Elf Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣪⠘⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⢤⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠴⡀⠀⢀⠀⢀⠀⢀⣀
⠀⠀⠀⠠⡤⠀⠐⢢⠖⠒⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢀⡀⠁⠀⠀⠑⠀⠸⠂
⠀⠀⠀⠀⠑⢂⠀⠀⠀⢀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡵⠃
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠒⠀⢀⣀⠀⠀⢀⠀⠀⠁⠐⠒⠀⠲⠎
⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠄⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠶⠖⠋⠀⠀⠀⠀⠠⡂⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠑⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⣴⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠉⠀⠀⢠⣶⡿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠨⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⣴⡶⠂⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⣼⣿⡇⠀⣯⣀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠙⠻⠶⢿⣿⣷⡾⠟⠃⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣦⣤⣀⣀⣀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣈⠙⠛⠛⠻⠛⠛⠛⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣿⣿⡆⢸⣿⣿⣀⣀⣶⣶⣶⣶⣶⠀⢀⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠛⣿⠋⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⠻⣿⣶⣦⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢿⠛⠿⣿⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠉⠙⠛⠻⠛⠃⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠀⠀⠀⠀⠈⢿⣿⣶⡶⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣇⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠟⠻⣿⣿⣷⣄⠀⠀⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣿⣿⣟⣁⣀⣀⣀⣀⣀
⣿⣻⣻⣛⣿⣻⣻⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣽⣭⣿⣿⣽⣽⣿⣯⣿⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠞⠛⠛⠛⠛⢶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⢀⣠⣤⣤⡶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⣸⣇⣴⡶⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⣶⣄⡀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⣠⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⡀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠈⣿⡄
⠀⠀⠀⣰⡏⠀⠀⣀⣤⠶⠶⠞⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠶⢶⣄⠀⠘⣷⡀⠀⠀⠀⠘⣧
⠀⠀⠀⣿⠇⣴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡆⢹⡇⠀⠀⠀⠀⢿⡄
⠀⠀⠀⢹⣶⡟⠀⠀⠀⠀⠀⢰⣶⣄⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⣇⠀⠀⠀⠀⠈⣷
⢀⣤⣤⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⠷⣤⣀⠀⠀⠀⠀⠀⠀⢸⣿⡋⣷⣤⣄⠀⠀⢿⡀
⣿⠁⠈⠛⠻⢷⡄⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⢠⣦⡄⠀⠀⠈⠛⢷⣄⠀⠀⠀⢀⣴⠿⠛⠃⠀⢹⣷⣴⠟⠛⢳⣦
⠹⣆⠐⠶⣶⣶⣿⣧⠀⠀⣸⠏⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⣠⣄⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠹⣧⡀⣠⣿⣿⣶⡶⠶⢀⣾⢿⣀⣠⣶⣾⡿
⠀⠹⣧⡚⠃⠉⢹⣿⣧⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠙⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⠉⠀⠛⣠⡾⠁⢾⣿⣟⠛⣁⣴
⠀⠀⠘⢷⣄⡀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠶⠶⠶⠶⠶⠶⠶⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠉⠀⣀⣴⠟⠀⠀⠀⠈⠛⠛⠛⠁
⠀⠀⠀⠀⠉⠛⠿⠿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⣠⡴⠶⢦⣄⠀⣴⠇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⣤⡿⠟⠃⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣠⣤⠾⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⣶⣦⣤⣤⣤⣤⣤⣶⡶⠾⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⣷⣦⣴⠟⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⣿⣄⢀⣴⡏⠀⠹⣦⣀⣀⡿⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠈⠙⠛⠙⠳⣶⠾⠋⠉⠋⠁⠀⠈⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢠⡞⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⢿⡀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⣸⡏⠀⢠⣾⠁⣿⣁⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣿⠈⢿⡀⠀⢻⣆
⠀⠀⠀⠀⠀⣰⡟⠀⠀⣾⠁⣸⡟⠛⠛⠛⠛⠛⠛⠛⣧⡼⠿⣩⣧⣼⣟⠛⠛⠛⠛⠛⠛⣧⠘⣷⡀⠀⢿⣄
⠀⠀⠀⠀⣰⡿⣿⣦⣾⡇⢠⣟⠀⠀⠀⠀⠀⠀⠀⣘⣿⣇⠀⠻⣿⣿⡟⠀⠀⠀⠀⠀⠀⣻⠄⢸⣷⠶⡾⢻⡆
⠀⠀⠀⠀⣻⡷⢶⣤⡿⠁⣰⡟⠛⠛⠛⠛⠛⠛⠛⠻⣷⡼⠿⠿⢷⣼⠟⠛⠛⠛⠛⠛⠛⢻⣦⠘⣷⣤⡶⢾⣇
⠀⠀⠀⠀⣟⠀⣤⣿⡇⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢸⣷⣶⠀⣿⡆
⠀⠀⠀⠀⣿⣀⣹⠟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠻⣿⣀⡿⠃
⠀⠀⠀⠀⠈⠛⠋⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢶⣦⣄⣀⣀⣀⣀⡀⣀⣼⡇⠀⢿⣄⣀⣀⣀⣀⣀⣀⣠⣴⣶⠞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠿⠿⠿⢿⣿⡇⠀⠀⣿⣿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠁⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣴⣿⠀⠀⠀⢹⣧⣤⣤⣤⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⢹⣿⠀⠀⠀⢸⡏⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣸⡿⠀⠀⠀⢸⣯⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⠛⢿⡇⠀⠀⠀⢸⣿⠛⠛⠛⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣀⣀⣸⡇⠀⠀⠀⢈⣿⣀⣀⣀⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠉⠉⠉⠉⢹⡿⠀⠀⠀⢸⡏⠉⠉⠉⠉⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢼⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠙⠷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠛⠁⠀⠀⠀⠀⢀⣀⣴⠟⠀⠀⠀⠀⠀⠸⢧⣀⡀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⡐⠶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣦⣤⣤⣤⡶⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠁⠈⠋⠉⢉⡉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣤⣤⣤⣤⣶⣶⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣠⠶⠟⠛⠙⢓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡏⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣧⡀⠀⠀⣠⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣠⠤⠶⠒⣲⠛⠉⠉⠙⠓⠒⠲⠦⢤⣀⡀⠀⠀⠀⡇⠉⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠋⠁⠠⡎⠑⠒⠙⣄⠀⠀⣀⣀⠤⢄⡀⠀⠀⠉⠓⠦⣴⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠙⢣⡀⠀⢠⠇⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠳⣄⡀⠀⣠⠖⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠉⡟⠁⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⠉⠉⡽⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡎⢳⡴⡆⢳⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⡇⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠏⠀⠀⢸⡗⠚⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣨⢟⣳⡄⠙⠲⠦⠤⢤⣄⣀⣀⣀⣤⠤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⡎⠀⠙⠲⢤⣀⣀⣀⡼⠀⠀⠻⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⠀⠀⠀⠀⠘⣧⡀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⣀⣀⣀⣂⠀⠀⠀⢀⣀⣧⠙⢦⠴⠶⣋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠤⣿⣀⣹⣯⣉⣉⠥⠴⢯⠀⠈⠲⣏⠀⠙⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢀⢿⡉⠀⠀⠀⠀⠀⠘⡇⠀⠀⢼⡼⠲⠤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢲⠊⠁⠀⣉⠒⠤⠤⠤⠒⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢷⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠘⡆⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡇⣀⣠⠇⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡍⠙⢿⠶⠉⠳⢊⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤⠊⠀⠀⡴⢁⠄⠠⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠴⠚⢧⣁⣤⣤⣄⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⡲⠖⠒⢒⡶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⡎⢀⠞⢀⠴⠊⠁⡠⠔⠊⠉⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡃⠸⣀⣁⣏⡤⠧⠴⠖⠛⠒⠶⠴⢯⣤⢿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⢌⣹⡋⠉⣀⣠⡤⠤⢴⠶⠶⠶⣤⠤⣤⡽⠛⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⣰⠋⡟⠋⡁⠔⠀⠀⡀⠁⢀⡀⠰⣦⠔⣿⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⢼⠰⡇⢊⣠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣺⣗⡷⠦⣆⠀⠀⠠⠀⠀⠀⠄⠀⠀⡰⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⢱⡄⣸⠿⣶⡤⣤⣤⡤⠤⢔⣺⣦⣀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠤⡾⠋⠁⣴⢻⠶⠳⣄⠔⠤⣼⢧⠀⠈⢳⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡾⠁⠀⡼⠁⠀⣠⠶⡛⠛⠛⠲⣬⣧⠀⠀⢹⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⣄⣞⠁⠀⠈⠆⠀⢠⣡⠟⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡸⣿⠢⣍⣑⣼⣴⠻⡜⡄⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣄⣜⡵⠃⠀⡟⢠⠋⣷⡀⠹⣌⣦⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠉⠙⠒⢮⣀⡆⠀⠑⢌⣩⠞⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⢤⡤⣬⣭⡿⠿⣭⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⠀⢸⡇⠀⣿⠀⠀⣇⠀⡠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢩⡞⠉⣽⣎⠉⠓⠒⠉⡓⠒⠋⣱⣬⡉⢹⡎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠉⠉⠁⠘⠚⠙⠼⢹⡛⠞⠑⠃⠈⠉⠉⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⣀⣀⣀⣀⣀⣼⣇⣀⣀⣀⣀⣀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⣿⠋⣳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⠉⢻⡅⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⠊⣭⡙⣾⣋⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢹⡟⠷⣬⣷⠉⠉⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠙⢒⡟⠦⠉⣐⣚⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠉⠉⠓⠶⠄⢽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠙⠒⠤⠄⢀⣀⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠷⣖⡒⠀⠤⠤⠤⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠳⠶⠤⢠⣤⣉⣁⣀⣨⣷⡀⠀⠀⢀⡀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⡤⠤⠴⠶⠶⠖⠒⠛⠋⠩⠭⢶⡀⠀⣼⠳⣶⠋⠉⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠋⢀⣀⣠⡤⢤⣤⣤⣤⣶⣶⣤⠤⣤⣤⡷⢠⡟⠀⠈⠳⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣀⡀⠀⢸⣶⣟⡉⠅⢠⡕⠶⠋⡍⢿⢩⠛⠶⢶⣤⣿⣿⣿⣃⣀⡀⠀⠹⣆⣀⡴⠃
⠀⠀⠀⠀⠘⠉⢻⡗⠚⢿⠉⠉⠉⠀⠀⠀⠊⠀⠐⠈⠃⢀⡀⠀⠘⡟⠛⢉⡟⠛⢀⡴⢿⣻⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⡄⠀⠐⢦⠀⠀⠀⡄⠤⠀⠀⢸⠉⠀⣰⠁⠀⣾⢡⠔⠉⠀⡈⠰⠟
⠀⠀⠀⠀⠀⠀⠀⣼⢧⡀⠘⣦⡀⠀⠑⠦⣤⣄⣠⡠⠔⢃⣠⠴⢇⣠⡾⠋⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⣀⠀⡼⣦⣭⣿⣟⡽⠛⠶⡒⠖⠲⠒⠶⣾⡍⠁⣠⠖⠁⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⣰⣥⠴⣿⣇⠈⣿⣙⢯⠶⣋⠼⡇⢀⣴⡆⣴⡆⣽⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠸⣿⠇⢾⠾⢻⣀⣿⠋⠀⢸⣧⣄⣧⠞⣿⢧⠸⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣶⡀
⠀⠀⠀⠙⠺⣶⡶⣿⡟⠁⠀⠀⡦⠈⠛⠁⠀⣿⠀⠉⠉⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣿⣡⡄
⠀⠀⠀⠀⠰⣘⢶⠏⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⠹⣤⣀
⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⣷⡀⠀⠀⣠⣴⠛⠉⠀⠀⠀⠀⠀⠉⠙⢲⣤
⠀⠀⠀⠀⠘⡟⠷⢦⣤⣤⣴⣥⠶⢄⣀⣀⡴⠿⢧⣄⣠⠔⠋⣷⠀⠀⠀⠈⠉⣶⠀⠀⠀⠀⠘⠏⠉⠁
⠀⠀⠀⠀⠀⢡⡀⠀⠀⣨⢿⣥⣀⠀⣉⠁⣀⣀⣀⣀⣡⠤⠶⠋⠀⠀⠀⠀⢀⡿⠀⣠⣴⣦⣀⠘⡄
⠀⠀⠀⠀⠀⠀⠉⠒⠚⠁⠀⠀⠀⠈⣿⠷⣿⣅⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠞⠋⠸⠀⠈⠓⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣼⠏⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠀⣰⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠉⢹⡧⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⣃⣹⡿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⣾⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣉⠉⠀⠀⠙⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⢸⡟⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⣀⣘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠛⢻⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠈⠉⠛
⠀⠀⠀⠀⠀⠀⡴⢚⣯⡓⡆⠀⢠⡦⠿⠶⣼⡷⠽⠧⢴⠀⢠⠞⣁⡉⢆
⠀⠀⠀⠀⠀⢼⡁⢏⡘⠋⠁⠀⣸⡧⣴⢦⣸⣇⣠⣄⣰⡀⠘⠷⠟⢱⠘
⠀⠀⠀⠀⠀⢸⣇⠀⠉⠉⠈⠐⠛⠓⠋⠀⢹⡏⠀⠘⢮⣁⡀⣀⣠⠾⢠⠇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠒⠒⠒⠒⠿⠷⣤⣤⣤⣤⠤⠤⠄⡒⣋⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⠄⠈⠉⠉⠉⠉⠉⠁⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠀⠐⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠗⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢄⡀⠀⠀⠀⠀⠀⡦⠀⠈⠓⢄⡀⠀⠀⣠⠞⠈⠱⠀⠰⡟
⠀⠀⠀⠀⠀⠀⢸⠀⢠⡟⠀⠀⠀⠀⠀⡖⠛⠠⠄⣀⠀⠀⠀⢀⡴⠁⠀⠀⠉⠓⢤⣀⠀⢰⠀⠀⠀⠀⠀⠙⠢⠔⠁⠀⠀⠀⠛⠃⠸
⠀⠀⠀⠀⠀⢀⡌⠀⣼⠗⠂⢤⡀⢀⡲⠀⠀⠀⠀⠉⠐⠒⠴⠋⠀⠀⠀⠀⠀⠀⠀⠉⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⡈⠁⠀⡿⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠄⠐⠒⠖⠲⠾⠟⢉⣉⣅⣉⡁
⠀⠀⠀⠀⣠⠃⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠤⠶⠶⠶⠖⠀⠉⠻⠛⠻⢄⡉⠑⠦⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⡜⠳⣄
⠀⠀⠀⣠⡍⠀⠀⡟⠀⢹⡟⠛⠻⡿⠛⠛⠛⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠰⣶⠀⠀⠀⠀⠙⠂⠤⠤⠀⣈⣉⣉⣉⣉⠓⢾⠋⠁⢄⠘⢆
⢠⠞⠃⠈⠉⠛⢾⠀⡘⡷⠀⠀⡐⠃⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⢦⠀⠀⠓⠘⢀
⣿⡇⠀⢠⡦⠀⠀⣾⢃⠀⠀⠀⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡆⠀⠈⡇⠘⡀
⠘⢿⣤⣯⣤⣾⠿⠏⢸⠂⠀⢰⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠇
⠀⠀⠈⠉⠉⠀⢸⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣭⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⣴⡀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⡌⠀⠀⢸⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⡰⣎⠳⠄⠀⠀⣼⠋⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠃⣀⠃⠀⢀⣾⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡘⠁⠈⠳⣼⠦⢴⣀⠔⠋⢸⠂⠀⣴⠊⢻⡃⠠⠤⣀⠀⠀⠃⠀⠀⠀⠀⢘⠃⠀⠸⣇
⠀⠀⠀⠀⠀⠈⢿⠋⢀⡴⠋⠈⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢛⣛⣃⣴⠏⣀⠼⡟⠻⠛⠛⢶⡏⠀⠶⣨⡅⣀⠤⢙⣇⠀⢸⡆⠀⠀⠀⡸⢇⠀⠀⢿⣄
⠀⠀⠀⠀⠀⠀⠘⠗⢛⠀⠀⠀⣧⠀⠀⢸⠇⢤⠀⠀⠤⠒⠘⡏⠉⡟⢶⡁⠀⠁⣌⡀⠀⢀⡼⠳⣴⠋⠈⣿⠒⠉⢹⡗⠸⡇⠀⠀⠀⡇⠀⠓⢄⢈⡟
⠀⠀⠀⠀⠀⠀⠀⡋⢸⠀⠀⠀⡿⠆⠀⠘⠀⠀⠉⣟⠂⣤⣀⣰⣤⠉⢢⣌⣶⣶⣏⠑⣶⣥⣤⠄⠿⠒⠐⢿⡔⢰⠞⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⠉⡇
⠀⠀⠀⠀⠀⠀⠀⡗⢠⠀⠀⢠⠀⢸⣤⡅⠀⠀⠈⣉⠉⠀⡿⠏⠉⠉⠉⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢩⠁⠀⠀⠀⢠⠀⡰⠀⡇⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⢀⠁⠰⠀⠀⡸⠀⠸⠁⠁⠀⠀⠀⠹⣶⣈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠞⠃⠀⠁⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠘⢴⡇⠀⡸⠇⠠⠇⠀⠀⠀⠀⠀⡠⠋⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⢸⠀⠰⡆⠀⢸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠧⣴⠃⣠⠏⠀⠀⠀⠀⢠⠞⠀⠀⠀⠟⢲⡶⠛⣶⠒⠒⠛⠛⠛⠷⢖⠂⠀⠀⠀⠈⡏⠀⠙⡄⠀⠀⠀⠀⠀⢧⠀⠰⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠁⠀⢠⠤⢤⡴⠃⠀⠀⢀⡴⠒⠉⠀⢠⣉⠀⣀⠀⠀⠘⡀⠈⠓⢤⣀⠀⢀⣇⡀⠀⠘⠷⠂⠤⠤⠄⠀⢧⣀⡿⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠐⠘⡏⠀⠀⠀⠀⠀⠉⣉⣭⡗⠀⠒⠃⠀⠀⠀⠈⢹⠉⠉⠉⠀⠀⠀⢂⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢹⡏⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⢈⡻⠛⠗⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠠⠤⣶⠤⠛⢠⡦⢄⣀⡀⠈⠻⠰⠞⠀⠀⢀⣀⡀⠤⠾⠦⡦⢴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⡤⠀⠀⠀⠀⠈⠀⠀⠂⠐⠛⠀⠀⠀⠀⠀⠀⣦⠤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠠⠶⠶⠊⣁⣄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠘⠛⢇⠀⠀⢀⡠⠟⣄⠀⠀⢀⠔⠁⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠖⠒⠒⠀⠈⠛⠁⠀⠀⠀⠈⢦⡠⠊⠀⠀⠈⠣⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⣻⢷⡄⠀⠀⠀⢸⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣠⡾⣟⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⡁⣸⠏⠀⠀⠀⢠⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠐⣀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠐⠀⠀⠀⢻⣆⢉⡠⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠈⠙⠃⠀⠀⠀⠀⢸⣤⠎⠉⠢⡤⠊⠑⢦⡴⠋⢢⡾⠋⠳⡤⠚⠉⠳⡴⠋⠙⢢⡟⠀⠀⠀⠀⠉⢋⠁⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠧⣀⡀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣀⣀⣀⠶⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠤⠄⠀⠠⠴⠧⠤⠤⠤⠤⠤⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⣀⠠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠶⠶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡯⠔⠒⠋⠉⠉⠉⠉⠙⠒⠢⢽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢤⡏⢀⣤⠶⣿⡟⠉⠀⠀⠀⠀⠙⢯⣿⠒⣤⡀⢹⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠙⡏⣿⠀⠙⠀⢀⠀⠀⠀⢀⡀⠀⠉⠀⢸⢹⠏⠁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⡇⣿⠀⠀⠀⠿⠃⠀⠀⠸⠗⠀⠀⠀⢸⢸⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣄⡇⣿⠀⠀⠀⠀⠀⠀⠙⠲⡦⠀⠀⠀⣻⢸⣇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢰⡇⢹⡇⢹⣄⣠⡤⠴⠒⠓⠒⠚⠳⠦⢤⣠⡟⢸⡏⠀⡇⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⢾⣉⠓⢦⣀⡇⢸⣀⠀⠀⠀⠛⢿⣾⠻⠭⣤⣤⡴⣖⠒⠓⣲⢶⡤⠤⠤⠽⣾⣽⠛⠃⠀⠀⣀⡇⢸⣀⡴⠚⣉⡷
⠀⠀⠀⠀⠀⠀⣴⠒⠚⠛⠂⠀⢁⡜⠙⢦⡀⠀⠀⠈⠻⡆⠀⠀⠀⠙⠮⠭⠭⠵⠊⠀⠀⠀⢠⡟⠉⠀⠀⢀⡴⠋⢳⡈⠀⠐⠛⠓⠒⣦
⠀⠀⠀⠀⠀⠀⠈⢉⡹⠶⠀⠀⡞⠀⠀⡼⠉⠉⠓⠒⠤⢽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣠⠤⠒⠋⠉⢧⠀⠀⢳⠀⠀⠶⢏⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠶⠖⠚⠛⠓⠒⠺⠧⣤⣀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⠀⣀⣠⠼⠗⠒⠚⠛⠓⠒⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣤⣀⡈⠳⣄⡀⠀⠀⠀⣠⡴⠋⣀⣠⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠂⠈⠳⢤⣠⠴⠋⠀⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣀⢀⣇⣀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣠⢴⡏⡿⠹⣍⣧⣀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⢴⡇⠷⠴⢋⡧⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⡩⠋⠻⢍⡀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢲⡖⠉⠁⠀⠀⠀⠀⠈⠙⢶⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢰⡶⢶⡒⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⣠⠎⠀⠀⠳⡄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⡰⠃⠀⠀⠀⠀⠙⣆⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢚⡲⢤⠀⠀⠀⠀⠀⣼⠃⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠸⣆⠀⠀⠀⠀⠀⡠⢖⡓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢰⠋⢩⣮⠇⠀⠀⢀⣰⠃⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠹⣆⠀⠀⠀⠰⣵⣏⠙⡆⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠓⠦⠤⢤⠴⣿⡉⠈⠓⢲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⠚⠉⢉⣷⣦⠤⠤⠴⠚⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠙⠲⢤⣄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣤⠴⠋⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣠⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠒⠒⠀⠲⠂⠀⠰⠐⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠐⠀⠀⠀⠒⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⢧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⢤⡖⠛⠉⠉⠩⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⢸⡄⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⠧⣀⣀⣴⡦⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡤⠔⣚⣉⣉⣽⠁⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣽⠁⠀⠀⡏⠀⠀⠀⠳⠤⠤⠴⠒⠒⠲⡄⠀⠀⠀⠈⣻⠗⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠈⠧⣄⠴⠃⠀⡤⠀⠀⠀⠀⠀⠀⠀⠠⡝⠲⠤⠤⣞⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⢷⡀⣀⢻⠀⠀⠀⣴⠆⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢰⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⠛⣯⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢤⣀⡀⢷⣤⣈⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⣩⠖⠻⢄⡀⠈⠉⠉⢹⠋⠉⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣄⠀⠀⠉⠙⠛⠛⠃⠀⠀⠈⠛⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠆⠀⠀⠀⠀⢸⣄⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⡀⠀⢀⡀⠀⠀⠀⠀⢸⡌⠳⣤⠤⠽⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⢭⣽⣉⢩⣗⣚⣛⣩⠭⡇⠀⠳⢶⡋⠉⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢈⣿⡉⠁⠀⠀⠀⠀⢸⠀⠀⠀⣧⠶⢤⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⡖⠉⠀⣙⠲⠤⠤⠤⢴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢷⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠘⡇⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣇⣠⣰⠇⣀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠁⢻⡛⠉⠙⢫⠟
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡴⠋⠀⢀⡞⠁⠂⠠⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠉⠳⠤⠤⠤⠤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠾⠛⠛⠛⠻⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠹⣧⡀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⣀⣀⣠⣶⣿⠟⠛⠉⠉⠉⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⣀⠠⠄⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠤⠄⠀⠀⠀⠀⠀⣿⠛⠉⢹⡟⠀⠀⠀⠀⠀⠀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⢀⣽⡂⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠿⣄⠀⠀⠀⠀⣠⣿⣤⡾
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⢶⣶⠟⠛⠛⠛⠛⠛⠛⠷⠶⠶⠶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠙⠛⠶⠶⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢻⣆⠀⢀⣠⣤⡶⠛⠛⣉⣉⣀⠀⣀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⣤⣄⣀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⡶⠛⠉⠀⠀⠀⢸⡏⠉⠙⠛⠛⠉⢻⣄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣤⡾⠃⣀⡀
⠀⠀⠀⠀⢠⡶⢶⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠳⢶⠶⠶⠟⠛⠋⠉⠉⠀⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⠛⠙⣿
⠀⠀⠀⠀⢸⡇⠀⠈⢻⣆⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢰⡿
⠀⠀⠀⠀⠘⣷⠀⠀⠀⠛⢷⣶⠶⠟⠁⠀⠀⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⡆⠈⠛⠛⠛⠛⢻⣧⠀⠀⠀⠀⣼⠇
⠀⠀⠀⢠⣶⣿⡇⠀⠀⠀⠐⣾⡆⠀⠀⠀⠀⠀⡀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⣠⡟
⠀⣸⣿⢿⣧⡘⣿⣄⣤⣶⡄⢸⡇⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⠟
⠻⣟⠻⣦⠛⠇⠈⠻⠁⣿⡁⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣤⣴⠟⠃
⣴⣿⣷⠉⠁⠀⠀⠀⠀⣿⠛⠛⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠈⠛⢷⣤⣀⠀⠀⠀⢰⠿⣦⣄⡀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠋
⠀⠀⠀⠈⠙⠛⢿⣗⣠⣴⠛⣉⣻⣷⠀⠀⠉⠛⠻⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢸⣟⠁⣠⡾⠛⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⣄⠀⠀⠀⠀⠉⠛⠷⣦⣤⣤⣤⣤⣴⠟⠀⠀⠀⠀⠘⠳⣤⡄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠐⠂⠀⠀⠀⠐⠂⠀⠀⣙⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠙⢳⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⢠⡄⠀⠀⠀⠒⠀⠐⠠⠒⠀⠒⠉⠈⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣾⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠣⠈⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢦⡀⠀⠀⠀⠈⢿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⠀⡔⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠛⢦⣄⣠⣤⣤⡿⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠳⠶⣦⣤⡶⣾⢿⢦⣄⣀⣀⣀⣀⣠⣤⠶⠾⣿⡄⠀⠀⣿⠁⠀⣠⣤⠶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢶⣤⣤⣽⡇⠀⠀⢴⣿⣁⣉⣉⣉⣁⣠⣤⡶⣿⠀⠀⠀⠙⠿⣿⣏⠁⠀⠈⠻⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠻⢿⣿⠿⠿⠟⠛⠋⠉⠉⠉⠁⠀⠀⠙⣧⠀⠀⠀⠀⠀⢿⠄⣠⣄⠀⠈⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⢀⣴⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣀⡿⠙⠳⣦⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣀⣤⡴⠟⠁⠀⠹⢦⣄⣀⣀⠀⠀⠀⣀⣀⣀⣀⣼⡇⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡀⠀⠀⠀⠀⣤⡄⠈⠙⠛⠛⠛⠛⠛⠋⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢘⣯⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⡏⠀⣀⣀⣸⡏⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠻⠿⢩⣽⣿⣁⣤⠾⢷⣤⣠⣼⢿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠙⣿⡛⠁⠀⠀⠈⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣴⠛⠀⠀⠀⢀⣼⠏⠀⡠⠤⡀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣋⡿⠏⠁⠀⠀⠀⢀⣼⠃⢀⠞⠓⠀⠛⢆⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⣀⣀⣤⡶⠛⢷⣄⠁⠀⠀⢀⡀⠈⠁⢨⡗
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣥⣤⣤⣬⣽⣿⣿⣿⣿⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠿⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣠⠴⠒⠚⠛⠉⠉⠉⠉⠛⠓⠒⠶⢤⣄⣀
⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣤⣀
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⢸⡇⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢸⣀⣀⡼⠃⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠉⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢀⡠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠁⠉⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢀⡔⠁⠀⠀⠀⢀⡠⠔⠒⠛⠃⠀⠀⠁⠐⠢⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠐⡅⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠤⢎⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠈⠛⠇⠀⠸⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠠⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⠴⢻⡤⠚⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⣸⠃⠀⠀⠀⢀⡏⠀⠈⢂⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⡶⠋⠀⠀⣀⡴⠛⠀⠀⢀⣠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡜⠛⠁⠀⠄⡀⠀⠀⠀⠈⠉⠹⠶⠒⢋⢁⣀⣠⠬⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⣉⣹⣄⣴⡆⠀⠀⠑⠰⠷⡦⠤⠞⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠺⢍⡈⠉⢸⠁⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠉⠓⠋⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢰⠄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡠⠊⠀⠀⠀⠠⠈⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢤⣴⠀⠀⣀⣄⠀⢰⡖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⣟⠁⠈⡇⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⣇⠀⣀⡇⢸⡀⠀⢀⠴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⣳⡄⢠⣎⠉⠀⢉⣹⣉⠁⢈⣉⣧⠘⣶⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠙⠧⠞⠛⠃⠀⢹⡄⡼⠀⠈⠉⠉⠋⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⠀⠀⠀⢀⣠⠤⠤⠞⠃⠙⠒⠒⠶⢤⣀⡀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⡀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⡇⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢃⣿⠁⠈⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣣⣾⠏⠀⠀⠹⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡿⠋⢀⣠⣤⣀⠈⠛⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⡟⢛⣫⣷⣶⣯⣙⠻⢿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠿⢞⣛⣛⣛⣛⣛⣛⡛⠾⠿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⣼⣿⡿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⣿⣿⡄⠀⠀⠀⠀⢀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣬⣉⠓⢶⣦⣬⣿⡿⣠⡶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣦⠹⣿⣣⣤⡶⠞⣋⣥⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠙⢿⣿⣦⣈⠻⣿⢃⣿⣾⣷⣦⣉⠂⠀⠀⠀⠀⠀⢊⣥⣶⣿⣾⡇⣿⡟⢉⡴⣾⣿⠋⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⣿⡇⢿⣷⣌⠀⠙⣿⡷⢦⣍⡳⣦⡄⠀⣠⡶⣫⣥⠾⣿⠟⠁⢈⣴⣿⠁⣿⢇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣷⣜⠿⠛⣧⡤⠈⢿⣌⣿⡟⠻⠇⠀⠿⠛⢿⣇⣼⠏⢠⣰⠟⠻⣋⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠛⡅⠀⠀⠈⣉⠁⠀⠀⠀⠀⠈⢉⡉⠀⠀⠀⡙⠷⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢌⠻⡟⠁⠀⠘⢢⡴⠊⠀⠀⠙⡿⣫⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠈⣙⣓⡉⠁⢀⣼⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⡀⠋⠉⠃⣠⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣽⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⡝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠘⠛⠉⠿⣿⣟⣿⣿⣿⣿⣿⣿⡿⠍⠉⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⣿⣿⣯⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⠀⠀⠐⠂⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠂⠈⠀⠀⠤⠜⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠈⠀⠀⠀⢈⣉⣉⠭⠭⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠂⢀⡠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⢻⣀⡤⠖⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠞⠁⠀⠀⠀⢀⣠⠴⡖⢏⠉⠙⣧⡉⡏⠉⢹⠋⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠿⠤⣄⡀⣠⠞⠁⠸⠾⠁⠈⠑⠲⠗⠙⠀⠀⢸⠊⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⡀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠺⠆⠀⡾⠃⠀⠀⠀⢸⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢠⠒⠂⢀⠀⠀⠐⠀⠀⢠⠊⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣧⡀⠉⠉⠁⠀⠀⠉⠀⠀⢀⣡⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢬⠙⢒⠒⠶⠖⠒⠒⢿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠈⠛⠉⠓⠖⢋⡓⠒⡇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢰⠃⠀⠀⠀⠙⢋⡀⣱⠀⢳⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣦⣾⡀⠀⣏⠒⠢⠴⢶⢩⠛⣍⣯⣧⣴⡇⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠙⠊⡟⠙⠒⠒⠞⢎⣓⣥⠏⣀⣀⣉⡦⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⣰⠃⠀⠀⠀⠀⢀⡴⣃⣍⣤⠂⠘⡳⢮⡈⠙⣆⠀⠀⠀⢀⡿⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠙⠒⣾⠻⣄⡀⠀⣀⣤⣋⠀⠋⡻⢿⣟⠦⣄⠄⡙⠲⣧⣀⣀⣤⠞⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣠⠟⠷⠟⠀⠀⠉⠉⠉⠀⠉⠳⢮⡀⠈⡨⠳⣮⠻⣤⡂⣠⢏⡩⠶⡄⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠤⠂⡨⢳⠾⠿⠛⠉⠀⠀⢹⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣈⡤⠋⠀⠀⠀⠀⠀⠀⢸⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⡠⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠶⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⢀

⠀⠀⠀⠀⠀⠀⣀⣴⣶⠿⠿⠿⠿⠷⣶⣶⣤⣀⡀
⠀⠀⠀⠀⢀⣾⡟⠋⠀⢀⡀⠀⠀⠀⠀⠈⠉⠛⠻⣷⣦⣀
⠀⠀⠀⢠⣿⠋⠀⠀⠀⠈⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⡀
⠀⢀⣤⣿⣧⣴⡾⠿⠿⠿⠿⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⣴⡿⠋⠉⠙⢿⣧⠀⠀⠀⠀⢘⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⣿⡇⠀⠀⠀⣸⣿⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠘⠿⣶⣶⣾⠿⠃⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣦⣤⣤⣈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⣀⣤⣶⡾⠿⠻⣷⣄⠀⢰⣿⠁⠀⠀⠀⠀⠈⠙⠛⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣾⠿⠛⠉⠁⠀⠀⠀⠈⢻⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣸⡿⠀⠀⢀
⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠈⠉⠙⠳⢦⣿⠇⣠⣾⣿⡆
⠀⠀⠀⠀⠀⠘⣿⣆⣴⠟⠃⠀⠀⣀⣤⣶⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⣷⣦⣄⣀⣸⣿⣾⠟⠁⢸⣿
⠀⠀⠀⢀⣴⣦⣌⣿⣇⠀⢀⣴⣿⠟⠉⢁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠈⠙⢿⡿⠛⠁⠀⡀⠀⣿⡆
⠀⠀⠀⣼⡟⠉⠛⢿⣿⣶⡟⠋⠀⠀⣴⡿⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢿⣦⠀⠀⠀⠀⠀⠀⢠⣿⠀⢻⡇
⠀⠀⢠⣿⠃⣠⡄⠀⠈⠻⠇⠀⠀⠈⠛⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠈⠛⠀⠀⠀⠀⠀⢀⣼⡟⠀⢸⡇
⠀⠀⢸⣿⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⢿⣿⡗⠀⠀⠀⠀⠀⠀⠠⣾⠟⠀⠀⣿⡇
⠀⠀⠘⣿⡄⠀⠘⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣀⣀⣀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠹⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠁⠉⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠿⠋
⠀⠀⠀⠀⠈⠙⠿⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠻⢿⣿⣿⠿⠿⠿⠿⢿⣿⡿⠟⠁⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠈⠉⠂⠀⠀⠒⠀⠁⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⣷⣦⣤⣀⣀⡀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠛⢿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⢿⣏⠙⠛⠻⠿⠿⠿⠟⠛⣻⣿⠋⠀⠀⣸⡿⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⡄⠀⠀⠈⠻⢷⣦⣄⡀⠀⠀⣀⣾⠟⠁⠀⠀⢠⣿⠃⠀⠀⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠘⢿⣦⡀⠀⠀⠀⠉⠛⢿⣶⣾⠟⠁⠀⠀⢀⣰⡿⠃⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠈⠙⠿⣶⣦⣤⣀⣀⡀⣿⣇⠀⣀⣠⣴⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠛⠛⠻⢿⣿⠟⠛⠋⠁⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⢿⣦⡀
⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣄⠀⠀⢀⣴⡿⠋⠀⠀⢹⣿⡀
⠀⠀⠀⢠⣾⠿⣿⣧⣀⠀⠀⠀⣠⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠻⣷⣤⣾⠟⠀⠀⠀⣠⣿⠿⢿⣦⣀
⠀⠀⠀⣿⠇⠀⠀⠉⠻⢷⣦⣼⡟⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠘⣿⣇⠀⠀⢠⣾⠟⠁⠀⠀⠙⢿⣦
⠀⠀⠈⣿⣷⣤⡀⠀⠀⠀⠈⢹⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠈⠿⣷⣴⡿⠋⠀⡀⠀⠀⠀⠀⢻
⠀⢀⣾⡟⠁⠙⠻⢷⣦⣄⣀⣾⠏⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⣿⡇⠀⠘⣧⡀⠀⠀⠀⣼
⠀⣼⡟⠀⠀⠀⠀⠀⠈⠻⣿⡏⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠹⣷⣄⣠⣿⠿⠷⣶⠿⠋
⠀⣿⡇⠀⠀⠀⠀⣾⡀⠀⢹⣷⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠉⠛⠛⠁
⠀⢻⣧⡀⠀⠀⣰⣿⣷⣤⣾⠟⠀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣇
⠀⠀⠙⠻⠿⠿⠛⠁⠈⠉⠀⠀⢰⣿⠉⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣀⣴⡾⠟⠁⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠙⠻⠿⣶⣤⣤⣀⣀⣿⣇⣀⣤⣤⣶⠿⠟⠋⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⣛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⢀⣾⣿⣧⠀⠀⠀⠀⢀⣠⣤⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡛⠛⠻⠿⢶⣶⣤⣾⣿⠀⢻⣷⣤⣶⡾⠿⠋⠉⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠈⣹⡟⠀⠸⣿⡉⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⣀⣀⣀⣰⣿⡃⠀⠀⣻⣧⣀⣀⣀⣀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠋⠉⠉⠙⣿⡇⠀⠀⣿⡟⠙⠉⠙⠙⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣀⠀⠀⣀⣿⡇⠀⠀⠀⢀⣿⡇⠀⠀⣿⡇⠀⠀⠀⠐⣿⣄⡀⠀⣀⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠻⠛⠛⠉⠀⠀⠀⢀⣈⣿⡇⠀⠀⣿⣇⣀⠀⠀⠀⠈⠙⠛⠛⠛⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣄⣀⣀⣠⣤⣾⠟⠛⠛⠁⠀⠀⠙⠛⠻⢿⣦⣄⣀⣀⣀⣤⣾⠟

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠙⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⢄⡀⠀⠀⠀⠀⠀⡠⡄⠀⠀⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠃⢀⡄⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠢⢄⡀⠀⠀⠀⢀⠇⠀⠀⠙⠦⣀⠀⣠⠎⠀⠘⢣⡀⠛⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠀⣼⡃⠀⠀⠀⢀⡘⠉⠁⠒⠤⠄⢀⠀⣠⠞⠀⠀⠀⠀⠉⠲⡤⣀⠟⠀⠀⠀⠀⠀⠉⠚⠁⠀⠀⠀⠈⠁⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠁⢠⣿⠉⠑⠒⢤⡼⠃⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⣤⣤⣄⠠⣶⠒⠒⠒⠊⠉⠉⠉⠉⠉⠀⠒⠒⠂⣾⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⣾⣿⣦⣤⣤⣄⣠⣤⣤⣤⡶⠒⠒⠒⠂⠈⠉⠉⠉⠋⣉⠀⠀⡉⠀⠉⠒⠤⣉⠁⠒⠒⠀⠤⣤⠤⣤⣄⣀⡶⢾⠈⠻⣿⣿⣿
⣿⣿⠿⠿⢠⣄⣸⣿⣿⡙⠀⠉⢹⠉⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠈⠋⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠲⡶⠒⠂⠙⢇⠀⠙⣄⠹⣿⣿
⣿⡅⠀⠀⡀⠀⢹⣿⠏⠃⠀⢠⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠘⡀⠀⠈⡆⢻⣿
⣿⣧⢀⡼⠃⣠⣼⡏⣰⠀⠀⡈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⢿⠀⠀⠃⠈⣿
⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢀⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡄⠀⢠⡀⢸
⣿⣿⣿⣿⣿⣿⡇⠀⠋⠀⠀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⡰⢶⡙⠛⠋⡱⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇⠀⢸⡇
⣿⣿⣿⣿⣿⣿⠃⢸⠀⠀⢀⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠎⠑⢆⡱⣀⣠⠃⣠⠜⢣⠀⠀⣠⠤⢤⠄⣀⡀⠀⠀⠸⠀⠀⠀⠀⠀⡇⠀⢸⡷
⣿⣿⣿⣿⣿⣿⣶⠇⠀⡠⠚⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠤⠄⢠⡟⠁⣹⡷⢴⠤⢼⣀⠎⠉⡄⢺⠀⢀⡸⢦⠀⠀⡆⠀⠀⠀⢰⣇⠀⠀⢿⡀
⣿⣿⣿⣿⣿⣿⣯⡤⠟⠀⠀⢰⡄⠀⠀⣷⢀⠀⠀⢀⠤⠘⡟⠚⡿⣤⠊⠀⠃⡐⠀⠀⠉⣳⣄⣸⠟⢲⡦⠄⠚⣧⠐⡿⠀⠀⠀⣶⠈⠦⡀⠘⣷
⣿⣿⣿⣿⣿⣿⣿⡇⢈⠀⠀⠸⣧⠀⠀⠇⠈⠑⢶⠦⢀⡀⢡⣀⠓⢈⡑⣤⣴⡛⢣⣤⣋⣀⠨⠧⠀⢴⣇⠀⣤⠏⠀⠁⠀⠀⢠⡿⠀⠀⢈⠛⣿
⣿⣿⣿⣿⣿⣿⣿⡇⢈⠀⠀⡀⠁⣆⣐⠀⠀⠀⠚⠒⠉⢽⡿⠛⠓⠈⠉⠉⠉⠙⠀⠁⠉⠁⠀⠀⠀⠀⠙⡀⠁⠀⠀⠀⠀⠀⠆⣳⠀⠀⢸⠀⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠄⠀⠀⡇⠀⡏⠉⠀⠀⠀⠘⣤⡀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡦⠿⠀⠩⠀⠀⢸⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣴⡆⠀⡸⠀⢰⠀⠀⠀⠀⠀⢀⠜⠉⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⡆⠀⡆⠀⢸⠀⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣀⡼⠁⡰⠃⠀⠀⠀⠀⡴⠃⠀⠀⠿⢲⣤⠒⢦⠒⠒⠒⠒⠲⡶⠖⠒⠀⠀⠁⢹⡃⠘⢦⠀⠀⠀⠀⠀⢡⠀⢰⡀⠘⣷⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣁⣀⣀⣠⣤⠞⠀⠀⠀⣰⠖⠊⠀⠀⡘⠀⠀⠀⠀⠘⡀⠈⠢⣤⠀⠀⢘⡃⠀⠀⢳⣦⣤⣤⣀⣀⣲⣀⣻⣤⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠂⡿⠀⠀⠀⠀⠀⠉⣉⣩⡖⠐⠞⠁⠀⠀⠀⠉⢷⠉⠉⠁⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢹⡏⠛⠛⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⣉⡛⠛⠓⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⠶⠛⠳⡶⠀⣀⡀⠈⠛⠰⠟⠀⠀⢀⣀⠀⠴⠿⢶⠆⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠤⠀⠀⠀⠀⠈⠀⠈⠃⠘⠉⠀⠀⠀⠀⠀⢸⡦⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠤⠤⡄⠀⠒⠛⠂⢀⠴⢄⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⢀⣀⡀⢠⣤⡶⠚⠀⠉⢣⠀⠀⣠⠖⠁⢆⠀⢀⡴⠃⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠀⠀⠉⠀⠀⠀⠀⠀⢳⡞⠁⠀⠀⠈⠛⠋⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠰⠏⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡏⠳⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠶⣴⣿⣿⣿⣿⣿⡂⣠⠶⣄⠀⡰⢦⡀⢀⡤⣀⣠⠶⣄⢀⡴⠦⡀⣠⠶⣄⢀⣿⣿⣿⣿⣿⣴⠖⠉⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠈⠉⠀⠀⠈⠁⠀⠘⠋⠀⠈⠁⠀⠀⠈⠁⠀⠀⢹⣿⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⢀⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢓⣀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⢀⠄⠼⠁⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⡆⠈⠀⠀⠀⠄⠑⠀⠀⠀⠀⠀⠀⠀⠀⡔⠠⠋⢀⢀⡰⠒⠋⠀⠀⢀⡎⡽⠃
⡔⠋⠉⠒⠢⢄⣀⠰⠀⢠⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢀⠘⠀⠒⠀⠀⢠⢄⣠⠒⢤⡍⠈⠗⣤
⢇⠀⢀⠀⠂⠤⢉⠙⠳⣌⡀⠀⠀⠀⠀⠄⠱⠄⠀⠀⠀⢻⡀⡄⣀⣠⠖⠁⠀⠁⠀⢠⠀⠀⠀⡾
⠘⣇⠈⠀⠀⠀⠀⠈⠠⠀⠙⠢⣄⠀⠀⠀⠢⠘⢆⠀⠀⠀⠀⢆⠈⠁⠀⣠⠂⠀⠠⠃⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠱⢶
⠀⠙⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠑⣄⠀⠀⠀⠀⠐⢦⣄⣀⠜⠀⠀⠰⠁⠐⠂⠀⠀⢀⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣦⣤⡴
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⢀⠞⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉
⠀⠀⠀⠳⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠓⠤⢀⡌⠀⠀⢀⠀⠀⠀⠀⢸⠀⠓⠂⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠘⢣⣄⠀⠑⢄⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠉⠁⢀⡔⠀⣠⠎⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡀⠈⢁⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠆⠁⢀⡾⠀⠀⠀⠐⠀⠅⠀⠀⣀⠲⣶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠟⢓⠖⠊⠀⠀⠀⠀⠀⠘⠀⠀⡘⠀⠀⠀⠠⠀⠀⠰⠀⠀⢩⡃⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠀⠐⠀⠀⠀⠘⠀⠀⡇⠀⢠⠁⠀⠀⠀⠀⢀⠴⠶⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠊⡀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡇⢠⡇⠀⠀⢸⡀⠃
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⢢⠀⠀⡇⠀⢡⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⡄⢠⠀⠀⠀⠀⠠⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠇⠀⠈⢆⠀⠀⠁
⠀⠀⠀⠀⠀⡘⠀⠄⢀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠇⣸⠀⠀⠀⠈⠣⠀⠈
⠀⠀⠀⠀⠀⠧⠀⢀⡠⢠⡔⠚⠉⡰⠂⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠱⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢀⠀⡰⠁⣀⠀⠀⢸⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⡇⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⡠⠁⢀⠀⠀⠀⠘⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⡁⠀⢠⠈⠳
⠀⠀⠀⠀⠀⠀⣠⠖⢁⡴⢰⠀⠀⡜⠀⠀⠀⠀⠇⠀⠀⣄⠀⠀⠀⠀⠀⠀⣀⠁⠴⡏⠦⠀⢳⡄
⠀⠀⠀⠀⠀⣘⡤⠊⠁⢀⠈⠀⠀⠇⠀⠀⠀⠀⢸⠀⠀⠈⠓⠖⠐⠒⠂⠈⢹⠀⠀⢠⠀⢆⠀⠹⡄
⠀⠀⠀⠀⠸⢿⠀⠀⠀⢸⠀⠀⢰⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠸⠀⠸⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣦⠀⢇⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠋⠀⠘⣀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠙⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⢀⡇⠀⠀⠃⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢀⠇⣼⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠃⠀⠀⠰⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡘⠀⢻⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠃⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠆⠀⠀⠀⢩⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡄⠀⠀⠀⠰⠀⠀⣀⠚⢣
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⠀⠀⠀⠀⡦⠄⠉⠐⣯
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠲⠜⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠠⣄⠀⠀⡀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⡀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⢠⢠⣜⠀⠀⢃⠀⠀⢡⡄⣤⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠒⠒⠠⢄⠀⣄⠐⠆⠨⡍⠢⢒⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡍⠠⢄⠘⡄⠀⠉⠆⢸⣂⢀⡀⠀⠀⢠⠂⠀⠐⠃⡀⠠⣄⠀⠡⠈⢂⡐⠒⢂⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⡶⠙⠀⢀⡾⢃⠉⢉⡙⠆⠀⠁⠀⣷⠒⠀⠀⠀⠀⠛⠦⠄⠋⠀⠛⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡏⠐⠂⠁⠈⠙⠲⢤⣤⣆⠸⡟⠁⠈⢂⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⠶⣶⣶⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠶⢟⣿⣿⠟⣋⣥⡶⠶⣦⣄⠉⢻⡏⠉⠉⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⢠⣾⣿⠃⠸⠋⠀⠀⠀⠀⠛⢀⣼⠇⠀⠀⠠⣤⣀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡵⠖⠂⠀⣾⣿⡇⠀⠀⡀⠀⠀⠀⠀⡀⠋⠁⠀⠀⠀⠀⠈⢿⣷⣆⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠉⠀⠀⠀⢰⣿⣿⡇⠀⠀⠘⠷⣶⣤⡶⠏⠀⠀⢲⣄⠀⠀⢀⣸⡇⠘⡆⢿⡿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣀⣀⣤⣴⡿⠀⣿⣿⣷⡄⠰⣄⣀⠈⠉⠐⠂⠀⣀⣼⡿⠀⠀⠈⠉⠀⣄⣤⣿⠇⣼⡏⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⠛⠉⠀⠀⠘⢿⣿⣿⣦⣈⠻⢿⣿⣿⣿⣿⡟⣋⣀⣠⡴⠂⠀⢠⡄⡉⢉⣼⡇⡇⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⣬⣉⣙⠛⠛⠋⠉⠁⠀⠀⢠⡟⡇⠈⡫⠟⠀⠁⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⠋⠀⢵⣦⣄⡀⠀⠀⢀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠤⠤⠀⠀⠀⠀⠀⠀⠉⣿⢣⠀⠀⢸⠿⣿⣿⣦⡐⠾⠛⢸⡇⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⡄⠀⠀⠀⠀⠶⠟⣉⣥⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠙⢊⣳⡄⢸⣶⣌⠙⣿⣿⣆⠀⠋⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣦⡀⠀⠀⠀⠀⠸⠏⠹⣿⠏⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿⠋⣠⠾⣿⣿⣦⠈⢿⣿⡆⢀⠶⣶⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⠃⠀⣿⡟⢻⣇⢸⣿⣿⡀⠀⢸⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣁⠀⣀⣿⠏⠈⣿⢸⣿⣿⡇⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⣰⠂⡏⣼⣿⣿⡇⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠛⠶⡿⠏⠀⣱⣿⣿⣿⠃⠰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠈⠁⢀⣠⣾⣿⣿⣿⡟⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣶⣶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⠿⠟⠋⣹⣿⡿⠁⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⢀⣀⡴⢀⣴⣿⡿⠁⠀⣀⣠⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣙⣻⡿⠟⣫⣴⣿⡿⠋⠀⠀⠀⠉⠉⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠟⠉⠀⠀⣠⡿⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⠉⠀⠀⠙⠶⠞⡿⢣⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣫⣴⠾⠋⠀⢰⡀⠀⠒⣉⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣾⠟⣱⣾⣿⠋⠀⠀⠀⠈⠻⢿⡿⠟⣻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⣠⡤⣾⡏⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⣠⣾⠏⠀⣿⣷⣿⠁⣿⡅⠀⠀⠀⠀⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⣼⣿⡟⠀⠀⢸⣿⡇⠀⠙⢷⣦⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⡿⣿⡇⠀⠀⠀⠘⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢸⠃⢿⣷⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⡄⣦⡙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠣⠈⠻⣶⡶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃


⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⠿⠿⢿⣿⠿⠿⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣽⣿⣿⣦⣶⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣿⣿⡛⠻⣭⠟⠉⠀⠀⠀⠀⠈⠛⢛⣻⣽⢿⣿⣟⣿⣿⡯⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠰⣿⣯⣽⣿⣗⣿⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡁⠙⢿⣿⣿⡟⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿⣿⣿⣿⠏⡐⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀⢤⢹⣿⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠈⢿⣿⠃⠾⣡⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡞⣿⣿⣿⣆⠈⣿⡏⣡⣬⣽⣷⣦⡄⠀⠀⠀⠀⢰⣿⣿⡶⢶⣶⢘⣏⢀⣴⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣆⠸⡇⠉⠓⠚⠟⠀⠁⠀⠀⠀⠀⠀⠀⠈⠒⠋⠁⢸⡇⣸⣿⣿⣿⣿⢻⣆
⠀⠀⠀⠀⠀⠠⣽⣿⣿⣿⣿⣿⣤⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣿⣿⣿⣿⣿⠈⣿⡆
⠀⠀⠀⠀⠀⠀⢛⡿⣿⣿⣿⣿⣟⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⡄⠸⡇
⠀⠀⠀⠀⠀⠀⣾⢹⣿⣿⣿⣿⡿⣿⡞⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧⠠⠁
⠀⠀⠀⠀⢀⢠⡿⣸⣿⣿⣿⣿⡟⣿⣿⣿⠀⠀⠀⠠⠤⠀⠤⣤⡤⠄⠀⠀⣠⣾⣿⣿⣿⣷⣿⣿⡎⣿
⠀⠀⠀⠀⡎⢰⣷⣿⣿⣷⣿⣿⣷⢻⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡆⡄
⠀⠀⠀⠀⢇⠘⢿⣿⣿⣿⣿⣿⣿⠈⣿⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡟⣧⠿⣿⣿⣧⢸⣧⠰⡄
⠀⠀⠀⠀⠈⡀⢘⠿⣿⣿⣿⣿⣿⠂⣾⣿⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣳⠏⣸⣿⣿⣿⣿⣿⠀⣇
⠀⠀⠀⠀⠞⢀⠂⡀⢿⣿⣿⣿⣿⣸⣿⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢩⡟⣰⠏⣸⣿⢻⣿⣿⢠⡟
⠀⠀⠀⠀⠀⠎⢠⢰⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢾⣿⡏⣐⣿⢻⡌⣿⣿⣿⡇
⠀⠀⠀⠂⠀⠣⠈⡘⣿⣷⠀⠻⣿⣿⣿⣯⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢸⣧⣿⣧⣿⡿⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠠⡻⣿⣆⠀⠙⢿⡟⠙⣿⡟⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⣿⣿⣿⣿⠁⠀⢻⡇⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣆⢸⡩⣿⣇⠀⠸⣇⠀⠹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡽⣿⢻⡄⠀⢸⣿⡇⢹⣿⠃
⠀⠀⠀⠀⠀⠈⠀⣿⢰⠀⢻⢿⡆⠀⣿⡀⢀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⣾⡇⠀⢻⣿⠇⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢻⠀⣸⣿⡇⠀⡹⠀⢸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⢀⣿⠏⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⠿⠃⠀⠃⠀⠻⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠹⠟⠀⢸⡿⠀⢸⢸⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⢰⡶⠿⣴⣦⣾⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⠺⡃⠀⠈⢿⣿⠏⢉⠁⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢂⠸⠂⠘⠻⡏⣄⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢱⣦⠀⠀⢑⣼⣿⣿⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⣿⣦⡤⣞⡹⣿⣏⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣸⣾⣽⣙⡯⢰⡉⣴⣿⣯⣯⣥⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢸⣿⣷⡟⣱⢈⣿⢹⣾⣩⣿⣿⡇⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢰⣿⣿⣿⣽⣣⣌⣿⣮⣵⡿⠃⢸⠷⣦⠀⠀⠒⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⢀⣿⡟⠙⣻⡛⣿⣤⡿⢿⠁⠀⢰⣧⣿⠀⠀⠀⠀⠙⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⢘⣿⠀⢴⡛⣷⣴⣿⣷⣤⠒⠀⠀⢹⡗⠀⢄⡀⠀⢀⡼⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡼⠉⣶⡸⣷⣴⠛⣿⣿⣿⣾⠀⣦⣿⠇⠀⠀⠉⢥⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⢀⣼⢧⡾⠃⠀⠁⠈⢻⣿⡿⢿⡿⠀⠀⠀⠀⡘⠒⠀⠀⠀⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠻⡾⠇⠀⢀⣄⣀⡀⠙⢿⣃⡇⠀⠀⠀⢸⡁⠀⢱⣆⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡀⠀⢸⣿⠀⣀⣽⣾⣾⣿⣿⣿⣷⡿⠿⣯⣀⡀⠀⣠⠏⠀⠀⢡⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣧⡀⠀⠈⢻⣆⡶⠦⠴⠴⠶⠾⠿⠷⠶⠾⠛⡿⡇⠰⢿⠀⡀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠁⠀⠚⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠒⡄⢠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠅⠀⠙⡇⠀⠀⠀⢀⣀⠀⢀⠀⠀⠐⡠⠀⠡⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⠘⢡⡈⢠⠀⠉⢙⣿⣿⣧⠄⠀⠀⢱⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣰⠎⡓⡄⠀⠀⣾⡏⢹⣧⠀⠄⠀⠈⠈⢳⡆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠶⠶⠁⠀⢀⢀⣿⠃⠀⢿⡄⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣢⡄⠊⠉⣼⠏⠀⠀⠈⢿⡀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠙⠀⠀⣰⡟⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⢷⠀⠤⣀⣿⡇⠀⠀⠀⠀⠀⠘⡳⠀⠀⠀⢿⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⣁⠀⣠⣤⠾⠃⠀⠀⠀⠀⠀⠀⢈⠙⠃⠀⢸⠓⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠂⠀⠵⠒⣒⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⢸⣇⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⣀⣀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⡿⢛⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠉⠉⠽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠾⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣭⠈⠉⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣺⠿⢙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣩⠉⣩⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⡾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⣉⠉⣩⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣀⣀⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⢛⠟⠊⢓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠚⠛⠁⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠶⠦⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡠⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⡄⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡀⢰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⡇⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢣⣿⠀⠀⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣳⣿⠃⠀⠀⠘⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣅⣠⢴⣲⢤⣀⣭⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣋⣭⣶⣿⣿⣷⣦⣍⣻⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣻⣭⣭⡥⠤⠤⠤⠤⢬⣭⣭⣟⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣄⡀⠤⣤⣀⡀⠀⣼⣿⡟⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡙⣿⣿⡄⢀⣀⣤⠤⢀⣤⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠻⣷⣦⣍⠛⢿⣾⡿⢸⣋⣭⡛⠦⡀⠀⠀⠀⠀⠀⡠⢞⣋⣝⣳⢸⣿⣾⠟⣉⣴⣾⠟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⢿⣏⢳⣄⡹⠇⢻⣿⣟⣛⠳⢤⡀⠀⠀⠀⣠⠴⢛⣻⣿⡿⠃⠟⣡⡾⢹⣿⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣜⣿⡘⣿⣿⣄⠀⠙⣿⡉⣿⣮⣿⡇⠀⣾⣷⣾⡏⣹⡟⠁⢠⣾⣿⠇⣼⣯⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣀⣼⠟⠀⠈⠛⠚⠓⠈⠁⠀⠉⠐⠛⠚⠋⠀⠸⢿⣀⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡀⢤⣴⠖⠀⠠⣄⣀⡴⠀⠐⢶⣤⢄⡰⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣷⡀⠀⠰⠬⠯⠴⠀⠀⣰⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄⠀⠟⠺⠇⢀⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣺⣿⣿⣿⣿⣷⣤⣤⣴⣿⣿⣿⣿⣿⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠷⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣞⣿⣿⣟⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠉⠉⠉⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢀⡔⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠴⠤⡀⠀⠀⠀⠰⠉⠢⢄⠀⠀⢀⠔⢂⠀⢸⠙⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠏⣀⠃⣸⠃⢄⡀⢀⠃⠀⠉⠒⠤⣀⠔⠁⠀⠀⠈⠒⢄⡠⠁⠀⠀⠀⠑⠔⠁⠀⠀⠶⠈⠀⠀⠘⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⢠⠍⢀⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡆⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣇⠖⠀⣼⣿⠂⠠⠄⡤⠠⠄⢠⠄⠐⠤⣒⠶⢒⡒⠒⠀⠐⠒⠊⠭⠭⠉⢀⣤⣬⠍⠛⢋⡩⢷⠀⠀⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣟⠉⠉⠉⢲⡟⠁⣀⠄⠊⢀⠠⠖⠁⠀⠀⠀⠀⢈⠑⠂⣬⡉⡉⢁⣀⣀⣈⣉⣁⣩⠅⠒⠞⠳⠶⢿⡀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡀⡼⢃⣠⣶⣦⠔⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢲⣾⣿⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠛⠋⢩⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⢀⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⠀⢹⣿⣿⣿⣿
⣿⣿⡿⠟⠃⠀⢀⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣆⠀⠀⠙⢿⣿⣿
⣧⣄⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠤⠀⠤⢀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⣀⡀⣀⣨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⢠⠊⠀⠀⠉⠢⡐⠃⠀⢀⣠⠴⣦⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣶⣶⣤⣤⡉⠤⣶⠿⠲⠞⢻⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠉⣩⠆⠉⠅⠀⠀⠑⢦⡶⢾⠗⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡇⠘⣩⠟⠈⠆⠸⣇⠀⠐⠢⣙⠈⣇⢠⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠠⠷⠊⠀⠀⠈⠃⢛⡴⠦⠖⠀⠈⠙⢿⠚⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⠀⠀⠀⠀⠀⣽⣙⠒⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣀⣀⣈⣄⣀⠀⠀⠀⢴⣉⡁⠀⠀⠀⠀⣠⢀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⠀⠀⠈⠹⠒⢲⡀⠤⠐⠂⢉⣻⣸⡗⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠆⠐⠒⠒⢷⠚⠁⠉⢻⠉⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠒⠒⠮⠠⢤⣤⣬⡖⠀⠀⠀⠉⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⡀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠁⠒⢄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠙⢄⡠⠊⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡔⠉⠉⠉⠉⠐⠒⠀⠒⠤⠀⠸⢎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢣⠤⠔⠢⢀⣀⣀⠠⠄⣀⠀⠀⢈⠜⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⠆⡄⠀⠀⠀⢠⡄⠀⠀⠉⣉⣹⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡧⠤⠤⠤⠤⠤⢿⡀⠀⠀⠠⠞⠤⠄⠀⠀⢣⢿⣿⣿⣿⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢌⣠⣿⣿⣿⡇⢀⣀⠀⢀⡀⠀⢈⠲⣤⡄⠀⠀⠠⠄⠚⠣⣼⣿⣿⣿⣄⡴⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⠊⠀⠙⠁⠈⠐⠉⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⠂⢀⠀⡔⠀⠈⠲⠀⠀⠙⡏⠄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⡻⠀⠸⠀⠄⠀⠀⠀⡇⠀⠀⠶⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣄⣤⠁⢀⣤⡤⠜⠀⠀⠀⣇⣠⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠄⠒⠈⠉⠉⠉⠉⠁⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⠀⠀⠀⠠⠏⠑⢄⠀⠀⠀⡠⠳⡄⢀⡇
⠀⠀⠀⠀⢠⠁⣰⣇⠀⠀⠀⡜⠁⠒⠤⢀⠀⣠⠞⠀⠀⠉⠂⢄⣀⠃⠀⠀⠀⠙⠦⠊⠀⠀⠘⠊⢀
⠀⠀⠀⠀⡌⢀⠙⡁⠉⠂⠶⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢎⡇
⠀⠀⠀⢰⠁⠈⠁⣇⣀⣀⣀⣀⣀⡀⣀⠀⢀⣠⡤⠤⠀⠤⠴⢒⠒⠂⠀⠉⠉⠉⠁⠀⠀⠉⢉⠻⠾⣿⢆
⠀⣀⠰⢇⣀⣲⠚⠉⠉⣩⠏⠉⢁⠼⠉⠀⠀⠉⠓⠶⣤⠤⢀⣈⣁⣚⡂⣠⣤⣤⠶⠶⣒⣒⠿⠟⠒⣹⠂⠣⢀⡀⢀
⣶⠀⢀⡄⠈⣇⠠⠔⠊⣀⠠⠐⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⢩⡉⠀⠀⠒⠒⠀⠀⠛⠃⢭⠍⠉⣉⣤⡜⠃⠀⠀⠀⣀
⠹⢦⣾⣤⡶⡟⢳⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡶⠤⠀⠤⠘⠁
⠀⠀⠀⠀⡐⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆
⠀⠀⠀⠀⠘⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠔⠁
⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠀⠀⠒⠤⣀⠀⠀⠀⡠⠒⠀⠁⠀⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠈⢣⠀⠎⠀⠀⠀⠀⠀⡀⢀⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⠁⠶⠢⣤⣀⠀⢀⡠⠿⠤⡀⢀⣠⣄⠢⠝⠃⡰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⣤⠄⠀⠀⡴⠓⠂⠩⠴⠶⠶⡿⠛⠶⣦⠀⠐⠿⠟⠳⠶⠶⠾⡟⢺⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⣀⣀⣀⡇⠀⠀⠀⠀⠀⣠⠗⠁⠀⠈⡀⠀⠀⠈⠑⢄⠀⠀⡇⢸⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡀⠀⣱⠒⠒⠒⠂⠾⠁⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠱⠞⡇⢸⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠠⠊⠁⠀⠈⠑⢄⠀⡌⠀⠀⠀⠀⠀⠀⠟⡄⠀⠀⠀⠀⠀⠸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢃⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠺⠀⢡⠀⠀⠀⠀⠀⢀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠈⢦⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⢰⡆⠀⠀⠦⠀⢀⠤⠚⠈⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠸⡏⠢⠼⠁⠀⠀⠀⠒⠁⠀⠀⠀⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠀⠀⢀⣀⣑⡄⠀⠀⠀⠀⠀⠀⠘⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⡠⠞⠁⠈⠉⢐⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠒⠒⠒⠂⠀⠀⢸⠀⠑⢄⡠⠊⠀⠀⠀⠰⢆⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡄⠀⠀⢠⡀⢲⡄⠂⠀⠀⠀⠀⠀⠀⠀⡇⢀⣘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠊⠙⢦⠀⠀⢀⠦⢜⠃⠂⠰⠳⣀⣿⣈⣿⠆⠓⠒⡒⠀⠐⠀⠂⠒⠘⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⠖⠁⠀⠀⠑⣶⠁⠀⠀⠀⠉⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣀⣀⡀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠀⠀⠀⠀⠀⠐⡇⠀⠀⢰⡿⠍⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠉⠁⡇⠸⢀⣠⣶⠶⣄⠀⣠⢄⡀⡇⠀⠀⠈⢳⠔⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠊⠳⠤⠚⠣⣠⠔⢆⢠⠀⠘⠋⢻⣈⡡⢎⠙⠁⠀⠙⠁⠀⠀⠀⠀⠄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⡉⠀⠈⠀⠀⠀⠀⢣⡀⡀⢀⠔⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⢀⣀⣠⠃⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠤⠀⠀⠀⠀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠄⣀⣀⣀⣀⠠⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠹⣆⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⣀⣤⣠⡴⣾⠟⠋⠉⠉⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣤⠰⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠀⠀⡀⠀⠀⣽⠉⠀⣿⠁⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠾⣄⠀⣿
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢻⡄⠀⠀⠀⠀⢠⡄⣰
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⡴⠶⠶⠶⠶⠶⢤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠙⠶⣤⣤⠶⠿⠟⠁
⠀⠀⠀⠀⠀⢸⡆⠀⢀⣠⣤⠶⠞⢋⡉⠁⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⣦⣤⡀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢻⡞⠋⠁⠀⠀⢸⡋⠙⠛⠛⠉⢻⣄⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠈⠙⠳⢦⣤⡟⣁⣄
⠀⠀⠀⠀⣶⠲⢾⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠳⠶⠖⠛⠉⠉⠁⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⣿
⠀⠀⠀⠀⣿⠀⠀⠻⣦⠀⠀⣠⠟⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢦⣤⣀⣀⣠⡾⠃⠀⠀⠀⣿
⠀⠀⠀⣀⣸⡆⠀⠀⠈⣻⡏⠁⠀⠀⠈⠉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠋⠀⠀⠈⠈⢩⡿⠀⠀⠀⣸⠇
⠀⣰⣦⣿⡹⣷⡀⣀⣄⠘⣷⠀⠀⠀⠀⢰⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⠂⠀⠀⠀⠀⣾⠁⠀⠀⣠⠟
⢶⡿⣷⡽⠷⠈⠿⠋⣟⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣄⣤⠾⠋
⣴⣿⡮⠁⠀⠀⠀⠘⡿⠛⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠈⠙⠶⣤⣀⠀⡀⠸⢷⣦⣄⠀⠙⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠋⠁
⠀⠀⠀⠀⠉⢻⣯⡴⠟⣡⣼⣿⡀⠀⠈⠉⠛⠳⠶⠶⠶⣶⠶⠶⠶⣶⠶⠶⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠈⣧⣤⡾⠃⠀⠈⠛⢶⣤⣀⡀⠀⠀⢀⣠⡏⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠉⠙⠛⡿⠋⠋⠤⠀⠀⠀⢀⡘⠋⠙⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠒⢢⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⡄⠀⠀⠈⠂⠀⠂⠐⠀⠒⠉⠉⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⣦⡀⠀⣀⣘⣷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠷⣤⣄⣠⣤⣶⣧⣀⠀⢀⣀⣀⣀⣠⣼⣷⠀⠀⢹⡟⠉⢉⣡⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⣤⣌⣿⠆⠀⠠⣿⣋⣉⣉⣉⣉⣤⣴⣿⠀⠀⠈⠳⣶⡏⠁⠀⠙⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠙⢿⡟⠿⠛⠉⠉⠉⠉⠉⠀⠀⠙⣆⠀⠀⠀⠀⢻⢀⣤⣀⠀⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣠⡞⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢿⣼⠏⠙⠳⢦⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⢶⡶⠚⠉⠀⠀⠙⠲⣤⣤⣤⣤⣤⣤⢤⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣼⣇⣠⣤⣼⡏⠀⢀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡙⠋⠺⣿⣧⡶⠉⠻⣦⠞⢛⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢘⡧⠀⢀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢳⡴⠋⠀⠀⠀⢀⡾⠁⡰⠯⠸⠦⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⣀⣀⣠⡴⠿⣄⠊⠀⠀⠀⠀⠑⢀⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣥⣤⣤⣽⣷⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣴⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣤⣤⣤⠄⠀⢿⡏⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡏⠀⣠⣴⢇⣀⠀⢀⣿⣶⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠉⢁⡹⠆⠿⢷⣀⣻⣿⣾⣿⣿⣿⣾⣷⡦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⢴⣯⢯⣀⣀⣹⡻⢿⣿⡇⣿⣿⠿⣿⡶⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣸⡟⠿⣿⣿⣿⡖⢶⠇⠉⢲⣭⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣽⣿⣿⣿⣷⡈⣴⣿⣾⣷⡿⠿⠚⠋⠀⠀⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡾⠟⠛⣿⣿⡟⣨⣿⣞⣯⡺⢿⣿⣿⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣾⣿⣿⡿⣿⣿⠿⠻⣿⣿⣯⣉⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⣥⣴⡟⡇⠀⢸⣿⡿⠦⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⡿⣹⠣⠒⠩⠋⠑⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡿⠚⠋⠁⠁⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⣀⣀⣀⠉⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣮⣥⣾⣿⣿⣿⣿⣭⣓⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣟⡛⢃⢳⣾⣿⣿⣿⣝⢿⣿⣿⣷⡜⢿⣿⡽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠍⣨⣿⡯⠥⢐⣠⡿⢸⣿⣿⣿⢗⣽⠼⠛⠻⣿⣿⣼⣿⣿⡘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡳⢃⣞⣯⣤⣤⣾⠿⠋⣴⣿⢛⢟⣷⣿⣿⡆⠀⠀⠈⠻⣿⣿⣿⡇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣭⣦⣴⣾⠟⣩⠤⣿⢿⡿⣶⢮⠀⠀⠀⢀⣿⣿⣿⢇⡼⣹⡆
⠀⠀⠀⠀⠀⠀⠀⡻⢿⠟⠛⠻⣟⣛⠿⢿⣛⢕⣫⣶⣿⣿⢿⣧⣾⡍⠀⢀⣴⣿⢿⣫⢗⡫⢰⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢻⡿⠚⠛⠛⠛⠛⠛⠋⠱⣿⣿⣿⣿⠏⢸⢿⣿⠃⣠⡿⣫⢖⡭⢚⣡⣶⣿⣿⣿⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢀⣀⠀⠀⠀⠀⠈⠛⠛⢃⣢⢨⢸⠏⣰⡟⣰⢕⣭⣾⣿⣿⣛⣯⣭⣥⣮⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠩⣥⠤⠀⠀⠀⠀⠀⠀⠀⠸⡡⠂⠀⣤⣿⢠⣿⣿⣿⣿⢟⣵⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸⣿⣸⣿⣿⡿⢡⢞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢿⣿⣿⠃⡞⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠖⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠘⣿⡸⣿⣿⢰⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢹⣷⣿⡏⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢥⡤⢄⣀⠀⠀⠀⠀⠀⢸⣿⣻⡇⢸⡇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⢚⣽⣶⡶⡻⠖⠚⠓⠿⠂⠀⠀⢀⣾⢃⣿⠇⢸⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢇⢸⣿⣟⣼⣅⣀⣀⣀⣀⣀⣠⣴⣿⠋⣼⣿⠀⣾⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⣓⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⠟⢁⡜⣽⠇⣰⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⢍⡛⠛⠛⠛⠛⠋⠁⠔⣫⡾⠃⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠳⠮⣵⠦⣤⣤⣤⣖⣚⣩⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢾⠿⠿⠟⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣽⣧⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣿⣯⣭⣽⣯⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠈⠉⠉⠁⠀⠒⠠⢀⢰⠈⢦⡀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⠀⠈⠳⠄⠠⠤⢀⠀⠀⠀⠀⠙⠄⠀⠈⠲⣄⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠬⠁⣤⠁⠀⠀⠀⠀⠀⠁⠲⠊⢣⠀⡄⠀⠀⢀⠜⠁⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢻⣤⡍⠄⠀⠀⢀⠀⠀⣀⠼⠖⠁⠀⠀⡌⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠁⠀⠀⠀⠐⣦⠀⠓⠀⠠⠤⣌⠀⠀⠀⠐⣀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⣠⠔⠉⠉⠀⠠⢃⠀⠀⠈⠑⠠⣀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⡠⠔⡋⠁⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠒⠋⠁⢠⢀⣈⣶⣤⣀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢄⠀⢄⠙⢦⡀
⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⠋⢁⠌⠉⠁⠚⠒⢄⡀⠀⠀⠀⠈⠆⠀⠀⠀⠙⢆
⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⣀⠚⠀⣀⠤⠋⠉⠃⠀⠚⠁⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠀⠘⠀⠸⡀⠀⠈⢣⡀
⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⢀⠄⠀⠀⠀⠀⠀⢈⠁⡅⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢇⠀⠀⠀⢳
⠀⠀⠀⠀⠀⢤⠀⠀⠀⠴⠒⠈⠀⣴⠀⠀⠀⠁⠀⠆⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣧⠀⢸⠀⠀⠀⠀⢣
⠂⣀⡠⢣⠀⠀⠀⠀⠀⠀⡀⠴⣄⠁⠴⠀⠀⠂⢑⣎⠀⠀⢀⠀⡀⢸⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠑⠀⣔⠒⠀⠁⠀⠀⡁⠀⠀⠀⠀⡘⠒⠊⠀⠁⠈⠀⠀⣾⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣼⠀⢸⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠁⢌⡀⠀⠐⢒⡠⠊⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠉⢰⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢀⠀⠧⠀⠼⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠌⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⠄⠛⠒⠆⠠⡔⠊⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠎⠀⢰⠒⢈⠡⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡀⠀⡠⠀⢀⠜⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⡐⢀⠀⠀⠀⢠⠃⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠊⠀⠀⠀⠀⠀⡀⠈⡠⢊⠔⣡⡤⠂⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠆⠐⠋⠀⠀⠀⠀⠀⠀⠘⠀⠒⠐⠃⠈⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠄⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠓⠤⠤⢀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠄⠒⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠤⠤⠄⣀⡀⠀⠉⠉⠀⠈⠉⠀⠀⠈⢉⣀⣉⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⣏⢉⡏⠑⠒⠒⠒⠈⠉⠁⠀⠸⠤⠴⠧⣀⠉⠉⠉⠉⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠈⠁⠀⠀⠀⠀⢀⠈⡆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⣳⡤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠦⠀⠁⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⣠⠞⠱⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠉⠀⠀⠀⠀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠑⠦⢀⣀⣀⠠⠖⠁
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠁⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⡄⢀⠈⠣⢤⣍⠒⠢⢴⡄⠀⣸⡄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⠷⠆⠠⠶⠦⠤⣤⠀⢰⠀⣖⣫⡇⠘⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠷⣦⣅⡀⠠⣶⣾⣿⠶⠆⠀⠹⢰⣏⣹⠁⠑⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠺⠄⢄⠈⠛⠛⠀⠀⠀⠀⠈⠉⢀⣧
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠠⠀⠑⠂⠠⣤⣤⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⡴
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠂⠀⠁⠐⠒⠟⠛⢃⡁⡱⣄⡀⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠄⠀⠁⣀⠀⠀⢀⠀⡎⢀⡴⣶⣦⠀⠀⠀⠈⡅⠀⠘⢿⣷⠢⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⡂⢼⠁⠛⠄⠀⠀⠀⠀⠀⢰⣣⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢉⡿⢇⠀⠃⢄⡤⠦⠄⠀⠴⠈⠙⡈⠘⠀⠐⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⡀⠀⡠⣠⡄⠀⠢⣯⣤⢀⣄⠀⠀⠀⠀⠀⠀⠉⢉⠀⡀⠠⡠⠴⠀⢠
⠀⠀⠀⠀⠀⠈⠺⠗⠋⢕⡰⣶⣾⠛⢋⠇⡀⢰⡖⠂⠀⠰⡆⠈⠓⢄⣶⣾⣷⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⡿⢧⣾⠇⠤⢺⡇⠀⡀⠠⠀⠰⢤⣾⣿⣿⠉⠉⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⢸⠓⠁⣠⣤⣤⣤⣤⢬⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⣷⣿⡿⣏⠀⠀⠀⠀⠀⠉⠛⢿⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠠⠄⠒⠃⠈

⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⣠⣶⣿⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣷⡀
⠀⠀⠀⠰⠾⠋⠻⣿⣏⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⣀⣸⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢋⣿⣿⠻⡷⠇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣏⠙⠛⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⣦⡀
⠀⠀⠀⠀⠀⢠⣿⠟⠙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⢹⡇
⠀⠀⠀⠀⣴⠟⠁⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⠉⠙⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⢸⣇
⠀⠀⠠⡾⠃⠀⠀⠀⠀⠸⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠇⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⣰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿
⠀⠀⠀⢀⣤⣶⣾⣿⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠶⢶⣿⣿⣤⣀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢮⣿⣷⡄
⠀⠀⡴⠋⢡⣿⣿⡿⠋⠉⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢸⣿⣿⡟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠶⢾⣿⣿⡄
⠀⠀⠀⠀⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣧
⠀⠀⠀⣠⣿⠉⠙⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠿⠋⠙⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠁⠈⠻⣧⡀
⠀⢀⡼⠋⠁⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⠛⠉⠀⠀⠀⠀⠀⠙⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣧
⠀⠙⠧⠀⠀⠀⠀⠀⠀⠈⠻⠖⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀
⠀⠀⠀⣠⣴⣿⣧⣤⣤⡤⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣘⣿⣦⣄⣠⠟
⠀⢠⡾⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠻⣿⣿⡅
⠴⠏⠀⣸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢀⣿⠏⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣃⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⢻⡇
⠀⠀⣼⡟⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠿⠛⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠸⡇
⠀⣠⡏⠀⠀⠀⠀⠘⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠶⠻⡿⠋⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⣿⢟⣦⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠙⠻⢃⣾⣿⣷⠀⢹⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠈⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⠏
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⣶⣆⠀⠸⣿⣿⡀⠀⢻⣿⣿⣿⣿⠋
⠀⠀⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠿⣿⣿⣿⣿⡆⠀⠙⠿⣧⠀⢸⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣤⣼⣷⣶⣤⣀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠑⢻⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣼⣦⣀⣀⠀⠀⡔⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠈⠉⠉⠉⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⣿⣷⡀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣼⢟⣽⣿⣿⣿⣿⣦⡀⠀⠀⠿⠗⠀⠀⠒⠒⠒⠲⢿⣿⣿⣽⣿⣿⣿⣿⢿⡐⠸
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⢾⣧⡀⠄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⡿⣼⣿⡟
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⡐⢦⣤⣽⣿⡿⠿⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠟⠻⢿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠟⠛⠋⠁⠀⠂
⣿⣿⠿⢿⢻⠿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡖⠋⢉⣩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡉⠉⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠩⠜⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠣⠭⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠖⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠒⢘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡚⢃⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣈⢑⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⣉⠞⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡻⣌⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣢⠞⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠓⣒⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠙⣩⣻⠖⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠲⣟⣍⠉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠙⠛⢛⣁⣤⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣈⡙⠛⠉⠻⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⢻⣿⣿⣷⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣾⣿⣿⡟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠻⣿⣇⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣼⣿⠟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⡹⣆⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣰⢯⠋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠙⣯⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⠃⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠘⣿⣿⣄⠀⠀⠀⠀⣰⣿⣿⠃⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⢿⣿⣦⠀⠀⣴⣿⡿⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⢻⣿⣷⣾⣿⠟⠁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⣽⣿⡿⠋⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⢁⣴⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁⣠⣎⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⣠⠞⠁⠈⠳⣄⠙⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⢁⡠⠚⠁⠀⠀⠀⠀⠈⠳⣄⠙⠻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⠋⣡⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⠛⢿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡿⠟⢋⣡⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣌⡙⠿⣿⣷⣦⣄⣀
⠀⠀⠐⠶⢶⣾⣿⣿⣛⣛⣩⠥⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢬⣙⣛⣻⣿⣿⣶⣶⠶⠂
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡏⡿⢹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢻⡇⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢸⠇⠀⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠁⠈⠀⠲⠶⢼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⣔⡒⠦⣤⣠⣿⣿⣿⡗⢾⠲⠶⣶⣶⣾⣷⣿⣿⣿⣄⣤⠴⢒⣢⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣟⣦⡘⣿⣿⢟⣠⣄⡀⠀⡀⠀⠋⣁⣤⣬⡻⣿⣿⢃⣴⠛⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣌⠻⣿⣾⡷⠺⡿⣿⣿⣶⡟⢳⣿⢿⡿⢿⡄⢿⣷⣿⡟⣡⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡸⡟⢧⠀⢉⣩⠞⡹⠃⠘⠟⠻⠧⠀⢀⡼⣻⢇⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡒⣪⣷⣤⡀⡠⢄⠂⠐⣐⠄⣠⣦⣿⣝⠂⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⡌⡟⠁⠈⠳⡞⠁⠈⣿⢿⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣥⡈⠀⢐⣊⡀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣶⣿⣿⣿⠀⠈⢙⡟⣽⣿⣷⣦⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢾⣿⣷⣽⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣯⣾⣿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠿⣽⡯⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠠⡀⠀⠀⠰⠆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠇⠀⠀⠀⠀⠀⢰⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢀⡀⠀⠀⠀⠀⣰⠟⠁⢠⣶⠀⢀⡀⠀⢀⠀⠈⠙⠻⢦⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠁⠀⠴⢁⠄⠀⠀⠀⠠⣼⡷⣧⠚⡡⠒⠂⠀⢀⠱⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡇⠀⠂⠀⠀⢀⠠⠂⠀⠀⠀⣼⡄⠀⠀⠀⣜⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠂⢱⡀⠀⣤⢠⣯⣶⡀⠀⠀⠀⣿⠁⡠⣆⠸⣸⡇⠀⠀⠀⠀⢰⡀⠀⠀⠀⠈⠐⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⢠⡄⣧⠔⠙⠺⠄⢷⡷⢄⣀⣠⠟⠀⠠⠄⠀⠙⠃⠀⠀⠀⡠⢋⣿⣵⣤⠀⣀⡠⠟
⠀⠀⠀⢀⡠⠖⠋⢀⣀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⢀⣴⡋⠉⠘⠳⠀⠀⠑⠲⠄
⠀⠤⠾⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠁⠂⣾⣤⠤⠤⠀⠀⠰⣿⡁⠀⠀⡴⠀⢀⠳⣄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠠⠦⣼⣿⣥⣤⡀⠀⣠⠀⢨⣿⠤⠤⢦⣤⠄⣀⣰⠷⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣬⠻⣦⣝⠻⡇⢥⣴⣿⣦⠀⠉⢻⣶⣶⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⣦⣿⠁⣟⣀⣘⣿⡷⠄⢿⢿⣿⣇⡀⣃⠀⢹⣿⣽⣧⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⣾⣿⡇⠋⠛⠿⣩⣟⠶⡦⠀⠈⢿⠽⠅⢸⠀⣾⢿⣿⣧⣼⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⢰⣥⣀⣼⣟⣿⠃⢚⠻⣿⣿⣿⡆⢁⡴⠐⠘⠛⢟⣿⡿⢻⣿⣿⣭⣽⣂⡹⣿⣴⣶
⠀⠀⠀⠀⠀⠐⢶⣷⠟⠱⠖⠋⠉⢀⣤⣤⣸⣿⣿⣿⣿⣟⠄⣬⣄⣼⣿⣿⡷⣬⣿⡄⢠⣶⢦⣸⣄⣉⠺⠿⣍⢻⣿⣿⣿⣿⡏⣠⣦⣤⡀
⠀⠀⠀⠐⠒⣀⣨⡽⠚⠉⠀⢀⣤⣾⣿⣯⣦⣽⡿⠛⢯⠇⢼⣿⣿⢛⡛⠋⠴⣿⣯⣤⣼⣷⣾⡶⣿⠿⠓⢰⣿⣿⣄⠉⠛⢿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⡸⣯⢿⣱⣿⢯⣶⠏⢩⣵⣿⣷⣟⣷⢈⠿⢻⡟⣛⠀⠸⠟⣰⣾⣿⠿⢻⡟⡄⠀⠀⠉⠛⠿⠟
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣷⣿⣥⣽⣮⣽⣏⡳⣾⣿⣿⣿⣯⣽⣧⡵⡶⢛⣿⠙⣇⠐⠂⣤⣨⣇⣤⣶⣿⣿⣇
⣿⣿⣷⣄⡀⠀⠐⠾⠿⠿⢛⣿⣿⣿⣧⣉⣉⣿⣟⣛⣛⣚⠻⣛⣿⣿⣮⣅⠀⠙⠦⡩⠶⣎⣸⣎⣍⣻⣧⣙⣋⣡⣶⡾⣀⣀⡀⠀⡀
⣿⣿⣿⣿⣿⣿⣦⣤⣀⣀⣩⠭⠉⠉⢙⠻⢻⡿⢿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⢿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠉⠉⠁⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣮⣱⠆⠄⠐⠂⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣶⣶⣶⣾⣶⣶⣷⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⣄⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣋⣤⠤⠶⠶⠶⠶⠤⣤⣍⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⠋⠀⠀⣀⣤⣴⠶⠒⠒⠒⠲⢶⣦⣤⣀⡀⠀⠙⡆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠓⢧⣠⢶⡏⢹⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⠉⢹⡶⣄⡷⠞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⣹⢰⡇⠀⠁⢀⣤⡀⠀⠀⢀⣄⠀⠀⠀⠸⡇⢿⡀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣿⢸⡇⠀⠀⠈⠛⠁⠠⣀⠘⠛⠀⠀⠀⢸⡇⣿⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⣿⢸⡇⠀⠀⠀⠀⠀⠀⠉⢙⡷⠀⠀⠀⢸⡇⢻⡼⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣤⣤⠀⠀⠀⠀⡇⠀⣿⠀⣳⣤⣤⠶⠚⠉⠉⠙⠉⠙⠒⠦⣤⣼⡁⢿⠀⢘⡇⠀⠀⠀⣤⣤⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⢯⣉⠓⢦⣨⡇⢸⣄⠀⠀⠀⠙⠻⣳⡟⠯⠤⣤⢴⡶⣗⠚⠛⣲⠟⡶⠦⠤⠼⢻⡏⡟⠛⠀⠀⠀⣨⡇⢸⣀⡴⠞⣉⡽⠂
⠀⠀⠀⠀⠀⠀⢠⡖⠒⠛⠓⠀⠀⢀⡞⠉⢦⣀⠀⠀⠀⠙⢷⠀⠀⠀⠀⠙⠮⠭⠭⠥⠚⠁⠀⠀⠀⣼⠋⠁⠀⠀⢀⡴⠉⢳⡀⠁⠀⠚⠛⠒⢲⡄
⠀⠀⠀⠀⠀⠀⠈⠙⣛⣷⠆⠀⢠⡟⠀⠀⣼⠋⠙⠓⠶⢤⣬⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣃⣤⡴⠖⠛⠙⣇⠀⠀⢻⡄⠀⠰⣾⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠯⠤⠶⠒⠻⠷⠶⠶⢧⣀⡀⠀⠀⠀⠀⠉⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠀⠀⠀⠀⣀⡼⠶⠶⠼⠟⠓⠦⠤⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣄⣀⠀⠹⣄⡀⠀⠀⠀⠀⢀⣠⠞⠁⢀⣀⣤⠔⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡶⠈⠻⢦⣀⢀⣠⠟⠁⣴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⡏⠁⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣰⡖⣚⣟⡲⣆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠋⢹⡇⣏⣈⣿⣟⠙⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠖⠚⠙⢧⡽⣭⠴⠟⠑⠲⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⣠⣤⠶⠛⠁⠈⠳⠦⣤⣀⣈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡽⠁⠀⢀⠀⠀⠀⠀⠀⠀⠘⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⣸⠋⠻⣇⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢀⡼⠃⠀⠀⠙⣦⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢠⡞⠁⠀⠀⠀⠀⠈⢧⡀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⣛⡓⢦⠀⠀⠀⠀⠀⢠⠏⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⢻⡀⠀⠀⠀⠀⠀⡴⢚⣛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣾⠁⣩⣦⠇⠀⠀⢀⣠⡏⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢳⣀⠀⠀⠀⠸⣵⣏⠈⣷⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠳⠬⣤⣤⣤⣾⡉⠀⠙⠲⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠴⠋⠉⢙⣶⣤⣤⡬⠤⠞⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠙⠶⢤⣄⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣀⣤⠶⠋⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⣀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠀⠀⠀⣀⣠⠤⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠉⠁

⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀⣀⣀⣀⣀⢀⣀⠀⠀⢀⣀⠀⠀⢀⡀⠀⣀⣀⠀⠀⠀⢀⠀⠀⣀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀⠀⡀⢀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⢀⡄⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣿⡉⠴⠄⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣹⣷⡄⠀⣼⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⠚⢿⣻⣷⠚⡋⠋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠺⠛⢻⣷⡟⢻⣄⢡⣎⣸⣻⣗⠀⠀⠀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣷⣶⡎⣳⣾⣿⣿⣿⣉⠙⠋⠀⠀⠀⠙⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡁⣻⣿⣿⡟⠿⣦⢠⣴⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠿⣿⠿⢿⢦⣞⠻⠻⣿⡄⣧⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠧⡿⢿⣿⣿⣿⣻⣷⣷⣬⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠟⠳⡄⢸⣿⡟⠘⣻⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣇⣠⣿⣿⣿⣷⣰⠯⢀⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠂⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⣿⣿⣿⣿⣿⣿⡾⣿⡿⠋⢸⠟⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⣿⡿⠿⣿⣿⣧⣀⣧⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⣿⣧⣀⠈⠀⣷⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠉⠁⠀⠰⣿⣷⡄⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣥⣌⢉⡒⢀⡨⠛⣿⣷⣿⡙⣷⣄⣤⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⠻⠿⠷⠯⢉⣹⣛⡿⠛⠛⡿⢶⣬⣽⡇⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⡼⠃⠀⠀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⢊⣴⠏⠀⠀⠰⠀⠀⠀⠀⠠⠤⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢳⣾⠃⠀⠀⡋⠀⢀⣀⣀⣀⣀⠀⢤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣽⡯⠹⠇⠀⠀⠂⣤⣋⡉⢴⠧⢹⡙⣿⠿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣴⡟⠁⠀⠀⠀⠀⠘⣸⣏⠁⠀⠐⣿⡇⠘⣆⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⡯⣿⡇⠀⠀⠀⢀⡴⠂⠋⠸⣆⠀⠰⣿⣷⣤⣿⣿⣿⣿⣿⡆⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢀⣿⡇⠀⡠⠔⠺⡀⠀⠀⠈⠙⠃⠚⠛⣩⣿⣿⣿⣿⣿⣿⣧⡤⣿⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⡁⢻⣿⡸⠿⣦⣄⠐⢄⡀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣭⣭⣤⣀⡀⠀⠀⠉⠁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣕⣹⣿⠇⠀⠈⣿⣿⣾⣿⣇⣀⠀⠀⠘⣿⣿⣿⡿⢿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠀⠭⢽⣷⡇⣠⣿⣿⣿⣿⣿⠁⠈⠛⠓⠋⠉⣻⠆⢁⡿⠛⣿⡇⣿⣿⣿⣿⣿⡟⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⡼⠃⢠⡞⠀⢀⣿⡷⡿⣿⣿⠿⠋⢹⡏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠋⠋⠬⣷⢻⣿⡈⠂⢤⡤⠊⢀⣴⣿⣿⣾⠿⣿⣿⣿⢿⣿⡀⢸⡛⣂⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⣚⢯⣿⣷⡴⠋⢀⣴⡿⣿⣿⣿⣿⠀⢸⣿⣨⣷⣼⣿⣿⠳⣾⣿⣿⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⠒⢲⣦⡀⠀⠀⠀⠀⠀⣾⣿⣿⣷⠞⠁⠀⣇⡿⣿⡿⢻⣄⣼⣿⣿⣿⣿⣿⣿⣶⣿⣿⡏⠀⡈⢀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣤⡀⠹⣧⠀⠀⠀⠀⣠⣾⣿⡿⠛⠒⠤⠠⠿⠦⠭⠶⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡐⢀⠂⣰⠟⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⣿⡄⠁⠀⠀⢀⠞⣡⣾⣿⠛⠂⠀⠀⠀⠠⠀⠤⠤⠶⣟⣻⣽⣾⣿⣿⣿⣿⣿⣿⢃⠄⠀⠂⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⣹⠇⠀⠀⢠⠎⣼⠏⢹⣿⠿⣿⣶⣦⠄⡀⠀⣀⣀⣀⣀⣉⣹⣿⣿⢧⡙⠿⣿⠅⠀⡄⣡⢔⣴⡏⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣱⡀⠐⠋⡄⠀⠈⠅⡈⠀⠀⢸⡇⠀⠘⣿⡟⠄⠀⠀⠈⠀⠀⠉⢻⣿⣿⣿⡄⠁⠀⣤⠀⢊⣴⠇⣾⣟⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢲⡈⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣿⣷⡈⠂⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⠀⠀⠈⣠⠞⠋⢉⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⡀⠈⠉⢉⣳⣤⣶⡰⠀⠠⠀⢠⣾⣿⡿⠛⠠⡀⠀⠶⣤⣀⠀⠀⢻⣿⣿⣇⠀⢀⣴⣾⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠋⠉⢻⣿⣿⣯⡀⣰⣿⣿⠏⠀⠀⠀⠀⠰⠀⠌⡉⠛⠻⠿⠿⣿⣿⢾⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⠀⠀⠀⠈⠛⣿⣿⣟⠇⠀⠰⢤⠴⠚⠀⡁⠀⠀⠀⠙⣿⣿⣿⡷⣋⠩⠽⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣤⢤⣿⡿⠛⠛⠳⡄⡀⠀⠀⠀⠘⣣⡴⢶⡼⢾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢳⠀⢀⡞⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⣤⡤⠈⠉⣷⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠞⠋⠘⣦⣠⡿⠀⠀⠈⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣤⣤⠀⠀⠈⠉⠛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⣀⣀⣀⠀⠳⡌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⣠⠔⠊⠉⠉⠉⠉⠛⠒⣤⡀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣠⠞⠁⠀⠷⡀⠀⠀⢀⡶⠀⠀⠹⢦⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣴⠋⠀⠀⠀⠀⠁⠀⠀⠘⠁⡄⠀⠂⠀⠺⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠃⣾⠟⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠀⠀⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠄⠛⣦⢤⢀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠛⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠖⠀⠀⣈⠛⠿⠉⠉⠙⠛⠛⠛⣏⠉⠉⠀⠶⠃⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⢠⣜⠏⡱⠄⠀⢸⣉⠲⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠠⣄⣴⠀⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⣹⢶⠀⠀⠀⠀⠈⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠐⠲⣤⣤⣼⠟⠀⠀⠀⠀⠀⠀⠀⣴⢡⡄⠀⠋⡀⠀⠀⠀⣀⣠⡔⠀⠀⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠒⠶⣿⣡⡎⠀⠀⢀⢀⡀⢀⣀⠀⠻⠆⠀⠀⢠⣄⠀⠀⠀⠛⢫⡀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⠉⠀⠈⠱⢦⣀⡀⠀⠉⠛⠁⠀⠀⡀⠀⠀⠀⡅⠀⠀⠀⠀⠈⣷⣤⣀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⣄⣀⠀⠀⠀⠀⠀⢀⡽⠓⠀⠄⢋⡁⠺⣇⠇⠐⢀⣡⠄⠀⠀⠀⠀⢘⣧⠀⠀⠀⡀⠹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⠛⠀⠠⢤⡤⠏⠀⠀⠀⠀⠈⢁⣀⡤⠀⢠⡿⢻⣦⡀⢀⣠⡶⢾⣿⠂⠀⠸⠇⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⢀⠄⣀⠀⠐⠀⠐⢄⡀⠀⢠⡌⢁⣤⣴⣿⣧⡈⠛⠛⠉⠁⠀⠀⢸⡇⣀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⡀⠀⠈⠛⠤⠀⠀⠀⠡⡔⠢⣤⣄⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢿⠈⠳⣄⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠚⠤⢀⡄⠊⠠⠆⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠘⣇⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢁⠔⢀⠐⢉⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠲⠈⠉⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⠁⠂⣠⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢠⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⡀⠀⢈⡠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢀⠈⠓⠤⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠐⢁⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠈⠃⠀⠶⣈⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣀⠀⠉⢀⡶⠆⠀⠀⠀⠀⠀⠀⢠⣄⣀⣴⡻⠄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠄⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠁⠘⢿⣮⠱⠄⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣄⡀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠁⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠿⠷⢤⣀⢺⡇⠀⠀⠀⠀⠀⠀⠀⢿⣖⣷⣄⡀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⣴⢿⣦⣍⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠈⠛⢢⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠃⠀⠀⣈⠋⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡍⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢿⠛⣷⠸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⠀⠀⢠⣦⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣀⣀⠀⠉⠀⠈⠙⠳⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠀⠀⣀⣾⠃⠀⢸⠀⠀⠀⡀⢰⣀⡀⠀⠀⠉⠉⠻⣦⡀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠰⠖⠀⣼⡏⠉⠑⠐⠆⠀⠳⣀⣾⠻⠇⠈⠓⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⢀⣀⠀⠀⠈⣿⠀⣼⠃⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⣶⠀⠀⠀⠀⠈⠻⢷⡄⣀⡀⠀⠙⢷⣄⠀⠻⢀⣀⣀⡀⠈⠑⠖⠲⠂⣀⠀⠀⠀⠀⠀⢠⣄⠀⠉⠙⠛⠻⣦⡀⢸⠂⠷⠛⠃⠘⡤⠂⠰⠇
⠀⢄⣠⣶⣿⣤⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠙⠳⢶⣤⡀⠀⠉⠀⠀⠉⠉⠀⠈⢳⠄⠀⠀⠀⠈⠛⠘⠉⠃⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤

⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣷⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣯⣿⡏⣿⣿⣵⣶⣶⣶⣲⣾⣿⣿⡿⣿⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠸⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⠴⠟⠚⠙⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣸⠀⡏⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣧⡀⠀⠀⣀⣀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⡤⠴⠶⢲⠖⠛⠛⠒⠒⠲⠶⢤⣄⣀⠀⠀⠀⠀⡇⠙⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠚⠉⠡⡖⠒⠒⠻⣄⠀⠀⢀⣀⣀⣀⠀⠀⠈⠙⠲⢦⣴⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠙⢧⠀⠀⢠⠇⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠳⡀⠀⠀⢀⡴⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠈⢳⡖⠉⠀⠐⣂⠀⠀⠀⠀⠀⠀⠀⠀⡒⠈⠉⠉⣿⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣟⢧⡤⡄⢱⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⢀⡏⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠝⠀⠀⠘⡷⠺⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣘⡿⢢⡌⠓⠦⢤⣀⣀⣀⣀⣀⣀⣀⣀⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⡞⠉⠑⠦⣀⡀⠀⠈⡹⠁⠀⢩⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠈⡉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⡀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠄⠀⠀⠀⠀⢸⣆⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⢀⡄⠀⠀⠀⠀⢸⣇⠑⢤⡤⠼⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⣉⣿⠛⢻⣗⠒⣒⣋⡩⣟⠀⠘⢦⡖⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⣿⠛⠉⠉⠁⠀⠀⢸⡆⠀⠀⢹⡰⢤⡀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢤⠴⠊⠁⠓⠤⣀⣀⣀⣤⡤⠇⠀⠀⠈⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣯⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠸⡄⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⢠⡇⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡓⠛⣿⣤⠒⢦⡼⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡴⠃⠀⣨⠆⡠⢀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣏⣀⣀⠴⣇⠀⠁⠉⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠉⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣖⣀⣤⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠀⠙⣿⡟⠙⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣁⠈⠙⠱⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣇⣠⠞⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣎⡿⢰⣡⣿⣿⣫⡴⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣰⣿⣿⡽⠼⣧⢿⣾⢻⢃⡀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⣿⡿⢻⡿⣿⣻⢿⠁⠈⣈⡇⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣿⠀⣞⠷⣼⣿⣦⠀⠀⢹⡇⠠⡄⠀⠜⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣳⣙⡿⠙⠛⢿⣧⣴⡿⠀⠀⠀⣢⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠿⡿⠀⢀⣀⡈⠻⣎⡁⠀⠀⢠⠀⢠⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢸⡇⢠⠿⠿⣿⣿⡿⣟⣿⣥⠀⣠⠃⠀⡠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⣷⡏⠀⠉⠉⠉⠉⠉⠉⠉⡁⣉⢀⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠙⡇⠀⠀⢀⡀⡀⠀⠐⠀⢉⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣁⡄⠄⠀⣸⣿⡷⠀⠀⠰⢾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠥⠆⠀⠀⣿⠀⢿⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢠⡄⠈⢸⠇⠀⠈⣧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⢰⡏⠀⠀⠀⠘⣇⠀⠀⢹⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠁⢀⣹⠇⠀⠀⠀⠀⠐⠲⠀⢸⠄⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠑⠰⠀⠀⠀⠀⠀⠀⢳⠀⠸⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡽⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡑⢒⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⣅⣼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠴⠾⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢤⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠶⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣻⣛⣝⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⢸⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠇⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⠿⠛⠋⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⠿⠛⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢺⣷⠉⢀⣤⣴⡆⢘⣀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠂⡈⠸⣷⠙⢫⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢾⣷⠿⣤⠙⠳⣤⠁⢠⠈⠀⠀⣠⡶⢶⡈⠳⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⡟⢀⡀⠙⣦⢍⣹⢋⣤⣾⣿⣿⣿⡿⠿⠶⠿⠿⠷⠤⢴⣶⣄⣤⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⡟⢛⡿⣿⡱⢿⣟⠘⠿⣬⣿⢦⣀⣀⣠⣤⣀⠀⠀⠀⠈⠉⠛⠿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⣏⡾⣿⣾⣿⣙⣿⢛⣻⠖⠀⠛⣿⣿⠟⠋⠉⠙⠻⣿⣿⣶⣦⡀⠀⠀⠈⠙⢿⣆
⠀⠀⠀⠀⢠⠞⣠⠘⣡⣤⠦⡄⣠⣿⠶⣶⣿⠋⣹⣿⣾⣿⣿⣿⣷⣶⣿⡿⢃⣤⣶⣶⣶⣬⣻⣿⢿⡿⠀⠀⠦⠤⣀⣙⣇
⠀⠀⠀⠀⠚⡾⠷⠀⣿⠋⠄⣴⡿⠃⠀⢸⣿⣿⡏⣽⣿⣿⣿⣿⡉⠉⠛⠑⠿⠛⠻⠦⢾⣿⣿⡆⠀⠀⠀⠀⠀⠀⣠⡤⠄⢢
⠀⠀⠀⠀⠂⠆⣠⣴⣿⢦⡸⣿⣷⣿⣿⣃⣜⣿⣿⡏⢩⣿⣏⡿⣷⣠⣤⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣡⣶⠶⣧⢸⣧⣄
⠀⠀⠄⠐⠛⢛⢿⡋⠁⣹⣯⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣯⣼⠇⣸⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⠀⠈⢻⣾⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⢀⡆⣿⣵⡄⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣺⣿⣿⣿⡿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠾⠿⢿⣮⣿⣿⣿⣿⣿⣿⡧
⠀⠠⢀⠀⠸⡃⠛⠇⢀⢿⣿⣿⣿⣧⣿⢿⣿⣿⠟⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠉⣡⣴⣶⣶⣤⣤⡙⠟⢿⣿⢏⣿
⠀⠈⠃⠀⠀⢉⢸⣻⣿⠾⢿⣾⣿⣿⣿⣜⢿⣿⣷⡉⣿⠛⢿⢇⡦⠈⡛⢿⠿⡍⠃⠀⠀⠀⠀⠀⠀⠀⠲⠛⠛⠟⠛⠛⠛⢿⠀⢸⣿⣟⣷⡀⠀⣀⡀
⠀⠀⢰⠂⠘⠫⠻⣭⣉⣵⣿⣿⣿⣿⣿⣯⡳⠦⢹⣶⣿⣤⡾⠟⣁⡀⠘⠻⣄⠘⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣽⣿⣟⣟⠛⠧
⣌⡄⠘⣢⣤⣶⠈⢛⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡼⣿⣿⡿⣴⡶⣿⣝⠺⢧⡈⠳⣠⣀⠀⠀⣀⣤⣴⣶⣶⣶⣿⣿⣿⣷⣶⣦⠀⢸⣿⣿⣿⡿⠛⠉⠁⠒
⢠⡶⢻⡿⠛⠜⠁⣼⣯⣿⢿⣯⣭⣭⣿⣿⣿⡿⣿⣿⣿⣷⡏⣻⣿⣿⣿⣷⣤⢦⡔⢿⠿⠋⠿⠿⠿⠛⠻⠛⠉⠹⢿⣿⢿⣿⢳⣼⣿⣽⣿⡗⠉⠖⠲⣄
⢨⣷⣄⣐⠀⠀⢰⣿⣿⣿⡼⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⠳⣽⣿⣿⡟⢿⢿⡴⡏⠉⠀⠀⠀⠀⢀⢀⣴⠀⠀⠀⠀⠀⠈⠉⢸⣾⣿⣿⣿⣧⡦⠠⡀⢼⣇
⠀⠀⠀⠈⠠⠃⠘⣎⢿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠄⠸⣿⡻⣿⣶⠸⠃⣇⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠁⢠⣷⣿⡏⣉⣉⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠁⠀⠀⡀⣠⣈⣛⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣇⣿⣿⠀⠀⣿⠇⢹⡿⢂⡗⡇⢀⠀⠀⣤⡀⢀⠀⠀⠀⠀⠁⠈⢁⣴⠟⠻⠿⠍⠔⠉⢀⠀⠀⠀⠸⠛⠁
⠀⠀⠀⠀⠸⣿⡀⠈⠁⠈⠻⣿⣿⣿⣿⢏⣾⣿⣿⡏⢹⠀⠀⠀⣿⠀⠸⡗⣿⣷⠇⢸⠀⢶⠁⠙⠋⠀⠀⢀⡠⠀⠂⢉⣡⠂
⠀⠀⠀⠀⠀⠈⠓⡦⠀⠀⠀⣈⣻⣿⣿⣾⣿⣿⣿⡇⢸⠄⠀⠀⢰⠀⢰⡇⣿⡧⠀⢠⡀⠆⠀⠀⠀⠀⠀⣎⣠⣄⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠘⠛⠋⠉⠁⠀⠈⢿⠙⠻⣿⣶⢿⣧⡀⠀⠈⠀⢸⡇⠁⠀⠀⢸⠀⢁⠀⢠⠜⠀⠀⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⠉⠙⠄⠀⠀⠸⡇⠀⠀⠀⢸⠀⠈⠀⠾⠀⠀⠀⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠈⣦⠀⠀⠀⢀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⠀⠀⠒⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠰⣄⠠⡄⠀⠀⠀⠀⠀⢈⠀⠀⠀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡙⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⢷⣿⡀
⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠛⠁⠀⠀⠀⠀⠘⠉⠛
⢻⣆
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢹⡄⠀⠒⠿⢧⡶⠤⣌⠁⢀⣉⠉⠀⠀⠠⠄⠈
⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣿⣿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠻⠿⣿⣿⡿⠟⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣧⠀⠀⢀⣧⡀⠀⠀⢰⡄⠀⣿⣦⣼⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢸⣿⢻⣦⣤⣿⣿⣿⣤⣤⣼⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⠃⣾⣿⣿⣿⡿⠚⢿⣿⣿⣟⢿⣿⡜⢿⣿⣿⣿⣿⠇⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⣮⣴⣿⣿⣿⣟⣀⣄⣠⠉⠻⣿⣆⠉⢿⣶⣝⣿⣿⣿⣧⣤⣾⠿⠃
⢤⣠⣤⣄⣀⣀⡀⠀⠀⢀⠀⠀⠰⣿⣿⣿⠶⠛⠋⢹⣻⣿⡷⣶⣄⠉⢃⣿⣯⣿⡏⠙⢿⣭⣍
⠀⠈⠙⠽⠛⠀⠀⠀⠀⠈⢆⡀⣀⣇⠀⢹⣄⠀⠀⢸⣙⡛⠄⣿⠀⢹⣾⡇⣻⣟⡇⢀⣼⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⡀⠈⠉⠋⠉⠁⠀⠀⠸⣦⡾⣿⡟⢻⡏⠉⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣦⣄⠀⢀⡝⠻⠿⠿⠓⠂⠀⠀⠀⠀⠀⠀⢻⣀⣼⡷⢾⠟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠀⠀⠀⠀⢠
⠀⠀⠀⣄⠀⠀⠀⡰⠃⢸⣟⢹⣿⡧⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠖⠀⠀⠀⢠⡏
⠀⠀⠀⠻⣀⢀⠋⠀⠀⡼⠙⠛⢿⠀⠑⠤⡀⠀⠀⠀⠀⠀⠉⠓⠒⣶⣶⠃⠀⣠⣿
⠀⠀⠀⠀⢨⣿⣆⣤⣾⡇⠀⣠⣾⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠘⠛⠃⠀⣰⣿⡿⠀⢀⣠⠆
⠀⠀⠀⠀⠀⢿⣿⡟⢿⣿⣾⣾⣿⠀⢲⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⣰⣿⠻⡧⢤⠾⠋
⠀⠀⠀⠀⠀⠈⢿⣇⠘⣿⣿⣿⡟⠀⠈⠀⠀⠘⠀⡰⠀⢠⣼⠐⠀⣤⢲⣿⣿⣤⢷⡜
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡻⠏⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⢴⡏⠀⢹⠟⠣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠚⠀⢀⣤⠀⢀⠀⠀⠀⠀⠀⡀⣠⣿⡀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠁⠶⠀⣇⠀⢧⣿⠿⠉⠁

⠀⠀⠀⠀⠀⠀⢠⣐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒
⠀⠀⠀⠀⠀⠀⠀⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡠⣠⣼⠟⠁
⢄⣄⠂⠀⠀⠀⠀⠀⠀⠙⢷⣄⠢⡀⣀⠀⠀⢀⡴⠀⡀⠁⠀⠀⣄⠀⠀⢀⢀⡠⣪⣾⠟⠁⠀⠀⠀⠀⠀⢄⣡
⠈⠙⢷⣦⡐⢄⠀⠀⠀⠀⠀⠙⣿⣞⠁⠀⠀⡽⠶⡆⢸⣿⠃⡴⠶⡅⠀⠈⣳⣝⠟⠁⠀⠀⠀⠀⠀⣂⣴⠟⠋
⠀⠀⠀⠈⠻⢷⣤⡢⠆⠀⠀⠀⠉⠙⠷⣦⣤⢠⣀⣶⢾⣿⢶⣦⣠⢀⣤⡾⠋⠉⠀⠀⠀⠸⢖⢤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠽⠻⣦⣀⠀⠀⠀⠀⠀⣸⣟⣳⣽⣿⢀⣤⠘⣿⣵⣛⣿⡀⠀⠀⠀⠀⣠⣴⠟⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡧⢬⡐⡀⣀⣿⣶⣺⣛⠾⠸⣻⠅⢗⣛⣲⣾⣇⠀⣀⣠⠬⣏⠀⠀⠀⠀⠀⠀⠀⣀
⠀⣶⣶⣶⠢⠤⠄⠀⠠⠜⣤⣴⣾⣇⢶⣶⣿⣟⠧⢴⣰⣶⣦⡤⢘⣿⣿⣄⣶⣦⣶⣤⡴⡆⠀⠤⠶⢖⣶⣶⡦
⠀⠉⠉⠛⠻⠷⣶⣤⣯⣥⣭⣛⡿⣿⣿⣟⣃⠀⢝⣿⣦⠤⣶⣟⠅⢀⣙⣿⣿⡿⣟⣫⣥⣭⣶⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠛⠻⣿⠄⣿⠉⣝⠿⢶⣿⣫⢦⣻⣷⠾⠟⣉⢹⡇⣼⡟⠋⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡗⡿⣧⣹⢟⣷⡿⠟⠀⠿⣿⣟⠿⣡⣾⡗⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠃⢸⣿⢛⠩⢧⣬⣦⣦⣶⢤⠭⢙⣻⣿⠀⢺⣿⡃⢀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⡄⢳⣋⠀⣼⣿⣦⣰⣇⢿⣿⣿⣿⢏⣴⣠⣼⣿⡆⢀⣱⢇⡜⠳⣦⣀
⠀⠀⠀⠀⠀⠴⠚⠉⠀⡤⠿⢮⣿⣴⣿⣿⠿⢿⣿⠋⠁⠀⠉⢻⣿⠿⢿⣿⣷⣴⣯⠾⠷⠄⠀⡉⠓⠦⠀⡀
⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠙⠲⣦⣤⣶⠚⠁⠀⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⣂⡠⢄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡋⣳⢾⠀⢰⣽⠀⠀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠉⢟⣳⣠⢤⣿⣦⡄⠈⢤⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣾⣯⣽⣯⣴⣶⠏⠒⢳⠴⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠿⣿⣿⣿⣿⢿⣿⣹⣧⣴⣆⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⣤⠴⣿⣿⣿⡟⣻⣾⣿⣽⣩⣷⡆⢀⠀⣀⣐⣷⣦⠖⠀⠂⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠟⠛⠓⣶⣶⣿⣏⣈⣟⣉⠳⣟⡟⣿⡡⡏⢬⣁⠹⡿⣿⣟⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⣼⠿⢉⣫⢿⣿⣿⠷⣄⣑⣍⢻⡏⢽⢿⣿⠛⡀⠡⠿⢯⡽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣠⣱⣤⣆⠀⡉⠠⠟⠀⠀⣌⠶⣯⡀⣖⣀⢼⠵⠊⣠⣀⠀⠈⠒⠓
⠀⠀⠀⠀⢀⣤⣤⠴⣤⠀⣶⣿⡿⢿⣿⣯⣻⣟⡁⠀⠀⠈⠁⠂⠀⠘⢿⡿⣍⠰⠂⠘⣿⣅⠂⣀⣦⡀⢐⠶⣀
⠀⠀⠀⠀⠠⠈⠂⠜⠛⢿⣟⠛⠊⠹⣿⣿⣿⠏⠁⠀⠀⠀⠀⢠⣠⠖⡾⠂⠀⢠⣬⢶⠀⠀⣺⣯⠏⠀⠨⣴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠉⠀⠀⢀⠁⠻⠭⣿⠵⠀⠀⠀⠀⠛⢻⣿⢉⣠⡦⠐⢯⠔⠀⣿⢯⠉⠀⠂⢈⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣳⠟⠘⠁⠐⠊⡀⣾⡁⠀⠁⠀⢀⢋⡋⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢀⡀⠀⠠⡾⠿⠟⠁⠀⠀⠀⢘⡽⣏
⠀⠀⣀⣠⣤⣄⣀⡀⠘⠃⡄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢸⠛⠗⢸⣇⣜⠀⢦⠀⠀⠀⠀⠀⠀⠘⠁⠈
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣄⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢠⠤⠃⢀⡟⠁⠀⠛⠂⠠
⢸⣿⣿⣿⡟⡛⢻⣿⣻⣿⣿⣿⠷⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠁⣞⠁⠀⠀⠀⠈⠀⣤⣀⠀⡀⠄
⢸⣿⣿⣿⡟⠉⣿⣿⢛⣹⣗⣓⣶⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠸⠄⠀⠀⠠⢂⣦⣄⣀⠀⠐⠆⠀⠀⠀⢀
⠠⡇⢸⠀⠀⠀⢿⣵⠟⢿⣿⣿⠛⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣆⠀⠘⠠⠛⠀⠻⠷⠦⣈⠋⠿⠏⢉⠋
⠀⠿⠆⠂⠀⠐⣤⣿⣷⣶⣶⠎⠀⢓⡶⣷⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢟⠰⡒⠀⠐⠄⠀⠄⠉⠉⠹⠂⠁
⠀⠀⠦⠆⢠⠀⢻⣿⣿⡛⠁⠀⠀⡘⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠉
⠀⠀⠀⠘⠾⢦⡄⠙⠉⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⢀
⠀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⡶⠶⠶⠤
⠀⠀⠉⠿⢿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⡿⠏⠉⠈⣁⣁⠈⠉⠉⠈
⠀⠀⠀⣾⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣩⣵⣷⣾⣿⣷⣿⣿⣿⣿⣿⣷⣶⣦⡄
⠀⠈⢿⣿⡻⢿⣻⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⠿⠛⠻⠟⠛⠛⠙⠉⠙⠛⠉
⠀⠀⠀⠉⠴⠟⠋⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⠀⠀⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠠⠄⠒⠀⠉⠉⠉⠉⠁⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⢠⠏⠑⢄⠀⠀⠀⡠⠛⡄⢀⡇
⠀⠀⠀⠀⢠⠁⣰⣇⠀⠀⠀⡜⠁⠒⢤⣀⡀⣠⠎⠀⠀⠉⠒⢄⣀⠃⠀⠀⠀⠙⠢⠊⠀⠀⠘⠂⢁
⠀⠀⠀⠀⡌⠀⡙⠁⠁⠂⠶⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢎⣧
⠀⠀⠀⢰⠁⠈⠁⣄⣀⣀⣀⣀⡀⠀⣀⠀⢀⣠⡤⠀⠀⠤⠴⢒⠒⠂⠀⠈⠉⠉⠁⠀⠀⠉⢉⠻⠾⣿⢆
⠀⣀⠰⢇⣀⣳⠚⠉⠉⣩⠏⠉⢁⠜⠉⠀⠈⠉⠓⠶⣤⡤⢀⣈⣁⣚⠂⣠⣤⣤⠶⠶⣐⡒⠿⠟⠒⣹⠂⠣⢀⡀⢀
⣶⠀⢠⡄⠈⣃⠠⠔⠊⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⢍⡉⠀⠐⠒⠐⠂⠀⠛⠃⢭⠍⠁⣉⣥⡜⠃⠀⠀⠀⣠
⠹⢧⣾⣤⡾⡟⢳⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡶⠤⠠⠤⠚⠁
⠀⠀⠀⠀⠐⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠂
⠀⠀⠀⠀⠘⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠊⠀⠒⠠⣄⠀⠀⠀⡠⠒⠀⠉⠀⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠈⢧⠀⠎⠀⠀⠀⠀⠀⠀⢀⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢤⠁⠖⠲⣤⣀⠀⢀⡰⠿⠤⡀⢀⣠⣴⠲⠙⠁⡰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⡤⠄⠀⠀⡴⠓⠂⠩⠤⠴⠶⡿⢻⠶⣦⠀⠐⠿⠟⠳⠶⠶⠾⡟⢺⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⣀⣀⡇⠀⠀⠀⠀⠀⣠⠗⠁⠀⠘⡀⠀⠀⠈⠑⢄⠀⠀⡆⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡀⠀⣳⠒⠒⠒⠂⠞⠁⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠱⢞⡇⢸⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠠⠊⠀⠀⠈⠑⢄⠀⡌⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢃⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠸⠀⢡⠀⠀⠀⠀⠀⢀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠈⢦⠀⠀⠀⠀⠀⠀⠘⢦⣄⠀⠀⢰⡇⠀⠀⠆⠀⢀⠤⠚⠈⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠸⡏⠢⠼⠁⠀⠀⠈⠖⠁⠀⠀⠀⡇⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠀⠀⢀⣀⣱⣄⠀⠀⠀⠀⠀⠀⠈⢆⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⣠⠚⠁⠈⠉⢐⠀⠀⠀⠀⠀⠀⠀⡇⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠑⠒⠒⠒⠂⠒⠀⢸⠀⠑⣄⡠⠊⠁⠀⠀⠰⢆⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡆⠀⠀⢠⡀⢳⡄⠂⠀⠀⠀⠀⠀⠀⠀⡇⢐⣈⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠎⠉⢢⡀⠀⢀⠦⢜⠃⠂⠰⠳⣀⣿⣈⣿⠆⠓⠒⠒⠂⠀⠀⠀⠒⠘⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⠖⠁⠀⠀⠐⣶⠁⠀⠀⠀⠉⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣀⣀⡀⠀⠀⠀⠀⠀⢠⠦⠤⠤⠄⠀⠀⠀⠀⠐⡇⠀⠀⢰⡿⠏⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠉⠁⡇⠸⢀⣠⣶⠶⣄⠀⣀⢄⠀⡇⠀⠀⠈⢣⠔⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠳⠤⠚⠳⣠⠔⢦⣠⠀⠘⠋⢿⣈⡡⢎⠙⠁⠀⠙⡁⠀⠀⠀⠀⠄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠠⠀⠀⠉⠀⠈⠀⠀⠀⠀⢣⡀⣀⢠⠔⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⢀⡀⣠⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠤⠄⠀⠀⠀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠠⠄⣀⣀⣀⣀⠠⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠒⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣰⠰⠄⠆⠖⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠄⠠⠠⣢
⠀⠀⠀⠐⠤⠠⣅⣰⡀⠂⣽⣀⠆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠡⣀⣯⠐⢀⣆⣀⠄⠰
⠀⠀⠀⠀⢀⠖⠩⠋⠁⠀⠈⠙⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠀⡄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠛⠁⠀⠈⠙⠍⠲⡀
⠀⠀⡀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⢀⠀⠐⢊⣀⡄⠀⣅⣙⠀⡄⠀⠀⠀⢀⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀
⠀⠐⢈⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⡐⠀⠀⠀⠁⡹⢋⠏⠁⠀⠙⠏⠻⡁⠀⠀⠀⠂⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡁⠂
⠀⠀⠀⢀⠑⠀⠀⠀⣀⡐⠈⠀⠀⠘⢂⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠐⠃⣀⡀⠀⠂⠀⢠⠀⣀⠒⡀
⠀⠀⠀⠠⢌⠀⠋⠀⠀⠀⠀⠘⠃⢨⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠⡅⠀⠈⠁⠀⠀⠈⠀⠀⡡⠄
⠀⠀⠀⠀⠀⠁⠀⢒⠀⠑⡂⠂⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠈⠀⠐⣔⠆⢄⡒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⡀⠀⠀⠀⠀⠀⠀⢀⠀⠑⢀⠈⠀⠀⠀⠀⠀⠀⠤⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠂⠄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⠀⢐⡃
⠀⠀⠀⠀⠀⠀⠀⣈⣀⢀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡨⡁⠀⠀⠠⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⡀⣨⣅
⠀⠀⠀⠀⠀⡀⠀⣛⡌⣟⢈⡃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⣧⣻⣡⣟⠀⢀⡀
⠀⠀⠀⠀⠀⠤⠤⠤⠤⠧⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠈⠃⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠼⠄⠤⠤⠤

⢀⡀⠀⠀⠀⠀⠀⠀⠀⡄⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠠⣄⣄
⢀⠀⠁⠀⠀⠀⠀⠀⠀⣏⠀⣠⢤⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢀⣀⣠⣤⣶
⠘⠻⣶⣦⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠙⠁⢸⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢧⣄⢀⣀⣀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠉⠿⠶⣤⠤⠴⢾⠁⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠿⠛⠛⡽⠋⠉⠁⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣤⣤⣤⣄⡀⠀⠀⢀⣠⣄⣤⣴⣾⠿⠋⠀⠀⠀⠸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠓⡦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠶⠋⠀⠀⠀⢈⣽⣾⣟⡛⠉⢉⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠭⠷⠦⠶⠶⠚⢻⡏⠀⠈⠙⣻⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠐⠒⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⣴⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⢸⣷⣦⣄⠀⠐⠙⠂⠺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⡉⢀⡀⠀⠀⣀⣤⠾⠃⠀⠀⢸⠇⣠⡖⠁⠀⠀⠀⠀⢀⡴⠋⠉⠓⠒⠋
⠘⠿⢿⣿⣿⣶⣴⣶⣶⣿⣶⡾⠿⣶⣶⣶⠿⠛⠛⠻⣄⠀⠀⠀⠀⠀⠀⡏⠀⣀⡀⠀⠀⠀⣠⣃⣀⣤⣤⠤⠤⠴⠛
⠀⠀⠀⠈⠙⠟⢿⣿⣷⣄⡑⣀⠀⠈⣻⣄⣀⣀⣀⡴⢇⣁⣀⣀⢀⣤⣶⠿⠛⠋⠙⠛⢛⣽⣋⠥⠖⠒⢀⡁⠈⠒⠲⡤⠶⠧⣤⣀⡀
⢸⣄⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣟⠛⠋⣯⣁⠉⠉⠡⠄⠉⠉⠉⢿⣀⣼⡀⣀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠠⣬⠆⠀⠀⠀⠀⠀⠀⠀⠙
⢀⡙⣷⡴⠾⢇⣠⣴⣿⠏⠉⠻⣿⣿⣤⡇⠉⠉⠙⢏⠉⠀⠀⣠⠞⠻⣄⠹⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⢸⣿⣿⣿⣿⡿⠿⣟⠁⠀⠀⠀⠘⠈⡿⢿⣷⣤⣄⣀⡀⠀⠎⠁⠀⠀⢘⣶⡖⣋⣁⡀⠀⠀⠀⠀⠀⣀⣀⡴⠊⠁
⠀⠀⠈⠿⣤⣀⠀⢨⠉⡦⠀⠀⠀⠀⠱⡄⠙⢻⣿⠛⠿⠿⣿⣿⡛⠛⣛⡉⠙⢯⡉⠉⠳⢦⣤⠴⠚⠁⠀⠀⠀⠠⢄⣀⣀⡀
⠀⠀⠀⢀⡴⠉⠙⢦⡉⠀⠀⠀⠀⠀⠀⠁⠀⠀⢻⣄⣀⡀⢀⣉⣿⢾⣅⡀⠀⠈⠷⠄⠀⠀⠙⢷⠒⠋⠁⡠⠀⠀⢲⣽⣿⣿⡆
⢀⣤⢞⣉⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⣶⢯⣬⣉⡉⢫⣄⡀⠀⢨⣿⣄⡀⠀⠀⢀⢼⣢⠼⢞⠉⠛⠒⠢⠀⢸⣿⣿⣿⣿⣤⡀
⠈⠀⣀⠀⠀⠀⠀⠀⢠⡏⠑⢦⡀⠀⠀⠀⢰⠃⠀⠀⡅⠉⠋⠉⠐⠂⠸⣇⣈⣿⣶⣶⣛⠉⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⣀
⠀⡰⠧⠒⢡⠘⣢⠀⣾⠀⠀⠀⢙⠙⠦⠀⢸⡄⠀⢰⠙⣆⠀⠀⠀⠀⠀⠈⠻⣅⠀⠉⢙⡿⠓⠶⠶⣤⡤⢤⠒⠁⠀⠻⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠈⠀⠀⠀⠀⠙⣋⣠⠋⣧⠀⠰⣭⠂⠀⠀⠀⢳⡀⠀⠀⠘⡦⡄⠀⠀⠀⠀⠐⠻⡄⠀⠉⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢀⢾⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⢰⡓⡌⠃⠠⠀⢷⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⢀⣿⡀⠀⡀⠀⠀⠀⠀⢠⣿⣿⣛⡛⠛⢿⣿⣦⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⡎⢱⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢸⡏⠉⠉⢳⡤⠤⠴⠟⠛⠛⣅⠀⠁⠀⠀⠹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⢸⠃⠙⢿⣦⣀⠀⠀⠘⢦⣀⠀⠀⠑⢢⠄⠄⠀⠀⠈⣷⣠⠀⠀⠀⠈⠻⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢸⠟⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢹⡀⠀⠀⠀⠀⢹⠛⠦⡤⠾⠶⢦⠤⡤⠂⢹⡙⠦⣀⡤⠒
⠀⠀⠀⠀⠀⣠⠤⠴⠚⢻⡉⠀⠨⡗⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠤⠤⡴⠃⠀⠀⠀⠀⠁⢸⡀⠀⠀⠀⢸⠆⠀⠀⠳⠄⠘⣇⠀⠀⠀⠙⢦⠄⠀⠉⡗⠂
⠀⠀⣠⢤⣼⠃⠀⠸⠀⢸⡷⠦⢤⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢠⠞⠳⡄⠀⠀⠀⠀⠀⢈⣈⠆⠀⣠⠞⠖⠒⡖⠀⠀⠀⠀⠉⠑⣖⢢⢾⣤⠄
⠀⠀⠿⠞⠧⠀⠀⠀⣠⠟⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⡄⠀⡏⠀⠀⠀⠀⠀⠀⠁⣠⠾⣀⢀⣀⣐⡇⠀⠀⠀⣀⠀⠀⠨⡏⢀⡯⣄
⠀⠀⠀⠀⣀⠜⠉⠟⠉⠀⠀⠀⠀⠀⡹⠤⢤⠤⠠⠄⠀⠀⠀⠀⠘⠛⠀⠀⠙⠀⠀⠀⠀⡵⠒⠟⠁⠀⠐⠀⠀⠉⡯⡙⢶⠚⣹⠇⠀⠀⠈⣎⠓⠻⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠀⠀⠀⠀⠀⠒⠂⠈⠉⠉⠉⠁⠒⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⡘⠀⡴⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⠁⣸⢧⡀⠀⢀⠎⠉⠲⣄⠀⠀⠀⡠⠊⠉⠢⡀⠀⠀⢠⠁⠈⠳⡀⠀⠀⢀⠴⠆⠀⣧
⠀⠀⠀⠀⠰⠂⢠⢻⠀⠈⠒⠎⠀⠀⠀⠀⠉⠒⠊⠀⠀⠀⠀⠈⠢⣤⠃⠀⠀⠀⠘⠦⠔⠁⠀⠸⡔⠋
⠀⠀⠀⢠⠃⢀⡌⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢀⠄⠒⠃⢄⡄⢡⠈⢹⠏⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠂⠐⠂⠀⠒⠀⠀⠀⠀⠒⠦⠤⣴⠚⢣⡀
⣿⠂⢀⡄⠈⣱⠄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡀⠀⢀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢦
⠙⢧⣯⣤⠶⢏⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠃
⠀⠀⠉⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡴⠃
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⣄⢀⣀⢀⣀⠀⢠⣤⡀⣀⡀⠤⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⣿⣶⣤⣼⠛⠳⢟⢿⡙⣅⣀⠠⢶⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢩⣿⡏⠉⡻⠒⣆⠠⠐⠚⣁⢤⣥⠀⠀⠀⡆⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠈⡷⠄⠊⠀⢠⣛⣤⡄⠒⢻⡀⠘⠢⢀⡀⣠⣀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠈⠉⠁⠀⠀⠀⠀⠠⣤⣾⣉⠉⠀⠀⠀⠀⠹⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠾⣤⣌⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠠⢤⡄⣀⠴⢤⣄⣀⡀⠀⠾⣤⣤⠀⠀⠀⠀⣀⣴⠀⢤⡔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣈⣯⠈⣉⡐⠲⠤⠦⡤⠒⠀⢋⣉⣄⣀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⠀⠀⢻⡇⠀⠀⣠⡄⠐⠲⣄⡀⠾⠁⢡⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠠⠼⣏⠉⠁⠀⢀⣀⣀⣬⡶⠾⠓⠛⠒⠒⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠻⠶⠚⠛⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡶⠊⠑⡄⠀⠀⡔⠙⢦⠀⠀⠀⡠⢄⠀⠀⢀⡴⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠲⠊⠀⠀⠀⠳⣤⠊⠀⠀⠣⡄⠊⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠀⠀⠀⣾⢯⠙⣄
⠀⠀⠀⠀⠀⠀⡴⢶⣦⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡆⠀⠀⣀⡀⠤⠀⠒⠁⠈⠃⠀⠀⠛⠦⡞⠈⠑
⠀⠀⠀⠀⠀⠘⣇⠉⣸⠇⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠀⠀⣀⠀⢀⠖⠠⣼⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠚⠀⢹⠁⠀⠀⠀⢠⡖⠋⠑⠶⠋⠑⠤⠊⠑⡆⠀⠀⠈⣄⠰⠤⡰⠉⠁⠋⠀⠀⠈⢄⠀⠀⢀⡰
⠀⠀⠀⠀⠀⠁⠀⠸⡀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⢀
⠀⠀⠀⠀⠀⢲⡀⠀⠉⠒⠂⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠆⠀⠀⠀⠀⠀⡄⠀⠀⣀⡀⠀⠒⠒⠐⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠤⠤⠤⠄⠀⢀⠀⡸⢄⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠶⠀⠠⢀⠀⠀⠀⠣⠀⠑⠄⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⡆⠈⠀⠀⠀⠁⠊⢡⣠⠀⠀⡐⠁⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠛⠓⠁⢀⠐⠄⢐⠉⡅⠀⠀⢆⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⡠⠎⠘⠀⠄⠀⠉⠐⠠⡈⠑⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢆⣀⣀⠤⠒⠩⡀⠀⠀⠀⠀⠈⠢⠀⢀⠈⠐⢄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣶⣶⠟⢳⠂⠤⠤⡀⠀⠀⠀⢢⠐⡄⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣀⢊⣀⠠⠊⠛⠀⠂⠁⠀⠀⠀⠀⠑⠂⠊⠈⠆⠰⠀⠈⢢
⠀⢀⡇⠀⠀⠀⠀⠀⢐⠄⢀⠀⠀⠄⡐⡀⠀⠀⠀⠌⢠⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⢣
⠀⠀⠀⡄⠑⠀⠈⠀⢀⣄⠛⡀⠀⠴⣬⠀⠀⠀⡀⢰⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⠀⡇⠀⠀⠀⡆
⠀⠈⠀⠀⠀⡔⠂⠀⠀⠉⠀⠀⠀⡔⠚⠁⠂⠁⢁⡎⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢸⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠠⠤⠭⠕⢃⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠰⠁⠀⠀⢀⣘⠒⠃⡀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡰⠁⡀⠄⢀⠀⠀⠈⢹⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⢀⠌⠀⠀⠈⠉⡀⡀⠀⠀⡈⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⢃⠔⠁⠀⠀⠀⢀⣈⣔⣔⠑⠚⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠘⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠤⠄⢀⣀⡉⠉⠀⠀⢒⡂⢀⡉⢁⡀⣀⣀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠒⠀⢇⠈⠉⠀⠀⠀⠐⠓⠚⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⢠⡀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⡠⠢⣠⠀⠀⠀⠀⠀⡆⢠⠀⠀⠊⠉⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠉⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠠⠄⠠⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⣄⡀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠈⢱⣏⣀⠟⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⡇⠀⢶⣶⠿⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡀⠀⠳⣶⣮⣿⡗⣤⣤⡄⠃⠀⡀⢡⣶⠖⡎⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢰⣶⡟⣵⣿⣿⢛⣭⢀⣥⣤⣿⡟⠉⣀⡁⠁⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣶⣿⡾⣿⣿⣿⣿⣷⣿⣿⣿⡿⠋⠉⢠⣿⣿⡟⠉⠉⠁
⠀⠀⠀⠀⠀⣸⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣷⣶⣶⣾⣿⣿⣥
⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢀⡀⠀⠀⠀⢀⣴⣿⣿⠟⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣄⣠
⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⣧⠀⠀⠀⢘⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⢸⣧⠀⠀⢠⣿⠋⠠⠀⣼⣶⣶⣦⡀⢀⠔⠁⢰⡇⠀⠀⠀⠀⠘⣿⣿⣿⠃
⠀⠀⠀⠀⠈⣿⠀⣸⡇⠀⢸⣿⣄⠀⣸⠃⠀⠀⠀⣺⣿⠉⠻⣿⡀⠺⣤⣿⡧⠤⠤⡀⠀⠀⠈⠛⣧⡀
⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⢸⣿⡿⠀⠈⠀⠀⠀⠀⠈⢓⣦⣠⣤⡙⣦⣏⠁⡀⠀⠀⠙⣦⠀⠀⠀⠈⠃
⠀⠀⠀⠀⡀⢹⠀⢹⣇⡀⠈⣿⣷⣄⠀⣿⠀⠀⠀⠀⣿⡏⠙⠷⢾⡿⠻⢷⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠀⢸⣿⡇⠀⢹⣿⣿⣿⣿⡀⠀⠀⠀⠉⢿⣄⣄⢸⣿⣤⣄⡀⠀⠀⠀⠸⣿⡄⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠂⢸
⠀⠀⠀⠀⠁⠀⠀⠀⠛⣇⠀⠈⣿⣿⠟⠛⠃⠀⠀⢀⣀⣀⣉⣛⣻⣿⣿⣿⣿⣧⡀⠀⠰⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣼
⠀⠀⠀⠀⣈⡁⠀⠀⣀⢿⡄⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⣰⡇
⠀⠀⠀⠀⢉⠙⠢⠀⠀⠈⢳⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣴⣿⠁
⠀⠀⠀⠀⠈⠁⠰⠟⣄⡀⠀⠁⢄⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠐⠀⠀⣀⠀⠀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠐⢠⣿⣿⣾⣿⣿⠏
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠁⠀⢀⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⡸⣿⣿⣿⣿⠏⠀⣀⣤
⠀⠀⠀⠀⠀⠉⠉⠓⠄⠀⠀⠀⢐⣾⠋⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣟⠁⢘⣿⣿⣿⣿⣿⣿⣿⠇⠀⢻⣧⣿⣿⡿⢋⠴⣪⠽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣰⣿⡧⠄⠀⠀⠀⠀⠻⣿⣿⣿⣿⡏⠁⣿⡿⠉⠻⢿⣿⣿⡟⠁⢀⣴⣿⣿⣿⡿⠃⠑⠉
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠚⠫⠄⠀⢤⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠐⡿⠁⠀⠀⠀⠀⠉⣠⠞⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⣛⣲⡦⣤⣀⣻⣶⣤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠉⢙⠛⠛⠛⠻⠿⠿⠿⠿⠶⠶⠤⢤⣤⠶⠶⠿⠟⢛⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠀⠒⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣀⠛⡻⣃⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣿⣨⣿⣯⣯⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠿⢿⣿⣿⣿⣿⣿⠷⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣭⣭⣝⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠉⠛⠓⣶⣭⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣼⠟⣡⠾⠝⣦⣴⣿⠋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣽⠛⣟⡵⠋⣡⣴⣾⣿⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⡁⠀⠛⢰⣿⣿⣿⣿⠿⢿⣿⣮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠒⠛⠇⠐⠒⣹⣷⣶⣶⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠿⠀⠀⠀⣀⣴⣯⣛⡻⢿⣿⣿⣿⣿⣿⣿⢃⡟⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠⣻⢷⣾⣭⣛⠿⣿⣾⣽⣿⣿⡿⣣⣾⠃⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠶⣯⣟⣿⣾⣿⣿⣿⣯⣾⣿⣿⡇⠀⠀⠀⠙⢿⣦⡀
⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⢟⣩⣶⡿⠟⠛⠛⠋⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡟⠋⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠻⣇⡀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣟⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠷⣊⣩⠷⣤⡀⠀⠀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡟⣱⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣋⣵⣷⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣿⣯⣛⣿⣟⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⣿⠇
⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣠⡴⠶⠚⠛⠋⠛⠉⠉⠉⠛⠓⠒⠶⢦⣤⣀⡀
⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀
⢹⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⣸⠁⠀⢀⡟⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠻⠤⠖⠋⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠤⠀⠠⠤⠤⠤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡀⠀⣀⡤⠄⠀⠤⡀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⡴⠋⠀⠀⢀⠀⠀⣠⠶⠒⠒⠓⠀⠀⠀⠒⠂⠂⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⠀⠀⠀⢾⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡠⠁⠀⠀⠀⠀⠀⢸⡦⠀⠀⠀⠀⠀⠺⠇⠀⠀⡿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⠀⠀⠀⠀⠀⠈⠑⠂⠀⠠⠄⠰⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢀⣠⠤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⢶⠏⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠋⠁⠐⠆⠀⠀⠀⠀⠒⠉⠛⠳⠶⠶⠗⠛⠛⠉⠉⠀⠴⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡼⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠧
⠀⠀⠀⠀⠀⠀⠀⠯⣄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⣤⣤⣤⣤⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢉⣧⠀⢰⣋⠈⠀⠀⣸⣇⡀⠀⢀⣈⣧⠘⣷⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠈⠿⡤⠾⠛⠃⠀⠸⡇⢨⠇⠀⠉⠉⠙⠛⠁⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣀⣀⣤⣴⠇⠘⠷⠤⠤⢤⣄⡀⠀⠀⢀⣤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⠀⠀⣀⡀⠀⠀⠀⡄⢀⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈
⠈⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⢈⠀⠀⠀⠀⢀⠆
⡀⠐⠀⠐⠀⠂⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠠⠀⠂⠈⠐⠀⣀⢄⠀⠀⡀⠒⠐⠀⣀⠀⢠⠄⠀⣀⡀⠀⠀⠀⠐⠂⠁⠂⢀
⠁⠀⠀⠀⡤⡄⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⢤⡄⡄⠀⠀⠀⠀⠀⠀⢰⠀⠡⠈⣉⡀⠀⠀⠀⠀⠀⠀⠐⡃⠰⣓
⠀⠀⠀⠤⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣀⠄⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒⠀⠀⠰⠶⣤⣂⠀⠄⠀⢀⠀⡔⠤⣠⡄
⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠁⠁⠀⠀⡐⣀⠱⣌⠴⠆
⣶⣶⡰⣁⡰⠃⣀⢀⠐⠐⠀⠀⠀⠂⣀⠐⠃⢱⡁⡳⠂⠐⠂⠀⡲⢤⡮⠠⠆⠨⠢⣔⠀⠒⠂⠀⠀⡀⠀⠈⠂⠀⠀⡾⠧
⣿⡿⠇⠀⠀⢾⣁⡷⣿⣿⡟⣿⣙⣧⣿⢹⢿⣷⢿⣘⣧⠀⣿⣹⣇⣿⠏⠠⣯⣍⡏⡿⣟⣇⣘⣧⠀⡇⠀⠀⠀⠀⠀⡇
⣿⣇⠀⠀⡠⣀⠁⠁⠉⠉⠉⠉⠈⠁⠀⠈⠈⠁⠀⠈⡕⢰⣭⣥⣭⢼⡆⠀⠈⠀⠁⠁⠀⠉⠀⠁⠀⠓⠒⠒⠒⠒⣲⠃
⣿⣿⠀⠈⠦⢤⠁⡀⠀⠀⠀⢄⡀⢺⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⡀⠀⠀⢨⠁⠀⡀⠀⢀⡀⠀⠅⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠀⡠⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠐⠈⠒⠀⢆⠀⠀⠀⡠⠈⠦⡁⠀⠀⠀⠈⡠⠄⠀⠀⠀⠈⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⢔⠀⡀⠄⠀⠀⠀⠀⠉⠀⠀⠄⠄⠀⠐⠀⣐⠂⠀⠂⠀⢀⠀⣀⠄⠃⠔⠂⢀⢀⠀⠂⠀⡐⠲
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠠⠀⠀⠁⡠⠆
⠀⠀⠀⠂⠀⠀⢀⠀⠂⠀⣀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠆⠆⠀⠐⠚⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⡈⠍⡈⣈⣀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡤
⠀⠀⣼⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠴⠢⠁⠀⠀⠀⠲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠒⠐⠂⡆⠀⠀⠀⠀⠄⠒⢂⠀⠀⠀⠀⠀⠀⠄⠂⢀⠂⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠷⠀⢀⠀⠀⢀⣀⣴⡲⢢⠇⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⣈⡁
⠀⠀⠀⠀⠀⡀⣀⠀⠒⠠⠔⢚⡛⣙⠰⠒⠒⠀⢀⡀⢄⠀⠀⠀⠀⢈⠚⣧⡇⢠⡀
⠀⠀⠀⠀⣠⣀⢹⡷⢤⡍⠨⠐⠦⠔⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠤⠄⠒⠛⠃⠀⠱⡀
⠀⠀⠀⠐⣗⡅⢨⠁⢵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠀⠀⠀⠀⠐⠲⢶⣄⣠⠜⠦⡀
⠀⠀⠀⢠⣿⡇⠈⠀⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠁⠀⠠
⠀⠀⠀⢨⠘⠁⠀⠀⠄⢤⣌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡏⠁⣒⣠⡦
⠀⠀⠀⠈⢡⠶⡤⠴⠂⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠿⠿⣛⠑⣃⠤⣀
⠀⠀⠀⠠⠃⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠊⠷⠊⠀⠀⠉⠁⠐⠒⢗
⠀⠐⠂⠥⣄⡀⠠⠤⣀⣀⡀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣒⣠⣤⣤⠤⠤⠴⠤⠤⠤⠤⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠃⠀⠀⠀⠁⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢓⢼⡲⢌⠉⢀⠂⠀⠀⠀⠣⠀⣰⣶⣾⠃⠀⠀⠀⠀⠀⢀⡄⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠨⡌⠋⢀⠈⠋⡠⠀⠀⠀⢦⣼⣿⣿⣿⣼⣂⡠⠃⠀⡠⠈⠻⣦⣘⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢣⡈⢀⠀⠀⠠⣺⣷⣭⡑⠿⡏⠀⢀⣤⠞⠀⠀⠀⣀⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⠀⠀⢌⠛⣿⣿⣿⠀⠀⠰⣿⣿⣿⠇⠀⠾⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⣿⡄⠀⠀⠹⣿⡿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠁⣾⣧⠀⠀⠋⢿⣇⣿⡇⢰⣇⠀⠀⠀⠀⣁⡀⢠⣿⣿⣿⣿⣿⣟⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠋⠹⠛⠻⠳⣦⡀⢸⣿⣹⣇⣿⢣⣧⣄⠀⠈⠁⢀⣾⣿⣿⣿⣿⣿⣍⠙⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⠏⠒⠒⠖⠐⣾⣧⣼⣿⡿⣿⡇⣼⡏⠉⠁⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠦⣄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣬⠁⣀⣱⣾⣄⣿⣿⠶⣿⣷⣿⣉⣴⣤⣀⣴⣿⣉⠘⠙⣿⠻⣿⣿⣇⠀⠀⠸⡅⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣏⠉⣷⡾⠻⠦⢶⣬⢿⣿⠀⣿⣿⣿⣛⣿⡟⠉⠩⠋⠈⠀⠀⢻⡄⠀⠉⠻⣧⣄⠰⢄⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣦⣤⣦⣄⡈⠳⣿⠀⣿⢻⣿⣷⣿⡙⠂⠐⣷⣿⠀⠀⢸⣅⠐⠀⠄⠸⣿⣷⣦⣀⠀⠀⠀⠠⡀⢾⠒
⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣸⣛⣿⣿⣯⣤⣤⡙⣿⣿⠃⢠⣿⣿⠏⢿⣎⡙⠂⠈⠀⣿⠀⢰⡆⣿⠁⠀⢺⣷⠻⣿⣿⣿⣷⣄⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⣀⠝⠻⠿⣟⡉⠉⠉⠙⢿⣿⣿⣿⣄⣾⡿⠗⠶⢼⣿⣧⣀⡀⠀⢻⣶⣼⣿⡏⠀⠀⢀⣅⠀⢹⡟⣶⠙⢿⡷⠦⠄
⠀⠀⠀⢀⣠⣾⡧⣀⢀⡀⠘⠀⣉⡀⠀⠀⢘⡛⢿⣿⣿⣿⠓⠦⠤⢼⣿⣿⣿⣇⠀⢸⣿⣿⣿⣿⣤⣠⣿⣿⣷⣌⣿⣾⣷⣀⣀⡄⠀⠀⠀⠀⠀⠀⠰⡖⠂⠲⢒
⠀⠀⠀⠸⣿⣿⣿⣽⢻⣟⠦⠛⠒⢋⠐⠂⣈⡁⠖⢈⣿⣿⣀⠘⢶⠐⣿⠟⣿⣻⣿⣯⣿⣿⣿⣿⢿⡿⠋⠙⣿⣿⣿⣿⡟⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿
⠀⠀⠀⠀⠙⢿⣿⣶⣿⣿⣶⣾⣿⣿⣶⣶⣿⣶⠟⠛⠁⡀⢻⠻⣿⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠈⠛⠈⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏
⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠁⢹⣦⡍⠘⣿⣽⣿⣝⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣾⣾⢻⣿⠋⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣧⣤⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣈⡿⣿⠷⠀⡰⠋⠀⣹⣿⣿⣿⣿⣿⣿⣾⡟⢉⢿⣼⣿⣿⡿⣦⡀⠀⣀⣤⣖⣼⣿⣿⠿⠛⠿
⠀⠀⠀⠀⢰⠿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣻⣿⣆⣐⡀⣴⣣⣿⣿⣿⣿⣿⣿⣿⣧⢆⠃⠹⠋⢻⣿⣿⣧⠀⣿⣿⣿⣿⣿⠃
⠀⠀⢀⣀⣀⣤⣾⣿⣿⣿⣿⣟⣩⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠁⡬⠛⢤⣘⣿⣿⡌⢉⣿⣿⣿⣿⣿⣧⡉⢀⠀⣸⣿⠘⠇⠀⢸⣿⣿⣿⣿⡄
⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢿⣛⣤⣦⣤⣽⢿⣿⣶⣾⠿⠿⣿⣿⣿⣿⣷⡯⢀⢧⣿⣤⣄⠀⣸⢿⣿⣿⣿⣧
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣈⡛⠛⠻⣿⣋⠁⠀⠀⠿⠽⠿⣉⠻⢿⣿⣿⣧⠀⠂⣼⠀⠀⠛⢿⣿⣿⣿⡗⠚⣠⣻⣿⣿⣿⣷⣎⣿⡏⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⣙⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠿⠋⣹⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣷⣤⡔⠉⠻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣻⣿⡝⣿⣿⣿⣿⣿⣾⡿⠿⠟⠛⠿⠛⢿⡇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣸⣷⣿⣿⣿⣿⡿⠿⠛⠛⠋⠙⣠⠀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⠀⠀⠀⠀⠀⠀⠼⣿⢏⠘⢿⡟⣀⠀⠀⠀⢀⢀⣸⣷⣶⣄⠀⣾⡜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⠲⠞⣿⡋⠲⡶⠆⠿⢿⣿⡏⠀⠈⠻⠏⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⢠⠀⠸⠀⢈⣋⢱⣄⢨⣫⣐⣧⣹⡇⠀⠀⣠⠂⠈⢻⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⣠⠴⡈⣶⣿⣦⣰⣧⣄⣽⣿⣏⠙⣻⣿⣿⣤⣤⠟⠀⣀⣼⣿⣿⡿⣿⢧⣤
⠀⠀⠀⠀⠀⠀⠀⠆⠹⣧⡬⠒⠒⠋⡀⠁⢱⢈⣿⣿⣍⣿⣯⣿⣿⣿⣷⣿⣿⣿⡿⠃⠀⢀⠙⢣⣿⣿⠀⠈⣴⡆
⠀⠀⠀⠀⠀⠀⠀⢲⠀⣽⣷⢀⣿⣌⣷⣼⣾⣾⣿⠿⣛⠦⣏⣿⣭⣿⣿⣿⣿⣿⣥⡀⢰⠋⠐⣿⣿⣯⣀⠛⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⠋⠁⠀⠉⠀⠀⠀⢰⣿⣿⣝⢃⠛⡿⢛⠻⣿⣿⣿⣿⡁⢦⡌⣳⣼⣿⡟⢿⡏⠀⢦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠨⢯⠓⡙⢆⡴⣿⣿⣿⡇⣈⣴⣳⣿⣿⠟⠢⠀⠙⢆⠎⠃
⠀⠀⠀⠀⠀⠀⠠⠒⠁⣹⣯⣐⣀⠀⠀⠀⠀⣿⣿⡟⡌⢢⡟⡹⢦⡐⣟⠉⠀⠈⠐⢻⣿⣿⣿⣦⡰⣦⡤⢈⠎
⠀⠀⠀⠀⢠⣪⠟⢻⢿⣿⣿⡍⠛⢷⣄⠀⢠⣿⣿⣟⡄⢣⡜⣤⢂⡝⡿⣀⠀⠀⠀⣸⣿⣿⣿⢜⡿⢟⡱⠁
⠀⠀⠀⠀⠋⠀⠀⠙⠀⢹⡟⠃⠀⠀⠈⠀⢸⢿⣿⣎⠲⡉⢎⢠⡟⡜⣳⣿⣿⣷⣶⣿⣿⠿⠿⠏⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠱⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⡌⢧⠙⡜⢣⣜⢦⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⣴⣿⣿⣿⣿⡶⣿⣴⣷⡾⣶⣿⡿⢛⣿⣏⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢰⣹⡿⣿⡯⠷⡿⢿⣿⢃⣾⢿⣿⢾⣿⣿⣏⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠛⠃⠀⠀⠀⢘⢣⣶⢻⡇⣿⣠⣾⡿⣿⡿⣼⣇⣼⣿⣻⣿⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠀⠀⠀⠀⠈⠸⠿⣾⣿⣿⣿⣟⡁⢻⡿⠛⣼⡿⠛⠛⣹⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣶⠀⠀⠀⠀⠀⣨⡬⣉⢿⣿⣿⣷⡌⠀⠀⢸⣇⠀⠈⢉⣹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⣻⠟⣿⣿⣿⣿⣿⣷⠀⠀⣾⠿⢫⣯⣟⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⠇⠀⠀⢹⣿⣿⣿⣷⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⠀⠀⢸⠻⣿⣿⣿⡟⠀⠀⠀⠈⠳⣨⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⡇⠀⠀⠀⠀⠀⣸⠺⢋⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⠟⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⢸⣿⠇⠀⠀⠀⢀⣰⣷⣶⣿⣿⣿⡇⠀⠀⢀⠀⠀⣾⡿⣻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠀⢀⣤⠄⣈⠉⣁⣿⣿⣿⣿⣿⣤⣤⣿⣧⣼⡽⠟⠉⠉⠛⣿⡇⣠⡀⠀⡀⢠⡔⠁⡼⠁
⠀⠀⠀⠀⠀⠀⢠⣤⣤⡞⠈⠻⠶⠿⠶⠞⠻⠿⠿⠿⢿⡿⠿⣿⣿⣿⣿⣿⣿⡗⠒⣠⣤⢤⡤⣿⣿⣿⠿⣶⣿⣧⣼⣇
⠀⠀⠀⠀⠀⢰⣆⣀⡀⠀⠘⠋⠠⠶⢷⣦⣤⣤⠤⢈⠉⣠⡍⠋⢍⣛⡉⠉⠉⠻⠟⠛⠻⣿⡟⣛⡛⠻⠟⠻⣿⣿⣥⡿⠁⣴⣆
⠀⠀⠀⣤⠂⠀⢿⣿⣇⡐⣤⠰⢠⣾⣀⠀⠀⠀⠀⠀⠈⢤⣄⣀⣀⣀⡀⠀⠀⠀⢢⣤⣄⣄⣬⣉⣩⣤⣶⣿⣿⣿⣿⠇⣼⣼⠃
⠀⠀⠀⣼⣧⣄⣿⣿⣿⣷⣻⡷⠉⢈⣙⢿⣶⣶⣤⣤⣴⣾⣿⣿⣿⣿⣷⣼⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⡷
⠑⢶⣶⣶⣿⣯⣛⣫⣥⣭⣿⣿⣾⣿⣿⢷⣿⣿⣿⣿⣿⣿⣍⣥⣤⣭⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣟⣭⣴⣽⣿⣿⣿⣤
⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣷⣴⣶⣶⣾⣶⣶
⣿⣿⣿⠹⣈⣻⣋⢸⡏⣙⣏⠇⣩⡟⢹⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡟⠛⠉⠹⣷⠑⢰⣖⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣃⣀⣀⡄⣹⡇⣄⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⢻⠉⠁⡀⠃⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣗⠀⠀⢁⠀⠙⠸⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⣄⡀⠀⠀⠀⠀⠀⣠⡆⠘⣿⣷⢶⣿⡄⠀⢀⣴⣯⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢄⠈⡓⡄⣰⠀⣠⣾⣿⡇⣼⡟⣿⣿⠿⡇⣀⣚⣿⠉⢻⣿⣶⡀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣿⣜⠸⠛⢿⣞⡿⡇⣿⢿⣟⣦⣤⠿⣇⢈⣯⣑⣤⣽⡿⣷⠰⡄⠷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⢤⣀⡿⢷⣧⣿⣿⢻⣫⣿⣿⣿⣿⣷⠝⣯⡭⠻⢽⡄⡥⡆⡝⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⣼⠻⢲⣄⠉⠻⣧⣿⣭⣻⣾⣬⣿⣞⢨⡮⠜⢳⣿⣿⣿⣠⢞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣴⣤⣿⣿⣷⣦⠉⣛⢻⣿⢉⣿⣉⣿⣷⣾⣿⣿⣿⢿⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣤⣌⡳⡽⣏⣟⣿⡿⣿⣿⡏⠀⣧⠀⠚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣛⠋⠹⣿⣿⡾⣿⣿⣿⣿⣿⣎⠛⢿⣿⣿⣿⣿⣿⣧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣯⡟⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣻⣷⣠⣍⠟⠿⢿⣿⡏⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⠔⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠙⢿⣶⣤⡬⠀⢠⠏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠄⠑⠿⡇⢰⣇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⡻⢿⣿⡿⢋⡤⠙⢿⣿⣿⢇⡄⠀⠈⠑⢼⡟⠲⠀⠒⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣯⣿⣿⣿⣿⢻⣿⣿⣻⣿⣿⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢟⣥⣿⣟⣿⢿⣟⣿⣿⠿⣿⣬⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣷⡇⣨⣿⡇⢸⠿⠁⠘⣹⠀⣿⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣯⣿⡿⢻⣿⣿⡇⣾⣿⣤⣾⡿⢰⣿⡿⡿⣻⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⢻⠁⠟⠙⢿⡇⡁⣿⣿⣿⢁⣸⠃⡇⡀⢉⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡂⠙⠿⣿⣿⣿⣿⣿⣿⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣴⣫⡦⣀⠀⣀⣀⣄⠀⠈⠻⣷⠈⡹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠿⠿⠛⠻⣷⢀⡴⠛⠒⠻⢸⣷⡄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣰⣿⣹⣿⣿⠛⠁⠀⠀⠀⠀⣠⠿⠋⠀⠀⠀⠀⠘⠀⣿⡆⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⢳⣍⡒⠴⢿⣿⡿⢏⣟⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢰⣠⣿⡿⠖⢊⠥⡞
⠀⠀⠀⠀⠀⠀⠀⢳⡈⠒⢄⡈⢻⡘⠋⣹⣿⣷⣦⡀⠀⠀⠀⣰⡾⢿⣍⠙⢢⠏⢁⠴⠊⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠸⠺⡇⠀⣿⣄⣰⡾⢿⡄⠀⣼⠷⣄⣴⣾⠂⢠⡜⠃⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆⠀⠸⣧⠀⠉⠛⠂⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⣠⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣀⣙⣆⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣴⣅⣀⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⢷⣤⠞⠀⠀⠰⢹⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⣄⢳⠀⠀⠶⠶⠆⠀⠀⢀⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣾⡁⠠⠟⠛⠃⠀⣰⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⣀⣀⣀⡴⠁⢸⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡇⢣⠘⢿⣿⠟⠻⠁⠀⠘⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢤⡀⠈⠏⠀⠀⢀⡠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⡿⠿⠿⢿⣿⣿⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⢰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡇⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠇⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣾⣯⣤⠝⣳⣦⡀⠀⠀⠀⠀⠀⣰⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣥⡴⢿⡿⣿⣭⣿⣿⣯⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡙⠻⠷⢻⣌⠻⣿⣿⣿⣿⣶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣰⢲⣿⣿⣗⠮⣝⣿⣿⣿⣿⡟⢿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢼⣿⠒⣿⣿⣶⣍⡛⢿⣛⢿⣆⠑⡍⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠄⠀⠒⠻⣿⣿⣷⣾⣿⣿⣤⣼⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⣠⡴⢀⣴⣿⠛⣭⠖⢀⠠⠼⣧⣽⣿⣿⣿⣿⣷⡘⡛⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣧⣾⣿⠇⢀⣸⣦⣿⣼⣮⣿⣿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣿⢀⡀⣋⣩⣿⣿⠀⠀⢉⡸⣿⣿⣿⣿⣿⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠹⢻⣿⣿⣿⣿⣿⣿⣧⡐⠟⣣⣿⣿⣿⣿⡏⣿⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⢿⣿⣿⣿⣿⣿⣿⠋⠛⠀⢨⣿⣿⣿⣿⡇⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠺⣿⣿⣿⡿⠛⢿⣟⠀⡷⣄⣡⢻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠈⠟⠛⠁⠀⠈⠉⠈⠱⠈⣿⠙⢰⣿⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⡞⣿⣿⣿⠶⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣯⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠿⣿⠁⠀⣶⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣇⢳⣄⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣌⠁⠙⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠈⠓⠦⠀⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠀⠒⢦⡀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⢷⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠈⣧⡀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣷⡄⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⢹⡇⠙⠈⠀⡰⠀⢨⣦⣶⣶⣶⣶⣤⣲⢯⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⠈⣯⠀⠀⣰⣗⣀⠜⣇⠀⠀⠀⠀⠂⠙⠤⠉⠉⠉⠉⠉⠉⠛⠛
⠀⠀⠀⢠⣤⣶⠶⠿⠛⠛⠙⡗⠀⠀⠀⠀⠀⠹⣆⢠⡏⢹⡿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠉⣺⠇⠸⣷⣶⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣰⡀⠀⠋⠀⠀⣿⠏⠀⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⡟⠷⠀⠀⢀⠀⠘⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⢸⡇⠀⢄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⠃⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂

⠀⠀⣀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣯⢽⣟⡛⣿⣛⡛⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠳⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠐⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠄⠀⢀⡠⠤⠖⠒⠒⠚⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⠞⠁⠀⠀⠀⠀⣀⣠⠤⠤⡖⠒⡦⠤⢼⡶⠋⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡁⠀⠀⢀⡤⠚⢏⡼⠈⠢⣄⣿⢢⠃⠀⢎⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⠉⠀⠈⠙⠒⠋⠀⠀⠈⠀⣤⠀⢠⡄⠀⠀⠀⢨⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠔⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⣆⠈⠀⠀⠁⠀⠀⠁⠀⠈⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⡍⠒⠒⠤⠤⠔⠒⢲⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⠋⠙⠢⠊⣑⠒⣏⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡜⠀⠀⠀⠈⣚⣀⣸⡀⠹⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠶⠯⡀⢰⣍⠙⠒⠒⡏⡎⢲⢳⠷⢶⣃⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠈⢉⠇⠉⠉⠙⠑⠬⣵⠎⠤⠤⢬⡇⣹⢦⡀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⣄⣀⣀⣞⠀⠀⠀⠀⢀⠞⢵⣶⣶⣄⠊⠍⠳⣄⢹⠆⠀⠀⢀⡿⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣠⡍⡿⠘⠲⢤⡤⠖⠧⣄⡀⠈⠙⠿⣝⠶⣅⠊⠙⢶⠤⢖⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠂⠘⢳⣌⣳⣴⡫⠖⠉⢳⡄⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢂⠎⠉⠁⠀⠀⡀⠀⣧⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠁⠀⠀⠀⠀⡠⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⢤⡤⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠤⠖⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠉⠲⣄⠀⢀⣤⣤⣄⠀⠱⠀⣠⠔⠉⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⢑⡆⠀⠀⠀⠳⣼⡋⢙⢿⣇⢀⠜⠁⠀⠀⢠⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢣⠀⠀⠀⠀⠀⠈⠃⡀⢰⣿⠁⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⣀⣀⣀⣀⡀⢰⡃⡄⢻⣇⣀⣀⣀⣀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠁⠜⠁⠖⠉⠉⠑⠠⣀⠀⠀⠠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⡰⠀⠀⢸⡀⣀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⣠⠇⡈⣀⠀⠀⣃⡀⢋⢧⡀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⢊⡡⠀⢉⣿⠀⠀⣿⣻⣿⣶⣍⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡇⠘⢿⣿⣿⣿⣿⠙⢦⡈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣻⣿⣷⠰⣾⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣾⣿⣋⣙⣿⣿⡿⠟⣀⡍⢻⣽⣿⣿⣛⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣽⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠳⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣸⣧⠀⠀⠀⠀⠀⢹⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡷⣤⣽⠿⣿⣼⣀⣧⣤⣤⣾⣿⣯⣤⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⣉⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣐⡻⣿⣿⣼⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⣿⠟⠛⠳⢿⣿⣄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⢽⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣯⡶⠲⣤⠑⣿⣿⣦⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⢮⣿⣿⣿⣧⢀⣀
⠀⠀⠀⣠⣴⣿⠏⠃⡼⢻⣿⢿⣿⣷⣆⠈⢷⡘⣿⣿⣄⠀⠀⢠⣿⣿⣿⣿⣿⡿⠁⣠⣮⢿⣿⣿⢫⣿⣿⣿⣷⣦⡀
⠀⢀⠞⢁⡾⠓⠩⠞⠁⣼⣸⣺⣟⣿⣯⣇⠘⣧⣿⣿⢸⠀⠀⣼⣿⣿⣿⣿⣿⠃⣰⣿⣿⣿⣻⢻⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⣾⢄⣾⢖⣩⣖⣥⣾⣿⡿⣾⢩⣿⣼⣽⠀⢻⣿⣿⢯⡇⠀⣿⣿⣿⣿⣿⡟⠀⣽⣿⣿⣿⣗⢌⢻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣿⣾⣷⣿⡿⠿⢿⣿⣿⣾⣿⣿⠙⡻⣿⠀⢸⣿⡟⡜⡄⠀⣿⣿⣿⣿⣿⡇⠀⣿⢿⠋⣿⣿⣷⡷⣿⣿⡿⢿⣿⣿⣿⣿
⠀⢛⣿⡠⠤⠞⠛⠋⠉⠛⠿⠿⠋⣢⠀⠙⠀⢸⣟⡜⣱⡇⠀⢿⣿⣿⣿⣿⡇⠀⠛⠈⣄⠹⠿⠿⠟⠈⠛⠛⠯⠥⣌⣿⠟
⠀⠘⠟⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠃⠁⠀⠂⣼⢏⡴⣻⠇⠀⠸⣿⣿⣿⣿⣇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠃
⠀⠀⠈⠀⠊⠵⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣯⢞⠜⡽⠀⠀⠀⢻⣿⣿⣿⣿⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠭⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡡⣣⣾⠃⠀⠀⠀⠈⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠐⢤⣤⠀⠀⠀⠀⢸⠀⠀⣼⢋⣼⡿⠃⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣆⠀⠀⡇⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡸⠀⣼⣵⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣆⠀⠇⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

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

⠀⠀⠀⠀⢀⣤⡶⠿⠿⠿⠿⠷⣶⣤⣄⡀
⠀⠀⠀⣰⡿⠋⠀⠀⣄⡀⠀⠀⠀⠀⠉⠛⢷⣦⡀
⠀⠀⣰⣟⣠⣤⣶⣶⣤⣽⡄⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⣴⡟⠛⠛⢿⡅⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⢿⡄⠀⠀⣸⡇⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠈⠛⠷⠿⠋⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠹⣧
⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⣀⣤⣴⡶⢷⣄⠀⢰⡟⠉⠉⠉⠉⠛⠻⠿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⣿⣥⡶⠟⠋⠉⠀⠀⠈⠻⣷⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄
⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠰⠶⢤⣀⣾⠇⢀⣤⡄
⠀⠀⠀⠀⠹⣧⣀⡴⠆⠀⠀⣀⣤⣶⠾⠟⠛⠛⠛⠛⠛⠛⠻⠷⢶⣤⣄⡀⢸⣿⣴⠟⠹⣷
⠀⠀⠀⣰⣦⣽⣯⠀⣠⣴⠟⠋⢁⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠈⠉⠻⡿⠛⠁⢀⠀⣿⡄
⠀⠀⣰⡟⠈⠙⠿⣿⡟⠁⠀⣼⡟⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⢻⣧⠀⠀⠀⠀⠀⣼⠇⢸⡇
⠀⠀⣿⠃⢾⣆⠀⠈⠁⠀⠀⠈⠀⢰⣶⡄⠀⠀⠀⠀⠀⢀⣶⣦⠀⠁⠀⠀⠀⢀⣴⡟⠀⣸⡇
⠀⠀⢿⡆⠈⠻⣦⣄⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠈⠉⠀⣠⡿⠁
⠀⠀⠘⢷⣄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠉⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⣤⣶⠾⠟⠁
⠀⠀⠀⠀⠙⠻⠷⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⡴⠋⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠉⠛⠋⠉⠉⠉⠙⠛⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠷⣶⣤⣀⣀⠀⠀⠀⠀⣀⣠⣴⡾⠟⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡅⠀⠘⣿⣍⠛⠛⠛⠛⠛⢋⣿⠟⠀⠀⣿⠛⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠻⣧⡀⠀⠈⠛⠷⣦⣄⢀⣴⠟⠁⠀⠀⣼⠟⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠙⠿⣦⣄⣀⠀⠀⢹⣿⠁⠀⣀⣤⡾⠋⠀⠀⠀⠀⠀⠈⠹⣷⡄
⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣿⡿⠟⠛⠉⠀⠠⣄⠀⠀⠀⠀⠀⠀⢈⣿⠿⣦⡀
⠀⠀⠀⢀⣀⣿⠃⠀⠀⠀⠀⣠⣴⠋⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⢿⣶⣄⠀⠀⣰⡿⠁⠀⢈⣿⡀
⠀⠀⢠⡿⠛⠿⣦⣄⡀⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣿⡙⢷⣾⠋⠀⠀⣰⡿⠋⠻⣷⣄
⠀⠀⢾⣇⡀⠀⠀⠙⠻⣿⣥⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⣧⠈⢿⣤⣀⣾⠏⠀⠀⠀⠈⢻
⠀⢠⣾⠛⠛⠷⣦⣄⣀⣿⢻⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠉⣿⡇⠀⢷⡀⠀⠀⢸
⠀⣿⠃⠀⠀⠀⠈⠙⣿⡇⢸⣇⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠸⣷⣄⣾⠿⠷⠾⠟
⠸⣿⠀⠀⠀⢠⣧⠀⢸⡧⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠈⠉⠁
⠀⠻⢷⣤⣴⡿⠛⠿⠛⠁⣼⠿⣧⣄⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⣠⣾⠟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠻⢶⣤⣄⣀⣀⣻⣇⣀⣠⣤⣶⠟⠋⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠈⠉⠙⠛⢛⣛⠛⠉⠁⠀⠀⠀⠀⣀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢶⣶⣤⣤⣀⣀⠀⢀⣾⠿⣦⠀⠀⣀⣤⣶⠾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠉⠉⠛⠻⢿⣿⠀⣿⡿⠛⠋⠉⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⣀⣼⡇⠀⠸⣷⣀⣀⣀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠛⠛⠋⣿⡇⠀⢸⣿⠛⠛⠛⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣄⣠⣼⠟⠀⠀⠀⣿⡇⠀⢈⣿⠀⠀⠀⠸⣧⣤⣠⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣉⠉⠀⠀⢀⣠⣤⣿⡇⠀⠠⣿⣤⣄⡀⠀⠀⠉⢉⣽⠟

⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠆
⠀⠀⠈⢻⣿⣿⣿⣿⣍⠙⠟⡈⢀⠀⣥⡀⠀⠀⢀⠀⠀⠀⠆⡀⠀⢀⠆
⠀⠀⠀⠔⠹⠉⢻⣿⠟⣀⠀⠀⠈⡀⠻⣣⠐⠀⠀⢀⠈⣤⣀⠁⠃⠸⣠⣤⣶⣶⣀
⠀⠀⠀⠀⠀⢠⣿⣿⣄⡉⣐⣂⣾⣷⣌⠡⣀⢀⢈⠼⣀⠋⠙⠈⢲⠸⡿⠉⠃⣼⣇
⠀⠀⠀⠀⠀⠀⢿⡟⠛⠋⠋⠉⠛⠉⢉⣘⣉⠉⠩⡾⠉⠀⠀⠈⢷⠖⠉⠋⠉⠉⠁⠙⠃⡀
⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠈⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠉⠁⠉⠁⠈⠉⠈⠉⠀⢸⣷⣶⣄
⠀⠀⠠⠀⠀⠀⣼⠃⠘⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠂⣹⣧
⠀⠀⠀⢀⠀⠀⢉⠀⠂⠀⠀⢠⣧⣤⣶⣦⣶⠀⠀⠐⠀⠀⠀⠁⢠⣰⠶⠿⣶⣶⡂⠀⠠⣿⠆⠙⠁
⠀⠀⢠⠀⢲⡦⠀⠀⡈⠃⠀⠿⢿⠋⠭⠩⡗⠀⠁⠀⠀⢀⡀⠀⠶⣿⠃⠄⡹⠙⠷⠀⠀⡔
⠀⣨⠁⠈⠀⠃⣠⠀⡁⠈⠀⢀⡴⢦⣆⠘⢿⣶⡀⢠⠤⠀⠀⢀⠀⠟⠁⠀⠀⠑⠀⠀⢀⡘⠀⠀⠀⠀⣠
⠀⠀⢁⠀⠀⠠⠛⢀⡁⢀⣤⡞⠀⢀⡈⣿⡸⢟⠟⣀⠀⢠⠄⢀⠉⠀⠘⣄⡀⠀⠀⠀⠠⣁⢀⡀⣰⣿⣿
⠀⠤⠈⠀⢚⣠⣤⠀⠁⣼⣿⣷⣶⡟⠀⠀⣿⣾⣾⡇⠀⠀⠀⠀⠀⠠⠀⠀⠛⠀⠃⢠⠂⣿⣿⣿⣿⣿⣿
⠀⠐⠀⠀⢈⡉⣠⡶⣾⣿⣟⣟⠛⣡⠀⠘⠛⠋⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢛⠟⠀⣿⣿⣿⣿⣿⣿
⠀⢀⠙⠃⡬⣿⣿⢵⣿⣿⣿⣇⣀⡤⢀⣦⠶⠀⢀⣄⠤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣜⢀⡀⣿⣿⣿⠿⣿⣿
⠀⠀⢄⣰⢠⣯⠛⠏⠁⢘⣿⣷⣾⠁⣤⣤⣼⡆⢠⣤⣤⠀⠀⢀⡀⢀⠀⠀⢊⣜⢉⠁⠀⡿⠋⠀⠀⠸⠋
⠀⠀⠈⠙⢸⣇⡄⠀⣠⣾⣿⣿⡿⠜⣻⣷⣿⣿⢾⢿⣏⢐⣲⡄⠻⡈⢃⠀⠨⡟⠀⠀⠀⡅
⠀⠀⠶⠾⡼⠿⠷⠐⠉⠙⠛⢿⣓⠋⠙⠉⠛⢷⡈⠑⠆⠀⠙⠿⠃⠍⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠖⠀⠀⠁⠀⠒⠀⠀⠀⠁⠀⠀⠀⠀⡀⠣⠀⠀⠀⠀⠀⠀⠰⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡴⠀⣀⠀⠀⠀⡀⢀
⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠁⠀⠀⢼⠁
⠀⠀⠋⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⣴⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⢈⠀⠀⠘⢨⣄⣀⡀
⢀⡖⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡸⢦⡴⠋⠀⠀⠈⠀⢠⠶⢇⢀⡈⣶⠀⢸⠋⠀⠀⠀⠈⠉
⠘⠃⠀⠀⠠⠀⢄⣠⠄⠀⠀⠀⠀⠠⠀⢈⠶⠄⠀⠀⠀⠀⠀⡉⠀⠀⠀⠘⠉⡃⢀⣬⡀⣀⣀⡀⠈
⠀⠻⠞⠋⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠉⠙⠃⠀⠀⠈⠉⠉⠈⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠘⠒⠶⠶⠤⠦⠰⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣄⢀⡀

⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠴⠂⠈⠉⠉⠉⠉⠐⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦
⠀⠀⠀⠀⠀⠀⠀⢀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⡌⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢰⠁⣼⠃⠀⠀⠀⢀⠦⣀⠀⠀⠀⠀⠀⢀⡰⢦⡀⠀⠀⠀⠀⢠⠋⢦⡀⠀⠀⠀⠀⢀⡀⠀⢸
⠀⠀⠀⠀⠀⠐⠃⢰⡟⠑⢤⡀⠠⠎⠀⠈⠑⠢⣄⢀⡠⠊⠀⠀⠙⠢⡀⠀⢠⠃⠀⠀⠑⢄⠀⢀⠔⠉⢡⠀⣸⠆
⠀⠀⠀⠀⢀⠃⠀⡆⠓⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠲⠎⠀⠀⠀⠀⠈⠓⠁⠀⠀⠀⠛⢡
⠀⠀⠀⢀⠎⠀⢰⠀⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⢀⠖⠉⠛⠢⣴⠂⡞⠀⡹⠌⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠁⠉⠈⠁⠉⠀⠉⠈⠉⠀⠈⠀⠀⠀⠙⠳⠦⢾⠛⠉⠆
⣿⡃⢀⣤⠀⢈⡶⠁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄⠸⣆
⠙⢷⣾⣥⡶⢾⣁⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈
⠀⠀⠉⠉⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡞
⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣄⠤⠤⣤⣤⣀⣰⣿⣦⡤⠄⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣷⣼⣿⡷⠶⠎⠀⠙⢫⡹⠆⢶⠤⠀⠘⠟⠁⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⣿⢯⠀⣨⠟⢹⣧⠐⠊⠙⣥⠶⢶⡄⠀⠀⢸⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡀⣿⠖⠈⠁⠀⠼⣭⣤⣖⢉⢻⡄⠀⠑⠢⣄⣀⣤⣤⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠉⠃⠀⠀⠀⠀⠀⠠⣴⣿⣉⠉⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠠⢶⣄⣈⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⢀⣀⣀⣀⣈⣆⣀⡀⠀⠀⠀⢶⣈⣭⠀⠀⠀⠀⠀⢀⣆⢀⣀⡠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⢻⡋⠉⠑⠶⣦⣠⣥⣠⠴⠆⠻⠉⣣⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠉⠉⠹⣷⠈⠉⠁⠀⠀⢠⣬⡄⢀⢠⡟⠙⡁⠁⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡀⣿⡖⠒⠂⠉⠁⠀⠀⠙⣃⣤⣥⣤⡧⠠⠤⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠹⣧⣀⣤⡤⠶⠚⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣠⢤⡀⠀⠀⠉⢉⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠀⠀⠙⣄⠀⡔⠁⠀⢳⡀⠀⢀⠜⠓⢆⠀⢀⠔⠁⠀⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⣶⠁⠀⠀⠀⠒⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⢸⡿⠇⢙⣄
⠀⠀⠀⠀⠀⠀⢠⠶⣶⣆⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢶⠀⠀⢀⣀⠠⠤⠀⠛⠁⠈⢇⠀⠀⠈⠳⢴⠋⠈⢳
⠀⠀⠀⠀⠀⠀⢹⡀⠁⣽⠂⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠉⠁⠀⠀⠀⠀⢀⠶⢄⣸⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⢰⠇⠈⠛⠁⠀⠀⠀⠀⣤⠚⠉⢦⠔⠉⠢⣤⠚⠲⣆⠀⠀⠈⢃⠀⡤⢀⡴⠉⠑⠊⠀⠀⠈⣅⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠈
⠀⠀⠀⠀⠀⠰⡄⠀⠈⠢⠤⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠠⠤⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠆⢀⣀⣀⣀⣀⡀⠴⠖⠀⠀⠂⠀⠀⠘⠁⠀⠀⠀⠀⠀⠘⠦⠴⠖⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣏⣙⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠛⠛⠋⠙⣿⣿⣿⡻⣖⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠵⣦⣤⣤⠆⢹⣿⣿⣷⣌⡲⣄⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣠⣀⣦⢠⡟⣿⣿⣿⣿⣿⣿⣧⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠤⣤⡖⠒⢦⣴⠿⣾⣿⣿⡾⢷⡿⠾⠛⠛⣿⣿⣿⡷⠊⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡘⢦⠙⢦⡈⢣⣀⣼⢿⠦⠈⣿⠁⣠⢾⣿⢤⣤⠞⠋⢉⡏⢀⡚⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣑⣄⣱⣶⠋⢳⠞⠀⣀⣉⠈⢤⠔⠛⢶⠉⣿⣶⠋⡰⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣈⣹⡟⣸⡀⢸⠀⠀⠁⣸⠀⠀⠀⠀⠀⠀⡿⢁⡜⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣽⠏⣸⣿⡷⣾⡉⢉⡛⠟⠉⠀⡈⠉⣿⠟⣠⣿⣍⡙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠾⣿⣿⣿⣿⣧⣸⣇⠉⠢⣼⠀⠠⢴⣾⣏⣴⣟⠿⠳⢭⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣽⠿⣥⣤⣭⣭⣭⣙⣷⣟⠶⠞⣻⡯⣿⣿⣻⡄⢰⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣤⣤⣤⣤⣤⣤⣼⣿⣯⣝⣳⣞⣿⡁⠄⠹⣧⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⢻⡿⠿⠛⡻⣿⣿⣻⡿⠟⢉⣠⣴⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠻⣷⣾⡿⡓⢈⣿⣿⣿⠿⠛⠋⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡉⠻⣿⣹⣯⣤⡜⠃⠀⢸⠃⠀⠀⡠⠞⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⡀⠉⠛⣿⠓⠞⠉⡇⠀⠀⠸⡇⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠛⠛⢶⣇⠀⠀⠀⡇⠀⠀⠀⣷⠄⠀⠀⠀⠈⠛⡆



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠀⠾⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢠⡀⢉⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢀⡴⠋⠉⢳⡀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⡀⡀⠈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢝⠃⣾⡀⠀⠀⠀⢱⣿⣀⣀⠐⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠧⢐⡒⠙⢒⣛⠛⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⠀⠀⣾⠿⠃⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣿⣅⢹⣎⡀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢒⢢⣭⠤⡆⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢶⣿⠉⠉⠿⠛⠛⢤⣿⢿⡷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢲⠂⠉⠀⠀⠉⠱⡗⠂
⠀⠀⠀⠀⠀⠀⢀⠎⠁⠀⣠⠉⢀⠀⢀⠀⠘⠈⢹⣇⢿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⢟⠀⢲
⠀⠀⠀⠀⠀⢀⠎⢀⡾⠀⠅⠶⠀⠀⠘⠇⠘⠇⠀⠈⠘⡄⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠈
⠀⠀⠀⠀⠀⣎⣠⡞⠁⠀⡆⢠⠀⢒⡀⠐⠂⠀⠀⣦⣾⢾⣷⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠀⠀⢀⡇⠠⠐⠃
⠀⠀⠀⠀⠘⢿⣉⡙⠲⢄⡀⣦⣀⢩⡍⠀⠰⠄⠀⢸⡈⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⠀⠐⠈⠁⠱⠀⡆⠐
⠀⠀⠀⠀⠀⠀⠈⠉⠑⠦⠍⠁⣿⣷⠀⢀⡀⢠⣼⡴⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠛⠘⠓⠁⠉⠀⠈⠳⣌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣀⠀⠀⠀⡀⢀⢹⡄⣈⡳⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠘⣿⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡜⠀⠀⠃⠠⠀⠀⡇⢸⠀⡇⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⡇⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⡼⠁⠀⠀⢀⠃⠀⠀⡇⠀⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣧⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢉⡔⠁⡴⠠⠁⡼⠀⠀⢀⠁⠀⡇⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⣿⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣏⠎⠀⡐⢡⠁⢠⠁⠀⠀⢰⠀⠀⠇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣞⠋⠀⠜⢀⠆⢠⠃⠀⠀⠀⠈⠀⠰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢺⠆⢄
⠀⠀⠀⠀⠀⠀⣼⠏⢀⠈⠀⠎⢠⠃⠀⠀⢠⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢀⠜⢹⢠⠂⠀⡜⢰⠃⠀⠀⢀⠆⠠⠀⠀⠿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠺⢄⠫⠆⠀⠜⠀⡇⠀⠀⠀⠘⢀⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡈
⠀⠀⠀⠀⠀⠈⣧⢠⡆⠀⢸⠃⠀⠀⠀⠃⠸⠀⠀⠀⡆⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠃
⠀⠀⠀⠀⠀⠀⢻⣸⡀⠀⡾⢠⠀⠀⢸⠀⠇⠀⠀⠀⡇⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠹⣧⢼⣇⣀⠀⠀⢸⣀⣸⠀⠀⠀⠅⠀⣰⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢴⠀⠉⠉⣅
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠺⢏⠉⠓⠛⠉⠉⠉⠢⠄⠔⢺⠟⢿⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠖⠒⢿
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣼⡦⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣸⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠚⠁⠈⡄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣼⣶⡀⠀⠀⠀⠀⠀⣴⣧⣴⣿⡟⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢉⣺⣧⣄⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠈⡟⠛⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡞⠿⣤⣾⣿⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⠻⣙⣀⣼⣮⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠿⣶⣼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢻⣿⣿⣽⣿⣿⡟⣙⣶⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⣿⣿⣿⣿⣿⣿⣿⣗⣶⣶⣶⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣻⣿⣀⣾⡿⠿⣿⣿⣭⣙⣿⣿⡗⠺⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠺⣿⣿⣿⠇⠀⣿⣿⡽⣿⡿⢏⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⣿⣶⣾⣿⣿⣷⣾⣸⣼⡿⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢟⣿⠈⠛⠛⠛⠋⠁⠀⠈⠙⠚⠻⣿⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣯⡽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣬⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠈⠉⠉⠁⠒⠀⠀⠀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡜⠉⠢⡠⠊⠀⠡⣀⠴⠊⠈⠢⣀⠤⠒⠉⢄⢀⠼⡆⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣀⣀⣀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡳⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠙⠍⣷⣧⠜⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⢡⢣⣜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠧⠄⠉⣉⣉⡉⠁⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣑⡄⠀⠀⡘⠀⠀⠀⠀⠠⠂⠉⠢⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣉⡀⢴⡀⢠⠁⠀⠀⠀⠀⡆⠀⠀⠀⠃⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢒⠈⣋⣀⣹⠀⠍⠃⢿⠐⠒⠂⠈⠉⠀⠀⠀⠀⠂⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠢⢄⠑⢢⡀⠈⣧⡅⠀⠘⠀⠀⠀⠀⠀⠀⢀⣀⣠⢃⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡀⠀⠒⠤⡀⠑⠄⠑⠈⡲⢂⠀⠀⣣⣠⡤⠐⠂⠉⠀⠀⠀⠀⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠀⠉⠒⠢⠈⠀⣀⠔⠈⠀⠹⡔⠊⣹⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢄⠉⠑⢆⡀⢀⠜⠀⠀⠀⠀⠀⠛⢯⣬⢀⠔⠱⡄⠀⡠⢄⠀⡠⠊⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣀⠀⠉⠑⠄⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠊⠀⠀⠛⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡀⠙⠢⡄⡔⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠰⡀⠑⠲⡀⡜⠀⠀⠀⠀⠀⠀⡍⢻⠆⠀⢸⢀⣤⣄⠀⠘⡤⠤⠤⠤⠤⠤⢧
⣀⠠⠤⢤⠒⠒⡈⢩⠀⢢⠈⢆⠈⢆⠀⡔⠀⠀⠀⠀⠀⠀⡈⠀⠁⠀⠀⢸⡞⢬⠜⠑⠚⣄⠤⣀⠤⣀⡠⣘
⢡⠘⡄⠀⢇⠀⢣⠀⢳⠈⠂⠈⠇⢀⠎⠀⠀⠀⠀⠀⠀⠀⢠⠀⠓⠤⠠⠼⠀⢰⠀⠀⠀⡇⠀⠀⠀⠈⠀⠈
⠀⢆⢸⠄⠨⠀⠈⠀⠙⠀⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⢂⠀⠉⠂⠊⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⠿⣯⣴⣿⡇⠀⠀⠀⠠⠘⣵⣆⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⣶⢾⣾⣶⠀⠀⠀⠀⠀⠀⢀⣴⠟⣠⣿⠿⣿⠃⠀⠀⠀⢠⣜⣏⡿⣄⠀⠀⠀⠀⠀⠀⠀⢂⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣯⡾⡗⣄⠈⢫⡀⠀⠀⠀⣠⡴⣟⡥⣶⣿⣿⠓⠀⠀⠀⠀⠀⡝⣉⣻⣝⡏⠀⠀⠀⠀⠀⠀⠀⡈⢲⡤⠒⠀⡀
⠀⠀⠀⠀⠀⠀⢴⣿⠋⢠⣷⣬⠝⠉⢁⠀⠐⠆⠀⠁⣘⠀⢤⢿⣾⣷⣄⠀⠀⣰⡏⢀⣿⠟⠘⣧⠀⠀⠀⠀⠀⠀⡈⢱⠉⠠⣿⡓⠆
⠀⠀⠀⠀⠀⠈⢷⡃⢀⣾⣿⠧⡀⠀⢻⡀⠀⠀⠀⠚⠁⢀⣾⣿⣿⣿⡿⢀⡴⢫⢀⣠⣿⣦⡀⢿⡇⠀⠀⠀⠀⠸⣿⠀⠀⠈⠋⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠱⠈⠿⡃⠀⢈⡔⣯⢻⡀⠀⠀⠀⠀⢘⣻⣿⣿⣿⠁⣽⢀⡿⣎⣙⣿⣯⠁⠀⢻⡄⠀⠀⠀⣀⡧⠦⠀⠖⣢⡀⣀
⠀⠀⠀⠀⡀⠒⢀⠀⠁⠀⠁⠀⠀⠁⠀⠁⠁⢀⠀⠀⠀⠈⠛⠋⠉⠛⠀⠁⣬⣾⣿⣿⣿⡿⡇⠀⠘⠳⠀⠀⢰⣿⠏⠉⢉⠁⢙⡇
⠀⠀⠀⠀⡧⠰⠆⢠⣶⠀⠀⠀⠀⠀⠀⠀⠠⢨⡆⠀⠠⠈⠀⠀⠀⠀⠀⢰⣿⣿⠇⣿⡏⠐⠃⠀⠀⠀⠀⠀⠿⢻⠀⠀⣷⠀⠘⣃
⠀⠀⠀⠀⠖⣀⡀⢸⣷⡫⣄⠀⠀⠀⠀⠀⠀⣼⡉⣄⠠⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠿⠆⠀⠈⠆
⠀⠀⣀⢰⣾⡿⡧⣆⡨⠓⢧⡀⠀⠀⠀⢀⡼⠛⢳⣌⣆⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⢸⠞⣹⣿⠛⢿⣥⣴⣶⠇⠀⠀⠀⢸⡇⢀⣴⠏⠀⢠⡀⠀⠀⣀⣀⣸⣿⠗⡄⠀⠀⠀⣀⣴⠀⠀⢰⡅⢀⠀⠀⠤⣾⢟⡀
⠀⢰⢏⣀⣤⣍⡀⢸⡿⡏⠤⠀⠀⠀⢠⢺⠋⠘⢿⣆⠐⣿⠁⠀⠀⠈⠉⣿⠉⣷⣀⠀⠀⠈⠁⠟⠀⠀⠘⠧⠙⠛⢀⠂⢳⣾⠇
⠀⠯⠾⠄⠿⠿⠿⠟⠀⠱⠤⠀⠀⠀⠆⢾⣶⢄⢈⡿⠤⢘⣆⠀⠀⠀⠳⣶⣶⡇⠹⣄⡀⠀⣠⠂⠀⠀⠀⠰⡄⠀⢀⣼⠺⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠀⠉⠀⠐⣶⣤⡀⠙⠇⢦⣼⣿⡶⡉⠀⠀⠀⠀⠀⠀⠰⠻⡿⠀⢹⡿
⠀⣴⠀⠀⠀⠀⠀⣠⡴⠊⠀⠀⠀⠀⠀⣠⣄⣀⣀⣠⢈⠱⠀⠀⠸⠿⢍⣡⣄⠀⢨⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣶⠛⢛⣷⡀
⠀⣿⡌⠟⠀⣀⢻⡦⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⣿⣷⠣⣤⠀⠀⠀⠀⠀⠉⠟⢷⡎⢷⣝⠷⠦⠀⠀⠀⠀⠀⠀⣰⠿⢿⡇⢀⣿⣿⡁
⠀⠈⢦⠀⠀⠈⠈⣁⠀⠀⠀⠀⠀⠀⠀⣿⣿⢷⢽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠈⠉⠉
⠀⠀⠈⣷⣾⣷⣧⢻⡀⠀⠀⠀⠀⠀⡠⣰⡟⠀⣾⢶⡄⠈⠁⠀⢠⡆⠨⠴⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⡀⠀⠢⡀
⠀⠀⠀⢸⢻⡿⣜⣧⢧⠀⠀⠀⠀⠘⢡⣟⣼⣷⡕⣏⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢳⡔⢩⡀⠀⠀⠀⠀⠀⠀⢸⠈⠓⠀⢨⣽⣫⣆
⠀⠀⠀⢁⣾⡿⠛⠏⠀⠀⠀⠀⠀⠆⣾⠉⠍⠙⣿⣹⠀⠀⠀⠀⠀⠈⢀⣤⠀⠉⢻⣷⡄⠁⠀⠀⠀⠀⠀⢰⢰⡄⠀⠈⣸⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠼⠛⠧⠀⠀⠀⠀⠀⠀⣾⡀⠀⠈⠏⢥⡆⢳⠀⠀⠀⢀⠟⠘⣧⣴⡞⢩⣿⠇⡏⡀
⢠⣴⣿⣷⣦⣀⣴⣿⣷⣦⣀⣴⣾⣷⣦⣄⣴⣾⣿⣦⡄⠀⠀⠀⠀⠀⠀⢻⡇⢸⠅⠊⠙⢿⣇⠇⠀⠀⠀⠀⢀⣿⣿⣷⡌⠃⠀⣰⢿
⢸⣿⣿⣿⣿⣿⣿⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⡇⠀⠀⠀⠀⠀⢴⠟⣡⣏⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢞⣽⠿⢋⠔⢁⡴⠃⠈
⠀⠙⠻⠟⠉⠀⠙⠻⠟⠉⠀⠉⠻⠟⠋⠁⠉⠻⠟⠋⠁⠀⠀⠀⠀⠀⠈⠈⠀⠛⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠞⠁⠚⠁⠠⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠂⠰⡄⠀⠀⣠⠄⣠⣄⡀⠀⣀⣐⣄⡈⠉⣹⡇⠀⠀⠀⢀⣶⣶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠃⠀⢼⡿⠏⠉⠙⢷⡀⠛⠛⢺⡇⣰⣿⣷⣶⡀⠀⠀⢉⣁⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠀⢀⣀⡀⠉⣀⣀⠀⣸⡇⢠⣴⢿⣴⣿⡏⣸⣿⠁⢠⣿⡟⠉⠁⠀⡀⠂⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⣭⣽⣷⡿⢿⣿⣿⡟⠋⣿⣟⢿⢿⣿⠿⢿⣆⣸⣿⣿⠀⠀⠀⠀⡀⠀⠀⠀⢠⡶⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣾⣓⡋⣉⣻⣾⣿⣽⣧⣸⣿⢁⣼⣿⣥⣄⣶⣿⡿⠟⠉⠀⢀⡔⠊⠉⠀⠀⠀⢡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡌⠙⠛⠛⠛⠿⠿⠻⣟⢻⣿⣼⡇⣿⠟⣻⠋⠀⠀⠀⠀⠂⠁⠀⠀⡔⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠴⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣷⣾⡆⠀⠘⢄⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣷⣿⣴⣿⠿⠇⢀⠄⠀⠤⠐⢁⡀⠀⢀⠀⠀⢀⠤⣄⢄⡁⠀⠈⠀⠀⡠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⠘⡉⠉⢩⣟⡟⠀⠀⠀⠀⠀⠀⣤⣀⠐⠻⣿⣿⣯⣻⡿⠀⡐⠁⠀⠀⠀⠀⠊⠀⠀⠉⢀⠀⠀⠀⠻⢂⡵⢄⠈⠀⣀⣤⣴⠖⠂
⠀⠀⠀⠀⠀⠀⡏⢻⣿⠆⠈⠛⠁⢀⠆⠀⠀⠀⠘⠛⠻⣶⣀⣿⡟⡇⢿⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠴⢓⠀⡀⠀⠡⠾⠋⠀⠁
⠀⠀⠀⠀⠀⠀⢿⣀⣁⣀⣰⣤⡞⠃⠀⠀⠀⠠⣤⡄⠀⣾⣿⣿⣠⣰⠊⡈⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣄⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢈⡉⠉⠉⢃⡀⢈⠀⠀⠀⠀⠀⠞⠁⠀⠉⣽⣿⣿⠏⢀⢻⡤⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠀⠀⡔⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠐⠸⣷⡀⠀⠘⢳⡎⠁⠀⠀⠰⠀⠀⠀⠀⠀⢸⠟⠃⠀⣸⠊⠁⠀⡀⠠⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡀⠘⠽⠇⠈⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⡿⠀⠀⠈⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡾⠃⠚⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠐⠺⢦⣀⠈⠛⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⣤⠜⢳⣤⡀⠀⠀⠀⠈⠀⠤⢠⠞⠀⠀⠀⣠⠊⠀⠀⠘⢳⠀⠀⠀⠀⠀⢰⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⢀⠀⠀⣾⠄⠔⢁⢸⡇⠀⠀⠜⠁⠀⣠⠃⠀⠀⢀⣀⠠⠤⣴⠃⡴⠃⠀⠀⠀⠀⠀⠘⠋⠀⠑
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠆⣡⠄⠀⢰⣂⠸⢃⡄⠀⠀⠀⢀⠜⠁⠀⢘⡁⠀⠀⣀⠞⣡⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢶⠔⠁⠀⢀⣼⠏⠀⠈⡅⣀⣤⠞⠳⠤⠄⠀⣿⠃⣀⠴⠆⢛⣏⣀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⠃⠀⠀⢰⣇⠀⠀⠀⠀⠀⣀⣼⠿⡾⠁⣀⠔⠋⣈⡛⣦⡀
⠀⠀⠀⠀⠀⠀⢠⠂⠀⠈⣱⣤⡀⠀⠀⠀⠀⠀⠋⠀⠈⢹⠆⠀⠀⠡⠟⢦⣤⡄⣀⣉⣁⣀⣴⡷⠾⠿⠟⠛⠉⠛⠛⠳⠶⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠱⣦⣤⡇⢹⣿⠀⠀⠀⡠⠀⠀⠀⠀⣿⢀⣀⣀⣠⣾⣏⡿⢷⣿⣛⣉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠣⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡆⠀⢠⠀⠀⠀⠁⠀⢀⢂⣰⣿⣾⣿⡿⠛⢿⡿⢦⠀⠈⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣵⣶⡿⠻⣿⠃⠈⠰⠀⠈⠳⠄⠀⡀⠀⠁⠀⠙⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⡴⠞⢉⣥⠿⠋⠀⢰⠁⠀⠀⢠⠀⠀⠀⠈⡄⠁⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠆⠀⠀⠀⠀⣠⡴⠋⠁⠀⠠⢈⠌⠀⠈⢀⡏⠀⠀⠀⠈⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡦⣤⣄⣠⣀⣤⣠⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠁⠀⠠⠂⠀⠀⠀⠈⠉⢠⡾⠀⠀⠀⣜⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠳⣤⠈⠀⠉⠰⠣⡧⠀⠀⡀
⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣠⣄⡐⣷⠀⠀⠀⠠
⠀⠀⠀⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠙⢷⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣄⠀⠁⠀⠀⠀⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠐⢤⣤⣤⣤⠀⠒⢺⡀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠀⠀⠀⠀⠀⠀⢀⣰⠀⠀⠈⠑⠖⠒⠀⠐⠂⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⠐⠀⠁⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⣠⡴⠈⠑⢤⡄⣀⣀⣀⣤⠊⠉⢄⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡟⢻⡏⢀⠌⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⠀⠀⠀⠙⠷⠾⠾⠟⠋⠀⠀⠀⠀⠀⠉⠉⠉⠀⠢⠀⠈⠑⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⠛⠓⠉⢤⣤⣀⡀⠼
⠀⠀⠀⠀⠀⣴⠒⠚⠇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠒⣄⠀⣀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠀⠀⡟
⠀⠀⠀⠀⠀⠙⠤⠀⠀⡤⢤⠀⠀⠀⠀⠀⠀⢠⠔⠉⣰⠀⠀⢸⠇⢉⡄⠀⠀⠠⠄⠀⠈⠣⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢀⣠⠖⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠂⢠⠒⠂⠀⠠⠆⣩⠀⠀⣾⢠⠊⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠈⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡀⠳⠖⠈⠣⠀⠻⠀⣠⡟⢀⠀⠀⠀⠠⠐⠂⠀⠤⠀⠀⠀⠀⠢⣀⠀⠀⠀⡀⠀⠒⢄⠀⠀⠀⠀⡋⠀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⡀⠀⡀⣼⣀⣴⣋⡀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢦⡀⠈⠳⡄⠀⠄⢰⠀⠀⠈⠀⡀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢗⣶⡾⠛⠀⠀⠛⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⢂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡞⠀⠲⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠇⢰⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠂⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⢀⣀⣀⠐⠀⠀⠀⠀⠈
⣀⣀⣄⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣄⣠⣀⣀⣀⣀⣀⣀⣄⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠇⢀⠀⢹⡟⠹⠟⠟⣿⠛⠿⡛⠿⢟⠙⡿⠿⠛⡿⠛⣿⢻⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⡟⢻⡟⠛⠛⠛⠛⠛⢻⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿
⣿⣦⣄⣴⣾⣷⣶⣿⣷⣾⣶⣾⣿⣷⣾⣷⣶⣷⣾⣷⣶⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿