Weight Emojis & Symbols

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⣶⣶⣶⣶⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢏⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⣾⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⠁⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⠿⠿⠿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⣿⡁⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⡠⣀⣤⣤⠀⠀⣤⣤⣀⠠
⠀⠀⠀⠀⠉⠉⠉⠉⣿⣿⣏⣉⣉⣉⣀⣀⣤⡴⠛⣿⡟⢶⣤⣀⣀⣉⣉⣉⣩⣿⣿⠏⠉⠉⠉⠁⠀⠀⠀⠀⣾⣿⣿⣻⠀⠄⣙⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡏⢀⣽⣷⣶⣯⡀⠙⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠘⢿⣿⣿⡿⠃⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣼⣿⣿⣧⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⣸⠁⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷
⠀⠀⠀⠀⢠⡏⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠸⣿⠄⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡀
⠀⠀⠀⢀⣀⣽⣿⣿⣭⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠏⠉⠙⠛⠿⠿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⢿⣿⡟
⠀⠀⠀⢹⣏⣉⣉⣉⣉⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡅⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢡
⠀⠀⠀⠘⠛⠛⣿⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣠⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣯⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢹⡀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣤⡆⠀⠀⠀⡇⠀⠀⠀⣦⡄⠀⠀⠀⢀⣴⣿⣦⡀⠀⠀⠀⣤⡜⠀⠀⠀⠃⠀⠀⠀⣢⣄
⠀⠀⠀⠀⠀⠈⢳⡞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⠿⠿⠟⠛⠁⠀⠀⠀⠻⣿⣿⣿⠇⠀⠀⠀⠈⠛⠿⠿⢿⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⣴⡟⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡃
⠀⠀⠀⠀⠀⠻⣷⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠛⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣀⣀⡀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠛⠉⠉⠛⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢠⡇⠀⠀⠀⠀⢸⡆⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⣇⠀⠀⠀⠀⣸⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠛⠛⠻⠿⠿⠿⠷⢾⣅⡀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⠶⠶⠶⠾⠿⠿⢿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣿⣯⣭⠽⠟⠛⢦⡀
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⢠⠇⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠶⢦⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢺⣁⣠⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡇⠀⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠙⠁⠀⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠐⡀⠀⠀⠘
⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠶⠶⠶⠶⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⣾⠀⠀⠀⠀⠀⠀⡤⠴⠶⠖⠒⠒⠒⢲⢀⡴⠒⠚⠓⢦⠀⠰⠒⠲⢆⣰⠦⡄⠀⠓⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢠
⢻⡄⠀⠀⠀⠀⠀⢷⣀⣄⠀⠀⣠⢤⣼⣾⠀⢀⣤⡀⠀⢇⡇⠀⠀⠘⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⡾
⠀⢳⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢁⠀⠀⢸⡀⠈⠋⠁⢀⡎⡇⠀⣄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣠⡾⠁
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠘⠧⠤⠜⠀⠀⠀⠙⠲⠶⠖⠋⠀⠳⠶⠋⠧⠤⢤⠗⠀⠀⠀⠀⠀⣠⠏⢀⣤⠾⠋
⠀⠀⠀⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⠾⠋⠁
⠀⠀⠀⠀⠉⠛⠓⠶⠶⠶⠦⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠶⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠙⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡴⠒⠙⠲⡄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡁⠀⠀⠀⡿⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠑⠦⠤⢞⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡞⢉⠉⠀⠈⡉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣚⡛⠉⠉⠉⠓⠚⠦⠾⠤⠧⠼⠓⠒⠲⠶⠦⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈⠉⠉⠉⠛⠒⠒⠒⠒⠒⠦⠤⣀⣠⠤⠒⠒⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢀⡔⠒⠒⣆⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠠⢮⣀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢰⠒⠒⠋⠀⠀⠉⠑⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠘⠒⠲⠦⠤⠤⣄⣀⡇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠂⠀⠁⠀⠀⠀⢸⡆
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣀⣠⠞
⠀⠀⠀⠈⠛⠲⠶⠤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠓⠒⠶⠶⠤⣤⣄⣀⣀⣀⣼⠶⠛⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠶⠛⢷⡄
⠀⠀⠀⠀⠀⢰⣿⡭⠴⠶⠚⠻⣦⣿⢶⡄
⠀⠀⠀⠀⠀⢸⠟⣠⠀⠐⠒⠒⠻⣿⣠⡇
⠀⠀⠀⠀⠀⣿⠘⣡⠤⠐⠒⠒⠶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠛⠉⠉⠉⠁⢀⣠⡴
⠀⠀⠀⠀⠀⡏⠀⠁⠀⣸⣷⠶⢂⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠁⠀⠀⠀⠙⠓⠒⠖⠛⠋⠁
⠀⠀⠀⠀⢸⡇⠀⡀⠀⠀⠘⣷⠛⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠸⡇⢠⣇⡀⠀⢀⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢿⡀⠉⠙⣛⣭⡴⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢠⠎⡭⢭⡭⠖⠈
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠑⠹⣆⠹⣦⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⠛⠛⠛⠶⢶⣾⣶⣤⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣿⠀⢳⡀⠀⠀⠘⢧⠘⣧⡀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⠹⣦⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢳⡀⠀⠀⠀⠀⠈⠻⣦⡀⢠⡟⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠘⣷⡀⠈⢿⣄⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠈⠻⣾⡃⡄⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠻⠶⣄⡙⠻⢶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⣷⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠄⠀⠀⠙⣦⠀⠈⠉⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢸⡇⣆⠀⠸⣝⢦⡀⠀⠀⠀⠀⠀⠀⠈⣿⠙⣷⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⢸⠀⣿⠀⠹
⠀⠀⠀⠀⠀⠀⠈⣷⠨⣃⠀⣿⠳⢽⣾⡭⠄⠀⠀⠀⠀⢸⣷⣼⢠⣀⣀⣀⣀⣀⣴⠟⠁⢀⡴⡀⠀⢸⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣀⣠⡾⢹⡌⠂⠈⠉⠉⠉⠉⠁⠀⣴⡱⠋⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠈⠻⢶⣤⣄⣀⣀⣈⣉⠁⠀⢸⡇⢀⡀⠀⠀⠀⠒⢌⣒⣃⢧⠀⠀⢀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠐⢦⡀⠈⠉⠉⠁⠀⠀⢀⣈⣤⠾⠃⠀⠀⠀⠀⠀⠉⠉⠑⠀⢀⣼⠇⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠙⢦⣄⣀⣀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⠁⠀⠀⠀⠉⠛⠿⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣌⡉⠁⢀⣠⣴⠶⠟⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠓⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠈⢫⢢
⠀⠀⠀⠀⠀⢀⠃⠀⣀⡀⠀⠘⡌⢆⣀⣀⣀⡀⣀⠀⠀⠀⠀⠠⢤⢤⡼⠀⠠⣎⢷⠀⠈⠀⣆
⠀⠀⠀⠀⠀⣞⡒⣺⢎⡇⠀⠀⡇⠘⠚⢳⠘⡏⠉⠉⠉⠉⡉⠉⢹⢆⡿⡉⠉⠓⠛⠀⢠⠀⠏
⠀⠀⠀⠀⠀⠘⡀⠀⠉⠀⠀⢀⠃⡎⠀⠀⡆⠘⣄⡴⠚⠉⠙⢿⠁⡰⠁⠐⢄⠀⢀⣠⣂⠜
⠀⠀⠀⠀⠀⠀⠙⠦⣀⣀⣤⠦⠊⠀⠀⠀⠐⡄⢹⣄⡶⠿⣻⡇⢰⠁⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢻⣧⢶⣾⣃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠊⠈⠒⢔⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡀⢈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠔⢳⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣈⠈⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢲⠊⠭⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢂⠀⡠⠮⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠓⡧⠘⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣼⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⣘⠦⢲⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡈⠿⢄⡀⢎⡘⠛⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⣀⣀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⢰⠋⠁⠉⢳⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢿⠀⠀⠀⢸⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡟⠚⠓⠶⠶⠿⢾⣁⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠚⠻⡿⠛⠋⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠁⠀⠀⠀⢀⡴⠒⠦⡄⠀⠀⠀⠑⠀⠀⠀⠀⢣⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠻⠴⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠤⠰⠃⠘⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⣀⣀⡀⠀⢀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⡾
⠀⠀⠀⠸⡆⠀⠀⠀⠀⣏⣉⠀⢀⣀⣠⡎⠀⣀⠈⢆⠇⠀⠹⠏⢹⠀⠀⠀⠀⠀⠀⡆⠀⠀⡼⠁
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠈⢹⠀⠀⠀⠀⢇⠈⠋⢀⠜⡆⢠⡀⠀⢸⠀⠀⠀⠀⢀⡼⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠓⠚⠀⠀⠈⠉⠋⠉⠀⠙⠃⠙⠒⠚⠀⠀⠀⢀⣞⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠙⠲⠤⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠴⠛⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣠⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣄⡀
⠀⠀⠀⢀⡴⣱⣷⣿⣿⣿⣿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣾⣎⢦⡀
⠀⠀⠀⡟⠻⠙⡏⠁⣿⣿⣿⣿⣇⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣸⣿⣿⣿⣿⠈⢹⠋⠟⢻
⠀⠀⠀⡇⢀⠀⡇⠀⣿⣿⢹⠛⢛⣉⣉⣭⣤⣤⣤⣤⣶⣶⣶⣤⣤⣤⣤⣭⣍⣉⡛⠻⡏⣿⣿⠀⢸⠀⡄⢸
⠀⠀⢘⡇⢸⠀⢣⠀⢻⣿⣏⣿⣿⡟⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢻⣿⣿⣹⣿⡟⠀⡜⠀⡇⢸⡇
⠀⠀⠘⢷⠈⡆⠘⡄⠘⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⢠⠃⢰⠀⡾⠃
⠀⠀⠀⠀⢧⣼⣤⣷⣲⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣖⣾⣤⣧⡼
⠀⠀⠀⠀⠈⠻⠿⢽⣷⣝⣿⡿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢿⣿⣫⣾⡯⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠟⠛⠛⠻⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣹⣿⣿⣷⣤⣄⣠⣤⣾⣿⣿⣯⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⢸⣿⡿⠁⠀⣰⣿⣿⠟⠉⠀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⢸⠟⠀⢀⣾⣿⣿⠃⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⢀⡀⠀⠻⣿⣿⣿⠀⠀⣿⣿⡇⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⣷⡄⠀⠙⢿⣿⡄⠀⠘⠿⣿⣿⠇⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⣀⣸⣿⣿⣆⣀⣈⣻⣿⣦⣄⣀⣀⣀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠿⠟⠛⠛⠉⠉⠉⠉⠛⠛⠻⠿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣦⡀
⠀⠀⠀⠀⢀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⡀
⠀⠀⠀⢠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡄
⠀⠀⣰⣿⡿⠃⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡞⠋⠙⢳⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠘⢿⣿⣆
⠀⢠⣿⣿⠁⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣇⠀⠀⣸⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠈⣿⣿⡄
⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⢶⣶⣶⣶⣶⣶⣶⣶⣾⠀⠀⠀⠀⣿⣿⠀⠀⠀⠐⣶⣶⣶⣶⣶⣶⣶⣶⡶⠀⠀⠀⢸⣿
⢸⣿⣿⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⣿⣿
⠀⣿⣿⡆⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⢰⣿⡿
⠀⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠘⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣶⣦⣤⣤⣀⣀⣀⣀⣤⣤⣴⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⠿⢿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠈⠙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡟⠁⠈⢻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣤⣤⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣩⣉⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣠⣶⣾⣿⣷⣄
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣞⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣐⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢰⣿⣿⣿⣿⣿⣯⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠁⣀⣤⣀
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⢻⣿⣏⠉⢻⣿⣿⣿⣿⣿⣿⣇⠀⠒⠛⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠘⠿⣿⣿⣿⣿⣟⣵⣶⣾⣿⣶⣿⣿⣿⣿⡿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠉⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠁⠀⣿⣿⣿⣿⣿⣿⡟⠋⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠛⠋⠉⣿⣿⣿⣿⣿⡿⠉⠙⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣤⣶⣦⠀⠀⠀⠀⠀⢰⣶⣦⡄
⣿⣿⣿⣿⠛⠀⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⠀⠛⣿⣿⣿⣿⠀⠀⠀⢠⣼⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣧⡄
⣿⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠈⠹⣿⣿⣿⠉⠉⠉⠉⠉⢹⣿⣿⡏⠁
⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠙⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⡶⠻⣤⡾⢶⣄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡏⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠿⣄⣀⣀⣀⡼⢋⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣼⠇⠀⠉⠉⠁⢰⠿⠥⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠉⠉⠉⠙⠛⠒⠒⠒⠒⠒⠒⠒⣶⠒⠛⠋⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠻⠤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠶⠶⠶⠤⠤⠤⠤⣤⣤⡼⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠑⡀⠀⠄⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠂⠀⠐⢀⠌⡀⠤⠄⠀⠴⠊⠀⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⣀⡠⠂⢐⠁⢄⠀⠀⢀⡖⠁⠀⠀⠈⠑⠤
⠀⠀⠀⠀⠀⠀⠰⢱⣶⣿⠆⡀⠀⠈⣷⣦⣤⠈⠠⣀⣈⠉⠀⠀⠀⠀⠘⠁⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⢀⣥⣤⡄⢈⡹⢻⠀⠀⣠⣿⡆⠀⠀⡄⠔⠁⢸⣿⣓⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⣿⣦⣁⠀⠘⣿⣏⣀⠀⠀⢿⣷⣶⡌⠿⠿⣿⡛⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠿⠿⠀⠀⠈⢉⣿⡧⠀⠀⠀⠁⠀⠀⣼⣿⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣖⠏⠀⠀⠀⢻⣿⣥⣤⠰⣿⣶⡄⠀⠘⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠉⠉⠃⠀⠀⠀⠉⢩⢻⡷⠀⠉⠁⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠶⢷⡦⡀⠀⠸⢤⡄⠹⣿⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⠀⠲⢤⠤⠄⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠈⠁⠀⠀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣩⠓⠃⠀⠀⢐⢀⢀⡀⠀⠀⠺⣿⣄⠙⠻⣿⣿⠇⠀⠀⠀⠸⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡆⠀⠀⠀⠿⠿⠿⡆⠀⠀⢙⠋⠀⠀⠘⢩⡿⢿⣷⡄⠀⢀⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⠵⠃⠀⠀⠀⠀⡐⠉⠐⢿⣧⡀⠀⢠⣴⣄⡀⠀⢾⣿⣁⠀⠘⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢊⠩⠀⠀⠀⠉⠛⠃⠀⠀⠛⠛⠀⠀⠈⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡽⠀⠀⠀⢀⠀⠸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠠⡈⡉⠀⠀⠀⠀⠀⠀⢀⠤⠔⠒⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠠⡀⠀⠅⠀⢀⠀⢀⡀⡀⠄⠂⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⢘⠛⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠃⠀⠀⠈
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⣭⣭⡑⠂⠉⠣⢀⡀
⠀⠀⠀⠀⠀⠀⡐⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠲⣷⣿⣿⡛⠷⣶⢄⣀⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠢⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠝⢛⠶⠿⠶⢷⣹⣿⡷⣆⡈⠠⠄⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠈⠘⠛⠷⡏⣰⠞⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠻⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠛⠉⠛⢿⡀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃⠀⠴⠾⣳⠄⠀⠈⠑⠐⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣎⣀⣠⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠁⠁⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠈⢙⠌⠁⠴⢀⠈⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⠄⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⣠⡿⠁⢸⠁⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠒⠃⠀⣚⠒⣺⢻⣶⣶⡿⢶⣶⣾⠇
⠀⠀⠀⠀⠀⠀⢀⠤⠀⢀⠒⠚⠉⠐⠒⠒⠦⢄⠤⠕⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⠁⠀⠉⠉⠀⠀⠉⠁
⠀⠀⣀⡤⠶⠏⠀⢓⠍⡭⠤⠀⠉⡤⠬⠉⠒⣄⠌⣐⡠⠀⣀⠀⠀⢀⠄⠀⢒⢢⡶⠃⠀⠀⢀⠀⢀⢰⣿⣶⣔⣶
⣠⡾⡿⠝⠧⠀⠀⠘⡆⢀⢈⣓⣎⡀⠀⠀⠀⠀⠀⠀⠘⠙⠒⠒⠂⠀⠀⠈⠥⣏⡀⠀⠀⢰⣽⠿⣷⣾⣫⣿⣿⣏⠀⡰⣶
⠑⢌⠈⠃⠀⠀⠀⢀⡘⠋⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠄⠈⠁⠀⠨⠽⠶⠀⠈⠋⢼⣵⣿⡿⣿⡾⢿⡘⠾⢃
⠀⠀⠀⠒⠤⣐⠒⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⢀⣠⣴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⣭⣽⠿⣛⡿⠻⠏⢰⣵
⠀⠀⠀⠀⠀⠀⠀⠁⠒⠷⣶⣄⡐⠂⠐⠒⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠏⠀⠺⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣞⡋⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣷⠀⠀⠀⣿⢿⡇
⠀⠀⠀⠀⠀⠀⣤⣔⡶⡷⣷⣿⡽⣽⣿⣿⣳⣦⠭⠭⠥⠀⠀⠀⣧⡌⠃
⠀⠀⠀⠀⠀⠀⠈⠉⢋⣛⣾⣿⣿⠟⠚⠚⠙⣺⣻⣿⡇⠀⠀⠀⣿⣿⠀⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠐⠻⠟⠿⢿⠿⣦⣀⡠⣾⣯⣻⡛⠃⠀⠠⡆⠁⠈⣆⠀⠀⠀⢁⠀⣀
⠀⠀⠀⠀⢄⠀⠈⠀⠲⡟⠀⠀⠀⠀⠀⢠⢭⡿⠯⠑⠉⠉⠀⠛⠆⠐⠛⠒⠀⠒⠲⢶⡾⠟⠁
⠀⠀⢀⠀⠻⠾⢧⣄⣀⣀⠀⠀⠀⠠⠀⢠⠀⠑⢄⣠⣤⣅⠀⠀⠀⠀⠀⠀⠒⢤⠀⠀⠀⠀⢀⣶⣤⡄
⠀⠀⠈⠙⠓⠶⢶⣶⣶⣶⣦⡀⠀⠰⠚⠞⠒⠚⠋⢩⠽⢟⣤⡤⠀⢠⣤⣤⣤⣾⡇⠀⣶⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢧⣤⣤⣿⣿⣿⣟⣉⣀⠀⠈⠛⠙⢂⡘⠓⣌⣘⣼⣃⣤⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠓⠚⠿⠶⣌⡀⣡⣾⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣬⣿⣿⣅⣤⣤⣴⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠿⢿⣍⣉⠉⠉⠛⣻⣿⣿⣟⠛⠉⠉⣉⣩⡿⠿⢷⣦⣀
⠀⠀⠀⠀⠠⢤⣤⣤⣼⣤⡴⠶⠟⠋⠁⠀⠀⠀⠀⠉⠉⠛⠛⢻⣿⣿⣟⠛⠛⠉⠉⠀⠀⠀⠀⠈⠙⠻⠶⢦⣤⣷⣤⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⣿⣿⣤⣤⣀
⠀⠀⢀⠀⣀⣤⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣤⣀⠀⡀
⠀⠀⠈⠀⠀⠀⠈⣿⠋⠲⣄⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⢠⠖⠋⣿⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠱⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠎⠀⠀⠀⣾
⢠⣤⣤⣤⣤⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣠⣴⣿⣤⣘⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⣃⣤⣿⣦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⢀⡤⠖⠛⠉⠉⠁⠀⠀⠀⢀⣈⡉⣹⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠛⠛⢿⣿⣿⣿⡇⣴⢋⣀⣀⡀⠰⠖⠓⠈⠁⠈⠁⡈⠉⣉⢠⣀⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠉⠀⢸⠰⡇⠈⢉⡀⣄⠀⣤⠴⠆⠘⠛⠨⠝⠀⡚⢠⢀⡄⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⠇⠴⠶⢦⢀⣴⠶⡾⠀⣷⠀⠈⠁⠁⣀⣮⠤⠀⠶⠒⠂⠘⠈⠟⠉⠀⠀⢠⣿⡶
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠿⠋⠀⠚⠻⠃⢸⠹⡾⠁⢀⡟⢀⣀⣀⣀⠸⠃⠀⠀⠀⠀⣀⣀⣀⣠⣤⡶⣾⣿⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢯⠀⠀⠀⠀⠀⠤⠴⢀⡇⠐⠟⠛⠉⠉⠈⠉⠉⠉⢉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣦⡀⠀⠠⢤⢣⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣨⣿⡀⠈⡽⠶⣤⣤⢏⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠏⠛⢋⡉⠛⢷⡏⢹⣯⣿⡄
⠀⠀⠀⠀⢀⡴⠊⠁⣿⣿⣾⡏⢶⣀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⡀⣼⣿⣯⠀⠀⠁⣸⣿⣿
⠀⠀⢀⡞⠁⠀⠀⠀⢻⣿⣿⣿⢄⡴⠉⢲⠛⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣈⠿⠯⠂⠀⠐⠀⠀⠀⡆
⠀⠀⢸⠀⠀⠀⠀⠙⠈⢿⣿⣿⡀⢿⡁⢸⠉⠁⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠟⣀⡴⠋⠒⢄⠀⢸⡄
⠀⠀⢸⠀⠀⠀⠀⠀⢄⠈⣿⣿⡇⢘⣧⣋⠤⢶⣿⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⢀⣤⣤⣬⡇⠀⢳⣤⣄
⠀⠀⠘⡆⠀⠀⠀⠣⠈⢂⢹⣿⠗⠀⠈⢻⣆⠀⠘⣆⠀⠈⠓⠦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⡄⢠⣾⣿⣿⣷⣦⡀
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢴⠙⠿⣷⣦⠼⣦⠀⠀⠀⠀⠀⠉⠉⠙⠲⣦⡀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⠀⠀⡠⠂⣸⠀⠀⠀⠙⠛⠛⠿⣶⠶⠶⣶⣶⣦⠴⠷⢮⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⢠⣾⡇⠀⡟⠀⠀⠀⢀⣠⡤⢴⣛⣋⣉⣡⣤⣤⣤⣤⣬⣽⣿⣷⣿⡿⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡿⠀⢀⣴⣩⠆⠀⡟⠉⠉⠉⠛⠶⣦⣤⣤⣬⣭⡽⠟⠋⠀⠀⠙⠉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠈⣻⣇⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⡇⠀⣴⣿⣫⣾⠧⠶⠶⠶⠒⠚⠛⠛⠛⠛⠛⠋⠉⠉⢹⣏⣉⣉⣉⣛⣛⣛⣛⠉⠉⠀⠀⠉⠻⣿⣿⡟
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣇⠀⢻⣷⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣶⣿⡟⠃
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⡇⠀⠈⠙⠛⠻⠿⢿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⠏⢿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⠟⠋
⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⡇⣀⡤⠤⠶⠒⢛⡉⣄⠰⠖⠃⠀⠀⠉⠉⠉⠁⠀⠘⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤
⠀⠀⠘⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⠁⣶⠄⠛⠒⢀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⣿⡀⠁⠀⠀⡶⠼⠃⠿⠾⠀⠷⡾⡇⠀⠀⢠⣀⢄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⡀⠀⠀⠀⠀⣿⣿⣿⡿⠖⠁⢷⡀⠀⢀⡀⠀⡀⠀⡀⢄⣀⣀⠰⡼⡿⠆⠉⠈⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⢻⣷⣶⣿⣿⣿⣿⠇⠀⠀⠀⠳⣄⡘⠛⠂⠙⠋⠙⢈⣉⣉⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠠⡤⠀⠀⠀⢀⠀⠠⠄⠀⢰⢁⡄
⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⢰⡒⢰⣂⠆⣏⣱⢿⠻⠀⠇⠀⠀⢠⢡⠄⠒⠄⠀⠙⣿⢠⡤⠁⠀⠀⠀⣀⣠⣤⡀
⠀⠀⠀⣴⣧⣪⡷⠊⠉⠓⡀⠀⠀⠐⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣾⢰⡇⠀⠑⠄⠈⠈⠳⠀⣀⠂⠘⣟⣮⡾⠃
⠀⠀⠀⢿⣷⡏⠀⣰⠒⠂⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣑⣀⢠⡎⢆⠈⣿⠷⢿⡀⠀⢸⡁
⠀⠀⠀⠀⠻⣇⠀⠈⠳⢤⣀⠁⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢜⢻⡉⣏⣿⡇⠳⠈⡄⢰⡖⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠻⢦⣀⠁⢢⣴⣊⠿⠛⢦⠀⠀⠀⣏⣳⣸⡾⢷⣿⡡⣆⠂⠀⡁⢨⣿⣁⣀⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡿⠁⡀⠪⠛⠓⠀⠀⠈⠉⠁⠀⠀⠻⡻⢾⣃⣠⠇⣸⠢⠭⠥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⣼⠋⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠞⠁
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣀⡀⠈⠻⣄⢻⣄⣀⡤⠂⠀⠀⠀⠀⠀⠀⡄⢼⠀⠁⠀⠀⠀⠀⠀⠀⡏⠡⠄
⠀⠀⠀⢀⣾⣭⡷⠾⢿⣿⣿⣿⣝⣿⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣶⢸⡇⣼⠀⡀⠀⠀⠀⠀⠀⢀⡇⠀⠐⠂
⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸⡇⣿⠀⣥⣤⣤⣤⣤⣤⣼⡇⠀⢠⠀⣏⡀
⠀⠀⠀⠀⣿⣾⣷⣶⣶⣶⣶⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢺⣇⣿⠀⠀⠀⠀⠀⠀⠀⢸⠁⡄⢸⠴⠏
⠀⠀⠀⢠⣿⣿⣿⣶⣶⣿⣿⣿⡿⢿⡄⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠉⣧⣤⠄⠀⠀⠀⠀⠀⢸⣀⡟⠋
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⡄⠀⠀⣴⣿⠋⠙⣇⣄
⠀⠀⢸⣿⣿⣿⣿⣿⣁⣨⣿⣿⣿⣷⣿⡇⠀⢸⣿⠃⠀⣰⣝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠈⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠙⠲⠭⠟⠆⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡝⣸⠁⣿⣷
⠀⠀⠀⠀⠻⢦⣽⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠇⡟⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠵⠾⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣩⣛⣭⣛⣿⣫⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⡠⠒⢁⣀⣀⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣤⣀
⠀⠀⠀⠀⠀⢀⡞⢀⡞⣠⡞⠋⠀⠉⢷⡄⠈⢆⠀⠀⠀⠀⠀⣠⠖⠉⡠⠖⢁⣀⣈⠳⡄
⠀⠀⠀⠀⢰⣾⠀⡼⢰⠏⠀⠄⠐⠀⠈⣿⠀⢸⠀⠀⠀⠀⡼⠁⠀⡜⢁⠔⠉⠀⠈⢳⡹⡀
⠀⠀⠀⠀⡏⡟⠀⡇⢸⠀⠀⣠⠶⠶⠦⠾⠤⠴⣤⣀⠀⣰⠁⠀⡜⢠⠏⠠⠒⡩⠉⠻⡇⣇
⠀⠀⠀⠀⡇⡇⠀⠃⢸⠀⠀⠻⣀⡀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⠃⠈⢠⠃⢠⠃⡤⠤⠥⠯⣤⡀
⠀⠀⠀⠀⢇⣇⠀⡆⠈⡇⠀⠀⠈⠉⢉⡟⠛⡞⠓⠒⠒⡇⠀⢸⠀⡂⢸⠀⣾⠀⡃⠀⠀⢰⠃⢹
⠀⠀⠀⠀⠀⠹⡀⠹⡄⠈⠢⢤⣤⠤⠞⠁⣸⠀⠀⠀⠀⡇⠀⢸⡀⢣⠸⡀⢣⠀⠀⢰⢒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠱⣄⠙⢄⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢳⡀⠈⣧⠈⠀⠙⠢⠅⠔⡟⡸
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠷⠦⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⢧⡀⠲⠤⠤⢞⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠦⢽⡳⠦⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣆⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⡟⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣯⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣄⣀⣤⣤⣀⣀⠰⣀
⠀⠀⠀⠀⠀⠀⠀⡌⢀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣟⣟⣿⣿⣿⣿⣿⣿⡤
⠀⠀⠀⠀⢀⣤⣼⣴⣾⣶⣶⣶⣶⣦⣤⣤⠄⢍⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠟⠋⠉
⠀⠀⢀⡀⠙⡿⡿⠿⠿⣿⣿⣿⣿⡿⠿⢿⣟⣼⢀⡷⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠙⠻⢷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⢀⣴⣿⣿⣻⠿⠿⣫⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣽⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠁
⣶⣶⡶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⠀⡀⠀⠀⢠⡆⠀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠛⠛⠀⠘⠜⠀⠘⠛⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠉⠉⠉⠙⠒⠦⣄⡀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢀⠘⠃⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣠⡏⠀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢠⡏⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸⡆⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠦⠤⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢧⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⢀⣼⠁⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⠤⠦⠤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⢤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣤⣤⣤⡤⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⢲⡟⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢀⡠⠞⠉⡠⠊⢀⣮⠴⠚⠋⠉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠈⠋⠀⢀⠞⠀⣠⠊⢁⣤⠴⠶⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⣠⠿⠖⠚⠉⠉⠉⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⢠⠒⠐⠤⡀⠀⠀⣤⣼⠃⢰⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⡠⠒⠐⡆
⠀⠀⠀⠀⠀⠈⠑⠒⠒⠛⠀⢰⠏⠀⡷⠋⠀⠀⢀⣶⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⠓⠒⠒⠁
⠀⠀⠀⠀⠀⡔⠋⣹⠖⠀⠀⡯⢀⣼⠁⠀⠀⠀⠈⣋⣥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡆⠈⣇⠀⠀⠰⢎⡋⢳
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⣏⡠⣿⠀⠀⠀⣰⠋⠁⠀⠀⠙⢦⠀⠀⠀⡀⠀⢀⡤⠲⢤⠀⢹⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣹⡀⠀⠀⣇⢀⣀⣠⣤⡤⢾⠀⠀⠀⡁⠀⢸⠧⠤⢤⠇⢸⡗⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⢧⠀⠀⠻⡉⠀⠀⠈⣠⠏⠀⠀⠀⢧⠀⠀⠑⠒⠊⠀⠀⣇⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⠦⣈⡄⠀⠀⠉⠓⠒⠛⠁⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⠀⠀⣠⠏⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⣠⣾⣿⣟⣲⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢀⣏⡹⢿⣿⡟⠀⠀⠀⢀⡴⠋⠀⠀⠀⡞⢙⡧⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠸⡆⠀⠀⠀⠀⠉⠙⠻⠁⠀⢀⡴⠋⠀⠀⠀⢸⠉⠑⣺⠀⡸⠓⡄
⠀⠀⠀⠀⣀⣤⣄⡤⠖⣋⡭⠖⠛⠛⢻⠖⠒⢷⡀⠀⠀⠀⠀⠀⠀⢀⣤⠏⠀⠀⠀⠀⠀⠸⡈⠘⠟⠚⣧⠜⠁
⠀⢀⡴⠋⠁⠀⠀⠀⢾⠁⠀⠀⠀⠀⠈⠛⢻⡟⠙⠶⢤⣤⣤⡤⠶⡋⢳⡄⠀⠀⠀⠀⢀⡴⢁⡤⠂⠈
⠀⠈⢣⡔⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠿⣦⣤⣤⣶⡭⠖⠋
⠀⠀⠈⠓⣾⡁⡼⠓⠁⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠈⠉⠁⢀⡀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡼⢋⢿⡄⠀⠀⠀⠀⣿⠶⢤⣤⣀⣀⡀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⣠⠟⡵⠃⣠⢷⡀⠀⠀⣀⣿⡄⠀⠀⠈⠉⠙⠛⠓⠶⠦⡾
⠀⠀⢀⡾⢫⠞⢠⣿⢷⡊⢧⠀⣺⣏⠈⠳⣄⣠⠀⠀⠀⠐⠢⢤⣾⠁
⠀⣰⠟⡴⢧⠀⡼⠋⠀⢷⡈⣋⡽⠛⠶⠚⠁⠉⠛⠶⢤⣤⡴⢾⣿⠶⢤⣀
⣾⣣⣎⣠⡼⠛⠁⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠆⣸⠇
⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢅⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠓⠞⢿⢿⠛⠓⢲⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣝⣒⣒⡒⠄⠤⠤⠏⣀⣈⣳⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣙⣛⣛⣛⣷⣶⣾⣶⣿⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠇⠀⠀⠀⠀⠀⠀⢀⣴⣶⡟⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠸⢲⢂⡇⠀⠀⠀⢰⣿⣿⣿⡟⣿⢟⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠎⠈⠀⠀⠀⠀⢿⣯⣿⣿⡱⣢⣿⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡗⢦⣶⣶⣦⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⠿⡿⠟⠀⠀⠀⠀⠀⠈⠙⠻⠷⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⠀⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⣦⠀⣿⢻⣞⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠹⣿⣷⡘⣷⡻⡷⢿⣟⣱⣿⣷⣶⣶⣶⣦⣤⣀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠻⣧⠈⠻⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠐⢄⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢅⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⣟⢶⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠉⠢⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠈⠳⡀⠀⠀⠀⠀⠱⡀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣚⢢⡘⢦⠀⠀⠀⠀⠘⣄⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢚⠥⠼⡓⠃⠈⠳⡀⠀⠀⠀⠘⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣊⠕⠊⠁⠀⠀⠈⠢⡀⠀⠹⡄⠀⠀⠀⠁⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢋⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠸⡄⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⡡⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⡦⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⡴⠊
⠀⠀⠀⠀⠀⢀⡴⠋⠀⢱⡀⠉⠢⣀⠀⠀⠀⢀⠔⢁⠴⠋
⠀⠀⠀⢀⡴⢏⠀⠀⠀⠀⢣⠀⠀⠈⠳⣄⠔⢁⠔⠁
⠀⠀⡠⠊⠀⠀⠳⡀⠀⠀⠀⠳⡐⡊⢰⣁⢴⡁
⠀⠀⢣⠀⠀⠀⠀⠘⢆⠀⠀⠀⠑⢜⠦⠬⠀⠱⡀
⠀⠀⠀⠳⡀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠑⢆⠀⠀⢳
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠑⢤⡀⢣
⠀⠀⠀⠀⠀⠀⣟⢦⡀⠀⠀⠀⠀⠙⠢⡀⠀⠀⣨⠟
⠀⠀⠀⠀⠀⢼⡩⠓⠛⠢⣄⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠶⠚⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⢸⣿⣧⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⣀⣀⣤⣶⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣾⣿⠿⠿⠿⠿⣿⣷⣦⣀⣀⣠⣴⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡇⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⢠⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡘⡍⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠛⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⠟⠀⠀⠀⠀⡇⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿
⠀⠈⠻⣿⣿⣿⣿⣿⡛⠛⠋⠉⠀⠐⣶⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡟⠛⠋⠙⠀⠀⢰⣶⣾⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠉⠛⠿⠷⠀⠀⠀⠀⠀⠀⠼⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⣠⣤⣇⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠆⠀⠀⠀⠀⠀⠴⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⣿⣿⣿⣇⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡦⠤⠤⠀⠀⠀⠀⠀⠠⠤⢤⢴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠟⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠉⠉⠉⠙⠓⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣀⣀⣠⡤⠤⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠈⣧⢤⣤⣀⣀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠘⢦⡀⠀⠀⢿⠀⠀⠀⠉⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⣀⣿⠤⠤⠤⠤⠶⠒⠛⠉⠀⠀⠀⠀⠀⠙⠲⠤⣾⣀⠀⠀⠀⠀⠀⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣼⠁⣿⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⣸⠈⡇⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⣿⠀⢠⡖⠮⠵⠚⡆⠀⠀⢰⠓⠮⠵⢓⡆⠀⢼⠀⡇⠀⠀⠀⠀⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠦⣿⠀⠀⠙⠒⠖⠊⢁⡖⢲⡀⠙⠲⠖⠋⠀⠀⣹⠴⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠓⠚⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠴⠒⠒⠦⠄⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⢷⡀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠏⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠒⠶⠤⠤⠶⠒⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢳⣀⣀⡤⠴⠒⠛⠹⡆⠀⠀⠀⠀⠀⠀⢰⠏⠛⠒⠦⢤⣀⣀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡸⠃
⠀⠀⠀⠀⠀⣰⠃⠀⠀⣀⣀⣀⣀⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⡴⠃⢠⠔⠋⠁⠀⠀⠀⠉⠓⢦⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⣸⠃⣰⠃⠀⠐⣄⠀⠀⢠⠖⠀⠀⢳⡀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⡏⠀⡏⠀⠀⠀⠈⢳⡴⠃⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⣧⠀⢧⠀⠀⠀⣠⠞⠙⢆⠀⠀⠀⢠⡇⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠸⣆⠈⢧⡀⠐⠁⠀⠀⠈⠓⠀⣠⠞⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠘⢦⡀⠙⠲⢤⣄⣀⣤⠴⠚⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠉⠲⠤⣤⣀⣀⣀⡤⠴⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⡄⢤⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤
⠀⠀⠀⠀⠁⠀⠀⠀⠁⠠⢴⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⡄⢠⠀⡔⠀⠀⢀⣾⢹⣆⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠐⠚⠉⠩⣝⡲⠄⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠰⠧⣀⣀⠀⠈⠀⣸⣷⣶⠼⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⠀⠀⢹⡦⠤⠀⠤⠾⢿⡆⠀⠀⠀⠉⠀⠀⠂⢨⡖⠏⢹
⠀⠀⠰⠲⢤⣀⠈⠉⠓⠠⠁⠉⣤⠃⠀⠀⠀⠀⣆⠀⢀⢠⢀⣀⡀⠈⢻⠀⠀⠀⠀⢸⣿⡦⠀⠀⠒⣾⡇⠀⠀⠀⠀⣭⣉⣛⢸⠀⠀⣸
⠀⠀⡃⢠⠀⠀⠰⠖⠀⢸⠒⠁⠒⢀⠀⠀⠀⠀⣩⠀⠀⠀⠆⠀⠴⠀⢼⠀⠀⠀⠀⠘⠧⠁⠀⠈⠰⣿⠂⠀⠀⠀⡆⠀⠠⡝⢾⣿⠻⣿
⠀⠀⣀⣈⡀⠀⠀⠀⠀⠚⠤⣶⠀⠘⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠐⣾⠀⠀⠀⠀⠰⣷⣶⢿⣷⣾⣿⡇⠀⠀⠀⡷⠄⠶⠉⣷⣿⣧⣼
⠀⠀⣷⡜⠀⠤⠀⣴⢀⢠⣞⡿⠥⢰⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⢸⣿⡛⠋⠙⢻⣿⠀⠀⠀⠀⡃⢀⡀⣠⣿⣏⢉⡿
⠀⠀⠃⠀⠠⢴⠄⠠⠈⠰⠾⣵⡞⠋⠀⠀⠀⠀⠿⣷⣤⣀⣀⣤⣴⣿⡿⠛⠀⠀⠀⠙⠷⣄⣀⠀⠠⠟⠃⠀⠀⠀⠈⠁⠋⠝⠿⠛⠁
⠀⠀⠊⠁⠦⠀⠀⢂⣀⣨⠀⠛⠀⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢘⡟⠋⢙⣏
⠀⠀⠀⠀⢰⠀⠀⠀⠂⠀⠤⣄⣀⡀⠀⠀⡀⠒⠒⠀⠄⢀⠀⠀⠀⠀⢣⣄⣀⣀⢀⡀⣀⠀⠀⣀⣀⣠⢴⡇⠀⠀⠀⢠⣔⠚⠉⠉⠉⠋⠂⣀
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢊⠀⠀⠀⠀⡹⣿⣿⣬⣀⣉⣉⣉⣀⣀⣰⣿⡷⠇⠀⠀⠀⢈⣉⠁⠀⠀⠂⢈⣿⣱
⠀⠀⠀⢸⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠀⢮⣟⠃⠀⠀⠀⠀⠀⠀⠁⠠⣼⣿⡼⠀⠀⠀⠘⠻⠶⠦⠤⠤⢿⣿⡟
⠀⠀⢀⡏⠺⡇⠀⠀⠰⣏⠀⡶⡆⢠⢰⡵⢐⠆⠀⢰⠀⠀⠂⠀⠀⠀⢸⣿⢯⡁⡀⠐⠒⢒⡀⠀⠀⣹⣿⠀⠀⠀⠀⢰⣀⠀⠀⠀⠀⣰⣿⣧
⠀⠀⡼⠀⠀⡇⠀⠀⠐⠤⢀⡀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⢸⢰⣿⠀⣌⣁⣈⣈⡉⣀⠀⢺⣿⣿⡆⠀⠀⠀⠐⠛⠿⠖⡖⠺⠿⣿⡗
⠀⠀⡿⠀⠀⡗⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣙⣿⢤⠀⠁⠀⠀⠉⠀⠀⣙⠿⣿⡉⠀⠀⠀⢸⣀⢇⠀⠇⠀⢰⣿⣷
⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠔⠷⠀⠀⡎⠀⠀⠀⠀⠈⠙⠢⠤⣀⣀⢀⢀⣀⡀⠤⠒⠉⠀⠀⠀⠀⠀⠙⠛⠓⠒⠒⠻⣿⡷
⠀⠀⡗⠀⠀⡇⠀⠀⠀⣴⠶⠀⠀⠀⠠⠅⠀⡀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⢠⣿⣧
⠀⠀⢱⠀⢰⠃⠠⡀⠠⠂⠀⠀⠠⠔⠀⢤⣔⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠈⠉⠉⠉⠉⠻⣽⣷
⠀⠀⠸⡀⡽⠀⠀⠈⠉⠀⠉⢶⣅⣆⡀⠉⠀⡗⠘⠁⠀⠀⡄⠀⠀⣇⡄⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⢻⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⣉⣀⠠⠊⢀⡀⠁⠀⢰⠄⠀⠠⢹⡇⠀⠈⠈⠈⠁⢙⠁⣌⣦⠄⠀⠀⣨⢁⡆⠀⢸⠀⠀⠀⠀⠀⠐⣿⡷⠂
⠀⠀⠀⠿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠁⠀⢠⣴⢺⠀⠀⢀⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⡇⠀⢸⢀⠀⠀⠀⢠⣽⣿⡟
⠀⠀⠀⠩⠥⠀⠀⠀⠀⠀⣀⣀⣀⣀⡂⣀⣈⣀⣀⣀⣂⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠤⣄⣀⣀⣠⣾⠿⠇

⣿⣿⠿⠻⡿⢻⠿⢿⠿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣒⣶⠶⠒⢒⣒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⣶⡶⢶⡎⢸⠋⠀⠀⢀⡄⠀⠐⡙⡆⠹⣶⣦⣤⣤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠕⠋⠀⠀⠀⠀⢣⠸⣄⠀⡤⣚⢧⡀⢀⡠⠃⣸⡇⠀⠀⠀⠉⠺⣝⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠏⠀⠀⠀⠀⠀⠀⠈⢦⣌⣉⠛⠛⢓⣋⣩⢴⡾⠛⠀⠀⠀⠀⠀⠀⠘⡜⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠙⠩⠙⠓⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⣽⣒⣒⣠⡤⠤⢀⣀⣀⣀⣀⣀⣀⡠⠤⣤⣤⣒⣲⣮⣷⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉



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

⠀⢹⡀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠉⠐⠒⣀⡀⠀⠀⡐⠀⢸⡃⠀⠗⠀⠀⣱⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠸⠀⠀⢰⠂⠸⠤⡀⠉⠩⠚⠁⠐⠲⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢺⠸
⠀⢸⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠧⠈⢀⠀⡄⠀⠀⠀⠀⠀⠀⠠⡄⠀⠁⠁⠀⠀⠀⠀⠀⢣
⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡈⠀⠀⠀⠀⣼
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⣀⡀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⡀⣠⣧⣠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢲⣺⠀⠀⠃⠀⣠
⠁⠈⠹⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠄⠛⠚⠛⠉⠀⠀⠉⢆⠀⠀⠀⠀⢀⡠⠔⠁⠀⢀⠠⠋⠀⢠⠖⠘⠓
⢠⠀⠀⠀⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠔⠊⠁⠀⠀⠀⠀⣱⠁⠀⠀⡇
⡉⠀⠀⠀⡟⠀⠀⢄⠠⠀⢀⢀⠀⠄⠈⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠄⡙⠀⢘⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢱⠠⡀
⣀⣀⣜⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⣀⣀⡠⠬⠔⠊⠀⠀⠈⢆⡃⠀⢀⣀⠴⠁⠀⠀⠀⠀⠀⠳⣵⣄
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠐⠒⠂⠠⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀
⠀⠀⠀⠀⠈⡆⠀⠈⠌⠂⡲⠄⠠⠔⢀⠀⠀⠀⠀⠐⠴⠋⠀⠀⠈⢳⡀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠐⠒⠋⠑⠒⠃⡀⠀⠀⠀⠀⠀⠂⠀⠀⠑⠀⠈⠀⠀⡇⠀⠀⡠⠊⢀⡃⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠐⠁⠀⢀⠃
⠀⠀⠀⠀⡠⠊⠀⢀⢀⡀⠀⢀⣀⠀⠀⣀⠀⠀⠀⠀⠈⢠⠄⡄⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠂⠐⠐⠈⠀⠀⠀⠈⠿⢏⠀⠜⠓⠀⡀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⣠⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠴⠠⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠋⠀⠀⠀⠀⠀⠀⠀⣀
⠆⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠎⠁⣀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠂⠀⠀⠀⠀⠀⡔⠈⢀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣄⣹⣥⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢢⠀⠀⠀⠀⡌⠁
⠀⠀⠀⣠⠁⠀⠀⠈⣀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠘⡨⠼⣫⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠱
⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⠻⠆⠄⠐⣀⠞⠀⠀⠀⢇⡀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠱⡀
⠐⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⠁⠀⠀⠀⠀⠀⠀⠉⠒⠛⠒⠚⠁
⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡡⠄⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⣀⣁⢱⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⢤⠖⠉⠀⠉⠳⡤⠒⠐⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣤⡤⠤⠤⠤⠤⠤⠤⢤⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⢾⠇⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠈⠛⠛⠛⠛⠒⠒⠒⠒⠚⠀⢀⣠⠜⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠓⠢⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠉⠑⠀⠓⠒⠓⠚⠉⠉⠁⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠉⠉⣺⣿⠟⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢿⡟⠁⠀⠌⠛⢿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠈⣦⣄⣐⣀⠵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡏⡇⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠓⠒⠒⠚⠀⠀⡇⡇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢠⣤⠀⢀⣧⢧⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡼⠀⠙⡉⢸⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢀⡞⠁⠀⠀⢱⢠⡅⠀⠀⠀⡄⢢⡜⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡴⠋⠀⠀⠀⠀⢸⣼⣀⠀⠀⣠⣀⠘⠇⠏⠜⣧⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠗⠀⠀⠀⠀⠀⠓⠚⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡟⡛⢛⡛⣻⢿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣛⣛⣛⣛⣛⣛⡛⣛⣛⣛⣛⣛⣛⡧⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠇⡀⠀⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⠟⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⣤⣴⢦⣤⣤⣴⠶⣶⡿⠆⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠿⠿⣿⣿⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣀⣀⣀⣀⣀⠤⣤⢤⡴⣤⡴⢶⣶⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⠿⣿⣿⠉⣿⣛⣿⣽⠽⠭⠽⠷⠶⢰⣦⣿⠒⠒⠉⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠀⠀⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡇⠀⠀⠀⠀⠀⠀⢸⡗⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣧⣤⣀⣰⢶⠀⠀⣸⣧⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡟⣿⠀⣿⣿⣿⣿⢿⡿⠏⠓⣒⠛⠛⡛⠿⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠏⢀⣤⣛⢔⠟⠖⣻⡾⠋⠁⠁⠀⣢⢀⡽⣤⣤⣥⣀⡀⡀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠴⣶⠌⣀⡾⠫⠀⠀⢢⡖⠚⠛⠀⠀⢧⠁⠉⠉⠛⠛⣦⣐⠄⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⣈⣟⡀⠟⠀⠀⠀⣦⠁⠀⠀⠀⠀⠀⢸⣄⣀⡀⠀⠀⠀⠉⢢⣀⡎⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇⢸⣯⣹⡄⠘⢲⡜⠀⠀⠀⢀⡼⠛⠀⠸⣇⠈⠙⠃⢦⠀⠀⠀⠙⡄⢀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢛⡇⢸⡿⣿⣶⣀⣰⠉⠀⠀⢀⡟⠀⠀⠀⠀⣿⡀⠀⠀⠈⢫⣀⠀⠀⠙⡏⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⣇⣤⣷⣿⡇⠂⢹⠀⠀⠀⠸⠁⠀⠰⠖⠒⣿⣷⡜⠀⠀⠀⠁⠀⠀⠀⠴⠆⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢋⡟⣧⣿⣿⡇⡤⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇⠀⠀⠀⠀⠀⠀⠀⢰⣀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⡇⢡⣽⣿⣷⡀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠠⠀⠀⢈⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢸⣟⠻⣿⡄⠀⠉⡁⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⢦⡀⠀⠀⠀⠀⠀⠟⠦⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⢸⠓⠘⢿⣷⡄⠀⣜⠇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⢦⡄⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢧⣜⠃⠀⣽⣿⣿⣷⣭⠀⠰⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⣁⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠇⠀⠀⠈⠙⠲⢾⣟⡻⣿⣿⣿⣆⠀⠀⣙⠒⠀⣦⠦⢤⡦⠐⢻⠁⢀⣽⣿⠟⠁
⠀⠀⢀⣤⣾⣿⡿⣶⣶⣤⡀⠀⠀⠀⠀⠀⡭⢛⢿⣿⣿⣿⣷⢮⣟⡦⢤⣴⣄⣤⣤⣤⣶⣿⣍⠮⡹
⠀⠀⠈⠛⠻⠿⣿⡿⣿⣿⣿⣦⣀⠀⠀⠀⠈⠘⣿⣿⣿⣿⣿⡄⠈⠉⠛⠓⠚⠓⠛⠈⠙⠃⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣎⡿⣿⣭⣽⣿⣿⣶⣦⡀⠈⠙⠿⣿⣿⡇⣤⣤⣤⣴⣶⠶⠶⠿⠟⢛⣛⣿⣽⣥⣤⣤⣴⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠿⣽⡻⣿⡿⢿⣶⣤⣠⣤⣭⡷⠷⠿⠶⠶⠶⠶⠶⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠻⣷⣭⣈⣀⣠⣤⡶⠶⠶⡶⠶⠶⠲⠾⣟⣛⠯⠝⠓⠛⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠂⠀⠶⠒⠒⠒⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡴⠞⠛⠷⣦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡾⠀⠀⠀⠀⠸⡇⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣧⠀⠀⠀⠀⢀⡇⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⣽⣦⣤⣤⣤⣼⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣴⠶⠏⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠶⠶⠶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⠛⠉⠁⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠐⠊⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠇⠀⠿⠴⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⡀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⣀⣀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠐⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠹⢠⠞⠉⠁⠈⠙⡄⢸⠉⠉⠹⣼⠋⢳⠀⠈⠁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠹⠶⢶⠀⠀⢰⠓⠚⢻⠀⠠⣞⡆⠀⢸⣼⠀⠀⠀⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢸⡄⠀⢰⠀⠀⠘⢦⡀⠀⢀⣠⠏⣜⠀⣶⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠙⠃⠘⠛⠒⠚⠀⠀⠀⠀⠀⣠⠏⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠓⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠟⠉⠉⠉⠙⠻⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢁⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠸⣿⡀⢿⣿⣿⣿⣿⣿⣿⣿⡏⢸⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⠀⠀⠀⠀⢠⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣷⠈⣿⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣼⡿⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⠻⣿⣦⣀⡀⠀⠀⠀⠀⢀⣠⣼⡿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣌⡙⠻⠿⠿⣾⡷⠿⠿⠛⢋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣌⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣡⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠋⠉⠉⠉⠉⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠲⢦⣤⠴⠶⠖⠒⢲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣧⠀⠀⠀⠀⠈⣷⠒⠲⠶⢤⣤⠶⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠘⢧⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⣀⣠⣤⡤⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠙⠳⢤⣄⣿⡀⠀⠀⠀⠀⠀⠀⠠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢠⠞⢹⡏⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣿⠙⢦⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⣀⣀⠿⠇⣀⡀⠀⠀⠀⠀⣀⡀⢿⠇⣀⡀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⡇⠀⠰⣏⠙⠳⠞⢉⣽⠀⠀⠀⠸⣏⠙⠲⠚⢉⡿⠀⠀⣿⠀⣸⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠳⢼⡇⠀⠀⠈⠛⠒⠚⠋⠁⣰⠛⢳⡄⠈⠛⠒⠚⠋⠀⠀⠀⣿⠴⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠋⠁⠀⠀⠉⠃⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠁⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠒⠶⠶⠤⠴⠶⠖⠚⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⣀⣀⣤⠴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠹⣶⠶⠚⠛⠉⠁⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠉⠉⠛⠲⠶⣤⣄⣀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣤⣀⣠⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⡾⠁
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⢠⡞⠀⢀⣤⠖⠋⠉⠉⠉⠉⠉⠓⢦⣀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢠⡟⠀⣠⠟⠁⢠⡀⠀⠀⠀⠀⢀⡄⠀⠙⢧⡀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⡾⠀⢰⠏⠀⠀⠀⠹⣆⠀⠀⣰⠏⠀⠀⠀⠈⣷⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⢰⡇⠀⢸⠀⠀⠀⠀⠀⠈⢳⡞⠁⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠈⣇⠀⢸⡄⠀⠀⠀⠀⣰⠏⠙⣦⠀⠀⠀⠀⠀⣼⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⢹⡄⠀⢳⡀⠀⢠⡞⠁⠀⠀⠈⢳⡄⠀⠀⣰⠃⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠻⣄⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠉⠛⠲⠶⠶⠶⠚⠋⠁⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣤⣀⣤⡤⠴⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⣀⠀⣠⣤⣄⡀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⡤
⠀⠀⠛⣿⣿⣿⣿⣷⠀⠀⠀⢺⣼⣿⣿⣧⠀⠀⠀⠀⠀⢠⣾⣿⣶⠋⠀⠀⠀⠀⠀⣤⡦⢤⣤⣤⠄⠀⠀⠀⢾⣛⢿⡿⡿⠊
⠀⠀⠀⠈⠉⠈⠉⠁⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⠀⠘⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋

⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⣀⣴⣿⡿⢦⠀⠀⠀⠀⠀⣄⢠⣄⢠⡄⠀⠀⠀⠀⠀⠄⣀⣢⣄⠀⠀⠀⠀⢀⣤⣶⣄⣀⡄
⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⠀⢸⡏⢀⡀⢰⣿⠀⠀⠀⠀⠺⡿⢿⡷⢿⠗⠀⠀⠀⠀⢷⣦⣿⠛⠛⠃⠀⠀⠀⠘⠿⣿⠟⠛⠃
⠀⠀⠲⠶⠛⠀⠀⠀⠀⠀⠀⠘⠓⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠻⣷⡿⠃⠀⠀⠀⠀⠐⣿⠀⠸⢿⣶⠀⠀⠀⠀⢰⣶⠛⠛⣻⠃⠀⠀⠀⠀⣰⣾⣷⣝⢦⠀⠀⠀⠀⠀⠀⣿⣿⠆
⠀⠀⠀⠀⣿⡗⠀⠀⠀⠀⠀⠀⠻⣶⣤⣾⠟⠀⠀⠀⠀⠈⠻⠿⣠⣿⠇⠀⠀⠀⠀⠻⠿⣿⡿⠟⠁⠀⠀⠀⠀⢠⠮⡟⡆
⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁

⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⢀⣤⣠⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⢰⡏⠁⠉⣷⠀⠀⠀⠀⠀⣼⠙⣿⠙⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⢀⢿⣿⡇
⠀⠀⠸⣇⣸⣀⡿⠀⠀⠀⠀⠀⠹⣄⣁⣰⠇⠀⠀⠀⠀⢠⣶⣶⣿⠟⠀⠀⠀⠀⠀⠀⣿⣿⣾⠃⠀⠀⠀⠀⠀⠸⣸⣟⡿
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⣿⣿⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣿⣿⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⣧⣤⣿⣿⣿⣿⣤⣼⣧⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⣤⠴⠂⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣄⣁⣀⣀⣀⣀⣀⣠⣤⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢠⠔⢢⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⣠⠒⣀⣤⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢦⣴⣠⠄
⠀⠀⠀⠀⠀⣠⢄⡸⠀⡞⠀⠀⠀⠀⠹⣶⢸⣿⠁⠀⢠⠾⠉⢓⡔⠃⠀⠀⠠⠀⠀⠀⢠⣾⡻⢟⡾⡱⠏⠀⠟⣢⠀⢀⡴⣫⠀⠂⠀⡬⢋
⠀⠀⠀⠀⠀⢣⠀⢷⣤⠖⢰⠀⠀⠀⠀⣇⣰⡶⠃⠀⢻⠐⠡⠀⠀⠠⠀⠀⢀⠇⠀⡔⠋⠂⠁⠈⠀⠀⠀⠀⡸⠁⠀⣸⠃⢀⡀⠐⠂⢠⠀⠄
⠀⠀⠀⠀⠀⠀⠳⢾⣿⠒⠁⠀⠀⢀⣾⡇⠀⣱⠀⠀⠈⠣⢀⠈⠀⠀⢀⣢⠎⠀⠀⠁⠉⠂⠄⠀⣀⣀⡠⠞⠁⠀⠀⢻⠋⠈⢁⡄⠒⢀⠜
⠀⠀⠀⠀⢀⡴⠒⠚⠓⠒⠒⢄⠀⢸⣏⠀⠀⡿⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⣄⠀⠀⠀⠀⠘⢘⣋⣈⡴⠛⠁
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡐⠀⢸⣴⠇⣼⠃⠀⠀⠀⠀⣠⠄⡶⣠⣫⠦⠤⠀⠀⠀⢠⣿⣯⠻⠾⠟⠋⣧⠀⠀⠀⠀⠀⠉⣁
⠀⠀⠀⠀⠈⠓⠂⢤⣀⠔⠈⠀⠀⢸⡟⢠⠃⠀⠀⠀⠀⣜⡕⡠⠁⠈⠡⢜⠀⠈⡆⠀⢹⠃⠾⡷⢘⡿⠫⡇⠀⠀⠀⠀⣀⠤⣿⠇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠯⠃⠀⠀⠀⠀⢰⡿⢠⡖⣣⠀⠀⣸⠇⢰⠇⠀⠀⠁⠀⠠⠤⠀⠀⠇⠀⠀⡴⣻⠀⠀⠻⠄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠀⠀⡀⢤⡠⢀⠀⠀⢸⡇⢸⣷⡇⠀⢠⡟⢀⡜⠀⠀⠀⠀⡀⠤⠂⠉⢲⣄⠀⢸⡫⣿⢣⢆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠟⢋⣴⠏⠐⣍⣽⡛⠛⡞⣁⠄⠘⢧⣿⡻⣧⣴⣿⣧⠞⠀⠀⠀⡴⢋⡤⣂⣦⠀⢸⢇⠀⠘⣿⠡⠢⠂⠉⠂⠀⠀⢠⠟
⠀⠀⠀⠀⠀⡠⢾⠂⣍⠉⠁⠀⠀⠳⡼⢿⣀⣸⣴⠀⠀⠀⠀⠙⠛⠘⠛⠁⠀⠀⠀⢸⡁⠸⣿⣇⠟⠀⡿⡸⠀⠀⠈⠑⠦⢤⣄⣤⠞⠖⠉
⠀⠀⠀⠀⣼⠁⠈⠀⢸⡀⠀⠀⠀⠀⣗⣻⡟⡾⠀⠀⠀⢀⣀⣤⡆⠀⠀⠀⠀⢀⣤⠘⡧⠀⢀⣀⣠⠞⡑⠁⠀⢀⣀⣤⣤⣌⡉⠁
⠀⠀⠀⢀⣿⠆⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⣛⡝⣋⣀⣀⠀⣀⣴⠏⡟⠀⠈⠁⠀⠀⠀⠁⠀⢠⣞⢁⠀⠀⠀⠀⠈⡆
⠀⠀⢀⣾⣿⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⣀⣀⣙⣷⣾⠿⠟⠛⠋⢁⢰⡿⠋⡸⠁⠀⠀⠀⠀⠀⢀⡀⠀⢻⡿⠂⡀⣀⣴⠊⠁⠂
⠀⠀⢸⢟⠟⣰⠀⠀⠀⠀⢸⠀⠀⢀⣶⠉⣸⠁⠀⠀⠀⠑⠄⠀⠀⢙⢻⡁⠈⠀⠀⠀⠀⠀⠀⣀⢸⡇⠠⡀⠙⠒⠀⣿⠰⣝⠔⣀⠀⣀⠀⢸
⠀⠀⠘⢯⡐⣥⠏⡆⣀⣠⠏⠀⢀⡏⢪⠀⡗⢨⠄⠀⠀⠀⢸⠀⠀⠘⠀⠉⠢⣀⠀⢀⠀⠀⣰⡇⣸⡇⠀⠱⠀⠀⠀⠈⠂⠨⠧⣀⣠⣡⠴⠋
⠀⠀⠀⠀⠉⠛⣾⡞⠋⠀⠀⠀⠘⡇⠁⢄⣧⠀⡤⠀⠀⠀⡼⠀⠀⢴⠠⣄⢼⢉⣁⣀⠇⠀⣿⡅⢿⣩⠀⠀⡇⠀⢀⡀⠀⠀⣤⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄⣹⣌⠐⢀⣠⠞⠁⠀⡇⣬⠀⡼⠈⠛⠚⠋⠀⠀⠸⡖⢸⣏⠆⣠⠁⠀⠀⢳⣯⠘⣯⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⠀⠀⠀⠀⠀⠀⠉⠁⠉⣉⡁⠀⡀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠿⠞⠁⠀⠀⠲⡸⢛⣛⡻⠋⡷
⠀⠀⠀⠀⠀⣴⣏⣿⠏⢻⣇⠀⠀⠀⠀⡠⠒⢸⠏⠉⢀⡶⡉⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠙⢢⣠⠅⠠⡀
⠀⠀⠀⠀⡔⢻⣹⠟⢳⣸⢾⡀⠀⢀⡾⣡⠀⠈⢹⢀⡼⢁⠵⠄⠈⠋⡆⠀⣠⣾⣏⠷⠚⠁⢰⣆⠀⠀⠀⠀⢀⣾⠗⢱⣂⠾⠬⡁⣋⠸⡀
⠀⠀⠀⠘⣧⣧⣿⢄⢸⠁⠀⡇⠀⢸⢡⡟⣀⠀⢻⠋⠀⠀⢡⠀⢸⠀⡵⠀⢿⣡⠀⠀⠀⠀⠀⠒⡘⠀⠀⠀⣼⣿⣡⣼⡣⢀⣠⠍⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠿⡴⠞⠀⠴⠁⠀⠸⡏⢿⡏⠀⡞⠘⠀⠆⠈⢀⡏⢠⠇⠀⢿⠁⠁⠀⠰⢆⠀⠀⡀⠠⠀⠀⣯⠑⣶⡏⢌⢦⡄⢠⢀⠤⡇
⠀⠀⠀⠀⠀⢀⠀⠿⠀⠀⠀⠀⠀⠀⠳⣌⢷⣸⢧⢀⡼⢀⠴⢋⡴⠃⠀⠀⠀⠁⠈⢱⠘⡣⠉⠀⠁⠀⠀⠀⠹⡶⢣⡮⢯⣹⣶⣇⢎⡼
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠏⠁⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠌⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠪⠷⣞⠿⢽⣿⠟⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠿⠿⠷⣶⣾⣿⣿⣷⣶⠾⠿⠿⢿⣷⣶⣄⡀
⠀⠀⠀⠀⢿⠀⠀⠀⣀⣠⣶⣿⠟⠋⠁⠀⠀⠀⠀⠀⠉⢻⣟⠉⠁⠀⠀⠀⠀⠈⠙⠻⢿⣶⣤⣀⠀⠀⠀⡿
⠀⠀⠀⠀⠈⠛⡿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⠛⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡐⠠⠀⠠⠐⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠴⠤⠄⣄⡀⠀⠀⠀⠸⣷⣤⢄⣒⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠡⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠈⠙⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡐⢦⣤⣤⢤⣄⠀⠀⠀⠀⣀⣠⣶⣦⡀
⠀⠀⠀⠀⢾⣽⠆⠀⠀⠀⠀⠀⠀⠘⠷⡆⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠑⠈⠉⠁⢈⠛⠇⠀⠀⡸⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠒⠁⠀⠀⠒⠀⠀⠀⠐⠙⠛⠁
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠐⡁⠀⠡⠈⠀⢀⠀⠀⠀⠀⠨⠉⠉⡀⠀⠀⠀⠀⠀⡊⠁⠐⠄⠀⠀⠀⠀⢀⣰⣶⣖⠒⠲⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⣾⣶⣶⢶⡆⠀⠀⠀⠀⠀⠘⠀⠂
⠀⠀⠀⠀⠀⠀⠛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡀⣤⣤⢤⡀⠀⠀⠀⡂⠘⠄⠀⠀⠀⠀⠀⠀⣀⣤⡤⣄⠀⠀⠀⠀⠀⠠⠠⢤⡀⠀⠀⠀⠀⢀⡐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠌⠡⠰
⠀⠀⠀⠀⠠⠀⠠⠀⡀⠂⠀⠀⠀⠈⠀⢈⠀⠀⠀⠀⠀⠨⠀⠠⠀⠀⠀⠂⠀⠀⠀⠅⠀⠀⠹⠷⠀⠀⠀⢸⣿⣟⣛⣿⣿⠀⠀⠀⢶⢴⢰⣶⣶⣿
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⣐⠰⠶⠶⠀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠖⠂⣤⣲⡄⠀⠀⠀⢠⢢⢀⡼⠁⠀⠀⢠⣴⣶⣿⣷⣶⣧⣄⣀⠔
⠀⠀⠀⠀⠑⠛⠻⠟⠛⠊⠀⠀⠀⠘⢤⣀⣠⠜⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠘⠚⠁⠀⠀⢀⣤⣶⡶⠀⠂⠀⠀⠉⠁⢈⡉⠀⢀⡄⠀⠉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⣰⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣐⠤⣀⠀⠀⠀⠘⠁⡤⠞⠓⠄⢀⠠⠂⠀⠀⢀⣘⣷⣻⢆
⠀⠈⠒⠦⠘⠄⠀⢀⡰⠀⠀⠘⡆⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣄⠀⠀⠀⠀⠀⠘⠛⠮⠻⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠈⠐⠃⠀⠡⠖⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⢠⠀⢄⡀⠀⠀⠙⠄⠀⠀⠀⠀⠒⡵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠈⠈⡀⢁⠀⠑⠀⠀⠸⣄⠴⠋⠱⡀⠀⠀⠂⠀⠀⠀⠀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠑⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢁⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠤⠙⠂
⠀⠉⠃⠀⠈⠉⠒⠠⣄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢴⣀⣁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠉⠴⠒⠂⠀⠀⠀⠠⠀⡠⠃⠂

⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢀⣀⡀⠀⠀⠀⡄⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠉⠁⢉⠀⠈⢸⢳⠂⠀⢀⠴⠋⢠⠄⠀⠀⠈
⠀⠀⠀⠘⠀⠀⠈⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠂

⢴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢤⡄⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡦⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀
⠈⠓⠤⠤⣤⣄⠀⢀⣤⣤⠤⠴⠋⠀⠀⠀⠀⠀⠈⠳⠶⠶⣶⣶⣶⣶⠶⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⢴⡆⠠⣤⠀⡀
⠀⠀⣠⠴⠒⣛⣉⣙⡓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣩⠭⠭⣍⡛⢦⡄⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠂⠈⠀⠒⢀⠀⠃
⢠⠞⣡⠞⠫⠅⠿⠀⠽⠛⢦⡙⣦⠀⠀⠀⠀⠀⢰⠏⣰⠋⠀⠀⡀⠀⠙⣆⠹⣆⠀⠀⠀⠀⠀⠰⠄⠃⠀⠀⢸⡀⠀⠀⠑⠠⠄
⡟⢰⣏⡙⠀⠀⢀⡴⠁⠘⣁⣷⠘⡇⠀⠀⠀⠀⡿⠀⡇⠀⠀⢰⡇⠀⠀⢸⡄⢻⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠇⢀
⣷⠸⡏⣡⠀⠀⠈⠀⠀⢠⡉⡾⢠⡇⠀⠀⠀⢸⡇⠀⢧⠀⠀⠀⠀⠀⠀⡼⠀⠸⡇⠀⠀⠀⠀⢠⠀⢄⢀⠀⠀⠀⠀⡀⡀⠀⡄
⠘⢧⡙⢦⣰⠂⢤⠀⢲⣠⠞⣡⠞⠀⠀⠀⠀⣼⠁⠀⠈⠳⢮⣄⣀⡤⠞⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠰⡄⠂⠀⠀⠐⠠⠂
⠀⠀⢙⣶⣬⣭⣉⣩⣭⣴⣾⠁⠀⠀⠀⠀⠀⠹⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠟⠀⠀⠀⢀⠉⠀⠀⠉⠀⠈⠀⠀⠉⠀⠀⠉
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠚⢛⡛⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠛⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠸⠇⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢠⠄⠂⣤⡤⠒⠋⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠛⠛⠀⠀⠀⢿⡀⢀⣼⠃⠀⠀⠚⠛⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠙⣷⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠳⣄⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠋⠀⠀⠀⠈⠻⢦⣤⣤⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⢠⡞⢲⡄⠀⠀⠀⠀⠀⠀⢀⣤⠴⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣷⠀⠀⠀⠙⢦⣀⠀⠀⠀⠸⡇⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣇⡀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠃⠀⠀⠀⠀⠀⠉⠛⠲⠶⢤⠤⠴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡏⠙⠳⣄⠀⠀⣯⠀⠀⠀⢠⡄⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣦
⠀⠀⠀⠀⠀⠀⢸⡇⢸⣇⠀⠀⠈⠳⣄⢿⠀⢠⡀⠀⠹⣄⠸⡆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡿⡆⠀⠀⠀⠙⣿⡇⠀⠻⣄⠀⠙⠶⠟⠛⠋⠉⢳⡶⠒⠒⠒⠶⠶⠶⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⠇⢸⠇⠹⣄⠀⠀⠀⠘⣿⣄⠀⠙⣆⠀⢀⣀⣠⡤⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠦⣤⡀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⢠⡄⢰⠀⠀⠙⢦⡀⠀⠀⠸⡟⢦⡀⠈⠉⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⣸
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠙⠶⣄⡀⢻⡈⠛⢦⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⢿
⠀⠀⠀⠀⠀⠀⢸⣧⣼⣧⣤⣤⣤⣤⣤⣤⣭⣿⣧⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣻
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⢀⠀⠀⠘⠐



⡏⠉⡷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⡏
⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠀⡔⠒⢤⠠⡄⠒⡖⠀⠰⡄⠀⡖⢢⠀⡄⠀⠀⠒⠂⠀⢠⡒⠂⠐⡖⠂⠴⠒⡄⣠⠒⠢⡄⢴⢠⠆⢰⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⡇⠀⡸⠠⡇⠀⡇⢠⠇⠾⠀⡇⢙⠄⡇⠀⠀⠃⠀⠀⠀⠉⡦⠀⡇⠀⠙⠘⡄⢧⠀⢀⠇⢾⠳⡄⢸
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣿⣿⣷⣿⣿⣶⣾⣷⣾⣷⣿⣶⣶⣶⣿⣿⣶⣿⣿⣿⣿⣿⣶⣾⣿⣷⣶⣿⣶⣿⣶⣿⣶⣿⣿⣶⣿⣶⣿⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣟⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣰⠟⢳⡀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠳⠏⠀⠘⠋⠉⠉⠁⠀⠉⠉⠉⠉⠛⠛⠲⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣇⠀⠀⠀⢀⣇⠀⠀⠀⣸⡀⠀⢠⡇⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⢳⢤⣸⠦⠤⠚⡀⠽⠶⠶⠾⠤⠛⠒⠒⠓⠹⠷⣯⠉⢹⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡧⠤⣄⠀⠀⠈⡆⠀⠰⠖⠁⠀⠰⠆⠀⠀⠀⠀⠀⠐⠆⠀⠀⠈⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠳⠬⠹⢦⣀⣀⠇⠀⠀⡠⡀⠀⠀⠀⠀⢀⠐⢄⠀⠀⠀⠀⠀⠊⡀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠉⠊⠑⠂⠐⠂⠁⠀⠀⠈⠁⠀⠀⠴⠪⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠺⣏⢳⡄⠀⠀⠀⠀⠀⢀⡼⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠀⠀⠀⢀⣠⠴⢋⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢀⣠⣾⠀⠀⠀⠀⡄⣠⠛⠦⢤⣀⡐⠶⠤⠤⣤⠤⠤⠶⠶⠒⠛⠉⠀⢀⡼⠳⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠋⠁⠘⠦⢤⠤⠞⣹⠃⠀⠀⠀⠈⠉⠉⠙⠛⣿⠦⣤⣤⠴⠚⠒⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠠⢀⣠⠴⠒⠒⡆⠴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢇⠀⠀⠀⢣⣰⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⡆⠀⠀⠸⡇⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠘⡆⠀⠀⢿⠀⢻⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣘⣆⠀⠸⠦⣤⠭⣽⣦⣤⡞⣷⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣠⠔⠊⠉⠁⠈⠓⠒⠒⠚⠒⠲⠦⠞⠀⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠷⠶⠶⠶⠶⠤⠤⢤⣤⣤⣤⣤⣤⣄⣸⠛⠋

⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⠆⠀⠀⠀⠀⠀⠀⢰⡖⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶
⠀⠀⠀⠀⠑⠄⠀⠀⢀⠀⠀⠀⡀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠛⢛⣛⣛⣛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⢀⠄⠘⣀⢀⣐⠃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣭⣭⣭⣤⡀
⠀⠀⠀⠀⠀⠀⢸⠂⠡⠀⠀⠀⠉⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡻⠍⠉⠽⡻⣿
⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⢤⠔⠁⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠂⠀⢀⣀⠔⠘⣼⡆
⠀⠀⠀⠀⠀⠀⡀⢁⠀⠈⠚⠀⠀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⡀⠀⠘⠊⠀⢀⢽⡇
⠀⠀⠀⠀⠀⢀⠔⢄⡐⠀⠄⠐⢀⠴⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢮⣢⣤⣤⣔⣮⢟⡀
⠀⠀⠀⠀⠀⢈⣀⣀⣈⣁⣀⣈⣁⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣭⣭⣿⣾⣿⣷

⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣀
⠀⢸⣿⣏⢻⣯⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠀⠀⠀⠀⠀⣿
⠀⢸⡿⠿⣶⢿⡿⢿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠄⠀⠀⠀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⣿⠀⠀⠀⠑⠀⠈⠀⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⠀⢰⠂⠀⣿⣀⣀⣀⣀⣀⣈⣀⡟⠀⠀⣀⠀⠀⣴⣿⣷⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⠀⣠⣤⡘⠃
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣠⣤⡀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⣽⣿⣿⣤⣾⠟⠁⠀⠀⠈⢧⣀⣀⢐⣿⣿⣷⣶⣦⡄⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠉⠁⠀⠀⠀⠀⠀⣶⣿⣿⣾⣿⣿⣿⠿⣿⣿⣦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣿⠻⠛⠻⠛⠟⢿⣶⡿⠿⢥⡤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⣿⠀⠀⠀⠀⢠⣾⠉⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠋⠀⠀⠀⠀⠈⠙⠀⠘⠀⠨⠄⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⡦⣴⣶⣶⣶⣶⣶⡆⠀⠀⡀⠰⠇
⠀⠀⠀⠀⠀⠀⡀⠀⣰⣿⣿⠿⣿⠇⠀⠀⠀⠀⠀⢠⣶⣶⣴⣶⣿⣿⣿⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠘⠃
⠀⠀⢀⣤⣤⣿⣿⣧⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⣿⡷⢿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢾⣿
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣮⠙⣿⡀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠙⠁⠀⠀⠀⠀⠘⠿⠟⠻⠿⠛⠸⠿⠿⠿⠿⠿⠿⠿⢿⡀⣾⡀⠀⠀⠀⠹⡿⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠐⠀⢀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠃
⠀⠀⠤⠀⣴⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠠⠀⢰⣾⣷⡄⠀⠀⠀⠀⠀⠀⢠⣤⢀⡀⠀⠀⠁⠁⠀⠀⠀⠀⠀⣀⡀⠀⠸⠄⠀⠀⠀⠠⠆
⠀⠀⠀⠀⣙⣿⣿⣤⣀⣠⣤⠀⠀⠠⠀⠀⠉⡈⠁⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣧⣄⣀⣤⡤⠴⠒⠛⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣆
⠀⠀⠀⣻⣿⣿⣿⣿⣿⣯⡁⠀⠀⠀⢠⣤⣼⣶⣤⣤⣤⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣍⠀⠀⣴⣶⣾⣿⣶⣶⣶⡆⠀⠀⠀⣤⣾⣿⣿⣿⣿⣶⣄
⠀⠀⣬⣭⣭⣭⣭⣭⣭⣭⣥⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣭⣭⣭⣭⣭⣭⣭⣭⣤⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⠛⠛⠛⠛⠛⠛⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠙⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠉⠉⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⣶⣶⣤⣤⠀⠀⢠⡶⢶⣶⠶⠶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣶⢶⣾⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠉⢀⣀⣀⣀⣀⡀⠈⠉⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⢀⣤⣶⠿⠛⠛⣯⡙⠛⠛⠿⣶⣄⡀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⢀⣴⡿⠋⠸⣷⡀⠀⠿⠃⠀⢰⡿⠀⠙⢿⣦⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⣾⠏⣶⣄⠀⠈⠀⠀⠀⠀⢀⣤⠀⢀⣴⡦⠻⣷⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡟⠀⠈⠉⠀⠀⠀⠀⠀⣰⡿⠋⠀⠈⠉⠀⠀⢿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⢸⡇⠶⡶⠆⠀⠀⠀⠀⠸⠟⠁⠀⠀⠀⠰⠿⠷⢸⡇⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢸⣧⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣼⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⢿⣆⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠛⠟⣰⡟⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠈⠻⣦⡀⢠⡿⠃⠀⣴⡄⠀⠸⣿⠀⢀⣴⠟⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠻⢷⣦⣤⣄⣻⣃⣠⣤⣴⡾⠟⠁⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⠉⠉⠛⠛⠉⠉⢁⣠⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣽⡿⠻⠿⠿⠿⠿⠟⠻⣿⣥⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⢶⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⠆⠘⢠⡌⠁⠰⠏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠤⠤⠤⠤⢤⣾⠉⠓⠀⠀⠀⣼⣧⠀⠀⠀⠉⠉⣷⡤⠤⠤⠤⠤⠤⠤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠀⠀⠀⢀⡿⢿⡄⠀⠀⠀⠉⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠃⠀⠀⠀⠘⢦⡴⠃⠀⠀⠀⠈⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣞⠁⢀⠀⠀⠀⠀⡀⠀⣳⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠤⢼⣧⠤⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡖⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⠿⣿⠿⢛⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣛⣻⣿⣿⠁⣾⣷⠀⣾⣿⣟⣛⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⢠⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣬⣥⣌⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣈⣙⣛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⠀⠀⠀⠈⠻⣿⣿⣿⣇⣻⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⣦⣄⠀⠀⢸⣿⣿⣿⡙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢹⣿⣿⣧⡀⠀⣿⣌⢿⣿⡌⢿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⢿⣿⣿⣿⡄⠸⣿⣆⢻⣿⣶⣶⣿⣿⠿⢠⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠻⣿⣿⣿⡄⠙⣿⣧⣝⡉⣩⣵⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠈⠻⢿⣷⠀⠈⠻⠿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠝⠙⠛⠛⠙⠿⠿⠿⠿⠿⠿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠀⠀⠠⠀⠀⠄⠀⠠⠀⠄⠀⠀⠀⠠⠀⠀⡄⠀⠀⠄⠂⢄⠀⠀⠠⠀⠀⠠⡀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠃⠀⠰⠀⠁⠀⠀⠀⢀⠀⠀⡀⢀⠘⡀⠀⡠⠀⠂⢂⠀⠀⢀⠄⠈⠀⠀⠂⢊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡞⢐⣆⣤⡸⢦⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡖⠐⠒⠾⢤⡈⡹⠖⠋⠀⠀⠀⠁⠈⠈⠀⠉⢓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠚⡲⡄⢈⠟⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⡉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣹⠁⠀⠀⠀⠐⢉⠤⠀⡀⠔⠒⠠⠀⠀⣠⠀⠀⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⠁⠀⠀⠀⠈⠑⢤⡀⠀⠀⢰⣲⠋⠀⠀⠀⣠⠖⢀⠂⠀⡀⠸⠧⠤⣔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠑⣀⣠⠚⠁⠀⠀⢁⣴⠋⠂⢨⡗⢚⠦⠄⠀⡀⡀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡟⠷⠀⠀⠀⠀⠀⠀⠀⣠⡴⢿⣿⢿⣿⣷⣦⣀⡾⠛⠉⠀⠉⠉⠉⠀⠐⠠⣄⠋⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠠⠀⢸⠐⡀⠀⠀⠀⠀⣼⡿⣧⡘⣧⣞⣩⣤⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐
⠀⢀⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⡀⠊⠀⠀⠀⢸⠀⡇⠀⠀⠀⢸⣿⣀⣡⢽⣴⢯⡐⢆⣻⡃⠀⠀⠀⠀⠀⠀⠰⡄⢸⢄⢀⠁⠀⡀
⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠊⠀⢀⡴⠀⢀⡏⢠⣀⣀⡤⠔⠚⠉⠉⣹⣿⢛⡄⠋⠀⠀⠈⢢⠀⠀⠀⢀⡀⠶⠛⢪⢀⠏⡠⣚⠁⡀
⠀⠀⠀⠈⡢⠄⠀⠒⠒⠀⠀⠁⠀⡀⠤⠚⠕⠀⢀⠞⡠⢚⠍⠉⠉⠁⠂⢀⡞⢡⣿⠋⠀⠀⣀⣠⠄⡪⠓⢏⠁⠘⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠠⢋⡎⠀⠂⠀⠀⠀⢀⢀⡎⢀⢀⠃⠀⢀⣞⡿⣮⡟⠀⠀⠀⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠖⠈⠀⠀⠀⠀⠀⢸⡄⠀⠂⣀⡀⠀⠀⡸⢀⠀⠸⡀⢀⡊⠐⠛⠉⠀⠀⠀⡐⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⡀⠀⠂⠀⠻⣦⣀⣀⣀⣀⠴⠣⠂⠀⠀⠙⢯⡀⠀⢀⡀⠠⠐⡈⠔⠁⠀⠀⠐⠠⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠈⠀⠀⠀⠀⠀⠀⠁⠀⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠉⠉⠉⠛⠳⣦⣤⣄⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠛⠹⣷⢿⡀⠀⠀⣀⠘⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣀⣤⠖⠋⠉⠻⣦⣀⡀⠀⠀⣼⠀⠀⠀⠉⢀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣀⠀⠀⢠⠞⠉⠀⢀⣠⠴⠶⠀⠉⢉⣿⣟⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣠⠈⢳⡀⢻⡀⠀⢀⡤⠴⠚⠃⠀⣴⢿⡙⣿⣆⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⣅⠘⣧⣼⠇⠀⠀⠴⠿⠁⢀⣴⡟⣦⠛⠉⣿⠀⠀⠀⠀⣤⡄⠈⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⠀⠀⠀⠀⠀⠈⠛⠁⠀⢸⡇⣷⠈⠁⠀⣿⠀⠀⠀⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣇⠀⠀⠀⠀⢀⣀⣀⠘⠛⠀⣿⡀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠓⠿⢦⣀⠀⠘⠋⣭⠉⠀⣀⡼⠋⠛⢶⡛⠉⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠓⠦⣤⡀⠙⠛⣶⡶⢶⣶⡞⡿⢦⡀⠀⠈⢳⡄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠙⠳⠞⠙⢿⣉⣿⠛⠁⠀⠙⢦⣤⣼⠇⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠙⡆⠀⠘⣷⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠸⡇⢸⡄⠀⠘⣧⡤⢤⣤⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣧⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣷⣤⠴⠶⠶⠚⠛⡻⣆⠀⡇⠀⢸⡆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⣷⠀⠀⠀⡏⠀⠀⠀⠀⠀⠘⣇⢻⠸⣇⠀⠀⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢻⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡘⡇⠹⣦⡼⠃⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⡇⠀⠀⢸⡆⠀⠀⠀⠀⠀⠘⡇⢻⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢹⡶⠒⠋⠻⣦⡀⠀⠀⢀⣠⡿⢸⡖⠛⢳⣦⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢰⡇⠀⠀⠀⠈⢿⣞⣋⣩⡥⠴⠛⠉⠉⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⣛⣻⣧⣀⣀⣤⣤⠾⠏⠁⠀⠀⠀⣠⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢹⡏⠁⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠀⠀⠀⣀⣀⣿⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣴⣾⣛⠉⠽⢿⣷⣤⡴⠶⠚⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⣠⠶⣿⡷⢤⣬⣁⠀⠉⡷⣦⣤⣬⣤⣬⠽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠛⠛⠛⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⢋⣥⣶⣦⣍⡛⣿⣿⣿⡟⣟⢿⡟⣿⢛⣿⢻⡟⣛⣛⣿⢻⡿⢻⢻⠻⣿⡟⡿⢛⣛⣿⣿⡟⣛⣿⢟⣛⣻⣿⠿⣿⡟⣿⡟⢛⣿⣿⣿
⣿⣿⡿⢰⣿⣯⣿⣿⣿⣿⡘⣿⣿⣿⣌⣌⣰⣿⠰⢾⣸⣆⠻⢉⣿⣰⣆⣸⢸⣰⣮⣁⣇⡻⢍⣽⣿⣿⠆⣹⡘⠟⣻⣏⣐⡘⡇⠿⣇⠰⢾⣿⣿
⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⠿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣦⡙⠿⠿⠿⠟⣃⣾⣿⣿⣕⢟⣿⢺⣧⣟⣣⣿⣟⣈⣳⣛⣼⣸⣷⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⠿⠿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⢶⠶⠶⠶⠖⠀⠀⠐⠶⠶⠶⢶⠶⠶⠶⠖
⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠰⠆⠀⠀⠀⠀⠀⠶⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⠴⠋⠉⠀⠀⢀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢤⠤⠤⣤⣤⠤⠤⡤⠀⠀⠀⢤⣤⡤⣤⡤⠤⢤⠤
⠀⠀⠀⠀⠀⠀⠁⠀⢀⡴⠁⠀⠰⠀⠀⠶⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⢀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⠄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠰⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⡀⠀⠀⠀⠰⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⠂⠀⠀⠀⠀⠀⠀⠀⠒⣶⣶⣶⡖⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⡄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⢲⡄⣰⠀⢀⡖⠀⠀⠀⣀⡀⣀⠀⣀⡀⢀⣀⢰⣆⡀⢰⠀⣀⠀⣀
⠀⠀⠶⠢⠤⠀⠦⠤⠀⠴⠤⠄⠀⠲⢤⠄⠀⠶⢤⠄⠰⠔⠦⡆⠀⠀⢸⠿⣯⡷⠚⢿⠂⠀⠸⣭⡗⡿⠹⣭⡿⣿⣽⢿⡏⢿⢿⠺⣭⠌⣿⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢠⣤⣤⣴⢿⣧⣤⣤⣤⣤⣿⣤⡤⣤⣤⣤⡄⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣭⣿⣝⣿⣿⣎⢷⣽⡾⣼⣿⡿⣩⣶⣯⢻⣿⣷⢻⣿⣿⣳⣿⣿⢯⣭⣯
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣼⣿⣿⣷⣶⣾⣿⣿⣃⣏⣿⣾⣏⣿⣿⣷⣶⣾⣿⣿⣏⣿⣿⣏

⠀⠀⢀⠠⠒⠀⠀⠀⠢⢀⠀⠀⠀⡠⠆⠤⣢⡀⠀⠀⡢⠄⣠⠄⠀⠀⠀⠏⠃⠡⠤⠈⡠⡄⢠⡀⢠⡀⠘⠑⠀⠀⠀⠠⠂⠛⠓⡆⠀⠀⠀⠀⠀⢀⣻⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡼⠁⣶⢐⡅⠁⠀⠀⢩⠋⠉⠐⠄⠀⠀⠈⢌⠀⠠⠅⢨⡆⠐⡂⢀⠔⡡⠀⠀⠀⠀⢢⡁⠀⠁⠇⠀⠀⠀⢠⡾⠥⠤⠔⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⡣⠀⢀⠞⡡⠂⠀⠀⡄⠀⠀⠀⠨⡀⠀⠀⠀⠑⠨⢐⠒⠒⠒⡚⠅⠊⠀⠀⠀⠀⠀⡠⠀⠤⠄⠐⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠
⠀⠀⠀⠈⠐⠒⠒⠈⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⡈⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⡀⠀⠀⢠⠄⠲⠀⢄⠀⠈⠢⡀⠄⡰⠁
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⠄⠀⠠⢀⠀⠀⠀⠀⠐⢄⠀⠀⠀⣠⠂⠀⠀⢀⠠⢀⣀⠠⢀⠀⠀⠀⠸⣀⡙⠦⣀⠀⠈⠀⠀⠉⡀⠀⠀⡠⠀⠀⠀⠈⠈⡀
⠀⠀⠀⠐⣧⡜⠀⠀⠀⠈⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠲⢦⣠⣐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⡪⡃
⠀⠀⠀⠀⣸⡇⠀⠀⠀⠠⢔⣲⠒⢂⡢⠌⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢽⠿⠴⢅⡀
⠀⠀⠀⢀⠋⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡂⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡒⠊⣉⡉⠃⠂⠀⠀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠁⠀⠀⠈⠂⠀⠀⠀⠐⠀⠂⢀⠀⠀⠀⢀⠊⠁⠢⠄⠀⠁⠀⠀⠐⠄⡀⠀⠀⠀⡀⠄⠀⠀⢧⡠⢀⢋⣁⡤⠸⠀⢀⠍⠡⡀⠀⢆⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠃⠙⠢⠀⠀⠑⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠍⠉⠀⠔⠁⠀⡣⢲⠣⣿⡆⠈⠢⡀⢀⠀⢀⠔⠁
⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠦⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⢀⠤⠒⣾⡈⣴⠦⡀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠰⢲⢑⡐⠆⠀⠀⠀⠀⡈⠀⠀⣀⠁⠀⠀⠀⠀⣀⠀⠀⢻⡷⠤⠀⠀⠂⠐⢻⠗⠒⢯⠳⠘⡀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⢀⠃⠀⠀⢰⣄⣰⠻⢎⡜
⠀⠀⢀⠄⠊⠉⠑⠠⡀⠀⠀⠀⠐⠂⠀⠀⠂⠀⠀⣠⠊⢀⠀⡲⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠃⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠸⢍⣸⠠⠤⠋
⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠺⠀⠀⢌⡤⠀⠘⠁⠀⠀⠀⠀⢀⡀⠈⠂⠄⣀⣀⠠⠔⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⡠⠒⠉⠒⢄
⠀⠀⠢⢀⠀⠀⠀⡀⠊⠀⠀⠀⠊⠁⣀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⢹⡇⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠁⢀⠊⠀⠀⠀⠀⠀⠘⠄⠉⠀⠀⣄⠀⠀⠰⡚⠘⠳⢤⡤⢞⣰⠃⠀⠀⠀⠀⠀⠀⠀⣤⢀⠀⠙⠢⠤⠤⠒⠉⢀⣠⡤⠒⡇⠀⠀⠈⢣⢀⠖⠁
⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⠈⠒⠬⠶⠖⠋⠁⠀⠀⡀⣤⣲⢒⢝⣧⠼⡎⠉⠁⠀⣀⣤⣲⢪⢏⣹⡰⠚⠁⠀⠀⠀⠀⣻
⠀⠀⡤⠀⢀⣀⣀⣀⣀⡠⠄⠊⠁⠙⢯⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⠎⠯⣱⡣⠶⠊⠁⠀⠀⠀⢰⡦⠚⣵⡰⠧⠛⢉⡀⢀⢀
⠀⠀⠈⠒⠠⢀⡀⠀⠀⠀⢀⡀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⠉⠁⠀⠀⠀⠀⠀⠀⠠⠓⠘⠇⠀⠀⠀⠀⠇⠼⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠉⠉⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠐⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⢄⠀⠇⠀⠀⠀⠀⠀⠀⠁⠢⠀⢀⠈⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠘⠀⠀⠐⠀⡴⠃⠀⠈⠀⠀⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣰⠇⠀⠑⢢⠀⠀⢀⡁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⢠⠔⠁⠠⠄⠀⠘⠀⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⢐⠀⠀⠀⣈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⡡⠋⠀⠀⠀⣿⠀⡐⢠⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⢒⠈⠀⠠⣠⡮⢄⢀⡀⡇⠀⠁⠀⠂⠄⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠁⠀⣀⠠⡀⠂⢈⣂⣀⣜⣩⠃⠀⠀⠀⠈⠒⠊⠀⠐⠧⠤⣔
⠀⠀⠀⠀⠀⠀⠄⠒⠀⠈⠀⠐⠂⠘⡿⠋⠁⢹⠈⠐⠤⠄⢀⠀⠀⢄⡠⠶⠫⠇
⠀⠀⠀⠀⠈⠀⠀⠂⠐⡀⢀⡀⠀⠂⣀⠀⠀⠌⠀⠀⠐⢠⡤⡐⠢⠡⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢒⠀⠀⢈⠀⢀⣇⡄⠐⠈⢠⠦⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢁⠔⠁⠄⠀⠨⠛⠺⡧⠀⠀⠀⠘⣀⠊
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠠⠂⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠄⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠠⠊⢀⡠⠄⠁⠀⠀⠀⠀⠘⠁⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠀⠀⢀⠀⠀⠀⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠇⠀⠀⠀⢀⠀⢀⡀⠠⣀⡈⠹⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣰⠇⠰⣎⠁⠀⠘⣉⢳⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⢁⡠⣤⣾⡏⠀⠀⣛⠋⠛⣿⠛⠛⠳⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡀⠊⠈⠛⣡⠟⠀⠐⠛⠷⠒⠀⠀⠀⠀⠀⠀⠉⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠀⣾⠁⠀⠀⠀⠈⠃⠘⠀⠀⠀⠀⠀⠙⣦⣦⣴⠶⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⣰⠟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠘⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⡀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠁⢠⣿⠂⠁⠀⠘⣷⠀⠀⠀⠀⠀⠀⠰⡾⠿⠾⠟⣩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⢻⣤⡶⣶⠶⠶⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠄⠻⣦⠀⠀⠀⣼⡟⢱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠹⣧⡀⣴⣯⣀⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡘⢿⣈⣿⣿⣿⣆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠈⣿⢻⣏⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡈
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡄
⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢾⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣠
⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣴
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠯⠄⠀⠀⣄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠶⠶⠶⠂⠻⣿⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠛⠿⢶⣤⣤⣤⣄⣰⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠶⠆⠀⠀⠀⣾⡄
⠀⠀⠀⠸⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⣀⡀⠀⠀⠀⠀⠀⢀⣠⣿⡏⢀⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠛⠿⠿⣶⣤⣤⣀⣀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣿⣏⡛⠛⠛⠿⠟⢳⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠿⠿⠿⠿⢿⣿⣿⣿⡿⠿⠿⠛⣸⠟⠛⠛⠛⠛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢸⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠑⣀⠃⠀⢻⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⣠⡾⢃⣰⣷⣾⡿⣫⠤⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢆⡙⠷⣦⣀⡀⠀⠀⠀⠀⠀⠙⠆⢻⣷⣾⣿⣯⣭⣴⣭⣤⣶⣤⣀⣀⣀⣀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠪⣷⡶⠋⢩⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠋⠉⠈⠉⠉⠉⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠶⠈⠳⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⡀⠁⠀⠀⠀⠀⢀⣀⣴⡆
⠀⠀⠀⠀⠀⠰⠦⣙⣉⣿⡇⠀⠰⠶⠶⠂⠉⠁⣀⣼⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢷⣶⣿⣶⣾⣿⣿⣿⣧⣄⣈⡁⠚⠛⠓⠂⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⣌⣙⣿⣿⡿⠛⣉⡛⢛⣛⣻⣿⣶⣶⣦⠠⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⡉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⠿⠛⠛⠛⠛⠉⠉⣉⣉⣉⠉⠙⠋⠙⠋⠉⠙⠉
⡇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢒⣠⠄⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠓⢠⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠚⠉⠀⠉⠉⠒⠲⣬⣍⡁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠄⠀⣀⠃⠀⠀⠀⠀⠀⢠⣦⡀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢿⣿⠃⠀⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⡀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠠⠄⠘⠛⠛⠛⠛⠛⠓⠦⠦⣄⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⡌
⡇⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡷⠀⠀⠀⠀⠉⣴⠂⠀⠀⠀⠀⠀⠀⠀⠉⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⣠⣦⡀⡸⠁
⡇⠀⠀⠀⠀⠀⠉⠀⠀⠒⠶⠶⠶⠶⠶⠶⠀⠈⠙⠛⠶⠦⣄⣠⣤⣀⠰⠶⠂⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⡷⠀⠀⠐⠒⠂⠀⠀⠀⠀⣶⠁⠀⠀⢹⣦⣤⠀⠀⠀⠀⠀⠐⠈⣿⣤⣤⣤⣤⣤⣤⣄⢀⡐⠂⠳⣆⣀⣤⡤⠀⠀⠀⡄
⣟⠀⠀⡇⠀⠠⡀⠀⠀⠀⠘⠶⠀⠖⠁⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⡆
⣿⠀⠀⡇⠀⡆⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠀⠀⠀⠀⠀⠀⠀⡇
⣇⠀⠀⡇⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇
⡿⠀⠀⡇⠀⠀⠇⠀⠀⠈⠳⡀
⣷⠀⠀⡇⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸
⡏⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣷⠀⠀⠘⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣷⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣿
⣿⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣿
⣿⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⣿⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣧
⡟⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣧⠀⠀⢸⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⡇
⡟⠀⠀⢸⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢁
⣧⠀⠀⢘⡀⠀⠁⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸
⣿⠀⠀⠈⠙⠶⠶⠶⠶⠶⠒⠶⠶⠶⠦⠤⣠⣄⣀⣀⣀⣀⣀⣀⡀⠴⠶⠖⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⣠⡴⠖
⣷
⣿⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣠⣤⣤⣤⣤⣄⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡿⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⢠⡀⠀⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣧⠴⣾⠀⣸⠁⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⠀⢸⠇⠀⡟⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⣠⠞⠃⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡟⠀⠘⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣼⣇⣠⠞⠁⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢤⣄⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠛⠓⠀⠀⠀⢻⣄⣨⠇⠀⠀⠀⠛⠓⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣟⣹⠟⠁⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⡏⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⣰⣯⡴⠞⠁⠘⢦⡀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⣠⣄⠀⠀⠉⠁⠀⠀⠀⠀⠀⢉⡛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣤⣤⡤⠴⠞⠋⠁
⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠛⢦⡀
⠀⠀⢈⡃⢈⡗⢦⣄⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢸⡇⢸⣇⠀⠈⠳⣄⠀⣿⠀⢀⠀⠈⢧⡀⢹⡄
⠀⠀⢸⡇⢸⣿⡀⠀⠀⠈⢳⣽⡄⠘⢧⡀⠈⢳⣀⣷⣤⡤⠶⣤⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢸⡇⢸⡏⢷⡀⠀⠀⠀⠻⣷⡀⠈⢳⡀⠀⠉⠀⠀⠀⣀⣨⠁⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠲⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢸⡇⢸⡇⠈⢳⡄⠀⠀⠀⢹⣷⡀⠀⠛⠦⢶⡶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⣤⡀⠀⠀⢸
⠀⠀⢸⡇⢸⡇⠀⠀⠙⢦⣄⠀⠀⢷⠙⢦⡀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢸
⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠈⠛⠶⣼⡆⠀⢙⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼
⠀⠀⢸⡏⠋⠙⠙⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⢻
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡷⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⢾
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸


⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠓⠚⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠈⠓
⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⠄⢤⠀⠀⠤⠄⢠⡀⢠⠤⡄⡄⠀⡤⠄⠀⣠⠀⠠⠤⠀⠄⢄⢠⠀⢄⢠⢠⠀⡤
⠀⠀⠀⠀⠀⠀⠀⢸⣀⢰⣁⡠⠀⠀⢀⢠⠃⢲⢸⣉⠆⣇⡀⣏⣀⠀⢀⣱⠀⠀⠀⠋⢃⠠⣀⡼⢸⠳⡄⣏⡀
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢤⠰⠘⠀⠀⠀⠀⠀⣴⣾⣿⣿⣶⡄
⠀⠀⠀⣰⠿⠿⠿⠿⠛⢳⡀⠀⠀⠀⣨⣿⠿⣷⣄⣿⠀⠀⢀⣠⣤⣿⣤⣀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡇⠀⠤⠀⠠⠄⠀⡇⠀⢸⡟⠉⢉⡶⣾⣟⣿⠀⠀⢸⣾⣿⣿⣿⣾⠷⡄⠀⠀⢹⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠘⢄⡀⠘⠀⣠⠞⠀⠀⢸⡇⠀⣼⠁⡠⠻⡄⠀⠀⠈⢿⣿⣿⣿⠟⠛⠁⠀⠀⠀⠛⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠉⠉⠁⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠀⢏⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⢠⡎⢩⡄⠀⢀⠙⢲⠀⠀⠀⢀⡿⣿⣿⢷⠀⠀⠀⠀⠀⡤⠤⠤⠤⣤
⠀⠀⢸⠀⢀⡞⣿⡟⣆⠀⢀⠀⢸⠁⠼⠅⣟⠼⠅⣸⠆⠀⠀⠘⢁⡿⣿⠘⠃⠀⠀⠀⠀⠣⠤⠤⠤⠟
⠀⠀⢸⠿⠿⢿⡿⣿⠿⠿⢿⠀⠀⣽⠲⢤⡤⠤⠴⠃⠀⠀⠀⠀⢸⡇⣸⠂⠀⠀⠀⠸⠰⠇⠁⠲⠥⠧⠸⠅
⠀⠀⠈⠀⠀⠈⠁⠁⠀⠀⠈⠀⠈⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠰⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣿⡿⣷⡄
⠀⠀⠀⣤⣤⣬⣤⣤⣤⡀⠀⠀⠀⠀⢸⣹⣿⡟⠀⠀⠀⢀⠀⠀⢲⢀⡀⣄⠀⠀⠀⠀⣠⣿⣷⣶⣶⣿⣿
⠀⠀⠀⢻⣿⢹⣿⣿⣿⣩⠆⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⡇⠀⠓⠸⡇⠓⠀⢀⠀⠀⠸⣿⣿⣿⣿⡿⠃
⠀⠀⠀⢀⡙⢿⣿⠟⢁⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⢻⠀⠀⠀⠉⢻⣿⣿⡇
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡁
⠀⠀⠀⠀⠀⣤⡿⡄⠀⢰⡆⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣄⠀⠀⠀⠀⢀⣼⣿⣧⡀
⠀⠀⠀⣀⣰⣿⡟⢻⣷⣸⡇⠀⠀⠀⠀⢿⢛⡿⠀⠀⠀⢀⣾⣿⠿⠟⠋⠈⠻⣧⠀⠀⠀⣼⢹⣿⡏⣧
⠀⠀⠀⣿⠀⠀⣾⢻⣿⣽⠇⠀⠀⢀⣴⣶⣾⣷⣦⠀⠀⢸⣷⠀⠐⢀⡐⠂⠀⣿⡇⠀⠀⠀⣸⠛⣧
⠀⠀⠀⣟⠀⠸⠇⢠⠊⢢⠀⠀⠀⣾⠃⣿⣿⣿⠿⠇⠀⠸⠿⠷⢄⣈⣁⡤⠾⠿⠃⠀⠀⠀⢹⠄⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠜⢃⠀⠀⠡⣰⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠃⠀⢿⢲⡦⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠸⡷⠤⢀⡒⠲⢽⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠢⡀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠠⢀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢹⡀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠉⠀⠀⠈⠀⣄⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⡗⠲⡄⠈⠀⠀⠠⠌⢀⣦⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣏⡄⠀⠀⠀⠀⠀⠀⠈⡄⢰⠀⠀⠀⠀⠀⣼⡇⡀⠘⠅⠙⠤⠤⢄⠤⠌⠂⢄⣉⣿⡋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠄⠀⠀⠀⠀⠀⠰⠧⠄⢷⣄⡀⣠⡖⢠⠀⠐⢃⡀⢊⣉⢐⡘⢒⡞⣓⣉⣍⣴⠃
⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⢀⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣯⣀⣰⠟⠙⣏⣲⢐⡶⠒⡯⢭⣯⣽⣧⣴⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠳⠶⢾⣶⣤⣤⣤⣄⣐⣂⣤⣦⣼⣿⡁⠀⣹⣶⡶⠟⠉⠉⠙⠋⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⢿⣦⡀
⣤⣄⡀⠀⠀⠀⣠⠎⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁
⠷⣼⡳⠀⢀⣼⡟⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣉⠀⠀⠀⠀⠀⣠
⢢⣻⡼⢿⠈⠉⠁⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⢀⡀
⣶⣿⣦⡄⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣇⠀⠀⠀⠀⢸⡇
⣿⣿⣁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢀⠀⠀⠈
⠈⠿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠃⢞⠃⠠⣤⣄
⡀⠀⢰⠁⠸⢹⡷⠤⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠸⣷⠀⢠⣤⢠⢻⡇⢠⣿⣿
⡗⠀⣿⣤⡆⠘⢁⣿⣄⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡬⣄⣁⡀⢿⡀⠈⣉⡉⠙⠧⡶⠋
⣇⠀⢿⠋⢻⡶⣿⣿⡏⠀⣴⡍⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⣫⣤⣿⢿⡆⠀⢻⡄⣿⣿⣿⠟
⢿⠗⠸⣿⢉⣴⢺⣿⣷⡀⢻⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⣹⡿⠿⣿⢇⣭⡀⠜⢃⣸⣿⠋⠀⣀⣤⡀
⢸⡇⠀⢹⡀⣿⣤⣟⣿⣧⡀⠻⣄⣀⠁⠄⢀⠀⢀⣄⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⢟⣿⡟⠁⠀⠀⠰⣌⠿⣇⠀⢈⡽⠁⠀⠈⢫⣿⣧
⠀⠹⣏⠀⠛⢼⣿⣿⣏⣿⣷⡤⠙⣿⡀⡠⢈⣿⡘⠣⠀⢨⡏⣼⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⣾⣿⣟⣵⣿⣿⠀⠀⠀⠀⠀⠙⣻⣯⢷⣿⠃⠀⠀⠀⣸⣿⡿
⣦⡁⢿⣕⣆⡀⠙⠿⠋⠁⢰⣆⠒⠂⡠⠀⢼⣿⣿⣿⡷⠋⣠⣿⢣⣯⣜⡡⠍⠒⠂⠀⢠⣿⢟⣽⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢾⣿⣧⣿⠃⠀⠀⢀⣾⣿⡇
⡉⣭⣈⢉⣉⣁⠀⠀⠤⣀⡀⠉⡦⠈⠠⣾⠀⠤⠋⡩⠴⢿⣋⣽⣿⡿⠤⠔⣖⡱⠶⣢⡎⠱⠋⠈⢈⣉⣋⠀⠀⠀⣀⠀⠀⠀⣿⣿⣿⣇⡀⠈⢲⣾⣿⡿⠁
⠀⢈⣽⠁⣍⠋⠐⠢⠤⠈⠛⠀⢠⠂⠀⣤⠀⢰⡶⣠⡰⣿⣿⣿⣿⡻⣷⣾⢿⣲⣶⣿⠠⣤⢀⣀⣀⣿⣿⣿⣦⠂⣿⣦⣀⣰⣿⣿⣿⠏⠀⠀⠀⠈⠁⠀⢀
⣿⣿⣿⣿⣿⣿⣾⣷⣾⣿⣷⣶⣾⣶⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣷⣶⣶⣶⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠉⠄⠊⡀⠄⠐⠂⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡠⠄⠀⠗⠁⠈⡠⢎⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢁⣀⠀⠀⢀⡴⠀⠀⢀⢀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⡁⠀⠀⠀⠀⠀⠘⠋⢉⠀⠀⠀⠀⠀⡁⠒⠒⢠⠀⠀⠠⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⠂⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠈⠀⠀⢈⠀⠀⢀⢀⣠⡅⠀⢰
⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠀⡀⠀⠤⡀⠠⠂⠀⠀⣠⠁⠀⡠⠁⠁⠂⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠘⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⢀⠂⡱⢤⠈⡡⠄⠐⠒⠛⡋⠀⠈⠁⠉⠁⠀⠀⠀⠉⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢠⠀⠀⠀⠂⠀⠀⠈⠐⠀⠐⠈⠤⠨⠀⠰⢊⠴⢦⠀⠀⠀⠆⠀⢀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⣀⣀⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⢀⠄⠀⠝⠁⠂⠀⠀⠀⢀⡀⠀⠀⠀⠀⠒⠛⡄
⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠀⠂⠀⡀⠉⠖⢚⠛⠛⠒⠒⠀⠤⠀⢀⡀⣀⣠⡤⠤⠶⠒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠤⠤⣀⣀⡀⣀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠀⠀⠴⠤⠀⠒⠈⠉⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒⠂⠀⠀⠀⠭⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⠀⠀⠀⣐⡂⠀⠭⠀⠀⠀⠀⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠄⠀⠒

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣬⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠄⠀⠰⠂⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⢰⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄
⠀⠀⠀⠁⠀⠀⠀⠀⢠⢰⢇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠠⡈⡞⣞⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠧⠀⠀⠀⢲⠰⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠆⠅
⠀⠀⠔⣉⡒⠀⠀⠀⣠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠐⡋⠁⠉⠂⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⡀⠎⠠⠀⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠐⡠⡈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⡂⠀⠀⠀⠀⠠⠀⠀⡀⠆
⠀⠀⠐⠦⠴⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⡀⠀⠢⠄⠀⠀⣀⣀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⢀
⠀⠀⢀⡄⠀⠀⠀⠀⠈⠐⠈⠁⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⡾⡴⢬⢱⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢠⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣴⢀⡤⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠤⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠰⡀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠛⠐⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠆⠀⠀⠒⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠀⠅⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠄⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢦⣀⣀⡄⠤⠋⠀⠀⠀⢠⠠⠤⡤⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠌⢀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠑⠠⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢠⡾⠀⠀⠀⠀⢁⡀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠐⠘⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⢠⠀⠀⠀⠀⠃⢠⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠆⠀⠀⠀⠠⠄⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠄⠀⠀⠂⡀
⠀⠀⠀⢀⠠⢶⠀⠀⠀⠀⢀⠐⣦⡀⠀⠀⠘⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠰⠀⠀⠀⢀⣶⣦⣤⣤⠀⠄
⠀⠀⠀⠀⠑⣌⠃⠀⠀⠀⠐⠃⠈⠀⠀⠀⠈⠉⠀⠀⠀⠰⡦⠀⠂⠀⠀⠀⠠⠂⠀⢀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠿⣛⣷⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠿⠿⠛⠛⠛⠿⠿⢶⣶⣤⣀⢀⣀⣀⣀⣤⣤⣤⣤⣶⣶⡶⠶⠶⠶⠶⢶⣦
⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⢀⣼⠟⠁⠀⠀⠀⠀⢀⣤⡶⠟⠛⠉⠉⠉⠈⠉⠉⠛⢷⣄⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⣾⠏⠀⠀⠀⠀⢀⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⠏⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠙⠻⣆⡀
⣸⡏⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⣿⠃⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⢹⡇⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠐⠂⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀
⠈⢿⣄⠀⠀⠀⠸⣆⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀
⠀⠀⠙⢷⣦⣄⠀⠈⢷⡶⠶⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠒⠒⠒⠲⠶⠶⠖⠒⠴⠶⠀⠀⠙⠻⠿⢶
⠀⠀⠀⠀⠈⠙⠻⢷⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⢉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⢳⡀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡞⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡴⠦⠀⣤⣠⡶⠚⢶⠶⠶⢦⣤⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⣄⠀⠀⠀⠀⡸⠃⠘⠷⣤⡀⠀⣠⣄⠀⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⣾⠛⠳⠶⢦⣀⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⢸⡇⠀⠀⠀⠈⠻⠾⠋⠙⠛⠛⠷⠶⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠉⣿⠀⣸⠛⠻⣦⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠘⢷⠤⠶⠤⠴⠒⠚⠛⠀⢿⡆⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⡀⠙⠶⠾⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠉⠉⠉⢙⣷⠆⠀⠀⠀⢰⡏⠙⠲⢦⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠇⢀⣴⣟⣁⡀⠀⠀⠀⢸⣧⣤⣀⣤⣿⠇⠰⣟⠉⠳⣆⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⣟⣁⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢈⣩⡝⢛⠛⣛⣛⡋⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠉⢿⠿⣿⡟⠉⠻⡆⠀⠀⠀⠀⢠⡞⠉⠉⠙⠛⠲⣤⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠛⠳⠶⣦⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣿⠛⣿⣿⣟⠁⠀⠀⠀⢀⡿⠑⠾⠟⠀⠻⠿⢿⡅⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠶⠟⠀⢹⡇⠀⠀⠀⠸⠷⠴⠦⠤⢀⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣶⢶⡄⢸⡇⠀⠀⠀⠀⢀⣀⣤⠶⠛⠋⠉⣸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠛⠾⠃⢼⣁⣸⡇⠀⠀⢸⡟⠛⠉⠀⠀⣠⣴⣟⡋⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡟⠙⢻⡇⠈⠉⠉⠁⠀⠀⠸⣧⣀⣀⠀⠒⠈⠉⠉⢻⡆⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⠷⠦⢤⣤⣄⠀⠀⠀⠀⠉⠉⠉⠐⠲⠶⠤⣤⣧⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢈⡇⢀⣠⣀⣀⡀⢀⣿⠀⠀⠀⢠⡶⠛⣷⢀⣤⣄⣴⡶⠤⣄⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣇⢸⣇⠀⠀⠉⠉⠉⠀⠀⠀⣿⠀⠰⣿⣿⠁⣾⣿⣧⠀⢹⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠉⣩⡿⠛⠛⢦⡀⠀⠀⠀⠀⠹⠶⠶⣤⣁⣀⡀⠉⠁⠀⢸⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢰⠄⢸⡇⠀⠀⠀⠀⠀⢀⣄⡈⠉⠉⠉⠛⠛⠒⠿⠃⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣀⣠⣾⠃⠀⠀⠀⠀⣠⡾⠉⠙⠒⠠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠶⠿⢯⣿⠻⢦⡀⠀⠀⠀⠛⠷⢶⣤⠀⠀⠀⠉⠙⢶⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⠁⣠⡾⠋⠀⠀⠀⠀⣴⠟⠁⠀⣾⠷⣦⡀⢰⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣇⠘⠀⠀⠘⠛⣷⠄⠀⠀⠀⠀⠙⠷⣤⣼⠏⠀⠈⠛⠟⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣻⠷⣶⣦⠴⠴⠿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡿⠁⠀⠀⠉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⢿⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠻⡦⠠⢤⣤⣤⣤⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠿⠿⢶⣶⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⡿⠟⠋⠀⣀⠀⣾⣾⣿⣿⣿⣧⠃
⠀⠀⠀⢀⣀⣔⣱⣶⣶⣶⣾⠛⠓⠛⢟⣫⣟⣉⣈⣒⣮⣙⣯⣭⣉⣙⣿⠏⣿⠿⣿⣿⠿⣿⡟⢻
⠀⠀⠀⠛⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠾⠉⠹⠁⣀⢴⡟⠓⣆⠀⠀⠀⠀⠀⣀⡠⠔⠒⠒⠒⠒⠒⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣐⣛⣒⣒⣲⠯⠓⢉⡶⠚⠁⠀⠀⠀⡴⠋⠁⠛⠓⠒⠒⠂⠀⠀⠀⢀⢨⡂
⠀⠀⠀⠀⠀⠀⣀⡠⠤⠶⠖⠙⢏⠙⢶⡁⠀⢨⠗⣒⣲⢾⣟⣛⣻⣍⡹⢦⠀⠀⢧⡄⠂⠰⠀⠄⠐⠰⠀⠀⠀⡠⠔⠁
⠀⢀⡤⢒⡉⣉⢀⠀⢀⣀⠀⣀⡈⠁⢀⠯⠍⠭⢭⣓⣍⣩⢤⠤⢦⡈⡙⢾⠀⠀⠀⠈⠉⠒⠒⠒⡎⠙⣇⠈
⠈⠓⠫⠿⠾⠯⠯⠽⠿⠟⠛⠓⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠾⠀⠀⠀⠀⠀⠀⠀⢰⡤⢀⡟
⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡴⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢿⣲⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠇
⠀⠀⠀⢀⡤⠒⢲⠾⡷⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢸⣼⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢀⠀⡠⡀⢀
⠀⢠⡖⢙⠀⢠⠊⣠⠃⠀⠘⡝⠢⡀⠀⠀⠀⠀⢰⠋⠀⠐⠗⠀⠈⡆⠀⠀⠀⠀⢀⠴⠊⠉⠉⠙⢠⢩⠄⠛⢠⡉⠉⠢⡀
⠰⠁⢘⠌⢢⠀⠑⢜⢦⡀⠸⣔⡅⢳⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠎⠀⠈⠁⣘⠡⠐⣠⠄⠭⠀⠀⠀⠀⢱
⡇⣴⢣⠞⠁⢀⡠⠊⣀⠇⠀⠈⡟⣾⠀⠀⠀⣠⢃⣀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⡆⡀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸
⢣⠈⣺⢦⡀⠸⣀⠸⣅⠀⠀⣎⢸⡝⠀⠀⣰⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠱⡄⠀⠘⢼⣦⠎⠑⠒⠒⠀⠀⠀⢀⣀⣠⠔⠁
⠀⠳⣧⣼⠁⠀⡨⠇⡼⠀⢀⣼⠟⠁⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠙⠓⠼⠤⡸⠤⠤⠛⠁⠀⠀⠀⠀⠘⢦⡀⡀⡀⠂⠀⠀⣀⡀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⡆
⠀⠀⣴⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠡⠠⠠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣰⢧⠇
⠀⠀⠹⡀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡻⠉⠁⠙⠋⢻⡇
⠀⠀⠀⠱⡀⢢⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡾⠿⢦⠩⠿⣶⢦⡀⠀⠀⠀⠀⠀⠀⡴⠃⠈⠋⠰⠓⠀⠶⠄⠙⡶
⠀⠀⠀⠀⠱⠈⢇⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡮⣀⠀⠸⠀⠀⢀⡱⠻⡀⡄⠀⠀⠠⣔⠃⠘⠁⠔⠀⠀⠤⠀⢠⠀⢵
⠀⢀⡤⠖⠲⢆⠘⠄⢱⣤⣀⣀⣀⠀⠀⠀⢀⣇⠁⠉⠪⡳⡇⣄⢔⠡⠐⠆⢧⠸⡄⠀⢀⡇⠀⡀⠀⠀⠤⠄⢀⡀⠀⠀⢘
⢠⠋⠀⡴⣲⡶⠷⣠⠞⣵⣲⣦⠈⠳⡄⠀⢸⣿⡀⠤⣰⠾⢀⠲⡀⠀⠀⡄⡞⠀⡇⠀⠈⡂⠀⠉⠀⠂⢀⡀⠀⠀⠚⠀⡞
⡞⠀⠀⡾⠉⢾⣠⡇⣸⠋⡇⠘⣆⠀⢹⠀⠀⢇⢎⠉⠀⠀⠂⠐⢌⠢⡔⡱⠁⡜⠀⠀⠀⡧⠐⠂⠀⠀⠀⠀⠀⠖⠀⡸⠇
⢣⠀⠀⣧⠀⠀⢳⣷⠉⣆⡇⠀⣿⡀⢸⠀⠀⠈⠣⣑⠦⠧⣀⣀⠤⣣⢞⡤⠊⠀⠀⠀⠐⠓⢄⡀⠘⠁⠀⠉⢀⣀⡖⠁
⠈⠣⣤⠗⠀⠀⠠⡼⠀⠀⠀⠀⠘⠶⠋⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠚⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⡠⠄⠒⠀⠀⢀⠂⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⠈⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢡⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠠⠐⠀⣠⠤⠤⢄⠠⠀⠀⠀⠈⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁⠀⠀⠀⠀⠀⠀⠀⡀⠂⠈⠀⡰⠂⠀⠀⠀⠀⡈⠀⠀⠠⢉⠀⠀⠀⠀⡁⠈⡀⠀⠀⠀⡠⠈
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠂⠠⠁⠀⠀⠀⠀⠀⠀⠀⠘⠠⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠈⠀⡠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⡰⠠⠤⠤⠤⠀⡀⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠀⠈⠁⠀⠀⢀⠄⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⢀⠀⠠⠄⠐⠒⠈⠉⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠐⠂⠈⠉⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠔⠁⠀⠀⢀⠀⠠⠂⠀⠀⠀⠊⠉⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⡀⠀⡂⠀⠀⠀⠀⠄⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠠⡱⠂⠠⠀⠠⠄⠒⠀⠀⠀⠀⠒⠀⠀⡀⢠⠠⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠈⠂⠀⠀⠀⠁⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠁⠀⠄⠀⠂⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠠
⠀⠀⠀⠀⠀⠠⣀⠠⠴⡋⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠢⡀⠀⠀⡀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠐⢄⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠈⢂⠀⠐⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠄⠀⣀⣀⠀⠘⠀⠀⠉⠉⠉⠈⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⢀⠀⠀⠀⢀⠠⠤⠀⢀⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⢀⢐⠀⣀⠀⠀⠀⠀⠀⠁




⡿⠿⠿⠿⠿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣟⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡤⠖⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⡀
⠀⠀⡞⠀⠀⠀⠀⠸⣆⣠⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣠⣶⣶⣶⣤⣀
⠀⠀⠳⣀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠈⠛⠛⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⠓⢤⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠈⠛⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢉⡷⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣰⣿⣿⣿⡿⡁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⡰⠋⠀⠀⢠⠞⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣠⣾⣿⣿⡿⣫⣾⣧
⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⣧⡞⠁⠀⢀⡴⡃⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠟⢰⣿⣿⣿⠟⢹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⢸⡇⠀⠀⣾⠀⢻⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠈⢧⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠈⢻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⢳⡀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠹⡄⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⡟⠀⠀⠀⠀⠀⠙⣆⣀⣠⠇⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⠤⠤⢤⡤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠒⠒⠒⠒⠒⠒⠒⠒⣒⡒⢦⠀⠀⠀⠀⠀⠀⢠⡖⠇⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸⠀⠀⠀⠀⠀⠀⠈⠳⡆⠀⠀⠀⠀⠀⠀⠀⠀⢧⡼⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠇⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠐⠲⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⣿⣿⡟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢸⣿⣿⡇⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣿⣶⣾⣿⣿⣷⣶⣿⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠛⠋⠙⠛⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠛⠋




⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠠⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣄⣀⣀⣀⣀⣠⣤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠭⣭⠭⠭⠭⣭⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠷⠶⠶⠶⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠚⣛⠓⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢫⠄⠀⢸⡇⠀⠀⡝⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠑⠈⠀⠀⠀⠀⠘⠁⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠁⠀⠀⠀⡆⠀⠀⠀⠈⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠒⠀⠀⠀⠘⠃⠀⠀⠀⠒⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠒⠀⠀⠀⠀⠃⠀⠀⠀⠀⠲⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢀⠀⠀⠀⠀⠀⠀⡀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢠⠄⠀⠀⠀⠀⠀⠀⠀⢠⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣮⣀⠀⠠⠄⠀⣀⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠠⠀⠀⡀⠀⠰⠄⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣭⣛⣛⣛⣛⣩⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣉⣙⣶⣦⣤⣴⣖⣋⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠳⣶⠻⣿⣿⡛⠒⠒⠒⠛⠉⠉⠉⡱⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣮⡿⣿⣶⣴⣤⣤⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣭⡭⢭⣭⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣿⣿⣿⣿⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢛⠁⠀⢰⡆⠀⠈⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣾⠿⠋⠉⠋⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⠀⠀⢸⡇⠀⠀⠁⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⠅⠀⠀⠀⡇⠀⠀⠀⠽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠦⠀⠀⠀⠸⠇⠀⠀⠀⠤⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⠀⠀⠀⠀⡇⠀⠀⠀⠀⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣜⠁⠀⢀⡀⠀⠀⣣⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⢀⠀⠀⠀⠀⢀⢀⣽⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣬⣙⠒⠶⠶⠖⣛⣡⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣦⣦⣶⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠉⠙⠛⠉

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠄⣀⡈⠁⠀⠀⠑⠀⠐⠂⠂⠐⠀⠠
⠀⠀⢀⠀⠀⠸⡄⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠠⢀⡀⠉⠀⠀⠀⠚⠐⠆
⠀⠀⠀⠀⠀⡀⢁⠈⠄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠓⠀⢠⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠠⠀⠄⠀⠀⠀⠈⠀⠀⠀⣠⠀⠀⠀⠌⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠁⠀⠂⠀⠀⠀⠄⠀⠈⠁⠄⠚⣦⣮⠀⠰⠀⠀⠀⠀⠀⠀⠑⠀⠈⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠥⠟⢀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡅⠀⠀⠀⠈
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⣔⡉⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠦⢄⡀⠀⠐⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠎⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⡀⠀⠀⠀⠇
⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁
⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠋⠀⠀⣀⠀⡀⠘⠀⡄⠉⠂⡀⠀⠀⠀⡠⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢀⣀⡜⠉⠀⠑⠀⡀⠔⠀⢄⠀⡸⠀⠀⠀⠀⠈⠀⠀⠙⠃⠀⠀⠀⠀⠁⠀⢠⠔⠁
⠀⢀⡈⠆⠀⠀⢠⡀⠢⠀⠀⠀⠀⠈⠒⢹⡆⢀⠃⠀⠋⡀⠀⢀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡀⠀⡀⠲⠀⡄⠀⡀⠳⠀⠀⠀⠠⠀⠘⠃⠈⡃⠠⠈⠁⠀⢀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢈⠐⠄⠐⠀⡇⠢⠀⠀⠀⠀⠀⠀⠑⠆⠀⡈⠁⠆⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠂
⠀⠐⠁⠄⠂⠀⢂⢡⠄⠄⠀⠀⠀⠀⠈⠐⠆⢱⢢⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠇⠁⠆⠀⠸⠁⠠⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⢀⠀⠀⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠀⠀⢡⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠠⠀⡖⠀⠊⠀⠀⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢦⠀⠀⠀⠱⡁⠀⠈⠂⠀⠄⠀⠈⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠰⠀⢀⡵⠀⠍⡠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢨⢡⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⠢⠀⠀⠀⠀⠀⠀⠒⠘⠃⠀⠀⠈⠁⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠈
⠀⠀⠰⠄⠀⠰⠆⠀⠀⠀⠀⠀⠠⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢴⠀⠀⢈⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡞⣩⠥⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠭⣍⢳
⠀⠀⠀⠘⠁⡇⡴⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠆⠠⠶⠶⠆⠰⠶⠲⠄⠰⠶⠲⠀⠀⠀⠀⠀⠀⠀⢠⠚⠙⡆⢸⠐
⠀⠀⠀⠀⠁⡇⠘⠒⠊⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠒⠒⠒⠒⠒⢒⣒⠒⢒⣒⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠁⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣀⣀⣀⣀⣀⣀⣘⣚⣀⣘⣚⣀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⣀⡀⡤⣄⡰⠊⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠑⠄⡠⢤⢀⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠖⢾⣄⡽⠤⠜⢇⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡸⠣⠴⢯⣠⡷⠲⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⣞⠳⠓⣊⠤⠔⠒⠒⠨⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠅⠒⠒⠢⠤⣑⢚⠞⡣⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠈⢩⠊⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠈⠹⡍⠁⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢸⠀⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⢠⠤⢄⠀⠀⠹⡀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢀⠇⠀⠀⡠⠤⡀⢸
⠀⠀⠀⠠⡀⡇⠣⣀⡼⠀⠀⠀⠙⠒⠶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠒⠋⠀⠀⠘⢆⣠⠇⢸⠀⡄
⠀⠀⠀⠀⢧⣙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣋⡼
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠶⠶⠶⠶⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣉⣉⣉⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣯⣿⣿⣿⠿⢿⣿⣿⣿⢹⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣞⠛⠛⠛⣛⡛⠛⠛⠛⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣄⣴⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⠀⠀⠀⠻⠋⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠻⣧⣼⠛⠻⣿⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⠀⣤⣤⠞⠣⣤⣄⠀⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡟⠿⠦⠶⠿⠿⠦⠶⠿⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢛⠀⠛⣀⡙⢋⣀⣀⣙⣋⣀⡷⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠂⠄⢀⠀⠀⠀⠀⠀⡀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠅⠀⠩⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣻⣿⣟⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠤⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠟⠛⠻⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠂⠀⡔⠁⠀⠀⣠⠎⢢⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⢀⠁⠀⡅⠀⠐⠁⠀⠀⢈⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠿⠁⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⡀⠀⠀⠀⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣯⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⣾⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⣿⣦⣀⣀⣸⣟⣀⣠⣴⡿⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣶⡾⠟⠛⠛⠛⠛⠛⠛⠛⠻⠿⠷⣶⣤⣿⣃⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣾⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⠇⠀⣠⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⣼⡟⢀⣾⠏⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠹⣷⡀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⣼⡟⠀⣾⠃⣴⡟⠉⠛⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠙⣷⡙⣷⣄⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋⣠⣾⠋⠀⠀⢻⣦⡿⠷⣦⣄⠀⠈⠛⠷⣦⣠⡶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⢿⣦⠈⣿⡈⠻⣶⣄⠀⠀⠈⠙⠻⠷⢶⣶⣶⣶⣶⡶⠿⠟⠋⢁⣠⣾⠟⠁⠀⠀⠀⠀⠈⠀⣴⡿⣿⣿⣶⣄⠀⠈⣻⡇⠀⣿
⠀⣀⣀⠀⣀⣤⡾⠟⠉⠀⢀⣤⣿⠀⣸⡇⠀⠈⠻⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣈⣿⡆⠙⣿⡶⠟⢁⣼⠏
⣼⠋⠹⡾⠛⠉⠀⣀⣴⣾⣟⡉⠙⠛⠋⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⠿⢿⣿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠛⠋⠀⠀⢿⣤⣤⠞⠁
⠸⣦⡀⢻⣤⣴⠿⠛⢿⣏⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣄⣼⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠈⢿⣦⣭⣿⠀⠀⠈⠙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠉⠉⠉⠉⠉⠉⠛⢷⣆⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠙⠿⣦⣄⣀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⣤⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⠷⣶⣦⣤⣤⣄⣄⣠⣤⣤⣤⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢹⡿⠉⢹⣟⠉⠉⠉⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣿⠇⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢰⡿⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣾⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⣤⣤⣤⣼⡏⠀⠀⠀⢸⣧⣤⣤⣤⣤⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⠉⠉⠁⠀⠀⠀⠈⠉⠉⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠛⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠋⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⣼⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿