Speech Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠁⠈⠉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠚⠉⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⣠⠤⣀⠀⠈⢧⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⣰⢚⠉⠀⠀⠈⢦⠀⠘⡆⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⠖⠁⠉⠀⠀⠀⠈⠉⡆⠀⢻⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠘⠧⡀⠀⠉⠀⠰⠀⠉⢀⡇⠀⢸⠀⠀⠀⠀⠀⡼⠀⠀⣀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠹⣄⠀⠺⡭⠗⢀⡜⠀⢀⠏⠀⠀⠀⠀⢠⡧⠖⠋⠁⠀⠙⢦⡀⠀⠀⠀⠀⢀⣠⠎⠀⠉⠛⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢤⡀⠀⢀⡼⠓⠒⠒⢶⡉⠀⣀⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠉⠙⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡭⠻⡟⠁⠀⠀⠀⠈⠛⣫⠤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣠⢾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠒⠤⠤⠤⠴⠚⠁⠀⠀⠙⢦⠀⠀⠀⠀⢀⡇⠋⠀⢙⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⣄⣀⣄⡀⠀⠀⠀⠀⠀⢳⡀⠀⢀⣼⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡀⠀⢀⠔⠱⠫⢕⡥⡿⠀⠀⢀⠀⠀⠀⠱⣄⠎⠀⠉⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⣠⣼⠀⠀⠀⢘⡾⠃⠀⠀⢸⠃⠀⠀⢠⠏⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡠⠋⠁⠀⠙⠲⣶⠉⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠴⠋⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⢸⠈⠳⣄⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⢤⠤⢤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠙⠛⠛⠻⠿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡿⠁⠀⢀⣤⡀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠛⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣧⣾⣿⠏⠹⣿⣦⠀⠀⠀⢀⣠⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣷⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣶⣾⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⡿⠟⢻⣿⣿⣿⣿⣿⠟⣿⣷⣼⣿⣿⣤⣿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣍⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢖⣂⡐⠒⠀⠲⠆⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢊⡡⠔⠢⠀⣀⣀⠀⠀⠈⠘⠰⢲
⠀⠀⠀⢀⠴⠊⠉⠁⠀⠀⠈⠙⠒⠛⠓⠶⣄⠀⠀⠀⠀⠀⠠⢿⡆⢨⢀⠤⠄⢉⡀⠀⠢⢀⠀⠀⠀⢈⣧
⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢀⣤⠖⠢⢠⢤⠙⠀⠀⣨⠀⠀⠀⣈⡆
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⣦⠋⢸⡀⠀⠀⢀⠃⠀⠀⢱⠀⠀⠀⡅⠁
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⡇⢆⠀⠫⠁⠀⠈⠀⢀⡀⢌⠀⢈⠙⡄
⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣛⠶⠆⠀⢳⡀⠑⠒⠠⣴⣶⡞⠁⠀⣠⠼⠤⠴⠃
⠀⠀⠙⣦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣙⢦⣤⣀⣉⣡⠴⠒⠯⣁⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⠞⠁⠀⢀⡤⢎⠉⢠⣧⠌⣀⠁⠀⠀⠀⣠⠞⠫⣆
⠀⠀⠀⠘⠷⣄⣀⠀⠀⣀⡴⠣⣄⣀⣤⠞⠁⠀⠀⢡⡀⢀⠘⣷⠫⠥⣄⣀⡅⠀⡀⡜⡇⠑⣄⠈⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⡿⠁⠚⠙⠣⠈⠀⢀⡭⡗⠿⢟⡡⠛⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣸⠁⠀⠀⠀⠀⠀⢀⠜⠀⠘⠋⠁⢀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣹⣄⣀⠀⠀⠀⠀⢣⣤⢆⠀⠚⡯⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣁⠀⠈⠉⢉⣉⠉⠁⠀⠀⠉⠒⠒⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠦⣀⡰⠉⢆⣀⡤⠴⡒⠺⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡮⡩⡇⠀⠀⠈⠁⠀⠀⠑⣄⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠉⠙⠻⣟⣯⡤⡺⠀⠀⠀⠀⠀⠀⠀⡸⢒⠚⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢍⣒⡛⠽⠋⠉⠀⠀⠀⠀⠀⠀⢀⡾⠓⠚⠻⠦⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢍⣒⠐⠒⢒⣊⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⠀⢀⣀⠀⣀⢀⡀⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠠⠄⠈⡐⣽⣻⣿⣿⣿⣿⣿⣿⡇⠈⠛⠷⣤⡙⢦⠽⠞⠃⠉⠉⠙⠓⠦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠰⣑⣿⣿⡿⠿⠛⠛⢻⣧⠘⠿⠶⣦⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠈⣿⡄⠻⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡓⠶⣤⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠚⠘⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠔⠀⢉⠀⠀⢀⠹⢷⡶⣲⢲⢺⡆⠀⠀⣬⣙⠳⣦⣌⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠄⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡴⠇⠺⠘⠁⣴⠻⣦⠙⢷⣄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣇⢈⣁⡉⠁⠀⠉⢄⣠⢤⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⢿⠔⠉⠈⣃⠀⠀⡀⠉⠈⠀⠀⠀⠀⠀⠁⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⡔⠈⠀⠀⠉⠙⠶⠚⡄⠀⠀⠀⠀⠀⠠⣿⣿⣿⢾⠖⢤⠸⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢾⡟⣿⣛⡃⢈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⢀⣀⣼⣟⣿⣉⡇⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⠀⢀⠀⠀⢠⠜⠋⢁⠐⢠⡊⠉⠉⣟⠋⠛⠀⠃⠀⠦⠄⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠘⠃
⠀⠀⠀⢀⡴⠀⠰⡀⠀⠛⡤⠶⣀⠶⠴⠤⠜⠁⠆⠀⢀⠀⡤⠃⠀⠀⢾⠉⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠈⢷⠀⠀⠈⢢⠀⠈⠂⠀⠤⠔⠒⠉⠙⠢⣄⡀⠂⠰⠤⠤⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⢀⣤
⠀⠀⠀⠀⠈⠒⠀⠐⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠘⠄⢀⠀⠀⠀⠀⠀⣀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠠⠀⡄
⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠛⠓⠆⠘⠃⠁⠀⠀⠀⢀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠈⢀⡀⢀⣤⣀⣥⢰⡄⣿⢰⡟⣰⠗⣠⣶⠖
⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⣄⢠⣉⡀⠤⣤⠀⠉⠀⠈⠉⠀⠈⢂⡀⠀⠀⠀⠀⠘⠻⠶⠋⠁⠀⠘⠛⠓⠾⣼⣫⠀⢋⣡⣶⠇
⠀⢠⠈⠓⣀⠀⠀⠀⠀⠀⠀⢀⠀⠃⣀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⢁⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢛⣉⣥⣤⡆
⠀⢰⠀⠂⠀⠈⠓⠦⠤⠦⠔⠠⠶⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡐⡂⠘⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠭⠤⢤⣤⡄
⠀⢸⣀⣇⣨⣨⡁⣏⢨⣁⡇⣸⢀⣙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣤⢼⡏⡁⠀⠀⢉⣷⣦⣄⣀⣤⣄⡀⣤⣴⣯⣽⠗⢶⣤⣤
⠀⠈⠀⠉⣿⡾⠖⠻⣼⡶⠗⢺⠊⠉⠀⠘⣧⡀⠀⢀⣤⡀⠀⠀⠺⠇⠀⠀⠀⠀⠈⠁⠠⠷⡀⠀⠰⢟⠹⠾⣻⠟⣿⣿⣿⡿⣿⢷⣌⠲⣷⣬⣍
⠀⠀⠀⠀⠀⠠⠀⠀⠈⠑⠀⠈⠀⠀⢠⠀⠀⠙⠋⠉⠀⠈⠉⠋⠀⠒⠓⠚⠛⠛⠂⠀⠀⠀⠙⠰⠾⠿⠏⠾⠿⠂⣰⡿⠿⠻⠟⠃⣛⢋⡙⢫⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠄⣤⠤⠤⠦⠴⠖⠆⠐⠒⠠⠀⠖⠛⠈⠁⠍⠻⠄⠀⠀⠀⢥⠼⠷⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡀⠀⢀⣀⡄⠀⠀⠀⣀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢲⣿⣧⣴⣤⣶⣴⣷⣶⡶⢶⣴⣆⠀⢘⣷⣼⣯⣿⣯⣿⣿⣴⣷⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠿⣿⠁⠀⠀⠈⢿⠿⠛⠿⣿⣿⣿⣿⣿⣇⣀⣸⣿⡇
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠈⢐⣄⡀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠘⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡷⠄
⠀⠀⠈⠉⠑⠢⠖⠚⠓⠂⠀⠀⠀⠀⠀⢣⡀⠀⠀⣀⠀⢀⣀⣀⣨⠁⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⣀⠀⡀⠀⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠡⠄⠀⠉⠙⠉⠉⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠁⠛⠋⠉⠛⢙⡓⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉⠉⠉⠓⠲⢄⡀
⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠲⢤⡀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠉⠓⠒⠒⠒⢲⠀⢀⡖⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡆⠀⡼⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠴⠊
⣠⠖⠒⠒⠛⠛⠛⠒⠒⠒⠒⠦⠤⣄⣀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⢤⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠤⠤⠤⠤⢤⣤⣀⣀
⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣀
⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠈⠓⠦⠤⠤⠀⠠⠤⡄⠀⣀⣀⣀⣀⣀⣤⠤⠴⠚
⠀⠀⠙⢦⣄⣀⠤⠤⠤⠤⡀⠀⢠⠖⠒⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠐⠀⠒⠂⠉⠀⠈⠉⠉⠉⠉⠉⠉⠀⠉⠀⠒⠒⠢⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠖⠛⠛⠳⠦⠤⣤⡀⠀⠀⠀⠀⠀⠈⠉⡝⢀⠔⠒⠒⠒⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⣠⡾⠛⠛⠛⠳⡦⠤⠶⣤⡄⠀⠙⡆⠀⠀⠀⠀⠀⡜⡰⠋
⠀⠀⠀⠀⢸⡏⠀⠀⠀⢂⢀⡇⠀⣰⠟⠛⠢⡀⢳⡀⠀⠀⢀⣼⠛⠁
⠀⠀⠀⢀⣼⠇⠀⣰⠋⠉⠉⠀⢀⡏⠀⠀⠀⢻⡄⡇⠀⠀⠺⠁
⠀⠀⢠⡟⠁⠀⠀⠣⣴⡟⠉⠛⠉⠀⠀⠀⣀⢼⡼⠁
⠀⠀⢸⡇⠀⠀⠀⣠⢾⡇⢈⣭⣭⠁⠀⠠⣶⣾⠻⣦⡀
⠀⠀⠀⠳⣄⠀⠀⢻⡈⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⣸⣿
⠀⠀⠀⠀⠀⢱⡆⠈⣹⣦⠀⠀⠀⣀⢀⣀⠀⣾⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠘⣧⣤⣿⣿⠀⠀⠀⠙⠛⠋⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡹⢿⡀⠀⠀⠀⠀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⡜
⠀⠀⠀⠀⠀⠀⠀⣼⣀⠀⠈⠁⠒⠢⣄⣈⣳⣄⡀⠀⠀⠀⠀⠀⠀⢠⠈⠋⠀⠘⢳
⠀⠀⠀⠀⣀⡤⠊⠀⠀⠉⠂⢄⡴⠂⠉⢿⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⡆
⠀⠀⠀⡾⠃⠐⠂⠀⢄⡀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠙⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠑⡄⠀⠐⠂⠀⠀⢀⠔⠁
⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠄⠁⠀⠀⠔
⠀⠀⠀⠈⠷⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠃⠀⠔⠁⠀⠀⠀⠌
⠀⠀⠀⠀⢱⡙⣆⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⢀⡾⠛⢣⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⢳⡈⢢⡀⠀⠀⠀⠀⠀⠘⣯⠒⠉⠱⡀⠀⢆⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠳⡀⠙⢄⠀⠀⠀⠀⠐⠿⠁⠀⠀⢁⢘⢪⣳⠁
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⢢⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠈⠑⢤⡀⠀⢀⡠⠖⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡆⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠓⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⡘⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⡁⡀⠠⠀⠐⡄⠀⠠⠀⠉⠀⠀⠀⠀⢁⠀⠀⠀⣤
⠀⠀⠀⠐⢀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠆⠀⠨⠂⠫⠀⠀⠉⠈⠉⠂⠚⡇⠀⠀⠀⠀⠐⠀⠀⠂⢀⠃
⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠄⠘⠂⠀⠢⠐⠐⠊⠀⡄⠀⠀⠀⠀⢀⣀⠀⠈⠈⠁⠈⠀⠐⠀⠀⠀⠈⠂⢀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠀⠒⠊⠁
⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠉⠙⠲⠶⢤⡀⠀⠀⠀⠀⠀⠀⠃⡔⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⣰⠟⠉⠉⠉⠻⠂⢛⣳⣖⡀⠙⡄⠀⠀⠀⢀⢣⠊
⠀⠀⠀⠀⣯⠀⢀⡤⠤⠾⠀⢸⠉⠀⠑⣄⢳⠀⠀⢠⠟⠁
⠀⠀⢠⡞⠁⠀⢾⢠⡴⠦⠤⠋⠀⠀⠀⣓⡼
⠀⠀⣿⠀⠀⠀⢀⣿⠁⢐⣒⡂⠀⢀⣩⡿⣧⡀
⠀⠀⠘⢦⡀⠀⢿⠘⠀⠉⠉⠀⠀⠀⠉⠁⢈⣿⠆
⠀⠀⠀⠀⠈⡦⠈⣻⡆⠀⠀⢀⣀⡀⢀⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠈⠷⣴⣿⣇⠀⠀⠈⠛⠃⢾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠌
⠀⠀⠀⠀⠀⠀⢀⠜⠣⢄⡀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢁⡈
⠀⠀⠀⠀⠀⣀⠞⠠⠄⠀⠀⢉⠶⣤⠛⠰⢄⠀⠀⠀⠀⠀⠎⠀⠀⠀⠸⡀
⠀⠀⢀⣴⢊⣀⡀⠀⠀⠈⠚⠀⠀⠙⡀⠀⠠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⢸⠁⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠈⠢⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠙⠀⠀⢁⡠⠀⠀⢀
⠀⠀⠀⢹⢄⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⡰⠀⠀⠉⠀⠀⠠
⠀⠀⠀⠈⣎⢧⠀⠀⠀⠀⠀⠀⢰⡀⠀⣠⠏⠓⠀⠀⠀⢠
⠀⠀⠀⠀⠘⣆⠳⣄⠀⠀⠀⠀⠀⣯⠉⠀⠢⠀⡡⡀⡄
⠀⠀⠀⠀⠀⠈⣆⠈⠢⣀⠀⠀⠈⠈⠀⠀⠀⠀⠁⡽
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠑⢤⡀⠀⠀⠀⢀⡔⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠈⠁⠂⠈⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⡄
⣶⣶⣶⢶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠈⠉⠳⣄⢀⣤⣶⣶⣷⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⢀⣤⣄⣀⣤⠴⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠙⢾⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠛⢿⣷⠋⠉⠉⠙⢆
⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⣀⣤⣤⣬⣽⣯⣭⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠬⠭⠭⠤⠤⠤⣄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⠀⢀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡀⠀⠀⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠠⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡄⠀⠠⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠇⠀⢰⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠙⠚⡇⠀⠀⡇⠀⠀⡗⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠀⠀⢸⠀⠀⢸⠛⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⡇⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⢸⡀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⣢⠀⠤⠀⠀⡠⠔⠈⠁⠀⠀⠀⠀⠀⠈⠁⠢⢄⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣰⠀⠀⠀⠀⢸⠀⠀⠅⠠⠃⢰⠁⣰⣀⡆⣶⠆⡤⠀⣠⠀⠀⣀⠀⠑⠀⠈⠂
⠀⠀⠀⠄⠀⠀⠀⠀⣤⡗⡼⠞⠉⠠⠂⠀⠀⠐⠀⠀⠀⡐⠀⠙⢄⠉⠈⠙⠓⠚⠷⠶⠧⠤⣾⣽⢖⠃⠀⢀
⠀⠀⠀⠀⠄⠀⣦⡷⠏⠉⠀⡠⠂⠀⠀⠀⠀⠁⠁⠂⠁⠀⠀⠈⠢⢕⠂⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠄⠀⠀⢀⠀⢀⡠⠐⠂⠀⠀⠀⡀⠀⠀⠈⠀⠐⠂⠠⠍⠩⠩⠅⠀⠂⠀⠑⢌⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⢤⠠⣄⢲⡀⣖⡆⢹⠀⠀⡁⠀⠀⠀⠠⡰⡄⣖⠰⡖⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠸⠸⠀⠏⠏⠃⠛⠁⠐⠀⠀⡂⠀⠀⠀⠐⠂⠃⠙⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠐⠀⡆⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠁⠠⠀⠀⠀⠀⠀⠂⠀⠃⢀⠀⠀⠀⠀⠀⠄⡂
⠀⠀⢎⠀⢄⠀⠠⠀⠀⠀⠐⠀⠀⠉⠉⠀⠀⠒⠀⠐⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂⠀⠀⠐⠀⡘⢸⢡⢲
⠀⠀⠄⠀⡀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠂⠂⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠴⠰⠖⠶⠆⠆⠰⠶⠷⠆⠀⠧⠇⠒⠀⠂⢀⡄⠀⠐⠑⠠
⠀⠀⠀⠀⠀⠂⠀⠀⠑⠀⠒⠂⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠒⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⢀⠠⠀⠒⠒⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠢⠄⠄⠄
⠀⠀⠀⠈⠔⠁⢀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⡐⢀⢄⡠⡠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⠀⠀⠚⠘⠀⠹⠀⠀⠘⢢⠀⠀⠀⠀⠉⠁⣁⠍⡘⢡⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠃⠀⢰⠏⠇⢶⢶⣏⠀⠀⢀⠀⠀⠸⠀⠆⠳⠉⠸⠃⠃⠠⠀⠀⠀⢀⠀⠀⠀⠀⠐⠈⠀⠀⠐⠁⠀⢀
⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠈⡁⣀⡂⠄⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠂
⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⠁⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣄⡀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢏⡀⠀⠈⠛⢿⣿⣿⡇⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⠟⠁⢠⣦⠀⠀⠀⢠⡄⠙⣣⠀⢀⠜⠀⠀⢀⠀⠀⠀⠤⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⡇⠘⠓⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠸⣷⣶⣶⠟⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⣀⣀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⣧⣀⢫⠰⢄⠀⠀⠀⠀⣀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠘⠠⠍⣃⠰⠾⠠⠤⡔⠈⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡍⠗⣾⣄⣀⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⣤⣤⣤⠈⠁⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⠛⡟⠛⡟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡐⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣴⡀⠀⠀⢸⣄⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠶⠂⠀⠉⠉⠉⠉⠉⠉⠉⠰⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠀⠀⠐⠒⠂⠀⠤⠄⠀⠀⠀⠀⠀⠀⠈⢖⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢄⠀⠈⠑⡀⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀⡠⠊⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠱⠀⠙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠴⠀⠁⡀⢁⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⡇⠀⢠⠀⢠⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⡀⠀⠀⡆⠈⡀⠀⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡄⠄⢰⠀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⡈⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⠀⠀⡆⠀⠀⠀⠁⢆⡹⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠈⡇⠀⠀⠀⡇⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⡠⠂⢠⠀⠀⢰⠂⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⣄⠀⠀⢁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣤⠐⠢⡔⠂⠁⠀⠀⠌⠀⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠹⠖⠀⣽⡄⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠀⣠⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴⠲⣆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢶⡀⠀⠀⠀⠀⠈⠀⠋⠀⢸⠵
⠀⠀⠀⠀⠀⠀⠀⢠⠙⢄⠀⠈⠘⠃⠢⠄⢰⠀⠀⡔⠀⠉⠐⠟⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢁⠐⠀⠐⠢⠀⠀⣀⡀⣼
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠘⠢⠄⢀⡇⠀⠀⠄⠀⢃⡀⠀⡀⠂⠀⠀⠀⠀⠀⠐⠢⠤⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠀⢀⠃⠀⠘⡀
⠠⠦⠤⠀⠀⢀⣤⠁⠀⠀⠀⠀⡠⠋⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠈⠀⠀⠀⠐⢄
⡀⠶⠆⠀⠉⣹⠋⠐⢄⠀⣠⠊⠀⢀⠆⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠰⠄⡀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠳⡀
⠁⠒⠂⠀⡝⠀⠈⠐⢠⡜⠉⠀⠦⣿⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⢈⠀⠀⠀⢀⡄⠊⠀⠀⠀⠀⠀⠐⡄⠀⠀⠈⡄
⠀⠀⢠⠂⠀⠀⠀⠀⣶⠀⠠⠊⠀⠀⠈⠁⠒⠤⠀⠀⠀⡀⣽⠲⡤⠐⢂⣉⣀⠙⠂⣠⣤⣀⡐⠢⠤⠐⠚⠡⠄⡀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠃⠀⠀⠀⠀⣌⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠾⠛⣀⣡⠐⠂⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⢝⡳⠦⢀⡀⠀⠀⢸⡇⠀⡠⠃
⠀⠀⠳⢄⣀⡀⠜⠈⢢⡘⠢⢄⡀⠀⠀⠀⠀⠀⠐⢁⠔⠃⠉⠠⢃⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⣐⡢⠔⠂⠀⠀⠀⠁⠘⠂⠺⢧⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠼⣷⠤⢀⡀⠀⠀⠄⠀⠀⠀⠀⠏⠀⠁⠤⠊⣉⣁⣀⣀⣉⠁⠠⠄⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠁⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠲⠐⠶⠄⠳⠦⠾⠻⠶⠶⠆⠶⠂⠶⠂⠶⠰⠖⠂⠤⠠⠤⠤⠤⠤⠐⠶⠲⠶⠰⠟⠶⠶⠂⠶⠂⠰⢦⠀⠀⠀⠀⠀⠷⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠆⢠⠔⠂⣤⠀⠀⣴⡀⢰⣀⠀⡖⠂⠀⡤⠒⡄⣶⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡆⠈⠃⠂⠘⠀⠘⠃⠃⠐⠙⠂⠊⠃⠀⠑⠀⠃⠛⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠇⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⢀⠔⠊⠉⠉⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠉⠒⢄⠀⠀⠀⣠⣶⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣶⣄
⡎⢀⡄⣄⣠⣠⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣠⢠⣀⡄⡄⠈⡆⠀⢰⣿⡟⣿⢿⢿⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⡿⡿⢿⢿⣿
⢇⠀⠋⠉⠉⠈⠁⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⠉⠉⠁⠋⢀⠆⠀⠸⣿⣯⣶⣶⣾⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣷⣶⣿⣽⣿
⠈⠳⢄⣀⣀⠠⣄⠱⡀⠀⠀⢀⠤⠒⠢⢄⠀⠀⠀⡰⢃⡤⢀⣀⣀⠴⠋⠀⠀⠀⠙⠿⣿⣿⣿⢿⣿⣆⠀⠀⠀⢀⣤⣶⣤⡀⠀⠀⠀⣰⣿⡿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⢈⠀⠀⠀⠈⡆⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⣿⣿⣿⣿⣿⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⢰⠋⠉⠳⠀⠘⠦⣤⡤⠞⠀⢰⠋⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣆⠀⠙⢿⣿⡿⠏⠀⣰⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠈⢢⣤⠞⡴⠒⠀⠀⠀⠀⠲⡜⢢⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡿⠋⣠⣴⣶⣶⣶⣦⣄⠙⢿⡿⠏
⠀⠀⠀⠀⠀⢠⠊⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡆⣿⣿⣿⣿⣿⣿⣿⢰⣶⣶⣦⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⡇⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⠸⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣈⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⢉⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠠⠖⠂⠠⡄⠘⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢐⠔⠁⡠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠒⢆⣮⠭⠄⠂⠀⠀⢠⠀⠀⠀⠀⣀⢠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠈⠪⡢⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢄⡲⣶⠖⠀⡀⠉⢸⠀⠀⠀⠰⠐⠃⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠁⠀⠈⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠘⠀⣀⠈⡢⠀⠀⢀⡴⡁⠀⡠⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠚⡀⠀⢀⣈⠑⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠙⠋⣓⠴⢗⢄⡀⡠⠔⢁⣄⡒⠲⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡅⡡⠬⡑⠒⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢴⠠⠀⠑⣜⣆⢀⣰⠜⠗⠉⠛⠛⣋⠀⠀⠀⠀⠀⠀⠀⠀⢎⢉⣉⡆⠀⠀⣔⡟⢠⠁⠀⡸⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⣂⠊⢆⠀⠀⠘⠛⠛⠁⠉⢀⣀⡤⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡍⠉⠳⢀⣼⠼⠥⠊⣀⠴⠋⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠬⠠⢂⠓⢀⣀⣀⡠⠄⠊⠀⠍⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⢀⡷⣤⠀⣀⣀⣠⠤⠚⠭⠊⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⠄⢀⡄⠈⠓⠒⠒⠒⠒⠤⢀⠠⠆⠀⠀⠀⠐⠂⠐⠂⠒⠒⣌⡡⠌⠒⠋⠉⠁⠀⢁⠤⡖⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡀⢠⠮⠀⠀⠀⠀⠀⠀⠀⠀⠌⢢⠀⠈⠛⠻⣿⣿⠿⠋⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢⣈⠐⠐⠐⠀⠀⠀⠀⠀⠀⠸⢀⠈⡆⠀⠀⠀⡟⡇⠀⠀⢀⠀⠀⠀⠙⠓⠂⢠⡒⠖⠴⠲⠚⠒⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⣈⣉⣉⡓⢒⠒⡄⠀⠸⠀⢀⢏⡀⠸⠀⠀⠀⡁⡇⠀⠀⠸⡀⠀⠈⠁⢡⣦⢸⣷⡆⠀⠀⠐⠒⠒⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢽⠀⠀⠀⠀⡙⠠⠁⠠⠀⠀⣚⠀⠄⠀⠀⠀⠀⠆⡇⠀⠀⠀⠣⠀⠰⠞⡋⡥⠀⢯⡇⠀⠀⠀⠀⢀⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠾⠀⠀⠀⡘⠀⠀⠀⢀⣈⠁⣀⡈⣈⡂⠀⠉⠉⠱⡅⠀⠀⠀⢰⠀⠀⠈⡐⠠⠸⣽⡇⠀⠀⠀⠀⢰⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢨⠀⠀⠀⠐⡄⠀⠀⡇⠀⠀⠀⠀⠀⢩⢖⡴⡂⣱⢋⢀⣀⣀⠎⣀⠔⢡⠃⠀⠁⢘⠄⠀⠀⠀⠀⠘⠂⡂
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠴⢘⠒⠠⣄⠠⢽⣦⣀⣰⠄⡄⠀⢘⠀⢉⡛⠫⢚⠃⠳⢚⠚⢫⡦⠠⣔⣁⣤⠴⢻⣭⡖⠢⢐⠀⢠⢀⣦⣇⢀⡀⢄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠉⠀⠉⠀⠈⠀⠈⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⠀⠀⣴⣴⠀⠀⠀⠀⠀⠀⢰⣶⣦⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⢄⠀⠀⡀⠀⠒⠂⠋⠁⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⣰⣶⣮⠁⠀⠀⠀⠀⠀⠀⠻⠷⠟⠀⡠⣤
⠀⠀⠀⠀⠀⡠⠀⠉⠁⠀⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡙⢿⠟⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠙⠛
⠀⠀⠀⠀⠈⠀⠀⡀⢀⠀⠀⠡⠀⠑⡀⠀⠀⠀⠀⣼⣿⣿⠿⢿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢴⠛⠛⢿⣿⡆⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠉⠉⠁⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣵⣭⣴⣭⣿⣵⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠉⠀⠀⠉⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⢂⠀⠔⠂⠀⠐⠒⠁⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠎⢐⠢⢀⡀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⡀⠀⠀⠀⠰⢄⡀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢤⠀⠀⡂⡄⠀⠀⠀⠀⠔⠀⠁⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠌⠚⠐⠠⡷⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠡⡟⠀⠂⡤⢤⠃⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠀⠀⠉⠐⠒⠃⠀⢅⢀⠀⠀⣸⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠦⠤⡬⠃⠀⠀⠀⢀⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠣⡄⠂⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠂⠁⠈⢂
⠀⠀⠀⠀⠀⠎⠈⠁⢀⠀⠀⠀⠡⠀⠀⠀⠈⠐⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠠⠐⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡴⢧⠀⢰⢦⡀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⡠⢰⡯⡔⠁⠀⠀⠠⠹⣂⠀⠀⠀⠀⠀⠀⡆⠀⠁
⠀⠀⠀⠀⢠⠰⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠔⠁⢀⡩⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⢃⠀⠀⠀⠀⣀⠤⠐⠂⠈⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠑⠄⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢀⠀⠀⢆⠀⠀⠐⠀⠀⣠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠠
⠀⠀⠀⠀⠈⠀⠀⠀⠢⠤⠓⠂⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⡀⠀⠀⠀⣀⣀⣀⣀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠉⠂⠈⠁⠀⠈⠙⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠲⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⢠⠑⠄⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠌⠁

⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⣀⡀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀
⠀⠀⠀⠀⠀⠸⣿⣿⠇⣾⣿⣿⡆⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃
⠀⠀⠀⠀⣠⣶⣮⣷⠶⣘⠻⢟⣑⢶⣮⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣈⣅⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣩⣡⣤⡀
⠀⠀⠀⠀⠛⠛⠛⢳⣿⣿⣿⣿⣿⣷⡙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡟⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣮⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⠶⢤⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣦⣶⣶⡄
⠀⠀⠀⠀⠀⢀⣴⣶⣴⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣟⡀⠀⠀⠀⠀⣘⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠘⠛⠛⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⢿⣋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⢿⣿⡇⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣍⣛⣋⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣦⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣠⣦⣭⣶⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⣿⡿⢏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠛⠛⠛⠛⠛⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠙⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠋⢉⣀⣀⣤⣤⣄⣀⡀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢁⡠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⡈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣀⠈⠣⡀⠀⠀⠀⣀⡠⠤⠤⠤⠤⣄⠀⠀⠙⢦⡀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢀⠎⠈⠳⣄⠈⠢⣠⠎⠁⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠱⡄⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⡏⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢹⡀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠋⠀⠀⢤⣄⣀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠠⣤⣬⣭⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⡎⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⣆⠀⣤⠶⠞⠋⠁⢹⣀⣀⡀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⡰⠁⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⢆⠀⠀⠀⠀⠀⠀⠈⠁⠙⢦⣀⣀⣀⣠⣄⠈⠢⡀⡰⠁⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠈⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡈⠑⠦⣄⣀⠀⠀⠀⠀⠀⠀⣀⢀⠤⠊⢁⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣄⣈⠉⠉⠀⠀⠈⠉⣀⣠⠤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣄⣀⣀⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣿
⠀⠀⠀⠀⣠⣾⡿⢻⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿
⠀⠀⠀⣰⣿⣟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿
⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠈⠁⢈⣍⡙⢻⣿⣿⣿⡇⣿
⠀⠀⠘⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠈⣿⣿⣿⠃⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣜⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⠙⣷⠀⠿⣿⡟⠀⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⢀
⠀⠀⠀⠈⣿⠿⠻⢿⣿⡇⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⢈⣸⠇⠀⢸⡇⣠⠞⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⡇⠈⠙⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⢸⠋⠀⠀⠈⠛⠁
⠀⠀⠀⠀⢙⣦⣶⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠈⠉⠀⠀⠀⣀⠀⠀⢸⣷⣶⣦⣤⣤⣀
⠀⠀⣠⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣤⣤⣤⢤⡤⠴⠚⠁⠀⠀⣾⣿⣿⣿⠃⠀⣽
⠀⣼⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠹⣍⠛⠛⣿⠀⠀⠀⠀⠀⣸⣿⣿⠛⠳⣤⠴⢧⡀
⢺⣿⣿⣿⣿⣿⣿⣿⠏⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠀⠀⠀⠀⣠⡾⠛⢿⡇⠰⢤⡏⢀⠞⢻⡆
⠙⠿⢿⣿⡿⠿⠛⠁⠀⠀⠀⠙⣳⡦⣤⣀⣀⣀⣀⣀⣀⣀⣤⠴⠺⣧⡀⠀⠈⣷⠀⠈⠙⠻⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠁⠀⠀⠀⠉⠛⠻⠿⠿⠛⠁⠀⢀⠀⢙⣦⡞⠁⠀⢠⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠹⣆⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠟⠀⠀⠀⣰⠏
⠀⠀⣀⣤⣤⣠⡤⠞⠁⠀⢀⡴⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢤⣤⠞⠁
⠀⠸⣇⠀⠀⠀⠀⠀⠀⡴⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠸⣟⣛⣻⢿⡄⠀⢹⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⠀⠀⠙⠛⠋⠀⠀⢻⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣶⡖⠀⠀⠀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⠚⠛⢻⡇⡟⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠀⡇⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⡼⢧⣤⣤⣼⠀⣷⣤⣤⣴⢧⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣉⣙⣃⣛⣁⣀⣀⣸⣆⣇⣀⣀⣈⣛⣘⣋⣉⣳⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣶⣶⣶⣶⣴⣶⣶⣾⣋⣷⣶⣶⣶⣶⣶⣶⣶⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣷⣇⠀⠀⠀⠀⣶⣶⠄⠀⠀⠀⣸⣿⣶⣦⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⠀⢀⣼⣧⡀⠀⠀⣴⣿⣿⣿⣿⡇⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⡇⠀⠙⠛⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣀⣀⣸⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠟⠛⠁


⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⠶⠶⠶⣶⣶⠶⠶⠶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠀⠀⠀⠠⠤⣀⠀⠀⠀⢀⡠⠐⠊⠉⠀⠀⠀⠉⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⡄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡴⠁⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠘⠠⡀⡌⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠰⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⠀⡠⠐⠒⠒⠢⠆⠉⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠉⢢⠄⢀⣀⣀⡀⡴⠀⡄⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢸⠀⠀⠀⠀⠘⡄⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠸⠀⠐⠂⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠢⡀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠁⢨⡳⠤⠤⣀⢸⡀⠙⡾⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠺⠉⠑⢄⣀⣀⠇⠀⡰⠀⠀⡠⠄⣀⠀⠀⠀⢀⡀⠤⠒⠁⠀⠀⠀⠁⠒⠠⠀⠤⠠⠄⠒⠂⠤⠤⠱⢀⠀⠀⠀⠀⠙⡜⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠑⠾⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡤⠂⢀⠰⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣆⠉⢁⣺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣋⢁⡠⠣⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢄⠈⠉⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡠⠠⡈⢆
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⢀⠀⠀⠀⠀⠈⡎⢢⠀⠀⠀⠀⡀⠴⡛⢽⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡔⠀⠀⠐⠈⡀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠱⡀⡠⠊⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡘⠮⠐⢄⠀⠀⠀⠀⠀⡇⢀⠃⠀⠀⠀⠐⠀⠇
⠀⠀⠀⠀⠇⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⡑⠂⠄⠀⡀⡇⡘⠀⠀⠀⠀⠄
⠀⠀⠀⢰⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠁⢣⠀⠀⠀⡜⠀⠀⢀
⠀⠀⠀⠘⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡤⠀⣂⠀⡔⠀⠀⠀⠘
⠀⠀⠀⡈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠶⠶⠶⠶⢶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⣀⣠⣤⡶⠶⠶⠶⠶⠶⠶⢶⣤⣄⣀⠀⠀⠀⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⣠⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⡀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⣰⡟⠁⠀⢀⣴⠟⠻⣦⡀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠈⢻⣆
⠀⠀⠀⠀⣰⡟⠀⠀⢠⡾⠁⠀⠀⠈⠻⣦⣶⠟⠉⠁⠀⠀⠈⠉⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⢻⣆
⠀⠀⠀⢠⡟⠀⠀⢠⡿⠁⠀⠀⠀⠀⣴⠟⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢻⡄
⠀⠀⠀⣾⠁⠀⠀⣿⠁⠀⠀⠀⠀⢰⡏⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠈⣷
⠀⠀⢰⡿⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡆
⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠠⣿⡀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⢸⣧⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⢿⡄⠀⠸⠏⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⡇
⠀⠀⠸⣷⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣶⠟⠁⠀⠰⠶⠶⠖⠀⠀⠀⠀⢸⡇⠀⠀⣼⡇
⠀⠀⠀⢿⡀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠸⣧⠀⠀⠀⠀⠀⢀⣿⠁⠀⢀⣿
⠀⠀⠀⠸⣧⠀⠀⠘⣧⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⠛⠛⠛⠈⠻⣦⡀⠙⠃⠀⠀⠀⠀⣼⠃⠀⠀⣼⠇
⠀⠀⠀⠀⠹⣇⠀⠀⠘⣷⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢀⣼⠋⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠹⣧⡀⠀⠈⢿⣄⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⡾⠁⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠈⠛⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠈⠉⠛⠻⠶⠶⢶⣦⣴⣶⠶⠶⠟⠛⠉⠁⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⠶⠶⠶⠶⠶⠾⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⣷⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠉⠉⠻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠃⠘⠛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⢛⣩⣴⣶⣶⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣿⡟⣡⣾⣿⣿⣿⣿⠟⠉⠉⢻⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⠿⠟⣰⣿⣿⣿⣿⣿⣿⡀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠒⠚⠛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣤⣤⣉⠛⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠟⠛⠿⣿⣿⣿⣿⣶⣝⢿⣿⣤⣤⣤⣤⣤⣤⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⠻⢿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠦⠤⠾⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣄⣀⣀⣀⣀⣀⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠛⠉⠈⠉⠉⠉






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⠿⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣝⢉⣭⣥⣤⣶⣶⣿⣷⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠉⠙⠃⠙⠿⣿⣿⡿⠿⠟⠻⣿⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢸⡏⢠⣶⡿⠷⣶⣄⠀⠀⣤⣶⣟⠛⠦⡀⢹⡿⣀⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⢯⠈⠛⠻⢿⠀⠀⠛⠟⠓⡶⠄⠀⣸⡿⢿⣿⣦⣩⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⢺⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣿⢦⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠙⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⠛⠿⢿⡿⠋⠉⡁⠀⠈⠂⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠓⠒⠛⠀⠀⠀⠀⠀⡼⠁⠀⠠⢤⡤⠾⠋⢀⡀⢳⡀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠒⠲⠦⠤⠤⢤⣀⣀⢶⣶⣧⣀⣴⣃⣀⡀⠀⠀⠁⠷⠄⠙⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠈⠉⠑⢲⣶⣿⣿⣵⣾⢿⣿⣿⣿⣿⣯⡴⠊⠉⢒⡄⠀⠀⠀⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⡀⠀⠀⠀⢸⣿⢿⣏⣿⣿⠿⣿⣿⡿⠋⠈⠳⡄⠈⣽⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢸⡟⢿⣷⣦⣤⣤⣀⣤⣿⣿⣿⣿⠷⠘⠹⡀⠀⠀⠀⠹⡄⢛⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠐⡄⠃⠈⠉⠙⠛⠛⠉⠉⠉⠁⠛⠚⡿⠀⠀⠙⠦⢀⡀⠀⠙⢾⠄⠀⠀⠀⠀⡼
⠀⠀⢀⣀⣠⠤⠒⠋⠁⢿⡀⠁⠀⠐⠠⣤⡀⠀⠀⠀⠀⠀⡀⠼⣷⢆⠀⠀⠀⠀⠀⠉⠀⠈⢧⠀⠀⠀⢀⣇
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢈⣦⢤⣴⣾⣇⠀⠀⠹⣻⡆⠀⠀⠀⠀⠀⠀⣠⣚⣦⠶⠒⠉⠈⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⣼⣿⣿⣿⣿⠟⣆⢀⡀⠙⣿⣧⠀⠀⠀⠀⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠃⠙⣿⣿⡏⠀⠈⠉⠀⠀⠘⢿⣷⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠋⠉⠉⢯⠉⠉⠉⠙⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⢷⡄⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠆⠀⠀⠳⡄⠀⢻⡄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠁⠀⠹⡆⠀⢻⠀⠈⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠇⠀⣼⠀⢀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃⠀⠀⠀⠀⣴⠃⠀⣼⠃⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⡞⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠤⣤⣀⣀⣀⣀⣤⠤⠖⠛⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⢉⣁⣤⣤⣤⣤⣤⣤⣈⡉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⣤⣖⣛⡉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⣤⡈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠋⠁⠉⠉⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡷⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⢷⡀⠹⣄
⠀⠀⠀⠀⠀⠀⣰⠏⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⢀⡴⠋⣠⠞⠁⠀⠀⠻⡄⠹⣆
⠀⠀⠀⠀⠀⢠⡏⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡴⠋⣠⠞⠁⠀⠀⠀⠀⠀⢻⡀⢹⡄
⠀⠀⠀⠀⠀⣸⠁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⣠⠞⠁⠀⠀⢀⡀⠀⠀⠀⠈⣧⠈⣇
⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⣾⠁⠀⣀⠀⢶⠀⢳⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁⢘⡇⠀⢸⡆⠸⡇⠸⡇⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⢹⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁⣰⡏⠉⠁⠀⢀⡇⢀⡇⢀⡇⠀⠀⢀⡟⢀⡏
⠀⠀⠀⠀⠀⠘⣇⠈⣧⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁⠘⢯⡉⠀⠀⠀⠀⡼⠁⣸⠃⣸⠃⠀⠀⣼⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠹⣆⠘⣆⠀⠀⢀⡴⠋⣠⠞⠁⠀⠀⠀⢠⠟⠀⠀⠀⠈⠁⣴⠃⣠⠏⠀⠀⣰⠃⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢷⡴⠋⣠⠞⠁⠀⣤⢤⣤⣀⡼⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⢀⡾⠁⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢾⣅⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡈⠛⠦⣤⣿⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠛⢁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣈⡉⠙⠛⠛⠛⠛⠛⢉⣁⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢹⣿⣿⠿⠛⠻⢿⣿⣿⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣶⣿⣇⠀⠀⠀⠀⠀⢀⣿⣷⣦⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⢈⣹⣦⣄⣀⣀⣤⣾⣉⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠉⠉⠉⠉⠙⢷⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⣦⣤⣤⣄⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣠⣤⣤⣶⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠿⠋⠛⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣔⢾⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠁⠈⠻⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣾⣿⣿⣿⡟⠁⠀⣴⣷⣌⠻⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷
⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⢰⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣷⣌⢿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣜⢿⣿⣿⣿⣿⣷⣝⢿⣿⣿⡄⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣷⣝⢿⣿⣄⠀⠀⠀⣤⣈⠛⣦⡀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣷⣝⠟⠀⠀⠰⣤⠙⣧⠘⣧⠀⠀⠀⣿⣿⣿⣿⠃
⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣷⣄⠀⠀⣸⡇⣹⠆⣿⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣷⡜⠋⣠⠟⢠⡟⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣦⡁⢴⠟⠀⠀⣼⣿⣿⣿⡿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠃⠙⢿⣿⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣦⡀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠒⠲⠶⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣠⢄⡀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡇⣞⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠋⣠⡤⠶⠛⠁⠀⠀⠀⠈⢇⠈⢧⡙⠆⠈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡁⠀⣗⡒⠒⠒⠂⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠈⠉⠛⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⡀
⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡀
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠙⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⢶⣤⣴⠢⢔⣄⡠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⡸⠆⡼⠙⢄⣠⢴⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⢿⠾⠃⠁⠈⠁⠀⠀⠉⠙⠻⢀⢀
⠀⠀⠀⠀⠀⠈⢢⣄⣀⠟⢄⠴⠋⠒⠁⠀⠀⣾⣋⣀⢸⣼⣆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠉⠀⠀⡀⠀⠠⠈⠂⠐⡀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢀⡅⠀⢀⣴⠶⠿⣟⣻⡆⣶⣤⡽⢿⡾⣿⠿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠁⠀⢤⠐⠦⠀⣀⠱⠀⣠⠈⣅⠃⠀⢼⣄
⠀⠀⠀⠀⠀⠈⠉⠲⠀⢸⣇⠀⢀⣿⠛⢶⡟⣙⢆⣘⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⡆⠀⠐⢦⡀⠑⡄⣀⡠⠧⠈⣀⠄⢀⡃⣾⡉⠄
⠀⠀⠀⠀⠀⠀⠀⢸⡠⠤⡝⠛⠛⡍⡠⣨⣶⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡅⠀⠓⠄⠔⠒⠡⠀⠠⠈⠘⠁⢀⡀⠛⠛⠓⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠋⠱⢿⠟⠠⠀⠹⠇⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣳⡦⡀⢀⣀⡀⠀⢀⣤⡤⡀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠊⠩⣝⠞⠋⠀⠐⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⡀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣸⠋⠉⢲⣆⠆⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣼⡃⠀⠈⠿⠀⠀⠘⠇⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠉⠳⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⡴⠛⠛⠷⠀⠀⠤⠠⠒⠔⠦⣠⡌⡁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢱⣟⣷⠄⠀⠀⠀⠀⠀⠀⠙⠆⠀⠻⡄⠀⠀⠀⠸⢧⡀⡀⠀⠐⡄⠀⠐⠊⠂⠙⠉⡇⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⡔⠈⠉⠱⢞⡴⣀⡀⢀⡀⡀⠀⠈⢦⠀⠀⠀⠀⠀⠀⡼⠛⠃⠀⠁⠀⢰⣄⣠⣄⣦⣤⢋⠉⠀⢀
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠐⠚⣷⣛⡾⣳⢱⡆⣦⢈⠀⠀⠀⠀⠀⠀⢧⡀⢈⣤⢨⣷⠿⠿⣿⡟⣉⠻⠀⢡⣆⠈⠀⠈
⠀⠀⠿⣇⡀⠀⠈⢱⡀⠀⠀⠀⠁⠈⠁⠉⠘⠀⢣⡏⠀⠀⠀⠀⠀⠀⠀⠉⠋⣏⠀⠁⣾⡇⠙⠀⣿⢆⠀⣆
⠀⠀⠀⠙⠷⠀⠀⠀⠛⠦⣄⢄⠀⠀⠀⠀⣀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠙⡟⢒⢚⠑⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠖⠒⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡾⠆⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⣀⡐⠀⠀⠀⠀⠀⠰⠶⢐⡄⢀⣠⠞⠛⠉⠉⠉⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢤⠟⠉⠉⠈⠉⠙⠆⠀⠀⠀⠀⠀⠀⠈⢀⣿⠁⠀⠀⠀⠀⠀⣀⢀⠀⠙⢠
⠀⢄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡽⠃⠀⠀⠀⣠⣄⣦⣶⡳⠀⣿⠤⠀⠀⠀⠀⠀⠘⡄⠀⠀⣤⣲⣀⡤⠀⠀⠂⠈⢱⡃
⠀⠘⠀⠀⠘⡈⣿⣷⣶⣄⠀⠀⣈⠤⠀⠀⠀⠀⠀⠀⠈⠇⢰⣶⡾⠽⠛⠋⠈⠀⢀⡶⠯⠤⠄⠀⠀⠀⠀⠀⠳⣀⠁⠈⢀⡠⢤⠀⠀⡶⠒⢋
⠀⠀⠀⠀⠐⠀⠙⠿⣿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠁⠀⠀⡀⢀⣴⠴⠚⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠋⠁⠀⠀⠃⣄⣧⠀⠁⠆
⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⠶⠞⠃⠏⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠂⠀⠀⠀⠰⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⢿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠏⠙⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠃⠀⠀⠀⠙⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⣰⣿⣿⣿⡟⣁⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣴⣿⣿⣿⢏⣼⣿⣿⣶⣤⡀⠀⠀⠀⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⠃⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣾⣿⣿⣿⢫⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⠟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢻⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣾⣿⣿⡿⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣿⡆⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⡟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠟⣴⣿⣿⣿⢏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣆⠀⠀⠀⣠⣿⣿⠿⠛⠛⠿⢋⣾⣿⣿⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣄⠀⠐⠛⠉⠀⠀⠀⠀⣠⣿⣿⣿⡿⠁⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣄⠀⠀⢀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣤⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣀⣠⣴⡾⠋⠉⠉⠉⠙⢿⠶⢦⣄⠀⠀⠀⠀⣠⣴⣾⡟⠋⠉⠉⠉⠉⠉⠉⠛⣲⣤⡀
⠀⠀⠀⢀⣾⠟⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⣸⡟⠈⢻⣄⣷⠀⠀⠀⠀⠀⠀⠀⠉⠙⣷
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣿⠇⠀⢻⡇⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡾⠋⠀⠀⠈⠻⢶⣶⠶⠾⠿⢿⣦⠀⣰⠟⠛⠋⠁
⠀⠀⠀⠀⠙⠷⢶⣶⠒⠀⠀⠀⠀⣀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠟⣁⣴⠏
⠀⠀⠀⠀⠀⢀⣸⣿⠿⠿⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⠛⠉
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⣀⣤⣶⠶⠿⠿⠿⠿⢶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⢶⣦⣄
⠀⠀⠀⢠⣾⣯⡅⠀⠀⠀⠀⠀⠀⠀⡈⢻⣿⣿⣦⠀⠀⠀⠀⠀⢠⣿⣯⣀⡀⠀⠈⠹⣿⣶⣦⣄
⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⠇⠀⢻⣇⠀⠀⢠⣾⠟⠉⠉⠈⠉⠀⠀⢀⡿⠀⠈⢿⡆
⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢸⡿⠀⠀⣿⡇⠀⣀⣀⠄⠀⠀⠀⠈⠀⢀⠀⣸⡇
⠀⠀⠀⠀⠙⠿⣶⣶⡶⡞⠀⢸⣷⣦⣄⣠⣴⣦⣾⠃⠀⠀⠘⢿⣾⡟⠀⠀⠀⠀⠀⠀⢀⣿⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢀⣾⠃⠀⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠹⣿⣤⣀⢀⣀⣠⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣶⣦⡀⠀⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⣀⡀⠀⠙⢿⣿⣆⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢿⣿⣦⡀⠈⢻⣿⣇⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠻⣿⣷⠀⠈⣿⣿⡄⠀⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⢸⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⢸⣿⡇⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⣼⣿⡇⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃⠀⢰⣿⡿⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁⠀⣰⣿⡿⠃⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠁⠀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠴⠒⠒⠂⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣭⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠤⡑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠶⢦⣄⡀⠘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⣴⣶⣿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢧⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢿⡀
⠀⡀⢀⠀⢀⣴⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠃⠎⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣼⣿⠁⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀
⠀⠀⢠⣿⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⣼
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠈⢃
⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠖⠁
⠀⠐⡀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⢣⡑⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢋⡀⣠
⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠞⠋⠁⠂⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⢶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⠦⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⠛⠛⢻⣦⣀⠀⠀⠀⢻⣆⢆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠻⢷⣦⣀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠉⠻⢷⣦⣙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠞⠛⠛⠛⠛⠳⠶⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠁⣀⣠⣤⡤⠶⠶⠶⠶⢤⣤⣄⣀⠈⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⣀⣴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⣦⣀⠈⠛⢦⣄
⠀⠀⠀⠀⢠⡾⠋⠀⠠⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠙⢷⡀
⠀⠀⠀⣰⠏⠀⣠⣦⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠻⣆
⠀⠀⣼⠃⢀⣼⠋⠀⠙⢦⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠙⣧
⠀⣸⠇⠀⣾⠁⠀⠀⠀⠀⠙⢦⡀⠈⠳⣶⠶⣤⣄⡀⢀⣠⣤⠶⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠹⣆
⢠⡟⠀⣸⠇⠀⠀⠀⠀⠀⣠⡾⠛⢦⡀⠈⠳⣄⠈⠻⠟⠁⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠸⣇⠀⢻
⣸⠇⠀⡿⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠙⢦⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⣿⠀⠸
⣿⠀⢸⡇⠀⣠⣴⣋⡁⠀⠀⠀⠀⣀⣠⣤⠿⢦⡀⠈⠻⣶⠶⠶⣤⣄⣀⠀⠀⠀⠀⢈⣛⣦⡀⠀⢸⡆
⣿⠀⢸⡇⠀⠉⠻⣯⡉⠛⠛⠛⠛⠿⣥⣄⣀⡀⠙⢦⡀⠈⠳⣄⣠⣬⠿⠛⠛⠛⠛⢉⣽⠟⠁⠀⢸⠃
⢸⡆⠀⣷⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⢦⡀⠈⠻⣄⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⣿⠀⢸
⠘⣧⠀⢹⡆⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠳⣤⡾⠋⠀⠀⠀⠀⠀⢰⡏⠀⣼
⠀⢹⡆⠀⢿⡄⠀⠀⠀⠀⠀⠀⠉⠻⢦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣽⣦⡀⠈⠳⣄⠀⠀⠀⠀⢠⡟⠀⣰⠏
⠀⠀⢻⣄⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠙⢦⡀⠈⠳⣄⠀⣠⡟⠀⣰⠟
⠀⠀⠀⠹⣦⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠻⠋⠀⣴⠏
⠀⠀⠀⠀⠈⢷⣄⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠂⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⡀⠉⠙⠛⠛⠒⠶⠶⠒⠛⠛⠋⠉⢀⣠⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠶⢤⣤⣤⡤⠶⠶⠖⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠻⣷⣄⣀⣀⣀⣀⡀⠀⠀⡀
⠀⠀⢀⣴⣶⠿⠿⠟⠋⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠘⠛⠛⠛⠛⠿⠿⠿⣶⣤⡀
⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣀⣀⣀⣀⣀
⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⣦⡄
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⢸⣿⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⣿⠀⠀⢀⣤⣶⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣄⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣸⡿⠁⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠸⠿⠉⠿⢦⣶⠾⠿⠿⣶⡄⠀⠸⡿⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣶⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣶⣦⡀⠀⠀⠀⠀⢀⣶⠿⠿⠟⠛⠁
⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠙⢿⣦⡀⠀⠀⢸⣿
⠀⠀⠈⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠁⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⢸⣿
⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣄⣀⣀⣀⣀⣀⣠⣤⣴⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣾⡿
⠀⠀⠀⠀⣿⣤⣶⡾⠟⠋⠉⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⢀⣀⣀⣀⣀⠀⣀⣀⠀⢀⣀⠀⣀⢀⡀⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠈⠒⢭⣳⣿⣿⣿⣿⣿⣿⣆⠈⠛⠷⣤⡙⢦⠽⠞⠃⠉⠉⠙⠓⠢⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠒⠷⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡱⣽⣿⢿⠿⠟⠛⢻⣧⠀⠿⠶⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠺⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣥⣤⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⠀⠚⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡓⠶⣤⣬⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⡴⢚⠐⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠔⠀⠉⠀⠀⢀⠹⢷⡶⣲⢲⢻⡆⠀⠀⣨⣝⠳⣦⣌⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠩⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁⠈⠀⠄⠀⠀⠀⠀⠀⠙⡰⠇⠺⠘⠁⣴⠸⣦⠙⢷⣄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣗⢈⣁⡉⠁⠈⠉⢄⡠⢤⠂⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⢿⠖⠁⠈⡣⠀⠀⡀⠁⠉⠀⠀⠀⠀⠀⠁⢄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⡔⠘⠀⠀⠉⠑⠢⠚⡄⠀⠀⠀⠀⠀⠠⣿⣿⡷⠺⠖⢀⠸⠁⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⠀⠀⠀⠀⠀⢸⣿⣿⢙⡃⢁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⢀⣀⣼⣿⣿⡉⡁⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⢀⠀⠀⢠⠟⠋⢁⠀⢠⡃⠈⠙⣟⠋⠛⠀⠃⠠⠄⠆⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠃
⠀⠀⠀⢀⠶⠀⠰⡄⠀⠛⡤⠶⡀⠶⠴⠤⠔⠁⠆⠀⠀⠀⡤⠁⠀⠀⠾⠉⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⢗⠀⠀⠈⢠⠀⠈⠂⠀⠤⠔⠒⠉⠑⢢⣀⡀⠀⠠⣤⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⢄
⠀⠀⠀⠀⠈⠒⠀⠐⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⣀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠂⣄⠁
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⣀⣥⢰⡄⣿⢰⡗⣰⠗⣠⣶⠖
⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡀⠀⣀⣄⣀⣈⡀⠤⣤⠀⠁⠀⠈⠉⠀⠉⢂⡀⠀⠀⠀⠀⠙⠻⠶⠋⠁⠀⠉⠛⠳⠞⣴⣫⠀⢋⣠⣶⠗
⠀⢀⠈⠓⣀⠀⠀⠀⠀⠀⠀⢀⠀⠁⣀⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣄⠀⠀⢁⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢛⣋⣥⣤⡆
⠀⢰⠀⠂⠐⠈⠓⠦⠤⠤⠔⠠⠶⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠂⠘⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠭⠤⣤⣤⡄
⠀⢸⣀⣇⣸⣈⡁⣭⢨⣀⡇⣨⢈⣙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡏⠁⠀⠀⢋⣷⢦⣀⣀⣤⣄⠀⣀⣴⢯⣙⠖⠲⣦⡄
⠀⠈⠀⠈⣷⡾⠖⠻⣴⡶⠓⢺⠎⠉⠀⠘⣧⡀⠀⢀⣤⡀⠀⠀⠐⠇⠀⠀⢀⡀⣠⠀⠀⠀⠀⠀⠀⠊⠑⠾⢋⠟⠸⢹⢳⡟⣎⠳⣄⠰⢶⣄⡁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠑⠀⠈⠀⠀⠀⠀⠀⠙⠛⠉⠀⠈⠉⠉⠀⠀⠓⠒⠛⠛⠀⠀⠀⠀⠀⠠⠶⠿⠀⠰⠟⠀⠀⠚⠈⠃⠙⠁⠉⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⢀⡤⠄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢳⣤⣧⣴⣤⣶⣴⣷⣶⡶⢶⣦⣌⠀⢘⣧⣴⣿⣿⣷⣴⣿⣴⣶⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠿⣿⠁⠀⠀⠈⢿⠟⠛⠿⣿⣿⣿⣿⣿⣇⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⢐⣄⠀⠀⠀⠀⠀⠀⢀⠘⠃⠀⠘⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠟
⠀⠀⠀⠉⠑⠠⠴⠚⠃⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⣀⠀⢀⣀⣀⣬⠁⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⣀⠀⡀⠀⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠈⠙⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠋⠉⠚⠙⡓⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⢀⡤⠖⠀⢀⠔⢛⡠⠄⠖
⠀⠀⠀⢠⠊⠀⢀⡰⠁⠀⠁⠀⠈⠉⢉⣙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠐⠒⠀⠐⠒⠒⠤⢄⡀
⠀⠀⠀⠆⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⡀
⠀⠀⢰⠀⠀⠀⠀⠀⣀⠃⢀⣠⠐⠄⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⡤⠚⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠮
⠀⠀⢸⠀⠀⠀⠆⠀⠀⠀⡎⠐⠊⠀⠠⠁⠠⢀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠀⠀⠀⠀⠀⠠⡉⠀⠚⠋⠙⠓⠃⠀⠀⡖
⠑⠢⢄⠀⠀⠀⠄⠀⠀⣺⠡⢾⠇⠦⠴⠺⡇⠈⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⠔⠁⠀⢄⠀⡄⠀⢐⡐⢢⠀⠀⠀⠀⠀⠀⠀⠆⠀⠦
⠀⢠⠜⠀⠒⠀⡏⠀⠀⠀⠑⠀⠀⠀⣀⡴⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣷⠀⠀⠀⠌⠤⣃⠘⢌⣁⠌⠀⣤⠄⠀⠀⠀⠀⠐⠀⠐
⠀⡇⠣⠀⠈⠐⠂⠀⠀⠀⠀⠀⡠⠠⠄⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⠀⠀⠀⠀⠸⣅⡀⠀⠀⠀⠰⠁⢡⡀⠀⠀⠀⠀⢸⠚
⠀⠈⠒⣎⠀⠀⠀⠀⠀⠀⠀⢸⢿⣶⠆⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣠⡾⡇⠀⠀⠀⠀⡜
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠰⡬⠋⠁⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠮⣤⡾⠋⣠⠅⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⠲⡶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣤⡤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠶⠠⠌⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⡠⠐⢠⠒⠁⣀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠂⡉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢡⠀⠀⠀⢀⡘⠀⠠⠊⠀⠐⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠓⠄⢂⠀⠀⠀⠷⠀⠀⠀⠀⠳⠈⠓⠈⠀⠀⠄⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣤⠀⠐⠀⠈⢄⣀⡤⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⢆⠀⠀⢄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢀⠜
⠀⠀⠀⠀⡀⠀⠀⠶⠀⠀⠈⠀⢰⢠⣤⠄⢳⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠊⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⢀⡴⠁
⠀⠀⠀⠀⡇⠀⢀⡀⠤⠐⠀⠀⠸⠀⠰⠘⠦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠈⠁⠐⠒⠒⠀⠈⠉⠫⣀⠀⡁
⠀⠀⠀⠀⢨⠉⠀⠒⠒⠒⠒⠀⠀⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⢤⠀⠀⢸⠀⠀⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠐⠚
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣙⠀⠀⠀⢇⣀⣀⠼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠣⢄⠀⠀⠀⠀⢘⠓⡆
⠀⠀⠀⠀⢠⠟⠒⠚⠛⠥⢀⡠⠂⠘⢍⠄⠀⠀⡙⠋⠈⠙⢦⠀⠀⠀⠀⠀⣄⠀⠀⠄⠀⠀⢀⡀⣤⠀⠀⢸⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⢧⡈⣀⡀⠀⠀⡘⠀⠀⠀⠈⡟⠠⠤⠖⣀⣀⠀⡠⣇⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠀⠈⠈⠉⠉⠉⠉⠁
⠀⠀⣀⠀⡉⠈⠈⣉⣁⣈⣁⣀⣀⣀⣐⣇⣀⣀⣈⣁⠀⣀⠐⡈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⡀⠀⠀⡀⠀⠀⡀⣀⣀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡶⠾⠿⠿⠿⠿⠿⠿⠷⢶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⢉⣀⣤⣤⣴⣶⣶⣶⣶⣦⣤⣤⣀⡉⠙⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⢁⣠⣴⡾⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢷⣦⣄⡈⠛⢷⣦⡀
⠀⠀⠀⠀⢀⣴⡿⠋⠀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣀⠙⢿⣦⡀
⠀⠀⠀⣠⣾⠋⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣧⡀⠙⣷⣄
⠀⠀⣰⡿⠁⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠘⢿⣆⠈⢿⣆
⠀⣰⡿⠁⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠈⢿⣆⠈⢿⣆
⢀⣿⠃⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠘⣿
⣸⡏⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠋⠙⠻⠷⠿⠟⢋⣿⠿⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢹
⣿⡇⢰⣿⠀⠀⠀⠀⣀⣤⣶⣿⣟⣩⣤⣤⣤⣤⣤⣄⣀⣀⣴⠟⠁⣠⣤⣤⣍⣻⣿⣶⣤⡀⠀⠀⠀⠀⣿⡆⢸
⣿⡇⢸⣿⠀⠀⠀⠀⢻⣿⡿⣿⣿⣭⣭⣍⣉⣉⣉⣉⣿⣟⣁⣀⣀⣈⣩⣭⣭⣿⣿⢿⣿⡟⠀⠀⠀⠀⣿⡇⢸
⣿⡇⠸⣿⠀⠀⠀⠀⠀⠙⠿⣦⣄⠈⠉⠉⠙⢛⣿⠟⠛⠛⠛⠛⠛⠋⠉⠉⢉⣠⣴⠿⠋⠀⠀⠀⠀⠀⣿⠇⢸
⢹⣇⠀⣿⡆⠀⠀⠀⠀⠀⠀⠈⠙⠿⠂⢀⣴⣿⣁⣀⣀⣀⣀⣀⣠⣤⣴⡾⠿⠋⠁⠀⠀⠀⠀⠀⠀⢰⣿⠀⣸
⠈⣿⡄⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⢠⣿
⠀⠹⣷⡀⠹⣷⡀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⢀⣾⠏
⠀⠀⠹⣷⡀⠹⣷⡄⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⢀⣾⠏
⠀⠀⠀⠙⢿⣄⠈⢻⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣠⡿⠋
⠀⠀⠀⠀⠈⠻⣷⣄⠉⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡈⠙⠻⢷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡾⠟⠋⢁⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⣈⠉⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠉⣁⣠⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠷⢶⣶⣶⣶⣶⣶⣶⡶⠾⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠚⠛⠛⠛⠛⠲⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⢀⡴⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⠏⢳⡄⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⢀⡾⠁⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⢻⡄⠀⠀⠀⠀⢸⠀⠀⠀⠛⠛⠛⠛⠛⠛⠂⠀⠀⣿
⠀⠀⠀⣀⣸⠃⠀⠀⠀⢸⡇⠀⠓⣲⡶⠒⠒⠲⡞⠛⠉⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⠀⠀⠀⣋⣛⣋⣉⣉⣛⡁⠀⠀⣿
⠀⢠⡟⠉⠉⢳⡄⠀⠀⢸⣇⠀⣼⠉⣾⠛⢳⡾⠛⢷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿
⠀⢸⡁⠀⠀⢸⡇⠀⠀⠀⠙⡏⠁⠀⠙⠓⠋⠙⡶⠋⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣿⣄
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⢻⣄⠈⠉⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢀⡼⠋⠁⣠⡼⠟⠛⠳⣄⣺⠛⠻⣤⠈⠛⢦⡀
⣤⢸⡇⠀⠀⢸⡇⡤⠀⢀⣤⢤⣍⠓⠲⢶⣿⡁⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⣴⠋⠀⠀⣼⠃⠀⢀⡄⠀⠈⠻⠀⠀⠈⣧⠀⠀⠹⣆
⠙⣦⡙⠲⠖⣋⡼⢷⡞⠁⠀⠀⠙⠳⣄⣼⠁⢻⡀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⣼⠃⠀⠀⠀⢷⣀⠀⠈⣷⣀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠘⣇
⠀⠀⠉⢻⡟⠉⠀⠀⠙⠳⣤⣀⡀⠀⠈⠃⠀⠈⣧⣀⣤⠶⠋⠁⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⢻⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹
⠀⠀⡶⠾⠷⢶⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠚⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣸⡄⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠈⠛⠛⠛⠛⠀⠀⠀⢀⣤⠶⠚⠛⠉⠉⠉⠛⠛⠶⢤⣄⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠘⣦⡀⠀⠉⢀⡿⠦⣤⡴⠏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⢀⣠⠶⠶⠶⠶⠶⠖⠶⢦⣌⠻⣦⡀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⣠⢿⠛⠛⢉⡴⠶⢦⣄⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠈⠻⣄⠀⠀⠀⠘⣧⡀⠀⠀⠀⢰⠏⠘⣧⡴⠋⠀⠀⠀⠈⠳⣄⠀⢀⡼⠁
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢸⠀⠠⣤⣤⣤⣤⣤⣤⣤⠴⠋⠀⠀⠀⠹⡆⠀⠀⠀⠈⠳⣄⠀⠀⡿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢹⣦⠟⠁
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢸⠀⠀⢰⠞⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠛⢾⣇⡀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠛⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢷⡴⠟⠀⠀⠀⣧⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠚⠛⠛⠋⠉
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠈⣇⠀⠠⠤⠀⠙⠀⣿⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣠⠾⠁⠀⠀⢸⠛⠛⠛⠛⠛⠛⣷⡤⠤⠶⠚⢻⡆
⠀⠀⠀⠀⠀⠹⣆⠀⠀⣠⠶⠚⠛⢦⡈⠉⢩⡿⣦⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⢷⣞⠁⠀⠀⠀⠀⠙⢶⣾⠁⠘⣧⠀⠀⣸⡀⠀⠀⠀⠀⢀⣿⢤⣤⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⠀⠀⠀⠁⠀⢀⣹⡤⠞⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠘⢦⠎⠀⠀⠀⡷⠒⠖⠲⠗⠤⠤⠤⠤⠤⠤⠤⣀⢀⣭⡉⢸⠀⣸⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⢀⡠⠤⠧⣼
⣤⣄⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⡟⠖⠁⢸⢸⠹⡀⠀⠀⠀⠈⠓⠤⢤⣤⣴⠃⠀⠀⠀⠈⢧⣠⣤
⠀⠀⡇⠀⠀⠀⣀⠞⠀⢀⠔⠊⠉⠉⠙⠒⠚⢡⠇⠀⠀⡇⠀⠀⢸⢸⠀⠉⠉⠉⡏⠉⢳⢰⣋⣁⣸⠀⠀⠀⠀⠀⣸⡇
⠀⠀⡧⠤⡀⢎⠁⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⡇⠀⠀⢸⠾⣄⣀⣀⣀⣧⠖⠹⡎⠀⠀⢸⡣⢄⣀⣠⠔⠁⡇
⠀⠀⢧⣄⠉⠛⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡧⣤⠤⠞⠀⠀⢧⣈⠁⡇⠀⠀⠁⠀⠀⠈⠀⠀⠀⢸⠀⠀⡇
⠀⠀⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠧⣇⠀⠀⠀⠀⡔⡩⠖⠚⠦⣄⡀⠀⠀⠀⠀⠀⡴⠓⢦⠀⡇
⠀⠀⡧⠤⠤⠤⠤⠤⠵⠤⣤⣀⣀⡀⠀⠀⢀⠞⠀⠀⠀⣠⡴⠃⠀⠲⢖⡛⠁⠀⠀⠀⢀⣉⣉⣉⠉⠉⠒⡡⢄⢀⠧⡇
⣀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠖⠚⠁⠀⠀⠀⢠⠁⡆⠀⠀⠀⠀⡟⡆⠀⠀⠀⡏⠀⠀⠀⢹⡀⠀⠑⠋⡇⠀⣇⣀
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠤⠤⠤⠤⠔⡣⠊⣧⠀⠀⠀⠀⡿⡇⠀⢲⠲⠃⠀⠀⠀⢸⡇⠀⠀⠀⡄⠀⠀⠈
⠀⣠⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⠤⠎⣠⡀⠰⣶⠲⠒⠁⠀⠻⠤⠤⠤⠤⠇⡇⠀⠀⠙⡆⠀⠀⠀⢸⠇⢠⠚⠉⠳⠤⣤⢤
⠔⠁⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⣀⡰⠃⠙⣄⢈⣙⣢⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢠
⠤⠤⣄⡀⡎⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⢆⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⡇⠀⠀⠀⢸⣀⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⣄⡀⠀⠀⠀⣠⠔⠓⠒⠺⡇⠀⠀⠀⢸⠤⠧⣄⠀⠀⡠⠚⠁
⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠉⠉⠀⠀⠀⣠⠔⣇⠀⠀⠀⡸⠀⠀⢸⠀⡜⠀⣀⣠
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱⠀⠰⠁⠀⠀⠉⠉⠉⢣⠀⠀⢸⠀⡅⠀⡇
⠤⠤⠤⠤⢤⠭⠟⡹⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⢣⡀⠳⠤
⣀⣀⡤⠖⠁⠀⠀⢇⠀⠀⠀⠀⠘⠤⠤⠤⠤⠤⠤⢤⠤⡤⠤⢄⠀⡤⠤⠜⠀⠀⠱⢄⡀⠀⣀⡠⠋⠀⠀⠸⣄⠀⠑⢤
⠛⠒⠲⠤⣔⠒⠋⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠲⡇⠀⠀⠻⠀⢠⠖⠒⠒⠒⠒⢧⠟⠳⢦⡀⠀⠀⢀⡤⠛⢀⣼⣀
⠀⠀⠀⠀⠈⢣⠀⠀⠈⠓⠦⠤⠤⡠⠤⢄⠀⡞⠁⠀⠀⣇⠀⠀⣀⡤⠼⠶⠶⠦⣄⠀⠈⠀⠀⠀⡇⠀⠀⡸⠀⠀⡇⠘⠁
⣀⣀⣀⣠⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠂⠀⠀⡨⠖⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡗⠒⡎⠀⠀⠀⠙⣆⣀
⠀⠀⠀⠀⠙⠒⠒⢲⠄⢠⠖⠒⠒⠒⠒⠒⠒⠒⠒⡲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣀⣀⣠⠗⣏⣁⠀⠀⠀⠯⠇
⠀⠀⠀⠀⠀⠀⠀⠺⢲⠁⠀⠀⡰⠒⠒⠒⠒⠒⠚⡃⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⡴⠋⠁⠀⠀⠀⠀⠉⢢
⠤⠤⣄⠀⠀⡤⠤⠴⡊⠘⠯⣉⠁⠀⠀⠀⠀⠀⠀⠙⠦⢤⡀⢀⣤⠔⢺⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⡤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⣿⣿⠿⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠿⣿⣿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠉⠛⠛⢻⣿⣿⡟⠛⠛⠉⠀⠀⠀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠙⠛⠛⠁⠀⢀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⡀⠀⠈⠛⠛⠋⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣦⣄⣀⣠⣤⣾⣿⠗⠀⠀⠀⠙⢿⣿⣿⡿⠋⠀⠀⠐⠾⣿⣷⣤⣄⣀⣤⣴⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⢀⣤⣤⡀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣴⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣦⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇







⣿⣷⣶⣾⣶⡷⢾⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⢷⡶⠾⠿⠶⢿⣶⠶⢶⠶⠶⢶⣾⠿⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠻⣷⡄⠀⠀⠀⠀⠀⣾⣿⣿⡿⠿⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢿⣧⠺⠗⣼⡿⠀⠀⠀⠀⠀⣿⣿⡏⣴⣿⣷⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣀⣬⡛⠿⠿⢛⣥⣀⠀⠀⠀⠀⣿⣿⣷⣝⠛⣋⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣛⣛⣛⣛⣛⣻⣿⣿⣿⡇⣠⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⡀⣿⡿⢫⣭⣭⣭⣍⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⠛⠛⠛⠻⢻⣿⣿⣿⡇⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣧⡛⠃⠻⠻⠛⠿⠿⠈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⠿⠛⡛⠿⣿⣿⣿⣿⣿⡿⢛⣛⣛⣛⣛⣃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⢿⡇⣿⣿⢡⣾⣿⣿⣷⡌⣿⣿⣿⣿⡇⣿⣿⣿⠿⣛⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⣿⣿⣿⣿⣿⣶⣤⣿⣇⢸⣿⣿⣿⣿⡇⣼⣿⣿⣿⡇⣿⣿⡇⣾⣿⣿⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣝⣛⣛⣫⣼⣿⣿⣿⣿⡇⣿⣿⣿⣬⣙⣭⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡟⣫⣭⣭⢩⡍⣭⣭⣝⢻⣿⣿⡇⣿⡟⢡⣶⣶⣶⣦⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⡿⢸⡇⢿⣿⣿⡆⣿⣿⣧⣙⣃⣙⣛⣛⡛⠛⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣌⣡⣿⣿⣿⣧⣽⣿⡟⣩⣭⣭⣭⣭⣥⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⡇⣿⣿⣿⢛⣩⣛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠃⣿⣿⡇⢿⣿⣿⢂⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⡿⠶⠭⠵⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⠀⠀⣿⡏⣰⣿⣿⣿⣷⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠁⠈⠉⠉⠉⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⠀⠤⠤⢀⡀⠀⠀⠀⠀⡀⠄⠂⠀⠉⠉⠉⠁⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⢰⠊⠉⠉⠀⠀⠈⠤⡀⠀⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢠⠁⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠃⠀⡠⠀⠀⠠⢤⠔⠒⠂⡄
⠀⠀⠀⠀⠀⠋⢦⡀⠀⠀⠀⠀⢀⠄⢸⠀⠀⠀⣁⣀⣀⣀⣀⣀⣀⣀⣀⠀⠈⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡆⡊⠀⠀⠀⠀⢠⡀⠀⠀⡘
⠀⠀⠀⠀⢸⠀⢀⠀⢨⠃⠩⠌⢘⠀⠘⢄⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⡼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡓⢄⣠⡀⠀⢸⠉⠲⡼⠆
⠀⠀⠀⠀⠀⡄⢅⠁⠘⡀⠀⢁⣰⠀⠀⡜⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠀⠀⠉⠒⠄⠠⢀⣀⡀⠠⠤⠤⢄⡀⢘⡄⠀⠀⠀⠀⠑⢾⡱⢠
⠀⠀⠀⠀⠀⠈⠺⡀⠀⠈⠹⢙⠃⠀⠘⠒⠒⠊⠁⠉⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⡢⠀⠀⣸⠕⠃
⠀⠀⠀⠀⠀⠀⢀⡏⠐⠂⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡖⠈⢨⣀
⠀⠀⠀⠀⢀⣔⡁⡈⠒⠒⠊⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠊⠁⠀⠱⡀
⠀⠀⠀⢠⠏⠁⠀⠈⡀⠀⠀⠀⠈⢛⢦⠀⠀⠀⠀⠀⢀⡠⢾⢷⡴⠏⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡐⠁⠀⢡⠰
⠀⠀⠠⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠱⡀⠀⢀⠔⠁⠈⢢⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡐⠀⠀⠀⠘⠀⡇
⠀⢠⠃⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡄⠀⠐⠔⠁⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠽⣓⠄⡀⠀⠀⠀⠀⠀⢰⠀⢠⠁⠀⠀⠀⠘⠀⡇
⠀⡌⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⡀⠈⠢⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡌⠀⡇
⠀⠁⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠈⠀⠐⠰⢤⠁⠀⠀⠀⠰
⢸⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⡀⠀⢁⠀⠀⠠⠁⠀⠀⢰
⠈⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⠆⠔⠁
⣧⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠟⠛⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠛⠻⠿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠈⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⣰⣿⠇⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣴⣿⢏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⢀⣿⡿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣛⣭⣭⢄⣾⡿⢣⣶⣶⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣵⣾⣿⣿⡿⢣⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣴⣿⣿⣿⣿⡟⣱⣿⠟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⠟⣼⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣴⣿⠿⠛⠿⢿⣧⢿⣿⣿⣿⢋⣾⡿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⡇
⠀⠀⠀⠀⠈⣿⣷⠀⠀⠠⠟⠉⠀⠀⠀⠀⠀⠀⠈⢻⡿⢣⣿⡟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣍⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠆⠀⠀⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣷⣄⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢷⣶⣦⣤⣀⣠⣤⣶⣶⣶⣾⣿⣷⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠉⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⣠⣴⣶⡿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⢀⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⣿⣿⣷⣶⣶⣦⣤⣀⡀⠀⢾⠟⠁
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣷⣤⡀
⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢠⣾⡟⣀⣤⣤⣶⣶⣦⣤⣄⣀⣀⣀⣀⣀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠠⣿⡿⠿⠛⠛⠉⠁⠀⠉⠙⠛⠛⠻⠿⠿⠟⠁⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣤⡀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠙⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠀⠐⠒⠲⠶⠖⠒⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡴⢧⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠜⠁⡀⢀⣀⠀⠀⣠⢄⣠
⠀⠀⠀⠀⠰⢮⠉⠙⠁⠐⠈⢉⡉⠁⣴⠋⠀⠀⠀⠘⠂⠒⠒⠀⡴⠋⠉⠉⠉⠉⢁⡀⠀⣦⣤⣿⣾⣿⣿⣾⣿⣿⣿⣶⡶⠂
⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⡠⠃⠀⠀⣈⣧⡄⠀⠀⠀⠀⠠⢀⠎⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠛⠉⠉⠙⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡜⠀⠀⠀⠀⠀⡎⠀⣤⠞⠉⠁⡈⡇⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠋⠖⠉⠙⢯⠀⣿⣿⣿⣿⣿⠟
⠀⠀⠀⡄⠀⠀⠀⠀⠀⡇⠀⠘⢄⡈⠀⠉⡁⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠠⠀⣸⠀⢻⡏⢋⣿⣇⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡄⠀⠀⣀⣀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠁⠀⠀⢁⡀⠀⠀⡸⠋⠀⢹
⠀⠠⢞⠁⢀⠀⠀⠀⠀⠀⠘⡄⠀⢉⡡⠾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⢤⣘⣁⡨⠔⠋⡀⠀⠀⡎
⠀⠀⠀⢉⡽⣠⠀⠀⠀⠀⠀⢸⢸⡁⢀⠀⡬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢹⠀⢰⠇
⠀⠀⠀⠉⠊⢭⣀⠴⢳⠀⢠⠏⠉⢿⢏⡉⠀⠀⡴⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢸⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠁⠰⡷⠚⣆⡜⠂⠒⠡⣎⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢀⡇⠀⠀⠀⠀⢀⡞⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡆⠘⠦⢾⡉⠀⠒⢢⣀⡖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡀⠀⠀⠀⠀⢘⣑⠒⠺⡃
⠀⠀⠀⠀⠀⠀⠀⢈⠗⢦⡴⠃⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⢳⠶⠶⠶⠶⢾⣀⣴⠞
⠀⠀⠀⠀⠀⠀⢌⡀⠀⠸⡇⠀⠀⠀⠀⢘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠲⣆⢀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠉⠚⠓⠛⡦⠒⠒⡻⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠀⡆⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠁⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣃⠀⠘⡆⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢦⣇⣹⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠉⠙⣦⠷⠋⠁⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢠⠋⠀⠳⡹⣶⡤⣤⡤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⣀⣸⣦⣄⣀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠼⢄⣀⣀⣱⣠⣀⣠⡶⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⢂⠄⠀⠀⠉⠀⠦⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢅⠀⠀⠐⠀⠀⠀⠄⠀⠀⠐⠂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠁⠀⢀⡠⠤⠴⠴⢄⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠁⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠁⠀⠀⠀⠀⠚⡠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡠⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢠⣀⣀⡄⠀⠀⠤⠈⠀⠈⠀⠀⠀⠀⠀⢇⠸⣒⡲⠀⡼⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⡀⠑⠚⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠄⡄⠞⠀⠁
⠀⠀⠀⠀⣠⡶⢁⠉⠸⢛⡕⣦⢄⠀⠀⠀⠀⠀⠀⠀⢠⠠⠄⠀⠂⠠⠄⡀
⠀⠀⢀⣾⢸⡇⣼⠁⡏⢸⠀⣿⢸⡏⢆⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⢀⠑⢤
⠀⠀⠀⠉⠈⠁⠛⠀⠃⠸⠆⠿⠘⠇⠋⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠚⠐⠂⠀⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡠⠤⢤⣠⠎⢧⡠⠤⠤⠤⠖⠋⣸⠀⠀⠀⠀⢳⡀⢀⣀⡀⠀⠀⣴⣤⣠⣤⣤⠤⠴⠒⠋⠉⡁⠀⣶⣄⣸⣶⣤⣀⣼⣿⣴⣿⣁⣀⣀
⠀⠀⠀⠀⠀⠈⣻⠆⠀⠀⠀⠀⠀⢀⡤⠄⠒⠺⡁⠀⠀⠀⠀⠀⠉⠉⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⣦⣀⣀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉
⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⢀⡤⠃⠠⢀⣀⡠⢿⣦⠀⠀⠀⠀⠀⠀⡘⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠋⠁⠀⡀⠈⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⡏⠀⢄⣴⠋⠉⠁⠀⢹⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠛⠶⠋⠉⠑⣟⠀⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⢧⠀⠘⠂⠙⠚⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣀⠀⠠⡀⠀⢸⠀⢸⣿⠩⠝⣿⡟
⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠉⠁⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣈⠤⠀⠀⠐⠋⠀⠀⠉⠁⢺⠟⠛⢳⡄
⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠰⣖⠴⢂⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⠀⠀⣠⠔⢢⠄⣀⡴⠋⠀⠀⢰⡇
⠀⠀⠛⢤⣀⣤⡀⠀⠀⠀⠀⠀⠀⢱⠀⣀⡤⠔⠋⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠲⠦⠴⠖⠊⠁⢤⡆⠀⠀⣼
⠀⠀⠀⠀⢀⠞⣁⡄⠀⠀⠀⠀⠀⢈⡇⣷⠀⢠⣀⣤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡟⠀⠀⠀⠀⠀⠀⣇⠀⢀⡏
⠀⠀⠀⠘⠛⠚⢹⣀⣠⠼⡇⠀⠀⡼⠉⠉⣿⣞⡋⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡤⠚⠁⣴⡤⠼⡄⢹⠦⠠⠴⢃⡾⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢸⠀⠀⠀⠀⠀⠀⢀⡏⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢰⠉⣧⢀⣳⠿⢤⢀⣀⡈⠉⣩⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢤⡏⠀⠀⠀⠀⠀⠀⡼⢤⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣁⠀⠀⣸⠀⠈⠉⠉⣇⠀⠀⠀⠙⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣷⣄⡀⠀⠀⠀⠈⠓⣲⠀⢈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠋⢳⡞⠁⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠉⠙⠛⠛⠋⠻⣤⡧⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠨⣄⠀⠀⠘⣇⠀⠀⠀⠀⠀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠈⠺⡧⠤⡄⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠳⠼⢦⠤⠤⠖⣚⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢲⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠇⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠇⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠧⣤⣀⠀⡇⢨⡧⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⢙⠿⢶⡾⣦⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠙⣿⠛⠉⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⡀⠀⡏⠀⠀⠙⡌⢟⢷⠶⠦⠶⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢀⣸⣧⣄⣀⣀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠾⢧⣀⡀⠀⢰⣤⣤⣤⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣤⠤⠴⠦⣤⣀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠉⠉⠙⠳⣦⡶⠋⠀⣀⣀⣀⠀⠉⢳⣤⠶⠛⠉⠉⠙⠛⢦⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⣰⣄⠀⠀⣼⠁⠀⠀⣿⣁⣹⠇⠀⠀⢻⡀⢀⣤⡤⣤⡀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣰⠿⢿⣄⠀⣿⠀⠀⠀⣿⠉⣹⡆⠀⠀⣸⠃⢸⡁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠐⠋⠀⠀⠹⠂⠘⣧⡀⠀⠉⠉⠉⠀⠀⣰⡏⠀⠈⠳⠶⠞⠃⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⡾⠋⠙⠶⣄⠀⢠⡶⠛⠉⢷⡄⠀⠀⠀⢀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢦⣼⡇⠀⠀⠀⠹⣶⡟⠀⠀⠀⢸⣧⣴⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⣀⣈⣀⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⢛⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣯⣴⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⣶⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠛⠂

⣿⡏⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡿⠿⠿⠿⠿⠿⠿⢿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⢠⣿⠏⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣾⠟⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⣼⡿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣭⣵⡶⣰⣿⢏⣾⣿⣷⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣶⣿⣿⣿⠟⣴⡿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⢋⣼⡟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇⣿⣿⣿⡿⢣⣿⢟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⠀⢻⣿⠀⠀⠠⠾⠛⠉⠀⠀⠈⠁⠘⣿⡿⣱⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣿⡟
⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⡿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠉⠙⠛⠛⠿⠿⠿⠿⠟⠻⢿⣿⡁⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣶⣶⣶⣶⣾⠿⠿⠟⠛⠉


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⡤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠦⠄⠆
⠀⠀⠀⠀⠀⣿⣿⣤⠘⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠸⢿⠾⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠠⠀⠀⣄⠀⢀⠀⠄⡀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠈⠀⠀⠀⠀⢂⠀⠀⠀⠀⠈⣀⣸⣅⡀⣡⠀⠛⠠⠤⠀⠀⢀⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⣴⢺⠀⠀⠀⠂⠀⢀⡀⠀⠘⢿⣿⡋⠀⠈⡀⠀⢄⠒⠖⡄⠀⠀⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠀⠀⢸⡇⡀⠁⣳⠦⠀⡀⠀⢨⡾⠟⠀⠑⣾⣷⣿⣷⠄⠰⢿⣁⣴⣤
⠀⠀⠆⠘⡆⠀⠀⠠⣀⡇⠈⠙⠒⠚⠠⡈⠂⠁⠀⠀⠁⠀⠀⠀⠚⠂⡀⠈⢉⡿⣿⣶⣷⣿⣿⣏⡿⠈⠃
⠀⠀⡄⠀⣧⠠⠐⢻⣽⣷⡰⠶⠤⠤⠴⠶⠶⠠⠦⠤⣴⣾⡷⠷⣶⣶⡾⠿⢾⣷⣶⡾⠟⠛⢿⠊⠀⠀⡃
⠀⠀⠈⠀⡿⠋⠀⢒⣹⢿⣷⠎⠁⠀⠀⠀⠀⠀⣠⣾⡿⠛⣿⡖⠒⡶⠒⢶⡾⠿⡿⠶⠂⢀⣨⢷⣤⠎
⠀⠀⢀⣀⡿⣄⡀⢐⢺⣿⣿⡄⠀⠀⠀⠀⣴⣾⣿⣿⠇⠀⠈⢁⡀⠀⠄⠂⠀⠀⠀⠀⠀⠈⠁⠛⠁
⢐⣤⣤⣿⣿⣟⣿⣷⣾⣿⣿⣥⣤⣤⣶⣿⣿⣿⣿⣿⠂⠀⠑⠄⡆⠄⢀⠄⠀⡠⢀⢂⠀⠓⠚⠐
⠀⢿⣿⣿⣿⣹⣿⣷⢺⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣀⡀⠦⠐⠑⠀⠓⡀⠛⠀⠛⠈⣷⡶⠃
⠀⢸⡟⢻⡿⣿⣿⢿⣎⣿⡟⠛⢻⡿⠋⠀⠀⠙⣿⣿⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠸⠇⠈⠇⠏⢚⠘⣇⣿⡇⠀⢸⠀⠀⠀⠀⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢷⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠄⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠁⡠⠄⠠⡀⢉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢰⣥⠀⢠⣬⡆⢂⢱⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠸⡇⠀⡄⠉⠁⡄⠉⢸⠀⠄⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⢇⠘⢍⣩⠇⣸⡄⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡐⠉⢀⣋⠅⡗⢤⣄⠔⣟⠩⠐⠀⢃⠀⠀⠀⠒⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠞⢢⡭⢧⣀⠘⠐⡿⢏⡙⣆⠈⠀⠀⠀⠀⠀⢠⢦⠁⠛⡄
⠀⠀⠀⠀⠀⠀⠀⡾⠈⠀⠋⡅⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠣⡀⢉⡩⠯⣀⠜⠀⠢⠑⡄⠀⠀⠀⠀⠐⠈⡠⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢁⡀⠀⠀⠀⠀⠀⢠⣴⣊⣉⠆⠈⢺⢖⣳⡏⣀⠄⡠⢄⡘⡄⠀⠀⣠⠃⠀⣁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⡀⠙⠦⣀⠀⠀⡾⣡⡼⣟⡉⢄⡂⠘⣷⡟⢠⠢⠐⣶⣠⣼⣿⢱⣌⢠⣀⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠁⠦⠈⠙⠒⠀⠸⣷⡿⣲⠵⠁⡔⣹⣗⣀⡉⠀⠈⠉⢹⠋⣉⣼⠇⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣁⠑⠀⢠⣟⡀⣗⡆⠈⢿⢫⣛⠿⡫⢄⡇⠄⣻⢾⣯⠭⢡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠉⠧⠼⠿⠧⠤⠥⠤⠴⠿⠶⠮⠮⠭⠧⠴⠿⠴⠯⠴⠅⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡠⠡⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠌⠀⠀⡁⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠌⡐⡁⡐⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⠂⠎⡐⡐⠀⡀⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⢂⠄⡔⡐⡐⡡⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠢⠁⢀⠎⡜⡐⡀⡐⡰⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠠⠠⢠⠃⠂⠄⡐⡐⠐⡐⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠁⠊⠜⠐⠐⠰⠡⠞⠄⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⠉⠉⠉⠉⢳⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣶⣶⣶⣶⣶⣶⣶⡝⣿⣿
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣸⣀⡀⣀⠀⡀⢀⡀⠀⢸⠀⠀⠀⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠇⠿⢿⢿⣿⢿⡿⢿⣿⡇⣿⣿
⠀⠀⠀⡏⠁⠉⠉⠁⡁⠁⡉⠀⠀⢙⡏⠈⠋⠘⠁⠀⣸⠀⠀⠀⣿⣿⣿⢱⣿⣿⣿⣿⢿⣿⢿⣿⣿⣦⢰⣷⣾⣷⣾⣿⠇⣿⣿
⠀⠀⠀⡇⠀⠀⠀⢺⠟⣺⠓⠀⠀⢠⡏⠉⠉⠓⢦⣿⠁⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⡍⣬⠅⣬⣿⣿⣿⢰⣶⣶⣬⡛⠡⣾⣿⣿
⠀⠀⠀⡇⠀⠀⠘⠟⠹⠟⠁⠀⠀⠰⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⢸⣿⣟⣛⣰⣖⣰⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠛⢦⡄⢀⣠⠦⠤⠤⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣮⣛⢻⣿⢍⣙⣉⣉⣉⣉⣩⣬⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣈⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠒⠒⠒⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠟⠉⠉⠉⠉⠉⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡇⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠁⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠷⠤⣤⣀⣀⣀⡠⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⣀⣠⢊⣽⠥⡿⠭⠧⣀⡀⠀⠀⠀⠀⢠⠂⠀⠀⡆⠀⠀⠀⠀⠀⠀⡞⠁⠐⡍⠓⠲⠤⠤⠤⠤⠤⠤⠒⠚⠉⠁
⠀⠀⠀⠀⢀⣰⡅⠚⡀⠙⠄⠓⠀⠀⠀⠈⠑⢦⡀⠀⠀⢓⠶⠚⠁⠀⠀⠀⠀⠀⠀⢙⠒⠒⠁
⠀⠀⠀⠀⣷⠁⠑⠄⠈⠀⠀⢀⡀⠀⠀⡀⠀⠀⠙⢆⠀⠓⠐⠁⠀⠀⠀⠀⠀⠀⠀⠐⠔⠀⠀⣦⣔⣪⣅⡠⠈⠀⡐⠤⣀⠀⠀⠔⠀⠉⠒⡄
⠀⠀⠀⢰⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠈⡎⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⡔⠁⠀⠈⠁⠀⠀⠀⡌⡆⠈⠳⣍⣿⢍⠙⡄⢸
⠀⠀⠀⢸⢠⠤⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠘⠀⢀⡀⠀⠱⡘⣄⠀⣈⢧⠈⠀⣇⡇
⠀⠀⠀⠘⣌⠉⠀⠀⠀⠀⠀⠈⠢⡄⠠⠈⠀⠀⠀⣠⠃⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠐⠇⠀⠈⠀⠀⠀⠘⢮⡓⠬⡎⡄⠀⡟
⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⣀⠔⠁⠀⢈⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⡀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠈⠁⠐⡇⠀⣇⠀⣀
⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⢄⢀⣀⠠⠤⡔⠊⠁⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠉⠒⠛⠁⠀⠀⠀⠀⠀⠀⠬⠇⡸⢌⠂⠈⡃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠙⠖⠃⡇⠳⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢓⠀⠀⠀⠉⠢⢀⠀⠀⠀⢀⣀⠀⣐⡠⠊⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⡿⠀⠐⠒⠃⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⡴⠐⢄⠜⢹⠏⠣⡉⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠁⢸⡇⠀⠈⠉⠁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠉⠀⠀⠀⠀⢸⠀⠀⢹
⠀⠀⠀⣤⣀⠔⠋⠀⠀⠀⢹⡏⠉⠉⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⢀⡞
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⡿⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠋⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⡀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠳⡎⠙⢞⠑⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⡀⠀⠀⢹⡀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣠⠇⠀⠀⠉⢿⠄
⠀⠀⠀⠀⠐⠒⠒⠒⠂⠀⠲⣤⣽⠆⠀⠀⠒⠲⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠙⠋⠁⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⣋⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢠⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⣿⣟⣛⣛⠛⣛⣛⣛⡛⢋⣉⣤⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿
⠀⠀⣠⣤⣄⠛⣿⣿⣿⣿⠀⣿⣿⡏⣡⡾⢋⣙⠻⢋⣈⠻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿
⠀⢰⣿⣿⣿⡇⢸⣿⣿⣿⣦⡙⢛⣡⣿⣇⠻⠟⢀⠸⠿⢃⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢨⣿⣿⣿⡇⢸⣿⣿⣿⣿⣧⢸⣿⡟⠛⣷⣾⣿⡗⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⣉⣉⠉⠙⠻⣿⡟⠉⠉⢉⣀⣀
⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣆⠻⢿⣿⣿⣿⠟⣡⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⣋⣴⣶⣿⣶⣈⠇⢸⣷⣦⡙⣿⣷⣄
⣦⠘⣿⣿⣿⡇⣸⠏⣿⡿⠟⢛⣙⠳⣶⣤⣤⠄⠺⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡏⣴⣿⣿⡟⢻⣿⣿⣧⣸⣿⣿⣷⠘⣿⣿⣷⡄
⠘⢷⣬⣉⣩⣴⠟⠀⠁⣴⣾⣿⣿⣷⣌⠻⡟⢰⣷⡘⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇⠻⣿⣿⣷⡘⢿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡄
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣤⣿⣿⣧⢹⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⢠⣤⣤⣤⣤⡄⢹⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⣭⣭⡌⠿⣿⣿⡿⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣌⠛⠿⠿⢛⣡⣶⣦⣤⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⢋⣤⣶⣶⣶⣶⣶⣶⣶⣦⡙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⢡⡄⣿⣿⠟⢋⣴⣦⣌⡙⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⢡⣿⣧⠘⣡⣾⣿⣿⣿⣿⣿⣦⠹⣿⣿⠏
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⣋⣥⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⠁
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⣿⣿⡟⢋⣴⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣄⠻⠿⢃⣾⣿⣿⡏⢻⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠁
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣘⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡌⢿⣷⣶⣾⣿⣿⡟⣠⣿⣿⣿⠟⠟⠛⠛⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⠿⣦⣍⣛⠛⣛⢡⣶⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⢠⣤⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⣉⣴⣶⣶⣌⠻⢿⣿⠇⣴⡘⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢾⣿⣿⣿⣿⣿⣷⣦⡙⢰⣿⣷⠘⣿⣿⣿⠸⣿⣿⡿⣿⣿⣿⡇⠈⠛⠛⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠛⠋⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡾⠿⠿⠿⠿⠿⠿⢷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⠿⠿⠿⠿⠿⠿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⢀⣴⣾⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⡆⠀⠀⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⢠⣿⠃⠀⠀⠀⠀⣴⡿⠋⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⢀⣤⣄⠀⠀⠀⠀⠙⣿⡄
⠀⢀⣿⠏⠀⠀⠀⠀⣼⡿⠁⠀⣠⣿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢁⣴⣿⠟⢿⣧⠀⠀⠀⠀⠹⣿⡀
⠀⣼⡿⠀⠀⠀⠀⣸⡿⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠈⢿⣇⠀⠀⠀⠀⢿⣧
⠀⣿⡇⠀⠀⠀⢀⣿⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⢁⣀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢸⣿
⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠘⣿⡄⠀⠀⠀⠸⣿⡄⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣿⠃⠀⠀⠀⢠⣿
⠀⣿⣇⠀⠀⠀⠀⢿⣇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣼⡿⠀⠀⠀⠀⣸⣿
⠀⠸⣿⡀⠀⠀⠀⠘⣿⣆⠀⠀⢿⣧⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⣰⣿⠃⠀⠀⠀⢠⣿⠇
⠀⠀⢻⣷⡀⠀⠀⠀⠘⢿⡿⠂⠈⠻⠗⠀⠀⣠⣾⠟⠁⣠⣤⣄⠀⠀⠀⠀⠀⣤⣴⣾⠟⠁⠀⣴⡿⠃⠀⠀⠀⢀⣾⡟
⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠈⠉⠻⣷⣄⠀⠀⠀⣿⡇⠀⠀⣠⣾⠟⠁⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⢠⣾⣟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⣿⡇⠠⣾⠟⠁⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣤⣀⣀⣀⣀⣀⡀⠈⠻⣿⡿⠃⠀⠀⠀⠀⠀⠀⣀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣶⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣶⣶⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠁

⣏⣹⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⡀⠀⠀⢀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠀⡨⢠⢢⠀⣶⠀⠈⠀⡌⠰⠢⢸⠅⠁⠸⠇⢟⠀⡀⠀⡇⢸⠂⠕⢎⠇⢸⠂⢏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⢀⠀⢀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⢀⠀⠀⠀⣀⣀⣀⢀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠈⠁⠁⠁⠉⠀⠉⠉⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⡋⢙⣿⠉⢹⣿⠉⢹⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣶⡦⠀⣶⡆⠠⣶⡆⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠘⢦⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡹⣿⣿⣿
⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⢈⡇⠀⠀⢸⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡷⣹⣿⣿
⠀⠀⠀⢀⡴⠿⠋⠁⠐⢤⡀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢸⣿⣿⣿⣿⢋⣠⣴⣿⣏⡻⣿⣿⣿⣿⣿⠟⣡⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠦⣷⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣭⣧⣙⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⢸⡏⠁⠉⠉⠁⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⣾⣿⡿⢻⣧⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⡇
⠀⠀⠀⠀⢰⣿⣿⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⡇
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠉⠻⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣴⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⣆⠀⢹⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⣧⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⡿⠀⠘⡝⢿⡇⠀⠸⣧⡀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⣻⠟⢋⠙⠻⣿⡿⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⠁⠀⠀⣄⣼⠇⠀⠀⣿⣂⣴⠟⠁
⠀⠀⠀⠀⠀⡟⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠻⣦⣀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⢀⠀⠀⠀⠀⣿⣶⣦⣤⡀⢀⣀
⠀⠀⠀⢀⣤⣶⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⡀⠀⠀⠀⠀⠀⣀⣠⡼⠃⠀⠀⢰⣿⣿⣿⣿⡿⠋⠙⢷⡀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣛⣿⠋⠉⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⢠⡿
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣬⣉⣉⣙⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠉⠉⢳⡴⠛⢷⡄
⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣠⡾⠋⠻⣿⠃⠲⢤⡾⠁⢀⡞⠙⣷
⠸⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠈⠛⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡋⠀⠀⠀⢻⣆⠀⠀⠻⠶⢿⣀⣰⠏
⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⠙⠛⠿⣶⣶⣶⣾⣿⣿⣿⠟⠉⠀⠈⠻⣦⡀⢀⡾⠋⠀⠀⣀⣄⣤⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⣆⠀⣰⠿⠟⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠘⢷⣠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠋⠀⠀⠀⢀⣼⠋
⠀⠀⠀⣀⣀⣀⣀⣠⣤⠟⠁⠀⠀⢀⡴⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⣤⣤⣤⠞⠁
⠀⠀⣾⠉⠉⠉⠉⠉⠀⠀⠀⢀⡴⠋⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠉
⠀⠀⠙⢷⣤⣤⣤⣤⣄⠀⠀⠸⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠘⢷⣤⡴⠟⠻⣆⣀⣠⡿⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠘⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣤⣤⠄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠇⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠀⣿⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡿⣦⣄⣀⣄⣿⠀⣿⣄⣀⣠⣠⡿⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⡵⠸⠦⠜⠉⠉⠉⢻⡀⣿⠉⠉⠉⠹⠴⠇⢼⠟⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⢾⡇⣿⠒⠒⠒⠒⠒⠒⠚⠒⠒⠒⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⢀⡀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣰⡿⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣾⠃⠀⣼⡟⠀⢀⣾⠟⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡿⠀⢠⣿⠀⠀⣾⠇⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡏⠀⢰⣿⠀⠀⢰⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⢸⣇⠀⠸⣿⠀⠈⣿⡄⠀⠀⠉⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠈⣿⡀⠀⢿⣆⠀⠘⣷⣄⠀⠀⠘⢻⣷
⠀⠀⠀⠀⠀⠀⠸⣷⡀⠈⢻⣦⡀⠈⠋⠀⠀⠰⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠙⠃⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠶⠶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠷⣦⣀⣴⠾⠛⠟⢷⣦⡀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡀⣾⠋⠀⠀⠀⠀⠀⠈⠛⠁⠀⠐⠠⡱⡙⣷⣤⣤⣤⣤⣀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐⠻⠹⠱⡱⡹⢙⢷⣄⠈⠆
⠀⢀⡤⠖⠂⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠡⠁⠁⢃⠜⣻⡆
⠰⠁⣠⡶⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⠛⠟⡷⣦⡀
⠁⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡐⠠⠊⣻
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣽
⠀⠀⠻⣦⡄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁
⠀⠀⢠⡟⠁⢠⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠸⣧⣸⣟⣗⣴⣼⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠙⠿⣾⣮⣿⣧⣿⣮⡵⣡⡖⡠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣵⣟⣼⣱⣳⣷⣆⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠁⢻⡆⠀⠀⠀⠀⣰⠟⠳⠶⠶⠶⠞⠉⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡾⠋⢀⠴⠤⠤⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⣤⡾⠋⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⠁⣀⠔⠉
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠚⠛⠛⠛⠛⠛⠛⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⣀⣤⠴⠖⠛⠋⠉⠉⠉⠉⠙⠛⠲⠦⣤⣀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠀⠈⠳⣄
⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠈⠳⣤⠶⠚⠛⠛⠛⠓⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⣦
⠀⠀⡼⠁⠀⠀⣰⠛⢦⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⢧
⠀⣼⠁⠀⠀⣼⠃⢰⠏⠙⢦⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⣧
⢰⠇⠀⠀⢰⠇⠀⡟⠀⠀⠀⠙⢦⡀⠀⠀⠈⠷⣄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢸
⣾⠀⠀⠀⡿⠀⢰⡇⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠉⢷⣄⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⡿⠀⠀⢠⡇⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⢷⣄⠀⠹⡄⠀⠀⢀⡙⠲⣄⠀⠀⠀⢸⡀
⣿⠀⠀⠸⡇⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⢷⣄⢙⡆⠀⣬⡙⢳⡌⢳⡀⠀⢸⠃
⢿⠀⠀⠀⣧⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⢿⣄⠀⠀⢹⡆⣿⠈⡇⠀⣼
⠸⡆⠀⠀⢹⡄⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⢷⣄⠞⢁⡟⢰⠇⢠⡇⠀⠀⢰
⠀⢳⠀⠀⠀⢳⡀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⠷⡉⢰⠏⢀⡟⠀⠀⢀⡟
⠀⠀⢧⡀⠀⠀⠻⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⠶⠞⠛⢦⡀⠀⠀⠘⢷⣤⠟⠀⠀⢀⡾⠁
⠀⠀⠈⢳⡄⠀⠀⠙⢦⡀⠠⣇⣀⣀⣀⣀⣀⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠁⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠙⠓⠦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠴⠚⠋⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⢤⣤⣄⣀⣀⣀⣀⣠⣤⡤⠶⠞⠋⠁
⣿⡿⠻⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿