⠛⠛⠛⠻⠿⠿⠟⠛⠛⠻⠿⣿⠿⠛⠛⠛⠻⢿⣿⠿⠿⠿⠿⠿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿
⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣿⡿⠚⠋⠉⠉⠙⠛⢦⣤⣤⣀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⡀⠀⠢⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣻⣶⡶⠒⠶⠛⠻⠿⠿⠿⢶⣼⣿⣄⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇⠀⠀⠀⠀⠰⣿
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣤⡀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣠⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⠿⠛⠿⣦⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣄⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣽⣿⣟⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⠀⢻⣿⡀⠀⣠⣾⠿⠛⠉⠙⠳⣦⡀⠀⠀⠀⠀⣸
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⣿⣧⣾⠟⠁⠀⠀⠀⠀⠀⣈⣿⣄⡀⠀⠀⣿
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⢹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠃⠀⠀⢸
⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⣿
⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⢀⣀⣠⣤⣤⣼⠿⠷⠶⠶⢶⣶⣶⣤⣤⣼⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⠟⢛⣉⣻⣷⣶⣶⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⡿⠿⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢻
⣿⡏⠀⠀⣀⣀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⠷⠿⠟⠋⢉⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⠋⠀⠀⠀⠀⠈⠻⣿⣶⣤⣀⠀⣀⣤⡶⠟⠁⠀⠀⢸
⣿⡇⠀⠀⠉⠉⠛⢛⣿⣿⣿⡿⣿⣿⣿⠇⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡁⠀⠀⠀⠀⠀⣸
⣿⣷⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠈⠙⠿⠄⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣙⣻⣿⣶⣤⣀⡀⢠⣿
⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠾⠛⠛⠉⠉⠉⣉⠉⠛⠻⠿⢿⣷⣿
⣿⣿⠁⠀⠀⠀⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢻⣿
⣿⣿⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣤⣀⣤⣤⣶⣶⣤⣤⣤⣤⣴⣶⣶⣦⣤⣤⣤⣤⣤⣶⣶⣶⣶⣤⣤⣴⣶⣶⣶⣶⣤⣴⣶⣶⣾⣿⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⢀⠂⠀⣰⠄
⠀⠀⠀⠀⡀⠢⠀⠊⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠋⠛⠇⠈⠡
⠀⠀⠀⠀⠣⡀⠐⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀⠘⠣⠤⠤⡀⠈
⠀⠘⣄⠐⢄⠡⠄⢡⠶⠈⠀⠀⠊⢠⠊⠀⠀⠀⠀⠀⠐⡀⠀⠀⠁⠀⡆
⠀⠀⠀⠑⣤⡁⠋⠀⡀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣀⣀⠀⠁
⠀⠀⠒⠀⠀⢠⡀⠈⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠈⢀
⠀⠀⠀⠀⠀⠀⠑⢦⣀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠆⣻⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⡀⣣⠀⢃⠀⠀⢀⣀⣀⣀⠠⠤⠔⠚⠉⠀⠀⠉⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⡄⠀⠀⠀⠀⢀⣠⠋⠉⠛⠀⠉⠉⠉⠉⠛⠻⢇⡀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠐⡀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⢙⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⣴⡀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠀⢀⠉⠠⣤⠀⠀⣠⣤⡤⠶⠒⠉⠁⣀⣼⣿⡇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢸⠀⠀⠡⣄⣀⠀⢀⣀⣀⣤⣴⣾⣿⣿⣿⡇⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⡦⢀⣀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⢸⠂⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡘
⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠈⠻⣿⣿⠋⠀⢈⠀⠸⠟⣿⠃⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠙⠶⠂⣀⣀⡠⠞⠁⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠊⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠛⢶⠖⠤⠴⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⣶⣶⠖⠚⢹⠉⠲⡄⣘⡀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⢸⢣⠀⡸⠀⠀⠀⠉⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠢⠤⠤⠤⠀⠚⠉⠀⠀⠀⠀⡄⠾⠃⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠈⠻⡀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⢀⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠚⠉⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⢰⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⣡⢶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠢⣄⣠⠎⣄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠖⢉⣡⠶⠛⢧⡀⠱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⣴⠃⢀⣰⠃⠀⠀⠀⡀⣠⣤⡴⠒⠉⠁⠀⠀⠀⠀⠻⡄⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⢆⠈⠉⠉⠉⠉⠉⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⠡⡀⠀⠀⠀⠀⢠⡆⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⠓⠤⠤⠴⠋⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠠⠤⠤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠈⠁⡰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⡹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣻⣿⣿⣾⣿⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣟⠿⣿
⠀⠀⠀⠀⠀⢀⣴⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣾⢷⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣯⡻⢧⡙⠳⠙⣧
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣻⣯⣻⣯⣿⣿⣿⣿⣼⡷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡳⢮⠻⣌⠓⣘⠐⣟⣆
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣟⣿⣿⣿⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⢦⣷⣼⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⢿⣿⣿⣿⣽⣿⣿⣟⣿⣦⣄⡀⠀⠀⠀⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣵⣿⣯⣯⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠿⠟⠛⣿⣿⣿⣿⣇⡀⢡⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣤⣿⣿⣿⡟⢉⣥⢨⡏
⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣯⣯⣵⢾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣶⡶⢥⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣶⣶⣦⣴⣶⣶⣶⣏⣙⠻⣿⣟⢫⣅⢮⣭⣯⠩⣟⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣧⡒⣀⡙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⡿⣯⣿⣿⡟⢿⣿⣙⢿⡷⠌⠳⡝⣿⣿⣿⣾⣿⣿⣿⠿⠈⠍⠹⠿⠛⣻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣆⣴⣿⣿⠿⠿⣦⠿⠂⠘⢿⠟⢿⡋⢅⣨⡤⠴⠔⠂⠒⠘⠛⠓⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠠⢉⣟⢣⡻⣅⣤⡀⠀⣸⣇⣄⣙⣿⠿⠅⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠓⠛⠓⠐⠞⠛⠛⠟⠛⠁

⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⢀⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⡍⠀⠀⠀⠀⠀⠁
⠫⢅⣢⡅⠀⢀⢀⡀⢠⢋⠂⠀⡔⠹⠀⠀⢀⠌
⠀⠀⠸⠉⢃⠹⠀⠀⡤⠊⠀⢀⡔⠁⠀⡠⠊⠀⠀⠀⢀⡄⡄⠀⠀⠀⠀⡠⡦⠀⠀⠀⡀⡄⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⡆⠀⠀⢷⠀⠐⠀⡠⠂⠉⢀⠤⠊⠐⡀⠀⠀⠀⡸⠷⠓⣀⢀⡠⠚⠯⠀⠀⠔⠈⠏⠸⣀⣀⠔⠉⡀⠀⠀⡰⠁
⠀⠀⠁⠀⠀⠈⠀⠸⢅⠀⠀⠳⡁⠀⠀⠀⠐⠤⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡄⢘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣶⢀⣀⠀⠀⡲⣄⠀⠀⠀⢀⡀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠁⡤⠀⠀⠄⠀⠀⠰⠂⠀⠀⣴⠄⢀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠸⢊⠆⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠒⠒⠒⠖
⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⣐⠀⠨⠥⠤⣀⣈⣓⣀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠉⠀⢀⡀
⠀⠀⠀⠀⠈⢇⣒⠀⠀⠀⠋⢐⡖⠂⠬⠭⠭⣝⣚⣒⣒⣒⣶⣒⠒⢲⡖⠒⠒⠒⡾⠉⠉⠀⣀⠄
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠒⠒⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠐⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⢀⣠⣧⣤⣬⣤⣴⣶⠮⠍⣉⣽⡋⠉⠉⠒⠒⠢⠤⠾⣀⣀⠐⠀⠀⠀⠀⠐⣒⠚⠉⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⠉⠑⠒⠢⠭⢍⣓⡒⠤⢄⣐⡀⠀⠀⣨⠉⠁⠀⠰⠋⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠛⠛⠓⠒⠒⠂⠀⢠⠀⠰⠃

⠀⠀⠀⠀⣀⣠⣤⡶⠶⠖⠛⠛⠛⠛⠛⠳⢶⣤⡀
⣤⣶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣴⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⢿⣄⠀⢸⠇⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠈⠙⠳⣿⡦⠤⠤⠤⠶⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⠀⠀⠀⠀⣠⡦⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠄⠒⠒⠒⠲⠖
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⡇⠀⣀⡴⣏⠀⠀⣀⢴⡟⠃⠀⣠⢾⡇⢀⡤⣾⠛⣧⡤⢾⣯⡴⣆⠀⣠⠾⢄⣀⣠⠴⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠙⠛⠁⠀⠈⠉⠉⠁⠘⠳⠞⠋⠀⠈⠛⠉⠀⠳⠞⠃⠀⠀⠉⠀⠈⠉⠁⠀⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢀⡄⠀⠄⢠⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠔⡄⠀⠀⠀⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠉⣛⠀⠀⠁⠠⣄⠈⢢⡀
⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⢀⡴⠃⠀⠂⢀⠠⠊⠀⠀⠀⠀⠀⢀⠠⠠⠅⢤⡀⠀⠀⠀⢀⢤⡖⠀⣰⡖⠀⠀⠀⠴⠊⡀⠀⠃⠀⠈⠄
⠀⠀⠀⠹⢶⡔⠚⠤⠰⡆⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠊⢙⡇⠀⠀⠘⠀⠀⠒⠉⠀⠀⠀⠴⠋
⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠉⠉⠀⠀⣤⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠠⢂⠀⠈⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢔⠀⠀⠠⠐⠊⠁⠀⢀⠄⠀⠀⠐⠀⠀⠈⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡰⠊⠉⠁
⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠬⠆⠊
⠀⠀⣤⠀⠀⠀⡇⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠰⠆⠀⠀⠀⠀⠚⢲⠇
⠀⠀⠙⣧⠀⢀⡼⢇⣀⣀⡀⠑⠀⠀⠘⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢠⠈⠂⠀⢰⢦⡀
⠀⠀⠀⠘⣧⠸⡅⠻⠩⡉⢻⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠉⠤⡞⠙⢃⠃
⠀⠀⠀⠀⠘⣧⠈⠒⢒⣚⡏⠀⢸⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠉⢻⡦
⠀⠀⠀⠀⠀⠘⣧⠀⠈⠛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠙⠁⠀⠠⡆
⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⡀⠐⠚⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⡤⠤⠔⠀⠀⠈⠛⣋⠉⠀⠀⠀⢀⣀⣠⠀⠐⠁
⠀⠀⠀⠀⠀⢈⠁⠀⠆⠨⠈⠡⠐⢈⠀⠄⠀⠀⠈⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⢰⢫⣤⣄⠇⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⢁⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⢠⣤⠀⠐⠀⠈⡉⠃⠀⠐⠀⠂⡄
⠀⠀⠀⠀⡏⡿⢈⡗⠂⠀⠀⠀⠀⠀⠈⠰⠆⠀⢀⡨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⡄⠀⠀⠉
⠀⠀⠀⠐⣧⣨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⡿⠀⢀⣀
⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡶⢦⡀⠀⠉⠉
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙

⡏⠭⠭⠍⠥⠅⠤⠬⠭⠌⠤⠤⠥⠤⠭⠬⠭⠭⠬⠭⠌⠍⠭⠩⠩⠩⠥⠥⠤⠤⠤⠤⠤⠠⠥⠨⠥⠡⠁⢀⡉⠉⠉⠭⠉
⡇⣠⠶⠞⠁⢈⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠓⠢
⡇⡇⢀⠈⠁⢠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⡧⠶⢤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⡐⡂⠀⡀⢰
⠀⢇⣙⣣⡼⣚⣽⣷⣦⣤⡤⠤⠤⠰⠴⠒⠂⠈⠀⠐⠋⢰⣁⠀⠀⠀⠁⠁⠀⠀⠐⠂⠀⠀⣀⣠⡿⣿⣦⣙⠦⣈⣃⠟
⠀⠀⠙⣷⢾⢿⡋⠛⠯⣝⡙⠛⠛⠂⣠⢰⣤⢊⣇⣶⣰⠷⣶⠟⣦⡖⠶⠖⣆⣤⡉⠁⠀⠀⣉⠥⠞⢋⣿⡿⣿⡏⠁
⡀⠀⠀⡟⢸⢿⠛⠢⣄⣤⡈⠀⠀⠠⡷⣅⣹⡰⢦⣤⠶⣄⡷⣠⠄⣤⣜⣈⣧⡨⢐⠀⠀⠉⠀⣀⠴⠊⠀⣍⠻⣧
⠀⠀⠀⡗⢸⣿⠀⠀⠀⢉⡑⠒⠒⠒⢳⡽⢿⣶⢬⣹⠶⠧⢾⠼⣓⣿⠤⢰⣿⢁⠚⠒⠒⠉⢁⡄⠀⠀⢠⡋⡄⣿
⠀⠀⢸⡏⢸⡿⡄⠀⠀⠘⠇⠀⠀⣦⠀⠉⢻⣥⣭⠒⣂⣻⣽⣏⣱⡖⡵⢚⡏⠀⠀⡆⠀⠀⠘⠀⠀⠀⢨⡅⡇⣟⡆
⠀⠀⠸⣇⠀⣿⡇⣠⠀⠀⠀⣤⢤⡤⡤⠤⣤⠿⠶⠿⠯⠿⢽⡽⢿⣿⣶⣿⣤⣤⣤⣤⣤⣄⠀⠀⠀⣴⠸⠃⡇⣏
⠀⠀⡴⣿⢠⢻⡇⠛⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠃⣸⠀⡇⣿⠀⡀
⠀⢀⡇⠂⠈⢸⣷⠀⠀⢰⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠃⠀⠀⡧⢸⠇⢿⠀⡇
⠀⢸⡇⡀⠘⢸⣿⠀⠀⠀⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⡇⠘⠀⠀⢰⡇
⠀⢸⠁⠇⠀⠀⣶⡀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣤⢠⣇⠀⠀⠀⠀⠇
⠀⢸⣀⣤⡤⡾⢟⡆⣻⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⢨⣍⠛⠲⣄⠃⡄
⠀⢸⠋⠃⠀⢐⡹⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⢬⣙⠲⢼⣈⡃⢸
⠀⣎⣤⠶⠛⠛⡇⠀⠀⢀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡀⠀⠀⢸⡛⠛⡳⣮⣳⣸
⠀⣿⣒⣒⣀⣀⡇⠀⠀⠘⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠈⠇⠀⠀⢸⣥⣤⣄⣈⠝⢹
⠸⢈⠻⠉⡍⠉⡃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⢸⡇⠄⢈⣀⡀⣸
⠀⠈⡀⠀⣡⡾⠁⠀⣶⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⣴⠀⠀⠙⢶⣬⡛⣇⣾
⠀⠘⣣⠞⠋⣀⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠹⠄⠀⢠⠀⠙⠿⡿⢿
⠀⢰⡏⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠀⠀⠀⡀
⠀⠈⠻⣦⣀⠁⠀⠀⠀⠈⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡴⠀⠀⠀⠀⢀⣠⡞⠃
⠀⠀⠀⠈⠙⠻⠶⣦⣤⣀⠁⠘⢧⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⡴⠋⠀⢁⣠⣤⡶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢦⡀⠀⠈⠉⠉⠉⡍⠉⠙⢦⣀⡴⠋⠉⠉⠉⡅⠀⠀⠚⠁⣠⡴⠟⠋⠉
⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⢶⠀⠙⠷⢦⣤⣤⣤⣤⣤⣄⣀⣙⣀⣀⣠⣤⣤⣥⣤⣤⠶⠞⠃⢰⠃⠈⠛⠚⠛⠛⠛⠓⠓⡆
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠀⠀⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⢤⠸⠤⠤⠤⠤⠤⠤⠤⠤⠾⠀⠂⠀⠀⠄⠂⠀⠀⠐⠀⠂⠂⠐⠐⠀⠀⠀⠂⠀⠀⠀⠀⠒⠘⠒⠒⠒⠒⠒⢒⣒⣒⣒⣃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣴⣶⡖⠒⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⠿⢿⣛⣛⣯⣭⣿⣿⣶⣟⡛⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠽⢿⣿⢟⣛⣛⣋⣩⣭⢭⣤⣖⣲⣮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣤⣄⣈⠉⠀⠁⠀⢠⣾⣿⣿⣿⣿⣿⣿⣣⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠿⣿⣿⣿⡖⠀⡇⣰⣿⣿⣿⣿⣿⣿⡿⣧⣟⡻⣫⣻⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣤⣭⡭⠀⣿⣿⣿⣿⣿⣿⣿⣿⣽⣟⣿⣿⣝⢿⣝⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣁⠘⣿⣿⣿⣿⣿⣿⣿⣽⢗⣾⣿⣿⣿⣮⣿⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⣽⢯⣿⣿⣿⣿⣿⣿⣿⡽⣷⡿⣿⣿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⡁⠀⣿⣿⣿⣿⣿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣽⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣬⣭⣙⣉⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠖⠮⡻⣿⣿⣿⣷⣿⣿⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⠏⠀⠀⠀⢰⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣽⣿⣿⣿⣿⣿⣿⡸⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⣿⣿⡛⠀⣿⢿⣿⣿⣿⣿⣿⣿⣯⡣⣀⣀⣴⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣙⠻⠿⢿⣅⠀⡿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⠖⠀⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠆⠀⣿⣿⣿⣿⣷⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣻⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠻⢿⣿⣿⣂⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣶⡤⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⣿⣿⣿⣿⠟⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠿⠿⣛⣛⣯⡤⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣝⠿⢿⣿⠀⢀⡇⠀⠙⢿⡿⠿⢿⣛⣻⣭⠽⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠤⠴⠧⠤⠶⠖⠛⠛⠉⠉


⣿⠿⢿⢿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢸⠀⡏⢠⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠀⠁⣀⣀⣀⡀⠀⠊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠊⠀⠀⠀⠈⢢⠀⢁⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢸⠀⠀⣀⡀⠀⡸⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠉⠑⠤⠤⡑⠤⠤⠤⠖⠁⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡀⠀⠀⠀⠀⢀⠤⠠⢀⠔⠚⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⠢⠖⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠲⢤⠞⠁⣀⠴⠒⠒⠒⠲⢀⡴⠚⠉⠀⠈⠉⠓⢄⠱⠤⠠⠤⣄⠀⠀⡀⠠⠤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠊⠀⠐⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠁⠉
⠀⠀⠀⢀⡴⠒⠒⠀⠀⢀⡤⣖⠋⠩⡍⠉⠉⠓⠲⣤⡀⢀⠔⠋⠁⠀⠀⠈⠙⠦⣄
⠀⠀⢠⠏⠀⠂⠀⠀⡰⠋⠀⠃⠘⠐⣇⠀⠀⠀⣍⠁⠛⣎⠀⠀⢀⠄⠚⢆⡀⠀⠘⣦
⠀⠀⢸⠀⠀⠀⠀⠰⠁⠀⠀⠐⠐⠉⠁⠀⠀⠀⠀⠓⠀⢸⡀⠀⠈⠢⣉⠔⠁⢀⠀⢸
⠀⠀⠸⡄⠀⠀⠀⢀⠀⠀⣀⠀⡀⠀⠸⡄⡄⠀⡀⠀⠀⢸⡇⠀⠀⠀⠀⡀⠀⠀⠀⡼
⠀⠀⠀⠙⢦⣄⡀⠸⡄⠀⡆⢂⡑⠀⠀⠣⠂⠈⠁⠀⠀⡼⠀⢤⣀⣀⣘⣑⣠⡶⠚⠁
⠀⠀⠀⠀⠀⠀⣄⠀⡙⢦⡙⠃⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⡀⢰⠀⢈⡭⠁
⠀⠀⠀⠀⠀⠀⠘⡆⠃⡴⠈⣝⠒⠒⠒⠒⠂⡀⠀⠀⠀⠀⢱⠈⢠⡎
⠀⠀⠀⠀⠀⠀⢀⡇⢰⡇⠀⠈⢦⠀⡏⢠⠞⠀⠀⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⡸⠀⢸⠇⠀⠀⢸⠀⢰⠋⠀⠀⠀⠀⠀⠀⢰⠃⢸
⠀⠀⠀⠀⡤⠖⠒⠒⠶⢄⠀⠀⢸⠀⢸⢀⣄⣀⡀⢀⡤⠤⡼⠀⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡟⠀⢸⠀⠀⠀⠉⠳⠄⠀⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠃⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⡀⠈⡄⠀⠀⠠⠀⡠⠆⢀⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠄⠀⠀⠀⠀⠈⠊⠀⠘⠁⠐⠀⠀⠀⢈⠀⠈⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡖⠠⠤⠀⠠⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠢⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⡀⠘⠂⣾⣿⣿⣿⣿⣿⣶⠀⠂⢂⣤⣤⣤⣀⡒⠀⠀⢁⠂⠄⡀⠀⠘⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⡠⠲⣹⣿⣿⣿⣿⣿⡿⣠⡀⣾⣿⣿⣿⣿⣿⣿⣶⢀⡟⠀⠌⠂⡀⠀⠀⠀⡠⢁
⠀⠀⠀⠀⠀⠀⠒⡂⠀⠂⠀⠠⠊⠀⠀⠙⠿⣿⣿⡟⠉⠀⠁⠁⠻⣿⣿⣿⣿⣿⣿⢋⠞⠙⠢⢄⡀⠈⠁⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠠⠈⠀⠀⠉⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠮⠭⠿⠭⠖⠁⠀⠀⠀⠀⠉⠲⢄⣀⣤⠁⠀⣀
⠀⠀⠀⠀⠀⠐⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠁⠋
⠀⠀⠠⢆⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠋⠐⢄⠀⡀
⠀⠀⠄⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⢂⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⢠⠊⠀⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢄⠠⠄
⠀⠀⠠⠀⠁⠀⢀⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠡
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠑⠀⢀⠃⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⣁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⢀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁⢀⠀⢨⠀⠀⠠⠌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⢠⠀⠀⠀⢀⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⡀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠂⢀⠠⠇⠀⠠⠒⡀⠀⠀⠜⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠁⠀⠈⠐⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠇⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠤⠄⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠂⠀⠈⠉⠉⠒⠂⠄⡀⠀⠀⠀⡀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠑⠶⠚⠁⢀⣄⠐⠁⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠐⠲⣄⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⢰⠁⡌⠀⠀⠆⠀⠈⠉⠉⠉⠀⠉⢉⠇⠀⠐⡏⠀⠀⠀⠀⠙⠢⠤⠤⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⡤⠼⢧⡇⠀⠀⣇⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠘⠦⣀⣉⣁⡤⠚⠁⠀⠀⠀⠀⢈⡴
⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⠶⠉
⠐⠀⡱⠐⠒⠈⠀⠀⠀⠁⠒⠠⠤⣩⣐⠠⢄⣀⡙⠦⣄⣀⠀⠀⠀⠀⠀⣀⣠⠞⠁⠀⠀⢀⠠⢐⡠⠆⠑⢄⡀⠀⡀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠥⣁⣉⡹⢛⣛⡻⠛⢛⡋⠀⠤⠤⠄⠒⠂⠉⠁⠀⠀⠀⠀⠉⠈⠃⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⢠⡄⢀⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠶⠶⠤⠤⠴⠖⠋⠡⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠠⠀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣠⣤⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢄⣴⣶⠐⢀⢀⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡏⢉⡀⣀
⠀⠀⠀⠀⣴⡿⢷⣶⣤⡀⠀⠀⠀⠄⠀⣞⣵⠄⠀⠀⠐⣺⣿⣿⠀⢁⡌⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⣮⡟⠁
⠀⠀⠀⢸⣿⣴⠾⠛⠋⠀⠀⠀⠀⠃⣴⣶⡇⠀⠀⠀⢰⣿⣿⣿⠀⠈⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣝⢸⢆
⠀⠀⠀⣨⠋⠉⠀⠀⠀⠀⠀⠀⠈⠿⠟⠿⡏⠀⠀⠀⠈⢿⣿⣿⠀⣀⠘⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣷⣾⣿
⠀⠀⢠⣿⣥⣘⢦⠀⠀⠀⠀⣀⣀⣬⣥⡌⠁⣠⣠⣄⣠⡬⣽⣯⡀⠀⠧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠎⠹⢿⣷⣿⣿⣿⣿
⠀⠀⣿⣿⢳⢈⡟⢇⠀⠓⠚⠓⠒⠒⠒⠓⠙⠓⠒⠒⠒⠒⠒⠒⠒⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣀⣀⣀⣿⣿⣿⣿⣿⣿
⢀⠀⢹⣿⣿⢰⡇⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⡃⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⢟⣿⣟⣿⡿
⠀⠀⢸⣿⣿⣾⡇⢰⡇⢀⢀⠀⠀⢀⠀⠀⠀⡀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿
⡄⣰⢾⣿⣿⣦⣻⡿⠃⣤⡖⠂⢴⣾⣃⣾⣛⣷⣶⢧⣄⣿⣿⣿⣿⣿⣷⣾⣿⣿⣶⣶⣤⣤⣿⣿⣿⣶⣽⣿⣿⣿⣿⡿⣿⣋⣿⣯⣿⣿⣿⣿⣿
⠀⢻⣿⣿⠋⠍⠳⠂⡀⣗⣍⣺⣟⣉⣙⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢛⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡠⣬⢿⣧⡖⠋⢀⣴⣰⣿⣟⡛⡛⠻⡛⡛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡿⠷⠾⢷⠶⣶⡶⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣴⣾⣿⡭⠆⢰⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢱⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⡿⣾⣿⡿⣷⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⡟⣿⣿⣿⢘⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡏⣽
⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⣻⢿⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⣿⣿⣧⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣳⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣙⡻⠿⡟⢿⣷⣮
⣷⠇⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠟⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣘⣟⢿⣿⡆⠘⢚⡛⠉⠉

⠀⠀⠀⠀⠀⡀⠐⠉⠩⠒⣄⠈⡀⢀⡀⠀⠂⠀⠀⠄⢀⠀⠀⠀⣀⠀⠀⠀⠄⠀⣀⠀⠠
⠀⠀⠀⠀⠘⠀⠀⢠⠊⠨⠱⠀⠂⠀⠀⠂⠠⠁⡠⠤⣿⡷⣼⣦⠈⠴⣦⡀⠀⣀⡁⢣⣦⡀⠀⢐⠀⠀⠀⠠⠀⠄⢀
⠀⠀⠀⢰⠀⠀⠀⡈⠧⠐⠈⠀⠢⠀⠀⠀⠀⠘⠃⠄⠈⠳⠈⡙⢤⣤⡿⠃⠀⣸⣿⢋⣭⣶⣤⠤⠈⢢⠀⠀⢀
⠀⠀⠀⢸⠀⠀⠀⠐⠇⠂⠀⠀⠀⣿⣿⣶⣦⣀⠨⡀⠌⠐⠦⠹⠐⢌⠠⠂⠀⡙⠂⣿⡿⠋⠀⠀⠀⠃⠡⠀⠀⠐⡀
⠀⠀⠀⠀⠁⠀⢀⠀⠐⠄⠀⣁⠔⠋⠏⡟⡿⡙⠷⣑⡄⠀⠄⠀⢴⠁⠀⠀⡀⠰⢃⡋⠀⢠⠐⠊⣀⡿⣰⠄⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠐⡄⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⢡⡀⠦⠙⠀⠄⡀⠼⢥⡀⠰⢀⠄⠈⠓⡀⠐⠊⡉⢡⠇⠌⠸⠀⠀⡰⠂⠂⠀⠀⠀⡀
⠀⡄⠀⠠⠀⠘⠀⠤⠀⠀⠀⠈⠀⠀⠀⠨⣄⢀⠰⠀⠄⢅⠘⠖⢄⠂⠐⣦⣈⠜⠀⠀⠀⠀⢰⠁⠈⠀⠀⠀⠛⢀⣉⡠⢀⠀⠠⠄
⠀⠐⠀⢠⡾⠋⡀⠀⣶⠆⢠⠀⡀⠀⠀⠤⠀⢦⠀⠀⡘⠂⠑⢀⣨⠬⠇⡫⢀⠔⠀⠀⠄⠀⢄⠘⠠⠀⢀⣀⣴⠏⣠⡘⠠⡑⠀⠀⠄
⠀⠀⠀⠜⠇⠀⠀⣾⣿⠉⠁⠀⢹⣿⣷⣷⡶⠂⠀⠉⠛⠓⢄⡘⠐⠉⠚⠉⠀⠐⠁⠀⠀⠀⠉⢈⠱⠚⠁⠈⠙⢂⣑⣠⣄⠇⠀⠈
⠀⠀⠀⠈⠈⠻⣶⣷⠎⢠⠒⢀⠀⠈⠉⡏⠀⠠⢁⠀⠠⠠⣨⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠁⠄⠁⠰⠈⠙⣋⠀⢀⡄
⠀⠀⠀⡀⣅⣼⣿⡿⡀⣸⠙⠀⠌⠠⣀⡀⢀⡰⠁⠀⢀⠀⠐⠈⠂⢀⡀⠄⠐⠈⠀⠀⠀⡀⠄⠂⠉⠀⠀⣠⠐⠁⠀⠇⢠⣴⣿⠟
⠀⠀⠀⠀⢈⡿⠋⠀⠉⠀⠀⠀⠀⠀⠆⠀⢤⡄⠀⠀⢈⠁⠠⠀⠀⠀⠀⠀⢀⠡⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠰⣿⣯⡿⠂
⠀⠀⡀⠘⡄⡰⢀⠂⢀⣤⣤⡔⣈⣬⠃⠈⠁⠁⡀⡠⠰⠀⠀⠀⠐⠄⠂⠉⠀⠐⠀⠀⠀⠙⠁⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠙⠁⠁
⠀⠀⠀⠀⠇⠀⡈⢠⡿⠛⠻⠀⢿⣤⠀⠀⠀⣾⠀⣟⠛⠒⠁⠠⢀⣐⡀⠠⠐⠰⠀⠀⣐⠆⠀⠄⠀⠀⠀⡀
⠀⠀⠀⢠⠉⠀⠀⠀⠻⠾⢃⡗⠂⠉⡃⠀⠀⠸⡄⠘⣄⠀⠤⠰⠊⠁⠀⠀⠰⢀⠐⡠⠀⢂⣵⣷⣦⠄⢀⠀⠠⠁⠀⠀⡀⠀⠠
⠀⠀⠀⠁⠀⠀⢀⡰⠀⠠⣿⠁⠁⠈⠁⠀⠀⢀⡎⠀⡀⠑⠒⢀⡶⠀⠀⠀⠄⠀⣼⡿⠟⠁⠀⣉⡣⡄⠊⣼⠃⡀⠠⠈
⠀⠀⠀⠀⠀⠈⠻⣿⣶⣟⡌⠀⠀⠀⠀⡠⣶⣟⣿⣦⣅⡀⠀⠀⠂⠀⣁⣀⡶⣓⡹⣶⣥⣀⡡⡌⠁⠀⢞⣛⣋⣠⣶⠄
⠀⠀⠀⡐⡏⠀⠀⡙⢿⣟⣂⣀⡀⡀⡉⠡⣸⠋⠉⠀⠈⠀⠁⠈⠩⠃⡸⡐⠒⠤⠐⣟⠟⠪⠀⠈⣢⠲⢞⡛⠛⠉⠁⠀⣀⡀⢀⠄
⠀⠀⠀⡆⠌⡠⠉⠀⢀⢻⣿⡟⠟⠋⠁⡩⠀⢀⠔⡠⣆⠘⠇⣀⢠⢹⣙⣤⡠⣤⢎⠑⠄⠻⢀⠖⠂⠁⡀⠀⢖⢂⣤⡾⠿⠓
⠀⠀⠀⡀⢠⠘⠀⠀⠐⠈⠀⠁⠰⢀⣠⣶⣿⡆⠒⣙⠔⢢⣀⢸⠀⢨⠫⠔⠨⠒⡡⢳⣄⣐⠋⠁⠄⣂⢁⡾⠻⠆⡿⠃
⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⠟⠛⢿⣷⣄⠈⠀⠐⠉⠉⣠⡖⠡⠊⠕⢊⠀⠀⣸⠆⠐⠸⡟⠁⠛⠿⠛⠰⠁⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠐⠠⠌⠋⠈⠙⠿⣿⣿⣿⣆⠑⢉⢆⠡⡈⢌⡜⠀⠀⣀⠀⠁⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠑⠁⠀⠙⠂⠡⠀⠔⠺⢿⣿⡿⠃⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⡶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢶⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠻⠿⠿⠿⠿⠿⠿⠶⠶⢶⣶⠶⠶⠶⠶⠿⠿⠿⠟⠟⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡄⠀⠀⡾⣕⢄⠀⠀⠀⣀⣤⠀⠀⠀⠀⣠⢧⡄⠀⠎⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⡁⠀⠙⢦⣼⠟⠀⠀⡐⠿⠿⠃⠀⠀⠻⣧⡴⠋⠀⣌⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠲⠶⠶⠥⣥⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠶⠶⠒⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣧⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡠⢑⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⣤⣤⣤⣤⣬⣤⣤⣤⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡍⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠄⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠠⠓⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠰⠁⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠷⢶⣶⣶⣶⣶⣶⣶⣶⠶⠾⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢢⡀⠀⠀⠀⢀⠇⠸⡀⠀⠀⠀⢀⠔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⢄⠀⠀⡼⠀⠀⢧⠀⠀⡰⠋⠀⡇
⠀⠀⠀⠀⠀⠀⢻⡒⠤⣀⠀⠀⠸⠀⠀⠈⠳⣠⠃⠀⠀⠘⣤⠎⠀⠀⢠⠁⠀⠀⣀⠤⢒⡟
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠙⠢⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠔⠋⠁⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⢀⠤⠒⠊⠉⠉⠉⠉⠑⠒⠤⡀⠀⠀⠀⠀⢀⠎
⠀⠘⠳⣒⠒⠒⠒⠂⠤⠵⠀⠀⣠⠞⡡⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⠀⠀⠮⠔⠒⠒⠒⠒⣨⠟⠁
⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⡔⢡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⢘⣢⠀⠀⣸⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣔⡁
⣀⡠⠤⠒⠒⠉⠁⠀⠀⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠒⠒⠤⢄
⠈⠉⠑⠒⠤⢤⣀⡀⠀⠀⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⡤⠤⠒⠚⠉
⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠛⢄
⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠱⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠔⠥⠤⠤⠒⠒⠒⡲⠀⠀⠈⠢⣑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⢖⠒⠒⠒⠒⠤⠬⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡀⠈⠑⠲⠭⠍⣉⣀⠠⠤⠖⠊⠁⢀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⡤⠒⠉⡇⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⢸⠉⠒⢤⣀⠀⢣
⠀⠀⠀⠀⠀⠀⡼⠔⠊⠁⠀⠀⢠⠁⠀⢀⡔⠙⡄⠀⠀⢠⠋⠢⡀⠀⠀⡆⠀⠀⠈⠑⠢⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠋⠀⠀⢱⠀⠀⡎⠀⠀⠙⢆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠜⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠱⣤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣐⣶⣦⣄⡀⣤⣸⣧⣠⠀⣠⣤⣶⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣷⣤⢤⣤⣈⠙⢿⣿⣿⣿⣿⡾⠛⢉⣠⣤⣴⣾⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣞⠉⠉⠉⠉⠙⠳⠈⠻⣿⡄⢿⣿⣿⣿⢁⣾⠟⠉⠐⠋⠉⠉⠉⢙⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠙⢻⣿⣿⣿⣿⣶⣶⣤⡀⠀⠈⢻⣼⣿⣿⡇⠞⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⡟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣖⠒⢛⣉⣤⡦⠍⠍⡉⢿⣿⣿⣄⢿⣌⣦⣿⣿⣠⢠⡾⣠⣾⣿⡯⠉⡉⠉⢬⣥⣉⡓⢒⢲⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⢿⣿⣿⣯⣴⣦⣴⣶⣶⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣶⣶⣶⣶⣮⣽⣿⡿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⡲⠄⠔⠒⣚⣿⣿⣿⠋⠙⣻⣿⣿⡿⠿⠿⠻⠿⢻⣿⣿⡟⠿⠿⠿⠿⢿⣿⣿⡟⠋⠹⣿⣿⣿⣟⠒⠦⠤⢖⣲⣶⣦⡀
⠀⠀⠀⠀⠀⢀⣿⣯⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣦⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣼⣿⣿⣿⣿⣟⠿⠿⠿⣿⣿⣽⣿⡀
⠀⠀⠀⠀⠀⢸⣿⠋⠁⠀⢀⣀⣤⣄⣶⣿⣿⣿⣿⣿⣿⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⣿⣿⣿⣿⣿⣿⣶⣠⣤⣀⣀⡀⠉⠛⣿⡇
⠀⠀⠀⠀⠀⣾⣇⢀⣴⣿⣿⠿⢟⡛⠝⠛⣩⣿⣿⣿⣿⣿⣶⣿⣿⣦⣀⠀⠀⠀⠀⢀⠀⣀⣴⣾⣿⣶⣿⣿⣿⣿⣿⣯⡻⢯⣟⡿⢿⣿⣿⣷⡄⣼⣷
⠀⠀⠀⠀⣸⣿⣿⣿⡿⠛⢁⠉⠁⠄⣾⡿⢟⣽⣿⣿⡇⠙⠿⣿⣾⢽⣿⣿⣿⠠⠤⢿⣿⣿⣿⣧⣿⠿⠃⢸⣿⣿⣮⡙⢿⣷⠠⠀⠁⠈⠛⢿⣿⣿⣿⡆
⠀⠀⠀⣰⣿⣿⡟⠁⠐⠈⠁⣀⣴⡿⠋⠀⢉⣵⣿⣿⣷⣄⠀⠀⠉⠻⣿⣿⠃⠀⠀⠈⢿⣿⡟⠋⠁⠀⣀⣼⣿⣿⣮⡉⠀⠙⢿⣦⣀⠀⠁⠂⠙⢿⣿⣿⣄
⠀⠠⠾⠛⣹⣯⡁⠀⠀⢠⣶⣿⠋⠀⠀⣤⣿⣿⣿⣿⣿⣿⣷⣷⡆⠀⣹⠃⠀⠀⠀⠀⠈⣯⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠹⣿⣷⣄⠀⠀⢈⣿⡏⠛⠦
⠀⠀⠀⢠⣿⣿⣷⣴⣶⣿⡿⠁⢀⣔⣴⡿⢟⣽⠟⣉⣽⣿⣿⣿⡇⠀⢿⣂⠀⠀⠀⠀⣀⣿⠂⢨⣿⣿⣿⣿⣍⡛⢿⡙⢿⣦⣲⡀⠈⢿⣿⣷⣴⣾⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⢀⢮⡿⢋⣴⣞⣥⣾⣿⣿⣿⣿⡿⠁⠀⠀⢫⣿⣶⣶⣿⡿⠃⠀⠀⢻⣿⣿⣿⣿⣿⣦⣹⣦⡙⢿⣷⡀⠈⢿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡋⣎⣿⣿⠟⣵⣿⣿⢟⣿⣿⣿⣿⣧⠤⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠤⣼⣿⣿⣿⣿⠻⣿⣿⣯⠻⣾⣷⡇⡹⡌⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⡏⣑⣸⣿⠃⣸⣿⡟⠁⣾⣿⣿⣿⣯⠉⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠈⣟⢿⣿⣿⣷⡈⠻⣿⣧⠸⣿⣿⠉⣻⣼⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣿⠏⠀⠈⣿⣿⣿⡿⣿⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣤⣀⠀⠀⣸⣿⣿⣿⣿⠁⠀⢻⣿⠃⢹⣷⣿⣿⣿⡟⣿⣿⡿
⠀⠀⠀⠀⢹⣿⡟⣾⣿⣿⡟⣿⢳⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⣤⣾⣿⡿⡟⠻⠿⠿⠛⡿⣿⣿⣷⣦⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⡎⣿⢻⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠈⣿⣧⢿⣿⣿⣇⣿⢸⣿⡿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡏⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠀⠀⢹⣿⣷⣿⣿⣿⣿⡇⣿⡿
⠀⠀⠀⠀⠀⠹⣇⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣄⠀⠀⢠⣀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⣿⠃
⠀⠀⠀⠀⠀⠀⠹⡄⢿⣿⣿⣿⣿⣿⡇⠀⠀⡿⢸⣿⡟⢻⣿⣿⡿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⡇⣽⠆⠀⢈⣿⣟⣿⣿⣿⡟⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⣿⣷⠀⠀⠁⢸⣿⡗⢸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⠛⢀⣿⠇⠸⠀⠀⢸⣿⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⣿⣿⣿⣯⣀⣀⠈⣿⣷⡀⠹⣿⣷⢠⣿⣿⣯⢻⡟⢿⣿⣿⠠⣈⣿⡟⠀⣼⡿⠀⣀⡄⣴⣿⣿⣿⢹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧⠸⣿⣧⠀⣿⣿⣿⣿⣿⣿⠀⢀⢏⣿⣿⣿⣼⣿⡑⣰⣿⣿⣼⣿⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣷⣽⣿⣦⢙⣿⣿⣿⣿⣗⠀⠀⣻⣿⣿⣿⣿⡥⣰⣿⣯⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⢿⣿⣿⣿⣿⣿⡛⢿⣿⣿⣿⣷⣾⣿⣿⣿⠿⢋⣾⣿⣿⣿⣿⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣉⠛⠿⢿⣿⣿⣿⣿⣿⣿⡷⠾⠛⣩⠴⠟⠋⠁

⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⢄⠀⠀⢀⠇
⠘⢻⣿⣾⠒⠒⠉⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⢠⣤⣤⣾⣷⣶⣦⡀⠀⠀⠀⠀⠀⡀⢠⡀⡀
⠀⣺⣿⣿⡇⠀⠀⠀⠀⠞⠛⠛⢿⣿⣧⡄⠀⣠⣼⣿⣿⣿⡿⠟⠛⢻⣷⣦⡄⠀⠠⣰⠿⠻⣿⣶⣦⣤⠀⢄⡀
⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇⠈⢿⣿⣿⠟⠃⠀⠀⠀⠀⣿⡟⠁⠀⠀⠁⠀⠀⠹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠋⠀⠀⠘⠿⢿⠧⢀⠑⠀⠊⠘⠋⠀⠀⠀⠀⠀⣄⣀⠀⢀⣽⠿⢋⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢃⠀⠀⠀⠀⠀⢀⣠⣾⣿⣧⣤⣦⡄⠀⠀⣴⡄⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣀⣼⠀⠀⠀⠀⠐⣿⣿⣿⣿⣽⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠻⠿⠀⠀⠀⠀⠀⠈⠛⠛⠿⠛⠛⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠲⢶⣶⣤⡀⠀⠀⠀⠀⠀⠀⣴⣶⠖⠛⠻⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠈⠁⢳⡦⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣹⡗⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⢀⣸⣿⠆⡀⣀⡀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⣉⠁⠐⠳⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⡿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣏⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠤⠤⠷⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠙⠻⠿⠟

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠤⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢧⣠⡿⠁⣠⣽⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⣸⣿⣿⣵⠀⣀⡀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠛⠃⠰⠋⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣺⡟⢋⣠⠼⢿⣿⣾⣿⢗⠝⠋⠁⠀⠀⠀⢀⠴⠿⠿⢛⠁⠀⠠⠖⠟⠉⣧⠐⠎⠓⠀⠀⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠋⡄⡶⣷⣿⣿⣿⣿⢧⡀⠂⠄⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⢰⠁⠀⡘⢧⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣥⠟⢹⡿⢹⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠐⠃⠈⠆⠀⠀⡄⠀⠀⣧⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⢼⣧⠌⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣡⠀⠀⢹⡀⠀⡀⠁⢰⠃⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠉⠀⠀⠀⠉⠁⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣧⠀⡀⠀⡞⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⢠⠃
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⣰⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢼⠦⣄⣀⣠⢿⠀⣿
⠀⠠⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠜⠓⠀⠾⢊⡴⢂⣉⣿⡄⣿⡀
⢀⣈⣽⣿⣷⣷⣤⠄⡀⠀⣠⡀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡰⠚⢙⠣⣃⠁⠚⣉⡐⢂⣔⣿⣿⣿⣷⠦⠄
⠀⢴⣿⣾⡿⠽⣿⡧⢹⣿⣿⣿⣿⡤⠀⡼⠙⠁⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣶⣿⣤⣴⣶⣮⣭⣥⣿⣿⣿⢿⣿⣿⢿⡟⠋
⠀⢸⣿⡟⠀⣀⡀⣸⣿⣿⠿⣯⠝⠂⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠻⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣭⢿⣄⠙⠆
⠀⣞⠏⠀⠉⠙⣿⠛⢉⡃⠈⠈⣀⠀⠀⡅⡀⢨⠀⠀⠀⠀⠀⠀⠀⢈⣉⣰⣾⡿⣽⣛⣉⡻⠦⠚⠛⠉⠉⠉⠉⠀⠹⣄
⠚⠁⠀⠀⠀⠀⠠⠀⠉⠀⠄⠠⢅⠀⠀⣧⡇⡈⠀⠀⠀⠐⠛⠙⠷⠿⠯⠶⠶⣿⣇⡱⠯⠻⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣉⡭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠖⡀⠀⠐⠂⠤⡀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⡏⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠑⠄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠉⠁⠀⠀⠈⠁⠀⠓⠠⢤⠂⠀⠈⠁⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⡐
⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⡠⠋⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠑⢄⠔
⢀⠔⠈⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⡰⠟⠛⠛⠛⠀⠀⠀⠀⠙⠛⠛⠻⢦⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⡀
⠃⠂⠉⠁⠢⡀⠀⠀⠌⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⡀⠀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⢡⠀⠁⠂⠀⠀⠊⢀
⠀⠀⠀⠀⠀⠱⡀⠈⠀⠀⠀⣤⣾⣿⢫⠃⣰⣿⣿⣿⣿⡟⠙⣿⣿⣿⣿⣿⡏⡏⢉⣷⣦⡀⠀⠀⠃⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠈⢻⣏⠇⢠⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⡿⡜⠀⣾⣿⠉⠀⠀⠀⢠⠀⢀⠠⠐⠁
⠀⠀⠀⠀⢀⠠⠄⡇⠀⠀⠀⠀⠈⠻⣄⣾⣿⣿⣿⡿⠋⠀⠀⠈⠿⣿⣿⣹⠁⣼⡿⠃⠀⠀⠀⠀⠸⠉
⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⢂
⠀⡔⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠰⣶⣦⡐⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⣠⣶⡞⠀⠀⠀⠀⢀⠂⠀⠡
⠈⢀⠂⠈⠁⠐⠢⠀⢱⡀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣶⡶⠶⠶⠶⢶⣶⣾⣿⠿⠋⠀⠀⠀⠀⢀⠎⠀⠀⠀⠑⠂⠄⠐
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠉⠛⠡⠤⠀⠀⠠⠤⠘⠋⠁⠀⠀⠀⠀⠀⢠⠮⢀⡀⠀⠀⢀⠠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠊⠁⢀
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢀⠤⠀⠀⠂⠁⠉⠐⠂⠀⠠⠤⠤⠄⠠⠀⠐⢅
⠀⠀⠀⠀⠀⠀⠀⠈⡄⢠⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠑⢄⠀⠀⠀⠢⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠜⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠘⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣐⣬⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠚⠛⠛⠛⠛⠛⠛⠳⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠖⠁⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⢠⠞⠁⠀⡴⠁⣰⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⣠⠞⣛⣉⠉⠉⠉⠉⠛⠳⢦⣄
⠀⠀⠀⠀⣰⠏⠀⠀⣸⠃⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⡍⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⢠⡏⠀⠀⠀⢿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢻⣄⡀⠉⠒⢦⣄⣀⠀⠀⠀⠀⢸⡷⠒⣟⣠⡴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠘⠓⠚⠛⢿⠛⣷⣿⣶⠶⢦⣤⣤⣤⣤⣤⣤⣤⣽⣶⡿⢿⣟⢦⠀⠀⠀⠉⠙⠛⠲⠶⢾⡷⢦⣸⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠹⢿⣿⣷⢟⡿⠁⣿⠀⠀⠸⣿⣿⣿⣷⣶⡟⡸⠃⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⣤⣤⠴⠞⠁⠀⠜⢷⣄⠀⠙⠻⠿⠿⢛⣜⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⡾⠋⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠠⡶⠂⠀⠀⠀⠝⠛⠶⣦⠶⠾⡋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢋⠀⠀⡆⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠳⢤⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⡇⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⡟⠀⣸⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣀⣀⣀⣤⣤⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⠃⢠⠇⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠦⠤⠤⠝⠉⣏⡿⠉⠙⠓⠶⠶⠖⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⡆⢀⠎⠀⡀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⣀⣀⠀⠀⣠⡀⠀⡠⢤⡰⡛⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢰⠀⡞⠀⢠⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⠉⢀⡏⠀⠑⠊⠀⠈⠉⠀⢸⡀⠉⠳⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡘⡄⣇⠀⠸⡀⢻⡀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⣠⠤⣊⠷⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣤⣴⡞⣅⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⡄⠀⠱⢤⣑⣦⣀⣀
⣔⠂⡀⠙⠀⠈⡂⠬⠴⠒⠋⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠙⢮⣑⠦⣄⡀⠀⢀⡴⢂⠄⢀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠶⠾⠿⠛⠋
⠠⢪⠄⡤⠀⠐⠁⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠈⠙⠲⠭⢭⡁⠀⠂⠘⢃⠶
⠀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⡐⡂⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣱⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣦⢤⠶⡶⣾⠛⠲⠶⠤⣤⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠸⢤⠇⡿⠀⠊⢋⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⢀⠋⠂⠑⡌⡰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢌⠀⠀⢠⡇⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢈⡑⠒⠲⠟⠶⣶⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⡾⣦⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⢩⡦⠀⠀⠀⠀⢀⡀⠀⡇⢸⠉⣻⠹⣇⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠤⢤⠞⠋⠁⠀⣠⠀⠀⣰⠇⠀⠀⠟⠁⢀⡇⢸⡴⢡⢀⣯⠟⣹⡁
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⣰⠇⠀⢀⡄⠀⠘⠀⠀⠀⠀⡀⣄⣤⠠⣼⡔⠉⠱⠙⢘⠅⡼⣉⢳⡀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢰⡇⠀⠀⠘⠇⠀⠀⣦⠀⡤⠂⢹⣾⢻⠀⡿⣌⡶⠤⣤⣬⡞⠁⠉⠀⣷
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠷⣦⣄⡇⠀⠀⡀⠀⠀⠀⠁⣼⢧⠀⠘⠛⠀⠀⠻⣯⡝⠻⣚⡟⠀⢰⠇⠀⢸
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠈⣷⠀⠀⣤⣤⡿⢶⠀⠀⠀⠀⠀⠀⠀⠉⢩⣥⠖⡇⠀⢀⠀⡼⠓⢆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣤⠶⠶⠌⣇⠀⠀⠀⠐⡇⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠋⢠⡇⠀⠀⢳
⠀⠀⠀⠈⣇⠀⠀⠀⣠⠟⠉⠀⠆⠀⠀⠘⢦⡀⠓⠆⠙⢮⡆⠀⠀⠙⢶⣆⠀⠀⢀⡴⠟⠂⢸⡇⠀⢀⡼⠀⢀⣀⣸⡄
⠀⠀⠀⠀⢹⣆⠀⣾⠉⠀⠀⣤⡀⠀⠓⠀⣀⣽⣦⣀⣤⠘⡇⠀⠐⢤⣀⡉⠀⠀⠠⣄⣤⠆⣸⢇⣠⠛⣦⠞⠛⠉⢉⡉⠓⢦⡀
⠀⠀⠀⢠⡿⠉⣿⠇⠀⣆⠀⠘⠃⠀⠀⡾⠁⠀⠀⠋⢼⣁⡙⢤⣀⣀⠉⠀⣀⠀⢠⣀⣠⠾⠛⠉⠀⠀⣿⠀⠺⠀⠘⠃⠀⠆⣧
⠀⠀⠀⣸⡇⠀⣯⠀⠀⠈⠁⠀⣠⣤⠀⣧⠀⠀⠀⠀⡆⠀⠉⠙⠺⠋⠉⢻⡙⠦⡼⠋⠀⠀⠀⠀⠀⢰⡏⠴⠆⠀⠰⠂⢠⣄⡿
⠀⠀⠀⢹⡇⠀⠻⣄⠰⠶⠂⠀⠀⠀⠀⠈⠙⢶⣤⣄⠠⣤⠤⠖⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⠘⠉⠉⠙⢢⣠⠞⠁⠀⡀
⠀⠀⠀⠘⣧⠀⠀⠙⠶⣄⣀⠐⠾⠃⠀⢀⣠⠎⠀⠉⠙⠮⡖⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⠀⣄⡀⠀⠀⠈⠙⠛⡿⠁
⠀⠀⠀⣼⠟⠣⡀⠀⠀⠀⠉⠛⠲⠶⠚⠋⠀⠀⠀⠀⠀⣠⠶⣦⡀⠀⣼⠃⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⣼⡴⠏⠙⠳⠶⠶⠖⠋
⠀⠀⣸⡏⢀⡲⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣷⠟⠁⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠸⣧⡈⠓⠀⢲⡌⠙⠲⢦⣤⣤⣤⣀⣀⣀⣀⣤⣤⣴⠿⠋⠁⠀⠀⢀⣠⣴⡾⠟⠋⠁
⠀⠀⠀⠘⠻⣦⣀⠀⠠⢤⠄⠘⠆⠀⠉⠉⠩⠭⠽⠿⢯⣥⣤⠤⠶⠶⠛⢋⡿⠃
⠀⠀⠀⠀⠀⠀⠙⠛⠲⣦⣤⣤⣀⣀⣙⣛⠀⠀⠶⠖⠀⠀⠟⠃⣀⣨⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣋⡉⠛⣯⣿⠙⠉⠙⣿⡿⠿⡿⢿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⠉⠛⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠃⠐⠲⠦⠤⢀⢤⣄⡀⢀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠈⠂⠀⠀⠀⣀⣴⣦⠳⠄⠀⠀⠀⠚⠓⠀⠀⠀⣰⣿⢋⠀⠀⠀⠠⠈⠒
⠀⠀⠀⠀⠀⠀⠙⠤⠄⡼⡶⠀⡄⠉⢻⠛⠻⣿⣿⠇⠀⠘⠛⠀⠀⣠⣴⠿⠛⢉⣀⣠⣤⣶⣶⣤⡀⠀⠀⠐⠶⠿⣡⠍⣿⢠⣄⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠒⠋⠀⠀⠈⠉⠀⠀⠀⢀⠀⠉⠁⢠⣖⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠈⠈⢃⣤⠞⠉⣤⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⢷⣄⠉⠻⣿⣿⣿⣿⣿⣿⣟⠻⣶⣄⡀⠀⠈⠉⠛⠁⠀⠉
⠀⠀⠀⠀⠀⢠⣖⣟⣋⣛⣛⡛⠋⠿⢩⠙⢷⡀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⣿⣏⠀⠀⠀⣤⠀⠀⠀⠝⣡⣞⣀⣼⣿⣿⡇⠀⠀⠀⠀⠆⠚⣆⠀⠀⠙⢷⣄⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠹⣿⣷⡤⣴⣶⣶⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠙⢷⣄⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠉⠉⠉⠉⣩⠉⠛⠻⣿⣿⡿⠀⠀⠉⠀⠀⠀⠂⠐⠀⠀⠐⠂⠀⠙⣷⣄⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⢉⣉⣤⣾⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠒⢄⠀⠡⠀⠀⠹⠋⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠛⠉⠁⢠⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⡠⠈⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡠⠎⠀⡀⠂⠀⠀⠒⠂⠀⠈⡠⣄⠀⠀⠀⠀⠀⠀⠀⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⡏⠁⠀⢤⠀⢑⡴⠆⠉⠀⠈⠅⠘⠂⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠋⠁⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠉⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⣤⡀⣀⠀⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋⠀⠀⠀⠀⠀⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉
⠀⠀⠀⠀⠀⠀⣠⣶⣛⣩⣅⣀⡀⠀⠀⠀⠖⡆⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡀
⠀⠀⠀⠀⣐⡾⠿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠈⢠⠒⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡀⠛⠛⠻⠿⠷⢿⣯⣽⡉⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠓⠒⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⠀⠀⢀⡴⠠
⠀⠀⠀⠉⢻⡒⠶⢷⣖⣦⣤⡄⣸⡟⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠁
⠀⠀⠀⠀⠸⠇⠀⠀⢸⠅⠉⠉⢸⡇⢀⡀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⡤⠤⠀⠀⠀⠀⠀⠠⡀⠀⣀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠸⢃⣜⠻⠖⠀⠀⠰⠀⠘⠀⠄⠀⠠⠄⠀⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣯⢠⡿
⠀⠀⠀⠀⠀⠀⠰⢥⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⡏⡤⣤⢈⣽⣏⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⡀⠠⠆⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠿⠓⠋⠻⠂⠰⣷⠆⠀⠀⢰⣿⣿⡟⠀⠀⠀⠋
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠘⠿⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⣶⣯⠸⣧⢟⣯⣜⠿⣿⢁⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡹⣿⣿⡟⣴⡌⢿⣿⡿⢣⣦⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⢀⣙⣛⣋⣛⡁⣁⣋⣉⣛⣙⡀⣘⣉⣋⣛⣃⡀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡙⢏⣼⣿⣿⣎⠿⣡⣿⣿⣧⡀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⢸⣿⣿⣿⣿⢃⣿⠛⣿⣿⠏⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⢠⣤⣤⣤⣤⢠⣤⣤⣤⣤⠄⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡆⣿⣿⣿⡟⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡆⣿⣿⣿⡏⣼⣿⣿⡿⠁⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⢸⣿⣿⢃⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⢸⣿⣿⢁⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⣿⡟⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⣿⡏⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠸⠣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢸⠡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣤⣀⡠⢀⣀⣤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⣤⣤⣤⡄⠠⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⠻⣯⣿⣿⣝⠇⠾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡜⣿⡟⣴⣷⡘⣷⢞⣼⣧⡀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⠰⣶⣶⣶⣶⢰⣶⣶⣶⡶⠂⠀⠀⠀⠀⠀⠀⠀⢐⣛⡛⣛⡊⢘⣛⣛⣛⠈⣚⣛⣛⣓⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡇⢿⣿⣿⡏⣼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣿⣧⢹⣿⣿⣿⢱⣿⣿⡿⠋⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠸⣿⣿⢡⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡎⣿⣿⡇⣿⣿⠟⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⢿⡟⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⢹⣿⢰⡿⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠸⠱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠎⠇⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠍⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢯⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣤⠀⠖⠡⠙⠛⠳⢽⣶⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠴⠣⠀⢃⠘⠂⠀⠈⣿⠁⠀⠀⠘⣿⣠⠀⠁⢯⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡰⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠠⣸⣿⣷⡀⠀⠀⡼⠡⠇⠀⠀⠇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣏⢷⣠⠛⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠘⢠⡏⣿⣻⣷⠀⠀⠇⠀⠀⠁⢠⢀⡀⠀⠀⠀⠀⠛⠀⠦⠌⣉⠻⢼⣏⣴⣣⡴⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡿⠀⠹⡟⠃⠀⠈⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⣩⡵⢈⣿⣶⣤⡙⣦⣤⣁⠀⠈⣿⠂⢀⠠⢒⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⠶⢾⡏⠠⠞⠈⢿⡈⢻⣸⣇⣙⣧⣴⣼⣿⡏⠶⢭⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⡀⠁⠀⠀⠘⣷⠀⠉⠉⠁⡰⢋⣴⣿⣿⣧⠘⡇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠹⠆⠀⠀⣟⡤⠋⣸⠛⠟⠘⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠀⢠⡏⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢉⣉⠁⠀⠀⡁⢀⡀⠀⠀⠀⠉⠉⠉⠉⢙⠃⠀⠀⠠⠤⠄⠀⠀⠀⠉⠉⠁⠐⠒⠠⠀⠀⠀⠀⠹⡇⠀⢾⠆⣾⢰⠚⠈⠉⡉⠈⠠⠤⠸⠋
⠀⠀⠀⠠⣤⡀⠘⠛⠻⠶⡶⣡⢤⣴⣷⣶⠀⠀⠀⠀⠀⣤⣤⣤⣤⠀⠀⠤⠈⠙⠁⢀⠀⠠⢀⣀⡀⠐⠒⠠⠄⠈⣷⡀⠐⢀⡟⠀⡛⠂⠀⠀⠀⠠⠿⠛⠃
⠀⠀⠀⠀⢴⣟⣩⣤⣶⣆⢀⡤⠀⠀⠈⠉⠁⢀⠀⠀⠀⠀⠀⠀⠛⠀⢤⣶⣆⣥⣄⣀⢀⢀⠀⠀⠀⠁⠀⠀⠙⠃⢙⠈⠂⢸⠃⠀⠶⠶⠶⠒⠂
⠀⠀⠀⠀⢠⣶⣤⡀⠀⠈⠸⣇⠀⠀⢴⣶⣶⣯⣤⣿⢠⢣⣶⡤⠄⡄⠀⠀⠛⠋⠉⠉⠀⠀⠤⠶⠶⣧⣶⣤⣠⣀⢈⠀⢄⣉⠀⢀⠀⣀⡀⢀⣤
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠴⡦⡴⠴⠆⢦⡄⢀⠀⣤⢀⣀⡈⠀⠈⠀⠀⠃⢠⠟⠛⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠈⠁⠉⠉⠁⠀⠀⣦⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠔⣧⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠁⠶⠶⠶⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⣠⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⣹⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠰⠆


⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣞⠁⣛⡦⠀⠀⢀⣠⠴⠚⠛⠳⢦⣄⡀⣴⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢋⣠⠴⠚⢉⣠⠴⣶⠀⠀⠀⠸⣏⡁⠀⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠚⢯⣀⠀⣿⠀⠀⠀⠀⠀⠻⣴⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠳⢦⣀⠈⠓⠿⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡟⠀⠀⣈⣷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠸⠷⠚⠋⠉⠀⠀⢠⢿⡄⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢶⡄⠀⠀⠀⠀⢠⡟⠀⢷⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠻⢦⣀⠀⢠⡟⠀⠀⠘⣇⠀⠀⠀⢀⡼⠃⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⢀⣤⣞⠁⠾⠤⠤⠤⠤⠽⠆⠀⣰⠏⠀⢠⡴⠋⠙⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⠏⠀⠈⠛⠒⠒⠒⠒⠒⠒⠒⠛⠁⠀⠀⠀⠙⢧⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣠⡄⣠⣶⡮⠙⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⠐⣶⣿⣿⣾⣿⡷⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠉⠀⠀⢿⣿⠙⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⡀⢀⣤⡴⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣇⡀⠀⠉⠂⠀⠀⡀⠀⠀⠀⠐⢡⣴⣿⣷⣽⣿⣷⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢰⣧⣬⣿⣿⡇⡀⠀⡰⠀⠀⢋⠀⠀⠀⠀⠀⠫⠉⠉⣿⣿⠃⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢿⡛⠋⠹⣿⣇⠑⢴⠃⠀⠀⠀⠈⠀⠠⠤⠀⠀⠀⠀⠸⣿⢀⣲⠊
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣿⡟⢀⠐
⠀⠀⠀⠀⠀⠐⠆⠀⠀⠐⠒⠀⠀⠀⠀⢲⡞⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠿⠇⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠸⠁⡜⡀⠀⠀⠀⠀⡠⠔⢀⣀⠀⢀⣄⣠⢄⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⡉⠀⠀⠀⠀⠐⡷⣿⣿⣦⣿⣿⡅⠀⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠉⠈⠁⣿⣿⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿⢃⢸⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠃⠀⠀⢀⣄⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠐⠋⣡⠾⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⡐⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠈⠙⠒⠦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠢⣄⠔⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⣄⡉⠻⣄⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠘⣿⠵⠤⣀⠉⠲⣌⠣⡀⠀⠀⠙⠦⠤⠤⠤⠒⠒⠉⠉⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⣄⣛⠶⢤⣄⡙⠢⡈⠳⡌⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠊⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠌⠟⢶⠘⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠬⠶⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠠⠤⠄⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢊⡡⠒⠒⣤⣤⣴⣮⣽⣶⣤⣀⣀⣀⣀⣠⣷⡄⠀⢤⢤⡤⣰⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠬⠴⠬⠭⠤⠤⠤⠤⠤⠬⠿⠤⠭⠤⠥⠭⠤⢽⣭⣤⣤⣤⣤⣭⠞⠉⢠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠚⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⢀⠀⢠⢧⡝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣴⣷⠀⡆⠀⠀⠀⠀⠀⡔⠹⡄⡠⠐⠁⠀⠱⡀⠀⠀⠀⠀⠀⢸⡠⠎⠛⠃⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠋⠙⠃⠀⡏⠉⠉⠉⡩⠿⣼⠂⠇⠀⠀⠀⠀⠀⢫⠉⠉⢉⡩⠛⠉⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠁⠀⠀⠀⠀⠀⣧⠔⠒⠉⠀⠀⠁⠀⢸⠀⠀⠀⠀⣀⡈⠳⠎⣁⠀⠀⡴⠶⣂⠤⣀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⣀⠀⠀⠀⣠⡃⠀⠀⡀⢃⠹⠱⡶⠟⣹⡀⣹⣅⡢⢍⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢠⢢⠀⢀⠤⢊⠀⠀⠀⢀⢂⠎⣜⡶⠊⠁⠳⣄⢰⠱⠈⡄⡇⢧⣰⠁⠈⠉⠺⣝⠂⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢧⠏⡾⢰⠃⠀⢸⠀⠀⢀⠇⠎⠜⣼⡎⠀⠀⠀⠈⢹⡄⠆⠇⠘⠘⣽⠀⠀⠀⠀⠈⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⣧⡇⠀⠀⠈⠓⢚⠟⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢱⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡻⡇⠀⠀⠀⠀⡎⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣤⡀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠋⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣿⣼⣿⣿⣿⣧⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣶⣦⣤⣭⣽⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣰⣁⣀⣀⣀⣀⣠⡬⠛
⠀⠀⠀⢀⣠⡴⠶⠛⠋⠉⣽⢏⣀⠈⢹⡄⠈⠉⠻⣿⣯⢿⡄⠀⠀⠀⢰⠋⢩⠟⠉⠉⠻⣌⣷⡶
⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠙⢦⣌⣭⣞⠁⠀⠀⠀⠈⢿⣿⠇⠀⠀⠀⣿⠀⢻⡒⣶⡞⢛⡟⠙⣇
⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⡀⠀⠀⠀⢈⣷⣄⡀⠀⠀⣿⠀⠈⠛⠶⠶⠋⠀⠀⠙⡆
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⢙⡷⠀⠀⢻⡄⠀⠀⢀⡔⠢⢲⠀⡟⠁
⠈⠛⠶⢤⣤⣤⣤⣤⣤⣶⠶⠚⠃⠀⠀⣠⠃⠀⠀⠀⠘⢷⡞⠁⠀⠀⠈⢷⡀⠀⠈⢗⠒⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠤⣤⠤⠖⠁⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠻⣄⡀⠈⠙⠉⠀⡇
⠀⠀⠀⠀⠀⠸⣿⣿⡟⢸⡇⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣬⠟⠂⠀⠀⠀⡟⣿⣿⣿⠃
⠀⠀⠀⠀⢀⣠⣼⠇⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣶⣤⣴⡶⠶⠛⠉⠀⠀⠀⠀⠀⠀⡇⠀⢺⣥⣀
⠀⠀⠀⡴⠋⠁⢀⣠⣼⣧⣤⣤⣀⣀⣀⣀⣤⣤⣤⣤⣾⠁⠀⠀⣼⠷⠶⠶⠶⠤⠤⣄⡀⢤⣀⡟⠶⣄⡀⠈⠙
⠀⠀⠀⠛⠛⠛⠉⢡⡏⠡⡱⡴⠋⠙⣄⣠⠋⡔⠁⠀⠸⡇⠀⢸⡇⠀⠀⠀⠀⢠⡄⠀⠉⠛⠛⠀⢀⡿⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠱⡙⣆⣀⠞⡡⠊⠀⠀⡆⠀⢿⢀⣼⠗⠀⡦⠀⠀⠀⠙⠶⠶⠞⠀⠀⢿
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠘⢌⡡⠊⠀⠀⠀⠀⣧⠀⢸⡟⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣃⣀⠀⠀⠀⠈⠀⠀⢀⣀⣀⣀⣿⣠⣤⣇⣀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⡿⠉⠉⠉⠉⠙⡟⠉⢿⡀⣰⣾⣿⡉⣻⠛⠛⢻⡟⠛⣿⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣤⢤⠤⣀⣧⠀⠀⠀⠀⢸⣇⠠⣬⣉⠉⠀⠈⠉⠁⠀⣀⣼⢇⢠⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠘⠳⠴⠤⠶⣌⠧⠀⠀⢠⣋⡶⠤⠤⠼⠃⠀⠀⠀⡔⣊⠭⠚⡼⠘⠼⠒⠤⠥⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢠⣼⣿⢯⠃⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢀⣼⣿⣿⣿⣷⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣿⣿⡿⣿⣶⣾⣿⣿⠛⢻⣿⣿⠇⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣶⠬⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣶⣿⣯⢶⡷⣦⠉⠻⣿⣿⣿⣿⠗⠰⠦⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠸⢿⣻⡖⠁⢙⡛⢭⣴⠂⠂⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠀⠚⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠐⢊⢁⡄⢠⡄⢀⠀⠀⡁⠀⠀⠀⠄⢄⠀⠀⠊⠀⠈⠀⢁⠀⠃⠠⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠐⢁⣶⠀⢠⠈
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⠆⠉⡄⠀⠀⠀⠀⠀⠆⠀⣠⣀⠀⠀⠀⠙⢆⠤⣴⣷⣾⣿⣿⣀⡐⠀⡀⠈⠀⠄⠀⠀⠀⠀⠀⡶⠋⠀⡰⢀⠔⠀⠠
⠀⠀⠀⠰⠀⠀⠠⠃⠀⣠⠑⠈⠀⠀⠄⠀⡤⠄⠀⠀⢸⠛⠻⠷⠆⠂⠠⡈⡄⠈⡀⢻⣿⣿⠟⣿⣤⠀⠀⠄⠐⣀⠉⠉⡀⠨⡆⠀⠀⠃⠂⠀⠀⠐⠄
⠀⠀⠀⠢⠀⢀⠄⢂⠀⠠⠀⠐⠀⠀⠰⠀⠐⠒⠀⠓⣤⣀⡃⠀⠮⠁⠀⢠⠈⠀⠀⠀⠠⠄⠀⢹⣿⣤⠀⠀⠊⠀⠤⢀⠀⡳⢈⡱⠀⠀⠰⣅⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢁⠀⠀⠂⠀⠸⠁⠐⢌⠘⢷⣄⣀⡀⡀⢂⢡⠀⢀⣀⡀⠀⣀⣼⣿⡇⠀⠀⠀⠐⠀⢀⠛⠁⡐⠄⠐⠀⠁⠉⠁⠒⠈⠦
⠀⠀⠀⠀⠀⠘⡀⠁⠀⠠⠘⡄⠈⢀⠀⠐⠀⠀⠁⣁⠀⢠⡀⠹⣿⡇⠀⠀⢂⠋⠁⠘⢄⠈⠟⡉⠉⠁⠀⠡⡀⠚⢁⡜⠐⡀⠀⠰⠘⡃⢄⠁
⠀⠀⠀⠀⠢⠀⡈⢄⠓⣼⣶⣷⡀⠀⡄⠀⠉⡀⠈⠁⣀⠈⡑⣄⠈⢷⠀⠀⠈⣆⠀⠀⠀⠙⡀⠀⢀⠜⠉⠢⣀⢉⠈⠀⡀⠀⠀⣐⢠⡅⢀⠁⠀⢈⠐
⠀⠀⠀⠀⠀⠀⠊⡰⣦⡙⣿⣿⣷⡁⠠⠄⠀⠉⠀⠀⠀⠀⢤⣿⠆⣸⣧⠈⠁⠀⠘⠢⠁⠀⠈⠀⡇⠀⡀⣘⢉⡍⠁⠠⢀⠔⠁⠀⠉⠊⠄⠀⠀⠀⡄
⠀⠀⢠⠠⠀⠈⠛⠀⢙⠷⡈⢩⣉⠃⠀⠀⠀⠀⠀⠄⣠⣾⠟⢁⠴⣿⡏⠀⠀⠠⠑⠠⣀⠙⠈⢈⡇⠀⠀⠈⡶⡁⢢⠖⣻⡀⠄⠀⠀⢠⠊⠀⠀⠃
⠀⠀⠀⠄⠀⠴⢀⣻⡏⣰⣿⣦⠙⢷⡈⣷⣶⣶⣾⣈⣋⣁⠾⢷⣄⠈⠻⣶⠿⠃⢈⣢⠀⠉⠐⠀⡇⠀⠀⢰⢅⢀⠗⣈⡴⣧⣀⣀⡐⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⣿⠘⣿⣷⣌⣛⣿⣷⡬⢡⣿⣿⣼⠟⠉⠉⠉⠁⠄⠈⡷⠛⡐⠢⡴⠻⣿⠃⠒⢠⡼⠀⠀⠐⠬⣒⠊⠙⠋⠀⢹⣿⠻⢀⠀⠄⡠⠄⠂⢀⠠
⠀⠀⠀⠁⠀⢨⣷⡘⠻⠿⠿⠿⠟⣡⡌⢻⡟⠁⠀⠀⠀⣀⠀⠀⠈⠠⡋⢈⣠⣀⣰⠏⣨⡆⠊⠀⠀⠀⢀⠂⠀⠀⠀⠄⣠⣿⠇⠘⣨⠊⠎⠄⡂⠀⠄
⠀⠀⠀⠀⠢⠀⠹⣿⣷⣶⣶⡶⠿⢿⡿⠀⠀⠐⠁⢠⠀⢀⠂⠀⠀⠀⠀⠀⡈⠁⠀⠚⠀⠆⠐⠄⠀⠠⢀⠁⠀⠀⢠⣾⡿⠁⠀⠐⠁⠘⢂⠀⠔⠂⣀
⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠃⢠⣤⡄⣇⠀⠣⠀⠀⠀⠢⠤⠀⠀⠄⠂⠠⠄⠈⠢⠄⠀⠰⡀⠀⢀⡈⠑⠀⠀⡘⢸⣧⡙⡅⠀⢀⢐⠂⡴⡃⠄⠒⢻⣄
⠀⠀⢀⠀⠈⠐⠁⣶⣾⡿⠀⠘⠿⠶⠿⠃⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⢀⡄⠀⠐⠀⠀⠀⠈⠃⠀⠀⠀⢼⣛⡇⢸⠄⣼⣈⣠⠓⠔⠀⠀⣼⣿
⠀⠀⠈⠀⠀⠀⠀⠾⡏⠀⢈⠀⠀⣤⣶⣾⡿⠦⠀⠀⠢⡀⠈⢄⠀⠀⡀⠀⡠⢊⠳⠄⠀⠁⠑⠀⠀⠀⠰⠐⠀⢸⣿⠇⣸⠓⣤⡛⣷⡎⠘⢣⡾⠋⠁
⠀⠠⠂⠀⠀⠀⢀⠂⠀⡐⠌⢂⠄⠸⠋⠁⠀⠀⠈⠂⢀⠀⠐⢄⡈⢁⠠⠌⢉⣤⢐⠨⢳⣄⡈⡂⠀⠠⡀⠀⡰⠿⢋⡰⣏⠀⠘⢝⡌⡷⠞⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠰⠈⢳⣾⡦⢀⣀⠴⠀⠀⠄⢠⣤⡁⠀⠈⡱⠖⠁⠄⠀⠰⣦⠖⠀⢈⠐⠥⢡⠩⢙⢒⠐⠘⠛⠒⠒⠉⠂⠀⠁⢲⡖⠻⢸⠂⠀⠀⣐⠄⠡
⠀⠀⠀⠀⠀⠀⠀⠘⠏⣾⠋⠡⢺⠀⠀⠂⡈⠛⠁⠀⡄⠀⠄⡀⠀⠀⠈⠉⠀⠂⢜⠓⠀⢣⡂⣠⢼⠀⠐⡀⢄⢠⠀⠈⢢⡴⢉⠙⢿⢸⠰
⠀⠀⠈⠀⠐⠀⠀⠀⢱⠁⠬⠑⠈⠣⣀⠐⠀⠀⠀⢀⢀⣷⡟⠁⠀⢀⠐⠂⠀⠀⠠⢤⠐⠀⠖⠁⠀⢈⠲⢇⠈⢱⣀⢔⠛⠘⠠⢠⠆⡌⢀⠂⡘⢠⡑
⠀⠀⠐⠠⢀⠐⣒⠢⡆⠨⠀⠀⣀⠃⣼⣷⣶⣶⢾⣏⢸⣿⠇⠀⠀⠀⠼⡿⠂⠅⠀⠀⠀⣉⠀⠁⠀⡉⣱⠀⠘⢻⠟⣐⣄⣁⡠⢋⡾⠃⢂⡐⡀⠠
⠀⠀⠀⠀⠸⡘⠠⠸⠆⠿⠄⠀⠄⢸⢿⣿⡿⠿⢾⣿⡀⢿⣧⡠⠀⠀⠒⢀⠈⠠⢀⡀⢀⠉⣶⣄⠀⠐⠁⠀⡠⠼⣄⠛⣛⣩⣴⡿⠁⢀⠢⡀⠀⠀⠃
⠀⠀⠠⢀⠆⢈⠀⢀⠶⠸⡀⠀⠁⡈⢞⠛⣀⡦⠈⣿⣷⡈⠋⠁⠦⠀⣀⣀⣠⣴⣾⡿⠃⣴⣿⣿⣿⣦⣀⠀⠣⠃⠄⡰⠙⠿⠏⠀⠀⠀⠠⠁
⠀⠀⠠⡈⠀⠤⠐⢁⠄⠁⠑⣀⠀⠀⠀⠉⠉⠁⠰⢿⠿⢿⣦⣈⠑⠶⠿⠿⠟⠛⢁⠄⠚⢿⡟⡻⠿⣟⣿⣷⣦⣠⡌⠀⠀⠀⠂⠁⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣀⡀⠄⢀⠀⠀⠈⠓⠒⠰⠀⠂⠈⠀⠀⠠⠼⠿⠛⠛⡋⠁⠐⠂⠀⠀⠀⠀⠠⠋⠁⠀⠈⠙⠻⠿⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠈⠠⠂⠐⠀⠀⠀⠈⠉⠀⠐⠄⠁⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢒⣶⣿⣶⡂⠀⠀⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⠏⢻⠉⠉⠷⠈⡿⠋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠈⠐⡀⠀⠈⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣾⢿⠓⠀⠀⠀⠀⢠⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⢠⡀⠀⠀⠀⠀⣆⡀⣠⣷⣴⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣴⣶⣽⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠑⠒⠓⠒⠉⠏⠉⠍⠉⢉⡙⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠒⠂⠂⣉⣁⣀⣤⡴⠒⠚⠀⠀⢠⢤⠖⠂⠁
⠀⠀⠀⠀⠀⢀⣀⣀⣢⣤⠤⠶⠶⠶⠾⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠈⠑⣶⣤⢤⠈⠁
⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠾⠿⠛⠛⠉⢀⡸⠀⠀⠀⠀⠀⣀⠔
⠀⠀⣀⠀⠤⠤⢄⡠⠤⠶⠞⣛⣩⣭⣤⠶⠶⡶⡶⠶⠶⢶⢦⣤⣤⣤⣤⣤⠤⢤⡶⢞⣉⣦⠀⣀⠠⠔⠈⠠
⠠⠠⠀⠒⠒⠒⠒⠚⠛⠉⠉⠉⠀⣀⣀⣀⡤⠤⠒⠒⠴⠒⠉⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠉⠀⣦⣤⣤⣶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣡⠤⠂⠀⠀⣀⡠⠤⠐⠀⠘⠇⠀⠀⠀⠀⠀⡠⢎⣤⡷⢖⣸⡟⢈⣩⡁
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠚⠛⠉⠉⠀⠠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠈⣑⡫⠔⠋⠀⠉⠁⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡀⠀⠀⠀⡀⠀⢀⡤⠀⠀⠀⠀⢀⠑⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠈⠀⠀⠀⠀⠁⠁⠀⠀⠀⠄⠅⡀⠀⠨⢀⠸⠦⡄⠆⡂⠠⠀⠀⠰⡀⡀⠘⠀⢠⡆⢀⣀⠀⠐
⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠡⢀⠂⠄⠀⠀⣿⢀⠀⠁⢪⠀⠈⠀⠀⠞⠁
⠀⠀⢀⡀⠀⣤⡔⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⡇⠀⢰⣖⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠉⠳⠆⢾⠀⠀⢀⠀⡀
⠀⠀⢹⡇⠀⢸⢳⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⢸⣿⠀⠀⠀⣤⢠⡄⠁⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠃
⠀⠀⠘⢇⣀⠸⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠃⠀⠀⠀⠸⠇⠀⢶⠀⠀⠛⠀⢐⠀⠀⣿⠀⠠
⠀⠀⠀⠋⣌⡓⣏⡄⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠉⠀⠛
⠀⠀⠈⠙⠃⢘⣏⠙⡆⢀⡀⠀⠠⢻⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⡟⠀⠀⠀⠀⠀⣒⠂⠀⣤⠀⣤⣤⣤
⠀⠀⣠⢸⣀⣸⢯⡯⠅⢸⡏⡷⣖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠟⠁⠈⠁
⠀⠈⣯⠟⣇⣟⠛⣿⢾⣶⣾⣷⡿⠰⠀⠀⢠⣤⡄⣤⢴⡆⠀⠀⠆⢰⠐⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠉⠈⠀⠉⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠁⠉⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡉⡳⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣼⠋⡇⠀⠀⠀⠀⠀⠀⠴⣉⡴⠃⠀⠀⠀⠀⡠⠤⣞⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠖⠒⠉
⠀⠀⠀⠀⠀⣰⠋⢲⡦⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⡀
⠀⠀⠀⠀⠀⠈⠉⠒⠢⠔⠃⠀⢠⣠⠀⠀⠀⠀⡠⠖⠊⠉⠙⠢⣄⠀⠀⠀⢀⡶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠜⢧⡀⢀⡎⠀⡀⠀⠠⠀⢀⠀⠱⡄⢀⠜⠉⠁⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⣶⣤⣤⣶⣶⣿⠀⢠⠗⠁⠀⠀⠀⠀⠑⢤⣿⠉⣲
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⡄⠀⠀⠀⠀⠣⣌⣛⣋⣉⣁⠥⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠧⠬⠗⠚⠁⠀⠀⠀⠀⠀⠀⢫⠈⡋⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡅⢃⠀⠈⡆⠀⠀⠀⣠⡒⢲⠦⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⣤⣀⡀⠀⠀⢸⠀⠈⠁⠀⠀⡇⠀⠀⠀⠀⠈⠓⠧⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠥⠴⠃⠀⠀⠈⠹⡋⠛⠒⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠢⡤⠲⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠧⠼⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠃⠀⠀⠀⠀⠀⠛⠁⠀⠀⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠀⠉⠉⢉⡉⠉⠁⠴⢄⣀⠀⢀⡠⠤⠄⣀⡀⠀⠀⠀⠀⠀⢀⣀⠠⠦⠀⠀⠀⠠⠴⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⢀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠙⠣⠀⠀⠀⠀⠀⠛⠀⠤⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠐⠒⠠⢄⡀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠀⠘⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⢀⠋⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⢣⡀⠀⠈⠑⢤⡀⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠘⠶⣄⡀⠀⠙⠦⡀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡀⠀⠀⠀⠀⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠿⣦⠀⠙⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠐⠳⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢇⠀⠰⠆⠢
⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠈⠀⠃⠀⠀⠀⠀⢸⠇⠀⢸⠀⡇
⠀⠈⠀⠀⠀⠀⠰⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⢸⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⢠⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡄⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⢀⡄⠸⡁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⡃⠰⡁⠢⡈⠁⠒⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠈⠀⠐⠛⠋⠀⠀⠀⠀⣀⠀⠀⠀⢀⡴⠁⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡄
⠀⠸⠀⡆⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠓⠶⠘⠁⠀⠀⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠠⢏⡀
⠀⠀⠃⣰⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠙⠲⠦⣀⡀⣀⠠⠴⠴⠞⠛⡀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡈⠃
⠀⠀⠀⢁⠶⠀⠀⠀⠀⠀⠀⠛⢄⡀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⢤⣄⠁⠀⠀⡀⠀⠀⠘⣠⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⣀⠙
⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠁⠘⠇⠈⠀⠀⠀⣀⡀⠀⠀⠠⠀⠈⠉⠒⠤⠿⣀⡴⠊⠡⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⣴⠛⡆
⠀⠀⠀⠀⠀⠘⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⠠⠂⡰⠀⠀⠀⠀⠀⠀⠉⠀⠸⡆⠃⡀⠙⠢⣤⣤⣤⡤⣤⠒⠉⠀⠄⢀⡌⢣⠀⡇⢀⡄
⠀⠀⠀⠂⠀⠂⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⢀⠄⢊⡐⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠆⠐⡀⠀⠀⠀⠀⠀⠈⠲⠐⠛⠒⠉⠀⠀⠢⡄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠀⠀⠀⠰⠎⠁⡀⠒⣁⠖⠁⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢡⠀⢆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⡠⠖⢉⡴⠋⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠣⡀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⡤⠖⠲
⠀⠇⠀⠀⠀⠀⠀⠀⣀⠴⠃⠙⠉⢁⡠⠞⠁⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⣄⠀⠑⢄⡀⠀⠀⢀⣆⠀⠀⢀⠂⠀⢦⡈⠲⡄
⠙⠀⠀⠈⠢⡀⠀⠀⠈⣀⡀⠀⡟⠁⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⠳⢄⡀⠀⠛⠀⣶⠘⢦⣠⠂⠀⠈⡀⠀⠂⠁
⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⢲⡇⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠙⠢⢤⣀⣤⠀⠀⠁⠀⠚⢁⡠⠔⠁
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠁⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣀⠘⠓⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠲⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⡀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠠⠤⢀⣉⣀⡠⠤⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠤⠈⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢸⡟⠛⠠⠤⠄⢒⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠀⠀⢹⡅⠈⠀⠀⠀⠐⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠸⠧⠀⠤⡿⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠀⠀⠛⠠⠀⠀⠀⠀⠀⠠⣧⠤⠤⢤⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡄⠒⠒⠒⠫⣆⡀⠀⠀⣴⠀⠀⠀⠀⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢣⡄⢀⠁⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢳⡄⠀⠀⠀⠀⠈⡄⢻⣆⠀⠀⠀⣠⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠙⢦⣄⣀⣤⡾⠁⢘⠇⠉⠲⠚⢁⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⢀⣼⣦⣀⣀⣤⠞

⠀⠀⠁⣀⠈⣤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡠⢄⡄
⠀⠀⠄⠀⠀⠀⠑⢷⢈⠀⠀⢠⠖⠉⠉⠛⢦⡀⢀⠀⠀⠀⠀⠁⠀⠐⠛⠀⠀⠀⠀⠁⠀⠀⠀⠠⠄⠂
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠨⡄⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⠁⠀⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⢀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣀⣨⢚⣿⣶⠀⠀⠀⠀⠀⠀⢄⠀⠰⡄
⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡟⠀⠠⣄⣠⠤⠙⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠂
⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣗⡞⣞⣾⣸⣵⣇⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣠⠴
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠞⢀⡉⢉⡉⠈⠁⠀⠀⠠⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡡⠔⠂
⠀⠀⠀⢠⣄⣀⠀⠀⠆⠀⠀⠀⠀⢤⣤⣤⠀⠀⠏⢠⣞⡵⡾⡋⠀⢀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡀⠉⠐⠀⠀⠀⠀⠀⠀⠈⢣⠞⠀⣀⣴⠻⡟⣸⢱⣷⣧⣷⣿⣴⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠘⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠙⠁⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢖⠒
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠
⠀⡆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⡠⡄⠀⠠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⢠⠀⢽⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠍⢡⣨⠆⠄⠀⠀⠘⠂⢰⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣂⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⢠⡇⠘⠄⠀⠀⠀⠀⠀⣝⠃⡞⢉⢹⣦⣐⣫⠵⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐
⢸⡇⠀⠀⠀⠀⠐⠀⣾⠏⠀⠘⠊⠹⡿⡿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⠄⠉⢉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠿⠿⠿⠻⠟⠋⠉⣽⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡫⠒⠂⠄⣀⠀⠀⠀⢤⣤⣀⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢡⠞⣁⠂⠀⠐⢈⣛⡉⠙⠋⠉⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣠⡄⠀⠀⠠⢁⠀⠂⡤⢂⡤⡀⢈⡽⠀⠠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣛⣭⣡⠥⠰⠒⢂⢄⣠⠼⣂⢘⣻⣶⠋⠀⠀⠀⠀⢀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⠯⡷⠏⠀⠀⣠⣼⣏⡷⠽⣚⡵⠚⠀⠀⠀⠀⠀⡀⣰⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢛⡛⣤⠀⣴⣶⣾⢿⣿⣿⣿⡿⠵⠋⠀⠀⠀⠀⠀⢀⠄⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣵⠦⢉⣉⣀⢛⣛⣛⣛⣛⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⣀⣰⠛⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠟⠻⢟⡿⠿⠟⣟⢋⡉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⡀⠀⣀⢔⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⣀⠔⠚⠁⠀⠉⠙⠒⠛⠓⠤⠀⠀⠁⠀⠀⠄⠀⠀⠀⠘⠋⠁⠈⠉⣿⡷⣡⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⠟⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢀⠾⣷⣾⣿⠇
⠀⠀⠀⠀⠀⢀⡠⢖⠝⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⣀⠀⠠⣐⡵⠟⠁
⠀⠀⢀⡠⣖⡭⣊⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣯⣶⣿⠁
⣠⣾⣿⣾⢕⣨⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⣴⡿⠿⠽⠟⠋⠁
⠀⠘⠿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠵⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣶⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⡏⠁⡠⠋⡼⢱⡈⠑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣖⠢⠤⢤⣹⣼⣥⣴⣷⣤⣣⡰⠃⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠈⢱⢶⠽⠓⠊⠉⠀⠈⠑⠪⢿⠯⣉⣩⠽⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡇⢀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣹⡄⠀⡼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⡼⡸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⡇⢰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⣻⠁⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⢧⡏⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⢻⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡇⣼⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⠃⠘⣾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⢠⠃⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣀⣀⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⡏⠀⢳⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡒⠒⠶⠽⢿⣷⣄⠀⠀⢀⣠⣤⣶⣿⡛⠶⢤⣘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡠⠋⠹⡝⢿⡾⠛⡻⠛⡍⠉⠉⣉⠵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠹⣰⣇⠞⢀⡠⠼⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠋⠁