Smug Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠶⠾⠷⢶⠾⣧⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣄
⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠇⣀⣀⡠⠤⠀⠐⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⢹⡟⠁⠀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠛⠻⢄⠙⡀⠀⠀⠀⠀⠀⠀⠀⢰
⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⠳⣶⣶⣶⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣈⣤⠡⠀⠀⠀⠀⠀⠀⠀⣼
⠀⢿⡀⠀⠀⠀⠀⠂⠀⠚⠀⠀⠀⠢⡄⠉⣡⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⠛⠛⢁⠝⠀⠱⣄⠀⠀⠀⠀⣰⡏
⠀⠀⠻⣄⠀⠀⠀⠣⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠈⠑⠒⠒⠁⠀⠀⠠⢀⡇⠀⠀⣰⠏
⠀⠀⠀⠙⢷⣀⠀⢀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠠⠌⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠋⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠙⠛⠋⠉⠻⠚⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠤⠦⠀⠀⠀⠀⠀⠀⠀⢀⡴⢧⣴⠿⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣤⣀⠀⠀⠀⣀⣀⣠⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠿⠇⠀⠀⠘⡿⣭⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠞⠛⠛⠀⠙⠛⠒⠒⠛⠃⠰⠿⠦⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣠⡴⠟⠲⢦⡄⠀⠀⢘⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⣠⣤⡤⠸⣧
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠈⢣⣄⣼⣯⣁⣀⠀⠈⠉⠉⠒⠢⣄⣀⣠⣤⡤⣤⡞⠉⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⠀⠀⠀⠀⠀⠙⢯⡀⢀⣄⣬⣶⠶⠂⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠙⠲⠋⠙⠋⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠸⢿⡶⢶⣄⣀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠞⠛⠢⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠲⢶⡶⠿⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⢀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠤⢤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣀⣼⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠛⠛⠀⠐⠚⠛⠒⠸⡷⠐⠛⠁⠉⠉⠉⠁⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⣶⣶⠶⠶⠶⠶⠶⠞⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⣀⠀⠀⢷⠀⠙⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠉⠶⢯⠅⢠⣀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⣀⠖⠁⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⡰⠋⢹⣇⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⢲⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠃⠀⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠜
⠀⠀⠀⠀⠀⠀⠀⢫⠚⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⢰⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣴⣿⣦⣄⣀⣀⡀⠀⠀⠀⣀⣀⣤⡼⣶⢾⣶⣶⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠙⢻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣇⣠⢾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣇⢛⣙⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣮⣙⣛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠋⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠁⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠃⠀⠀⠀⠀⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠁⠀⠀⠀⠀⠀⠀⠸⢿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠲⠤⠚⠉⠉⠛⠲⠚⠉⠛⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠛⠓⠦⡄⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⢀⣠⠴⠖⠚⠛⠛⠛⠓⠾⢇⠀⡀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜
⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⠉⠁
⣠⣏⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣔⢲⡄⠀⢰⡛⠓⣲⠈⠙⠶⢤⣤⣤⣤⠤⠶⠚⠙⠻⠶⠶⠶⠶⠖⠋⠁
⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠞⠁⠀⠈⠉⠀⠀⠘⠳⠚⠉
⢸⠁⠀⠀⠘⢄⠀⠀⠐⠂⣈⠉⠉⠉⠹⢋⡀⠘⣧
⠈⢧⡀⠀⠀⠀⡇⠀⠀⠼⠛⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠸⡇⠀⠀⠀⢁⠀⠀⢀⠀⠀⠀⢆⠀⢶⠀⠀⡆⠘⣆
⠀⠀⢿⠀⠀⠀⠈⡀⠀⠈⢧⡀⠀⡸⠀⠀⠳⠔⠁⠀⠘⣆
⠀⠀⠸⡆⠀⠀⠀⠃⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⢷⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⣰⠛⠁⠀⠠⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⡧
⠀⠀⣇⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠁⠈⣏
⠀⠀⠙⠦⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⠀⠤⠀⠀⢶⠞
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⣹⠆
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠼⠦⣤⠀⢀⡀⠀⠀⠀⠀⠀⢀⡾⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢯⣀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡽⠻⢤⣀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣀⠀⢀⡴⠋⠀⠀⠀⠈⠉⠁
⣶⣶⣤⣶⣶⣤⣴⣶⣶⣶⣶⣶⣶⣬⣿⣿⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣴⣦⣶⣤⣤⣶
⣿⣿⣇⣀⣁⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠲⠶⠛⠋⠙⠓⢦⡴⠖⠒⠦⣄
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⣀⠤⠖⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⡄
⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣀⠴⠒⠦⣄
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠸⠖⠒⠒⠒⠒⠲⠦⣄
⠀⠙⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠛⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣀
⠀⠀⠀⠀⠈⠉⠻⣄⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠲⠞⠛⠋⠉⠉⠓⠲⠶⠤⠤⠶⠖⠛⠁⠀⣟⡉⠉⣷⠀⠀⢎⣙⡆⠀⠀⣱⠂⠠⡤⠤⠀⠀⣀⣀⠀⠀⣠⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠉⠀⠀⣰⠃⠀⠁⠀⠀⠀⠸⢦⠀⠀⠰⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⡀⠱⡄⢀⡚⠀⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣁⣁⠇⠸⡚⠀⠃⡸⠀⠀⠀⠠⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢦⠄⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠠⠄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣈⢶⡐⠒⠂⠀⠈⠉⠁⠀⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠙⢦⡀⠀⣀⡤⠔⠚⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⣿⢿⣦⣄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣱⣶⣿⣿⣿⣦
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁⠀⡀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⣴⣿⣿⣿⡟⠻⠿⠿⠛⣉⣀⣀⡀⢲⣾⣿⣷⣯⣙⢿⣿⣿⣿⡿⠿⣿⣿⣧
⠀⠀⣿⣿⣿⣿⠀⠀⢤⣶⠛⠉⠉⠉⠉⠉⠙⠛⠿⣿⠿⡿⠛⠛⠁⠀⠀⠹⣿⣿⠇
⠀⠀⣿⣿⣿⠃⠀⠛⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⣿⣿⣿⣶⡾⠖⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟
⠀⠀⣿⣿⣿⣿⣷⣶⣷⣾⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⣼⠿⠿⢉⣻⣿⣿⡿⠋⠀⠀⠀⢀⣤⣤⣠⣤⣤⣤⠀⠀⠀⠀⢠⣤⣴⣶⣿
⠀⠀⡇⢾⠿⢮⣻⣿⣿⠀⠀⠀⠀⠴⠟⠛⠛⠛⣛⠛⠁⠀⠀⠀⢀⣬⣭⣥⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣧⣸⢠⡞⠁⠘⠋⠀⠀⠀⠀⠀⠠⠾⣿⣿⠙⠃⠀⠛⠀⠀⠋⢨⣟⡁⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢀⡀
⠀⠀⠸⣟⢯⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⡇
⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⡄⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡋
⠀⠀⠀⠀⠀⠙⠷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⡀⠈⠁⠀⢹⡄⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡽⠊
⠀⠀⠀⠀⠀⣄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠈⠋⠉⠀⠶⠛⣁⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣼⠁
⠀⠀⠀⢀⣾⢿⣎⣧⠀⠀⡀⠀⠀⠀⣆⠀⠟⠶⡶⡶⠶⠶⠶⠶⢶⠞⠁⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⠤⣬⠿⡷⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⢀⣠⠞⠁⠀⢻⡎⢷⡄⠹⣄⠀⠀⠹⣆⠀⠀⠀⠉⠛⠛⠒⠒⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠻⠛⠲⣄⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠓⠒⠀⣼
⠐⠋⠁⠀⠀⠀⠀⢻⡌⠻⣦⡈⠳⣤⡀⠙⠂⠀⠀⠀⠓⠶⣶⠶⠃⠀⣼⠁⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⢀⣤⣄⣀⢘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⠻⣦⡈⠻⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣴⠟⠁⠀⠀⢀⡴⠾⠛⠉⠁⠀⠈⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠻⣦⡈⠻⣿⣿⣶⣤⣤⣤⣤⡴⠟⠀⠀⠀⠀⣿⠀⠀⣠⡄⠀⠀⠀⠀⢀⣠⡴⠾⢧⣽⠒⠒⠒⠒⠒⠒⠲⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⠛⢶⣌⣻⠟⠁⢈⡏⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⠃⠀⠀⠀⠀⠘⠉⠁⠀⠀⣸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢀⣿⣿⣷⣆⢸⡇⠀⠀⠀⠀⠀⠀⣾⠀⠀⡈⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠴⠛⢿⣿⣿⣿⣼⡇⠀⠀⠀⠀⠀⢰⠇⡀⢸⣧⡞⠀⣄⠀⡀⠀⠛⠋⠉⠀⢀⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⢨⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⡾⢰⠇⠘⠿⠁⠀⠙⠛⠁⠀⠀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣾⡇⠀⠀⠀⣠⡼⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠛⣿⠇⠀⣠⠞⠉⠙⣮⢳⣄⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠈⢳⡝⣦⡀⢀⡤⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⡎⠻⠞⡵⠟⠉
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⡶⠶⠶⠶⢶⣶⠶⢶⡶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⣤⣰⣄⣠⣠⡄⣄⣄⢀⣤⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢄⡢⡐⢦⣕⢾⣿⣿⣟⣟⢯⡿⣾⣿⡽⣿⣷⣷⣿⡄⣄⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠹⣿⠿⠛⠂⠈⠛⠃⠙⠛⠈⠀⠙⠙⠋⠁⠙⠻⠟⠿⠇⡿⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣤⣶⡄⠀⢀⠰⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣼⣿⣿⣿⡄⠈⠹⣄⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⡀⠰⠀⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣰⣿⣿⣿⣿⣷⡀⠀⠙⣆⠳⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠂⡐⠀⠈⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡯⡟⠈⢿⣿⣿⣿⣿⣦⡀⠙⢦⣈⣉⣉⡉⠛⢦⡀⠀⠀⠀⠀⠒⠀⠂⠈⠅⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠁⠀⠀⢻⣿⣿⣿⡟⢿⣦⣀⠉⠀⠈⠻⣦⡀⠙⢦⡄⠀⠀⠀⠈⠐⠠⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠙⠿⣿⣷⠀⠈⠻⣷⣦⣀⠀⠈⠙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣀⣀⣠⣤⣤⣤⣾⣿⣿⠏⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢀⣤⣄⠀⠀⣷⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠉⠁⠸⠟⠁⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠘⢠⣿⠀⠀⣴⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⣿⡿⠀⢠⡏⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠻⠃⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠉⢉⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⠟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡾⢻⡄⠀⣀⣤⣤⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢈⣤⠆⠀⠘⠉⣴⠆⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣰⠟⠁⠀⠀⣠⠞⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⣤⠞⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠘⠷⠚⣯⠵⠶⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣆⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⣠⣿⠏⠀⠀⢀⣴⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡀⠀⠀⠀⠀⠸⣷⣀⣠⣴⣿⠟⠁⠀⣠⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣀⠀⠀⠀⠙⠛⠛⠉⠀⣀⣴⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⣦⣤⣀⣀⣠⣴⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⣄⣠⣶⣶⣶⣶⣶⣶⣶⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣋⣀⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠉⠻⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⢀⣴⠛⠛⠛⠛⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⣽⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣴⣦⡀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠈⠛⠀⠂⠰⠶⠶⣀⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⣀⡴⠟⠀⠀⢹⣷
⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡴⠋⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡶⠶⠶⢶⣄⠀⠀⢻⣇⣀⣤⣄⡀
⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠋⠉⠙⠻⢷⣤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣴⡏⠀⡀⠀⣀⣹⣆⠀⠈⣿⡟⠉⠙⠿⣧
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⠏⠘⢿⠇⠀⢸⡿⠀⠀⠸⣷⡄⠀⠀⣿
⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢀⡀⣤⣶⢴⣿⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠻⣦⡄⠀⣴⠟⠀⠀⠀⠀⢻⣇⣀⣶⡟
⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣠⣶⡦⠼⣿⣿⡇⠀⠀⠀⣿⡆⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠸⣿⡟⠁
⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠙⠷⣾⣿⡶⠖⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠸⠏⣹⣧⡀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣄⣀⣀⣤⡴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿
⠀⠀⠀⠀⠀⣀⣤⣤⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢠⣾⠟⠁⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⢰⣷⠀⠀⠀⢠⣿⠃
⠀⠀⠀⢸⣿⠀⡁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠛⢿⣶⣦⣄⣀⣀⣠⣶⡿⠋⠀⠀⣀⡟⠀⡆⠀⠀⣾⡏
⠀⠀⠀⠸⣿⣄⠁⠀⢀⣠⣾⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠁⠀⠀⠀⣴⠟⠀⠀⠇⠀⣸⡟⠁
⠀⠀⠀⠀⠈⠛⠿⠿⠟⠋⠁⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⢸⠋⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠉⠀⠀⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠙⠿⢦⣤⣤⣀⣀⣀⣤⣤⣤⠾⠛⠉⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⡀⠠⠄⠉⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣧⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⠟⠛⠛⠛⣷⣶⣀⣀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠟⢡⣀⠀⢩⣻⣿⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⣦⣤⣾⣿⣧⣀⣴⣶⣿⡇⠉⠛⠛⠿⢿⡿⠿⠟⠛⠛⠁⠀⠀⠀⢻⣷⣤⣿⣿⣿⣷⣤⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢆⣉⡻⣷⠾⠛⠉⠙⠛⠛⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠈⠛⠹⡶⣟⡛⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣾⡇⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠙⣦⠀⠀⠀⠀⠀⠀⠀⢼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⢰⡏⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠙⢿⣄⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⢠⣾⠁⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠛⠛⠛⢶⣤⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠞⠿⠿⠃⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢨⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣆⠀⠀⠀⠀⠀⣿⡏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⣿⡿⠿⢶⡶⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠰⢶⣶⣿⣟⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠁⠀⠀⠀⠀⠀⢰⣿⣇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠈⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⠶⠶⠶⠶⠾⠟⠛⣿⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠷⢶⣦⣤⣤⣤⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠏⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡄⠀⠀⠀⢠⣤⡤⠀⠀⠀⣾⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢸⣏⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⠉⢹⣿⠉⢹⣿⡋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢸⡿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢸⣧⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢸⣏⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⢹⡿⠉⠙⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣼⣷⣶⣦⣾⣿⣦⣶⣶⣿⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⠿⠿⠿⠿⠻⣿⣿⣿⣿⠏⠀⠀⢸⡟⠀⠁⠈⣿⣿⣿⣿⠟⠛⠻⠿⠿⢷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⠉⠙⢻⡄⠀⠀⠀⠀⠈⠛⠉⠛⠀⠀⠀⢸⡇⠀⠀⠀⠘⠉⠉⠉⠀⠀⠀⠀⢀⡼⠋⠉⠙⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⣠⣤⣤⣤⣼⣷⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⡄⢼⣷⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣷⣤⣤⣤⣄⣨⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣄⣥⣠⣿⣼⣧⣀⣹⣧⣀⣤⣤⣀⣥⣄⣀⣿⣄⣸⣿⣨⣧⣄⣡⣀⣤⣄⣤⣀⣠⣧⣀⣄⣀⣤⣄⣸⣧⣤⣿⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠉⠁⠈⠉⠉⠉⠉⠉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⡄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠶⠶⠲⠦⠄⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠾⠛⠻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠘⠃⠀⢀⣤⣶⣶⠀⠀⠀⠀⠀⢀⡴⢜⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡄⠀⠀⠀⠀⣿⠛⠉⠉⡀⠀⠀⢳⡀⠸⡷⠎⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⡀⠀⠀⠀⠀⠁⢰⡾⠛⠁⠀⠀⠀⠉⠓⠶⢤⣄⣀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡿⡃⠀⠀⠀⠒⠒⠛⠒⠚⠋⢁⣀⣠⣴⣀⣀⣤⠴⠟⠁⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠗⠀⠀⠀⠀⣀⣤⣔⣚⣉⡉⠀⠀⠀⠀⠀⠀⢀⣀⡦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⣿⡱⠋⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢢⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢥⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠸⣟⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢱⡀⠈⠛⢦⣄⠀⠀⢀⡴⣟
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⢳⣀⣀⣀⠀⠀⠀⠀⠀⣀⣤⠴⢿⡀⠀⠹⡄⠀⠀⠈⠛⢶⣄⠀⢿⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣷⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡷⠚⠉⠉⢀⣈⠙⣦⡤⠖⠛⠉⠀⠀⠘⣇⠀⠀⠹⣄⠀⠀⠀⣿⠟⣧⣼⠙⠶⠍⠛⠶⢦⣤⡀⠀⠀⠀⣴⠋⣴⠟⢁⣤⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢿⠀⡴⠚⠉⣁⣀⣈⢳⡄⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠘⢦⡀⠀⡿⣾⣇⢹⡆⠀⠀⠀⠀⠲⣤⡙⢦⡄⠀⣧⣼⢃⣴⠏⣁⣤⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠒⠋⠉⠉⠉⠙⣇⠀⠀⠀⠀⠀⠀⠸⢇⡀⠀⠀⠈⢻⣾⢸⣇⣾⢾⡇⠀⠀⠀⠀⠀⠈⢻⡞⠿⣦⡈⢙⠺⣇⡼⢋⣀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡈⢳⡀⠴⠚⠉⠉⡉⠰⡟⠀⠀⠀⠀⠙⢿⣗⠲⠿⠷⣦⣤⡀⢳⣸⡿⠁⣼⣿⠀⠀⠀⠀⠀⠀⠈⣷⢤⡀⠹⢾⣀⠹⣇⠈⣷⠋⠀⣾⠁
⠀⠀⠀⠀⠀⠀⣴⠃⠸⣇⠀⠈⠙⢿⣷⡼⠃⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢻⠃⣼⠋⣿⠀⠀⠀⠀⠀⠀⠀⢿⡄⠹⡆⠀⠀⠙⠈⢹⠇⠀⢸⡏
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠙⣆⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢿⡏⢀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠷⣄⡀⠀⠀⢸⡀⠀⣾
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠘⡇⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠈⢷⣾⢿⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢻⡄⠀⠈⠃⢰⠏
⠀⠀⠀⠀⡿⠀⠀⣠⡴⢾⡶⣇⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠈⢷⡘⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡿⢧⣄⣀⣀⣞
⠀⠀⠀⢀⣧⠶⠛⠉⠀⠀⢣⡘⢦⡀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠳⣤⣈⣉⡉⠉
⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠙⣄⠙⢲⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢠⣿⠋⠉⠙⠛⠛⠛⠒⠒⠖⠲⠿⠶⠿⠦⠤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣉⣉⣙⣲⡀
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⢉⣉⣤⣤⣤⣶⣦⣤⣄⡉⠙⢲⡄
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⣀⣴⠾⠿⠿⣿⣿⣿⣿⣿⣿⣿⠆⠀⣸⠇
⠀⠀⠀⠀⠀⢿⡆⠀⠀⢘⣟⣠⣾⠤⠶⠶⠛⠉⣠⣴⣿⣷⠈⠀⠛⠉⠿⠛⡟⣀⡴⠋
⠀⠀⠀⠀⠀⢸⣿⣚⣉⣭⣤⣴⣾⡀⠀⠀⠀⣾⠿⣋⡥⢤⡀⠀⠀⣼⡿⣿⣿⠁
⠀⠀⠀⠀⣠⡴⠿⠿⣿⣶⣶⣛⣿⡟⠀⠀⠀⠀⠀⣡⣾⠛⠳⠀⠀⢹⣏⠊⢇⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⢿⠀⠶⠲⢦⡀⠙⢿⣿⠅⠀⠀⠀⠀⠀⠁⠥⠐⢀⣀⡠⠋⠉⢳⠘⡄⢀⡠⠖⢋⣥⠶⠾⡷
⠀⠀⠀⠀⠈⠛⢦⡄⢠⣿⠀⠹⠛⠃⠀⠀⠀⠀⠀⢀⡀⠀⠘⠅⠶⠂⠐⠋⠀⣹⣏⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠸⣇⠘⠿⠂⢠⠀⠀⠀⠀⠀⠀⣠⠯⣤⣤⣤⠤⠤⠤⠖⠚⠉⠉⠀⠀⡍⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⡶⠶⡟⠀⠀⠄⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠚⠛⠲⣤⡀⢈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢧⠀⠀⢀⠀⠂⠀⠈⠀⠀⠀⠉⠀⠀⡄⠁⠀⠀⠈⢁⡀⠈⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣆⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⣴⣶⠚⣦⡴⠾⣦⡀⠠⠄⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⠂⠀⠘⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠓⣿⡟⢲⣿⡟⢶⣤⣤⣄⣈⣁⣀⣈⣀⣠⣤⣤⣶⣾⠛⢹⣿⠗
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⢻⣷⠾⢿⣧⣨⣿⣷⡖⢻⣿⠉⠹⣿⣧⠠⣾⣿⡁⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣜⣿⣦⣤⡿⠿⠤⠿⠷⢼⣿⣦⣀⣿⣟⣀⣿⣿⣇⡴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠤⠔⠒⠒⠋⠙⠶⠋⠑⠦⡀⠀⠀⠀⠀⣀⠤⢤⡀
⠀⠀⢀⡠⠔⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⡰⠁⠀⠀⡸⠖⠒⠒⠒⠢⢤⡀
⢀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣄
⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⡰⠁⠌⠋⠉⠉⣄⠀⠒⡐⠁⠀⢸
⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣔⣈⣣⠀⡔⠐⢢⡀⠈⠀⠀⠀⠀⠀⠇
⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⣡⠀⠜⠈⢏⣀⣑⠇⠀⠀⠀⠀⠀⢸
⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⠀⠀⠠⡞⠋⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠉⠙⠒⠲⠶⠶⠒⠒⠋⠉⠉⠁⠈⠓⠦⣄⡙⣄⠀⠀⠀⣮⣀⣀⡀⢀⠀⠀⠀⠀⠀⠠⠤⠀⡘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠀⠀⡼⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠀⠀⠤⠰⠀⠀⠀⠀⠀⢀⡎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠂⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡇⠀⠀⠀⠀⠀⠀⢀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⠼⠲⡀⠀⠀⠈⠀⢀⣠⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣢⣤⣒⣋⣁⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣿⣿⣦⠰⣿⡀⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡏⠀⠸⡿⠃⠈⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣿⣷⡀⠀⢳⡤⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣵⠿⣿⣿⡇⠀⠘⢏⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣾⣿⡿⠋⠀⠀⠠⠀⠱⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡟⠛⠛⣿⣇⠁⢀⠱⡴⠒⣲⣇⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣽⣿⣦⣀⣉⠃⠀⠀⠘⠉⠉⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠄⣸⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠙⠷⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⠟⢻⠅⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈⠀⣐⣡⣆⣾⡆⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢹⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣳⣈⣄⠀⠀⢁⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⣿⡿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠛⠽⣿⣿⡄⢃⠈⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠘⠀⣿⣯⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢩⡿⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢽⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⢿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠂⠀⠀⠀⠀⠀⠀⠀⢹⣴⣤⣿⠆
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠂⠂⠀⣶⣽⣿⡿⠁
⠀⠀⠀⠀⠀⢠⠾⠛⢁⣀⣀⡀⡤⠀⠀⠀⠀⠀⠀⠀⡄⢀⣠⣾⣿⠏⠀⠀⠀⢀⣠⣄⠼⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠰⠥⠷⠿⠛⠛⠉⠉⠁⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠔⠚⢉⣤⣴⣛⠋⢉⣉⡀⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⠛⠁⠺⠟⠈⠹⣿⣿⣿⣿⣿⣯⣤⣈⣻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢦⡄⢠⢀⣴⣶⣦⣤⠀⠙⣿⣿⣿⢿⣿⣿⣿⣿⣏⠉⠓⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠏⠁⠘⢻⣿⣿⣿⣿⣿⣦⣀⣤⣴⣿⣀⠹⢿⣿⣿⣷⣦⣤⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠈⠉⠻⢻⣿⣿⣿⣿⣿⣿⣟⣋⣘⢶⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⡟⠡⠴⣃⣄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠛⠛⠁⢙⣿⣿⡿⢻⣿⠿⠟⣻⡄
⠀⠀⠀⠀⠀⠀⣜⠁⣶⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⢿⡿⠃⢘⣿⣿⡿⢛⡿
⠀⠀⠀⠀⠀⠀⢳⡄⢿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⡏⠈⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠈⢻⣿⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⢀⡇⢸⣿⣿⣳⡇
⠀⠀⠀⠀⠀⠀⠔⠚⠂⣿⠃⠀⣤⣶⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡇⠸⣿⣧⠟
⠀⠀⠀⠀⠀⠀⠈⠷⡀⠘⠀⠐⠋⠁⠈⣉⣙⣟⠿⣿⠟⣀⠀⣶⣴⣾⠿⠿⠟⠻⠷⠀⡿⢃⡄⡄
⠀⠀⠀⠀⠀⠀⠆⢰⣿⠀⠀⠀⢰⣶⠿⣿⣿⠻⣟⠷⠀⠁⢀⣾⣿⣿⢾⣿⣷⣿⣄⠀⢠⡈⢱⠁
⠀⠀⠀⠀⠀⠀⠐⡈⠓⠀⠀⠀⠈⠀⠀⠀⠴⠟⠃⠀⢸⠀⢸⣿⣧⠀⠀⠤⠾⣯⡍⠀⢸⠇⣸⠀⠀⠀⣠⠔⠒⠒⠂⠈⠉⠉⠉⢉⣉⣁⣀⣀⣀⣀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠻⢿⡆⠀⠙⣼⠁⠀⠀⠀⣿⣶⣤⣅⣐⡓⠒⠒⠂⠚⠛⠛⠛⠋⢉⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⡆⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⢻⣯⣭⣻⡇⠀⢀⣴⡇⢀⣠⠋⠀⠀⣠⠤⣿⣿⣿⡿⣿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠐⣿⠟⠋⠀⠲⠤⣤⣀⣼⣛⣛⡙⠿⣿⣿⣿⠇⡌⠀⠀⡴⢋⡤⠖⢻⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠈⢣⣦⣄⣀⣀⣀⡀⠉⠙⠛⠻⢿⣧⠸⣿⡿⢠⠃⠀⢸⠀⡏⠀⠀⢸⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣠⠀⠈⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠈⠃⠀⠈⠉⠉⠛⠛⠛⣟⣴⣿⡿⣠⣿⠀⠎⠀⠀⢸⠀⣇⠀⠀⢸⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣦⣤⡶⢿⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⢳⡀⠀⠀⠀⠀⠈⣫⣘⣿⣾⣿⠟⠀⢠⡿⢁⣾⡆⠀⠀⠀⠣⣌⠑⠢⢼⣿⣿⡇⢻⠀⠀⢀⣠⣴⡀⠀⠀⠀⠀⣶⠈⠉⠂⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⠀⠹⣆⠀⠀⠀⠀⠀⠉⠁⠘⠃⠀⢰⣿⡡⣅⣿⣿⡄⠀⠀⠀⠈⠉⠲⣼⣿⣿⡇⢸⠀⠐⡇⣀⡠⣃⣀⣀⣤⣤⣿⣦⡀⠀⣸⠇
⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⡄⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣶⣴⣯⣪⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢸⣿⣿⣷⢸⠀⠀⠀⣙⠿⠟⠛⢹⡟⠉⠉⠉⠉⢻⠁
⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⠳⣦⣴⣾⣶⣤⡿⠛⠁⠉⠙⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠘⢿⣿⣿⣾⠀⠔⠛⢋⠗⢀⡀⠀⠀⢀⠶⠶⠶⠺⢆
⠾⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣽⡟⠿⢶⣤⣾⣷⣾⣟⣿⣮⣍⣉⣀⣤⣤⣿⠀⢱
⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢠⣾⢟⣿⡆⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⠟⠛⢿⣿⠦⡀⠀⢨⠉⢹⣯⡍⢓⡼
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⣷⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠘⣿⠇⠀⢸⠿⢄⡙⢄⣘⣦⣼⢋⠴⠋
⠀⠘⠛⠉⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢀⡜⠁⣸⣿⡿⠈⢷⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣠⠆⠀⠀⠁⣠⣿⠟⣡⠋
⠀⠀⠖⠛⠛⢩⣿⣿⣿⣿⣿⣿⣿⣿⣇⠴⠋⠀⣴⣿⣿⣿⣦⠈⢧⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⣾⡿⣡⡞⠁
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠰⣿⣿⣿⣿⣿⣆⠈⠣⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠒⠛⠋⠀⠉
⣶⣶⡶⠶⢶⠶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠚⠛⠛⠓⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢁⠀⠀⠀⣀⡠⣴⣒⣮⠭⠭⠭⠭⠭⡶⣦⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⠎⠀⡠⣪⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⡙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠐⠊⡰⣼⠎⠀⠢⠤⠤⠀⠒⠒⠒⠒⠀⠀⠐⡲⠃⡙⢼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⠀⢱⠇⠀⠘⠀⠒⠀⠀⠀⠀⠒⠒⠀⢉⣀⡀⢠⠁⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠁⠈⠆⢸⠇⠀⠀⡀⠀⠐⣉⡀⠀⠀⠀⠀⠀⡠⢥⡈⣄⡀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃⠀⠀⠸⠞⢠⠀⠀⠀⠉⠀⠉⠀⠀⠀⠠⠀⠀⠀⠀⠱⡸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠃⠀⡀⠀⢳⠲⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢺⡀⠄⠀⠣⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢄⡈⠢⣉⠒⠤⠤⢀⣠⠀⡀⡤⢄⣒⡭⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡵⠉⠒⣲⠒⠲⠶⠶⢿⡟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡇⡠⡪⢤⡤⠔⠓⢂⡲⠌⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣇⠣⣉⣁⠠⠔⠎⠑⢢⡤⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⣑⣖⡇⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠚⠛⠛⡆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣥⠎⠀⠀⠀⠸⣬⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣶⣶⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⢉⣉⣭⣭⣍⡉⠉⠉⠉⠁⠀⢀⣠⣤⣤⣬⣍⠉⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠿⣿⠀⠘⢉⣀⣤⣉⠁⠀⠀⠀⠀⠀⠈⣉⣤⣄⣀⠉⠀⣸⡿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠀⢿⡄⠀⠈⠹⣿⠏⠀⠀⠀⣾⠀⠀⠀⠉⠻⠟⠉⠀⠀⣿⠀⣠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠶⠶⠶⢶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁⠙⢿⣿⠟⠉⠙⠿⠋⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⣿⣿⠿⠿⠿⠋








⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠘⠀⠀⠀⠐⠐⠂⠘⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠁⠠⠀⠃⠀⠀⠀⠌⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠉⠲⢄⠀⠀⢀⣀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣀⡀⢀⡤⢴⣶⣖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⢶⣛⣋⣍⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⢀⣈⡁⠸⣿⣿⣿⢳
⠀⠀⠀⠀⠄⠄⠐⠐⠐⠒⠀⡼⣉⣁⡚⠁⢿⠀⠀⠈⢱⠄⠀⠀⠀⠀⠀⢀⠔⠉⢹⠀⠀⠀⠉⣒⡀⠉⠅⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡹⠿⢿⣟⠀⠉⠀⠒⠒⠉⢀⣤⣶⣶⠌⠀⠁⠑⠒⢊⠉⢉⣹⡿⠿⠿⠃⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠳⢦⣤⣀⡀⠀⠀⠀⢹⠀⠀⠀⢀⣠⣤⠶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⠼⠴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⣀⣀⣀⣰⣆⣠⣤⠞⠃⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⢀⣄⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⢛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠊⢿⠀⠀⠀⠈⣿⣿⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠸⣇⠀⠀⠀⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣿⡄⠀⠀⢻⣿⣧⢠⣤⣤⣴⣶⣶⣶⣦⣤⣤⣠⣄⣀⣀⣀⣤⣴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣏⡀⣀⡸⠿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠉⠁⠼⢤⣭⠝⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⢀
⠀⠀⠓⠁⠁⠉⠀⠈⠀⠃⠀⠀⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡻⣯⣝⣳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢿⣇⣴⡴⠿⢿⣿⠟⢿⣿⠿⢯⣿⣯⠙⢋⣼⣏⣽⣿⣿⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣏⣓⣿⡿⢿⣷⣶⣷⢿⣷⣛⡿⣷⣶⣿⢷⣶⢾⣶⣿⣷⣤⣼⣶⣷⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢛⠉⣿⣫⣴⣿⣶⣤⣾⠟⣹⡿⠏⠸⢿⢘⣡⣻⣿⡇⢻⣏⣽⣯⣹⣎⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⢙⣿⠿⠿⠶⠛⠛⠚⠛⠛⠋⠉⠛⠙⠛⠛⠛⠛⠓⠛⠿⠿⢯⣍⣹⣏⠹⣿⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡦⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠏⠙⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣙⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣦⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⣀⡼⠀⢸⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⣠⡶⠿⠛⠛⠛⠻⢿⣶⣶⡆⠀⠀⠀⠀⣠⣶⣶⣶⣾⡿⠿⢿⣄⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢻⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠈⡉⠀⠀⠀⠀⢸⣿⠛⢉⣉⣀⠀⠀⠀⠈⠁⢸⣿⡷⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣀⠘⡿⠃⠀⠀⠀⠀⠀⠀⠉⣑⣂⠀⠁⠀⡱⠀⠀⠀⠀⢸⡏⠘⠉⣉⣈⠉⠂⠀⠀⠀⠸⣿⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠏⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣽⡠⢺⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡈⢧⣸⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡘⠦⢸⡆⠳⡄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠐⢧⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠈⠳⠀⠒⠀⠀⠀⠆⠀⠀⠑⠦⣄⠀⠀⠀⠀⡾⣯⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠐⠂⠒⠒⠒⠂⠐⠒⠂⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠃⠀⢢⡉⠒⠒⠀⠀⠀⠐⠂⠉⢀⠀⠀⠀⠀⢀⠆⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⣦⣤⣤⣤⣴⣶⠃⠀⠀⠀⢀⡎⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠄⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⢠⣟⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⠤⠤⠤⠤⠴⠖⠚⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢤⠤⠂⠚⠓⠉⠉⠉⠉⠉⠓⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⠾⠉⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⡀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⢠⠃⢀⠠⠄⡀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂⠄⠀⠐⠀⠀⠀⠈⣆⡄
⠀⠀⠀⠀⠀⡆⠐⠁⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⢀⡴⠊⣠⣶⠀⠄⠀⠀⠃⠀⠀⠀⠀⠳⣏
⠀⠀⠀⢀⡀⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠄⠀⠉⠿⠀⠀⠤⡄⢸⠀⡤⠒⠒⠒⠲⡋⠁
⠀⠀⠀⠈⠂⡀⠀⠈⠂⠀⠀⠀⠀⠀⠀⣀⡕⢺⠀⠀⠀⠀⠀⠀⡠⠉⠘⠀⢸⠀⠀⠀⡀⠀⢱
⠀⠀⠀⠀⠈⠀⠀⠉⢷⣶⣶⠀⡐⠺⣼⡿⠁⠈⠀⠀⠀⠀⢀⠔⠀⡀⠀⢀⡜⠀⠀⠸⠉⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠛⠂⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠸⠀⠀⠀⢰⠚⠉
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁⠀⠀⠀⠀⠘⠀⠀⢰⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠘⢯⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⢉⠦⠀⠤⠀⠀⠀⠀⠀⠀⠤⠄⠄⠀⠀⠤⠁⡀⠀⢀⡶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣧⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢇⣀⠀⢲⣶⣶⡖⠀⠀⠉⠉⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⡟⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⠀⠀⠀⠉⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⣼⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠛⠶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠄⠀⠀⠈⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⢀⠟⠻⣏⠉⠉⠙⠛⠛⠛⠛⠳⠶⠶⠶⠶⠶⣦⣤⣤⣤⣀⣠⣤⡶⢾⡿⠟⠁⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⣴⡿⠤⠔⠛⠃⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠉⠉⠉⠉⢁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡘⢷⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣆⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠛⠷⣄⠀⠀⠀⠀⠀⢀⣴⠶⠟⠛⠛⠻⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⣸⣏⣀⣀⡀⠀⣀⡀⠩⣧⡀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⣿⡉⢉⣿⣥⣤⣭⣅⣄⣼⡇⠀⠀⣾⠷⠲⡶⠶⠶⠲⢶⡶⠶⠺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡦⠴⠷⠶⢦⡶⣾⡧⠤⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢹⣧⣤⡀⠀⠀⣠⣴⠟⠁⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⣼⡏⠉⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠙⠿⣶⣤⣀⣀⣀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⣀⣤⡴⠤⢾⣿⣦⡀
⠀⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣠⠶⠶⠛⠁⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠙⠛⠛⠛⠻⣿⡿⠛⠿⠶⢶⣦⣤⣄⡀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣶⣤⡀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣬⠇⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⢀⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⣠⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⠿⠿⠛⠋
⠀⠀⢿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠙⠻⣿⡟⠛⠛⠒⠲⠤⢤⡤⠴⠾⠛⠛⠋⠉⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⣴⡿⠿⢷⣤⡤⠤⠤⠔⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⢿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠉⠻⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⢀⣤⣶⣶⡿⠿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣤⡀
⣰⣿⠋⠁⠀⠀⠀⠀⠘⢷⣄⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⡴⠶⠶⠾⠟⠛⠛⠋⠉⠉⠉⠈⠙⢿⣧⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⡛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢹⣿⡀⠀⢀⣀⣤⣴⣾⠟⠋⠉⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⡿⠃
⠀⠻⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣤⣶⡾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠙⠲⣄⠀⣀⣀⣿⠒⠒⠒⠒⠒⠒⢰⠃⠀⡔⠁⡞
⢰⠚⠙⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⣉⠉⠉⠀⠀⢀⡔⢦⠀⠀⠀⠀⠀⠸⠦⠀⣧
⠹⣄⣠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠰⠛⠁⠀⠀⠀⠀⢈⡩⠤⣄⠀⠀⠀⠀⠀⠀⣳⡄
⠀⠀⠹⡍⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠒⠢⡄⠀⠀⠀⠀⢎⠀⢀⣸⣀⠀⠀⠀⠀⠀⠈⠧⠤⠤⠤⠤⣤
⠀⠀⠀⠙⢦⡀⠀⠉⠙⠒⠢⠤⣤⣀⣀⡀⠀⠀⠀⠀⡏⠀⠀⠀⣹⡀⠄⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠉⠲⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠦⠀⠀⠀⠀⠀⠀⠉⣉⡭⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⡾⢭⡶⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠤⣄⣀⣀⣀⣀⣀⣀⠠⢤⡔⠒⢨⠛⠢⠼⢀⠜⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣗⠺⠷⣜⣙⢦⣠⠇⠉⢢⣜⣀⡤⠚⠁⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡶⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢞⣡⣦⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⣀⡀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⣸⣠⢾⡄⢀⡀⠀⠀⠀⠀⠀⠈⢯⡉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⢳⠏⢳⠀⢰⡀⠀⡶⠦⣄⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡞⢳⢠⡇⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⣾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣤⣄⣠⣶⣶⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠉⣨⣿⣿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠿⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠻⠇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⠀⠀⣠⡟⠁⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠰⣿⣀⣀⣀⣀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⢻⣦⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠳⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣴⠞⠉⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠟⠁⢀⡀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠙⠿⣦⡀⠀⠘⣿⣤⣤⣤⣤⣄
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠟⠉⠉⠉⠉⠻⢿⣤⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣿⣇⢠⣤⣤⣤⣼⣿⠀⠀⢸⣿⡍⠉⠉⠿⣷
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⠈⢿⠋⠁⢈⣿⠀⠀⠀⢻⣧⠘⠀⢰⣿
⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⢀⣀⢀⣴⣶⢾⣿⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠈⢷⣤⣀⢀⣴⡾⠁⠀⠀⠀⠘⣿⣆⣠⣾⠟
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣀⣀⣠⣶⠶⣾⣿⣿⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢹⣧⡀⠀⠠⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢿⣿⠛⠁
⠀⠀⠀⠀⡈⢿⣷⠠⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠁⠀⠀⠈⠛⠃⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠸⠷⣸⣿⣦⣴⠄⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢻⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠀⠀⠀⢀⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢠⣶⡿⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⣤⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⢰⣿⢋⡄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣀⡄⠀⠀⠀⠀⠀⣀⣴⡿⠁⠀⠀⣸⢿⡆⠀⠀⠀⣾⡏
⠀⠀⠀⢸⣿⡘⠇⣀⡰⠄⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣷⣾⣶⡾⠟⠋⠀⠀⠀⣼⠏⠀⡇⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠹⣷⣤⣉⣠⣤⣾⠟⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⡇⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠁⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠛⠿⠦⣤⣤⣤⣤⣤⣤⣤⠿⠟⠉⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣀⠰⠖⠋⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠿⢿⣆⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠻⠿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠛⠛⠉⠀⢀⣀⣠⣤⠛⢿⣶⣶⣶⣦⣤⣤⣴⣶⣶⣿⣿⣿⡇⠘⣶⣄⣀⡀⠉⠛⠛⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠋⠁⠀⣀⣠⣴⣶⡾⠿⠛⣿⡇⠀⠈⢈⠻⣿⣟⠛⢿⣁⢀⣠⡄⢻⡏⠰⠀⠹⣿⠟⢿⣷⣶⣤⣀⣀⠈⠛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⣶⣿⠟⠋⠉⠀⠀⠀⠀⣿⡇⠀⠀⠹⠀⠈⢻⣦⣀⠋⠉⠙⣿⡟⢀⡸⠃⠀⢻⣧⠀⠀⠈⠛⠛⠿⣷⡆⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⣰⡟⠑⠎⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢰⣿⠃⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠘⢦⣀⠀⠹⢿⡟⠁⡠⠊⠀⠀⠀⠀⢹⣿⠀⠀⠀⢠⣿⠃⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⣤⣾⣿⡟⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠛⠂⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣸⣿⣦⣤⣀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⠀⠉⠛⠛⠛⢷⣤⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠛⠛⠋⠉⢹⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⢹⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⠀⠀⠀⠀⠀⢸⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢿⣶⠄⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢶⣶⣿⣿⣙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠏⠉⠀⠀⠀⠀⠀⠀⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠟⠛⢷⣦⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⠉⠻⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣤⣤⣤⣤⣤⣤⣼⠟⢿⣏⠉⣉⡉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣉⣿⣧⣤⣤⣤⣤⣤⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠈⢿⣷⡉⣠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⣀⣀⠀⠀⠀⣀⢀⣀⡀⢀⣠⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢻⣿⠀⠘⡿⠋⢹⣿⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⣿⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣿⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡟⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣼⣿⣤⣤⣴⣿⣤⣤⣤⣿⣧⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⡿⠿⠿⠿⠿⣿⣿⣿⣿⠋⠉⠉⢉⣏⠉⠉⠉⢹⣿⣿⣿⣿⠿⠿⠿⠿⠿⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⠛⢻⣄⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠀⠀⢸⡇⠀⠀⠀⠈⠻⠛⠛⠋⠀⠀⠀⠀⠀⣠⡟⠋⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⢀⠀⠀⠀⣙⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢈⣏⡀⠀⠀⠀⠀⢀⣀⡀⣀⠀⣀⠀⢠⣿⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠻⠿⠿⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣠⣤⣤⣤⠶⠶⢶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠿⠛⠛⠉⠀⠈⠱⢆⡀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠛⠋⠉⠀⠀⢠⡞⠓⢤⡄⡀⠀⠈⠳⣀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⣿⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⣙⠂⠀⠀⠙⣆⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠻⠀⠀⠀⠀⠰⣇⣀⠀⠀⢀⠻⠀⠀⠀⠀⠘⣧⣀⣀⣠⣤⣴⣿⣦⣄⡀
⠀⠀⠀⠀⠀⢀⣿⠁⠰⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠈⠉⠲⠦⠾⠀⢀⣤⣴⠶⠿⠛⠉⠉⠉⠁⠀⠀⠀⠉⠻⠶⣦⣄
⠀⠀⠀⠀⠀⣼⡇⠀⡄⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢀⣠⡶⠟⠛⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣀⣈⣙⣷⡄
⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢧⡀⠀⢀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⣀⣤⡾⠿⠛⠉⠉⠉⠹⣧⠈⠉⠉⠙⠋
⠀⠀⠀⠀⠘⣿⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⢧⣴⠟⠋⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⠀⢠⠞⠉⢙⡦⠀⠙⣷
⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠀⠀⠀⠀⠰⣆⡰⠋⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠘⣷⡀⠈⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⡰⠶⠒⠶⣄⠀⢸⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠘⣿⣶⣾⣦⣤⣶⣿⣤⣄⣀⣀⣀⣀⠀⠀⢀⣴⡟⠀⠀⠀⠀⢠⠀⠀⠀⢸⡧⣤⣤⣤⣿⠀⠀⢿⡏⠉⠙
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣹⠏⠉⠛⠛⠛⠛⠛⣉⣿⡄⠀⠀⠀⢠⡄⠀⠀⠘⣦⡈⠁⢠⡾⠀⠀⠈⣿⠁⣠
⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⣇⣀⠠⠶⠶⣶⡟⠃⠀⠈⣷⠀⠀⠀⠈⢷⡀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⢻⡟⠋
⠀⠀⠀⠀⠀⠀⢀⡘⢿⡆⠀⠀⠀⠀⠀⢻⡍⠀⠀⠀⠉⠁⠀⠀⣠⠋⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠙⡿⣿⠋⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠋⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⢀⣰⠟⠀⠀⣸⡇⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠙⠷⠿⠷⠶⠟⠁⠀⢀⡼⠀⡇⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠟⠋⠙⣷⠀⠀⠀⠀⠀⠀⠟⠳⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠏⠁⠀⠃⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⡧⠀⠀⠉⠒⠂⢤⣤⣤⣤⣤⡤⠺⠋⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠐⠈⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⠾⢿⣷⣶⣤⣄⣀⣴⠟⠻⣦⣴⣾⣟⠉⠙⢿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣶⣿⣿⣠⣤⣾⠿⣦⡉⠉⢹⡇⠀⠀⢸⣷⠟⠉⢿⣦⣾⢿⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⢛⡷⠾⠛⠋⠀⠙⠋⠀⣿⠀⠀⠉⠛⠛⣿⣆⣴⡾⣷⡀⠀⠀⣿⠁⠀⠀⠙⠹⣶⠛⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣌⠻⣦⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⢸⡇⢨⣿⣇⠘⢿⡀⠀⢹⣧⠀⠀⠀⢀⡾⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠘⢧⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢸⡇⣸⠛⠻⣧⠸⡇⠀⠈⢿⡄⠀⠀⣸⠇⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⠛⠛⠲⣦⣼⡟⠀⠀⠀⠀⢸⣧⠟⢀⡆⠈⠻⣷⠀⠀⢸⣷⠚⠛⠋⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠘⣿⠇⠀⠀⠀⠀⠘⠉⠀⢈⡃⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣷⣦⣤⠀⠀⣿⣇⣀⣀⣀⣀⣀⣠⡶⠛⠛⠳⣦⣀⣀⣀⣠⣿⢀⣠⣴⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠁⠀⠉⠀⢰⣿⡉⠉⠉⠉⠉⠉⠛⢧⣤⣤⣤⠞⠉⠉⠉⢹⡇⠀⠉⠉⠙⢾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠶⠋⠻⣧⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠳⠶⣶⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣾⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣿⠉⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠿⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⢶⣶⠶⣿⠶⢶⣶⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠐⣿⠖⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣿⣤⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣀⣀⣯⣀⣠⣭⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠿⠿⠿⢿⣿⣿⡟⠉⠉⣿⠉⠉⢙⣿⣿⣿⠟⠛⠳⠦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⣆⠀⠀⠀⠈⠉⠉⠁⠀⠀⡿⠀⠀⠀⠉⠉⠁⠀⠀⠀⣠⠞⠈⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢤⡶⢦⣴⠻⣦⡄⢤⡄⢴⡄⠰⣦⠀⣿⢶⡆⠤⠴⠂⣴⠖⢶⠦⠿⠂⣤⠠⢦⡝⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠛⠛⠛⠒⠛⠓⠚⠛⠚⠛⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠛⠛⠛⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠉⠀⠈⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠒⠒⠒⠒⠒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⣀⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⡀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢀⠴⠖⠦⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡇⠀⢠⣾⣿⣷⡸⡆⠀⠀⠀⢠⡇⣴⣾⣶⡄⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣿⣿⣿⣧⡇⠀⠀⠀⢸⣼⣿⣿⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢧⠀⢻⣿⣿⣿⣿⠃⠀⠀⠀⠈⣿⣿⣿⣿⡿⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠈⢧⣀⠉⢋⡵⠃⠀⠀⠀⠀⠀⠘⢎⠛⠋⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⡦⠤⢤⡤⠤⢤⡤⠤⢶⠶⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣿⣀⣀⣇⣀⣀⣇⣀⣸⣇⣀⣸⣀⣀⣽⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢿⠉⠉⡟⠛⠛⡟⠛⢻⡟⠛⢻⠉⠉⣻⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠈⠳⠶⠧⠤⠴⠧⠤⠼⠧⠤⠼⠦⠞⠁⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠤⠤⠤⠤⠤⠤⠶⠒⠋⠉

⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⠀⠀⠀⣤⡀⣠⣶⡀
⠀⠀⠀⣠⣀⠀⡿⠙⢶⠷⡄⢠⣄⡛⠛⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠋⢳⠚⣦⣸⠉⣿⠋⠈⣇⣶⠂
⠀⠀⠀⣾⠏⣷⠃⠀⠈⠀⠈⡏⠈⠁⠀⠀⣏⢀⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣄⣿⠀⠀⠀⠀⠸⠇⠀⠙⠀⠀⠟⣿
⠀⠀⠀⣿⠀⠘⡀⠀⠀⠀⠀⢁⠈⠀⠀⠀⢸⣿⠁⠈⣿⣶⡾⡿⣶⣄⣰⡾⠿⣷⣤⣴⠋⠉⣹⣿⠀⠀⠀⠀⡀⠀⢠
⠀⠀⠀⠃⠀⠀⢳⡀⠀⣧⠀⠸⣦⠀⠀⠀⣿⣿⡆⠀⢸⣿⡄⠀⠀⣿⣯⠀⠀⢸⣿⡄⠀⠀⣿⣿⡇⠀⠀⠸⡇⢀⣾⣄⡔⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⣷⣤⣿⣷⣤⣿⠀⠀⢠⣿⣿⣿⣆⢸⣿⣷⡄⠀⣿⣿⡄⢀⣿⣿⡇⢀⣼⣿⣿⡇⠀⠀⢠⣿⣿⣿⣿⠁⠀⣐⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣾⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣼⣿⡿⠛
⠀⠀⠀⠀⠀⠀⢤⣼⠁⢸⣿⣿⣿⣷⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⡿⠻⡇⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢿⣿⣇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢠⣿⠇⠀⢀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠾⠀⠀⢸⡿⠛⣦⣼⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⣿⡄⠈⢿⠀⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⢿⡿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢻⠃⠀⢸⣤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⡀⠀⣈⡇⠀⠀⢹⡟⠈⢻⡟⠈⢻⡟⠁⢹⡋⠀⢹⣇⠀⠀⠸⣤⣤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠛⠀⢀⣿⣧⣀⣼⢇⣀⣼⣇⣀⡼⢷⣤⡼⣿⡀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⠟⠀⠈⠁⠈⠉⠀⠈⠋⠀⠀⠁⠀⠘⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⡇
⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠸⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠹⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡇
⠀⠀⢹⣿⣿⣿⣼⣿⣿⣿⠃⢿⣿⣿⣿⣿⣿⣿⣿⠉⢻⣿⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⡇
⠀⠀⢸⣿⣿⣿⣿⣿⠻⡏⠀⠸⠙⣿⣿⣿⣿⣿⡟⠀⠈⢿⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⢸⣿⣿⣿⣿⣃⣤⣥⣤⣤⣀⡹⣿⣿⣿⣿⡇⠀⠀⣠⣵⣶⣶⣶⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⢸⣿⣿⣿⢱⠋⠀⢸⣿⣿⣿⡗⠘⢿⣿⣿⠀⠀⠚⢹⣿⣟⣿⣿⠀⠈⡷⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⢸⣿⣿⣿⡇⠀⠀⠸⠟⠒⠛⠁⠀⠀⠙⠿⡄⠀⠀⠈⠉⠉⠉⠉⠁⠀⢸⣿⣿⣿⣿⣿⠉⠉⠃⠀⡇
⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⣠⣷⠀⡇
⠀⠀⠈⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣴⣿⣿⡇⡇
⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠘⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠒⠒⠈⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⢹⣿⣿⣿⠟⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠙⢿⣯⠛⢿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢙⡤⠖⠛⠉⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠉⠙⠻⣤⠀⠀⢀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠿⠛⠿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠻⠷⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠐⠣⣤⡴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠉⡽⠻⡉⠛⠛⠛⠛⠒⠒⠢⠤⠤⢤⣤⣄⣀⣀⣠⠤⢲⡿⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠠⠾⠄⠘⠋⠀⠀⠀⠀⠀⢠⠾⣄⠀⠀⠀⠀⠉⠉⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⢽⣦⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⣠⠶⠂⠸⢿⣀⠀⠀⠀⠀⣀⡴⠶⠶⠶⢦⣄⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⣾⣁⣀⠀⢀⣄⠸⣆⠀⠀⣴⠋⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⣿⠤⠿⠦⠴⠧⠤⢾⡇⠀⣿⠙⣟⣛⠛⠛⡟⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠘⣦⡀⠀⠀⠀⣠⡞⠁⠀⢿⡁⠈⠉⠉⠉⠉⠙⢻⡇⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⢠⡿⠉⠛⠛⠛⠁⠀⠀⠀⠀⠻⠶⣤⣄⣀⣤⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⣸⡏
⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⡿
⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⣀⣤⠴⠶⠾⢿⣄
⠸⣿⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠖⠉⠉⠀⠀⣀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⢸⡏⠀⠀⠉⠉⠙⠶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠂⠀⠀⢀⣾⠃
⠀⠀⣠⡼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⣤⠾⠃
⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⠉⠁
⠀⠀⠛⢻⣷⠒⠢⠤⠤⠤⠤⠤⠖⠛⠋⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⣠⡾⢧⣀⣀⣀⠠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⢿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢀⣤⡶⠶⠿⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡿⣧⣄
⣿⠃⠀⠀⠀⠀⠈⠳⣤⣤⣤⡤⠤⠴⠖⠒⠒⠒⠒⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠙⣷
⢿⡄⠀⠀⣀⣠⣴⠾⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿⠁
⠘⠿⠾⠟⠛⠉⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⣀⣤⣶⠾⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣻⣿⣟⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⣀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣦⡶⠿⣷⣴⣾⣿⣄⣀⣀⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠉⠙⠛⠻⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡄
⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣉⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠢⠧⡀⢀⣀⡠⠤⠄⠀⠚
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⠀⠐⢶⡾⠋⠉⠉⠑⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡴⠶⢶⣇⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣄⣀⣀⣀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠛⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠈⣿⡀⠀⠀⠀⠀⠀⠀⣀⠀⣸⡀⠀⠀⢀⠈⠻⢿⣿⠟⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⢶⣶⣶⡶⠚⡇⠘⢆⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠘⣧⡀⠀⠀⠀⠀⢊⠁⠀⠉⠀⠀⠀⠀⠑⠶⠀⠤⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠉⠉⢀⡴⠁⠀⠈⠷⣀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠈⢷⡄⠀⠀⠀⠈⠣⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠄⣸⠆⠀⠀⣠⡿⠁
⠀⠀⠀⠈⠹⢦⡀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠚⠁⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠙⠶⠶⠛⠻⢶⡴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠤⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⢀⣶⣦⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠛⠛⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣶⣤⠄⠀⠀⠀⣀⣀⣠⣴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⠻⠿⠀⠀⠀⠀⢿⠯⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠿⠷⠀⠰⠾⠶⠶⠶⠾⠟⠀⣸⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠚⠳⣤⡀⠀⠀⠀⡁⣀⣀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⠤⢹⣷
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠙⣦⣀⣼⣿⣉⡀⠀⠀⠈⠉⠉⠙⠶⠦⣀⡀⠀⣠⣤⣄⣤⣴⠛⠉⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⣼⣟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠛⠦⣀⠀⠀⠀⠀⠀⠀⠘⢿⣏⠁⠀⠀⠉⣛⣤⣤⡦⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡄⠀⠀⠀⠀⠀⠀⠙⢦⣄⡿⠞⢿⡯⠉⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⢰⣿⣁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠿⣶⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⠴⠚⠛⠉⠉⠛⢦⠄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⢶⣶⠿⠿⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣼⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⣀⣀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⡄⠀⠒⢾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣀⣿⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠿⠷⠀⠀⠒⠲⠶⠶⠠⣷⡦⠐⠒⠂⠉⠉⠉⠉⠉⠀⠈⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣤⣤⣠⣤⣤⣤⣤⣀⣀⣤⣼⣧⣠⣤⣤⣤⣤⣤⣤⣤⣤⣶⡾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠶⠶⠞⠧⠴⠒⠒⠒⠒⠶⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠤⠀⠀⠀⢸⡄
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣈⣧⠰⡆⠁⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡴⠂⠒⢤⠀⠀⠐⠒⠒⠒⠀⠈⠉⠀⠋⠁⠈⠃⡐⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠐⠀⠀⠾⢤⣤⣤⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠿⠿⠋⠇⠈⠦⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡠⠀⠧⠀⠀⠢⠍⠩⠔⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠒⠂⠁⠀⢀⠀⠜⠀⣠⠏
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠃⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⣠⠊⣤⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⣀⣈⢂⡀⠀⠀⠀⠀⠈⠉⠀⠒⠀⠒⠒⠀⠀⠀⠀⠀⣠⡾⠋⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠈⠉⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⣄⠀⠀⠀⠀⣶⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠖⣻⠀⠀⠀⢀⡏⢹⡦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠉⠀⠀⠈⠀⢈⠁⠠⠀⠀⠀⠆⠀⠀⠈⠉⠙⠛⠛⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢢⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣤⣤⣤
⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠘⠀⠀⠀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠴⣿
⠀⣤⣤⣤⣾⠁⠀⠀⠀⠀⠀⣀⣠⠴⠞⠋⢹⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠛⠳⠶⣤⣀⡀⢰⠐⠀⢀⣺
⢀⡿⠆⠀⡀⢡⡀⢀⣤⠶⠛⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠙⠛⠳⣤⣿⡿
⢿⣖⡡⡀⢠⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠈⠻⠶⠻⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣶⠇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣗⠀⠀⠈⠉⠉⠛⠀⠈⠀⠉⠁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠧⠤⠤⠤⠤⡴⠀⠀⠤⠜⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠤⠤⠤⠤⠶⠶⠶⠷⠶⠶⠶⠤⢤⣤⣿
⣿⣿⡿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢰⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⢿⡇
⠀⠀⠘⡏⢻⣯⣻⢶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⢾⣻⣿⣿⠃⠘
⠀⠀⠀⠀⠀⢿⣿⣷⣬⣛⠫⠛⠻⠭⢝⠛⠛⠛⠯⣙⠳⢦⣄⣀⣀⣠⡴⠛⡫⠽⢒⣛⣭⠭⠟⣛⠭⠟⢛⣩⣶⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣦⡄⡀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠀⠀⠈⠉⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢧⢻⣿⣿⣄⠉⢻⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⡎⣿⣿⣿⣶⣴⡿⠀⠈⠙⣿⣿⠿⠶⣾⣶⣶⣶⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣉⣽⣿⣿⣿⣿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢷⢹⣿⣿⣿⣿⣿⣦⣀⠀⢛⡟⠀⠀⠀⢹⣿⠏⠉⠛⠛⣿⡿⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⡏⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⣀⣼⣄⣀⣀⣠⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣽⣿⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣿⣿⡟⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢹⣿⣬⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣠⣿⣿⣿⠇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠹⣿⣆⣸⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⢠⣽⣿⣿⣿⡟⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣗⠐⠛⢿⣿⣆⠀⠀⠙⠉⠉⠛⠻⠛⠛⠉⠙⠁⠀⢠⣾⣿⣿⣿⢻⣿⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠙⢿⣿⣾⣶⣆⡀⢠⣾⡆⠀⣀⣿⣷⣾⣿⣿⡿⠛⢡⣿⢇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠋⡿⠛⠛⠛⠻⠿⠿⠿⠿⢿⡟⠛⠙⠁⠉⡠⣿⣟⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠙⠀⠀⠀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣩⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣿⣶⣦⣤⣤⣴⣷⣶⣶⡾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢠⣿⠋⠉⠙⠛⠛⠛⠒⠒⠖⠲⠿⠶⠿⠦⠤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣉⣉⣙⣲⡀
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⢉⣉⣤⣤⣤⣶⣦⣤⣄⡉⠙⢲⡄
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⣀⣴⠾⠿⠿⣿⣿⣿⣿⣿⣿⣿⠆⠀⣸⠇
⠀⠀⠀⠀⠀⢿⡆⠀⠀⢘⣟⣠⣾⠤⠶⠶⠛⠉⣠⣴⣿⣷⠈⠀⠛⠉⠿⠛⡟⣀⡴⠋
⠀⠀⠀⠀⠀⢸⣿⣚⣉⣭⣤⣴⣾⡀⠀⠀⠀⣾⠿⣋⡥⢤⡀⠀⠀⣼⡿⣿⣿⠁
⠀⠀⠀⠀⣠⡴⠿⠿⣿⣶⣶⣛⣿⡟⠀⠀⠀⠀⠀⣡⣾⠛⠳⠀⠀⢹⣏⠊⢇⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⢿⠀⠶⠲⢦⡀⠙⢿⣿⠅⠀⠀⠀⠀⠀⠁⠥⠐⢀⣀⡠⠋⠉⢳⠘⡄⢀⡠⠖⢋⣥⠶⠾⡷
⠀⠀⠀⠀⠈⠛⢦⡄⢠⣿⠀⠹⠛⠃⠀⠀⠀⠀⠀⢀⡀⠀⠘⠅⠶⠂⠐⠋⠀⣹⣏⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠸⣇⠘⠿⠂⢠⠀⠀⠀⠀⠀⠀⣠⠯⣤⣤⣤⠤⠤⠤⠖⠚⠉⠉⠀⠀⡍⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⡶⠶⡟⠀⠀⠄⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠚⠛⠲⣤⡀⢈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢧⠀⠀⢀⠀⠂⠀⠈⠀⠀⠀⠉⠀⠀⡄⠁⠀⠀⠈⢁⡀⠈⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣆⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⣴⣶⠚⣦⡴⠾⣦⡀⠠⠄⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⠂⠀⠘⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠓⣿⡟⢲⣿⡟⢶⣤⣤⣄⣈⣁⣀⣈⣀⣠⣤⣤⣶⣾⠛⢹⣿⠗
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⢻⣷⠾⢿⣧⣨⣿⣷⡖⢻⣿⠉⠹⣿⣧⠠⣾⣿⡁⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣜⣿⣦⣤⡿⠿⠤⠿⠷⢼⣿⣦⣀⣿⣟⣀⣿⣿⣇⡴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢿⣿⣿⣿⠟⠀⢠⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢄⣴⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢀⣀⡒⣾⣿⣿⣿⣿⣿⠛⠁⢿⣄⢡⡽⠆⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⠁⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⣿⣄⠀⣀⡤⠞⠳⣄⠀⠀⠀⠀⢀⡠⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠈⠛⠁⠀⠀⠀⠈⠙⠲⠤⠖⠋⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⣄⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⢀⠀⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠙⠓⠶⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠃⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡼⠋⠀⠀⠀⣿⣿⣿⣷

⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠻⠿
⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣶⣖⣒⣒⠚⠻⠷⣶⣶⣒⠶⠶⣶⣶⣮⣍⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠈⡄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣶⢦⣬⣉⡓⡤⡀⢰⣿⠏⣴⡿⠛⢋⣉⠉⠛⠛⢦⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣅⣀
⠀⣇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠇⣼⣿⣿⣿⣿⣿⣷⣾⣿⣿⣼⣿⣥⣴⣾⣿⠿⠿⠷⣎⢳⣄⠀⠀⠀⠀⠀⠸⣿⠹⣿⠿⠌⣁
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣼⣟⣿⣿⣿⣿⣿⡿⠻⠮⡉⠙⠛⠛⠻⣿⣿⣿⡦⠀⠘⣷⠌⠳⡄⠀⠀⠀⠀⠙⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣼⣿⣿⣿⣿⡿⠛⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠙⠻⣄⠢⢐⣿⠀⡄⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡼⢻⣿⣿⡟⢸⠀⠀⠀⠈⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⡿⢳⣾⣿⠀⠘⣧⠈⢻⣆⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠈⣿⡘⠱⡆⠀⢀⣠⣤⣄⣸⡇⠀⠀⠿⠿⠿⢶⣶⣧⣼⢿⣿⡇⠀⢻⣆⠀⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠈⠃⣤⢐⡞⠉⠀⠀⠈⣿⠁⠀⠀⠀⣀⣀⣀⡿⣽⠻⣎⣿⣧⠀⠀⢿⡆⢁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⣦⡀⢀⣾⠁⠙⢏⣿⣿⣣⣴⠋⠀⠀⠔⢫⠝⠁⠸⡻⣇⠀⠙⢿⣿⡀⠀⠈⣿⣿⢸⡇
⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⢡⣿⠁⠀⠀⠀⠀⣩⠟⠁⠀⠀⠀⠀⠘⢶⣶⣤⣁⠘⠇⠀⢸⣿⡿⣄⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣕⡀⠀⠀⣾⡏⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠞⠿⢦⣀⠀⣿⣿⡿⢳⣄⠽⣿⡇
⠀⡇⠀⠀⠀⠀⠀⢀⣾⣧⠈⡄⢰⣿⠃⣤⠀⠘⣿⣿⣷⣦⣤⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⢝⣦⣿⡇
⠀⡇⠀⠀⠀⠀⣠⣿⡿⠏⠀⠃⢸⡿⠀⠘⢦⣀⠈⠉⠈⠛⠋⠛⠉⠀⠀⠀⠀⣂⣶⣶⣤⠀⠀⣼⣿⣿⠃⣻⣯⠿⣿⣧
⠀⠀⠀⠀⢀⡴⣫⠎⠀⠈⡄⠀⣾⡇⠀⠀⠀⠈⠙⠓⠒⠐⠀⠀⠒⠒⠚⠛⠋⠉⠀⠀⠀⠀⣰⣿⣿⠇⠐⣩⣿⡆⢻⡟
⠀⠀⠀⠀⣨⠔⠁⠀⠀⠀⠐⢠⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣨⠖⣹⣿⡁⣀⣴⣏⣿⠟⣾⠇
⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠊⢁⣾⠛⣸⣿⡏⢱⡿⢋⣼⠟
⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⢠⣟⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡾⠋⠀⠀⢀⣴⡟⠁⠀⣿⣿⢰⡟⠁⣿⡏⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣤⣦⣄⣀⡄⣠⣴⣿⣿⣿⣥⡴⠶⠶⠶⠿⠋⠀⠀⠀⣿⣿⡶⠁⡀⢹⣷⣄⠀⠀⠀⠀⢠⠇
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣷⢲⣦⣄⣿⡏⣦⠀⠀⠀⡾
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢹⣷⡀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣿⡀⢸⣍⣿⣿⠃⠄⠀⢸⡇
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣡⡾⠁⢹⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢀⠼⣿⣹⡙⢻⣾⣿⢦⣄⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣠⣴⢿⣿⣦⣴⣄⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣿⠀⠀⢹⣯⣷⠈⢟⣠⣶⡀⣿⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣷⣾⣧⠀⠀⠀⠀⢠⠖⠋⠀⠀⣿⠀⠀⢀⣻⡟⢠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⡟⠀⣀⣼⢻⡿⣿⣿⣿⣿⣻⠇⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣟⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡀⡇⣾⣿⣿⣿⣟⣿⣿⣧⡞⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⠃⠸⣷⢸⣿⣣⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⠿⣿⡿⣿⣿⣿⣿⠟⣭⣿⣿⢿⣿⡟⠻⡙⠓⢦⡀⠀⢤⠼⠀⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠷
⡀⠀⠀⠀⠀⠀⠀⠨⠿⢿⡟⢉⢀⠀⣼⣿⣿⡟⢋⣥⣾⡿⠋⣙⣾⣿⣿⡄⠈⢂⣀⣿⠀⠀⠀⠀⢸⡇⣾⣻⣿⣿⣿⣿⡿⠋⣡⡇⡆
⣝⠀⠀⠀⠀⠀⠀⠀⠠⣦⡽⠗⠋⠠⣴⣿⣡⢾⣻⠿⠋⡀⠐⡿⠋⢹⣿⣿⡄⠀⠹⢿⡀⠀⠀⠀⣿⢃⣿⣿⣿⣿⠟⢉⣴⣾⣿⣵⡇
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣟⣿⣤⣌⣀⠀⠀⠀⠠⠿⢿⣿⣦⡀⠘⣷⢀⠀⠀⢹⣸⣿⣿⣛⣥⣾⣿⣿⣿⣿⢻⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⢶⡿⠉⠀⠀⠀⠄⠀⠀⠹⣮⣳⣦⡙⣄⠱⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠈⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠂⢀⣀⠀⠀⠈⠻⢦⡙⢿⣷⣄⣿⣣⠾⠛⠛⢿⣿⠿⠛⠁⠀⡼⠁
⠀⠀⣀⢀⣀⡤⠒⠒⠀⠀⠀⢀⣀⠀⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢴⡀⠉⠀⠈⠛⢿⣄⠀⠀⠀⠛⠀⠀⠀⠀⠀⢻⡄
⠀⠰⠀⠀⡼⢡⣌⡴⠚⠉⠉⠉⠁⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠟⠛⠿⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠴⠃⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢠⡾⠷⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⣠⣼⡷⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠈⣿⠇⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠖⢺⣟⠁
⠀⣿⠀⠀⠀⠀⠀⠈⠻⢝⠓⠶⠶⠶⠖⠛⠛⠛⣉⠉⠉⠉⠉⠉⠁⣴⡉⢢⡀⠀⠀⢻⣆
⠀⢿⡄⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⣴⢋⡆⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⢻⣆
⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢰⡄⠀⠀⠀⢻⠀⠀⢻⡆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⢻⡄⠀⠈⢿⡀⠀⠀⢸⡇⠀⠀⢻⡄
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢸⡇⠀⠀⠈⢷⣄⣠⡾⠁⠀⠀⠈⢿⡄
⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠻⣦⣀⢀⣠⡾⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⣠⣿⠤⢤⣀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠙⠳⠶⢤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣽
⠀⠀⠀⢸⡏⠀⠀⠒⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢤⠖⠚⠛⠉⠉⢹⣯⠉⠉⠉
⠀⠀⠀⠸⣧⠀⠀⠀⠸⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠹⣧⣄⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠓⠶⠦⢤⣤⡤⠤⠴⠶⢶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⢤⣤⡼⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠷⠶⠤⢤⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠙⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⢻⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠉⠛⠷⣦⣤⣄⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣤⣄⣀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣶⣤⣠⣴⠟⠁

⠀⠀⢠⣾⣷⢶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣷⠄
⠀⠀⢺⣯⡏⠀⠀⠉⠙⠛⠿⣶⣤⣄⡀⠀⠀⡀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡶⣀⣠⣶⣴⢇⣶⣃⣀⣀⠀⣀⣠⣴⡶⠿⠛⠉⠁⠀⢀⣽⡟
⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠉⠙⠻⣷⣦⣽⣿⣿⣿⣿⣷⣶⣿⣶⣷⣿⣸⣷⣿⢱⣿⣿⣿⣳⣿⣿⣿⣿⣷⡾⠟⠋⠁⠀⠀⠀⠀⢀⣴⣿⣿⠋
⠀⠀⠀⠀⠘⢯⡙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠻⢿⣿⠛⠉⠉⠛⠛⠟⠻⠇⠛⠋⢿⡿⠿⠻⠟⠻⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠾⣡⡟⠁
⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣥⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣶⣦⣦⣶⣤⣤⣤⣄⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠈⠛⠛⠉⠉⣉⣉⣉⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠐⣾⣿⡄⠀⠀⠀⠀⣀⣴⡾⠟⠛⠛⠛⠻⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠉⠉⠙⠻⣿⡄⠀⠀⠀⠀⠀⣰⠏⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣰⣿⡟⠀⠀⢀⣄⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⢠⣄⠀⠀⣻⣷⠀⠀⠀⠀⢰⣿⣾⡋
⠀⠀⠀⠀⠀⠀⣤⣿⠛⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠉⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⠀⠈⢻⡿
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣤⣤⣤⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⣤⣴⣾⣿⣿⡏⠀⠀⠀⠀⠀⣼⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢻⡃⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣧⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⣶⠽⣷⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠘⢧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢋⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣯⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣿⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⢀⣰⢿⣧⠀⠈⢹⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⣻⠟⠁⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣤⣤⣰⣿⠿⣿⣿⣿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⢋⣠⣼⣧⡀⣠⣿⡃
⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠉⠙⠋⠀⠈⠙⣿⠃⠀⠉⠙⠛⠷⢦⣤⣄⣀⣀⣤⣤⣄⣤⣾⠟⠋⢠⣶⠟⠋⠁⠈⣿⠉⠈⠙⣶
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠷⣤⣀⡀⠀⠀⠀⠈⠉⠈⠛⠛⠛⠛⠉⠀⠀⠀⢸⣧⠀⠀⣶⣴⡿⠂⠀⢀⡏
⠀⠀⢰⠟⠀⠀⠀⠀⣴⣾⡿⠀⠀⠀⠀⠀⠀⣶⡶⠀⠀⠉⠛⠻⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠁⠀⠀⢹⡗⠀⢠⡾⠋⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠉⠁⠀⠀⠀⠀⠀⠀⠙⠃⠀⣴⣄⠀⠀⠹⠿⣦⣄⠀⠀⠀⠀⢀⣠⣶⠟⠋⠁⢀⣤⣀⣈⠳⣶⠊⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠻⣧⡀⠀⠀⠈⠙⠶⡄⢠⡶⠟⠋⠀⠀⢀⣠⡿⠋⠉⠉⠙⢿⣶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⢠⣀⣀⣀⣠⡄⠛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⠈⠀⠀⠀⠀⠂⠀⠈⠀⠀⠁⠈⠈⠁⠐⠀⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢟⣛⣭⣭⣻⣷⣌⠻⣿⣟⠇⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠔⣋⣉⣉⠙⢿⣿⣿⡆⠹⠹⡆⢰⢸⡔⡌⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡟⢉⣼⣿⣿⢿⣿⣆⢻⣿⡇⣶⠀⣧⣾⣾⢃⡇⣸⣿⣹⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣶⣿⡿⠿⠭⡷⢌⢻⣞⡏⢁⣿⣼⣿⠟⣡⠞⣴⡏⠁⢌⡻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣭⣽⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣮⡳⣤⣀⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠯⣭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡦⣍⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣠⣤⣄⣄⡀⠀⠀⠀⢀⣠⣤⣴⣶⡄⡀⠘⣏⢿⣦⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠁⢉⣉⣙⢻⡄⠀⠀⠈⠉⣉⣭⣀⠀⠀⠀⠻⣧⠡⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣄⢸⠀⠀⠀⠀⠀⠈⢸⡇⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢸⣾⣴
⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡗⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢿⠄⠀⠀⠔⠊⠀⠀⠀⠂⠊⠀⠲⢄⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢄⣀⣀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⠀⠀⠈⢿⠿⠿⠿⠿⢿⠏⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠄⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠌⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠤⠤⠤⠔⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠀⠀⠐⡀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⠁⠀⢀⣀⡀⠀⠀⠀⠐⠙⠛⠛⠉⠀⠂⠀⡇⠘⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡀⠀⠀⠁⠐⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⡄⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⡾⠿⠧⠀⡘⠀⢀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢰⣿⣎⣄⠀⠀⠈⠳⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠑⠂⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡉⣻⣿⣷⡲⣤⢾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣛⣳⣮⠑⣼⣷⣄⣾⣷⠀⣠⡀⠀⣥⠀⠀⢀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣭⣷⡸⢟⣙⣿⣋⣻⣃⢿⣇⣠⣿⠀⣾⠛⣹⣗⢈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣭⣭⣵⣷⣙⠷⠶⢟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠀⢀⣠⣶⡾⢀⣤⠖⠋⢩⣯⣤⠴⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⣠⠞⣿⣧⠶⠛⣡⣿⠶⠋⠁⠀⠀⠛⠁⠀⠾⠓⠛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣹⡧⠚⠁⠘⠋⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡟⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠶⠒⠚⠛⠛⠛⠻⠿⢿⣿⣶⣀⠀⠀⠀⢹⣦⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡿⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⠀⠀⢿⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢸⠃⠀⠀⠀⠀⢀⡠⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣰⡿⠁⣠⣊⣭⠽⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣒⡲⢤⢹⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡿⠁⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠓⠷⢿⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠇⠀⠀⣠⡶⠛⠉⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠛⢢⡸⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢸⠀⠀⢰⣿⠁⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⠀⠀⠀⢧⢻⡀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⢶⡄⠀⢸⠀⠀⢸⣇⠀⠀⣿⣥⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⢻⡿⣷⠇⠀⢀⡟⠈⣿⡟⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢾⢦⡄⠙⣆⢸⠀⠀⠀⠻⣦⡀⠈⠁⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣠⠞⠁⠀⢹⣷⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠈⠀⠹⡆⠈⠻⠇⠀⠀⠀⠀⠉⠛⠛⠋⠁⢀⡠⠀⠀⠀⠀⠀⠀⢠⠀⠀⠉⠀⠀⠀⠀⠀⣿⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠁⠀⠀⠀⢠⡾⠛⠋⠀⠀⠀⠀⠀⢸⣁⣠⣤⡀⠀⠀⠀⣨⣇⣼⠀⠀⠀⠠⢶⡆⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⡄⠀⠀⢿⠀⠀⣶⣿⡂⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠋⠀⠀⠀⠠⣤⡀⠀⡇⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠛⣿⣿⡶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⠇⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠙⣿⡷⣦⣉⠉⠛⠓⠒⠶⠶⠶⠖⠚⠛⣩⣿⡟⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠈⠻⢦⣉⠉⠓⠒⠒⠤⠤⠄⠒⠒⣫⡿⠋⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠶⠶⠶⠶⠛⠉⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⡀⠀⠀⠀⠀⠒⠶⠦⠤⠤⠶⠂⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡏⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡼⠀⢰⣷⠀⠀⢀⣿⠁⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⢠⡇⠀⢠⠎⢸⠀⢀⡼⠉⠀⠀⢠⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⣤⣾⣿⠖⠁⠀⠈⢷⣿⠃⠀⠀⣠⡼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠙⠓⠚⢋⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣀⣀⣠⣤⣴⣶⡄⣀⣠⣤⣼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⢩⣾⣿⣿⡟⠀⠈⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠀⠀⠀⣿⣿⡿⠋⠀⠀⠀⠀⠙⠻⠿⡿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠉⠉⠀⣴⡲⠄⠀⠀⠀⣾⡅⠑⢄
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡆⠀⠀⠀⠀⠀⢿⣇⡀⠀⠀⠀⠚⢇⠀⠀⠑⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣨⠟⣛⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠤⠞⠀⠐⣦⡀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠰⡇⠘⠉⠓⠀⠀⠀⠀⠀⠀⠀⡠⠚⠣⠀⠀⠠⠒⢊⠁⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠖⠓⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢾⣅⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣖⡒⢒⣶⣾⣿⡧⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣞⡿⣿⣿⣄⡹⣿⣷⣸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⣷⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣤⣈⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⡀⠀⣼⣿⣿⣿⣿⣿⣿⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⣸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⢰⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠋⠉⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠉⠙⠛⠿⠿⠿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠻⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠉⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⣏⣶⣦⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢧⡻⠵⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⠈⠉⠉⠁⣀⣀⣠⣤⣤⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⢻⣿⣶⣿⣭⣭⣭⣭⣭⣶⣶⣿⠏⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠙⢿⣿⡿⠿⠿⠿⢿⣿⡿⠃⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠉⠒⠦⠤⠖⠚⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠒⠋⠁

⠀⠀⣦⣿⣿⣿⣋⠀⠀⣘⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣷⡟⣫⣾⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢐⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠙⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠀⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣯⠄⢀⣤⣶⣶⣾⣿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠀⣾⣿⣿⣿⣿⣿⠟⠁⠀⠻⣿⣿⣿
⠀⠀⠀⡇⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣛⣋⣠⣤⣾⣿⣿⡿⠋⠁⢀⣾⣆⠀⠙⢿⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⡀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣴⣿⡿⣿⣧⠀⠈⣿
⠀⠀⠀⢸⠄⢀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢠⠿⠛⠋⠀⠀⠹⣧⠀⢸
⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡇⠀⠀⡎⢧⣄⠀⠀⠀⠀⢻⡀⢈
⠀⠀⠀⢸⡻⣿⣿⣿⣿⣿⣿⣭⣍⣙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢹⣦⡀⠀⢸⠁⢸
⠀⠀⠀⠀⠙⠦⡈⠉⠩⢿⣽⣧⠈⠉⢉⡿⠒⣲⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡀⠀⠀⠀⢠⣼⣿⡇⠀⠏⠀⣼
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠈⠫⢽⣇⡴⠋⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧⠀⠀⠀⠻⠿⠋⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠛⠉⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⢴⣿⣿
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⣆⠀⠀⠀⢀⣤⣾⣿⡿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠙⠻⠿⣿⠿⠟⠉
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⣄⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⠀⣀⡀⠀⢸⠟⠒⣶⣿⣁⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠐⠛⠋⠁
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠔⡛⠛⠁⣀⣀⣤⣶⡿⠃⠀⢉⡓⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠳⠦⡤⠶⢺⣴⣿⣿⣿⡿⠛⠉⢀⢤⡶⠛⠁⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣉⡉⢀⣀⠔⢊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠳⠴⠋⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠇⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⣠⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⣰⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢻⣿⣿⣿⣿⣿⣿⣷⣦⣄⣤⣀⣤⣴⣿⣼⡆⠀⠀⢀⣴⣿⡏⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣴⣿⣿⣿⡇⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣤⣾⣿⣿⣿⣿⡷⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⡟⠋⠁⠈⠉⢻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣈⡁⣼⣾⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢠⡞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⢻⡶⢤⣄⣀⠀⠀⠀⠀⢀⡀⢰⣄⣠⣴⣴⣿⣿⣴⣤⣿⡆⡀⣰⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠸⣿⡄⠈⠉⠙⠛⠳⢦⡈⠿⠛⠋⠁⠙⠟⠈⠀⠀⠉⠉⠉⠃⠹⠿⠿⠿⣶⣿⣀⣠⣤⡶⠟⠛⠉⢁⣾
⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢯⠛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠞⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣆⠀⠐⢺⡿⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⢶⣦⣶⡀⠀⠀⠠⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⠇⠀⠀⢘⣧⣠⣼⣿⠇⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢘⣿⡏⠀⠀⠀⢺⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠿⠿⠇⠀⠀⢰⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠖⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠙⢶⠄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⡄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠙⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠱⢶⣶⣄⣤⣠⣀⡀⣀⣠⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⠿⢿⣏⣉⣽⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢭⣿⣤⣼⡿⠏⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣦⣄⣀⣀⣀⣀⣀⣤⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢿⡿⠉⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡾⢻⠃⠀⣠⡤⣦
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠓⠶⢦⡴⠋⠙⠶⣄⡴⠚⣡⠞⢀⣯⡴⠋⢸⠀⣿
⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⣠⠏⠀⠀⠀⣠⡟⠁⡰⠋⠀⡼⠋⠀⣠⠞⠛⠛⠳⢤⡀
⠀⠀⠀⠀⢀⡞⠁⠀⠀⣰⠏⠀⠀⢀⡼⠋⠀⢠⢃⣀⣼⠁⠠⠀⠻⣄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⣜⠁⠘⡄⣰⠏⠀⠀⢠⣾⢢⠀⣀⠾⠛⠉⠈⠙⠒⠶⢤⣈⢷⡀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⢩⠀⢰⢠⡟⠀⠀⢠⣿⠜⣠⠞⠁⠀⢠⣾⣿⣷⡄⢸⡷⠟⣸⠃⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⢀⣸⠦⣈⡿⠀⠀⢀⠧⣘⡿⠁⠀⠀⠀⠀⠀⠈⠙⠃⠀⠉⠉⠁⠀⠀⢰⣾⠗⠀⠀⠹⣆
⠀⠀⣴⠋⠀⠀⠈⡇⠀⢀⡜⢀⡿⠁⠀⠀⠀⠀⢀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠐⢦⡀⠹⣤⠾⢻
⠀⠀⣇⠀⠀⠀⠀⢷⡀⣼⡧⢸⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠙⢦⠀⠀⠀⢀⠀⠀⠿⠲⠖⢚⡃⠀⢿⡄⢸
⠀⠀⠙⢦⣄⣀⣠⠞⠉⢿⢃⣼⣇⠀⠀⠀⣸⠁⣀⣀⣤⡆⠘⢻⡇⠀⠀⢻⡄⠀⠈⠃⠤⠊⠀⠀⠘⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⠀⢹⡄⠀⠀⠸⡇⠀⠀⠉⠁⠀⣸⠃⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠋⠹⣄⠀⠀⠙⠦⣀⣀⣠⠾⠋⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣧⣀⡘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⣠⡿⠀⢠⠇⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣭⣤⡼⢧⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠈⠙⠋⠛⠁⠀⡰⠃⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠁⠉⠒⠤⠄⣀⣀⣀⡀⠴⠋⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡟⢉⣙⡷⢶⣤⣤⣄⣤⠶⠋⢡⣌⣹⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢒⣬⠟⠛⠙⠛⠙⣯⠀⠀⠀⠀⠨⡁⠀⠀⠀⢿⡋⠉⠙⠛⡵⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠳⣄⠀⠀⠀⢰⡟⠀⠀⠀⠋⠀⠀⠃⠀⠀⠘⣇⠀⠀⢠⠇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙⢦⣤⣀⢸⡇⠀⠀⠀⡀⡀⠀⠐⠄⠀⠀⢹⣄⣤⡟⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠈⣿⠄⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⠘⡇⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠶⠄⠀⣿⣤⣤⣤⣤⣼⡛⠉⢙⣦⣤⣤⣤⡏⠠⠶⣓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣀⣀⣀⡴⢿⡄⡀⠀⠀⠈⠉⠛⠉⠁⠀⢀⣼⣅⣀⣀⣨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⡄⠀⠀⢀⡀⠀⢀⣴⠞⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣄⣄⣤⣀⣀⣨⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⡿⠖⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣀⣀⣀⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣏⣉⣭⣉⣩⣇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⠛⠛⠛⠿⠿⠇⠀⠉⠀⠈⡿⠿⠏⠙⠛⢲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣁⣀⡸⢄⡀⣀⢀⡀⢠⡀⣤⣀⢀⡀⢀⡀⣀⣠⡇⡀⣀⡹⣆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⠓⠓⢚⢓⡚⢓⣚⣒⣓⣚⢓⣛⣛⣓⣒⣛⣒⣒⡓⣒⠚⠛⠓⣋
⣿⣿⣿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣤⣀⠲⣄⠀⣠⠖⣛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⡯⢿⣿⣻⢿⡿⣷⣽⠿⠖⠛⠛⢿⡽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣮⣝⡻⡾⠟⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⢿⣿⣿⣿⠿⠗⠿⠋⠁⠁⠀⠀⠀⠀⠀⣀⠀⢸⡆⣀⣤⡀⠀⠀⠀⠀⠀⠀⢀⡤⡞⡏⡇
⠀⠀⠀⠀⠀⢀⣿⡿⣟⣫⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣿⣋⣤⣽⡆⠀⠀⠀⢀⣴⣿⣆⣷⠟⠁
⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⢸⡏⠀⢘⡇⠀⣀⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠡⢿⣇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⣷⠖⠀⠀⠀⠀⠈⣷⠀⢸⣧⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠈⢟⣧⠀⢀⡤⠶⠒⠋⣉⣿⣦⣄⡀⠀⢼⣤⣤⣄⠀⠀⠀⠀⢹⣦⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣼⣧⡀⠀⠀⠀⠀⠉⠀⣠⡼⠋⠀⠀⢀⣤⣼⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⢠⡿⠒⠶⢮⣷⡄⠀⠀⠀⢠⠞⠙⢧⣐⣦⣀⣸⡯⠶⠖⠛⠛⠉⠁⣿
⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠈⠻⣆⠀⠀⣏⣀⣤⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⣠⣤⣿⣴⡟⠛⠛⠛⠛⣿⠛⢛⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠘⣧⡀⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣿⠀⠀⠈⡇⠀⠀⠀⠀⣿⣠⢼⣄⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡇⣿⠀⠀⠀⡇⠀⠀⢀⣤⡿⠿⢷⣬⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⡀⠀⠀⡇⠀⢰⡚⠦⢬⡙⢢⠘⠇⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠶⠶⢤⣤⢞⡽⠚⣋⣉⡽⠀⠀⡇⠀⠈⠉⠳⣄⠱⠀⢂⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⢰⡟⢫⡴⠿⠛⠛⠦⣄⣀⣛⣻⡟⠛⣻⠟⠐⢊⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⠛⢻⡇⠁⠠⠶⠶⢾⣓⠋⣿⠿⢿⠷⡾⠃⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠀⠈⠉⠙⢳⣼⡇⠀⣠⣤⡤⣤⣈⣷⡟⠀⢻⡘⣇⠀⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢹⡅⢀⣌⡛⠓⠒⣾⣾⠃⠀⠀⠙⣎⣧⡬⠷⠚⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢠⡀⠀⠀⠀⠀⢸⠇⠀⣿⠏⢹⣿⣁⣿⠀⠀⢆⠀⠈⢿⣇⣀⡤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣧⠀⠀⠀⣠⡾⠀⠀⣿⠀⢸⣿⣿⡿⠀⠀⠈⢷⡀⠀⣿⠁⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣆⣴⠞⠙⣧⣠⡴⠋⠀⠀⢹⣿⣧⠀⠀⠀⠀⢳⡀⢹⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⣤⣠⡾⠋⠙⣆⠀⠸⡇⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⢳⡸⠆⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⣠⠞⠁⠀⠀⠀⣹⡶⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢀⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢖⣶⡿⠿⠿⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠻⠿⢿⣶⣦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠶⡶⠶⢶⣶⣶⣶⣶⣶⣶⠶⢶⡖⠀⠰⢶⠶⠶⣶⣶⣶⣶⣶⣶⣶⠶⢶⠂⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢱⠀⠈⢿⣿⣿⣿⣿⣿⣷⡿⠀⠀⠀⠈⣆⠀⠹⣿⣿⣿⣿⣿⣿⣷⡏⠠⣾⡤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠳⣄⠈⠙⠿⠿⠿⢟⠿⠁⠀⠀⠀⠀⠈⢦⡀⠈⠻⠿⠿⠿⣻⠏⠀⠀⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⢠⡀⠈⠑⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣁⠀⠀⢂⢠⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠁⣀⣤⣤⠄⠀⠀⠀⠉⠉⠉⠀⠈⠂⠙⠘⠀⠀⠀⢉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⢠⠀⠀⠀⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣦⣄⠀⠀⠀⢨⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⠀⠀⠀⠱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠉⠀⢇⡀⣸⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⠿⣿⣦⣄⣀⣬⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⠿⣭⡣⣹⣯⢁⣶⡆⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⡄⠿⣹⣿⣛⢿⢸⣇⢀⣿⠀⣶⢛⣳⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣵⣧⣛⣶⣾⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠀⠠⠤⠤⠀⠀⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢀⣴⠾⠛⠛⠛⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠛⠛⠿⢶⣄⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠉⠀⣀⠠⠤⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⠄⣀⠈⠁⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⡠⠊⠀⠀⢀⣀⡀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⣀⡀⠀⠀⠑⢄⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢰⠁⠀⢀⠞⣁⣀⡈⢢⠀⠀⢡⠀⠀⠀⠀⠀⠀⡌⠀⠀⡔⢁⣀⣈⠰⡀⠀⠈⡆⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠂⠀⠀⡎⣼⣿⣿⣿⡆⡇⠀⢈⠀⠀⠀⠀⠀⠀⡃⠀⢸⢰⣿⣿⣿⣧⢱⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠘⡀⠀⠇⣿⣿⣿⣿⢧⠁⠀⡜⠀⠀⠀⠀⠀⠀⢣⠀⠘⡼⣿⣿⣿⣿⢸⠀⢀⠃⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠖⠋⠉⠉⠛⠷⣅⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣸⠿⠛⠉⠉⠙⠲⢧⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡴⠞⢍⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡩⠳⢦⡀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⠋⠁⠀⡞⠀⠑⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠊⠁⢳⠀⠈⡙⡆⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉⠸⠀⢸⠁⠀⠀⡜⠀⠉⢲⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡖⠉⠀⢳⠀⠀⠈⡇⠀⠃⠉
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣄⣾⠀⠀⢸⠃⠀⠀⡼⠀⠀⠉⠀⠀⢰⡆⠈⠈⠁⠀⠀⢧⠀⠀⠘⡇⠀⠀⣷⣰⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢹⡿⣄⠀⣼⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⣷⠀⣠⠿⡏⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢧⠈⠓⢿⡀⠀⠀⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⠀⠀⢀⣿⠚⠁⡜⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠣⡀⡾⠉⠉⠓⠚⢦⡤⠤⣤⣀⣸⣇⣀⡤⠤⠤⡴⠓⠚⠋⠉⠹⣄⠜⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠘⠣⡀⠀⠀⠀⡼⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⡄⠀⠀⢀⡤⠉⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠈⠢⢄⣰⠇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣧⡠⠔⠁⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠤⠤⠸⠇⠤⠤⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠀⠐⠒⠒⠂⠀⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⠀⠀⠸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⣸⠀⣴⢀⠹⡆⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⠺⣥⠞⣰⠇⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⢀⠲⣾⠃⢀⣠⣴⣾⣿⣿⡿⣻⣿⣿⣿⣿⣿⢯⢍⡉⠉⠛⣹⣿⠿⠟⠛⣻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢽⢸⣤⣴⣿⣿⣿⣿⣿⢿⡏⣰⢿⣿⠛⣿⠟⢁⣴⣾⣷⣨⡇⢉⡵⠞⣻⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⢿⣽⣿⣏⣨⣉⡭⠠⡟⣤⡀⠈⡁⢸⡏⣼⣿⣿⣿⣿⣿⢧⠞⠁⠈⢩⠾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⠃⣾⣿⣼⣿⣆⡱⣾⢁⣿⣿⣿⣿⣿⢏⠃⢀⠆⣠⣴⠾⠛⠻⠆⣈⣹⣿⣿⣧
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣧⣿⣼⣿⢿⠟⠋⠁⣠⣼⡷⢛⣉⣤⣤⣶⢀⠀⠀⢹⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠈⠁⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⡿⠏⠂⣀⣰⣶⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⡛⢿⣆⠻⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣄⣒⣒⣉⣩⡭⣽⣿⣿⣿⣿⣿⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⢿⣿⣿⡟⠙⠛⠶⣭⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣟⣁⠀⠙⠛⢿⣿⣿⣇
⠉⠉⠉⢉⠍⠉⢭⣭⣉⡽⠛⢻⡍⢩⡌⠁⣭⠃⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣟⠛⠛⠭⣤⣤⣬⣿⣿⣿⣿⣿
⠀⢤⡤⠞⠀⣠⣄⢀⡡⢤⢴⣦⣝⠋⣀⣴⣫⣤⡼⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣶⣶⣆⣤⣙⣿⣿⣿⣿
⠒⠢⠒⠺⠋⠁⠉⠉⠀⠀⠈⠙⠛⠻⠿⠚⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⢬⣟⡻⠿⠟⢛⣉⣴⠟⢉⣀⡉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⣶⣦⣄⡀⠀⠀⠀⠀⢀⣀⣠⣶⣿⠟⠉⠁⠀⠙⠄⠀⠀⠙⢻⣿⣿⣿⡟⠁⡴⣿⣿⣿⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⣽⣿⢿⣟⡀⠁⠀⠀⡙⣻⣿⣿⣷⣦⡤⠂⠀⠀⠀⠀⠀⠈⠉⢽⣿⡇⠈⠷⣌⠻⠙⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠛⠉⣣⣿⠀⠀⠘⠁⠉⣩⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢠⣷⡿⢀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢨⡟⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢲⣿⣿⡆⠁⠰⠟⣰⣿⣿⣿
⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠀⠀⠀⡴⠒⠒⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⠛⠁⢸⣿⣿⣄⣠⣴⣾⣿⣿⣿⣿
⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⣿⠀⠀⠀⠀⣀⡀⠜⠓⠶⠶⠟⠣⣄⠀⠈⠀⠀⠀⠀⣴⣿⣿⡿⣻⣿⣿⡿⢻⣿⡿
⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⣿⣦⣤⢤⣾⣡⣤⣴⣦⣶⠶⠛⠀⢹⠀⠀⠀⠀⠀⠀⣩⣿⡿⠁⢹⡏⠙⢷⣿⠛⠁
⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣸⣷⡟⠉⠿⡥⠴⠿⠿⠏⠙⠋⢁⣤⠀⠀⠈⠀⠀⠀⠀⠀⠀⢉⠁⢠⠃⢸⡇⠀⠀⠉⠓⢦
⠿⠟⠛⠛⠛⠛⠛⠒⠒⠀⠉⠉⠉⠉⠉⣉⣉⣀⣀⣀⣀⣀⡤⠤⠼⣷⣄⠐⣤⡶⠶⠶⠶⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣰⠋⠀⣾⠁⠀⠀⠀⠀⠈
⣶⣶⣶⣶⣶⣶⣶⣎⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⠀⠀⠀⠀⢀⣀⠤⣄⠀⣀⣀⢠⡆⠀⣠⣶⡟⣡⠞⠁⢀⣾⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⢋⣿⡖⠒⠋⠉⠀⠀⠀⢇⠀⠀⠩⢉⣤⣾⣿⣿⠞⠁⠀⣠⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢀⣀⣀⡐⢲⣶⣶⣿⣿⡿⠋⠀⠀⢀⣴⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⡿⣿⢻⣄⠂⣀⡀⠘⣛⣋⡹⠿⣿⣿⣿⠿⠋⠀⠀⠀⢠⣿⡿⣹⡏⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣾⡿⢿⣿⣿⣿⣿⣿⣿⡇⠙⣳⡛⠛⠉⠉⠉⠀⠀⠀⠀⠀⡕⠀⠱⡀⠀⠀⠀⠈⠋⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣰⡟⣿⡇⠀⢿⣿⣿⣿⣿⣿⠀⢸⡏⠀⠀⠀⣠⡤⠔⠒⠒⠒⠛⢄⠀⠀⣼⣄⠀⠀⠀⢠⡿⠠⣀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⠁⠀⠘⢿⣿⣿⣿⡏⢠⠈⠁⢀⣤⠀⠀⠀⠀⠀⠀⢀⣠⣎⣠⣾⣿⣿⣷⣤⣀⣿⠁⠀⠈⠙⠶⣤⣤⠤⠔⠂⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡀⠀⠀⠈⠛⠿⢿⡇⠘⠀⢀⠸⠂⠀⠀⠀⠠⠶⠛⠋⠁⣨⠛⢿⡉⠁⠀⠀⣽⠇⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋⠙⠃⠀⣿⡇⠀⠀⠀⠀⠀⡼⠁⠀⢸⡇⠀⠀⠰⠀⠀⠀⣀⣤⣴⣾⣽⡇⠀⢷⡀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⠀⠀⠀⠀⢸⣧⠀⠀⣠⣴⣿⡧⢇⠀⠸⠀⠀⠀⠀⣠⣤⣴⣿⣿⣿⣿⣿⡇⠀⠘⡇⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⢶⣧⠱⡄⠀⠀⠻⢀⢀⡿⠻⣏⡔⠈⢆⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⡟⠁⠸⠃⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⣿⣿⣿⣿⣿⠟⠛⠋⠛⠀⠀⠈⠳⣽⣦⣀⠀⠁⢸⡇⠀⠹⣎⠄⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⡭⠋⠀⠀⠀⢧⠀⠀⠹⣆⠀⠀⠰⢒⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⡿⠛⠁⠀⠀⣀⣤⣶⡶⢶⣦⣄⠴⠋⠀⠀⠀⠀⠀⠈⢳⡄⠀⠈⠣⣠⡼⣿⣿⣿⣿⣧⠛⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀
⣶⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢹⣿⣿⣿⢇⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⢿⣿⣿⣾⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠈⢿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣿⣿⡆⢿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠈⣿⣿⣿⣿⢱⣿⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢹⣿⡇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⢠⣿⣿⣿⣿⢸⣿⣿⠛⡿⠀⠀⢸⠉⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠻⣇⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣸⣿⠋⠀⠃⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⠀⣀⣀⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⡿⢣⡴⠿⠛⣿⣿⣶⣶⣄⡙⣿⣿⣿⣿⡟⠇⠀⢀⡴⣾⣿⡟⣿⣿⣏⠉⠻⣮⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⡇⣟⠀⠀⢰⣿⣿⠼⣿⡏⠁⠈⠻⣿⣿⡇⠀⠀⠈⠀⢿⣿⣋⣻⣿⡏⠀⠀⠽⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠘⣿⣿⣿⣿⣧⠀⠀⠀⠀⠿⠓⠚⠛⠃⠀⠀⠀⠈⠻⣷⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠉⠉⠉
⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⣰⡆
⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⡇⠀⣴⣿⣿

⠀⠀⠀⠀⠀⢀⣠⡤⣴⣶⣦⣤⣀⣀⣠⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠈⠑⢾⣿⣿⣷⣾⣔⣤⣈⣀⣉⣿⣯⣯⣤⣬⣍⣉⠙⠒⢢⡤⠂⠀⠀⠘⢶⣿⣏⣋⠁⢸⠉⠉⠛⣲⣶⠃⠀⠀⠀⣤⡐⠀⠠⠀⠀⢀⡀
⠀⠀⠀⠀⠹⣿⣩⡟⠛⠛⠛⠿⠻⠛⠛⠛⠛⠋⠋⢛⣟⡽⠋⠀⠀⠀⠀⠀⠀⠙⣯⣿⡲⡾⠶⣶⣿⠿⠃⠀⠀⠀⠀⠈⠛⠓⠲⠾⠛⠋
⠀⠀⠀⠀⠀⠈⠻⣷⣴⣦⣤⣤⣀⣴⣤⣾⣶⣿⣲⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡤⠤⡤⠤⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⡶⠶⠋⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⣀⣀⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠔⠋
⠀⠀⠰⣶⣶⣾⡿⢿⣿⣿⣥⣬⣽⣭⣭⣭⣬⣴⡾⠻⣻⡿⠀⠀⠀⠀⠀⠀⠀⣎
⠀⠀⠀⢿⣿⣏⡀⡰⠁⢀⠟⠉⢹⡏⠉⠙⡟⠀⢣⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠂⠀⠂
⠀⠀⠀⠘⣿⣿⣿⣿⣶⡞⠀⠀⠘⠁⠀⠀⣳⣶⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣥⡀
⠀⠀⠀⠀⠘⣿⢿⣿⣾⣿⣿⠷⣿⡶⡶⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⣏⣳⣢⣤⣀⡄
⠀⠀⠀⠀⠀⠈⠻⣍⠻⢿⢿⣷⣿⣾⣿⡿⣯⣱⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣤⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣦⣧⣕⣯⣷⣻⠟⠁⠀⠀⠀⠀⢀⢠⠶⠶⠂⠀⠀⠀⠀⠙⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠭⣭⣭⣭⠬⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⣀⣀⣴⡤⠖⠛⠉⠉⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⢞⣫⣥⣽⣿⣿⣴⣿⣶⣶⣾⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⢠⣀⣀⣴⣿⡷⣿⠿⠿⠿⠛⢿⠛⠛⠛⠻⡟⠛⢿⣿⣷⠶⠃⠀⠀⠀⢀⣀⣠⣄⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠈⠛⢿⣿⣷⣤⣼⣄⣀⣀⣀⣸⣀⣀⣠⣤⢷⣶⣛⣿⠃⠀⠀⠀⠀⠀⠘⣏⡛⡖⠲⠖⠞⢛⣙⣿⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣾⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣤⡤⣤⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡙⣿⢃⠟⣿⣿⠟⣸⣫⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⣛⣛⡫⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠚⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠐⠒⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡀⢤⣄⠀⠀⠀⠀⣁⢀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣷⢰⣿⠀⠀⠀⠀⣿⠸⣒⣿⠇⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢠⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠙⢦⣄⡀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠠⡶⠚⢋⠉⠐⠢⠀⠀⠙⠳⢤⣤⣀⣀⣀⠠⠐⠈⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡝⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⢀⡟⠙⠓⠶⠦⢤⣤⣤⣤⣤⣤⣤⣴⠶⠖⠚⠋⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⣀⣀⡀⣀⣸⡁⣀⣀⢀⣀⢀⣿⣤⣤⠀⣤⣤⣤⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣼⣄⣀⣀⣀⣠⣼⡤⠶⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⠛⠛⠛⠻⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⡾⠛⠉⠉⠀⣈⠀⠙⠶⠶⢄⣀⡀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⢀⠃⠀⠀⡠⠜⠃⠤⠘⠓⠀⠀⠈⠙⠛⠳⣤⣀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠉⠀⠰⠊⡀⠆⠁⠀⣀⠀⣄⡀⠀⠀⠚⠓⠀⠛⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⢀⠈⠀⠜⠁⠀⢀⣠⣴⠶⢿⡛⠠⢿⣿⠿⠿⠿⠶⠶⣦⣼⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡈⢀⣤⡴⠟⠂⡟⢋⡁⠒⢀⠉⠒⣄⡀⠀⠒⠲⠶⣶⣦⣤⣬⣽⣿⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠛⠛⡄⠁⣰⣿⠞⠃⠠⣀⠤⠘⠁⠠⠄⠘⠋⠀⠀⣀⠀⠀⣀⣀⠼⠛⣽⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⡧⠀⡧⠀⢢⡙⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠚⠁⠻⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢸⠀⠀⢘⣥⠘⡈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢦⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⣳⡄
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢱⡏⢹⡄
⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⣼⢿⡄
⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⢹⡆⣿⡹⣦⡀
⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⣿⡘⣷⠸⣷⣸⣷⡀
⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣄⠀⠀⠀⠀⠀⠘⣧⢹⣧⠸⣷⡹⣷⡿⣷⡄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠰⠾⠃⠠⠄⠀⠀⠀⠀⠀⠙⣶⢻⣧⠙⣷⡘⣿⣋⣿⡆
⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡈⣷⠀⠉⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠘⢷⣿⣷⡘⣷⡍⣿⢃⣿
⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣤⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠻⣿⣿⣸⣧⠿⠛⠛⠳⢦⡄
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣶⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⡟⠁⠀⠀⠀⠀⠀⠹
⣤⣄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣦⡼⠧⠛⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢻⡇
⢉⣿⣦⠋⠈⣷⣶⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠇⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠈⠿⠿⣷⠛⠟⠻⣷⣿⣷⡤⡀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣄⡄⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡠⠀⠤⠀⠉⠁⠀⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠈⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋
⠀⠀⠀⠀⠳⠶⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠃
⠀⠀⠀⠀⠈⠑⠂⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠾⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⣀⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⡇⠀⠀⠀⠀⠀⠀⡟⠰⣀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁
⠀⢱⡄⠀⠀⠀⠀⠀⠁⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣦⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠶⠆⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠉⠉⠈⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠀⠀⠰⠀⠀⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠘⠣⣤⣤⠀⠀⠿⣤⣤⣤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⣡⣴⣿⣧⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣶⣮⣻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⡔⠂⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⢠⣤⠶⢶⡜⠷⣧⡀⠶⠎⠛⠻⢋⣹⠷⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⣦⡘⢷⣤⣔⠠⠉⠙⠿⠐⢿⡄⠚⣿⢶⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠿⠀⠸⠷⢶⠚⢠⡿⠖⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣿⠳⠄⠻⣶⠇⠘⣷⢻⣧⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣽⢷⣤⣀⣿⡾⣷⢠⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣼⡿⣿⢛⣄⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⢳⣿⣾⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⡽⠋