Genome Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣄⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠈⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠻⠿⠿⠂⠀⣿⡇⠀⠀⠀⠀⠀⠀⣠⣶⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⣴⣶⣶⠄⢀⣿⡇⠀⠀⠀⠀⠀⠀⢿⣇⣀⣼⡿⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⣀⣴⡿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡟⠁⢀⣾⠟⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢁⣴⡿⣿⣦⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠈⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡿⠋⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠿⠿⠿⠂⠀⣿⡇⠀⠀⠀⠀⠀⢀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⣶⣶⣶⠄⢀⣿⡇⠀⠀⠀⣠⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⢀⣤⣾⠟⠀⠀⣠⣾⠟⠁⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣏⠁⠀⠀⠀⠛⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠉⠛⠃⠀⠀⠀⠺⣷⡄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠿⠿⠂⠀⠀⠀⠿⠿⠇⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣶⣶⠄⠀⠀⠀⣶⣶⡆⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⢀⣠⡀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣷⣿⣏⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⢠⠆⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠄⠂⠠⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠀⠂⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠄⢦
⠀⢸⣀⡀⠀⠀⠉⠓⢤⡀⠀⠀⠀⡠⠖⠉⠀⠀⢀⣀⣀⣀⡀⠀⠀⠉⠢⣄⠀⠀⠀⢀⡤⠊⠁⠀⠀⣀⣀⣀⣀⠀⠀⠈⠑⠦⡀⠀⠀⠀⣠⠔⠋⠁⠀⢀⣀⣸
⠀⠀⠈⡏⡏⠒⢤⡀⠀⠙⢦⡠⠊⠀⠀⡠⠖⠙⠉⡇⠀⢸⢹⠑⠢⣄⠀⠈⠳⣄⡴⠋⠀⢀⡤⠒⢹⢹⠀⠀⡏⠙⠓⠦⡀⠀⠈⠢⣠⠞⠁⠀⣠⠔⠊⡏⡏
⠀⠀⠀⡇⡇⠀⢸⢹⢦⠀⠀⠑⣄⡠⡎⡇⠀⠀⠀⡇⠀⢸⢸⠀⠀⣿⢱⣄⠀⠈⠣⣀⢴⢩⡇⠀⢸⢸⠀⠀⡇⠀⠀⠀⡏⡦⡀⠀⠙⢆⣠⢾⢹⠀⠀⡇⡇
⠀⠀⠀⡇⡇⠀⢸⢸⡴⠓⣄⠀⠈⠣⡇⠇⠀⠀⠀⡇⠀⢸⢸⠀⠀⣿⢸⠞⠣⡀⠀⠙⢾⢸⡇⠀⢸⢸⠀⠀⡇⠀⠀⠀⡇⡧⠛⢆⠀⠀⠱⣼⢸⠀⠀⡇⡇
⠀⠀⠀⡇⡇⣀⠼⠋⠀⢀⡼⠣⡀⠀⠈⠣⣄⣀⠀⡇⠀⢸⢸⢀⡠⠟⠁⠀⢠⠟⢦⡀⠀⠙⠧⣀⢸⢸⠀⠀⡇⢀⣀⡤⠋⠀⠀⡠⠳⣄⠀⠈⠻⢄⡀⡇⡇
⠀⢰⠚⠛⠉⠀⠀⣀⠴⠋⠀⠀⠈⠢⣄⠀⠀⠉⠙⠓⠒⠚⠋⠁⠀⠀⡠⠞⠁⠀⠀⠙⠦⡀⠀⠀⠉⠛⠒⠒⠛⠉⠁⠀⢀⡤⠊⠀⠀⠈⠳⢄⡀⠀⠈⠉⠛⢲
⠀⠸⡄⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⠄⡀⠠⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⠀⠀⠤⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⠄⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠰⣶⣄⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠈⠻⣷⣄⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢀⣀⠀⠈⠻⣷⣄⠀⠈⠛⢃⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣬⣭⣭⣄⣛⣃⣀⣀⣈⣛⣠⣤⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⡋⠀⠀⠻⣦⡀⠀⠀⠻⡄⣶⣶⣦⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⣤⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⢡⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠈⠻⣦⡀⠀⠈⠻⡦⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⡸⣿⣿⣇⠻⠆⠀⠀⠈⠻⠆⢀⣠⣴⣿⣿⡿⠃
⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⢸⣿⣿⠿⠛⠉⢱⣄⠀⠀⠀⢽⣅⢻⣿⣿⠘⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠉⠁⢶⣄⠀⠀⠙⢷⣄⠀⠀⠉⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠙⢷⣄⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠙⠷⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⢠⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣰⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠤⠤⠤⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡒⡲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣊⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠥⠤⠤⠽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠂⠀⠀⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡉⣉⡉⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠑⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠒⠂⠒⠊⠇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢄⠀⠀⠑⢦⣀⠂
⠀⠀⠀⠀⠀⠀⠠⠔⠀⡠⠀⠁⢉⠀⠀⠀⠀⢋⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⠃⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠂⠄⢈⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⡀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⣄⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠀⠐⣄⠀⠀⡅
⠀⠀⠀⠀⠀⢸⡄⠞⠀⠀⠀⠀⠒⡀⠰⠓⡀⠙⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⠀⠀⠁⠀⠈⠀⠀⠀⠁⢤⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⢶⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡦⡀⠀⠀⠀⠈⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢈⢦⡀⠀⠀⠀⠀⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠠⠄⠀⠀⠀⠁⡰⠋⠀⠙⢄
⣿⡿⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿
⠀⠀⠀⠀⠀⠻⣿⣿⣍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡛⠛⠛⠛⠛⠛⠛⣻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⡄⠀⣠⣴⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣡⣴⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠿⠋⠁⣴⣿⣶⣄⡀⠀⠀⠀⠀⠀⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣛⣁⣀⣀⣀⣀⣙⣻⣿⣿⣦⣀⠀⠀⠀⠈⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣷⣄⠀⠀⠈⢿⣿⣿⣦
⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⠀⠀⠈⢿⣿⣿⣧
⠀⠀⠀⠀⢠⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣧⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⡀⠀⠀⠸⠿⣿⠀⠀⠀⠀⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠘⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠈⠻⣿⣿⣿⣿⣶⣶⣤⣄⣀⣀⠀⠀⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⠷⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣄⣀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠈⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⡶⠀⠀⠈⠉⠙⠛⠻⠿⠟⠛⠿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣆⡀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣭⡉⠉⠉⠉⠉⣩⣽⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀⠀⠈⠉⠉⠛⠛⠻⠿⠾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠓⣠⣴⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠛⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠛⠁⠀⠻⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠉⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⣴⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣾⠿⠁
⠀⠀⠀⠀⢰⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣷
⠀⠀⠀⠀⠘⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⢐⣒⣀⣐⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢈⡵⢤⠾⠛⠻⢏⣿⣿⣿⣷⣄⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣈⣥⡤⠒⠉⠀⠀⠈⣤⠄⠠⣼⣿⣿⣟⣹⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡀⣤⣤⣤⢤⣤⣤⣤⣤⣤⣶⣶⠟⠋⠀⠀⠘⠁⠀⢠⠞⠁⠀⠀⣿⣿⠉⠉⠁⢹⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡼⠞⠉⠀⠀⢉⡈⢾⣉⢱⣿⣟⠻⣿⣿⡀⠀⠀⣤⡀⠀⠀⠁⠀⠀⠀⢘⣿⠁⠀⢤⣤⣹⣿⣿⣿⣧⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣲⣯⡄⠠⠀⠀⠀⠈⠠⠀⣸⣿⣧⣿⡿⣏⣿⡆⠀⠀⠸⠿⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠩⣿⢻⣿⣿⡆⡇
⠀⠀⠀⠀⠀⠀⠀⢆⡏⡗⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣇⣬⣿⣪⣽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠈⡙⣿⣿⣿⣇⣧⣀⡀
⠀⠀⠀⠀⠀⣴⣾⢹⡄⣙⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠈⠟⣻⡿⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣖⣿⣿⣿⡿⢸⣿⣿⠉⠳⣄
⠀⠀⠀⠀⠀⣼⣿⠘⣯⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠋⠁⢠⣿⡁⠀⠀⣳⣤⢀⣀⢀⣀⣀⣤⣴⣤⣿⣿⣿⣣⠿⢋⡽⠃⣾⣿⣿⣧⠀⠈⡄
⠀⠀⠀⠀⢸⣿⣿⣇⠹⣿⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣿⠃⠀⠈⠛⠋⣽⣿⠿⠟⠛⠛⣻⠃⢠⣾⠃⣼⡟⠉⠀⢸⠀⠀⠃
⠀⠀⠀⠀⠈⢿⣿⣿⣆⠉⠷⣴⠱⣇⠀⠀⠀⠀⠀⠀⢒⡻⠄⠀⢿⣿⡇⠀⠀⢻⠀⠀⡀⠀⣠⣿⡏⠀⠀⠀⡾⠁⠀⣼⢃⣾⣿⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠐⠀⠀⠋⠿⡜⣷⣤⠁⠂⢭⡄⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⠃⠀⠠⠋⠀⠀⠁⠀⠀⣿⣿⣆⣴⠞⠁⠰⣾⢃⣼⣿⠁⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣧⣄⠳⣆⠀⠀⢀⣀⡀⣤⣴⡿⠋⠉⠉⠈⠁⠀⠀⠀⠀⠀⣾⣿⣿⠟⠁⠀⣠⣾⢣⣾⡟⠃⠀⠀⠀⢹⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣷⡄⠻⢧⡈⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠁⠀⠀⣶⡾⢳⣿⠏⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣈⡀⠀⠀⠈⠉⠋⢦⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣀⣾⡿⠶⠟⠋⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠀⠙⠲⣽⣧⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⡈⠁⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⡀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣼⣾⣷⣿⠿⠿⣷⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⢺⣿⣗⣄⢀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣼⡇⠀⢰⣿⡇⠀⠀⠀⣦⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⢸⣿⣿⣷⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣿⣿⣿⡁⠀⣸⣿⡷⠀⠀⠀⣅⠀⠀⠀⠀⠀⢨⣿⣧⡀⠀⢸⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡿⠀⠀⠀⠀⢀⢸⣿⣿⣿⣷⣾⣧⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠿⢻⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⢸⡆⠀⠀⢘⠁⠀⠀⠀⠀⠀⢸⡄⠀⠉⠻⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠁⣧⠈⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⠳⣄⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⢤⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡠⠚⠛⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠉⠉⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⢿⡿⠃⠀⠀⢀⣀⣼⣄⣀⣀⣧⣀⣀⣸⣇⣀⣀⣼⣀⡀⠀⠀⠀⠈⠻⡿⠃⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⡇⠀⠐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡇⠁⠉⠉⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣅⣀⣀⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡧⠤⠤⠤⠤⠤⢴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⠿⠟⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⣤⣶⣄⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣠⠞⠀⠀⣤⣷⣄⠀⢸⡇⠀⠀⠈⠉⠉⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠒⠚⠁⠀⠀⠀⠿⣿⠟⠀⠘⢧⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⠉⠛⢿⣦⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⢠⣿⠛⠉⠙⠛⠿⣶⣶⣶⡾⢿⣿⠁⠀⠀⠀⠀⠀⠀⠈⣿⡿⢷⣶⣶⣶⠿⠛⠛⠉⠛⣿⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣤⣄⡀⠀⠸⣿⡀⢀⡀⠀⠀⠀⠀⢀⣿⠇⠀⢀⣠⣤⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⣿⡄⠀⠀⠀⣿⣏⣹⡇⠀⠀⢹⣿⠟⠻⣷⡄⠀⠀⣾⡏⠀⠀⢸⣏⣹⣿⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠈⠻⢷⣶⣶⡿⠿⠛⠁⠀⠀⢸⣿⣄⣀⣾⡇⠀⠀⠛⠁⠀⠀⠈⠛⠿⢿⣶⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠙⠛⠉⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢀⣴⡶⢶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⢸⣏⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠈⠛⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⣿⡄⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣤⣿⠃⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⠁⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡆⠀⠀⢸⣿⢸⣿⠀⠀⠀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣴⡆⣿⣿⢸⣿⣿⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⣄⣴⣿⣿⣸⣿⣿⡷⠶⠾⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣏⣽⣿⣯⠿⠟⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣰⣿⣼⡟⣿⣩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠿⠿⣺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣹⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣯⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⣷⡖⣻⣿⡇⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡇⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⡇⣿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡖⣶⣼⡏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⣿⣶⣯⣿⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⢸⣏⣻⣿⣟⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠛⠸⣿⣿⢹⣿⣿⡷⠶⢶⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠻⠇⣿⣿⢸⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠸⠃⠀⠀⢸⣿⢸⣿⠀⠀⠈⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⢠⠄⠴⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠠⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡔⠁
⠀⢀⠤⠒⠒⡋⠉⠉⠉⠉⠉⠉⠍⠉⠩⠍⠉⠉⠉⠙⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠊⠉⠉⡉⠀⠠⠄⠀⠉⠠⠀⠉⠀⠉⠑⡆
⢰⠁⠠⠀⠤⢀⠠⠀⠀⠐⠀⠂⡠⡤⠀⠄⠀⡠⠁⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⢀⠔⠁⠀⠐⠀⣀⠐⠄⢀⠠⠀⠀⠀⣀⠈⣀⠘⠂⠐⠀⡆
⠘⢆⠀⠀⠀⠀⠉⠀⡀⣀⣁⣠⣁⣀⣰⣂⣀⣐⡈⠀⠀⠂⠀⠀⠂⢀⠀⠙⠦⠎⠀⢄⠄⠠⠀⠁⠄⠀⠀⢉⣠⡤⠤⠤⠴⠒⠒⠒⠒⠋
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠓⠒⠬⣄⠀⠀⢀⠀⠀⠀⠈⠀⣀⡠⠴⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⡀⠀⠀⠣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠋⠁⠀⠡⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⣑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢉⠑⢦
⠀⢀⠤⠖⠒⢒⠒⠒⠲⠲⠒⠒⠒⠒⠒⠒⠂⠉⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⣠⡄⠀⠀⠀⠠⡠⠐⠀⠀⢀⡀⠢⡀⣀⠠⠀⡔⠄⣀⠀⡀⢸
⢀⠇⠀⡀⣀⠃⠐⣤⢀⠒⠁⠀⣄⠠⠀⠃⠂⠀⠀⠀⠀⠀⠐⠂⠂⠀⣀⠔⣡⠈⠒⢤⡀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠈⠀⠈⠀⠠⠀⣀⡠⠎
⠀⢇⠀⠈⠀⠀⠸⠋⠋⠁⠚⠷⠚⠏⠱⠲⠍⠀⠂⠠⠀⠠⠂⣀⠤⠊⠁⠀⠂⠀⠀⠀⠈⠉⠒⠒⠰⠶⠶⠶⠖⠒⠒⠒⠒⠉⠉⠉⠁
⠀⠈⠑⠺⠤⠤⠤⢌⣡⣤⣀⣀⣨⣂⣠⣄⣀⠠⠤⠤⠒⠒⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠄⠀⠄⠂⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠉⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⡾⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠀⠀⠉⠙⠣⢤⣀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠁⠈⠙⠷⠶⠶⠶⠦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⢀⡶⠖⠶⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠶⠶⠞⠋⠀⠀⠀⠀⠀⠈⠛⢦⡀⣠⠎⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⡤⠖⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣠⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠒⠒⠂⠀⠀⠀⠀⢰
⠀⠀⠀⢻⣤⣀⣉⣉⣉⣉⣉⠛⠛⠛⠛⠶⠶⠶⠶⢦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠶⢶⡀⠀⣠⣄⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠧⣄⡀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢹⣄⢀⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋⠀⠀⠀⠉⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠙⠿⠁⠀⠀⠉⠉⠉⢉⣻⠶⠿⠿⢶⣶⠾⠛⠉⠈⢳⡄⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⡀⠀⢠⠄⠀⠀⣴⠌⠁⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠙⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⢰⡄⠀⣠⠇⠀⠀⠀⣴⣦⠀⠀⠀⢦⠀⢀⣀⣤⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⢸⡀⠀⠀⠀⠙⠋⠀⠀⠀⣼⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⠶⠖⠒⠛⠛⠛⠋⢻⣄⣀⣀⣀⣀⡀⠀⠀⠘⠳⢾⢧⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠠⠤⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⡀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⡄
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠓⠚⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⢠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⡀
⢀⡤⠻⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠟⠶⣄
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠻⠦⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡈⠻⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢷⣌⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⣿⠈⠻⣦⣈⣻⣶⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⠀⣷⣄⠙⢷⠌⠉⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠻⣯⡙⠃⣿⡏⠻⠿⠛⠛⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣄⠰⣶⣾⡄⢻⣤⣤⣤⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋⠻⣦⡈⠻⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢶⣄⠙⢷⣌⠻⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣦⡈⠻⣦⣨⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣦⡙⢷⣄⠙⠷⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣤⣤⣶⣤⣸⣿⢠⣌⣻⣦⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠛⠛⠛⣧⠘⡿⠿⠆⠙⠃⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠿⣯⡉⠻⣦⡀⣿⡈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣀⡐⢷⣄⠙⢿⠀⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣦⡈⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⡙⢷⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢿⡿⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢻⠳⢤⣖⠉⠀⠠⡇⠀⠉⣲⠤⠚⡟⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⢸⠀⠀⠈⢱⠀⠈⡇⠀⡏⠀⠀⠀⡇⠀⣤⣀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⣿⣿⣿⣦⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⣇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠢⣄⡀⢸⠀⢀⡇⠀⡇⠀⣀⠼⠿⠧⡈⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠔⠊⠀⠀⠀⠀⠀⢉⠿⠀⣨⣏⠰⣷⣉⠀⠀⠀⠀⠀⠑⠠⢄⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢁⠀⠀⠀⠀⢀⡠⠔⠊⠀⣠⠔⠻⠟⠢⣄⠈⠑⠢⣄⡀⠀⠀⠀⠀⡨⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠤⠒⠁⢀⡠⠖⠉⠀⣀⢴⣦⣀⠀⠉⠲⢤⣀⠉⠲⠤⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄⠊⠁⢀⡠⠔⠈⠀⠐⡇⠈⠙⠢⢄⡀⠈⠑⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡉⠀⠀⢠⠂⠁⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠉⠲⡄⠀⠀⢙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢼⠀⠀⠀⡠⠔⠊⠉⠉⠙⠢⢄⡀⠀⠀⡧⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⢀⠠⠐⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠋⠛⠛⠛⢹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠐⣿⠛⢿⠛⠛⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⣷⠀⣺⡁⠀⠀⠘⡷⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣠⠞⠛⠳⣆⠈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⣾⠀⠀⣧⠀⠀⠀⣿⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡿⠀⣽⠂⠀⠈⢻⡖⣿⠁⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⢸⣦⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠛⠲⢤⣀⣿⠀⠀⠀⠀⠀⠿⣤⠿⠦⠴⠦⠾⠁⠈⠙⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣦⣄⡀⠀⠈⠙⠶⣤⣀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⢻⣿⣷⣦⣄⡀⠀⠉⠙⠲⢦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠙⢿⣿⣿⣿⣿⣶⣤⣀⠀⠈⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣰⠟⠻⢿⣿⣿⣿⣿⣿⣶⣤⣀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠶⣤⣀⠀⡴⠃⠀⠀⠀⣼⡏⠉⠉⠉⢹⡏⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⡹⢿⡛⠳⣄⢀⡞⢸⡇⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠾⠋⠀⢸⣧⠤⠤⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠛⠛⢻⡆⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⠀⠀⠳⢶⢶⡶⠶⠾⠃⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣇⣀⣀⣀⣀⣼⣸⣇⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠓⠒⠒⢾⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢲⡶⠟⠁⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⡤⢤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⢀⡴⠛⠉⡉⠉⠛⢦⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠉⠙⠓⠦⣄⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡾⢠⡞⠉⠉⢳⡄⠈⣧⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⣄⠀⢠⠟⠀⠀⠀⠀⠀⠀⣧⠘⣧⡀⠀⣸⠇⠀⡿⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠒⠒⠒⠒⠒⠃⠀⠈⢳⠏⠀⠀⠀⠀⠀⠀⠀⠘⢧⣈⠉⠉⢁⣠⡾⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⠤⠤⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣰⣏⣀
⠀⢠⡶⠖⠒⠚⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠶⠒⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠙⠛⠓⢦
⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠸⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣷⣁⠀⠀⢀⣤⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⡄⣸⣿⣿⣿⣿⣿⣖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣿⣿⣿⣿⣿⣿⣿⣇⢞⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡼⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⢷⡿⡽⣿⣿⣿⣿⣯⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣽⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣼⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⣿⣿⣿⣿⣿⣿⣯⣸⣿⣿⣿⣿⣿⣿⣏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣟⣾⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣽⣿⡻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣦⢿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢻⣷⣿⣿⣿⣿⣿⣏⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣹⣿⣿⣿⣿⣿⣿⣿⣿⡈⣿⣿⣿⣿⣿⣷⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠋⠘⢿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠻⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠰⡆⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢏⠀⡗⠀⠀⡇⠀⢸⠂⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠳⡈⢢⠀⡇⡴⠋⡔⠋⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠉⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⡔⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣟⡍⣭⢹⣭⡍⣭⣩⣩⡝⣿⣿⣀⣀⣀⣀⣀⡤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⣀⣀⣀⣀⣿⣿⣯⣥⣭⣬⣭⣥⣭⣭⣬⡅⣿⣿⣀⣀⣀⣀⣀⠔⠉⠉
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠉⠉⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠉⠉⠉⠉⠉⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠔⢁⠜⠀⡇⠳⣄⠣⡀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣇⠀⡧⠀⠀⡇⠀⢸⠄⢹⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠁⠀⠀⠀⠰⠇⠀⠀⠀⠈⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⡲⢖⠶⠖⠒⣉⡩⠿⠟⢳⠢⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢀⠀⣴⠃⠀⠀⠀⠹⡄⠘⢲⡎⠈⣿⣝⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⢀⠄⠐⠁⠀⠀⣀⡄⢚⠁⠼⡁⠀⠈⠀⡠⠐⠁⠀⢠⣧⠉⢁⣿⡄⠛⠉⢻⣿⣶⣄⡀
⠀⠀⠀⢀⠰⠋⠀⠀⢠⡄⠀⠈⠱⠊⡀⠀⡼⠂⠀⢰⠃⠀⠀⢰⠃⢸⠃⠉⠘⣷⣀⣷⣿⣿⠉⠙⢿⣦⡀
⠀⢠⡞⠁⠀⡀⠀⡀⠀⠈⠢⡀⡀⠈⠀⡌⠀⠀⠒⠛⢄⠀⠀⡄⠀⣿⠀⠀⢂⣼⣿⡏⣈⢿⡶⡄⡈⢿⣷⡀
⢀⣾⠀⠀⢈⠁⢀⡇⠀⡀⢀⡟⠀⠀⢠⣇⣀⣀⣤⡤⠤⠤⡴⠇⠀⣻⠐⢇⠈⢹⡿⣇⠅⣼⣿⡷⠌⣾⣿⣷⡀
⣿⡻⢀⣴⡞⠀⠈⠀⠀⡈⣿⣧⠤⠔⠋⠉⣅⠀⠀⠀⡀⠀⠠⠴⠚⠻⣦⣤⣤⣼⠧⠛⣙⣿⡍⠀⣴⣿⣧⣿⣷
⣿⡇⠀⠈⠉⠓⠄⠀⠀⠀⠟⠁⡆⠀⢀⡶⠛⠀⠀⠈⠀⠠⢤⡤⠀⠀⠀⠈⣇⣠⣶⡿⢟⠻⣄⣠⣿⣿⣿⣿⣿
⠹⠷⢤⠀⠘⢤⡄⠊⢁⣼⠀⠀⡛⠲⢋⠀⠀⠀⠰⣄⠀⠤⣂⡼⣳⣀⣐⣀⠘⠋⢉⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢢⠀⣠⣤⣬⣤⣴⣿⣷⠀⠀⡇⠐⢁⣴⡀⠁⡠⣋⣴⣾⣿⣾⡿⠿⠿⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⡟⠿⣿⠏
⠀⠀⠑⠻⢿⣿⣿⠿⠛⠉⢄⣠⣉⣉⣭⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠉⠁⠒⠶⣤⣀⣀⣲⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢽⣫⣴⣾⠟⠀⠙⠿⢿⣿⡿⣿⣿⣦⣀⠰⣶⣜⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠙⠦⣍⡙⠿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠿⠟⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣛⣱⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣙⣛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢥⣿⡄⠀⠀⠀⠀⠀⠀⠀⣼⣦⢭⡟
⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢛⡉⠁⢳⠀⠀⠀⠀⠀⠀⣸⠁⠍⣙⠁
⠀⠀⠀⠀⠀⠸⡇⡻⡄⠀⠀⠀⠀⠀⢀⡼⡃⠽⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣻⡿⣆⠀⠀⠀⠀⢰⠯⣟⣲⠃
⠀⠀⠀⠀⠀⠀⢳⡾⡿⡄⠀⠀⠀⠀⡾⣷⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣶⡾⡄⠀⠀⠀⡿⣶⣦⠏
⠀⠀⠀⠀⠀⠀⠀⠡⡀⠚⠀⠀⠀⣸⠦⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠶⢷⠀⠀⢸⠶⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠀⠀⠀⠀⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⣤⠀⢠⣦⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠤⢇⢠⠧⢄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣃⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡞⠆⡞⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⠼⠮⣄⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠄⠰⢳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣷⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⢻⡟⣛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢦⠂⢱⠾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠈⡍⢸⠉⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣥⠀⢀⣬⠆
⠀⠀⠀⠀⠀⠀⠀⢠⣞⠷⡞⠀⠘⡶⢟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠉⢻⠁⠀⠛⠉⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠆⠀⠀⠰⣞⣦
⠀⠀⠀⠀⠀⠀⢀⢾⡿⡿⠀⠀⠀⢹⡿⣛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣈⠉⠃⠀⠀⠘⡉⢁⢡⠀⠀⠀⠀⠀⠀⠀⠀⣘⢩⠋⠀⠀⠀⠀⠹⡉⣢
⠀⠀⠀⠀⠀⢀⣾⡶⣤⠁⠀⠀⠀⠀⢣⡴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠦⣤⣜⠀⠀⠀⠀⢡⣤⡴⢆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣼⣥⣹⠃⠀⠀⠀⠀⠀⠀⢛⣡⣽⡄⠀⠀⠀⠀⠀⠀⠀⢰⠧⣤⣠⠁⠀⠀⠀⠀⠀⣄⣤⡼⣆
⠀⠀⠀⠀⠀⣷⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⡇⠀⠀⠀⠀⠀⠀⠠⠯⠝⣓⡆⠀⠀⠀⠀⠀⠀⢹⣚⡫⠽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣛⠲⡶⠁⠀⠀⠀⠀⠀⠀⠀⢲⠶⢛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣬⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣥⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤
⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⠿⣛⣅
⠀⠀⠀⠀⠀⠀⠈⠿⣛⣭⣶⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣩⣍⣙⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡌⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⣡⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠋⢿⡿⠿⢛⣩⣴⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⠁⠀⣠⣴⣾⣿⡿⢟⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠀⠀⠀⠛⠛⠋⣵⣆⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⣤⣤⣶⣾⣿⣿⠿⠛⠉⠁⣀⣤⣶⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠛⠛⠛⠛⠻⣿⠉⠀⢀⣤⠶⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣶⣿⣿⣿⣿⣦⡀⠀⣀⣬⣶⣿⡆⣻⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⡈⠿⠟⠋⠥⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢻⠓⣄⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠢⡔⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⡋⠀⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⢁⡐⡜⠁⠀⠀⠀⠀⠀⠰⠶⡄⣣⠀⣠⠚⢉⡁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠜⣧⠀⠀⠀⠀⠀⠀⠀⠑⠻⣟⠔⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⢹⠴⠋⠀⠀⠀⠀⠀⠀⢠⡷⠺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢹⡀⠀⠀⠀⠀⠀⠀⠀⡆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⡎⢿⠭⠛⠉⠭⠍⠉⣽⢃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣍⠹⡄⠀⠙⣌⠳⣖⡒⠒⣒⡶⢃⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣌⢩⣠⢡⠶⡢⠀⠑⢮⡵⢚⣭⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣅⠈⢾⣰⠁⢀⡔⣡⠞⠓⢌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠻⡇⢠⠏⡼⠿⠤⠴⠾⠷⡘⢦⠀⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢳⠇⣼⣢⣤⣀⣀⣖⣲⣿⡈⣇⡀⠀⢀⠖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣧⠀⢀⠀⠀⠀⢀⣠⡇⣿⢸⡖⣡⠴⣫⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢻⡉⠁⠀⠁⠀⠈⡿⢁⠇⣈⡜⠛⣞⢟⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡙⠯⡥⢤⡽⠛⣡⡯⠎⡥⡀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢜⣩⢴⡋⠁⠀⠈⠢⢭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣰⡯⠹⢦⡘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡾⢁⣀⠀⢀⣹⣆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⡟⠉⠉⠁⠀⠀⠁⠙⡆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡸⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡤⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⣣⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢬⢣⣵⣄⡀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠔⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡯⣤⣝⠓⠀⠀⠀⠀⠀⠀⠀⠐⢥⠤⡪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡇⡰⢈⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈
⡿⠿⠿⢿⠿⡻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣠⡉⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢀⠰⠀⠒⠄⠀⠀⠀⠀⠀⠠⠰⠀⠀⠄⠆⢀
⠀⠀⠀⠀⠀⠀⠸⠤⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⢃⢀⠞⡍⠌⣂⠄⡀⠀⠀⠀⠀⠐⢯⢨⠭⠅⠭⠝⠹⡆
⠀⠀⠀⠀⠀⠓⠖⠀⢀⡤⠃⠀⠀⠀⠀⠀⠀⠀⡤⢁⡀⣤⠀⠀⡖⠒⠀⠀⠀⠀⠀⢀⠉⠲⡠⠔⣇⡄⠴⠀⠀⠀⠀⠀⠇⡀⠀⢆⠂⠀⠠⠇
⠀⠀⠀⠀⠀⠰⠉⠈⠉⠡⠀⠀⠀⠀⠀⠀⠀⠨⠀⠨⠤⠄⠅⠀⠑⠀⠀⠀⠀⠀⠀⠈⠚⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠱

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠀⠀⠀⠈⠀⠠⢀⠄⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡉⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠛⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠠⢥⠄⠀⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠊⠀⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠰⣤⠚⡖⣰⠒⠀⠀⠀⠀⠀⠀⢠⣾⠒⢠⡘⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠈⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⠀⠀⠀⡐⠁⠠⡢⠀⣠⠀⡔⠄⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠐⠀⠀⠈⢬⡱⡀⠰⠹⠀⠀⠀⠀⡔⠁⠀⠈⣤⡀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠰⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⣀⠄⣈⠉⠀⡀⠡⠂⠀⠀⠀⠀⡄⠀⣠⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⢀⠁⠐⢄⠀⡠⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⢦⠀⢤⠂⠀⠀⠀⠀⠀⠀⢀⡔⠓⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⢀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⡶⠢⢄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠘⡇⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣴⣴⣶⣦⠴⡟⠻⢗⣸⣯⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢿⣼⡿⡿⠦⣼⡧⠀⢨⠿⣿⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢿⣿⣾⢿⣶⠤⣼⡧⠔⡿⣯⣵⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⣯⢽⡷⣤⢺⢃⡼⣳⢷⣢⡿⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣽⢷⣻⣽⡭⢿⣳⡛⣟⣞⡾⡼⡗⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢿⣞⣯⣶⣻⣿⣿⣯⣽⢷⡿⣌⣿⣵⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⣷⣿⡿⠋⡗⠊⠐⢚⡟⢾⣯⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⣷⠀⠀⢀⡇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣇⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠱⡄
⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢹
⠀⢨⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣀⠀⠀⠀⡌
⠀⢸⠀⠀⠀⢙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠙⠿⣶⣤⡇
⠀⠀⣆⣠⣶⠟⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡹
⠀⠀⠸⡏⠁⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠹⡀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⣤⣄⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠰⣦⣰⠃
⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠈⢯⠁⠀⠐⠂⠀⠠⠤⠀⣀⣠⡇⠀⠀⠀⢀⠇⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠉⠉⢀⡟⠂⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⣠⣴⣾⣿⡆⠀⠀⠀⠀⠀⠀⢰⢿⣷⣦⣄⡜⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠛⠉⠀⢹⡀⠀⠀⠀⠀⠀⡏⠀⠈⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢣⠀⠀⠀⠀⢸⠃⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠈⡆⠀⠀⠀⡗⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠸⡄⠀⢰⠁⠀⢠⠎⠀⠀⠀⠀⠀⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣀⣹⣤⣁⡀⠀⡎⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⣉⣿⡀⠤⠀⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⡀⠀⠀⢈⣉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠉⡟⢻⠉⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡘⠀⠀⢣⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢠⠃⠀⠀⠈⡄⢺⠄⢣⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡞⠀⠀⠀⠀⠙⣰⢖⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢸⠃⠀⠀⠀⠀⠘⣏⢈⡥⠘⡆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢻⡯⢉⣀⠘⡄⠀⠀⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠈⡗⠆⣼⡄⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢠⡽⢁⣴⠋⠱⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢶⠿⢷⡀⠙⡄
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠰⠚⠟⠓⠀⠘⡄
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⣡⠶⣋⣷⡀⠘⡄
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢨⡿⠯⢙⠧⠤⡘⡄
⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⢁⡞⣡⠴⠚⠅⠘⡄
⠀⠀⣰⠃⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⣹⠻⠞⠛⢷⠀⠸⡄
⠀⢠⠃⠀⠀⠀⠀⠀⢠⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⣧⣔⡬⢛⠖⠄⢹⡄
⠀⡇⠀⠀⠀⠀⠀⢠⠏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⢮⡴⣤⠒⣆⠀⠣
⠀⡇⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣿⠛⠏⡜⢀⡎
⠀⠑⢆⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠖⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠛⠛⠛⠛⠛⠛⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠦⠴⣦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠉⠉⢉⣉⡉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⠀⠀⠀⠤⠟⠘⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡄⠀⠀⠀⢘⡷⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡿
⠀⡤⠶⠞⠿⣿⣟⠲⢦⣽⡆⠀⠻⣆⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠔⠁⠀⡼⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀
⣼⡁⣠⡶⠒⠶⠬⢿⣷⣄⣙⣦⡀⠈⠳⢤⣤⠞⠛⠋⠉⠉⠉⠙⠛⠁⠀⠉⠉⠉⠀⠀⠁⠀⠈⠉⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⠉⠉⠉⠙
⠈⢹⣯⣤⣀⠀⠀⠀⠈⠉⠛⠦⠽⢦⣄⡀⢹⣦⣤⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠙⠢⢥⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠻⠷⣶⡶⠦⠤⢤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠒⠤⣤⣤⣤⣤⡤⢤⣤⣄⣠⣤⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⡀
⣿⣿⡟⠛⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠒⠋⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢄⡀⠀⠀⢀⡴⠋⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡎⠀⠈⢀⠀⠐⢄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠋⠉⠉⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢿⠀⠠⡄⢡⡀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠨⠖⠒⢀⠀⠙⣆⠡⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⡁⠀⠀⠀⠀⠄⢸⠀⠡⠀⠈⠂⠀⠀⠀⢠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⢹⠄⠀⠄⠚⠳⠞⡀⠀⠁⠀⠈⡦⣈⣦⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠶⢤⣄⣀⠀⠀⠈⠉⠁⠀⠀⠀⣘⠁⠀⡀⠂⢀⡸⠁⣨⠁⠀⠀⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠦⠲⠤⣀⡀⠉⠛⠶⣄⡀⠀⠀⠀⡀⠉⠒⠦⡐⡴⠊⠁⠀⠉⡐⠠⡐⡀⠀⠀⠀⠀⠈⡀⢷⣄⣀⣠⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠳⠄⠠⠀⠀⠉⠑⠦⣌⡀⠈⠓⠦⣴⣤⣄⠀⠀⠜⠁⠀⠀⠀⠀⠀⠰⠟⠂⠁⠂⢀⡠⠀⠀⡡⠚⠛⠥⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠁⠐⠠⢀⡀⠉⠓⠤⣀⠠⢙⠻⢷⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⢀⠀⠤⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⣦⠲⠦⣱⣀⠀⠀⠉⠂⣬⡀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠒⠢⣄⡉⣏⠙⠢⢀⠀⠉⠳⢄⡀⠉⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣽⣶⣭⣋⠿⣶⣄⡉⠂⠔⠈⠛⠳⠀⠐⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣙⠻⢿⣿⣶⠀⡙⢿⣦⡈⠳⠀⠤⡀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣌⡛⢷⠦⡽⠋⢿⠂⠀⠀⠀⠐⣄⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡤⠶⠶⢤⣌⠀⡇⠀⠀⠌⠉⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠈⡶⠑⢠⠈⠀⠀⠀⠱⢶⣀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⡖⠠⠂⠠⠂⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠠⠤⠤⠶⠖⢶⡶⢶⡤⠤⠤⠤⠤⣀⠀⠀⠙⣦⣤⣤⡤⠞⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠲⠖⠚⣛⣂⡉⠑⢾⣝⠛⠒⠲⠶⣼⠀⠀⠀⠀⠀⠌⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠀⣱⠆⠀⠀⠌⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠏⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠁⠀⢠⡄⠀⠈⠉⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠰⢿⡿⠗⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡾⢛⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀⣈⣁⣀⡀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢀⣴⠟⠉⠻⣿⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡇⠀⠀⠀⣀⣼⠟⠁⠀⠀⠀⢈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⣤⣤⣾⠟⠁⠀⠀⠀⢠⡾⠛⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣅⠙⢿⣦⡀⠀⠀⡀⠸⣷⣤⣴⡿⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠻⢷⣤⣀⡀⢀⣀⣠⣾⠟⠀⢈⣿⣷⣄⠙⢿⣦⣾⠟⠀⠀⠉⠁⣤⡈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⣿⠿⢷⣄⡉⠙⠛⠛⠛⠉⠀⠀⢰⣟⢁⣼⡿⣷⣄⣹⣷⠀⠀⠀⠀⠀⠹⣧⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢁⣾⢋⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠋⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⣿⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠘⠿⣿⠇⠈⢿⡄⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠘⠻⣶⣄⠈⣿⡄⠀⢸⡟⠉⠉⠉⠉⠉⠉⠉⠉⣿⡆⠀⠀⠀⠀⠀⠀⠀⣼⡏⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠿⣾⣷⡀⠈⠛⠿⠿⣿⣿⠿⣿⣿⠻⠛⠀⠀⠀⠀⠀⢀⣠⡾⠋⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠙⠃⠀⠀⠀⠀⠈⠻⣷⣄⠙⠻⠂⣰⡾⠻⢷⡄⠙⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡟⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣙⢻⣶⡀⢻⣧⣀⣼⠏⢠⡟⣋⢁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣇⣀⣠⣿⠇⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠓⠀⠉⠉⠁⠀⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢠⢞⡉⣉⠙⣆⠀⠀⠀⠀⠀⠀⠀⣰⠋⡉⣉⡑⣄⠀⠀⠀⢀⡞⣉⢉⡉⢲⡀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡉⣉⡳⡄
⠀⠀⠀⠀⣿⠀⣻⣯⡀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⣹⣞⠁⢸⠀⠀⠀⢸⡀⢙⣾⡁⠈⡇⠀⠀⠀⠀⠀⠀⠀⡇⢀⣽⣞⠀⣿
⠀⠀⠀⠀⢹⠀⢿⣸⠇⢸⠀⠀⠀⠀⠀⠀⠀⡇⠘⣇⡼⠀⡿⠀⠀⠀⠸⡇⠹⣕⡟⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠸⣇⡿⠀⡏
⠀⠀⠀⠀⠘⡆⢸⠻⣦⠈⡇⠀⠀⠀⠀⠀⢰⠇⣰⠏⣧⢠⠇⠀⠀⠀⠀⢧⢠⡏⢷⠀⢧⠀⠀⠀⠀⠀⠀⢸⠁⣴⠛⡇⢰⠇
⠀⠀⠀⠀⠀⢱⡈⢛⡿⣄⢻⠀⠀⠀⠀⠀⡜⣠⢾⡞⢁⡞⠀⠀⠀⠀⠀⠘⡆⠙⣿⢦⡸⡄⠀⠀⠀⠀⠀⡞⣠⢿⡟⢁⡞
⠀⠀⠀⠀⠀⠀⠳⡈⠷⣽⣈⢧⠀⠀⠀⣸⣁⣿⠼⢃⡞⠀⠀⠀⠀⠀⠀⠀⠘⣆⠻⢵⣇⡹⡀⠀⠀⠀⡼⣁⣧⠾⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠙⣆⢿⣉⣏⢣⡀⡼⣹⣏⡽⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡸⣍⣷⠹⡄⢀⡜⣹⣉⡿⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢯⣻⡻⣿⣹⢃⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⢻⣹⡟⡟⣟⡽⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢦⡸⣍⣏⠳⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣶⢚⣙⢶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣞⣽⣦⣿⣹⡈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢻⡼⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠎⣾⣉⣏⡜⠁⢳⣹⣏⣳⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣞⣹⠆⡇
⠀⠀⠀⠀⠀⠀⡰⢃⡶⢻⢉⡞⠀⠀⠀⢳⡉⡟⢲⡌⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⢿⡀⡇
⠀⠀⠀⠀⠀⣰⢁⡬⣷⠋⡼⠀⠀⠀⠀⠀⢧⠙⢺⣧⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢳⣞⠁⣇
⠀⠀⠀⠀⢠⠇⢸⣤⠟⢀⡇⠀⠀⠀⠀⠀⠸⡆⠹⣆⡟⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢯⣹⠇⢿
⠀⠀⠀⠀⣸⠀⣞⢹⡆⢸⠀⠀⠀⠀⠀⠀⠀⡇⢠⡯⣳⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡼⢳⡄⢸
⠀⠀⠀⠀⣿⠀⡽⣟⠁⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⣹⢯⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢻⣿⠁⢸
⠀⠀⠀⠀⠘⢮⣁⣈⣠⠏⠀⠀⠀⠀⠀⠀⠀⠙⢄⣉⣉⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣋⣈⣣⠞
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⢀⣿⣤⣧⠀⠀⠀⠀⣼⣇⣼⡄⠀⠀⢠⣧⣼⣇⠀⠀⢀⣿⣤⣧⠀⠀⠀⣼⣇⣿⡄⠀⠀⢠⣧⣼⣇⠀⠀⢀⣿⣠⣧⠀⠀⠀⣼⣄⣿⡀
⠀⠀⠈⣿⢙⡿⠀⠀⠀⠀⢹⡟⣿⠏⠀⠀⢙⣿⢻⡟⠀⠀⠈⣿⠹⣿⠃⠀⠀⢹⡟⣿⠇⠀⠀⢹⣟⢻⡟⠀⠀⠈⣿⢹⡿⠁⠀⠀⢻⡟⣿⠃
⠀⠀⠀⣩⣮⠁⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠉⠀⠉⠀⠀⠀⠈⠁⠉⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠉⠈⠁⠀⠀⠀⠈⠁⠉
⠀⠀⣼⠋
⠀⠀⡇⠀⠀⠀⢀⣆⣰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣰⡀
⠀⠀⠐⣄⡀⠀⣸⣟⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⢻⡟
⠀⠀⠀⠀⠙⠛⠛⠏⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠃
⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⣴⣷⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣄⣤⣼⡇⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⣿⠻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣴⣶⡿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⣇⠀⠀⠀⣀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⠀⠀⠀⠀⠐⠒⠛⠁⣠⡝⣥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⣄⣀⣀⠀⢀⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡿⣯⡶⠿⢾⣝⢿⠠⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠰⣷⠀⣿⢾⣷⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡿⢦⣤⡼⣿⣾⠁⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠻⣿⠛⠉⣋⡼⣿⡽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⡃⢶⠄⣄⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⣷⣶⣾⡿⠂⢄⠀⠀⠙⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠁⠄⠀⠘⢦⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⠄⠐⢃⣀⠭⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠔⠚⡈⠁⠀⠐⠢⣶⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠄⠁⠘⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠄⢉⣿⠻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢈⣣⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢷⣵⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢨⢾⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠐⣕⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢪⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡆⠀⠓⣘⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⣠⣤⠮⣅⢀⡀⠘⠿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣫⡁⢰⠒⠿⣦⡿⠤⠤⠦⢤⠄⠀⠉⠳⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣼⣿⣤⣤⣶⣟⠋⡄⠠⣔⡆⠀⠀⠀⡒⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠜⡄⠁⠙⢿⡙⣿⣦⠃⠀⣽⣧⠀⠀⠀⣽⣟⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⢤⡳⠄⡇⠌⢠⡄⠱⢸⣿⣶⠀⢸⡣⠀⠀⠀⣟⢏⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣬⣦⡝⢆⢣⠘⢦⣄⠠⢸⣟⢿⠀⠠⡿⡄⠀⠀⠲⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢈⢢⡰⣮⣷⠄⠐⢤⠤⣴⡿⠋⣭⡄⠀⠿⡂⠀⠀⢰⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⡳⡝⢊⡑⢉⠴⢶⣶⣼⣧⠀⠻⡇⠀⠸⠁⠀⠀⠣⣤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠩⠂⡑⣊⢁⠲⠀⣿⡇⠁⠀⢸⡇⠀⠘⢁⠀⠀⠘⠊⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠟⢂⣾⡟⢁⠛⠻⠚⠿⣷⡾⠄⢈⣁⠀⠀⣦⠀⠀⠀⡟⠅⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠰⡛⣾⣯⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢽⡄⠀⠀⠸⣂⣤⣺⡷⣿⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡧⣹⢌⣿⣽⡃⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣃⢨⠳⠶⠚⠛⠉⠀⠘⠂⣉⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣽⢢⣿⡵⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠔⠒⢐⣀⠁⡠⢤⣶⣾⣥⠈⢀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⢻⣎⣧⠀⡆⠀⠀⠀⠀⠀⠀⠈⠓⠉⠈⢻⡏⠁⢹⣄⡌⠈⠉⠳⡟⢧⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢮⡺⣽⠿⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠏⢹⡄⠀⠃⠀⠀⠀⠙⣄⠑⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢰⢟⠻⣟⢿⡆⣱⣞⣓⡆⢀⡀⣤⣚⣉⣀⠄⢻⠀⠇⠀⢸⠃⠠⠤⠄⢀⣃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢣⢷⠙⢷⣾⣿⡋⠰⣿⣢⣬⠚⠭⠷⠿⠄⠀⠒⠀⠐⠊⠀⠀⢶⡷⠄⢀⣤⣿⡥⠶⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⢫⣷⣈⣁⠙⠥⠴⣟⠉⠂⠔⣦⣄⣀⣰⣂⣀⣀⣀⠶⠀⣀⢠⠴⣾⠛⢛⡣⠐⢂⣤⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠿⣽⣿⣿⡻⢶⣶⣤⣭⣡⡰⠞⠻⢿⣛⣛⣋⣼⡶⢟⠏⣳⡤⠔⣋⣤⣴⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠔⣒⣮⠉⣨⣀⡈⣡⣭⣮⣥⣉⣍⣠⣿⣮⣿⣿⣥⣰⠿⢛⣭⣶⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡌⣄⠢⠉⣽⠀⣭⣹⡄⣨⣿⣿⣿⡏⢹⣟⡛⠿⢿⣿⣿⣏⣾⡿⠛⠉
⠀⠀⠀⠀⠀⢀⡤⠒⠉⠈⠁⣾⠇⠀⢼⠃⢿⣹⡇⠸⣟⣿⢿⣇⠈⣉⠩⢟⡛⠶⠉⠛⠛⠛⠻⣤
⠀⠀⠀⠀⢀⠉⠀⠀⠀⠳⢰⠝⠃⢀⡲⡀⢸⡎⣇⢨⡄⣳⢦⣤⠋⡡⠈⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠰⡀⣤⡄⢀
⠀⠀⠀⠀⠈⣄⠀⠀⠼⡿⢦⣄⡀⠈⠱⠜⢸⣿⣿⢸⣧⡝⢺⣿⣎⠀⠀⠀⠀⠀⢀⣀⠤⠒⠈⢀⣟⣈⠑⠙⠋⠢⠍⡑⠢
⠀⠀⠀⠀⠀⣿⢦⡀⠀⠉⠃⠯⠟⣛⡦⢢⣤⠔⠛⡞⠋⢀⣀⠡⠙⢻⡚⡍⠉⠁⠀⠀⢀⣤⠆⡁⢀⡀⠈⠉⠀⠂⠀⠀⠂
⠀⠀⠀⠀⢀⣟⢢⡈⠢⣄⠀⠀⠀⠀⠐⠑⠟⠀⠀⠟⣹⠿⠜⢛⣛⡻⣥⣼⡒⢶⡺⢟⡿⣷⠷⣴⣮⣽⣟⣳⠶⣶⣤⣤⢠
⣀⠠⢄⡙⠦⡝⢷⡴⢂⠈⠳⢄⠀⠀⠀⠀⠶⡑⠄⠈⠣⡀⠀⠉⠛⠙⠊⠑⢉⠢⠜⠊⣁⡤⠒⢌⠉⢁⡘⡙⢻⡿⣿⡿⠞
⠠⠍⡒⠬⢐⠢⢄⡻⢯⡐⡠⡀⠑⠢⣀⠀⠑⠈⡢⢕⠀⠈⠂⢄⡀⠀⠀⠀⠀⠉⠀⠉⠊⠁⠊⠐⠊⠙⠁⠊⠁⠁⠈
⠀⠀⠀⠈⠠⢌⠣⢌⠛⠻⢧⣜⠖⡀⠈⠑⠦⡀⠀⠀⠓⠊⡠⣀⠈⠲⢄
⠀⠀⠀⠀⠀⠐⠪⢐⠨⣑⢤⡙⢾⢦⡕⠔⠀⢀⠁⠂⠤⢀⣈⠈⠓⠂⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⢎⡰⣬⠛⡵⣦⡞⠱⡓⢦⡀⣀⠀⠀⠈⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠾⠋⢝⠶⢠⣀⡙⠋⠙⠏⠘⢉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠹⢟⠛⠲⡴⢾⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢶⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⢇⠈⢧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣧⠈⢇⠈⢆⠈⢇⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠠⣦⠘⢇⠘⣇⠘⣇⠙⢧⠈⢆⠈⢆⠈⢆⠈⣦⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢿⣈⣿⣿⣿⣿⣿⣿⡙⡇⠘⣄⠈⣆⠘⣆⠈⣆⠸⡧⠸⠇⠸⠇⠸⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠛⠋⠈⠛⠛⠿⣿⣿⣿⣿⣧⠘⡄⠈⡄⠘⡄⠈⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⢋⣬⠐⡄⠘⣄⠸⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣿⡷⠞⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡠⠞⠋⠁⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⣀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⠀⢶⣦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠈⣿⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠟⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⣀⣀⣀⣀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⢉⣉⣉⣉⣉⠉⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣌⡉⠉⠉⣉⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣋⣴⢞⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⣀⢈⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢁⣀⣉⣉⣉⣁⣀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⣭⣭⣭⣭⣭⣍⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⣤⣤⣤⡄⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠩⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣀⣀⣀⣈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⣤⣭⣭⣭⣥⣄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢸⠆⠀⠀⠀⠀⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⢸⠁⠀⠀⠀⠠⢜⠋
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠣⠱⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢐⡆⠀⢡⢪⠀⠀⠉⠙⢄⠀⠀⡸
⠀⠀⠀⠀⠁⠀⠀⢣⠀⠀⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡀⣰⠁⠀⠂⣘⡄⠀⠀⠀⠀⠀⠘⣧⠈⠀⢸⠃⠀⠀⢠⠀⠀⠀⠀⢄⠁⡞⠁⠀⣀
⠀⠀⠀⠀⠂⠀⠀⠘⣆⠀⠀⠀⠸⠁⠀⠆⡤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠘⠇⣠⠜⠀⠀⠀⠀⠀⠀⠀⠐⠁⡷⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢠⠇⠀⠀⠀⢓⣄⠀⢀⠠⢠⡀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⡀⢠⡣⣀⠀⠀⠀⣴⠀⠀⠐⠀⡜⠁⠀⠀⢱⡀
⠀⠀⡀⠀⠀⠘⠲⠄⠁⢀⡤⠊⠀⠀⠰⠀⠀⠀⠀⠈⠀⠀⠱⡀⠐⠀⠀⣾⠀⠀⠀⠀⠀⠈⠀⡜⠃⠈⢧⡀⠀⠀⠀⠐⠤⠴⠃⠀⠀⠠⡼⠃
⢁⣶⠀⠀⠀⠀⡀⠀⢰⡄⠂⢤⡀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠱⡄⠀⢸⠇⠀⠐⠂⠀⠠⠀⠸⠀⠀⠁⠀⠳⡀⠀⠀⠀⠀⠀⠀⠈⣑⡇
⠸⠄⠄⠀⠀⠈⠀⠀⢇⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄⠃⣠⠟⠀⠀⠀⠀⠠⠀⢠⡇⠀⠀⠀⠀⣰⠇⠀⠀⡀⠈⢷⠀⠐⢀⠐
⠀⠀⠀⠀⡈⠀⠀⠀⡎⠀⠃⠀⠀⢹⡀⠀⠀⢠⡰⢖⡀⠀⠀⠈⢲⡗⢤⡀⠀⠀⠀⠀⠓⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠂⠀⢈⡀
⢠⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢣⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⣆⠀⠀⠀⠀⠀⠀⡠⡝⠁⠀⠀⠀⠀⠀⠀⣸⠇⠁⢀⠴⠞⠁
⠀⠀⢀⠀⠀⠀⣠⠃⠀⠀⠀⠐⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⢀⠀⠘⢆⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⢀⠄⢂⡀⡞⠚⠁
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠰⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⣷⠀⣷⠀⠀⡠⢠⠀⠀⠀⡀⠁⠀⣼⠃⣧
⠀⠀⠘⠢⠔⠃⠀⠀⠊⠀⠀⠀⠀⠀⠤⠤⠃⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠈⠀⠤⢸⡆⠸⣄⠈⠀⡼⠀⠀⠊⠀⢠⠜⠩⠀⠘⡆⠀⠀⠀⢢⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠐⠒⠚⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⢴⠛⠁⡐⠁⠀⡠⠋⠀⠀⠀⠀⢹⠀⠀⠜⠉⠅
⠀⠀⠀⠀⠀⠀⠀⡠⡘⠀⠀⠀⠀⢠⠂⠐⡀⠁⠀⡃⠀⠀⠀⠀⠀⢀⣴⠄⠀⠀⢀⠈⠀⡿⠊⢦⠀⠓⠴⠋⠁⠀⠀⠀⢀⡀⣘⡆
⠀⠀⠀⠀⠀⠀⠀⣹⠁⠀⠀⠀⠀⠀⠙⢦⡈⠃⣸⠃⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠆⠀⡞⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⢒⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣿⡄⠂⢀⠀⠀⠀⠀⠀⠀⠀⠌⠀⣠⠃⠀⠀⠠⠀⠸⡆⠀⠘⠁
⠀⠈⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⢢⡀⠀⠄⠀⠀⠀⠀⠀⠓⠞⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⡇⠀⠀⠙⠶⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠘⠋⣡⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢈⣤⡀⠈⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡼⠟⠿⣿⡀⠈⠑
⠀⠀⠀⠀⠀⠀⡛⠁⠀⠀⢠⣤⣤⡌⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠁⠈⠀⠢⡀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣈⣥⡦
⠀⠀⠀⠀⢀⡀⡁⠀⠀⠠⠿⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣄⠋⠁⠀⠀⠻
⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⡂⠀⠀⠀⠀⠀⠀⠃⠻⠀⠀⠀⠀⠀⢰⣶



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣙⣋⣍⣹⣟⣽⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠀⠀⢀⠀⠀⠀⢀
⠀⠈⠀⠀⢀⣴⠾⠗⣣⡄⠀⠀⠊⣠⡞⠛⠛⠀⠀⡀⠠⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷
⠈⡁⠀⣴⠃⠀⠀⠀⠉⠁⡦⠀⠘⠿⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡿⢿
⠀⠀⣼⡇⠀⠀⠀⠀⣠⠾⠃⠀⠀⠀⠤⠀⠀⠈⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢾⣻⡞
⠀⣸⣿⣠⣤⣤⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⢿⢿⣸⠟⠉⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⣀⠀⢀⣄⠂⠀⠀⠑⠀⠀⠀⠀⠈⠓⠆⡤
⠀⠀⠀⠀⠀⠘⠟⣤⠌⠀⠀⠀⠘⠋⠀⠙⠒⠉⠁⠁⠀⠀⢀⣿⠁⠁⠀⣠⡾⠃⣀⡀⠀⠀⠀⠀⡀⢠⡟
⠀⠀⡀⢀⣄⠀⠀⠀⠐⠂⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠘⠿⢣⣾⠟⢡⣀⠀⠀⠀⡠⠈⡁
⠀⠘⠏⠈⠟⠀⠀⠀⠀⠀⠐⡏⠈⢳⠀⢀⢀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠲⣧⠔⠛⠅⠀⢻
⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⣇⣠⠜⠀⣼⠬⡇⡀⠀⠀⢸
⠀⣰⣿⣷⡀⠀⠀⠀⠀⠀⠀⡇⣴⣳⣆⢸⠀⡏⢹⢴⢹⢼⣴⢛⡦⡞⣳⡘⣼⠃⠈⠀⢀⡀⠀⣀⢀⡃
⠀⣿⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠁⠀⠈⠉⠈⠀⠉⠀⡽⣻⠀⣸⠀⠀⠀⠈⢙⣛⡉⠀⠂
⠸⣛⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠉⠁⠤⠇⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠸⠷⠶⢲⠖⠾⠟⢋⣀⣀⡤⢀⠤⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠠⠆⠁⠀⣄⡐⠀⠀⢿⡀⣴⣶⣶⠆⠀⡜⠡⠤⢤⡀⣸⢸⢀⡴⢂
⠀⠀⠀⠀⠀⠀⠀⠒⠃⠢⡄⠃⠀⠀⣄⠀⠀⠀⠈⠛⠀⠀⠘⠃⠠⠶⣶⣶⣶⠌⠒⠦⠤⠭⠕⠃⠈⠱⠎
⠀⠀⣀⠰⠈⠈⠀⠀⠀⣠⠋⠃⡀⣨⡏⠀⠀⠀⢠⢤⡆⠀⠀⢀⢠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠲⠐⢣⠄⣀⠀⢀⡼⡄⠁⠀⠀⠈⠀⠀⢀⡴⠂⠁⠀⠀⠀⣿⠆⢸⡇⢸⡇⢰⣿⠀⣿⡆⣿⠀⠀⣾⡇
⠀⠀⠈⠙⠐⠁⠭⠤⠓⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠉⠀⠈⠁⠈⠀⠈⠁⠀⠈⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⢳⣥⡀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢿⣦⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡹⣷⣄⠈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣻⣷⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣶⣯⣈⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣤⣤⣬⣽⣿⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⠛⠻⢻⣿⣿⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡒⠀⠀⢸⣿⣿⣿⣿⣿⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⠈⢀⣠⣾⣿⣿⣿⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠏⠁⠉⠁⠈⠉⣛⣛⢓⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠀⢀⡴⠶⠓⠀⠹⠟⠴⠀⢙⠛⠓⢶⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡏⢠⡈⢀⡠⣀⠈⠰⠤⢴⠆⣀⣀⣠⣤⠾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣇⣤⠴⠛⢛⡁⢹⣿⣿⣿⣿⠁⠀⣿⣿⢴⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣽⢷⣿⣿⣶⣮⣵⣀⣘⣛⣛⣛⣋⣱⣬⣵⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠛⢿⣿⣿⣿⣼⡶⣶⣿⡶⣿⣿⣶⣿⣿⡿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡀⠀⣸⣿⣿⣿⣿⡇⣰⡄⡅⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⠇⣠⣖⠒⠲⢤⡀
⠀⠀⠀⠀⠀⡀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣌⠳⣄⠀⠹⣄
⠀⠀⠀⢀⣾⣧⣄⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣌⠳⢤⣸⣿⣿⣿⣶⣶⣶⣶⣦⣤⡄
⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣻⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣋⠛⣂⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣾⣭⣟⣿⣿⣿⣷⣦⣄⡀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣮⣽⣿⠿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣶⣯⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠁⠀⠀⢀⣄⠰⣆⣤⣀⣠⣤⣄⣠⣤⣦⡶⠶⠾⠓⠒⠈⠟⠉⡠⠃⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⣶⣤⡶⠶⠿⢻⡟⠋⠏⠀⠁⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⣤⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠸⠏⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⣤⣄⣬⣿⣤⣀⡀⠀⠙⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣃⣤⣶⢿⣿⣿⣿⣿⣿⣦⣀⣈⣠⠀⣽⣿⣿⣿⣿⣿⣿⣶⣄⠀⠐⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢿⣷⣼⣿⣿⣿⣿⣿⣿⠋⣁⡀⠙⠒⣿⣿⣿⣿⣿⣿⣿⢣⣿⠆⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⡏⣸⣿⣥⣤⠀⢿⣿⣿⣿⣿⡟⣯⣿⣿⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⡀⠀⠠⢽⣿⠿⠿⠟⠉⠀⣿⣿⣿⣿⣅⠀⠐⣍⣹⡿⢃⠛⠑⠛⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⡇⣀⣠⡶⣤⡀⣀⣤⡖⠀⢸⣿⣿⣿⣿⣿⠇⠰⣾⣧⣴⣬⣀⣄⣀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠥⣄⡙⢿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⡏⠀⠘⠧⠼⢟⣻⣿⣿⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⡌⢻⡿⠀⠀⠛⠿⠛⡇⠿⠟⠑⠀⠀⡁⠂⣾⣿⠏⣿⣄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢐⡇⠀⠂⠀⠀⠀⠙⠀⣿⡿⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⡷⠖⠖⡖⡲⢆⣴⣮⣷⣷⡤⢴⡶⣶⡶⠘⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⠈⢿⣇⢤⡤⡄⡀⢀⣿⣿⣿⣿⣷⠼⠷⢚⠇⠀⡿⠁⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⡆⠀⠀⠀⢤⣿⣧⣸⣿⣿⣿⣈⡇⢀⠀⠘⠃⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⢀⠀⠸⡏⣿⣿⣿⣿⣿⣗⠮⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣀⠀⠀⠀⠀⠙⠹⠟⠿⠀⠋⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠼⡄⠀⠀⠀⢀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣶⣦⣴⣿⣆⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠈⠙⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣴⣋⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠙⣧⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⡨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣾⠟⢯⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠳⣧⡀⠀⠀⣨⢾⣦⣞⠟⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡈⢻⡦⠚⠁⠀⠀⠈⢢⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⠃⠀⠀⠀⠀⠀⠈⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⢠⠇⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⣶⠶⠶⣖⠒⠒⠋⠀⠀⠀⡠⠋⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠉⠒⠒⠒⠒⠉⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡑⠲⢤⡔⠓⣦⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠏⠉⠉⠉⠉⠉⠙⠒⠉⠉⠉⠉⠉⠉⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⣀⣀⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⢶⣶⡶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⠛⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⠿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣤⣤⣬⣥⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣉⣉⣩⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣉⣩⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣉⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣶⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⠋⠛⠛⠋⣛⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣤⣼⣿⣿⡀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣻⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⣿⣿⡅
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠚⠻⠿⠟⠳⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠠⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠦⣕⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⣊⠁⠀⠀⠀⠙⠐⠀⠀⠀⠀⠀⠀⠀⠾⠋⠁⠀⠀⠀⠃⠀⠈⢮⡢⠘⠳⣄⣤⡴⢶⠒⢒⣿⠿⠯⠽⠗⠲⢄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠀⠒⣶⡈⣷⡅⡀⠘⢿⣟⠂⠥⠚⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢀⣷⣾⡷⣾⣿⡔⠀⠸⣧⣀⡆⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠻⣿⣧⢤⡩⣝⣧⠁⠀⠀⠀⡀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡾⢀⣍⠐⢉⡙⢿⣆⠀⢻⣿⢿⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢿⠮⡀⠁⣻⠘⡠⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢠⠞⠴⠋⠉⠉⠅⠁⠀⢹⣟⢸⡏⢰⠀⠀⠀⠀⠀⠀⠀⠃⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⠆⠛⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠘⠁⠀⠀⠀⠐⠀⡿⣿⣷⠛⠀⠀⠀⠀⠀⠀⠀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠹⡷⣀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠁⣿⠏⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣀⡀⠀⠀⠽⣦⡀⡀⠀⠈⠀⠀⠀⠉⠀⠉⠀⠀⠀⡤⠄⣠⢀⡼⠋⠀⠀⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠛⢆⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠭⠟⠊⠀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⣤⠴⠟⠳⠀⠀⠀⠀⠀⠀⠀⣠⢯⡤⢤⣀⡀⠀⠀⠀⢀⡠⢚⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠛⠻⠃⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⡗⠸⡀⠀⠀⠀⡗⡴⡶⣾⣆⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⢺⣧⡀⢥⣄⠀⠀⣃⣀⣤⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⡞⠙⢿⡒⣙⢿⣿⠍⠁⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠙⠻⣤⣿⣷⣃⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐

⠀⠀⠀⠀⣠⣤⣤⣄⠀⢀⡀⣀⣠⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣿⣟⣿⣷⣿⡇⠘⠓⠛⠛⠚⠐⢠⣺⣷⡄⡀⠀⠀⠀⠰⢆⣂⣂⠈⠀⢀⣀⣠⡶⠋⠀⠀⠠⣀⠀⠀⠀⠀⠳⢎⡸⠀⠀⠀⠀⠄
⠀⠀⠈⠛⠿⠟⢿⠋⠀⠀⠀⣠⠶⢄⠀⠀⠉⠉⠀⠀⠀⠲⢣⣞⠁⠈⢻⣛⣆⠛⠋⠁⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠸⠟⠻⠇⠀⣠⡄
⠀⠀⠀⣠⣴⣤⣌⡀⠀⣀⡤⣷⣤⡾⠃⠀⡀⠀⢠⠀⠀⠀⢾⣿⡄⠀⠈⣿⡏⡇⣰⠆⣠⡦⠀⡴⡄⠀⠀⢠⡶⣦⢠⣶⣦⡸⣧⣸⣿⡀
⠀⠀⢰⠻⣿⣿⡿⠟⠋⠑⠀⡼⠁⡀⠐⡤⡾⣿⡯⠑⢏⠀⠘⢏⡁⣠⣤⣍⡠⠃⠋⠊⠁⠀⠐⣃⡀⢠⣦⠘⢷⡿⢻⣧⣴⣷⠘⠋⠹⣧
⠀⠀⡏⠀⠘⣯⣠⣄⣀⣀⣾⣇⠀⣧⠈⠳⢤⣮⣦⠴⠋⠀⠀⠀⠉⠙⠛⡉⣠⡄⢠⣠⣦⢠⣿⠛⣿⡌⣿⣤⣴⡆⠀⠻⢿⠿⠀⣀⡀⢀⡄
⠀⠀⢿⠀⢤⠁⠉⠻⢮⣿⣿⣿⡇⠈⠷⣶⣆⠀⠀⠀⢤⣰⠚⢣⣤⣄⣺⣧⣸⣷⢻⣿⣿⣎⣿⣦⣿⠇⠻⠿⠛⠋⠀⠀⢰⣾⣿⢿⡇⠈⠇
⠀⠀⠈⢷⣤⡀⠀⠀⠀⣈⡽⠟⠁⠀⠀⠉⠀⣀⣤⣴⡄⠈⣀⣿⡏⠙⠛⣿⡏⢿⡞⡿⠙⠟⠈⡉⠁⠀⠀⠀⣧⠀⠀⠀⣸⣷⣤⣄⣧⣀⡀
⠀⣤⣶⣾⣍⣛⣻⣿⡛⠉⢀⢤⡒⠄⣇⡀⠹⠛⣿⡏⢺⣿⣽⡜⢿⣶⣿⠻⠗⠈⠀⠀⠀⡀⠛⠁⠲⠄⠀⠤⠞⠀⠀⠘⣿⣷⣾⣿⣿⣿⠃
⠀⣿⣟⠛⠛⠛⠛⠛⢻⠃⡜⠀⠀⠀⠀⠂⢀⠀⢹⣿⡘⣿⣷⠶⠀⠉⢁⠀⠀⠀⢹⡀⠀⠸⡄⢠⠤⠒⠀⣀⣠⣤⣤⣀⠀⠙⢿⣿⣿⠃
⠀⣿⠿⡀⠀⠀⠀⢀⡏⠀⢷⠀⠀⠀⠀⢈⡆⠀⠀⠉⠀⣀⡀⠀⠀⢰⣬⣟⣲⠤⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣟⣿⣿⡟⠀⢠⡀⠙⠁
⠀⠻⣧⣇⣀⣀⣀⡼⠁⡀⠀⠑⠂⠔⠐⠀⢀⣠⠤⠒⠉⠙⣁⣀⣤⣼⣿⣿⡿⢿⣿⣶⡆⠀⡁⠀⠀⠈⠙⢿⣿⣿⣿⠀⠰⡏⠘⠳⠆⣀⡀
⠀⠀⠀⠉⠉⠉⠁⠀⢄⠀⠠⣀⡀⠰⠞⠛⢉⣀⣤⣶⣺⣿⣿⣿⣻⣿⣿⣾⣿⣿⡿⠃⠈⡥⠁⠀⠀⠀⢀⡀⠈⣵⡆⠀⣼⣿⣦⣦⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣆⡔⠀⠸⣃⣤⣶⣾⣿⣽⣿⠿⠿⠟⠛⠛⠛⠉⠹⣿⡿⠋⠀⠂⡠⠔⠉⡁⠀⢤⣿⣷⡄⢸⡆⠀⠻⠿⠛⠁⠐⣿⠂
⠀⠀⠀⠃⠁⡔⠀⠀⠉⠉⠀⠈⠛⣛⠋⠉⠉⠀⠀⠀⠀⠀⢈⣶⠃⠀⠀⠉⠈⠆⡆⠀⠀⣀⣀⢿⡿⠄⢉⣩⠥⠔⠧⢄⡀⠀⠀⠀⠬⡁
⠀⠀⠀⠂⠀⠀⠀⠀⠂⣤⡞⢉⣻⣅⣈⠑⣤⠢⠐⠀⠻⢷⣾⠋⠀⠀⠑⠈⠢⣠⠖⣆⠀⡇⠈⡎⠁⣰⠋⣠⣶⣶⣶⣦⣙⡄⠀⠠⣤⣄
⠀⠀⠀⠀⡀⠰⡆⠀⣾⣤⣾⣿⣽⠶⠶⠟⠾⠷⣉⡴⠠⠐⠀⠀⠀⣤⣦⣤⣤⡘⣦⣈⠛⣷⣤⠇⢰⡇⣬⣿⣿⣿⣿⣿⣿⣇⠀⢨⣷⣾
⠀⠀⣠⣾⣷⡀⠉⠘⡿⢋⠉⠀⢻⡀⠀⠀⢤⣤⢸⣷⡀⠐⡅⢰⣟⣛⣿⠻⡟⣿⣎⠙⢻⣿⣿⣆⠈⣧⠸⣿⣿⣿⣿⣿⠏⢹⡀⠘⠿⠟⠂
⠀⠀⠛⣿⡟⠓⠀⢸⣿⡾⠆⢠⣾⣿⣷⣦⣄⣀⠞⠙⣿⡀⢇⢸⣿⣛⣿⣠⣇⣿⣿⠀⠘⣏⡕⡻⣷⣿⡦⠄⣉⣉⣉⡤⠒⠉⢿⡵⣦⡀
⠀⠀⠀⣿⡇⠀⢠⣿⠉⠓⢦⣾⣿⣿⢿⣿⠟⢅⠀⠀⢸⣧⢪⢸⣿⢙⠏⡿⡏⣿⣿⠀⠀⠙⣾⣱⣹⣿⢴⢒⣍⣩⣭⡍⡭⣽⠸⣿⣒⣳
⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠘⢯⣛⣛⠛⣋⡠⠆⠀⠀⣴⣛⢦⡀⣿⡞⡿⣷⣗⢿⣿⡆⢰⠄⠈⠛⠿⣿⡸⣙⠞⠙⢿⠇⢉⡼⠀⣿⣿⣾⡆
⠀⠀⢀⣿⡇⣴⠋⢿⣳⡄⠀⠀⢨⠏⠛⠂⢲⠀⠀⢸⣿⡦⡌⢷⢹⣷⣤⢿⡼⠼⣿⣧⠘⡄⠀⠀⠀⢸⣦⣄⣉⣘⣛⣋⣥⣴⣾⣿⡿⠙
⠀⠀⢸⣿⡇⢹⡀⡏⢿⡷⠀⠀⣸⣀⠀⠀⡇⠀⠀⠈⠛⠓⠘⠉⣿⣿⡆⣾⡿⠾⣿⣿⡙⡄⠀⠀⠀⠈⣿⠿⠿⢿⣿⣿⡿⢟⣹⡞⢷⣰
⠀⠀⣾⣹⣷⠀⠉⢡⣴⣶⠆⠀⠀⢸⠁⠀⢉⠇⢀⣀⣾⣀⡀⠀⣾⡿⣻⢻⡟⢿⢻⣿⡧⡇⠀⠀⠀⢀⣿⣭⣭⣭⣧⢸⣿⣿⣶⡷⣶⣄
⠀⢰⣿⣹⣿⣧⡀⢀⡏⠀⣾⠆⠠⠚⠚⣧⠎⠀⠀⢹⢿⡏⠁⠀⢻⣷⠹⢿⡇⣆⢘⣿⠇⡸⠀⠀⢰⣯⣛⣿⣿⣿⣿⠸⣿⣿⡿⠿⠿⢿⡆
⠀⠈⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⠁⠀⠀⠈⠛⠿⠿⠿⠿⠿⠋⠀⠁⠀⠀⠘⠋⠋⠛⠛⠛⠉⠀⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣒⠒⡖⣢⠨⣡⣶⡔⠦⠃⢀⠀⠂⠀⡤⠀⠒⠒⣂⣀⡐⠒⠒⠂
⠀⠀⠀⠀⣀⠤⣺⠁⢠⣤⣤⣤⣶⣶⣶⣾⠀⠀⣾⣦⣴⣿⣿⡧⠀⠨⢎⠌⠜⡼⠠⣖⡻⠷⣔⠃⡎⠀⠀⡀⢰⣷⠀⠁⢻⣿⡇⠀⠿⠿⠿⠿⠗⠐
⠀⠀⠀⠸⢐⠥⠚⡁⢸⣿⣿⣛⣟⣻⣿⣿⠀⠀⠙⢿⣿⣿⡟⠁⠀⣐⣠⣂⣊⣾⡀⠢⠄⠉⣹⡶⣄⠀⢀⠁⠸⠿⠀⢠⠘⠛⣁⣤⣤⣦⣤⣄⠀⠀⡀
⠀⠀⠀⡋⠤⢂⠥⡇⠈⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠄⠙⠉⠀⠀⣸⡿⠿⠿⠿⢿⣧⣀⠀⢀⡟⠃⠙⠆⠘⠀⢠⣶⣦⠸⠀⣼⣿⢟⣩⣍⡝⠻⣿⣄⠁
⠀⠀⠀⡇⠊⡡⠔⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢀⣼⡟⠁⠀⢀⡀⠀⡉⢻⣧⠈⠁⠆⠀⠀⠀⠀⠈⠻⠋⠀⢸⣿⡇⣾⣿⣿⣿⣇⢹⣿
⠀⠀⠀⡇⣈⠔⣊⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠹⣿⣇⠀⠀⢘⡁⠉⠁⣸⡟⠀⠀⡃⠀⠀⠀⠀⠀⠀⢠⠀⠘⣿⣧⡹⠿⣿⠿⣣⣿⡟⡀
⠀⠀⠀⣋⡠⢊⠅⠘⠼⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣼⠀⠀⠙⢿⣷⣤⣨⣁⣠⣴⠟⠁⠀⠀⠃⠀⠀⢠⠀⠀⠀⠈⠀⠀⠈⠻⢿⣷⣶⣿⠿⠋⠀⠂
⠀⠀⠀⠧⢈⣤⢊⣠⡀⠀⠀⠀⠀⠀⢀⢼⠀⠀⡒⠄⡁⠠⠐⠀⠀⠈⢻⡻⠻⠟⣿⠀⠀⠀⠀⡆⠀⠀⠀⢄⣀⣀⣀⣠⡄⣠⡠⠠⠤⠤⠤⠤⢠⠲
⠀⠀⠀⠀⡀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠐⠊⠀⠀⠈⠢⣿⠤⠚⠀⠀⠀⠘⠒⠀⠒⠞⢀⡴⣤⠀⠘⠀⠠⡶⣦⠈⠓⠓⠚⠓⠒⠒⠂⠋⠂⠈⠈⠁
⠀⠀⠀⠋⠀⠀⠀⣀⠄⣆⣂⣖⡠⡀⠀⠀⠀⡌⠁⡀⡀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠚⠿⠂
⠀⠀⠀⠀⠀⠠⡎⢰⠊⠉⠙⢿⣿⣮⣢⠀⠈⠀⠁⠠⣿⡴⠂⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣥⣥⣤⣤⡀⢱⡀⠀⡊⠁⠰⠈⠀⣧⢀
⠀⠀⠀⠀⣐⢃⡡⠁⠀⠀⠀⠀⠹⣿⣿⣇⠀⠀⣠⣤⣿⢀⡀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠉⠉⣉⣉⡉⠉⠙⠛⢻⣿⡇⢸⡇⠀⠀⡀⠀⠀⣰⠏
⠀⠀⠀⠀⡉⢀⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⢸⣿⣿⣿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⢷⣆⠀⢸⣿⡇⢸⡇⠀⢀⠀⢠⡴⠃
⠀⠀⠀⢠⡁⣈⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⠀⠈⠛⠛⠁⠀⠀⢀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⠀⠸⣿⡇⢸⡇⢀⠀⠀⣸⣿⠀⠻⠃
⠀⠀⠀⠀⡁⢌⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⠀⠀⢠⣄⣰⡆⢀⠀⣿⣿⠀⠀⠀⠀⠐⠘⢿⣷⣿⣽⣿⡛⠋⠀⠀⣿⡇⢨⡇⢸⠀⠀⣿⣿⡆
⠀⠀⠀⠀⢅⠈⣭⣉⠛⢶⣄⠠⠐⠂⢩⣙⢻⣿⡄⠀⠈⢿⡿⠃⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣿⡇⢠⡇⠸⠂⠀⣿⣼⠇
⠀⠀⠀⠘⠂⠁⠈⢿⣷⣄⠻⣧⠔⢠⣿⣿⡆⣿⣷⠀⠀⠀⠀⠀⠀⢄⠻⠿⠿⠿⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣾⠿⠿⠿⢃⣸⠇⠀⠀⠄⠀⡀
⠀⠀⠀⠀⢰⣾⣿⣆⠻⣿⡄⢻⣇⣿⣿⣿⡇⣿⣿⠀⠁⠀⠀⠀⠄⠂⠀⠀⠈⠉⠁⠁⠈⠻⢿⣿⣿⣷⣾⣟⣃⣉⣀⣀⣉⣑⣒⡄⠀⠀⠸⠀⠰⠁
⠀⠀⠀⠀⣿⣿⣿⣿⡆⢹⣷⢸⣿⣿⡿⠏⣰⣿⡟⠀⠀⠠⠀⠀⢷⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠠⠀⠐⠀⠈
⠀⠀⠀⠀⠻⣿⣿⣿⡏⢨⣟⣿⣧⣤⣴⣾⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⣲⠄⣀⡲⠄⢠⠰⠤⠀⣀⣀⡀⠀⠀⠀⢀⠠⠐⣒⣒⣒⡀⠄⠁
⠀⠀⠀⠀⠄⠈⠛⠛⠴⠛⠋⠀⢀⠉⠁⠀⠈⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠘⠛⠀⠀⢹⣿⣷⠀⠁⠀⠓⣿⣿⠿⣿⣿⠿⢷⣾⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢠⠂⠚⠀⠠⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⡆⠀⠀⠀⣿⡇⠀⣿⡏⠀⣾⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠨⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠡⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠘⣿⠀⢹⣷⠀⢸⣇⣼⠏
⠀⠀⠀⠀⠀⠀⠒⣴⣶⣶⢽⣿⢿⡍⠃⣉⣶⣶⣿⡙⠻⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠀⢀⣿⠀⣸⣿⠀⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣼⣿⣽⡧⠧⣿⣿⣿⣿⡷⠀⠀⢐⢂⠀⠀⠀⠀⠀⠉⠉⠈⣈⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣴⣾⣧⣀⣿⣧⣸⣿⣷⣦⡀
⠀⠀⠀⠀⢀⠀⠀⠙⠛⠛⡉⢩⣉⣠⡂⠙⠻⠿⠛⠁⢠⠀⠈⠀⠀⠀⡦⠠⠤⠀⠤⠤⠤⠀⢀⠀⠀⠀⢸⣿⣿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⠒⠐⠂⠂⠒⠒⠒⠓⠚⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⢠⢖⠉⡑⢆⠀⠀⠀⠀⠀⠀⡔⣊⢑⢢⠀⠀⠀⢠⢖⢉⡓⢄⠀⠀⠀⠀⠀⠀⡰⢊⠉⡲⡄
⠀⠀⠀⢸⢈⡿⣅⢸⠀⠀⠀⠀⠀⢸⠀⡼⣯⠀⡇⠀⠀⢿⢈⡿⡅⢸⠀⠀⠀⠀⠀⠀⡇⣨⢿⡁⡇
⠀⠀⠀⢸⠀⣳⣇⠸⡀⠀⠀⠀⠀⢸⠀⣳⡏⢰⠀⠀⠀⢸⠈⣳⡇⢸⠀⠀⠀⠀⠀⢀⠇⣸⣞⠀⡇
⠀⠀⠀⠀⢇⠳⣽⡀⢇⠀⠀⠀⠀⡎⣸⡧⠇⡎⠀⠀⠀⠈⣆⠣⣽⡀⡇⠀⠀⠀⠀⡸⢀⣧⠞⡸
⠀⠀⠀⠀⠈⢆⠻⣽⡘⡄⠀⠀⡸⣹⡯⢇⡜⠀⠀⠀⠀⠀⠘⢆⠻⣽⡘⡄⠀⠀⢠⢃⣯⠟⡰⠁
⠀⠀⠀⠀⠀⠈⠣⡸⢽⣝⣆⣼⣹⠽⣣⠊⠀⠀⠀⠀⠀⠀⠀⠈⠣⡻⢽⣙⣄⣰⣫⡯⢇⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢮⡻⡯⣿⣞⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣺⣻⣟⣟⡵⠃
⠀⠀⠀⠀⠀⠀⠀⢠⢞⡿⣟⣿⢬⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣞⡯⡇
⠀⠀⠀⠀⠀⠀⡴⢳⣺⣫⠏⠻⣹⣲⡟⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣞⣧⢧
⠀⠀⠀⠀⢀⠞⣴⢻⢡⠃⠀⠀⢱⣹⣷⡎⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡼⢧⢸
⠀⠀⠀⠀⡞⡴⢻⠁⡎⠀⠀⠀⠀⢇⢹⡗⡌⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡽⣧⢸
⠀⠀⠀⢸⠁⡽⣏⢠⠁⠀⠀⠀⠀⢸⠀⡽⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣹⣏⢸
⠀⠀⠀⢸⢈⣷⡃⢸⠀⠀⠀⠀⠀⢸⠀⢳⣟⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢳⡞⠘⡇
⠀⠀⠀⠘⠮⣀⡡⠎⠀⠀⠀⠀⠀⠀⠣⢍⣨⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣁⡩⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣉⣡⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣆⠘⢿⣿⣿⠟⢀⣾⣿⣿⣿⣶⣤⡀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⡈⠿⠋⣠⣿⣿⣿⣿⣿⣿⣿⡏⢰⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⠀⣶⣆⠀⠀⠀⢀⣶⡆⠀⠀⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠀⠀⠀⣿⡏⠀⠀⠀⠈⣿⡇⠀⠀⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⢘⣿⠃⠀⠀⠀⠀⢻⣟⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⣾⠏⠀⠀⠀⠀⠀⠈⢿⡆⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢿⣁⠀⠀⠀⠀⠀⠀⠀⠀⢙⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢨⣿⡄⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠿⠏⠀⠀⠀⠀⠀⠀⠹⠟
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⠢⣄⠀⠀⠀⡠⠖⠋⡍⠀⣸⡉⠓⠦⡀⠀⠀⢀⡤⠚⠉⠀⡀⠁
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣇⠀⠈⢣⡤⠊⠀⡆⠀⣇⠘⢧⠗⠀⡆⠈⠢⣤⠋⠀⡼⢦⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠻⡼⠃⣠⠞⠙⢦⡀⠇⠸⣬⠃⢸⠀⠀⠇⣠⠞⠙⢦⡀⢹⠋⠀⡇
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠚⠁⠀⠀⠀⠈⠓⠤⣇⣀⣘⡤⠔⠋⠀⠀⠀⠀⠙⠢⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⣀⠀⡀⡀⢀⡀⣀⠀⣀⢀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠟⠻⠟⠟⠥⠻⠳⠻⠧⠟⠵⠺⠼⠻⠿⠗⠧


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡧⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣤⠀⠀⠀⠀⠀⣤⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣤⠀⠀⠀⠀⠀⣤⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡆⠀⠀⠀⢠⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡄⠀⠀⠀⢠⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠉⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠻⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⠀⢠⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡷⠀⠀⠀⠀⠀⠀⠀⠾⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟


⠚⠛⠛⠛⠛⠻⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠛⠛⠛⠻⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⠛⠛⠛⠛
⠀⢀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⣿⠀⠀⠀⣸⡇⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⡀
⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠈⢷⣄⠀⠀⢀⣴⠟⠁⢠⡄⠀⠀⠀⣿⠀⠀⣾⠋⠙⣷⠀⠀⣤⠀⠈⠻⣦⡀⠀⠀⣠⡾⠁⠀⠀⢸⡇⠀⠀⠀⣿
⠀⢸⡇⠀⠀⢀⣿⡀⠀⠀⠀⠀⠙⢷⣤⠟⠁⠀⠀⢸⡇⠀⠀⢀⣿⡀⠀⠻⢶⡾⠋⠀⠀⣿⠀⠀⠀⠈⠻⣦⡾⠋⠀⠀⠀⣶⠛⠻⣆⠀⠀⣿
⠀⢸⡇⠀⢰⡟⠉⢻⡆⠀⠀⠀⣠⡾⠻⢦⡀⠀⠀⢸⡇⠀⢰⡏⠉⢻⡆⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢀⣴⠟⢷⣄⠀⠀⠀⠻⣦⣴⠏⠀⠀⣿
⠀⢸⡇⠀⠈⠻⣶⠟⠁⠀⣠⡾⠋⠀⠀⠈⠙⢦⣄⠘⠃⠀⠈⠻⣶⠟⠁⠀⢸⡇⠀⠀⠀⠛⠀⣀⡴⠟⠁⠀⠀⠙⢷⣄⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀
⠤⣤⣤⣤⣤⡴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣤⣤⣤⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⣠⠀⢠⡀⣤⣤⡀⢀⣤⣄⠀⣤⡀⢠⣄⠀⣠⣤⡀⢀⣤⡄⢀⣠⣄⡀⢠⣄⢀⣤⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠀⠀⣿⠶⢾⡅⣿⢼⡃⣿⠀⢸⡇⡿⢧⡞⣿⢸⡇⠀⣿⠘⠷⣄⢾⡁⢈⡷⣼⢿⣼⢿⠀⣿⠶
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣷⣿⣤⣾⣦⣿⣼⣧⣼⣿⣿⣤⣷⣼⣧⣿⣬⣿⣿⣧⣾⣿⣧⣬⣿⣿⣥⣿⣼⣧⣼⣦⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠷⠄⢀⡄⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠈⠀⠔⠀⠄⠙⠂⠀⠀⠀⢄⣈⠁⠀⠀⣀⡂⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠻⡤⠄⠀⠀⠀⠀⠀⢳⣶⣾⠇⢠⠆⠀⠠⣄⠀⠀⠀⠀⢹⡍⠓⢒⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢷⠀⠘⢷⣾⢃⠤⠂⠀⢀⠠⠀⠀⠈⢩⠁⡀⠀⠰⠄⠀⢿⠀⠀⠀⠀⠀⠙⠲⠦⠞⠉⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⣄⠘⢧⠀⠂⠀⠉⠀⠀⢰⣇⠀⠀⠀⠀⠈⠃⠁⠀⠀⣀⡀⠈⠀⠀⠀⢠⡄⠀⠀⠀⢀⠃⠀⠀⠀⢠⠘⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢻⠀⠈⢧⡀⠀⠀⠄⠀⠀⠿⠆⠀⠀⠀⠀⡀⠀⠀⠢⠘⠇⠀⢀⣴⣀⠀⠻⣥⠀⠀⠘⠀⡀⢠⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡀⠈⣷⠀⠘⢧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢹⣷⡀⢴⣶⡀⠀⠀⠀⡠⣟⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠽⡟⠀⠘⣧⠀⠘⢣⠀⢠⠀⠀⠀⠌⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⠀⣶⡦⠀⠀⠀⢾⠉⠀⠀⢠⣄⠀⠀⠂⠐⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠈⡁⠀⠀⡀⢹⠀⠀⠘⡄⠀⠘⣇⠀⠀⠙⠂⠀⠀⠀⠀⠤⣤⣄⣉⡉⠋⣉⠀⠀⠙⠃⠉⠁⠀⠀⠘⢲⠀⣤⡄⢾⣀⠂⠀⠀⠀⠀⠀⠸⡦
⠀⠀⠀⠀⠀⠀⠸⢿⠦⠀⠀⢀⠀⠀⠹⡇⠄⠈⠀⠘⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠙⠛⠻⠎⡀⡀⢀⠀⠀⠀⠀⠠⠋⡀⠘⠋⣠⣄⡀
⠀⠀⠀⠀⠐⠂⢀⢴⣶⠀⠀⠀⠀⠀⠀⠱⣄⠠⠌⠄⠀⠀⠀⠀⢀⠀⢆⠘⠄⠀⠀⠀⢀⠀⢤⡁⠈⠉⠀⠀⠀⠀⠸⠿⠆⠘⠛⠂⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠄⠀⠙⢆⠰⠈⡆⠀⠀⠀⠻⠤⠀⠈⠀⠀⠀⠀⠘⠠⠈⠙⠀⠀⠀⠀⣰⠁⢂⠠⠐⣷⡄⠀⢀⠐⠀⠖
⠀⡀⠀⡀⣄⠀⠀⠀⠀⠀⡠⢀⡀⠀⠀⠀⠀⢈⠓⠤⠄⠀⠀⠀⢀⠀⠀⠘⠀⢀⠐⡄⠀⠐⠠⠾⠀⠛⠀⢸⡃⠀⡘⠛⣤⡤⣀⠠⣏⣄⢠⡆⠓⢆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠁⣈⠃⠀⠹⠀⠀⠀⠁⠀⠀⠀⠰⠀⠂⠀⠀⡔⠊⠁⠀⠀⠘⠀⠀⠀⠀⠀⠓⢲⡶⣥⠀⠀⢿⣷⠟⣤⠈⠁⣤⠀⣠⣈⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠇⠤⠀⠸⠄⠀⠀⠀⠀⠀⠀⢀⠀⢠⠇⢠⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠷⠋⣠⣶⡆⠰⣶⣶⡉⠰⠟⢀⣤⣉⡉⠀⢁
⠀⠀⢀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡌⠀⠄⠈⠀⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⣙⡋⠰⡇⠄⠉⠀⣞⡀⠈⠓⠳⠃⢾⠂
⠀⢠⡀⠀⢀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡜⠀⠀⠀⠙⠀⠀⠃⠰⡀⠀⠀⠘⡰⠂⠀⠀⠀⠀⢴⣿⡁⣾⣵⣠⡄⣀⢈⡁⠀⢸⠟⠀⠈⠁
⠀⠈⠙⢦⠀⠑⠀⡀⠀⠀⠀⠀⢀⣠⣤⢸⣶⣔⠄⠀⣿⠀⠁⠀⠀⣰⠁⠈⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠠⠀⠻⠀⣉⠻⠆⣩⣄⢻⡿⠆
⠀⠀⠀⠀⠒⢦⠀⣄⠀⢸⡣⢄⠸⡌⠛⣠⣬⡁⢶⢀⡃⡀⠀⠀⠀⠷⣀⠀⣀⡀⣤⢂⡄⢀⠀⣰⠀⠀⠀⠐⠀⠀⠀⠠⣉⠀⠘⠗⠃⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠎⠁⠀⠑⠀⠀⠛⠛⠴⠈⢺⠀⠀⠰⠀⠀⢀⠁⠘⠁⠙⢀⡀⠀⣨⡦⢠⣆⡀⠀⠀⠀⠀⠀⠀⠈⠁⢉⠁⠀⠀⠠⣀⠔⠁⠀⢠⠄
⠀⠀⠀⠀⢤⡀⠁⠀⠑⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⣸⠀⠀⠀⣠⠃⣄⣄⠐⠆⣹⠘⠟⠁⠙⣁⣈⠋⡴⡀⠀⠀⣂⣀⠀⠀⠀⠀⠀⠠⢀⠄⠀⢀⣠⡼⠇
⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠐⢻⣄⠀⠀⠀⠀⠼⠢⠀⠀⠀⠀⠻⠀⠷⠀⢠⢠⡏⠳⠀⣏⠀⠘⢣⡆⠀⠘⠯⠈⠀⠀⠀⠀⠀⡠⠀⣠⣴⠟⠁⠆
⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⢀⠀⠀⠀⠑⠂⠀⠀⠀⠐⠀⠀⠀⠰⠀⣴⠒⠀⢀⡾⠀⠁⢰⡇⠀⢰⠖⠈⠘⡄⠂⠀⢰⡇⠀⠀⠀⠀⢀⣾⡟⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⢀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠂⣠⡄⠀⠁⠃⠈⠻⠃⠸⠀⠀⠀⠉⠀⠀⠈⠃⠐⠁⠀⠂⠛⢡⠦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠹⡇⡈⠢⠁⢰⡀⢦⠀⠀⠀⠀⢀⣤⣝⠙⠚⠻⠟⢐⡂⠀⠀⠀⠐⠟⠐⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡴⠀⠿⠄⢈⣃⠀⠀⠀⠐⠀⠀⠻⡉⠃⠀⠀⣠⢿⣠⠿⡀⠀⠀⣀⠀⠈⣐⡄⠿⠀⠔⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⢀⣾⣿⠦⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⢀⣴⡯⢤⠘⢁⡋⠀⠉⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠏⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠀⠛⠀⠀⠀⠈⠁

⣿⣿⡿⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠂⠀⢀⠀⠀⠀⠀⠀⠘⢣⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠉⠉⠀⠂⠀⠀⠠⢄⣀⠤⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⢤⣀⠡⠄⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡠⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠒⠛⠂⠐⢢⣀⢈⢁⡒⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠭⠀⠀⠀⠀⠄⠐⠀⠒⠐⠶⢶⢎⠀⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡉⠉⠙⣫⣍⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡟⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⡦⠄⠠⠠⢠⣄⡀⠀⠀⠀⠀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣔⣤⣀⣩⠥⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠦⠉⠤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠚⠋⠁⠐⠶⣀⠂⢀⢙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠎⠥⠌⠀⠀⠀⠸⠀⠀⠈⠠⡥⡍⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡠⠟⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢹⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⣃⣥⡠⠤⠤⠀⠀⢀⡤⠔⠋⢠⠤⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⣔⣓⡶⠒⠒⠀⠀⡄⠄⠦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁⠀⠀⢀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠰⣯⡽⠇⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠓⠒⠐⢊⠔⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢉⢣⡀⠀⠀⠀⠀⢀⠈⠒⠄⠠⠀⢑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠒⢆⠀⠀⠀⠀⠀⠀⠁⠀⠲⢜⡀⡃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣚⡂⣀⣀⢀⣀⡀⠀⠀⠀⢙⡓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠕⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⢠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⣸⣿⡇
⠀⠀⣶⣦⡀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠹⣿⣷⡀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠘⢿⣷⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠈⠻⣿⣦⡀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠈⠙⣿⡿⣷⣄⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣿⣦⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢹⣿⣷⡀⠀⣼⣿⡇⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⣠⣶⠟⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠹⢿⠇⠀⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⠀⠀⢀⣾⡟⠁⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠙⠋⠀⣠⣿⡟⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠁⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠂⠀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⡯⠈⠀⣾⣧⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⠇⣨⡝⠂⣸⣿⡏⠀⠀⠀⢋⠁⢤⠤⠒⢂⣷
⠀⠀⠀⢀⡀⠀⠀⠐⠀⢠⠥⠄⢹⣿⣿⣿⣿⣿⡿⠀⠀⠒⣞⠀⠈⡀⢠⣿⣿⠁⠤⢤⣄⡐⢠⢯⢐⣦⣍⡿
⠀⠀⠘⡄⠀⠀⠀⢀⣂⣀⣭⣵⣀⣿⡿⠟⠋⠁⣀⣀⣀⣀⠈⠉⠒⠋⠚⠛⠁⠀⠀⠀⠀⠫⠉⠭⠜⠛⠉
⠀⠀⠀⠙⠛⡛⢛⡻⠛⠉⠀⣹⠛⠛⢋⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠀⠀⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⢠⠉⣐⢎⠐⠁⠀⠀⠀⠀⡇⢠⣿⣿⣷⣶⣶⣶⣶⣦⣤⣬⣤⣤⡀⠀⢿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⢸⢰
⠀⡼⠣⢍⡁⢱⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡿⠟⠻⣧⣐⣲⣆⡼⣸
⡘⢠⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣨⠛⣙⣤⣶⣿⣿⣿⢟⡠⠋⠔⠁
⠇⢸⡀⠀⠀⠀⠀⢀⣀⡀⣸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⢾⣟⠁⣿⣿⣿⣿⡟⠉
⠈⠀⢱⢤⡒⣈⣭⣤⠄⠀⢀⠗⠘⠛⢛⡿⠿⠿⢿⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣶⣿⣿⣿⢤⣤
⠀⡔⠁⠀⠚⠛⠋⣁⡤⠖⠉⠛⢶⣤⣤⣀⣈⣉⠉⠙⢌⠻⣿⣿⣿⣥⠖⣺⣿⣿⣿⢿⣫⣷⣿⣿⠀⣀⡀
⠀⠈⠒⠒⠒⠛⠛⠳⠤⠄⠀⣶⠻⠿⣒⣚⠛⠛⠛⠓⣚⣓⣍⠻⣿⣿⣷⠞⠛⣫⣵⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣐⡒⠫⠉⠩⠉⣙⣓⠒⠓⠾⠛⠿⣶⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠒⠠⠬⠥⢉⣑⣒⠒⠂⠨⠥⠄⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⡠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡰⠠⢍⣉⠑⠒⠒⠤⠬⣭⣉⣉⣉⠀⢻⣿⣿⣿⠟⠁⣀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣈⣉⠛⠓⠶⠤⣤⣄⣀⣀⠀⠀⠉⠁⠀⢸⣿⡿⢃⡴⠊⠑⣢⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⡂⠭⢉⡓⠒⠦⠤⢤⣉⣉⡙⠛⠻⠶⠶⠾⠋⠠⡏⠑⢢⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣚⠁⠀⠀⠈⢉⠚⠂⠀⠨⠉⠐⠂⠀⠀⠉⣍⣩⢿⠒⠒⢂⡄⠈⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣤⣀⣀⡉⠒⠒⠀⠀⠀⣠⢮⢟⣰⢃⢞⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣦⣤⣽⣷⣶⡖⣡⡾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⢠⣾⣿⣿⣿⡟⠉⠻⣿⣿⣿⡿⠋⠙⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⢠⣿⣿⣿⣿⣿⣷⡀⠀⠙⣿⡟⠁⠀⣴⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠁⠈⢿⣿⣿⣿⠏⠀⢹⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣆⠀⠈⢿⣿⠏⠀⢀⣾⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⠛⠀⢀⣾⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢀⣾⣿⣿⣿⣿⡄⠀⠸⣿⣿⣿⣿⣿⡿⠃⠀⣠⣿⣧⡀⠀⠻⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠙⣿⣿⣿⣿⣇⣀⣰⣿⣿⣿⣷⣀⣀⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣄⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣤⣠⣼⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⣿⣿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⡀⢹⣿⣿⣿⠀⣠⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠏⠀⠀⠀⠀⠀⠀⠀⡀⡆⢣⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⡀⢄⠳⠘⡄⢂⠐⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⢿⣿⣿⣿⣧⡄⠆⠳⠘⡄⢇⠱⡈⣄⢡⡘⡄⢦⠱
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠿⠳⠾⣿⣿⣿⣷⠱⡈⠄⠡⠘⠀⠦⠉⠈⠀⠁⠉
⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⢻⡿⢟⣥⡠⡈⡄⠣⠈
⠀⠀⠀⠀⠀⠀⢿⡃⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⠋⠁⡀
⠀⠀⠀⠀⠀⠀⠘⣷⣤⠀⠀⠀⠀⠀⠋⠁⣀⣤⣶⣿⣿⣷⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡄⠀⠀⣶⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡐
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡖⠀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣿⣿⡿⢿⠟⢉⣿⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⣹⡇⠀⠀⠀⣿⣿⣷⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢁⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣿⠀⣿⣿⡏⠛⠋⣠⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣁⣀⣈⠙⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣉⣉⣿⠆⠸⢿⠟⢉⣿⣯⣀⣀⣀⣀⣀⣀⣀⣄⣀⡀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⢻⡏⢠⣷⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⢸⣿⣿⣿⠿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⢸⣧⡉⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢸⣇⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⢀⣀⣽⣷⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠻⢿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣅⠀⠀⠉⠛⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢩⣿⣿⣿⣿⣶⣄⡀⠀⢈⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣿⣿⠏⠈⠛⠿⣿⣿⣷⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠉⠻⢿⣿⣿⣦⣄⣰⣿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⢀⣾⣿⡿⠛⠉⠉⠉⠙⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠁⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠉⠛⠿⢿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣦⣄⣀⣀⣠⣴⣿⣿⣿⣀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣿⣿⣿⣿⠿⠟⠿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠉⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣷⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷⣦⡀⠀⠀⠈⠻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⠙⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠉⠻⢿⣿⣿⣶⣤⡀⠀⠀⣠⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆⠀⠀⠈⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣶⣤⣀⠈⠙⠿⣿⣿⣷⣶⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⠀⠀⠀⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠻⣿⣿⣷⣦⣀⠀⠙⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⠉⠻⢿⣿⣿⣾⣿⡿⠉⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣦⣤⣶⣿⣿⠟⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⣸⣿⣿
⣠⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⢀⣿⣿⡇
⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠀⠀⠀⣼⣿⡿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠁⠀⠀⣸⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⣴⣿⣿⠃
⠈⠉⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣿⣿⣁⣀⣀⣀⣾⣿⣿⣃⣀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠈⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⡀
⠀⠈⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷
⠀⣼⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣐⣀⣠⡖⠂⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠘⢿⣿⣿⣿⣿⡋⣇⡀⠀⢀⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠖⠂⠀⣀⣥⡄⠀⠀⠀⠘⠿⠀⠙⣿⠇⠉⠀⢀⣄
⠀⠀⠀⠀⠀⣠⣾⢟⠒⠀⠀⠀⠀⣻⡏⠀⠀⠀⠀⠀⠀⠀⣀⠈⢣⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣼⣿⡟⠈⠁⠀⠀⠀⠀⣿⣅⠀⠐⠀⠀⠀⠀⠀⠘⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⡅⠀⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢠⣿⣿⣅⡄⠀⠀⠀⠀⢸⣿⠄⠀⠀⠀⠙⠦⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⣇
⠀⠀⠀⢸⣿⣿⣾⣿⣧⣀⣀⣴⣿⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠆
⠀⠀⠀⢻⣧⠾⣾⣿⣿⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣾⡄⢀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈
⠀⠀⠀⠀⠙⠢⠌⠙⢛⣼⣿⣿⠿⠛⠓⠒⠒⠀⢉⣉⣁⣰⡿⠡⢶⣿⣷⠬⢷⡦⡤⠾⠛⠋⠉⠀⠒⠖⣿⡷⠶⢶⣶⣦⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⣫⣤⣿⣿⠛⠞⢋⣁⣬⣥⣾⢹⣿⣤⣾⠿⠟⠷⠷⢶⣶⣶⣤⣀⡄⠀⢀⠐⢠⣀⠀⢀⠉⠋⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣾⣟⣛⡹⢆⣼⣿⣿⡿⠁⠛⠈⠉⠁⠁⠀⠀⣀⢀⣠⡄⢹⠿⢿⣧⡀⠀⠀⠐⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣷⡿⣿⢁⣿⣿⣿⡟⠀⠀⠀⢠⣿⣄⠤⠒⠽⠋⢠⡾⠁⠀⡶⠈⠛⢻⣷⡀⠀⠀⢘⡷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣿⣾⠋⣿⣿⣿⣏⣀⣀⡠⠀⠙⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣧⡄⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⢻⣿⢻⣿⣿⣿⣿⣿⣥⡃⠀⡀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠠⡚⣿⢿⡇⢰⡆⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⢼⣿⣿⣿⢁⡝⢩⣿⣟⠳⣷⣴⣾⣵⣶⣶⣦⣤⣴⢷⣦⣤⣀⣹⣿⣶⣤⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣷⣿⡿⢷⣿⣅⣈⠀⠟⠸⠙⣿⣛⣿⢿⡿⣿⡋⠛⢟⠉⣯⢣⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠷⣌⡛⠻⣶⣶⣦⣄⣈⣀⣀⠂⢀⠉⢠⣄⣼⠁⣦⣹⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣯⣘⠳⠤⣌⡉⠝⠛⠛⣛⠛⠛⠛⠟⢯⠘⢇⡈⠻⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⣿⣿⡉⠃⣀⡈⠁⠀⠰⠃⢉⠙⢳⡶⠿⠞⢣⢈⡛⢶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢸⣿⣿⣿⣿⣿⡍⢀⡺⠤⠖⠲⠶⠆⢹⡄⠘⣦⠶⢶⣤⠘⣁⣼⣿⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿⣿⣿⣿⡇⠳⣌⣁⠲⠶⣶⡤⠞⠀⠳⣄⠘⢀⣀⢸⣇⢻⣿⣽⠇⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⣆⣠⠟⣉⠛⣦⠈⣷⡆⠀⠀⠉⡄⣸⠿⣿⣿⡿⣻⣿⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢹⣿⣧⣿⣿⣿⣷⣋⣤⠈⢉⣉⣉⣀⣤⠾⣷⣇⣰⣿⣯⣽⡿⠁⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⡀⢿⣿⣿⣿⣿⣿⣿⣿⣃⣏⠉⣩⡿⢀⣤⣿⣿⣷⣯⣿⡿⣡⡾⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡄⡙⢿⣿⣿⣿⣿⣽⣿⣯⢵⣿⣷⣿⣟⣷⣿⣿⡟⢡⣾⠋⠀⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣮⡙⢿⣿⣿⣿⣿⡗⠆⠉⢻⣿⣿⣿⠟⠉⣠⠟⡀⠀⠀⠀⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣦⣈⡙⠛⠛⠳⠆⠀⠘⠛⠉⢠⣤⠞⠃⠀⡁⠀⠀⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⠃⢽⣿⣷⣶⣦⣤⣶⣴⣿⣿⡿⡁⠀⡀⠎⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣇⠀⠀⠀⠀⠈⢿⣿⣍⠙⠏⠈⣿⣿⡀⠉⢄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣤⣄⠀⠀⠈⢛⣿⣿⣷⡖⣿⣿⣿⡀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⠾⠿⠿⠿⠖⢠⡈⣿⣿⣿⡇⢹⣿⣿⣧⣰⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠹⣿⣿⣿⣤⡀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⢰⡄⢁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣾⡖⠠⣀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡞⠳⠿⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣂⣸⠤⠤⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣁⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢻⣦⣤⣀⠀⠀⠀⠀⢠⡴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠞⠁⠀⠈⠳⣄⠀⠀⢻⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⢹⡷⢦⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⣠⠞⠉⠉⠉⢷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⣄⠀⠈⠙⠋⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠸⣧⣤⡟⠀⠀⠀⠀⢀⣸⣀⣀⣀⣸⣀⣀⣀⣸⣇⣀⣀⣸⣃⡀⠀⠀⠀⠀⠘⢷⣤⡿⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠉⣻⠀⠀⠀⡇⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦⠀⠀⠀⠀⠀⡇⠀⠀⢼⡋⢙⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠋⠀⠀⠀⠓⠒⠒⠒⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠃⠀⠀⠈⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣅⣨⡷⠒⠒⠒⠒⠒⠒⠒⠲⣿⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⡘⠃⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠒⣿⣀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣻⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣀⣀⠀⠀⠀⠀⢿⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⡀⠀⠀⢸⡇⠀⠀⠸⠧⠤⢴⡏⢉⣷⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⣦⣀⣀⣠⠞⠁⠀⠀⣾⠋⢹⡆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠟⠀⠀⠁⠀⠀⠀⠀⠘⠛⠛⠁⠀⠈⠳⣄⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠺⣇⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



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

⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠂



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡠⠤⠤⠤⠤⠤⠤⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠘⢷⣄⠤⠤⠤⢄⣴⠟⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣠⣼⣿⣿⣤⣠⣶⠍⣻⣶⢾⡋⠁⣤⣿⣿⣧⣄⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⢏⣴⠟⠭⠤⠬⠝⢷⣌⢻⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⡟⠁⠀⠀⠁⣾⠣⠤⠤⠤⠤⠤⠤⢹⡆⠀⠈⢻⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣧⡀⠀⠀⡀⢿⡔⠒⠒⠒⠒⠒⠒⣸⠇⠀⢀⣼⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣎⠻⣦⣒⠒⢒⣢⡾⢋⣼⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠙⢻⣿⣿⠛⠙⠿⣂⣽⠿⢿⣅⡀⠛⣿⣿⡟⠋⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⢠⡾⠋⠓⠒⠒⠊⠻⣦⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠑⠒⠒⠒⠒⠒⠒⢹⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢈⡅⠈⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢀⠔⠉⠁⠀⠀⣴⠆⠀⢱⣶⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢠⠀⠠⢤⡏⠀⢰⣿⣿⣿⣟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠆⠀⠀⠀⡐⠃⠀⣠⠟⠀⠀⣿⣿⣿⣷⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⢂⣵⣿⣤⡀⠀⠈⠀⠀⢀⡟⠀⠀⡼⠉⠁⠀⠀⠁⠈⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣂⣩⣤⣶⣞⠉⠁⠀⠈⠻⣧⡀⠀⠀⣾⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠈⠅⠉⠻⢿⣷⣄⠀⠀⠀⠈⢻⣄⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣄
⠀⠀⠀⠀⠀⠐⣿⡏⣠⠀⣆⠀⠁⠀⠀⢀⡀⠨⡀⣿⡆⠀⠀⠀⠀⢿⣿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⣷⡄
⠀⠀⠀⠀⢠⢾⣿⠀⣟⠈⡷⠄⡀⠀⠀⡀⢀⡼⠁⢹⡇⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⢻
⠀⠀⠀⠀⣆⢀⢻⡠⣏⣨⠀⠶⠆⠀⢀⠶⠉⠀⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⣾⣿⣿⡇
⠀⠀⠀⠀⢿⣿⣿⣧⢹⡭⡁⠂⠐⠓⣶⢶⠆⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⣴⣿⣿⣿⡟⠀⢠
⠀⠀⠀⢄⠘⣿⣿⣿⣧⡙⠣⣄⠀⠀⠈⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣡⡾⢟⠉⠉⢿⡧⠔⠃
⠀⠀⠀⠀⠈⠉⠙⢻⣿⣿⣦⣄⣉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣊⣴⡿⠛⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢈⠛⣿⣿⣿⣯⣟⠣⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⣡⣾⠟⠋⠀⠀⠀⠀⢠⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢂⣉⠉⠁⠀⠀⠀⠀⣸⣷⡄⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⢡⣿⣿⠆⠀⠀⠀⠀⠀⠀⣠⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⠄⠀⠀⠀⠓⢼⡧⠀⠀⠀⠀⠀⢰⡟⣿⣿⢁⢿⡿⣟⠀⠀⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⠉⢤⣤⣶⣾⠋⠛⣧⠀⠀⠀⠀⣾⠃⢹⣿⠀⠀⠀⠀⠀⠀⢀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣻⢿⣇⠀⠐⠿⡄⠀⠀⢸⣿⡗⠛⣿⠈⢠⡀⠀⢀⣿⣇⣰⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⣿⡂⣿⣿⡆⠀⠇⠀⠀⠀⣿⡃⠹⣿⡌⣿⣽⣦⣬⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢃⠀⡅⠀⠀⠀⣿⠀⠀⣿⡞⠻⢿⣿⣿⣼⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣟⡠⠀⠁⠀⠀⠀⣿⠀⠀⣹⡇⠃⠀⠻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢠⡿⠀⠈⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠼⠻⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⡄
⠀⠀⠀⠀⠀⢴⠋⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠙⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⡤⠊⠉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡸⣄⠀⢠
⠀⠀⠀⠀⢀⠖⡋⡻⣯⡉⡟⠓⠒⠒⠒⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⣸⠀⣸⣗⠠⢄⡀
⠀⠀⠀⠀⢸⢨⠀⡸⣸⡇⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⣿⠀⢅⣀⠇⠀⠈⠢⢄
⠀⠀⠀⢀⡞⠳⢭⡴⠟⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡇⠀⠀⡇⣿⠀⢨⡿⣀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⡞⢀⡔⣻⠳⡄⠀⡇⠀⠀⠀⠀⢸⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣈⠑⠣⢄⡀⢹⠉⠉⠉⠀⠈⠑⢄⠀⠀⠀⠑⡀
⠀⠀⢰⠇⡼⠀⡇⠀⠙⠒⠳⢤⡤⠔⣚⣭⠝⢒⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠻⣿⣶⣤⣈⠙⠢⢄⡀⠀⠀⠀⠀⢃⠀⠀⠀⠃
⠀⠀⢸⠀⡇⢸⡃⠀⠀⠀⠀⢿⣔⣋⣯⠟⢿⣏⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢅⡀⢠⠋⠙⣻⠛⠻⡖⠠⠃⠀⠀⠀⠀⠈⡄⠀⠀⢸
⠀⠀⢸⠀⣇⠸⡇⠀⠀⢀⣀⣀⣯⣀⡿⣆⢈⡆⠙⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠘⠯⣲⠜⢉⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠸
⠀⠀⠘⡇⢹⠀⣷⣶⠶⣶⣛⣋⠛⠿⠷⠾⠿⠒⠒⢲⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⡆
⠀⠀⠀⢳⠘⣏⣉⡓⠒⠚⠛⠻⠿⠿⠤⢴⠒⢚⣉⡥⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢰⣐⡒⣲⠀⡰⠁⠀⠀⠀⡇
⠀⠀⠀⠈⢇⣼⠟⢝⣿⡏⠉⡏⠉⠉⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣯⣥⡤⠊⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⡼⠁⠂⠊⣿⡇⣠⠧⠤⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⠄⣀⣀⠀⠀⠀⠀⠀⢀⠜⠁⡠⢊⣉⠢⡄⠀⠀⡄
⠀⠀⣠⠞⠀⠀⠀⠀⣿⡇⠻⢯⣭⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢱⠄⡀⡔⠁⠀⠀⡇⢅⣠⠃⡅⢀⠜
⠀⣸⠥⣀⠀⠀⠀⠀⣿⡧⠶⠀⠉⠉⠛⠛⠛⠻⠿⠶⠶⠶⠶⣦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⡜⠀⠀⠀⠀⠈⠒⠒⢊⠔⠁
⠀⢺⠀⠀⠉⠀⠒⠂⠠⠭⠙⣓⣒⡒⠢⠤⢤⣤⣀⠀⠀⠀⠀⢸⢸⡇⠀⢀⠀⠤⠤⠤⠀⠒⠒⠒⠂⠈⠉⠈⠉⠉⢉⣀⡀⠤⠤⠤⠀⠒⠒⠃⠤⠤⡀
⠀⠈⠓⠲⠤⢤⣄⣀⣀⡀⠀⠀⠀⠀⠈⠉⢹⠀⡏⠉⠓⠒⠲⠞⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠲⠤⢤⣼⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⠀⠀⡣⢿⡄⣼⣿⡄⢸⡇⡄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⡀⢠⣾⣿⣿⠀⢸⣧⢸⣿⣿⢧⡄⣿⣷⣶⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠈⠀⣬⠉⣀⣻⣿⣿⣿⡿⣤⣷⣿⣯⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⣣⡄⠀⠈⢀⣸⣿⣿⣿⣿⠙⢻⣧⣿⣧⣿⢯
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠓⠶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠒⣀⣮⣿⣿⣿⣻⣿⣿⡘⣿⣯⡿⢿⣿⣿
⠀⠀⠀⠀⠈⣿⣤⡀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠤⡐⠻⠟⠉⠻⠆⢾⣿⣿⣿⣽⣿⡏⣻⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣌⠉⠲⣶⣤⣤⣀⣀⡀⠹⣷⣤⠀⠀⠀⣀⠀⠀⠀⠘⠀⠀⠀⠀⣠⠁⠀⠀⠘⠃⠀⢠⣤⡼⠃⠀⣨⣽⣿⣿⣿⣿⣿⣿⢻⣿⣿
⠀⠀⠀⠀⠀⢿⡟⠳⣦⣤⣬⣤⣭⣬⣭⣽⣿⣿⠿⣶⣶⡦⠼⢦⣄⣀⣀⣀⣘⡃⠈⠁⠀⣠⣤⣀⡀⠈⣡⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠘⢿⣶⣾⣾⣿⣿⣾⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⢱⣿⣿⣿⡿⠿⣿⣿⣿⣀⣤⣤⣸⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡼⣿⣿⡁⡀⢠⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠆⠀⠞⠀⣤⣤⠀⢰⣿⣿⣧⡀⣁⠀⠙⠿⢀⣤⣴⣶⣾⣿⣿⣷⣿⣿⣷⣿⣷⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠂⠀⠀⢠⡦⡀⣀⡀⠁⠀⠸⣿⣯⣷⣿⣯⡈⣹⣁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⠀⠀⠀⠺⠗⠀⠁⠇⠀⡰⠂⣶⠢⣿⣿⣿⣿⣿⡏⠹⣿⣿⣿⣿⠿⣿⢷⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠛⠋⠙⠉⠿⠷⠈⢿⡿⠿⠿⠆⣿⣵⣿⣿⣿⣟⣿⡟⣿⣙⣿
⣿⠿⠿⠿⡿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣦⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡄⢀⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡞⠁⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⢀⣀⣽⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠷⣿⡟⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⣿⣿⣿⣿⣿⡀⠸⣷
⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠈⢀⡟⠐⣹⠿⠛⠛⢿⣿⡀⢹⣇
⠀⠀⠀⠀⠀⢀⣾⢿⡿⢿⣿⣿⡧⣐⠁⢠⣀⣀⡀⢹⣷⡀⢻⣧
⠀⠀⠀⠀⠀⣸⣿⡷⢊⣿⣿⣿⣿⣾⠀⣹⣦⠠⣴⡀⣿⣧⠈⢻⣆
⠀⠀⠀⠀⢠⠏⣿⠁⣾⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⣰⣿⣿⣧⠀⢿⣦⣤⣶⣶⣦⡄
⠀⠀⠀⢀⡟⢸⠏⣾⣿⣿⢿⣿⣿⣿⣿⣿⡟⠓⢾⣿⣿⣿⣿⣆⠈⢿⣿⣿⣝⣿⡑⠀⠤⠄
⠀⠀⠀⣼⢻⡇⢸⣿⣿⣿⣿⣿⣿⣯⣿⣯⡀⠀⠈⢻⣿⣿⣿⣿⣆⠀⢿⣯⣿⣯⠂⣰⠶⠶⠆
⠀⠀⢠⡏⣾⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠈⢿⣿⣿⣿⣿⠄⢀⡾⡏⠛⡣⣟⢧⣄
⠀⠀⣾⢷⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠘⢿⣿⣿⣿⢛⣧⡾⢡⣾⣷⣯⣄⡈⠙⠶⠄
⠀⠀⣿⢸⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⢻⣿⣿⣿⣟⡾⠀⣠⢤⣭⣿⣿⣶⣤⡄⣠⡀⢀⣀⣄⣀⣀⣀⡀
⠀⢠⡿⣸⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⠙⠿⣿⣿⡋⢷⣞⣿⣿⠶⣾⡵⠻⣿⣿⣿⣹⣷⣯⣽⡏⢱⣟⣷
⠀⢸⡇⠛⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢨⣿⡇⠈⠛⣿⣧⣶⣿⡶⢦⡟⠳⠀⠐⣫⣾⣿⣿⡯⠛⠉
⠀⢸⣧⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣠⣴⣻⣤⣶⠿⣷⠄⣀⣰⢤⣼⠟⠃⠀⢠⠖⠀⢻⣿⣿⠟⠃
⠀⠈⢿⣿⠈⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣠⡴⢋⡽⢫⣿⣿⣧⣶⣿⡿⠈⣻⠟⣁⣴⣿⣾⣇⠄⠚⠛⠛⡁
⠀⠀⠋⣿⡀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠰⠟⠉⠑⢋⣥⣾⣿⣿⣿⣿⠟⣭⠶⢩⣴⣿⣿⣿⣯⣠⣽⠓⠠⠦⠼⠏
⠀⠀⠙⠻⣧⢸⣿⣿⣿⣿⣿⢠⣴⣮⣵⠄⠒⣤⣭⣭⣒⡂⢈⣛⡥⡽⠒⠚⢿⣟⣿⡿⣿⣿⠿⠻⢿⡁
⠀⠀⠐⡀⢿⡚⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣽⡃⠠⠀⢩⡆⣠⠾⠇⠓
⠀⠀⠀⢀⢸⣷⢹⣿⣿⣿⣿⣿⡄⠀⠉⠉⠉⠉⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣤⣿⠿⠀⠀⢀⠀⠛⠀⢀
⠀⠀⠀⠀⠉⠻⣷⣿⣿⣿⣿⣿⣿⣦⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣯⢺⣿⡷⠀⠑⠈⠘⠓⠰⠟
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⠿⣿⠟⠛⠻⠿⣿⣿⣟⣫⡁⢿⡿⠃
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣏⡀⣷⣀⡠⢶⠶⠿⣿⣿⣿⣷⣿⡅
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠏⣶⣤⣌⢰⡶⢾⣿⣿⣄⠈⠻⠎⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠉⠻⠷⣾⣤⣿⣿⣿⣿⣶⣤⡄⠀⠀⠸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟⢹⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣾⣷⠶⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣶⣾⣿⣿⣿⣿⡇⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢛⣛⣉⣭⣬⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣟⣛⣛⣯⣽⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢾⣦⣬⣬⣿⣿⣿⣯⣽⣿⣷⣾⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣟⣛⣿⣽⣷⣬⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠐⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠁⠀⠀⠀⠀⠀⠀⠉⠉⠭⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠄⠀⠀⠒⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⢂⣀⡐⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢾⣧⣌⠈⠙⠿⠣⠑⠶⣬⡑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣤⠞⠡⠴⠌⣠⣧⣉⢠⣆⢠⡄⠀⠘⠛⣦⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣼⠃⠐⢄⠀⠸⣛⣿⣫⠁⠈⠽⣉⡀⢳⣦⣌⢳⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣸⡏⠂⠀⠸⡇⠈⠃⠈⣄⢁⠄⣀⣿⣿⣞⣧⣟⣯⣷⡹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣿⠁⠰⠃⡀⠓⠘⠊⠀⢃⡈⣶⣹⣽⣟⣿⣯⣿⣿⣿⣗⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⡄⠘⠐⢤⣜⡀⠀⠀⢲⣼⡏⣴⢸⣿⣟⣿⣿⣿⡟⢻⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⣿⡇⠀⢠⡄⢿⠀⠀⠀⠈⢋⣽⣿⣾⣿⣿⣿⣿⣿⣇⠈⡆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⣿⡇⠀⢀⠀⠂⠀⠀⠀⠀⠿⢿⡉⠉⠉⠉⠉⠛⢿⣿⣦⢳⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⠗⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠙⢿⣾⡆⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⡀⣀⣀⠤⣴⣒⡒⠀⣐⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣟⣠⡭⢳⡆⠀⠀⠀⠀⠀⠈⣀
⠀⠀⠀⠀⢀⣠⠔⠛⠛⠛⠉⠀⠑⠠⠃⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣯⢿⣽⣛⡇⠀⠀⠀⠀⠀⡄⠙⡀
⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠟⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣿⢿⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣿⠟⡏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃
⠀⠀⠀⢹⡇⢯⠻⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⡇⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠁⠀⣷⣌⣧⠘⢆⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⡆⠀⠀⠀⠀⡠⠃⠀⢠⡟
⠀⠀⠀⠙⠋⠉⣻⢷⣟⠠⠀⠀⠀⠀⠘⢦⠄⠠⡀⠀⣠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠲⣿⣿⡀⠀⠀⠀⠀⠃⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡅⠀⠀⠀⠀⢻⡼⡄⢹⠋⠉⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢧⣧⢿⡄⠀⠀⠀⠰⠄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠈⠳⣄⣼⡀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣿⡼⠁⠀⠀⠀⠀⠀⣁⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⡄⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⣠⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠰⠄⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠇⠀⠀⢀⣼⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⣀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⣠⣴⠿⣋⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢄⣠⣴⢫⢃⡀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿⣿⡿⣿⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⣠⠟⠘⠀⣾⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠋⠀⠙⡿⢣⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⢻⡄⠀⠀⠀⠂⠈⣰⠦⠀⠀⠀⠠⠶⣾⣿⣽⡷⠤⢎⠘⢿⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣾⣦⢠⣴⣾⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣉⣀⣤⡴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⠉⠁⠛⢯⣿⠀⠀⠈⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠠⡀⣨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⠀⠐⠄⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⢶⣶⣿⣿⡇⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠈⡗⠀⠀⠀⠀⣰⣿⣾⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⡇⢀⠀⠀⠘⠀⠍⠛⠿⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢷⣾⠂⠀⢸⣷⡀⠐⠀⣽⢹⠘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⣸⣿⣿⣥⣆⢯⣿⣷⣦⢶⠟⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⢰⣿⡏⣿⢿⡟⠛⠚⠓⠿⣸⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⢻⡿⣷⠒⣀⠀⠀⣿⠁⠀⠃⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⣿⠞⣿⣟⣷⠿⢷⡇⣿⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠿⣿⣿⣕⠪⡼⣿⣿⡁⠂⠠⣿⣿⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢷⣤⡀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠴⠛⠛⠛⠻⠿⣾⡽⣿⣧⣶⣶⢻⣍⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠂⠀⡀⠀⠀⣀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠚⠛⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣙⢟⡀⢴⡥⠄⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣿⣿⣥⣴⣼⡿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⢰⡿⣿⡏⠐⢩⠊⣵⡾⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠾⣷⣿⣸⢿⣹⠆
⠀⠀⠀⠀⢀⠔⠈⠀⠀⡀⠀⠀⢸⣵⣋⠁⠷⠓⠀⢨⡤⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣴⣧⡿⢈⠁⠃
⠀⠀⠀⢀⠂⠀⠀⠀⠠⠀⠀⠀⢺⡕⠁⠣⢌⡐⠓⠶⠁⠺⢗⡖⡀⠀⠀⠀⠈⠑⢿⣶⣿⣿⣿⠅⠓⠂
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠾⡁⠀⠀⠂⠀⢀⡉⠵⣴⠁⠀⠀⠀⠀⠀⠈⣿⣿⡿⢫⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠠⠔⣶⡞⢻⡏⠑⢟⣿⣷⣎⣴⣶⣄⣼⢿⠛⠃
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢿⣾⣄⣜⣀⣐⢟⣛⣫⣽⡿⠟⠉⠉
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⡛⠋⠉
⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠠⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡿⠟⠋⠀⠀⠀⠀⢀⣠⣴⠾⠋⠀⠀⠀⠀⢀⠤⠾⠿⠿⢿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛
⠀⠀⠀⠀⣀⣤⣶⠟⠋⠁⠀⠀⢀⣠⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠰⣶⣶⣤⣄⣀
⠀⣠⣴⣾⠿⠋⠀⠀⠀⢀⣠⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠿⣽⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣷
⣿⠟⠋⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⠁⠀⡿⠟⠋⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀
⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣋⣽⣿⣿⣿⡿⠟⢿⣷⣶⣤⣀
⣿⣿⣿⣿⣿⣾⠉⠉⠛⠻⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢈⠍⢺⣿⡇⠀⢸⣿⠟⠉⠙⠛⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⣠⣶
⣿⣿⣿⣿⣿⣿⣦⣤⣶⡾⠛⠃⣀⠖⢶⣄⠀⠀⠀⣤⠿⠟⠃⣰⣄⣀⣴⡾⠛⣁⠀⠛⠃⣀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣶⣦⣶⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣼⣿⡿⠄⠀⢿⢿⡾⢛⣽⠟⢶⣴⠿⠀⠐⢋⣠⣍⣿⠏⢀⣼⠋⠑⠂⠀⢀⣬⣿⠶⢦⣤⣼⣿⠏⠁⢸⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡘⠁⡿⢣⣾⠿⠁⠰⠿⣶⠛⠋⢀⣄⣉⣿⠿⣿⣷⣾⣿⣿⡧⢠⣾⣿⣦⣄⣠⣿⣦⡾⣿⣿⣿⣷⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⣃⠀⠀⢠⣤⣉⣷⣶⣾⡿⣿⠷⣶⣤⣾⠋⠀⣉⡳⢾⠛⢍⣙⣿⠛⠻⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣩⣿⣷⣶⠎⣛⢉⣩⡿⠀⠀⠤⢱⡌⠛⠁⣧⣀⢨⣶⡿⣷⣶⣀⣻⣿⣷⣦⣤⣩⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣭⠤⣿⣦⣩⣶⣿⢤⠄⠀⣻⡀⠈⣥⣼⣿⣭⠟⠻⠟⢋⣿⣿⣿⣿⣿⣿⡿⢻⣷⣽
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⣽⠶⣿⣉⢛⣿⣿⡟⣯⣾⠿⣿⣿⣿⣽⣯⣀⣀⠀⠀⠀⢉⣿⣿⣦⣭⣿⣿⡟⠻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⣿⣿⡳⡝⠲⣶⣬⣽⠟⠁⣠⣵⣏⡃⠺⣦⠟⢈⣿⠉⣿⣿⣿⣴⣾⣿⣿⣿⣿⡿⢷⣼⣿⣷⣾
⣋⠿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⡳⠏⣧⣉⣙⣷⣶⠿⠷⣶⣾⣿⣿⣤⣙⢛⣛⣶⣿⠟⣷⣿⣽⣿⣿⣿⠻⠖⣿⣿⣷⣿⡋⠘⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣼⣿⠀⣿⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⣘⣛⣓⣃⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡯⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⠀⣽⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣿⣿⣿⣿⣇⡠⡟⣧⠉⠙⣦⣭⣥⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣾⣿⣿⣿⠀⠳⣶⣶⣿⣃⣀⠀⠀⠬⣝⡳⢦⣄⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⡀⢰⣿⣷⣤⣾⣿⣷⣶⣤⣠⡙⢷⠙⢷⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣿⣿⣿⠇⠞⢿⣿⣿⣿⣿⣿⢿⣿⣿⣟⢠⠳⡈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⡆⠶⢽⣿⣿⠉⢈⣹⡆⠀⠈⠙⣧⡆⠁⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣟⣶⣿⣿⡿⠷⠵⠛⠀⠀⠀⢀⠘⣿⠃⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠇⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⣿⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⢀⣶⣿⠃⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠠⣿⣿⠿⠂⠀⠀⠀⠀⠀⣀⠤⠊⢀⣼⣣⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣠⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣟⡻⠝⠻⢿⣿⣿⡿⣟⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣿⣿⣏⣿⣷⣞⣳⡀⠈⠻⣿⣼⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⣦⣤⣴⢶⠶⠶⢶⣿⣿⣿⣿⣿⣿⣿⣧⣰⢄⡀⢹⣿⣿⣿⣿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠻⠛⠉⠁⠀⣿⣿⣿⣿⣿⣿⣻⣾⣿⣣⣾⣶⣿⣹⣿⠿⠉⣹⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣃⣀⣀⣀⣀⣈⣛⡛⣋⢉⡋⡉⢉⡉⣉⠉⣉⢉⡁⣈⠁⣀⢀⣙⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⡀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣨⡳⣶⡮⡕⠆⠀⠀⠀⠀⢐⡦⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠇⢰⣻⣾⣿⣿⣿⣆⠀⠀⣀⡗⣦⣭⣦⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠷⣿⣿⣿⣷⣾⣿⣿⡽⠀⢡⣿⣿⣿⢿⣭⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠆⢛⣷⣿⣿⣿⣿⣟⢟⣶⡀⢻⣿⣿⣿⣿⢯⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣏⢿⣿⣧⣿⣿⣿⡏⢿⡁⢹⣿⣿⣯⣻⢷⣿⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⣩⣾⣿⣿⣿⣽⣿⡿⡾⠀⢻⣿⣿⣿⢿⣏⣽⣜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⣿⣏⣿⣿⣿⣿⡇⠀⣹⣿⣿⣽⣿⣷⣛⣎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠆⢠⣿⣿⣿⣯⣿⣿⢏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⡙⢿⣿⣿⣿⣯⡀⠸⣿⣿⣿⣭⠿⡷⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣾⣿⣶⡄⡀⣿⣿⣿⣿⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⢿⣿⣿⣿⣷⡮⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠶⢉⠙⡛⠛⠷⣤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣏⣾⣿⣷⣮⣤⣮⣲⠘⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣧⢿⣿⣿⣿⣿⣿⣿⡀⣿⣧⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢱⡽⣿⣿⣿⣿⣿⣿⣷⢁⣿⣿⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣬⣿⣿⣿⣿⣿⣿⣏⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⣿⣹⣿⣿⣫⡿⢳⣿⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⢫⣴⠿⠋





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿