Jelly Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠤⠶⣿⣍⣉⡛⠲⢤⡴⠚⠋⠉⠛⢲⣤⠖⠚⠛⢛⣲⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⣀⣤⡤⠤⢤⣬⣉⠓⢦⣙⢦⠀⠀⢠⠞⠁⢀⡴⢚⣉⣡⣤⣤⣄⣈⣉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⣠⠶⠋⠁⠀⢀⣠⠶⠚⠛⠛⠛⠛⢾⡶⠛⠉⠛⠲⣿⡛⠉⠉⠙⠓⢦⣄⠈⠉⠙⠲⢤⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠞⠁⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⢦⡈⢳⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⡞⠁⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠈⠳⡄⠙⣦
⠀⠀⠀⠀⠀⡼⠁⠀⣠⠏⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠹⣆⠈⢷⡀
⠀⠀⠀⠀⡼⠁⠀⣰⠃⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠘⣆⠀⢳⡄
⠀⠀⠀⣸⠃⠀⡼⠃⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠘⢧⠀⠹⡄
⠀⠀⣰⠇⠀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠈⢧⠀⠹⡄
⠀⢠⠏⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠈⢧⠀⠹⡄
⢀⡟⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠘⣧⠀⢻
⢸⡁⣰⠃⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠘⣇⢨
⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠸⣟
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠹
⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⢷⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣀
⠀⠉⠙⠛⠒⠒⠒⠚⠛⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠙⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠧⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠈⠛⠶⢤⣄⣀⣀⣀⣀⣤⡤⠶⠛⠁⠀⠀⠈⠉⠉⠛⠛⠛⠛⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣖⣀⠀⠐⣾⡀⠀⣩⠆⢀⣀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⠤⠖⠤⣬⠓⠚⠉⠉⠉⠉⢓⢻⡤⠶⠶⣌⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡼⠁⠀⣠⠖⠉⠉⢱⠖⠀⠐⣎⠉⠉⠑⢄⠀⠀⠳⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡼⠀⠀⣰⠃⠀⠀⠀⠏⠀⠀⠀⠘⡆⠀⠀⠈⢧⠀⠀⠱⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢰⠁⠀⢠⡏⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡆⠀⠀⠂⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡇⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣸⠁⠀⠀⠇⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡆⠀⠀⢱⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿⠀⠀⢸⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠘⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠇⠀⠀⣼⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢀⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⢿⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣸⠆⠀⠊
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠉⠢⢀⣀⢀⣀⣀⠤⠚⠀⠀⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠀⠠⠤⠄⠀⠠⠤⠄⠀⠀⠒⠈⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠢⡠⠊⠉⠁⠠⡠⠂⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⢙⡄⠀⠠⡀⠀⡤⠐⠠⠀⠀⠠⠀⠀⠜⠈⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠐⠇⠀⠀⠀⠀⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠠⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠇⠐⠀⠀⠀⡎⠀⢨⠀⠀⠀⠀⡄⠀⠀⠀⠀⢂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠘⠀⠐⠀⠀⠀⡗⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠠⢠⠌⠈⠀⠱⡀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣰⠁
⠀⠀⠀⠐⠀⢸⠂⢀⠂⠎⠉⣐⢲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠉⠃⠐⠏⠀⠀⠀⣀⡤⠊
⠀⠀⠀⠐⣄⠀⠙⠒⠺⠧⠀⡄⢬⡈⠙⠒⠲⠤⣀⣀⡀⢀⣀⣄⠀⠀⢀⠀⠀⠀⠀⣀⣤⠀⣒⠻⠃⠐⠀⠀⢀⣦⣀⠀⠀⠈⠀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠙⠢⢤⣀⠀⠀⠀⠀⠈⠒⠂⠴⠤⢄⣉⡉⠉⣉⣉⣙⣑⣒⣒⠒⠚⣙⠩⠀⢁⣀⣠⠁⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠤⠤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠉⠛⠛⠁⠊⠉⠁⠀⠀⠀⢀⣀⡀⠤⠤⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠉⠉⠉⠁
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠶⢶⣶⡟⠛⠛⠷⢶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⠷⠦⢤⡀⢀⣨⣷⣄⡀⠀⠸⢉⣠⡤⠴⠾⠿⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣡⡴⠶⠚⠲⣶⣃⠀⠀⠀⠀⠀⠀⠀⣀⣹⣶⠚⠙⠛⠶⢌⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠞⠁⠀⠀⢀⡴⠛⠉⠉⢳⡶⠛⠁⠰⡟⠁⠈⠙⢧⡀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢠⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢀⣠⣶⢿⡿⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢶⣄⡀
⠀⠀⢀⣴⠟⠋⢀⣼⠇⣼⠁⠀⠀⢀⡀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⡆⠀⠀⠘⡆⢹⣆⠈⠙⠶⣄
⠀⣰⠟⠁⠀⢀⡾⣿⠀⡿⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢷⢸⣿⢦⠀⠀⠈⠳⡄
⣰⠋⠀⠀⠀⡾⠁⢹⣾⣇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣾⠏⠈⢧⠀⠀⠀⠘
⡏⠀⠀⠀⠀⡇⠀⠀⠹⣿⡀⠀⠀⣇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⠀⠀⠀⡼⠋⠀⠀⣸
⣿⡀⠀⠀⠀⠻⣄⠀⠀⠈⠻⠶⢶⣿⡀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⣿⣦⠤⠊⠀⠀⠀⣠⠏⠀⠀⠀⢀
⠘⣷⣄⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠉⠛⠶⠶⠶⠶⠿⢿⣦⣄⣀⣀⣀⣀⣀⣤⡴⠿⠶⠶⠶⠚⠛⠉⠀⠀⠀⢀⣠⠞⠃⠀⠀⠀⣠⡾
⠀⠈⠻⣷⣤⣀⠀⠀⠈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠋⠁⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠈⠛⢿⣿⣶⣤⣀⡀⠀⠉⠙⠛⠷⠶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠖⠛⠉⠁⠀⠀⢀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣷⣶⣦⣤⣀⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣠⣤⣴⣶⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⢷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠿⠟⠛⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠚⠛⠛⠛⠛⠻⠶⠶⣦⣤⣤⣄⣀⣀⣀
⠀⠀⠀⣠⡶⠛⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠶⠶⠶⣤⣤⣄⣀
⠀⠀⡾⠁⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⢸⡇⠀⠀⠀⠈⠓⠲⠶⠶⠦⠤⢤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⢹⡄
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠒⠒⠲⠶⠶⠶⠒⠒⠛⠋⠀⠀⠀⠀⠀⣼
⢀⣼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⢸⡇⠀⠙⠒⠒⠒⠒⠶⠶⠶⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⢶⣏
⠀⣻⡦⠤⠤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣿
⢠⡏⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⢹⠆⠀⡏⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⣤⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⡟⠀⠀⢧⣀⣀⡤⠞⣻⠁⠀⠀⠀⠀⠀⠀⢹⢦⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢸⡇⠀⠀⠀⠈⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⣴⠛⣦⠀⢀⣤⣀⠀⠀⢀⡶⣦⡀⠀⠀⠀⢿⡀
⣿⠀⠀⠀⠀⠀⠀⣴⠶⠶⠞⠛⠛⣻⠛⠛⠛⣿⡟⢻⡇⠀⢸⡟⢻⡏⠐⣿⡁⠙⠳⡶⠋⣠⡾⠁⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠛⠛⢻⡄⣿⠛⣿⢰⠋⠛⠉⣧⠸⡇⠀⢸⡇⠘⣇⠀⠈⠻⣦⠀⠀⣴⣿⢦⠀⠀⠀⠀⠘⣇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡀⢻⠘⠖⣲⠀⢻⠀⣷⠀⢸⡇⠀⣿⠀⠀⠀⣿⠀⠀⡇⠳⠼⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⢴⠶⣆⣀⣼⠃⢸⡇⢸⡄⣯⣉⣀⣸⡄⠻⠶⠶⣧⠀⠛⠛⠛⣧⣿⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢻
⢻⠀⠀⠀⠀⠘⢧⣌⣉⣁⣤⠟⠀⠸⣧⣬⣭⣤⣼⣧⣤⣤⡴⠾⣤⣤⣴⠶⠾⢹⣦⣤⡿⢹⡀⠀⠀⠀⠀⠀⢸
⢸⡆⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠙⠃⠀⠀⠉⠑⢆⠀⠀⠀⢸
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢸
⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⣀⡀⠀⠀⠀⠀⢠⡇⠀⠀⢸
⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡇⠀⠀⠀⢀⡼⠀⠀⠀⢸
⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠉⠀⠀⠀⢠⡿
⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷
⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠟⠁
⠀⠀⠀⠀⠉⠙⠛⠶⠶⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣴⠶⠶⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⣤⣴⣶⡾⠿⠟⠛⢛⣛⣿⣷⣶⣦⣤⣀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠁⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠁⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⢾⡷⠀⠀⢀⣿⣿⣿⡏⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣇⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⣶⣤⣤⣬⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠈⢠⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠈⠿⠃⢹⣿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⠀⢸⢀⣤⣤⣄⠀⠀⠀⠀⡏⠀⠀⣀⣤⣤⣀⠀⠀⢸⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⣿⡀⣹⣿⣷⠀⠀⠀⡇⠀⢸⣏⣀⣽⣿⡆⠀⠸⢸⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡤⠀⠀⢀⣾⣷⠀⡟⢿⣿⣏⣹⠏⢀⠀⠀⠛⢀⠘⢿⣿⣏⡿⠃⠀⠀⡜⣷⠀⠈⠋
⠀⠀⠀⠀⠀⠀⣠⡶⠚⡩⠯⠃⢀⡇⠀⠈⠉⠀⠀⠘⠿⠶⠾⠋⠀⠀⠈⠁⢀⠀⢧⠀⠹⣿⣦⠀⠀⢠⡀
⠀⠀⠀⠀⠀⢰⣏⢀⠀⠀⢀⡼⡸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⡌⠳⣄⠈⠫⣻⣦
⠀⠀⠀⠀⠀⠘⣿⢰⡀⣴⠋⠀⠀⠀⠀⠐⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢦⡀⠀⠀⡌⢻⡇
⠀⠀⠀⠀⠀⠀⠈⠳⢭⣭⡶⢤⣶⠁⠀⡀⠀⠀⣠⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⢀⡴⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⠬⠷⠞⠙⢷⣄⣀⣴⡿⠶⣤⣤⡾⠿⢶⣿⣿⣯⣭⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⡴⣾⠛⠶⠂⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡴⢾⠻⠦⠁⠾⣷⢂⡈⡙⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡀⢡⣼⣧⣦⡀⣭⣘⡃⣹⠟⢳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢣⠞⡻⢈⡁⠀⣙⡒⠛⡓⢻⡏⢉⡿⠲⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢲⡤⠿⡌⠈⣟⢷⣷⠙⣏⡉⢉⡹⣞⠋⢙⠁⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡎⠑⣤⣀⢬⣙⣛⣁⣏⣤⠊⢙⣮⣵⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⣿⠛⠀⠙⢩⡿⡆⢻⡏⢡⣾⣀⠉⠠⡤⠀⠈⢷⡠⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢖⣿⣿⠀⣾⣧⡦⠲⣶⢦⠞⠠⢞⠘⠋⠓⠀⠂⠠⣄⡀⠳⠂⠹⠆⡁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⢏⣀⣤⠈⡂⠿⠃⢻⠎⣶⣇⠈⣑⠢⠄⡈⠐⠤⡈⠑⢢⡈⠠⠀⠙⠦⡐⠤⠄⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⢷⡹⣿⣻⠁⣿⠷⠈⠳⢭⡀⠠⠼⠷⠲⣄⡉⠐⠿⡤⢄⠐⠀⠉⣆⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣯⢁⡸⠟⠀⠀⠠⠀⡈⠀⠂⠀⣀⠈⠑⠢⠀⠐⢄⠑⠆⠀⠄⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠁⢄⠀⠀⠀⠀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠈⠆⠁⠀⢠⡄⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠄⠀⣤⠀⠃⡀⢠⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⡀⠀⠂⠃⠀⠀⠀⠀⡁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⠀⠠⠀⡇⠀⠘⠁⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢰⠀⠀⠈⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠀⠀⠐⠀⠘⠀⠀⢘⡇⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣆⠀⠀⠀⠀⢀⡿⠈⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡄⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡰⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁





⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣿⣹⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠚⠛⠛⠛⠛⠛⠷⠶⠶⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠒⠶⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⠉⠛⠒⠒⠒⠲⠶⠤⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣐⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠒⠲⠶⠶⠦⠤⠤⠤⢤⡤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠴⠶⠒⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠙⠛⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠘⡇⠀⠀⢀⡴⡿⠀⠀⠀⠀⠀⠀⢲⣄⠀⠈⣿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠙⠒⠚⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⢦⡀⠀⠀⡀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢀⣤⣤⣬⠷⠶⢶⡶⠶⠶⢶⡷⢶⡆⠀⢰⡷⢶⡇⢰⡟⠙⢷⣤⡴⠋⣸⠇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⠷⠶⣦⢰⡖⢺⣇⡶⠲⠚⣧⠸⡇⠀⢸⡇⢸⡇⠀⠛⢦⡀⠀⢠⣾⣥⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡇⢸⡏⠧⢶⡀⢸⠀⣷⠀⢸⡇⠈⡇⠀⠀⢸⡇⠀⢸⠻⣬⠃⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⢶⡄⢀⡿⠀⣿⠈⡇⢺⡉⣁⣸⡆⠻⠶⠾⡇⠀⠿⠖⢻⣼⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠹⣦⣉⣉⣠⡼⠃⠀⣷⣬⣭⣥⣼⣧⣤⣤⡤⢿⣤⣤⡤⠶⠏⣷⣤⣼⠟⣆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠛⠀⠀⠉⠙⢦⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⠀⠀⠀⠀⢀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣀⡜⠁⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠁⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠲⠶⠶⠶⠶⠦⠤⠤⠤⠤⠶⠶⠶⠶⠶⠾⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠴⠾⢯⣍⣛⠓⠦⣴⠚⠋⠉⠛⢲⡤⠖⠛⠛⣛⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣠⠤⠤⠤⠤⣌⣉⠲⣌⢷⡀⠀⣰⡏⠀⣠⢖⣋⣡⣤⠤⠤⣤⣈⡉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⡴⠚⠉⠀⠀⣠⠶⠚⠉⠉⠉⠙⢻⠟⠋⠉⠉⠛⢿⡉⠉⠉⠉⠓⢦⡀⠀⠉⠛⠦⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⡴⠋⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠉⢳⡀⠀⠀⠈⢷⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣠⠞⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠹⣄⠘⢦
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣰⠋⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠘⣦⠈⢳⡀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⡼⠁⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠈⢧⠀⢳⡄
⠀⠀⠀⠀⠀⢰⠃⠀⡼⠁⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⢧⠀⠹⡄
⠀⠀⠀⠀⢠⠏⠀⡼⠁⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠈⢧⠀⠹⡄
⠀⠀⠀⢀⡟⠀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠈⢧⠀⢻⡀
⠀⠀⠀⣾⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⢧⠀⣷
⠀⠀⠀⠙⣶⠃⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠘⣶⠏
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠈⠛⠒⠲⠶⠶⠒⠒⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠒⠒⠒⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⡼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢤⣀⣀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠙⠲⢤⣤⣀⣀⣀⣀⣤⠤⠖⠋⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠉






⣿⣿⣶⣶⣷⡿⣾⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣾⡿⣶⠿⠶⠶⠾⣶⡾⠷⢶⠶⠶⠿⠷⡶⠷⠶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢤⣄⡀⢀⣠⠤⠤⣄⡀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⣉⣉⣛⠒⠦⣍⡻⡄⠀⠀⢠⠟⠉⢀⡤⠖⠛⠛⠛⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣀⠴⠚⠉⠉⢉⡭⠽⠗⠲⠿⣾⡤⠖⠛⠦⣼⠷⠒⠛⠻⢭⣉⠉⠓⠲⢤⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣠⠞⠁⠀⠀⢀⠞⠁⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠈⠳⣄⠀⠀⠉⠳⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢀⡼⠁⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠙⣆⠙⢦
⠀⠀⠀⠀⠀⠀⢠⡟⠀⢠⠟⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠈⢧⠈⢳⡀
⠀⠀⠀⠀⠀⢀⡞⠀⣰⠋⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⢣⡀⠳⡄
⠀⠀⠀⠀⠀⡾⠀⣰⠃⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢳⡀⠹⡄
⠀⠀⠀⠀⡼⠁⣰⠃⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢳⡀⢹⡄
⠀⠀⠀⣸⠃⣰⠃⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢳⡀⢻⡀
⠀⠀⠀⡇⢠⠏⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢷⠀⡇
⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠘⣆
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠈⠳⢤⣀⣀⣀⣀⣠⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢤⣤⣤⣤⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠶⠶⠶⠒⠚⠋⠁⠈⠳⢤⣄⣀⡀⠀⢀⣀⣀⡤⠖⠋⠈⠉⠓⠒⠶⠶⠶⠶⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠒⠀⠘⠛⠀⠀⠙⠛⠀⠐⠒⠢⣄⠀⢠⣄⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠃⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠁⠁⠀⠀⠒⠒⠒⠒⠒⠂⠀⠁⠈⣹⠀⠸⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠈⠀⠀⣐⡒⠒⠒⠒⠒⢂⣀⠀⠀⠓⢤⠔⠁⠉⠁⣠⠶⠂⠀⠀⠀⠙⠲⢤⠉⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢘⠂⠀⠀⢀⡐⠀⠀⠈⠃⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠑⡄⠀⠰⠈⠀⢅⠀⠀⠀⠆⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⡘⠀⠀⠀⢀⠀⠀⣶⢨⡇⠀⢰⡦⠀⠀⠀⢸⠁⠀⠀⠀⠀⠿⠀⠀⣸⠺⠇⠀⢀⠇⠀⠉⠂⠀⠸⠀⢀⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠠⠐⢲⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⢰⡀⠀⢀⡀⠀⠀⢀⡇⠨
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⠟⠁⠀⠁
⠀⠀⠀⠀⠀⢀⣼⠟⠋⢹⠀⠀⠀⢰⣷⣶⠂⣶⢰⣶⣶⣲⢺⡃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢠⣾⠃⠀⠀⢸⠀⠀⠀⢈⣤⣛⣘⣉⣛⣛⣋⣛⣬⠁⠀⠀⢸⠀⠀⠀⠀⣀⣿⣟⢲⠰⡆⢷⡏⠀⠀⠀⠀⢸
⠠⠀⠀⠀⠈⠁⡀⠀⠀⢸⠂⠀⠀⢸⠟⢿⡏⢽⠈⡇⢹⡾⡟⠆⠀⠀⢸⠃⠀⠀⠀⠙⠙⠛⠙⠛⠛⠉⠁⠀⠀⠀⠀⢸
⠇⣠⢾⠀⠀⠐⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⢒⣚⣛⣛⣒⣲⣶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣤⣶⣿⣿⡿⠿⢿⣿⣿⠻⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠘⢿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣿⣀⡄⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠻⢯⣭⣿⢸⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⣛⣯⣭⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣉⠉⠛⣛⡛⠻⠿⠿⠿⠿⠿⠟⣛⣛⣛⣫⣭⣭⣶⣶⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢈⠉⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠀⠀⠀⣾⣤⣿⣿⡆⠀⠀⠀⠀⠀⠀⣾⣤⣿⣿⡇⠀⠀⢀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡇⢸⠀⠀⠀⠙⠿⡿⠿⠁⠀⠀⠀⠀⠀⠀⠙⠿⣿⠿⠃⡇⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⣼⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡼⠃⠀⠀⠀⠀⠀⠀⡇⠀⠸⡄⢿⣀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⣄⣁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣁⠀⢀⣠⣖⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢰⠀⠀⠀⢠⡿⣄⡀⠀⣨⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠳⣤⣀⣀⣠⡴⠚⠹⡏⠀⠀⠸⡆⠀⣀⡾⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⠼⠋⠋⠉

⣠⣤⣤⡶⠶⠶⠶⠶⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠷⠶⠶⠶⠶⣶⣤⣤
⣿⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠈
⣿⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁
⣿
⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠉⣿⡟⠛⠛⠛⠛⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠛⠛⠛⠛⠛⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⡇⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿
⠀⣿⡇⠀⣿⡛⠛⠛⠻⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠟⠛⠛⠛⣿⠀⠀⣿
⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣰⡏⣷⡴⠞⠋⣉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⠙⠁⠈⠁⣴⢟⣭⣿⣷⣄⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⢀⣀⠘⣮⣷⣀⣴⠾⢻⣟⣁⠀⠈⢻⡏⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠘⣯⠙⣾⠿⠛⠻⣶⡟⠋⠙⠻⣦⣼⣇⠀⣸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⣿⡇⠀⣿⠀⠀⢀⡴⠞⠛⠛⠱⣿⢀⣶⠟⠛⠻⣦⡀⢀⣿⠁⠙⣿⠋⠈⣿⣤⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⣿⡇⠀⣿⠀⠀⠙⠶⣄⣀⣀⣴⠿⢿⣧⡶⢶⣤⣸⡷⠟⢻⣆⣴⠿⣦⣴⡟⠀⠙⣧⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⣩⠿⣿⠀⣿⠁⠀⠀⠘⣿⣀⠀⣸⡟⠁⠀⠘⣿⣀⣀⣰⣿⡄⠀⠀⠀⠀⣿⠀⠀⣿
⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠛⠛⢹⣷⣿⣦⣄⣠⣴⠏⠙⣿⠻⣧⣤⣤⣾⠟⢿⡏⢉⣼⠇⠀⠀⠀⠀⣿⠀⠀⣿
⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⣀⣻⣏⡀⠀⣠⣿⣀⣤⣿⣁⠀⣀⣾⠟⠛⠁⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⣿⡇⠀⠻⣧⣄⣀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠛⠛⠋⠉⠉⠁⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⣀⣠⣴⠟⠀⠀⣿
⠀⣿⡇⠀⠀⠀⠉⠙⠛⠳⠶⠶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠶⠾⠛⠋⠉⠀⠀⠀⠀⣿
⠀⠸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟
⠀⠀⠈⠙⠻⠶⢶⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣴⡶⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⡶⠶⠿⠿⠿⠿⠿⠿⢷⣶⣶⣦⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠿⠿⠷⠶⣶⣶⣦⣤⣤⣄⣀⣀⡀
⠀⠀⠀⢀⣴⡾⠋⠁⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠻⠷⠶⣶⣤⣄⡀
⠀⠀⢀⣾⠋⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣆
⠀⠀⣾⡇⠀⠀⠀⠀⠈⠻⠶⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠷⠶⠶⠶⣶⣦⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣬⡿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⣿⠃⠈⢷⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⢿⡄⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠲⠶⠶⠦⠤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⡤⠶⠶⠛⠻⣷⡄
⠀⠈⣿⡶⠤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⡇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⣼⠏⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏
⠀⢻⣦⡀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⣿⠁⠀⢸⠁⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢶⣦⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⢠⠃⠀⠀⠘⣦⠀⠀⢀⣠⠶⠋⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣄⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⡏⠉⢷⠀⠀⢀⣤⣤⡀⠀⠀⠀⢀⣾⠿⣦⡀⠀⠀⠀⠀⠈⣿⡆
⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣶⡿⠾⠿⢿⡿⠿⠿⠿⠿⣿⣿⠞⣿⡇⠀⠀⢰⡿⠿⢿⣿⠀⢰⣿⠉⠉⠻⢷⣤⣴⠟⠁⣠⣿⠃⠀⠀⠀⠀⠀⢿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣭⣤⣤⠀⢠⣤⣶⣾⡇⢀⣤⣤⣤⠿⣿⠀⢻⡇⠀⠀⢸⡇⠀⢸⣧⠀⠈⠻⣷⣄⠀⠀⠉⠁⢀⣼⢟⠁⠀⠀⠀⠀⠀⠀⢸⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⡆⢸⣇⠀⢸⡇⣾⠁⠀⠀⠀⢸⡇⠸⣧⠀⠀⢸⣷⠀⠈⣿⠀⠀⠀⠀⠹⣷⠀⠀⢠⡿⣿⠛⢷⠀⠀⠀⠀⠀⠀⠘⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⡇⠸⠶⠚⣧⠀⢸⡇⠀⣿⡀⠀⠈⣿⠀⠀⢿⡄⠀⠀⠀⠀⣿⠀⠀⢸⣇⠹⣤⡾⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⣸⡇⠀⣿⡆⠘⣧⠀⣶⠛⠋⠀⠀⣿⠀⢸⣧⣤⣤⣿⡀⠀⢸⣷⣶⣶⣦⠀⣿⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢻⡄⠀⠀⠀⠀⠀⢸⣏⠙⢷⣤⣾⠟⠀⢠⣿⠁⠀⣿⠀⠹⠶⠶⠶⠶⣿⠀⠈⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⢸⡇⣿⡀⠀⠀⣿⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠻⣶⣤⣄⣀⣤⣴⠿⠁⠀⠀⢻⣦⣤⣤⣤⣴⣶⣿⣷⣶⣶⡶⠶⠿⢿⣶⣶⣶⠶⠿⠿⠃⣹⣧⣦⣤⡿⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠛⠀⠀⠀⠈⠉⠻⢦⠀⠀⠀⠀⠀⢸
⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⡟⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢠
⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢀⣀⣤⠞⠁⠀⠀⠀⠀⣸
⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠙⠿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⢶⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣶⣶⣶⠾⠿⠟⠛⠛⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⣛⠓⣶⢞⣋⣉⡙⢳⡶⢋⣉⣉⠳⣶⠛⢉⣉⣛⢶⡖⢛⡓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠱⠛⠉⢻⠃⠛⠉⠈⠉⣿⠀⠋⠉⠉⠁⠘⡗⠋⠁⠉⠁⢿⠋⠙⠓⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡆⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⡇⠀⠀⡟⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣤⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡟⠀⠀⢸⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣸⠁⠀⢀⡏⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠈⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠿⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶⠶⠾⠷⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣯⠉⠉⣹⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠾⠷⠶⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⣤⣤⣤⡼⠃



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣤⢠⢀⡄⣀⢠⣤⡀⣤⡄⢠⡤⠀⣤⣄⠀⠀⢠⣤⢠⢤⡀⣤⠤⢀⡤⣤⢠⡄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⡏⢸⢼⡇⣿⢸⣤⠇⣧⡿⢸⠶⠀⡇⢸⠀⠀⣿⠈⢸⣼⠃⣿⠆⢸⣧⣼⢸⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⠃⢸⠘⠇⠿⠸⡇⠀⠯⠀⠸⠤⠄⣧⠞⠀⠀⢻⡴⠸⠘⠆⠿⠤⠸⠇⠸⠸⠇⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠆⠀⡆⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇⠀⠇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡇⠀⠘⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⡻⠀⠀⠀⠁⠀⢰⠀⠀⢰⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠥⢀⡔⠁⠀⢀⠌⠀⠀⠀⠀⠀⢸⠀⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡥⠚⠁⠀⢀⠠⠋⠀⠀⠀⠠⠀⢀⠌⢠⠃⠌⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⢀⠔⠁⠀⠀⠄⡀⠔⠁⡠⠂⡴⠃⢠⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⠉⠑⠀⠀⠀⢀⠠⠔⠊⣀⠄⠊⣠⠀⠀⠀⢠⠀⢰⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⢀⠂⠀⠈⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⠊⠀⢀⠠⠐⣈⣉⣀⡠⠤⠔⠊⠁⠀⠀⠀⡀⠔⠁⠀⡠⠀⢀⣼⡅⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡥⠚⠉⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠉⠀⢀⡠⠊⡠⠞⠁⢸⡇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠉⠀⠀⠠⠒⠀⠒⠒⠒⠀⠉⠀⠀⠀⠠⠔⠊⡁⠀⠈⠀⠀⠀⡸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠄⠐⢒⡀⠀⡈⢉⣉⠥⠄⠒⠒⠒⠀⠁⠀⢀⠔⠀⢀⡴⠁⠀⠀⠀⣾⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠘⠃⢀⣠⠔⠚⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣠⠴⠋⠀⠀⠀⢀⡜⠁⢇⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠒⠒⠒⠈⠀⠀⢀⠤⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠖⢹⠀⠀⡸⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠒⠀⡀⠀⢠⠞⠉⠉⠓⠒⠢⠤⠤⠤⠔⠂⠁⠒⠁⠀⠀⢀⠀⠀⠁⠀⠀⠀⣠⠃⠀⣠⠃⢀⠂⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠐⠈⢙⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊⠁⠀⠀⢀⠤⠊⠀⠀⠀⠀⣀⡤⠚⠁⢀⠔⠁⡐⠁⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠒⠁⢀⣠⠴⠒⠋⠉⠀⠀⣠⠆⠀⠀⠈⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠀⠀⡠⠒⠉⠉⠉⠉⠁⠀⢀⣠⠔⠋⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⣠⡤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣔⣋⡠⠴⢾⡉⠙⠂⠀⠛⠉⠀⠀⠀⠀⣠⣴⣒⣋⣉⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣄⣀⣀⣀⣤⠞⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣀⠀⣿⣿⣇⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣴⠂⠀⠠⠈⠀⣼⣿⡟⢋⣁⡈⠻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠙⣏⠀⠀⠆⠀⠀⢈⣭⡖⠚⠿⠃⢨⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣄⠀⠀⢸⣿⠃⢸⡿⠀⠀⠃⠀⠀⠙⢿⡿⢣⣶⣾⡝⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⠉⠙⣆⠀⠈⢿⠂⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⡂⠜⠀⠀⠀⠀⠘⠄⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢺⣿⣿⣿⣠⣶⣦⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠛⢄⣁⠈⠹⢿⣿⠇⠀⠀⠀⠀⢸⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡧⠀⠾⣿⡗⠀⣻⣦⣄⠀⠀⠀⠀⡍⠀⠀⠀⠋⠙⢋⣈⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⡔⢀⣀⣤⣿⣿⡟⠀⠀⠀⢠⠉⠀⠀⠚⠀⠲⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⢿⣿⣿⡷⠀⠉⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⡜⠀⢠⠆⠀⠀⣼⣷⣿⣷⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡠⠃⣴⣶⣤⣹⠿⠿⢻⡥⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠄⠀⢾⣿⣿⡿⠈⢣⣄⣌⠀⠙⣻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠐⠀⠀⢉⣩⡖⠒⠺⠿⣷⠀⢉⣽⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠰⣿⣿⣿⣆⣴⢀⣀⣰⢼⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡀⠀⠀⠀⡀⠂⠘⠿⠟⠉⢼⣿⣿⣿⡇⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠔⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠈⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⢿⣶⣄⠀⡘⡏⠀⠀⠁⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣄⣀⣈⡀⠙⠛⢠⡿⠁⠀⠀⢀⠖⠉⠛⠛⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⠿⠷⠒⠐⢮⠃⡄⠀⠀⠉⠀⠐⢶⣶⡄⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡥⠀⠀⠀⠀⠀⠀⠈⠻⣆⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢆⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢳⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠳⣄⠀⠙⠳⠤⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⡶⠶⠶⠶⠶⠞⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⠲⠶⠶⠶⠶⣦⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠉⠙⠻⣦⡀⢀⡀⠀⠘⠟⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠗⠀⣀⢀⣠⡾⠛⠉⠉⠙⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⢀⣴⠾⠛⠛⠛⠻⢶⣴⠟⠛⠷⢶⡄⠀⠀⠀⠀⠶⠟⠛⠛⢛⣴⠾⠛⠛⠛⠻⢶⣄⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⣀⣴⠿⠟⠛⠿⣶⡶⠿⠛⠻⠶⣶⡶⠛⠛⠿⢷⣄⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⣴⠟⠁⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢰⡟⢠⣿⠀⠀⢸⡟⠀⣤⠀⠀⢸⡏⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠈⣿⠀⠀⠀⠀⠈⣿⠀⠀⢸⡆⠘⣷
⠀⠀⠀⠀⠀⠀⣸⠃⣸⡇⠀⠀⢸⡇⢰⣿⠀⠀⣿⠃⣠⡀⠀⠐⡟⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠀⢀⣄⠀⢻⡆⠀⠘⣷⠀⣿⠀⠀⠘⣿⠀⢿⡀
⠀⠀⠀⠀⠀⠀⡿⠀⣿⠁⠀⠀⣼⠀⢸⡇⠀⠀⣿⠀⣿⠁⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⣿⠀⢸⡇⠀⠀⣿⡄⢸⡄⠀⠀⢻⡇⠸⡇
⠀⠀⠀⠀⠀⢸⡇⢸⡿⠀⠀⠀⣿⠀⣾⡇⠀⠀⡿⠀⣿⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⠀⢻⡇⢸⡇⠀⠀⠸⣷⠀⣿
⠀⠀⠀⠀⠀⣾⠀⣼⡇⠀⠀⢠⡇⠀⣿⠀⠀⢀⡇⢀⣿⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡀⠀⣧⠀⠀⢸⣇⠀⣧⠀⠀⠀⢿⡆⢸⡆
⠀⠀⠀⠀⢠⡏⢀⣿⠀⠀⠀⢸⡇⢸⡿⠀⠀⢸⡇⢸⣿⠀⠀⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⣿⠀⠀⠈⣿⠀⣿⠀⠀⠀⠸⣧⠈⣧
⠀⠀⠀⠀⣸⠃⠈⠋⠀⠀⠀⣾⠀⢸⡇⠀⠀⢸⡇⢸⡟⠀⠀⢸⡇⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡇⠀⣿⠀⠀⠀⣿⡆⢸⡄⠀⠀⠀⠻⠀⢻⡀
⠀⢀⣠⡾⢿⡄⠀⠀⠀⠀⠀⣿⠀⠻⠃⠀⠀⣾⠁⢸⡇⠀⠀⣾⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢸⡇⠀⢻⡆⠀⠀⢹⡇⢸⡇⠀⠀⠀⠀⠀⣼⠟⢷⣤⡀
⢠⡾⠋⠀⠈⠻⣦⣀⠀⠀⢸⡟⠀⠀⠀⠀⠀⣿⠀⠘⠃⠀⠀⣿⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⡧⠀⢸⡇⠀⠀⠸⠗⠸⣷⠀⠀⢀⣠⡾⠋⠀⠀⠙⢷
⣿⠁⠀⠀⠀⠀⠈⠙⠳⢦⣼⣇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣿⣤⠶⠛⠉⠀⠀⠀⠀⠀⠈
⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢦⣴⣧⣀⣀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣀⣨⣷⣴⠶⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠻⠟⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡴⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠶⠶⠶⠶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠶⠶⠶⠶⠶⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⢠⣿⣧⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣌⣿⣿⡿⣀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢰⣶⠿⣸⣿⣿⣯⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⢌⣱⡞⢿⠛⠟⢻⣿⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠁⠀⠐⡚⠛⣋⠩⣴⡗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠀⠀⠀⠛⢷⣶⣴⣾⠖⢠⣼⢹⡇⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠐⢴⠃⢌⣭⡾⠷⢺⣛⣱⡿⠈⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⡕⠊⠉⠀⠀⢘⣠⡞⢁⣀⣤⡾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠉⠉⠙⠲⠖⠒⠉⠙⠛⠿⠛⠉⢶⣶⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠆⠐⠐⠄⢠⠀⢀⡀⠀⡀⠀⡠⣦⡔⢾⢙⣧⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣠⠀⣆⢔⣀⣀⡆⠀⡇⣰⡆⠹⠳⡿⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⢇⢻⡀⣄⠙⣿⣀⡿⢁⡗⣾⢦⣷⣼⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⢼⡾⡏⣭⠓⢿⣀⣧⣾⡧⣼⣾⣿⣿⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣧⣷⢻⣧⣼⡍⣿⣸⣏⣿⣹⣿⣿⡗⣥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⢼⣧⣹⣟⣿⣻⣿⣿⣿⣿⡿⠽⣋⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⡗⠻⢭⠄⠊⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡤⠖⠒⠲⣤⠴⢒⠲⢤⡴⠒⠒⠦⣤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠚⠛⡿⠘⠋⠉⢳⡇⠚⠉⠙⠀⣷⠚⠉⠙⠸⡟⠛⠊⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣶⠀⢠⡇⠀⠀⠀⢸⠃⠀⠀⠀⠀⢻⠀⠀⠀⠀⣧⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠇⠀⢸⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢿⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⠀⠀⣸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾⠀⠀⡿⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠸⡇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡏⠀⢠⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣧⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢻⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⡿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡇⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢀⡤⠤⠾⠤⠤⠤⠴⠧⠤⠤⠤⠤⠤⠧⠤⠤⠤⠤⠤⠤⠷⠤⠤⠤⠤⠤⠷⠤⠤⠤⠼⠦⠤⣄
⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠾⠤⠤⠤⠤⠷⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠟



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠠⣄⣄⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠲⠋⢀⣀⣤⡀⠀⠀⠾⠁⠀⠀⠉⠉⠈⠒⠒⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠃⢠⠀⠀⠀⠙⠛⠛⡡⠀⠀⢰⣶⡴⠶⠀⢀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⣀⠰⠂⠐⠈⡀⠠⠜⠁⠀⠀⠀⠀⠤⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡞⠉⠀⡠⠐⠒⠀⠀⠀⣀⣀⠄⣀⡔⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⡀⠀⣇⠀⠀⠀⠀⠀⠙⠿⠷⢀⣀⣀⣀⣀⣠⠴⠖⠋⠉⠫⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠉⠉⠉⠑⠒⠒⠒⡉⠉⠀⠀⠉⠁⠀⠀⣀⠀⠐⠄⠀⠙⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠤⠔⠀⢀⠀⠀⠉⠉⠀⠠⠾⣿⡶⠀⠀⠀⠀⢡⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠋⠁⠴⠿⠗⠠⠖⣶⠂⠀⠀⠀⠀⠀⠀⠈⠉⠱⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡍⠉⠙⠒⠦⠤⠤⣤⡤⢄⡁⠀⠈⠀⡀⠠⣶⠶⠆⠀⢀⣀⣼⡒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡠⠚⠓⠀⡀⠀⠐⠻⠽⠿⣀⠈⠒⠲⣒⡛⠢⠤⠒⡫⠝⠋⠀⣠⣍⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡀⠀⠀⢠⠖⡋⠁⠀⢀⠄⠘⠛⠦⠀⢉⠉⣠⣶⣽⣷⠀⠀⠉⠀⠈⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠳⢤⢦⣀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠂⠸⠲⡌⢙⠓⠶⣿⣿⠷⣄⡀⢀⣀⣴⣀⣠⠤⣄⣀⣄⣀⡤⠶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⢀⠸⠷⠿⠃⠀⠄⠀⢀⣒⠒⠾⠋⠉⠤⠚⠁⠀⠀⠀⠈⡀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡶⢯⡀⠀⠀⠀⢂⡁⠈⢹⠁⠈⠐⢄⠀⠀⠈⠳⡠⠁⠀⠀⣿⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡀⠀⠉⠓⠶⠶⡾⠗⣶⣤⣀⣀⣠⡾⢓⣶⡦⠴⠷⠴⠶⠚⠁⠀⠍⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⠁⠀⢾⠁⣀⠈⠃⠠⣙⠆⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢰⣿⡇⠀⠘⣿⣳⡈⠁⢀⣤⣤⡤⠀⠀⠀⠐⠤⡄⠉⠛⠛⠃⠀⡸⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⠄⠀⠀⠒⠲⠶⠬⣄⡀⠀⠈⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⣔⣀⣇⣀⡀⠴⠚⠉⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⣬⠉⠁⠛⠛⠓⠲⠖⢒⠋⠀⠀⠚⠉⠁⣀⡀⠀⠀⢀⣀⠂⣠⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡀⡀⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡈⠉⠀⠀⠀⠺⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠙⠒⠊⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⣀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⠭⣽⢯⠥⠤⣹⢋⠥⠬⠝⣏⡥⠭⠿⣯⠭⡗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡄⠀⡟⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢻⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠁⢀⡇⠀⠀⠀⡏⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡖⠀⢸⠃⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠘⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⡇⠀⣸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⣇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠸⠁⠀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡇⠀⠀⠀⢰⡇⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⡇⠀⠀⠸⡆
⠀⠀⠀⠀⠀⢀⣀⣼⣀⣀⣀⣼⣀⣀⣀⣀⣸⣅⣀⣀⣀⣀⣠⣇⣀⣀⣀⣀⣧⣀⣀⣀⣧⣀⡀
⠀⠀⠀⠀⠀⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠙⡏⠉⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠉⠉⠉⠉⠉⠋⠉⠉⠉⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠛⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠉⠁⠀⠀⠹⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡤⣤⣄⣄⠀⢠⠀⣀⡼⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣤⣄⣀⡀⡀⢀⢠⢀⣼⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠉⠉⠛⠋⠉⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠈⠉⠙⠛⠚⠛⠋⠀⠚⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠛⠓⠶⠶⠴⠤⣤⣦⣦⡤⠼⢳⡄⠀⠀⠀⠀⠀⠀⠀⢀⡿⠷⠷⢶⣴⣴⣴⣠⣀⣆⣄⣶⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢸⣸⣠⢀⢀⡃⣀⡰⠈⠉⠉⠩⠅⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠯⠍⠉⠉⠛⠛⠛⠳⠶⡖⣴⣴⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⢸⣯⡝⠛⠛⠛⠿⠿⠿⢿⣷⣶⣶⣶⣶⠏
⠀⠀⠀⠀⠀⠀⠀⢼⣁⠀⢀⡀⠀⠀⠀⠀⠸⣅⣸⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣾⠍⠀⢀⣀⠀⠀⠀⠀⢸⣇⣿⠀⡉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣉⡽⢿⡚⠳⢿⡵⣦⣤⣄⣒⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠛⠛⠻⠿⠿⠿⣶⣶⣯⣁⣀⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢙⣄⡤⠷⡖⠛⠻⣅⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠁⠀⢀⠤⠄⠀⠢⠈⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⢉⣧⡤⠞⢾⠉⣙⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠉⢀⠀⠐⠔⠀⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠻⡉⢀⣳⣤⠴⠷⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⢀⡀⠀⢤⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠛⢯⠁⣘⣦⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠒⠀⠁⠀⣐⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡖⠚⠻⣅⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠢⠀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣈⣦⣴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠐⢀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣤⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠴⢾⣯⣉⡛⠓⠦⣤⡶⠛⠋⠙⠛⢦⣤⠖⠚⠛⠛⣲⣦⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⢀⣀⣤⣤⣤⣤⣉⡙⠲⢬⡳⡄⠀⠀⣠⠏⠀⢀⡴⠚⣉⣉⣀⣀⣀⣉⡉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡤⠞⠋⠁⠀⢀⣤⠶⠚⠛⠛⠚⠳⣿⠶⠚⠛⠓⠶⣿⠚⠋⠉⠛⠳⢦⣀⠈⠉⠛⠶⣤⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⢀⡴⠋⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣴⠋⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠳⡄⠙⢦
⠀⠀⠀⠀⠀⣰⠃⠀⢀⡾⠁⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠙⣆⠈⢳⡀
⠀⠀⠀⠀⣰⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠘⢇⠀⠹⡄
⠀⠀⠀⢠⠏⠀⢠⠏⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠈⢦⠀⠹⣆
⠀⠀⢠⡟⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠈⢧⠀⠹⣆
⠀⢀⡞⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠈⢧⠀⠙⡆
⠀⡼⠁⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠈⢧⠀⠹⡄
⢸⠃⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠈⣧⠀⣻
⠈⢷⡞⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠘⣧⠏
⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠸⡆
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢹
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢈
⠘⠶⢤⣤⣀⣀⣀⣤⣤⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⣤⣤⣤⣤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⣄⣠⣤⣤⠤⠶⠒⠋⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠉⠓⠶⠤⢤⣤⣤⣀⣤⣤⠤⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣶⣶⣶⣶⣶⣞⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠙⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⣶⠿⠿⠿⣶⣄⡀
⢀⣿⡇⣰⠖⠂⠈⠻⣿⣦⡀
⠈⣿⡇⣇⠀⠀⠁⠂⠈⢿⣷⡄
⠀⢹⣿⠹⣄⠀⠀⠀⠈⠀⠻⣿⣆
⠀⠀⢻⣧⡘⢦⡀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠻⣿⣦⡙⠦⡀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠈⠻⣿⣦⡈⠲⣄⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣌⡙⢦⡀⠘⣿⣦⠀⠀⠀⣀⣀⣤⣤⣶⣶⡶⠿⠿⢿⠿⠿⣿⣿⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⣬⣀⠘⣿⣷⣾⣿⢟⡛⠉⠉⠀⠀⠀⠀⠀⢻⣷⠀⣸⣿⣿⣟⣻⣿⣭⣭⣭⣭⣭⡍⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣷⣼⣷⣶⠾⢿⣶⠀⠀⠀⣸⣿⣟⣛⣟⠛⠛⠛⠛⠛⣉⡉⠙⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡿⠟⠋⠉⣀⣤⣼⣿⣷⣾⣿⣶⠾⢿⣿⣿⡟⢿⣿⡇⠀⠀⠀⠀⠻⣝⣦⠘⣿⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⣿⣛⡉⠅⠀⠀⠀⠀⠀⠀⠑⢦⣿⠘⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣗⠠⠀⠀⠀⠀⠸⠃⣠⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣏⣀⣀⣸⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣠⣬⣥⣤⣿⣿⣿⣶⡶⠾⠿⢿⠿⠟⠛⠛⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠯⠈⠍⠉⠉⠉⠀⠻⣶⣶⠶⠋⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠁⠀⢠⡾⠋⠙⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠹⣦⣤⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢹⡄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⡾⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⢘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⢀⡤⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⣤⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⣤⠀⠀⡟⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣀⠀⢷⣄⣠⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣠⡾⠃⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣄⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠟⠛⣻⣿⠟⠛⠛⢻⡟⡿⠉⠉⠙⣿⣡⣦⣴⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣷⣶⣦⣼⣷⣤⣤⣶⡿⠿⠷⠾⠿⠿⠛⠛⠋⠉⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⡡⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⣿⣤⣤⣶⣶⢶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⢶⣶⣴⣶⢶⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢦⡀⣴⡆⣿⡛⠀⣷⣴⣷⠀⣴⡆⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢰⡄⣿⠉⢹⠸⡟⢹⡇⢿⣥⠀⡏⠁⡇⣸⢿⠁⣸⠁⣷⣠⠗⠦⠀⠀⠀⠀⠀⠀⢀
⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠻⡌⠛⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠚⢱⣏⠃⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⣴⡆⣴⡰⠂⡶⠆⣶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⢶⠗⡇⢿⠹⡄⠯⡅⢿⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠂⠀⠂⠐⠒⠒⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀⣴⣶⢶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⢶⣶⣶⣦⣠⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⣠⠀⠠⠀⠀⢀⠀⠀⠀⣼⣧⣽⢾⠋⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠐⡆
⠀⢠⣿⠀⢃⢸⡀⣆⠀⣦⠀⣟⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣀⠨⣽⣿⣿⣷⣿⠂⠀⠀⠀⠀⠀⠀⢀⣿⣇⢀⡇
⠀⠀⠛⠻⠊⠻⠿⣿⠶⠿⠶⣿⡟⠛⢿⣶⣶⣾⣭⣡⣾⣿⣃⣌⣠⣄⣬⣬⣿⣧⣼⣴⣿⣿⣿⣿⠏⢿⠀⢰⠀⢠⡄⢰⣆⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠻⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⢿⠇⠀⠈⠀⠻⠆⠘⠃⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠴⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠠⢄⠀⠑⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣸⣿
⢀⡰⣃⠀⠄⠀⠨⠀⣀⣼⣿⢇⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠙⠒⠾⠛⠁
⠈⠛⠿⣿⣄⣠⣴⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢂⡀⠠
⠀⠀⠀⠹⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣼⡿⠀⠀⢠⠀⠦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⠃⠀⠀⣀⣴
⣦⣤⣤⣦⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣧
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠢⠤⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠈⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢰⣤⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⣿⡏⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠛⠉⢲⠇⣼⢸⠿⡂⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⡩⢷⡰⠀⠂⢐⣿⢸⡆⠘⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣼⠁⠀⡹⠁⠀⢾⣿⠈⡆⠀⠠⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢐⢺⡇⠀⠀⡇⠀⠀⠘⣿⠀⢿⣄⠀⠊⠉⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠄⠐⣄⠀⠳⣄⠀⠙⠹⣧⡈⢿⣼⣄⣀⡼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⢠⣶⠶⢈⡉⠒⠺⣟⣶⣶⠶⢷⣆⡈⠃⠒⠒⠺⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠘⠿⣼⠁⠀⠀⢀⢀⡀⠈⠉⠛⠁⠠⠤⢀⡀⠀⠀⠒⠊⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢘⢧⣄⠈⠢⠤⡔⠋⠀⠙⠲⢄⡀⠀⠀⠀⠀⠉⠲⣏⣿⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⢀⠉⠙⠲⠤⢤⣤⣀⣈⢙⡛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢀⣀⣀⠙⠓⠾⣦⣠⡆⡀⠈⢉⠑⡒⠂⠢⠤⠈⠉⢉⣡⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠛⣿⡦⠤⣤⣭⣭⣿⣷⣦⣤⣤⣤⡴⠶⠞⣮⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⡇⠀⡇⠀⣿⠀⣾⠉⢹⠇⠀⡿⠀⢸⡇⣼⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⣿⠀⢁⠀⣿⠀⣿⠀⢸⠀⢸⡇⢀⣿⢠⡏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹⡇⠘⠀⢹⠀⡿⠀⡼⠀⣼⡅⣸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠈⢧⠀⠀⢸⠀⣷⢠⣧⣰⣿⢡⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣻⡆⠓⢺⣶⡿⢸⡟⣿⣿⣿⡿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠘⠋⠓⠚⠚⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠆⣠⠶⠛⠻⢷⡆⣠⡶⠛⠛⠶⣄⠰⣾⠟⠛⠲⡄⠰⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣴⣶⡶⢠⣷⣾⣿⣿⣶⢠⣿⣶⣿⣿⣶⣿⡆⣶⣾⣿⣷⣾⡄⢶⣶⣦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣿⣿⡇⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⣿⠇⣾⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣷⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣾⣿⣿⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣿⣿⣿⢰⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣿⣿⠇⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢶⣶⣶⣶⣶⣴⣶⡶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡿⢱⡯⣿⣼⢿⡟⣿⢽⣷⢘⢿⡽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢹⡧⣿⢽⡏⣿⣿⡇⣿⣿⣿⣦⣿⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⣿⡇⣿⢽⡇⣿⣿⡇⡇⣿⡟⡇⣿⣿⢹⡄
⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⣿⣿⢧⢻⣿⢷⣿⣿⣷⣿⣾⡇⡇⣿⣷⣇⣿⣻⡜⣧⢿⣿⣿⣶⣶⣦⣤⣀⡀
⠀⠀⣴⣾⣿⣿⣿⠿⣛⣽⣿⣷⡆⡜⣾⣿⣸⣿⣿⢿⠟⣻⡇⡇⣿⣿⢹⢻⣯⣧⢸⡞⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠻⣿⣿⣿⣯⣺⣿⣿⣿⣿⡳⣷⣿⡏⣿⣿⣿⢸⣷⣿⡇⡇⣿⣿⡼⣸⣿⣿⡞⠿⢹⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠉⠛⠻⠿⢷⣾⣿⣯⣛⣺⣭⣭⣟⣛⣛⣻⠿⠿⠷⣛⣻⣿⣿⣋⣻⣯⣵⣿⣭⣽⣶⡿⠿⣟⣻⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠯⠭⣭⣭⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣿⣯⣭⡭⠽⠶⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠭⠭⠭⠭⠭⠭⠍⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡔⣀⣀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠁⢀⠀⠀⠀⢀⡠⢊⣀⡠⠂⡀⠀⠀⣚⡫⣇⣤⣠⣄⠨⠄⠀⣃⡀⠀⢀⣀⣀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⢦⡐⠖⡉⠛⣷⠀⠀⠻⣅⠈⠩⢅⠈⠓⠲⠌⠉⠀⠈⠙⠀⣞⠁⠀⠂⠀⢍⠲⠭⠉⠉⠉⠩⢏⠂⠋⡇
⠀⠀⠀⠀⠀⢀⠉⠀⠉⠐⠒⠁⡀⠀⠘⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠠⠃⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠠
⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠨⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠄⢁⠀⢘⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠈⠀⠠⣀⠀⠀⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠬⢥⠴⠤⠤⠤⠮⠤⠤⠴⡿⠄⠀⣶⣾⡆⠀⠀⠤⠤⢒⠒⡶⢐⠿⣄⠀⠀⠙⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠉⠉⠀⡐⠀⠀⠀⠠⠔⠉⠐⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⣤⡴⠞⠛⠛⠳⢦⣴⠞⠛⠛⠳⢦⡴⠞⠛⠛⠳⢦⣤⠶⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣵⠶⠶⣶⠏⠴⠛⠛⠻⢦⡟⠁⠾⠛⠛⠳⠈⢻⡤⠞⠛⠛⠷⠹⣶⠶⠶⣮⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡟⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢠⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠏⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⡏⣰⠄⠀⠀⣸⠃⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⣸⠇⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⣿⢠⡏⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢰⡏⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⣼⠃⠛⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢿
⣠⠶⠶⠾⠿⠶⠶⠶⠶⠶⠿⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶⠶⠶⠾⠷⠶⠶
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠛⠛⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠿⠷⠲⠶⠶⠶⠶⠿⠳⠶⠶⠖⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠦⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢠⣶⣤⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠈⢹⡆⠀⠀⠀⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣸⣿⠃⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢦⡀⠀⠀⠀⢀⡶⠋⠁⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡿⠃⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠀⠀⠙⢶⡀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⢀⣀⣴⠇⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⣤⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠞⠃⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⢤⣄⠀⠀⠀⢰⡇⠀⠀⢰⠏
⠀⠀⠀⠀⠀⢀⣤⡤⠤⠶⠶⠶⠶⢶⡞⠁⠀⠀⠀⠙⢷⠒⠒⠒⠒⠚⢳⡟⠁⠀⠀⠀⠈⢷⡖⠒⣿⠀⠀⢠⡿⠶⠶⠶⠶⠤⣤⣤⡀
⠀⠀⠀⠀⠀⣿⠉⠛⠛⠛⠒⠒⠲⠾⠶⠶⠶⠀⠶⠶⠾⠷⠶⠶⠶⠶⠾⠶⠶⠶⠶⠶⠶⠾⠷⠶⠷⠶⠶⠛⠒⠒⠚⠛⠛⠛⠛⠉⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⢹⠀⢀⡾⠋⠉⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⢳⡄⢠⡇
⠀⠀⠀⠀⠀⢸⡇⠘⣇⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣰⠇⣸⠃
⠀⠀⠀⠀⠀⠈⣧⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠈⠳⠞⠁⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⢀⡟
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢀⡶⠛⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⣦⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠘⢧⣄⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣠⠞⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠘⣧⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⣸⠇⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣻⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠿⠿⠿⠛⠛⠿⠿⠿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣠⣾⡿⠋⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⢠⣴⡾⠿⠛⠛⠛⠛⠛⠛⠿⢷⣦⣾⡿⠋⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⢠⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡄⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣾⡟⢿⣶⣤⣤⣀⣀⣀⣠⣾⣿⣿⡿⢻⣷⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⣸⡿⠀⢠⡄⠉⠉⠛⣛⣿⡿⠋⠉⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⢠⣿⠃⢠⣿⠇⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⡏⠀⣾⡟⣠⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠘⣿⣄⡀⢩⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠃⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠈⣹⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡿⠿⠋⠁⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣠⣾⡿⠋⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢷⣶⣶⣤⣤⣤⣤⣶⣶⡾⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⢀⡈⠉⠉⢉⠉⠁⣀⣀⡀⡀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢩⣅⢸⣯⣭⢸⡇⠀⢀⡞⣧⠀⠉⡏⠁⣿⢸⡿⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠛⠘⠷⠶⠘⠷⠖⠚⠉⠙⠃⠐⠃⠀⠛⠘⠃⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠆⢰⠶⣦⢰⡶⠶⢠⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⣸⠻⣏⢸⣟⣛⢰⣟⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠈⠀⠉⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡌⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⢀⡹⠁⠄⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠠⠀⠈⡟⠁⡀⠐⡤⠀⠠⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣉⠀⡌⠁⠀⢳⡥⡞⠀⡥⢠⠆⠀⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⢤⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠠⡱⣇⣅⣂⢴⠡⣀⢄⣠⠄⠐⢈⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠘⠀⠀⠀⠀⠅⣠⠖⠉⠀⠀⢹⠟⠉⠉⠙⡄⢈⠋⠠⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⠁⡀⢀⣡⡄⠀⠀⡞⠁⠀⠀⠀⢀⠃⠀⠀⠀⠀⠙⡅⠰⠂⠀⠠⠀⠀⢤⢀⠀⠄⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡲⠛⠁⠀⠀⣰⠷⠓⠂⢄⡀⠀⢸⠀⠀⡀⢠⣀⠴⠁⠀⠁⢂⠀⣰⠰⠟⠓⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠃⡠⠤⠀⡜⠁⠀⠀⠀⠀⣤⢖⠟⠙⠛⠫⡙⢥⣀⠀⠀⠀⠘⡌⠉⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢀⡄⠄⠀⠐⢉⠁⠀⡀⠀⠀⢸⣷⠁⡠⠀⡀⠀⠀⢂⢳⠠⣀⣠⠴⢧⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠘⠓⠚⠀⠀⠀⢼⣀⠀⠐⠂⠈⠐⢀⣼⣿⠚⠁⠀⠀⡼⠀⡐⠈⠐⠲⢤⣂⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⢀⡲⢆⠀⠀⠀⠎⢿⣶⡤⣄⢠⣤⡾⠏⠀⠈⠐⠒⣹⠰⣂⡀⢀⠂⠀⢢⠀⠙⠉⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⣤⣄⣆⢴⠁⠀⢀⡥⣖⠀⠀⠀⠀⠩⠋⠙⠈⠀⠀⠀⢹⠒⠊⠪⠼⣆⣲⠋⣘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠠⠜⠃⠚⢆⠀⠈⠀⠈⢄⡀⢀⣠⠂⠀⠀⠀⢀⣀⣠⣼⠳⣷⣞⠁⠈⠇⠋⠱⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠈⡄⠀⠁⠀⠄⠐⢩⡆⠠⠀⠐⠂⠀⠈⠁⠁⢠⡖⠒⢲⠉⢽⠅⡛⠀⣿⢫⠩⠁⠶⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠀⠀⡤⠈⠀⢀⠣⣌⡾⢀⠤⢀⡠⡀⠀⠑⣷⡖⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠐⠀⢀⡀⠀⠁⠀⠛⢻⠀⠀⠀⠀⠇⠒⠀⠀⣳⣀⠠⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠈⠛⠂⠀⠀⠀⠐⠀⠀⠀⣆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⢶⣶⣤⣤⠖⠛⠛⣶⣤⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣛⣯⣭⣭⣽⣶⣶⣭⣙⢿⣧⡀⣾⣵⣿⣚⣭⣟⣛⣻⣭⣭⣟⣒⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠋⠉⢡⣤⣴⡶⠾⠿⠿⣿⣿⣩⣭⣭⣽⡿⠿⠿⡿⠿⠶⣤⣬⣭⣍⣛⠺⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣁⠀⣠⠴⢚⣉⣥⣶⣶⡾⠟⠛⣩⡵⡹⠁⡝⣌⢿⣿⣲⢬⡉⠛⠲⢮⣍⡛⠮⣝⢦⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣼⡵⠊⣡⣶⣿⣿⣿⠟⠁⢀⣴⣿⠋⣼⠁⠀⢹⡜⣆⠻⣿⣷⣮⡓⢄⠀⠉⠻⣷⣌⠳⣽⣦⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣾⡟⢁⡜⠉⢈⣿⡟⠁⢀⣴⣿⣿⠃⣸⠇⠀⣤⣼⣷⠸⡆⢹⡿⠻⣿⣎⠳⡀⠀⠘⣿⣷⡌⢿⣧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⡾⠁⣾⡟⢀⠎⠀⠀⣼⡟⠀⢠⠋⠀⢹⠇⢠⡿⠀⠀⠀⠈⣿⡄⣿⠈⠁⠀⠘⣿⣇⢹⡀⠀⠘⣿⣿⡌⣿⣧⠹⣆
⠀⠀⠀⠀⠀⠀⣼⠃⣼⡿⠀⡞⠀⠀⣸⣿⠁⢀⠇⠀⠀⣿⠀⣼⡇⠀⠀⠀⠀⣿⡇⢹⡆⠀⠀⠀⢹⣿⡄⢷⠀⠀⠸⣿⣷⠘⣿⡆⢻⡄
⠀⠀⠀⠀⠀⢰⠏⢰⣿⠃⡼⠀⠀⢠⣿⡇⠀⡼⠀⠀⢸⡇⠀⣿⠇⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⠀⠀⢿⣷⠸⡆⠀⠀⢻⣿⣇⠹⣿⡀⢿⡀
⠀⠀⠀⠀⢀⡟⢀⣿⡏⢰⠃⠀⠀⣼⣿⠀⠀⠁⠀⠀⣾⠀⢸⣿⠀⠀⠀⠀⠀⣿⣇⢸⣿⠀⠀⠀⠀⠸⣿⡆⢻⠀⠀⠈⣿⣿⡆⢻⣷⠘⣷
⠀⠀⠀⠀⣼⠁⣾⡿⢀⡏⠀⠀⢠⣿⡇⠀⠀⠀⠀⢠⡏⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⣿⣷⠘⡇⠀⠀⢹⣿⣷⠈⣿⣇⢹⣇
⠀⠀⠀⠀⠙⠶⣿⠃⣼⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⣸⠇⠀⣿⡟⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠙⣿⡆⣿⠀⠀⠘⣿⣿⣇⢻⣿⣾⠏
⠀⠀⠀⠀⠀⢀⡟⢠⡇⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇⠀⠀⠀⠀⠈⣿⣅⢘⡅⠀⠀⢻⣿⣿⡜⣿
⠀⠀⠀⠀⠀⠈⠳⣾⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⢰⡟⠀⢠⣿⡇⠀⠀⠀⠀⠀⢸⣿⠸⣿⡇⠀⠀⠀⠀⠀⢿⣿⠀⣧⠀⠀⠘⣿⣿⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠛⣿⠀⠀⠀⠀⠀⣸⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⢹⡶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢤⣄⣀⣀⣿⣃⣀⡼⢿⡇⠀⠀⠀⠀⠀⣾⣿⠀⣿⣧⣀⠀⠀⠀⣀⣸⣿⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠙⠲⠦⢤⣤⣤⣿⡿⠞⠃⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣞⠻⣶⠟⠛⠛⣷⡶⠶⢶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣴⠟⠛⠻⢶⣤⣼⣯⣴⠞⠛⠷⣤⡀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢠⡿⠁⠀⠀⢰⡟⠁⠀⠀⠘⣷⠀⠀⠈⢷⡄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢀⡿⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢿⡄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣼⠃⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠈⣷⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢠⡟⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢹⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣿⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢠⡟⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹⡆⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣸⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⣷⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣿⣄⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢀⣠⡿⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠛⠳⠶⢤⣤⣤⡤⠶⠞⠛⠉⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣤⣶⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠈⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣶⣶⣶⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⣴⡾⠿⠿⣶⡿⠋⠀⠀⠀⠈⠻⣷⡾⠿⠿⣶⣤⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣠⣾⠿⠛⠛⢿⡿⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠹⣿⠟⠛⠻⢿⣦⡀
⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠹⣷
⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⣿⠇⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢀⣿⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢹⣇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⡿⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡄⠀⠀⠀⠸⣿
⠀⠀⠀⣾⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⣿⡆
⣴⣶⣶⣿⣷⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣿⣷⣶⣶
⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⣀⣙⣻⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣛⣁
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠃⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⢾⣄
⠀⠀⠀⠀⠀⠀⠀⣰⢻⠻⡐⠀⠀⠀⢠⡟⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠰⣟⣲⠿⣋⣩⡝
⠀⠀⠀⠀⠀⠀⠀⠉⣥⣀⣤⡀⣤⣤⣠⠁⠀⣠⡮⣭⡁⠀⠘⡆⠀⢸⣿⣿⣫⣭⣽⡿⣻⡆
⠀⠀⡠⠒⠚⠓⠶⣄⠉⣿⣏⠉⣉⣿⠟⢠⠞⠉⠀⣠⠼⠶⠴⢧⡀⠐⣿⣷⣶⣶⣶⣿⣉⠀⠀⣸⢷⣤⡀
⠀⢸⡁⠀⠀⠀⠀⣽⡆⢻⣿⠿⣿⣿⠀⣮⠁⠀⣴⠃⠀⠀⠀⠀⠨⡀⢻⣞⣽⣟⣩⣿⠇⠀⢸⡼⣊⡿⢽⣄
⠀⠀⠇⠀⠀⠀⠖⢾⡀⢸⣿⡟⣿⡟⠀⠀⡏⣶⠻⡄⠀⠀⠀⠀⢸⡇⠀⣿⣯⣭⣿⡏⠀⢲⡟⣓⢻⡆⣽⣷⡆
⠀⠀⢰⣠⢀⣶⣤⡞⠁⠈⣷⡟⣿⠇⠀⠈⠳⣿⣤⢈⣖⠒⢘⣿⡿⠀⠀⢸⣟⠟⣿⡇⠀⠙⣷⣷⣦⣭⣽⣶⠃
⠀⠀⠘⣭⢻⣽⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠋⠀⠀⠀⠘⣯⡿⣿⠀⠀⠀⢻⣿⡿⣷⠺⡟
⠀⠀⠀⢆⣹⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣉⡏⠀⠀⠀⠀⠀⢿⠿⣿⠀⠀⠀⠸⣾⡷⢿⢿⠇
⠀⠀⠀⢸⠷⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣶⢛⠃⠀⠀⠀⠀⠀⢸⣶⡏⠀⠀⠀⠀⢹⣿⢾⡟
⠀⠀⠀⠀⢂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣶⣟⣻⣿⣿⣿⣿⣿⣿⡷⠶⠶⠆⠀⢀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠿⢛⣩⣥⣶⠾⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⣹⠟⣡⣾⣿⣿⣉⣤⣴⡿⠿⢟⣛⡛⢻⡛⠛⠛⠛⠛⠛⠛⠻⠷⠶⣦⣤⣄⡀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⡄⣿⢸⣿⣿⣿⡿⢛⣩⣴⠀⢭⣽⣷⡂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠠⣌⡙⠻⣷⡄⠀⡀⠀⢰⣯⣻⣄
⠀⠀⠀⠀⠀⠀⢀⣾⡏⢠⡄⠳⣝⠿⣿⣿⣧⣶⣿⣿⣿⠀⢭⣿⣟⡂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⠨⣿⣿⣷⣮⣿⡾⣃⠴⠋⢈⣁⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⡀⠀⠈⠳⢮⣽⣻⡿⣿⣿⣿⣀⣭⡿⠓⠂⠘⠀⠀⠀⠀⠀⠀⠀⢀⣀⣒⣨⣿⡿⠿⢛⣉⣬⣷⣾⣿⣿⣿⡟⣸⡇
⠀⠀⠀⠀⠀⠀⣾⣴⣌⡻⣷⣄⡀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⣫⣾⡟⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠶⣽⣻⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⢿⣿⣶⣄⣤⠾⢟⣫⣵⣾⣿⣿⣷⣿
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⣹⣿⡷⠾⢯⣿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣄⣀⣀⣀⡀⣠⣤⣤⡤⠴⢖⣛⣛⣭⣥⣶⠞⠛⢹⣿⡏⣿⣿⣿⣼⣆
⠀⠀⠀⠀⣼⢿⣿⣿⣿⡟⠀⠀⣿⣿⡇⠀⠀⠀⠈⠉⠉⠉⠛⠀⠘⠛⠛⠛⠻⣷⡶⠒⠒⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⡄⠀⢸⣿⣧⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣿⠸⣿⣿⣿⡇⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⣛⣻⣻⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⢇⣿
⠀⠀⠀⠀⠘⣧⡙⢿⣿⠁⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣛⣿⣿⣛⣻⣏⠀⠀⢙⣛⣿⣿⡿⢫⣾⠃
⠀⠀⠀⠀⠀⠈⠻⢦⣉⠻⢦⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⣀⣸⡿⠟⣉⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣬⣉⠛⠶⢦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠟⠛⣉⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠶⣬⣭⣍⣛⣛⣛⠲⠶⠶⠶⠶⠾⠿⠷⠶⠶⠶⠖⣛⣛⣛⣩⣭⣥⠴⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣳⠘⠁⠀⠀⠠⠂⢠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠑⠀⠀⠀⣷⣶⣾⡓⠀⠀⢡
⠀⠀⡀⠄⠀⠀⠛⢀⡌⠀⡴⢁⠔⠉⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠴⢂⡀⠀⠴⠖⠈⠈⢹⡿⠏
⠀⠀⠀⠀⣘⣿⣀⠀⠀⠈⠐⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⣸⠃
⠀⠀⠀⠀⣿⠟⠻⡁⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠈⡇⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⣐⣿⣧
⡀⠀⢀⣌⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠙
⠿⣿⣿⣍⠀⠀⠘
⠀⢨⡿⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠆⠤⣄⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠑⠲⣀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣈⡂⢇
⠀⣴⣄⢀⡠⠄⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈
⠉⢹⣿⣿⠉⠀⢤⠀⠀⠀⣀⠔⢁⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠀⣶⣤⣴⠯
⣠⡾⠋⠁⠀⠀⠐⠁⢀⡔⠡⠠⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠙⣿⣿⠆
⠁⠀⠀⠀⠀⠂⣺⣶⣾⡋⠁⠉⢖⠒⠂⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠉⠁⠀⢹⡟⠁⠀⠀⡀⠀⢀⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⣶⣶⡆⢀⡀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠶⠾⡟⠙⠛⠳⣶⠶⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢋⣉⣙⣲⠀⠒⠛⠒⠀⠈⠰⢖⣂⢩⣍⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠡⠚⠉⠀⣠⠿⠒⢦⣤⠴⠀⣴⠒⠚⢧⡀⠀⠉⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⡾⠁⠀⠀⡼⠁⠀⠀⠉⠀⠀⠀⠙⣄⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠘⠀⠀⠀⢰⠁⠀⠀⠀⠀⢠⠀⠀⠀⠘⡆⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢀⣠⣾⠃⡞⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀
⠀⠀⢀⡴⠟⢩⡟⢸⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣆⠀⠀⢠⢸⡏⠛⢦⣀
⠀⡴⠋⠀⢠⢿⡇⡟⠀⠀⢸⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢹⠀⠀⠸⡄⣿⢆⠀⠈⠳⡀
⡼⠁⠀⠀⡏⠈⢿⣧⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡄⠀⠀⣷⠇⠈⡆⠀⠀⠹
⣇⠀⠀⠀⣇⠀⠈⠻⣄⣀⣿⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⣇⣀⠐⠃⠀⢀⠇⠀⠀⢀
⠹⣆⠀⠀⠈⢦⣀⠀⠀⠈⠙⠷⣤⣤⠤⢿⣦⣄⡀⠀⠀⢀⣀⣼⠧⣤⡤⠶⠛⠋⠁⠀⢀⡴⠋⠀⠀⢀⡾
⠀⠙⢷⣦⣀⠀⠈⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⣀⣠⠴⠞⠉⠀⢀⣠⡴⠋
⠀⠀⠀⠉⠻⢿⣶⣤⣄⣀⠉⠙⠛⠓⠶⠶⠤⢤⣤⣤⣤⣤⠤⠤⠶⠖⠚⠋⠉⠀⣀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠷⣶⣶⣤⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⣤⣶⡶⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠁⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣀
⠀⠀⠀⠀⠀⠀⡏⡇⠈⠙⣒⠤⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⡇⠑⠖⠋⠉⡷⠀⠀⠉⣉⡒⠲⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⠀⠣⣤⠴⠋⠀⠱⣀⡠⠔⠒⢦⠉⠉⠙⣒⣶⠦⠤⠤⣄⣑⣄
⠀⠀⠀⠀⠀⠀⢧⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠑⠒⠉⠀⠀⠙⠒⠶⠤⠖⠛⡇
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⢾⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡤⠒⣈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠏⢸⠀⠀⢱⢄⠀⠀⠘⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡂⠀⡟⠀⢸⠀⠀⠈⠚⠝⡲⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⣆⣇⣂⡇⠀⠸⠁⠀⠉⠑⠄⠘⡌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⣇⣆⣲⣿⡿⠉⠀⠀⠀⠀⠀⡀⢠⠀⠀⡇⢸⣶⠀⢀⠀⢀⠀⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⡶⠒⡟⣷⣿⡟⠀⠀⣀⠤⠀⠀⠀⡠⠋⠀⠜⠀⡿⣒⣾⠒⢀⣰⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣿⣷⣶⣿⣿⡇⣠⠞⠁⣀⣠⠔⠊⠁⠐⠋⠐⠚⡇⢻⣿⣶⣸⠘⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠛⣿⣧⣄⣿⡉⠉⠉⠉⠁⠀⠀⠀⣠⠀⠀⠀⠀⡰⠁⢀⣿⣿⣿⣺⡛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡧⢄⣿⣟⣀⣀⣠⣤⡴⠖⣋⠠⠀⠀⣠⠞⠁⢠⢿⠹⣾⣾⠻⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠍⠟⢿⡁⠀⠀⠀⠉⠀⠀⠀⠀⢀⣠⠞⠁⢀⡴⠃⣸⠀⣹⠻⢿⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⣥⣿⣃⠀⠀⠀⠀⢀⣀⠤⠖⠋⣀⣤⠖⠋⢀⡴⠋⢀⣿⣦⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⡌⠉⠙⠛⠛⠻⠶⠶⠒⠚⠋⠉⣀⡤⠖⠋⠀⢀⡞⠀⡏⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⢽⣇⠀⣀⠀⠀⢀⣀⣀⡤⠖⠋⠉⡁⠀⠀⢀⡠⠋⢀⡼⠁⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⣿⠻⢭⣝⣲⠶⠖⠒⠚⠈⣀⣠⣄⣒⡯⠶⠚⠁⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⡄⠀⠀⠉⠉⠉⠉⠙⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣓⠶⠤⠤⠤⠤⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠉⠁⡀⠒⠂⡀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠂⣀⠐⢀⡀⠐⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠐⠀⣀⠐⢂⣀⠐⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚⢁⣀⠒⢂⣀⠉⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣁⠤⠒⢂⠤⡉⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣁⣉⣉⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⣿⣿⣿⣷⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⠿⢿⠿⢿⣿⠿⠿⡿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠶⢶⣒⡢⣤⠒⠒⠒⡤⠖⢒⣒⡦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠔⠒⠒⣒⠶⠭⠾⣧⡠⠾⢤⡴⠭⠶⢶⡒⠒⠤⢍⡲⢄
⠀⠀⠀⠀⠀⠀⢠⠊⠀⡰⠋⠀⠀⡰⠋⠀⠀⠀⢠⠏⠀⠀⠀⠙⡆⠀⠀⠈⠳⣄⠀⠙⢦⡑⣄
⠀⠀⠀⠀⠀⣰⠃⢀⠞⠀⠀⠀⡸⠁⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢦⠀⠀⠱⡌⢦
⠀⠀⠀⠀⢠⠃⢠⠋⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⢇⠀⠀⠙⡄⠳⡀
⠀⠀⠀⢠⠇⢠⠃⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡄⠀⠀⠘⡄⠱⡀
⠀⠀⢀⠎⢠⠃⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠘⡄⠱⡀
⠀⠀⡜⢠⠃⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠘⡄⣣
⠀⠀⢈⠏⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠸⡃
⠀⠀⡞⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢱
⠀⠀⢧⣀⣀⡀⣀⣀⡼⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⠼
⠀⠀⠀⠀⠀⠉⠀⠀⠹⡀⠀⠀⠀⠀⠀⢀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠁⠀⠉⠒⠤⠤⠤⠤⠤⠒⠋⠀⠈⠉⠙⠓⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠂⠀⠒⠒⠀⠀⢠⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠘⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠂⠀⠀⠀⠀⠒⠄⠀⠑⡆⠀⠀⠐⠉⠀⢀⠀⠀⠉⠂⠀⢈⠆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢠⣰⠀⠀⡄⠀⠀⠀⡇⠀⠀⠀⣤⠀⠀⠴⡆⠀⠠⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⣦⣀⣀⡀⢀
⠀⠀⠀⠀⠀⣠⣶⢿⠀⠀⠀⣀⣀⢀⢀⢀⣀⣀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠋
⠀⠀⠀⢀⡼⠋⠀⢸⠀⠀⠸⠛⠧⠾⠾⠿⠿⢋⠆⠀⠀⡇⠀⠀⠀⣤⡤⡄⡄⣤⡄⠀⠀⠀⡇
⠠⠀⠀⠘⠁⠀⠀⢸⠀⠀⢸⠿⣿⢻⠙⡏⣯⠿⠄⠀⠀⡇⠀⠀⠘⠛⠓⠓⠓⠚⠀⠀⠀⠀⡇
⠠⠔⠂⠀⠐⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⢶⣤⠖⠛⠛⠛⢶⡴⠚⠛⠛⠲⣴⠞⠛⠛⡓⢦⣴⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠶⠛⠛⣾⠑⠟⠉⠙⢃⡟⠀⠞⠉⠙⠃⠘⣧⠞⠋⠙⠓⠹⡞⠛⠳⠌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⡿⢠⠀⠀⢠⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣼⠁⠋⠀⠀⣼⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⡿⢸⠃⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣼⢠⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢀⡇⠸⠇⠀⠀⣾⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣾⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢀⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿
⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻
⠻⣄⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠾⠷⠶⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠟
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢉⠀⠳⣄⡀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢢⡀⠀⠈⢳⣦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⢀⢤⠀⠤⢀⣀⡁⢐⢿⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠁⠛⠁⠉⠙⢒⠉⣠⠞⠉⠀⠑⣄⣀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡞⠀⡇⠀⠀⠀⠀⡟⢻⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠠⠴⢃⣀⣠⠴⠚⠀⣘⡇⣰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡖⠒⠴⠒⠒⠂⠀⣀⡤⠖⠋⠉⠉⠀⣀⣤⣴⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠲⢤⣤⣤⣴⣿⣅⣀⣀⣤⣶⠶⡿⠛⠝⣃⢈⣨⣤⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢤⣀⡀⢀⠀⠀⠈⠀⠈⠈⠎⠁⡡⠀⣀⣢⣤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡘⡌⡏⢹⡗⢺⠓⡟⠛⡟⢻⠿⣿⠛⡏⢹⣡⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠑⢹⡈⣇⠨⡇⢳⠠⡇⢸⠀⡇⢰⡇⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣾⡇⣿⠀⣧⣸⢰⣇⣸⣀⣧⣼⣹⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⡇⣿⣿⣝⡟⣿⣼⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡔⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⣶⢿⣿⣿⣿⣿⣿⣶⣶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡟⣻⣯⠊⠀⢙⢙⣛⣻⡟⢋⣭⣿⣿⣿⣿⣶⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠛⢷⣿⣏⣼⢜⣵⣯⣤⡼⣿⣿⣿⠿⣯⣕⡛⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢯⡙⢦⣄⠙⣿⣧⣼⣿⣿⡿⢽⣿⣿⣛⠿⢦⣌⣿⣶⣿⣷⣿⣔
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⢿⠳⣝⠢⣌⠳⣼⢿⢿⣿⠿⢿⣼⣏⣩⠬⠭⣳⣬⣿⣿⣯⡍⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⡄⡙⠪⠿⠷⣿⠃⠔⡅⢀⣰⢻⢫⢞⡿⣶⣤⣼⡿⠭⠭⠭⢝⠔⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣷⣬⣙⢒⠢⣴⡇⠈⠘⠁⣴⢃⣳⡿⢣⢊⢴⡿⢿⠿⠍⠉⠉⠵⣦⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢛⣻⣯⣤⣸⣿⣿⣿⡀⠀⢀⢿⡎⣿⠧⡳⣣⠉⢀⠟⠙⡲⢶⡤⡬⣑⢽⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠟⠻⠿⣿⣿⣿⣶⣿⣼⣧⣿⢠⢣⠃⢀⣞⣀⣀⣯⡵⣾⣮⣾⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣟⠲⠶⢤⣤⣙⣿⡟⣻⡟⣿⣿⣾⣿⣤⣾⡅⠉⠻⣗⣝⢞⢊⣦⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡷⣮⣔⢤⣹⣿⢠⣿⣿⡏⢹⣿⡟⡿⣿⣿⠆⠀⠈⡉⡀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠺⣵⣾⣿⢨⡇⠸⡇⣾⡿⢣⣇⢊⣿⡁⢀⠀⣯⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⣿⣿⠾⣧⣦⣇⣿⣷⢸⢉⢸⣿⣿⣼⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⡦⣭⣿⣿⣿⣷⣿⡿⠻⠟⠙⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣛⣿⣿⣻

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠀⢴⣾⣿⣿⡄⠈⠂⠙⠛⠀⠀⠐⠀⠀⠀⠀⠎⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠋⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣐⣀⣀⠀⠀⠀⠀⠀⠀⠋⡿⠃⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣼
⠀⡠⠒⠃⠀⠒⢀⠀⠀⠀⠀⠀⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢄⢀⣾⣟
⠞⠀⠀⠀⠀⠀⠈⠁⡄⠀⠀⠀⠉⠉⠤⠤⠤⠄⢀⡴⠒⠒⠒⠠⡀⠀⠀⠈⠃⡀⠀⠀⠀⠄⠠⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⣀⣶⣿⡟⢉
⠂⠒⠂⠐⠄⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⣄⡀⢈⢢⠤⢤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣷⣷⣿⣿⣷⣾⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠐⠒⠄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠩⢯⣿⣿⢿⣿⣿⣯
⠷⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⠈⠃⠀⠀⠀⡠⠒⠀⠀⠐⢉⣿⡿⠄⠉⠏⠛
⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢵⣤⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠰⢄⡀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⣆⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣶⠇
⠀⠀⠀⠀⠈⡟⠒⠠⠀⠀⠀⠣⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠉⠉⠙⠧⠀⢀⠀⠀⠀⠒⠤⢤⣿⣿⣿⡇⠀⣿⣇⣠⣴⢅⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠀⠠⣤⡼⣿⣿⣿⣸⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⢈⡄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠠⠀⠀⠀⠑⠀⢀⡎⠀⠄⠺⠃
⠀⠚⠀⢤⡀⠀⠀⠀⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⢔⣾⠿⠛⢿⡇⠐⠦⠀⠀⢀⣀⡀⠀⠀⠀⠃⠀⠀⠤⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀
⠀⠈⠐⠛⢧⡀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠄⣀⣀⣀⡀⠤⢰⠞⠋⡠⠀⡀⠀⠙⣄⠀⠀⠀⠀⠀⠉⠁⠀⠊⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠁⠀⠀⠀⠀⠀⠀⠀⠸⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠖⠒⠻⣷⢶⣷⠀⡀⠀⠀⠀⠀⠀⠀⠄
⠔⠋⠀⠀⠀⠀⠀⠀⠀⠂⠣⣄⣀⡄⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠄⠀⠀⠀⠀⠀⠁⠉⠉⠉⠚⢆⠀⠀⠀⠀⢀⠀⠀⠀⣠
⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠉⠘⠈⠁⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⢸⣇⣸⣀⣠⣾⠿
⠀⠀⠀⢀⡇⠉⠉⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢱⣄⣆⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⣽⣧⣾⣿⣿⡿⠿⠃
⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⢿⣿⣆⡀⠀⡜⠁⠀⠀⢀⣴⣞⡥⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢿⡏⡁⠀⠀⠀⠐
⠀⠀⠀⡀⠠⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⢤⣀⠀⠀⠀⠀⢻⣿⣷⣀⣠⡄⠀⣤⣿⣿⠟⢀⡀⠀⠀⠀⠀⠢⡄⢀⠍⠀⠂⠉⠷⣁⠂⠤
⠒⠀⡉⠁⠀⠈⠀⠀⠐⠄⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠈⠙⢿⣿⡛⠃⢰⣿⣿⣩⠆⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⠓⠒
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣿⣿⣶⣶⣉⡄⠓⢦⣹⣿⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠁⠀⠀⠀⠒⠋⠉⠉⠈⠞⠛⠙⠋⢻⡿⢿⠅⣠⣆⠋⣼⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣠⣤⠠⠤⣤⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠞⠛⢁⣬⡄⠀⠤⣄⣀⣀⣀⣈⠉⠉⠷⡿⠬⢝⣂⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣶⣼⣿⠿⠥⣴⠊⠉⠀⠁⠀⢠⣏⡽⠋⠀⠠⠿⠦⠀⠀⣠⠞⠛⠳⠀⠐⢄
⠀⠀⠀⠀⠀⣰⣿⣿⣼⣀⡀⠀⠈⠟⠛⠓⢢⠄⠀⠛⠁⠀⣼⣦⣠⠀⢀⠀⢻⡾⠛⢲⣤⣴⣨⠷⡀
⠀⠀⠀⠀⢀⠻⣿⡿⠃⢀⠾⠛⠏⠀⣀⣾⣅⣀⣤⣤⣄⠀⠀⡄⠁⠀⡸⠛⠈⠁⠀⢿⣭⣿⣿⣸⡇
⠀⠀⠀⠀⠈⢦⡙⢤⠰⣅⡄⢀⣀⠀⠙⠋⠟⠁⠀⠀⡿⠟⠛⠀⠀⠸⢧⣤⢤⣄⣀⣶⡿⢏⣴⡏
⠀⠀⠀⠀⠀⠈⠻⣦⢑⡚⠶⡄⣹⣳⣶⣀⠀⠀⣰⣾⣶⣾⣶⣦⣄⠀⣤⣜⡶⠋⢁⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣭⣅⡒⠂⠌⠉⠥⡄⠈⠉⠚⣉⣉⣉⡋⡉⠀⠀⣀⣒⣭⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠏⢻⠓⠖⡶⢶⢤⣤⣤⣤⣦⣶⣶⣶⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢹⠀⠀⠀⠀⠀⠈⠀⠏⢹⡟⡟⢻⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠍⠣⠘⠏⠛⠙⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡤⣤⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣭⣿⣹⣿⣽⣿⣾⣿⣽⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⣠⣴⣶⣲⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣏⣠⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣷⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡷⣿⣿⣇⣼⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⡜⢿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⣴⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣷⣄
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠙⠋⠛⠙⠃⠘⠃⠈⠉⠉⠉⠛⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣿⡇⠀⣶⣶⡀⡖⢀⢀⣶⡆⢰⡖⠃⢰⢶⡀⠀⠀⣴⣆⠀⢷⣄⢰⡏⣶⠀⠀⣷⢷⡆⣾⢴⡇⣾⠀⢿⠗⢰⡶⠂⠀⣿⣿
⠀⠀⣿⡇⠀⠉⠉⠈⠉⠉⠈⠉⠉⠈⠉⠁⠈⠈⠁⠀⠈⠁⠉⠀⠁⠁⠈⠉⠁⠀⠀⠀⠈⠀⠉⠈⠁⠉⠀⠉⠁⠈⠉⠉⠀⢹⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠘⠒⠀⠀⠐⠀⠂⠐⠃⠀⠀⠀⠀⠂⠀⠀⠒⠂⠐⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠁⠈⠀⠀⠁⠀⠈⠀⠁⠀⠈⠉⠉⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⣿⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠿⠀⡀⠀⠀⠀⣴⠋⠙⡃⠀⡀⠀⣿⠀⣀⠀⣰⡀⠖⢀⢀⠀⠀⢀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⣿⠻⣆⠈⡟⢹⣦⡿⢹⣧⠹⣾⠋⠀⠺⣯⡀⠀⠀⣿⠀⠘⣿⣾⡽⠃⣿⣨⠽⡇⣿⠁⣿⢼⠛⣿⣼⡽⠗⠀⢸⣿
⠀⠀⣿⡇⠀⠛⠀⠻⠓⠗⠘⠃⠛⠞⠃⠛⠘⠗⠀⠒⠚⠃⠀⠀⠘⠓⠚⠋⠙⠓⠊⠛⠙⠚⠓⠛⠒⠛⠚⠃⠻⠘⠓⠚⠀⢸⣿
⠀⠀⣿⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⢀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⡀⠈⠁⡈⠉⡀⠀⠀⠀⢀⠀⢁⠀⢀⠈⡁⠉⠀⠉⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⣄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠰⠄⠀⠠⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠄⠠⠀⠠⠀⠀⠠⠀⠠⠄⠄⠀⢸⣿
⠀⠀⣿⡇⠀⠒⠂⠐⠀⠆⠀⠐⠀⠒⠐⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠂⠂⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠑⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠃⠀⠐⠈⠛⠀⢸⣿
⠀⠀⣿⡇⢀⡉⡁⢈⠁⠁⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⡀⡀⠀⠀⡀⠀⠀⡀⠀⠀⢀⢀⣉⠉⣁⡀⢀⠁⠉⠀⢸⣿
⠀⠀⣿⡇⠈⠁⠠⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠄⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠄⠠⠀⠄⠀⠀⠀⠀⠠⠠⠀⢸⣿
⠀⠀⣿⡇⠀⠐⠒⠀⠐⠀⠀⠀⠐⠀⠈⠂⠀⠀⠀⠀⠀⠀⠂⠀⠂⠐⠒⠀⠐⠀⠀⠐⠀⠀⠂⠐⠐⠀⠛⠘⠛⠛⠃⠚⠀⢸⣿
⠀⠀⣿⡇⠀⠉⠀⠈⠀⠁⠁⠀⠀⠀⠁⠀⠀⠈⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⡸⣆⠀⢸⠥⢿⢰⢲⢸⡂⠀⡺⣇⣶⣴⢶⢶⢴⣶⣷⣤⠀⠀⡇⠁⣷⡗⡿⢉⣿⢉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⡀⠀⢁⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠈⠀⠀⠀⢀⠁⢈⡀⠁⠀⡉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠋⠀⢸⣿
⠀⠀⣿⡇⠀⠀⣀⠀⢀⠀⢀⡀⠀⢀⢀⣀⠀⡀⠀⡀⢀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⣸⣿
⠀⠀⢿⣷⣄⣃⡙⠘⡈⡀⣈⠃⠀⠘⠈⠉⠀⠁⡀⠁⠘⢛⡛⢛⢁⡀⠘⠉⠙⠐⠀⠛⠛⠛⠛⠛⠋⠛⠛⠂⠈⠁⠈⢃⣠⣿⡟
⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣓⡲⢦⣄⢴⣶⣶⡶⢀⣴⡶⠶⣂⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣛⣭⣽⣶⠯⣉⣓⣊⠣⠻⣛⣑⡻⠋⣒⣋⣉⡩⢭⣭⣝⡻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢟⣵⣿⣿⡿⣫⣶⣿⣿⣿⣿⢇⣾⣿⣿⣿⣦⡹⣿⣿⣿⣷⣝⢿⣿⣷⣌⢳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣱⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣷⡙⣿⣿⣷⡝⣷⡀
⠀⠀⠀⠀⠀⠀⣰⣿⢟⣼⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⢇⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣜⢿⣄
⠀⠀⠀⠀⠀⢰⣿⢏⣾⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣎⢿⣆
⠀⠀⠀⠀⢠⣿⢏⣾⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣎⢿⣆
⠀⠀⠀⢀⣾⢏⣾⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣎⢿⣆
⠀⠀⠀⣼⢏⣾⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣎⢿⡄
⠀⠀⠀⠉⣼⣿⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⡌⠁
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⡄
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠙⠛⠛⠛⠛⠛⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡎⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⠿⠃⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠘⠿⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣊⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣼⣃⠖⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡿⠩⠂⠀⠀⠀⠀⠀⠀⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡗⠵⠀⠀⠀⠀⣄⡀⠀⢾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣹⡄⡠⠀⠀⠀⠀⡄⠈⠆⢸⠛⡡⠄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⠥⡄⠀⠀⣷⣤⠐⢄⣰⢃⡼⠀⠔⠃⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡨⡄⠀⠀⠻⣿⡇⠀⡹⢁⣤⣤⠀⠀⡀⠄⠚⠅⣀⠄⠒⡩⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠣⢄⡀⠛⡄⠀⠀⠀⢠⢿⡂⢅⠀⠀⠠⢌⠣⣀⢣⣿⣿⠏⠀⠌⢡⠀⢰⢊⡀⠤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠠⠌⣦⡘⡄⢰⣶⠀⣐⣗⠧⠀⠀⠐⡄⢱⣸⠼⠛⠁⠀⡐⠀⢀⡴⠋⠁⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠈⠑⣆⠀⠁⠀⠃⣿⢎⠄⠀⢀⠜⠁⠀⠀⠉⠐⠒⠇⣀⣼⠤⠐⢒⠞⠀⢀⠤⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠤⠀⣈⣤⡀⠘⡟⠪⣷⣂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⡔⠋⠀⢠⠁⢀⠌⢀⡠⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠈⢧⠀⡷⠊⠀⠇⠀⠀⠀⠀⠀⢀⠄⠒⠒⠄⡀⠀⠀⠈⠐⠤⣴⠖⠃⠒⠮⣀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠠⠄⡉⠒⠤⠄⡤⠋⠀⠀⠀⡀⠤⠤⠤⠄⣀⡎⠀⠀⠀⠀⠈⢊⠀⠀⠀⠀⠈⢋⢣⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠥⣦⠊⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢀⣠⠊⠀⠈⠢⠀⠀⡤⢀⡀⠀⠀⠸⠄⣉⣁⡉
⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠄⣀⢰⠖⢹⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡔⠉⠉⠑⠄⠀⠀⡄⢀⡅⠀⠈⠢⡀⣰⠢⡘⠦⠈⠆
⠀⠀⢀⡤⠒⠫⡀⠀⠀⠱⡄⠀⠈⠑⠺⡀⠀⠀⠀⢨⠀⠀⠀⠀⠀⢣⠀⠀⠶⠂⠀⢠⠗⠈⢡⠀⠀⠀⠘⡅⠣⣈⠄
⠀⡴⢋⡠⠄⠒⠒⠓⣒⣦⠬⠶⣄⣀⡀⢡⡀⠀⠀⠈⠄⠀⠀⠀⠀⡠⠥⡀⠀⣀⠔⠁⠀⠀⡘⠀⠀⠀⠀⢰
⣰⣀⠁⠀⢀⡔⠊⠉⠀⠀⢀⠠⠂⠉⠉⠀⡱⣄⠀⠀⠈⡶⠠⠄⠚⢄⠀⠈⠈⠀⠀⠀⠀⡠⠁⠀⠀⠀⢀⠃⢀⣠⡟
⠁⠀⠀⠀⠀⠀⠁⠒⠂⠴⠥⠄⠤⠄⠒⡨⢤⣫⠶⢆⠈⠀⠀⠀⠀⠀⠑⠤⣀⣀⣀⡤⠊⠁⠀⠀⠀⢀⠌⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣬⢛⡡⠑⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠤⠀⠐⠒⠂⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡾⢏⠢⠀⠀⢀⠔⠊⠱⢄⠀⠀⠀⢀⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡽⡟⠌⠊⠀⢀⠜⢁⣀⠤⣴⠋⠉⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡸⡗⡌⠀⠀⢀⠊⠉⠀⢀⠔⡇⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣔⣾⠣⠈⠁⠀⠀⣈⠀⡀⢴⠏⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⣆⣽⢣⠉⠀⠀⠀⠀⡏⠁⠀⠎⠀⠀⡄⠔⠁
⠀⠀⠀⠀⠀⠀⠮⣽⣉⡍⠀⠀⠀⠀⠀⢀⠀⠐⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠰⡰⠀⠀⠀⠀⠀⠀⠀⠈⠆⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠉⠊⠑⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠚⠋⣁⡠⠶⠖⠳⣄⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠒⠊⠉⠀⠀⠀⠀⠀⠣⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡆⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢡⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡀⠀⠄⠉⠀⠀⠈⠳⡖⠒⢦⣄⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠀⠀⣠⠞⠋⠙⠳⣄⣧⣴⠤⢤⣀⠀⠀⠀⡇⠀⠀⣨⣅⡀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡗⠀⠀⣸⢁⣀⣤⠤⠤⠤⣄⠀⠀⠀⠈⠉⠛⠳⣇⡴⠛⠁⠀⠙⢦⣤⣬⣇
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠠⣿⣤⡀⣠⠟⠋⠁⠀⠀⠀⠀⠈⠳⣤⣤⠴⠶⣏⢉⡽⠛⠋⠉⠉⠛⠳⢦⡀⠸⡇
⠀⠀⠀⢀⡴⠛⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⣷⠒⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⠇
⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⢠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣤⡾⢧⣄⣀⣀⣀⣀⣀⣈⣹⣆⣀⣏⣀⣀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠘
⠀⣠⠞⠉⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣦⣤⣴⠚⢉⠉⢻⡄
⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠇⠀⠀⠀⢰
⣠⠟⠀⠀⠀⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠰⣾⡁⢠⡄⠀⠀⢀⣤⠀⠀⠀⢠⣄⠀⠀⠀⢀⣤⠀⠀⢠⣿⠀⠀⠀⠀⣼
⣏⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠈⢷⢾⡇⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⢸⣿⣆⠀⠀⠀⢹
⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢸⡇⠀⠀⢸⡿⠀⠀⠀⢸⡇⠀⠀⠀⠈⣿⠀⠀⢸⡷⠈⢳⡄⠀⠘
⠀⠈⢹⡶⠶⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⢺⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⣠⢶⡄⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠄⠀⠀⢠⡆⢹⡇⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠳⣄⣠⠾⠁⠀⠹⣶⠶⠶⠓⢦⣀⣤⣄⠀⠀⣠⣤⡶⠋⣧⢸⡇⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⡜⢸⠀⢠
⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠢⡀⢀⡠⠌⠃⢾⠀⣾⠋⣼⠁⠀⣿⠘⣧⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⡇⢸⣤⠞
⠀⠀⠀⠀⠸⡇⠀⠀⢢⡀⠀⠒⠉⠃⠄⠀⢠⡟⢠⡇⣼⠃⠀⠀⢸⡆⢻⡀⠈⣷⠀⠀⠘⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣼⠃⢸⠁⣿⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠐⢂⣀⡀⠠⠈⠉⣹⠃⢸⣷⠏⠀⠀⠀⠈⣷⠘⣧⠀⢻⡀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢠⡇⠀⢀⡟⢠⡏⢸⡟
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠉⠢⠀⠀⣰⠏⢀⣼⡏⠀⠀⠀⠀⠀⠸⣇⠸⣇⠈⢣⠀⠀⠸⡆⠀⠀⢸⠇⠀⠀⠀⡈⠁⠀⡘⢀⡾⢠⡿⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠐⢤⡀⠀⠶⠈⠉⠙⠓⢛⡿⠀⠀⠀⠀⠀⠀⠀⠹⣆⠹⣦⠈⢣⠀⠀⠙⡀⠀⠀⠀⠀⠀⠰⠁⠀⡜⢁⡾⢡⡿⠁
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠑⠠⡀⡀⠄⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡘⢷⡄⠑⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠊⣠⠏⣠⡿⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠘⠢⠀⠈⠉⠁⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⣀⡴⠞⢁⣴⠏
⠀⠀⠀⠀⠀⠀⢸⡆⠐⢀⠘⠓⠂⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣈⠉⠓⠒⠶⠖⠚⠛⢉⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠸⡇⠀⡀⠑⠶⠆⠈⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⠶⠶⠶⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠦⠤⠀⢁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠢⢤⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢤⣀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⢠⠀⠀⢀⡀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁⠀⠀⣴⠋⠀⠀⠈⢧⠀⠀⠀⠀⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⠀⠀⠚⠁⠀⠀⠀⠀⠈⠣⠀⠀⠀⢀⣀⣤⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣛⣳⣦⣤⣶⣤⣤⣶⣶⣶⣚⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⡆⢰⠀⠀⠘⡄⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⢸⡄⢳⡄⠀⠻⡷⠀⠀⡷⠀⢻⠀⠀⠀⠈⠳⠀⡏⢸⠀⡀⣸⠀⡇⢸⠀⠀⠀⠙⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⡇⠀⠀⢻⠀⡇⠀⢰⡆⠀⣦⠀⣤⠀⠀⠀⠀⠛⠀⡄⢰⠀⠃⠀⠀⠀⡁⠸⠃⢆⠀⠀⠀⠀⠀⠀⠈⣀⠀⠸⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠘⠀⠃⠀⠘⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡆⠀⣶⣶⣾⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠿⣿⣷⣦⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⣿⣿⡿⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⣾⣿⣿⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⢠⣾⣿⡟⠉⣿⣿⣿⠀⢸⣿⣿⡇⠀⣼⣿⣿⠀⢀⣠⣤⣤⣤⠀⢀⣿⣿⡟⠀⠀⣸⣿⣿⡇⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣰⣶⣶⡄⠀⣼⣿⣿⠇⠀⣾⣿⣿⠛⠛⣛⣻⡋⢀⣿⣿⡿⠀⢰⣿⣿⠇⠀⠿⠿⠿⠿⠋⠀⢸⣿⣿⣇⠀⣠⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⠀⠀⢿⣿⣿⣧⣾⣿⣿⠏⠀⠀⠻⣿⣿⣤⣾⣿⠟⠁⠸⣿⡿⠇⠀⠿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⡀⠀⠘⣦⠀⠈⠁⠀⠈⠙⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠸⣦⠀⠹⣧⠓⠀⠈⢷⣤⡤⠙⠛⠋⠀⢀⡀⠀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣄⠀⠀⠀⣴⣆⣹⡇⠀⠹⠗⠚⠂⠀⢁⣄⡀⠀⠀⠶⣉⡉⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠿⢿⡆⠀⠀⠈⠉⠉⠁⠀⠀⠀⣠⡴⠿⣉⠰⠗⠈⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡲⠆⠾⣿⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣛⡿⠿⠛⠉⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣍⡛⠷⢶⣿⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿
⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠿⠿⢿⠿⣿⠿⠿⢿⠿⣿⠿⢿⠿⣿⡿⢿⣿⡟⠻⠛⠿⠿⠿⠿⠿⠿⣿⡿⠿⣿⠿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣠⣿⣷⣀⣿⣿⣀⣿⣿⣦⣾⣧⣴⣿⣿⣤⣶⣾⣶⣿⣴⣦⣤⣖⣶⣞⣧⣼⣿⣆⣸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣴⣿⣷⣤⣿⣤⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠒⠖⠒⢶⡞⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⢒⣿⡟⠀⠀⠤⠀⠂⠀⠀⠀⣼⣿⢿⣦⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⡉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢀⡜⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠠⠀⠀⠀⠀⣴⣶⣶⣦⣠⣬⣿⣄⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⢰⡿⡿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠆⠀⠀⠀⠘⣯⣝⣻⣿⣿⣿⡛⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣴⠟⡷⠁
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⣿⣿⣿⣿⢿⣙⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣹⣦⠞⠁
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠟⠉⠘⠻⠿⠿⣷⣤⣄⣀⣀⠀⠀⠀⠀⠀⣀⣀⣤⠄⠘⣭⣶⠏⠉
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠈⠛⠛⣻⣿⣿⣿⣿⣿⣿⣯⣀⡤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠴⢿⣿⠟⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣦⣤⣤⣀⣠⣤⣤⣤⡖⠋⣁⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡈⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠄⠀⠀⠀⠀⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡐⠐⠂⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⣤⣠⣀⣄⣀⡀⡄⣤⠆⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⡐⠀⠀⠈⠁
⢀⠄⠀⠀⠠⠢⠀⠀⠀⠀⠀⠀⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈
⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⠀⠀⠀⠀⠈⠱⠶⠴⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀
⠀⠳⣄⠀⢀⠠⠀⠙⠦⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠰⠃⠉⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⢀⡀⠤⠈⠁⠰⠀⠀⠀⠀⠀⡀⠙⠓⡆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠏⠀⠀⠀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠂⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⡠⢀⣠⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣮⣵⣦⣴⣧⣼⣯⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠀⡀⢀⠀⢀⡀⢀⠀⢀⠀⠀⣀⣤⣤⣄⡀⢀⣀⣀⣀⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠐⣧⠀⣧⠀⡇⠀⠁⢸⠀⢸⠀⡼⠋⠀⠀⠀⠉⢉⠁⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⢠⠀⠀⠀⡔⠙
⠀⠀⢸⠀⠀⠙⣠⠇⢸⠄⣸⠀⡌⠀⠀⡀⠀⠀⠀⠁⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⢀⢀⠀⠀⠀⠀⠀⡏⠉⢹
⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⢀⠀⠠⡀⠀⠀⠀⣄⠀⠀⣄⠀⠈⠀⠀⢀⡀⣀⠀⡀⠀⣀⢀⣀⠔⡁⠀⠀⠀⡉⠂⠀⠰⠋⠉⠉⠴
⠀⢰⠇⠀⢈⣀⠀⢀⠀⠀⠈⠢⠀⠁⠺⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠻⠄⠿⠵⠛⠖⠋⠈⠇⠀⠁⠂⠁⠃⠩⠀⠈⠀⡀⢀⠀⠀⠉
⠀⠸⠀⠨⠉⠁⠀⠈⠨⠅⠂⠂⠐⠀⠠⠄⠀⠤⠄⠠⠠⠀⠤⠀⠀⡂⠀⠀⠀⠂⠂⠐⠐⢁⡀⠀⠲⠤⠖⠀⣠⠙⠀⢰⡂⠀⡀⢠
⠀⠀⠀⠀⠀⢙⢚⡁⠂⠀⠀⠀⠀⠀⠄⠀⡀⢰⠆⠰⠇⡀⢀⡤⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠅⠉⠀⣁⡔⠂⠀⠇⠀⠉⢉
⠀⠀⠀⠂⠀⠘⠐⠀⠂⠈⠀⠄⠀⠀⠀⢠⡄⠀⣀⡀⠀⠁⠈⠀⠀⣤⣤⣤⣬⣤⣥⡄⠀⠐⠄⠀⠂⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠁⢀⡸⡄⠐⢈⣆⡠⠀⠀⠠⠠⠀⠀⣹⠀⠿⠆⡤⠤⣄⣠⠤⣀⢠⡤⠤⢤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠄⠀⠄⠰⠀⠀⠀⢰
⠀⠀⢀⠀⢨⡀⠀⠀⡄⠀⠀⠀⠀⠠⠀⠀⣿⠀⠀⠀⡇⣤⡈⠁⣤⢸⡆⢹⠀⣿⣶⢰⡟⠁⣠⢶⣄⣴⣦⠀⠀⠀⠀⠀⠀⠀⠒⠘
⠀⠀⠁⠀⠰⠂⠀⠀⠂⠀⠀⢃⣀⣀⢈⡇⠀⢷⠀⠀⡇⣼⡇⠸⡿⢸⠇⢸⠀⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⢸⡁⢀⠀⠸⠀⠁⠰⡀⠀⢸⠀⡂⢸⠀⠀⠀⣿⡀⣇⣩⠴⣤⣠⠞⢰⣏⣀⣹⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⣀⡉⢠⠉⢀⢈⢀⠉⠀⠉⠉⠉⢈⡀⣀⣀⣬⢥⣄⣀⣀⣛⣠⣤⣀⠀⣀⣀⣈⣋⣀⣀⡀⣀⣈⣀⡀⠁⠁⡀⠀⠀⠀⠀⠨⠀⠨
⠀⠒⠈⠤⠤⠤⢹⡀⠰⠂⠀⡄⠐⠀⢯⢹⡏⢠⢀⣤⡀⠙⣿⠀⠀⣶⡌⠛⣥⡙⢠⣶⣿⢸⡇⡇⠲⢾⡇⠀⠀⠀⠠⢀⠀⠀⠠⡀
⠀⠀⣀⣀⣀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠈⢧⠁⣿⢸⣿⠃⠀⣿⠀⠀⡉⢶⣧⣟⢿⡄⣭⣿⢸⡇⡇⣿⠙⠃⠐⠀⠀⠂⠘⠂⡝⢽⡇
⠀⠀⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣄⣉⣴⡀⠛⢀⡀⣿⡈⣇⠛⢀⡄⣛⣿⣘⣶⣇⣻⠠⠮⠿⠧⠀⣠⠀⢳⡿⠾⠟
⠀⠀⢠⡴⠀⠀⠀⡀⡀⠀⠸⣿⣿⡇⠀⠘⠛⠃⠈⠉⠀⠙⠛⠉⠛⠻⠉⠛⡟⠋⢉⡩⢉⠉⠉⠙⠙⠀⢰⣀⠀⠰⠇⠀⠀⠀⠠⣀⠄
⠀⠀⠀⠁⠀⠀⠐⠒⠒⠀⠀⠀⡀⠀⠀⠀⠘⠈⠉⠉⠁⠀⡀⠀⠀⠀⠀⠀⠁⠀⢬⠀⡠⠃⢀⣀⠀⠀⠀⠟⠀⢶⣶⣶⡖⠀⠀⡯
⠀⠀⢁⢨⡤⠀⠄⠤⠤⠤⢬⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠓⠆⠀⣀⢱⠎⠀⠀⠁⣤⣾⣿⣿⣿⣿⠀⠀⠀⠈⠸⠏⡁⡄⠀⠟
⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⡂⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡠⠀⢀⠁⠂⠈⠁⡀⠀⠠⡄
⠀⠀⠀⠀⠀⠟⡁⠀⠉⠃⠀⠃⠀⠀⡀⠀⢀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⠀⠮⠆⠭⠂⠀⠀⠂⠈⡇⠀⠁⠐⠀⠀⠄⠤⠈⠀⠁⡀
⠀⠀⠀⢠⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠨⡅⠈⠯⠀⠀⠀⠀⠀⠀⠀⢰⡂
⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠄⠀⢰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⡟⠀⠀⠀⠀⠀⠀⠓⠀⠀⠴⠛⠂⠀⠛⠳⠆⠀⠁
⠀⠀⠐⠐⠃⠈⠀⠀⠀⠉⢙⠠⠄⠀⠸⠃⠀⡀⠐⠀⠰⠀⠀⠀⠸⠈⠀⠀⠀⠀⡇⢰⡋⢸⠀⢸⡅⣶⠚⠛⢲⠰⣤⠂⡖⠛⠋⢷
⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⠁⠀⠀⠀⠈⠀⠉⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⡽⠞⠛⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢼⣀⣠⣼⣥⣀⣠⠼⠷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⣇⠀⢀⡿⠁⠀⠀⣹⣅⠀⠀⠀⢸⠏⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠉⠛⠉⠷⣤⡤⠞⠁⠘⠦⠤⠶⠋⠀⠀⠀⣸⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠓⠶⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠴⠛⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢀⡤⢶⡟⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠹⡄⠀⠀⠸⣧⣤⡀
⠀⠀⠀⠀⣴⡟⠀⢾⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⡿⠈⠻⣆
⠀⠀⠀⢸⣿⠁⠀⠘⠦⣤⡴⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠇⣠⡾⠁⠀⠀⣿⡄
⠀⠀⠀⢸⠘⣇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣼⣄⣀⠀⠀⢀⣠⠾⠛⠁⠀⠀⠀⣼⢻⡇
⠀⠀⠀⠘⣇⠈⠳⣄⡀⠀⠀⠀⠀⠈⠙⠦⠤⠤⠴⠞⠉⠳⠶⠤⠤⠶⠚⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⣠⠞⢁⡾
⠀⠀⠀⠀⠘⢧⡀⠈⠛⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⢁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⡀⠉⠙⠳⠶⠤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠞⢛⣉⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠦⠤⣤⣤⣀⣈⣉⣉⣉⣉⣉⣉⣩⣭⣭⠭⠴⠶⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⣿⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⣹⢣⢎⣿⣝⢯⣝⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢖⡌⠀⢛⡠⡿⠋⢻⠘⣯⠈⣷⠈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⣧⣦⡿⣤⣤⠴⢋⣰⣟⣠⡿⠀⡿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠋⢛⣡⠤⡶⠖⠓⠺⠯⠉⣀⡀⠡⠚⠗⣠⣿⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠤⠊⠁⠀⣀⣤⣤⣶⣶⠿⠷⠶⢦⣶⠶⣾⠟⣿⣿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⣤⠶⠛⠋⠉⣟⢡⠤⠤⠒⢻⣋⡡⠾⡟⢆⣿⣿⡇⣿⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣡⡼⠋⠁⣠⠖⠋⠉⣀⡤⠖⠚⠉⠁⠀⣀⣘⣠⠾⣻⣿⢃⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢇⣿⠀⣰⠋⠀⢀⡴⠊⠁⢀⡠⠴⠒⠈⠉⣁⣠⡴⠞⣉⠴⠞⠛⣾⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠈⢿⡄⣧⠀⠀⡎⠀⢀⡖⠉⢀⣠⡴⠚⠋⢉⣠⡖⠃⠀⣀⣠⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠦⣄⣀⠈⠚⠳⣤⣇⣀⣿⢁⣴⣿⣃⣀⣤⣴⠿⣿⣏⣻⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠈⠉⠑⠒⠲⠬⣬⣭⣭⣯⣹⣯⣿⣷⣿⡶⣿⢿⢻⡏⡿⢹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠈⠄⠇⠸⠄⢸⠠⡇⢸⡇⣿⡞⢸⢡⣇⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡆⢸⠸⡇⢸⢠⡇⣇⣿⣾⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠐⡈⠂⠀⠀⠀⠀⠀⣧⢸⢠⠀⡟⢸⠰⢼⣇⡏⣿⣿⣟⣿⣷⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⣦⣯⢪⢺⠈⠀⢢⠁⢸⢸⣸⢰⣇⣿⣿⣿⢸⣿⣿⣿⣿⣯⣿⢿⣏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠺⣏⣏⣿⣧⢱⣸⢸⣼⣿⣿⣼⣿⣷⣿⣟⣿⣿⣿⣿⣿⢿⡿⢯⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠾⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄⠉⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠹⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡀⢀⡠⠔⠒⠂⠈⠉⠉⠉⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⠂⠀⠘⠀⠛⠛⠻⢾⣶⣶⠄⣴⠀⠀⡀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠘⢖⠐⠁⠺⠆⠐⠀⠀⣠⡀⠀⢁⡴⡜⠁⠀⠀⠒⠀⠐⠛⠀⠀⠀⠀⡰⠋⠂⠀⠛⠁⠀⠀⠠⠤⠀⠑⢤⡀⡀
⠀⠀⠀⡠⠊⠀⠀⢁⠅⢠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢁⡐⠶⠀⠀⠀⠐⠛⠂⠀⣄⠀⠀⣷⡈⠙⠦⣀
⠀⢠⠊⠀⠀⠀⠀⠀⠀⡏⠀⠐⠘⠀⠀⠶⠀⠀⠀⠀⠀⠁⠁⠀⢉⡽⡓⠀⠀⠀⠀⠀⠀⢠⣄⠋⠀⠂⠂⠀⠀⠀⠀⠀⠀⠌⠁⠀⢻⡇⠀⠀⠈⠢⡀
⢠⡏⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠘⠃⠀⠙⠃⠈⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠇⠀⠀⠀⠀⠈⢦⡀
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⢳
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣴⣆⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⠿⢿⣁
⣟⣠⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⢀⡉⠉⠓⠛⠐⠶⠦⠤⢿⣀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⢀⢠⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠐
⠸⣿⣶⠀⠀⠀⣠⡼⠋⠀⠀⠀⠋⠁⠀⠀⠀⠀⣀⠀⠀⠀⠉⠓⠦⣤⣴⣄⣰⣀⡄⠤⠤⠔⠂⠂⠈⠉⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢀
⠀⠈⠻⣷⣿⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡀⣸
⠀⠀⠀⠀⠈⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣀⡀⣼⣿⠋
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡠⠟
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣘⣆⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠶⢄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣌⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣦⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠧⠄⢤⡴⠤⠶⠤⠐⠊⠁⠀⠈⠓⠴⢦⣄⣤⣀⣤⣤⡤⠴⠒⠚⠉
⣿⣿⡿⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡋⠻⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠸⣻⣿⣷⠏⠁⢶⡷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣸⡂⢀⣤⣾⣷⢠⣤⢤⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣿⣾⣿⣿⣿⡷⣌⣁⣉⣼⠟⠛⢩⢛⣳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡀⠈⠉⠉⠁⠈⠻⠏⠀⠀⠀⢠⣜⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠿⠟⠻⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⢦⣄⡀⠀⠠⠀⠀⠀⣄⣠⣴⣾⠿⠟⠉⠉⠁⠀⠀⢤⣄⠀⢰⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠙⠙⠛⠚⠘⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠏⠆⠚⠋⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⡟⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠉⠁⠐⠒⠐⠦⠤⢦⡤⣤⣤⣤⢶⠶⠶⠞⠛⣋⠉⠳⣤⣧⢉⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢄⣄⡀⢀⡀⠀⢀⠀⠀⢀⡀⠄⠉⣤⢸⣷⣦⣄⢇⡉⣰⣖⣼⣿⣶⠻⢿⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢁⡯⠵⠛⣯⣫⠾⠛⢦⣤⢶⠄⣰⠟⣿⢿⠋⠛⢾⣿⣿⡀⠙⠻⣿⠆⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠁⠀⡞⠚⠁⣼⠒⠁⠀⠈⠛⣶⠀⠀⠉⣿⡇⠀⣰⣿⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣷⠀⢰⡆⠀⠀⠀⠀⠀⣿⠀⣤⡀⠀⢰⣿⠐⠀⢸⣿⡧⠀⣿⡏⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠸⣷⠀⠀⣷⠀⠀⣿⣸⣷⣵⠀⢸⣿⣣⠀⣾⣿⠇⣀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿⡆⢡⣼⣀⢠⣿⣿⣿⡿⠀⣿⣿⡏⢀⣿⣿⡇⢸⡿⣿⣿⣷⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⣾⣇⣸⣿⣿⡸⣿⢮⣿⣧⢈⣿⣿⡃⢰⣿⣿⡆⣿⣻⣿⣿⡧⣞⣶⣾⠭⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⣿⣿⢰⣿⣿⡆⢿⣿⣿⣿⣾⣿⣿⡇⢹⣿⣿⡁⣷⣿⣿⣿⣷⣶⣴⢶⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠤⠿⣿⣅⣍⣽⣷⣏⣘⣿⣾⣧⣽⣿⠽⣿⣿⣿⡿⣿⣷⡿⣿⣷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⢿⠿⠿⢿⡿⠿⠷⠛⠒⠛⠛⠁⠤




⣿⣿⡿⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⡆⣴⢳⣴⣦⣶⢰⡶⢰⣦⣶⢰⣶⢰⡖⡆⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠈⠉⠈⠁⠈⠈⠉⠈⠀⠉⠉⠈⠉⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠰⣍⡷⢾⣹⢾⣏⡷


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⣉⢷⢊⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢠⠖⠁⢱⠟⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⡇⠀⢀⠞⠁⢀⠔⠁⢨⠀⡧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠀⠀⢧⣸⡳⢀⠔⢡⡀⠀⠀⢠⠁⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⢾⠴⠯⠥⠬⢁⣈⡠⠔⠁⠀⠀⠀⠀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⡠⠤⡀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠘⡝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⣀⠞⣡⠔⠁⠀⠀⠀⢀⡠⠤⠤⠛⠛⠠⠬⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢾⡼⠋⢁⣎⡥⣤⠀⠀⠀⣤⡏⠁⠈⢉⡩⠍⡉⠉⠉⢹⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣭⣭⣭⡍⠉⢙⣆⠀⠀⠀⡇⠀⢰⢡⠴⡒⡌⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣍⠉⠉⠉⠉⠉⢉⡝⠈⠳⡀⠀⡇⠀⠘⢜⠚⠿⣣⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⣒⣒⣒⣊⡹⠀⠀⠀⠘⠋⠷⣶⣶⣶⣿⣿⣶⣶⣆⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣭⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⣀⣤⣤⣾⣶⡶⠾⠷⠶⣶⣶⣤⣤⣀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠁⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀⠀⠀⢠⣿⣿⣿⠏⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠘⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣿⣿⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣛⣛⣛⠛⣛⡓⡒⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⡟⠟⢻⠛⠛⠛⠛⠛⠛⠛⢻⣿⠟⠛⠛⠛⠛⠛⣿⡿⠻⣿⡟⠛⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⡧⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⣇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⣧⡴⠻⢿⣶⡄⠀⠀⠀⣿⠀⢀⣴⠟⢿⣷⡄⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡌⢀⣿⣷⣤⣾⣿⡿⠀⠀⠀⣿⠀⢸⣿⣤⣾⣿⣿⠀⠀⢇⣿⡀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠐⠂⣀⣠⢾⣿⠇⢸⠏⠻⠿⠶⠟⠁⢤⣀⣀⣀⣴⠀⠛⠿⠷⠞⠁⢇⠀⢸⣹⣇
⠀⠀⠀⠀⠀⠀⣴⡟⠉⠑⠋⠊⡠⣸⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⡀⡘⣆⠀⠻⣿⣦⡀⠀⠶
⠀⠀⠀⠀⠀⠰⣿⢂⠄⠀⢀⡴⡣⠃⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢳⡈⠳⠄⠈⠺⢽⣦
⠀⠀⠀⠀⠀⠀⢻⣞⢆⢰⡋⠀⠀⢀⠀⠀⠁⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢦⡀⠀⢰⠈⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠿⠒⢶⣏⠀⠰⣄⠀⢀⣤⠫⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⢀⣠⣶⣇⡠⠞⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠾⠛⠋⠉⠳⣤⣤⣶⠟⠳⠶⠶⠾⠟⠻⠿⣿⣿⣯⠶⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⣂⠐⢁⣰⠐⡐⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡮⠿⣧⠟⡈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠲⠊⠁⠘⠠⠢⠄⡖⠒⠆⢹⠊⠁⠈
⠀⠀⠰⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢨⣄⠘⠓⢂⠀⠀⢀⡀⣾⣗⠦⢀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⢀⡴⠈⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠐⠀⠀⠀⠀⠘⣏⠑⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣰⡦⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠉
⠀⠀⠀⠀⠀⣀⠩⠉⠀⣀⡿⡄⠀⠠⣤⡄⠀⠀⢀⣄⠀⠀⣀⠀⠀⠀⠀⢠⣄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠤⠛⣿⡆⠀⣤⣾⣷⠟⡴⢾⣿⢀⣠⠾⠋⣠⠁⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⢈⣀⠀⠀⢀⡀⠂⠺⢃⡈⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠑⠊⠁⠀⠘⡂⠈⠡⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠤⠀⠒⠒⠒⠠⠒⠒⠤⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠤⠖⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠈⡉⠉⠲⠤⢀
⠀⠀⠀⠀⢀⡤⢚⠁⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠈⠀⠀⠤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠂⠑⢦⡀
⠀⠀⠀⢠⠎⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠠⢀⠀⠀⠀⢀⠀⠀⠀⢇
⠀⠀⠀⢸⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠊⠀⢀⡀⠘⠀⠤⠀⠀⠀⠀⠐⠀⡄⠁⠀⢸
⠀⠀⠀⠀⠣⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⡀⠁⠀⠀⠀⠀⠈⣠⠂
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠤⠀⠀⠀⠂⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⠤⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⡀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢠⡆⣤⠈⡄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⡛⢀⠔⠀⠀⠀⠀⢤⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢄⠀⠐⠁⠀⠀⠘⠁⡀⠀⠒⠲⢄⢀⠠⠀⠀⡥
⠀⠀⠀⠀⠀⠀⠣⣀⠂⠠⠆⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠡⠈⣿⣵⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠳⢄⠤⣤⣀⣁⠐⠤⠀⠐⢞⣻⠎⡁⠀⢭⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠠⠅⠀⣼⣿⡿⠀⠒⢄⠀⠀⠉⡆⠈⠀⠀⠉⠸
⠀⠀⠀⠀⢀⠄⠀⠀⠼⠀⠀⠀⠐⢇⠀⠋⠹⠁⠀⠀⠀⠑⠀⠁
⠀⠀⠀⠀⠂⠀⠀⢀⡜⠒⢀⣤⡂⠀⠀⠀⢰⡀
⠀⠀⠀⠁⠀⠀⠁⢁⣤⣴⠊⢨⠀⠀⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠁⡾⠆⢀⠀⡆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠲⠀⠈⠀⠰⠀⠀⠀⠀⢰⠄⡃⠰
⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠈⠂⠤⣤⣷⣠⠀⠀⠀⢐⠂⡀⠐⠁⠁
⠀⠀⠀⠀⠀⠀⠃⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⢠⠰⠆⠀⠀⢀⢿⣾⡯⠓⠒⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡐⠀⡀⢈⠠⠁⠀⡏⡐⠒⠠⢀⣰⠿⡐⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠈⠆⢘⠀⠳⡀⠀⣰⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢢⠀⠀⠀⠴⠐⠀⢠⠃⠀⡬⣿⣹⠶⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠈⠀⠁⠢⣤⡷⠛⣮⣀⠤⠚⡁⢀⡄⠀⠀⠻⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠰⠄⠱⠤⢴⣀⡀⠠⢽⣷⣤⡿⣇⠀⠀⠀⢨⠃⢶⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢁⡄⠒⠁⣘⢹⡀⢀⠀⠀⡘⠙⠀⠘⢆⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠆⣀⠠⢶⠻⠠⠂⣡⠀⡛⠰⠀⠈⢃⡀⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢠⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠀⠀⠸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠶⠶⠾⠿⠿⠷⠶⢶⣦⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠳⠶⠶⢶⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⢠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠘⠷⢤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠒⠶⠶⠶⠦⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠾⠃⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠙⢦⣄⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠓⠒⠲⠶⠶⠶⠶⠤⠤⠤⣤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠒⠛⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⠛⢛⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⡀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣏⠀⠀⠀⠀⣠⣾⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢶⡶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠘⢦⣤⠴⠋⠁⡇⠀⠀⠀⠀⠀⠀⠀⠘⠳⠦⣤⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⣀⣀⣀⣀⠀⣀⡇⣀⡀⠀⠀⢰⣏⣹⡆⢀⣠⣦⣀⠀⠀⣠⡾⢷⡄⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢴⠶⠖⠒⠛⢛⣋⣿⠉⠉⠉⣻⡟⢹⡇⠀⠈⣿⠋⣿⠀⠸⣧⡀⠉⠳⠾⠋⣠⡾⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠛⠛⣷⠸⡟⠉⣿⢸⠋⠉⠉⣿⠀⣧⠀⠀⣿⠀⢹⡄⠀⠈⠳⣄⠀⠀⣴⣿⢦⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⣿⠀⣿⠸⠖⢲⡀⢸⡀⢻⡀⠀⣿⠀⠸⡇⠀⠀⠀⣿⠀⠀⡏⠳⡼⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⡶⣦⡀⢠⡿⠀⣿⠀⣿⠀⣏⠉⣀⣸⡇⠸⠷⠶⢿⡄⠀⠿⠚⠻⡆⣿⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠈⢷⣌⡛⠋⣁⣴⠏⠀⢸⣇⣈⣉⣉⣹⣧⣀⣤⣤⣼⣧⣤⣤⣤⣤⠿⢻⣄⣀⣿⢹⡀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠉⠉⠉⠉⠉⠈⠉⠉⠀⠀⠈⠉⠁⢀⡤⠚⠋⠛⠋⠁⠈⠙⠓⢄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⢾⠂⠀⠀⠀⠀⢠⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣠⠴⠋⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠶⠶⠶⠶⠶⠦⣤⣤⣤⣤⣤⣤⣴⠶⠶⠶⠶⠾⠿⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⠉⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣦⣤⣤⣴⣿⣿⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣍⣉⣉⣉⣉⡀⠀⠉⠛⠛⠛⠛⠉⠀⣀⣉⣉⣉⣉⣩⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠙⠛⠻⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠈⣿⣷⣄⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠿⣿⣷⣤⡀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⠿⣿⣷
⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣶⣶⣾⣿⠿⠛⠋⠉⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠈⠻⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠰⠦⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡄⣉⣄⣠⡀⠁⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣼⡏⠉⠉⠛⣷⡄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⢻⣦⣤⠄⠀⠘⣧⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⢀⡿⢀⣴⠾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⣟⠐⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣶⢶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢰⡿⠋⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢷⣄⠀⣿⠁⠀⠘⠦⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⡁⢙⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢠⣶⠿⠿⢾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⢻⣏⠀⠀⢀⠹⣷⠀⠀⠀⠀⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⣈⠙⠛⠁⡰⠂⣿⠀⠀⠀⣠⡤⠞⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠟⢰⡇⠓⠀⢀⡿⠀⣠⠟⣩⠖⠛⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠁⠂⠀⣾⠃⣼⣣⠞⡕⢺⠇⣧⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢀⠖⠁⠀⠈⠛⠀⠀⠈⣸⣾⣏
⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠦⠶⣤⣄⡀⠀⠀⠀⠙⣶⣦⣴⡿⠀⠀⠀⠀⠘⢃⣼⠟⠉⠀⠈⠳⢄
⠀⠀⠀⠀⠙⢿⣄⠰⣖⢛⡿⣶⢶⣬⣇⠀⠀⢀⣀⣽⣿⣿⡷⢦⣤⠶⠖⠋⠛⢿⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠛⣷⡈⠫⡻⣄⡈⠉⢛⡷⠀⠉⠁⠉⢻⡏⠁⣾⠁⠀⠀⠐⠀⢼⣿⣆⣤⣤⣶⠷⠚⠓⠓⢆
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⣌⠊⠗⣦⣌⠀⠀⠀⠀⠀⣈⡳⠂⠉⠀⠂⣤⣤⣴⣾⣿⣏⣹⣿⡿⠂⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠆⠉⠘⢦⡀⠀⡀⣤⡿⠁⠀⠀⠀⠀⠁⠈⠙⠛⣻⣿⣿⣿⣷⣄⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⢿⣿⣂⠠⠤⠤⢤⡁⢀⠊⠉⠁⠀⠙⢯⡀⠙⢣⣤⣜⣓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢚⣿⠶⣿⣏⡿⠀⠀⠀⠀⠙⢬⡄⠀⠀⠀⠀⠀⠉⠀⠀⠛⢉⠻⢿⠦⡙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢻⣌⡇⠀⠀⠀⠈⠀⣸⣷⠀⠀⠀⣀⡀⠂⠠⣶⣁⠦⠤⠴⠴⠙⣣⡘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⣿⠗⣀⡀⣤⣤⣤⣼⣿⡷⢿⣾⠉⠀⠀⠀⠀⢩⣤⣥⣉⣉⣉⣉⣥⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⢻⡄⠀⠀⠀⣠⣳⠞⠉⠁⠀⠀⢀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢡⠉⠈⠉⠁⠉⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢁⡼⣻⠟⠁⡋⢃⡤⡆⠀⠀⠀⠀⠀⠀⢸⡿⠁⢻⠿⣦⡀⠀⠀⠀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢠⡿⢱⣯⣴⠶⠟⡫⠜⣣⡀⠀⠀⠀⠀⣠⠏⠀⠀⠈⠀⠈⠟⡹⢿⣿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣿⠧⠖⠒⠒⣉⣡⣴⠞⠛⠁⠉⠒⠀⠂⠹⠀⠀⠀⠀⠀⠀⠈⡅⠉⢋⢷⡹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣠⣤⣴⡶⠾⠿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀⠀⢀⣤⣾⡓⢦⣈⡌⣧⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠙⢷⣝⠦⣍⡃⣿⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣮⣙⠛⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⡿⢿⠿⠟⠛⣉⡉⠀⠀⠀⠉⠻⠿⢿⣿⣿⣿⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠘⠿⠟⠀⢺⣿⠆⠀⠀⠀⠈⠻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⢀⣀⡀⠀⣴⣾⣷⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠘⠿⣃⣀⠙⠿⠛⠁⠀⠀⣶⣄⡀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⣀⣀⠹⠿⠀⣾⣷⡴⣿⡿⠙⠛⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⣶⣤⣍⠀⠀⠀⢀⣀⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠛⠛⠛⠿⣷⣤⡿⠛⠁⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢀⣿⣿⣇⣤⠀⠀⢠⣾⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡿⠿⠛⠻⣿⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⣠⣤⣤⣶⡿⠏⠀⠀⠀⠀⠀⣼⡏⠀⢹⣿⠀⢠⣿⣿⣿⣿⣿⠇⣀⠘⣿⣿⣿⣿⠁⠀⢠⣦⠸⣿⡄⠀⠀⠀⣠⣶⣿
⠀⠀⠀⠛⠛⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⡇⢀⣾⡇⠀⣾⢿⣿⣿⣿⠟⣾⣿⣷⣿⣿⣿⣿⡄⠀⢸⣿⣤⣿⣧⣤⣴⡾⠿⠛⠉
⠀⠀⠀⠀⣾⠿⠃⢸⣿⠀⠀⠀⠀⠀⠀⢿⣧⡈⠉⠀⣼⣿⣿⣿⣿⡏⠀⠹⠿⠿⢻⣿⣿⣿⡇⠀⠈⣿⡟⢻⣿⠛⠁
⠀⠀⢀⣀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠈⠉⠀⠙⢿⠟⠀⠀⢀⣀⡀⢈⣿⣿⣿⣿⣆⠀⠙⠃⢸⣿
⠀⢠⣄⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠟⠻⣿⡕⣿⢷⡀⠀⣾⡏
⠀⠀⢿⣿⣶⣿⣿⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⣉⣏⡁⠀⠀⠉⠁⠀⣈⠁⣼⡿⠁
⠀⠀⠀⠉⠉⠛⠛⠿⢿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⢿⣿⣶⣄⠀⠀⢸⣿⣿⣿⠀⠀⠀⢀⣤⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣹⣿⣿⠏⠁⠀⠈⠉⠛⠻⠷⣾⣿⡿⢿⣶⣶⣾⠿⠟⠋⢻⣷⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣶⣶⣶⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⢿⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⡿⠿⠿⠿⢷⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡙⠋⢹⣿⠀⣠⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣟⡏⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢴⡄⢠⣶⣦⣾⣻⡄⢿⣃⣦⡙⠛⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣠⡤⠭⠉⠈⠈⠁⣸⠯⠛⠁⢸⣿⣧⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠏⠁⠀⠀⣀⣀⠤⠓⠃⠀⢀⣠⣿⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠴⠶⠶⠿⠅⠀⣰⡶⠟⠛⠙⣥⣴⣾⣿⣿⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⣷⣤⡴⣶⠞⠁⠀⢠⣈⣷⣿⣿⢿⣿⣿⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡴⠞⠛⠿⣿⡿⠟⠁⠀⢀⣀⡠⢦⡿⢛⣴⡿⢛⣿⣋⣭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣴⡚⢿⣿⣇⣾⠗⠉⢁⡴⣃⣴⣿⣿⠶⠟⠛⣻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣆⣺⣿⣿⣿⡟⠃⠈⢀⣶⠏⣴⡿⣿⣿⣷⣴⣶⣾⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡟⠻⣿⣿⣇⣀⣀⣠⣿⣁⣾⣷⣤⣾⣿⣿⣿⣿⣿⣿⣻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣻⣿⡎⠃⠠⢨⠻⣿⠻⠏⠉⣉⣥⣭⣤⢨⡿⣫⣍⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠍⢹⡿⣷⣾⢧⣌⣁⣈⠃⣄⠀⣿⡿⣏⣩⣙⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠛⣿⣆⣾⣿⣿⣠⠀⢤⣼⣄⣾⣿⣿⢿⣿⣿⣿⣿⡋⢾⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡀⢹⣿⠟⢿⣿⡋⣤⣾⣿⣿⣿⣿⣿⡇⠹⣿⣿⡟⠁⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⣧⠈⣿⡇⡉⠁⣿⣿⠈⢙⣿⡏⠀⣼⣿⡇⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢸⣿⡆⣿⡇⣧⠀⣿⡏⠀⢸⣿⠁⢠⣿⣿⢁⣿⣿⣆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢻⣧⢸⡿⣿⢠⣿⡇⠀⣾⣿⠄⣾⣿⣇⣼⣿⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠈⢳⢻⢸⣿⣧⠀⣿⡿⢰⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣼⣛⣘⣿⠟⠀⣿⣯⣾⣿⣿⣿⣿⣿⣿⣗⣲⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡛⠉⠁⠠⠭⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠃⠀⠀⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀⣠⡴⢶⣶⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠸⡆⣡⡴⠶⠿⠿⢷⣿⠶⣶⣮⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠻⠶⠿⠿⣏⡻⢶⣾⣛⠛⠚⠛⢛⣿⣾⣿⣿⣥⠙⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⣅⠬⠓⠒⠢⠬⢌⠛⡛⠛⡛⠂⡁⠠⠙⠷⠶⠏⢍⠫⠐⢠⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢂⣝⣳⠶⠦⣤⣴⣮⣬⣤⣇⣂⣑⣀⣀⣉⣀⣬⣤⣼⣶⣾⡯⠟⡀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠄⠩⢳⠀⠹⠳⠶⠈⣤⢈⡉⠍⠉⠛⠛⠋⠉⠉⠉⠉⠀⠉⠁⠂⢀⠘⢺⡦⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠹⣧⣀⣬⣮⣁⢍⠄⡉⡰⠀⠕⢀⠀⡀⠠⡀⠀⠠⠐⡐⢀⡀⠀⣀⣀⢠⡿⠐⢹⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠀⣁⢀⡿⠋⣾⠫⠝⢷⣀⣵⣶⡿⠶⣦⡀⠀⣡⡶⠶⢶⣄⢀⢆⡴⢟⠉⠛⢻⡇⢈⠆⣷
⠀⠀⠀⠀⠀⠀⣼⡇⣤⢠⡿⠙⠀⠇⠀⢈⢸⣿⡝⢉⠆⠙⠈⣿⣾⠟⢠⠁⢀⠹⣧⣸⣷⠀⠀⡀⠘⣷⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⣿⢣⠈⣼⠇⡇⢸⠀⠀⠀⣾⢋⢱⡁⠄⠀⠀⢰⣿⠀⡆⠀⠈⠀⠹⣿⡇⠄⠂⠄⣡⣿⡄⡀⢸⡇
⠀⠀⠀⠀⠀⢰⡇⠛⠘⣿⠀⡿⠸⠀⠀⢰⡏⡎⣧⠀⠀⠀⠀⣸⣿⠀⡇⠀⠁⠀⠀⢿⡇⠀⠀⠐⠀⢸⡇⠀⠸⡇
⠀⠀⠀⠀⠀⢸⣧⡇⣸⡇⠄⠃⡀⠂⠀⣾⠃⣧⡟⠀⠀⠀⠀⣿⢻⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣿⠀⠈⣿
⠀⠀⠀⠀⠀⢸⣿⡇⣿⢃⠠⢾⠁⠀⢰⣿⠀⣿⣧⠀⠀⠀⠀⣿⠘⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡀⠆⢿
⠀⠀⠀⠀⠀⢸⣿⢳⣿⢀⡇⢸⠀⠀⢸⡇⠀⣿⠋⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⢠⠹⡇⠆⢸⡇
⠀⠀⠀⠀⠀⢸⡏⢳⣿⠀⠃⢸⠠⠀⣼⡇⠀⣿⠁⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠙⢶⣿⠂⡼⣇
⠀⠀⠀⠀⠀⠈⣿⠘⣿⢘⠀⠘⠀⠀⣿⡇⢰⣷⢈⠀⠀⠀⠀⣿⡀⣿⠀⠀⠀⠀⠀⠄⣿⠀⠀⠀⠀⠀⠸⣿⡀⠀⢻
⠀⠀⠀⠀⠀⠀⠙⠷⢿⣄⠖⠊⠰⠄⣿⡁⠂⣾⠀⠀⠀⠀⠀⢿⡇⣿⠀⠀⠀⠀⠈⠀⠹⠇⡀⠁⠀⠀⠂⢺⠇⢂⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣥⣥⣼⣧⡐⠀⠂⠀⠀⠀⠀⠸⠇⠟⠀⠀⠀⡀⠈⠄⣠⣦⣤⣤⣤⣤⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⢶⣦⠶⠶⠚⠛⠶⠶⠶⢦⡴⠶⠞⠋

⠀⠀⠀⠀⠁⠀⢤⣧⣤⣀⡀⠀⠤⡃⠀⠿⣿⡷
⠀⠀⠀⠀⠀⠉⠿⢿⡿⠏⠀⠀⠐⠚⠁⠀⢸⡇
⠀⠠⡄⠰⡂⣠⠀⠀⠀⠀⠀⠈⠙
⠀⠈⠹⠀⠀⠨⠀⢰⡀⠆⠀⠉
⠀⢀⣰⣀⡀⠀⠀⢀⡡
⠐⢲⣿⣿⣿⠉⠀⠀⠸
⠀⠀⠀⠉⠀⠀⠀⠒⠆⢀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⣤⣾⣇
⠀⡇⠀⡄⢠⡀⠐⠒⢀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠀⠛⡆⠀⠀⣀⠄⠉⢻⡏
⠀⠀⠀⠉⠈⠠⢦⣀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠎⠀⢀⡀⠸⠃
⠸⣿⣶⠯⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⠀⠈⠛
⢀⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⣠⣀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣤⣾⡃⠀⠈⢐⣆⠘⡀⢴⡀⠠⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⢀⠄⠉⢻⡇⠀⠀⠀⠸⠀⠀⠈⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣄⠀⠀⠀⠸⠃⠈⠀⠠⣼⣤⣀⠀⠀⠠⢕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢀⡀⢄⠀⠀⠀⠈⢹⡿⣿⠿⠉⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⣦⠴⣛⣛⠓⣦⠞⣛⣛⠲⣴⠚⢛⣛⢦⡴⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠒⠋⢹⡏⠚⠉⠉⢹⡇⠘⠉⠉⠃⢸⡞⠋⠉⠉⢹⡏⠙⠃⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰⠆⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⡇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⠀⠀⣾⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣷⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡆⠀⠀⡟⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢻⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠇⠀⢸⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣸⠀⠀⢸⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣇⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣸⠁⠀⠀⠀⠀⢐⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⠶⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠶⠾⠧⠶⠶⠶⠾⠷⠶⠶⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠃


⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠘⠯⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀⢀⢀⣀⢀⡀⠀⣀⡀⡀⠀⣀⠀⠀⣀⡀⢀⣀⢀⣀⡀⣀⣀⠀⡀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢐⠀⡹⢠⠠⡇⡼⢱⡀⡓⡇⣇⠀⣗⠂⠀⠙⣦⠀⡇⢸⠲⡙⡅⢀⠇⡿⣄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠁⠀⠈⠁⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠘⠀⠀⠀⠐⠀⠂⠘⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠘⠂⠘⠁⠃⠈⠂⠁⠊⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣌⠻⣿⣍⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠛⢿⣿⣯⡃⠘⢟⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⠿⠿⠛⠻⠶⣄⠀⠀⠈⠻⣿⣷⡀⠈⢣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⢸
⠀⠀⠀⡖⢒⡆⣶⠀⠀⢀⡶⡄⢀⡖⠛⠂⣶⣠⠖⠀⠀⠀⢸⣿⣿⣇⣴⣾⣷⣶⡄⠀⠀⠉⠁⠀⢀⣤⣶⡾⠻⣷⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⢸
⠀⠀⠀⡏⢙⡆⣿⠀⠀⡼⠒⢧⠸⣄⠀⠀⣿⠹⣄⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠶⠶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠛⢷⣧⠸
⣤⣤⣤⣭⣭⣤⣬⣭⣥⣥⣤⣬⣤⣬⣭⣥⣬⣤⣬⣤⣤⣤⣼⣿⣿⣿⣧⣤⡞⠿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡆
⣿⣿⢗⢣⣿⡼⣇⣸⢧⡇⣻⡇⣿⣿⡇⣾⡇⣻⣿⣿⣿⣿⣿⣿⣿⡇⢿⡏⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡇
⣿⣿⣼⣧⣼⣧⣼⣧⣾⣧⣿⣧⣿⣿⣧⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣷⠀⠀⠀⢦⣀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿
⠀⠀⠀⢠⢠⢀⠀⠀⡀⣄⢀⠀⠀⢀⡀⠀⣀⠀⣀⠀⠀⠀⢸⣿⣿⣿⡄⢿⡆⠀⠀⠘⠛⠁⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠘⠸⠸⠤⠈⠆⠣⠸⠰⠸⠘⠄⠙⠤⠟⠇⠇⠀⠀⢸⣿⣿⣿⡇⢸⣿⡄⠀⠀⣼⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡏⡃⢀
⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠘⣿⣿⣆⠀⠻⣿⣿⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⢿⣿⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⣿⣿⣿⣦⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠀⠀⡼⢿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⠁⠀⢀⠞⠀⢸⣧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣶⣤⣤⣤⣴⣾⣿⡿⠋⠀⠀⠀⠀⠊⠀⠀⠀⠹⣆