Lifesaver Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⠖⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠐⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢈⡭⠊⠙⠶⣀⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢰⠎⠀⠀⠀⠀⠀⠙⠐⠒⢬⡀
⠀⠀⠀⠀⠀⢀⢈⢀⡠⠤⢄⢀⡀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣄⡘⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⡆⠆
⠀⠀⠀⠀⠀⢛⡟⠀⠀⠀⠀⠀⠘⠢⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⢀⢃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠐⠉⠳⣀⠄⢠⠃
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⡀⠀⠀⠀⣿⣶⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠹⣆⠀⣰⠏⠙
⠀⠀⠀⠁⡃⠀⠈⠄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⣴⡋
⠀⠀⠀⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⡞⠜⣠⣞⣿⠟⠁
⠀⠀⠀⠐⢄⡀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠠⠄⠀⠄⡤⠁
⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⣀⡓⠤⠦⣀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠈⠑⠓⡦⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠐⠉⠑⠲⠬⢕⣐⠐⠠⠠⢀⣀⡀⠤⠘⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣷⣶⣤⣀⣠⣴⣶⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠈⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣰⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣷⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣠⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣧⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠈⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡏⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠘⢿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⠟⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣀⣀⣈⣉⣻⣿⣿⣿⣿⣿⣿⣋⣉⣀⣀⣠⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠛⠛⠉⠀⠈⠙⠛⠻⠿⠿⠟⠛⠋⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⡟⠀⠀⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣋⡥⠤⠖⠒⠚⡇⠀⠀⣸⠛⠓⠲⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠉⠹⡄⠀⠀⠀⠀⠙⠦⠴⠋⠀⠀⠀⠀⠀⠈⠉⣳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠋⠁⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠈⠙⢶⡶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠋⠉⠉⢉⡽⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠙⢦⡀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽⠖⠒⠛⠛⠛⠛⠒⠶⠤⣄⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢻⠀⠀⣼⠉⠳⣄⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⣈⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠸⡇⢠⠇⠀⠀⠈⠑⢤⡀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣠⠤⠖⠚⠉⠁⠸⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣷⢸⠀⠀⠀⠀⠀⠀⠙⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡇⢰⡇
⠀⠀⠀⠀⠀⢠⠚⠉⠉⠉⠳⡄⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢠⠞⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠘⢤⣀⣀⣠⡴⠃⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⢦⣄⣀⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⣿⢹⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀⠀⠀⠀⠀⠀⠀⡏⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡟⠘⡆⠀⠀⠀⠀⠀⣀⡠⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⠓⠦⢤⣀⠀⢰⠃⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⡀⣠⠴⠖⠋⠁⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠈⢙⡟⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣲⠦⠤⢤⣤⣤⡤⠤⠶⢶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢀⣠⠞⢁⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⣄⣀⣀⠀⠀⠙⠲⢤⣀⣰⠃⠀⠀⠀⠀⣠⠖⠲⣄⠀⠀⠀⠀⢧⢀⣀⡤⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠦⢬⣉⠛⠒⠦⠤⢤⡇⠀⠀⢹⣤⣤⠴⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⠤⣧⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡔⠊⠉⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠲⣾⢹⡘⣿⣾⣿⣿⣶⣿⣴⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠘⣷⣌⠢⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠸⣧⢳⡌⠺⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢻⡌⣇⢧⡙⢿⡿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣎⠀⠀⠀⠀⠀⠀⠀⠀⡠⠓⠝⠚⠋⠊⠀⠀⠀⠉⠛⢄⠀⠀⠀⠀⠀⠀⠀⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣦⣤⣀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⣀⣤⣴⣾⣿⣿⣷
⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣃⣀
⠀⠀⠀⠀⡜⠁⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡟⠈⢣
⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸
⠀⠀⠀⠀⠘⠤⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧⠤⠃
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠟⠛⠉⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠉⠛⠻⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢍⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣀⡀⠀⠀⠀⠀⢀⣀⣴⠊⠀⠀⠀⠀⠀⠀⠀⠀⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢻⣿⠿⣿⣿⠿⣿⠻⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⣀⣀⠠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣤⣶⣶⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣡⠔⠊⠁⡾⠛⠛⠛⡻⡇⠉⠑⠬⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣿⠏⠁⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠈⠻⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⡾⠃⠀⠀⠀⣠⠖⠋⠉⠉⠉⠉⠛⢦⡀⠀⠀⠀⠘⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣇⣀⣀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⣀⣀⣘⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠧⠼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠦⢽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠈⢤⠖⠒⠉⠛⠒⠢⢤⣀⡀⣀⡾⠁⠀⠀⢀⡿⠁
⠀⣤⣿⣿⠒⢦⡀⠀⠈⡿⣧⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⢀⡬⠋⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⣠⠀⠤
⠘⠀⠈⠀⠀⠀⡇⠀⠀⢳⡘⢷⡀⠀⠀⠀⠈⣹⣷⣶⣶⣿⠉⠀⠀⠀⠀⣠⡼⠏⠀⢀⡀⠀⠀⢯⠀⠀⠂
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠙⢫⣝⡷⣦⣀⠀⣿⣷⣷⣶⣿⠀⠀⢀⣠⣶⡕⠀⠀⠀⢿⡈⠀⠀⢀
⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⣠⠼⠯⣿⡶⣭⣷⣿⣿⣿⣿⣷⣒⠈⢁⡘⣥⡀⠀⠀⠀⠀⠉⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠁⠀⠀⠀⠀⠈⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣀⣤⣤⠶⠶⠶⠶⠶⠶⣤⣤⣈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡏⠀⠀⠀⠀⠈⠻⣦⣀⣤⠴⠶⠶⢦⣤⣄⣴⠟⠁⠀⠀⠀⠀⢙⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣿⠋⠻⣦⡀⠀⠀⣠⡶⠋⠉⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⢀⣴⠟⠙⣿⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⣼⠃⠀⠀⠈⠻⣦⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣴⠟⠁⠀⠀⠘⣧⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢠⡟⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢹⡄⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⡇⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡇⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢸⡇⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⣧⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣸⠇⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⢻⡄⠀⠀⢀⣴⠟⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠻⣦⡀⠀⠀⢠⡟⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣿⣄⣴⠟⠁⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⠾⠋⠀⠀⠈⠻⣦⣠⣿⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣇⠀⠀⠀⠀⢀⣴⠟⠙⠛⠲⠶⠶⠞⠛⠋⠻⣦⡀⠀⠀⠀⠀⣸⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣎⠉⠛⠛⠶⠶⠶⠶⠶⠶⠟⠛⢉⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠛⠛⠋⠉





⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠊⠉⠙⠳⢦⣄⡀⢀⠀⠀⠛⠒⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⣀⠠⠤⠤⠄⣀⠀⠀⠀⠀⠐⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢾⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠘⠢⠄⣀⣀⣀⣀⠤⠦⠀⠀⠀⠀⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡤⠀⣀⠀⠀⡀⠠⠤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣠⣴⣦⣭⣥⣶⣶⣬⣭⣶⣦⣄⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡽⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⢷⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣯⡀⠀⠀⠈⠻⣿⠿⠟⠛⠛⠛⠿⣿⠟⠁⠀⠀⢀⣵⡀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⡀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⢀⣴⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⢿⣿⣿⠟⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠈⠻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠁⠀⠀⢀⣴⣿⣶⣤⠤⠤⣤⣶⣿⣦⡀⠀⠀⠈⣻⠁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⢳⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡠⣞⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠙⠻⠟⠛⠿⠿⠿⠿⢛⠿⠟⠋⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄⠀⣀⠀⠀⠀⠀⣀⠀⢠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠁⣄⡁⢈⣢⠈⠀⠀⠀⠀⠉⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠠⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠄⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⣶⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠐⠒⠒⠂⠰⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠂⠀⠈⠻⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡈⠀⠀⠀⠀⠈⠻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠟⠁⠀⠀⠀⠀⢁⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠘⢀⠄⠐⠁⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡋⠢⠘⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⡄
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢃⠀⠀⠀⠀⠀⣠⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣄⠀⠀⠀⠀⠀⡘⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠈⠄⠀⠀⣠⣾⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷⣄⠀⠀⠠⠁⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠁⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠄⠀⠀⠀⠀⠰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠙⠿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠋⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠁⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠐⠈⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⡀⠀⠀⠀⠈⠉⢩⡀⢀⠍⠈⠁⠀⠀⠀⣀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠀⠉⠢⠤⠠⠔⠋⠀⠀⠛⠉⠉



⣾⠓⠛⣿⠶⠶⢾⠿⠶⣿⡟⢿⠷⠿⢶⠾⠶⠶⢿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⡷⣷⡶⠾⠿⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡠⠒⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⠆⢄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣷⣤⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣰⣿⣿⣷⣆⠀⢻⣿⣿⣿⣿⣿⣿⠁⠀⣠⣼⣶⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⠈⠙⣿⣧⠀⠀⠀⠀⠀⠀⣼⣿⡿⠛⠉⠻⣿⣿⣿⣿⣿
⠀⠀⠀⢰⣿⣿⣿⣿⣿⠤⠚⠛⠛⠛⠛⠻⠇⡀⣼⣿⣿⣿⣿⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣀⢿⡏⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢿⣿⡿⠋⠀⠈⣱⣤⣄⣀⡀⠀⣀⣀⡠⠛⠛⠻⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⢀⣴⣿⣿⡾⠛⠻⠿⢿⣿⣿⠿⠿⣧⣶⣄⡀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠻⡅⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢸⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⡸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠑⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠤⠊⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣝⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⣥⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣠⣴⡚⠉⠑⠒⠚⠛⠉⠉⠛⠓⠒⠊⠉⢓⣦⣄⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣱⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⣴⣮⡻⣿⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣿⢟⣿⣿⣿⣿⣿⣿⣧⠀⣀⣀⣀⣀⣀⣀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠱⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢀⡞⠈⠓⢾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠛⠁⢳⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢱⠀⠀⠀⠉⠛⡿⠁⠀⠀⠀⠀⠀⠰⣏⣹⠆⠀⠀⠀⠀⠀⠈⢿⠛⠉⠀⠀⠀⡎⣇⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢶⣾⣷⡶⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣶⡄⠀⣸⣇⠀⢠⡶⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡿⢸⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢹⡸⠀⠀⠀⣀⣤⣷⡀⠀⠀⠀⠀⠙⠻⢿⡿⠟⠋⠀⠀⠀⠀⢀⣾⣤⣀⠀⠀⠀⢇⡏⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢧⢀⣤⣾⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⢤⡀⡼⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⢇⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠉⠉⠉⠉⠉⠉⠀⢻⣿⣿⣿⣿⣿⣿⣵⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡻⣿⣮⡻⠟⣻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⢏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠙⠻⢥⣀⡠⠤⢤⣤⣀⣀⣤⣤⠤⢄⣀⡬⠟⠋⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠒⠒⠋⠉⠁⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⣴⠶⠶⠶⠶⣦⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⢀⡴⠞⠋⢹⠠⣞⣳⠄⡏⠙⠳⢦⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣴⠋⠀⠀⠀⣸⣇⣉⣉⣸⣇⠀⠀⠀⠙⣦⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⣰⠟⠁⠀⠀⠀⠀⠈⠻⣦⠀⠀⠘⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣉⣛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣛⣉⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠸⢧⠿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠿⡼⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠒⠚⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠓⠲⣶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣧⠀⠀⠈⠳⢦⡤⢤⡤⢤⡴⠞⠁⠀⠀⣼⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⠳⣄⡀⠀⢸⢃⡶⢶⡘⡇⠀⢀⣠⠞⠁⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣈⣹⡶⣾⣀⣙⣋⣀⣷⠶⣏⣁⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠄⠀⠀⠀⢀⠀⠀⠄⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣴⣄⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣠⡀
⠀⠀⠀⠀⣠⣾⣿⣿⣷⣄⠀⠈⠻⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⠟⠁⠀⣠⣾⣿⣷⣄
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣠⣾⣿⣿⣿⣿⣿⣆
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡆
⢀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿
⠈⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⣠⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⣄⠀⠈⠻⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁⠀⣠⣾⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣷⣄⠀⠈⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⣄⣰⣿⣀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠞⠋⠉⠀⠀⠉⠙⠳⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠃⠀⠀⠀⣿⣿⠀⠀⠀⠘⣧⣴⠄⠀⠀⠀⢀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡇⠀⠀⢀⣤⣼⣧⣤⡀⠀⠀⢸⣇⡀⢀⣤⣀⣿⠷⠶⠿⠶⣾⣟⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣸⠋⣡⣀⡽⠋⠀⠀⣠⣤⡀⠀⠈⠻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢹⣆⠀⠸⠿⠿⠿⠿⠇⠀⣰⠟⠿⠉⡿⠁⠀⠀⠀⠻⡿⠃⠀⠀⠀⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⢻⣦⣤⣤⣠⣤⣴⠛⠿⠆⠘⠻⡇⠀⠀⢠⣾⣿⣿⣿⡆⠀⠀⢈⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠙⢳⣦⠿⣣⡈⠛⠀⠀⠀⠀⣶⣿⡀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡴⠟⠛⠛⠛⠛⠿⣇⣠⣄⠀⠀⠀⠀⣠⣿⣤⡈⠋⠛⠛⠛⠃⣠⣾⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣴⠋⠀⠀⢠⣶⣦⠀⠀⠈⢻⣅⣀⠀⠀⠀⠈⠀⢸⡿⠛⣶⡶⠺⣿⡁⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⣈⣛⣋⡀⠀⠀⠀⢻⡋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⣸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢷⡀⠀⢸⣿⣿⣿⣿⡇⠀⣠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢦⣈⡈⠉⠉⣈⣠⠾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠙⣿⠋⠹⠷⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠿⠿⠿⠷⠖⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣺⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠋⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣿⠿⣿⢿⣷⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠀⠀⠀⢀⣤⣤⣶⣿⣿⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⣿⠟⠁⣠⣧⡀⠙⢿⣧⣀⠀⠀⠀⠀⠀⠀⠀⣀⢼⣭⠶⠷⢮⣷⣄⡀⢨⣿⣿⠟⠉⢉⡟⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠠⣤⣶⣿⣷⣶⣶⠸⢿⣿⡟⠛⣿⣿⡟⠛⣻⣿⠿⠁⣤⣀⣤⡄⠀⠘⣏⣟⠙⢦⣴⠋⢹⣼⢡⣾⣿⡏⠙⢲⣿⣄⣀⣿⣿⡿⠇
⠀⠀⠀⠀⠀⢀⣰⣿⠯⣤⣞⠹⣿⣦⢀⣿⣷⣦⣀⣇⣠⣶⣿⣇⢠⣾⡿⡛⣻⣿⣶⢄⣦⢷⣴⠋⠉⢢⣼⣹⡏⠹⣿⣿⣄⣼⠁⢀⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠙⣿⣦⣜⣉⣽⣿⠃⠈⣿⠋⠛⣿⣿⡟⠋⢻⡏⢸⣿⣤⠛⢣⣼⣿⠀⣤⠙⣭⡛⠟⣭⡟⣡⡄⠀⢾⡿⣿⣿⣿⣿⡿⠟⠟
⠀⠀⠀⠀⠀⠀⢶⡟⠛⢿⡿⠛⢹⡷⠀⠸⣇⠀⣨⣭⡀⢀⡾⠀⢶⡉⢿⠿⠿⠟⢀⡖⠹⣇⠈⣤⣤⡀⢀⡿⠀⠀⠐⣆⠀⠈⠛⠋⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠈⢧⢰⣿⣿⡄⡼⠁⠀⠀⠹⣆⠿⠿⢇⡾⠁⠀⠈⣧⢰⣿⣿⠄⡾⠁⠀⢹⡌⠿⠿⢃⡾⠁⠀⠀⠀⠹⣆⢼⣿⣷⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣭⣭⣼⠃⠀⠀⠀⠀⢻⣷⣶⣿⠃⠀⠀⠀⠘⣧⣭⣭⣼⠃⠀⠀⠀⢻⣷⣶⣿⠁⠀⠀⠀⠀⠀⢹⣬⣭⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠀⠸⣿⢿⡏⠀⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠸⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠶⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⢁⡴⠖⠲⣄⠀⠖⠒⢦⡀⠐⣶⠒⢶⡖⠲⣆⠰⣖⠀⢲⡖⠀⣲⠆⣠⠖⠲⢦⡀⢲⡶⠖⠀⡇⣰⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣟⠚⠒⠛⢰⡏⠉⣻⡇⠀⡇⠀⢸⠇⠀⡿⠀⠹⣤⡟⢹⣰⠏⠀⣧⠀⠀⣸⠇⢸⡇⠀⠀⡟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠉⠛⠛⠁⠈⠙⠛⠛⠃⠘⠛⠀⠛⠃⠘⠛⠀⠀⠙⠀⠀⠋⠀⠀⠈⠛⠛⠉⠀⠛⠃⠀⠘⠛⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣤⣶⣦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣿⣿⣿⣯⡀⠀⠀⠀⢻⡗⠀⠀⠀⡀⠈⣻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣟⠀⠀⢾⡷⢾⣿⣶⣿⠚⠿⠚⢿⣿⣿⠟⠀⠀⠀⠀⣤⣶⣶⣄
⠀⠀⠀⣁⠀⠀⠀⣠⣴⣶⣤⣿⣿⡿⣿⠟⠻⠿⠃⠀⠀⠀⢴⣿⠿⣿⠆⠀⠐⢾⡟⠡⣿⣦⡀⠀⢸⠏⠉⠛⠻⠄
⠀⠀⠀⠙⣷⡀⢸⡿⠋⠉⠉⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠈⢷⣦⡻⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣴⣦⣀
⠀⠀⠀⠀⠀⠀⢻⣿⣧⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠻⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣴⡿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣤⡄
⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⣤⡉⠁
⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣥⣤⣤⣤⣤⣼⣿⣧⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣯⣬⣿⣿⣧
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⣚⣩⠩⠭⠭⢽⣿⡿⠿⠛⠿⠟⠛⠻⢿⣖⠲⠦⢭⡙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡵⠚⠉⣀⠴⠒⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⣄⠉⢧⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡵⠁⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢳⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣠⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡼⠁⠀⠀⠀⠀⠀⠀⠀⢀⣦⡤⠤⠴⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡋⠀⠀⠀⠀⠀⠀⠀⡼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠹⡄⠀⠀⠀⠀⠀⠀⠀⠉⣳⠶⠦⠶⠶⠒⠋⠉⠀⠁⠢⠀⠀⠀⠀⠀⡸⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡇⠹⣄⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⡴⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡹⣄⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⢁⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣳⠦⣄⣈⡙⠲⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⢁⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠭⠭⠿⠿⠿⠿⠿⠻⢿⡿⠟⠉⠙⠛⠛⠛⠛⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠿⣯⠉⢉⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⣨⢷⡋⠁⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣷⣷⣦⡴⢿⣿⣁⣀⣻⣾⣿⢡⣤⣼⣷⣴⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⣡⡜⢻⣿⡎⠉⠛⠻⠋⠋⢠⣶⡿⣫⣽⣭⡹⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣶⡄⣾⣿⣿⡠⢻⠃⣴⣿⠇⠀⠀⠀⠀⠀⢠⣿⡇⢿⣯⣽⡿⣹⣿⡁⣠⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⣱⡿⢋⣿⣿⣿⡿⡟⣯⣴⣦⣴⣀⡀⠀⠀⢹⣿⣦⣭⣯⣷⣿⣿⣇⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⡿⠋⠀⠀⣀⣿⢻⣧⡶⣾⣿⣷⡆⠀⠀⠀⠛⠉⠻⠻⣿⣬⣿⣷⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢼⡇⢾⣏⠶⣩⣷⣻⣟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⣆⠀⠀⠀⠀⠀⠐⢿⣾⡛⠻⣛⣷⣿⠉⢀⡀⠀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠈⣿⠃⠀⣀⣀⣶⣤⣼⡍⠛⠻⠏⣛⣸⣷⣾⠿⢿⣿⣶⡆⠀⠀⢰⡿⠋⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⢰⡿⢠⣤⣿⣿⣿⣿⣿⣿⡟⠀⣀⣿⡿⠻⣷⡀⠀⣨⣿⣿⣿⠀⢸⣿⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⣶⣿⣶⣿⣤⡄⢠⣿⣿⣿⠁⠀⣿⣿⣿⠟⣹⣿⠁⠀⣈⣷⡾⠋⠁⢹⣿⠗⠈⠛⣠⣀⣌⢛
⠀⠀⠀⠀⠀⠀⠀⠠⣴⡿⠋⠀⠀⠀⠈⢻⣷⡆⣹⣿⣿⣷⣾⣿⣿⠿⠃⢙⣿⣶⡿⠋⠁⢻⣦⣀⣾⡿⠦⣤⡿⠛⠛⠛⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣇⡀⠀⠀⠀⠀⠀⣿⡷⠀⠈⠿⠛⣿⠛⠛⠀⠀⠈⠙⣿⣷⣦⣤⣤⣿⣿⠟⠇⢺⣿⠀⠀⠀⠀⠀⣿⡗
⠀⠀⠀⠀⣴⣿⣿⣦⢿⣿⣇⠀⠀⠀⣀⣼⡿⠇⢻⣧⢀⣴⣶⣄⠀⣿⠇⠀⠀⠁⠈⠛⠙⣿⡏⢩⣴⣦⣝⣿⣄⢀⣄⠀⣠⣿⡋
⠀⠀⠀⠀⠻⢿⡿⢋⣼⣿⢻⣿⣿⡿⣿⠛⠋⠀⢸⣿⠸⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⣿⡿⠈⢿⣿⠿⡿⠛⠃
⠀⠀⠀⠀⢠⣶⣶⣾⣿⣥⣾⡿⠋⠀⠀⠀⠀⠀⠈⠻⣷⣬⣭⣵⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣭⣭⣤⣶⡿⠃
⠀⠀⠀⠀⢸⣿⣿⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢀⣿⠇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣿
⠀⠀⠀⣼⡿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿
⠀⠀⠀⠙⠁⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠘⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡴⠶⠶⠶⠶⠲⠲⠶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣛⣯⣭⣶⡶⠶⠶⠶⠶⣶⣦⣤⣤⣀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣯⡷⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣦⣄⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠶⠶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣷⣄⡀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⡀⠀⠀⠀⠀⢀⣠⣶⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣽⣿⣶⣄⣀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣦⣀⣤⣾⣿⣿⣾⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢻⡿⠻⣿⣿⣻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣾⠿⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⣾⠃⠀⠈⠙⠿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠟⠋⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⢰⣿⠀⢀⣿⠀⠀⠀⠀⠀⠀⣹⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠀⠀⠀⠀⠀⠀⢸⣿⢳⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡏⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣧⠀⠸⣷⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢀⣿⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⣿⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢸⡿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣹⡆⠀⠀⠀⢀⣠⣴⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣤⡀⠀⠀⠀⠀⠀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣤⡶⢞⣻⣽⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⠶⣤⣄⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡾⣟⣫⣿⣿⠿⠛⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠻⢿⣿⣿⣿⢿⣶⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣄⣀⣀⣀⣀⣀⣠⣤⣴⠾⠛⠁⠀⠀⠀⠀⠉⠛⠿⣷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠈⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠈⠛⠿⢶⣦⣤⣄⣀⣀⣀⠀⠀⣀⣀⣀⣤⣤⣶⠾⠟⣫⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⢉⣉⣥⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠶⣤⢤⠤⠶⠴⠶⠶⠞⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣿⣿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠙⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠟⠛⠛⠛⠛⠻⠷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⢰⣿⣶⣶⣦⣤⣤⣀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⣤⣤⣴⣶⣶⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⠿⠿⠟⠛⠛⠉⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⠛⠛⠻⠿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣤⣤⣤⣤⣴⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⣠⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⢿⣿⣿⡿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠿⣿⣶⣷⠄⠀⣠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⠛⠛⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⣡⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡄⣿⣿⣦⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠆⠀⠀⠊
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⢻⣿⣿⣧⠀⢀⠘⢋⣿⣿⣿⣿⣿⠃⠀⢻⣧⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠐⠈⡃
⠀⠀⠀⠀⠀⣸⣿⠿⣿⣿⣿⢸⣿⣿⣿⣾⡿⡷⠟⠛⠋⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠸⠄⠀⢠⡃⠀⠠
⠀⠀⠀⠀⠀⣿⣿⣤⣿⣯⣤⠈⣿⣿⠙⠋⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠉⠙⣋⣿⣿⣿⠀⡿⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡯⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⡗⠿⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠸⡷⠾⠿⠿⢿⡇⠙⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠛⠛⣿⣷⣦⣤⣀⣈⠄⠇⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣤⣴⣶⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⢿⣿⡿⠁⠈⠈⣗⣀⣤⣴⣾⣧⣤⣤⠤⠤⠀⠐⢲⠒⠂⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣿⠟⠁⣼⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⡥⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣿⣿⠙⠿⣿⣻⡿⠟⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡃⠀⠀⣿⡉⠀⠀⠀⠀⠿⠿⠿⠿⢿⡟⠋⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣿⡆⠀⡸⣿⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠻⣿⣴⣧⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠇⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠶⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡷⠒⠒⠒⠒⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣧⠀⠀⠀⠀⢠⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⠀⠀⠀⠀⣸⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠛⠛⠛⠳⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠿⠛⠛⠛⠚⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠙⣿⠋⠀⠀⠀⠀⠀⠀⠙⣿⠛⠉⢹⡇⠀⠀⠀⠀⠀⡼⢹⠁⠉⠛⣿⠁⠀⠀⠀⠀⠀⠀⠘⣿⠟⠛⠙⡏⢦
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⢸⡇⣀⣠⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⣀⠀⢸⡇⠀⠀⠀⠀⠀⢧⣸⣀⣠⣤⣷⡀⠀⠀⠀⠀⠀⠀⢀⣯⣄⡀⢠⣇⡞
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣶⣤⣀⣀⣤⣶⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⣄⣀⣀⣠⣴⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⠇⠀⠈⠀⠀⢻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡏⠀⠀⠀⠈⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢦⣤⣤⡴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡤⢤⣤⣤⠤⢼⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠵⠖⠒⠒⠲⠾⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⣧⠀⠀⠀⠀⢠⠏⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠈⣇⠀⠀⠀⠀⢠⠏⠈⠳⢤
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⣸⡦⠶⠶⢤⣏⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⢀⡼⠶⠖⠲⠶⣯⡀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⣸⠛⠶⣤⣠⠞⠁⠀⠀⠀⠀⠈⠳⣄⣀⣤⠼⣇⠀⠀⠀⠀⣠⢞⡏⠓⠶⢤⡴⠋⠀⠀⠀⠀⠀⠀⠙⣆⣀⡤⠴⢿⠳⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⢹⠀⠀⠀⠀⡏⢸⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⡇⢻
⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣸⠀⠀⠀⠀⣇⢸⡀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢠⡇⣼
⠀⠀⠀⠀⠀⠀⢹⡖⠛⠉⠙⢦⡀⠀⠀⠀⠀⢀⡴⠋⠛⠲⣤⠏⠀⠀⠀⠀⠘⢮⣷⠖⠛⠉⠹⣄⠀⠀⠀⠀⠀⠀⣠⠏⠛⠶⢤⣼⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⣽⠓⠒⠒⠺⡏⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠈⣿⠲⠶⠶⢶⡏⠁⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣴⠃⠀⠀⠀⠀⢻⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⣸⠃⠀⠀⠀⠀⢻⡀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡲⠶⠦⠴⠶⣺⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣄⡴⣢⠖⡴⢢⠶⣢⣴⣶⣶⣦⡴⢢⠖⡴⢢⠴⣢⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢰⠧⠏⠉⠀⠀⢀⣀⣤⠴⠶⠒⠛⠛⠛⠛⠓⠶⠦⣤⣀⡀⠀⠈⠉⠒⢻⣤⠄
⠀⠀⠀⠀⠀⣀⣶⠛⠈⠀⠀⣀⣴⣾⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣦⣀⠀⠀⠈⠛⣧⣀
⠀⠀⠀⠀⣤⠿⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣄⠀⠀⠐⠲⠦
⠀⠀⠀⢰⡜⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢀⣀⣀⣀⣀⡀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠈⢹⡂
⠀⠀⢀⢷⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠛⠉⠉⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢯⡀
⠀⠀⢈⠷⠀⢰⠿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠿⠻⡆⠀⠳⡅
⠀⠀⢨⡓⢀⡟⠀⠀⠉⠙⠻⠿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠟⠋⠁⠀⠀⠀⢻⡀⣙⠆
⠀⠀⠰⣍⣸⠃⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠘⡇⣜⠃
⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⢰⡝⢸⡄⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢠⡏⣙⠆
⠀⠀⠰⣍⠈⣧⠀⠀⠀⢀⣠⣴⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣦⣄⣀⠀⠀⣼⠁⢬⡃
⠀⠀⢘⢦⠀⠸⣦⣶⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⠇⠀⢲⡁
⠀⠀⠈⣳⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⣀⣀⣤⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢀⢷⠁
⠀⠀⠀⠨⣇⡀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠈⠉⠉⠉⠉⠁⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⡜⠇
⠀⠀⠀⠀⠲⠧⠄⠀⠀⠙⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⣶⠛
⠀⠀⠀⠀⠀⠉⢻⣤⠀⠀⠀⠉⠻⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠟⠉⠀⠀⡀⣤⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠛⣧⠤⣀⡀⠀⠈⠉⠛⠲⠶⢤⣤⣤⣤⣤⠤⠶⠖⠛⠉⠁⠀⠀⣀⣰⢰⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠫⠞⠣⠞⠴⠣⠞⠻⠿⠿⠟⠋⠶⠣⠞⠴⠫⠞⠘⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⡖⢊⣙⣉⣙⣛⠒⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣴⢚⣫⠴⠒⠋⠉⠉⠁⠀⠀⠈⠉⠙⠲⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠲⠶⠶⠶⠲⠖⠒⠢⠤⣤⣾⣤⣭⠛⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⣆
⠀⠀⠀⠀⠀⣠⡴⠞⣋⣩⡤⠤⠒⠐⠓⠋⠉⠉⠉⠉⠙⢒⣶⣶⣷⣿⣶⣿⣷⣿⡿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢸⣆
⠀⠀⠀⢀⡼⢃⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢿⣿⡟⢟⠳⡜⠌⢛⠜⣈⣱⣎⣧⠛⠛⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠘⡇⣻⡀
⠀⠀⢠⠏⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⡋⠔⠀⠈⠻⣦⣡⡘⢎⡎⠘⡿⣿⣿⣿⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⡇⢸⡇
⠀⢠⠟⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠊⠀⠀⠀⠀⠀⠸⣷⣿⠸⣟⠐⠄⠿⢻⣿⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⠀⠀⢰⡇⣾⠃
⢀⡿⡼⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⡆⡆⠀⢼⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⣸⣿⡏
⣼⢧⡇⠀⠀⠀⠀⠀⠀⠀⣠⣞⣿⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢡⣤⡄⣾⡸⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠹⣦⣿⣿
⣿⢼⠃⠀⠀⠀⠀⠀⠀⣼⢯⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣿⣿⣿⣿⣿⣿⣦⣠⠀⠀⠀⠀⠀⠀⢀⣼⡃⠀⢹⣿⣿⣆
⣿⠘⡄⠀⠀⠀⠀⠀⣼⠇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠟⠋⠉⠉⠉⠛⠿⣿⣿⣿⣷⣄⠀⠀⣠⣴⣿⣿⣧⠜⡡⢻⣿⡾
⢹⡆⢃⠀⠀⠀⠀⣸⠇⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣽⡖⡫⠉⡩⣿⡾⣿⣖⡀⢈⣿⣷⣄
⠘⣷⠘⡄⠀⠀⢀⡟⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡧⢀⠡⠐⡻⢿⣽⣋⠀⡁⣿⠿⣿⣦⣀
⠀⠹⡆⠰⡀⠀⢸⠁⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⢋⡀⠐⠁⢐⢾⣿⣿⣷⣴⣿⡇⠀⠑⢌⡱⢄
⠀⠀⢹⡆⢳⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⣧⠑⢊⡁⠐⣋⣼⡟⠁⢻⡇⠀⠀⠀⠙⢮⠳⡄
⠀⠀⠀⢻⣞⣧⣿⡄⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⣍⣢⣠⡵⠞⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⢳⠙⡆
⠀⠀⠀⠀⠹⣿⣿⣷⢀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢽⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠘⡇⢹
⠀⠀⠀⠀⠀⢻⣿⣿⣯⣸⣀⣀⣀⣀⣤⣴⣶⡞⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⣇⢸
⠀⠀⠀⠀⠀⢸⣿⣿⡟⠉⠉⠉⢽⣿⡿⣿⡭⠞⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⢿⢸
⠀⠀⠀⠀⠀⢸⣟⣿⣿⠀⠀⠀⠚⢩⣷⣿⣷⠄⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠧⠊⡝⢦⡀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⢀⠏⣹
⠀⠀⠀⠀⠀⡏⢿⣿⣿⣤⠀⢐⠀⢈⣿⡿⣣⢎⣤⠀⠫⢿⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⢏⢢⠑⣕⣜⣦⣻⡄⠀⢰⣿⠃⠀⠀⠀⠀⠀⢀⡾⣴⠇
⠀⠀⠀⠀⠀⠹⣤⣉⣉⣽⣇⠀⢂⠤⠋⣸⢟⣾⡿⢠⡤⢠⠟⠻⢿⢷⣶⣶⣶⣾⣿⣿⡿⢿⣫⣇⠨⣦⢳⣬⠻⠿⡟⢷⣼⣿⠏⠀⠀⠀⢀⣀⣴⣯⡽⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡗⠂⢁⠠⠂⡴⢻⡿⣹⣿⣾⠆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠳⣊⠷⣻⣦⢻⣆⠹⠗⠀⠀⢸⣿⢏⣠⡴⣖⡾⠯⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣽⡝⣿⣏⣤⠀⢀⠐⢀⣼⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⢡⡿⠿⢿⠀⠀⠀⣠⣾⢻⡥⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⢼⠁⠘⣿⣿⣤⣀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⡟⢣⡄⠘⠀⣰⣾⢿⡵⠋
⠀⠀⠀⠀⠀⠀⠀⡟⡼⠀⠀⠈⠻⣿⣟⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣧⣼⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⢠⠃⡇⠀⠀⠀⠀⠈⠻⣿⣶⣤⡔⢦⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣷⡿⣿⣟⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢦⣶⣤⣉⡛⢛⣖⣶⢶⡤⣴⣶⣶⣿⡿⠟⠋⣰⣿⣼⠃
⠀⠀⠀⠀⠀⠀⠘⡇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⣰⠛⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢷⡈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⡑⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⢋⡰⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠴⢍⡙⠒⢤⠤⣤⠤⠤⠤⡴⢶⠚⣫⡭⠖⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣶⣷⣶⣷⣶⣧⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡿⠁⠉⣉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠂
⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⡀⠂⠀⠁⠖⠒⠛⠛⠛⠟⠁
⠀⠀⠠⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⠐⠻⢦⠀⠀⠀⠀⠀⠀⠑⠂⠄
⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⡤⠐⢤⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣼⡻⠀⢀⡞⠀⡠⣶⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢾⡄⡔⠀⢻⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡿⠃⠀⠀⠈⣧⢁⠀⠀⢻⡞⡆⠀⠀⣠⣤⣶⣦⣤⡤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢦⠀⠀⢀⡤⡞⠋⠀⠀⠀⠀⠀⠀⢸⡆⡆⠀⠀⣷⣷⣶⣾⠏⠙⢩⣿⠟⠟⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠸⣶⡺⠟⠉⠀⠀⠀⠀⠀⠀⣀⣴⣾⣷⣶⡶⣿⣿⡟⠿⠻⣷⠀⠈⠳⣌⢂⠀⠉⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⢸⣁⣀⢀⣀⠀⠀⠀⠀⣰⡿⠿⠊⠋⢻⣿⣿⢸⣿⢂⢀⢡⡀⠀⠀⠈⠈⢢⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠺⠛⠛⠉⠁⠀⢀⣤⣼⣯⡀⠀⠐⠀⡀⢻⣟⣾⠻⣧⠠⠀⠡⠡⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠈⠀⣆⠀⠀⠀⠀⢀⣼⣿⡿⣿⣷⣄⠀⠀⠀⠈⠉⢉⠣⡘⣧⠠⠠⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⠤⢤⣌⢳⠀⠀⠀⣾⣿⠉⠀⠉⢿⣝⣆⠀⠀⠀⠀⠀⠀⠁⠘⢧
⠀⠀⠀⠀⠀⠀⠀⢠⢟⡆⠀⢰⣏⡏⠀⢀⣴⣿⡉⢆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⢀⡟⣸⠀⢠⡟⣸⢀⣶⡿⡟⡩⢷⡱
⠀⠀⠀⠀⠀⠀⣸⢣⠃⠀⠈⣹⣷⣾⣿⡄⠘⡓⡀⠧
⠀⠀⠀⠀⢀⣤⣿⣶⣶⣶⣿⣿⡿⠛⣿⠙⡄
⠀⠀⠀⢀⡾⣿⣿⣿⢂⠙⢝⣿⠣⠨⡿⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣖⣍⣵⣽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣚⠕⢛⣡⠖⠋⠉⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⠙⠿⣝⠢⢿⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢮⠖⣁⡴⠋⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠻⡑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢃⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠹⡽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡻⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠿⠿⠛⠛⠿⢿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠻⡾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣶⣤⣤⣄⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⣀⣠⣤⣴⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠛⠛⠉⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠙⠛⠻⠿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠙⣷⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢽⣧⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⡏⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⡀⠀⠀⣀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠹⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢣⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⢧⠀⠀⠈⠙⠦⣄⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⡤⢖⡩⢴⡯⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣳⠤⡤⣠⢤⠤⣭⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⡾⢿⣾⡭⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠙⠛⠛⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣊⠭⠤⠖⢺⠀⠀⡟⠳⠦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠉⢣⠀⠀⠀⠈⠢⠔⠁⠀⠀⠀⠀⠈⡱⠢⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⢤⡶⠋⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠙⢯⡓⠒⠤⣄
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⣧⡠⠤⠤⠤⠤⢄⣀⠀⡰⠁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠇⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⢣
⠀⠀⠀⠀⠀⡇⠀⢀⣎⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⢱⡀⠀⢸
⠀⠀⠀⠀⠀⢇⠀⡜⠀⠑⢤⡀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣀⡠⠤⢧⠀⢸
⠀⠀⠀⠀⠀⢸⣠⠇⠀⠀⠀⠙⠦⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠊⠉⠁⠀⠀⠸⡄⡜
⠀⠀⠀⠀⢠⠚⠛⠒⠢⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠔⠒⠓⠓⡄
⠀⠀⠀⠀⠘⠤⣤⠤⠔⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠢⠤⡤⢤⠃
⠀⠀⠀⠀⠀⢰⠟⡆⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠤⣀⡀⠀⠀⢰⠃⠘⡆
⠀⠀⠀⠀⠀⢸⠀⢣⠀⢀⡠⠴⠒⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠓⠦⡜⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠈⢏⠁⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⡜
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⡝⠑⠒⠒⠒⠒⠊⠹⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣠⠞⢁⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⡀⠀⠙⠢⢄⣀⡞⠀⠀⠀⢀⠔⠢⡀⠀⠀⠘⡆⢀⡠⠔⠛⠓⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠲⠬⣉⡒⠒⠦⢼⠀⠀⣧⡤⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠚⠤⠤⠋

⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⢀⣀⣀⠀⠀⣼⣿⣿⣷
⠀⠀⠀⠀⠀⣀⡻⣿⣿⠟⣁⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣤⣬⣛⣛⣥⣤⠀⠀⠀⣠⣬⣛⣟⣫⣤⠰⣿⣿⣿⡆⣠⣬⣻⣟⣫⣄
⠀⠀⠀⠀⣾⣿⣿⣶⣶⣿⣿⣷⠀⠀⠀⣰⣶⣭⣿⣵⣶⡽⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⢿⣵⣽⣛⣯⣶⣝⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠿⠿⠿⠿⠛⠁⠀⠀⠻⠿⣿⣿⠿⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⠉⠛⠿⠿⠿⠟⠛⠉⠀⠀⠀⠙⠛⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣦⠀⠀⠀⣴⣿⣶⠀⠀⠀⢰⣾⣷
⠀⠀⠀⠀⠀⠀⣶⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣴⣽⣿⣵⡄⠀⢠⣮⣿⣯⣦⠀⢠⣬⣿⣯⣦⡀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⣼⣿⡄⢠⣼⣿⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣺⣿⡛⠛⣻⣿⡛⠛⣻⣿⡀⠀⠀⠀⠀⠉⠉⣵⣾⣽⣷⣮⠉⢹⣾⣽⣷⣾⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡗⠸⣿⣿⡿⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠀⠸⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡲⠉⠉⠉⠑⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⣀⣤⢴⣤⣤⣀⡀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣰⠋⠀⠀⠀⠈⢻⡕⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠇⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠘⢷⣄⣀⣠⣴⠟⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠈⠉⠉⠀⠀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠀⡀⠀⠸⠷⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤⣔⠒⠚⠛⠭⠉⠀⠀⢀⣀⣀⠀⠀⠀⠊⠉⠉⠛⠓⠲⠤⠤⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⡏⠉⠀⠉⠉⠉⠉⠑⠒⠶⠀⣀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠄⠤⢀⣀⡴⢾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠙⠧⣄⡠⠦⠤⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠦⠶⢦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⢀⡽⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣄⠀⠀⠀⠀⣀⣤⣶⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣾⡾⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣶⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠠
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⢿⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡇⠀⠐
⠀⠀⠀⠀⠀⠈⣷⣶⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸⠇
⢀⡀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀
⢸⠉⠲⡃⠀⠀⠀⢻⣿⣿⣿⣿⣿⠿⠛⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠾⡿⠃⠀⠀⣠⠖⠉
⢸⠀⠀⠁⠘⢶⣤⡈⢿⡿⠛⠉⠀⠀⠀⠀⠉⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠁⠀⠀⠀⠀⠀⠀⢀⡴⢀⣠⣶⠟⠁
⠈⡰⠀⢹⣦⣈⠻⣿⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣠⣾⡗⣿⣿⣃⣴⡟⠀⢆
⠀⢣⢧⡄⢻⣟⢷⣮⠉⠟⠛⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⡀⠀⣠⠾⠛⠛⠓⣭⡟⣿⡟⢠⡸⡜
⠀⠈⢎⢿⣷⠙⢦⡉⠳⣦⣀⠀⠈⠻⢦⣄⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡷⠚⠁⠀⣀⠄⠟⢁⡼⠛⣾⡿⢱⠁
⠀⠀⠘⣾⣿⠀⠀⠻⣦⡀⠉⠛⠶⣦⣤⣬⣽⣿⢍⢤⣤⣀⣀⣀⣀⣀⣨⣿⡿⣿⢿⣯⣥⡤⠴⠶⢛⠁⢀⣴⠟⠁⠀⡿⣧⠃
⠀⠀⠀⠈⠧⣵⠀⠀⠈⠙⢶⣄⡀⠀⠀⠉⠉⠉⠀⠀⠂⠄⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⠤⢚⣡⡶⠛⠁⠀⠀⣬⠼⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢂⣥⠶⠋⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠠⡀⠀⢀⠀⠀⠀⢀⣠⠔⢋⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣮⡓⢼⣦⣠⡴⠋⣡⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠋⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣽⣿⣷⣞⣩⣯⣹⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⡐⠲⣠⠞⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡴⠊⢉⡞⠁⢦⡄⠠⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⢧⡀⢸⣄⡀⢠⠇⣰⠀⣙⠓⠒⣄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠉⡂⢸⠀⠘⣦⡈⠓⠈⠹⡆⠹⢤⣀⡀
⠀⠀⠀⡜⠁⠀⠐⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⡼⢠⡋⠀⣠⠚⣹⠛⠲⣄⠈⠀⠀⣀⡇
⠀⠀⠀⢧⣀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠙⠚⠁⠀⣇⡀⠀⠈⢧⠉⠉⠉
⣀⣀⣀⣀⡉⠙⠋⠁⢀⣀⣀⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⣰⠛⣆⠀⠘⡇
⠁⠀⠀⠀⠉⠙⠛⠋⠉⢁⣀⠀⠀⠀⠀⠀⠀⡠⠶⠒⠚⠛⠛⡻⠛⠛⠒⠛⢂⡿⠀⢀⣇⡀
⠀⠀⠠⠒⠤⢤⣄⣤⠞⠉⠁⠀⠀⠀⠀⠀⠀⢆⡀⠀⠀⢠⢎⠄⠀⢀⠴⠚⣉⣀⣤⢞⠅⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠓⠿⠶⢋⣠⣮⣍⣁⣠⣶⣋⣀⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠙⢢⣀⡴⠖⠒⠒⢆⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⡀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⡄⢳⡀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⡴⠋⠀⠀⢳⡄⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⡾⠁⠀⠀⠀⠀⠹⣆⠀⠀⠘⠿⣦⣄⡴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠉⠉⢀⡟⠀⠀⣾⠋⠉⠛⠓⠂⠀⠀⠀⠀⠀⠉⠛⠛⠶⠶⣤⣄⡈⠻⣄
⠀⠀⢀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢌⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡣⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢀⡀⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠐⠢⣛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠛⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⡰⠋⠀⠀⠈⠙⠲⣥⣁⣒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠤⠤⠤⠤⠠⣄⠀⠀⠀⠀⢼⠃⠀⠀⠀⠀⠀⠀⠀⠢⡈⠙⠢⡑⢄
⠀⠀⠀⠀⠀⠀⣐⣠⠠⠖⠒⠲⠖⠤⣀⠀⠀⢸⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠌⡀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡀⠀⢹⠈⡆
⠀⠀⠀⠀⠀⠀⣹⠏⠀⠀⠀⠀⠀⠀⠈⠣⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣠⣀⠄⠠⠄⢄⣀⠂⠐⠀⠘⠀⡇
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⡘⡐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⡈⠁⠓⣐⡔⠁⡜
⠀⠀⠀⠀⢠⠃⢀⠀⠀⠀⠀⠀⠀⢳⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠤⣀⠀⠀⠀⠀⢸⣦⡞
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢳⡄⠀⢁⡟⠉⠃
⠀⠀⠀⠐⢸⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢿⣠⡞⠁
⠀⠀⠀⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡕⠂⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠐⠄⠄⠄⢀⣼⣟⣤⠔
⠀⠀⠀⠰⡀⠣⠀⠀⠀⠐⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠐⣈⠔⡟⣱⠟⠁
⠀⠀⠀⠀⠈⠢⣌⣐⠂⠀⠠⠑⢀⠀⠃⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠊⠀⠈⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠢⢤⣉⡞⠓⠤⣀⡀⢀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⠀⠀⠄⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣆⠀⠀⠈⠉⢳⡂⠠⢀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠈⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠐⠀⠋⠩⢁⠶⠬⢍⣐⡒⠢⠤⠄⣀⣀⣠⠄⠘⠁⡠⠊⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠛⠒⠖⠤⠤⠲⠒⠖⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠾⠻⠶⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠶⠟⠷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣴⣷⣮⣙⠿⣿⣿⣿⠿⠛⠛⠛⠙⠛⠻⠿⣿⣿⣿⠿⣋⣵⣿⣦⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⣤⣝⠻⣿⣿⣷⡮⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢵⣾⣿⣿⠟⣫⣦⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣸⣿⣿⣿⣦⣝⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣫⣴⣿⣿⣿⣇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢹⣿⣿⣿⠟⣫⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣝⡻⣿⣿⣿⡏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠻⣋⣵⣿⣿⡿⢗⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡺⢿⣿⣿⣦⣝⠛⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠻⣿⢟⣫⣶⣿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣶⣿⣿⣿⣷⣍⡻⢿⠟⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣴⠶⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠶⣦⡶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠈⢳⠀⠀⡠⠄⢿⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠸⡆⠉⣹⠀⡜⠀⠀⠈⡦⠊⠈⠱⡀⠀⢠⠊⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠐⠻⠖⠁⠀⠈⢢⠀⢰⠀⠀⡀⠀⡇⠀⣊⠑⣠⣎
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⢢⠀⡀⠀⠀⠀⣄⠀⠳⡏⢀⠀⡇⢠⠃⡔⠞⠉⠀⣁⠀⢀⠐⠂⠲⡄
⠀⠀⠀⠀⡠⠤⢤⣸⠤⡴⠁⠇⠁⢄⠀⠀⡏⡧⡀⠱⡟⢠⠇⡼⠀⡧⠂⢀⠀⡏⠀⢘⢄⣀⡀⡇
⠀⠀⠀⢸⠀⠀⢠⢻⢢⢨⠏⠀⣠⣀⠑⠲⢇⡇⠱⣀⡅⡘⢀⡧⠊⣣⢔⣋⣰⠥⠆⠀⣉⠈⣨⠁
⠀⠀⡔⠉⢂⣤⠎⣤⣀⡉⠀⠠⣇⠀⠑⠢⠴⣑⣤⣶⡿⠟⢍⠉⠭⠤⠳⠤⢤⠤⠄⣼⡀⠀⠈⡆
⠀⢸⠁⠀⠀⠀⠀⢀⡨⠐⠒⠛⠊⠀⠀⠈⢉⠽⠫⡀⠕⢶⡰⣉⠒⠧⢀⣀⡎⠦⣀⣡⡭⠦⢼⠃⠈⢆
⠀⠈⢇⠀⠀⠀⢀⡎⠀⠀⣼⡤⠤⣔⡲⠚⣣⠊⢠⢃⠘⡆⠉⠊⠑⢦⡀⠈⠉⠉⣵⠋⠐⢄⠀⠀⠀⡸
⠀⠀⠈⢆⠀⢰⢡⢣⠖⢉⠭⠔⠈⠁⣠⡞⡡⢤⠃⠈⡄⠘⡄⠀⠘⣄⣙⣯⣐⠒⣿⡄⢀⣸⡄⡀⣰⠁
⠀⠀⠀⠘⡗⢶⢸⡇⠀⠀⠀⠀⣠⢊⠎⡜⠀⠀⡇⠀⢱⠀⠘⢄⢰⠋⠀⠈⡆⠀⠀⠀⠀⣇⡇⣧⠇
⠀⠀⠀⠀⡇⠈⠄⢷⣄⠀⠀⠀⣧⡇⠀⢧⣀⣠⠇⠀⠈⡆⠀⠈⠛⣦⣀⡴⡃⠀⠀⠀⠀⠉⢀⡏
⠀⠀⠀⠀⡇⠀⠰⡆⢹⠓⠂⠉⢸⡇⠀⠀⠉⠒⠈⠐⢄⢸⡀⠀⠀⠈⠉⠀⠑⢦⠀⠀⠀⢹⠋⠇
⠀⠀⠀⠀⠃⠀⠀⠀⡼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠸


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⣒⣂⠰⠦⢞⣒⡦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡊⠅⠂⠉⠁⣀⣀⡀⠀⠀⠈⠉⠒⠮⣑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⣨⣿⣭⣾⣷⣿⠉⠁⠀⠀⠀⠀⠈⠉⠒⠠⢀⠀⠈⠑⡌⢦
⠀⠀⠀⠀⠀⠀⠂⠕⠉⢀⢼⣿⡛⠁⠁⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠘⡎⢆
⠀⠀⠀⢀⢔⡔⠁⠀⠰⠋⠀⠙⢄⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡸⡘⡄
⠀⠀⢀⣠⠊⠀⠀⠄⠀⠀⠀⠀⠀⠻⣆⣠⠴⠟⠒⠛⠒⠒⠲⢄⡀⠀⠀⠀⣀⠤⢺⣷⣿⣿⣄
⠀⠀⡘⡈⠀⠀⡘⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡤⠞⠁⠀⠀⢹⣿⣷⡛⣄
⠀⠀⣟⠀⠀⠀⠁⠀⠀⠀⠀⠀⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢀⠀⠀⢀⣷⣿⠓⢎⣆
⠀⠀⡞⡀⠀⢀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠋⠉⠉⠉⠁⣽⠀⠈⣆⡄
⠀⠀⢰⣱⡀⢸⠁⠀⠀⠀⢀⣠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣼⠀⠀⠸⠖
⠀⠀⠀⠳⡳⣼⢶⡞⠉⠉⠉⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⡃
⠀⠀⠀⠀⠘⣿⣯⣷⡀⠀⢀⡴⠞⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠁⠀⠀⠀⠀⠀⡸⠀⠀⠠⠰
⠀⠀⠀⠀⠀⠹⡛⣾⢿⡤⠊⠀⠀⠀⠀⠈⠑⠢⠤⣤⠤⠤⢾⠋⠁⠘⠓⢤⡀⠀⠀⡴⠁⠀⡰⡡⠁
⠀⠀⠀⠀⠀⠀⠱⡘⡝⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⢀⣽⣦⠞⠁⢀⢐⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢳⠹⡀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣤⣴⣾⣿⡇⠀⠚⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣙⢄⡀⠈⠒⠠⠀⣀⣀⣀⣀⣀⣀⣠⡤⠼⡿⢏⠵⠛⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠬⢑⡦⢤⢀⣀⠀⠈⠉⣉⡡⠤⣒⠭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠯⠩⠣⠐⠒⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣕⣯⣭⣭⣭⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⢴⣿⣿⠇⠀⠀⠀⠀⢨⣉⣉⡅⠀⠀⠀⠀⠀⠰⣾⡯⢀⣠⡀
⠀⠀⣠⣞⣵⡿⣻⠏⠀⠀⠀⠀⠀⡸⢻⡟⡆⠀⠀⠀⠀⢠⣤⡄⠀⠈⠛⠓⠀⠀⠀⣠⣶⣶⣶⣊⣽⣦
⠀⠀⣏⠿⣯⠞⠁⠀⠀⠀⠀⠀⢠⡋⠤⠄⢻⡀⠀⠀⠀⠈⠛⠁⢰⣾⡦⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠉

⠀⠀⠀⠀⢀⣤⣶⡟⠀⠀⠀⠀⢸⣷⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠆⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⢀⣴⣿⣿⠟⠀⠀⠀⠀⠀⠼⠿⢿⠿⠿⠄⠀⠀⠀⠀⣠⣾⣿⣿⠁⠀⠀⠀⠀⠀⢈⡿⠿⠍⠁
⠀⠀⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⣘⣉⣻⣋⣛⡀⠀⠀⠀⠀⠙⠿⡿⠟⠀⠀⠀⠀⠀⠀⠂⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉

⠀⠀⠀⡀⠀⠀⡀⢠⠆⠀⠀⠀⡞⠉⠉⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠇⠀⠀⠀⠃⡞⠀⠀⠀⠀⡇⠘⠒⠡⠃⢸⠀⠀⠀⢠⣴⣶⣿⣿⣿⡟⠀⠀⠀⠀⡐⢚⣿⡓⠂
⠀⠀⠀⠰⣤⡤⠞⠁⠀⠀⠀⠀⣇⣀⣉⣁⣁⡽⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠄⠥⠤⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⣠⣶⣶⣦⠀⠀⠀⠀⠀⠠⠀⠀⠀⠢⡀⠀⠀⠀⠀⣀⣀⠀⣀⣀⠀⠀⠀⠀⠀⣠⡀⣄⢀⣄
⠀⠀⣼⣿⣿⣽⠮⠀⠀⠀⠀⠀⠀⢀⣴⠿⠁⠀⠀⠀⠀⢸⣿⡏⣭⡉⣿⡇⠀⠀⢀⠞⠛⡟⠿⢻⠛⠳
⠀⠀⣿⣿⣿⣿⣦⣤⠀⠀⠀⠀⢀⡈⣉⠀⢀⠀⠀⠀⠀⠘⠿⠷⠾⠶⠿⠇⠀⠀⠀⠓⠒⠁⠀⠈⠂⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢒⣲⣾⣿⣶⣶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣫⠾⢯⠀⠀⠀⠀⣩⠿⢮⣳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⣠⠞⠁⠀⠈⣧⠀⠀⢰⠇⠀⠀⠙⢯⡙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣼⠙⠦⣄⣠⡞⠙⠉⠉⠛⠓⣄⣠⡴⠚⢳⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣯⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⣀⡤⠾⡇⠀⠀⠀⠀⢀⡷⠤⣄⡀⢸⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⢿⡁⠀⠀⠉⣿⠓⠒⢻⡋⠀⠀⠀⡽⣯⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢽⣦⣀⡴⠃⠀⠀⠈⢳⣀⣠⠞⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠙⠳⠦⣤⣤⡤⠖⠛⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⡤⠟⠛⠙⠒⠚⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⣦⠖⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠤⠒⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢈⡭⠊⠙⠶⣀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠙⠐⠒⢬⡂⠄
⠀⠀⠀⠀⠀⢀⢈⢀⡠⠤⢄⢀⡀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⡆⠆
⠀⠀⠀⠀⠀⢛⡟⠀⠀⠀⠀⠀⠈⠢⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡀⠀⡀⣀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⢀⢃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠐⠉⠳⣀⠄⢠⠃
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⢄⡀⠀⠀⠀⣿⣴⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⣠⠏⠉
⠀⠀⠀⠀⠁⠀⠈⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣴⠋
⠀⠀⠀⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡳⠔
⠀⠀⠀⠐⢄⡀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⣀⡓⠤⠦⣀⠀⠀⠌⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢄⠀⠈⠑⠓⡦⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠐⠉⠑⠲⠬⢝⣐⠐⠤⠤⢀⣀⡀⠄⠘⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠲⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠣⠤⠔⢑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠒⠁⠀⢀⡀⠀⠈⠢⠤⣄⣀⠖⠤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡔⠁⠀⠱⠀⠀⡴⢪⡇⢸⡱⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉⡀⠀⠘⠤⠤⠊⠀⠀⠳⣄⣘⠖⢤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⢈⡆⠀⣀⡀⠀⡴⢊⡶⠝⠀⠁⠉⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠕⠈⠉⠁⠘⠵⢮⠀⠈⠆⠂⢀⠅⠀⡎⢉⠆⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⡐⠉⠑⠀⠀⡎⠀⠀⠀⠀⠈⠑⡄⣈⡁⡠⠬⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠤⠀⢠⠃⠀⠐⠤⠔⠁⢀⢉⠆⠀⠀⠀⢀⡀⣍⠀⠑⠁⠀⠀⠀⢠⠤⠋⠑⠊⡆
⠀⠀⠀⠀⠀⡌⠜⡠⡾⡍⢑⠀⡤⠤⣀⡅⢔⢄⠀⠀⠀⠘⣃⡬⠀⠀⠀⠀⠀⠀⡴⠀⢎⢉⠆⢨⠃
⠀⠀⠀⠀⠀⢣⠑⢴⠁⠟⠀⠉⠀⠀⠀⠀⠈⠪⡒⠀⠐⠂⢸⡆⠒⠤⣀⣀⢔⡲⠀⢇⠤⣀⡔⠚⢠⢖⣄
⠀⠀⠀⠀⠀⠀⠱⡀⢸⢈⡑⠠⢐⡩⠃⠀⠀⠀⠀⠀⠘⠀⢸⠁⠘⠐⠒⠒⠁⠀⠀⠀⢔⠤⡀⠀⠳⣰⡃
⠀⠀⠀⠀⠀⠀⠀⢃⡼⢠⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠆⡄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠨⠭⠥⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡳⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢐⠠⡈⢢⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⣉⡡⠂
⠀⠀⠀⠀⠀⠀⠀⡠⢃⠄⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⡄⢈⢹⠰⡄
⠀⠀⠀⠀⠀⠀⣎⠔⠁⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠘⡈⠢⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠃⠀⠈⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣀⣂⠲⣐⣂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⠪⠓⠊⢉⣀⠀⠠⠀⢀⣈⡉⠚⠕⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣮⣊⠥⠒⠉⠉⢱⠚⠛⠉⠉⠛⢲⠊⠉⠑⠪⢳⣭⡢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡲⣧⠞⠋⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠈⠛⢮⡵⣲
⠀⠀⠀⠀⠀⠀⠀⡠⣢⡟⠪⡢⡀⠀⠀⠀⠀⠀⠀⣹⣶⠶⢶⣾⡁⠀⠀⠀⠀⠀⠀⡠⡪⠚⣧⡓⣄
⠀⠀⠀⠀⠀⠀⢜⣽⠋⠀⠀⠈⠪⡢⣀⣠⠔⠊⠉⠀⠀⠀⠀⠀⠈⠉⠒⢤⣀⡠⡪⠊⠀⠀⠈⢫⠪⡣⡀
⠀⠀⠀⠀⢀⢎⢾⠃⠀⠀⠀⠀⠀⡸⠮⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠾⡀⠀⠀⠀⠀⠀⢣⠑⡱
⠀⠀⠀⠀⢆⠃⡏⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⡇⢡⢃
⠀⠀⠀⠸⡆⢰⣱⠒⠶⠤⣄⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣠⡤⠤⠖⢲⣰⠀⡵
⠀⠀⠀⡇⠇⠌⡟⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠀⠀⠈⡏⠀⠇⡇
⠀⠀⠀⠇⡆⢠⣷⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⢰⣧⠀⡇⡇
⠀⠀⠀⢰⠅⠘⡘⠒⠚⠉⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠉⠙⠒⠚⡘⠀⣣
⠀⠀⠀⠈⡜⡄⢇⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⢀⠇⡜⠎
⠀⠀⠀⠀⠘⡱⡈⢆⠀⠀⠀⠀⠀⡸⣫⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡻⡀⠀⠀⠀⠀⢀⢎⣜⠜
⠀⠀⠀⠀⠀⠈⢪⡲⣧⡀⠀⡠⡪⠊⠀⠈⠑⠢⢄⣀⢀⣀⣀⢀⣀⠤⠒⠉⠀⠈⠪⡢⡀⠀⣠⡿⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠁⢊⡷⣪⠊⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⠛⡄⠀⠀⠀⠀⠀⠀⠈⢪⡲⣯⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠫⡻⣦⣀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢷⠀⠀⠀⠀⢀⢠⡾⣫⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢝⢍⡒⠦⢄⠼⢶⣶⣤⣴⣶⠾⢄⠤⠒⢊⢵⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠫⢖⡤⢄⣀⣀⣀⣀⣀⡀⢤⣐⠬⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠂⠁⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⢉⣩⣿⣶⣶⠐⡆⢲⣶⣿⣍⡉⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⣴⣾⣿⣿⣿⣿⣿⠂⡗⢸⣿⣿⣿⣿⣷⣦⡀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠠⣇⣼⣿⣿⣿⣿⣿⣿⣿⣦⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⣾⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣷⡄⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠒⠒⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⣭⣭⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣶⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡌⢻⣿⣿⣿⣿⣿⣿⣿⡶⢲⠒⣶⣿⣿⣿⣿⣿⣿⣿⡿⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠙⠻⣿⣿⣿⣿⣿⡇⢸⠀⣿⣿⣿⣿⣿⣿⠟⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣤⣭⣿⡿⠿⣧⣸⣄⣿⠿⣿⣿⣭⣤⡴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣒⣯⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⡀⠠⠐⠒⠊⠉⠙⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠀⠡⠤⢤⡄
⠀⠀⠀⠀⡠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡣⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⡼⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢱⣤⠀⠀⠀⢀⡇⠀⠀⠀⠀⠘
⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣢⠄⠀⠰⠈⠛⠒⠒⠛⠁⢀⡀⠀⠀⠀⠀⠀⠉⠰⡄
⠀⠀⠀⠀⠠⠀⠀⠄⠀⠒⠒⠂⠀⠈⢹⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠊⠈⣉⡀⠰⠂⠀⠀⠀⠀⠉⠂⢤⡀⠀⠀⠠⠞
⠀⠀⠀⠀⠀⠄⠀⠀⠐⠂⠐⠒⠒⠒⠚⠰⢾⡦⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⢒⠒⠓⣚⢙⠁⠐⠒⠒⠒⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⣞⣢⣄⣰⠀⢧⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⠀⠀⣇⣨⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠑⠄⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠶⠒⠰⠢⠄⠗⠐⠐⠀⠀⠐⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⡆⠘⠠⢤⣀⣀⣀⠠⠤⠤⠄⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡰⣇⠀⠠⣔⠉
⠀⠀⠀⠀⠀⠀⡀⠀⠆⠁⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣇⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⡋⢀⡠⠒⢠⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⡿⠁⠴⠁⠀⠀⢣⠈⢹⡄⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⠀⡟⡘⠀⠀⢸⣦⠁
⠀⠀⠀⠀⠈⡇⠀⠁⠀⠀⠀⠈⠀⢦⡃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣸⣇⠃⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⡇⠀⠁⡆⠀⠀⠀⠀⢸⡁⢸⣄⡀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⡤⡄⠀⠃⢉⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢇⠀⢧⡟⣉⣉⡀⠀⢸⠀⢸⠆⠀⢠⣭⣉⠈⠹⠷⢤⠀⡁⠀⠄⡁⢀⠔⠁⠉⠉⠈⠉
⠀⠀⠀⠀⠀⢸⡀⠈⢀⡀⣸⠄⠀⡎⠀⢸⠃⠈⠉⠀⠀⢀⣀⣀⠸⠀⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⡘⠡⣄⠀⠉⠁⢄⣸⠁⠀⢸⠆⠀⠒⠉⠉⠀⠀⢸⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠆⠀⢸
⠀⠀⠀⠀⠀⢇⠀⠉⣷⡂⠒⡙⢹⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⢰⠀⠀⠀⠈⠁⠒⡶⡖⠚⡿⡇⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠓⠋⠀⠀⠀⠀⠀⢰⠀⡇⢀⠃⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠦⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠈⠀⠉⠉⠒⠀⢄⠀⢌⠘⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠒⠘⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⠐⠐⠒⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠘⠛⠛⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣆
⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠒⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣶⣾⡄⠀⠀⠀⠀⠀⠀⠀⢾⣧⠀⠒⢿⡷⠂⠀⢀⣼⡿⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡶⣶⣾⣾⣏⣇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠈⠀⠀⣴⠛⠋
⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⣻⣿⡏⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⠀⠰⠇
⠀⠀⠀⠀⠀⠀⡟⢹⣷⣶⣶⣿⣼⣧⣤⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣤⣤⣤
⠀⠀⠀⠀⠀⢸⠇⢸⡇⠀⠀⠀⢸⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡿⠀⢸⡷⠶⠶⠶⢾⡷⠶⠶⣧⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠇⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⣿⡿⢩⣿⣿⠁
⠀⠀⠀⠀⡿⠀⠀⢸⡟⠛⠛⠛⠻⡟⠛⠛⠛⣧⠀⠀⠀⠀⠀⢸⣿⡇⠀⢿⣿⠀⠀⠀⢀⡀
⠀⠀⠀⢸⠇⠀⠀⠸⠇⠀⠀⠀⠀⡇⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⣿⣷⠀⣾⣿⡆⠀⣰⣿⣿⡄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢹⣿⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣸⡇⠀⢸⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⠃⠙⠃⠀⢸⡿⢿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⣤⣔⣲⠽⠾⠾⠿⣬⠁⠀⠈⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡿⠔⢋⣩⠥⠖⠒⠯⠋⠉⡇⢄⠀⠂⣕⠀⡀⠀⠀⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⠔⢉⡤⠞⠉⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠿⠠⢥⡒⢍⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠣⣯⣴⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢦⡙⢎⢦
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣺⠃⠀⠀⠀⠠⢀⡀⠄⢂⢤⣤⣴⣦⣀⣀⠀⠀⠀⠀⠀⠀⡐⠁⠉⠶⢽⣎⢿⣇
⠀⠀⠀⠀⠀⠀⢰⠃⠀⢈⠓⠀⠀⠀⠀⠀⠀⡙⣲⠗⠋⠉⠀⠀⠈⠉⠙⠒⠦⣀⣀⠞⠁⠀⠀⠀⠠⡜⣿⡜⣆
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠂⠀⠀⠀⠀⠀⠀⡔⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣆⠀⠀⠀⠀⠀⠈⢺⡜⣞⡆
⠀⠀⠀⠀⠀⠸⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠆⣇⢹⢷
⠀⠀⠀⠀⠀⠀⣷⡟⣧⠈⠀⠀⠀⠀⢠⢓⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠂⠂⢀⡀⠀⢰⣸⢸⣾
⠀⠀⠀⠀⠀⠀⡯⡇⢿⠇⠀⠆⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠠⠩⢻⣼⢿⡀
⠀⠀⠀⠀⠀⢸⣿⡇⢸⡶⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⡇
⠀⠀⠀⠀⠀⠘⡏⣇⠈⢿⢀⡀⠀⠀⠀⠀⠘⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⣽⠁
⠀⠀⠀⠀⠀⠀⠸⡜⣄⠘⢧⡐⠂⠀⡀⠀⠊⠀⠊⠑⠢⠤⣀⣀⡀⢀⣀⡤⠶⠋⠐⢄⠀⠀⠀⠀⢠⠄⠀⠀⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠙⣎⢦⡈⢷⡸⠇⡔⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⡉⠁⠀⠀⠀⠀⠀⠈⠒⠤⡀⠐⡁⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢯⡓⢽⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠤⣨⢞⡽⡹⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⠓⠢⠦⡀⢀⠀⠀⠀⠀⢠⢀⠠⠀⠀⠄⡨⢖⣢⠴⠊⣱⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠑⠋⠓⠬⣦⡤⠼⡧⠴⠦⠗⠚⠉⣠⣴⡿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣐⣀⡿⣶⣦⣤⣀⣀⣠⣔⡯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠃⠉⠁⠁⠁⠉⠉⠚⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠉⢀⠀⠠⠀⠀⠀⠄⠀⠀⠠⢀⡀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣚⣤⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣬⣒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣲⣶⣶⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⣶⣒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠈⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢡⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⠿⠀⠘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠸⠀⠀⠀⠉⠙⠛⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠛⠋⠉⠀⠀⠀⠇⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢰⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠸⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠰⠀⠀⣀⣠⣤⣴⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⣤⣀⣀⠀⠀⠆⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣄⠀⢷⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡾⠀⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⢃⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠠⠤⠀⠀⠤⠄⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠛⢻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠟⠛⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣙⠻⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠟⢋⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠉⠁⠀⠒⠒⠒⠒⠂⠈⠉⠀⣀⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡤⣀⣀⣀⣀⡤⠴⠒⠉⠁





⣿⡟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠛⠛⠛⠛⠿⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣵⣶⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⣶⣮⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣀⣀⣀⣀⣀⣀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢡⣿⠋⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠙⣿⡌⠻⢿⣿⣦⡀
⠀⠀⠀⠀⣠⣿⣿⠏⠀⠀⣿⡇⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⢹⣿⡀⠀⠙⣿⣿⣆
⠀⠀⠀⣰⣿⡿⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⢿⣿⣆
⠀⠀⠀⣿⣿⠃⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠘⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⠆
⠀⠀⠀⣿⣿⡆⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠸⣿⣿⡄⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠙⣿⣿⣦⡀⠀⢿⣇⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⣸⡿⠀⢀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣘⣿⣆⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣰⣿⣃⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠉⠉⠉⠉⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡟⠿⢷⣶⣤⣤⣤⣀⣀⣀⣀⣤⣤⣴⣶⡾⠿⢛⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠉⠉⠉⠙⠛⠛⠋⠉⠉⠁⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣶⣤⣤⣶⣶⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠛⠉⣈⣉⡉⠲⠤⣄⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⡰⠖⠉⠀⣀⣀⣭⣶⠶⠶⠾⣿⣿⣭⣿⣽⣟⣛⣿⣷⣶⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠋⣀⠝⠉⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣽⣿⣾⢷⣤⣘⣀⠉⢒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⡼⠀⡠⢁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠙⢿⣿⢿⣿⣿⣿⣿⠀⠈⠙⠳⣬⡁⠃⠤⡀⠗⠢⣀
⠀⠀⠀⠀⠀⢘⠀⠸⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣶⣌⣡⣄⡛⡉⠛⡇⠀⠀⠀⠀⠀⠙⢷⣄⠈⠑⠄⣈⠳⡄
⠀⠀⠀⠀⠠⠃⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠙⢄⠈⣆
⠀⠀⠀⣠⢥⣀⣿⣁⣀⠤⢀⣀⠀⠀⠀⠀⣀⠤⠒⠂⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠘⠀⢸
⠀⠀⢠⡿⣾⣿⣿⣿⣿⠀⢠⣿⣿⣦⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⡇⢰⡆
⠀⠀⣾⣻⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡠⠀⡇
⠀⢸⡟⣼⣿⣿⣿⣿⡏⠁⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⣠⣴⣖⠒⠐⣶⣶⣾⣷⣶⣧
⠀⣈⠈⢹⡏⠉⠉⠀⠀⠀⠉⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣭⣄⠈⢿⣿⣿⣿⡯⣿
⠀⠇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣾⣿⣿⣿⡇⡿
⠀⣄⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣏⢠⣿⣿⣿⣿⣧⡇
⠰⡇⠀⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠉⠉⠉⠀⠛⠻⣿⣿⣿⣿
⠀⡇⠀⠁⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏⢙⠟
⠀⡿⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⡠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢀⡞
⠀⢱⡀⠘⡀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠃⡰⠁
⠀⠈⢧⡀⠘⢄⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡀⠈⠉⠹⠻⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢋⠆⢸⠃
⠀⠀⠈⢧⡄⠀⠑⢄⡀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⢰⣿⣾⣷⣿⣦⣶⣶⣾⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠡⠂⢰⠇
⠀⠀⠀⠀⠙⠲⣆⣠⠈⢁⠒⠄⠈⠟⠦⠄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢁⠐⠁⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠲⠶⠗⠦⠵⠤⣤⡀⠀⢻⣿⣿⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⣀⣠⡴⠞⡋⠄⠊⠁⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣦⣽⣻⣻⣟⣛⡟⠒⠛⠛⠛⠟⣁⡀⠉⣀⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠒⠐⠘⠒⠒⠐⠋⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡤⣬⣀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠩⠀⠀⠂⠰⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠠⠀⠀⣀⠀⢐⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠠⠐⠈⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠠⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠰⠁⠀⠐⠊⠀⠈⠀⠈⠁⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⠠⠐⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⡠⠊
⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣽⣽⣽⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣷⣾⣿⣷⣶⣶⣶⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠒⠒⠒⠒⠒⠒⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢚⡟⠁⠀⠀⠀⠀⢀⣠⡴⠚⠁⠈⢧⡀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠰⣶⠒⠒⠒⠚⢫⠍⠀⠀⠛⠒⠒⠒⠒⠒⠋⠉⠀⠀⢠⣀⠀⠀⠹⡄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠔⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢊⠉⠢⡀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢈⠐⣙⣷⡡⠀⠀⠀⠀⠠⡤⣄⠀⡇⠁⣼⣷⡇⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠉⣉⠳⠶⠋⠀⠀⠀⠀⠘⠄⣌⠿⠃⠀⠀⠀⠀⠀⣠⠏⠀⠘⠶⠶⠟⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⣴⠟⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠠⠈⠉⠠⠘⡹⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣸⠇⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⣀⡰⠃⠀⠀⠀⠀⠀⢀⡼⠁⢀⠜⢻⣦⣤⡶⠒⢄⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣀⣴⡯⢴⡞⠁⠀⠸⣷⡜⣷⡀⢸⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⢤⣀⣀⡀⠀⠀⠀⣀⣀⣀⣠⡤⠴⠖⢋⠿⡏⠀⠸⣇⠀⠀⠀⠹⣟⡮⠟⠉⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡇⠀⠀⠀⠸⣅⡀⠀⠀⠀⠀⠸⡄⠱⡀⠀⠙⢷⣤⠤⣀⣙⣷⣄⡀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⡁⠀⠙⢦⡀⠀⠀⠈⠏⢳⣦⣀⠀⠀⠙⠒⠛⠒⡞⠻⡷⡄⠀⠀⠈⠉⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⢱⠀⠀⠀⠉⠢⠤⠚⠀⠀⡆⠈⠳⣄⠀⠀⠀⣠⢧⢀⡫⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠈⠳⠤⠞⠁⠀⣻⢯⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠙⠳⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠘⣦⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⢤⠔⠒⠒⠒⣧⠀⠀⠀⢾⣓⠲⠦⠤⢤⣀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡞⠉⠤⣴⡇⠀⢸⠿⠛⠀⠀⠉⠛⠛⠓⠛⠋⠁⠀⠛⠷⠶⠯⠙⠲⢤⡖⢿
⠀⠀⠀⠀⡇⢸⡆⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⠇⠈⢧
⠀⠀⠀⠀⡇⢸⠇⠀⣣⠀⢸⡀⠀⠀⠀⠀⢀⣀⣤⠤⣤⣀⠀⡴⠿⣛⣩⠵⠋⠁⠀⠀⠈⢧
⠀⠀⠀⠀⢧⡀⠐⠛⢿⠀⠘⡟⠛⠂⠀⠘⠛⣒⣛⣋⣩⠭⠴⠒⠋⣹⢧⡀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣁⡽⠗⠦⢤⣀⡀⠀⣰⠋⠀⠻⣄⠀⣀⣀⣀⣤⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠉⠁⠀⠀⠀⠀⣠⠟⠉⠁⠀⠀⠀⠉⠉⢿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⢠⡶⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⠽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣶⣶⡾⠿⠿⠿⠿⣶⣶⣦⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⢟⣻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣛⠿⣷⣄
⠀⠀⠀⠀⠀⢠⣿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣌⢻⣧
⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣸⡿⠛⠛⠛⠛⠛⠛⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠛⠛⠛⠛⠛⠻⣿⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣶⣶⣶⣶⣶⣾⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿
⠀⠀⠀⠀⠀⠈⢿⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠟⠛⠋⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣕⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣔⣉⡥⠤⠤⠤⢄⣀⡉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠺⣮⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢾⣯⣼⣿⣿⠋⣡⢤⡉⢻⣷⠀⡀⠀⢀⠀⣠⣾⣿⢿⣿⣿⣿⣿⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⢡⡎⠹⢿⣿⣿⠀⠧⡴⠇⢠⣿⣇⣑⣒⡋⣰⣿⢋⡔⠲⣌⠹⣿⣿⣿⣌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⡟⠀⠀⠀⠙⠻⣷⣦⣤⣶⣿⠟⠁⠀⠀⠈⠙⢿⣌⠳⠾⠟⣰⣿⣿⣿⣿⣿⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⠀⢸⠀⠀⠀⠀⠀⠀⠈⢛⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣾⣿⡿⠟⠋⠉⠙⡆⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠋⠁⠀⠀⠀⠀⠐⠃⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡇⠀⠀⠀⠀⠀⠀⢀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢱⠀⠀⠀⣀⣤⣾⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⡀⠀⠀⠀⠀⠀⢨⠇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⢹⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⡤⢞⣿⣿⣿⣿⣶⣄⠀⠀⠀⣬⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠙⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠛⠉⠉⠹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⢁⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢸⠈⢦⣻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠻⣬⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠃⠸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠱⣀⠀⠈⠙⢿⢻⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣯⡵⠞⠉⠀⡠⠃⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡙⡏⠀⠀⠀⠁⢹⣅⡈⠁⠒⢲⣶⠒⠒⢒⡩⡿⠋⠀⠀⢤⡤⠊⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣦⡛⠂⠀⠀⠀⠀⢸⠀⠈⠉⠒⡏⢹⠒⠉⠁⠀⣿⠀⠀⠀⣮⡄⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠃⢸⠀⠀⠀⠀⢹⡀⠀⠀⠀⠷⣋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣸⠀⢸⡇⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡀⠀⢨⡏⠀⠀⣷⠦⠀⠺⢘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠉⠛⠛⣧⠀⠀⡟⡛⠋⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢶⣿⠉⢹⠋⠁⠀⠀⢙⣿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⢲⣝⣷⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣽⣻⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣦⣚⣛⣛⣛⣛⡯⠭⠗⠀⠀⠀⠈⠩⣛⠿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉

⣿⢿⠿⠿⡿⡟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣛⣿⣿⣟⣿⣿⣿⣟⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⢲⠖⠒⠀⢈⡶⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠊⠉⠀⠀⠀⢠⠃⠀⠀⣰⠋⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠊⠁⠀⠀⠀⠀⠀⢀⣀⣤⣾⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣨⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⡠⢐⣢⣴⣾⣷⣿⣿⠿⠦⠀⠀⣹⠀⠀⠀⠀⢀⡤⣾⣿⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠚⠓⠠⢤⣀⣀⢀⣠⣤⣴⣺⡿⠛⠋⠉⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢘⡇⢈⣼⣷⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡮⠤⠤⣤⣄⠀⠀⠈⢿⡿⠟⠋⢡⡝⠀⠀⠀⠀⠀⠀⢀⣠⠚⠁⠀⠀⠀⠀⢀⣸⣷⢦⣽⠟⠳⣤⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⣤⣽⣷⡤⠀⡸⠀⠀⣰⠟⠀⠀⠀⠀⡠⠐⠀⠉⠁⠙⠒⢄⡀⠀⠀⠀⣸⣿⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⣥⠟⢉⣽⠶⠊⠀⣠⠶⠁⠀⠀⢀⣴⠾⠶⠶⢤⣄⠀⠀⠀⠀⠹⣆⠀⢠⢟⡿⠣⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠫⠀⠀⠀⠀⠐⣸⢏⠜⠁⠀⣀⣔⠗⠁⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣀⣿⣿⡷⠊⠀⠒⡇
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⡨⠿⠾⢠⣴⣿⡍⠤⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⠀⠀⠘⠟⠉⠀⠀⢀⣻⠁
⠀⠀⠀⠀⠀⠀⢸⣣⠄⠀⠀⢀⠎⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⡎⠐⢄⡠⣀⠀⠀⢀⢶⣿⡿⣿⡇⠀⠀⠀⠀⠀⣴⡽⠁
⠀⠀⠀⠀⠀⠀⢈⣿⣧⠀⢀⡆⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⡔⠃⡤⢄⢰⣿⡊⣠⡼⠟⠍⠚⠉⠀⠀⠀⣀⣴⠿⠊
⠀⠀⠀⠀⠀⣴⡻⠛⢿⡻⣾⡇⠀⠀⠸⣷⡤⠄⠀⠀⢰⡶⢟⡿⣳⣌⣡⣾⣿⠿⢟⣧⣠⣤⣤⢤⠦⠖⠚⠉
⠀⠀⠀⠀⢰⡻⠀⠀⠀⠙⠻⢻⡀⠒⠂⢻⣶⠶⠞⠠⢀⠴⣫⡞⠭⠐⠊⠉
⠀⠀⠀⠀⣯⢇⠀⠀⠀⠀⠀⠀⠈⠂⠀⠠⠹⢗⡲⣖⠽⠋
⠀⠀⠀⠀⠘⣞⢢⡀⠀⠀⠀⠀⠀⣀⣠⠤⣆⠽⠚⠁
⠀⠀⠀⠀⠀⠈⠓⠼⠥⠶⢰⠦⠬⠄⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠟⠛⠛⠛⠛⠛⠛⠷⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⠞⠉⠁⠀⠀⠙⢷⡀⠀⠀⢀⣠⠤⣤⠀⢀⡀⠀⢀⣹⣧
⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⢰⣴⡄⣴⢀⢳⣸⣷⣤⠞⠉⠀⠀⠙⠊⠁⠀⠀⠈⠉⠹⣧
⠀⣾⠁⠀⠀⠀⢀⠀⠀⠀⠀⢀⣤⠀⢀⡠⠀⣠⣾⢀⠛⠀⠟⣿⠎⡀⠋⣥⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⢸⡏⢀⣠⠖⠋⣏⣀⠠⠂⠁⠀⠓⠒⢉⣠⠴⢷⢽⣿⠀⠺⠀⠀⢄⠛⢰⠂⠀⠉⠉⠙⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⣿
⣾⠗⠋⠀⠀⠀⠀⠁⠀⠀⢀⡠⠖⠉⠉⠀⠀⠘⣚⣆⠀⠀⣈⡓⠾⠃⡼⠓⠒⠲⢤⡴⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢹
⣿⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⢀⣠⠴⠚⠛⢧⡘⠦⣀⠀⠉⢀⣼⠁⣠⠴⠊⠁⠀⠀⣀⣀⠤⢦⣀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⡴⠋⠠⠼⠶⢾⣉⣀⣀⣀⡀⠀⠙⡄⠈⠉⠛⠋⢹⣼⠁⠀⢶⣚⠋⠩⠥⠤⢤⡀⢫⠑⡄⠀⠀⢸
⢻⡄⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⢱⣀⣁⣠⡤⠴⠶⠶⢄⠀⢀⣘⠁⠀⠀⠀⠀⠀⠙⣾⠇⢹⠀⠀⣼
⠸⣧⠀⠀⠀⠀⠀⢈⡗⢲⠖⠒⠂⠁⢤⣀⡇⠀⠀⣯⠀⠚⠉⢉⣁⡠⠼⠒⠁⠀⠙⢦⡀⠀⠀⠀⠀⣿⠂⡾⠀⢀⡿
⠀⢻⡄⠀⠀⠀⠀⡸⡇⡏⠀⠀⠀⠀⣰⠋⠙⠛⠛⠛⠛⠉⠉⠉⠳⡄⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⢠⠏⣼⠃⠀⣼⠃
⠀⠈⢿⡄⠀⠀⡞⠀⢳⡁⣶⣤⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣠⣶⣶⠆⠈⠀⢀⡴⠁⣼⠃⠀⣼⠏
⠀⠀⠈⢻⣄⠀⢷⣀⣀⣽⠿⣄⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⣿⣿⣦⣴⡾⢋⣠⡾⠃⢀⣼⠏
⠀⠀⠀⠀⠻⣦⡀⠉⠉⠁⠀⠳⡉⠓⠾⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⠴⣿⠛⠉⠛⠛⠛⠉⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢀⣀⣤⣼⡿⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣛⣷⣶⣶⣶⣶⣶⣶⣟⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡈⠉⠃
⠀⠀⠀⠀⠀⢀⢀⠁⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠸⠿⠿⣿⣼⠛⠿⠃⢸⣿⠟⠦⠾⠿⠃⠀⠀⢀⣿⣿⣧⠀⠀⠀⢸⣿⣿⡀⠀⣸⣿⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣤⣤⣤⣤⠀⠀⠀⣾⡿⠘⣿⣆⠀⠀⢸⣿⢻⣧⢀⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢀⡆⠀⣡⣿⠀⠀⠀⢸⣿⠉⠉⠉⠉⠀⠀⣼⣿⣷⣤⣿⣿⡄⠀⢸⣿⠘⣿⣼⡿⢸⣿⣿⣏⠘⠃
⠀⠀⠀⠀⠸⡿⠀⣿⣿⢀⡤⠂⢸⣿⣶⣶⡶⣶⡆⣰⣿⢏⢱⣤⠬⢻⣿⡀⢹⣿⠀⢿⣭⡇⢸⣿⡿⠇
⠀⠀⡀⠀⠀⠀⠀⠀⢠⠞⣀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠
⠀⠀⠀⢠⣦⠀⠀⢠⣤⠄⠀⠀⣠⣤⠀⠀⣠⣴⣶⣦⣤⡀⠀⠀⣤⣤⣤⣤⣤⣄⠀⠀⣠⠈⣇⡀⢀⣤⣤
⠀⠀⠀⠸⣿⡇⠀⡾⠁⢀⡀⢀⣿⡏⢀⣾⡿⠉⠁⠉⠋⠹⡄⢀⣿⡏⣩⣭⠙⣿⡇⠀⣿⣶⢉⣴⣿⠟⠁
⠀⠀⠀⠀⢿⣷⢸⣶⠀⢿⣧⢸⣿⠁⢸⣿⡇⠀⠀⠑⢄⣾⡇⠀⣿⣷⣾⣿⣿⠟⠃⠀⣿⣿⣿⢿⣷⡀
⠀⠀⠀⠀⠸⣿⣿⡏⠀⠸⣿⣿⡏⠀⠈⢿⣷⣀⣀⣀⣼⣿⠃⠀⣿⡇⠀⠙⢿⣷⡀⠀⣿⣿⠁⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠛⠛⡡⠄⠀⠛⠛⠁⠀⠀⠀⠙⠛⠿⠟⠛⠁⠀⠀⠛⠃⠀⠀⠈⠛⠛⠀⠛⠛⠀⠀⠀⠛⠛⠂
⠀⠀⠀⠀⠀⠐⠉⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠉⠛⠻⠿⣿⣿⣿⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢴⡆⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⣤⣤⣦⣄⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⡟⡟⠻⢿⣿⣿⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⣄⣠⡀⠀⠀⠀⠀⣼⣅⣁⣁⠛⠘⢻⡟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠫⡻⣿⣿⣿⣯⣆⠀⠀⢠⣿⣿⣿⣿⣿⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠉⠉⠉⠛⠛⠿⠿⣿⣿⡿⠿⠛⠛⠿⠁⠀⠉⠉⠉⠁⠀⠀⠀⠀⠛⠛⠿⠿⣿⠿⠿⠛⠛⠻⠿⠿⠛⠛⠻⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⢒⣒⣛⣛⣳⣶⣶⣾⠷⠶⠶⠾⠿⣿⣯⡭⢭⣍⣙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡭⠖⠚⢉⣩⣥⡶⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⣄⡉⠳⣄⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡷⠋⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠙⢦⡘⣆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⠁⢠⠞⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⡆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡏⡴⢃⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⠁⠂⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠉⠉⠉⠙⠻⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣧⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡄⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⢀⣀⡴⣿⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢧⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠛⠓⠒⠛⠋⠉⠁⠀⠀⠀⠢⠀⠀⠀⠀⠀⡼⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢸⡀⠹⣄⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢀⡾⠁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣳⡀⠈⠓⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣳⠦⣄⣈⣙⡛⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠋⣠⣲⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠭⠭⠿⠿⠿⠿⠿⠟⠿⣿⠿⠋⠉⠙⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠛⢻⣿⠇⠙⠉⠠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠃⢀⣶⠂⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⣴⣿⣏⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢡⣿⠛⠋⠀⠀⠘⠛⠋⠩⣶⣿⡿⠿⢋⣀⣀⢀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⣤⠛⠛⠓⢤⣤⣤⠀⠀⠀⠀⠀⢠⣼⠿⠻⣿⠛⠛⠙⢿
⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⣠⡿⠁⠀⠀⠀⠈⣿⠁⠰⠛⠛⠛⠻⠆⠀⠀⠀⠉⣤⠀⠀⣿
⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣰⣿⣷⣶⣤⠀⣤⣤⣶⣦⣤⣤⠀⣿⣀⣴⠟⣄
⠀⠀⣴⠟⠁⠀⠀⠀⠀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠹⣿⣿⣿⠇⠀⠀⠘⡿⠁⢀⡿
⠀⣾⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣧⣤⠟
⣶⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁
⠿⣄⠀⠀⠀⠀⠀⠀⠛⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⠀⠀⠀⠀⠀⣠⡴⠛⣿
⠀⠈⠳⣤⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⣀⣤⡴⠛⠁⠀⠀⣿
⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠂⢤⣤⣤⣤⣤⣾⠛⠿⠁⠀⠀⠀⠀⠀⠛⢃⣤⣤⡀
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠋⠁⢸⠟⠛⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠉⣀⠻⣄⠀⠀⠀⢀⣤⡜⠛⠁⠀⠀⠀⣾⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀⠀⠀⠀⢀⣿⠟⣷⠀⣿⠀⠈⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡄⠀⣸⠛⠀⠈⣦⠈⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⣿⠀⠀⠀⠉⣤⠉⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⣶⠉⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⠉⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡆⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠻⣤⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠛⣦⡀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠛⢦⣄⠀⠀⣿⠀⠹⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡈⠛⢤⣿⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣠⣿⠀⠀⠈⣤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠛⠀⠀⠀⠿




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠘⠋⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣤⡴⠛⠁⠀⠀⢠⡚⠓⠀⢀⣤⣤⣿⡄⠀⠀⠀⠀⣤⠀⠀⠈⠙⠛⠛⠛⠛⠃⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠛⠛⡿⠛⠻⡿⠿⠷⣤⣤⣤⣶⣿⣶⡶⠺⠿⠀⠀⠿⠛⠻⡄⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣠⠀⠀⠀⠀⠀⣩⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⢀⣤⣤⣶⣦⢠⣿⡄⠀⠀⠀⠀⣟⠀⠀⡋⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣄⠀⠈⠛⠀⣤⣤⣴⣿⣷⣤⣶⡾⠛⠛⠛⠛⠛⠛⣂⡀⠀⣁⠀⠀⣿⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠛⠂⠀⢤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠙⠛⠛⠛⠛⠛⠀⠀⢿⣤⣴⠋⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⠀⠀⠀⠀⠀⢀⣶⠠⣶⡀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠛⠛⠛⠛⠛⠓⠒⠒⠂⠀⢠⣤⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠁⠀⠙⠁⠀⠀⣿⠃⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠘⠀⠀⠀⠀⡆⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⢻⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣾⠀⠀⠀⠀⠸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠁⠀⠀⠈⣄⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⣿⠀⠀⠀⠟⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠛⠀⠀⠀⢀⣤⠔⠛⠁⠀⠀⣾⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡋⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⣤⣀⣄⠛⠁⠀⠀⠀⠀⠀⠀⠀⢛⣀⡘⠛⠛⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⣛⣷⣿⣿⣿⣟⣲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣋⡼⠿⣿⣿⣿⣿⣿⣿⡿⠷⣌⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⢁⡾⠉⠀⠀⠹⣿⣿⣿⣿⡟⠀⠀⠈⠻⣍⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣸⣿⣄⡀⠀⢠⡴⠻⠛⠛⠿⠳⣄⠀⢀⣠⣼⣷⡀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⡿⠟⠋⠻⣄⣀⡀⠀⣀⣀⡼⠋⠛⠿⣿⣿⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠻⣄⠀⠀⠀⢈⣿⣿⣿⣿⣏⠀⠀⠀⢀⡾⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⣷⣄⣠⣾⣿⣿⣿⣿⣿⣆⣀⡴⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⠛⠿⠿⣿⣿⡿⠿⠟⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣀⡿⠿⠓⠦⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠾⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸⠅⣸⡀⣷⢼⠅⣿
⠀⠀⠀⠀⠀⠀⡏⠀⠙⠁⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⢉⡁⠈⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠙⣆⡀⢾⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡰⠟⠋
⠀⠀⠀⠀⢀⡠⣀⠀⠙⣀⡁⢀⡠
⠀⠀⠀⠀⢸⠀⡿⡁⣧⠏⣇⢸⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣄⣠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⢀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠲⠆⠙⠚⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⢹⠛⣽⡏⣿⢿⡟⣟⡿⢿⢟⢿⠉⣿⡿⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣼⣜⣛⣴⣜⣮⣤⣧⣤⣼⣮⣾⣰⣽⣦⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡰⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠉⠙⠛⠛⠛⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⠀⣿⡈⠉⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⢟⣳⠶⣞⣻⣿⣿⡿⠂
⠀⠀⠀⡄⣠⢀⣀⢩⡅⡄⣀⢩⡁⣀⡄⢠⠔⠖⢀⣄⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠻⣿⣼⣿⣿⣿⣷⣿⠟
⠀⠀⠀⠃⠉⠈⠛⠂⠙⠃⠋⠐⠙⠉⠃⠊⠃⠃⠋⠈⠁⠙⠀⠀⠀⠀⠀⣀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢉⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⡴⢰⡀⡆⢠⡶⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣏⣹⡉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠁⠈⠁⠁⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⡀⢄⠠⢄⣀⢠⡄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠰⠁⠏⠘⠉⠚⠸⠣⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣷⢀⣴⣿⡿⠿⣿⣦⡀⣾⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣻⣻⣟⣿⢸⣿⣿⡶⢃⣿⣿⡇⣿⣻⣟⣿⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣿⣿⣿⣿⣿⠘⢿⣿⣿⣽⣿⡿⠃⣿⣿⣿⣿⣿⣿⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⠀⠋⠉⠙⠛⠉⠁⠀⠀⠉⢻⡟⠉⠀⠀⠈⠙⠛⠋⠉⠙⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣿⣿⣿⣿⣿⡟⠻⣿⣷⣄⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⣠⣾⣿⠟⢹⣿⣿⣿⣿⣿⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠿⠟⠁⠀⣿⣿⣿⣿⣿⡇⠀⠈⠻⠿⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠛⠛⠁⠀⢸⣿⣿⣿⣿⣿⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⣴⣿⡿⠋⢹⣿⣿⣿⣿⡏⠙⢿⣿⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢹⣿⡇⠀⠀⠀⠙⠋⠀⠀⢹⣿⣿⣿⣿⡇⠀⠀⠙⠋⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠸⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠘⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⣠⡟⢀⣀⣀⣤⣤⣤⡤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣤⣄⣀⣀⠘⣧⡀
⢀⣤⠶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⢦⣄
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⠀⠀⣀⣀⣤⣤⡴⠶⠶⠶⠾⠿⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠺⠿⠶⠶⠶⢶⣤⣤⣄⣀⡀⠀⢸
⣿⡶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⣾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟
⠀⣿⠙⠷⣄⡀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡶⠖⠛⠛⠛⠛⠛⠿⠿⠿⠛⠓⠶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⣀⡴⠟⢹⡇
⠀⢹⡄⠀⠈⠙⣶⠦⠶⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠶⠦⢴⡞⠋⠀⠀⣼⠁
⠀⠘⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠖⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢦⡀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡟
⠀⠀⠹⣆⠀⢀⣿⠀⠀⠀⠀⢰⡅⠠⣶⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢰⡆⠀⡷⠀⠀⠀⠀⢸⣇⠀⢀⡾⠁
⠀⠀⠀⢹⡾⢫⣿⡄⠀⠀⠀⠀⠻⠤⢤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣤⠼⠃⠀⠀⠀⠀⣼⣯⠻⣾⠁
⠀⠀⠀⢸⣇⠈⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢀⣿
⠀⠀⠀⠀⠙⠳⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠛⠁
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠘⠲⣤⣄⣀⣀⣀⣀⣀⣀⣤⡴⠚⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠉⠉⠙⢍⠉⠉⠉⠉⠉⢉⠝⠉⠉⠉⠙⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣴⠂⢠⣤⠌⠳⣄⢀⢀⡴⠃⢤⣤⠀⢲⣄⡀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣠⡶⠛⢉⡟⠀⠀⠀⠀⣰⣟⠻⢛⣷⡀⠀⠀⠀⠘⣏⠙⠳⣦⡀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣰⠏⠀⢰⡏⠀⠀⢸⡇⠀⠀⠀⠀⠹⢿⣤⠘⢸⡇⠀⠀⠀⠀⣿⠀⠀⠈⣷⠀⠀⣧⡀
⠀⠀⠀⠀⢠⡞⠁⢀⡀⠈⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⠦⠾⠃⠀⠀⠀⠀⢸⡇⠀⢰⡏⠀⣀⠀⠙⣦
⠀⠀⠀⢠⡟⠀⡴⢻⣿⣀⣈⡷⠀⣿⣛⢛⡛⠛⣛⢛⣛⡛⣛⡛⣛⠛⠛⡛⢛⣻⡇⠰⣏⣀⣼⣿⢳⡄⠘⣧
⠀⠀⠀⠀⠙⠛⠳⠛⠀⠉⠉⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠈⠉⠁⠘⠳⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢀⠀⢀⣾⠛⢻⣆⢀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠾⠶⠾⡏⠀⠀⡿⠶⠶⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⡇⠀⠀⡏⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠞⠛⠛⠛⠶⢾⡷⠀⠀⣇⠀⢀⡇⠀⠰⣿⠶⠞⠛⠛⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣀⣀⠀⣀⣀⣀⣀⡀⣀⡿⠀⠸⣇⠀⣀⣀⣀⣀⡀⢀⣀⣀⡈⣿
⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⣈⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢂⣉⣩⣉⡁⠒⠤⣄⣠⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠔⢁⡠⠒⠉⣀⣠⡤⠴⠗⠓⠓⢻⣾⣿⣿⣿⣭⣿⣿⣷⡄⠀⠄⠀⡀
⠀⠀⠀⠀⠀⡐⠁⡴⠈⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠛⠷⣮⡀⠈⡍⠐⠠⣀
⠀⠀⠀⠀⡘⠁⠔⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⣄⠛⠋⠻⠿⠿⣿⠀⠀⠀⠙⠲⣄⡈⠀⠆⠑⠦⡀
⠀⠀⠀⠠⠃⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣶⣶⣶⠇⠀⠀⠀⠀⠀⠈⠙⢦⡀⠈⢆⡙⢄
⠀⠀⣠⢄⣰⣏⣀⡀⣀⡀⠀⠀⠀⢀⡠⠄⠒⠒⠀⠉⠉⠙⠛⠻⠿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠃⠸
⠀⢰⣇⣿⣿⣿⣿⡁⢰⣿⣷⣄⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠰
⠀⣾⢽⣿⣿⣿⣿⡀⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠀⠀⠀⠀⠀⠀⠀⠸⣶
⢸⢧⣿⣿⠿⠿⠏⠀⢼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣴⣶⣯⠁⢨⣿⣿⣿⣿
⡰⠀⢸⠁⠀⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣗⠀⣿⣿⣿⡟
⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⢰⣿⣿⣿⡇
⡏⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⠛⠛⠉⠺⢿⣿⣿⣿
⡏⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⣽⠉⡿
⢿⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣶⣤⣤⣀⣀⣀⠀⠠⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃⡴
⠸⡄⠀⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⠿⣿⢿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢠⠃
⠀⠱⡄⠀⠂⡀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣠⡀⣈⣉⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠐⢠⠆
⠀⠀⠙⠦⣀⠈⠐⠀⢀⡙⢶⣄⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠂⢠⠎
⠀⠀⠀⠀⠈⠙⠓⠤⠤⠴⠤⣀⣄⣀⠀⠐⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣀⣠⠴⢋⠁⠂⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠿⢭⣽⣟⣟⣿⣛⡗⠶⠶⠞⠟⠉⠐⠀⣠⠤⠒⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣧⣦⣬⣶⣶⣴⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡷⠶⠆⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢛⣩⣤⠶⠖⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⡟⠛⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠁⣀⠀⠐⠶⣶⡈⣓⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠈⢻⣟⣋⠀⠁⠈⢳⣄⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠳⢤⣤⠯⠀⠈⠛⠶⣤⣀⣻⡆⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡏⠀⢻⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⡇⢴⣅⣀⡀⠉⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣄⣸⡇⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠉⣩⣍⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⢀⣟⣉⣻⡠⠾⣧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠻⠷⠶⠶⠖⠛⠋⣉⣉⣭⣭⣉⣠⣿⣄⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⢀⣀⣤⠶⠛⢙⡏⢸⣤⣼⣇⠈⠋⢤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠛⠿⠾⠛⠛⠉⠀⠀⠀⠀⡇⠈⠡⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣤⣤⣤⣤⣤⣤⠤⠴⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿
⠠⡛⠛⠛⠛⠛⠛⠛⠛⠓⠒⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣤⣤⣤⣤⣤⣤⣤⡼⠁⠒⢚⠋
⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⠉⠁⢠⡇⠀⠀⠀⢸
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⣁⠀⠀⠀⣼
⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠰⠶⠶⠶⠤⣤⣤⣤⣄⣀⣀⣰⡟⠀⠀⠀⣸⠇⣿⠀⠀⠀⣿
⠀⣀⣀⣿⣀⣀⣀⣀⣀⡀⠀⠀⠀⢹⠛⠛⠛⠒⠲⠶⠶⠦⣤⣤⣤⣀⣿⠃⠀⠀⣰⡟⠀⣿⠀⠀⢰⡇
⢤⣯⣤⣤⣈⣉⣉⣉⣉⠙⠛⠛⠛⠛⠻⠿⢷⡆⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⣰⡟⠀⢸⡟⠀⠀⣾⠃
⠀⡇⠀⣸⠉⠉⠉⠉⠉⠉⡙⠛⠛⠛⢳⠖⢺⡇⢠⣄⣀⣀⣀⠀⣾⠏⠀⠀⣴⡟⠀⠀⣾⠇⠀⣸⡟
⠀⡇⠀⣿⠀⣤⠴⠖⠊⠉⠀⠀⠀⠀⠘⡆⠀⣿⠀⠀⠀⠀⠉⣼⡏⠀⠀⣴⡟⠀⠀⢸⡟⠀⢠⣿⠁
⠀⣿⠀⣿⠐⠒⠶⠶⠶⠦⠤⢤⣤⣤⣄⣷⠀⢻⡀⠀⠀⠀⢀⣿⠁⠀⣰⡟⠀⠀⠀⣿⠃⠀⣼⡇
⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⡇⠀⠀⠀⢸⣿⠀⢠⣿⠁⠀⠀⠀⣿⠀⢠⣿
⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⡇⠀⠀⠀⢸⡿⠀⣼⡇⢀⣠⠖⠁⣿⠀⢸⡷⠶⠶⠦⠤⠤⢤⣤⣤⣤⡄
⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣿⠀⠀⠀⠘⣷⢀⣿⠖⠋⠀⠀⠀⢹⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢃
⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⠀⠀⠀⠀⡇⢸⠇⠀⠀⠀⠀⠀⠘⡇⡟⠀⠀⠀⠀⠀⠀⢀⡴⠚⣡⠖⠉
⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⣧⡴⠚⠀⡇⡼⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁
⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣏⣠⣤⣠⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣠⠞⠁
⠀⣿⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢻⠀⢀⣀⣈⡉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠓⠾⠋⣠⠞⠁
⠀⡯⠀⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡄⠀⠀⠉⠉⠉⠙⠛⠛⠛⠒⠶⠶⠶⠶⠶⠞⠁
⠀⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢏⡁⠈⠳⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣃⡀⠀⢐⡿⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣶⣿⣠⡤⢶⠶⠶⢒⡲⠶⠶⣿⣿⣿⣿⡏⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢀⣴⠿⠿⠛⠿⢥⣄⣀⣤⡤⠶⠶⠾⠟⠀⠉⠙⠉⠳⡀⠀⠀⠀⠘⣧⡀
⠀⢀⣴⠖⠚⠋⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠈⣿⢷⡄⡀
⠀⢸⠇⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢹⠂⠘⣼
⢀⡇⠀⢸⡀⠀⢀⠴⠊⠩⠙⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⣸⠀⠈⣧
⠈⢮⡂⢘⣇⣴⣃⠀⠀⠀⣠⠔⠊⠉⠓⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠔⠛⠥⣀⠐⠉⡱⢆⠀⠀⢀⡏⣀⢀⡞
⠀⠀⠉⢢⡿⣿⣿⣷⣤⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣤⣶⣾⣷⣶⣾⣾⡴⠋
⠀⠀⠀⠀⠻⣮⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣹⣟⣛⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⠙⠳⢌⡫⠓⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⣻⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⢀⠈⠉⠛⠒⠲⢢⠤⣤⣤⣤⣤⣤⣤⠶⠖⣿⣿⠷⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⢀⣀⣤⣄⣀⡀⠀⠈⠀⠀⢀⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠠⠂⠀⠂⠀⠀⠀⠂⠂⠀⠀⠉⠉⠁⠁⠈⠁⠁⠁⠉⠀⠈⠈⠉⠈⠉⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠀⠠⠤⠠⠀⠠⠀⠠⠠⠀⠀⠂⠀⠀⠂⠀⠀⠈⠉⠈⠁⠁⠁⠈⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢁⠐⠀⠀⡀⠈⠁⡄⢀⠀⠀⠀⠀⠉⠨⣆⠀⠀⠀⠀⠀⡃⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠀⠤⠠⠀⠀⠂⠀⡀⠀⠀⢀⣤⣿⡏⢇⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠰⠒⠂⠐⠀⠉⠄⠀⠐⠀⢀⠠⡤⠈⠁⠂⠀⣀⣰⠂⡘⡦⠍⠘⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣤⣤⣀⣀⢈⠁⠀⠀⠚⠁⠀⠂⠀⠐⢀⠴⢰⢢⠀⠃⡏⠀⠂⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠛⠛⠀⠁⠁⠉⠀⠈⠉⣖⠀⢤⣀⠀⠀⠋⠘⠉⠉⠀⣦⠀⠀⠀⠀⣠⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡌⡇⡀⢀⠀⠢⢅⠆⠀⠽⠃⠀⠀⣸⢭⣀⠄⠸⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢽⣇⣧⠀⠐⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠋⠃⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠠⠀⠀⠀⠠⠤⠤⠤⠄⠤⠤⠤⠀⠀⠒⡄⢀⠀⡔⢀⠀⠒⠐⠢⡄⠀⡶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⡄⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⡡⢀⣄⣠⠘⣁⣄⢄⡤⡅⢀⣆⣤⡤⡤⢤⠄
⠀⠀⠀⠀⠀⠤⠀⠒⠖⠒⠂⠰⢾⠱⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠈⠉⠉⠀⠨⣇⣭⣴⣱⢀⡀⠀⡅⢶⡥
⠀⠀⠀⠀⠀⠀⢰⠆⣀⠀⠀⠘⠉⠉⠁⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘