9 Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡄⠀⠀⠀⢠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠛⠛⠁⠀⠀⠀⠘⠛⠛⠿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠓⠲⢦⡀⠀⠀⣠⠴⠖⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠙⣆⡼⠁⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⡀⠀⢸⡇⠀⠀⠀⠀⣠⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⠀⠀⠀⠀⠀⠉⠁⠀⣸⣇⠀⠀⠀⠀⠈⠁⣰⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠹⣆⠀⠀⠀⠀⠀⢀⣴⠋⠘⠷⣄⣀⣀⣤⠞⠁⠈⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣠⠄⠀⠈⠛⠶⠦⠴⠶⠛⠁⠀⠀⠀⠀⠉⠁⠀⢀⣠⠗⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠈⠙⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⠿⠷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠁⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠶⠶⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿
⠀⠀⢀⡀⠀⠀⣀⣤⡴⠶⠖⠛⠺⠿⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⢀⡴⠋⠈⠁
⣠⣾⣿⣿⣶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⢀⣠⡶⠋
⢻⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⠓⠛⠛⠋⠁
⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣀⡀
⠀⠀⣀⣠⣴⣶⣿⣿⣿⠿⠛⠉⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⢀⣿⡏⠀⠉⠉⠛⠳⠶⣤⣄⡀
⠀⠀⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠈⠙⠛⠿⢿⣶⣦⣄⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⠿⢿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠆⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠐⡔⠉⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⠀⠀⡀⡅⠀⡄⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠈⠢⢀⠠⠣⢀⣀⠔⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡀⠔⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⢀⠊⢈⡁⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠈⠳⡀⠀⠀⠠⡤⠈⠄⠀⠀⠀⠀⠀⠀⢢⠀⣁⠠⡘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠘⡀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⡠⠃⠀⠀⣰⡇⠀⠀⠃⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⠓⠒⠒⠊⠁⣀⠠⣾⣿⠁⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡟⠀⠀⠀⠨
⢠⠀⠄⣊⠘⢄⠀⠀⠀⠀⣀⡠⠜⠓⠤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠁⠦⠈⠀⢀⡳⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠁⢐⠞⣠⠀⡎⠀⠀⠀⠀⠀⢀⠖⠢⢄⣀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠒⠁⠦⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⢤⠤⠤⠤⠤⢴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠘⡄⠀⢸⢀⣀⣀⣀⣀⡀

⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣻⣶⣦⡀
⠀⠀⢀⣿⣿⡿⠛⠛⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡉⠛⠛⠉⣿⣿⡄
⠀⠀⠸⣿⣿⣇⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⢸⣿⡀⠀⢻⣿⣷⠀⠀⠀⠀⠀⢸⠃⣀⣠⣄⢀⣤⢦⠀⠀⠀⠀⠀⠸⣯⣿⠇⠀⠀⠸⣿⣽⡇
⠀⠀⠀⠹⢿⣿⡷⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⡀⠙⢿⣆⢸⣿⣿⠀⣾⠋⣿⢣⣿⢸⡟⢸⡟⣾⠧⠋⠀⠀⠀⠀⠀⠀⠻⣿⣶⣿⢿⣶⣿⢿⡧
⠀⠀⠀⠀⠀⠈⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣿⠟⣼⣿⡟⢸⡿⠠⣿⣼⣧⡿⠃⠸⠷⠛⠓⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⣻⣿⡋
⠀⠀⠀⠀⠀⠜⠛⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣼⣿⡟⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠿⠷⠾⣏⠸⣿⣽⡧
⠀⠀⠀⢠⡄⣤⡤⣄⡄⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣤⣶⣶⣤⣿⡿⠁
⠀⠀⠀⢸⢿⣿⣿⡿⡇⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠾⠛⠋
⠀⠀⠀⠈⠈⠉⠉⠁⠁⠉⠁



⠀⠀⠀⠀⠀⡠⠤⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣤⣤⣀⣀
⠀⠀⢠⡶⣿⣿⣿⣿⣷⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⣯⣡⣭⣛⡻⣷⣄
⠀⠀⣿⢹⣿⡝⠀⠈⣹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⣤⡄⠀⠀⡀⢐⢈⡄⣴⡆⣾⢸⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⣿⠟⠛⠻⣿⣟⡌⣿⡂
⠀⠀⢻⢸⣿⡱⢄⣀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⢸⣷⢸⢸⡇⣿⢿⣿⢸⡟⠃⠀⠀⠀⠀⠀⠀⠀⠠⣿⢿⣿⠀⠀⠀⢸⣿⣃⣿⡇
⠀⠀⠀⠣⠙⠿⣿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠸⠿⠃⠀⢸⠹⣿⢸⡇⣿⠘⣿⠸⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣻⡄⠀⠀⣼⣿⢻⣿⠁
⠀⠀⠀⣠⣤⣀⣀⣤⣾⣿⠏⠌⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⣠⡆⠀⠸⠀⠛⠈⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⣀⠀⣀⡉⠣⠀⣰⡿⢻⡟⠃
⠀⠀⠀⠣⣻⠿⠿⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⠛⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⣏⡷⠠⢔⣼⣋⣾⠛⠁
⠀⠀⠀⣤⣀⣤⣤⣄⢠⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠁⠁⢉⠴⣫⠖⠋
⠀⠀⠀⠟⠙⠿⠿⠋⠻⠻⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠟⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡆⠀⠀⠀⠀⢠⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⠻⠇⠀⠀⠀⠀⠸⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠛⢶⣤⣤⡤⠤⣤⣀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡶⠟⠁⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣠⠞⠁⠀⠀⠀⠀⠈⠳⣄⠀⡴⠛⠁⠀⠀⠀⠙⢦⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠁⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⣿⠀⠰⡷⠆⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣆⠀⠀⠀⠀⠀⠀⣰⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠻⣄⠀⠀⠀⠀⠀⢀⣠⠟⠀⠘⠳⣤⣤⣤⣤⠞⠋⠀⠀⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣠⡄⠀⣀⣈⣻⡶⠶⠶⠶⠛⠁⠀⠀⠀⠀⠀⢠⠤⠒⢤⣠⡾⠃⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠙⠶⣧⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠉⠁⠉⠙⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣸⡇⠀⠀⠀⠀⢀⠠⢄⡀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⣿⠀⠀⠀⠀⢀⡇⠀⠀⣳⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⣿⠀⠀⠀⠀⠀⠓⠤⠔⠋⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⢀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⢻⡆⠀⠀⠀⠀⠀⠀⠀⣴⠒⢲⠀⠀⠀⠀⠀⠙⠛⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏
⠘⣷⡀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠈⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣶⣤⣄⣀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠞⢛⠛⠛⠛⠛⠛⠷⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⠀⠀⠄⠀⠁⠀⠂⠀⠘⠀⡛⢄⠀⠀⢉⣛⠷⣤⣀
⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠠⠄⠐⢦⠀⠀⡈⠈⠁⠲⡆⠀⠀⢀⣿⣎⠢⠀⠀⢻⣷⡌⠙⢷⣄
⠀⠀⠀⠀⣠⡾⢋⡄⣶⠸⠀⠐⡤⠈⠀⠀⢲⣦⡄⠀⠀⠄⠀⠚⠛⠛⠇⠱⣌⡠⠄⠈⠀⠰⠝⢷⣄
⠀⠀⠀⣴⠏⣶⡬⠲⢋⠳⢡⣾⡇⠆⢔⣂⠁⣨⡶⠶⠾⠶⢶⣄⠈⠀⠀⠂⠠⠟⢠⠌⠐⠾⣿⣧⠹⣧⡀
⠀⢀⣾⠃⢈⠛⠀⠀⠀⠂⠀⠙⡉⠛⢻⠃⣼⠋⠀⠀⠀⠀⠀⠙⣷⡀⢂⣤⣾⠃⠀⠀⠀⠀⠈⠃⠀⠘⣷⡀
⠀⣾⠃⠰⠀⠄⢀⡆⣶⡠⡄⠠⠀⠰⠈⢸⡏⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⠛⠿⠀⠀⢀⡶⠄⠰⠀⠐⠠⠘⣷
⢸⡏⢀⠀⠀⠴⣀⣸⠁⣷⣀⡤⠄⡀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠐⡡⡀⠀⠀⠀⠀⠐⠠⣴⣶⠹⣇
⣾⠁⠢⠀⠐⣻⢥⣬⠄⣤⣬⢿⡓⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠠⢀⢀⠀⠀⢐⠻⠀⠐⢀⢃⠂⢿
⣿⢰⣄⠀⠀⡉⠀⡜⣦⢏⡄⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⣿⠻⣐⡂⢀⢰⣌⢠⠉⣾⡆⢸
⣿⠈⠇⠈⠠⠀⠀⠁⠉⢠⠀⠀⡄⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠅⢀⢰⣍⣠⠮⣿⡧⠋⠁⠙⠈
⢺⡆⠀⠻⣾⣷⠶⠈⢁⠀⠤⣴⡖⣖⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⡖⠠⠀⠀⠡⠿⠡⠀⠀⠀⠀⠰
⠘⣷⠀⠀⠛⢁⠀⠀⠀⠀⠸⣦⣅⠃⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⡈⠡⢝⠻⣿⡇⠀⢁⠠⣤⣲⠀⢰⡀
⠀⠹⣇⠈⠀⠄⠠⣤⠀⠀⢠⡿⠛⠛⠈⠀⢻⣄⠀⠀⠀⠀⠀⣰⠟⣰⡇⠅⢐⣂⣭⠂⠐⢀⠀⠷⠳⣌⠳⢉
⠀⠀⠹⣦⠀⠀⠀⠀⠐⣀⡄⠀⠀⠀⠀⠁⢠⡙⠷⠶⠶⠶⠟⠁⠀⢻⡇⡀⠬⠉⣙⠄⠐⠈⠀⡤⡰⢋⣴⣌⢰
⠀⠀⠀⠘⢷⣄⠣⠀⢡⠀⠀⠀⢶⠀⠀⠀⣼⣷⡜⢷⣤⡄⠈⠀⠀⠈⠀⠁⠐⣨⣴⣿⡇⠀⡄⠐⠋⠭⠀⠩⣸
⠀⠀⠀⠀⠀⠙⢷⣄⠚⠿⣷⠄⠂⠐⠄⠈⠉⠉⠛⠊⠯⢄⡁⢀⠀⠀⠐⠃⠂⠀⠒⡐⣶⡐⠀⠁⠀⠀⠰⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣍⡁⠂⠄⠀⢀⣀⠀⠀⠐⠀⠄⠀⠉⠃⢀⡂⠀⣀⡂⡈⠸⣋⠹⢖⣿⡷⣌⡀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⣬⣥⣤⣤⣤⣥⣬⣴⠶⠾⣿⠇⠈⢦⣭⣤⣍⡀⠴⠈⢋⠈⣀⢱⡟
⠀⠀⠀⠀⢀⣤⡶⠶⣛⠛⠳⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⢻⡟⢉⠀⠀⠀⢨⣄⢠⡿
⠀⠀⠀⣴⠟⠡⢀⠈⠹⣍⡴⠤⠈⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠶⠀⢐⠀⠀⠀⢀⠀⠀⠀⢡⡿⠁
⠀⠀⢸⡏⡘⠀⠐⠈⢠⣛⠳⠐⢂⠀⠘⢷⡀⠀⠀⠀⠀⠀⢠⡾⠃⢀⣀⠀⠀⠀⠁⠈⠙⠆⠀⣰⠟
⠀⠀⢸⡇⣀⠀⡀⣴⡷⢣⣾⡀⡂⠀⢠⡌⢷⣤⣀⣀⣠⡶⠋⢀⡐⢛⠛⠛⢃⡀⢤⣄⠀⢀⣾⠋
⠀⠀⠸⣧⠠⠀⢀⡟⣰⠿⠿⠷⠈⢀⠈⠀⢀⣬⣭⢩⠁⠀⢈⢀⠈⠂⣤⢀⠁⠄⡁⢊⣴⠟⠁
⠀⠀⠀⠹⣧⡚⠉⢠⠀⠀⠀⠂⠠⠈⠀⠀⢿⠫⠐⠠⠄⣠⢸⣧⣹⡼⠿⣤⠎⣨⡴⠟⠁
⠀⠀⠀⠀⠈⠻⣦⣄⡁⠀⠀⡶⠁⠀⠀⡀⠚⠀⠣⠹⡷⠼⠚⠛⠃⢂⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣄⣀⣘⣀⣀⣀⣈⣐⣁⣠⣤⡶⠾⠛⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠋⠉⠉⠉⠉⠉⠙⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠓⠲⡄⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⠙⠛⠒⠒⠚⠛⠉⠁⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠖⠚⠋⠉⢉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠙⠓⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠒⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠀⠒⢒⣀⣀⣤⣤⣤⣤⣄⣀⡒⢀⠉⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠀⠄⣂⣤⣤⡶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⣂⡠⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡠⢀⣠⣴⣾⣿⣷⣿⣿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣶⣯⣿⣦⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣴⢛⣽⣿⡿⠟⠋⠁⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣯⣝⡻⢿⣿⣿⣿⣧⡰⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢂⠐⢡⡾⣻⣾⡿⠟⠁⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣟⢿⣿⣿⣿⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢂⣴⣟⣾⡿⠋⠁⣠⣾⣟⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣾⢷⡔⡀
⠀⠀⠀⠀⠀⠌⠀⢡⣾⣵⣿⠟⠁⣠⣾⣿⣷⣿⣿⣿⣿⣿⢟⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣷⣻⡄⠀⠆
⠀⠀⠀⢀⠀⡐⢀⣿⣸⣿⠏⠀⣴⣟⡿⠇⣸⣿⣿⣿⣿⢋⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⢻⣿⣿⣾⣿⣿⣿⣻⣆⡄
⠀⠀⠀⠄⠠⢀⡞⣾⣿⠏⠀⣼⣿⡏⠰⢰⣿⣿⣿⣿⢃⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⡿⣿⡤
⠀⠀⠰⠀⠀⣾⢱⣿⡟⠀⣼⡿⣞⠁⠀⣾⣿⣿⣿⡟⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⠘⣾⣿⣿⣿⣿⡇⣿⣷⡎⢀
⠀⠀⡄⡄⢰⡇⣾⣿⠀⠠⠿⢟⠠⢁⡀⣿⣿⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⢹⡟⢿⣿⣿⡇⣿⣿⣧⠃
⠀⠀⠁⠁⣸⣷⣿⠃⠀⣰⣎⡏⠠⠈⠀⣿⣿⣿⣿⢁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⣷⣼⣿⣿⣷⠹⣿⣿⢀⠇
⠀⠀⢰⠀⣇⣼⡟⠀⠀⣿⠟⠀⠀⠀⠀⣿⣿⣿⡿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣸⢻⣿⡟
⠀⠀⠀⠀⠉⠹⠁⠁⠈⠉⠀⠀⠀⠀⠘⠿⠿⠿⠃⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⡇⢾⣿⡇⢰
⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⢀⢸⣿⣿⣷⣾⣿⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⢻⣿⣿⣿⡇⣻⠛⡇⠘
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣄⠉⣈⣾⣿⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠀⠀⣸⣿⣿⣿⡇⣿⣴⣧⡄
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠸⣆⠸⣽⣿⣿⣿⣿⣿⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡏⠀⠀⢹⣿⣿⣿⡗⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠠⡌⠣⣏⣿⣿⣿⣿⣿⣿⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⢉⠦⣹⣿⣿⣿⣿⣿⣿⣷⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⠀⠀⠈⡳⠈⣿⣿⣿⣿⣿⣿⣿⣷⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢸⣿⣿⡏⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⢘⡤⢛⢿⣿⣿⢿⣿⣿⣿⣿⣦⣴⣊⣉⣠⣤⣤⣤⣶⣿⣿⣵⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⣾⣿⣿⡏⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠳⣾⣚⡿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣿⣿⣿⠇⣸⣿⣿⡌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⠘⢿⣿⣿⢿⣿⣷⣬⣛⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣬⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⡇⠀⠀⠀⣼⣿⣿⡟⢰⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠈⣿⣿⣿⡇⠀⠀⢀⣿⣿⣿⣷⣾⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⣸⣿⣿⣿⣿⣿⣿⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠁⠀⠀⣻⣿⣿⠛⠛⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⢀⣬⣵⣾⣿⢠⣀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⢀⣼⣿⣿⡿⢁⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣷⣾⣿⣷⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⢠⣿⣿⣿⡿⢡⣾⣿⡿⠂
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣭⣿⣿⣯⣝⣻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢧⣿⣿⣿⣿⣃⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢠⣿⣿⣏⣼⣿⣿⠿⠿⢿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣼⣿⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⢸⣿⣿⢻⣿⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣦⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠟⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⣋⣭⣶⣬⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⡿⣿⣾⣿⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡿⠋⠈⠀⠀⠀⠉⠉⠛⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣶⣤⡀⡀⠀⠀⠀⢠⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣶⣿⣾⣿⣿⣿⣿⣿⠟⢿⣿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠟⠙⠛⠛⠿⠿⠉⠉⠀⢀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣷⣦⣤⣦⣤⣀⣀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⡿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⣉⠑⢦⠀⠀⢠⠖⢋⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠸⠀⣳⢸⡇⠀⡏⢰⠁⠽⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠁⢸⡇⠰⡇⠈⠓⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠓⢯⡀⢀⡷⠚⠉⠙⠲⢄
⢀⡤⠶⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢳⡞⠀⠀⠀⠀⠀⠈⣇
⡾⢀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠶⠀⢸⡇⠀⠿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠲⢤
⢧⠀⠀⠈⠀⠀⠀⠀⠙⢦⡀⢀⣠⡴⢷⡀⠀⠀⠀⠀⠀⢀⡼⠻⣄⠀⠀⠀⢀⡴⠛⠿⣶⣄⠀⠀⣠⠴⠋⠁
⠈⢧⣀⠀⠀⡀⠀⠀⠀⢠⡽⠛⢁⡀⠀⠙⢦⣄⣀⣠⡴⠞⠁⠀⠈⠙⠛⠛⠉⠀⣤⠀⠈⠙⣷⣞⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠛⠶⢤⣤⣀⣼⠋⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠊⠁⠀⠀⠀⠈⢿⣧⠀⠀⠀⠄⠀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡶⠒⠒⠛⠋⠉
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠶⠶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⡾⠀⠀⢰⠊⢱⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠑⢉⠔⠢⡄⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠻⡀⢀⡜⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠿⣦⣀⣀⣀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⡆⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⣿⠁⠀⠀⠀⠀⡤⠤⡈⠉⠁⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣸⠃⠀⠀⠀⠀⠘⢄⣠⠇⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠰⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢤⣀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣶⣦⣤⣀⠀⣾⡿⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠶⡶⠲⠖⠶⢶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠋⡁⠀⠑⣄⣠⡤⠶⠶⠷⠶⠷⠶⠿⣽⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢋⠀⠈⠢⣀⣨⠷⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢡⠀⠀⠳⣄⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠑⢆⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⢄⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠑⢤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠋⠹⣏⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠢⣀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⢿⣈⠑⠚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠂⡀⢨⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⣀⠙⠢⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢄⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⡈⠳⣄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠉⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣸⠇⡈⠳⡌⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠢⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣰⠟⡀⠙⡄⢈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⢋⠱⡀⠸⡄⢘⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠾⠤⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠙⠳⠦⣤⣄⣀⣀⣀⠀⣀⣀⣀⣤⡴⡖⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠻⠋⠉⠷⣄⠀⠀⠀⠉⠉⠉⠉⠉⢹⡟⠀⠱⡈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡋⠍⡐⠀⢁⠃⠀⠀⢀⣉⣿⣦⠀⠀⠀⠀⢀⣰⡏⠀⠰⡀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡝⢶⣀⣀⣤⡴⠶⠛⠉⠉⠀⠈⠳⣤⠤⠶⠋⡇⠀⠀⠀⣱⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣄⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣜⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠶⠶⠶⠦⠤⢤⣀⡀⠀⢠⠞⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠏⠀⠀⠀⠀⠀⡏⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠖⠒⠒⠶⠤⣄⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢰⠋⠁⣀⡤⠴⠒⠚⠛⠛⠛⠚⣧⠀⠀⠀⠀⠀⠀⡾⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠸⡄⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢰⠇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠙⢾⣄⠀⠀⠀⠀⠀⣀⣠⠞⠁⠀⠀⠀⠀⠀⣼⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⡄⠀⠀⠀⠀⢸⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠯⢭⣉⣉⣉⣉⣉⣁⣸⡇⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⢸⡇⠀⠀⠀⠀⢸⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣧⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠸⡄⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡷⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣠⠼⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⣠⡤⠞


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠀⠀⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠂⠀⠀⠀⠀⣒⣾⣤⣴⣤⡼⢶⣄⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⡄⣀⠀⣀⠒⠭⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⢉⠀⣬⣴⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣄⠂⠥⣛⣠⣾⣿⣿⣿⣿⠟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢸⣟⣿⣿⣾⣿⣿⠁⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⣷⣟⣻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⢀⣼⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣘⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⢴⡿⢿⣟⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠻⠿⠿⠿⠿⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠘⠃⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠸⠠⠅⠰⠌⠀⠏⠀⠀⠀⠘⠘


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

⠀⠀⠀⢀⣴⣶⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣻⣷⣶⣄
⠀⠀⢀⣾⣿⡿⠛⠛⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣍⠙⠛⠋⣹⣿⣧
⠀⠀⠸⣿⣿⣇⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⣴⠸⣿⡅⠀⢹⣿⣿⡀⠀⠀⠀⠀⢐⡟⣀⣀⣤⠀⣠⠴⡄⠀⠀⠀⠀⠀⣿⣽⡿⠀⠀⠀⢿⣯⣿
⠀⠀⠀⠹⢿⣿⣷⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⣿⡄⠙⢿⣦⢸⣿⣿⡇⣼⡟⣻⡇⣿⢃⣿⠁⣿⢸⡯⠜⠁⠀⠀⠀⠀⠀⠙⢿⣧⣼⢶⣦⣼⡿⣷
⠀⠀⠀⠀⠀⠈⣰⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣾⡿⣸⣿⣿⢀⣿⠀⣿⣴⣿⣼⡏⠘⠿⠞⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⢹⣿⡟
⠀⠀⠀⠀⠀⠔⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣰⣿⡿⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣾⣿⢾⣿⠁⣿⣏⣿
⠀⠀⠀⢠⡄⣤⡄⣄⣄⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣦⣴⣶⣦⣼⣿⠇
⠀⠀⠀⢸⢻⡟⡇⡿⡇⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠟⠛⠁
⠀⠀⠀⠘⠈⠉⠁⠁⠋⠙⠃



⠀⠀⠀⠀⠀⣀⠠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢀⣀
⠀⠀⢀⣶⣾⣿⣿⣷⣶⣝⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣿⣋⣽⣟⠛⣿⣆⡀
⠀⠀⣾⢳⣿⡛⠁⠉⢽⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⢠⡄⠀⠠⠀⢐⡂⣀⢠⣇⢰⡆⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣾⡿⣶⣿⠟⠾⡏⣿⣊⢻⡷
⠀⠀⢻⢸⣿⡡⣀⠠⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢸⡷⠀⢸⣿⢸⡇⣿⢸⢿⣼⡇⣿⠚⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⡅⠀⠀⠈⣿⢟⢸⣿
⠀⠁⠈⠧⡻⢿⣿⣿⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⣿⠏⠀⢸⡟⣿⡇⣿⢸⠀⣿⡇⣿⡴⠆⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⣆⠀⠀⢀⣿⡟⣿⡏
⠀⠂⠀⢀⣄⣀⣀⣠⣼⣿⡟⡐⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢀⡤⠀⢸⡇⠻⠇⠛⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢀⡙⠛⠀⢀⣾⠟⣿⠟
⠀⠀⠀⢧⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠷⢢⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣼⣿⢸⠄⡀⢤⡾⢫⡾⠋
⠀⠀⠀⣤⠈⣅⣤⡄⢠⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠘⠘⢢⢴⣭⠾⠉
⠀⠀⠀⠿⠻⠿⠿⠟⠿⢿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠟⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⠴⠒⠂⠀⠉⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠉⣠⠔⠊⣀⣀⣈⢉⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠞⣁⡴⢧⣄⠠⠶⠶⠤⠰⠶⣄⡀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡾⢁⠀⢀⣿⠷⠆⠀⠤⣭⠏⢴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⠸⠋⠀⣀⣉⣫⡥⠩⣟⣓⡉⠙⠓⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢳⠈⢸⠘⠀⡄⠀⢀⠐⢒⡂⣄⣓⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⠸⠰⣇⠋⠰⠈⡀⠡⠍⠰⠸⢀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣇⠁⡀⡀⠃⠘⠧⠙⢳⣤⣤⣤⣤⡾⢋⡹⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠸⡀⢃⡅⢰⡘⠦⠠⠀⡿⠁⠈⢻⠧⠠⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠁⣸⣿⣄⠉⠂⡴⠀⡇⠀⠀⠨⠗⠄⠞⢃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠻⣷⣾⠁⠀⠐⢦⡴⠂⠀⣸⣾⠛⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣾⣿⡉⢦⡀⠈⠳⢦⠀⠚⠃⢠⡴⣋⣠⢊⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡟⢳⡄⠁⠀⠀⠈⠛⠾⠟⢩⡾⠋⢉⡡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⠘⠻⣦⡀⠠⣀⣀⠀⠠⠏⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⣄⡀⠀⠙⢦⡈⠙⠛⠃⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠢⡄⠁⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣀⠀⠀⠀⠁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⢰⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣈⣠⡿⠐⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⡙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⡴⠶⠶⣦⣄⡀⠀⠀⠀⣠⡾⠋⠉⠉⠉⠙⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⠀⠉⠻⣦⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⣠⣶⢦⢸⡇⠀⠀⠀⠀⢀⣴⡶⢦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣿⣧⣼⢾⡇⠀⠀⠀⠀⢸⣿⣧⣼⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠻⠿⠟⠸⣇⠀⠀⠀⠀⠘⢿⣷⡿⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⣴⠿⣄⠀⠀⠀⠀⠀⠉⠀⠀⢠⡾⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠙⠷⣤⣀⣀⣀⣴⠾⠋⠀⠙⢧⣀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⠛⠳⠞⠛⠋⠁⠀⠀⠠⣶⣶⣦⣄⠉⠻⣦⡀
⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⠙⠷⠀⠈⠻⣦
⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⢿⡁⢸⣿⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣶⣤⣤⣤⣤⣤⣤⣤⣴⡶⠾⠟⢋⠁⠀⠀⠱⢈⣿⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢻⣦⠀⠀⠈⢦⠀⠀⠀⢹⡿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⢇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠶⠶⠶⢶⣦⣤⡀⠀⠀⠘⣷⡀⠀⠈⠆⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠉⠻⣷⡄⠀⠸⣧⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⢻⡇⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢈⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠰⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣄⣀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣶⣤⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⡶⠾⠛⠋⠉⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⣿⠿⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⢠⣿⠀⠀⠈⠙⠛⠿⢶⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠉⠻⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣴⠶⠟⠋⠉

⠀⠀⠠⣄⣠⣟⠀⠀⠀⠀⠀⣀⣀⣺⣦⠀⠀⠀⠀⢀⣼⣃⠀⠀⠀⠀⢀⣟⣀⠀⠀⠀⠀⠘⢶⣷⠞
⠀⠀⠈⠉⡿⣍⠁⠀⠀⠈⢿⣿⣟⠁⠀⠀⠀⠠⣤⡿⠷⢤⡄⠀⠀⠀⢸⡯⡅⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠼⠑⠃⠀⠀⠀⠀⠚⠑⠋⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠏⠁⠀⠀⠀⠀⠀⠞⠘⠇
⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠻⣮⡉⣫⡾⠃⠀⠀⣠⣶⢿⣦⡄⠀⠀⢠⢴⣿⣿⡦⡄⠀⠀⠀⣀⣴⣀⡀
⠀⠀⠀⠙⣿⡛⠁⠀⠀⠀⣿⣿⣈⣿⡿⠀⠀⠈⠲⢿⡿⠗⠁⠀⠀⠀⢹⠿⣯⠁⠀⠀⢠⣤⣿⣽⣇⢀
⠀⠀⠀⠈⠳⠃⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠚⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠚
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⣤
⠀⠀⢉⣶⡟⡛⢦⡀⠀⠀⢰⢿⡆⠀⣄⠀⠀⣤⠠⡆⢰⠄⣤⠀⠀⠀⣀⠚⣿⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠸⣇⠘⠃⣸⠃⠀⢠⡟⠛⣿⡈⠛⠁⠀⠈⢷⣿⣿⡾⠁⠀⠀⠘⠻⢀⣿⠀⠀⠀⠀⠈⢠⣿⡄
⠀⠀⠀⠈⠙⠋⠁⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⢀⣤⠶⣤⡀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⣠⠴⢦⡄⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢠⣄⣀⣀⡀
⠀⠀⢸⣿⣴⣿⣷⠀⠀⠀⣠⣾⣿⣦⡦⠀⠀⠀⣿⡀⣠⣿⡀⠀⠀⠰⠐⠖⡶⠟⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⢹⣷⣿⣿⠁⠀⠀⢸⣏⣛⣹⡇⠀⠀⠀⢈⡉⠉⠉⠉
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⣠⡀⠀⠀⢀⡀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⡶⠆⠀⠀⢠⡀⠠⣶⠄
⠀⠀⠀⠀⠘⣿⡿⠂⠀⢸⡇⠀⢀⡼⠻⠀⠀⢀⣠⡍⣤⡉⣷⠀⠀⠀⠐⣡⣰⣶⠀⠀⠀⠓⠦⠥⠤⡶
⠀⠀⠀⣀⣴⠟⠁⠀⠀⢸⣇⣞⣉⣀⣀⠀⠀⠋⠈⢦⣍⣤⠟⠀⠀⣺⣇⣿⣟⣿⠀⠀⠀⠀⠰⣷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⡌⠂⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢣⣸⠀⠈⠙⠃⠀⠐⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠴⣍⡆⠀⠀⠀⠠⠀⠀⠈⠀⡏⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⠉⢸⠆⠀⠀⠀⢁⡦⠀⠀⢇⡏⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡤⢿⡷⠐⣏⣠⣖⡒⠉⡁⠀⣠⠊⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡖⢠⡾⢋⠅⡼⢟⣡⣄⠄⣰⠃⡀⠀⡀⠉⠳⣄⠐⠁⠀⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣮⡰⣇⠈⢧⠸⢊⣴⢋⡜⠁⣼⠃⢸⠀⠀⠈⠛⢆⠈⠢⠄⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⡲⣆⠹⣿⠇⢸⢠⢺⡎⢸⢀⢸⡎⢀⡟⠀⠈⠉⠒⠄⠀⠀⡀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⡉⠟⣸⠇⠠⠟⠈⢯⠃⣟⢸⠸⡆⠸⠁⠀⠀⢀⠀⠀⠀⠴⣿⠏⠀⠰
⠀⠀⠀⠀⠀⠀⠀⣀⢤⠞⠁⠀⠀⠓⣀⠀⠈⣦⠀⠀⠀⠈⢧⠙⠮⠧⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣠⣤⣦⠀⠶⠆⠛⠛⠋⠀⣤⣤⡤⠀⠙⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⢀⣄⠤⠠⠖⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣇⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢋⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠛⠋⠁

⣠⠤⣴⣶⣶⣶⣶⣶⢒⡤⢤⡀⠀⠀⠀⠀⢀⣠⣠⣤⣤⣤⣤⡤⠤⣄⣠⣀⠀⠀⠀⡀⣠⠴⣄⣀⣀⣀⣀⡀⠀⢀⣀⠦⢄
⢧⠀⠘⣿⣿⣿⣿⣿⠼⠂⢀⡇⠀⠀⠀⠀⢎⠀⠉⣿⣿⣿⣿⣿⡇⠍⠁⢨⠀⠀⠀⠉⠣⡀⢹⣿⣿⣿⣿⣿⣿⣷⠹⢀⠜
⠀⠑⠠⣽⣿⣿⣿⣿⡤⠖⠋⠀⠀⠀⠀⠀⠈⠢⣀⠘⣿⣿⣿⣿⡝⢀⠤⠃⠀⠀⠀⠀⠀⠠⡈⠛⠿⣿⣿⣿⠿⠗⢏⠅
⠀⠀⠓⠟⢿⣿⣿⡟⠳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠃
⠀⠀⠀⠀⠈⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⡀
⠀⠀⠀⠀⠰⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠆

⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣾⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡴⢀
⠀⠐⢶⣬⣉⡋⠠⠓⢒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢊⠨⣭⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡄⣤⣌
⠀⠀⠀⠹⢉⣾⠘⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⡗⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿
⠀⠀⠀⠴⠛⢻⡓⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⣠⣶⣾⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⣴⠿⠿⠿⠿⠿⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⣠⠆
⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠛⠓⣦
⠀⠀⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠒⠒⠛⠃
⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⡀⠀⠀⢀⣀⣀⡀
⠀⠀⢠⣾⣽⡿⠿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣶⢀⣪⣿⣿⡿⠟
⠀⠀⣽⣿⣿⠆⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣾⣿⣶⡶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣲⣿⣿⣿⠏
⠀⠀⠈⠿⣿⣿⣿⣯⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⠉⣯⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠻⣵⣿⣿⡿⠏
⠀⠀⠀⠀⣿⣭⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⣀⣿⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⠷⢁
⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠛⣷⣔
⠀⠀⠀⣾⣿⡟⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣺⣿⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣤⣿⡏
⣤⣤⣤⣯⣥⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⡀⠀⣤⣤⠄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠮⡥⠉⢡⡽⠷⡜⠁⣤⢾⣷⡈⠟⡃⣸⡷⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⢈⣧⣶⢶⡰⣟⣽⢳⣧⢻⣿⢯⣶⣷⣾⡽⣿⡿⠈⠽⠛⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣏⣏⡬⢿⣿⠛⠓⠈⠀⠀⢀⣀⣠⣤⣀⣉⡉⠁⠻⣿⣃⣴⣿⣼⠏⢤⡀
⠀⠀⠀⠀⠀⠀⣼⣃⡏⢰⣿⣶⠊⠁⠀⢀⣤⢞⡟⢋⣹⠯⣌⣿⣷⣿⣂⣦⠀⠐⡍⢻⣟⠁⢃⠷⡀
⠀⠀⠀⠀⠀⢸⡩⢮⡕⠆⣻⠁⠀⣠⡾⡹⢿⣭⡽⣶⣿⣿⡶⠟⡛⠈⠀⠁⠀⠀⠐⠀⠘⣿⣯⠚⡧
⠀⠀⠀⠀⠀⢺⣶⢄⡟⢈⣿⠀⠀⣿⣟⣷⢭⣍⡁⠻⣿⣿⣷⣋⢽⣀⡀⠀⢀⡔⠀⠈⠀⢸⡷⢾⡶
⠀⠀⠀⠀⠀⢨⡹⠾⣷⡈⠿⣦⡴⣿⡋⠀⠠⣄⠴⠷⣌⠳⣤⡘⢦⣈⠙⠓⠋⠀⠀⠃⠀⠐⡷⢶⡁
⠀⠀⠀⠀⠀⠀⢻⡄⠾⣛⠛⣫⣽⡏⠂⢀⣲⡃⠀⠀⠀⠙⠚⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⣼⡯⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠨⢙⠘⠛⠛⠊⠁⠀⣤⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣸⡷⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠆⠐⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡞⣩⣹⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣻⢯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣠⠆⠳⣿⣵⡸⠏⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣤⣴⣴⢶⣾⣾⠉⢿⣿⣣⣾⣾⣿⣿⣿⣿⣿⡶⢦⢤⣀⡀
⠀⠀⠀⠀⠈⠀⠀⢠⡤⣦⢦⣼⠿⢛⡻⠟⣷⣀⣪⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣟⡷⢼⠯⠇⢀⣦⣄
⠀⠀⠈⢀⣤⣶⠿⠿⠳⣷⣿⠝⠓⠉⠉⢁⡴⢻⡟⣿⣿⡿⠟⠋⠉⡁⠀⠠⠠⠠⠠⠡⠉⠛⠷⣤⣶⢿⣮⣛⣉⣡⡄
⠀⢀⠶⣿⠯⣍⢿⣿⠟⠉⠀⠀⠀⠀⢠⣯⣷⣮⣛⡿⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢿⡃⠛⠋⢩⣿⡆
⠀⣼⠲⠞⠧⠙⣼⡏⠀⠀⠀⠀⠀⠀⣾⡌⢻⣿⣿⣇⡀⠀⠀⢀⣀⣠⣤⡔⢲⣖⣶⣶⣶⡄⣀⠀⠀⠈⡿⣿⣿⡷⣿⢛
⠀⣿⠛⡿⢷⡞⢿⠃⠀⠀⠀⠀⠀⠀⢺⣵⡆⢥⡄⣿⣷⠶⢋⡽⠛⢛⣯⠒⣩⡭⣬⠉⣷⠀⣾⣗⣄⠀⣷⣮⣁⠸⢿⡞
⠀⢹⣺⢳⣿⢏⡐⣜⡦⠀⠀⠀⠀⠀⠈⢷⣶⣗⣲⢿⡶⢃⣴⣲⣮⣅⣀⡼⠿⠿⠯⢿⣯⠈⢀⣡⣥⣿⣭⣻⣭⣧⡍⣠
⠀⠈⢿⣿⣥⢾⣿⢩⢮⢀⣷⣄⡀⠀⠀⠀⠉⠻⠿⢮⢡⣼⠽⠿⠛⠉⠁⠀⠀⠀⠀⢀⣨⣿⢮⡽⢿⣿⣆⣫⣭⠟⢠⠇
⠀⠀⠀⠻⣿⢛⣏⡘⢫⡠⣽⡶⢶⣝⠒⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣯⠿⣉⡧⠔⠘⣇⢻⡟⠋⣰⠃
⠀⠀⠀⠀⠈⠙⡯⣿⣌⣱⣮⠿⣑⡛⢈⡳⢠⣾⢻⣿⠱⡴⢇⣾⣉⣻⣻⣿⣿⣿⣿⣿⣷⣯⣼⢀⡜⣿⣶⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠍⡙⠓⠾⢿⣿⣌⣡⣿⡴⢟⣙⡃⢀⣾⣟⣻⣿⣓⡿⣿⣿⣿⣿⠿⠞⢹⠌⢻⡾⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⢀⡉⠉⠁⠙⠛⠻⠿⠛⠿⠟⠟⠇⠃⠛⠉⢉⡀⠄⠒⡧⢀⣼⡃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠂⠀⠈⠈⠀⠀⠀⢜⡑⢿⢐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠃⡀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⢦⣤⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢀⠶⣆⣰⣯⡿⠟⢻⣽⣿⣦⣅⠀⢀⡠⢲⣭⣵⣟⣻⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢽⣿⢿⣁⣀⣀⣀⣀⣩⣥⡿⣸⡆⢣⣵⣟⣻⣾⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣻⣖⣎⣝⢣⠮⣝⣿⣿⡆⡾⣷⡸⣃⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇⣉⡛⣩⣟⡿⠛⠉⣿⣿⢚⡷⠟⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⣀⣼⣫⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡴⠶⡶⠶⠶⠶⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⡉⠉⠁⣄⣀⣨⣦⠤⠷⠶⠶⠦⣭⣿⣳⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⡋⠁⠐⢄⠀⣈⡶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡁⠀⠘⢦⣀⣤⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠈⠦⡀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢅⠀⠀⠠⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠑⢄⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⣶⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠢⡀⠀⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⢹⡆⠢⢝⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⢲⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢸⡗⠢⢄⡈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠁⠢⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠗⢦⡀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠲⢄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣾⠑⢤⡙⢦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣸⠏⠡⡀⠙⢤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⣠⣾⠋⠱⡀⠑⣄⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠑⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣥⣄⢳⡀⢳⣀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠄⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣜⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠉⠛⠲⠦⣤⣤⣀⣀⣀⣀⣀⣠⣤⣶⠾⡋⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠟⡻⠉⠉⠻⣦⡀⠀⠀⠀⠀⠉⠉⠉⢈⡟⢀⠈⢦⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡡⠃⠀⡜⢠⠁⠀⠀⣀⣨⣿⣦⡀⠀⠀⠀⠀⣠⢾⠁⡀⢆⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠳⣦⣀⣠⣧⠶⠞⠛⠉⠁⠀⠈⠳⣤⡤⠶⠏⠙⢨⠀⠀⢈⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠶⠶⠶⠶⠶⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀
⠀⠀⠀⠀⠀⡴⠊⠉⣉⠉⠑⢄⠀⠀⠀⠀⠀⠀⢠⡴⠊⢉⣉⡉⠛⢧⠀⠀⠀⠀⠀⠀⠀⠐⠀⣉⡉⠑⠆
⠀⠀⠀⠀⡼⠀⠀⠀⣿⠀⠀⠀⠃⠀⠀⠀⠀⢰⡿⠀⠀⠈⢀⡇⠀⠈⣧⠄⠀⠀⠀⠠⠁⠀⠀⢁⡷
⠀⠀⠀⠀⢳⠀⠀⠀⣿⠀⠀⠀⡞⠀⠀⠀⠀⠈⢧⠀⠀⢸⠋⠀⠀⠀⡿⠀⠀⠀⠀⠐⠀⠀⢀⡈⡷⠀⠀⠘
⠀⠀⠀⠀⠀⠳⣄⣀⠉⣀⡠⠞⠀⠀⠀⠀⠀⠀⠘⣷⣄⣘⠛⣃⣠⡞⠁⠀⠀⠀⠀⠀⠔⢄⣀⠉⢁⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢀⠴⠚⠉⠓⠲⣄⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠋⠛⠳⢤⡀
⠀⠀⠀⠀⣰⠃⠀⢸⠀⡇⠀⠈⢳⠀⠀⠀⠀⢠⡟⠀⠀⣾⠉⠀⠀⠹⡆⠀⠀⠀⠀⣰⠃⠀⢰⡏⠃⠀⠀⢳
⠀⠀⠀⠀⢿⠀⠀⠸⠶⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠙⢹⠀⠀⠀⡇⠀⠀⠀⠀⢻⠀⠀⢸⡟⡆⠀⠀⢸⠃
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠇⢀⡠⠋⠀⠀⠀⠀⠀⠳⣄⠀⠙⠞⠀⣀⡜⠁⠀⠀⠀⠀⠈⢧⣀⠘⠷⠃⢀⣠⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠛⠋
⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠲⢦⣦
⠀⠀⠀⠀⣰⠃⠀⠐⢲⠂⠀⠘⡆⠀⠀⠀⠀⠀⡜⠀⠀⡖⣦⠀⠀⢣⠀⠀⠀⠀⠀⣼⠏⠀⠀⡞⣦⠀⠈⢦
⠀⠀⠀⠀⣯⠀⠀⠀⡾⠀⠀⠀⣹⠀⠀⠀⠀⠸⡅⠀⠀⡷⣿⠀⠀⢀⡆⠀⠀⠀⠀⢻⠀⠀⠀⠧⣿⠀⠀⢸
⠀⠀⠀⠀⠘⣄⠀⠰⠇⠀⠀⡠⠃⠀⠀⠀⠀⠀⢣⡀⠀⠧⠟⠀⢀⡜⠀⠀⠀⠀⠀⠘⣧⡀⠀⠦⠟⠀⣠⠏
⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠏⠉⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⠶⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⢀⣤⡈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢸⠀⡇⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⡇⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣈⠉⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠋⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠺⠭⠀⣀⠀⠀⣀⠙⢽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢠⣵⡆⠈⠉⠀⠀⠀⠀⠈⠻⣶⣿⣦⣾⣿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠙⠈⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣛⠁⠀⣤⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢧⡠⠐⣲⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡿⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⣿⣿⣿⣿⡆⢀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣧⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⡿⣿⢓⢀⠘⠂⠀⠀⠀⠸⣿⠟⡠⢿⡟⠛⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠛⣿⣼⣿⠻⠶⣾⣿⣶⣶⣿⠆⠀⠀⣿⢿⣿⡻⢏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠂⠀⠘⠋⠉⠉⠁⠀⠀⣴⠀⣸⣿⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣫⠄⠀⠾⣶⣤⣀⣀⠀⠀⠀⢀⣠⣾⣶⣿⣿⣿⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡂⣩⣿⣿⣬⣿⠗⢾⣿⢿⢿⣿⣿⣟⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣶⣿⠏⠛⣿⠿⣧⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠁⠛⠉⠀⠉

⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠂⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣐⡀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠁⠀⠀⠀⠀⠀⠀⠀⠠⢂⠀⠁
⠀⠀⢀⡴⠚⠈⡇⠀⠀⠀⠀⣠⠊⠉⠋⠙⠛⡄⠀⠀⠀⠀⠀⣏⠈⠑⠛⠙⡽⠃⠀⠀⠀⠀⠀⡜⠑⠁⢺⠀⠀⠀⠀⠀⠀⢸⠁⢉⣻⣋⡑
⠀⢸⢁⣠⠀⠂⡇⠀⠀⠀⠀⠑⠦⠊⢙⠄⠀⡇⠀⠀⠀⠀⠀⠀⡠⠋⠀⣜⠁⠀⠀⠀⠀⢀⡜⢀⠜⡁⢸⠀⠀⠀⠀⠀⠀⢸⠀⠧⢄⡀
⠀⠈⠁⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠞⠀⡰⠃⠀⠀⠀⠀⠀⠈⠦⠤⡄⠀⠱⡄⠀⠀⢀⠞⠀⠘⠚⠁⠘⠲⡄⠀⠀⠀⠀⠘⠒⢢⡀⠈⢦
⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⡔⠁⠀⠘⠓⠒⡆⠀⠀⠀⠀⡤⠤⠴⠃⠀⢠⠇⠀⠀⠘⠒⠒⠒⢲⠀⢰⠊⠁⠀⠀⠀⠀⡴⠤⠞⠁⠀⡼
⠀⠀⠀⠸⠦⠤⠇⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠴⠇⠀⠀⠀⠀⠧⠤⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠸⠤⠼⠀⠀⠀⠀⠀⠈⠲⠤⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡀⠈⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠄⡀⢦⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢤⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⡨⠴⠶⠔⢲⠀⠀⠀⠀⡔⠒⠒⠴⢗⣂⠷⡄⠀⠀⠀⠀⡠⢤⢧⣴⡀⠀⠀⠀⠀⠀⡠⠶⡠⢗⡰⣄⠀⠀⠀⠀⠀⢀⠤⠶⢷⡨⣄
⠀⢠⠋⠀⣠⣒⡒⠋⠀⠀⠀⠀⠧⠤⠤⣤⠀⠀⣀⠇⠀⠀⠀⡎⠀⣔⢐⠄⢸⠄⠀⠀⠀⢸⠁⢠⠊⠑⡄⢘⡆⠀⠀⠀⡴⠁⡠⠒⠒⢄⠈⢧
⠀⡎⠀⠀⡀⣀⠉⢢⠀⠀⠀⠀⠀⢀⡞⠁⣠⠞⠁⠀⠀⠀⠀⣱⠄⠈⠉⠠⢎⠀⠀⠀⠀⠸⡀⠈⠒⠚⠁⠉⡇⠀⠀⢰⠇⢰⠃⠀⠀⠘⡄⢸⡀
⠀⡇⠀⢈⠀⢈⠆⠈⡇⠀⠀⠀⢀⠎⠀⣰⠃⠀⠀⠀⠀⠀⡼⠁⢠⠒⠐⡆⠀⢣⠀⠀⠀⠀⠉⠒⣲⠆⠀⢰⠃⠀⠀⠸⡆⠈⢧⣀⣀⡼⠀⢸⠁
⠀⠙⢄⠀⠉⠉⢀⡼⠁⠀⠀⠀⣾⠀⢀⡇⠀⠀⠀⠀⠀⠀⠹⣄⠈⠑⠊⠁⣠⠏⠀⠀⠀⠰⡏⠉⠀⢀⠔⠃⠀⠀⠀⠀⠱⣄⡀⠉⠁⢀⣠⠏
⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠠⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠀⠀⢶⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⣀⣀⣼⠃⠀⣄⠠⠤⢶⢀⡉⠀⠀⢀⣦⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⢿⣿⠀⢰⣿⠀⠀⠀⠋⠈⠁⠈⡤⠟⢻⣿⡇⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⡀⡙⣄⢼⣏⡀⠀⠀⠠⠑⢲⡁⠠⠾⠾⢿⡇⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠸⡇⠀⠘⣿⣦⣤⡀⠀⠨⠦⠀⠄⠀⠀⠉⠙⠁⠉⡏⣶⠂⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⣭⣇⠀⢷⡀⠀⠘⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠱⣦⣤⣄⠀⠀⠀⢀⣾⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⢸⣿⣤⣀⠹⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀⠰⠟⠛⠁⠀⢀⣠⡾⠟⠛⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠄⠀⠈⠛⠷⣝⣟⠻⣷⣟⣷⣤⡀⠀⠂⠁⠀⠀⢀⣠⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣀⠀⠈⠛⠿⣿⣿⣏⣉⠀⠀⠀⢀⣠⣴⣿⣿⠋⠀⢀⣤⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠑⢦⣀⡀⠈⠛⢿⣿⣷⣤⣀⣀⣀⣴⡿⢁⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⠀⠁⣽⣦⣤⣤⠛⣩⣿⡿⠿⠿⠛⠂⠉⡉⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠙⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠚⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⣀⠀⠀⠀⡤⠖⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠈⢳⣀⡞⠁⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠀⠀⠀⠀⠀⣻⡇⣿⠀⠀⠀⢸⣽⡆⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠈⢧⠀⠀⠀⠀⠻⠃⣼⡄⠀⠀⠈⠛⠁⠀⣰⠋⠙⠳⣤⡀
⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣠⠞⠁⠙⢦⡀⠀⠀⣀⡴⠃⠀⠀⠀⠈⠛⢦⡀
⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠹⣆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⠀⠀⢻⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢷
⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⣠⠼⡇⠀⠀⠀⠀⢸
⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠒⠒⠒⠚⠋⠙⣇⠀⣷⠀⠀⠀⠀⠈
⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⢀⡀⣀⣼⣄⠇
⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠓⠦⠞
⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⡾⠁⠈⠙⠳⠦⣤⣀⣀⠀⠀⠀⠀⢀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠠⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠉⠛⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠶⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡉⡤⢤⠀⠀⠀⠀⠀⡠⢔⣬⢖⡐⡀⠀⢨⢙⡟⠂⡇
⠀⠀⠀⠀⢀⣠⣄⣤⣖⡔⡔⠀⠀⡠⠨⡠⡞⠁⠀⠈⠙⠾⢶⠀⠀⠀⠀⢠⠇⠀⡥⣕⣋⠉⣷⠃⢁⠈⢹
⠀⠀⢀⡴⠋⢀⡤⠤⡑⠗⢣⢀⡬⠣⢽⡁⠘⠚⠉⢹⠀⠀⢸⠀⠀⠀⠀⣼⠀⠘⠧⣀⠀⡹⠁⢠⢺⠀⢸⡄
⠀⠀⡞⠀⣰⠃⠀⠀⠘⣆⠼⠋⠀⢀⠼⡇⠀⣀⠴⠋⠀⣠⢏⠀⢀⡀⠀⠉⠉⢢⠀⠈⢻⠁⠐⠓⠚⠀⠘⠓⣦
⠀⢸⠃⠀⣇⠀⠀⠀⢰⠛⢦⡖⢻⠀⠀⡗⠉⠁⠀⣤⡚⢁⣮⣴⣥⣵⠖⠲⠤⠞⠀⠀⢸⠦⠤⠤⢤⠀⢠⡖⠟
⠀⠘⣇⠀⠘⠦⠤⠖⠃⠀⡼⠁⢸⠀⠀⣧⣄⣀⠀⠀⡸⠉⢈⠥⢍⠉⢳⣄⣀⣀⣠⠴⠃⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠈⠲⣄⣀⣀⣀⣠⠞⠁⠀⢸⠀⠀⢿⠀⠈⠉⠓⢧⡀⠘⠲⠚⢀⡼⠀⠀⠀⠀⡀⢦⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠘⢳⡖⠚⠀⠀⠀⢠⠖⠉⢀⣀⣀⠉⠲⣄⡠⠶⠶⣤⡸⣤⡀⢠⠚⢄
⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠲⠴⢞⠥⢹⣀⠀⠀⠀⣏⠀⠐⣇⣀⣸⠆⡰⠋⠀⣠⠴⠦⠤⣧⢆⡺⠫⠎⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠾⣤⣀⣀⡀⠈⠉⠁⣸⠤⢤⢡⡛⣐⣇⠈⠉⠁⣰⠁⠀⠀⠉⠉⠓⢤⡇⠀⡔⠉⠙⡆⡸⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣠⠔⠿⠤⣀⡉⠑⠚⡩⠋⠉⠉⣿⠀⢠⠋⠉⢱⠀⠘⣇⠀⠙⠶⠞⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⡴⠋⠀⠀⢠⠔⠊⠀⣠⠊⠀⠀⠀⠀⢹⡀⠈⠳⠴⠊⠀⣸⠉⠲⠤⡤⠴⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡞⠁⠀⠀⠀⠘⠶⢤⡀⠈⠱⡄⠀⠀⠀⠀⠙⠲⠤⠤⠴⠚⠁⠀⠀⣀⡤⠚⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠻⠤⢤⡞⠀⠀⠀⠀⡴⠤⣤⠼⠃⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢥⣀⣀⣀⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠰⢞⣱⠆⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⡐⠠⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣦⡀
⠀⠀⠀⠀⠀⣀⣿⣿⣀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠉⣿⣿⡄⠀⠀⠀⠀⢀⡃⡐⠁⢠⡆⠀⢡⠐⡀⠀⠀⠀⠀⠀⣰⣿⣧⢸⣿⢹⡆
⠀⠀⠀⠀⠀⠙⠁⣿⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⣤⣿⡿⠁⠀⠀⠀⠀⠐⡎⢷⡀⠠⠇⠀⣸⢅⡖⠀⠀⠀⠀⠀⢿⣿⣿⡾⢋⡼⠃
⠀⠀⠀⠀⠀⢀⣰⣿⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⣿⣶⡀⠀⠀⠀⠀⠀⠹⣞⡅⠀⠀⠈⣃⠾⠁⠀⠀⠀⠀⠀⢀⣙⣲⣖⠋⡀
⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣿⠶⢿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⢤⣀⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆
⠀⠀⠀⠀⠰⠻⢶⡾⠛⠁⠀⠀⠀⢠⣤⣄⣴⣿⡿⠙⢦⣠⣤⠀⠀⠀⢸⡁⢹⣿⣿⣿⣿⠸⠁⣹⠀⠀⠀⠀⠀⠀⠀⣷⣿⠃
⠀⠀⠀⠀⠀⣠⣬⣧⣤⡀⠀⠀⠀⢀⣿⣿⡞⢋⣰⣿⣯⣿⣿⡀⠀⠀⠀⠑⠶⣿⣿⣿⣿⡧⠖⠁⠀⠀⠀⠀⠀⠀⢀⣸⣟⣀
⠀⠀⠀⠀⢸⣿⠛⡿⢻⣿⠀⠀⠀⠀⠀⠀⠈⠚⠙⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠻⢿⡇
⠀⠀⠀⠀⠸⠿⠾⠷⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠾⠇

⠀⠀⠀⠀⢀⢔⣮⣗⡢⡀⠀⠀⠀⠀⠀⠖⢸⣿⣿⣿⡆⠲⠀⠀⠀⠀⠀⢀⠠⠀⢀⡀⠀⠠⢀⠀⠀⠀⠀⠀⣄⣀⣀⣀⡀⣀⣀⣠⡀
⠀⠀⠀⠀⡦⣿⡛⢛⣿⢰⠀⠀⠀⠀⠀⠈⠊⢿⣿⡿⠔⠁⠀⠀⠀⠀⢐⠂⡆⠀⢸⡇⠀⠈⠄⠂⠀⠀⠀⠀⣿⣿⣿⢟⣵⣿⣿⣿⡇
⠀⠀⠀⠀⠰⣝⡿⢿⣫⠖⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠨⡄⢷⡀⠾⠷⠄⡼⠁⡍⠀⠀⠀⠀⣿⡿⣣⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢺⣿⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣻⣷⡆⠀⠀⠀⠀⠀⠀⠛⣦⡅⢀⣀⠠⣤⠞⠀⠀⠀⠀⠀⠉⠀⠁⠉⠻⣄⣼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⣿⠛⢛⣛⠇⠀⠀⠀⠀⠀⠀⣔⣉⣯⣉⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⠁⠀⠀⠀⠀⠀⠰⠒⠄⣶⣶⣶⣶⠠⠒⡆
⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⠀⠀⠀⠀⣆⢹⠿⡏⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠿⢧⠀⠀⠀⠀⠀⠀⠀⢂⠀⣿⣿⣿⣿⠀⡰
⠀⠀⠀⠀⢀⣀⣼⣁⣀⡄⠀⠀⠀⠀⠀⠀⠈⢑⣶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡏⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠏⠔
⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⢀⣤⣖⠋⠉⠉⠉⠉⠉⠉⠉⠑⠢⢄⠀⠀⠀⠀⠀⠀⣀⠀⣠⣴⡖⠂⠀⠀⠀⠀⠖⠀⠀⠀⢄⡀⢀⡀
⠀⠀⣤⣠⠤⣿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠤⡤⡄⠀⢻⠻⣤⣽⡿⣦⣤⣤⣤⣀⣠⣤⣤⣤⣶⣿⣥⠿⣻⠁
⠀⠀⢻⡇⠈⠙⠀⠈⠉⠃⠀⠀⠒⣤⣤⡤⣽⣾⣾⠉⠁⢸⠇⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠟⢻⠏⠀⠀⣠⠃
⠀⠀⠀⠳⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠇⢀⡠⠋⠀⠀⠀⠀⠱⢄⡀⠀⠀⠀⠀⠀⠀⠀⣴⣴⢆⣠⠔⠁
⠀⠀⠀⠀⠀⠉⢈⡦⡀⠀⠀⠀⠀⠀⣠⣮⣿⢿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢠⣧⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⢤⣤⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣤⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠁⠼⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠀⠽⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠳⠅⣀⣀⡨⣽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣏⣀⠀⢀⣙⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠉⠀⠐⠂⠚⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⣉⣉⢉⣥⡀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠐⣂⣒⣒⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣹⠟⣛⣻⣻⣡⠿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢠⡞⠁⠐⠂⠬⢍⣒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣋⠭⠄⠂⠈⠙⣦
⠀⠀⠀⢸⡖⠚⢋⣉⣓⡒⠦⢝⡮⠷⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠴⠭⣞⠥⠖⣚⣋⣙⠛⠒⣿
⠀⠀⠀⠈⢷⡉⣉⠀⠨⢭⣍⣻⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣼⣋⣭⠍⠀⣈⡉⣸⠇
⠀⠀⠀⠀⠘⢧⡠⠤⠖⠒⠒⠒⠛⠦⣄⠀⠀⠙⢦⠀⠀⠀⠀⣠⠞⠁⠀⣀⡴⠞⠒⠒⠒⠒⠦⠄⣰⠏
⠀⠀⠀⠀⠀⠈⢷⣐⠊⠉⢉⣉⣉⡙⠛⠛⠒⠦⣄⡳⣐⣢⡴⣁⡤⠖⠛⠛⠛⣉⣉⣉⠉⠉⢒⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠧⣤⣤⠬⣭⣭⣭⠯⠭⢽⣏⢘⣿⠭⠭⣯⣭⣭⡭⢤⣤⡬⠿⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣁⠤⠄⠐⣶⠤⠒⠊⡝⣿⢾⡿⡑⠒⠤⢴⡒⠀⠤⢄⣉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⣉⠤⠒⢈⡏⠀⠀⣠⢾⢀⡏⡘⡇⠹⢆⡀⠀⠙⣏⠐⠢⢌⠓⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠤⠂⡩⠳⡴⢺⠁⠀⣸⢀⢃⢹⠸⠀⢻⠲⡴⠫⡁⠢⣄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⢠⠊⠀⡜⠀⠇⠀⢰⠇⠈⠃⠈⣿⠀⠈⡄⠑⡀⠈⢦⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣇⠀⡘⠀⠀⡆⢀⡞⠀⠀⠀⠀⠸⣆⠀⠇⠀⢸⠀⢈⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠦⠶⠚⠋⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣷⢠⣤⠄⠀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠃⠀⠀⠀⠀⠀⠈⠁⠚⠘⠁⠀⢀⣉⣶⣙⣃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠆⠀⣠⣶⣀⣤⣤⣾⣶⣠⡁⠀⣘⣙⢽⠟⣻⣿⣦⣤⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⢠⣼⣀⠟⢛⠿⣟⠻⠿⢿⡿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣄⢰⣼⣿⣤⣤⣤⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣶⢶⡀⠀⠂⠀⣠⠟⢉⣾⣾⢡⣟⡙⠛⣻⣿⡟⣿⣿⣿⣿⣽⣿⣿⢿⡯⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢈⣯⣤⣤⠼⠆⠓⠃⣰⡶⠦⠠⢉⣻⣿⣿⣿⣄⠀⠀⠁⠰⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣠⣴⣿⡾⠟⠋⠙⠛⠛⠺⣟⠃⠈⠈⠒⢊⣭⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣦⠐⡚⠀⠀⣤⣸⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢛⣻⡈⠛⡻⠄
⠀⠀⠈⢋⣍⣀⠀⠶⠷⠀⣥⡔⠀⠷⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⣿⡿⣽⠶⣷⣆⡈⣷⣆⢀⠀⠀⣰⣴⡟⠀⠀⠀⠀⢤⢤⠀⠀⢠⡄⢈⠄
⠀⠀⠀⠀⠛⠀⢸⠀⢈⣽⣿⢀⣏⢡⣤⣿⣿⣿⣿⠓⠀⢀⣀⣤⡾⠓⠛⠁⠈⠉⠀⣀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠈⠧⢰⣿⠇⢹⡿⠛⢹⠿⡷⠄⣈⣰⣀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡧⠀⠀⠀⣀⡀⠀⣠⡾⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⣿⠇⣀⣶⡶⠿⠿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣼⡿⢺⣿⠋
⠀⠀⠀⠀⢀⠄⣫⡵⠟⠁
⠀⠀⠀⠀⠀⣘⡿⠁
⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⢚⠷⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠥⡀⠀⣠⠴⠦⡶⣥⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠉⠓⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠰⣧⣛⣇⡞⠁⠀⣀⣀⠀⠁⠸⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⡴⡻⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢹⡉⠲⠋⠁⢸⡆⠀⢠⡇⠀⠀⠀⣀⠊⣰⢊⡙⢆⠀⠀⠀⢀⠔⠁⠀⠐⠣⠤⠃⠀⡟
⠀⠀⠀⠀⠀⠀⠀⡖⠁⢀⡄⠘⠉⣇⠀⠀⢀⠞⠀⢀⡾⠀⣾⠉⠉⠉⠑⠉⠮⠴⠋⢹⠀⠀⠈⠛⠲⠦⢤⣀⠀⠀⠀⠉⡶
⠀⠀⠀⠀⠀⠀⠀⠹⠶⠋⢹⡀⠀⢸⣄⠴⠃⠀⣠⣾⣤⣀⣿⠤⠤⠤⣤⠄⠀⠐⢁⡼⠀⠀⢀⠀⡠⠀⠀⡾⠀⢠⡟⠒⠃
⠀⠀⠀⠀⠀⢀⠄⣀⠴⡆⠈⣇⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠃⠀⢀⡖⠉⣠⣅⣶⢇⠷⣸⡀⠘⠧⣤⡼
⠀⠀⠀⠀⠀⣢⣤⢥⣄⡰⣤⢹⡄⠀⢸⡗⠒⠒⠒⠒⠒⠒⠃⢀⡞⠀⠀⣰⠋⢠⠞⠁⢀⡤⠤⣍⠀⠙⢦
⠀⠀⠀⣰⠋⠀⠀⣀⣡⣀⡏⠀⠓⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢰⠇⢀⡏⠀⢰⠃⠀⠀⠀⢳⠀⠘⡇
⠀⠀⣼⠁⠀⠠⠯⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡟⠀⢸⠃⡀⣿⡐⠆⠀⠀⢸⠀⠀⣿
⠀⠀⡇⠀⠀⢀⣠⣄⠀⠈⢳⠀⢠⠀⢀⣰⠄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⡼⣇⢤⡋⣌⡦⢤⣠⠞⠀⠀⡟⠀⠀⠀⢀
⠀⠘⡇⠀⠠⡏⠀⠀⣷⢀⡬⠧⡄⢮⡡⢮⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢁⣭⣬⣀⣸⠁⠀⢀⡼⣃⡤⠶⢶⠡⣰⠐⢶
⠀⠀⢷⡀⠀⠓⠦⠖⠃⢸⣇⣀⣈⡁⠈⠉⡴⠁⠀⠀⣀⠀⣠⡀⠀⠀⠀⠀⡇⠀⢸⣳⣦⣤⣤⠤⠶⠋⡼⠁⠀⣀⢬⣉⠫⠥⣎
⠀⠀⠀⠳⢤⣀⣀⣀⣤⠖⠃⡠⠊⠁⢀⡞⠀⠀⠀⣰⣤⣭⣤⣵⠀⠀⠀⠀⠧⠤⣄⡀⠈⠑⣆⠀⠀⢸⡃⠀⢸⡁⠀⢈⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢺⣀⣀⡀⠀⠉⠳⣤⠊⠁⣉⠭⡉⠉⢳⡀⠀⠀⠀⠀⠀⡷⠀⠀⢸⠂⠀⠈⢧⡀⠀⠙⠒⠋⠠⠔⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⣿⠀⠀⠧⡤⠟⠀⣸⠃⠀⡸⠉⠙⠉⠀⠀⢀⡼⢠⠶⠤⣤⣩⠷⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠛⠁⠀⠀⣠⣯⠷⠀⠀⠀⠀⠾⢅⠀⠀⠙⠶⠤⠤⠤⠖⠋⠀⢿⣀⡀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⢤⣤⠤⠴⠚⣽⠁⠀⣰⠚⠛⢦⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠹⠤⠤⠞⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⡤⠞⠁

⣿⣿⠟⠛⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⠻⠷⠶⠶⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠶⠶⣤⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⠀⠀⠀⠈⠙⠻⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣦⠀⠀⠀⠀⢸⡇
⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠘⣿
⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠖⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⣸⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⣀⣀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⣿⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡋⠉⣈⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣿⡄⠸⡄⠀⠀⠀⠀⠀⠀⠀⠰⣤⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡶⠿
⢹⡇⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠻⢾
⠈⣿⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠹⣧⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠹⣧⡀⠀⠀⠀⠉⠛⠶⣤⣤⣄⣠⣇⣀⣀⣇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠙⢷⣄⠀⠀⠀⠊⢠⡄⠀⠉⢹⡏⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣶⣤⣤⣤⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣀⣤⠀⠀⠀⠀⠀⠰⠤⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⡤⠤⠤⠤⠤⠶⠶⠶⠒⠒⢺⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣴⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡄⠀⣠⡶⠛⠛⠷⣤⠀⠀⣿⠀⣿
⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⢸⡇⠘⣇⢰⡏⠀⠀⠀⠀⢸⡇⢸⡇⢰⡏⠀⠀⠀⣠⣤⣤⣄⡀
⠀⠀⠀⢠⡟⠁⠀⠀⠙⣷⠀⠀⠀⣿⠀⢿⡈⢷⡀⠀⠀⢀⣼⠃⣾⠀⣾⠁⠀⠀⣾⠋⠀⠀⠈⢻⡄
⠀⠀⠀⠸⣧⠀⠀⠀⢀⡿⠀⠀⠀⢹⡆⢸⣇⣀⣙⣛⣛⣋⣁⣰⡏⢠⡟⠀⠀⠀⢿⡀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠈⠻⠶⠶⠛⠁⠀⠀⠀⠈⣷⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⣸⠇⠀⠀⠀⠈⠛⠶⠶⠟⠁
⠀⢀⣀⣤⣴⠆⠀⠀⠶⣤⣤⣀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣤⣤⠶⠀⠀⠰⣦⣤⣀⡀
⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⣿⠀⢠⡄⠀⠀⠀⠀⠀⠀⢠⠀⢈⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡁⠀⡄⠀⠀⠀⠀⠀⠀⢠⡄
⢻⡆⢸⡇⠀⠀⠀⠀⠀⠀⣾⠀⣼⠇⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠸⣧⠀⣷⠀⠀⠀⠀⠀⠀⢸⡇⢰
⠀⠻⣾⡇⠀⠀⠀⠀⠀⠀⣿⡾⠋⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇⠀⠙⢷⣿⠀⠀⠀⠀⠀⠀⢸⣷⠟
⠀⠀⢸⡇⠀⠀⢠⡀⠀⠀⣿⠀⠀⠀⠸⡇⠀⠀⢠⡶⢶⡆⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢀⡄⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣷⢸⡇⠀⠀⣸⠇⠀⠀⠀⢿⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⣿⠀⠀⢸⡇⠀⢀⡗⠀⠀⠀⠀⢻⡄⠀⠀⣿⣼⠁⠀⠀⣿⠀⠀⠀⠀⢸⡀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⢿⣿⠀⠀⢰⡏⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿
⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⣷⠀⠀⢸⡿⠀⠀⣸⠇⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿
⠀⠀⠀⢸⡆⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢿⡀⠀⠸⠇⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢰⡇
⠀⠀⠀⢸⣇⣀⣸⣇⣀⣼⠇⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣠⡏⠀⠀⠀⠀⠀⠸⣧⣀⣸⣇⣀⣸⡇
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⢴⣦⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⣰⣿⣇⣀⣀⠀⠀⠀⠀⠀⠀⠉⢿⣮⡛⠛⠻⣶⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⡄⠀⢶⠆⠀⣶⠄⢀⣄
⠀⢰⣿⣿⣿⡯⣳⣿⠃⠀⠀⠀⠀⠀⢸⡇⠉⣇⣷⠄⢸⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⢻⣄⣸⣧⣠⣿⣠⡾⠉
⠀⢸⣿⣿⣿⣯⣿⡏⠀⠀⠀⠀⠀⠀⠘⣷⣄⠈⠉⣀⣾⠏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠃
⠀⠈⠉⠛⠛⠓⠛⠂⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠒⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⣤⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠻⣷⣆⠀⠀⠀⠀⠀⠀⠀⢒⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄
⠀⠀⢀⣝⠉⣀⣀⠈⢻⣆⠀⠀⠀⠀⢸⣿⣄⠀⢀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢹⣿⡄⠀⠀⣴
⣠⠾⠻⠟⠁⠿⠿⠃⢸⡿⠀⠀⠀⠀⠀⢻⣿⣀⣾⣿⣿⡗⠀⠀⠀⠀⠀⠀⠉⢠⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣴⣿⣷⠀⠛⣿⠛
⠃⠀⠀⠺⣶⣤⣤⣴⠿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⢀⣸⣟⠀⠀⣸⣿⣇⡀⠉
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣜⣛⡴⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⣰⡄⠀⠘⢀⠀⠀⠀⠀⣀⣀⣚⣛⣛⣛⣂⣀⡀
⠀⣼⣹⣿⡅⢸⣿⣏⣧⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠀⠲⣶⣿⣿⡶⠂⠀⠄⠀⠀⠀⢻⡈⣯⣿⣿⣿⠏⣼⠃
⠀⢿⣚⣿⡇⢸⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠿⡇⠀⠠⠀⠀⠀⠀⠀⠙⠚⢿⣿⠟⠚⠁
⠀⠈⠻⣾⣿⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⠇⠈⢿⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈

⠀⠀⠀⠀⠀⣠⡿⡇⠀⠀⠀⠀⠀⠀⣤⡄⠀⣤⣿⣤⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣘⠿⠃
⠀⠀⠀⠠⠚⠁⢠⣤⣄⠀⠀⠀⠀⠀⠘⢷⣄⣀⣛⣀⣠⣄⠀⠀⠀⠀⠀⠀⠬⠿⣾⣿⢷⡴⠆⠀⠀⠀⠀⠀⠀⢰⡶⢶⣶
⠀⢀⣀⡀⣿⣷⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠙⣛⠛⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⠿⣦⠀⠀⠀⠀⠀⣴⣤⣤⣾⣿⣼⣿⣀⣀⣀
⠀⠸⠿⠇⠿⠿⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢼⠃⠀⠀⠀⠀⠀⣿⡗⣼⣿⣿⣿⣿⣿⢂⣿
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠸⠂⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢹⣿⡇⠀⠀⠄
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡴⠉⠀⠀⢿⡿⠀⠀⠙⢦⠀⠀⠀⠀⡄⠀⠀⠀⢀⣯⠀⠀⠀⠀⢀
⠀⢰⢰⠀⢀⢄⣀⣀⣀⡠⠀⠀⡖⡆⠀⠀⢸⠁⠀⠀⠀⠼⠇⠀⠀⠀⠘⡆⠀⠀⢸⡇⠀⠀⠀⠘⠛⠃⠀⠀⠀⣿⡇
⠀⠸⣔⠣⣀⠀⠻⣿⠟⠀⣀⡜⡰⠁⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠠⠀⢀⣿⠁
⠀⠀⠈⠳⠄⠀⣴⣿⣦⠀⠡⠞⠁⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠀⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣿⠇⠀⠀⡀
⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⢠⢞⠏⠀⠀⣴⣷⣤⠀⠀⠻⡳⡀
⠀⠀⡇⡄⠀⠀⣶⣶⣶⠀⠀⢠⢸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠝⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠧
⠀⠀⠳⡈⠲⠀⠹⣿⠋⠠⠔⣁⠞⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠹⣌⠦⡄⠀⠀⠀⠀⠀⢠⠴⣱⠇
⠀⠀⠀⠙⠃⠀⣴⣿⣦⠀⠘⠋⠀⠀⠀⠀⠈⢧⡄⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠈⠳⠀⠂⠄⠄⠠⠐⠐⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠄⠰⠖⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⢀⡀⠸⣿⠇⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠈⣿⡿⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⣿⠃⢀
⠀⠀⠀⡐⠝⠀⠠⣷⠄⠀⠺⣂⠀⠀⠀⠀⢠⡞⠀⠀⠀⣼⡆⠀⠀⠀⢳⠀⠀⠀⠀⢀⢣⠂⠀⢠⣿⡄⠀⠐⡵⡀
⠀⠀⣸⣺⠀⠀⠀⠀⠀⠀⠀⣟⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⡂⠀⠀⠀⠀⠀⠀⠀⣀⢪
⠀⠀⢸⡘⡄⠀⠀⠀⠀⠀⢠⢃⡇⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⢳⠴⡀⠀⠀⠀⠀⠈⢠⠆⡏
⠀⠀⠀⠻⣜⢢⠀⠀⠀⡖⣣⠟⠀⠀⠀⠀⠀⠙⢿⣶⣤⣀⣠⢤⣶⡟⠋⠀⠀⠀⠀⠈⠧⣕⡢⡄⠀⢠⢔⣫⠞
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠒⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡴⠶⠶⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠁⡰⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢀⣤⠒⠋⠉⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡀⠬⠳⡀
⠀⠀⠀⠀⡼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⡇⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢤⡀⠀⠀⣠⡇⠀⠲⣄⠀⠀⣠⡟⣡⡞⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⣇⢠⣀⠀⣀⣹⢦⡀⠻⣄⣴⠟⡇⠀⠀⠹⡳⡜⡝⡴⢹⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⣿⢀⣙⠻⡦⣹⣾⣌⡦⠾⠴⠴⠣⠶⣦⣄⣸⢠⡷⢁⠟⣿⡤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢹⡀⠉⢳⣝⣜⢿⠏⠀⠀⠀⠠⠊⠀⠈⢷⡇⢸⡇⡜⢸⣿⢡⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⢱⠛⠍⠉⢻⣾⡀⠀⠀⠀⠀⠀⠀⣀⣼⣇⣸⣷⠋⠉⠉⣹⠉⢹⡄⠀⡇⠀⢀⣀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠚⠁⠀⠉⠑⠚⠉⠉⠉⠉⠀⠀⠈⠁⠈⠓⠒⠚⠉⠀⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⠀⡀⡀⡀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⣠⢀⣻⢿⣿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡂⠁
⠀⠀⠀⠠⠋⣹⣿⣿⣵⣿⡏⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣿⠠⣦⠁⠁⢠⣾⣏⢤⣴⡶⠄
⠀⠀⠀⠀⠀⡟⠀⠉⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠅⠀⡀⠈⠀⠙⢄⠀⠁
⠀⠀⠀⢀⠌⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⡀⠑⠢
⠀⢀⡤⠋⣀⡔⠁⠀⠀⠀⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠐⢤⡄
⠀⠈⠠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⢀⣠⡼⠾⣿⢻⠷⢷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠿⠛⠿⣤⠄⠀⠀⠀⡀⣤⣷⡾⢿⣷⢷⣾⣤⢀
⠀⠀⢀⣾⡋⠀⢰⣇⣼⣆⣀⣙⣿⡄⠀⠀⠀⠀⠀⠀⣠⠀⣀⢈⣿⡐⠿⢀⣿⡄⠀⠀⠠⣽⣿⡁⢀⣸⣇⡀⢈⣻⣿⡄
⠀⠀⣾⠧⣍⣹⡟⠁⠈⠹⣏⣩⠽⣿⠀⠀⠀⠀⣺⣶⠿⠿⢿⣮⡯⢿⠟⠿⠇⠀⠀⠀⣿⡏⠙⢻⡿⠉⠉⢻⡟⠋⢹⣿⡀
⠀⠀⢿⡄⠈⢹⣧⣀⣀⣴⣯⠀⢀⣿⠀⠀⠀⢻⣿⠃⣠⣦⡀⢹⣷⠦⠀⠀⠀⠀⠀⠀⢾⣧⣠⣼⣷⣄⣀⣼⣷⣄⣸⡿⠂
⠀⠀⠈⢷⣴⢋⡬⠟⠙⢧⡘⣦⣾⠃⠀⠀⠀⠛⣿⣆⠈⠋⢀⣼⡿⠆⠀⠀⠀⠀⠀⠀⠉⢿⣿⡁⠀⢹⡏⠁⢈⣽⡿⠍
⠀⠀⠀⠈⠛⠿⣶⣤⣤⢶⠿⠛⠀⠀⠀⠀⠀⠈⠋⢻⡿⢿⡟⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⡷⣿⣿⢾⡟⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣀⣤⡀⣶⣶⡀⣤⣄⠀⠀⠀⠀⠀⠀⢲⡆⠀⠀⠀⠀⠀⠀⣰⠖⠀⠀⠀⠀⠀⠀⢀⣠⣴⣴⢴⣤⣄
⠀⠀⢠⣤⣼⣿⣿⠿⠿⣿⣿⣯⣠⣄⠀⠀⠀⠙⠺⠿⣄⡶⠼⠦⣶⣠⠿⠗⠋⠀⠀⠀⠀⣼⣟⡿⢾⡋⠙⣻⠾⡟⣶⡀
⠀⠀⠙⣿⣿⠋⠀⠀⠀⠀⠙⣿⣿⡋⠀⠀⣠⢤⢤⢶⣿⣶⡶⣶⢦⡿⣶⣤⣤⣄⠀⠀⢼⣯⢯⣄⣈⣷⣴⣇⣀⣽⣟⣗⡀
⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠿⢦⣼⣼⣬⣿⣤⣿⣬⣧⣿⡧⡿⠟⠀⠈⣿⣽⣀⣨⣿⣁⢈⣿⣉⠀⣟⣿⠂
⠀⠀⢠⣿⣿⣦⡀⠀⠀⢀⣠⣿⣿⣦⠀⠀⠀⠀⠀⣠⠿⣦⣤⣤⣴⠿⣄⠀⠀⠀⠀⠀⠹⣼⣏⠁⣸⠟⢻⡅⠉⣻⣿⡿
⠀⠀⠈⠉⣹⣿⣿⣿⣿⣿⣿⣯⠉⠁⠀⠀⠀⢄⢹⠋⠀⠀⠈⠁⠀⠀⠹⡏⡢⠀⠀⠀⠀⠹⢾⣟⣿⡤⣤⡗⣾⣽⠛⠁
⠀⠀⠀⠀⠈⠉⠀⠻⠿⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠛⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢭⣷⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⣼⠏⠹⣧⣠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢿⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠈⣠⣤⣤⣄⠁⢠⡿⠁
⠀⠀⠀⣠⣤⠶⣤⣄⢰⣗⣾⣿⡖⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠛⠀⣾⠋⠀⠀⠈⣷⠀⠻⢶⡆
⠀⠀⣼⣇⡽⣶⠿⣨⣿⡷⣾⣹⡼⠃⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⠀⢻⣄⠀⠀⣠⡿⠀⣴⠾⠃
⠀⠀⣧⡼⣟⣻⣻⢧⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣾⠃⣀⠉⠛⠛⠉⣀⠈⣷⡄
⠀⠀⠘⠷⣾⣉⣿⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⢻⣆⣰⡟⠉⠻⠋
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡶⢠⡂⢲⡀⣴⣶⣦⡀⣶⣒⠂⢰⠶⢶⡖⠀⠀⣰⠐⢲⡴⠶⣶⡖⢠⠰⣆⠆⡐⣄⠀⣴⠖⠒⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⡀⠈⢁⣼⠹⡿⠀⣽⡇⢬⣉⡗⢰⠀⣾⠁⠀⠀⣿⢠⣬⠀⠾⢿⣄⢐⢸⣿⠇⠒⢸⣧⣹⡗⠒⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⣿⣿⢀⡀⢹⡄⢀⣾⠟⠀⣠⣷⣦⡀⢸⣿⠀⠀⠰⣿⠀⢶⠐⢶⡾⠇⢨⢘⠿⡀⠍⣸⣿⠘⢓⣒⣚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆⢸⣿⠈⠃⠚⠰⠿⠿⠿⠇⣿⡇⡘⠃⢨⠭⡄⠀⠀⢸⠀⣿⢸⡇⢸⡇⠑⢾⣶⣷⣾⣿⠏⠸⠿⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣾⠟⢰⣋⣵⠐⠒⠂⠈⠀⣿⡟⢻⡇⢸⡀⡿⠀⠠⢖⡦⡾⡌⢛⢻⡗⠷⣶⢪⣍⢹⠓⠀⠐⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠭⡀⣰⠞⢻⠀⡆⠀⠀⠰⠂⣿⣇⣼⡇⢸⠛⣧⠀⢀⡘⣧⠙⠁⡀⡨⡁⢰⡟⠈⠛⣈⠙⡆⢸⣿⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡽⠙⣒⡋⢀⣥⣝⡀⣾⢸⡎⣉⣋⣀⢸⡀⣿⠀⠀⢩⠱⣢⢀⡇⢀⡇⣿⠃⣾⠃⠤⠴⠃⣐⣦⢠⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⢸⣉⡁⣿⠋⣹⣷⠈⠼⠃⢉⣿⠃⢈⣛⡁⠀⠀⠸⠴⠥⠨⡌⠉⠧⣖⠂⣿⠻⣦⡆⣷⠀⣿⣸⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⣸⠇⣿⠿⢻⣿⢸⢿⡂⠈⢹⡇⠸⢤⡄⠀⠀⣺⣼⠃⡜⢱⡾⠀⣿⠃⣿⠀⣿⠤⠟⠉⠁⠘⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠅⠀⠻⠿⠿⠏⠸⠼⠑⠦⠾⠃⠯⠼⠃⠀⠐⠧⠼⠁⠐⠸⠯⠘⠤⠏⠻⠾⠟⠰⠂⠀⠀⠄⠅⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⢀⣀⠀⣀⠀⣀⣀⢀⣀⠀⣀⣀⢀⡀⠀⠀⢀⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡤⠘⢧⣜⠸⣿⡌⣑⡆⡏⢹⢺⣿⢨⡕⣺⣭⠃⠀⢰⡋⠀⢹⠤⣿⡇⠈⠀⠐⠛⠸⠟⠛⣿⡆⣭⣍⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠣⢰⡟⢹⣧⠈⣴⠒⠀⣥⠾⢀⠋⠰⠲⠶⣽⠧⠀⠀⡑⣲⠋⠀⢺⢶⡶⠶⢸⣻⣴⡟⠻⠿⠳⠆⠹⠁⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⠄⠠⣼⡟⢰⣿⢹⡆⣧⣴⠸⠠⠇⢁⠎⣿⣷⠀⢀⡴⣧⠀⣦⡉⠸⠟⣷⠰⡿⠻⠿⡿⠿⠇⢈⡟⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣦⣄⢸⣿⠀⠻⠞⢡⣶⠀⡄⢦⢀⣿⢀⣭⡍⠀⠘⠧⣿⠸⠷⡷⠰⣦⣿⠸⣿⢰⣿⠅⢀⠘⠋⢸⡟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠈⠻⠿⠋⢸⠀⡇⢻⣿⣦⣅⡜⠀⠋⠞⣶⡄⠀⠀⡉⠵⢚⣶⢴⠶⠶⠶⣶⣴⣆⠀⠀⠔⠀⠤⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣶⠑⢹⡴⣿⢈⡐⡇⣴⠙⠻⢿⣿⣆⠺⠴⠟⠀⠀⢀⡅⢘⣿⢅⡆⠈⣱⢞⣁⣿⣟⣩⣭⠙⢬⠋⣈⣛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡶⡾⢿⡃⣿⢈⣌⡈⢿⡐⠇⠀⣿⠇⠀⡖⠒⣇⠀⠈⠲⠛⠋⢈⠋⢺⡇⢸⢲⣬⠁⢋⣿⠀⢢⢸⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠇⠿⣿⣸⡇⠀⢁⣀⠈⣥⡄⠀⣇⣀⡄⠀⠐⠟⣿⠈⢸⠰⣿⣿⡏⣭⣐⢠⣆⢻⡗⡃⠸⢷⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣿⢠⡞⠙⡆⣿⢻⡇⠀⠅⠄⠄⠰⡇⠀⡇⠀⡇⠀⢠⡾⢋⠀⢸⢼⣏⢹⡧⡾⢠⡏⡛⡋⢸⢾⠰⣦⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠙⠛⠁⠈⠛⠁⠀⠂⠈⠀⠛⠃⠘⠓⠚⠃⠀⠈⠛⠛⠀⠛⠈⠛⠛⠁⠃⠘⠁⠁⠈⠈⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠛⣠⡿⠽⢧⣘⠋⠻⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣠⠏⠁⠀⠀⠀⠈⠑⣤⣶⡌⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢸⠃⠀⠀⠀⠀⠀⠀⠀⠘⡄⠁⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡴⢿⡆⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠟⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠃⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠃⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠶⢶⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢟⠷⠀⠘⠛⠇⢀⣿⠿⠚⠞⠛⠇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠀⠀⢠⡀⠀⣠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⣦⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠊⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⣧⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⡛⣡⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠉


⣿⣿⠟⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠐⠒⠂⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠃⠀⠀⡄⠀⠀⠠⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠐⠀⠐⠀⠣⠤⠔⠀⠀⠘⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⡀
⠀⠀⡠⠂⠀⠀⠀⡀⠀⠀⠠⡀⠐⠦⠒⠉⠀⠀⠐⠁⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⡔⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⢆⡔⠀⠀⠤⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⢀⣀⣀⠀⢀⡀⠀⠀⠀⠀⡀⠀⡠⠀⠉⠉⠀⠠⠐⠒⠠⢧⡀⠀⠀⠈⢆
⠀⠀⠀⠀⢀⡤⠒⠺⠃⠈⠀⠢⠄⠀⠄⠀⠀⡀⠀⠠⠂⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠂⠀⠀⠀⠀⠀⡀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⡔⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⡌⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠠⡀⠀⠀⠀⠈
⠤⠀⡧⠀⣀⠈⠀⠚⠿⠀⠀⠀⠀⠉⠐⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⣴
⡔⠀⠡⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠑⢄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡀⠤⠀⠈⠐⢄⠀⠀⠀⠀⠀⠤⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠈⠄⠀⠐⠊⠀⠀⡄⠠⠔⠋⠁⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠤⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⠀⠀⢀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠂⠉⠁
⠀⠈⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⢠⠎
⠀⠀⠈⠄⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠐⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠰⠂⠀⠘⠀⠀⠀⢄⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠁
⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠄⠀⠀⠈⠢⡀⠀⡇⠀⠀⠸⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠁⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠘⠦⠞⠃⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡠⢀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⢀⣀⡠⠆⠀⠐⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠒⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠂⠠⠄⠀⢀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⡰⠀⠀⠀⠀⠀⠀⠁⠀⠚⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣟⠀⠀⠀⠐⠀⠀⡀⠢⠊⠀⠀⠀⠄⠀⠂⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠤⡀⠀⠀⠀⠀⠀⡀⠂⠐⠀⠄⠀⠀⠘⠂⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠃⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠐⠀⠀⠀⠀⠢⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡴⠚⠓⢦⠀⠀⠀⠀⠃⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠋⠀⠐⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠃⠀⠀⠐⠦⠄⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⣉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⡄⢀⠀⠠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡐⠿⢀⣄⠀⠐⠈⠁⠴⠈⢉⠐⠀⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢿⣿⣿⠇⠀⠈⠉⠀⠀⠀⠀⠰⣿⠆⠀⠂⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⣭⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢛⣀⣄⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣿⣿⡿⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡄⢾⣿⣿⡇⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠌⠉⢴⡆⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⣈⠙⠋⠀⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠈⡀⠰⠆⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠰⠄⣰⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠛⠁⢿⣿⠆⢀⠛⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠰⠷⠀⠙⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢃⣤⣤⡀⢾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⣀⠙⠻⠛⣁⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢸⣿⣿⡟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢛⡁⣶⡆⠛⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠈⠀⢠⡌⢡⣶⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠶⢠⣾⣿⣶⠐⠿⠃⠀⣠⣤⡀⠉⠠⢈⣤⣄⠂⠀⠄⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠻⠿⠛⠀⠃⢀⠀⢿⣿⠇⠀⠀⣿⣿⣿⠇⠛⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⠌⠉⢡⠀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡃⠛⠂⠠⠌⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⠐⣀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠂⠀⢁⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣶⣿⣶⠀⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⠄⡈⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡘⠃⠰⠆⣴⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠃⠶⢀⡅⠶⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⠿⠻⠿⠛⠿⢿⠿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⣿⠿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣶⡶⠶⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡗⢻⣟⡀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⠆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠄⠀⢀⣀⢀⠘⠀⠀⠀⠀⠀⢠⣿⣿⠍⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣶⣶⣿⠟⠃⠀⠀⠀⠀⠀⢸⣟⣿⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠁⠀⠀⠀⠀⡆⠀⠀⠀⢸⣿⣿⡅⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⡀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇⣿⣧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣥⠀⠀⠀⠀⠈⠀⠀⠀⢸⣇⣾⣿⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣬⣷⠀⠀⢀⣀⠐⢦⣤⣼⣿⣿⣿⣿⠷⡤⢾⣇⠀⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⣶⣿⣶⣿⠿⠈⠀⠀⠀⠀⣰⣬⣭⣴⣶⡞⠁⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢉⡀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⡏⣀⡀⠀⠀⠀⠀⣤⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⡗⠙⠁⠀⠀⠉⠉⠙⠋⠉⠉⠀⠀⠙⠀⠀⠀⢰⣿⣿⠿⠀⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⡀⠄⠀⣰⣦⢀⣴⣿⠟⠁⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣧⠼⠋⠀⠀⠀⠀⠀⠀⠀⡠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠻⣦⣶⣦⣴⣶⣶⣾⣿⣿⣟⠛⠁⠀⠀⠀⠀⠀⠀⢀⣀⣼⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠛⣋⡉⠉⠉⣾⠏⠀⠀⠀⠀⣸⣦⣶⣾⣿⡋⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⠃⠈⠉⢳⢀⣼⠋⠀⠰⠀⠀⣠⣿⣿⣿⣿⣿⡿⠓⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡇⠀⢀⡴⠋⣸⣇⣠⠾⠅⢀⣴⣿⣿⣿⠻⠿⠏⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡄⠀⠐⠒⠀⠀⠀⣠⣼⣿⣿⣿⣿⣦⣭⣤⣾⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠛⠛⢻⣶⣿⣿⣿⣿⠟⠹⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⡁⢸⡿⠿⠟⠋⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⡿⠿⠋⠁⠀⠀⠉⠙⠛⠛⠛⠋⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠋⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠘⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢞⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⣻⣿⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢽⣿⣿⣿⣽⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢷⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡪⠽⣿⣿⣿⣯⡔⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣸⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡐⠻⡿⣿⣽⠾⣋⣴⠆⡀⠀⠀⠀⢀⣤⣿⢡⡄⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⢻⣷⣾⣯⣵⡾⣋⣠⡾⢀⣼⡣⢁⣾⠃⠀⠀⠀⣈⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣦⡿⣿⣿⣷⣿⣿⣯⣶⣿⣿⣥⡿⣯⣴⣶⣾⠷⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣏⣹⣿⣿⣿⣿⣿⣿⠿⠋⠙⢻⣿⣿⡯⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠙⠋⠿⠿⣟⣋⠁⠀⠀⠀⠈⢱⣬⣁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⣶⣶⡾⢿⡏⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠾⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣿⣿⡷⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣻⣿⣿⣿⣿⣃⠀⠀⠀⠀⠀⢬⣿⣿⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⣭⣽⡯⠅⠀⠀⠀⠀⠀⠈⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣦
⠀⠀⠀⠀⠀⣰⣷⢸⡿⠋⠀⠀⠀⣠⣴⣶⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣦⣄⠀⠀⠀⠙⢿⡇⣾⣆
⠀⠀⠀⠀⠀⣿⡟⣠⣾⡗⠀⠀⢰⣿⡏⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠹⣿⡇⠀⠀⢺⣶⣄⢻⣿
⠀⠀⠀⠀⣾⡜⠳⢿⡋⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢰⣿⡇⠀⠀⠀⠙⠿⠞⢋⣾
⠀⠀⠀⠰⣿⡇⣼⣿⠇⠀⠀⠀⠀⠻⣿⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢠⣿⠟⠀⠀⠀⠀⠘⣿⣦⢸⣿⠇
⠀⠀⠀⢠⡙⠇⠿⣋⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣴⠿⠃⠀⠀⠀⠀⠀⠀⣙⠿⠼⢟⡄
⠀⠀⠀⢸⣿⡄⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢠⣿⡇
⠀⠀⠀⠈⠿⣇⣿⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣿⣸⡿⠃
⠀⠀⠀⠀⣷⣦⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣨⣾
⠀⠀⠀⠀⠹⣿⣧⢿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡿⣰⣿⠏
⠀⠀⠀⠀⠀⢨⣝⡋⢱⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡎⢁⣫⡅
⠀⠀⠀⠀⠀⠀⢻⣿⣦⢻⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡿⣱⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⣛⣓⠁⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠈⢚⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣝⠿⠸⣷⣤⡀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣠⣶⡇⡿⣫⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢭⣥⣤⣙⡻⣿⡀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⢀⣾⠿⣋⣤⣬⣭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⣲⣿⣿⠇⠈⠈⠙⠀⠀⠂⠀⠀⠐⠁
⠀⠀⢀⠄⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⣿⡟⠀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠾⡿⠁⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢀⠄⠒⠀⠂⡀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⣿
⡋⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠠⠒⠉⠉⠐⠄⠀⠀⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⡀⠀⣀⣀⣄⣄⢀⣤⣾⣿⡋
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠒⢦⠀⠊⠙⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⢿⣿⣿⣿⣯⣤
⠦⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⡠⠄⠀⠀⣻⡿⠜⢻⠻
⠀⠀⠀⠀⠈⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠠⣼⣦⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠱⡦⠀⡀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠑⠀⠀⠀⠐⠈⠻⠆⠀⠀⠀⠀⠤⢤⣾⣿⣷⠀⣿⣀⣠⣀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠤⠾⠿⣿⣾⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄⠀⢀⠃⠀⣀
⠀⡄⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⡅⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠉
⠀⠀⠼⢆⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⡂⡬⠞⡉⠀⠉⢧⡀⠀⠀⠀⠐⠂⠀⠄⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠎⠀⠀⠀⠀⠀⠀⠳⢀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠷⠶⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠢⠄⠤⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⣤⠀⠀⢀⣴
⠀⠀⠀⡃⠒⠂⠴⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢆⣠⠀⠀⠀⢰⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢻⣤⣿⣶⣿⠿⠁
⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⣿⣧⡀⢠⠋⠀⠀⣠⣶⡥⠀⠀⠀⠀⠀⠀⣸⡿⣿⡁
⡀⠀⠤⠀⠄⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠒⠤⡀⠀⠸⣿⣿⣤⡆⢀⣾⣿⠋⠠⠀⠀⠀⠀⠐⢀⠂⠀⠀⠙⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⠀⠀⠀⠀⣀⣈⣠⣤⣤⣙⠛⠠⡘⣿⡿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠠⠒⠒⠉⠭⠛⠛⢛⡿⡿⢉⣤⠚⣽⣯⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠸⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠗⢆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⢫⠏⠀⠀⠀⠀⠀⠀⠀⠠⠈⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠃⠀⠀⠀⠂⠀⠈⠉⠁⠀⠀⠀⡞⡸⠀⠀⠀⠀⠀⠀⠀⢀⣞⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠠⠁⠀⠀⠀⠀⡀⠀⠀⠀⠈⠙⢢⠀⡀⠀⠀⠀⠀⠀⣴⡇⠀⠀⡆⣥⣀⡀
⠀⠀⠀⠀⠁⠒⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠐⠁⠀⠀⠀⠤⠄⢠⡤⡀⠀⠀⠀⠀⢱⢙⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⡀⠧⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⢀⣴⣡⠊⠀⢀⡀⠀⠀⠀⠀⠀⠢⠷⠄⠀⠀⠀⡖⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣮⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠉⠉⠈⠈⠉⠁⠀⠁⠀⠉⠉⠀⠀⠀⠀⠉⠐⠐⠚⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠊⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠂⠀⠀⠀⠀⠀⠀⠐⠂⠀⠖⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢒⠆⠀⠀⠀⠀⠀⠀⠒⢀⡤⡀
⠀⠀⠀⠀⠀⢀⡤⣠⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⡎⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡌⠀⠀⠀⠀⡠⡴⠀⠀⠀⢱⢹
⠀⢀⠄⠀⠀⠀⠀⠀⠉⠙⢢⠢⡀⠀⠀⠀⠀⠀⠀⢀⣎⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠜⠀⠀⠀⠀⠀⠢⠧⠄⠀⠀⡨⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠱⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⡖⡶⠀⠀⠀⠀⠱⡈⡄⠀⠀⠀⠀⠀⠀⡜⠐⠁⠀⠀⠀⠀⠀⢀⠔⡶⠀⠀⠀⠈⢢⠱
⠀⠀⠀⠠⠤⣤⣀⠀⠀⠀⠀⠎⡐⠀⠀⠀⠀⠀⡇⢇⠀⢀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡸⠠⠁⠀⠀⠀⠀⠀⠀⡟⢸⠀⠀⠀⠀⠀⢨⠆⠃
⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⢜⡴⠁⠀⠀⠀⠀⠀⠉⠐⠓⠁⠀⠀⣰⢁⠆⠀⠀⠀⠀⣰⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⠚⠂⠀⠀⠀⢀⠔⡸
⠀⠀⠀⠀⠀⠀⠴⠶⠾⠷⠂⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⣠⡤⠾⠵⠁⠀⠠⠤⠤⠤⠯⠾⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⠤⠤⠤⠴⠶⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢢⡠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠑⢬⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠋⠀⠀⠀⠀⠀⠇⠇⠀⠀⠀⠀⠀⡠⡺⠋⠀⠀⠀⠀⠀⠀⠀⠱⡘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣱⣄⠀⠀⠀⢀⢇⡇⠀⠀⠀⠀⡘⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠊⡘⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠎⠀⠀⠀⠀⠀⠀⢡⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⠤⠤⡄⠀⠀⠀⠀⢀⡌⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⢀⣠⣣⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠿⠛⠛⠻⠿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⡛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢀⡶⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠸⣧⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣀⣀⣀⣀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⢸⠃⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⣿⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡟⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣀⣀⣀⣠⣴⣿⣴⣶⡾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠋⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣕⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣰⡶⠀⠲⣶⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⢸⢉⣉⢹⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠢⠀⠄
⢽⣇⠀⢀⣿⠅⠀⠀⠀⢈⣿⣻⡅⠀⠀⠀⠀⢸⠐⠒⠀⠀⠀⠀⠀⠀⣹⣿⣿⡏⠀⠀⠀⠀⠀⠄
⠀⠛⠓⠛⠃⠀⠀⠀⠀⠈⠁⠘⠓⠀⠀⠀⠀⠈⠉⠉⠛⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛

⠀⢶⣦⣶⠂⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⣀
⠀⢠⡿⣥⠀⠀⠀⠀⠸⣧⣾⣷⣴⡇⠀⠀⠀⢲⡞⠰⠠⠤⠀⠀⠀⠀⠢⣿⣿⠕⠀⠀⠀⢰⣶⣾⣿⣷⣶
⠀⠻⣤⡾⠀⠀⠀⠀⠘⠁⠀⠀⠈⠑⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⡜⢣⠀⠀⠀⠀⠀⠀⠈⠛⠁

⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣤⢠⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⡇⢀⣤⣀⠖⠀⠀⠀⠀⠀⠀⢀⣤⣾⠀⠀⠀⠀⠐⠿⠈⠀⠀⠀⠀⠀⣤⣴⣧⣤⠀⠀⠀⠀⣼⣯⡊⠙⡆
⣇⢋⣀⣁⣀⠀⠀⠀⠸⠇⠿⠻⠟⠿⠀⠀⠀⠀⠐⣿⡖⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠀⠀⠙⠢⣝⠴⠃
⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⡖⠒⠒⣒⡒⡄⠀⠀⠀⣠⣴⣮⡭⣄⠀⠀⠀⠀⣙⡿⣏⠀⠀⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⢀⣼⣿⣿⣷⡶
⠧⠭⠤⠭⠤⠇⠀⠀⠘⢔⡻⠿⠃⠁⠀⠀⠀⢸⣿⠴⣻⡇⠀⠀⠀⠰⠮⠽⠯⠽⠆⠀⠀⠀⠻⢿⣿⡿⠃
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣤⣤⣤⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣷⣿⣿⣿⣿⣿⣷⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣿⣿⠿⢛⡩⠭⠅⠒⠂⠭⠭⣙⠻⢿⣿⣧⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢟⠵⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢝⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⣿⡟⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡙⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡎⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣷⡙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣱⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣑⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣡⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⣿⣿⣷⣦⣭⣓⣒⣒⣒⣂⣩⣥⣶⣿⣿⡟⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⣷⣦⣾⣬⢉⡉⣉⢩⣴⣦⣶⣦⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣼⣿⣿⣿⡟⢸⠇⢿⠘⣿⣿⣿⣿⡄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣿⣿⣿⣿⠃⣿⠀⠸⡇⢻⣿⣿⣿⣇⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠿⠛⣿⡟⢸⡇⠀⠀⣿⠘⣿⡟⠻⢿⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⣴⡾⣷⣌⠁⣿⠀⠀⠀⢸⡇⢉⣴⡿⣶⣤⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠈⠻⣷⡇⠀⠀⠀⠀⣷⡿⠋⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣄⡀⠀⠠⠀⠄⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣽⠇⠀⠀⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⢀⡀⠀⠀⠀⠦⠹⠴⠋
⠀⠀⠀⠀⢷⡶⠃⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠻⡆⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠖⠀⠀⠀⠀⣴⡀⠀⠀⠓⠚⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠉⠉⠤⠄⠈⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠟⡆⠀⠀⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⡰⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠐⠂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠆⠀⠀⠀⠀⢄⣶⣿⣿⣤⣰⣿⡇
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣟⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣿⣿⡏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡷⣿⣿⣿⣷⣶⣶⣦⣿⠏⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣞⡛⢿⡇⢻⡿⣿⣿⠿⡎⠉⢁⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣠⣦⣤⣬⣽⣿⣷⡠⡁⠐⠶⠿⠃⠀⣰⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣿⣗⡒⠻⢿⣿⣿⣾⣦⣄⣀⣀⣼⣿⣿⣿⣿⣿⣟⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣧⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠉⢠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡆⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣯⢳⣦⣾⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⡿⡏⠙⢻⣿⣿⣶⣀⡀⡀⣉⠙⠛⠛⣻⣿⠘⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⡇⠁⢀⡀⣿⠿⣿⣿⣿⣿⣿⣧⣴⣿⣿⢿⣸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⣹⡿⣿⣟⣿⣿⣿⢿⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⢩⣿⣿⣷⣿⣿⣿⡻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⢸⠛⠛⢛⣿⡟⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣇⠀⠰⣤⣴⡦⡰⠛⠁⠀⠀⠈⠐⠄⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⢀⠀⠀⡀⠀⠀⠀⠀⠎⠀⣿⣿⡷⢲⣶⣿⣷⠁⣤⡀⡀⠀⠀⠀⠀⠈⠹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢷⣿⣿⣟⣦⣍⣩⣾⣿⣶⣾⣿⣷⣆⡀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢾⣿⡀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡈⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣯⠞⠙⠈⣽⢃⠁⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠸⡟⠿⠿⣿⠟⠟⠙⠃⠀⠀⠀⠀⡁⠈⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢰⣶⣧⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣾⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠼⣿⡃⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣷⠄⠀⠀⠐⠂⠠⠀⠀⠄⠀⠀⠀⡀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⢴⠾⠋⠶⣴⣦⣀⣠⠀⠀⠶⣦⣔⣋⣽⣆⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣤⣄⡀⠀⠀⠀⡼⣏⢀⣀⠀⢰⣮⣻⣿⣿⣿⣆⠸⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣦⡀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠿⠿⠿⠿⠶⠤⠀⠀⠠⣿⣦⣼⣿⣿⣿⣿⣿⣿⣆⠰⣦⠀⠀⠀⠀⠀⠀⢃⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄⣀⣀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣌⡃⠀⠀⠀⠀⣴⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠂⠘⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⣦⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠸⣆⠀⠐⢾⣿⣦⡀⣄⠀⣀⠀⢀⡄⠀⢀⢀⣠⣶⡿⠋⠀⢀⡴⠋⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠻⣟⠃⠀⠀⠈⠻⣿⠟⠖⠋⠑⠋⠗⠗⠻⢋⣴⠟⠁⠀⠀⠠⣷⠄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣧⠀⠘⣯⣁⠀⠀⠀⢀⣿⣤⣤⡀⠀⠀⠀⠀⢠⡭⠷⠶⢦⣤⠄⠰⢺⠏⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠂⠀⠙⠳⠄⣠⣴⠟⠋⠉⠀⠙⠆⠀⠀⠀⠀⠀⠰⠶⠿⣥⣄⡐⠁⠠⠴⠛⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠈⠧⠀⠀⣀⡤⢞⣫⣤⠶⠂⢠⡈⠲⣤⡀⢀⣠⡾⠃⣰⠀⠀⠀⠈⠙⠛⣶⠖⠀⠈⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣽⡓⠀⢉⣥⣾⡿⠋⣥⣤⣤⣌⣷⡄⠈⢻⡿⠋⠀⣴⣧⣤⣤⣤⡄⠀⠈⠻⣦⡀⠘⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠉⢁⣴⣿⠟⠋⠀⠀⠈⠛⢶⡄⠙⢿⡆⠀⠀⠀⣾⠏⢁⣴⠟⠋⠀⠀⠀⠀⠈⢻⣆⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢣⣿⣿⡿⠟⣛⣋⣀⠘⠛⠻⢿⣦⠈⣿⠀⠀⢸⡇⢠⡿⠟⠛⠛⢂⣀⣀⡀⠀⢀⣹⣧⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⡾⠛⠁⡀⠀⢉⡼⠋⠰⣞⢟⣦⠈⠠⠃⢰⣆⠀⠃⠉⢀⡶⢖⡆⠈⠻⣄⠁⠚⠉⠉⠉⠓⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣌⠁⣠⠞⠁⣒⡉⠀⢰⠀⠈⠉⠛⠂⠀⢀⣿⣿⡆⠀⠀⠞⠛⠋⠀⢀⡄⠈⠙⠂⠀⠘⠶⣤⣤⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣴⠟⣫⣶⠟⠉⠤⢤⣼⡆⠀⢰⠀⢦⡀⠈⣿⣿⡇⢀⡞⠀⢠⠆⠀⣸⣧⠤⠄⣀⣤⣤⣀⡀⠉⡙⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⣫⣴⢟⡿⠁⠀⠀⠀⠀⠈⠉⣀⠘⣧⠈⣿⡄⠘⠿⠁⣼⡇⢠⡟⢀⡈⠁⠀⡴⠛⠉⠉⠉⠻⣿⣦⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⡟⠁⡾⣠⠇⡶⠀⠀⠀⣠⠞⢹⣆⡼⡇⢸⣷⣠⣤⣄⣿⡄⡿⣄⣾⠙⠢⡀⠀⠀⠀⠀⠀⠀⠈⠻⣟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⡷⢋⣼⠃⠀⠀⠘⠁⠀⠈⢿⠁⠀⣸⠿⠛⠛⠛⠻⠇⠀⠸⡇⠀⠀⠈⠀⠀⡀⠀⠀⠐⣆⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⡾⠃⣴⠇⠀⠀⠀⠀⡀⠀⠀⠀⢡⣶⣿⣿⣿⣷⡄⠀⠀⢀⡄⠀⠀⠀⠀⣿⡄⠀⠀⠘⣧⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠞⡉⠀⠀⣿⣇⠀⠀⠀⠀⠹⡆⠀⠀⢹⣯⣤⣴⣤⣹⡿⠀⢠⡿⠁⠀⠀⠀⣠⣿⡇⠀⠀⠀⣿⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢰⡇⢀⠀⠻⣿⣧⡀⠀⠀⠀⠻⡀⠀⠀⠙⠿⠿⠿⠛⠁⢀⠜⠁⠀⠀⢀⣴⣿⠟⠁⣿⡀⠀⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣸⡄⠀⠈⢿⡻⢦⡀⠀⠀⠉⠢⣔⠶⠶⠶⠶⣢⠔⠁⠀⠀⢀⡾⢋⡼⠃⠀⢀⣿⡇⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⠀⠀⠻⠀⠹⡆⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢠⠏⠀⠘⠀⠀⣠⡞⣿⡇⡼⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠳⠦⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⢠⡿⠁⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡑⠦⣄⡀
⠀⠀⠀⠀⠀⠲⠖⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡤⢿⡷⠀⢠⠚
⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠃⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⢀⣿⠇
⠀⠀⠀⠀⠀⠀⣀⠠⢊⣤⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠈⠀⢰⣦⣭⠼
⠀⠀⠀⠀⢩⣉⠻⣷⣾⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⢛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠛⣿⣷⢠⠄⠀⠀⠀⠀⠈⠝⢷⣤⡀
⠀⠀⠀⠀⠙⢿⣷⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣽⢒⡐⠂⠼⠷⠖⣻⣿⣿⣦⠀⡐⣺⣶⡄⠀⠀⠸⠗
⠀⠀⡀⠀⠀⢢⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠲⢴⣿⣿⣿⡜⡗⠀⠀⠀⠀⠀⠀⠀⠠⣾⠏⢉⣽⣾⡿⠶⣿⣿⣿⣿⡿⠀⠉⡿⠋⣱⣿⣿⣷⡆
⠀⠀⣶⣤⣄⣝⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⣾⣿⣯⣿⠟⠃⠀⠀⠀⠀⢠⣿⣇⢀⣰⣯⣘⣾⣿⠏⠀⠰⣿⣿⡿⠟⠁⠀⠀⠀⠀⣸⠉⢉⣿⡇
⠀⢠⣳⣛⢟⣿⢻⣿⣿⡟⠃⠀⠀⡆⠀⠘⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⣿⣿⣿⡀⠀⠘⣿⣟⠀⠀⣠⣶⣶⡟⠛⡏⠀⠸⠟
⠀⣈⣿⣭⣿⣿⣘⣿⣿⣧⡉⠷⠾⣯⣤⠲⣿⣿⣿⣿⡟⠀⠀⠀⢰⣾⣿⣼⣿⣿⡋⠛⢳⣶⣿⡿⠛⢿⣿⣿⣤⡞⢻⣿⠹⣿⣶⡃
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣎⠀⢀⡄⠀⢻⣿⣿⠁⠀⠀⣆⣨⣿⠀⣿⣿⡿⠁⠀⢸⣿⣿⠗⣸⣿⣿⣿⣿⣿⣾⣇⣾⡏⢻⣷⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣶⣿⣿⣷⣄⠙⢿⡀⢸⠈⣿⣹⣽⣷⣿⡿⣁⣺⠿⣇⣿⠟⠉⣻⣾⣿⣿⣿⣿⣿⣿⣏⠃⠄⠙⠻⣄
⠀⠠⠊⠉⠉⠛⠛⢟⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣤⠸⣷⡄⠈⡼⢿⣿⣿⡿⣲⣿⢿⣿⣿⡟⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠉⠳⣄⣾⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣨⡾⠋⠁⠀⠀⠀⠀⠀⠘⠛⢻⣦⡉⠁⡤⡴⠛⠛⠛⣀⣼⠃⠘⣿⠋⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⣀⣸⣿⣿⡗⠸
⣴⣾⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⣀⡈⣳⣟⣲⣿⣿⣷⣾⡟⠁⠀⠀⠘⠉⣡⠟⢃⣠⠴⠞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⢿⣿⣿⣿⠿
⡾⠿⠛⠻⠻⠻⢿⣿⣿⣿⣿⣷⣤⣉⠿⣿⣿⣿⣟⣿⣿⣿⣷⣶⣶⣶⣤⣾⡇⡀⣿⣶⠾⣶⣴⣿⣿⣷⣽⢻⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⡆
⣆⣴⣶⣦⣤⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣄⢾⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡟⢷⠤
⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠠⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⣀⠀⠀⠀⣶⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⣛⣠⣿⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⡀⠀⠀⣀⢀⣴⣶⣿⡿⠄⠀⠀⠀⠀⠈⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠈⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠈⠿⠋⠁⠀⠀⠀⠲⣆⠁⠀⠀⠀⠀⠀⢀⣀⣸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⠀⠀⠀⠀⢰⠄⠄⠀⠀⠀⡄⠀⠀⠀⠀⢀⡀⠉⢹⡏⠉⠁⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣷⣦⡀⠀⠀⢠⣿⠃⠀⠀⡀⠀⠀⢀⡠⠎⠀⠀⠑⠄⠀⠁⠀⠀⠀⠀⠘⠁⠀⢸⡇⠠⠰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠤⠄⠼⢤⠦⠤⠤⣤⠄⢠⡀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⣿⠀⠀⣀⠠⠄⠉⢀⣠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠷⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣵⣿⡶⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣷⡀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂⠀⠀⠀⠘⣷⡄⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢿⠿⠿⠿⠿⣿⣿⣿⣿⣦⡈⢻⣦⠀⠀⠀⠀⠀⠀⣿⣿⠟⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣿⣦⣀⠀⠀⠀⠀⢻⣿⠀⠀⠀⢠⣾⠁⠀⠀⣀⣤⣶⣦⡴⠄⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣷⣶⣦⣤⣼⣿⡀⠀⠀⢸⡟⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠉⠀⠀⠀⠈⠉⠙⠛⠻⣶⣦⣼⣷⡀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣿⣿⣷⣾⣿⣆⠀⠀⠀⠀⠀⠀⡀⠀⠈⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠛⠁⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣦⣶⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣿⣿⣦⣄⣠⡛⠳⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠓⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡆⢸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠰⠊⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡅⠀⠀⠀⢠⣿⡇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⠀⠀⢀⣠⣤⣄⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢸⠇⠲⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⣀⣼⡁⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡹⠃⠀⢠⣿⣿⣿⣯⡙⠀⠙⢿⣿⣿⣿⣿⠃⠠⠀⠀⠀⠀⢀⡦⠰⣿⠆⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⠁⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣟⣥⡴⠋⠁⠀⢀⣿⣿⣿⣿⣿⡇⠀⠐⠀⠉⠻⣿⠛⠓⠀⠀⠂⠀⣤⠎⠁⠂⣽⣶⡆
⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡏⠀⠂⠀⠀⠀⠀⠈⠉⠉⠒⠉⠁⠀⠀⠀⠀⠈⢹⠇
⠀⠀⠀⠀⠈⢻⣦⡀⠠⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠌⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠹⣷⣶⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⠀⠀⠀⢰⠗⢀⠀⢠⡄⠀⠀⠀⠀⠄⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠈⣧⣔⣀⡀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣶⠈⣻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡄⢀⣴⣾⣿⣏⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⡃⠈⠿⠿⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⡽⠛⢛⣿⡷⣿⣿⡟
⠀⠠⠤⣴⣤⣄⡀⠀⠀⠀⠀⣾⡟⠀⠇⠀⢰⡄⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⣎⣹⣷⣿⠋
⠀⠀⠀⠀⠉⠻⣿⣦⠀⠀⣸⣿⠁⠐⠀⠀⠸⡇⠀⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⣼⠆⢀⣻⡅⣼⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣇⣠⣿⡇⠀⠀⠀⠀⠀⠻⣄⢀⣄⠙⠀⠋⠉⠁⠉⠈⠉⠑⠆⠉⠙⠂⠉⠉⠈⠉⠁⢠⣤⡄⠘⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠀⣀⣴⡆⠀⠀⠀⠀⠛⠈⠿⠓⣶⢴⡆⢠⡀⠀⠀⣀⠀⠀⢘⡀⠠⡧⠴⠆⠁⡸⠏⠃⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡾⠟⠛⠃⠀⠀⠀⠀⠀⠀⠙⠀⡤⢠⠀⠈⠁⡀⠉⠉⠉⠈⢉⠁⠀⠀⢠⡄⠞⠃⠀⠀⠀⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠃⠀⠂⠀⠃⠘⠀⠘⠃⠀⠀⠀⠀⠀⠠⠂⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠹⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠠⠈⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠈⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢀⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣦⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


⠀⠀⠀⠀⠀⠀⠰⠂⣶⡄⢴⠆⢠⡆⢰⡄⢰⡆⠲⠀⢰⡆⢲⣦⢰⡆⢰⣦⠰⠶⠀⠐⠆⢠⡦⢴⡆⠲⠀⠰⡆⢰⡦⠰⡆⢴⠀⡆⠀⢶⡶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠸⠇⠸⠀⠈⠀⠟⠀⠽⠂⠙⠀⠇⠈⠟⠀⠀⠸⠂⠀⠟⠀⠿⠀⠿⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡀⢀⣄⠀⣠⣀⣀⡀⢀⡀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠻⣏⠙⠛⣿⠋⠈⣿⠛⠛⣿⠙⡯⢉⣿⡿⢻⠏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣾⣷⣶⣷⣶⣶⣿⣶⣷⣿⣷⣶⣾⣿⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠈⠻⣦⡀
⠀⠀⠀⠀⢀⣴⣿⡁⠀⠀⠀⠀⠈⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿⠁⠀⠀⠀⠀⢈⣿⣦⡀
⠀⠀⢀⣴⠟⠁⠈⠻⣦⡀⠀⠀⠀⠀⠹⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⠏⠀⠀⠀⠀⢀⣴⠟⠁⠈⠻⣦⡀
⠀⠀⠀⠻⣦⡀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠈⢻⣦⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣴⡟⠁⠀⠀⠀⢀⣴⡟⠁⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠙⢷⣄⠀⠀⢀⣹⣷⠶⠟⠛⠛⠉⠙⠛⠛⠻⠶⣾⣏⡀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠙⣷⡶⠟⠉⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠉⠻⢶⣾⠋⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⢠⣾⠋⠀⠀⣠⡶⠟⠋⠉⠁⠀⠀⠈⠉⠙⠻⢶⣄⠀⠀⠙⣷⡄⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡿⠁⠀⣠⡿⠋⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⢿⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⣼⠏⠹⣧⠀⠀⠀⠀⠀⠀⢻⣧⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⢰⡿⠀⢀⣀⣤⣤⣤⣼⠏⠀⠀⠹⣧⣤⣤⣤⣀⡀⠀⢿⡆⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠈⠻⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠟⠁⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠈⣿⡀⠀⠀⠀⢸⡟⠀⠀⢀⡀⠀⠀⢻⡇⠀⠀⠀⢀⣿⠁⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡄⠀⠘⣷⡄⠀⠀⣼⣧⣴⠾⠛⠛⠷⣦⣼⣧⠀⠀⢠⡾⠃⠀⢠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠈⢿⣄⠀⠈⠻⣦⣄⠙⠉⠀⠀⠀⠀⠀⠀⠉⠋⣠⣴⠟⠁⠀⣰⡿⠁⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣹⣷⣄⠀⠀⠙⠻⠶⣦⣤⣤⣤⣤⣴⠶⠟⠋⠀⠀⣠⣾⣏⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠶⠶⢶⣾⠋⠀⠙⠷⣦⣄⣀⠀⠀⠀⠈⠁⠀⠀⠀⣀⣠⣴⠾⠋⠀⠙⣷⡶⠶⠶⠿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣠⡾⠟⠛⠛⠷⠶⠶⠶⠶⠾⠛⠛⠻⣷⣄⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⣠⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⣀⠔⠂⠉⠉⠉⠒⠦⡀⠀⠀⠀⠹⣿⣿⣿⡄⠀⠀⠜⠀⠀⡠⠁⠀⠀⠀⣀⠴⡒⣒⢐⡒⡢⢄
⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠙⣿⣿⣿⡄⠌⠀⠀⡐⠁⠀⠀⢀⠜⣵⠝⠋⠀⠀⠈⠛⣴⠑⡄
⠀⢸⠀⢴⠀⠀⠀⠀⠀⠀⠀⡆⠈⡄⠀⠀⠀⠘⣿⣿⣿⡄⠀⡐⠀⠀⠀⠀⡘⣸⡃⠀⣠⣴⣧⣄⠀⠘⣧⢸
⠀⢸⠀⢿⡀⠀⠀⠀⠀⠀⢀⡯⢀⠃⠀⠀⠀⠀⠘⠿⠿⣿⡔⠀⠀⠀⠀⠀⢇⢽⡄⠀⢈⡿⢿⡇⠀⠀⣯⢸
⠀⠀⢣⡈⠳⢤⣀⠀⣀⡴⠟⢁⠎⠀⠀⠀⠀⠀⠘⣓⠄⢖⡋⠀⠀⠀⠀⠀⠘⣔⢿⣄⠀⠀⠀⠀⣠⡼⢂⠎
⠀⠀⠀⢙⢦⣄⣉⣀⣁⣠⡴⡁⠀⠀⠀⠀⠀⣴⡽⢓⠀⢐⠻⡷⡄⠀⠀⠀⠀⠈⠲⢍⣛⠃⠀⢛⣨⠔⠁
⠀⠀⠀⡜⣾⣿⣿⠛⣿⣿⣷⢣⠀⠀⠀⠀⢸⣽⠐⠀⠀⠀⠑⢸⣽⠀⠀⠀⠀⠀⢠⣿⣷⣾⢿⣶⣿⣧
⠀⠀⢰⠹⠛⣿⠏⠀⢻⣿⠛⠏⡆⠀⠀⠀⠸⣻⡔⢄⠀⢀⠔⣸⡻⠀⠀⠀⠀⢀⣿⠿⣿⠇⠈⣿⡿⢿⡆
⠀⠀⠀⠀⠀⠙⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠑⠿⣲⣀⣲⡺⠟⠁⠀⠀⠀⠀⠈⠀⠀⠛⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⢔⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⡂⠀⠀⠀⠀⣜⡁⠔⠻⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⡦⠀⠀⠀⠀⠀⠀⠀⠱⢶⡏⢣
⠀⠀⠀⠀⣠⠦⠿⠯⣛⡸⢄⠀⠀⠀⠀⠀⢠⣐⢇⣚⡇⠀⣠⠖⠋⠛⠛⠾⠟⣆⠀⠀⠀⣤⠤⠴⣰⣎⠶⡼⢤⠀⠀⠀⠀⢠⠞⠒⠓⠲⡎
⠀⠀⢀⡞⠁⢠⠴⠒⠦⡀⠀⢳⡀⠀⣀⡴⠋⠀⠀⢸⡇⠰⢇⡀⣠⠶⢤⠀⠀⢸⡆⠀⠀⢧⣀⣀⣈⠀⠉⣰⠏⠀⠀⠀⢠⠏⠀⣰⠂⠀⡇
⠀⠀⡾⠀⢰⠇⠀⠀⠀⢸⡄⠀⣧⠀⣇⣀⣤⡄⠐⢹⡇⠀⠀⠉⠁⠀⣸⠀⠀⣸⠁⠀⠀⠀⣠⠔⠁⠀⣴⠃⠀⠀⠀⣰⠃⢀⡜⢸⠀⠀⡇
⠀⠀⡇⠀⢸⡀⠀⠀⠀⢸⠇⠀⣹⠀⠉⠁⢸⡇⠀⢸⡇⠀⠀⠀⢀⡴⠁⠀⣰⠃⠀⠀⠀⠈⣧⣠⣀⠀⠀⠙⢦⠀⣰⠃⠀⠈⠉⠉⠀⠀⠙⢻⡆
⠀⠀⢷⠀⠀⠳⣤⣠⡴⠋⠀⢀⡟⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⡔⠋⠀⠀⠚⠓⠒⢲⡆⠀⠀⠀⠀⢈⡇⠀⠀⢸⡖⠳⠲⠶⠶⠶⢶⠀⠀⡖⠚⠃
⠀⠀⠈⢧⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢸⡇⠀⢸⡇⠀⠐⣇⣀⣀⣀⣀⣀⣀⣸⡇⢀⡏⠙⠛⠉⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠈⠙⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠈⠓⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠒⡤⠒⠄⠀⠀⠀⠀⠀⠀⠠⠁⠐⠈⢣⡀⠀⠀⣠⠤⠤⠴⣐⣾⠱⠄⡧⣤⠀⠀⠀⢠⠀⡠⢦⠀⠀⠀⠀⠀⠀⡐⢁⣶⡉⢃
⠀⠀⠀⠀⡏⠑⠙⠣⠭⠏⠉⡇⠀⠀⠀⣠⠞⠛⠛⠓⠊⣹⠀⠀⣿⠀⠀⠀⠀⠀⠙⠋⠅⢸⠀⠀⣠⣧⣼⣶⣴⣄⠀⠀⠀⡴⠋⠙⢒⣙⠒⠋⠳⡀
⠀⠀⠀⠀⡇⠀⢰⠋⠉⠉⠉⠁⠀⠀⡼⠁⠀⣠⣒⡛⠙⠃⠀⠀⠛⠒⠚⢛⡷⠃⠀⢀⡴⠛⠀⡞⠀⢠⠴⢬⡀⠘⣇⠀⢸⠃⠀⣸⠉⠈⢱⠀⢰⢷
⠀⠀⠀⠀⡇⠀⠈⠑⠒⢦⡀⠀⠀⢸⠁⠀⠀⠉⠉⠉⠓⢆⠀⠀⠀⠀⣠⠏⠀⢀⡴⠋⠀⠀⠘⣇⠀⠙⠦⠞⠁⢠⠏⠀⠸⡄⠀⠘⠲⠶⠋⠀⠉⢸
⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⠹⡄⠀⣿⠀⠀⡴⠉⠉⢣⠀⠘⡇⠀⠀⣰⠏⠀⢀⡾⠁⠀⠀⠀⣠⠞⠃⠀⠀⠀⠘⠳⢄⠀⠀⠙⠦⢤⣤⣄⠀⠀⠀⡟
⠀⠀⠀⢠⠤⠤⠤⠞⠀⠀⢠⠇⠀⢸⡀⠀⠳⣄⣠⠞⠀⢠⡇⠀⢠⡏⠀⠀⣼⠁⠀⠀⠀⢰⡇⠀⢠⡎⠉⠙⡆⠀⠈⡇⠀⣠⣤⠤⠖⠁⠀⢀⡼⠁
⠀⠀⠀⢾⣀⡀⠀⠀⣀⡤⠋⠀⠀⠀⠳⣄⣀⠀⠀⣀⡤⠏⠀⠀⠸⣥⣤⣤⠇⠀⠀⠀⠀⠘⣇⠀⠀⠓⠒⠚⠁⠀⣠⠇⠀⣷⠀⢀⣀⣠⠴⠋
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠤⠴⠚⠁⠀⠀⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠊⠥⣄⠀⠀⠀⢠⡄⠀⠀⢙⣀⢐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠢⠠⣤⠀⠀⠀⠀⠀⠰⠚⢽⠀⠀⠀⢀⠄⠐⡀⠀⠈⠀⠙⢀⡀⡠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡀⣐⢾⠁⢀⣀⠐⡀⠀⠀⣀⡠⡄⠈⠣⡀⠀⢐⡄⡠⠃⠀⠀⠈⠂⢀⢄⡢⠄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠞⠁⠉⠃⠈⡃⣔⠤⠓⠀⡕⢔⡕⠂⠀⡇⣠⠜⢅⡂⠀⠀⠀⠘⠊⠉⢪⠀⠀⠀⠐
⠀⠀⠀⠁⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠋⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⢀⣤⣒⠲⢤⡀⠀⠐
⠀⠀⠐⠀⠀⡰⡩⠖⢮⠱⣀⡀⡒⢒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⢀⠀⠀⠄⠈⠐⣀⣠⡗⠁⠀⠈⢂⢱⡀⠀⠂
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠓⠛⠛⢊⢈⠀⠀⠀⠒⠀⠐⠐⠒⢃⢐⢂⠂⠀⠀⢀⠒⠚⠉⠉⠁⠀⠀⠀⢸⢀⡇⠀⠒
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢸⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢊⠔⠁⡠⠪⠀⠀⠀⠀⠀⠀⠀⡌⣸⠁
⠀⠀⠀⠀⠀⠔⠇⠀⠀⠀⠀⠈⡀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⢂⠕⡠⢊⠄⠔⣡⠀⠀⠀⠀⠀⡐⢠⠃
⠀⠀⠀⠀⠀⠐⡹⡀⠀⠀⠀⠀⠁⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠕⡡⢘⠅⡡⠊⠅⠀⠀⠀⢀⠜⡰⠁
⠀⠀⠀⠀⠀⠀⠘⡳⣄⠀⠀⠀⠀⡄⠀⡀⠀⠂⠀⠀⠀⠀⠀⠤⡊⢐⠅⡨⠂⠍⠀⠀⠀⡠⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠺⡧⣄⠀⠀⠘⢄⠀⢀⠀⠀⠀⠀⠀⠀⡨⢀⠅⡪⢂⠎⠀⢀⡠⣊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢕⡢⣄⡈⠣⡀⠀⠀⠁⠀⠀⠠⣢⡴⣛⣖⣡⣤⡶⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠻⠻⠷⢤⡀⠀⡠⢞⡩⠖⠟⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠀⡚⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⡚⡺⠵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠃⠓⠰⠆⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠈⠌⡈⡀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣷⣆⣀⣀⣀⣀⣐⣷⣶⣤⣶⣶⣶⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡴⠤⠦⠤⠤⠤⠤⠤⠼⠤⠬⠬⠥⣌⢊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣴⡷⢝⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⡠⠀⣄⣀⣀⣈⡉⢀⣉⢉⢋⢉⢫⡋⡓⣪⢊⣵⣡⣇⡀⣀⠀⡀⢀⠀⡀⢀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠈⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠉⠁⠉⠀⠁⠊⠁⠁⠈⠉⠁⠉⠈⠈⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠄⠠⠠⠤⠀⢀
⠀⢀⣰⣶⣟⠲⢶⢖⠳⣘⣫⣶⠿⠿⠞⠛⢓⢹⠄⠂⡈⣡⢰⣺⣦⣷⣠⡄⠒⠚⠚⠇⡴⡂⢄⣃⣀⡀⢀
⠀⣠⡼⢖⣀⣤⣯⣥⠴⠂⣿⠃⠀⣀⠀⣄⣾⢻⠀⣀⠝⠨⠧⢸⡽⣏⠉⠳⢄⣄⣤⢤⡀⣤⣴⣛⣃⢼⣌⡙⣤
⢀⠏⡀⠈⠁⢈⢇⣘⣆⢀⡼⡆⠒⠉⢣⣀⢯⡘⡆⠀⠠⠤⠀⠈⢃⡷⠲⠆⠀⠀⠐⠊⠏⠁⠀⠉⠹⡀⠀⢈⣛⠁
⠊⢿⡾⠠⣿⣾⣹⡽⣝⢧⣀⣈⣀⠀⣉⡠⠀⠘⣇⠀⠐⢠⣯⠴⣟⢲⣄⢀⣐⣆⡀⢀⠶⠂⠈⠛⠂⠀⠃⣀⠉
⠀⢀⠇⠀⠀⡞⡩⠙⣶⡾⠟⠛⠻⡛⠻⣟⡣⢾⣿⣭⣶⣿⠋⠛⠉⣠⣟⠦⢌⠿⣻⣿⣾⠛⡯⡀⠠⠂⠀⠀⢀
⠀⢨⠔⠊⡰⣠⠇⢡⣿⠷⣶⡶⠀⠹⢷⡟⠙⠷⢼⣮⢻⣻⣧⠄⠞⢻⣯⣿⣃⣼⠇⠀⠈⠛⣧⣼⡁⠀⠀⠀⢨⢰⡀
⠰⠖⠶⠂⠤⠀⠀⠈⣩⡗⠈⠀⢴⡄⠀⡓⢊⠒⠶⠶⣾⡽⡈⠋⣄⣀⡏⣩⣷⡋⠀⢀⣀⣀⢸⣿⣖⡆⢸⣄⠱⣣⠂
⠀⠰⠀⠀⢘⠄⠀⠀⠀⠱⢤⣀⠤⡄⣰⡀⠙⢂⠀⠀⢪⡄⠇⡠⡤⠀⣤⢸⡂⠛⠃⢀⠁⢉⠾⣟⣿⠇⠀⠈⣄⠉
⠀⠃⣈⠉⠩⠓⠒⢲⡞⢁⠘⢿⣟⣻⠷⠖⢶⣿⣦⣐⣿⣧⣤⣀⢀⡼⠗⠛⠗⠢⡄⠛⠋⠜⠃⣁⣀⡀⠀⣀⡄⢀
⠀⢶⠉⡚⡶⠞⠶⡖⡽⠎⠃⠼⢸⣵⠴⠚⠛⠛⠿⢏⡙⠛⡛⠛⣻⠆⠀⠀⠀⠀⠙⢶⣵⣧⣼⡿⣭⣙⢷⣄⣀⠆⣃
⠀⠈⢻⠉⠀⠀⠃⠞⠀⠀⠀⠀⢈⢻⣾⣷⠄⠺⣿⡆⣿⡇⠉⣉⣿⠆⠂⠰⠶⠈⠐⢀⣏⣽⠻⠋⣹⡛⠿⠻⡆⠀⣹
⠀⠀⢩⠁⠀⠀⠀⠀⣀⡀⡵⠖⣶⠞⣇⠀⢀⣀⠀⢀⡋⠀⠘⠛⠣⡄⠷⠀⠀⠀⠀⠊⠉⠈⡆⣀⣭⣩⠀⠜⠀⠙⠃
⠀⠀⢨⡇⠤⢖⠨⠱⢠⠀⠀⠀⠀⠀⣨⣁⠠⣠⠀⡈⠀⠀⠀⠀⣰⣬⠀⠂⢀⠀⠀⠀⠀⠀⣙⢏⣁⠤⠀⠰
⠀⠀⣚⠦⣿⡴⣴⠂⠟⠀⠸⠁⠀⠀⣸⣭⠓⠻⢺⣾⡃⢤⢠⢤⣼⣿⣲⢶⠃⠘⠿⠗⢀⣶⠳⠾⠲⣄⠀⠒⠿⢻
⠀⠀⠀⡞⢿⣠⣤⠠⠒⠓⡆⠈⠀⣜⢟⣁⡈⠚⠓⢋⣷⠻⠇⡆⠨⡿⢿⠀⠀⠀⠀⠀⠀⣹⠈⠁⠰⠟⠛⡖⣺⣿⢧
⠀⠀⠀⢀⠈⠎⠘⢲⣥⠀⢈⢶⣺⢶⠊⠛⡛⡁⠸⡈⢻⠀⠀⠓⠼⠀⢸⡏⣡⠶⠄⠤⣸⠛⠟⢀⡄⠀⠀⢰⠿⠛⠈
⠀⠀⠀⠉⢠⡔⠓⠂⣶⠒⡡⡶⠀⠛⠀⠀⢀⡁⠀⢉⢠⠀⠀⠚⠻⢂⡞⠼⢿⡷⢶⣟⡹⢠⡀⠀⠀⠊⡠⠀⠆⠄
⠀⠀⠀⠀⠠⠱⡤⠀⠄⠀⡄⠀⠉⣥⠀⡠⢬⣁⣀⡠⠈⠀⠀⠤⠀⠈⠊⠉⠸⠉⠓⠛⠀⠀⠘⠃⠄⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣙⡏⠀⠀⠀⠀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⡉⠉⢙⡇⠀⠀⠀⠀⢧⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⠀⠀⠀⠀⢀⡤⢒⡚⠉⠙⠯⣙⠋⠉⠙⡇⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢣⠀⠀⢀⠔⠁⠰⣇⡸⢠⣞⢲⠀⠉⠉⠈⠑⢄⠀⠀⠀⡸⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⢦⣠⠋⠀⠀⣠⠔⠒⠢⡉⠉⠀⠀⠀⠀⠀⠈⠆⠀⢠⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡏⠀⠀⢰⠁⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠘⡤⢃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢇⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡄⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠙⢄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠙⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢄⣀⣀⣀⣀⢤⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠳⡀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢇⠀⠀⠀⠀⢰⠃⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠳⠤⠤⠔⠁⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠒⠒⠚⣙⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⢏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠼⢿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠔⠒⣏⡉⠉⣒⣻
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣷⣫⠟⠉⠛⢦⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠛⢁⣤⣤⡀⠀⠀⠀⢀⡀⣠⠴⠛⠁⡌⠉⢁⡀⠀⠀⠈⠑⢲⠜⠁⣤⠔
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠟⠉⠀⠙⢶⡄⠄⡉⣟⠡⠀⠊⢀⣄⠀⣸⣷⠀⠀⢀⣴⠇⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠀⠀⠀⠀⠀⠉⢶⣄⡤⠀⣠⠄⢈⣿⣀⣿⣿⡖⠊⠀⠁⠀⠠⢇⣼⣿
⠀⠀⠀⢠⣦⣀⡀⢈⡴⠁⠀⠀⠀⠀⠀⢈⣿⠋⠀⠀⠀⠀⢼⣿⣿⣿⣿⣷⠄⠀⢀⢡⡶⢋⢉
⠀⠀⠀⠀⠈⠙⠻⣮⣼⠷⢤⣀⠀⠀⣠⣾⠗⠀⣀⠈⠀⢨⣿⣿⣿⣿⣿⣿⣶⣿⢠⣶⣤⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠲⣬⡁⠀⠀⠉⠷⢶⣿⣃⡀⣀⣿⣧⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⢀⣴⠄⠈⠙⢶⡀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣉⣀⡠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠛⠋⠲⠦⣀⡀⠉⠲⢦⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠈⠛⠉⠀⠀⠀⣠⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⠷⣶⣤⣽⣿⣿⣿⣿⡿⠿⠟⠿⠛⠋⠀⢀⡄⢀⡀⠀⠀⣠⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⢁⣿⣿⠿⠟⠛⠀⠀⠀⠀⠀⠀⠴⠿⠟⠉⠀⠀⠈⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⡞⠀⠀⢰⣞⡟⠀⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⢀⣠⢞⣵⣤⣾⠃⠀⠀⠐⠹⣷⣜⣷⠀⠀⠈⣶⠂
⠀⠀⠀⠀⠀⠀⢀⠾⣧⡀⠀⠠⠀⠉⠛⠿⣿⣧⡀⠀⠀⠀⠀⠈⣙⣿⠀⠀⠀⠸⣆
⠀⠀⠀⣠⢴⡏⠁⠀⠉⠓⢠⣄⢀⠀⠀⠀⠈⠓⢷⡀⠀⠀⠀⠀⠉⣿⡶⡀⠀⠀⠈⠂⠀⠀⠀⠠⣤⠜⠀⣠⡎
⠀⠀⠈⠛⠻⢧⡀⠀⠀⠀⠀⠈⠙⢦⣤⣀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠃⠧⣿⢦⣀⠀⠀⠀⠘⢧⣤⣄⠀⠴⠋⠁
⠀⠀⠀⠀⠀⠈⠙⣆⠄⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⡼⠃⠀⠀⠘⠁⢣⡟⡠⡀⠀⢀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠚⠀⠀⠀⠀⡆⠀⠋⣠⣼⠇⠀⠀⠁⠀⠐
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⢀⣤⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⣤⠀⠀⠀⣰⡀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠠⢺⠏⢀⣬⡾⠃⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⢠⣿⡇⠀⠀⣿⡇⠀⠀⡀⠈⠙⠛⠒
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠾⢿⣧⢀⢠⣿⣇⠀⢸⡗⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠠⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠾⣿⡧⠀⣽⣿⣸⣿⡆
⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠻⠏⢼⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠞⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠦⠊⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣄⠅⠉⠀⠀⠀⠠⠙⡄⠀⠀⠖⠙⢴⠈⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠂⢠⣉⡭⠋⠀⠀⢀⡄⠀⠈⠁⠀⠉⠀⠠⡄⠐⠂⠀⠘⠀⢠⡄⠀⠀⠐⢖⠒⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢖⡠⠼⣷⡀⢲⠄⢈⠠⣄⠀⠰⡆⠺⣄⠀⠈⢦⠀⡀⢾⢵⠉⠀⠀⠀⢀⣈⡠⡞⢶⠛⠁⠀⠀⡀
⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢠⡥⠯⣒⠁⠀⠰⠕⢵⠋⠁⠀⢸⡨⠚⠑⢦⡇⠀⠀⠀⠈⠁⠀⠈⠾⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⣒⡒⢦⡀⠀⠈
⠀⠀⠀⠁⠀⠀⠞⡱⠒⠢⡙⣆⣀⠊⡉⣉⠉⠉⠉⠀⠉⠀⠀⠀⠀⡈⠈⠀⠁⠠⠠⢠⠈⠄⠑⣀⣠⠿⠉⠀⠀⠈⢆⢳⠀⠀⠉
⠀⠀⠄⠀⠀⢸⣸⠀⠀⠀⠒⠚⠛⠒⠐⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⡓⢒⠒⠂⠀⠂⠐⠐⠒⠚⠉⠉⠁⠀⠀⠀⠀⢻⢸⠁⠀⠒
⠀⠀⠄⠀⠀⢠⢸⠀⠀⠀⠀⠀⠀⡄⢈⠅⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⢊⠔⡡⢀⠔⡡⣂⢸⠀⠀⠀⠀⠀⠀⠀⡌⢸
⠀⠀⠀⠀⠀⠘⢸⡀⠀⠀⠀⠀⠀⠃⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡠⢊⠔⡡⢊⠔⡁⠊⠀⠀⠀⠀⠀⠀⡘⢠⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠡⢇⠀⠀⠀⠀⠀⠸⠀⢂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢊⠴⢈⠔⡡⢊⠔⠃⠀⠀⠀⠀⢀⠜⣠⠏
⠀⠀⠀⠀⠀⠀⠐⢟⣆⠀⠀⠀⠀⠀⢆⠀⡊⠀⠢⠀⠀⠀⠀⠀⠀⠀⠐⡡⢚⡡⢊⠔⡡⠆⠀⠀⠀⠀⣠⢊⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢮⡷⡀⠀⠀⠀⠈⡆⠀⠆⠀⠑⠀⠀⠀⠀⠀⠀⣫⢔⡡⢊⠄⡊⡐⠁⠀⠀⢀⠔⣠⠊⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡢⣄⠀⠀⠈⢀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠔⡡⢂⠕⣪⠊⠀⢀⡠⢐⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢝⣢⢤⣀⡑⣄⠀⠂⠀⠀⠀⠀⢐⣡⡺⣿⣭⡥⣴⣶⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠷⠦⣄⠀⢀⠬⢓⡥⠚⠛⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠕⢐⡺⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣀⡰⠖⣟⣦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠸⠇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠀⠀⠢⢒⠔⠉⡑⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡁⠀⠀⠀⠀⠀⢀⠕⡈⠢⢀⢀⢈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠺⠿⠧⠄⠾⠋⠭⠤⠿⠿⢿⠿⠿⠿⠿⠿⠧⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠟⠛⠛⠒⠒⠒⠒⠒⠒⠚⠛⠚⠚⠛⢺⠌⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢌⢼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣿⣯⣺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠄⣀⠸⡀⠀⠈⠀⠀⠀⠰⠛⠛⠃⠚⢒⢂⠶⠖⡷⢋⡞⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠋⠙⠉⠛⠐⠛⠓⠓⠓⠓⠛⠟⠺⠞⠟⠛⠛⠟⠩⠓⠩⠂⠩⠒⠝⠊⠔⠚⠊⠓⠉⠊⠂⠒⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠿⢿⣶⣝⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣫⣶⡿⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⣤⣴⡿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿⣦⣤⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠈⠉⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠁⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣤⣀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣾⣿⣷⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣾⣿⣷⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠉⠹⣇⠀⠀⠀⠀⣸⠏⠉⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠹⣆⠀⠀⣰⠏⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⠁⠈⠳⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠸⠀⠁⠈⠀⠀⠀⠀⠡⠀⠀⠀⠀⢠⠃⠀⠄
⠀⠀⠀⢠⣄⣀⠀⢪⣩⣆⢠⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⡇⠰⠀⠀⠀⠐⠀⡔⠀⠁⠀⡁⠀⠆
⠀⣀⠐⢦⡀⠙⣷⣘⢳⠏⢰⢀⠏⠰⠀⠀⠀⠀⠀⡔⠡⡐⢤⠤⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢣⠀⠀⠈
⠀⠸⣆⠀⠙⠄⠈⣿⢼⠇⠰⣎⡴⠓⠄⢀⢔⣤⡞⠀⢀⠇⠈⡆⢠⠔⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸
⠀⠀⠈⠑⢄⢸⣆⢻⣶⡆⠐⣽⠁⠀⠀⢈⠘⣿⡇⠀⢨⡆⠀⢿⢀⡟⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⢦⠻⡈⣻⢃⢣⡏⠀⠀⠀⠀⠄⠻⣷⡄⠸⣆⣀⡞⣼⢁⠇⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⡄⠀⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠣⣷⣿⣾⠋⠀⠀⠀⠀⠀⠀⠣⠼⡿⠿⠋⠜⡿⠑⠁⠀⠀⠀⠀⠀⠀⠈⠯⣤⠀⢳⠆⠀⢰⡟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⠘⠀⠀⠀⠀⠀⠀⡴⠒⠂⣴⠶⢄⡀⢳⣼⣇⣼⣿⠷⠚⠃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠑⣄⡀⠈⠷⣇⡙⢶⣍⢯⠉⣤⣤⡶⠿⢛⡋⡳
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⢀⠀⠹⢦⣦⡭⣷⣶⠽⠷⠦⣤⠤⡤⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣉⣀⣀⣑⣀⣐⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⣟⡛⢻⡟⣻⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⣰⣿⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⢀⣿⡟⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠸⠋⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡆⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡎⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⢐⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢻⣽⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄⠈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣷⣿⣧⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣆⠻⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⣛⣷⣾⣻⣿⣿⣟⣶⣤⣀⠀⡀⢀⣀⠁⡀⠀⠈⣻⣬⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣉⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡃⢂⠀⠈⠀⠉⠀⠀⢀⠀⠈⠬⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠼⠛⠉⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣿⣷⣦⣲⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠠
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣠⡴⠖⠚⣛⣩⣭⣶⣶⣿⣿⣿⡿⠟⠛⠛⠛⠻⠟⠋⠀⠉⠃⢀⡦
⠀⠀⠀⢀⣲⣬⡵⢒⣩⣴⣶⠟⠉⢀⣤⣾⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣉⣀
⠀⢠⠀⢳⡟⣅⠀⠈⠉⠉⠀⠀⠚⠛⠛⢻⣿⠿⠋⠁⢀⡀⠀⠀⠀⠀⢀⣴⠦⡀⠀⠂⠀⠀⠊⢀⢻⣿⣿
⠀⠈⠓⠊⠃⠀⠀⠺⢷⣦⣶⣶⣲⠀⠀⠙⠁⣢⠀⢸⣛⡻⡦⠀⠀⠀⠟⠁⠀⡇⠀⣀⣴⣤⣤⡆⣾⠟⠁
⠀⣀⣀⡀⣬⡛⠏⢑⣶⣶⣶⣶⣦⣤⣄⣀⣒⣉⢠⣘⣫⣜⠁⠈⠀⠀⠀⣀⣾⣵⣿⣿⣿⣿⣿⠟⠁
⣾⣿⣿⣿⣿⣷⣦⣤⣀⣐⣲⣦⣤⣉⠉⠉⠉⠉⠀⠉⠀⠀⠀⠀⠄⠀⠀⣻⢹⣿⣿⣿⣿⠟⠁
⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣶⣧⣄⡄⡶⠃⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢰⣾⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠛⡛⠂⠈⠙⣷⡀⠰⣦⡴⢄⢀⣤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣀⣀⣠⣶⣶⣿⣿⣿⣿⣏⠂⢄⠀⠈⠁⠀⠈⢿⣼⠋⠙⠙⢢⣀⣀⣼⣿⢷⣤⣀⠀⢀⡾⠿⢧⡄
⠀⠀⠈⠻⠻⠌⠻⠿⢉⣽⣿⣿⡄⠀⠀⠀⢀⣀⣤⣤⢀⡀⠁⠀⠀⢀⣁⣈⠉⠀⢀⣥⣾⣟⠣⠀⠀⠀⠐⠳⡤⠴⠤⢦⣀
⠀⠀⠀⠀⠠⡶⠾⣆⠈⠻⢿⠁⠀⠀⢀⣴⣾⣿⣿⣿⣇⠙⠳⠀⠀⠀⠉⠉⠁⠀⠁⠀⢀⣿⠀⠀⠂⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠿⣿⣿⣿⢿⡟⡩⠛⣿⣾⣷⣦⡀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⣼⣿⡁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠐⠂⢀⠀⠀⠀⠀⠀⠙⠯⠀⠰⠁⠀⣫⠿⠛⠙⠁⠀⠀⠀⠀⠲⣿⣿⡌⠀⠀⠀⢠⣿⣿⣇⣀⡄⠸⠁
⠀⠀⢸⣿⣷⣤⢤⣀⡀⠀⠀⠁⠀⢀⡀⠀⠀⠁⠤⠐⠁⢀⣶⣶⣼⣦⡀⠀⠀⠀⠀⠱⠿⢾⠂⠀⢸⣿⣿⣿⣋
⠀⠀⠀⠛⠉⠉⣻⣿⣿⣾⣷⣀⣄⢸⣿⡄⣠⡀⠀⠀⠉⠚⠿⣿⡝⠋⠙⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⡀
⠀⠀⢀⡀⣠⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣀⠀⢠⣶⣀⡄⠉⠙⠦⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣦⣦⣀⣤⡀
⠀⠀⠀⠉⠙⠛⠛⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣠⡄⠀⣠⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣟⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠓⠉⠙⠋⠋⠉⠛⠛⠛⠿⠶⠿⠇⠀⠀⢠⣦⣼⡿⠟⠛⠛⠻⢿⡿⠟⠉

⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢈
⠀⠀⠀⣠⠞⠃⢿⠀⠀⠀⠀⢀⠔⠉⠙⠑⠛⢦⠀⠀⠀⠀⠀⢸⡁⠁⠚⠊⢩⠏⠀⠀⠀⠀⠀⣠⠃⠈⠒⡇⠀⠀⠀⠀⠀⢸⠉⢈⣋⣏⣈
⠀⠀⡏⣀⡄⠀⢻⠀⠀⠀⠀⠈⠢⠞⠉⡇⠀⢸⠀⠀⠀⠀⠀⠀⣠⠜⠁⢠⡃⠀⠀⠀⠀⠀⡰⠃⡰⠃⠀⡇⠀⠀⠀⠀⠀⢸⠀⠸⢤⣀
⠀⠀⠉⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⠏⠀⠀⠀⠀⠀⠀⠳⠤⢤⠀⠈⡆⠀⠀⠀⡰⠁⠘⠓⠃⠀⠃⡆⠀⠀⠀⠀⠘⠓⠲⡄⠀⢱
⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢰⠋⠀⠐⠛⠒⢲⠀⠀⠀⠀⢠⠤⠤⠞⠀⠀⡎⠀⠀⠀⠓⠒⠒⠒⡆⠀⡖⠁⠀⠀⠀⠀⠠⠦⠴⠃⠀⢰⠁
⠀⠀⠀⠀⠧⠤⠞⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠾⠀⠀⠀⠀⠺⠤⠤⠤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠇⠀⠀⠀⠀⠀⠓⠤⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀

⠀⠀⠀⠀⠠⢀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⡄⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣠⠴⠶⠦⢓⠄⠀⠀⠀⢰⠒⠒⠢⠾⣒⡴⢦⠀⠀⠀⠀⣀⢤⠼⣤⣆⠀⠀⠀⠀⠀⢀⠔⠄⠿⡂⢦⡀⠀⠀⠀⠀⢀⡤⠲⠟⣆⢅⡀
⠀⠀⡜⠁⢠⣔⡒⠚⠀⠀⠀⠀⠸⠤⠤⣤⠄⠀⢈⡼⠀⠀⠀⢰⠁⢰⡀⡂⠈⡇⠀⠀⠀⢠⡇⠀⡖⠉⢦⢀⣷⠀⠀⠀⢠⠋⢠⠔⠒⢢⡀⢱⡀
⠀⢸⠀⠀⣀⢀⡈⠑⡄⠀⠀⠀⠀⠀⡰⠃⢀⡴⠉⠀⠀⠀⠀⢈⠦⠀⠉⠀⢶⡁⠀⠀⠀⠘⣇⠀⠑⠒⠃⠈⢸⠀⠀⠀⡏⠀⡎⠀⠀⠀⡇⠀⡇
⠀⢸⠀⠠⡁⠀⡱⠀⢹⠀⠀⠀⠀⡼⠁⢀⡞⠀⠀⠀⠀⠀⢠⠃⠀⡖⠁⢢⠀⠘⡀⠀⠀⠀⠈⠑⢒⡖⠀⢀⡎⠀⠀⠀⣇⠀⠱⣄⣀⡠⠃⠀⡇
⠀⠈⠣⡀⠈⠉⢀⡠⠃⠀⠀⠀⢰⡃⠀⡸⠀⠀⠀⠀⠀⠀⠘⢦⡀⠉⠒⠁⢀⡼⠀⠀⠀⠀⣏⠉⠁⢀⡠⠊⠀⠀⠀⠀⠘⢦⣀⠈⠉⠀⣀⠜⠁
⠀⠀⠀⠈⠉⠉⢉⠀⠄⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠁⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠄⠆⠀⠠⡀⣀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠲⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⣀⣤⠀⠚⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡙⠃
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿