Punching Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⢀⡄⠀⠀⣀
⠀⠀⠀⢀⣴⡄⠀⠀⠀⢻⣇⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡿⠁⣠⠏⠀⣠⣾⣿⡦
⠀⠀⣀⠈⠻⣿⡄⠀⠀⠀⢻⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠞⠁⠜⠁⣠⣾⡿⠟⠁
⠀⠀⠉⠁⠀⠈⠻⣆⠀⠣⠀⠓⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⣠⡾⠟⠉⠀⠀⣠⡀
⠀⠀⢀⣀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡭⢗⠂⠙⣿⠉⠙⠉⠉⠀⠚⡩⠽⠷⢦⡀⠀⠈⠀⠀⠠⠖⠛⠉
⠀⠀⠈⠉⠛⠢⠄⠀⠀⠀⠀⠀⠀⣀⡤⢶⡟⠁⠀⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⣀⣀⣤⣤⡄
⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⣠⠊⢓⠂⠀⣷⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢠⡾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣆⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠘⡟⣆
⠀⠀⠀⠀⠀⠀⡿⠊⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⢷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⣀⡤⠤⠄⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⢀⡀⠀⢹⣧⡀⠈⢀⣠⠤⢂⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠘⣿⡄⠀⠘⠉⣉⡤⠀⣸⣿⣷⣦⣤⣴⣶⠿⠋⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠠⠤⢖⠀⢻⣆⠐⠒⠒⣊⡁⢸⣿⣦⣬⣭⣥⣴⣾⣿⣿⠿⠟⠉⠁⠀⣠⣰⡏
⠀⠀⠶⠒⠀⠀⠀⠀⠀⠀⠙⠳⢦⣥⠶⠿⠛⢷⣤⣬⣥⣴⠿⢟⡿⠿⠽⠿⠓⠲⣶⡀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⢀⣀⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠸⡇⠀⠀⠀⠀⣀⣿⣇⣀⣀⣤⣤⡶⠟⠁⠀⠀⢀
⠀⠀⠈⠛⠋⠁⠀⠀⣠⡶⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠙
⠀⠀⠀⠀⠀⢀⣴⣾⠟⠀⢠⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠰⡀⠀⠘⣷⣄⠀⠙⢷⣦
⠀⠀⠀⣠⣶⣿⡿⠁⠀⣰⡿⠀⠀⢸⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢿⡄⠀⠈⢻⣷⣄
⠀⠀⠀⠙⢿⠋⠀⠀⠀⠈⠁⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⠀⠹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠶⢶⠒⠒⠒⣚⣛⡿⠶⠖⠒⠂⢱⡖⠒⠒⠒⠒⢲⠛⠂⠀⠀⣀⡀⡤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⢷⡶⠶⠶⠶⠛⠓⠓⠈⢫⡍⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⠀⣠⣄⡀⠐⠛⡄⠀⣀⣀⣀⣤⣴⣶⣾⣿⡷⠊⠑⠀⠀⠀⠀⠀⢀⡀⢄⡷⠀⠀⠀⠈⠓⢦⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠁⠀⠈⠙⠛⣿⣄⣀⡩⠿⠿⠿⠿⠋⠉⠑⡀⠀⠀⣠⣶⣶⣿⣯⣥⣎⡀⠀⣀⡀⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⡪⠷⠦⣄⡀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠚⠛⠉⠻⣿⣿⣿⣿⣿⣷⣾⠿⠷⠶⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢙⠋⠉⠉⠀⠀⠀⠀⢈⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢻⣿⣿⣿⢱
⠀⠠⢤⡤⠤⠤⠤⠤⠾⠼⠤⠤⠤⢤⣀⣤⠞⠉⠀⢹⡀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⡃⠀⠀⠖⠴⠶⠤⠤⠤⢤⡄
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⢇⡀⢠⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠳⣦⣤⣤⣤⠤⠄⠂⠉⢹⠈⠉⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣃⣼⠀⠘⣿⠶⠶⢶⠤⠤⠤⠼⠦⠤⠄
⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⣀⠀⢹⠀⠀⢸
⠀⠀⢀⣀⣾⣇⣀⣀⣤⣤⠤⢤⡤⠔⠒⣞⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢿⡁⠀⠈⠁⠁⠉⢩⠓⢤⣀⣀⡀
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠘⢧⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠉⢸⠁
⠀⢀⣼⣀⣀⠀⠀⠀⣀⣀⣤⠄⠸⡆⠀⠀⢸⡄⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠑⢢⣶⣶⣾⣦⣄⠀⢸⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠉⠙⠉⢻⡄⠀⠈⣷⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⣀⡀⢠⣦⠞⠀⠀⠀⡼⠀⠈⢳⠒⠒⠓⠊
⠀⠀⠀⣀⣀⣀⡠⠤⠾⣇⠀⠀⠀⠀⢻⡄⠀⢻⡄⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠈⠻⢶⣶⣾⣟⣿⣽⣾⣋⠉⠀⠀⠀⢀⣧⣤⣤⣬⣀⣀⣀⣀
⠀⠀⢹⡏⠉⠀⠀⠀⠀⠀⢨⢦⠀⠀⠈⠉⠂⠸⣷⠀⠀⠈⢻⣤⣀⣀⣠⣠⠖⠀⠀⣠⠴⠛⠉⠀⠀⠀⠈⠁⠀⠀⠀⢸⠋⠉⠛⠛⠛⠛⢩⠁
⠀⠀⠈⣁⣀⡀⠀⠀⢀⡴⠁⠈⡆⠀⠀⠀⢈⡄⠹⣧⡀⣀⢀⡉⠛⢻⣿⣷⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠉⠉⠉⠉⠉⢹⠋⠀⢀⠐⠁⠀⣀⡴⠟⠛⠉⠀⢣⠘⣿⣶⣾⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⠛⠛⠛⠓⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣆⣐⣁⣠⠒⠉⠁⠀⠀⢀⣀⣤⣾⣶⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⡰⠀⠀⣀⣀⡀⣀⣠⠞⠁⣚
⠀⠀⠀⠀⠀⠀⠀⠒⠋⠉⠙⠙⢇⣠⣤⣶⡿⠿⠿⠿⠿⠙⠛⠋⠑⠀⠸⡷⣀⡄⢀⠀⠤⢞⣛⡛⠛⠻⢯⠟⠛⠋⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣁⡀⠀⠀⠀⠀⠀⠀⠈⠉⠐⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠧⠤⠤⠤⠤⡲⠖⠒⠛⠛⠻⠿⠿⠿⠿⠷⠶⣤⠤⠀⠀⠀⠀⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣾⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠉⠉⠙⠶⠋⠉⠛⠲⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢠⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠈⠳⣿⣿⣷⣤⣀⣀⣀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠒⠁⠀⠀⠀⠀⠀⠙⣻⠿⠛⠛⠛⢥⣀⣀⣀⣀⣠⣀⠀⠀⠀⠀⣠⣾⣏⢈⡗⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣷⣄⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠙⢶⣶⣾⠿⠿⠟⠋⠀⢈
⠀⠀⠀⠀⠀⣀⠔⠁⠈⠙⠿⣶⣶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⢀⣠⠞⢀⡴⠞⠉⠉⢹⠇
⠀⠀⠀⠀⠜⠀⠀⠀⠀⢀⣼⣿⡏⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⠙⢿⠻⠿⠷⠚⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⣿⡟⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠰⣶⡧⠴⢦
⠀⠀⠀⠀⠀⠀⣲⣤⣤⣿⡿⠀⠀⣇⢹⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣀⣠⠞⠉⠀⠀⠈⢿⡇⠀⠀⡼⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⡇
⢀⠖⠁⠀⠀⠀⠀⢀⠠⣸⠁⠀⣾⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⠀⠀⠀⡟⠀⢀⠀⠀⠀⢀⣀⡤⠔
⢾⡀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣼⠀⠀⠀⡎⠀⠀⣰⠃⠶⠾⣷⣿⣯⢁⡀
⠀⠱⢲⠂⠀⠀⠀⠀⠀⢱⠀⠀⠈⢧⣦⡀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠃⠀⢠⡇⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠁⠀⠀⢀⢀⣤⣶⣾⡇⠀⠀⠀⣨⠽⠷⠦⣤⣭⠿⠋⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡞⠀⢠⡟⠁⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⢳⡀⡼⣿⠋⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣀⡀⠀⢒⡤⠖⠁⠀⠀⣸⠁⢀⣿⣁⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠙⠀⠀⠀⢠⠊⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⣄⣀⣀⣤⣴⢃⣰⠋⠀⣿⣿⡉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠘⢦⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⡶⡿⠁⠀⠀⠘⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⢉⣩⣽⣷⠖⠺⠦⢄⡀⢀⣀⡿⢿⣿⡛⠋⠁⠀⠀⠀⠀⠀⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⡴⡿⠉⢠⡏⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠑⢤⡀⠀⠠⣄⣀⣀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣸⠛⠋⠀⠀⠀⠀⠈⠉⠉⠉
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⠢⠄⣹⠓⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠤⢤⡀
⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⢀⡇⠀⣼⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠉⠉⠁⠀⠀⠀⠉⠛⠲⣄⡀
⠀⠀⠀⠀⣾⠏⠀⠀⣤⣼⣿⣿⠋⠉⢀⡼⠁⠀⢈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣾⠀⠀⠀⠀⡆⠀⠀⠀⢧⣀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⣿⣿⣿⣶⣿⡁⠀⣠⡟⠀⢙⡄⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⡿⠀⠀⠀⢸⠀⠀⠀⢠⠀⠉⢳
⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠹⣷⠈⢿⣿⣿⣿⣿⣄⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⢀⡇⠀⠀⠀⣿⠀⠀⢀⡟⠀⠀⡾
⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠹⡇⠀⠙⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠀⠀⢸⡇⠀⠀⢠⡿⠀⠀⣼⡇⠀⣰⠇
⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣌⡳⠿⠿⠛⠶⠿⠿⢷⣾⠿⠟⠷⠿⠛
⠀⠀⠀⠀⠀⣸⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣾⣷⣶⣶⣶⣶⣿
⠀⠀⠀⠀⠰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡏⠉⢢⡤⢤⡀
⠀⠀⠀⠀⠀⠀⣴⡿⠋⠙⠉⠀⠀⠈⠓⠒⠲⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠉⠂⠒⢠⠀⢸⣄⡀
⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⢠⣄⣀⡤⠔⠊⢠⠂⠀⢙⡄
⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⡾⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠠⣾⣿⣿⡁⠀⠠⠃⠀⢀⠌⠉⢳
⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠉⠻⣿⣿⣶⣧⣀⣴⠋⠀⢀⣼
⠀⠀⠀⢠⡿⠋⣧⠀⠀⠀⣧⠀⠀⠀⢸⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠘⢏⠻⢿⠿⠿⣶⠶⢻⡏
⠀⠀⠀⢸⡅⠀⠘⠳⢶⣶⣿⣦⣀⣠⣾⣄⣀⣴⣃⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠈⢻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣤⣤⠄⠀⢀⣀⣠⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣴⣶⣾⣦⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣿⣿⣵⣾⣏⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⣷⡓⢄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠻⢦⣿⣟⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⣰⣿⣿⠛⡃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢻⣿⣿⣿⣿⣿⡧⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠏⠉⠹⢠⠷⣄⡀⠀⠀⠀⠀⢀⡀⠀⠀⢸⡎⣿⣿⣿⣿⣿⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣷⣄⡀⠀⢠⠏⠀⠀⠉⠙⠢⠄⠀⢸⣷⡶⣿⣿⡇⣿⣿⣿⣿⣿⣿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡧⢙⠈⠛⠟⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢸⣿⣿⣿⣿⣿⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣥⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣿⣿⣿⣿⣿⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢺⣿⣿⣿⣿⣿⣟⡅⢿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⣿⣿⣿⠿⢟⡏⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣷⣳⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡷⠶⠚⣋⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⡇⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠉⢿⣿⡁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠀⡨⠀⡔⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢸⡀⠇
⠀⠀⠀⣴⣷⣾⣄⠀⠀⠀⠀⠀⠀⠀⣇⣠
⠀⠀⠀⠛⠛⠿⠿⡷⠂⠀⠀⠀⠀⢀⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠺⠿⠿⠿⠿⠻⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠶⠶⠶⢦⠀⠀⣀⣠⠷⠶⠖⠒⢲⠀⠀⠀⢀⣾⣒⣶⠦⣔⣠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠀⠀⠀⠀⣀⠈⠟⡉⠉⠁⠀⠀⠀⠀⠈⢙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢱⣤⠒⠠⠤⣼⣄⠀⠲⣶⣾⣿⣿⣿⣇⠀⠈⠀⢀⣤⣤⠶⢂⡠⠁⠀⠀⠉⠢⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣸⣦⣀⠀⠀⠀⠈⠐⣺⡿⠛⠛⠻⠿⣿⣦⣠⠤⠙⠒⠺⣿⣿⣦⣵⣶⣤⣀⣀⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠉⠉⠁⠀⠈⢛⣲⣤⣤⣾⠹⠁⠀⠀⠀⠀⢀⢁⢻⠀⠀⠀⠀⠈⢻⣿⢋
⠀⠀⠀⡴⠒⠒⠒⠚⠒⠂⠀⠀⠤⣤⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⡇⠐⠒⠒⠒⠲⠦⡆
⠀⠀⠀⡇⢀⠀⠀⠀⢀⣀⠤⡄⢄⣿⡇⠀⠀⠀⠀⠰⣧⣇⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⢧⣄⣀⣀⣀⡀⢸⡁
⠀⠈⠉⠀⢻⠛⠋⠉⠁⢠⠞⠋⠙⠋⠁⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⢸⡏⠉⠹⠉⠉⠉⠉⠁
⠀⠀⣤⣤⣼⣀⣀⣀⣠⡏⠀⠀⠀⠢⣌⣄⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⣿⠀⠈⡖⠀⣀⣀⡀
⠀⢀⡅⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠸⣧⡀⡇⠀⠀⢨
⠀⠚⠓⠒⠀⠀⠐⢲⡾⣿⣦⡀⠀⠀⠀⠙⣿⣷⡄⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢿⣿⣿⣂⣄⡼⠶⠦
⠀⠀⠀⠀⠀⠀⣀⣸⡇⠘⣯⠳⣄⠀⠀⠀⠈⠛⢻⣆⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢰⣧⣀⠀⠀⠀⠀⣸⢿⡇
⠀⠀⠀⡟⠋⠉⠀⠀⠈⡰⠻⢷⣽⣧⡀⠀⠀⠀⠀⠙⣷⣤⣤⣤⣴⣿⣷⣀⣀⣠⣾⣿⣯⣭⣥⡴⠞⠋⠀⣿⠛⢽⠉
⠀⠀⠀⣧⣤⣀⣀⣀⠞⠀⠀⠆⠙⠿⣿⣦⣀⣀⡄⠼⠋⣉⣿⣿⣿⣿⣿⣯⡟⣩⠤⠤⣄⠀⠀⠀⠀⠀⠐⣟⡀⣸
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⣀⣾⣿⣿⡿⠟⠛⠛⢋⠙⣿⣿⣿⣿⣿⡟⣸⠁⠀⠀⢘⡆⠀⠀⣀⡤⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠼⠶⠶⠾⠿⡿⠛⠉⠁⠀⠀⢀⣠⣼⠤⢻⠇⠤⠬⣹⣇⢹⡀⠀⣀⣼⡥⠶⠯⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣄⣠⡴⠖⠛⠉⠀⠀⢀⣼⣀⠀⠀⠀⠙⠦⠿⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠰⠲⠢⡴⠒⠈⠉⠉⠉⠩⠿⠷⠶⠆⠀⠀⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⠶⣶⠶⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⣶⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⠿⠟⠛⠿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⠿⠁⠀⠀⠔⠚⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠉⠉⠋⠉⠛⠿⣿⣿⣿⣿⣿⣿⣁⢀⣀⣠⣶⣶⣿⣤⠄⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠿⠛⠻⣿⣿⣿⣿⣯⣍⡛⠀⣀⣴⠏
⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠦
⡶⠶⠿⠿⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠈⠉⠛⠻⠿⠟⠷⠶⣦⣶⠶⠤⠤⠤⠤⣄⣀⣴⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡠⠂⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠘⢄⣀⣀⠀⠀⠀⠀⠃⠀⠈⠀⢰⠀⠀⠀⣀⣀⡔⢡⠞⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣷⣶⡶⠿⣿⠏⠀⠈⣀⠀⠀⢸⠀⠀⠀⡏⠠⠁⠁⠀⠹⣿⠻⢶⣶⣾⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠋⠈⢤⣆⠀⠠⡏⠀⢸⠶⠷⠜⠀⠠⠠⠽⠿⡇⠀⢙⠄⠀⣰⡤⠁⠘⠛⢻⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⣿⣱⠶⣄⣴⣿⣿⡄⢾⠂⠂⠀⣤⡶⠃⢠⡄⠘⢶⣤⢀⠐⠐⡷⢀⣿⣿⣦⣠⠶⣎⣿⣦⣄
⠀⢀⣀⣀⣀⣀⣠⣤⣾⡿⣹⣿⣿⠿⠏⠚⣿⣿⡇⡋⠀⠀⠀⣿⡿⠶⣿⣿⠶⢿⣿⠀⠀⠀⢙⢸⣿⣿⠓⠹⠿⣿⣿⣏⢿⣷⣤⣄⣀⣀⣀⣀⡀
⠛⠛⠛⣛⣻⡟⢉⣿⣿⣁⣿⢿⡙⠓⠀⢸⣿⡿⠧⣽⣆⢠⡤⠁⣀⠄⢘⡃⠠⢀⠈⢤⡄⣰⣯⠼⢿⣿⡇⠀⠚⢋⡿⣿⣈⣿⣿⡉⢻⣟⣛⠛⠛
⠤⠐⠂⠒⣛⣇⢼⣿⣿⣿⣿⣦⣀⠀⣠⣿⣿⣧⣠⣀⣚⣃⣐⣃⣠⣖⣼⣧⣲⣄⣈⣒⣚⣓⣀⣄⣼⣿⣿⣂⠀⣀⣴⣿⣿⣿⣿⡯⣼⣛⠂⠐⠂
⠀⠀⡀⠀⠀⠀⣼⣿⠹⣏⢿⣷⠶⠖⢾⣿⣿⣿⠛⠉⠉⠁⠉⠁⡈⠉⢹⡏⠉⢁⡈⠉⠈⠉⠉⠛⣿⣿⣿⡷⠲⠶⣾⡿⣹⠏⣿⣧
⣀⣄⣙⣓⡦⣷⡜⣽⡆⣿⡄⢛⣳⣿⢿⣿⣿⣿⣷⣷⣶⣤⣤⠺⠇⣠⣾⣷⣄⠸⠗⣤⣦⣶⣾⣾⣿⣿⣿⡿⣿⣟⡛⢀⣿⢰⣯⢣⣾⢳⣚⣉⣀
⠛⢛⠋⠉⠙⠛⢿⣻⣦⣹⣏⣽⣿⠋⠘⠁⣰⢿⣍⠉⠉⠉⢋⠟⠋⡿⢹⢹⣯⠛⠻⡝⠉⠉⠉⣩⡿⣆⠈⠁⠙⣿⣯⣹⣟⣴⣟⡿⠛⠋⠉⠙⡛
⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠟⠻⢤⢾⣯⡁⣸⠟⡿⠷⠖⣀⣀⢸⠃⠈⠈⢿⠀⣀⣀⠲⠾⢿⢻⣇⢈⣽⡷⡤⢟⡻⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡐⠉⠀⠉⠍⠀⡎⠀⠀⠀⠀⠇⠀⠉⠉⠈⢫⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠴⠖⠉⠀⠀⠀⠀⠙⠶⠦⠆⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢢⣀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢰⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡏⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡏⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣏⢴
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⣇⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⡯⡘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢾⣷⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣿⠃⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⣼⣿⣆⠀⠀⣼⣿⣅⠀⠀⣤⣾⣿⠀⠀⠀⠀⢀⣼⣿⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠉⣹⠟⠋⠛⠛⠳⠿⠷⠾⡿⠟⠋⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠖⠁⠀⠐⠄⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⢴⣧⣀⣀⣀⣀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠶⡟⠁⠀⠀⠉⠑⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠋⠉⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠋⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⢠⠞⠉⠉⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⣿⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⢠⣿⠀⠀⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⡰⠟⠟⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⢸⣶⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢧
⠀⠀⠀⠀⠀⢿⡆⠄⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠈⢣⡀
⠀⠀⠀⠀⠀⠈⠳⢀⡀⠀⢐⣿⣷⡄⠀⠀⢀⣠⣴⣿⣿⣦⡀⠀⠀⠀⠄⢶⣿⣿⣷⣤⣤⣀⣀⣀⣤⡿⠟⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠛⠋⠘⠛⠛⠛⠛⠋⠈⠙⠻⠷⠶⠤⠶⠿⠿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⢁⠄⠀⠤⠀⣀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢠⠃⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⡀⢀⣀⣀⣀⠀⣀⣀⣀⣀⣀⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠁⠀⠀⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠀⣠⣦⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⣼⣿⣿⣿⡇⠀⢀⣤⣄⠛⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⣠⣤⠓⣮⠎⢹⣿⠻⠧⠼⢿⡿⢿⣿⣿⣕⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⣤⠽⠿⠿⢃⣚⡁⠀⣠⣖⣶⠙⣯⣿⡹⠣⡻⣁⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢠⡞⠓⠀⠠⠶⠛⠉⠳⠶⣭⣛⡿⣴⣿⠻⢦⣼⠃⣽⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⢭⡿⢄⣠⠊⠀⠀⠀⠀⠀⠀⠉⠀⠈⠑⠃⣴⣾⣧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⣺⣿⣫⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣹⠿⠿⢄⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠠⢈⣼⢏⡿⢆⣦⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠃⠀⣩⢤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣛⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢄⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣼⣿⢻⡿⠿⣿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠨⠘⡇⠰⠾⠏⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⣿⡇⠀⠀⠀⠀⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣵⠀⠛⣿⣿⢭⣤⣤⡄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣴⢧⣴⠟⠈⠀⠀⠀⠀⠀⠀⠙⠇⣀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠁⠉⠛⠏⠀⣿⠇
⠀⠀⠀⠀⠀⢀⣁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⢐⡲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢽⠄
⠀⠀⠀⠀⢠⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠀⠠⠀⠀⢈⣡⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⢀⡴⢳⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⢧⠀⠀⠀⠀⢀⡞⠀⠀⢧⠀⠀⠀⠀⢀⡴⠋⣷
⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⢸⠁⠀⠈⢳⡀⠀⠀⡾⠀⠀⠀⠸⡄⠀⠀⡴⠋⠀⠀⢹
⠀⠀⠀⠘⣏⠙⠦⣄⠀⠀⣾⠀⠀⠀⠀⠹⣄⣸⠃⠀⠀⠀⠀⢳⣀⡞⠁⠀⠀⠀⢸⡄⠀⠀⠀⠀⣀⡤⠖⡆
⠀⠀⠀⠀⠘⡄⠀⠈⠙⢶⡇⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⣇⠀⢀⣠⠞⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡶⠋⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⣠⠴⣶
⠀⠀⠀⠀⠀⢀⣽⠤⠖⠒⣆⡠⠐⠀⠉⠙⠓⢤⠀⢀⣀⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⣠⠖⠋⠀⢀⡏
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢳⠉⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⢸⠃
⣶⢤⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⠒⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⡤⠴⠒⢻⡇
⣿⢀⣈⣛⡆⠈⣧⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠘⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⡀⠀⢇⠀⣀⢤⠀⠀⠀⣠⠴⣄⠀⠀⠀⠀⠀⠉⠁⠀⠀⣠⠏
⠉⠉⠉⠉⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠺⣭⠇⠀⠀⠀⠀⢰⠁⠀⠘⠊⠁⠘⣇⡠⠊⠁⠀⠈⢦⠀⠀⠀⠀⢰⠶⠤⣰⡇
⠀⠀⠀⠀⠀⣀⣼⡄⠀⠀⠀⣾⠶⠦⠤⢶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣱⡄⠀⣰⠃⠀⠀⢈⡿
⠀⠀⢀⠴⠚⠁⠀⣇⠀⠀⠀⢹⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⡼⠁⠀⠀⡴⠋⠀⢀⡴
⢠⣞⣥⣤⣤⣄⡀⢻⣤⣤⣤⡾⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣤⣤⠴⠾⡆⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡾⠁⠀⢀⡞⠁⢴⣞⠉⢀
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⡀⠄⣀⠀⠀⠀⣀⠤⠀⢀⡀⠀⠀⠸⣆⣠⠶⠋⠁⠘⣆⠀⣠⠖⠋⠀⠀⠀⠀⠈⠓⢦⣤⢿⠀⠀⠀⠈⠓⠾
⠀⠀⠀⠀⡼⠁⠀⢀⣀⡤⠴⠶⠊⠀⠀⠀⢡⡤⠋⠀⠀⢀⡀⠈⠑⠄⡀⠈⠁⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⣠⠤⣤⠀⠀⠘⡇
⠀⠀⢀⣾⡥⠶⠚⠉⠀⠀⡔⠀⠀⠀⠀⠀⠊⠀⠀⠐⠀⢠⠒⠂⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠞⠀⢀⡀⠹⡄
⠀⠀⠀⠁⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⣠⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠙⠲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⣧⠞⠁⠈⣇⠀⠀⠀⠀⣴⠻⡄⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⠒⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡆⠀⠀⡰⠁⠀⠙⣆⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠹⣄⡼⠁⠀⠀⠀⠈⠳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⡠⠶⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⢠⠖⠁⠀⠈⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠱⡂⠀⠀⠀⠀⠈⠒⢄⢀⡀⠤⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⡶⠋
⣀⣀⣀⣀⣀⣀⣀⣈⣙⣂⣴⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠈⢷⡀⠀⠀⠀⠀⢀⡞⠁⠀⢿⠀⠀⠀⠀⠀⢀⡴⠛⣷
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠻⣄⠀⠀⠀⡾⠀⠀⠀⠘⣇⠀⠀⠀⣰⠋⠀⠀⢹⡀
⠀⠀⠀⠘⣟⠳⢤⣀⠀⠀⠀⣼⠀⠀⠀⠀⠙⣆⠀⣸⠃⠀⠀⠀⠀⢹⡄⢀⡾⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣤⡄
⠀⠀⠀⠀⠘⣆⠀⠉⠳⢦⣀⡏⠀⠀⠀⠀⠀⠈⢷⠏⠀⠀⠀⠀⠀⠀⢳⡞⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣀⡴⠛⠁⠀⣷
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣠⠞⠋⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣀⡀⠀⠀⢀⡀⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣠⡴⠚⠉⡟
⠀⠀⠀⠀⠀⢠⡴⠛⠊⠉⠀⠷⠖⠋⠀⠀⠀⠀⠉⢳⣀⡠⠶⠒⠒⠒⠲⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣠⡴⠛⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣀⣀
⣿⠶⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣤⠴⠚⠛⢉⡟
⣾⣀⣀⣉⣷⠂⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡸⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢸⡀⠀⢷⠀⣀⡤⣄⠀⠀⠀⡠⠲⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠏
⠉⠉⠉⠉⠁⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠸⣭⡿⠀⠀⠀⠀⠀⢸⠁⠀⠘⠛⠁⠀⢹⣀⠴⠊⠀⠀⠈⠳⡀⠀⠀⠀⠀⣴⠶⢄⣰⡏
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢰⠶⣤⣤⣤⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢙⣆⠀⢀⡼⠃⠀⠀⢈⣿⠆
⠀⠀⠀⣀⡴⠚⠉⠸⡇⠀⠀⠀⢸⡄⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⢀⡞⠁⠀⠀⣠⠏⠀⠀⣀⣴
⢀⣤⠚⠁⠀⠀⠀⠀⣧⣀⣀⣀⣸⡇⠀⠀⠀⠀⠈⠳⣤⣀⣀⠀⠀⢀⣀⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢠⡟⠀⠀⢀⡼⠃⣀⡴⠞⠉⣸
⠉⠉⠉⠉⠉⠉⡿⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠈⢧⡀⢀⣠⡴⠛⠛⣦⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠿⣄⡀⢠⡞⠁⠀⠈⠛⢦⣄⡏
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⣀⡠⠂⠁⠉⠢⡀⢀⠔⠊⠉⠉⠉⠒⠤⣀⠀⠈⠻⠛⠁⠀⠀⠀⠘⣧⡴⠋⠀⠀⠀⠀⠀⠀⢀⡀⠈⠙⠋⢻⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⡞⠁⣀⣤⠴⠖⠛⢉⠝⠁⠀⠀⠀⠀⡿⠁⠀⢀⣀⠐⢃⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⢙⡇⠀⠀⠘⣧
⠀⠀⠀⠾⠖⠋⠉⠀⠀⠀⢀⠎⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⢸⠈⠉⠁⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⢀⡦⣄⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⣰⠞⣧⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣸⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢳⠞⠁⠀⢸⡆⠀⠀⠀⢀⡾⠙⢧⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⠁⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢻⡀⠀⢀⡾⠁⠀⠈⢳⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢷⣀⡾⠀⠀⠀⠀⠀⠹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⢠⡏⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⣀⠜⠁⠈⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠳⡄⠀⠀⠀⠈⠢⡑⢄⡀⠀⠀⠀⠀⢀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠈⠢⣑⣦⠒⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠑⠲⣄⡀⠀⣠⠎⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣤⣶⣶⣶⣶⣶⣶⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣝⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⣉⣛⣮⣭⣽⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⠿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⡿⠟⠋⣁⡴⢾⣻⣽⠿⢝⣀⠤⢤⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣭⣭⣭⠹⣿⣷⡀⢽⣷⠶⠋⠁⠚⠉⠁⠀⠀⠀⠀⠀⠀⠇⢿⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⡾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⣧⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣷⡸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⢿⣿⣆
⠀⠀⠀⠀⠀⣡⣾⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢿⡸⣿⣿⣷⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣎⢹⣿⣿⣿⣿⣿⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣇⣿⣿⣿⣧
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⡄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣷⡄⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⢁⣤⣀⡐⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠁⠙⠋⣿⣦⣀⢀⣀⣤⡶⠛⠁⠀⠐⠿⠛⢧⢳⣿⡏
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣫⡾⠋⠀⠀⠀⠀⢸⡟⣿⣿⣩⡥⠒⠀⠀⠀⠀⠀⠀⠈⢷⣸⡇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣿⣿⡿⢿⣿⡯⠽⠞⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧⠹⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣵⢯⠃⠀⠀⠀⠀⠀⢀⣦⠗⠋⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡆⠙⢷⣻⣿⣿⣿⣿⣿⣿⣟⣿⢿⣿⠯⠻⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡯⢀⡰⠋⠻⣿⣿⡛⠭⠽⠞⠛⠛⠉⠀⠀⠀⠈⠙⠿⠿⣿⣲⣶⣃⣠⡶⢟⡩⢄⠀⠀⠀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⣩⠴⠛⠁⠀⠀⠉⠿⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣤⣀⠀⠀⠀⠀⠙⠉⠳⣄⠀⠀⢀⣀⣀⣠⣤⣴⣿⠿⠛⠉⠉⠻⣤⣤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠀⠀⢀⣠⠟⠉⠉⠋⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣄⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⣀⠀⠀⡏⠀⠈⠉⠓⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠢⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⢒⣥⢤⠔⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵⠛⠀⢙⠀⠀⠀⠀⠀⣠⠶⠒⠦⣄⣤⠞⢉⣀⡈⠳⣄⣀⡤⠤⢤⡀⠀⠀⠀⠀⠠⠴⠶⠇⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠈⢻⡗⠒⠒⠛⠉⠀⠀⠀⠀⣠⠴⠶⢶⣾⠇⠈⠁⠀⠈⠋⠀⠀⠀⠀⠀⠛⠁⠀⠒⠂⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠔⠀⠠⡠⠆
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⡾⠁⠘⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠰⠾⢧⣤⣀⡀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡟⣷⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⣀⣉⠙⠳⢦⣀⠀⠀⠘⢧⣀⣀⣠⣾⣧⡀⠀⠀⣠⣿⣇⠀⠀⠀⢀⣿⡀⠀⠀⣀⡾⠁⣿⢀⣤⠶⠒⠒⠒⠶⠦⢷⡀
⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠠⢤⠙⣦⠀⠀⠀⠈⠉⠁⠀⠈⠙⠛⠛⠉⠀⠉⠛⠛⣻⠟⠉⠙⠛⠛⠉⠁⠀⣿⠏
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠒⢢⣀⣀⣀⣀⣀⣰⢿
⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⢨⡝⠓⠂⢀⣠⡤⠶⢤⣄⡀⠀⠀⠀⠀⠀⠣⢔⣾⡯⠭⣭⣉⠉⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠙⠒⠈⠓⠊⠀⠀⣿⣠⡴⠞⠋⠁⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠈⠳⣦⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠁⠀⠀⠀⣀⡤⠀⠀⠀⠀⠹⣆⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡀⠀⠀⠐⠀⠄⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠚⠑⠀⠀⠊⠋⠀⠀⠀⠀⠀⠀⢻⣄⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠢⣀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠈⠙⠻⣷⣦⣤⣴⡾⠿⠿⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠶⢶⡶⠾⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠁⠀⠀⠀⠀⠈⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡞⠀⠀⠀⠀⠀⠀⠀⡀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⡴⠀⠀⠀⠀⠀⠀⠀⢦⢸⣷⠀⠃⠀⠀⠀⠀⠀⠀⠀⣧⣷⢇⠆⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⢠⡴⠞⠛⠛⠛⠛⢋⡀⡇⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣄⠸⣷⡀⡇⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠠⣬⣿⡿⣷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿
⠀⠀⠀⣸⡏⡄⠀⠀⠀⠀⠀⠈⢿⣧⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⢷⠀⠀⠀⠀⠀⠀⠈⢿⣆⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢻⡄⢠⣷⡄⠀⠀⠀⠀⠀⠈⢿⣧⣽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠘⣷⣄⠉⢻⡆⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⣿⡄⠁⢨⣿⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⢻⣇⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⠀⠀⠀⠀⠀⠀⣰⣿⣍⠀⠀⠀⠀⢀⣠⣾⡏⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠈⢿⡄⠀⠀⠻⣧⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣶⣤⣤⠄⠀⢀⣾⣯⠁⢤⣉⣉⣉⣴⡟⠀⠀⠀⠀⠀⠀⢀⣴⠟⠙⣧
⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠈⠛⠷⣶⣶⣦⣤⣾⠿⠿⠿⣿⣿⣖⣂⣤⣴⣿⣿⣿⣷⠀⠈⣉⣻⣿⣿⣦⣄⣈⣉⣿⣿⣿⣿⡂⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠿⢿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣴⣦⣠⣄⡀⠀⠀⠀⠀⣠⣾⠟⠁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠛⠦⢤⣤⣤⣴⣿⠏⣰⠏⠁⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢻⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠘⡄⠀⠀⠀⠀⣰⡇⠀⠀⠀⠠⣄⣀⣤⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⣀⣴⣾⣿⣶⣶⣶⣶⣶⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠟⠛⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠆⠰⠂⠐⠐⠀⠐⠐⠀⠀⠐⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠘⠐⠀⠂⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⡀
⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⢾⡿⠿⠟⠉⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⡀⠀⣼⡇⠀⠀⠀⠀⣠⣴⠾⠛⠉⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⢿⣿⣦⡀⠀⠀⠀⢀⡀⠀⠀⢿⣿⣿⣿⣷⣾⣿⣇⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⠀⢸⣿⣷⣶⣼⣿⣿⣿⣿⣿⣿⣿⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⣦⡀⢷⣾⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⣷⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢤⣀⣈⠛⣶⣿⡆⠀⠀⠀⣠⠶⠛⠉⠈⢻⣼⡆⠀⠀⠀⢠⣦⠞
⠀⠀⠀⠀⠤⢤⣤⣤⣤⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣿⡄⠀⠰⡇⠀⠀⠀⠀⠀⢿⣿⠶⠛⠛⠛⠻⣄
⠀⠀⠀⣀⣀⣤⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⢳⡀⠀⠀⠀⠀⣸⢿⡇⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠈⠙⠿⣿⠿⠿⠿⠿⠿⠿⠟⠛⠛⢹⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣻⠿⣷⣄⠙⠶⠶⠶⢚⣡⠟⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⣼⠇⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡙⠀⠀⠙⠛⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⣴⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⣿⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⠴⠿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣿⣿⣿⠟⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣟⠉⠀⢸⠀⠀⠀⠀⠀⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⢠⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠒⠂⢀⣀⠀⠀⢀⣴⠟⢁⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠛⠉⠀⠀⠀⠈⠁⠀⣸⢻⣿⣿⠿⠶⣦⣤⣤⣤⣀⣀⣀⣀⣼⣥⡴⠟⠁
⠀⠀⠀⠀⠐⠛⠛⠿⠷⢶⣦⣤⣤⣀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣿⠙⠀⠀⠀⠀⠈⠉⠉⠉⡏⠉⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣶⡄⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣶⣶⣤⣽⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣶⣄⡀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣼⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠁⠉⢛⣉⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⢴⣶⣶⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠉⠛⠻⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⢹⣿⣿⢿⣿⣿⣿⠛⠻⠿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⣿⣿⠀⠹⣿⡿⠀⠀⠀⠀⠉⠙⠛⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠸⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠴⠚⣍⠀⠹⠀⠀⢀⠉⠓⠦⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡋⠁⠀⠀⠀⠀⠸⡇⠁⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⣠⠔⠒⠺⠭⢔⠟⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠹⡿⡆
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⢹⡄
⠀⠀⠀⢸⣇⠀⠀⠀⡄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡱⡘⣄⢳⡀
⠀⠀⠀⡾⢻⠀⠀⠀⢷⡏⠀⠀⠀⠀⠀⠰⣠⠀⠀⠀⠀⠀⠀⠀⠈⣦⡗⠁⠀⠀⠀⠀⠀⠀⠀⣌⢿⣿⢈⣧
⠀⠀⢠⡇⠈⠀⠀⢈⣿⡇⠀⠀⠀⠀⢀⢺⡇⠀⠀⠀⠀⠀⠀⠀⣝⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⣸⢻
⠀⠀⣸⠀⠀⠀⠀⢐⣿⠆⠀⠀⠀⠀⢈⣻⡇⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⡷⠁⣾
⠀⠀⢻⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⢽⡇⠀⠀⠀⠀⠀⠀⠀⢽⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣽⣧⡞⡏
⠀⠀⠸⡆⠀⠀⠈⢽⡷⠀⠀⠀⠀⠀⠀⣽⣇⠀⠀⠀⠀⠀⠀⠠⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣼⣟⠀⠇
⠀⠀⠀⢷⣄⠀⠐⢾⣧⡄⠀⠀⠀⠀⠀⢺⡂⠀⠀⠀⠀⠀⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣼⣿⠃⢸
⠀⠀⠀⠸⣧⣄⠀⢽⣿⡿⠶⠦⠀⠀⠉⢹⣯⠉⠉⠑⠂⠀⠀⢚⣿⡄⠀⠀⠀⠀⠀⠀⢠⣀⡸⣿⡿⠂⣿
⠀⠀⠀⠀⠹⣭⣀⣹⣿⣿⣿⣤⡀⣀⣫⣽⣿⣦⣲⣶⠄⢤⣤⣼⣿⡮⣄⣈⡓⠒⠦⠤⢙⣾⣿⣿⠃⠰⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠀⠉⠛⢻⡷⠛⠛⠿⠿⡿⢶⡾⣷⣶⣤⣿⣿⡿⠛⠆⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣀⠤⠦⠤⢴⡀⠀⠁⠀⠀⠉⠻⠃⠉⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⡇⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠀⡀⠀⡀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⣦⣾⢆⠄⠀⢀⠀⢰⠃⠀⢱⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢾⡏⣜⠀⡀⠀⢦⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣼⣧⣾⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⣠⠞⠋⠁⠀⠉⠻⣆⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣤⠀⠈⠛⣿⣿⠏⠀⠒⠛⠛⠓⠀⢹⣶⣶⠞⠋⠉⠉⠙⢷⣄
⠀⠀⠀⠀⠀⢀⣴⠶⠛⠛⠻⢿⣿⡟⠀⠁⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠈⠉⠉⠁⠀⢻⡄
⠀⠀⠀⠀⢠⡿⠁⠠⠖⠂⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⢿⡄
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⣼⡇⢸⡇
⠀⠀⠀⠀⠀⠈⢷⣄⣀⢀⣀⣤⣾⣿⣧⡀⠀⠀⠀⠀⣰⣿⣿⣀⠀⠀⠀⠀⠀⣸⣿⣄⠀⠀⠀⠀⢀⣼⠟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠁⠈⠛⠿⠶⠶⠶⠿⠛⠁⠙⠻⠷⠶⠶⢶⣿⡿⠛⠛⠿⠶⡶⠾⠟⠛⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠵⠒⠛⠳⣄⣀⣀⣀⣀⡀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⣀⣤⠴⠞⠛⠋⠉⠉⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠄
⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡏⠙⠦⣄⣀⠀⢀⣀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢠⡇⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠐⠛⠋⠀⠘⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢄⡤⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣠⠖⠒⠲⣄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⡝
⠀⠀⠀⠀⠀⠰⣤⣤⣤⡤⠖⠋⠀⠀⠀⢀⣠⣄⣠⣴⠋⠄⠈⠙⠿⠁⠈⠉⠁⠘⡷⠋⠉⠍⠳⡄⠀⠀⠀⠈⠀⠀⠀⠀⠠⣀⡀⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⣴⠋⠤⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠚⠿⠶⣤⣄⡀⠀⠀⠀⠀⢷⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⡇⠀⠀⠀⢀⡿⠀⠀⠀⠀⣼⢻⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⢠⠤⡈⠙⠳⣦⡀⠀⠈⠳⣤⣤⠴⠿⣦⣀⣀⣠⡿⣧⣀⣀⣀⣼⣷⣄⣀⣀⣼⠃⢸⣀⡴⠚⠋⠉⠛⠓⠚⠷
⠀⠀⠀⠀⢀⡔⠀⠀⠀⠐⠓⠌⢷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⢉⡝⠋⠀⠈⠉⠁⠀⠀⣼⡏
⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠼⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⣉⡵⠤⠤⠤⠤⠶⢻⡆⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠨⣀⡀⣄⣀⠄⠀⣸⠋⢉⣠⡴⠞⠛⠙⠛⢦⡄⠀⠀⠀⠀⢉⡽⠛⠛⠛⠳⢦⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠞⠉⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠙⢦⣸⡇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠴⠞⠁⠀⠀⠀⠀⢻⡄⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠈⠁⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠄⢀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⣀⡄⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⠛⠿⣿⣿⠋⠙⠿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⠀⠀⢤⣴⠂⠀⠀⣀⣀
⣿⣿⣿⣿⣿⠀⠄⢹⣿⡀⠀⠀⣻⣏⠀⠈⠙⠿⠉⢿⣿⡇⠀⢀⣼⣾⠀⠄⠘⣝⡆⠀⠀⢹⡻⠀⠀⠘⠿⠃
⣿⣿⣿⣿⡇⠀⠀⢸⣿⡃⠀⠐⣿⡿⠀⠈⢰⡄⠀⠸⣿⡇⠀⣼⣿⣧⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠈⢱⣆⠀⠰
⣿⣿⣿⣿⣿⠀⠀⣼⣿⠃⠀⠀⢻⡇⠀⠀⣼⡇⠀⢰⣿⡇⠀⢻⡟⣿⡄⠀⢼⣽⡇⠀⠀⢸⣿⠀⠀⣼⡏⠀⠰
⣿⣿⣿⣿⣿⠀⠀⢹⣿⣷⠀⠀⢸⣿⠀⠀⣿⠁⠀⣿⣿⡇⢀⣾⣿⣿⡆⠀⠸⣿⣷⠀⠀⢸⣿⠀⠀⢻⡇
⣿⡟⢻⣿⣿⡄⠀⠈⣿⣿⠀⠀⢸⣿⡀⠀⣿⣦⣴⣿⣿⣧⡟⠻⣿⣻⡇⠀⠀⡏⣿⠀⠀⢸⢿⡄⠀⡞⣧⡤
⣿⣧⠀⠻⣿⣿⣦⣴⣿⣿⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⣷⡀⠻⢷⣛⣶⣴⣿⡹⣧⣤⡜⠂⠉⠛⠉
⣿⣿⣷⣤⡌⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠐⢝⠦⡆⠉⢻⣿⣿⣿⢏
⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠑⠝⠷⠿⠋⠉⠒⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠤⠄
⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⢻⣷⣄⠀⡄⠀⠀⣠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠙⢢⣀⠀⣶⣤⣼⣿⣿⣿⣽⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⣠⠤⢤⣧⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠠⠤⣤⣤⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠘⠛⣆⠀⣎⠀⠀⠀⢿⡖⠒⠚⢧⡀
⠀⠀⠀⠀⠀⠲⣶⣶⣿⣿⡿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⡘⠤⣀⠤⣺⠇⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⡏⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠉⠛⠒⠛⠁⠀⠀⢀⣆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⣶⡶⠖⠒⠒⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣉
⠀⠀⠀⠀⠀⠀⣀⣤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠉⠀⠀⠀⠀⢹⠗⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣆⡀⠀⢠⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣶⣶⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢶⠤⣄⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣀⣠⡤⠤⠤⠠⠤⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠾⢻⣷⣽⣿⣿⣿⣿⣾⣿⡁⠹⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⢿⣟⢿⡵⠛⠛⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠈⣿⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠻⣧⡀⠀⠀⠀⢀⣼⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⢿⣦⣀⣴⠟⠁⠈⣿⡆⠀⠀⠀⣀⣤⣶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡷⢶⣶⣤⣤⣼⡇⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠸⣷⣴⡶⠟⠛⠉⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣸⣿⠀⠀⠀⣀⠀⣠⣶⠿⠶⠾⠟⠉⠙⠻⢶⠶⠶⢶⣤⠀⠀⠀⣼⣯⣄⣀⣀⡀
⠀⠀⠀⠀⠙⢿⣟⠛⠛⠛⠛⠛⠀⣰⡿⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠉⠉⠙⢛⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⢠⣿⠁⠀⠀⡀⠀⠀⠀⣴⡀⠀⠀⠀⣤⠀⠀⠀⢸⣿⠀⠀⠀⠀⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢈⣿⡷⠀⠀⢸⣿⠀⠀⢰⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠐⢾⣿⣁
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⣿⠇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠛⢷⣦⣀
⠀⠀⠀⠀⠐⠻⣿⣤⡀⠀⠀⠀⠈⣿⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⣀⣤⣾⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⡀⠀⢿⣇⠀⢸⣿⠀⠀⠀⣿⡆⠀⠀⢀⣿⣄⣀⣀⣼⠟⣿⡇⠀⠸⣿⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠈⠛⠻⠟⠛⠿⠶⠿⠛⢿⣶⠶⠿⠋⠙⠛⠛⠉⢀⣿⠃⠀⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⢀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣤⣤⣤⣤⣤⣴⡾⠏⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⣴⣿⣷⡶⠿⠛⠋⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⣸⡿⠶⢶⣶⣦⣿⣷⡀
⠀⠀⠀⠀⠀⠚⠉⠉⠀⠀⠀⠀⠀⢸⡇⠀⢀⣠⣴⡾⣷⡀⠀⠀⠀⠀⣰⣦⣀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣾⠟⠋⠁⠀⠹⣷⠀⠀⢀⣼⠟⠉⠛⢷⣦⣀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⢹⣧⣠⣾⠏⠀⠀⠀⠀⠈⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⢹⣿⡿⠻⠿⢿⡬⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⠀⢹⡛⣷⠻⠿⠿⡦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠉⣻⣿⣿⣿⣿⣦⣴⣀⣀⡈⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣆⡉⣻⢻⣿⣧⢿⡦⠤⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣄⠈⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢰⡻⣿⣿⣿⣇⢿⣿⣸⣿⠷⣾⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⠋⠀⠀⠈⠛⠉⠁⣸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⣷⣽⣷⣽⣿⣮⣿⠋⠁⠀⠈⠛⠋⠁⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⠛⣻⣿⣿⡇⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡘⠛⣻⣿⢫⡇⠀⠀⠀⠀⠀⣰⡾⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡽⠿⣿⣿⣦⣄⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣦⡽⠿⣎⠿⣦⣄⣀⣤⣾⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣈⣛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣷⣌⣑⡾⣿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⢿⠋⠙⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⣿⡿⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣿⣿⣿⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⣤⣿⣿⡿⠀⠀⠀⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⢹⡿⣿⣿⠇⠀⠀⠀⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣼⠁⠘⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣧⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠿⠃

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠐⠃⢡⠈⡀⠄⠀⠀⠀⡴⠃⣠⡀⠀⢤⡀⠀⢠⠶⠛⠛⠚⢻⡧⠄⠀⠀⠀⢀⡤⢮⠀⣈⡤⠲⡄
⠀⠀⠰⠀⢷⣄⣿⣯⣤⣀⠂⠀⠰⢷⡀⠘⡿⠚⢠⠃⠀⢠⠃⢈⡃⠉⢸⣦⡰⠀⠀⣞⠵⣦⢀⣇⣓⣴⣂⢱
⠀⠀⠀⠇⠀⠀⠄⠈⠀⠈⠠⠀⠀⢸⠁⠀⠀⢀⠇⠀⠀⠀⠉⠀⣌⠉⠘⠉⠀⠀⠀⠸⡆⠈⠋⠰⠋⠀⠉⡸⠃
⠀⠀⠀⠈⢄⠀⠀⠀⠀⠠⠂⠀⠀⣇⠀⠀⠀⡌⠀⠀⠀⠀⠘⣄⠈⠀⠛⣦⠆⠀⠀⠀⢫⠀⠀⡀⠀⠀⣸⠃
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢹⠀⠀⠀⠀⠈⠃⠚⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⣄
⠀⠀⠀⠀⠀⢀⠠⠒⠒⠀⡄⠀⠀⠀⠀⠀⠀⡀⠂⢺⠀⢱⠀⠈⡀⣀⠀⠀⠀⠀⢠⡶⠟⠓⠒⠚⠻⣿⣤⣄
⠀⠀⠀⠀⠐⠀⢀⠀⠠⠀⠁⠄⠀⠀⠀⠀⢰⠐⢷⡀⠂⢾⡶⠊⢀⡈⢢⠀⠀⠀⢸⠖⠒⠀⠀⠒⠢⣏⡀⢠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⡆⠀⡄⠀⠀⠸⡀⠙⠚⠽⠛⠟⡻⠟⠶⡄⠑⠀⠀⠘⡤⠤⠦⠤⠤⢠⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠤⢦⣴⣷⠶⠶⠖⠁⠀⠀⠀⡇⠀⠀⠐⠄⠈⠀⠀⠀⠀⠔⠀⠀⠀⢁⠀⠐⣦⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠠⠟⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠈⢧⠀⠀⠈⠉⢳⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠚⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣾⣿⠿⠛⠛⢛⣛⠿⣿⣶⣶⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠿⢿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠳⡌⠿⠋⠁⠀⠈⠉⠛⢿⣿⣆
⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⠏⠀⠀⠀⠙⢦⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⣠⣶⣿⠿⠿⠛⢿⠏⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⡄
⠀⠀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇
⠀⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⡟⢀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣿⣿
⢰⣿⣇⡏⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿
⢸⣿⣿⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⢸⣿⣿
⢸⣿⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡿⠘⣿⣿⡀
⢸⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⡇⠀⠸⣿⣧
⠸⣿⣧⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠁⠀⠀⢿⣿
⠀⠹⣿⣦⣤⣀⣀⣠⣿⣆⡠⣀⣀⡤⠔⣸⣷⡀⠀⠀⠀⢀⣀⣤⣾⣿⣷⣶⣶⠤⠀⠀⢀⣀⣾⠿⠳⠀⠀⠀⠀⢸⣿
⠀⠀⠈⠛⠿⢿⡿⠿⠿⢿⣿⣷⣶⣶⣿⣿⠟⣻⣿⣛⠛⠛⠛⠉⠁⠀⠉⠉⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⣿⢰⠃⠀⠈⠙⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⣤⢤⣤⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣾⣧⣀⢠⡄⠀⠀⡼⠀⠀⠀⢀⣮⣵⣾⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣷⣤⣄⣥⣤⣶⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠉⠉⠛⠒⠲⠤⣖⠒⠂⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢤⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠷⣆⠀⠀⡔⠀⠀⠀⠀⢸⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⠤⠤⡴⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣶⡇⠀⠀⠀⠀⢸⠉⠙⡆
⣀⣤⠤⠤⠔⠒⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠈⠹⣄⠀⠀⠀⠘⣆⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡀⠀⠀⠀⠈⠳⢤⣀⣀⣹⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠉⠉⠉⠙⠛⠒⠒⠛⠛⢹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⣶⠤⢤⣄⣀⣀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠶⠦⠵⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠝⠒⠀⠶⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠖⠚⠉⠁
⠀⠀⣀⣠⠴⠒⠋⠉
⠖⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⣛⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⢰⠀⠀⢌⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠦⠀⠈⠁⠠⢁⠠⠐⠀⠀⠀⠠⠆⠀⠰⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⢰⣦⠴⠺⠀⢠⣤⢄⣀⠀⣤⡄⠈⠗⠶⣤⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠀⣾⣧⠄⠀⣴⢦⡆⣴⣤⠄⠠⣼⣷⠀⣽⣦⡤
⠀⠀⣴⣶⠶⠶⣶⠶⣞⣿⣯⡿⠀⣸⣛⢳⣄⣠⣀⢥⣐⣀⣠⣞⣻⣇⠐⢿⣽⣿⣳⠶⣶⠶⠶⣶⣦
⠀⠀⠶⠂⠃⠐⣀⣉⣩⡹⢿⠗⣰⣿⣯⣍⣉⣩⣍⢶⣩⣍⣉⣩⣽⣿⣆⢺⡿⢏⣍⣉⣀⠂⠘⠐⠶
⠀⠀⠰⠶⢟⣿⣻⡯⠿⠿⣷⣉⠉⠛⠿⣭⣉⠉⢉⠀⢉⣉⣉⣭⠿⠛⠉⣉⣾⠿⠿⢽⣟⣿⡻⠶⠆
⠀⠀⠀⠙⠋⠉⠀⠀⠥⠀⠤⠊⠉⠁⡠⠘⠈⠀⡤⠀⠐⢌⠀⠁⠀⡈⠉⠀⠀⠄⠀⠀⠀⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡝⠁⠀⠶⠸⠃⠘⠀⠈⠎⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠠⢄⣀⠀⠀⣠⣀⠀⠀⠐⠶⠾⠷⠆⠀⣿⣿⣿⠀⠀⠀⣀⠀⣀⡄⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⡀⠀⠀⠡⠀⣿⣿⣿⡆⠀⠀⢈⠀⠤⢀⠀⠘⠄
⠀⠀⠀⠐⢁⡀⠰⠀⠀⠀⠀⠀⠀⠀⠈⢙⡷⢄⠰⣦⣼⣿⣿⣧⣴⣾⣿⡄⣠⠛⠿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠈⠙⠳⣦⣈⠈⣿⣿⡟⢻⣿⣿⠃⢸⡷⣄⠠⠘⡆⡸⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣄⣀⣤⣄⣀⠀⠉⠓⢈⣿⣷⣾⣿⡟⣰⠀⢿⡜⠀⡀⢁⠇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄⠀⣼⣿⣿⣿⣿⣷⣿⡀⣸⣿⢀⣷⣶⣆⣾⠁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠁⢥⠟⡉⣻⢰⡀⣿⢿⣿⣿⣿⣿⣿⣿⠃⢸⣎⠙⠿⠛⣋⣠⣴⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣆⡀⠀⢀⣴⣶⠲⣿⣿⣿⢃⣿⣿⣿⣿⣿⡿⣿⡇⠀⠀⠈⠋⠛⠋⢩⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠿⣼⢛⣡⣿⣾⣿⣿⣿⣿⣿⣶⣿⡅⠀⠀⠀⠀⣤⡄⢺⣿⣿⣿⣿⣿⡆⢀⣼⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠿⠿⠃⠀⢙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡄⠀⠀⠀⠀⠛⣮⣿⣿⣿⣿⣿⣿⣾⣿⡿⠏
⠀⠀⠀⢀⠀⠀⠀⣴⠟⠋⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⣴⣿⣷⣶⡾⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠙⣿⣿⣿⣿⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠞⠀⠀⠈⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠃⠀⠀⠀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡀⠀⠀⠘⣿⣿⣿⣿⣿⠇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣶⣿⣿⣿⣧⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⢿⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣿⡟⢹⣿⣷⡟⢻⣿⣿⣿⣿⣠⡄⠀⠀⠈⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢘⣿⣿⣿⣿⣿⣻⣗⣽⡟⢿⣿⣿⡿⢟⡉⠈⢆⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣤⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠈⢻⣿⠿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣦⠀⠀⠀⣰⢃⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢦⠀⢸⣄⠀⠀⢉⡜⠙⠀⠀⣰⠣⠚⠃⠀⠀⠀⡰⠛⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣄⠀⠀⠈⢷⠤⡀⠃⠀⠀⠑⢤⠜⠀⡀⠡⡴⠋⠀⠀⡄⠀⠀⢀⣤⡞⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⡈⠑⠀⠀⠈⠄⠀⠉⠛⠀⠰⠄⣀⠎⠑⠤⠤⠂⠡⡀⡧⠐⠈⠁⢀⠀⠀⠀⠊⡠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠦⠄⡀⠀⠈⠻⣟⡲⠒⠂⠱⠂⠀⠀⠒⠡⠶⢤⡔⠁⠤⠔⠐⠄⠁⠀⠲⠀⠀⠢⡄⠀⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⢸⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⢀⣀⡴⠁⠀⠀⠙⠻⣷⡀⡞⠀⠁⠖⠐⠲⢏⠶⠒⠐⠂⠁⠈⡆⣰⡿⠟⠁⠀⠀⠱⣬⡝⠛⠛
⠀⠀⠐⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⢻⡇⠀⠀⠀⠀⢐⣟⠓⠉⠀⠀⠁⠀⠀⢨⠀⠀⠀⠀⠀⠈⠑⢻⣗⠀⠀⠀⠀⠀⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣇⠀⠀⠀⠀⣱⣿⠖⠒⠒⠒⠒⠓⠛⢧⠟⠛⠒⠒⠒⠒⠒⢾⣷⡁⠀⠀⠀⢀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢷⠀⠠⠞⠁⣸⡷⠤⣀⠀⣀⠀⢀⠜⢄⠀⢀⡀⣀⣠⠴⣿⡀⠙⠦⠀⠰⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣢⣟⠀⠀⠀⠀⠀⠙⠦⡀⠀⠈⠉⠉⣅⠀⠠⣍⠉⠁⠀⠀⡠⠟⠁⠀⠀⠀⠀⢘⣧⣀⣀
⠀⠀⠀⠀⢀⣀⡠⠤⠒⠊⠉⠀⠀⠀⠉⠉⠒⠦⡄⢀⡀⢀⣨⠅⠴⠒⢁⣌⠲⠀⠈⠑⢲⠄⣭⣀⠀⠀⣀⠤⠒⠊⠉⠛⠀⠀⠈⠉⠒⠢⠤⣀⣀
⠀⠀⠀⠀⠈⠀⠀⠀⠐⠋⣁⠀⠀⣀⡴⠊⠀⠐⠀⠀⠉⣀⡀⠱⡀⠐⢡⡂⠱⠎⢉⡉⠃⠀⠀⠀⠈⠑⠚⠧⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠁⠰⠞⠁⠀⠀⢠⢁⡠⣴⠊⠀⢀⠀⠀⡠⡿⠑⢄⢀⠃⠘⡆⢄⢰⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⣡⡞⠁⠀⠀⢸⣤⠊⣼⡇⠀⠈⡿⠀⡀⠸⣆⠙⠆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠃⠀⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⡷⠶⡼⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣀⣶⣄⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣟⡐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣎⣽⣿⡎⠐⠿⡿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣸⠻⠙⣮⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠒⠻⠷⣈⢷⣿⠛⠁⣀⣦⡀⣸⣿⣳⣄⠀⠀⠀⠀⠘⢋⣤⡒⢋⣵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣆⠀⠀⣄⠀⠈⠈⠙⠛⠉⠩⠈⠙⠿⢦⣼⣥⣀⡀⢘⡟⠀⠻⡅⣚⣀⢁⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡏⠑⠂⠹⣧⠀⡄⠀⠀⢠⣶⢠⣖⣻⣷⢶⡀⠉⠛⠛⠻⣷⣄⣽⣶⡛⠘⠁⠘⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢷⣆⠀⠈⠉⠛⠛⠛⠚⡛⠉⢸⣿⣿⣷⣿⡇⠀⠀⠀⠀⠈⣛⡋⣿⣿⠲⡄⠀⢰⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣙⢷⣤⣀⠀⣀⣀⣠⡿⠜⠘⣿⣛⡛⢻⡀⠀⠀⠀⠀⠀⠙⠻⠿⣿⡗⠀⠙⠻⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⣧⣽⣿⣿⣟⣉⣥⠀⠀⠀⠈⠉⠳⢦⣻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢦⣀⣤⣄⣀⣠⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⠣⠤⠋⢉⣽⣿⣯⣈⡁⠀⠀⠀⠀⠀⠈⢻⣟⣿⣿⠇⠀⠀⠀⠀⠀⠀⠡⠛⠛⠻⢿⣿⣦⡜⡂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠿⠇⠀⠀⠀⠈⠉⠛⠛⣿⠅⠀⠀⠀⠀⠀⠀⠘⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠚⢋⠉⠀⠀⢐⡉⠉⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⢐⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠄⣶⠋⠕⠁⠠⢀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠤⠒⠉⣠⠄
⠀⠀⠀⠀⠀⢀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡏⠀⠀⠀⠀⠀⠖⠁
⠀⡀⠀⠀⢀⠾⠿⠀⣀⠀⢀⣀⠀⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠠⣄⡀⠀⠈⠀⣼
⠀⠂⠀⠀⠀⠀⠀⠀⠈⠠⠛⠷⠀⠀⠀⠘⢄⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⢠⡿⠓⠀⠰⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣤⣤⣶⣿⠟⠛⠛⠿⣿⣦⣤⣠⣤⣶⣿⠿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋⠁⠀⠉⠉⠉⣿⠃⠀⠀⠀⠀⠀⠉⢻⡿⠛⠋⠁⠀⠀⠉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠿⠿⠿⠛⠹⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⣼⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣦⢸⡇⠀⠀⠀⠀⠀⡇⢿⡆⠀⠀⠀⠀⠀⢸⣸⡇⠀⠀⠀⠀⠀⠀⣧⣷⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⢿⢸⣇⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⢸⣏⣿⡀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⢀⣿⣿⠄⠈⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⢴⣿⣿⣿⣦⡀⠀⠤⣽⣿⣿⣤⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣀⣀⣹⣿⣿⣿⣦⡀⠀⠀⣐⣿⣿⣿⣦⣄⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⡫⠍⠁⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠟⠛⠉⠉⠉⠉⠛⠻⢿⣿⠁⠈⠁⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⡿⢁⡴⠶⠶⠶⣦⡄⠀⠀⠀⠈⠃⠀⠀⠀⢀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣇⣾⠁⠀⠠⣤⣼⡇⠀⠀⠀⠀⣆⠀⣠⣾⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣤⣭⣽⣿⣇⣀⣀⠀⠀⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠻⠿⠿⠟⠋





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣹⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⢀⣤⢠⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡅⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠈⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢹⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢻⣷⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣇
⠀⠀⠀⠀⠀⢻⣷⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠙⣷⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠙⠛⠿⣿⣿⣿⣿⣿⡧⠐⢶⣬⣯⣥⣾⢶⣷⠟⠡⣮⣽⡻⠿⣿⣿⣿⣿⣿⢋⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠌⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠈⠉⣙⣛⣋⣭⣵⣾⣷⣌⠛⠿⠟⠛⢋⣠⣴⣶⣤⣙⠻⠿⠿⠿⠿⣟⣥⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠋⠁⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣛⣛⣻⣟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⢿⣿⣿⣿⡿⢟⣵⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠏⠛⠉

⠀⠀⠀⠀⠀⠀⠀⡠⣤⠶⢄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠢⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⢴⡒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠲⡤⢤
⠀⠀⠀⠀⠀⡉⠙⠀⢸⠀⢀⠄⢤⠀⠀⠀⠀⢠⠞⠣⣐⣄⠀⠁⠢⣄⠀⠀⠀⡶⠛⠃⠉⠉⠙⢻⡷⠴⠄⠀⠀⠀⠀⣀⡤⠞⢦⠀⣄⣡⡔⠛⣆
⠀⠀⠀⢠⠃⣹⢄⡁⣺⣟⣁⣠⠀⠣⡀⠀⠠⣫⣄⠀⠘⢿⡷⠎⢀⡼⠀⠀⠀⡝⠒⠚⠓⠒⠓⢚⣥⡀⡼⠀⠀⢠⡏⣑⣤⡀⠸⡆⠓⢤⣤⠀⢸⡄
⠀⠀⠀⠈⣆⠀⠉⠓⠋⢋⠝⠋⠛⠄⢀⠀⠀⠀⣿⠀⠀⠈⠁⠀⡸⠁⠀⠀⠀⢳⠖⠚⠓⠲⠒⢼⡿⠙⠰⠀⠀⠀⣷⡀⠙⢯⠞⢛⡷⠟⠛⠓⠀⣷
⠀⠀⠀⠀⢻⠀⠀⠈⠃⠀⠀⠀⠀⠀⠊⠀⠀⢠⡇⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⡄⠠⠵⣆⠀⡀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠘⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢸⡃⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠈⠉⢷⡴⠀⠀⠀⠀⠀⢳⠀⠀⠀⡀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠋⡅⠀⠀⠀⠀⠀⠈⢷⠐⠚⠃⠀⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣰⡿⠁⠀⠀⠀⠉⠳⠦⠴⠊⠉⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠉⠀⠈⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⢧⠀⠀⠀⠀⠀⢰⣸⡆⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡐⠉⠉⠉⠉⢰⡎⢀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⣼⡀
⠀⠀⠀⢠⠁⠀⠀⠀⢬⣾⢿⡌⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠹⣇⠻⡄⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠇⠀⡄⠀⠀⠀⠀⠹⣆⣿⡀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢿
⠀⠀⠀⢸⠄⠉⢆⠀⠀⠀⠀⠙⢿⣷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⡄⠀⠸⡄⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢨⡆⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢿⡀⠀⠱⣀⠀⠀⠀⠀⢀⣿⣦⣤⣤⡤⠀⢩⣿⠀⣄⣀⣀⡜⠀⠀⠀⠀⢀⣠⠞⢦
⠀⠀⠀⠀⠈⠳⣄⠀⠈⠛⠶⠶⠶⠟⠛⠛⠿⠿⠴⠶⣿⣿⣧⣤⣬⣿⣿⣷⣦⡬⠿⠿⠷⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠈⠓⢤⣄⣀⡀⠀⠀⢀⣀⣠⣄⣀⠀⠀⠀⠉⣩⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠈⠉⠂⠠⠤⢼⡏⢰⠋⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠸⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⢀⣤⣾⣥⣤⣤⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⢸⠀⢣⡀⠀⠀⠀⠀⣀⡴⠋⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠒⢤⣀⠀⠀⢀⡏⠀⠀⠙⢷⣶⡾⠟⠋⠀⠀⢀⡟
⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⣠⠔⠒⠢⡄⠀⣾⣇⠀⠀⠀⠀⡀⠀⠀⠂
⠀⠀⠀⠙⢣⡂⢄⡀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢀⠒⠁⠈⠉⠛⠛⠁⠀⠀⠀⠘⣆⠻⠿⠗⠒⠋⠁⣠⠎
⠀⠀⠀⠀⠀⠈⠲⣄⠉⠒⠲⠦⠶⠾⠟⠀⣀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⠀⠀⢠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣄⠀⠀⣸⣿
⠀⠀⠀⠀⣀⣀⣀⣠⠤⠾⠃⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠈⠢⡀⠘⠿⣷⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠉⠓⢶⣄⡀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠙⢦⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⢓⠀⠀⠀⢠⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⡏⠀⠀⠀⢁⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢯⣠⠀⠀⠀⢣⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⡇⠀⠀⠀⣴⡿⠀⠀⢰⠃⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠙⣦⡀⠀⠀⠳⠀⠀⠀⠀⢁⠀⠀⠀⠀⣻⣶⠶⠿⠛⠀⠀⢀⠏⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠑⢤⠀⠀⠈⢄⠀⠀⠀⣶⣤⣴⣾⠋⠀⠀⢀⣀⣀⣠⠎⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⣠⣤⠶⢶⣶⡵⢤⣀⣼⠗⠒⠋⠁⠀⠀⠙⠦⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⡜⡡⠖⠋⠁⠀⠀⠀⠈⢿⡄⠀⠀⢀⣠⣴⣶⡶⠶⠶⠶⠶⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠞⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠄⣀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⡏⠁⠀⠀⠀⠉⠑⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄
⠀⠀⠀⠀⠀⢀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⡀
⠀⠀⠀⢠⡟⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣰⠀⢧
⠀⠀⠀⡸⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⢸
⠀⠀⢀⡇⠀⠀⠀⠀⠠⣿⡇⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠂
⠀⠀⢸⡇⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⡀⢡
⠀⠀⠈⣧⠀⠀⠀⠀⢐⣿⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⡇
⠀⠀⠀⢻⡂⠀⠀⠀⢸⣿⠂⠀⠀⠀⠀⠀⠀⣻⡗⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢰⠃
⠀⠀⠀⠘⣷⠀⠀⠀⢼⣿⡇⠀⠀⠀⠀⠀⢀⣿⣷⢤⠀⠀⠀⠀⠀⠀⢴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⣼
⠀⠀⠀⠀⢹⣇⠀⠀⢸⣿⣿⣶⠀⠀⠀⠀⢐⣿⣿⠓⠀⠀⠀⠀⠀⢀⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠲⣾⣿⣿⡟⠀⢿
⠀⠀⠀⠀⠀⠙⠤⠤⢾⠿⢿⣿⣀⡀⠀⣠⣾⣿⡿⣧⣄⠀⠀⢈⣿⣿⣿⣿⣷⡀⠀⠀⠈⢐⣒⣀⣠⣿⣿⣿⠿⠁⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠁⠀⠀⠉⠛⣻⠟⠛⠙⠛⠛⠻⠿⣿⢿⣿⣶⣾⣿⣿⡿⠻⠉⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣠⠄⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡟⠁⠀⠀⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠤⣄⣠⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠂⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠷⢶⣶⣼⣶⣶⣆⣄⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠀⠄⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣤⠈⠄
⠀⠀⠀⢔⠐⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⡱⠀⠀⢠⣿⣿⠿⠛⠉⣇⠸⡀
⠀⠀⠀⣀⣽⣿⠦⡁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠸⠱⠀⠀⠘⣿⣇⠀⠀⠀⡇⠀⠈⡄
⠀⠀⢠⣿⣿⡁⠀⠈⠢⡬⠒⢄⠀⠀⠀⠀⠀⠀⡄⢣⠀⠀⠈⢻⣷⡖⠈⢳⠐⢄⡁
⠀⠀⠀⠙⢿⣥⣀⣀⠠⠧⡄⠀⠵⢄⠀⠀⠀⠀⠡⠘⠁⠀⠀⠀⡜⠁⠈⠁⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢸⣄⠔⠁⠀⠀⠀⠈⠢⡀⠀⠀⠙⠢⠄⠐⠘⡀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠔⠁⠐⡀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠐⡀⢀⠤⠒⠐⠇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢈⠦⠀⠀⠀⠀⠀⠘⡳⠀⠂⠀⠒
⠀⠀⠀⠀⠀⠀⠃⠂⠈⠉⠢⡀⠀⠀⢀⡠⠄⠂⠁⠀⢡⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠎⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⡘⠀⠀⠀⢀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡌⠀⠀⡠⠊⠉⢄⠀⠀⠃⠀⢀⠁⠀⡠⠂⠁⠈⠢⣀⠀⠘
⠀⠀⠀⠀⠀⠀⠰⠢⠄⠀⠀⡘⠀⢀⠎⠀⠀⠀⠀⠡⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠈⠢⡀⠆
⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⡰⢀⠔⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⡆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠱⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠂⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠐⠦


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣿⡆⠀⠆⠁⠠⡆⠀⣶⢶⣀⡀⠐⠀⠀⠀⠀⠀⡇⠘⠘⣿⣿⣿⣿⣿⡇⣿⣿⣿⡿⠿⣿⢸⣿⡇
⠀⠀⠄⠀⠀⠀⣿⣇⣀⣀⣀⣀⡃⠀⠙⠋⠁⠀⠀⠀⠀⠀⢠⣄⣷⣠⣀⣿⣿⣽⣭⣭⣥⣾⣿⣿⣶⣾⣶⣼⣿⡟⠻
⠀⠈⠉⠉⠉⠈⠛⠿⢿⣿⣿⡿⠁⠂⠀⠂⠄⠀⠀⠀⠀⠀⠘⠛⡇⢰⡟⣿⣿⣿⣿⣿⡏⣿⠟⡋⠙⣿⢻⢸⣿⠁
⠀⠀⠀⠀⠰⠀⠀⠀⢀⣩⡀⠀⠀⠀⠀⢈⠀⢀⣠⠀⠀⠀⠀⠾⠖⠈⣿⣿⣿⣿⣿⡿⠇⣿⡀⣠⣟⣿⣧⢿⣿⡄
⠀⠀⠀⠘⠀⡖⢐⣂⣠⣤⣤⣀⡂⢀⡀⢸⠃⠀⠉⠀⠀⠀⠀⢸⠥⠒⣿⣿⣿⣿⠃⠀⠰⣤⠖⣵⡎⣿⡿⢿⣿⠃⢀
⠀⠀⠀⢤⣠⡒⠛⠛⠿⣿⣿⣿⣿⣿⣵⠄⠀⠀⠀⠀⠀⠀⢾⢬⡴⠆⣼⣶⡶⠒⠒⢶⣶⣿⢋⣟⢻⣷⣷⣾⣿⠉⠻
⠀⠀⡴⡜⠛⢿⠂⠀⠀⠀⠹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢼⡤⠄⣿⣿⣿⠀⠀⡠⢎⣡⣾⣟⠸⡿⡏⣿⣿⠃
⠀⢀⣻⡀⠀⢺⠶⠄⠀⠀⠀⢹⣿⠟⢉⡷⢟⣦⡄⠀⠀⠀⠀⢸⡟⢆⣿⣿⣿⡿⠋⠁⢠⣿⣯⡿⢿⠁⡷⣿⣿
⠀⠈⢻⢉⣴⡈⠀⠈⠁⠀⠀⢨⡇⡴⠋⣴⣿⡘⣧⡀⠀⠀⠀⠸⠷⠂⢿⣿⠿⢷⠤⣶⢺⣯⢭⠣⠞⠀⠇⣿⣿⠀⡀
⠐⢿⡾⢯⣐⡁⠀⠰⢶⣿⡶⣾⣷⣃⣼⢳⡙⠛⣿⠁⠀⠀⠛⢹⡟⣓⢾⣿⣟⢿⡉⣉⣾⣯⢿⣍⣿⢻⣷⣿⣿⠙⠛
⠀⢾⣿⣿⣿⡇⠀⢀⣀⢉⣠⣹⠏⢈⡇⣺⣿⡿⠟⠀⠀⠀⠀⠻⠿⠏⠛⠛⠛⠒⠓⠛⢹⠿⠿⠇⠋⠛⢻⠿⠿
⠀⢘⡁⠋⡌⠁⠀⠀⣹⣛⣳⣿⠖⠿⣷⢹⣿⢿⣶⣤⡀⠤⠀⠀⠀⠀⠷⠀⢀⣶⣷⡆⢼⡄⠀⠀⠀⠀⢸⡆
⠀⢸⣿⣰⡼⠀⠠⢾⣳⣿⣿⣿⣻⣦⣿⣾⣿⠾⢿⣿⣇⠀⠀⠀⣀⣤⣶⠛⢋⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠸⣿⣿⣿⣰⣿⣾⣿⣿⣿⡿⠿⠿⣿⠏⠁⠀⠀⢈⣯⣤⣶⣿⠿⠻⠎⡩⠉⠀⠀⠀⢀⣀⢀⣀
⠀⢰⡟⠉⠀⣿⣶⣿⣼⣿⣷⡄⢀⠀⠀⣴⣾⣦⣻⣯⠑⣻⣥⣤⣾⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⢸⣷⠶⣦⡀⠈⣿⡿⠿⡥⣥⡀⠀⣻⣿⣿⠟⠋⠁⠀⣿⣿⣿⣿⣿⠇
⠀⢀⣿⣭⣯⣤⣤⢨⣳⣾⡷⠿⢷⡰⡿⠟⠁⣤⣤⣶⠋⠉⠉⠁⠀⡀
⠀⠀⢿⣶⣾⢿⣿⡷⣿⣿⣷⣄⠈⠓⠀⠀⣼⣿⣿⣿⣆⠀⠀⢀⣿⣵⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠐⣾⡿⣁⣨⣥⣶⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⠀⠀⢨⣿⣮⣟⣯⣶⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠀⠀⢸⣿⢹⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⣠⣿⣿⣿⣿⣍⠋⠓⠐⠒⠦⠀⠄⠀⠀⠀⠀⠀⢀
⠠⠶⠴⠆⠀⢋⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⣏⠀⠀⢀⣀⣤⡀⠀⠀⠀⢘⡁⢈⣉⣰
⠀⠀⠀⠀⠈⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣯⣿⣷⣿⡿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⡿⠏⣀⢠⠀⣸⡅⠀⢈⢉
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣈⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠃⠀⠈⠙⠟⠋⠉⠛
⠀⣀⠀⠘⢶⣿⠗⣿⣿⣿⣿⣿⣿⣿⣟⢸⡏⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡁⠂⠤⠤⢄⡀⠀⠀⠀⠈⠉⠓
⠀⠁⠀⠀⣸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣆⣼⣿⣿⡿⣯⣷⣿⣿⣿⣿⣿⣿⣾⠇⠀⠀⠀⠀⡀⠀⢠⣦⣤⣤⣄⣀
⠀⠀⠀⠀⢾⣿⣿⣿⡇⣿⣿⢿⣿⠏⢩⣤⣮⣿⡇⢸⣀⡽⢿⣿⣿⣿⣿⣿⣿⣉⡁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣿⡟
⠀⠀⠀⠀⣹⠀⣿⣿⠒⣿⠟⠀⠀⠀⠀⠘⠛⣯⠁⣸⣿⣷⣿⣿⣿⡛⠛⠋⢠⣈⣉⠿⡲⣦⣀⡄⠀⠀⠿⠛⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠛⠀⠀⠠⢤⡄⢰⣿⡿⠷⡿⠿⢻⡟⠛⠉⠁⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⠀⠈
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢀⣀⠀⢀⠀⠳⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣼⣿⣿⢿⣿⣿⣍⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⠷⠶⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⠁⠀⠀⢁⣭⣫⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠷⢶⣾⠋⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣍⡉⢡⣼⣿⣿⣿⣿⡀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣤⣀⣀⠀⣀⣸⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⢻⣿⢿⣿⠿⠇⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⣿⠋⠀⢠⡀⢱⠿⠟⠛⠻⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠃⢀⣾⣿⠟⢿⡎⣠⣤⣤⣤⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣤⣿⣧⠀⢀⡠⠽⠮⠖⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠿⠀⠸⣿⡿⣿⣮⣛⣭⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠙⠛⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣤⣄⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⢒⣒⡒⠒⢋⠉⠀⠀⠀⠙⠲⣤⣀⣤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⢚⣡⠀⠀⠀⢢⡉⠀⠈⢻⣷⠰⠀⠀⠸⠀⠀⣤⣤⠀⠀⠈⢷
⠀⠀⠀⠀⠀⣰⠞⠉⠉⠉⠁⠀⠉⠻⣄⠀⠀⠘⠁⠀⠀⠀⠉⢧⠀⠀⠀⠀⠀⠉⢿⡲⠀⠀⢸⠳⡄
⠀⠀⠀⠀⣸⢣⢠⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⡼⠁⢸⣯⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠘⣧⡆⣿⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⢹⠁⢿⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠘⢷⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠸⡎⡄⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣿⣴⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⣸⣤⣀⣀⠀⠀⠀⢀⣿⣿⣷⣄⣀⠀⠀⣠⢸⣇⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠈⠛⠷⣬⣍⢉⣘⣿⣿⣿⣿⣿⣓⣒⣾⣿⣿⣿⣯⣭⣽⣿⣿⣞⡻⠯⣽⠭⢂⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠁⠙⠛⠿⠿⠛⠋⣡⣿⢟⣿⣿⠿⠿⠛⠛⠛⠛⠿⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣋⠀⠈⠙⡆⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣣⣤⣤⣴⣦⣤⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⢀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⣜⣿⣿⣿⣿⣧⣿⣿⢿⣛⣻⣯⣿⣷⣶⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠞⠉⠐⡆⠀⠈⠙⠓⢦⣄⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠂⠈⠉⠁⠀⢠⣦⠂⠀⠁⠀⠀⣢⠀⠀⠀⠀⠈⠉⠉⠁⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⢛⡅⠀⠀⠀⠀⠀⠀⠀⠈⡧⠄⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠁⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢠⠸⡆
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⡀⢻⡄
⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⢴⢸⡥⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢦⢳⡄⢿⡀
⠀⠀⠀⠀⠀⠀⡿⢻⠀⠀⠀⢸⣾⡃⠀⠀⠀⠀⠀⢀⢷⡰⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢬⡻⣿⣧⢨⣧
⠀⠀⠀⠀⠀⠸⠇⠈⠀⠀⠀⣹⣿⠁⠀⠀⠀⠀⠀⠺⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣱⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⡇⡾⣿
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢾⣷⠀⠀⠀⠀⠀⠈⠳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠿⠃⡏
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡻⣮⢀⡶⡇
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣾⡅⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⡝⢷⠟⢡⡇
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⣀⣿⠅⠀⠀⠀⠀⠀⠀⢀⣾⠂⠀⠀⠀⠀⠀⠀⠀⣀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣴⣾⡗⢸⠁
⠀⠀⠀⠀⠀⠀⣿⡶⠀⠀⢚⣿⣿⣤⠀⠀⠀⠀⣀⣰⣿⣥⣀⣀⣀⡀⠀⠤⠴⣺⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⣟⠀⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣶⠀⣐⣿⣿⣟⡻⠗⠂⠀⢀⣀⣿⡷⠂⠀⠀⠉⠀⠀⠈⣽⡿⣄⠀⢀⡀⠀⠀⠀⠀⣲⢤⣦⣿⣿⠻⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠙⢾⣤⣤⣿⢿⣿⣿⣶⣄⣠⣭⣷⣾⣿⣦⣽⣿⠗⢲⣤⣴⣿⣿⡷⠤⣈⣉⡙⠓⠒⠒⣨⣿⣿⡿⡛⠀⠘⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠀⠀⠉⠛⢛⡿⠟⠛⠛⢿⠿⡟⢿⡿⡿⣷⣶⣼⣿⣿⡿⠛⠃⠘⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⣠⠤⠤⢤⣴⠀⠀⠁⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⡦⠀⠀⠀⠀⠀⠑⣦⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠤⣄⢀⢄⣴⡣⠀⠀⠀⠀⠀⠀⠀⠀⢳⢠⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢶⣃⣤⢀⡄⠀⠋⠀⡀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣼⡖⢰⠁⣤⣽⣤⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠚⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠒⣍⡀⠱⠀⠀⡈⠙⠲⠤⠤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⠉⠀⠀⠀⠀⠀⣏⠀⠀⠀⣨⠇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢀⠔⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢣
⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢇
⠀⠀⠀⠀⣸⡇⠀⠀⢣⣶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢄⢸⡄⠀⠀⠀⠀⠀⠀⠐⢤⣽⠈⡆
⠀⠀⠀⠀⠇⠀⠀⠀⢸⡟⠀⠀⠀⠀⠠⣿⠃⠀⠀⠀⠀⠀⢀⢼⡟⠀⠀⠀⠀⠀⠀⠀⠈⢮⣿⢠⢿
⠀⠀⠀⢸⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⡑⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠌⣸
⠀⠀⠀⢸⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠨⣿⠀⠀⠀⠀⠀⠀⠀⣺⠇⠀⠀⠀⠀⠀⠀⠀⠀⢝⣯⡴⠃
⠀⠀⠀⢸⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠠⡽⠀⠀⠀⠀⠀⠀⠀⠀⢨⣢⣿⠁⡆
⠀⠀⠀⠀⣷⠄⠀⠀⣿⣀⠀⠀⠀⠀⠀⣇⠀⣀⡀⠀⠀⣠⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣻⡇⢰⠁
⠀⠀⠀⠀⠸⣷⡄⢐⣿⣦⣀⠀⠀⢀⣀⣿⡂⠀⠈⠁⠀⠀⣽⣷⡄⠀⣀⠀⠀⠀⣠⣤⣽⣿⠃⣾⡀
⠀⠀⠀⠀⠀⠙⠲⠶⠟⠻⢿⣶⣤⣶⡶⠿⠿⣶⣿⣋⣰⣶⣿⣿⣗⣀⣀⡉⠉⣃⣼⣿⡟⠉⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠋⠁⣈⠙⡉⠉⠋⠙⠛⠿⣿⠛⠏⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡖⠁⠀⠈⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⠤⣤⣤⣮⠀⠀⠀⠀⢠⠀⠀⣄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠧⡶⢸⠀⡁⠀⡄⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣼⣥⣶⡿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠉⠉⠁⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⣈⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⡠⠚⠁⠀⡏⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢂⠘⡶⠉⠀⠀⠘⠃⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠑⣤⡟⠲⠀⠀⠀⠀⣀⣀⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠁⠀⠀⠻⠀⠀⠀⠀⠀⠀⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⠠⠤⠀⢤⣤⠁⠀⠀⠀⠀⡞⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠋⠁⠀⠀⠀⠘⠀⠹⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢩⠁⠒⠛⢛⠲⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⣀⡄⠐⠒⡇⢀⡴⠃⠀⠀⠀⠣⡀⠀⡸⠀⠲⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠏⠉⠡⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢃⠘⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠘⡄⠀⠑⢄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠣⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⡇⠀⠀⠀⠀⠀⠀⡰⠀⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠀⠀⢰⠀⠀⠀⢀⡰⠊⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⡄⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠰⡀⠀⠀⠀⡆⣠⠔⠉⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⢢⠀⡘⠑⠂⠤⠄⢀⣀⣀⠀⠀⣠⠆⠀⠀⠱⠀⠀⠀⠉⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⠀⠀⠀⠱⣄⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠤⣬⡷⢀⣀⣀⣀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠋⠀⠀⠀⠀⠀⠀⠈⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠈⠙⠢⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠒⠒⢲⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢇⠀⠀⠀⠀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠑⢄⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠎
⠀⠀⠀⠀⢠⠏⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠾⠶⠻⡆
⠀⠀⣠⠒⣇⣀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠈⢦
⢠⡖⠉⠈⠉⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢨⠴⠂⣀⠀⠀⠑⢤⡀
⠣⣝⠦⡀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠋⠵⠶⣤⣭⠆
⠀⠀⠑⢌⣦⡀⠈⢆
⠀⠀⠀⠀⠉⠻⢦⣌⡧
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠙⣿⣿⣿⣿⣷⣤⣴⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⣿⣿⠟⠶⠛⠉⠀⠀⣀⠀⠘⠋⠉⠙⠛⠛⠛⠋⠉⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⡾⠿⠋⠁⠀⠙⠋⠀⠀⠀⠀⠀⠛⠉⠙⠛⠃⠀⠀⠀⠀⠀⣀⣀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠉⠻⠟⠁⠀⠀⠀⠠⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⣼⠏⢀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⢸⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⢸⣿⣄⣿⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠻⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⣀⣀⣀⣼⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣷⣄⣀⠀⠀⠀⣀⣀⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⣀⣀⣀⣠⣼⡿⠿⣷⣦⣤⣤⣴⣾⣿⣿⠿⠿⠿⠿⣿⡿⠟⠉⠉⠁⠀⠈⠉⠛⠻⠶⠄⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠻⠷⠶⠶⠶⠶⠶⠯⣭⣍⣉⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣤⣼⣿⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢠⡀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡄⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⣸⠇⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⣷⣶⣿⣤⣤⣀⣀⣀⣁⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠂⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄
⠀⠀⣄⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀
⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⢿⣿⣦⡀⠀⢠⠀⠀⠀⠀⢀⣠⡤⠖⠛⠟⠛⠁⠀⠀⠈⠳⡄
⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠘⣿⣿⣿⣤⣿⡇⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠉⠉⠓⢄
⠀⠀⠀⠀⠙⠳⣄⡀⢸⣿⣶⣽⣿⣿⣿⣿⣧⡼⠃⠀⠀⠀⠀⠀⡀⡆⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠘⢦⣻⡇⠀⠀⠀⣠⠴⠶⣼⡆⠀⠀⠀⣄⠏
⠀⠀⠀⠠⣤⣤⣄⣤⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⡄⠀⡎⠀⠀⠀⠘⣿⠶⠖⠚⢯⡀
⠀⠀⠠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⠹⣄⠀⢀⡠⣿⠃⠀⠀⠀⢳⡀
⠀⠀⠀⠈⠛⢭⡉⠉⠉⠁⠀⢀⡟⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠈⠳⠦⠭⠭⠖⠋⠀⠀⠀⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⢸⡇⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⣀⣀⣀⣀⣀⣠⡤⠤⠶⠷⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠶⠘⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣿⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠂⠀⠀⠀⠀⠀⠈⠙⠒⠶⠄⢀⡀⠉⢀⡼⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⢀⣴⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠀⠀⠁⠀⡼⢿⣿⠛⠲⠶⠤⠤⢤⣤⣼⡴⠚⠁
⠀⠀⠀⠀⠈⠉⠉⠙⠛⣶⣀⡀⠀⢸⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣆⠀⠀⠀⠀⠀⠸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⣤⣬⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣶⣤⣀⡀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢟⣿⣿⣿⣿⣿⣷⣶⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣁⣁⡬⠥⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠟⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠙⠳⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢻⣿⠻⣿⡿⠉⠙⠛⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠓⢦⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣄⣀⣀⣼⣁⣀⡀⢳⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠙⠒⠈⢀⡇⣾⠀⠀⢹⡆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠐⢄⣀⣀⣤⠴⠋⢀⡇⠀⠀⣸⠁⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⣠⣿⣿⡇⠀⠀⣸⠃⠀⢀⡏⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠔⠊⠁⠀⠙⠷⢶⣤⣿⠀⠀⢸⡁⠀⠀⣼⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣽⡿⠧⠴⠾⠛⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠒⠀⠀⠀⠀⠀⠈⣉⣉⣛⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠿⠉⠁⠤⠤⠀⠉⠩⠭⠍⠛⠿⢿⣅⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡴⠖⠋⠈⠁⢀⣤⣤⢤⡆⣴⠶⣶⢰⣶⢰⡦⡤⣤⣤⣀⠀⠉⠉⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠘⣷⡂⢸⡧⠻⠦⠿⠸⠯⠽⠇⢰⣯⢀⡟⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⢀⠔⠛⢧⠀⠀⣀⣀⠁⣀⣠⠤⠄⠀⡠⠤⠀⡀⠤⢤⣄⡀⢀⣀⡀⠀⢠⠞⠓⣄
⠀⠀⠀⠀⡘⠉⠀⠀⠘⢿⣿⠉⠁⣿⠁⠀⠀⠀⠀⡇⠀⢰⠀⠀⠀⠈⣿⡇⠉⢹⣿⡟⠀⠀⠈⠙⡄
⠀⠀⠀⠘⠁⠀⣄⠀⠐⠒⠛⠁⠀⣿⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⣿⡇⠀⠙⠓⠒⠆⢀⡄⠀⠙
⠀⠀⠼⠘⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠸⠆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠛⠛⠛⠻⠿⣷⣦⡀⢀⣴⡾⠿⠿⠿⠿⣶⣶⣦⣠⣶⠿⠿⠿⣷⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⠏⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀⠀⠀⠀⠀⠈⠙⣿⣦⣀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠈⢻⣿⡏⠉⠉⠉⠛⠛⢿⣷⡄
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⣿⡇⢦⡀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⣿⡇⢀⣙⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⢠⡿⠟⠛⠛⠛⠛⠻⠷⠶⠶⠶⠶⠶⠶⠿⠿⣿⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢹⣆⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇⠀⠀⠀⠀⢀⣀⣠⣼⣿⠁⠀⠀⠀⠀⢀⣠⣼⣿⠀⠀⠀⠀⠀⠠⠶⣾⣿⠃
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠙⢷⣦⣤⣀⣀⣀⣀⣀⣤⣤⣶⡾⠟⣿⣿⠁⠀⠀⠀⠀⠀⠉⠉⣽⡟⠀⠀⠀⠀⠀⠀⢩⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⢿⣯⣭⣤⣤⣴⡾⠟⢿⣆⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⢀⣾⣟⠀⠀⠀⠀⠀⣀⣴⡿⢻⡇
⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡿⠉⠉⠉⠀⠀⠈⠻⢿⣶⣶⣶⣶⣾⠟⢻⣧⣀⠀⠀⠀⣠⣾⡟⠛⠿⠿⠾⠿⠿⠟⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠛⠛⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇

⠀⠀⠀⠰⠧⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⡂⠀⠀⢀⣤⣦⡄⠈⠑⠒⠒⠆⠀⢤⣠⡄⣀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠀⠑⠤⢤⢀⠚⠀⠀⠉⠀⠉⡒⠙⢛⣈⣁⡀⠛⠉
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠳⣤⣴⡇⠀⠀⠀⠀⠀⠄⠀⠙⡁⢀⡒⠬⢱⣄⣀⣄⣰⣞⠀⣦⡄
⢠⣤⣀⣨⣿⡆⣀⠔⠁⠀⣴⣿⣿⡗⠄⣼⣿⡖⢾⣒⠂⢰⡄⠀⠀⠀⢀⣽⣿⣿⣿⣿⠀⠻⣏⣩⡆⠃⠀⣀⣀
⠀⢷⡦⠄⠺⡯⢅⣀⠀⠀⠈⣹⠿⣷⣴⡡⣻⣿⡦⣈⠀⠀⣷⠀⠀⢂⡿⡛⢹⡏⡻⠋⠀⠀⢘⣧⣟⠛⢤⣿⡅
⠀⠀⠹⣿⡁⣶⣦⠅⠀⠄⢹⣎⣣⣿⣿⠷⠯⠥⠤⠈⣓⠀⠕⠀⠞⠉⠀⢀⣀⠰⣾⣇⠀⠔⣒⣯⣽⡖⢀⠖
⠀⠀⠀⠀⡧⠈⠹⢷⡀⠠⠞⠻⣷⠙⢩⣧⠀⠀⠈⣣⣨⣁⢹⣅⣤⠀⠠⢌⣉⡴⠞⠷⠆⠀⣠⠀⠄⣇
⠀⢠⣤⠀⡧⠀⠀⠀⢍⠀⠀⠀⠉⠀⢀⡙⠷⣤⣴⠿⢻⡟⢸⣯⠛⢀⡀⢈⡵⠋⠀⠀⢰⠀⠀⢀⣀⡇⢠
⠀⠀⣀⣀⠇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⣽⠥⠭⣭⣭⣽⠿⠀⢸⠐⠁⠀⠀⠀⠀⢸⠀⠀⠀⢀⠉⢀
⠀⠀⠉⠁⠀⠀⠁⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⠘⢀⣀⠈⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠐⠈⠀⠀⠀⢠⣤⠀⠄⠒⠄⠀⠀⠀⢠⣤⠄⠀⠒⡄⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⣒⡢⠤⣒⣫⣭⣭⣭⡓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⣵⠾⣛⣿⣿⣿⠟⣩⣶⣶⠿⣿⣷⡕⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣫⣭⣶⣿⡁⠾⠟⠋⠀⢻⣧⠉⠀⠀⠀⠹⣿⣿⡶⠟⢿⣶⣍⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⣵⠟⣫⣶⡾⠻⣷⡀⠀⠀⠀⠀⢿⣧⠀⣠⡴⢞⣫⠁⠀⠀⠈⠹⣿⣧⢣
⠀⠀⠀⠀⠀⠀⢀⡴⣫⣶⢾⣿⣄⠋⠉⠀⠀⠹⣷⡄⠀⠀⠀⠈⢿⣿⠉⠊⠉⢹⡆⠀⠀⢀⢠⣿⣿⣇⢇
⠀⠀⠀⠀⠀⢠⢎⣾⢋⣴⡿⠉⢿⣦⡀⠀⠀⠀⠘⣿⡄⠀⠀⠀⣈⣿⡿⣵⠖⠋⢀⣠⣴⢏⣾⣿⣿⣿⢸
⠀⠀⠀⠀⠀⣼⢸⡇⠘⠉⠀⠀⠀⠙⣷⡄⠀⠀⠀⠈⢿⡝⣶⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠛⠡⣿⣿⣿⡆⡆
⠀⠀⠀⠀⠀⢸⢸⡇⠀⣘⣦⡀⠀⠀⠈⣻⣖⢦⣤⣶⣿⣿⣬⣛⣛⣭⣾⣿⣿⣿⣿⣿⣿⣓⠀⢾⣿⣿⣿⡘⡄
⠀⠀⠀⠀⠀⢸⢸⣇⢀⣤⣽⣿⣟⢶⣾⣿⣿⣦⡻⠿⢟⣫⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠷⠭⡉⠂⠀⣿⣿⢣⠇
⠀⠀⠀⠀⠀⠘⡇⣿⢠⣤⡈⣩⢿⣷⣭⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠂⠀⠀⠀⢀⣠⣾⡿⣡⠋
⠀⠀⠀⠀⠀⠀⢧⢻⡇⢿⣇⢀⡾⠟⣉⣬⣭⣛⢿⣿⣿⣿⡟⣾⠟⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⡴⠁
⠀⠀⠀⠀⠀⠀⠸⡌⣿⡘⣿⡄⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⠑⡟⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⢏⡞
⠀⠀⠀⠀⠀⠀⠀⢳⡹⣧⠹⣿⡄⠀⠀⠀⠀⠀⠀⢻⢹⣿⡆⣧⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠹⣧⠙⣿⡄⠀⠀⠀⠀⠀⠘⢸⣧⠃⠘⣿⣦⣤⣤⣤⣾⣿⣿⡿⢋⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡙⢷⣌⠁⠀⠀⠀⠀⢀⣠⡿⠻⢷⣤⣈⣛⣿⣿⣿⣿⣿⡏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⣿⣿⣶⣬⣭⣛⣋⣉⠀⠀⢀⢉⣉⣩⣭⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡆⣉⠙⠛⠶⢤⣀⡀⠀⠸⢸⣿⣿⣿⣿⣿⣿⣿⣇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡟⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⡇⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢻⢸⡟⣿⣿⣿⡇⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢦⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣾⢻⣷⣧⣀⡀⠀⠀⠀⠀⠀⢀⣶⣶⣆⢇⠀⠈⢆⠀⠀⠀⠀⣾⣿⡿⣿⣷⣾⣿⣿⣿⣷
⠀⠀⠀⠀⢠⣿⣿⣄⣀⣀⣀⣠⣿⣿⡄⠀⠀⠀⠀⠀⣀⣈⡿⢿⡉⠉⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠘⣶⣾⣿⡄⠀⠀⠀⠻⡿⣧⣿⢿⣼⣿⢿⡿⠏
⠀⠀⠀⠀⠈⢿⣿⠉⠉⠉⠉⠉⣿⡿⠁⠀⠀⠀⠀⠈⠉⠉⢁⣼⠃⠀⠀⠀⠀⠀⠀⡜⠀⠈⡝⠀⠀⠹⣿⣿⠇⠀⠀⠀⠀⠙⢿⡇⣾⣿⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣬⣥⣤⣤⡄⠀⠀⠀⠀⢧⣀⡼⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟

⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢈⣿⡏⠹⠉⠉⠉⠍⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⠿⠿⠿⠀⠀⠀⠀⠀⣿⣧⣼⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣴⣿⣿⣶⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢈⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣀⡀⣀⣾⣽⣷⣜⣿⣃⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠀⠀⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠆
⠀⠀⠀⠀⠀⠀⣶⡶⠛⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⠀⠀⠸⡇⠀⠀⠀⠀⠀⠘⣻⣿⣾⣿⣷⣿⣷⡇⠀⠀⠀⠀⢀⣀⠖⠀⠀⢰⡀⠀⠀⠀⠀⠀⣶⣾⢿⣧⣤⣤⣾⢿⣷⡄
⠀⠀⠀⠀⠀⠘⣧⡀⠀⠈⠁⠀⣸⠇⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠎⢠⡍⢾⣄⠘⠁⣥⠀⠄⠀⠀⠀⣈⣁⣘⣛⣛⣛⣋⣀⣙⣋
⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠊⠉⠀⠄⠁⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠀⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⣷⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣴⠀⠀⠀⠀⠀⠀⣀⣠⢻⢀⡼⠇⢀⣀⣀⡀⠀⠀⠀⣧⡇⢠⣾⣯⣿⣝⣦⢸⣿
⠀⠀⠀⠀⠀⠻⠟⣶⣶⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⠃⢸⡼⠀⠀⠀⠀⠀⠀⢸⠁⢺⣺⣿⣿⣿⣿⢂⡏
⠀⠀⠀⠀⠠⠤⠤⠬⠭⠭⠭⠤⠤⠤⠄⠀⠀⠀⠀⠀⠶⣴⣤⣤⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡇⠀⠀⠀⠀⠀⠀⠸⠀⠈⠳⠯⠯⠷⠋⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠈⠁⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠒⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⠟⠋⠉⠓⠲⢄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢖⠋⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⢳⣦⡀
⠀⠀⠀⠀⢠⠖⠉⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢀⠘⣇
⠀⠀⢀⡿⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡆⢹⡄
⠀⠀⣸⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢁⠘⡇
⠀⠀⡟⠀⠀⠀⠀⢐⣿⠁⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⣰⠃
⠀⠀⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠔⢹
⠀⠀⢹⡄⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣼⣿⡆⡞
⠀⠀⠘⣇⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢠⠇
⠀⠀⠀⢻⡆⠀⠀⢹⣿⣇⠀⠀⠀⠀⠐⣿⣿⡍⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡇⢸⡀
⠀⠀⠀⠈⢻⣀⣀⣼⣿⣿⡇⠀⠀⠀⣰⣿⣿⣆⠀⠀⠀⠠⣤⣾⣿⣿⠁⠀⠀⠀⠠⠀⠀⢀⣼⣿⣿⡟⠀⠈⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠙⠛⠳⠶⠾⠛⠛⠉⠙⠛⠶⣶⡶⠿⠿⠿⢿⣷⣶⣶⣶⣤⣴⣿⣿⣿⠟⠟⠁⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⡀⢀⣀⠀⠀⠈⠉⠉⠉⠙⠛⠉⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡾⠁⠀⠀⠀⠉⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠷⠤⣄⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠘⢘⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⢶⣴⣼⣶⣶⣄⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠢⢄⣀⣀⡀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢱
⠀⠀⠀⠀⢤⣤⣤⠤⠞⠁⠀⠀⣀⣀⣠⡞⠍⠉⠲⠋⠀⠂⠘⡶⠚⠉⠲⡀⠀⠀⠀⠉⠁⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⢠⠏⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⣧⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢰⡇⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⢇
⠀⠀⠀⣠⣴⡋⠀⠀⠀⠀⠸⡄⠀⠀⢸⡄⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⣿⡆⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠈⣉⠓⠶⣄⠀⠀⠳⣀⣀⣼⣇⠀⠀⣸⣇⡀⠀⢠⣇⠀⠀⣰⠃⡇⣠⠤⠶⠤⠤⢷
⠀⠀⠀⢀⠄⠀⠀⠀⠊⢳⡀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⣹⢋⠉⠉⠉⠁⢀⡿⠁
⠀⠀⠀⠘⠄⠀⠀⠀⠀⢘⡧⠄⠀⣀⣀⣀⡀⠀⠀⠀⠀⣋⣤⣴⣒⡒⠒⠊⡇
⠀⠀⠀⠀⠐⠂⠐⠂⠀⣼⡤⠖⠋⠁⠀⠀⠙⣆⠀⠀⢠⠞⠁⠀⠀⠉⠢⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠄⠀⠠⠔⠀⠀⠀⠘⡆⢠⠏⠀⠀⠀⠀⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠉⠙⠻⣶⣤⣀⣴⠶⠶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⠒⠲⠖⠚⠻⠋⠀⡀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠈⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢠⡀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⣀⡀⠀⣠⣤⠞⢸⡀⠀⠀⠀⠀⠀⢳⣸⣇⠁⠀⠀⠀⠀⠀⠀⣿⣿⡌⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⢀⠎⠀⠀⠀⢀⠀⣿⡈⣇⠀⠀⠀⠀⠀⠈⢿⣟⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⡘⠀⠀⠀⠀⢤⣱⣾⣷⡙⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢰⠁⠀⠀⠀⠀⠀⠘⣿⡈⢿⡄⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣾⠀⠂⠀⠀⠀⠀⠀⠘⣷⡈⣿⡀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⡇
⠀⢸⡄⠐⢧⡀⠀⠀⠀⠀⠘⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣿⠀⠀⣳⠀⠀⠀⠀⠀⠀⠛⢿⡇⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢿⡄⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⣠⣿⡀⠀⠀⠀⠀⣠⡼⠀⠀⠀⠀⠀⠀⢠⣗
⠀⠀⠘⣷⡀⠀⠹⣷⣄⠀⠀⠀⠀⢠⣿⣿⣶⣶⡦⠤⠀⢀⣾⣏⠠⢤⣤⣤⣴⡁⠀⢀⣀⣀⣀⣤⠟⠙⣆
⠀⠀⠀⠈⢷⣄⠀⠈⠙⠻⠿⣿⡶⠟⠛⠛⠻⠿⢿⣶⣶⣿⣿⣿⣧⣤⣬⣿⣿⣿⣶⣦⣼⠿⠿⠿⠏⠀⢿⡄
⠀⠀⠀⠀⠀⠈⠳⣦⡀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠈⠙⢛⣿⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠚⠗⠒⠶⠒⠉⠉⠉⠻⠦⠀⣀⣀⣰⡿⢡⡖⠐⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢹⡇⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠈⠆⠀⠀⢀⣠⠇⠀⠀⠀⢤⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣤⣴⣿⠿⠿⠿⠿⠿⠟⠛⠁