Traffic signs Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡒⠒⠖⠒⠒⠒⠒⠒⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠐⠒⠳⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⡟⠛⠻⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡽⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠄⠤⠠⠤⠗⠶⠶⠓⣀⢀⣀⠒⠄⠤⠤⠔⠦⢴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⠶⠤⠤⠤⠤⠄⠠⠤⠆⠽⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢁⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠿⠀⠇⠀⠀⠀⠀⣀⣠⠤⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣁⡧⠤⠒⠊⠉⠀⠈⠀⠠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣤⣀⣂⣠⠤⡴⠶⠛⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⢰⠙⣇⠀⠀⣿⠖⣩⣇⡠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⢬⣳⡰⠓⢢⡇⠉⠙⠙⠗⠀⠅⠉⠃⠴⠺⢥⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠋⠙⠉

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

⢀⡴⢚⣩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣙⠲⣄
⡞⣰⣿⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣷⡘
⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⢸⣿⠛⠛⢿⣷⡀⠀⢸⣿⠟⠛⠛⠛⠀⠛⠛⣿⡟⠛⠛⠀⢠⣿⠟⠛⢿⣧⠀⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⣿⡟⠛⠻⢿⣆⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⢸⣿⠀⠀⠀⣿⣇⠀⢸⣿⣄⣀⣄⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡏⠀⠀⠈⣿⡇⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⣿⣇⣀⣤⣼⡟⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⢸⣿⠀⠀⠀⣿⡟⠀⢸⣿⠛⠛⠛⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⣿⣿⠀⠀⣿⡟⠛⢿⣯⠀⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⢸⣿⣀⣀⣴⣿⠃⠀⢸⣿⣄⣀⣀⣀⡀⠀⠀⣿⡇⠀⠀⠀⠹⣿⣄⣀⣼⡿⠁⠀⠸⣿⣦⣀⣠⣿⠇⠀⠀⣿⡇⠀⠘⣿⡆⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⠘⠛⠛⠛⠋⠁⠀⠀⠈⠛⠛⠛⠛⠛⠃⠀⠀⠙⠃⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠛⠁⠀⠀⠘⠛⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣇⢻⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⢃
⠘⢦⣙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢛⣡⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠚⠲⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠂⠒⠂⠀⠀⠀⠀⠭⠩⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠙⠈⠀⠀⠀⠀⠀⣻⡄⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠙⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠟⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⢂⡒⣀⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⣀⠀⡊
⠀⠀⠀⢀⡀⢒⣀⠀⠀⠀⠀⠀⠀⡁⢀⡀⠀⠀⠀⠀⢠⢾⣠⡏⢳⠀⠀⢸⡿⡉⣿⣐⢾⡇⠀⠀⣰⠃⣴⠃⠀⠀⠀⠀⢘⣱⠀⠀⠀⠀⠀⢡⣜⠙⡿
⠀⠀⠀⠘⣲⢼⣈⠇⠀⠀⠀⢀⣾⡇⣇⢹⠀⠀⠀⠀⠉⠙⠁⠓⠃⠀⠀⠸⠶⠦⠾⠭⠾⠃
⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆⠀⢰⠶⠶⠒⠶⢶⡄
⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⢸⠦⣓⡄⣓⢺⡁⠀⠀⢤⠤⠄⡄⠀⠀⠀⠀⣆⡠⢦⠀⠀⠀⠀⠀⣠
⠀⠀⠘⠿⠾⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣛⣛⣛⣛⡀⠀⢀⣀⡈⠐⢚⣀⡀⠀⠀⠘⠐⠂⠀⠀⠀⠀⠀⠚⠑⠚⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠰⠤⠄⠀⠀⠀⠀⠨⡄⡀⣀⠐⢠⡅⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠠⣤⠀⠀⠀⣴⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠒⠂⠀⠀⠀⠀⡴⠰⠶⠶⠶⠄⠀⠀⠀⢰⠄⠀⠀⠀⠰⠎⠉⠀⠀⠀⠉⠈⠁⠀⠀⠀⠿⠈⠃⠀⠀⢧⣤⣤⣤⡇
⠀⠀⠀⠠⡦⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠛⠛⣿⠂⢻⣿⣿⣿⡗⠀⢠⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⢀⣀⠀⠀⣴⣿⣿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⠘⣦⡼⠃⠀⠈⢿⣀⡟⠀⠀⠸⠣⠟⠀⠀⠀⠂⠁⠀⠀⠀⠸⠇⠉⠀⠀⠀⠀⠊⠄⠀⠀⠀⠛⠋⠛⠀⠀⢿⣿⣿⣶⠇
⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠁⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠈⠙⡆⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⣄⢀⡴⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⣤⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠻⣔⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈

⠀⠀⠀⣠⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣦⡀
⠀⣀⣼⡿⠛⠛⠻⢷⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠉⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⣿⣿⣦⡀
⣾⣿⣿⡧⠸⠿⠇⣸⣿⣷⠀⠀⠀⠀⢰⣿⣿⣿⣥⠀⣜⣿⣿⣷⠄⠀⠀⠀⠀⢠⣾⣏⠉⠉⢉⠉⠉⢉⣿⡆⠀⠀⠀⠰⣿⣏⣉⣉⠀⣉⣉⣹⣿⠆
⠈⠻⣿⡗⢰⣷⣿⣿⠟⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⣿⣿⠟⠁
⠀⠀⠈⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⣠⢾⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠛⠉⠛⠻⢷⣄⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠋⠙⠛⢷⣤⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⢿⣿⢧⡀
⠀⠀⣠⣾⣿⠀⠉⢙⣦⡀⠀⠀⠀⠀⢠⡿⠙⢷⣄⠰⡷⠀⠀⢻⣆⠀⠀⠀⠀⢀⣾⠋⠻⣦⣴⠀⠀⠀⠹⣧⠀⠀⠀⠀⢠⣴⣿⣤⡇⠿⠾⠇⣿⣦⡀
⠀⣾⣿⣿⢟⡃⣤⡈⣿⣿⡦⠀⠀⠀⢸⡇⠀⢀⡙⢷⣿⡄⠀⢨⣿⠀⠀⠀⠀⢸⣯⠀⣰⣶⣿⢶⢆⡀⠀⣿⠀⠀⠀⠀⠈⠻⣿⣿⣇⢢⣤⢢⣿⠟⠁
⠀⠈⠻⣧⣬⣧⣿⣿⡼⠋⠀⠀⠀⠀⠘⢿⣄⠘⠃⠀⣹⢷⣤⣾⠇⠀⠀⠀⠀⠀⢿⣆⠙⠋⠉⠈⠒⢧⣼⠟⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠁
⠀⠀⠀⠈⠻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣴⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣤⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄
⠀⠀⠀⣠⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⢴⡄⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⠿⠿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠛⢻⣷⣄
⠀⣠⣾⣿⡛⠛⠛⣻⣷⣄⠀⠀⠀⠀⢸⡏⠀⠀⠒⢺⠒⠂⠀⠘⣿⠀⠀⠀⠀⠀⢰⣿⣯⣀⠊⠋⠑⣀⣿⣷⡄⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠌⣿⣿⣷
⠘⢿⣿⣿⣿⡄⣾⣿⣿⡿⠃⠀⠀⠀⠸⣷⠀⠘⢆⣸⣀⠼⠃⣰⡿⠀⠀⠀⠀⠀⠀⠙⢿⣃⣀⣀⣀⣹⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠛⢰⣿⡿⠋
⠀⠀⠙⢿⣭⣥⣭⡿⠋⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⠀⢀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋

⢠⡶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⣦
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠛⢿⣦⠀⢸⣿⣿⣄⠀⢸⣿⠀⠀⣿⡟⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠈⣿⡇⢸⣿⡟⢿⣦⢸⣿⠀⠀⣿⣷⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣀⠀⣀⣼⣿⠃⢸⣿⡇⠈⢻⣿⣿⠀⠀⣿⣇⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁⠀⠈⠛⠃⠀⠀⠙⠛⠀⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⣸⣿⣷⠀⠀⣿⡏⠀⠀⣰⣿⣿⡄⠀⠀⠙⣿⣆⠀⢠⣾⠟⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⢿⣷⢠⣿⠋⣿⡇⢸⣿⠁⠀⢰⣿⠇⢻⣿⡀⠀⠀⠈⢿⣷⣿⠋⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⡟⠀⠸⣿⣿⡏⠀⢠⣿⠿⠶⠾⢿⣷⡀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠻⠛⠀⠀⠛⠛⠀⠀⠀⠈⠛⠗⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠹⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣠⠊⢀⣤⣄⠈⠳⡄⠀⠀⠀⠀⠀⢰⣋⣉⣉⣉⣉⣉⣢⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⠀⣠⣴⣶⣤⠀⣧
⠀⠀⠀⡇⠀⣿⣿⣿⡤⠀⡇⠀⠀⠀⠀⠀⢨⠋⢩⣩⣩⡍⠉⢇⠀⠀⠀⠀⠀⠀⡇⠀⢶⡄⣶⡀⠀⡇⠀⠀⠀⠀⠀⢸⢰⣟⣛⣛⣛⡇⣿
⠀⠀⠀⠳⣄⠻⠿⠟⠁⡠⠃⠀⠀⠀⠀⠀⠸⣄⠘⠛⠘⠃⣀⡼⠀⠀⠀⠀⠀⠀⠱⣄⠘⠁⠛⢀⠜⠀⠀⠀⠀⠀⠀⢸⠀⠻⠿⡿⠟⠁⣿
⠀⠀⠀⠀⠈⠓⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠓⠁


⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠲⠦⣄
⠀⠀⠈⢧⠐⣿⣿⣿⡷⢀⡞⠀⠀⠀⠀⢰⡏⢉⣉⣉⣉⣍⡉⠉⣷⠀⠀⠀⠀⢀⠔⠁⣀⣤⣦⡌⠳⡄⠀⠀⠀⠀⠀⡼⠱⣾⡷⣦⠞⠉⣧
⠀⠀⠀⠀⢣⡈⢿⡿⢁⠞⠀⠀⠀⠀⠀⢸⡇⠘⠛⠛⠻⠿⠛⠀⣿⠀⠀⠀⠀⠣⡀⠀⡏⠀⠉⠀⢀⡼⠀⠀⠀⠀⠀⣇⠀⣠⠞⢡⡇⠀⣸
⠀⠀⠀⠀⠀⢳⡈⢠⠏⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠈⠢⡁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠘⢾⣁⠀⠈⣃⡴⠃
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢀⡴⢚⣉⣙⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢈⠳⣄
⠀⠀⢀⡏⢸⡮⣅⠉⣷⠈⣇⠀⠀⠀⠀⢀⡴⠋⠠⢾⡷⠄⠈⠳⣄⠀⠀⠀⠀⣠⠞⠁⠺⣦⡴⠃⠙⢦⡀⠀⠀⠀⢀⡔⠁⣠⣾⣿⣇⠈⠳⣄
⠀⠀⠈⣇⢺⡷⠈⠳⣛⢀⡏⠀⠀⠀⠀⠈⠳⣄⠠⠟⡷⠀⢀⡴⠃⠀⠀⠀⠀⠘⢦⡀⠀⢙⠁⠀⡀⠚⠁⠀⠀⠀⠈⢷⣄⠿⠿⠿⠿⣧⡾⠋
⠀⠀⠀⠈⠳⢤⣀⣠⠽⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⣁⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉⢷⣄⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⣀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠤⡄
⠀⠀⠀⣰⢯⣡⡤⢤⡉⢳⡀⠀⠀⠀⠀⠀⢀⡠⢊⣤⡄⣈⠢⡀⠀⠀⠀⠀⠀⠀⣠⠞⣡⣀⣈⠳⣄⠀⠀⠀⠀⠀⠀⠈⢻⡇⢿⡿⠀⡟⠃
⠀⠀⢠⡇⠀⢙⢦⣀⡿⠀⣷⠀⠀⠀⠀⠠⡎⠀⠏⠀⠀⢸⡆⠈⡦⠀⠀⠀⠀⣞⠁⢀⡬⠉⢩⠀⠈⣳⠀⠀⠀⠀⠀⠐⢾⡇⣶⣷⠀⡿⠇
⠀⠀⠀⢧⡀⢸⠇⠙⢦⣠⠇⠀⠀⠀⠀⠀⠙⢦⡙⠶⠴⠈⡡⠊⠀⠀⠀⠀⠀⠈⠳⣍⠀⠀⠈⣡⠞⠁⠀⠀⠀⠀⠀⠠⣼⡇⣼⣥⠀⣧⡄
⠀⠀⠀⠀⠙⠒⠶⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠬⠭⠤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢸⣤⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⢰⣶⣿⣭⣻⣶⠄⠀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⠀⢀⠰⣚⣿⠙⡀⡀⠀⠀⣴⠟⢯⣠⣤⡉⠻⣦⠀⠀⠀⠈⢻⣿⢿⣿⡟⠀⠀⠀⣰⡟⠉⣡⣤⣉⠻⣧⡀
⠀⠀⢸⠀⡟⢹⣄⡇⣿⠀⠠⣿⠀⠈⣿⣤⡿⠀⣿⡇⠀⠀⢸⣿⣵⣶⣿⣿⠁⠀⠀⣿⠀⠀⣿⣠⡿⠀⢹⡇
⠀⠀⣾⠀⡇⢸⢹⡇⣿⠀⠀⢿⣄⠀⠿⠘⢦⣠⡿⠁⠀⠀⢠⣽⡿⣻⢿⣥⡀⠀⠀⢻⣆⠀⠿⠀⠀⣠⡾⠃
⠀⠀⠻⠀⠇⠸⠘⠇⠿⠀⠀⠀⠙⠻⠶⠶⠟⠋⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠀⠀⠀⠀⠙⠛⠶⠶⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⣠⠴⢦⣄
⠀⠀⠀⢀⡴⠋⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⣠⠞⠁⢀⣌⡢⡀⠀⢠⡖⠋⠁⠀⠀⠈⠙⢲
⠀⢀⡴⠋⠀⣽⡦⡀⠈⢲⡀⠀⠀⣤⣶⣿⣿⡿⠟⠀⠀⠀⡞⠁⢠⡞⠛⠟⠋⠈⣦⠸⡇⢶⢰⡆⡖⣴⡶⢸⡀
⠀⠈⢧⡀⠀⡞⣇⠙⢀⡴⠃⠀⠘⠛⠉⠉⠉⣄⣀⣀⡆⠀⠙⢦⣸⡇⠀⠀⢀⠔⠁⠀⣇⠈⠀⠀⠈⠀⠀⣸⠇
⠀⠀⠀⠙⠦⣀⢈⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⠦⣤⠔⠁⠀⠀⠀⠈⠙⠲⣄⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠉⠉⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⢲⡶⠆⠀⠀⣠⢖⠋⠉⠙⠲⡄
⠀⠀⠿⠿⠿⠿⠿⠿⠿⠆⢠⡇⣈⠢⣀⠴⢋⠈⣇⠀⢀⣴⣶⣶⣶⠀⣤⢤⣼⣿⠿⣆⢰⠃⣀⣷⣤⣤⣤⢸⡄
⠀⢠⣶⣶⣶⣶⣶⣶⣶⡆⠈⣇⢉⠔⠋⠢⡈⢁⡏⠀⣿⡿⠛⠁⠀⠈⠙⠞⠉⠉⠒⠁⠘⡄⠉⠉⠉⢫⠉⣸⠃
⠀⠈⠿⠿⠿⠿⠿⠿⠿⠃⠀⠈⠳⢤⣀⣠⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠤⠟⠁


⠀⠀⠀⡄⠐⠒⠒⠒⠂⠄⠀⠀⠀⠀⠄⠤⠀⠀⢰⡆⠌⡆⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠
⠀⠐⣶⣾⣿⣷⣤⣾⣿⣦⡶⠀⠰⣦⣶⣿⣦⣰⣶⡶⢶⣶⡆⢀⡾⠻⣿⣧⡀⠀⠀⣴⣶⣿⣿⠃⠀⢠⣦⢤
⠀⠀⢭⠽⣏⢛⣙⣹⢹⠍⡇⠀⠀⣭⣽⣭⣭⢹⣿⣇⣸⣿⡇⠘⣇⠀⠀⠀⣹⣤⠈⢧⣾⡾⠿⠁⠠⠾⡙⢂⠇
⠀⠀⠸⣷⠈⠉⠉⠉⠡⣿⠁⠀⠀⢹⡷⠈⠉⠁⠉⢹⡿⠉⠀⠀⠈⠛⠖⠚⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣽⣿⣽⣯⣭⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣯⣿⣿⣭⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠄⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⢚⣭⣴⣶⣿⣿⣿⣿⣿⣶⣦⣭⣓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣌⠢⡀
⠀⠀⠀⠀⠀⢀⠔⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠢⡀
⠀⠀⠀⠀⡰⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢦⠀⠀⠀⠀⠀⠀⢀⠔⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠢⡀
⠀⠀⠀⡸⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢇⠀⠀⠀⠀⠐⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⠂
⠀⠀⢀⢃⣿⣿⠟⣛⠻⢿⡟⣛⠛⢛⣛⣿⠿⢛⡛⠿⣿⣿⠛⡛⡛⢿⣿⡘⡂⠀⠀⠀⠀⢸⣿⡿⢛⡛⢿⣿⡛⠛⠛⡛⣿⡿⠟⣛⠻⢿⣿⠟⢛⡛⠻⣿⡇
⠀⠀⢸⢸⣿⣧⡘⠿⢿⣾⣿⣿⡇⣿⣿⡏⣼⣿⣿⣷⢸⣿⢸⣿⡿⢂⣿⡇⠇⠀⠀⠀⠀⢸⣿⣘⠻⠿⣶⣿⣿⡇⢸⣿⣿⢰⣿⣿⣿⡌⣿⠀⣿⣿⠗⣸⡇
⠀⠀⢸⢸⣿⡟⢿⣿⣶⢘⣿⣿⡇⣿⣿⣇⠻⣿⣿⡟⣸⣿⢰⣶⣶⣿⣿⡇⡇⠀⠀⠀⠀⢸⣿⠻⣿⣷⡆⣿⣿⡇⢸⣿⣿⡘⣿⣿⣿⢃⣿⠀⣶⣶⣾⣿⡇
⠀⠀⠈⡆⣿⣿⣶⣤⣴⣾⣿⣿⣷⣿⣿⣿⣷⣦⣴⣾⣿⣿⣾⣿⣿⣿⣿⢠⠃⠀⠀⠀⠀⢸⣿⣷⣦⣤⣾⣿⣿⣷⣾⣿⣿⣿⣶⣤⣶⣿⣿⣶⣿⣿⣿⣿⡇⡀
⠀⠀⠀⠹⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⠎⠀⠀⠀⠀⠠⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⠆
⠀⠀⠀⠀⠑⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⠎⠀⠀⠀⠀⠀⠀⠈⠢⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⠔⠁
⠀⠀⠀⠀⠀⠈⠢⡙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⣙⡻⠿⠿⣿⣿⢿⠿⠿⠟⣋⠥⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁


⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⢀⣠⡶⠞⠛⠛⠛⠷⣦⣄⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠻⢦⣤⣄⣀⣠⣤⠶⠋⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣤⠶⠶⠳⠶⢶⣤⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠳⣤⣄⣀⣀⣀⣠⡴⠟⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⢦⣤⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠈⠙⢶⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠘⠷⣤⣀⡀⠀⢀⣀⣠⡶⠋⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡟⠛⠛⠛⠛⠛⠋⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠤⠤⠤⡤⠤⠤⠤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠰⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⢲⡶⢶⣳⣶⣶⠀⠐⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢐⢠⣭⡭⠽⠭⠭⠯⠯⠦⠿⠯⠽⠭⠯⣯⠭⠽⠿⠭⠭⠭⠭⠯⠯⠭⢯⠦⣥⣒⠢⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣕⠢⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⠟⡨⠃
⠀⠀⠀⠀⠀⠀⠀⢐⢼⣧⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣴⠟⡫⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠥⠭⠭⠭⠭⠭⠭⠽⠭⠭⠭⡭⣭⢭⣭⣭⠭⠭⠭⠭⠭⠭⠩⠭⠽⠉⠅⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣘⣯⣾⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠅⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⠈⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣀⣀⣀⣀⣀⣀⣀⣰⣀⣀⣀⣀⡀⠀⠀⠀⢀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠻⡯⢿⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠐⠒⠒⠂⠀⠐⡒⠒⠒⠚⠒⢒⠚⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⠀⠤⠤⠤⠤⣤⣀⣐⣀⣀⡀⣤⠄⠄⠀⠠⢠⠤⡔⣤⡼⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⡔⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡊⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡊⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢖⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣡⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠅⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢯⢸

⠀⠀⣠⣾⣿⣦⡀⠀⠀⢀⡴⡄⠀⠀⠀⠀⣰⢦⠀⠀⠀⠀⢀⡶⡄⠀⠀⠀⠀⣰⢆⠀⠀⠀⣰⠚⢉⡳⡄⠀⢀⡶⣋⡙⢦⠀⠀⣰⡾⣿⣷⡄⠀⢸⣿⢟⢿⣿
⠀⠀⠿⣿⣿⣷⠇⠀⢀⣞⣵⣙⣆⠀⠀⣰⣃⣊⣣⡀⠀⢠⣞⣢⣝⡄⠀⠀⣼⣑⣊⣧⠀⠀⢻⡐⠊⣠⠇⠀⠘⣆⠈⢻⡼⠀⠀⢻⣿⣄⣹⠇⠀⢸⠃⢸⢈⢿
⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉
⠀⠀⢶⠲⠲⢲⠆⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⣠⣶⣶⣄⠀⠀⢠⠖⠒⢦⡀⠀⠀⣴⣒⡲⣄⠀⠀⣠⢶⡶⢦⡀⠀⢰⡖⢒⠲⣶
⠀⠀⠈⢣⣠⠏⠀⠀⠀⡼⢥⠻⡄⠀⠀⣠⠋⠜⢦⠀⠀⢀⡼⠤⠳⡀⠀⠐⣯⣭⣭⣽⠃⠀⢿⠒⠒⢒⡇⠀⠸⡅⠗⢦⣸⠀⠀⢯⡼⡿⢦⡇⠀⢸⡇⣩⣥⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠈⠛⠋⠛⠋⠀⠀⠛⠛⠛⠛⠁⠀⠘⠛⠛⠛⠋⠀⠀⠈⠛⠛⠁⠀⠀⠈⠙⠒⠋⠀⠀⠀⠉⠓⠚⠁⠀⠀⠈⠙⠒⠉⠀⠀⠘⠓⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⠤⠤⣄⠀⠀⢀⡤⠤⢄⠀⠀⠀⣠⠤⢤⡀⠀⠀⢀⣤⣤⣄⠀⠀⢠⡤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⣳⡀⠀⠀⢠⣞⡙⣄⠀⠀⠀⡴⠫⢣⠀⠀⠰⡇⡦⢴⢸⠄⠀⣾⢰⢴⡂⡇⠀⢰⡋⠒⠀⢹⠀⠀⣿⣿⣻⣿⡇⠀⢸⣿⡆⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠓⠓⠀⠀⠛⠛⠛⠚⠂⠀⠘⠓⠛⠚⠓⠀⠀⠙⠶⠶⠋⠀⠀⠈⠳⠶⠚⠁⠀⠀⠑⠦⠞⠋⠀⠀⠘⠻⠿⠛⠀⠀⠸⠿⠷⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣀⡀⠀⠀⢠⣤⣤⣤⣄
⠀⠀⠀⢀⡴⠂⠀⠀⠀⢠⠏⣧⠀⠀⠀⢀⡜⢳⡄⠀⠀⠀⣰⠛⣆⠀⠀⢠⡏⠐⠆⢹⡄⠀⣾⠁⠀⠈⡇⠀⢰⠋⢢⡜⢳⠀⠀⣾⣏⡯⢿⡆⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠉⠀⠀⠀⠀⠰⠧⠼⠬⠷⠀⠀⠾⠶⠿⠿⠄⠀⠰⠧⠭⠬⠧⠀⠀⠳⢬⡤⠞⠀⠀⠙⢦⣤⠴⠃⠀⠈⠳⣥⡴⠞⠀⠀⠙⠿⣿⠿⠁⠀⢸⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⣰⣷⡀⠀⠀⠀⢀⡞⣆⠀⠀⠀⠀⡴⢣⡀⠀⠀⠀⢠⢿⡄⠀⠀⢀⡞⢩⡉⢳⡀⠀⢸⢁⠆⢸⡆⠀⢠⠞⠉⠉⢳⠀⠀⣴⣿⢻⣿⡄⠀⢸⢿⣽⢿⢿
⠀⠀⣴⣁⣄⣹⡄⠀⢠⣏⣹⣈⣦⠀⠀⣼⣡⣀⣳⡀⠀⢠⣏⣼⣿⣆⠀⠈⢧⣀⣀⡼⠁⠀⠀⠑⠒⡊⠀⠀⠘⢦⣀⣁⡼⠀⠀⠻⣿⣾⡿⠃⠀⢸⣏⣸⣘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⢠⢦⠀⠀⠀⠀⢀⡴⡄⠀⠀⠀⠀⣠⢦⠀⠀⠀⠀⢀⡶⡀⠀⠀⠀⡴⠛⡙⢦⠀⠀⣠⠞⠛⠲⡄⠀⠀⠀⢀⣴⠆⠀⠀⣿⡿⠻⣿⡇⠀⢸⡿⣿⡿⣿
⠀⠀⣠⣋⣈⢳⡀⠀⢀⡞⣱⡛⣄⠀⠀⣰⢣⣞⣣⡀⠀⢀⣾⣀⣹⡄⠀⠘⣇⠁⠙⣸⠃⠀⢻⡀⠈⣀⠇⠀⠀⣴⠟⠁⠀⠀⠀⣿⣿⢰⣿⡇⠀⢸⣇⣶⣆⣿
⠀⠀⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠁⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣉⣉⣁⣀⣈⣉⣉⣉⠉
⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⣠⠖⠲⢄⠀⠀⢀⠴⠒⢦⡀⠀⠀⣤⣶⣶⣄⠀⠀⠀⣿⡟⣿⣟⣛⣿⢋⣙⢻⠁
⠀⠀⢠⠞⣌⣧⠀⠀⠀⡼⠱⠳⡄⠀⠀⢠⠟⠞⢆⠀⠀⢀⣼⠥⢳⡀⠀⠐⡇⠘⠇⢸⠂⠀⢿⠘⠛⢃⡇⠀⠸⣯⣭⣨⣿⠀⠀⠀⣿⡄⣾⣿⡟⣿⡮⠿⣾⠁
⠀⠀⠛⠛⠛⠛⠃⠀⠘⠛⠛⠛⠛⠀⠀⠛⠛⠛⠛⠃⠀⠘⠛⠛⠛⠋⠀⠀⠉⠓⠚⠉⠀⠀⠈⠑⠒⠋⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⣿⣾⣾⣿⣷⣿⣷⣤⣿
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡤⢤⡀⠀⠀⢀⣠⠤⣄⠀⠀⠀⣀⣤⣤⡀⠀⠀⣤⣤⣤⣤⡄
⠀⠀⢀⡜⠝⣆⠀⠀⠀⣰⢫⢷⡀⠀⠀⢀⡞⣹⡄⠀⠀⠀⡴⣉⣧⠀⠀⠠⡧⡴⢶⢸⠄⠀⣾⢀⣀⠈⡇⠀⢰⣿⢣⣠⣿⠀⠀⣿⢾⣿⢸⡇⠀⢈⠁⠀⠁⠁
⠀⠀⠺⠶⠶⠾⠆⠀⠰⠷⠶⠶⠗⠀⠀⠾⠶⠶⠿⠂⠀⠸⠷⠾⠶⠗⠀⠀⠑⠦⠴⠊⠀⠀⠘⠲⠤⠞⠁⠀⠀⠻⠾⠿⠋⠀⠀⠿⠾⠿⠾⠇⠀⠈⠁⠈⠉⠈

⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣴⠛⠛⠛⢶⣄⠀⠀⠀⠀⢀⣴⡞⠛⠛⠳⣄⠀⠀⠀⠀⠀⣠⣖⣟⠛⠓⢦⡀
⠀⠀⠀⣾⡭⣿⢹⢫⡝⠩⣻⠀⠀⠀⠀⡿⢈⣳⣶⣶⣶⢹⡆⠀⠀⠀⣼⠁⠹⣦⡘⠄⠘⡇⠀⠀⠀⢰⡇⠙⢿⣯⠀⠀⣷
⠀⠀⠀⢿⣽⣿⣼⣮⣽⣽⡿⠀⠀⠀⠀⢿⡙⠋⠉⠻⣏⣼⠁⠀⠀⠀⢹⣄⠀⠀⠙⢦⣸⠇⠀⠀⠀⠘⣇⠀⢰⠛⣷⣤⡟
⠀⠀⠀⠀⠙⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠛⠁⠀⠀⠀⠀⠀⠙⠳⠶⠶⠞⠁⠀⠀⠀⠀⠀⠈⠛⠶⠶⠞⠋


⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⢀⡾⣯⣭⣭⣝⢷⡀⠀⠀⠀⠀⣠⢾⣍⠉⠉⠻⣦⠀⠀⠀⠀⢠⡾⣏⣉⣩⣙⢦⡀⠀⠀⠀⠀⡴⣯⣩⣭⣍⠳⡄
⠀⠀⠀⢼⠁⢸⡷⣤⠟⠈⣷⠀⠀⠀⠀⣟⠘⠻⢿⣿⠀⢸⡇⠀⠀⠀⣾⠀⢸⡿⣝⠉⠈⡇⠀⠀⠀⢸⡇⢸⡿⣦⣿⠀⣿
⠀⠀⠀⠘⣧⡘⠇⠈⢳⣼⠃⠀⠀⠀⠀⠻⣄⠀⠀⠿⣦⡾⠀⠀⠀⠀⠘⣦⡸⠃⠈⢳⡼⠃⠀⠀⠀⠈⢷⣙⠋⠈⢿⣶⠏
⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁

⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⠛⠛⠳⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⢀⣴⠞⠛⠛⠛⠛⠛⠛⣿⡇⠀⠀⠀⠀⠀⢀⣴⠾⠛⡛⠛⠻⢶⣄⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⢶⣄
⣾⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣴⠟⠀⠀⠛⠛⠻⠷⣦⡙⢷⡄⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹
⢿⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠈⢻⠘⣷⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸
⠸⣇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠻⠶⠶⠶⠶⠶⠶⠶⣿⡇⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⣿⠷⠶⠶⠶⠶⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⣀⣤⡶⠟⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣀⣀⣠⣤⣤⣤⣄⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣤⣤⣤⣄⣀⡀
⣰⡟⠋⠉⠉⠉⠉⠉⠉⣿⡇⠀⠀⠀⠀⢀⣴⠞⠋⣭⣭⣍⣉⠻⢶⡄⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠛⣷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣾⠃⠀⠀⠀⠈⠉⠙⢷⣄⢻⡆⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⢻⡆⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠌⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾
⠘⣷⣀⣀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣰⡟
⠀⠙⠛⠛⠛⠛⠛⠛⠋⣿⡇⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠙⠷⠶⣦⡶⠶⠟⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣀⣤⡶⠶⠶⠶⠶⢶⣿⡇⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣿⣶⠶⠶⠦⠶⠶⣦⣄⡀
⣼⠏⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⡾⠋⠁⠶⢶⣶⣤⣉⠻⣦⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠈⠛⣧⠹⣇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡆⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾
⠈⣿⣤⣤⣤⣤⣤⣤⣤⣿⡇⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠻⣤⣀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⡄⢰⠂⠀⠀⠀⣦⣿⢰⡇⠀⠀⠀⢠⡄⢰⢰⠆⠀⠀⠀⢠⡄⡶⣠⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⣴⣶⣿⠀⠀⠀⠀⣀⣧⣾⠀⠀⠀⠀⣸⣿⣾⠀⠀⠀⠈⢦⣧⣿⡟⠀⠀⠀⠳⣜⣧⣷⡏⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⠄
⠀⠀⠀⠈⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢹⣿⣿⣷⡾⠋⠀⠀⠀⠈⣿⣿⣿⠏
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⢈⣿⣿⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠐⠿⠿⠟

⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠘⣦⣤⣦⠀⣠⠆⠀⠀⠀⣠⣤⢶⣶⣦⡀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⢿⣿⣿⣿⠋⠀⠀⠀⠀⢠⡾⠿⢿⣿⣷⠀⠀⠀⢤⡀⣰⣿⣧⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣄⡀⠀⠀⠀⠀⣾⣿⣿⡀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠺⣦⣀⣼⣿⣿⣷⠀⠀⠀⠙⢿⣿⣿⣿⣦
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠋⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣶⣶⣾⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣀⣠⣤⣀
⠀⠀⠀⠀⠀⣰⡿⠛⠛⠛⠛⢉⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⠒⠒⠂⣀⣠⣼⣿⣷⣦⣤⣤⣄⡀⠈⠉⠉⠛⠻⠿⠿⢿⣿⣷⡀
⠀⠀⢀⣤⣤⡋⠀⠀⠀⠀⠀⠈⣿⣄⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⡿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣄
⠀⠀⠈⢻⣿⣿⡀⠀⠀⠀⠀⢀⣿⣿⡄⠀⠘⠿⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠆
⠀⠀⠀⠈⢿⣿⣷⣄⡀⠀⢠⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠃⠘⠿⠿

⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣰⣷⣆⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠸⣿⣷⣤⣤⡀⠀⠀⢀⣠⣤⣶⣾⠇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠛⠿⣿⣿⡄⢠⣿⣿⠿⠟⠃⠀⠀⣤⣾⣿⣿⣷⣦⣤⣴
⠀⠀⠀⠀⢀⣾⣿⣿⠟⣿⣿⣧⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠉⠙⢸⣿⣿⣿⡿⠿
⠀⠀⠀⢠⣾⣿⡿⠁⠀⠈⢿⣿⣷⡄⠀⠀⠀⢀⣾⣿⣿⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⣸⣿⣷⣸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⠀⣀⣤⣶⣶⣤⣀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⣿⣯⡀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢀⣾⡟⠋⠁⠈⠙⢻⣷⡀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢀⣽⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⣀⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣼⣿⠀⠀⠈⢿⣦⡀⠀⠀⢀⣴⡿⠁⠀⠀⣿⣧⣴⡿⠟⠋
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣉⣻⣿⠀⠀⠀⠀⠙⠿⠿⠿⠿⠋⠀⠀⠀⠀⣿⣟⣉⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⠛⠛⠛⠛⢿⣿⠀⠀⠀⢀⣴⣾⠿⠿⣷⣦⡀⠀⠀⠀⣿⡿⠛⠛⠛⠛⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠻⢿⣦⣄⡀⠀⠀⠀⢸⣿⠀⠀⢠⣿⠏⠀⠀⠀⠀⠹⣿⡄⠀⠀⣿⡇⠀⠀⠀⢀⣠⣴⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⡀⢸⣿⠀⠀⢸⣿⡀⠀⠀⠀⠀⢀⣿⡇⠀⠀⣿⡇⢀⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⠀⠀⠀⠻⣷⣤⣀⣀⣤⣾⠟⠀⠀⠀⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⠀⠈⣙⣛⣛⣋⠁⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⣰⣿⠟⠛⠛⠻⣿⣆⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⣀⠀⠀⢸⣿⠀⠀⢰⣿⠁⠀⠀⠀⠀⠈⣿⡆⠀⠀⣿⡇⠀⠀⣀⣴⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣾⣿⠀⠀⠸⣿⣄⠀⠀⠀⠀⣠⣿⠇⠀⠀⣿⣷⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⠙⠿⣷⣦⣴⣾⠿⠃⠀⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⠉⣿⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠤⠴⠞⠋⠁⠀⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠋
⠀⠀⠀⠀⠀⢠⡞⠚⠛⠋⠁⣹⣿⣤⣤⡴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠶⠶⠞⠛⠙⠛⠒⠲⠶⠶⢤⣤⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠑⠒⢄
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠛⠲⣤⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣄⣀⣀⣀⡀⠠⣤⣀⠀⢀⠜
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠰⠀⠀⠀⢤⣀⠀⠀⣀⣀⣀⣤⣶⠀⠀⠀⠈⠉⠉⠉⠋⠙⠛⢯⡉⠉⠉⠉
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠉⠩⠛⠛⠻⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣀⠀⠀⠄⠈⠙⠛⠶⠶⣄⣀⣠⡼⠃
⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢻⣄⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⡅⠀⠙⠳⣦⣀⡀⠀⠀⠀⠀⢺⡏⠁
⠀⠀⠀⠀⠀⠀⢷⣄⣀⡠⠤⠤⠿⠋⠉⠛⠳⠦⢤⣤⣄⣀⡀⠀⠀⠳⣄⡀⠀⠀⠈⠉⢻⡦⠴⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠚⠚⠻⠶⣤⣤⡴⠏


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠙⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠈⠉⠉⠙⠛⠒⠒⠖⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣬⣽⠷⠶⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⠀⠀⣠⣶⠒⠶⢤⣤⣤⡀
⠀⠀⠀⠀⠀⣀⡤⠠⠶⠖⠒⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠛⡆⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢨⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠙⠁⠋⠉⠉⠉⠉⠓⠒⠛⠛⠒⠒⠒⠚⣻⡿⠿⠶⠶⠶⠟⠉⠀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡗⠶⠶⢶⠶⠶⠛⠁⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⢀⣠⣤⡴⠶⠛⠁⠀⠀⢨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣾⠉⠁⠁⠀⠀⠀⠀⣀⡴⠛⠛⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣀⣠⠴⠖⠛⠉⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⣶⡟⢀⡀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣄⡀⠀⢀⣤⣾⣯⣤⣤⣴⠾⠾⠛⠋⠉⠁⠀⠀⠀⠈⠉⠉⠉⠓⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁

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




















































⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣶⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡿⣿⠿⢿⣿⡿⢿⣿⣿⣿⠿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦
⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠈⠛⠛⠉⠉⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⣤⣴⣶⣦⣤⣴⣤⣤⣤⣶⣶⣄⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡛⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⣠⡴⠚⠛⠛⠛⠲⢦⣀⠀⠀⢹⠛⠛⠛⠛⠛⠛⠛⣛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀⠀⠀⠀⢸⡇⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⢸⠀⠀⠀⢀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄⣸⡇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⣦⣀⣀⣀⣀⣠⡴⠛⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠶⠶⠶⢾⡇⠀⠀⠀⣠⣬⠿⠿⠯⣥⣀⠀⠀⠀⢸⠶⠶⠶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⢸⡇⠀⣴⠟⠁⠀⠀⠀⠀⠀⠉⠳⣄⠀⢸⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠀⢸⡇⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢸⠀⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢾⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⢸⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⣦⣀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠉⢛⣲⣶⣞⣛⠉⠀⠀⠀⢸⣄⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⡀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡴⠞⠉⠉⠀⠉⠉⠻⢦⡀⠀⢸⠁⠀⠀⠀⠀⠀⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀⠀⠀⢸⡇⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⠀⠀⢀⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣼⡇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠳⠦⣤⣤⣤⡴⠞⠋⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⣿⠛⣿⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡰⡾⢩⡍⢷⢄⠀⠀⠀⠀⠀⠀⢀⣶⣴⣧⠀⠀⠀⠀⣠⢀⡤⠀⢄⢠⡀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠆⡇⢰⠆⢸⢸⠀⠀⠀⠀⢐⣤⣸⣿⣿⡟⠀⠀⠀⠀⣿⡈⠑⢶⠀⣸⡇⠀⠀⠀⠀⠘⠿⠥
⠀⠀⠹⣿⣻⣗⣿⠋⠀⠀⠀⠀⠘⠿⢿⣿⣯⡦⠀⠀⠀⠀⠈⠿⣦⣤⣶⠟⠀⠀⠀⠀⠀⠲⠶⣶⣶⣶
⠀⠀⠀⠁⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⡀⢀⣾⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⡢⠤⠠⠀⠀⠀⠀⠀⣸⣃⠀⣺⡂
⠀⠀⣧⡈⠻⠷⣠⡿⠀⠀⠀⢀⡞⠛⣇⣸⠛⢳⡀⠀⠀⠠⣤⠄⠀⡇⠀⣤⡄⠀⠀⠀⠸⣿⡟⠼⣿⣿
⠀⠀⠘⣿⠀⢰⣿⠁⠀⠀⠀⠈⠳⠴⠋⠙⠦⠞⠁⠀⠀⠀⠀⢀⣴⣧⣄⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠁⠀⠀⠉
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣰⠞⢹⡏⠛⣦⠀⠀⠀⠀⠀⠀⠀⣀⢴⣤⠀⠀⠀⠀⡴⡛⠩⡍⢯⢦⠀⠀⠀⠀⡴⣏⣉⣉⣙⢆
⠀⠀⡇⣀⣼⣧⣀⢸⠆⠀⠀⠀⣤⣶⣿⣟⢭⣿⠁⠀⠀⠐⠀⠆⠤⠤⢼⠴⡆⠀⠀⢰⣇⣟⠿⣏⡉⢸⡄
⠀⠀⠹⢯⣸⣇⡽⠏⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠳⢳⣓⣂⡒⠖⠀⠀⠀⠀⠻⣭⣀⣈⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢲⣷⠂⠀⠀⠀⠀⠀⠀⢐⣖⠛⠿⠟⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠂⠀⠀⠀⠀⢰⣿⡟⣿⣷
⠀⠀⣠⣤⠂⠈⡢⡤⡀⠀⠀⠀⠀⣿⣿⠧⠀⠀⠀⠀⠀⠀⠶⠶⢰⣿⡇⠲⠶⠀⠀⠀⠀⠀⢙⣿⣿⠁
⠀⠀⠻⠟⠁⠁⠳⠷⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠾⠟⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠊⠹⣦⠈⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡇⠈⠆⠀⡿⠀⠈⣿⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⠀⢀⣴⡿⠀⠃⠀⠀⡔⠁⠀⠀⢷⡀⢆⣠⠞⣿⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢣⠀⠀⡴⠟⠁⢀⣴⠀⠨⠀⠐⣄⠀⠀⠀⡇⠈⠻⡀⠙⣦⠀⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁⠀⠠⠊⠀⣠⠤⠞⠃⠀⠀⠀⢀⠜⠀⢠⠊⢣⠀⠀⠈⡀⠀⠀⠀⢧⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠋⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⢀⣴⡿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠛⠋⠀⠠⠊⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⡇⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠄⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠒⠂⠁⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⣺⣿⣷⡀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⣿⡂⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣗⠀⠐⣿⣿⣿⣠⠿⠿⠇⠀⠀⠀⠀⠀⠀⢀⠀⠑⠀⠀⠀⠈⠀⠀⢄⠀⠀⣽⣿⣿⡄⢠⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⣽⠿⣿⡿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠀⢐⣿⣿⣿⣟⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣽⠿⠿⠿⠿⠍⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠾⠦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⠀⢆⠀⢙⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢨⣿⠅⠀⠀⠀⠀⠀⢸⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠰⠣⠅⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡖⠦⠄⡀⠀⠀⠀⠀⠀⣁⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠂⢀⠀⠀⠸⠋
⠀⠀⠀⢀⠀⢠⠀⢀⣀⡤⠀⠀⡀⣀⠠⢀⣩⣠⡿⣿⣿⣶⣶⣶⣤⡈⠑⠄⠀⠀⠆⣧⠀⠂
⠀⠀⠀⠀⠟⠺⠃⢈⢉⠰⢶⣴⣾⣶⣶⣾⣿⣿⣷⣿⡿⠟⠋⠋⠉⠉⠂⠀⢠⠀⠀⠉⠁
⠀⣀⡖⢀⣀⣶⠤⢾⡷⢶⣾⣿⠞⠿⠿⠟⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠐
⠠⠟⠂⢀⣸⣗⣶⣟⡱⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡽⠥
⠀⠈⢹⢂⡺⣿⣿⡍⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠨⠡⡘⣀⡀
⠀⠰⠰⠻⠇⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠨

⠀⠀⠀⠀⣤
⠀⠀⠀⢀⡿
⠀⠀⠀⣸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣟⠁⡀⣰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠄
⠀⠀⣻⡄⢤⢤⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡯
⠀⠘⠛⠓⠄⣼⡧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⣩⣀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠈⠻⣄⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣯⣯⣼⣟⣿⣷⣦⣤⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣴⣾⣄
⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⣷⣿⠭⣟⣿⣿⡿⠛⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⢀⡤⢚⣿⠿⠋⠀⠀⠀⠀⡀⠐⠾
⡡⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠁⠴⣿⣛⠋⠀⠀⠀⠀⠀⠈⢢⣤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠳⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠒⣤⡀⠀⠀⠀⠀⠘⠣⡀
⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠉⠁⠺⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣶⡶⠤⠤⠤⠄⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠈⠉⠙⠿⠛⠛⠿⣧⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣤⡄
⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⠀⠀⠀⠉⠙⠳⠄
⠀⠀⠀⠀⡄⠤⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠻⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠐⠒
⠀⠀⠀⠀⣠⡀⠐⠄⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠐⠒⠐⠒⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠘⠀⠀⠀⠉⠈
⠀⠀⠂⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⡀⠢⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⠀⢇⠠⡣⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠰⠀⠀⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠂⠀⠀⠉⠁⠀⠐⢦⡀⠀⡀⠰⠄⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⢀⠀⢀⡎⠀⠀⠀⢀⡤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⢳⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠶⠊⣠⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⠀⡠⠊⠀⠈⠁
⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠮⠊⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠐⠁⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠐⠱⣖⣢⠈
⣀⣀⣀⣀⣰⣖⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⠶⢤⣄⡀⠀⠀⠀⠀⠀⣿
⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⣏⠀⠀⠀⣠⡾⠋⢁⣤⡴⠶⠶⠦⣤⡈⠙⢷⣄⠀⠀⠀⣹⠶⠶⠶⠶⠶⠶⠶⠶⠶
⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⠋⢠⡾⠋⠀⠀⠀⠲⠶⣤⡙⢷⣄⠙⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⣿⠀⣸⠇⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢹⡆⠸⡇⠀⣿⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⢹⠷⣤⡀⠀⠀⠀⣿⠀⡿⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⠀⣿⠀⣿⠀⠀⠀⢀⣤⠾⡏
⠀⠀⠀⠘⡇⠀⠙⣦⠀⠀⣿⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿⠀⠀⣴⠋⠀⢸⠇
⠀⠀⠀⠀⠹⣆⠀⣿⠀⠀⣏⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣿⠀⠀⣿⠀⣠⠏
⠀⠀⠀⠀⠀⠈⠳⣿⡀⢀⣿⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⣿⡀⢀⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⠶⠶⢦⣤⡀⠀⠀⠀⠀⠀⣿
⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣯⠀⠀⠀⣠⠾⠋⣀⣤⠴⠶⠶⠶⣤⡀⠙⠷⣄⠀⠀⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶
⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⠋⣠⠞⠉⠀⠀⠀⠒⠶⣤⡙⠷⣄⠙⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⣿⠀⣸⠃⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠹⡆⠘⣇⠀⣿⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⢸⠷⢦⡀⠀⠀⠀⣿⠀⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⠀⢿⠀⣿⠀⠀⠀⢀⡴⠾⡇
⠀⠀⠀⠘⡇⠀⠙⣦⠀⠀⣿⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿⠀⠀⣴⠋⠀⣸⠃
⠀⠀⠀⠀⠹⣆⠀⣿⠀⠀⣇⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⣿⠀⠀⣿⠀⣰⠏
⠀⠀⠀⠀⠀⠈⠻⣿⣀⣀⡿⠀⠀⠀⠀⠀⠙⠶⣤⣄⣀⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⣿⣀⣀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠶⠶⠦⣤⣀⠀⠀⠀⠀⠀⣿
⡶⠲⠶⠶⠶⠶⠶⠶⠶⠶⣟⠀⠀⠀⣠⠞⠋⡀⠠⠔⠒⠒⠀⠄⣀⠙⠳⣄⠀⠀⠀⣹⠶⠶⠶⠶⠶⠶⠶⠶⠖
⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡼⠃⣠⠞⠁⠀⠀⠐⠒⠶⢤⡈⠳⣄⠘⢧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⣿⠀⣸⠃⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠹⡆⠘⣇⠀⣿⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⢸⠿⢦⡀⠀⠀⠀⣿⠀⠿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣿⠀⠿⠀⣿⠀⠀⠀⣀⡴⢿⡇
⠀⠀⠀⠘⣇⠀⠙⣶⠀⠀⡿⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣿⠀⠀⣶⠋⠀⣸⠃
⠀⠀⠀⠀⠹⣆⠀⣿⠀⠀⣷⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⣿⠀⠀⣿⠀⣰⠏
⠀⠀⠀⠀⠀⠈⠻⢿⣀⣀⣿⠀⠀⠀⠀⠀⠙⠳⣤⣄⣀⣀⣠⣤⠞⠋⠀⠀⠀⠀⠀⣿⣀⣀⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣦⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣴⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣏⢉⣉⡉⠉⠉⢉⠉⠉⣉⠉⣹⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⣿⠉⠉⣿⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣷⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠙⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣴⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣤⣴⠶⠶⠾⠷⠶⢶⣤⣄⡀⠀⠀⠀⠀⠀⢸⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⢀⣠⡾⠛⠉⣀⣠⣤⣤⣤⣤⣀⡀⠉⠻⣶⣄⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠘⢿⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠀⠀⢠⡾⠋⣠⣴⡿⠟⠋⠉⠉⠉⠉⠛⠻⣷⣤⡈⠻⣦⡀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⠟
⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⢰⡿⢁⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠘⣷⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⣼⣿⣿⠀⣿⠃⣾⡟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⢹⡇⢸⣿⣿⡄⠀⠀⠀⣤⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣴⣿⣿⣿⠰⣿⣾⣿⠁⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⡿⢸⣿⣿⣿⡄⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⣿⣿⣿⣿⠀⠀⠘⣿⡀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⣿⣿⣿⣿⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⢻⣧⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⢸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⠀⠀⠀⠈⢻⣧⡀⠙⠿⣿⣷⣦⣄⣀⣀⠀⠀⣠⣾⠏⠀⠀⠀⢸⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣽⣿⣶⣤⣀⣉⠉⢉⣉⣠⣴⣾⣿⡅⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠶⠾⠶⠶⢶⣤⣄⡀⠀⠀⠀⠀⠀⢸⣿
⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⣠⡾⠛⠉⢀⣀⣤⣤⣤⣄⣀⡈⠙⠻⣶⣄⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠘⢿⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠀⠀⢠⡾⠋⣀⣴⡿⠟⠋⠉⠉⠉⠉⠛⠻⣷⣤⡈⠻⣦⡀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⠟
⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⢠⡿⢁⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣆⠘⣷⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠈⠙⢿⣦⣄⠀⠀⠀⣼⣿⣿⠀⣿⠃⣾⡟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⢹⡇⢸⣿⣿⡄⠀⠀⢀⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣰⣿⣿⣿⠰⣿⣾⣿⠁⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡿⢸⣿⣿⣷⡄⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⣿⣿⣿⣿⠀⠀⠘⣿⡀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⣿⣿⣿⣿⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⢿⣇⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⢸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⠀⠀⠀⠈⢻⣧⡀⠙⠿⣿⣿⣦⣤⣀⣀⠀⠀⣠⣿⠏⠀⠀⠀⢸⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣽⣿⣶⣤⣀⣉⠉⠉⣉⣡⣴⣾⣿⡅⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠶⠿⠷⠶⣶⣤⣄⠀⠀⠀⠀⠀⠀⢸⣿
⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⣠⡾⠛⠉⣀⣀⣤⣤⣤⣄⣀⡈⠙⠻⣦⣀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠘⢿⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠀⠀⢠⡾⠋⣠⣴⡿⠟⠋⠉⠉⠉⠉⠛⠿⣷⣤⡈⠻⣦⡀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⠟
⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⢠⡿⢁⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠙⣷⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠈⠛⢿⣦⣄⠀⠀⠀⣼⣿⣿⠀⣿⠃⣾⡟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⢹⡇⢸⣿⣿⡄⠀⠀⠀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣴⣿⣿⣿⠠⣿⣼⣿⠁⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⡿⢸⣿⣿⣿⡄⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⣿⣿⣿⣿⠀⠀⠀⣿⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⣿⣿⣿⣿⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⢿⣇⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⢸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠀⠀⠀⠈⢿⣧⡀⠙⠿⣿⣿⣶⣤⣀⡀⠀⠀⣠⣾⠏⠀⠀⠀⢸⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣽⣿⣶⣤⣀⡉⠉⠉⢉⣀⣤⣾⣿⡅⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣠⣾⠏

⠀⠀⢀⡴⣺⣿⣗⠦⡀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣠⢖⣫⣭⣗⢦⡀⠀⠀⠀⢀⣤⠀⠀⠀⠀⢠⣄
⠀⠀⡏⣾⢫⣶⡝⣷⢹⠀⠀⠀⢸⣿⢸⠰⣿⠆⡇⣿⠆⠀⠀⢰⣧⣿⢩⣦⡙⣷⢻⠀⠀⠀⢸⣇⠾⢸⣿⠺⢆⣿
⠀⠀⢧⠻⠎⣿⠱⠟⡼⠀⠀⠀⠀⠛⠀⢠⣿⡀⠈⠛⠀⠀⠀⠈⢧⡻⢊⢉⠸⢏⡞⠀⠀⠀⠀⠋⠀⡐⠐⡀⠈⠋
⠀⠀⠀⠙⣸⣿⣇⠊⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⢂⠉⠀⠀⠀⠀⠀⠀⠠⠥⠤⠥
⠀⠀⠀⢠⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠂

⣿⣿⣦⣀⣈⡧⢳⣟⠓⢧⠆⠉⣾⠻⣷⣿⣿⣿⣷⣽⣯⣯⠉⠀⠸⠏⠰⠀⠻⠈⠛⠈⠋⢿⣶⠼⠃⠀⡟⠀⢸
⣿⣿⣿⣿⣿⣶⠮⢍⡀⠈⠙⠆⠸⠛⠻⣚⣻⣿⣟⣝⡶⢿⣳⠉⠁⣯⣤⣀⠒⠀⠀⠀⠀⠈⡏⠀⠘⠠⡄⠔⢋
⣿⣿⠟⠛⣳⢾⣿⢸⢟⠖⠂⠀⠁⢸⡟⠯⠉⠛⠉⠉⠛⠃⠀⠰⠏⠛⡿⡻⡡⢀⠂⠄⠀⠢⠠⠀⢁⣠⡷⠖⠛
⣿⣿⣿⣿⣧⣼⡿⣛⣋⣴⠶⢀⣘⡓⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⡞⠀⠋⠁⠰⠆⣠⠀⠑⢰⡄⠠⠀⠸⠁⠀⠰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢤⠟⠘⠁⠀⠙⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡅⣿⢸⠃⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣁⣼⣴⡶⠀⠀⠀⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⣶⠞⡿⣵⣿⢿⡇⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣴⣷⠀⠀⠀⠈⣅⡀⠀⠀⠀⠄⠈⠁⢄⠒⠋⠸⡆⠉⡟⠁⢀⠀⠀⢀⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣭⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⢀⣄⢤⠋⠁⠃⠀⠀⠀⠀⠀⡽⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⡴⣢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⠴⠠⠄⠀⠀⣀⠀⠀⠚
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣈⣾⠁⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠀⠈⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣧⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⣰⣿⣅⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠏⣡
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣬
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡶⡟⠃⣄⡀⠀⠀⠀⡶⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠗⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⠿⠶⠀⣷⢹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣿⠏⣹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣰⢉⣶⣴⠀⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠰⢤⣼⣿⣭⣿⡗
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠀⠀⠀⠀⠀⠀⣨⠚⢻⣣⣾⠿⠅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⡠⠬⠍⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣧⠀⠠⠂⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣠⣴⢾⣿⣿⣿⣿⣿⣿⣄⣸⣤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣿⡇⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⣇⠀⠀⡀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡿⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⠴⠕⠂⠁⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠸⠿⠿⢟⠛⠋⠉⠉⠀⣀⣀⣤⣶⢂⠀⠀⠀⡄⡀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⢀⡀⠀⡄⠀⠀⠀⣀⠀⣀⣠⣼⣤⣶⣶⣾⣿⣿⣿⣿⠃⣾⡅⠒⢸⡇⠀⠾⠿⢇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⣀⣧⡀⠒⢾⣃⡀⠀⠸⠦⠴⠒⡛⠦⢭⡿⠿⠛⢉⠛⠛⢟⡉⡀⠐⠻⠇⣈⣼⣿⡰⠎⠀⠈
⠀⠀⠀⠀⠀⠀⢸⣿⠠⠛⠉⠀⡠⡊⡈⠘⠃⠘⠀⠠⠧⠡⠒⠀⢀⠀⠰⠀⢸⡄⠀⠈⠘⠓⠢⣧⡀⠀⠈⢢⠔⠈
⠀⠀⠀⠀⠀⠀⠘⠁⠔⠀⠀⡀⠀⠀⣇⣰⠀⠒⠀⠀⠀⢠⡀⠊⠀⠄⢀⡇⣿⣇⠀⣁⡀⠀⠀⠀⢁⡄⠀⠰
⠀⠀⡄⠀⠀⠀⠀⠄⢀⡀⢀⣤⣤⣤⣝⠀⠀⠤⠀⣠⡄⣸⡇⠲⠥⠠⠚⠛⠛⠉⢁⠁⠂⠀⠀⠈
⠀⠀⠀⠉⠀⢀⡁⠀⡀⣀⣀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠻⠟⠀⠀⢠⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠺⠖⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⢀⢀⣀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣈⣈⣤⠤⠄⠀⠒⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠀⠑⠒⠂⠐⠒⠤⠄⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣄⢀⡄⢀⡀⠀⠀⠐⠀⠀⠀⠀⠖⠐⠴⠖⠐⠂⠀⠀⠂⠀⠈⠉⠀⠀⠀⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠴⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠐⠆⠠⠀⡤⠤⡤⠀⢀⣀⠄⠀⠂⠀⠀⠱⠆⠀⠓⠤⣀⠄⠀⠀⠀⠀⠀⠀⠀⢈⠡⡄⠄⠀⡀
⠀⠀⠀⠀⠀⠄⠐⢀⣈⠠⠀⠀⠀⠀⠀⠀⣠⠶⠒⢛⡤⠖⠚⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠢⠤⠒⠒⠀⠊⠀⠀⠀⢂⢀⠀⠀⠀⠀⠀⠁⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠁⠀⠀⠀⠘⣠⣀⢀⡘⠄⠠⠄⠀⣀⡀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠐⠠⠠⠒⠒⠉⠉⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠲⠤⠀⠀⠈⠁⠉⠉⠁⠀⢺⣄⠀⠉⠉⣉⠚⠓⠂⠠⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣐⢄⠂⢂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⠈⠂⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⡴⠄⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢤⠒⢳⠚⣧⣼⠅⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣼⠙⠳⠈⡆⢰⠁⡰
⠀⠀⠀⠀⠀⠀⠀⠀⢺⢠⢳⡅⠸⣽⠘⠀⠅⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢀⢘⠒⢿⡄⢀⣇⣀⠘⡗
⠀⠀⠀⠀⠀⡆⢀⣄⠀⡃⠋⠁⠀⠁⠀⠙⠓⠂⠑⠊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⢈⠋⠁⠒⢠⠀⡉⠘⠋⠓
⠀⠀⠀⠀⡄⠒⣀⠰⠄⠐⠀⠀⠀⠀⠀⠀⢀⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⢄⠒⢈⠀⡀
⠀⠀⠀⠀⠠⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⡈⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣭⣭⡒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⠟⠁⠈⠻⣦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⠟⠁⠀⠀⠀⠀⠈⠻⣦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⠟⠁⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠻⣦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⠟⠁⠀⠀⢸⣿⣿⣿⠟⠉⠀⠀⠀⠀⠉⠻⣿⣿⣿⡇⠀⠀⠈⠻⣦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⠟⠁⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠈⠻⣦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡑⢄
⠀⠀⠀⠀⠀⡠⢊⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡑⢄
⠀⠀⠀⡠⢊⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⠋⠉⠉⠙⠻⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡑⢄
⠀⡠⢊⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡑⢄
⡜⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⢣⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠑⢌⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡡⠊
⠀⠀⠀⠑⢌⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣤⣀⣀⣤⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠑⢌⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⣦⡀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⣦⡀⠀⠀⢸⣿⣿⣿⣦⣀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡇⠀⠀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⣦⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⣦⡀⠀⠀⠀⠀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⣦⡀⢀⣴⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⣛⣛⠥⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⣀⡀⠒⠲⡤⠊⠉⢶⣄⡀⠀⠀⠀⠀⠀⡔⠈⠻⠄⠈⠯⠀⠐⣤⡤⠤⢄⠀⣀⣀⡀
⠀⠀⠀⠀⡤⠞⠀⠀⠀⠉⠁⣀⠚⠁⠀⣘⡅⠀⠀⠀⠀⠙⠲⠤⠮⠦⡠⠒⠚⠹⢄⠀⠀⢹⣭⠀⠈⠦⢄
⠀⠀⠀⠀⠉⠀⠐⠂⠠⠄⠴⠾⠧⠴⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣄⠀⠀⣤⠄⠀⠀⠀⠀⠀⠈⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠚⠓⠛⠂⠐⠒⠒⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠉⠳⢬⡹⡄⠀⠀⠀⣠⠤⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠻⢧⡈⣶⡌⢧⣿⣞⢦⣀⣴⡿⠀⠈⠻⢯⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⣓⡒⠠⢿⡶⣿⣄⣉⣙⡓⣽⣻⣶⡦⠙⠦⢄⠙⢿⣧⡀
⠀⠀⠀⠀⠀⣠⣦⢠⣾⢟⣀⣤⣉⠃⠀⠀⠙⠮⠻⣦⡈⠻⣮⡟⣮⣿⣶⣤⠬⢷⣛⠿⢿⣆
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣴⣷⠄⠀⠀⠀⠀⣈⣡⣴⣾⣿⡿⠯⠉⠉⠁⠀⠈⠉⠒⡮⢷⣄⠀⠀⢀
⠀⠀⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣴⣾⣿⡿⢿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣷⠶⣾⣗⡀
⠀⠀⠀⠈⠉⠉⠛⢿⠛⠟⡛⠟⠋⠉⠉⠋⠉⠁⠀⡽⠂⢿⣁⣀⡀⠀⠀⠀⠀⠀⠀⣠⣵⠿⠏⠁⠘⣻⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢰⣿⣆⡀⠀⠙⠒⠦⢶⣄⡉⠙⣲⡤⠴⢿⡞⠻⠅⠀⠀⠀⢸⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢄⣾⣿⣿⣿⣿⣿⡄⢀⣤⣤⡔⠷⠞⠛⠛⠷⣮⡍⠀⣈⢱⣤⣀⡀⠀⠚⠛⣿⡟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⠓⠀⠈⠀⠉⠉⠁⠀⠀⠘⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣶⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠖⠒⠉⠉⠀⠉⡀⠹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣠⡀⣴⡦⠶⢤⣤⣄⣠⣽⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⢿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢈⣴⡆⠙⣷⣿⣿⣿⣿⣿⣿⢿⣿⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣇⣿⣿⢸⣿⣥⣿⣏⣿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣶⣾⣿⡿⠇⣠⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⣠⣽⣿⣿⣿⣿⣿⣿⣯⢽⡷⠟⠛⠈⢿⡛⣿⠻⠿⠛⠻⠟⠻⠟⠛⠋⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛⢉⣉⣀⣤⣤⣾⣿⣿⣿⡿⢿⣻⠝⠉⢀⣴⠟⠁⢀⡾⠻⠋⠰⠋⣿⠀⠀⠻⠀⠀⠀⠈⠓⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠒⠊⠉⢀⣤⠶⠋⠀⠀⣠⡿⠁⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈





⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⠠⠴⠭⠦⠤⡀⠀⠀⠀⠀⠀⣤⠛⡼⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠉⠉⠉⠉⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀⢔⢆⣀⠀⠀⠀⠀⠀⢰⡇⠉⠉⠁⠈⠐⠀⠀⠀⠀⠂⠤⠤⠠⠠
⠀⠀⠀⠀⠀⠀⢁⢀⠤⣀⡈⠁⠀⠀⠀⠀⣤⡥⠀⠀⠀⣧⡇⠀⠀⠴⠀⡠⢬⢻⡑⢛⡟⡠⢤⠐⠆⠀⠈⢁⡀⠀⠈⣀⠀⠀⠀⠀⡼⡀⠀⠀⠂⡧⢆
⠀⠀⠀⠀⠀⠀⢤⠤⠡⠡⡤⠄⠀⠀⠀⠸⣐⣃⣀⣁⣀⣒⡈⠀⠀⠀⠀⠑⠊⠀⠈⠁⠀⠀⠊⠀⠀⠀⠰⣌⡁⠈⠀⣉⡄⠀⠀⠘⢄⣇⠀⠒⢀⣚⡰
⠀⠀⠀⠀⠀⢀⣊⣀⣀⣀⣹⡀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠈⠋⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠠⠤⠄⡀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⡄
⠀⠀⠀⠀⠀⠀⠂⠠⠂⠠⢰⠀⠀⠀⠀⠀⠠⠁⢠⠶⡄⠊⠂⠀⠀⠀⠀⠀⢀⣀⠀⠀⠡⣉⠅⠀⠀⠀⠀⠀⡾⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⣀⢄⡠
⠀⠀⠀⠀⠀⠀⠣⠈⠒⠁⡈⠀⠀⠀⠀⠀⠐⢄⠀⠓⠂⢀⠂⠀⠀⠀⠀⢐⢰⣢⢐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠱⠀⠀⠀⠀⠀⠠⡃⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠳⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠞⠁⠀⠀⠀⠀⠀⠀⢂⣠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⠀⢃⠀⠀⠀⠀⠀⠐⡀⢀⠎⠂
⠀⠀⠀⠀⠀⠀⢠⣀⣁⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠀⠉⠲⠀⠀⠀⠀⠀⠀⠽⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠤⣀⠀⠀⠀⠀⠀⡀⡤⡤⡄⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠘⢦⡩⡩⠀⠀⠀⠀⠈⠀⠀⠀⡧⡀⠀⠀⠀⠀⠀⢠⠃⠈⠄⡰⠣⠀⠀⠀⠀⠎⠀⠀⢁⣤⠀⢣⠀⠀⢀⠊⠀⠀⡇⠰⠈⢂
⠀⠀⠀⠀⠀⠐⢹⠐⠖⠶⠂⡏⠂⠀⠀⠀⠀⠠⠀⠀⢠⣇⠤⠃⠀⠀⠀⢠⠋⠉⢄⠘⠉⠢⠣⠀⠀⠀⡀⢀⡴⠗⠁⠀⢸⠀⠀⠸⠂⠛⠐⡇⢘⠃⠺
⠀⠀⠀⠀⠀⠀⢸⠀⡊⢱⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠘⡖⠒⠃⠀⠀⠐⢌⡀⠂⢀⠤⠃⠀⠀⠀⠑⢄⣄⣇⣄⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠈

⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⣠⠲⠶⣤⠀⠀⠀⠀⡆⠄⠠⠐⠠⠠⠔⡄⠀⠀⠀⠀⠀⠒⠒⠒⡄⠀⠀⠀⠀⠀⠀⢀⠒
⠀⠀⠀⠀⠀⠀⢸⠁⣴⡦⠈⡂⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠀⠁⠀⡇⠀⠀⠀⠀⣀⠀⢅⡀⠀⢆⣀
⠀⠀⠀⠀⠀⠀⢸⠀⠨⢁⠀⠃⠀⠀⠀⠀⠀⢀⠜⠷⠤⠤⠞⠁⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠃⠀⡇⠀⠀⠀⠈⠐⠄⠌⠀⠀⠔⠊
⠀⠀⠀⠀⠀⠀⠘⠐⢲⡖⠚⠀⠀⠀⠀⠀⡔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⢀⠤⢀⠀⡠⠃⠀⠀⠀⠀⠐⠀⠂⠀⠇⠀⠀⠀⠀⠀⠀⠑⣠⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡀
⠀⠀⠀⠀⠀⠀⠀⡰⢆⢀⠶⡀⠀⠀⠀⠀⠀⠀⢰⣐⠠⡍⠧⠀⠀⢀⢄⣀⣀⠀⠀⠀⢀⣀⣀⡄⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⡋⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠑⠎⠈⠖⠁⠀⠀⠀⠀⠀⠀⢸⠈⠀⡇⠀⠀⠀⠑⠖⠀⠐⡟⢂⠐⡛⠂⠐⠖⠉⠀⠛⠤⠄⠀⠀⠈⠃⠁⠀⠐⠎⡭⠬⠿⡭⢬⡡⠂
⠀⠀⠀⠐⠎⡭⠭⡁⠀⠀⠆⡉⠭⡉⠖⠀⠀⠀⢸⢀⠀⣧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠆⠶⠀⠀⠀⢈⡁⠆⠀⠀⠀⠠⠀⡦
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠙⡜⠁⠓⠂⠀⠀⠀⠀⠀⠀⠣⠄⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣼⡇⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣴⠋⠀⠀⠀⠈⢳⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⠀⠀⠀⢸⡇⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣿⠀⠀⠀⢀⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⣸⡇⠀⠀⠀⢳⡀⠀⠀⠀⠀⣰⠇⠀⠀⠀⣿⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠙⠓⠶⠒⠛⠁⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⢸⠋⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⠀⢀⣤⠶⠒⠲⢦⣄⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠋⠛⢻
⠀⠀⠀⠀⠀⠀⠘⠷⣤⡀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡞⠁⠀⠀⠀⠀⠙⣆⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀⠀⢸⡇⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣿⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢾⡇⠀⠀⠀⠹⢦⣀⡀⣀⣠⠞⠁⠀⠀⠀⣿⠞⠋⠁
⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⣼⡇⠀⠀⠀⠀⠀⢉⣉⡉⠁⠀⠀⠀⠀⠀⣿⡤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠞⠋⠉⠉⠛⢦⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⡀⠀⠀⢸⡇⠀⠀⢠⡏⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣿⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣸⡇⠀⠀⠈⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣿⣠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠈⠳⢦⣤⣤⠶⠋⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠚⠃

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

⣶⣿⣿⣿⣿⣿⣿⡿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠿⠿⠶⠶⠶⠿⠿⠿⣿⣶⣦⣤⣀
⣿⣿⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠀⠀⠉⠛⠛⠛⠻⠿⠿⠶⠶⠶⢶⣶⣶⣶⣆
⣿⣿⣟⠁⢀⣀⣀⣀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡆
⣿⣿⣟⡁⢸⣿⣿⣿⣿⣿⣿⣧⣤⣤⡀⠀⣤⣤⡄⠀⢠⣤⡄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣿⣽⡿⠁
⣿⣿⠟⠉⠉⠀⣻⣿⣿⠓⢺⣿⣟⣻⣿⢉⣿⣿⣿⡀⣻⣿⡏⣿⣿⠋⠉⠉⢙⣉⣀⣒⢒⣶⣶⡿⠛⠋⠉⠁
⣿⣿⣛⠃⠀⠀⣿⣿⣿⠀⢸⣿⡿⣿⣏⣸⣿⣿⣿⣧⣿⣿⡇⣿⣿⣤⣤⠀⠉⠉⠉⠛⣿⣯⣾⡇⡀
⢹⣿⡿⠂⠀⠀⠻⠿⠿⠀⠸⠿⠇⠈⠿⠿⠟⠉⠛⠻⠿⠻⠃⠛⠻⠿⠟⠀⠀⠠⠭⣭⣿⣯⣿⠇
⢸⣿⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢚⣿⣿⣿⠟
⢸⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⣺⣿⣿⣿⠉
⢸⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠠⠀⠀⠠⠤⠄⠀⠀⢀⣿⣿⣿⡇
⢸⣿⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣼⣿⡿⠋⠁
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⢻⣿⡏⢻⡟⢿⣿⣿⡟⠛⠛⠛⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠘⠇⢸⣏⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣟⣿⡇
⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⡆⢸⣯⣿⡇
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⢸⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠸⣆⠀⢳⡀⢠⡄⠀⠀⠀⠀⠀⣿⡇⠀⡇⠈⣏⣹⡇⠀⠀⠀⠀⠀⣀⢴⠇⠀⠀⠀⠀⣠⠞⠉
⠀⢀⠹⡄⠈⣷⠀⢷⠀⠀⠀⠀⠀⣿⡇⠀⠁⠀⣿⢻⡇⠀⠀⠀⠀⠜⣱⠃⠀⣠⠄⣠⠞⠁
⠀⠀⠂⢨⡀⠸⣇⠘⣇⢃⠀⠀⠀⣿⡇⢸⠀⠀⣿⢾⡁⠀⠀⠀⠀⣼⠃⠀⡴⢃⡾⠃⢀⡀
⠀⠐⡄⠈⢷⠀⢻⣶⡘⣼⡄⠀⠀⣿⡇⢸⠀⠀⣿⣾⡁⠀⠀⠀⣼⠃⠀⡼⢡⠟⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢻⡄⠈⣦⠀⣿⣷⢹⣷⢰⠀⣿⡇⢸⡀⠀⣿⣿⠇⠀⡘⣰⠏⣼⡿⣡⣿⢀⠜⡁⢀⣀⣠⢴⡶⠛⠁⠀⢀⡀
⠀⠀⠈⣷⡀⠘⣦⠸⣷⡌⣿⣿⡇⣿⡇⢸⠁⠀⣟⣿⠀⣰⢣⠏⣼⡿⣱⡿⣡⠋⢊⣴⢟⣩⠶⠋⠀⠀⢀⠐⠋
⠀⠀⣆⠘⣿⣦⠹⣷⠙⣿⣿⣿⣇⢹⡇⠀⡈⢠⣏⣿⣦⢳⡏⡼⣻⣷⠟⣱⠟⣴⢟⣵⠛⠁⠀⠀⠀⠀⠁⠀⣀⣀⡄
⠀⠀⣿⣦⣿⣿⣧⢘⢷⣌⢿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣇⡾⣰⢳⣿⢋⣼⣫⡾⣣⣿⠿⠃⢀⠴⣀⡄⠀⢀⣉⣉⣩⣤
⠀⠀⢸⣿⣿⣿⣿⣿⣧⠻⣆⢻⣿⣿⣇⠀⠀⢸⣷⣿⣿⣴⣧⣸⣧⣿⣿⣿⣿⣿⣧⣤⣴⣿⣾⣿⣿⣦⣉⣀⣀⣠⣄
⠀⠀⢸⡿⠿⠿⠿⠿⠿⢦⠿⠿⠿⢿⣿⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶⠾⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠸⠃⠀⠀⠀⢀⣀⣀⣲⣾⡿⠛⢋⣭⣭⣭⣭⣭⡍⠩⠭⠥⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⡶⠦⠴⠶⠶⠾
⠀⠀⠀⠛⠛⠓⠒⠒⠛⠃⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠁⠀⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⠋⠁⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⠉⠻⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⣀⣤⣴⣶⣶⣤⣄⡀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⠿⠿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⡤⠂⠀⠰⠁⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⢺⡗⠀⠀⢴
⢀⢠⠄⠀⠀⠈⣠⠤⠒⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠉⡤⠉⠀⠂⠀⠸⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⢀⡀⢠⡆
⠁⠀⠀⠀⠐⡆⠀⠀⢊⡥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⢁⡤⠔⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠋⢀⡸⢋
⠀⣀⠀⠀⢺⣿⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣤⡿⣤⣆⣿⣾⣿
⠀⠛⠀⠠⣿⣿⡧⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠛⠛⠛⠛⠻
⠀⠀⠐⣰⣾⣿⣗⠂⠀⠀⠀⣠⠀⠲⢌
⠀⠀⠀⣸⣽⣟⣧⠁⠀⣀⣚⠟⠀⣤⣈⡓
⠀⠀⠐⠟⢰⡟⠁⠀⠄⠀⠉⠙⠈⠉⠁
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠷⠤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣄⠀⠇⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⣶⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⠟⢛⣛⡛⠻⢿⣿⣿⡟⢛⢻⠟⣛⢻⡟⣿⡟⢻⠟⣛⣻⠛⣛⣿⣿⣿⠿⠛⣛⣛⠛⢿⣿⣿⡄
⠀⣿⣿⣿⡿⢡⣾⠟⠛⠻⣷⡄⢻⣿⣇⣴⣾⣄⣋⣼⣇⣛⣧⣸⣄⣋⣽⣀⣛⣿⣿⢃⣴⡿⠛⠻⢿⣆⠹⣿⡇
⠀⠈⠛⠻⠇⢾⣏⠸⣿⡇⣸⣿⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⢸⣿⠰⣿⡿⢸⣿⠀⠛
⠀⠀⠀⠀⠀⠈⠿⣷⣤⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣴⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠀⠰⣟⠋⠉⠉⠉⠉⠙⣳⠆⠀⢿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣤⣤⢠⣄⢻⡶⠖⠛⠛⠶⢶⡟⣠⡄⣤⣤⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡟⣧⠀⠀⠀⠀⠀⠀⣼⢻⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣇⣿⠀⠀⠀⠀⠀⠀⣿⣸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣤⣀⣀⣀⣀⣀⣀⣤⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣦⢰⡆⠀⠀⠀⠀⣿⢀⣿⡁⠉⠉⠉⠉⠉⠉⢈⣿⡀⣿⠀⠀⠀⠀⢰⡆⣴
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠻⣦⡀⠀⠀⠻⣄⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⣠⠟⠀⠀⢀⣴⠟⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⣙⠛⠀⠀⠉⠛⠷⠶⠶⠶⠶⠶⠶⠾⠛⠉⠀⠀⠛⣋⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⢀⢀⠀⣀⠀⡀⠀⢰⢀⠀⢀⢰⢴⢂⢀⠄⠀⣤⠀⢀⣀⢀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠘⠈⠂⠁⠃⠘⠀⠈⠈⠐⠘⠈⠈⠀⠙⠀⠘⠈⠃⠈⠀⠀⠈

⡟⠉⣻⣛⣻⣛⣛⣟⣻⣛⣻⢛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣻⣿⣟⣻⣟⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠶⠾⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠷⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⡇⢠⡶⠶⠶⠶⠶⠶⠶⠶⡶⠶⠶⠶⠶⠶⠶⡆⢸⡟⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠷⠾⠿⠿⠿⠿⠿⠿⠷⠿⡇⢸⡇⢸⡇⠀⠀⣠⡶⠟⠛⠛⠛⠛⢷⣦⡀⠀⠀⡇⢸⡇⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡄⠀⠀⠀⠀⡇⢸⡇⢸⡇⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⡇⢸⡇⣿⠀⠀⠀⢀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⡇⢸⡇⢸⡇⠀⠙⢷⣄⡀⠀⠀⠀⢀⣠⡴⠟⠀⠀⡇⢸⡇⣿⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠻⣇⣸⡇⢸⣇⣀⣀⣀⣉⣙⣛⣛⣛⣛⣉⣀⣀⣀⣀⡇⢸⣇⣿⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣽⡇⢠⡭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⡅⢸⣯⣭
⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡇⢸⡇⢸⡇⠀⠀⣀⣴⡶⠾⠿⠿⠶⣦⣄⠀⠀⠀⡇⢸⡇⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⡄
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⡇⢸⡇⣿⠀⠀⠀⠀⣠⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⡇⢸⡇⢸⡇⠀⠹⣦⣀⠀⠀⠀⠀⠀⢀⣠⡾⠁⠀⡇⢸⡇⣿⠀⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⢾⡇⢸⡇⢸⡇⠀⠀⠈⠙⠻⠶⠶⠶⠞⠛⠁⠀⠀⠀⡇⢸⡇⣿⠶⠶⠶⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⡇⢈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃⢸⡟⠛
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡟⢻⡇⢸⡇⠀⠀⠀⣀⣤⣴⣶⣦⣤⣄⡀⠀⠀⠀⡇⢸⡟⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⢸⡇⢸⡇⠀⣠⡾⠋⠁⠀⠀⠀⠀⠉⠻⣦⡀⠀⡇⢸⡇⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡇⢸⡇⣿⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⡇⢸⡇⢸⡇⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⡇⢸⡇⣿⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣀⣀⣠⡇⢸⡇⢸⡇⠀⠀⠙⠻⢶⣤⣤⣤⣴⠾⠛⠁⠀⠀⡇⢸⡇⣿⣄⣀⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣧⣼⡇⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠇⢸⣧⣿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠃⠀⠀⠀⠀⠀⣤⣤⣤⡄⣤⢠⠤⣤⠄⣤⣤⣤

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⢀⣾⣿⣦⡀⠀⠀⠀⠀⠙⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣶⣦⣄⡀
⠀⢀⣴⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠈⠉⠉⠉⠉⠙⠛⠛⠃⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠃⠀⠀⠈⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉
⠀⣠⣤⡊⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⡀
⠀⠀⠉⢻⣿⣿⣷⣶⣄⠀⠀⣿⣿⣯⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣴⡿
⠀⢀⣤⣿⣿⣿⣿⣿⣿⡇⠀⠈⠛⠿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠄⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⣠⣴⣿⣿⠋⠀⠀⠠⣶⣾⣏⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⡿⠋
⠀⠀⢠⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢻⣿⣿⣷⣶⣤⡀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⡏
⠀⢰⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⣿⣿⣿⡧⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠛⠛⠛⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢉⣈⣍⣉⠉⠀⠀⠉⠉⠉⠉⠉⠁⠉⠉⠙⠛⠛⠉
⠀⠀⠀⠀⢴⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠛⢿⣷⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣾⣿⣿⣿⣗
⠀⠀⢰⣶⣶⣿⣿⣿⣿⣶⣶⣶⡆⠀⠀⠀⠀⠘⣿⣧⡀⠀⠘⣿⡇
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶
⠀⠀⢰⣶⣦⣄⠀⠀⠀⢀⣠⣴⣶⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠸⣿⣿⣤⣤⣤⡀
⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠃
⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣯⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⢹⣿⡆⠀⠀⠀⣿⣿⠀⠀⠀⢰⣿⡏⠀⠀⠀⢀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⢻⣿⡀⠀⠀⢿⡿⠀⠀⢀⣿⡟⠀⠀⠀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠿⣿⣶⣤⡀⠀⠀⠀⠛⠛⠀⢀⣠⣴⣾⣿⡿⠿⠿⣷⣦⣄⡀⠀⠛⠛⠀⠀⠀⢀⣤⣶⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣶⣤⣉⠻⣿⣦⡀⠀⠀⣴⣾⡿⠟⠋
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠉⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⢻⣷⡄⠀⠉⠁⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠻⠿⢿⣿⣷⣶⣶⣦⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣷⠀⠀⣴⣶⣶⣾⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣤⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⠣⡟⢇⠙⢯⣷⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠋⢻⠦⣈⠶⣟⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠁⠀⢀⣧⠀⠛⠈⠉⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠄⡂⢁⡀⠁⠀⠂⠄⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠑⠢⣂⡠⠐⠀⢀⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⢀⠠⠐⠡⠊⠀⠀⠀⠀⠀⠀⠠⠂⡩⡄⠀⢸⡦⠀⠁⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡀⠄⠂⠁⠀⠐⠀⠀⠀⢀⠠⠀⠀⠀⠒⣆⡸⠃⡀⠜⠑⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣾⣶⡡⠜⠖⠒⡒⠄⠓⠺⢇⠀⠀⠆⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⢿⡄⠀⠠⡄⣄⠈⢀⡸⠂⠁⠀⠈⣄⣀⡀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⢤⠃⠃⠀⠀⠀⢠⡀⠛⠁⠀⠀⡠⠀⠀⠢⠈⠆⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡇⠐⠄⠀⠀⠀⢈⣚⠗⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠖⠂⡉⢁⡀⠠⠀⠚⢱⡆⠀⠈⢠⡀⠈⠟⠟⣀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠈⡄⠑⠠⠤⠔⢲⣾⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠠⠀⢀⠘⠆⠤⢠⠖⠉⠘⠘⠀⣤⠐⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣧⡔⠉⠐⠄⡀⠀⠀⡇⢀⠀⠀⠀⠀⠊
⠀⢀⠌⠀⠀⠀⠀⢀⡤⢂⡂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⢹⠀⠀⠀⠀⠀⠒⢠⡇⠘⠀⢀⠔
⠀⡌⠀⠀⠀⠀⠀⠞⠀⠠⡇⠀⡀⠀⠀⠤⠀⢀⠀⠀⠆⠆⠀⠀⠀⠀⠀⠆⠸⢃⡆⠐⠁
⠐⠀⠀⠀⠀⠀⠀⠈⠄⢠⠇⠊⠀⠀⠁⠀⠀⠀⠀⡉⣿⠀⠀⢀⢤⡄⣸⠤⣴⡐
⠠⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠃⣀⠔⣋⢀⠀⢇⢠⣾⠁
⠀⢂⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠂⢋
⠀⠀⠑⠀⠐⠁⠀⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⢡⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠱⢄⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠠⠦⠄⢀⣀⠀⣀⠠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢸⢿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠘⡜⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠘⢮⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠑⢽⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠙⢮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣮⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⠶⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⡄⠀⢰⢸⢸⢰⠀⠀⡀⠀⠀⡄⠀⡄⢸⣸⢰⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡀⡇⢠⣿⢸⣿⢸⣤⢀⡇⣀⠀⡇⣄⣿⢸⣿⢸⢰⢸⡄⡀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢠⣇⣿⣾⣿⣾⣿⣼⣿⣾⣇⣿⣦⣧⣿⣿⣿⣿⣾⣿⣼⣇⡇⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠸⡟⣿⣿⣿⢿⣿⢿⣿⣿⡟⣿⡿⡿⣿⣿⣿⣿⢿⣿⢿⡟⡏⠀⠀⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠃⡇⠘⣿⢸⣿⢸⠿⠘⡇⠉⠀⡇⠋⣿⢸⣿⢸⠹⢸⠇⠃⠀⠀⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠃⠀⠹⢸⢹⢸⠀⠀⠁⠀⠀⠃⠀⠇⢸⢻⠸⠀⠈⠀⠀⠀⠀⠈⠙⠛⠛⠛⢻⡄⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣤⣤⣤⣤⣤⣬⣷⡄

⣶⣿⣿⣿⣿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠿⠿⠿⠿⠷⠾⠿⠿⠿⣶⣶⣤⣀⡀
⣿⣏⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠓⠒⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣆
⣿⣿⣯⡀⣤⣤⣤⣤⣤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿
⣿⣿⣷⠄⠿⢿⣿⣿⣟⣿⣷⣶⣶⡄⣴⣿⣧⠀⣼⣿⣤⣿⣧⠀⠀⣀⣀⣀⣀⣀⣀⣀⣴⣶⠿⠾⠋
⣿⣯⣥⠀⠀⢸⣿⣿⠀⢸⣿⣷⣿⠇⣿⣿⣿⡆⣿⣿⢸⣿⡇⠀⠀⠰⠶⠶⣶⣼⢿⣿⠁
⣿⣿⣦⠀⠀⢸⣿⣿⠀⢸⣿⡍⢿⣶⣿⠛⢿⣿⣿⣿⢸⣿⣿⣿⠀⠀⠀⠲⢲⣿⢻⡿
⢸⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢽⣿⣟⣿⠇
⣼⣿⣟⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⠤⠤⠤⠤⠤⠤⠤⠶⣶⣿⡟⠋⠁
⢻⣿⡿⠶⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠤⠤⠤⠀⠀⢀⣻⣟⣿⠃
⢸⣿⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣄⣤⣼⣿⡿⠋
⠈⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠙⣿⢹⣿⣿⡟⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠙⠀⣧⣼⡇
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡿⣿⡇
⠀⠘⠓⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⠀⣿⢿⡇
⠀⢤⠀⠸⡄⠀⡀⠀⠀⠀⠀⢸⣿⠀⢸⠀⣿⢻⡇⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⣠⡶⠋
⠀⠈⢧⠀⢻⡄⢳⠀⠀⠀⠀⢸⣿⠀⠘⠀⣿⢾⠇⠀⠀⠀⢀⠎⠝⠁⢀⠀⣠⡾⠋
⠀⠘⠈⣂⠈⣧⠈⣇⢂⠀⠀⢸⣿⠀⡄⠀⣿⣼⠀⠀⠀⠀⢀⡞⠀⣠⢋⡴⠋⢀
⠀⢠⡀⠹⡄⠸⣷⡘⣼⡀⠀⢸⣿⠀⡇⠀⣿⣼⡀⠀⠀⢀⡞⡀⢰⢃⡞⠁⡰⠋⠀⠂⠀⠀⠀⠀⢀
⠀⠈⢷⠀⢱⡀⢻⣷⢸⣧⣦⢸⣿⠀⡇⠀⣿⣿⠃⠀⠆⡾⢰⣷⢣⡿⢀⠎⠀⣀⣀⠤⡶⠟⠁⠀⣀
⠀⠀⠘⣇⠀⢳⡌⢿⣆⣿⣿⠀⣿⠀⡇⠀⣿⣿⠀⡜⡼⢡⣿⢣⡿⣡⠃⣠⡾⣫⠴⠋⠀⠀⡠⠈⠁
⠀⠀⣆⢹⣿⡌⢿⡌⢿⣿⣿⣇⣿⠀⠰⠀⣿⣿⡟⣽⢣⢯⣿⠟⣵⢋⡾⣫⣞⠁⠀⠀⠀⠀⠀⣀⣤⡤
⠀⠀⣿⣿⣿⣿⣦⡽⣧⠻⣿⣿⣿⠀⠀⢸⣿⣿⣷⢃⡏⣿⢋⣾⣷⣿⣼⡟⠋⢀⣔⣡⣦⡀⠤⠤⠴⠶⠆
⠀⠀⣿⣿⣿⣿⣿⣿⡈⣧⣽⣿⣿⡆⣄⣰⣿⣿⣿⣿⣧⣿⣿⣿⣟⣛⣛⣳⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶
⠀⠀⢿⠍⠉⠉⠉⠀⠉⣭⣭⣭⡿⠿⠿⠛⡛⠛⠛⠟⣛⣛⣛⣛⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣿
⠀⠀⠠⠤⠤⠤⠤⠽⠟⠻⠿⠶⠀⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠏⠉⠉⠀⣉⣉
⣿⣿⡿⢿⡿⠿⢿⠿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠉⠑⠀⠀⢀⣤⣤⣄⠀⠀⠐⣿⣶⣿⠂⠀⠀⣠⣤⣤⡀⠀⠀⠊⠉⠍⠑⠀⠀⢀⣤⣤⣄⠀⠀⠐⣿⣶⣿⠂⠀⠀⣠⣤⣤⡀⠀⠀⠊
⢀⣘⠀⠀⣼⣿⣿⣿⡇⠀⢘⡙⠛⢛⡇⠀⢸⣿⡿⣿⡇⠀⠀⣃⣀⢀⣘⠀⠀⢸⣿⣿⣿⡇⠀⢸⡛⠛⢛⡇⠀⢸⣿⡿⣿⡧⠀⠀⣃
⠛⠛⠀⠀⢣⣤⣤⣤⠃⠀⠘⠛⠛⠛⠃⠀⠘⣤⣤⣤⡜⠀⠀⠛⠛⠛⠛⠀⠀⢣⣤⣤⣤⠃⠀⠘⠛⠛⠛⠃⠀⠘⣤⣤⣤⡜⠀⠀⠛
⣀⡀⠀⠀⢸⣷⣶⣿⠀⠀⠀⣀⣀⣀⠀⠀⠀⣿⣶⣾⡇⠀⠀⢀⣀⣀⡀⠀⠀⢸⣷⣶⣿⠀⠀⠀⣀⣀⣀⠀⠀⠀⣿⣶⣾⡇⠀⠀⢀
⣿⣿⠀⠀⣞⠛⠛⠛⡆⠀⢸⣟⣿⣻⡇⠀⢰⠛⠛⠛⣳⠀⠀⣿⣿⣿⣿⠀⠀⡞⠛⠛⠛⡆⠀⢸⣟⣿⣻⡇⠀⢰⠛⠛⠛⢳⠀⠀⣿
⣄⡉⠂⠀⠸⠿⠿⠿⠁⠀⠚⠁⠀⠈⠗⠀⠈⠿⠿⠿⠃⠀⠐⢋⣤⣤⡙⠂⠀⠹⠿⠿⠿⠁⠀⠚⠁⠀⠈⠑⠀⠈⠿⠿⠿⠇⠀⠐⢋
⣼⡧⠀⠀⠀⣀⣀⡀⠀⠀⠠⠓⠒⠚⠄⠀⠀⢀⣀⣀⠀⠀⠀⢼⣧⣼⡧⠀⠀⠀⣀⣀⡀⠀⠀⠠⠒⠒⠒⠄⠀⠀⢀⣀⣀⠀⠀⠀⢼
⠿⢷⠀⠀⣸⣿⣿⣿⡄⠀⢰⠄⠀⠠⡆⠀⢠⣿⣿⣻⡇⠀⠀⡾⠿⠿⢷⠀⠀⢸⣿⣿⣿⡆⠀⢰⠤⠀⠠⡆⠀⢰⣿⣻⣿⡇⠀⠀⠾
⠷⠿⠀⠀⢏⣉⣉⣉⠇⠀⠸⠿⠿⠿⠇⠀⠸⣉⣉⣉⡹⠀⠀⠿⠾⠷⠿⠀⠀⢏⣉⣉⣉⠇⠀⠸⠿⠿⠿⠇⠀⠸⢉⣉⣉⡱⠀⠀⠿
⠀⠀⠀⠀⢸⣯⣭⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣭⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣽⡇
⡿⣷⠀⠀⣼⠿⠿⢿⡄⠀⢰⣟⣿⣿⡆⠀⢠⠿⠿⠿⣧⠀⠀⣾⢿⡿⣷⠀⠀⡼⠿⠿⢿⡄⠀⢰⣿⣿⣿⡆⠀⢠⠿⠿⠿⣧⠀⠀⣾
⠉⠻⠄⠀⢳⣶⣶⣶⠃⠀⠼⠋⣉⠙⠧⠀⠘⣶⣶⣶⡞⠀⠠⠟⠉⠉⠻⠄⠀⢳⣶⣶⣶⠃⠀⠼⠋⣉⠙⠧⠀⠘⣶⣶⣶⡎⠀⠠⠟
⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣛⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⣠⠀⠀⢰⣿⡿⣷⡄⠀⢠⣿⣿⣿⡄⠀⢀⣾⢿⢿⡆⠀⠀⣄⠀⠀⣠⠀⠀⢰⣿⡿⣷⡀⠀⢠⣿⣿⣿⡄⠀⢀⣾⢿⢿⡆⠀⠀⣄
⣤⣾⠀⠀⡼⠛⠛⠛⡇⠀⢸⣆⣤⣴⡇⠀⢸⠟⠛⠛⢷⠀⠀⣷⣤⣤⣾⠀⠀⡾⠛⠛⠻⡇⠀⢸⣦⣤⣴⡇⠀⢸⠟⠛⠛⢯⠀⠀⣷
⠁⠀⠀⠀⢸⣟⢛⣷⠀⠀⠀⠈⠉⠁⠀⠀⠀⣾⡛⣻⡃⠀⠀⠀⠉⠉⠀⠀⠀⢸⣟⢛⣷⠀⠀⠀⠈⠉⠁⠀⠀⠀⣾⡛⣻⡆
⣶⣦⠀⠀⣸⣿⣿⣿⡀⠀⢠⣶⣶⣶⡄⠀⢀⣿⣿⣿⣇⠀⠀⣴⣶⣶⣦⠀⠀⣸⣿⣿⣿⡀⠀⢠⣶⣶⣶⡄⠀⢀⣿⣿⣿⣇⠀⠀⣴
⠛⢿⠀⠀⢯⣤⣤⣬⠇⠀⣸⠗⠛⠺⡇⠀⠸⣥⣤⣤⡝⠀⠀⡿⠛⠛⢿⠀⠀⢯⣤⣤⣬⠇⠀⣸⠗⠛⠺⡇⠀⠸⣥⣤⣤⡽⠀⢀⡿
⢶⡆⠀⠀⠈⠉⠉⠉⠀⠀⠀⣀⣀⣀⠀⠀⠀⠉⠉⠉⠁⠀⠀⢰⡶⢶⡆⠀⠀⠈⠉⠉⠉⠀⠀⠀⣀⣀⣀⠀⠀⠀⠉⠉⠉⠁⠀⠀⢰
⣿⣏⠀⠀⢠⣤⣦⣤⡀⠀⠈⠀⠄⠀⡁⠀⠀⣤⣤⣦⡄⠀⠀⣹⣿⣿⣏⠀⠀⢠⣤⣤⣤⡀⠀⢈⠀⠄⠀⠁⠀⠀⣤⣴⣤⡄⠀⠀⢹
⣉⣽⠀⠀⣼⠿⠿⢿⡇⠀⢸⣅⣀⣨⡇⠀⢸⡿⠾⠾⣧⠀⠀⣯⣉⣉⣽⠀⠀⣼⠿⠷⢿⡇⠀⢸⣅⣀⣨⡇⠀⢸⠿⠾⠾⣧⠀⠀⣯
⠛⠉⠀⠀⢱⡶⠶⣦⠁⠀⠈⠙⠛⠋⠁⠀⠈⣴⠶⢶⡎⠀⠀⠉⠛⠛⠉⠀⠀⢱⡶⠶⡦⠁⠀⠈⠙⠛⠋⠁⠀⠈⢴⠶⢶⡎⠀⠀⠉
⣤⣄⠀⠀⢸⣿⣾⣿⠀⠀⢀⣤⣤⣤⡀⠀⠀⣿⣿⣿⡇⠀⠀⣠⣤⣤⣄⠀⠀⢸⣿⣾⣿⠀⠀⢀⣤⣤⣤⡀⠀⠀⣿⣷⣿⡇⠀⠀⣠
⢯⣿⠀⠀⢞⣉⣋⣙⡇⠀⢸⣿⠿⣿⡇⠀⢸⣋⣙⣉⣻⠀⠀⣿⡽⣯⣿⠀⠀⣟⣉⣋⣙⡇⠀⢸⣿⠿⣿⡇⠀⢰⣋⣙⣉⣳⠀⠀⣿
⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠈⣤⣤⣤⠃⠀⠀⠛⠛⠛⠃⠀⠀⠡⠀⠀⠌⠁⠀⠘⠛⠛⠛⠀⠀⠈⣤⣤⣤⠁⠀⠈⠛⠛⠛⠃⠀⠀⠁
⡿⣿⢷⢶⠶⠶⢶⠶⠶⣶⣿⣿⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣿⣿⣶⡶⣶⡶⣶⡶⣶⣾⢿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡤⠤⠴⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠟⠛⠋⠉⠁
⠀⠀⠀⠀⣠⣴⣿⣿⠟⠁
⠀⠀⡠⣾⣿⣽⣿⣇⡠⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⣀⣀⠀⠀⠀⣀⡀
⠀⠀⡇⢿⣿⣿⣟⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣷⣶⣾⣭⣤⣖⣂⡠⠄⢀⣀
⠀⠀⠘⠪⣝⡻⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⣛⡛⠿⢿⣿⣿⣷⣶⣾⣭⣔
⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠛⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠿⠭⠭⢭⣟⣛⣛⠿⠿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠮⢭⣛⡻⠿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢬⣍⣛⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲



⠀⠀⠀⠀⠀⠔⠀⢀
⠀⠀⢘⠀⠀⠀⢀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠠⠄⢀⡀
⠀⠀⠈⠢⣀⠀⠀⠉⠑⠂⠀⠀⠀⠀⠒⠒⠒⠒⠀⠀⠀⠀⠀⠒⠒⠒⠒⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠉⠑⠒⠂⠤⢄⣀
⠀⠀⠀⠀⠀⠉⠓⠒⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⣀⣀⣀⠀⠀⠀⠀⠈⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠢⠤⣄⣀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠤⠠⠴⠦⠄⠤⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢛⠉⠀⠀⠀⠈⠉⢹⠆⠀⠀⠀⠉⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠸⢄⠁⠀⠀⠄⠀⢀⣀⠀⠠⠄⠀⠊⠠⠂⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⡨⠒⠀⠁⠀⠀⠀⠀⠉⠐⠀⢄⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⣸⢃⢖⠠⠄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠐⠙⠻⢂
⠀⠀⠀⠀⠀⢰⠃⠈⠐⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠈⠉⠀⠘⡆
⠀⠀⠀⠀⠀⡍⠀⠀⠀⠄⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠠⠀⠀⠀⢩
⠀⠀⠀⠀⢀⡆⠤⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⡠⢰⡀
⠀⠀⠀⠀⠈⠇⠀⠐⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠉⢸⠁
⠀⠀⠀⠀⠀⣃⠀⠀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠐⠀⠀⠀⣘
⠀⠀⠀⠀⠀⠸⡄⠀⣤⠄⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣀⡀⢀⢠⡏
⠀⠀⠀⠀⠀⠀⢹⡮⡔⠀⠀⠀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢖⡀⠀⠓⢫⣩⡟⠌
⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠈⠑⠤⠄⣀⠀⠀⠀⠀⣀⠀⠀⠊⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠪⢆⡄⠀⠀⠂⠀⠉⠁⠀⠐⠀⠀⠀⠃⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⣀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⣀⡥⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠒⠐⠲⠖⠂⠒⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⣯⣝⣛⡓⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠉⠉⠙⣷⢋⡟
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠾⠤⠤⠤⢤⣤⣄⣠⣏⣸⠁
⠀⠀⠀⢀⣠⠴⠛⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⠙⠛⠛⠛⠛⠿⠿⢿⣿⣶⣶⣤⣤⣄⣀
⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠿⣿⣶
⢾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡾
⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣂⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⠇
⠀⠀⠀⠈⠻⢦⡤⠴⠶⠒⠚⠛⠋⠉⠉⠉⠉⠉⢀⣀⣏⣉⣉⣉⣉⣉⣉⣿⣿⣉⣛⣛⣛⣓⣲⡶⠦⠤⡤⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠿⠛⠛⠛⠛⠛⢻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣛⡒⠒⠒⠶⣶⠶⠶⠧⠤⢤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⢤⣤⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⣠⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿
⠀⠀⠀⠀⠀⢾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣀⣠⣤⡤⠤⠤⠤⠤⠦⠤⢤⡤⠥⠤⢤⣤⡤⠤⠤⠤⠤⠤⠤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠶⠶⢿⡀⠀⢸⣿⠶⠶⠤⢤⣤⣀⡀

⠀⠒⠊⠉⠐⠒⢤⡀⠀⣀⡠⠤⠐⠐⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠁⣭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠀⢀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠉⠉⠉⢹⣻⣿⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠠⡤⡤⢼⡤⠤⢤⠤⣾⠀⢀⣀⡀⡈⠀⠀⠀⠀⢸⠦⠴⠶⢴⣿⢯⣽⠉⣹⣛⢻⠉⠉⡗⠒⠒⣦
⠀⠀⠀⠀⠀⢰⡿⠶⢿⠶⠶⠷⠶⢿⠀⢼⣭⣽⠇⠀⠀⠀⠀⢸⠶⠶⠶⠾⠇⠤⠤⠤⠬⠬⠿⠤⠤⠷⠒⠒⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⢸⣀⣸⣭⣧⣁⣀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠤⠄⠀⠠⠀⠀⠤⠄⠀⡿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⢀⣀⣀⣀⣀⠀⠀⡀⠀⢀⠀⡀⢀⣀⣀⣀⡀⣀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠰⡶⡇
⠀⠀⠀⢀⡀⣸⣇⣀⠈⢀⣴⣻⠭⠭⠭⠭⣩⣡⡀⠀⢸⣛⣆⠀⠀⢹⣺⢂⠀⢈⠁⣠⣤⠄⡟⠂⠤⢤⡀⡀⡏
⠀⠀⠀⠀⠠⠤⠀⠀⠀⠚⣟⣉⣃⣀⣀⡘⠛⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⣬⡧⠆⠐⠐⠂⠀⠀⠈⠒⠤⣭⣅⣀⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠐⠂⢗⣶⠶⠶⠮⠶⢤⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋
⠀⣀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠘⠈⠉⠉⠉⠉⠉⠁⡤⣀⠀⠀⠀⠀⣄⠁⠱⠂⠂⠀⢠⣤⣍⠀⠀⠀⡀
⠀⠀⠀⢀⣠⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⢌⠆⢀⠈⣨⣤⠀⢠⡖⠐⠠⠀⠀⠀⠠⠀⢀⡀
⠀⠀⠀⠀⠉⠉⠉⠉⠁⠉⠉⠉⠉⠁⠀⠁⠁⠀⠀⠸⠀⣙⠃⠂⠼⠃⠀⢰⠂⠀⠀⢶⠀⠈⠁⠄⠈
⠀⢐⣀⣀⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⣠⢲⠀⠀⢸⡇⠀⠘⠀⠀⠀⢰⢲⡟⠀⠀⠀⡈⠀⠀⢠
⠀⠄⢀⣿⡒⡒⠚⢛⢛⣍⠀⠀⠀⠀⣼⠈⠂⠠⠸⠃⠀⠀⠀⠠⠀⠙⠀⠃⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠘⠙⠛⠛⠛⠛⠛⠛⠛⠛⠃⠐⠆⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⡄⠀⠀⠁⠀⠀⠀⠀⢀⢀⠘

⠀⠀⠀⠀⣠
⠀⠀⠀⠐⣷⡆
⠠⢤⠀⠹⣭⣇⡀
⡭⠴⠶⢺⣧⠼⠉⠀⢸⠄⢠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠂⠊⡸⢶⡤⠠⠏⠀⢼⣷⡖⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡁⠘⢇⠈⡢⡀⠀⠐⢾⢹⡃⡠⢍⢀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣄
⠁⠈⣥⣴⠉⢉⡀⠈⢀⠬⠼⠂⠉⠐⢫⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠂⠀⠀⠀⠠⢀⠖⠘⠌⠟⠷⡍⢨
⡈⠑⢛⡿⢶⡃⠳⣀⢀⠂⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠆⠀⠃⡀⠂⠉
⠁⠰⣋⡀⠈⢰⢦⡈⠀⠀⠀⡾⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⠀⠀⠀⠁⡠⠏
⠁⠽⠛⣿⣀⣠⠀⠉⠓⠀⠘⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠃
⣽⣤⡴⢋⣿⡏⠳⣔⠛⠃⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁
⠐⠋⠀⠈⠋⠛⠦⠘⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠄⠀⠀⠐
⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠛⠉⠁⠈⠉⡉⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠉⠀⠀⠀⠀⠀⠐⠲⠽⠷⣼⣆⡀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠃
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⢀⣀⣠⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣥⣴⠶⠞⠒⠒⠒⠒⠒⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⢿⡏⠀⠀⢀⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣦⣤⣀⣀⣀⣰⣆⠀⠀⢀⡨⠓⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢸⠃⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⢉⣿⣿⠋⠉⠩⡉⠉⢦⣄⣀⢁⣒⣲⡶⠐
⠀⠀⠀⠀⠀⠀⣾⣴⣿⣿⣧⣿⣾⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣓⠶⢤⣧⣀⣀⣀⣩⣭⣭⣽⣤⠶⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣧⣄⣀⠀⠀⢀⣀⣀⣠⠤⣪⣿⣿⣿⣿⣉⢈⡆⠀⠈⠉⠉⠀⠀⠀⣠⢀⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⡍⠉⠛⠊⠉⠉⢠⠴⠞⣿⣿⣿⣿⡁⢩⡛⠦⣤⣀⣀⣀⣤⣤⣤⣬⠾⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄⠀⣄⣲⠸⣿⡶⣄⡀⠀⠀⠀⠀⠀⠠⣽⣿⣿⣿⣾⣅⠀⠀⠉⠉⠁⠀⠠⢠⠇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣇⣀⣙⣉⣀⡿⠀⠈⠛⠶⠶⣦⣦⠴⠾⠿⠛⠋⠁⠀⠈⠛⠛⠚⠒⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣯⣥⣤⣤⣤⣤⣤⣤⣤⣽⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢫⠉⠉⠉⠉⠉⣿⠀⠀⠀⣀⢤⣲⣤⣴⣖⡤⡀⠀⠀⠀⢸⠉⠉⠉⠉⠉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⣿⠀⠀⣜⣵⠟⠉⠉⠉⠉⠛⢮⣢⠀⠀⢸⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣿⠀⠸⣼⠁⠀⠀⠀⠀⠀⠀⠈⣧⡇⠀⢸⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⣿⠀⢰⢻⠀⠀⠀⠀⠀⠀⠀⠀⡘⠇⠀⢸⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣿⠀⠀⠣⠦⡀⠀⠀⠀⠀⢀⠴⠟⠀⠀⢸⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⣿⠀⠀⠀⠀⠀⣁⡀⠀⣈⠀⠀⠀⠀⠀⢸⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⣿⠀⠀⢀⢔⣽⠶⠿⠿⠷⣯⡢⡀⠀⠀⢸⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⣿⠀⢠⢣⡟⠁⠀⠀⠀⠀⠀⠹⣞⡄⠀⢸⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⣿⠀⢸⢾⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⠀⢸⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⣿⠀⠘⣜⡀⠀⠀⠀⠀⠀⠀⢀⣬⠃⠀⢸⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⠀⠀⠈⠉⠂⠄⠀⠀⠀⠐⠉⠁⠀⠀⢸⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡤⣔⣒⣒⣢⠤⡀⠀⠀⠀⢸⠒⠒⠒⠒⠒⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⣿⠀⠀⡔⣵⠟⠋⠉⠉⠉⠻⣮⣢⠀⠀⢸⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣿⠀⢰⣸⠃⠀⠀⠀⠀⠀⠀⠈⢧⡇⠀⢸⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⣿⠀⢰⢹⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⢸⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣿⠀⠀⠣⠦⡀⠀⠀⠀⠀⠀⠰⠟⠀⠀⢸⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣐⣀⣀⣁⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⢤⣤⣤⣤⣶⣶⣶⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣤⣤⣤⡤⣤⣀⣀⡀
⠀⠀⠀⢀⣀⣀⣤⣤⠴⠶⣒⣻⣭⣽⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣭⣽⣛⣒⠶⠤⣤⣤⣀⣀⡀
⠾⠟⠛⣋⣩⣭⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣭⣍⣛⠛⠿⠷
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡤⠶⠛⠉⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠞⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠁
⠀⠀⠀⠀⠀⢀⡶⠲⠶⠛⠃⢹⣿⣄⣀⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠤⠶⠞⠛⠛⠃⠴⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠒⠒⠠⡀
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠶⣤⡀⠀⠀⢀⣀⣄⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢠⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⣼⡀⠀⠀⠈⠉⠛⠛⠛⠹⠿⢯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⢛⠟⠿⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⠀⠀⡀⠉⠛⠶⢶⣤⣄⠀⢀⣠⠟
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⡏⠉⠙⠶⣄⡀⠀⠀⠀⠀⠀⢨⡿⠉⠁
⠀⠀⠀⠀⠀⠀⢄⡀⢀⣀⣀⣀⣿⠟⠓⠲⠶⣥⣄⣀⡀⠀⠀⠀⠀⠻⣄⠀⠀⠈⠉⠛⠲⣤⣤⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠶⠶⠶⠾⠷⣤⣄⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠈⠛⠳⠲⠶⠶⠤⠤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢦⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣀⠀⠀⠀⠀⣀⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⡤⠴⠶⠶⠶⠶⠶⠖⠛⠛⠋⠉⠁⠈⠙⠛⠛⠛⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠾⠷⣇⠀⠈⠉⠉⢻⡄
⠀⠀⠀⠀⢰⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠈⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣤⣴⣤⣤⣄⣤⣴⠾⠋⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠈⠀⠀⠀⠀⠀⠀⣠⣷⣦⣤⣀⣀⣠⣀⣤⠆⠀⠀⠀⠃⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⢀⣤⡴⠟⠋⠉⠉⣆⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⣤⡶⠚⠛⠋⠁⠀⠀⠀⢀⣴⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡂⠀⠀⠀⢀⣠⣤⠾⠋⠀⠀⣼⠄⠀⠀⠀⠀⠀⠀⢰⢀⣀⣀⣀⣼⠇⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⢾⡉⠀⠀⠀⠀⣠⡾⠋⠀⣀⣠⣤⣤⡴⠶⠚⠋⠉⠁⠙⠻⠴⠲⠤⠤⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⣦⠴⠛⠛⠛⠛⠛⠋⠉⠉




⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡈⠳⣖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋⠙⢦⡘⢿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⠞⠀⠀⠀⠀⠱⣄⠙⢯⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢡⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⡌⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢃⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⢷⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢴⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿⡦⠿⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡤⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡶⠶⠶⠶⠶⠶⠶⠶⡄⠀⠀⠀⡇⠀⠀⣴⠶⠶⢶⠶⠶⢶⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡞⠉⠉⠉⠉⠙⠀⡇⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⣿⠒⠒⢻⠒⠒⢺⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⠀⠀⣀⠀⠀⡇⠀⠀⠀⡇⠀⠀⣿⣀⣀⣸⣀⣀⣸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⣇⣧⣀⣀⣀⣀⣨⣠⣇⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇

⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⢻⣿⢛⣿⣿⠿⠟⠛⠛⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣾⣿⣿⣦⣶⣶⡆⡀⠀⠀⢠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⣿⣿⣿⡟⠀⠉⠀⠀⠀⠀⠀⢬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠾⠿⢿⣿⣤⣤⣤⣬⣤⣤⣾⣿⣿
⣤⠶⠶⢶⣾⣿⠟⠛⢿⣷⣶⣶⣤⡀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣄
⠁⠀⠀⠈⠛⠻⣽⣷⣄⠻⣿⣿⣿⡘⡄⠀⠀⣿⠘⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣆⣀⣀
⠀⠀⠀⠤⠀⠤⠀⠀⢘⠻⠿⠅⣿⣏⡇⠀⠀⣿⡄⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣀⣀⣀⠀⡐⠁⠀⠀⠀⣿⡏⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠤⢤⣬⣿⣿⣿⡖⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠉⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⢸⡟⢻⣄
⠀⠀⠀⠀⠀⠀⠠⣿⠟⠛⠛⠛⠛⠛⠛⢿⡇⠀⣿⠀⠀⠀⠀⠀⠀⣠⠶⣮⡙⠛⠿⠿⠛⣩⡶⢦⡀⠀⠀⠀⠀⠀⢸⡇⢸⡿⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⡰⢿⡼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⡦⡀⠀⠀⠆⢸⡇⢸⡷⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣮⡹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢫⡄⠀⡆⢸⡇⢸⡇⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢰⡜⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢡⠀⠀⢸⡇⢸⡧⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⢸⡇⠀⣿⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⢸⡇⢸⡧⢀⣠⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡇⠀⣿⠀⠀⢸⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠀⠀⢸⡇⢸⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣘⠃⠀⠆⢸⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠳⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠁⠀⠀⡄⢸⡇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠈⠑⠾⣤⡴⠒⠀⠀⠶⣤⡽⠞⠉⠀⠀⠀⠀⡄⢸⡇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣠⣿⠀⢆⡀⢀⣀⠀⠀⣀⣀⠀⠉⠉⠉⠀⢀⡀⠀⣀⣀⢀⡀⣀⠆⢸⣇⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠙⣿⠀⠎⠉⠁⠉⠉⠉⠉⠉⠉⣉⣉⣁⡈⠉⠉⠉⠉⠈⠉⠈⠉⠆⢸⡟⢻⣆
⠀⠀⠀⠀⠀⠀⠐⣿⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⣿⠀⠀⠀⠀⠀⢀⡤⣞⡻⠶⠤⠉⠤⠶⠾⣳⡤⣀⠀⠀⠀⠀⠃⢸⡇⢸⡿⠛⠛⠛⠛⠛⠛⠛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⡴⣯⠗⠉⠀⠀⠀⠀⠀⠀⠀⠉⠘⢯⣷⡄⠀⠀⠃⢸⡇⢸⡯⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣶⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣱⡆⠀⡄⢸⡇⢸⡇⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢱⠀⠀⢸⡇⢸⡇⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⢨⡇⠀⣿⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢸⡇⢸⣏⢀⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⠀⣿⠀⠀⠸⢛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠀⠀⢸⡇⢸⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢿⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠄⢸⡇⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⠀⠀⠀⠀⠱⣦⠂⡀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢢⡟⠁⠀⠀⠂⢸⡇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠛⠶⢞⣁⣠⣤⣤⣀⡙⠶⠞⠃⠀⠀⠀⠀⠂⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠃⢸⣧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⠀⠀⢀⠀⠀⠀⠀⡀⠀⣀⠀⢀⠀⠀⢀⡀⠀⣀⠀⡀⠀⠀⠀⢸⣏⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡟⠙⣿⠀⠂⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡏⢻⣦⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⣿⠉⠋⠉⠉⠉⠉⠉⠻⡇⠀⣿⠀⠀⠀⠀⠀⢠⣴⢳⡬⠙⠛⠛⠛⠛⢩⣶⣢⣄⠀⠀⠀⠀⠃⢸⡇⢸⡿⠉⠉⠉⠉⠉⠉⠉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢀⣼⢿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡷⣄⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢿⡝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣡⡆⠀⡄⢸⡇⢸⣧⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⠀⠀⢸⡇⢸⡿⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⢸⡇⠀⣿⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⢸⡇⢸⡗⢀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡇⠀⣿⠀⠀⠸⢋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⠀⠀⢸⡇⢸⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡼⠃⠀⠃⢸⡇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠹⢮⡄⠀⡀⠀⠀⠀⠀⠀⠀⡀⠄⣠⠟⠁⠀⠀⡁⢸⡇⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠘⠛⠴⠶⠖⢒⠒⠴⠦⠜⠛⠀⠀⠀⠀⠀⠁⢸⡇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣿⠀⠃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠃⢸⣧⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠤⠶⠟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠶⠶⣶⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣭⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⣠⣤⣶⣶⣾⣿⡿⠿⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡿⢿⣽⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⣿⣭⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣟⣫⣷⣾⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⠿⣯⣵⣾⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⠟⣻⣷⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⢉⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣅⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣄⣽⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣨⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣈⠙⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⠆
⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⣿⠁
⠛⣿⠃⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠿⠀⢠⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢀⡉⠀⡇⢠⠀⠀⠀⠀⠀⡼⠷⠀⠀⡇⠘⠸⡇⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⢸⠀⣾⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⢿⣿⠀⠀⠸⡇⠀⠀⠀⠀⠄⠀⠀⠀⠃⢀⠀⠉⠀⠀⠀⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠁⠀⠴⠀⠉⠀⠓⠀⠐⠓⠀⠐⠀⢠⡀⠏
⠸⠿⣤⡀⠀⠀⠀⢠⣴⣿⡖⣀⠀⠀⠀⠉⠛⠶⠀⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠶⠖⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠻⠁⣠⠃⠀⠠
⠀⡀⠀⠀⠀⠒⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠸⠇⠀⠀⠸⠇⠀⣴⣦⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠙⠋⣠⠆⠀⠀⠀⣤⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠈⠛⠉⣀⣰⣷⡀⠀⠉⠓⠀⠀⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠐⠂⠀⠀⠴⢦⠄⠀⢤
⡄⠀⠀⠀⠀⠈⠉⠉⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⢀⠀⠀⠀⠐⠦
⣷⣴⣤⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢻⣿⠗⠂
⡿⠟⠉⣀⣾⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣱⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠀⣈⣟⣙⣋⣋⣫⣛⣏⣏⣋⣛⣟⣙⣙⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠀⠀⠐⠀⠐⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⣿⠛⢿⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠇⠀⠀⠀⠀⠁⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠠⣤⠄⠀⠘⠀⠀⠀⠀⠀⢀⠀⠈⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⢀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠒⠋⠉⠀⠈⠒⡀⢀⡔⠁⠀⠐⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠊⠁⠀⢶⠀⠀⠀⠀⠀⢀⣰⠆⠀⢦⡀⠀⠀⠀⠀⠈⠁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠋⠛⠢⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⠶⠀⠀⢉⠀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠞⠳⣊⠁⠀⠙⠶⣄⡀⠙⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⡟⠀⠈⠄⠁⢹
⠀⠀⠠⠀⠈⠀⠁⠀⠀⠠⠀⠖⠀⠈⠛⠀⠈⠀⠀⠀⠀⠉⠂⠐⠢⣄⠈⠉⠐⠺⡏⠠⠄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢼⡇⠀⠀⡄⠀⠈
⠆⠀⠀⠀⠀⠐⠒⠀⠀⢀⡀⢀⡀⣀⣤⣤⣤⣤⣤⣤⠤⠤⠤⠖⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡷⢈⠀⠸
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢸⠇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠘⢼⣦⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣿⣿⣿⣦⣤⠀⠀⠀⠀⠀⢀⣀⣤⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⢼⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠛⠛⠘⠿⠟⢿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⣶⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⣿⡟⠀⠀⠀⢀⡞⢹⡆⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢠⡾⠀⠈⣧⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢀⣴⣿⠃⠀⠀⠸⡆⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⢀⣾⣿⠋⠀⠀⠀⠀⢿⡀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠁⢀⣾⡟⡀⠀⠀⠀⠀⠀⠘⣇⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⡔⠸⠿⠰⠇⠀⠀⠀⠀⠀⠀⢹⡆⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠿⢃⣠⠞⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠙⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣶⡌⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠈⠓⠠⠌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠎⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠶⠛⠛⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡰⣆⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣱⡤⠂⠁⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣈⡀⠘⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⣾⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠶
⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣁⣀⣀⣀⣀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⢀⡉⠀⠀⠀⢨⣐⠲⢎⣵⣂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠂⠀⢘⣰⣏⣤⡄⠐⠀⣷⡈⢻⣿⣦⣇⡸⠗⣰⢲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣿⣇⠀⣄⣿⣿⣿⡟⠀⠀⠈⣬⣭⠛⢇⢹⡄⠈⠁⠶⣸⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡙⠘⠟⣽⣦⡈⠊⠙⠛⠟⠀⣠⠾⠻⠻⣓⠦⡌⣉⢀⠐⢫⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣘⡵⠛⡉⢲⠶⠐⠃⠁⠀⠀⠀⠀⠘⠛⠛⣌⡬⠕⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⡋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠈⠣⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢠⡄⠀⢀⡔⠁⠀⣀⣤⠤⣄⣆⠀⠘⠂⠀⣸⢤⠜
⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠠⡄⠈⠀⠀⠀⠑⠬⡉⠙⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⣠⣀⣀⣠⡕
⠀⠀⠀⠀⠀⠀⠠⡄⠀⠐⠀⠀⢠⠞⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣈⡀⠃⡸⢆⠀⠀⠀⠀⠀⢦⣄⣀⣀⣀⣀⣤⣧⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣘⠎⠛⣷⢿⡀⣀⠀⠀⠶⡆⢀⣄⣀⢙⠉⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡶⢾⡘⢷⣀⣬⢟⠀⠀⠁⠀⠛⠛⠻⠷⣄⡀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡸⠁⠀⡡⠒⠺⡠⠋⠀⢀⠀⠀⠀⠀⠀⠀⠈⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⡵⣀⠀⢛⠈⠂⠀⠐⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣀⡃⢿⠉⡋⠹⠅⠉⢠⡄⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣁⡿⢠⡄⠤⢂⠸⣋⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡈⠉⠉⠉⣉⠈⠀⠀⠂⢀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡹⠁⠀⣠⠞⣏⣉⣉⣉⠳⠦⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⢢⠔⠇⠀⣼⣽⡿⠟⢟⠟⡿⣷⠷⣝⣧⡀⢸⡌⢳⠄⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣴⠢⣶⠀⢸⡆⢨⡟⠎⠠⠡⠁⢊⠌⠔⡝⣾⣿⡇⢸⡇⢸⡦⠤⠦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢸⡇⠰⡇⠀⠀⠀⠀⡠⠆⢄⡫⢨⣿⠀⢸⡥⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⢠⠇⠀⠻⣦⡀⠀⢐⡒⣈⡅⣃⡞⠉⢠⠸⡂⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⡄⠀⠀⠈⠉⠑⢲⣶⣾⣛⠉⠀⠀⠘⠀⡖⢽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⠀⢀⡠⠞⣋⣍⣭⣭⣍⣟⠷⣄⠀⠀⡷⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⢼⠀⢸⠇⠀⡟⣤⣿⠟⠛⠛⠳⠦⣍⢻⣿⡧⠀⣇⢽⠂⠐⠂⢂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠾⠭⣼⠁⠘⢰⠀⡾⠛⠀⠃⠀⠁⢎⢔⠜⣧⣻⡿⠀⣟⢼⣮⠒⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⢸⠀⢹⡀⠀⠀⠀⠀⢖⠅⡪⢎⣾⡇⠀⣗⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠘⠀⠈⢿⣤⣀⠀⠨⠔⠚⣫⡾⠋⠁⠀⣗⢽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠙⡛⡛⠚⠉⠀⠀⠀⠀⣗⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡇⠀⡀⠀⢀⣴⣶⢿⣿⣯⡯⠭⣓⣦⡀⠀⠀⣟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢶⣒⣇⠀⡇⢠⣍⠞⠿⠛⠉⠁⡩⡙⢮⡻⣿⡄⠀⡧⣸⢣⡉⡉⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡀⠐⢻⠚⠀⠁⠀⠀⢁⣈⠔⣷⣿⡟⠀⡿⢹⡟⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⡆⠀⠨⣆⠀⠀⠀⢀⣺⠷⠶⠛⢿⠇⠀⠿⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⡇⡀⠀⠙⠷⢥⣀⣐⣫⠈⠤⠄⠀⠀⠀⠦⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⣂⣀⣀⣀⣀⣉⣉⣁⣀⣀⣀⠀⠴⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠺⠷⠖⠶⠾⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠠⡊⠊⢔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢑⢔⠥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢕⠥⣪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠕⡨⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠦⢊⠄⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠄⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠐⠓⠢⠴⡄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⢲⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠊⠁⠀⠀⢀⠀⠀⠀⠀⠀⠋⠀⠀⡁⠶⢶⣾⣿⣧⣬⣷⣾⣶⣾⣷⠾⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠃⠀⠀⠀⠀⠀⢲⣷⣶⠦⠦⠤⠤⣀⡘⠀⢀⣼⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠦⡀⢀⣀⣀⠉⢉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢛⣿⣿⣿⡇⠀⠀⠀⠀⠉⣙⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⣾⠄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡟⠁⠠⠤⣀⣐⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⣦⣀⠀⠀⠀⠀⢀⣇⡀⠀⢀⣮⣽⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣤⣽⡿⠿⠿⠷⠷⠿⣿⣿⣿⣿⣿⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠘⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢀⡞⣉⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⢠⣿⠿⠒⠀⠀⠐⠒⠛⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⠀⠀⠙⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣿⠀⠸⠁⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⣼⣿⣆⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⡐⢁⣀⠘⠄⠎⠐⣠⣿⣿⣆⣈⠐⠠⢀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⡋⠉⠀⠀⠉⠙⣛⣯
⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠈⠂⠛⠙⠛⠻⠿⣶⠀⠀⣈⣤⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⣣⠄⠀⣀⡀⣸⣿⡿
⠀⠀⠀⠀⠀⢰⠀⠄⠀⠀⠒⠢⢄⡀⠀⠀⠀⢠⡆⡇⠀⠸⢻⡟⠀⠀⠀⣿⣿⣿⣿⡇⠀⢃⡀⠀⠀⣰⣇⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⣨⣽⣿
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠑⠒⠴⣾⣧⢱⠀⢀⠌⠀⠀⠀⢠⣿⣿⣿⣿⣿⣄⣸⣿⠿⠶⠾⠿⣷⣤⣀⣀⣀⣀⣠⣾⣿⣶⣦⣿⡿⠃
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠆⢄⣀⣀⠀⠀⣰⣄⣀⢀⠅⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡶⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⢀⡈⠉⠉⠈⢻⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠾⠿⠾⠶⠿⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⣠⣤⣤⡀
⠀⠀⠀⣴⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠂⠀⡿⢷⠿⡷
⠀⣿⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡙⠿⠞⠁
⠀⣿⣄⡀⢩⡍⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣭⣭⣥⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠋⣴⠿⣿⣾⣶⣿⢿⣆⢀⣀
⠀⠈⠉⠐⠊⠁⠀⠙⠛⠿⠿⠶⠆⠀⠀⠀⠀⠀⠀⢰⡟⣹⣿⣿⣿⣿⣷⣾⠏⠀⢠⣤⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠉⠩⣶⣿⠿⣿⠛⠻⣿⢿⣿⣮⠉
⠀⠀⠀⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡁⠈⠉⢠⣤⡞⣻⣤⡄⣤⣤⠄⣤⣼⠟⣤⡄⠀⠀⠀⠀⠀⠀⠙⣿⣾⣧⣀⣠⣿⣾⡟⠁
⠀⠀⠀⣀⣀⣄⣀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢠⣴⠟⣹⣿⢿⣿⡇⠀⠀⠸⢯⡴⠿⠏⠼⠿⠏⠼⠿⠏⢼⡿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣟⣿⣭⣝⣟⣿⣿⡀
⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢼⠿⠂⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⢸⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⠛⢻⣿⣿⠟⠻⣿⣿
⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⡟⣿⡿⣷⣾⣿⣿
⠀⠀⠀⣿⣿⢿⣿⠀⠀⢰⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸⣿⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⣿⣷⠉⠉⠉⠉
⠀⠀⠀⠈⠁⠈⠁⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⢀⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⢸⣿⣿⡟⢠⣶⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠜
⠀⠀⠀⠀⣴⣦⣭⣉⣁⣸⣿⣿⠿⠛⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠐⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⡏⠀⠀⠀⣼⣡⣤⣾⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠉⠀⠀⠀⠀⠀⠀⣰⣁⣤⣼⣏⣆⠀⠀⠀⡇
⠀⠀⠀⠀⢻⡿⣿⡇⠀⠀⠀⠀⡀⠀⣷⠿⠿⠿⢷⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⠀⢷⡾⠿⠿⢷⡾⢀⡀⠀⠇
⠀⠀⠀⠀⢸⣿⣿⣿⣆⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢸⣿⡇⢹⣿⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⣸⣿⠃⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤
⠀⠀⠀⢰⣿⡏⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⢀⡙⢿⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣈⣛⣛⣁⣀⣰⣿⣷⣿⣾⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣶⠶⠿⠷⢶⣶⣶⣤⣤⠀⠀⠀⠀⠀⢀⣘⣿⣿⣃⡀
⠀⠀⠀⠀⢀⣀⣉⠉⠉⠉⠉⠉⣿⣿⣿⣿⠉⣿⢩⣯⣿⣿⡆⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⣴⣶⣦⡈⢿⣇⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿
⠀⠀⣼⠋⠉⠉⠉⠙⢳⡀⠀⠀⣿⣿⣿⣿⠀⢻⣿⣿⣿⡉⣱⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⣀⣀⣀⡻⡿⠟⠁⠘⣿⡄⠀⠀⠙⠉⣿⣿⣿⣿⠛
⠀⣰⠷⣶⣶⣶⣶⣶⡾⢧⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠿⣿⡄⠀⠀⠀⣿⣿⣟⠛
⠀⢹⣾⣿⣿⣿⣿⣿⣷⣿⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⣼⡇⠀⠀⠀⣿⣿⣿⣿
⣤⣸⢿⣠⣤⣤⣄⣤⣸⣯⣠⡀⣿⣿⣿⣿⠀⣤⢸⣿⣯⣿⡇⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡅⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠈⠛⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⣿⣿⠁⠀⠀⠀⣿⣿⣿⣿

⠀⠀⠀⣴⡆
⠀⠀⠀⣿⡇⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⢸⣿⠀⣿⡇⣴⡆⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠛⠃⠘⠛⠀⠛⠃⠚⠃⠀⠀⠀⠀⠀⠘⠛⠀⠛⠃⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡦
⠀⠀⠀⠀⠀⣰⡾⠟⣉⣥⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤
⠀⠀⠀⢀⣾⡟⣠⣾⠟⣉⣭⡅⠀⠀⠀⠀⠀⠀⣠⣾⠟⣉⣭⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⢸⡿⢰⣿⢁⣾⠟⣩⡁⠀⠀⠀⠀⠀⢠⣿⠃⣾⡟⣩⣅⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣫⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠹⠃⠘⠟⠈⠛⠀⠛⠃⠀⠀⠀⠀⠀⠘⠟⠀⠛⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉