Bowling Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣸⣦⣄⣀⠜⠀⠀⠀⢠⠖⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⣧⣤⣼⡀⠀⠀⠀⠀⠀⣰⣯⣙⡻⠃⠀⢀⣀⣀⠃⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠘⢄⢀⣤⣾⡄⠀⠀⠀⠀⠀⢰⠋⠉⠉⢣⠀⠀⠀⡠⠊⠉⠛⢿⡇⠀⢠⠋⠀⠈⢿⣶⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⣡⣼⣦⡀⠀⠀⢀⠇⠀⠀⠀⠀⠣⡠⠊⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⣶⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠋⠀⠈⠑⠢⡜⠀⠀⠀⢀⣀⣠⣀⣀⣀⠀⠀⠀⢠⢀⡤⠺⠿⢿⡟⢙⠁
⡠⠒⠒⠤⡀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⡼⠋⠀⣾⣿⣶⡆⢸⠀⠀⠀⠀⠀⠀⣠⠒⠉⠈⠙
⢇⠀⠀⠀⣸⣦⣀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠈⢁⠸⠤⠤⠤⣤⣶⠻⣿⡆⠀⢀⡴
⠀⠑⠢⠴⣿⢋⣼⡿⠒⠒⠂⠤⠵⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡧⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠙⢿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠛⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣀⣀⣀⣀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠉⠉⠉⠉⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢀⣠⣴⣶⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠙⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⣀⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡏⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣴⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡸⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣀⣀⣀⣀⣀⡼⠁⠀⠀⠉⠛⠿⠿⢿⣿⣿⡿⠿⠿⠛⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣟⣛⣫⣟⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣭⣭⣽⣿⣯⣭⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠳⡄⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢸⠀⠀⠁⠀⠀⠀⠀⢀⡔⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠉⠳⡄⠀⢪⣣⠘⡇⠀⠀⠀⠀⠀⠀⠀⢧⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⠢⣀⣻⡁⠀⠙⣿⠿⢦⡀⠀⠀⢠⣤⠒⢢⣦⣾⣄⠀⠀⠀⠀⠀⠀⢀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⠈⣿⠛⠓⢦⡇⠀⠀⠉⢢⡀⠀⢸⣠⣿⠋⠁⠈⠳⡄⢀⡤⠒⢠⡃⡠⠋⠀⠃⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡞⠁⠙⢆⠀⢄⠀⢸⢀⠀⠈⢀⡀⠀⠀⠀⢳⣠⠞⠹⡇⠀⠀⠀⠀⣿⣤⠴⠒⠛⡟⢁⡔⠚⠉⠉⢙⣯⠠⠜
⠀⠀⠀⠀⠀⠀⠙⢶⠄⠈⢧⠀⠑⠈⢎⠆⠀⢇⢣⡀⠀⠀⠈⡏⠀⠀⢳⡀⠀⠀⢸⣿⠁⠀⢀⠄⡇⠘⣿⣀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠐⢽⣦⣿⣧⣅⡀⠀⠐⠀⡀⢣⠑⠀⢀⣀⡇⠀⠀⠠⠓⢤⣾⢿⣟⠀⣠⢞⠼⠁⠀⣀⠀⢀⣤⡤⢤⡀
⠀⠀⠀⠀⠒⠦⠄⠀⠀⠀⡹⠋⠀⠉⠓⠢⣄⠈⠁⠈⠚⣋⣽⣧⡴⠶⠦⣀⠀⠉⠃⠙⠓⡊⠡⠤⠬⡥⣴⣿⣟⣁⣠⠴⠁⠀⢀⣀
⠀⠀⠀⠠⣄⠀⠀⠐⠤⠀⠳⡇⠀⠀⠀⠀⠈⢣⠠⠀⣪⣿⡟⣿⠀⣀⠀⠀⠙⣗⢀⠖⠡⠖⠊⠀⢀⣀⣸⣏⠁⣀⣀⣀⠔⠊⢁⣠⡿
⢠⡔⠒⠲⠤⣉⡒⢄⠀⣀⣀⣿⢦⠀⠀⠀⠀⠀⢷⣾⣿⣿⢯⡁⠘⠛⠀⠀⠀⠈⣇⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⢹⠿⠓⠋⠉⠉
⠃⠳⢶⣶⣦⣤⣹⣿⡟⠀⠀⠉⠑⢥⡀⠀⠀⠀⠘⣿⣿⣿⣼⠋⠀⠀⠀⠀⠀⠀⢸⣄⣠⡏⠁⠀⠀⢀⣀⠀⣀⠔⢑⣫⠴⠒
⠀⠀⠀⠒⢒⣀⣀⣀⣙⣦⡠⡀⠀⠀⠈⠓⢤⣠⡶⡿⣿⢿⣿⣷⣄⠀⠀⠀⢐⣠⡜⠀⠐⠷⠤⠤⠤⠴⠖⠊⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠲⠤⠴⠖⠚⠒⠒⠿⢝⠧⣝⡻⠿⣿⣷⣾⣿⣿⣁⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠩⠟⠁⠒⠒⠒⠂⠀⠒⠒⠿⠶⠭⠽⠿⠽⠾⠵⠂⠀⠀⠈⠩⢭⣭⣭⣍⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⣀⢀⣠⡄⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡽⠀⠀⠀⣸⠀⠈⠉⢰⠃⠘⠛⠀⠀⡾⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⡀⠐⠋⠤⡄⠀⢀⣈⣣⡀⠀⠀⠀⣧⣼⠃⡔⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠉⢉⣽⣦⡀⠀⠰⡀⠀⠀⠸⡶⠃⢠⠀⣀⠀⠀⣰⠧⣿⠊⠀⢶⠋⠈⡴
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⣴⣯⢫⡾⠓⠦⣤⣥⣤⣀⡀⠀⠀⡌⠘⠿⠃⣰⠃⠀⢸⠀⠀⠐⠒⠓⠆
⠀⠀⠀⠀⠀⢀⣠⠜⠳⢤⡄⠉⠻⣅⠀⠀⠀⠈⠉⠙⠻⢷⣤⡃⠀⢀⡜⡏⠀⠀⠸⡇⠀⡠⢀⡥⠖⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢹⣀⣄⣮⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠘⠀⡇⠀⠀⣸⠇⠈⣠⠎⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢠⣴⣮⠅⢀⡉⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⢷⣀⣠⠏⠀⣰⠃⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠋⠙⠀⠀⠈⠉⠑⠒⠄⠹⣧⣀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⣠⠩⠅⠀⣰⣇⠀⠀⠀⠀⣠⠞⢁⡀
⠀⠀⠀⠀⣠⠒⠢⣄⣀⠀⠀⠀⠺⠷⠀⠈⠻⢿⣶⣤⣤⣤⣶⣿⣺⢯⡇⠤⠄⣀⡞⠹⣽⣶⣤⣠⠞⠁⣀⡰⣆
⠀⠀⠀⠀⠈⠲⠤⣾⣩⡟⠒⠒⠒⠒⠲⠦⣄⡀⠈⢉⠛⠛⠿⠿⠗⢋⣠⡴⠞⠻⣻⣦⣀⣩⠟⠁⠀⠀⣹⣀⣰⠋
⠀⠀⠀⠀⠀⢀⣠⠤⠌⠙⢦⣀⠀⠀⠀⠀⠈⠙⢶⣄⠀⡔⠘⣠⠖⠋⠁⠀⠀⠀⠙⠻⢿⠏⠈⠉⠉⠐⠂⠀⠉
⠀⠀⠀⠀⠀⢴⡆⠀⣠⣄⠀⠙⠷⣦⣤⣀⣀⣀⣼⡿⡼⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⠀⠀⠀⢠⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⡰⡄⠀⠈⠉⠉⠉⠉⡼⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠈⠑⣠⣤⠘⠳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⠚⠁⠙⢒⡦⠀⠀⠀⡐⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⢠⣌⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣀⣀⣻⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢃⡀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠉⣤⣄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⢿⣧⣀⣀⣀⣠⣴⠾⢛⠁⠀⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠛⠋⠁


⣿⣿⠛⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠉⠉⠉⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⠀⠀⠀⠀⠀⠀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⢿⣿⣿⣿⣿⠿⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣤⣤⣤⣤⣤⣤⣤⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠙⢆⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠁⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢀⣀⣤⡄⠀⠀⢠⣾⣶⣤⡀⠈⢳⡀⠀⠀⢎⠀⠀⠱⣈⠀⢀⠔⠁⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣰⠟⠉⠉⠉⠀⠀⠀⠀⠀⠙⠧⠀⠀⢷⠀⠀⠀⡑⠀⠀⠀⠀⠁⠀⠐⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠁⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠐⢤⠀⠀⠀⠀⠀⢠⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢐⠀⠀⣤⡄⠀⠄⠀⣤⡄⠀⠀⠀⠀⠀⣷⠀⠀⡰⠁⠠⠂⠀⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣀⠀⠀⠀⠀⠉⠀⠀⠄⠀⠈⠁⠀⠀⢀⡀⠀⢹⡔⠊⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈⠉⠀⢸⣇⡠⠂
⠀⠀⡀⠄⠤⢄⠀⡠⠒⠉⣁⣼⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⣼⠉
⢀⠴⠒⠒⠀⠀⠙⡤⠊⠉⠀⠀⣧⠀⠈⣷⣶⣄⣀⣀⡀⠀⢀⣀⣀⣤⣶⣾⡟⠀⠀⡟
⡇⠀⠀⣀⡀⠀⠀⠀⠁⠀⠀⠀⢹⠀⠀⠘⢿⠿⠛⠛⠻⠛⠛⠛⠻⣿⣿⡟⠀⠀⢠⠇
⠈⠒⠟⠹⠀⡆⠀⢂⠁⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠘⠤⠔⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⡿⠷⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣴⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⡾⣆⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠷⢦⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠷⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣾⠟⠁⠀⠈⡇⠀⠀⠀⠀⠀⣠⠔⠻⢷⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢳⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⡏⠀⠀⠀⢠⠇⠀⠀⣠⢔⡯⠂⠀⠀⠀⡇⠀⠀⢀⣤⣶⣶⣦
⣠⡾⠉⠓⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣿⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢸⣦⣤⣤⠏⠀⣠⣪⣵⠟⠀⠀⠀⠀⠀⢸⠀⢠⠟⠋⠁
⢿⣧⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⢰⣷⣿⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢸⣄⣉⡍⢀⡜⣵⡿⠃⠀⠀⠀⠀⠀⠀⢸⢀⣇⠀⠀⠀⢀⡜
⠈⠻⣇⠀⠀⢀⣾⠧⡀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢱⣠⠋⠉⠉⢣⡜⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⣼⡞⠻⢷⣤⠔⠉
⠀⠀⠀⠉⠛⠻⣅⣴⡿⠑⠒⠒⠶⠤⢯⣿⣇⠀⠀⠀⠀⠀⠀⢀⣜⡵⠀⠀⠀⠈⢿⣿⠁⠀⠀⠀⠀⣀⡤⠔⠚⠉⠻⢶⠊
⠀⠀⠀⠀⠀⠀⠈⢫⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡀⠀⠀⢀⢎⡾⠁⠀⠀⠀⠀⠈⡇⠀⠀⡠⢔⣯⡷⠋⠁⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⢏⣿⡇⠀⠀⠀⠀⠀⠀⢸⡤⣪⣾⡿⠋⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⡏⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡻⣿⣦⣄⠀⠀⠀⠀⢀⣷⣿⣷⣾⣿⣷⣶⣦⣤⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢽⡻⢿⣦⣤⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠳⡄⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣾⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⣿⣶⣿⣆⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢿⡖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣌⣉⣛⣛⣻⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⡄⢀⡴⠶⠦⡀
⠀⠀⠀⠀⠀⠀⣴⠞⠓⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⠁⡏⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⣾⠀⣧⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⣶⣶⡇⠀⢸⡀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣠⣤⣀
⠀⢀⠀⠀⠀⠀⠀⢳⣤⣴⣾⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣏⢈⣉⣉⡇⠀⠈⣷⢰⣶⣿⠀⠀⠀⠀⠀⢸⠃⠀⠀⡆
⣾⠁⠈⠱⣄⠀⠀⠈⣇⣥⣾⣧⣠⣴⣾⣿⣿⠿⠿⣿⣿⣷⣾⠿⠸⠿⠿⢧⠀⢸⣷⢰⣦⣿⠀⠀⠀⠀⠀⠸⡆⠀⢠⡇⠀⠀⢀⣀
⠙⢄⠀⠀⠙⣆⠀⢀⡟⢈⣵⣿⡿⠿⢿⣿⣇⠀⠀⣽⣿⣿⠏⠀⠀⠀⠀⠀⢇⡾⠉⠈⠉⠉⢇⠀⠀⠀⠀⠀⣧⣤⣼⠁⣠⠞⠉⠉⠛
⠀⠀⠙⢦⡼⢟⣧⣼⢣⣾⣿⣿⡀⠀⣠⣿⡿⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⢆⠀⠀⠀⢸⡧⣿⣿⠞⠁⠀⠀⠀⣠
⠀⠀⠀⠀⢻⡞⠋⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⣹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠘⣄⣀⣤⣿⡶⣋⢹⣶⣀⣤⠴⠚⠁
⠀⠀⠀⠀⠈⣧⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢿⡷⠟⡏
⠀⠀⠀⠀⠀⠸⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠜⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠹⣜⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣸⠀⠀⢠⠊⠀⠀⠀⡿
⢀⠔⠒⠚⠛⠒⠒⠺⣿⡟⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⡟⢀⡴⠁⠀⠀⠀⢰⠇
⣾⠀⠀⠀⠀⠀⠀⠀⢿⡇⢸⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣼⠟⠁⠘⡄⠀⠀⣠⠏
⠘⠷⣤⣤⠤⠶⠒⠚⠋⠉⠉⠛⠳⢶⣤⣄⣀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢠⡿⠷⡀⠀⠀⠀⣰⠃⠀⠀⠀⠙⠓⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠟⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠛⠓⢦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡴⠖⠛⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⣉⠙⠳⣼⠃
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠖⠛⣉⣉⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣉⣉⡙⠳⣿
⠀⣠⣶⠿⠛⠛⠛⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡴⠛⣉⣩⣭⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⠉⠳⣿
⣰⡟⠁⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠋⠉⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⢠⣟⣁⣀⣀⣀⠀⠀⠀⠀⢻⡀
⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢈⣻⣴⡶⠿⠛⠛⠋⠉⠉⠙⠛⠛⠷⢶⣼⣇
⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣀
⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⣠⡾⠃⣸⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⡾⠋⠀⢠⠖⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣟⣁⣠⠞⠁⣠⡏⢹⡦⣄⡀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢀⣼⠟⠀⠀⠀⢇⢸⣿⡿⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣠⣤⠾⠋⣠⡟⠁⠈⠛⠶⣤⣀⠀⠀⠀⠈⣿⠀⣾⠏⠀⠀⠀⠀⠀⠉⠉⠀⠀⢰⠋⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡶⠟⠋⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⡀⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢾⣿⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⢠⠊⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠸⣀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⣀⣀⣀⣀⣠⣴⣿⠟⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠉
⣀⢀⣀⡀⣀⣀⡀⣀⣀⢀⡀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⠛⠛⢻⠛⣋⣁⣀⣀⣀⣀⣉⣛⠛⠷⣶⣶⣶⣶⣶⣶⠟⠛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡐⠁⠀⠀⠑⠀⠀⠅⠀⠀⠀⠀⢸⠀⠀⠊⠀⠀⠈⢂⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⠀⠘⣄⣀⣀⣀⡎⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠘⣤⣤⣤⡌⠀⠀⠀⡟⠛⠛⢛⠀⠀⠀⢣⣤⣤⣤⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠶⠶⢶⠂⠀⠀⣭⣉⣩⡇⠀⠀⢀⠿⠿⠿⠿⡄⠀⠀⢸⣍⣩⣭⠀⠀⠐⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠿⠶⠾⠄⠀⠰⠉⠉⠉⠡⠀⢀⠎⠀⠀⠀⠀⠘⡄⠀⠌⠉⠉⠉⠆⠀⠠⠷⠶⠷
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⢡⡌⡰⠀⠀⠀⠀⠀⠰⡌⠀⠀⠀⠀⠈⢤⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⢸⡠⠇⠒⡒⠒⠒⠠⢄⣇⠀⠀⠀⠀⠐⠸⠆⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠰⠀⠀⠀⢀⠔⣡⠖⠉⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠆⡄⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⢃⠞⠁⠀⠀⠀⢠⣦⡄⠸⠶⠀⠀⠘⡄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⡞⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠳⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠂⠄⠀⠂⠀⢣⠀⠄⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠠⠀⡜⠀⠐⠀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⢄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡠⠃⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢌⠂⠀⠀⠀⠀⠀⠠⠔⡫⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠙⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⢀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣟⣒⡒⡞
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡥⢒⣳⡀⠀⠀⠀⠀⠀⠀⣰⣓⣒⢬⠇
⠀⠀⠀⠀⢀⡶⠋⠉⠉⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣮⠵⠒⠳⡄⠀⠀⠀⠀⡴⠁⠀⠀⠙⡄
⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⢦⢀⣀⣼⣥⣤⣤⣀⣀⡇
⠀⠀⠀⠀⠀⠳⢤⡀⠀⠀⠀⣱⢄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡴⠞⠉⠁⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⢞⣡⠎⡷⣤⡀⠀⠀⠀⠀⡇⠀⣰⠋⠀⢊⣿⠆⠀⢀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡖⣊⡴⠃⠈⠓⢦⣀⠀⢿⣼⠁⠀⠀⠈⠁⠀⢈⣾⠇⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡅⠀⠀⠀⠀⠀⠈⠙⠾⣇⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⢁⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠾⠛⠉⠙⠲⠤⣤⣄⣀⣤⡤⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣧⣤⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢀⣿⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⢠⠎⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇⡰⠁⠀⠀⠀⢱
⠀⢠⠔⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⠁⠀⠀⠀⠀⢸
⠀⣇⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⠀⠀⠀⠀⢠⠇⠀⢀⣠⠄
⠀⠘⣄⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡠⢃⡤⠒⠉
⠀⠀⠈⢶⡾⠟⢧⠀⠀⠀⠀⠀⠀⠀⣠⢔⣒⣈⣉⡉⠐⠒⠤⡑⠂⠸⣏⠥
⠀⠀⠀⠈⢷⡾⠟⠳⣄⠀⠀⠀⣠⣾⣾⣿⣿⣿⣿⣿⣿⣷⣦⡈⠳⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠈⠳⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠹⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣧
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⡇⡟⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠑
⠀⠀⣀⠤⠤⢜⣆⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠗⠒⠤⠤⠤⠤⣾⡝⣿⡆⠀⢠⡶
⠀⡸⠁⠀⠀⠀⠙⡧⡤⠤⠒⠒⠦⢿⣿⣿⠘⣿⢣⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⠟⠛⠊⠉
⠀⠱⡀⠀⠀⠀⢠⣧⡇⠀⠀⠀⠀⠀⠻⢷⠀⠀⡼⠋⠁⠀⠀⠀⠀⠀⠀⢀⡠⠔⣠⠞⠁
⠀⠀⠉⠒⠒⠚⢻⠛⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢃⠀⠀⠀⠀⠀⠀⠲⠾⢓⡡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣓⢤⣀⣀⠀⠀⢀⡴⠃⠀⠈⠒⠒⠒⠒⠒⠒⠒⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣷⣶⣾⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⡀⠀⠀⠀⠀⢀⠔⢢⠀⠀⠀⠀⠀⠀⠀⢘⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢠⢄⠢⡀⢁⡀⠀⠀⠀⠈⡀⠀⠇⠀⠀⠀⠀⠀⠀⠺⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠘⡯⠟⠢⢀⠀⠀⠘⣤⣶⡄⠀⠀⠀⡔⠁⠀⠀⠀⢀⢤⠊⠱⠀⠀⠀⠀⡁⡱⣾⡤⠂
⠀⠀⠀⠀⠀⢀⣀⣼⣲⠀⠀⢁⠀⠀⠀⢣⠀⢸⡏⠀⠈⠢⡀⠘⠀⠀⠀⢠⠀⠃⠀⡇⠸⠀⠀⠀⡠⠋⠀⠀⠁
⠀⠀⠀⠀⠀⢇⠀⠑⢄⡃⠀⠈⢢⠀⠀⠀⢣⠁⠃⠀⠀⠀⠐⡀⠀⠀⠠⠃⣸⣤⡄⠘⠿⠇⠀⡐⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠑⠢⢞⡝⠀⠠⢀⠑⠄⠀⣈⠀⠐⡀⠀⠀⢀⣡⣤⣤⣤⣀⠈⠉⡇⠠⠀⠀⢀⠁⠀⡠⠊
⠀⠀⠀⠀⠀⠀⢁⠀⠈⠣⠀⠀⠀⠈⠢⠈⠲⠀⠀⠀⡄⣰⣟⣱⣿⣿⣿⣿⣷⡄⣰⣶⠀⠀⠈⠖⡎⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠁⢄⠀⠀⠀⠑⠌⣆⠀⣸⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣡⣾⡆⠀⠀⢸⢇⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠑⠀⡺⣻⡿⡻⣷⣴⣖⢧⢮⡩⡉⠀⢿⣿⣿⣿⣿⣿⣿⣿⢏⠛⠉⠈⢀⠀⣸⢿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢈⠈⠈⠊⠃⡑⠁⠈⠈⠲⣝⢿⣿⣿⡿⢿⣿⣿⣿⡄⠀⠀⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡀⠀⢀⠈⠂⠀⠀⠀⠈⠻⢷⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢌⠀⠀⠀⠈⠢⡀⢀⠀⠀⢁⠉⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠢⡐⢌⠂⠐⢄⠢⡀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠑⠀⠢⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⠱⡔⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⢘⣇⠀⢁⠀⠀⠀⠀⠀⠀⠀⣾⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠹⣄⠀⠻⣆⣸⡄⠀⠀⠀⠀⠀⠀⠙⠄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⡑⢤⣷⣅⡀⢸⠟⠛⠢⣄⠀⠀⠚⡏⢨⣾⠿⠷⣄⠀⠀⠀⢀⡀⢠⠚⢳⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⠈⡏⠁⠉⠻⠀⠀⠀⠈⢳⡀⢀⣷⣿⠃⠀⠀⠈⣷⡀⠞⢉⣠⣧⡴⠁⢀⣀⣀⣀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠈⢦⠈⠢⡀⢇⢄⠈⢠⢠⠀⠀⠀⠀⣷⡎⠀⢹⠀⠀⠀⠀⣿⡷⠛⠉⠀⢻⠀⣮⠀⠀⠀⣠⠟⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠈⣧⡠⠀⠈⠪⡂⠈⢆⢣⡀⠀⠀⢸⠀⠀⠈⢧⡀⢀⣠⣿⠇⠀⢠⠂⡞⠀⠝⠢⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠛⠻⣿⠿⠶⢤⣀⠈⠲⡤⠷⣕⣤⡶⣞⠀⠀⢠⠁⠙⢿⣿⠿⣄⣰⠯⠊⠀⠤⠔⣠⣤⡶⠋⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢠⣽⠀⠀⠈⠉⠙⢦⠀⠀⢀⢠⣶⣿⡿⠉⠉⠒⢦⣄⠀⢀⡨⢔⣂⠚⠒⠒⠚⣿⠧⠶⠒⠊⠀⣠⠤⠲⢤
⠀⠀⠀⠀⢀⣀⣙⠢⢄⡀⠈⠁⠀⠹⣦⠀⠀⠀⠀⠀⢣⡠⣵⣿⣿⠼⢃⣴⡆⠀⠑⠝⣤⠋⠀⠉⠀⣀⡤⠶⠒⠃⠀⠀⢤⣶⣎⣠⠤⠶⠟
⠀⠀⠀⡼⣅⣀⠈⠙⠲⢬⣷⡤⠒⠺⠿⡳⠀⠀⠀⠀⠈⢿⣿⣿⡏⣷⠄⠈⠀⠀⠀⠀⠸⡄⠀⣴⡾⠁⠀⠀⠀⠀⠀⢀⣴⡯⢄⣀
⠀⠀⠀⠀⠈⠛⣛⡛⠓⠛⠳⣇⠀⠀⠀⠈⠓⢤⣀⠀⢀⣸⣿⣿⣿⣧⡀⠀⠀⠀⠀⡀⢀⠗⠒⣻⡀⠀⠄⣐⣒⣠⠔⠋⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠲⠾⠷⢔⣂⣀⣀⣀⣈⣗⣛⣹⡽⣯⡻⣿⣿⣦⣤⣠⣬⣶⡟⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠠⠤⠤⠭⣍⣥⣄⣀⠀⢈⣉⣪⣝⣲⠭⣽⣿⣿⡿⡛⠒⠒⠒⠒⠶⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠴⠤⡍⠛⠛⠛⠒⠒⠲⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢛⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠲⣾⣿⡶⢂⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠀⠟⠛⠃⠨⠿⠏
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠰⣷⣄⣸⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠻⢿⢿⡿⣿⢿⣿⠟⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣧⣤⠀⠀⠀⠀⠀⠀⣰⣾⣿⣷⠶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡾⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠠⣴⣶⣿⣿⣶⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡁⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⣴⣯⣿⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣽⡿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣿⣧⣤⣤⣾⠇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠀⠈



⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⣿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⣿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⣿⢿⣿⣆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⢿⡏⠏⡿⠉⣿⠹⢻⡟⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣟⡻⡝⢎⣣⣶⡵⠷⠶⠿⠶⠷⣾⣵⡏⣴⡿⣻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⣿⣮⣣⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣾⣿⣿⡿⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⠀⠀⣰⣾⠿⣷⣄⠀⣰⣾⢿⣷⡄⠀⣴⡿⢿⣆⠀⢠⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⠀⠀⣿⡏⠀⢻⣿⠀⣿⡏⠀⣿⣿⠀⣿⡅⢸⣿⢀⣾⠃⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⣸⣿⠀⣿⡇⠀⣿⣿⠀⠙⠿⠿⠃⣾⠇⣼⡟⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣤⡄⢿⣷⣤⣿⡟⠀⢿⣷⣤⣿⡟⠀⠀⠀⠀⣼⠏⠀⢻⣧⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡃⢀⢷⡈⣻⠃⡖⢾⣓⢰⣟⡃⣸⣆⢰⠛⠒⠘⡟⢰⣦⡞⢳⡔⣧⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢮⠟⠼⠛⠧⠸⠀⠇⠢⡽⠳⠏⠠⠟⠻⠜⠦⠖⠀⠇⠸⠟⠧⠼⠹⠏⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣤⣿⣿⣿⢳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡟⣭⢻⣧⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣥⣿⢃⣿⣻⡟⡷⠶⡶⠶⡿⣛⠛⣥⢷⣹⣷⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣿⣴⣼⣷⣰⣯⣤⣽⣷⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣰⢣⠀⢀⢶⠀⠀⣴⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⡀⢰⠈⢣⣠⠃⠬⢧⠎⠌⢧⡞⠸⡆⢔⢽⠀⢀⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢄⡀⢸⠀⠉⠾⠀⠀⠉⠀⣈⣀⣀⣈⣀⣀⠀⠋⢐⠝⡶⣫⡪⡇⣠⠔⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢂⠀⠉⠛⠀⣀⠤⣔⣦⠽⠼⠧⠗⠓⠚⠓⠣⠿⠿⣶⣢⢮⣊⠂⢙⢕⠡⣇⡠⢖⡛
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠉⠑⠚⠀⢀⣔⣻⠔⠋⠁⠀⢀⣠⠄⠀⡄⠀⡠⢤⠀⠤⣀⡈⠙⠲⢯⣖⡭⣈⠫⢞⢕⣧⠔⣺⠃
⠀⠀⠀⠀⠀⠰⡒⠢⠼⠇⠀⢀⢔⡳⠊⠀⠀⢴⣂⠀⠈⣤⠄⢰⠾⢆⢇⠀⠀⠀⠇⠀⠀⡄⠀⠈⠙⠷⣯⠲⣥⠪⢪⣌⡤⢴⠆
⠀⠀⠀⣀⣀⣀⣑⡄⠀⠀⣴⡻⠊⢀⡀⠈⢦⠀⠀⣙⠄⠐⡀⠈⠀⠈⣈⣋⠀⠘⠀⠀⠘⢠⠊⢹⠆⢀⠀⠙⢮⡵⣴⠑⣰⣎⣠⠤⡤
⠀⠀⠀⠘⢮⠢⠀⠀⢀⣞⡞⠁⠀⠋⠢⡀⠀⠑⠀⢀⣠⡤⣖⣟⣯⣿⠭⠽⠯⢿⣶⣤⣀⠈⠓⢁⡔⢻⠀⣀⠀⠙⢿⣻⠐⡕⣠⡞⠁
⠀⢤⠒⡒⠒⠓⠀⢠⣳⠎⠀⠘⢗⡢⠤⡈⠁⡐⡼⣎⠷⠋⠩⠄⠄⡔⡔⠰⠐⠰⢠⡉⠻⢗⣦⡀⠀⠖⠊⠀⠀⠀⠒⢳⡻⡞⢌⡉⣉⡽⠃
⠀⠀⠙⠦⡠⠀⢀⣷⠏⠠⡃⡀⠈⣱⠄⠀⢠⣾⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠆⠑⣽⣆⠀⠀⠀⠀⡀⠀⢂⢻⡵⡪⢺⠯⠤⠤⣤
⠠⣖⢊⠉⠁⠀⠸⡍⠀⠀⠀⠀⠀⠜⠀⠐⣼⡿⠀⣀⠤⢄⠀⢠⢖⠒⢄⠀⢀⣎⢏⢶⡀⠀⠀⠈⢿⣇⢀⣀⡰⠱⣀⣀⡺⣎⣧⠹⣢⡾⠋
⠀⠈⠑⣮⡀⠀⣗⡇⠀⠀⠀⠀⠀⠀⠀⣀⣷⠇⡔⢁⢖⣿⠀⡏⣽⢷⡽⣇⠸⢠⡟⢇⣣⠀⣠⡀⠘⣿⡈⢣⣀⣴⣷⠋⢀⣷⣺⠘⢟⣭⢏⣲⠄
⠐⠯⣩⢁⠀⠀⣻⠃⠀⠀⠀⣤⠀⠀⠈⢸⣿⠀⠙⠙⡯⣺⠀⡇⢽⠈⡗⣿⠀⡖⡇⠨⣾⡄⠓⡏⠀⣟⡇⢸⡧⠛⠾⠀⠠⢸⢸⠐⡱⢿⣉⡁
⠀⢀⣈⡲⠄⠀⣿⡄⢀⡀⡴⠀⠦⢤⡄⠺⣻⡀⠀⠀⣯⢜⡄⢷⢝⣆⣕⢾⠀⢥⢝⣾⢜⡇⢰⠻⠕⣿⡇⠀⠁⠀⠀⠀⠀⢸⢽⠞⣴⢟⣴⡭⠗
⠛⠫⢔⣒⠠⠀⡿⡇⠀⠱⠄⣰⣻⡋⠀⢁⢣⣇⠀⠀⠗⣧⣇⠘⢯⣚⣫⡾⠀⠈⠢⠭⠾⠁⠀⠀⢰⡾⠀⠀⣠⢄⡀⠀⠃⣎⣿⢞⢎⠟⢅
⠀⠀⣠⠜⠃⠀⢹⢥⠀⠀⠔⠉⠑⠇⠀⠀⢌⣯⢦⠀⠐⠚⠋⠀⠀⠉⠁⠀⠀⠄⢄⠀⢀⢀⠐⣠⣿⠃⠀⠘⠲⢄⡹⢀⣰⣶⡡⢣⣎⡮⠮⠷⠂
⠀⠚⠓⠒⢢⡍⠀⢟⣧⠀⠀⠀⠀⠀⠀⠀⠀⢊⠫⣷⣄⡀⠀⠀⠀⢀⡀⡈⠎⡈⡌⠐⠈⣠⢾⡷⠁⠀⠾⣠⠂⣀⠀⢎⣏⡯⣰⡫⡙⡦⡀
⠀⠀⠀⡰⠋⠀⡄⠈⢶⢣⡀⠀⠀⠀⠀⡠⠂⠂⠀⠈⠛⠻⢯⣖⣶⣢⠤⡤⡤⣤⣶⣶⡻⠚⠉⠀⢠⠂⡀⠈⠛⠁⢈⡞⡞⡐⡑⢟⠋⠑⠉
⠀⠀⠘⠁⠉⢩⠏⠀⠀⠻⡕⣄⠀⠀⢰⣁⠆⢠⠃⡰⠀⢀⠀⠀⡈⠉⠉⠉⠉⠉⠉⠁⠀⢀⢀⠂⠀⠙⢥⠚⣠⣾⣟⡿⣼⣯⣾⣿⣷⡄
⠀⠀⠀⠀⢰⡧⠼⠤⣷⠀⠈⠣⣗⡤⡀⠀⠀⢇⠴⠁⡴⣯⠀⢨⢭⠇⠀⠸⡆⠀⡶⢄⡧⠀⠈⢦⠀⠀⣠⢖⡿⣧⠏⣾⢿⡾⡆
⠀⠀⠀⠀⠀⠀⠀⣰⠣⣀⢦⡄⠈⠑⠯⣷⢦⣄⠀⠈⠀⠀⠀⠂⠘⡄⢀⠏⠹⡀⢠⠀⠙⠀⠀⡀⠀⣢⣔⣻⢋⢼⣎⢻⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢸⠁⢀⣤⠀⠀⠉⠚⠽⣓⡶⡤⢄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣂⣤⢤⣖⡯⢗⠩⢢⢪⠶⠧⣅⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠚⠁⣼⢀⡮⢶⠀⡪⢒⢎⢍⡝⡚⡳⠿⠿⠽⢽⢯⠿⠞⡗⡋⡩⡡⣣⢊⢺⠓⢤⣚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠊⠀⢸⢈⡵⠛⡤⢪⡮⢾⠐⡕⣱⡣⢪⢪⡎⢜⠔⠿⣔⢵⠋⠳⣍⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⡧⠊⠀⠈⣏⡴⠁⢳⣁⠎⠈⢮⡸⠀⠘⠶⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠛⠀⠀⠈⠁
⠀⠀⠀⠀⠔⢔⠕⡢⢔⠤⢂⠤⢂⠔⡉⠔⣁⠔⡠⠂⡀⠄⡠⢀⡠⢀⡄⢀⠄⢀⠄⡠⠀⡠⠂⡠⢀⠔⡠⢀⠔⡠⢀⠄⣠⢀⡄⡠⢀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⠊⠀⠊⠀⠊⠰⠊⡠⢊⠴⢋⠔⣁⠔⣁⠔⡡⢊⠵⢊⡤⢊⡴⢊⠔⡡⢊⠔⡡⢊⠔⠡⠊⡴⢪⠊⠔⠱⠋⠊⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣖⠛⠭⠉⠒⠀⠀⠀⠀⠚⠉⠍⠛⣲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠯⠂⠉⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠈⠐⠙⡳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠊⠁⣀⣀⣾⣦⣀⡀⠀⠈⢿⣿⣿⡿⠃⠀⢀⣀⣼⣧⣀⡀⠀⠐⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠫⠊⠀⠀⠀⠈⢻⣿⣿⡏⠀⠀⠀⠼⠛⠛⠇⠀⠀⠀⢹⣿⣿⡟⠁⠀⠀⠀⠀⠝⣆
⠀⠀⠀⠀⠀⢀⡜⡑⠀⠀⣀⣶⡀⠀⠋⠁⠉⠃⢀⡀⠄⠀⠀⠀⠀⠠⠀⠀⠘⠉⠈⠙⠀⣀⣶⡀⠀⠈⢈⢧⡀
⠀⠀⠀⠀⠀⡞⡌⠀⠀⠙⣿⣿⣿⠋⠀⠀⡠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠙⣿⣿⣿⠋⠀⠀⠡⢳
⠀⠀⠀⠀⣼⠑⠀⠀⠀⠐⠛⠉⠛⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠋⠉⠛⠀⠀⠀⠀⠋⣧
⠀⠀⠀⢀⡇⠂⠀⠀⣠⠤⣄⢠⡄⠀⣤⢀⡤⢤⡀⠤⣤⠤⠀⣠⢤⡄⢠⣄⠀⣠⡄⣤⠤⠤⠀⣤⠤⣄⠀⠀⠐⢸⡄
⠀⠀⠀⢸⠹⠀⠀⠘⡇⠀⠀⢸⡇⠀⣿⠈⠓⠲⡄⠀⣿⠀⢸⡇⠀⣿⢸⡟⣦⢿⡇⢿⠒⠀⠀⣿⠶⣞⠀⠀⠀⠆⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠙⠛⠉⠀⠙⠛⠁⠈⠛⠛⠁⠀⠙⠀⠀⠙⠛⠁⠈⠁⠉⠈⠃⠙⠛⠛⠀⠋⠀⠙⠀⠀⠀⠀⡇
⠀⠀⠀⢸⣰⠀⣞⡓⠀⢠⢶⡀⠐⢻⠃⢰⠀⣞⡒⠀⣞⠛⠂⢀⣶⡀⢠⡞⠓⠀⢻⡟⠀⡆⣴⠛⣦⢰⣆⢰⠀⣄⡇
⠀⠀⠀⠈⡇⠀⠤⠼⠂⠟⠛⠷⠀⠸⠀⠸⠀⠤⠽⠂⠏⠀⠀⠞⠛⠳⠘⠦⠤⠀⠸⠇⠀⠇⠻⠤⠟⠸⠈⠻⠠⢸⠃
⠀⠀⠀⠀⢹⡠⠀⠀⠀⠀⣤⣀⣤⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣤⣀⣤⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⢧⢂⠀⠀⣠⣿⣿⣿⣄⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⣠⣿⣿⣿⣄⠀⠀⡐⡼
⠀⠀⠀⠀⠀⠈⢳⡡⡀⠀⠉⠿⠉⠀⣀⡀⣀⡄⠈⠁⠀⠀⠀⠀⠀⠒⠈⠀⢀⣀⢀⣠⠀⠉⠿⠋⠀⢀⢜⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣔⢄⠀⠀⠀⢀⣼⣿⣿⣇⠀⠀⠀⢰⣤⣤⡆⠀⠀⠀⣸⣿⣿⣧⡀⠀⠀⠀⡠⣢⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⠄⠀⠉⠉⢿⠟⠉⠁⠀⢠⣾⣿⣿⣷⡄⠀⠈⠉⢻⡿⠉⠉⢀⠠⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠄⡀⠀⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⣀⠠⣢⠵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠮⢤⣐⣀⣠⠤⠀⠀⠠⢄⣀⣀⡠⠵⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⢻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣶⣿⣿⣿⣶⡄⠀⣀⣉⠛⠛⣉⣀⠀⢠⣴⣿⣿⣿⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣶⣿⣷⣦⡶⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢶⣶⣾⣿⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣇⠰⣶⣶⣿⣿⣶⡶⠆⠀⢰⡆⠀⠰⢶⣶⣿⣿⣶⣶⠆⣹⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠙⠀⠀⣽⣿⣿⣍⠰⣶⣶⣿⣿⣶⣶⠆⣩⣿⣿⣯⠀⠀⠋⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠈⠙⠂⠈⣽⣿⣿⣿⠁⠐⠋⠁⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠈⠛⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠉⠙⠲⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢤⡎⠂⣾⡁⢸⠤⣤⠖⢂
⠀⠀⢀⠴⠒⠂⠠⢀⡀⢀⠔⠁⢀⡤⠖⠒⢤⠀⢹⠀⠀⠀⠀⣀⠉⢳⡼⠀⢸⣷⣀⠓⣇⠀⠀⠊⠠⠁
⠀⢠⡏⠀⣀⣀⣀⠀⠈⠃⠀⡰⠋⠀⠀⠀⠈⡇⢸⡇⠀⡠⢼⣄⠑⠌⠑⠂⠀⡿⣷⠀⠈⡄⠀⠀⢦
⠀⢸⡀⠐⡅⠀⠈⠉⢻⠀⠰⠁⡄⠀⠀⠀⢰⠁⢸⠇⠀⠑⢦⡙⠇⠀⠀⠀⠀⠐⡽⠄⠀⠀⠀⠀⠈⡄
⠀⠈⣇⠀⠰⡀⠀⠀⠘⡄⠀⢠⡀⠀⠀⠀⡌⠀⣸⠀⠀⠀⠀⠉⢢⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠘⢆⠀⠹⣄⢠⠖⠀⣠⡂⠉⠒⢄⡰⠁⢠⠇⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⡆
⠀⠀⠀⠈⠳⡀⠈⠁⠀⡴⠁⠈⠢⡀⠀⠁⢠⠎⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠘⢆⣠⠊⠑⠢⠤⠖⠉⢂⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⠤⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣀⡠⠤⠴⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⢀⡠⠄⠲⠊⠉⠉⠉⠩⠉⠉⠛⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢀⡴⠊⠉⠉⠉⠉⠉⠉⠉⠑⠒⠤⣀⡀
⠀⠀⢠⠏⠉⠈⠉⠈⠉⠁⠁⠈⠉⠉⠉⠈⠉⠁⠀⠀⢳⠀⠀⢠⠏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄
⠀⠀⢨⠧⢀⠄⠀⠀⠀⢀⠀⠀⣀⠀⢀⡀⣀⣀⠀⠀⢺⡀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠠⠤⠀⠀⠀⠈⢧
⠀⠠⡚⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⢀⠀⠀⠀⢹⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠧⡄⢀⡠⠬⡈⢀⡀⠀⠀⠁⣀⣁⢈⣐⠀⣠⠌⣹⠋⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠑⠤⣀⣀⣀⠀⠀⡀⠀⠊⠀⠀⢀⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⠤⠤⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⢀⣤⣶⣶⣤⡀⠀⠀⢀⣤⣶⣦⣤⡀⠀⠀⣀⣤⣶⣦⣄⡀⠀⠀⣠⣤⣶⣦⣄⠀⠀⠀⣠⣴⣶⣦⣄
⠀⢰⣿⣉⣿⣿⣉⣿⡆⢰⣿⣙⣿⣿⣉⣿⡄⣼⣏⣹⣿⣿⣉⣿⡄⣼⣏⣹⣿⣟⣹⣷⠀⣾⣏⣹⣿⣏⣹⣷
⠀⢸⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⠇⢿⣿⣿⣿⣿⣿⣿⠇⢿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣯⣿
⠀⠀⠙⢿⣿⣿⡿⠋⠀⠀⠻⢿⣿⣿⠿⠋⠀⠈⠻⢿⣿⣿⠿⠋⠀⠈⠻⢿⣿⣿⠿⠃⠀⠘⠻⣿⣿⣿⠟⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠁⠀⠀⠐⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠰⠦⠶⠴⠆⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠆⠀⠀⣀⣀⡀⠀⠠
⠀⣠⣴⣾⠟⣿⡇⠀⠀⠀⠈⠀⠀⠆⠀⠑⠒⠚⠂⠚⠀⠀⠀⠔⠀⠀⠀⠈⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐
⠀⣿⣬⡼⠶⠛⠃⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢀⣠⣞⡇⠈⠀⠫⠩⠥⠼⠃⡀⠀⣶⣳⠐⠂
⠀⠉⠁⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠔⠀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣣⠟⠁⠀⠀⠀⠰⠇⠀⠀⠀⠀⠘⠋
⠀⢀⠀⠀⠀⢀⡀⠀⠀⣰⣂⣂⣲⣀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠀⠀⢸⠟⢁⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠻⠭⠿⠭⠝⠇⠀⠰⡏⣋⣹⣷⣽⠇⠀⠀⠀⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠉⠉⠁⠐⠀⢀⠀⠀⠀⢀⠀⠀⠦⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠩⢐⠀⠀⢨⠀⠀⠀⠀⠸
⠀⠀⠴⠀⠀⠀⣈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠀⠀⢨⣲⠀⢨⠀⠀⢄⣀⠈
⠀⠠⠀⠀⠀⠀⢀⣈⠔⠀⠀⠀⢰⡶⠶⠄⠀⠀⠀⠀⠀⢠⡤⣤⣤⠀⠀⠀⠁⠀⠈⣤⡤⠆⠀⠀⠀⠀⠀⠈⠐⠀⠐⠌
⠀⠀⠀⠀⠀⠁⢉⠄⠐⠀⠀⠀⣼⣇⠀⠲⠤⠤⠄⠀⠤⠘⠀⠈⣻⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣽⣉⣁⣄⣀⣀⣀⣀⣀⣁⣀⣀⣙⠀⠀⠀⠀⠀⠀⡈⢲⣠⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⡏⣠⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠂⠀⠤⠀⠀⠁⠀⠀⠈⠂⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⢿⡟⠃⡀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠶⡶⡢⡀⠀⠁⠂⠐⠒⠀⠀⢀⢀⠀⠀⠀⡀⡀⡀⠀⣰⠖⠀⠒⠀⠐⠐
⠀⠈⠀⠀⠀⠝⠂⠀⠀⠀⠀⠀⠀⡎⣭⢉⣑⢂⣜⣣⡥⠏⠀⡄⡀⠀⠀⠘⠀⠛⠘⠀⠂⠀⠃⠓⠃⣼⡏
⠀⡀⠀⠀⠀⠀⠀⠁⠀⢀⡆⠀⠀⠛⠾⠿⠿⠛⠋⣽⣀⣠⠀⡁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀
⠀⣀⠀⣀⣀⣀⣐⣒⣀⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢡⠐⡴⣆⠀⠍⠠⢂⢠⠀⠀⠀⠀⠀⠀⠀⠇⠇⠀⠄⠘
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⡂⡤⡒⠂⠀⠂⠄⠀⢂⢢⠀⠂⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠈⠐⠀⠠⠀⠀⠀⠤
⠀⠀⠀⠀⢠⡀⠤⠀⠀⠀⠀⠀⣀⠁⠀⠁⠁⠀⠁⠀⠀⠁⢀⡀⠀⠈⠂⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠓⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡴⠶⢶⡄⠆
⠀⠀⠀⠠⢠⠀⠀⡠⠀⠀⠀⠀⠀⠀⣾⣿⣭⣽⣽⣽⣿⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠃⠂
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⡀⢀⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠁⠐⠀⠀⠁⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⡀⢠⠀⠀⢄⠀⠀⠀⠀⠀⠀⡄⠀⡠⠀⠠⠐⠀⣀⢡⡀⠆⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⡄⠀⠸⠀⠔⠘⠘⠊⠄⠀⠉⠀⠀⠡⠀⠀⠀⠀⠀⡀⡀⣧⡄⣧⡗⡠⡐⠳⡘⠆⡅⠄⣦⢄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠘⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡈⠁⠀⠀⠀⠀⠀⠁⠆⠋⣁⠘⡰⢇⠉⠆⠙⢸⠀⠄⡟⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠳⠾⣄⣅⠣⠈⣤⠆⣻⡼⢢⡄⢠⠀⠀⠀⠀⠀⠤⠀⠠⠀⠀⡀⢠⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠴⠥⡌⠀⠄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⣷⠀⠀⠀⠀⢸⠀⠐⠠⠄⠀⠔⣠⠀⡄⣶⠀⠘⠁⠀⠀⠀⠀⠀⠀⠉⠀⠐⠲⠶⠶⠞⠴⠟
⠀⠀⠀⡈⠓⠠⠤⢄⠀⣀⣀⣀⠀⠀⠰⠞⠃⣈⣾⠀⠀⠀⠀⠀⢇⠀⠁⠀⠁⠈⠉⠉⠁⠒⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⡖⠒⢤⣀⡀⢀⡀⠀⠀⠀⠀⢀⡄⠉⠭⣲⠆⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⢠⠉⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣰
⠀⠀⠀⡗⢦⡈⠉⡀⠀⠀⠁⠈⠉⠊⠉⠀⠀⠀⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⢀⠘⠷⠤⠀⠀⠀⠠⠴⠔⠒⠀⡠⠖⠁⠀⠀⠀⠀⠀⡄⠈⡀⠀⠀⠄⣄⠀⠠⠃⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠄⠠⠴⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠃⠂⠘⠂⠒⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠁⠄⣠⣄⠐⠒⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⢀⠙⠲⠄⠀⠢⠴⠀⠀⢫⣙⢃⠀⠑⠒⠶⠆⠀⠀⠢⠐⣦⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠓⠋
⠀⠀⠀⠀⠀⠈⡈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠐⠂⠈⠠⠔⠠⠄⠩⠗⠠⠔⠰⠒⢠⡀
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠊⠀⠒
⠀⠀⠀⠀⠀⠀⠉⠾⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠋⠀⠐⠁⠀⠀⠀⠠⠘⠁⠀⠤⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣍⠀⠈⣩⣿⣿⣿⣯⡉⠀⢉⣽⣿⣿⣿⣍⠀⠀⣩⣿⣿⣿⣯⡉⠀⢈⣽⣿⣿⣿⣍⠁⠀⣩⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⣴⣦⣹⣿⣿⣿⣿⣤⣶⣌⣿⣿⣿⣿⣏⣴⣦⣹⣿⣿⣿⣿⣡⣶⣤⣿⣿⣿⣿⣏⣴⣦⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠛⠉⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠉⠛⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄⠀⠙⠿⣿⣿⣿⣿⣿⣿⠿⠋⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣓⣀⡀⢀⡉⣉⢉⡉⣀⣀⣀⣚⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⣷⠟⠉⠳⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⣰⡟⠁⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢿⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣺⣿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⣘⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡾⠋⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠿⠻⡷⣄⣀⣤⣾⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣾⡀⠈⠻⡉⣁⣨⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢣⠀⠀⢹⠛⠋⠁⠑⢦⣀⠀⠀⠀⠀⠀⠀⢸⠛⠿⣷⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⠀⠀⠀⠀⠀⠳⡀⢸⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⢀⣿⣦⣤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⡟⠀⠀⠀⠀⠀⠀⠱⣼⢰⠀⠀⠀⠀⠀⠀⠀⠈⠢⠔⠃⠀⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⣿⠃⠀⠀⠀⠀⠀⠀⠀⠉⢸⡄⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣾⣿⣶⣶⣶⣶⣶⣤⣄⣀⠀⢸⣧⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⡆⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⡀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⣿⢱⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢸⡁⢹⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠦⣀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠈⠉⠓⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⡶⠖⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠖⠛⣏⠉⠙⣇⣀⣀⣷⣤⠴⠿⣦⡀
⠀⠀⢀⣀⣤⢶⡞⠛⢻⠁⠀⢹⣀⣤⠼⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠙⣦⡀
⢰⡟⠋⢯⠀⢀⣧⣤⠼⠗⠛⠋⠁⠀⠀⠀⠀⠀⡀⣄⠰⣄⠀⠀⠀⠀⠈⠻⣄
⠈⣷⡴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠉⠈⠂⢀⣀⡤⠴⠒⠛⣦
⢰⠿⡟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠉⠉⠀⠀⠀⠀⠀⠸⣆
⣿⠀⣿⠈⢳⡄⠀⠀⠀⠀⠀⣀⣠⠴⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⣿⠀⠸⡇⠀⢻⣄⡠⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⣿⠀⠀⢿⠀⠀⢷⠀⢠⠞⡴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⣧
⢸⡄⠀⠸⡇⠀⠘⣧⠀⠚⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⡀⢹⡄
⠈⣧⠀⠀⢿⡀⠀⢹⡆⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢷⣤⠀⣿⣁⠀⢷⡀
⠀⢹⡆⠀⠘⣇⠀⠀⢿⡀⠀⣟⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⠏⠁⠈⣿⣦⠀⢉⡇⠘⣧
⠀⠀⢷⠀⠀⢻⡄⠀⠈⣧⠀⠈⠓⢦⣄⠀⠀⢀⡀⠀⣼⠋⢹⣆⠘⠷⠤⠆⠈⠁⠀⠀⠀⠀⢹⡆
⠀⠀⠸⣇⠀⠈⣷⠀⠀⠸⡆⠀⠀⠀⠉⢷⢹⡏⠻⣆⠻⠦⠾⠙⠀⠀⠀⣀⣠⠴⠒⠀⠀⠀⠀⢿⡀
⠀⠀⠀⢻⡀⠀⢸⡆⠀⠀⢻⡀⠀⠀⠀⣸⠇⠓⠀⠙⠀⢀⣀⡤⠖⠚⠉⠁⠀⢀⡀⢠⠀⠀⠀⠈⣧
⠀⠀⠀⠈⣧⠀⠀⢻⡀⠀⠈⣧⠀⠈⠛⠉⠀⠠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⢸⠇⠀⠀⠸⡆
⠀⠀⠀⠀⢸⡆⠀⠈⣧⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢿⡀⠀⠘⣇⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠋⠉⠹⡇
⠀⠀⠀⠀⠀⠈⣧⠀⠀⢹⡆⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢻⡀⠀⢻⡀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⣷⠀⠘⣧⣠⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠸⣇⠀⢻⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢴⡖⠛⣏⢁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⢻⡄⢸⡇⠀⣓⢌⠑⠆⠀⠀⠀⣀⣀⣤⢶⡞⠛⠹⡅⢀⣠⣷⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⣷⢸⡇⠀⠈⢁⣠⣤⢶⠞⠛⠙⣇⠀⢀⣧⣤⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡤⢶⡟⠙⡇⠀⠀⣧⣠⡤⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⣠⣷⠴⠾⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⠀⣈⢳⠀⠀⠀⢀⡀⠀⠀⠀⢀⡴⡄⠀⠀⠀⢀⡀⠀⠀⠀⡰⠋⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠒⢸⡀⠀⠀⠎⠘⢆⠀⡰⠋⠀⠘⡄⢀⡴⠋⢱⠀⠀⢠⠇⠀⠀⠠⠀⢸
⠀⢀⠴⠚⡛⠲⣄⠀⠀⠀⠀⡘⡆⠀⠀⠈⢸⡍⠒⠎⣠⠖⠒⠛⢦⡀⠀⠀⠘⠋⠀⠀⠈⡄⢀⣼⠀⠀⠀⠂⣰⠃⠀⠀⠀⢀⣠⠤⣄
⠀⡎⠀⠀⠀⠦⠘⡆⠀⠀⠀⠀⠹⣤⣤⣤⣾⠁⠀⠀⡇⠀⠀⢀⡂⢣⠀⠀⠀⠀⠀⠀⠀⠓⢁⣿⣶⣤⣄⡴⠁⢀⠀⣠⠖⠉⠀⠀⠀⠹⡄
⠀⠸⣄⠀⠀⠀⠂⢹⡀⠀⢀⡇⠀⡿⠿⠟⠛⣇⠀⠀⡇⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⣮⣙⠛⠿⡿⠃⣉⣿⣞⠁⠀⠀⢀⢄⠄⢠⠇
⠀⠀⠈⢦⡀⢀⣠⣼⣇⠀⠉⠀⠀⣿⣾⣿⣿⣿⣆⠀⢷⠀⠀⠀⢠⠇⠀⠀⠀⠀⣀⣠⠴⠻⢿⣿⣿⣷⣿⣿⡈⢿⣿⣧⠀⢀⣡⠤⠒⠉
⠀⠀⠀⠀⣙⣿⣿⠿⢛⣧⡀⠀⢠⠟⠉⠉⠉⠀⠈⢣⢸⣿⣿⣿⣿⠀⢀⡠⠖⢋⡵⠃⠀⠀⠀⠀⠉⢻⠿⣿⣿⣌⠿⠛⢻⠃
⠀⠀⠀⠀⠐⡸⣤⣾⣿⣿⠿⠢⣜⠀⠀⠀⠀⠀⠠⠀⢻⣭⣭⣭⣽⡖⠉⠀⢠⠋⠀⠀⠀⠀⠀⠀⠉⠈⡇⢹⠟⠁⠀⡴⠃
⠀⠀⠀⠀⢀⣱⡿⠛⠉⠀⠀⠂⠈⠓⢄⠀⠀⠀⠈⣡⠿⠿⠿⠿⠿⢧⠀⢰⠃⠀⠀⠀⠀⠀⠀⠐⠂⢀⡧⠋⠀⠀⠘⠒⠠⠤⣀⡀
⠀⠀⢖⠉⠀⠀⣿⠀⠀⠀⠀⠀⠱⡂⠀⠳⡄⠀⢀⡏⠀⠀⠀⠀⠀⠈⢦⠏⠀⠀⠀⣀⣀⣀⣤⣍⣀⣸⣁⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠑⢄⠀⢹⠀⠀⠀⠀⠀⠀⠈⠔⡀⠘⣄⡜⠀⠀⠀⠀⠀⠀⡂⠘⣆⡴⠞⠋⠉⠀⠀⠀⠀⠀⠉⠉⠙⠳⢦⣀⠀⡔⠁
⠀⠀⢀⡤⠚⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⢐⠄⢸⠇⠀⠀⠀⠀⠀⠀⣤⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⠄⠉⠳⣬⠑⠢⣄
⠀⠮⢁⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢂⢸⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠈⢳⣴⠊
⠀⠀⠀⠈⠑⠆⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦⠀⢸⣿⣿⣯⠀⠀⢹⡄
⠀⠀⠀⡠⠊⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠀⠀⠙⠛⠃⢂⡀⠀⢿
⠀⠀⠚⠥⠤⠤⠄⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⢀⣇⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠉⠠⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠉⠒⠄⠚⠉⠈⣆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠄⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠉⠉⢩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣆⠠⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠒⠉⠑⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠔⠁⠘⡄⠀⢀⡔⠑⢄⠀⢀⠇⠀⠙⠻⣛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⡤⠋⠀⠀⠀⠑⠊⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡂⢄⠀⡀
⠀⠀⠀⠆⠀⠀⠀⠠⠆⢩⠁⠀⠀⠀⢠⣴⣶⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡤⢀⠈⡄⠀⠀⠀⠠⢠⣉⣩⠝⠀⣤⠤⠛⠂⡀
⠀⠀⠀⠈⠃⠁⠀⠀⠀⠗⠀⠁⠀⠀⠀⠙⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠻⣶⣦⣟⣀⣴⡾⠃⠀⠀⠤⠾⢖⡽⠇⠀⠌⠇⠘⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃

⠀⢀⣤⣤⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⣡
⠀⠈⢯⣽⣿⠏⣹⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⣤⡁⣤⠀⣴⡀⢠⡤⣄⢠⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⢫⠸⠀⣾⣿⡏
⠀⠀⠈⢻⠙⠛⠓⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣝⡷⣿⠻⣿⣰⠿⢷⠻⣤⡤⢸⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢹⣿⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢺⢿⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠀⡀
⠀⠀⠀⠀⠈⠁⣀⣀⢤⣶⣀⠀⠀⠂⢀⡤⠀⠐⢯⠿⠀⠶⡆⠀⠀⠀⠘⠛⠈⠉⠉⠃⡀⠈⠓⠢⠬⠅⠀⠀⠠⠄⠐⠀⡀
⠀⠀⠀⠀⢸⠯⠛⢟⠋⠁⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠂
⠀⠀⠀⠀⠈⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⠀⣠⣤
⠀⠀⠀⠀⠘⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠘⠖⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣀⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠠⠄⠠⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠤⠴⠷⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢘⡃⠃⠙⠈
⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡤⣠⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⣍⣀⣀⣩⡄
⠀⠀⠀⠂⠀⠓⠒⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠘⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠭⢀⡉⣤⣄⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡿⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣫⣭⣝⡻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢻⣽⣩⣴⡆⡇⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣾⣿⣿⣿⣷⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡹⠿⠸⣿⣧⡇⣱⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣙⢿⣿⡿⢏⣾⡏⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⢻⡜⣶⣭⣣⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⠾⡫⣶⣶⢰⣾⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⣥⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣞⡃⣿⣿⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣝⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠃⠐⣦⠀⠉⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡿⠿⢷⢍⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⠿⠥⠤⠄⠘⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡀⣰⣟⠍⣻⣟⠉⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⢀⣤⣤⢰⣿⣕⣸⣿⣅⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢣⣿⣿⣇⣿⡿⣛⣭⣭⡻⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣚⣇⢿⣞⣿⡸⣯⣰⣶⣶⣶⢆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠞⠃⠙⠲⠛⠘⢛⢻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠀⣩⠏⣿⣭⡷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⡴⠋⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣢⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣴⣿⣾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⣷⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁






⣿⣿⡿⠿⣿⠿⢿⠿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣶⣿⣶⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⡆⢀⡴⠖⠢
⠀⠀⠀⠀⠀⢀⡶⠋⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢸⠇⡟⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣼⠀⢻⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣷⣶⡟⠀⠸⡄⠀⠀⢸⠁⠀⠀⠀⠀⢀⡤⠤⠄
⢀⣀⢀⠀⠀⠀⠀⢳⣰⣶⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⣾⢠⣤⣤⣷⠀⠀⡿⠸⢿⣿⠀⠀⠀⠀⠀⢸⠀⠀⢸⡆
⣿⠀⠀⠙⣆⠀⠀⢸⡇⣶⢾⣦⣴⣿⣿⣿⠟⠛⢿⣿⣿⣿⠟⠘⠛⠛⠻⡀⢠⣷⢰⣶⣾⠀⠀⠀⠀⠀⢸⠀⠀⣸⠀⠀⢀⣀⣀
⠈⠦⡀⠀⢘⣦⠀⢸⠃⣠⣾⡿⠛⠛⢿⣿⣄⣀⣸⣿⣿⡏⠀⠀⠀⠀⠀⠱⡾⠁⠀⠀⠈⢣⠀⠀⠀⠀⢸⡧⣶⡟⢀⠔⠉⠀⠈
⠀⠀⠈⢲⣟⣩⣷⣾⣼⣿⣿⣷⣄⣀⣾⡿⠛⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢃⠀⠀⠀⣼⢗⣿⡿⠁⠀⠀⠀⣠
⠀⠀⠀⠀⢹⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠈⣦⠴⠾⠻⢻⣥⢻⣷⡴⠖⠋⠁
⠀⠀⠀⠀⠈⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢻⠟⠙⡄
⠀⠀⠀⠀⠀⠸⣆⢹⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⠀⠀⠀⡰⠁⠀⠀⡇
⠀⠀⣀⣀⣀⣀⣘⣾⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡿⠀⢀⠜⠀⠀⠀⢰⠇
⡠⠈⠀⠀⠀⠈⠉⢹⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢰⣧⠴⢏⠀⠀⠀⢀⡞
⢳⡀⠀⠀⠀⢀⣀⣨⡭⠥⢽⣷⣀⡀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⣰⣇⠀⠀⠀⠀⢀⡟⠁⠀⠈⢄⣀⣠⡞
⠀⠉⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⣤⣤⣤⡤⠤⠴⣀⠀⠀⢀⣰⠏⠉⠢⠤⣤⠤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢰⠉⠢⡀⡜⠀⢣⣀⡴⠉⠃⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠆⠀⠀⠈⢀⣀⣀⠉⠀⠀⠠⠔⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠐⠒⠀⢀⣠⣴⣶⡿⡿⡿⢿⢿⣿⡿⣶⣦⣄⡀⠀⠗⠂⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠄⠀⡄⠀⣠⣶⣿⣿⡿⣪⣧⣾⠷⠾⠶⢾⣽⣣⡿⣛⣿⣶⣄⠀⠰⠄⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢀⣼⣿⣿⣛⡽⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⢯⣞⣽⣿⣷⡀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠃⠀⠈⠉⣉⠉⠉⠀⣀⣀⡀⠀⢀⣀⣀⠀⠀⢀⣀⡀⠉⢉⡉⠉⠁⠀⠘⠂⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠰⢾⣿⠀⠀⣾⡟⠹⣿⠀⣾⡏⢻⣷⢠⣿⠉⣷⠀⣾⠃⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⣿⠃⣿⡁⢸⣿⠀⠿⠾⠏⣼⠃⣾⠛⣧⠀⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⢠⣴⣿⣦⡄⢻⣧⣼⡿⠀⢿⣧⣼⡟⠀⠀⠀⣸⠏⠀⣿⣤⡿⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⣀⡀⢀⠀⣀⣈⡁⣀⡀⣀⡈⢁⠀⣀⡀⣀⣀⡀⢀⣀⢉⠀⠀⠀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⡤⠀⠀⠀⣓⡆⡼⣧⢸⠀⡏⣓⡄⡗⢀⡿⡆⣇⣀⢸⡁⡷⣇⣸⢸⢷⡀⠀⢠⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢠⣬⣤⣤⣤⡀⠀⠀⠈⠀⠀⠈⠀⠀⠈⠁⠀⢀⣤⣬⣥⣤⡄⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⢐⠂⠀⠹⣿⣿⢷⣽⢶⣤⣀⡀⠀⠀⠀⣀⣀⣤⢖⣫⣲⣽⣿⠟⠀⠐⡂⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⠀⣀⠈⠛⢿⣧⣾⣡⡏⣿⡻⣟⢻⡟⢹⣅⣷⣿⡿⠛⠁⢀⠀⢀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⣀⠈⠙⠻⠿⠿⢿⣿⡿⠿⠿⠟⠋⠁⢀⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠊⠁⢸⠀⢀⠤⡄⠀⠀⣀⠀⠀⢀⠤⡀⠀⡀⠈⠑⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠰⣀⠞⠀⠱⣀⠌⠀⠈⠲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⣿⣿⠗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣿⠋⠀⢀⣠⡟⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠁⠀⠀⣼⡟⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣶⠶⠿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⣀⣈⡃⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⡾⠋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠿⠿⠿⡷⢄⣀⣀⣤⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⣴⡇⠀⠙⣟⠋⠉⣈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠉⠉⢧⠀⠀⠈⣷⡿⠿⠋⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣶⣦⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠈⢣⡀⠀⢸⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⣿⡀⠉⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡎⠀⠀⠀⠀⠀⠀⠳⡀⣾⢀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⢀⠞⠻⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⣿⠀⠀⠀⠀⠀⠀⠀⠀⠱⡏⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡁⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣾⣿⣷⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⣿⣷⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢹⣿⡆⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⡀⢀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⡇⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⣿⣄⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠉⠓⠦⣤⣄⣀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⡿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣀⠀⣷⡆⢢⡠⠀⠀⠀⠀⠀⠀⢀⡼⠁⠄⠀⣻⡆⠀⠀⠀⠀⠀⠀⣀⡀⠀⣸⡅⠀⣀⡀
⠀⠀⠀⠀⢰⠆⣸⠛⣰⠏⠁⡜⠛⢐⡷⠀⠀⠀⠛⠋⠠⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⠿⢳⡿⠋
⠀⠀⠀⠀⡇⠀⡏⠀⣿⠀⠀⠃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡷
⠀⠀⠀⠀⡇⠀⣇⣴⣿⣿⠉⢀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠁
⠀⠀⠀⠀⠁⠀⢿⣿⣿⣿⣿⣿⡧⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡿⠟⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢆⠀⠀⠀⠀⢀⡴⠀⠀⠀⠁⠉⡆⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡆⠀⠣⡀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠁⠀⢀⣩⡴⠗⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠫⢄⡒⠂⠰⠏⣷⠖⠋⢀⣤⣴⡒⠦⠀⠈⢳⣄⡀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣦⡀
⠀⠀⠀⠀⠠⠦⠄⠀⠒⢒⣀⠉⠀⠀⢤⣿⣄⠀⠸⣿⣿⣿⣾⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠳⢤⣀⣀⡈⠻⣦⡀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡄⣰⠇⠀⠀⣁⠀⠂⠄⠲⠠⡀
⠀⠀⠀⠀⠀⠀⢙⣦⣄⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⣤⣤⢀⡄⠁
⠀⠀⠀⠀⢰⣿⣿⣡⣿⣷⡄⣠⠶⠿⠁⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣾⣮⡻⣿⠿⣁
⠀⠀⠀⠀⠻⣿⣿⣿⡟⠛⠁⣽⡄⠀⠀⠀⠀⠒⡀⠀⠀⠀⣀⡀⢄⠀⢄⠀⠀⠀⠀⠀⠘⣻⣿⣿⣾⣾⠃
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣸⣷⣦⣵⣾⠇⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢧⡏⡾⠙⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣸⢸⣁⣀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡜⣤⢂⠀⠀⢀⣀⡀⠀⠀⠀⠀⢸⣿⣿⣦⣼⡀
⠀⠀⠀⠀⠀⣈⣣⡽⣦⣉⢠⣿⡀⣁⣼⢧⣊⣀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⠑⢠⠘⣿⡛⠙⢻⠟⢁⠄⠊⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣤⡀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⡿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣼⣿⡆⣴⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠟⠉⠀⠺⠂⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⠄⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠘⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠠⠄⠀⠀⠀⢀⠀⠁⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠒⠦⠀⠀⠀⠀⠀⠀⠀⡁⠐⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⡀⠀⠐⠂⠀⠀⠀⠈⠁⠀⠸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠄⠀⠀⠀⢈⠀⣀⠀⠁⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⡌⠀⠀⠀⠀⠀⢈⠉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠒⠀⠠⡄⠀⠀⠁⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⡁
⠀⡁⠀⠀⠀⠀⠠⠐⠀⠠⠀⠀⠀⠀⡰⠒⠲⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠐⠁⠀⢀⣴⠞⠉
⠀⢇⠀⠀⠀⠀⠈⠂⠒⢰⡆⠀⡠⠘⠁⠀⠀⠉⢁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢢⠄⠀⠀⠀⠀⠀⠀⣏⠉⠉
⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠠⠖⠁⠂⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⠀⠀⣀⠴⢀⠀⠈⣱
⠀⠀⠀⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⢀⠚⠛⠊⠉
⠀⠀⠀⠀⠀⠌⠀⠀⣀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠅⢀
⠀⠀⠀⠀⠌⠀⠀⢠⡞⠈⠑⢀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠈⠀⠀⠱⠏⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⡌⠀⢀⠜⠀⠁⢄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⡒⠂⠀⠶⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠈⠀⠠⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⡠⠂⠀⠀⠀⠀⠀⠀⡠⠊
⡰⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⣀⠀⠀⠀⠀⠀⡠⠊
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠚⠉⠉⢻⠂⢄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⣀⠀⣀⡀⠀⠀⠀⠀⠀⠉⠀⠐⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⢀⣼⣿⠇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢺⣿⣿⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠚⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣆⠀⠀⠀⣼⣿⣆⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⢠⣿⣧⡀⠀⠀⣰⣿⣿⣄
⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⣛⣛⣉⣭⣼⣿⣿⣿⣦⣤⣀⡀⠀⣼⣿⣿⣧⠀⢀⣀⣠⣤⣿⣿⣿⣧⣭⣉⣛⣛⢻⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⡙⢿⣿⣿⣿⣿⣿⢩⣭⣴⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⣭⠝⣿⣿⣿⣿⣿⡿⢋⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠟⠻⢶⠙⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣾⣿⣿⣿⣿⡋⡴⠟⠻⢿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⣿⣿⣿⠿⣿⣿⣷⠙⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿⡿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠉⠀⠀⠈⠙⠻⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆⠟⠋⠁⠀⠀⠉⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠟⠋⠀⠈⠙⠻⢿⡇


⣤⣤⣤⣤⣤
⣿⣽⠭⣭⣿
⣿⣿⠤⣿⣿
⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀⢀⠠⠄⢀
⠀⠀⠀⡀⠀⠀⠀⡀⠀⠠⠀⢉⠑⢆⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡁⠀⢁⠀⢀⠀⢠⡀⠠⠀⠈⠁⢆⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠢
⠀⠀⠀⠘⠀⠈⠀⠀⡀⠠⠀⠐⠄⠁⡀⢨⡀⠀⠀⠀⣄⠀⠈⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠸⢀⠈⠀⢀⣀⢀⣠⠊⠓⢀⠀⠜⠁⠀⠀⠀⠈⠂⢀⡀⠁⠀⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠈⠀⠉⠀⠉⢄⡀⠠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠈⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠠⢄⡂⠐⣈⣖⡓⢼⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠉
⠀⠀⠐⠀⠀⠒⠀⣱⠠⢆⡡⢉⣡⡀⠃⡴⢠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠁⠈⠀⠈⡉⠂⠘⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⢈⠂⠀⡘⠁⠀⠀⠹⠄⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡰⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠜⢀⠼⠄⣀⣀⡠⡀⠀⠢⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠒⠈⠉⢀⠣⡀⠀⠒⠂⠀⠀⠀⠃⠠⠆⠀⣠⣬⡄⠀⢥⣶⠆⡞
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣅⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣤⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒⠁⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠔⣀⡀⢈⣙⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠺⠶⠷⠶⠤⠿⠤⠀⠔⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠒⠚⠁⠀⠀⢸⠀⠀⠀⠀⠀⢈⠕⢢⡀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢡⠀⠀⠀⡀⠀⠐⢹⠄⠀⠀⠀⠀⠀⠀⡠⠐⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⠢⠀⠀⡄⠀⠂⠁⠀⠀⠀⠀⠀⠀⢰⠐⣐⠨⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠠⢈⣐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠖⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⡀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢠⣆⠀⠐⠒⠒⠒⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⣀⣠⣴⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠉⠋⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠀⠀⠀⢶⣿⣿⣿⣿⣿⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣹⣿⣿⣿⣆⣀⣀⡀⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣦⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣿⣿⣿⣿⣿⣿⠿⠿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⣠⠄⢠⢠⣠⡄⢠⡄⡤⢤⢠⣠⡄⣤⢠⣤⠄⢠⠄⣠⡄⣤⢠⢠⡄⢠⡄⣤⣀⡤⠀⣤⢠⣠⠤⣄⣄⡄
⠀⠀⠀⠀⠙⠒⠘⠚⠐⠛⠘⠃⠓⠊⠛⠛⠓⠛⠚⠛⠂⠐⠃⠋⠓⠛⠘⠒⠛⠛⠐⠋⠋⠓⠂⠛⠘⠘⠒⠉⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠘⡆⠀⠀⠀⠀⠀⠀⡸⠤⠼⡄⠀⠀⠀⠀⠀⠀⡼⠃⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣠⠾⡀⠀⠀⠀⠀⣰⠁⠀⠀⠹⡀⠀⠀⠀⠀⣰⢧⣀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠚⠉⠲⣄⠀⢀⣧⣠⣤⣤⣀⣣⠀⢀⡤⠊⠙⠲⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣱⣾⣿⣿⣿⣿⣿⣿⣶⡏⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣣⠀⢸⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅⡀⢳⡹⣿⣿⣿⣿⣿⣿⣿⠇⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠲⠭⠭⠯⠽⠛⠓⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⡄
⠀⠁⠀⠀⠀⢈⠀⠀⠀⠀⡄⠀⠠⠀⠰⠟⠛⠚⠆⠀⠀⠀⠀⠠⠅⠀⡤⠈⠀⠐⠤⠀⠀⠀⠀⠀⠠⠆⠁
⠀⣤⣶⢿⢻⣻⠀⠀⠀⠀⠉⠑⠚⠀⠘⠛⠛⠃⠛⠀⠀⠀⠎⠀⠀⡀⠈⠀⢀⡀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠠⡶
⠀⣿⡮⠿⠛⠛⠀⠀⡈⠁⠀⠀⠀⠀⠀⣀⢀⠀⡀⠈⠀⠀⠀⠀⠀⠀⣠⡶⣿⡇⠁⠉⠃⢻⠿⠋⠀⠀⣼⣽⡀⠂
⠀⠁⠄⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠰⠀⠇⠐⠀⠂⠀⠀⠀⠀⠀⠀⢸⣿⠾⠋⠀⠀⠀⠈⠛⠄⠀⠀⠀⠉⠉⠀⠐
⠀⣤⣀⣀⣀⣠⠀⠀⢠⣯⣿⣿⢧⡄⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠘⠁⠴⠄⠀⠀⠂⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⢰
⠀⠉⠚⠛⠒⠹⠀⠀⠻⣽⣷⣿⣿⠟⠀⠀⠀⠚⠉⠀⠁⠀⠀⠠⡀⠀⢀⠠⠀⠀⢀⠀⠀⢀⢤⡀⠀⠀⡄⠀⠀⠀⢈
⠀⠀⠀⠀⢀⠤⠰⠀⠂⠄⠀⠀⠠⠐⠀⠂⠄⠀⠀⠠⠂⠀⠉⠐⠀⠦⠀⠀⠀⠄⠂⠄⡀⠀⠣⠅⠀⠰⠀⠀⠀⠀⠘
⠀⠀⠉⢀⠀⠄⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠄⠀⢀⡀⠀⠈⠭⠂⠰⣲⠉⢑⢠
⠀⠘⠐⠀⠠⠀⠴⠞⡈⠀⠀⢨⡿⠉⢙⠀⠀⠀⠀⠀⢸⠛⢿⣿⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠁⠁⠉⡀
⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠸⡿⠤⠈⠉⠉⠉⠀⠉⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠜⡑⠐⠀⠖⠀⠤⠠⠤⠠⠄⠐⡇
⠀⠀⠀⠀⠀⢀⡾⢀⠀⠀⠀⠘⠓⠖⠚⠒⠒⠒⠒⠒⠶⠳⠒⠂⠀⠀⢀⡀⠀⠐⠀⠳⠋⠀⠀⢄⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⣾⣥⠋⢀⠀⠀⠆⠀⠀⠀⢀⣀⣠⣤⡀⠀⠀⡄⠀⡀⣀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠀⣀⠠
⠀⠠⠀⠀⢀⣿⠡⠐⠀⠀⠀⠃⢠⢳⠻⠛⡝⠫⢯⠖⡷⠀⡃⠀⠀⠀⢐⠰⡆⠄⢠⠀⢸⠰⡸⢀⣾⠋
⠀⠠⡄⢠⠀⠠⠀⠄⠀⢠⠀⡀⠹⣭⣶⣾⣶⠿⢟⣋⣁⠀⡇⠂⠒⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠒
⠀⠅⠦⠀⠀⠤⠴⠤⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⢃⠠⣭⡢⠖⡲⣠⡆⡆⠀⠀⠀⠀⠀⠀⢌⣒⠀⢱⢱
⠀⠋⠉⠉⠉⠉⠉⠉⠉⠓⠀⢀⠠⢀⠠⢄⠀⡤⠀⠐⠤⢄⠈⠄⠙⠛⠐⠛⠒⢁⠃⡀⠀⠀⠐⠂⠄⢀⡀⠁⠀⣀
⠀⠀⠀⠀⢀⢠⡄⠀⠀⠀⠀⠈⠈⠀⠈⠊⠀⠑⠑⠀⠊⠈⠀⠀⠐⠄⣀⡠⠒⠁⠨⠀⠀⠂⠀⠀⠀⠆⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠌⠈⠤⠀⠀⠀⠀⠈⠉⠁⢀⣀⣀⣀⣀⡀⠀⠈⠁⠀⠧⠀⠀⡀⢴⠀⠀⠀⠀⠀⠀⠂⠀⠀⣠⡤⣤⣄⡄
⠀⠀⠀⢄⡀⢀⠀⡀⠀⠀⠀⠀⠀⢰⣿⣟⣽⣻⣭⣿⠀⠀⠀⠀⡄⠁⠀⠁⠐⠀⠀⠀⠀⡀⠀⠀⡀⡎⠛⠶⠞⠃⠂
⠀⡀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠀⢀⠄⠀⡀⢀⣛⠃⢰⠀⠀⠀⠀⠀⠀⠀⠄⢁⠀⠀⠀⢀⡅
⠀⠣⠤⠤⠤⠤⠤⠤⠤⠽⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠒⠒⠃⠀⠀⠄⠀⠄⠊⠀⠈⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⡀⠒⠄⠀⠀⠀⠀⠀⠠⠂⢄⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⡀⡡⡀⠀⡀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠺⠽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈
⠀⠀⢀⠀⠀⠈⢲⠽⠂⢀⠀⠀⠀⣧⡤⣦⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡆⠐⡀⠀⠀⠀⠎⢈⣴⣧⡀
⠀⠀⢸⣶⡆⠀⠘⠀⠀⠀⠐⡄⠀⣵⠉⠁⠑⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⣇⠀⠇⠀⠀⠀⠘⠀⠀⠉
⠏⠀⠙⢍⠁⠀⠀⠣⠀⠀⠀⠘⡠⠉⠀⠀⠀⠀⢠⠀⠀⠀⠀⠂⣰⣀⡐⠈⣶⠧⠀⠀⠀⠀⠀⢀⠆
⠀⠰⠤⣴⣳⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠉⠛⠇⠀⠃⠀⠀⠌⠀⠀⡠⠊
⠀⠘⠀⠈⠣⠀⠀⠀⠁⠂⡀⠢⡇⠀⠀⢂⠀⣠⡞⣻⣿⣿⣿⣿⣦⡀⢠⢀⠀⠀⠀⢢⢤⠂
⠀⠘⡀⠀⠀⠱⢄⠀⠀⠀⠐⢄⠰⠀⠀⣸⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⣼⡆⠀⠀⢸⢸⠀⠀⠘
⠀⠀⠡⠀⢠⢶⣦⣷⣶⡄⣀⣴⢈⡒⡲⠛⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠼⡻⠙⠀⠀⠀⣶⠦⠐⠁
⠀⠀⠀⠀⠁⠙⠙⠙⠝⠛⠊⠫⠳⠕⠌⠢⠠⣝⢿⣿⣿⣿⣿⣿⣟⣵⣦⡀⠀⠈⠂⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⣭⣯⣽⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⡻⠟⠟⠯⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠒⠚⠋⠉⠉⢉⣉⡉⠉⠙⠓⠒⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⢀⣠⣥⣴⣶⣦⣬⡀⡀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⠀⢀⣒⣀⣀⣐⠠⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠠⢊⣴⣾⣿⣿⣿⣿⣿⣷⡐⠀⠀⠈⠙⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⡀⠄⣀⣀⣠⣄⣀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠘⠻⠿⠿⠿⠟⠛⠉⠀⠀⠔⣡⣶⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⣠⠶⠿⠿⠿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠈⠁⣀⠀⡀⢀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠈⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡠⠒⠉⠁⠈⠁⠢⢎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⣾⣻⡆⠀⠀⠀⠈
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣶⡄⠀⠀⠀⢸⠀⠈⠉⠁⠀⠀⡠⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠙⠛⠁⠀⠀⢀⠋⠒⠄⠀⠀⠔⠊
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠐⠀⠈⠢⢄⣀⣀⣀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠂⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠉⠒⠒⠀⠀⠐⠒⠀⠉⠀⠀⠀⣀⣀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⣀⣴⣶⣶⣾⣿⣿⡿⠛⠉⠉⠉⢙⡶⠔⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠓⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡁⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⠤⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢸⣩⣝⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢀⡞⠙⠛⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⣰⠋⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⡄⢀⡞⠁⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⣇⡞⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⣠⠖⠚⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣾⠃⠀⠀⠀⠀⠀⠀⠀⡿
⠀⢰⡇⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢸⣿⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠈⣇⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢀⡤⠖⠋
⠀⠀⠘⣆⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⡴⠃⣠⠴⠚⠉
⠀⠀⠀⠈⢧⣤⣤⣶⣞⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠖⠒⠒⠶⠤⢄⣀⠘⠢⢤⣀⣀⣾⡵⠋⠁
⠀⠀⠀⠀⠈⢻⡟⠋⠁⣘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣛⣩⣴⣶⣶⣶⣶⣦⣤⣀⣀⠈⠙⠦⣀⠀⠐⠉⠖⠋⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠋⠳⣄⠀⠀⠀⠀⠀⢀⣼⣿⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠈⢣⡀
⢄⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⠳⣄⠀⠀⢠⣿⣿⣿⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠹⡄
⠀⠑⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠳⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣝⣿⣿⣿⣿⣿⣧⠀⢻⡀
⠀⠀⠈⢂⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⡇
⠀⠀⠠⢀⠑⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠶⢤
⠀⠀⠀⠀⠑⠠⡀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⣿⡇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣋⠁
⠀⠀⠀⠀⣀⣤⠤⣤⣘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣿⣿⣿⣿⣿⡧⠿⠶⠶⠶⠤⠤⠤⠤⠤⠴⣻⣅⢻⣿⣧⠀⠀⢀⡠⣪
⠀⠀⢠⠞⠁⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⣀⣀⣀⡀⠸⣿⣿⣿⣿⡘⣿⣿⣿⢃⣾⣿⡿⢟⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡎⣿⣿⣧⡤⠴⠚⠁
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢸⡟⣶⠛⠉⠉⠉⠉⠉⠙⠻⣿⣟⠿⡇⠘⣿⠃⣼⢟⡭⢞⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡼⠟⠋⠉
⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣸⢇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⢷⠀⠀⠀⢀⠴⡋⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡊⢀⡔⠋
⠀⠀⠈⠳⢤⣀⣀⣀⣠⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢳⠀⠀⢸⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⢖⡫⠕⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠉⡀⠀⠈⢯⢈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣆⠀⠈⠀⠀⠀⠀⠀⠬⠭⠵⠒⠋⣉⡤⠞⠉
⠀⠀⠀⢀⡂⠉⠀⠀⠀⠀⠈⢧⡓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠈⠲⠶⠤⠤⠤⠤⠤⠶⠶⠖⠚⠉⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠬⣭⣝⣛⣀⣤⠤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢦⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⡇⠀⠁⠀⠀⠀⠀⠀⡴⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠈⢳⡀⠘⢟⡆⢳⠀⠀⠀⠀⠀⠀⠀⠳⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢝⠢⣀⣯⡂⠀⢹⠿⠳⢄⡀⠀⠀⣦⠎⢰⣷⣾⣄⠀⠀⠀⠀⠀⠀⡠⠖⡄⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠁⠘⡟⠉⠓⡾⠀⠀⠀⠙⣆⠀⢸⣤⣾⠋⠀⠀⠑⣄⠠⠖⠋⣰⣁⡔⠁⠈⣁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣇⠀⠙⢆⠐⢄⠀⡇⡄⠀⢁⢄⠀⠀⠀⠘⣦⠞⠉⣏⠀⠀⠀⢀⣿⡤⠚⠋⠙⡏⢠⡎⠉⠀⠀⣹⠧⠐⠃
⠀⠀⠀⠀⠀⠀⠙⢷⡀⠈⣆⢀⠁⠘⢜⠀⠘⡜⢆⠀⠀⠀⣿⠀⠀⠸⣄⠀⢀⣸⣿⠀⠀⢠⢀⡇⢘⠷⠤⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠫⢺⣿⡷⠤⣀⡀⠁⠤⣈⢎⢂⣀⣤⡏⠀⠀⠀⠉⠲⣿⣻⣯⡀⢴⡯⠊⢀⡠⠄⣀⣤⡶⠒⢢
⠀⠀⠀⠀⠉⠒⠀⠀⠀⢠⣹⠀⠀⠈⠋⠲⣄⠀⠀⠈⣩⣴⣿⡾⠛⠓⠢⣄⡀⠀⢀⡩⠅⣒⠒⠒⠒⢾⣿⠥⠤⠖⠋⠀⣀⠤⢤
⠀⠀⢀⠑⠢⣀⠀⠈⠒⠀⢻⣀⠀⠀⠀⠀⠈⢆⡂⣴⣿⣏⡽⢂⣤⡄⠀⠊⢫⡠⠋⠐⠉⠀⣀⣤⠤⠾⠁⠠⠤⣤⣴⣉⣠⣤⡾
⡰⣏⡉⠉⠓⠦⣍⣲⡤⠴⠶⣯⡢⡀⠀⠀⠀⠘⣿⣿⣿⡟⣦⠈⠉⠀⠀⠀⠀⢳⠀⠀⣴⡞⠁⠀⠀⠀⠀⠀⢀⢟⡭⢁⡀
⠁⠉⠛⢛⡛⠒⠚⠷⣇⠀⠀⠀⠈⠲⢄⡀⠀⠀⣸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⠒⠒⣇⠀⠀⠤⠤⣒⣠⠖⣡⠵⠚⠉⠁
⠀⠀⠀⠈⠑⠒⠶⠦⠬⠷⣔⡀⠀⠀⠀⣉⣖⣛⣻⡻⣟⠿⣿⣷⣤⣀⣀⣬⣾⠏⠀⠈⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠠⢭⡭⢉⣉⣁⣀⠉⢉⣙⣮⣓⠮⢭⣿⣿⣟⡿⣛⠶⠲⠶⠶⠶⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠠⠤⣬⡛⠛⠻⠖⠒⠲⠶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠉⠉⠉⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣤⣤⣄⣠⣤⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠛⠛⠻⠿⠿⠛⠛⠋⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠉⠉⠉⠛⠋⠉⠉⠉⠀⠀⠹⣄⠀⠀⠀⠀⢀⡠⠄⡀⠀⠄⠀⠀⢰⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⣠⣤⣤⡄⠀⠀⠠⠿⣿⣶⣄⠀⠘⢧⠀⠀⠀⢇⡀⠀⠈⠦⠄⡀⠤⠁⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⡾⠋⠉⠉⠉⠀⠀⠀⠀⠀⠈⠻⠄⠀⠘⣇⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠘⠀⠀⠁⠀⠁⢀⡐⠈⠀⠀⠑⠀⠀⠀⠀⢸⡄⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠰⠂⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⢠⣄⠀⠐⠀⠀⠀⠀⡇⠀⠀⡠⠂⢀⠔⠒⠀⠀⠊⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠃⠀⠈⠋⠀⠀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣿⡠⠊⠀⡠⠂⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠋⠁⠀⠈⠂⢀⣀⠠⠊⠀⠄⣀⡀⠀⠁⠀⠛⠂⠀⣿⡁⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⣿⠛⠁
⠀⢀⣊⣁⡀⠉⠑⠖⢁⠤⠖⠒⠉⣇⠀⠀⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⡏⠀⠀⡇
⡰⠁⠀⠀⠀⠀⠀⠐⠡⡀⠀⠀⠀⢻⠀⠀⠘⣿⣿⣷⣶⣶⣦⣤⣴⣶⣶⣾⣿⣿⡟⠀⠀⢸⠃
⢣⣀⣠⡴⡖⢠⠀⠀⡠⠃⠀⠀⠀⠸⡆⠀⠀⠘⠏⠁⠀⠀⠉⠀⠀⠀⠈⠻⣿⠟⠀⠀⠀⣼
⠀⠀⠉⠐⠁⠸⡀⠀⠂⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣦⣧⣶⣤⣿⣴⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⠇⠀⢸⠀⢀⠘⠀⠀⠀⠀⢠⣤⡤⠤⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡓⢄⠀⠀⡎⠣⠀⠀⠀⢈⣅⣠⣤⠤⠴⠶⠶⠚⠒⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢈⣀⡀⠀⠠⠤⠧⠴⠖⠒⠓⠛⠛⠀⠀⢈⣀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠄
⣀⡀⠀⠤⠤⠐⠒⠀⠀⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢐⠀⠠⠀⠀⠀⠀⠐⠀⠐⠒⠀⠌⠁⠀⠉
⠀⠀⠀⠂


⣠⣠⡄⠀⠤⠠⣤⣶⣶⠦⠴⣶⣖⣄⣶⡈⠀⠉⠀⣭⣩⣁⣭⣉⠉⠀⣉⠉⢈⡽⠋⢸⡉⠩⢍⠉⠍⣉⠉⠁⣀⡀⠀⡀
⠿⠛⠁⠀⠀⠀⠄⠒⢚⡃⠀⠀⠀⠀⠟⠇⣀⣰⣶⠄⠘⠉⠉⠐⠚⠉⠀⠴⠟⠀⠀⢸⣧⣄⠀⠙⢦⣄⠉⠓⠶⢭⣉⠒⠢⠤⢄⣀
⠀⠀⠀⠰⠶⠶⠟⠛⠋⠁⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠛⠲⡄⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠹⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⣀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠆
⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⣴⣿⣿⣿⣿⠟⠛⠛⠉⠉⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡄
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⣿⣿⣿⣶⣴⠶⠞⣡⣶⣿⣿⣿⣿⡿⠛⠛⠉⠉⠉⠁
⠀⢠⣾⣿⣿⡿⣿⣿⣿⡟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢻⡏⠀⠀⣠⣾⡿⠿⠿⠿⠿⣿⣷⣤⣤⠄⠀⢀⣤⣴⡶⠶⠒⠒⠒⠂⢀⣀⠀⢀⣀
⢠⣿⣿⣿⣏⢾⣿⣿⣿⣧⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢿⣶⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣯⣀⣠⣤⣶⣿⠿⠛⠛⠉⠉⠉
⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⡾⠿⠟⠛⠿⢿⣿⣿⡿⠿⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡟⠋⠉⠉⠉⢀⣤⣤⣦⣤⣤⣄⣀⣀⣀⣠⣤⠖⠒
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣿⣿⣷⣦⣤⣤⣶⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠉
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣸⠁⠀⠈⢉⣛⡛⠛⢛⠙⠻⣿⣿⣿⡶⠶⠒⠂
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⣿⣷⣶⣶⣿⠿⠛⠛⣋⣤⠶⠒⠂
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠟⠘⠿⣿⣿⣿⣿⣿⣶⣶⠾⠋
⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠈⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠶⠶⠶⠶⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠭⢟⡶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠈⢷⣦⡈⠢⡙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣉⣔⡂⠀⠑⢦⠀⠀⠀⠙⠻⢧⠷⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄⠈⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠠⠢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢊⢖⠀⡀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡒⠒⢄⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠐⡃⣠⠆⠀⠡⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠃⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⡜⡁⢂⠖⠀⢂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠄⡁⢫⠞⡥⢨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢞⡄⣡⠞⡄⡀⢢⠆⠴⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠜⡡⢢⠞⠅⣡⠋⢐⡀⡅⡡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⠕⡡⢂⠞⡥⡠⢋⢀⢞⠑⡁⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠱⢣⢎⢞⡵⠡⠀⢊⢎⢞⢞⡴⠁⡃⠈⠜⢀⡴⠀⠀⡀⡰⡀⡀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠐⠢⢊⠞⠔⡥⠀⡴⠃⢪⢞⠜⡁⡠⠊⠊⢠⠆⢺⠎⠈⠄⠔⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠊⠂⠈⠀⠰⠃⣩⢊⠘⠁⠕⠅⠁⠀⡠⠀⠂⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠂⠀⠀⢈⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠓⠚⠛⠛⠛⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⢀⠔⢹⠟⠁
⠀⠀⠀⠀⠀⢀⠔⢉⠖⠁
⠀⠀⠀⢀⠔⢉⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⠀⣠
⠀⢀⡔⢩⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⡷⠟⠛⡛⠛⠻⠶⣇⣠⡄
⣔⢁⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡟⠁⢄⣨⣴⣶⣦⣬⡂⠌⠻⣶⠆
⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣟⢹⡟⣉⢿⢫⠻⢫⢹⢟⣿⣿⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡟⠸⣇⠟⣰⡸⢰⣷⣞⣎⠆⣿⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣯⣟⣿⣿⣿⣿⣻⣿⣿⣻⣛⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣏⠉⢝⠻⠿⣿⡿⠟⡋⠉⣩⣟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡷⣦⣌⣀⣀⣈⣡⣴⢾⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠻⠋⠿⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⣦⠀⠀⠀⠀⠀⠀⢀⡴⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢹⠀⠀⠈⢷⡀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⠒⠦⢼⠀⠀⠀⠀⠳⡴⠋⣡⣾⠀⡁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡀⠐⠀⠀⠀⠀⣀⣤⣾⢿⠟⠁⠀⠊⠙⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣖⡲⠦⢤⣤⣀⣀⡈⣧⠹⣿⣧⣀⡴⠟⠋⠁⠀⢀⣀⣀⣀⡄⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠈⠉⢉⠁⢹⣷⠉⠀⠀⠀⣠⣾⣿⣿⣧⡀⠙⢯⠀⠀⠀⠈⢿⣷⡦⠤
⠀⠀⣠⣤⣤⡤⠴⠶⠖⠒⠒⠛⠛⠉⠉⠁⠀⠀⢇⣀⣬⣿⣄⣀⠀⠂⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠂⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⢀⣽⣶⣆⡀⠀⠀⠀⠀⠉⠛⢿⣿⣦⠀⠉⠉⠙⠻⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠻⢶⣤⣀⣀⠀⢀⡈⠻⣿⣷⣄⠀⠀⠀⠀⠉⠻⡟⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠈⠉⠛⢷⣤⣀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡏⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠰⣷⣦⠀⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⠿⣿⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣀⡀⠹⣿⣿⣿⣿
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣷⡀⢈⣿⡄⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠻⣿⠄⢹⣿⣿⣿
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀⠀⠀⠀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠁⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠛⠁⢀⣀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⢀⠐⠀⡤⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠈⠅⠀⠀⠀⠀⢈⣀⡄⠐
⠀⠀⠀⠠⠠⠄⠀⠀⡀⠀⠀⠀⠁⠀⠀⠔⠀⠉⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⡅⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀
⠀⠀⠠⠠⠴⠶⠾⣶⡖⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢟⠉⠛⠿⣯⣙⠲⢄⡀
⠀⠀⠀⢣⠁⠀⠀⠹⣟⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠙⠳⡦⣈⠑⠢⣀⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠘⢎⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠈⠛⢖⢄⡉⠓⠢⢀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⢧⠂⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠈⠈⠢⢄⢀⠉⠂⢠⡄⢠
⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢣⡡⡀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⢀⠈⠙⠛⠟⠒⢬⠗⢾⢶⣦⣀⡞⢹⡆⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠱⡐⠄⠀⠀⠀⠢⡀⢂⠈⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⢂⠀⠀⠐⠋⠀⠈⠂⠈⣹⣷⣾⣤⣀⣞⢁⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢀⠀⠀⠀⠀⢄⠁⠐⠢⡀⠀⠈⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⣺⠀⠀⠀⠏⡹⠉⠈⡅⠕⠋⠹⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠈⢆⠢⠀⠀⠀⠀⠡⡀⠠⠤⢤⡀⠀⠀⠀⢱⠂⠄⠀⠀⠀⠀⡠⠐⠀⠉⡐⠠⣤⡐⠀⠠⠊⠀⠀⢠⣤⠂⢳⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠢⠱⡀⠀⠀⠀⠈⠳⠠⣀⠓⠤⠀⡀⠙⡀⠠⠀⠎⠈⠋⠀⠀⠀⡅⠈⠀⠈⠣⠐⠁⠀⠀⡠⢊⡇⣠⢖⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⡘⠄⠀⠀⠀⠀⠑⠄⠑⡀⠀⠀⠈⡣⠘⠄⠐⡀⡇⠀⠀⠀⢐⠀⣄⠀⠀⣯⠤⠤⣊⠔⠁⠀⠁⠀⢈⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠐⠌⢂⠀⠀⠀⠀⠈⠢⠈⠢⡀⠀⠇⠀⠈⢤⠃⠈⠄⠀⠀⢸⠀⠈⢂⡠⠀⠀⡠⠁⠀⠀⣠⠴⡁⢀⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠈⠀⠡⡀⠀⠀⠀⠀⠐⢄⠀⠉⠇⠀⠀⢸⠀⠀⠈⣆⠀⢸⠀⠀⠀⣅⠔⠈⠁⠀⠀⣠⠁⠛⠍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢂⠐⡀⠀⠀⠀⠀⠀⠢⡀⠘⠀⠀⢰⡄⠀⠀⠀⠀⠈⠄⠀⠀⠘⠂⠈⠳⠤⠚⠁⠀⠀⠀⠈⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠡⠈⠄⠀⠀⠀⠀⠀⠀⢄⠈⠈⠁⠐⠄⠠⠂⠀⠀⠀⠢⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠷⠶⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⣠⠤⠒⠂⢼⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠑⠦⠤⣤⡯⠤⠖⠁⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢤⣤⣤⣀⣀⣤⣤⡴⠶⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠴⠶⠶⠶⠶⠤⠤⣀⡀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠁⡀⠀⢠⠀⠀⠁⠀⢀⠀⠀⠉⠲⣄⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣄⣀⠀⠀⠀⠤⠀⠀⠉⠀⠘⠀⠀⠈⠂⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠬⠤⠤⠤⣤⣄⣠⣤⣤⣀⣠⣤⣤⣤⣀⣀⡤⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠶⠟⠳⢶⣤⣭⣿⣍⣀⣀⣀⣠⣴⢦⣤⣤⡶⠂⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣶⢦⣀⡀⠉⠉⠉⠉⠉⠉⠁⣉⣠⣴⠖⠋⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⠄⠀⠀⠀⣿⡀⠀⠉⠉⠛⠓⠚⠛⠛⠛⠉⠁⢀⣠⠄⠀⠀⠀⠈⠻⠷⣦⣤⣤⣤⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⢠⡏⠀⠀⠀⠀⠈⠻⢶⣦⣤⣤⣤⣤⣤⣤⣤⠶⠞⠋⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢁⣠⠤⠒⠛⠋⠉⠉⠉⠉⠛⠛⠻⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⢋⣁⣀⣀⣀⠠⢤⣤⣄⣀⣀⡀⠀⠀⠀⣸⣿
⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⣀⣈⣙⠛⠛⠛⠉⠻⢶⣄⡈⠉⠉⠁⠀⠀⢠⣿⣿
⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⢶⣶⣤⣈⠉⠛⠿⣷⣦⣄⠀⠀⠉⠻⠷⠄⠀⠀⢀⣾⣿⠇
⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄⡀⠀⠀⠀⣴⠟⠁⠘⠳⢦⣄⡉⠛⠿⢷⣦⣄⡉⠛⠿⣶⣄⠀⠀⠀⠀⣠⣿⣿⠏
⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠒⠀⠈⡙⠷⣤⣀⠈⠙⠻⢷⣦⠈⠙⠋⠀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠈⠙⠛⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣀⣀⣀⣠⣤⣤⣤⣴⣶⣾⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⠀⠀⠀⢠⣤⣶⣶⣿⣿⣿⣷⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠸⠟⠋⠁⠈⠛⠿⠀⠀⠀⠀⠀⣰⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣏⣾⣿⡿⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢨⣵⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡼⠿⢿⣛⣻⣭⣿⣶⣷⡄⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣛⣯⣭⣽⣶⣶⣷⠀⠀⠀⠀⢀⣀⣰⣿⣿⣆⣀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⠿⢟⡋⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁

⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⢀⠀⢀⡀⠀⠀⢀⡀⠀⡀⠀⡀⢀⡀⠀⠀⢀⢀⣄⠀⠀⡀⣠⣄⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⡀⢠
⠀⠀⡈⡇⠀⠀⠉⠢⡀⠀⢠⠎⠀⢀⡀⠀⠑⣄⢹⠀⠀⢀⠀⡠⢫⠀⠀⣧⠾⠀⠀⡆⠀⢐⡏⠀⠀⠀⢸⠀⠀⠀⠀⠀⡈⡇⠀⠀⡇⠀⠀⣠⠎⠁⠀⠀⠀⡄
⠀⠀⡂⡆⠀⡴⣄⠀⠙⣤⡟⠀⠀⡏⠹⡄⠀⢹⢸⡀⠀⢸⠀⡇⢸⠀⠀⣿⣺⠀⠀⡇⠀⠸⣇⠀⠀⣌⢼⠀⠀⠀⠀⠸⡇⠄⠀⠀⡇⠀⡼⠁⠀⠀⣠⣶⠞⠁
⠀⠀⠆⡇⠀⠛⣿⠀⠀⣟⡇⠀⠀⠁⡄⡇⠀⠘⡊⡇⠀⠘⠀⡯⢸⠀⠀⡯⣸⠀⠀⣧⠀⢨⠆⠀⠀⡿⢘⠀⠀⠀⠀⠀⣇⠅⠀⠀⡇⣺⡇⠀⢀⠞⠁
⠀⠀⡁⠇⠀⠒⠁⠀⣰⠡⡇⠀⢸⠀⡇⢳⠀⠀⡇⠆⠀⠀⡆⠉⢸⠀⠀⡿⣹⠀⠀⡎⢣⣔⡇⠀⠀⡿⠊⠄⠀⠀⠀⠀⢸⡆⠀⠀⡟⣸⠀⠀⣼
⠀⠀⠄⡇⠀⠀⠀⢾⡛⠐⡇⠀⢸⣀⡅⣻⠀⠀⡇⣧⠀⠀⣿⢿⢺⠀⠀⡇⣽⠀⠀⡇⠀⢸⠖⠀⠀⣿⠀⡇⠀⠀⡇⠀⠀⠇⠀⠀⡇⣿⠀⠀⠆⢀⡄⠀⠤⢤
⠀⢸⠂⡇⠀⡴⡄⠀⠹⠈⡅⠀⢸⡅⡋⣺⠀⢰⡇⢡⠀⠀⠛⠀⠛⠀⠀⢃⢿⠀⠀⡇⠀⢸⣊⠀⠀⣿⠄⡇⠀⠀⢿⡀⠀⠀⠀⠀⡇⢻⠀⠀⢺⢈⣃⡀⠀⠰
⠀⠸⠁⡇⠀⠷⣹⠀⠀⡇⢇⠀⢸⡇⣏⣅⠴⠚⢙⠺⣄⠀⠀⠀⠀⠀⢸⢐⣺⠀⠀⡇⠀⢸⢶⠀⠀⣿⠂⡇⢀⠴⠛⠓⢤⡀⠀⠀⡇⢸⡀⠀⠈⢧⡄⢻⠀⢸
⠀⢸⢂⡇⠀⡴⠃⠀⣰⠇⢹⠀⠈⠣⠿⢿⠀⠀⠀⡒⠘⡄⠀⠀⠀⠀⢸⢸⢸⠀⠀⠧⠤⣼⣻⠀⠀⠽⡅⣣⡏⠀⠀⢀⡀⢹⠀⠀⡟⡄⢳⡀⠀⠀⠉⠁⠀⢸
⠀⢸⡁⡇⠀⢀⣠⡾⠧⣴⡀⠳⣀⠀⠀⢸⡄⠀⠀⠀⠀⡇⣀⣧⠀⣀⣼⣼⣸⡀⢀⣀⣀⣿⣾⠀⢀⡄⣆⣿⠀⠀⠀⠘⢂⣎⣀⣀⡏⠑⢄⠘⠦⣄⣀⣀⠀⢸
⠀⠸⣅⣑⡩⢽⠉⠀⠀⣄⠹⡤⠬⠩⠟⠋⢳⠀⠀⣀⣠⡿⠛⠺⡟⠁⠀⠀⡈⣯⠭⠭⠏⠸⠭⠬⠽⠁⢥⣯⣀⡀⠀⢠⠎⠩⠭⠌⡡⠖⠋⠉⠓⢦⠌⣩⡽⠋
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠘⡄⢳⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⡇⠀⠀⠸⠁⡔⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⣿⣿⡿⠃⠀⢀⡴⠋⠀⠀⠀⢀⠠⢸⡀
⠀⠀⠀⠀⠀⣀⡹⣄⠀⠀⠀⣀⡇⠀⠀⠀⠀⣩⣥⣤⣶⣾⡄⠀⢧⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣦⣬⣧⣴⠺⣿⣷⡄⠀⠐⠀⣁⡤⢊⠝
⠀⠀⠀⠺⣉⠀⠀⠈⠳⣴⣿⣿⡿⢆⠀⠀⠀⡿⠿⠟⠛⠛⠙⣄⢸⣦⣴⣶⣾⠁⠀⢀⡠⠔⣺⠟⠁⠀⠉⠛⠻⢿⣿⣿⣦⠹⣿⠿⠒⠚⠉⣡⠞⠁
⠀⠀⠀⠀⠈⠑⢤⡀⠀⠹⣋⣡⣶⣿⣷⣄⣸⠁⠀⠀⠀⠀⠀⠈⢾⠛⠛⠛⢻⣠⠖⠉⢀⡜⠁⠀⠀⠀⠀⠀⠂⢸⡌⢿⡿⠋⠁⠀⠀⠀⠀⠈⠓⠢⣄⡀
⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⣿⡿⠟⠋⠁⠀⠙⢦⡀⠀⠀⠀⠀⠂⣼⣿⣿⣿⣿⡇⠀⢠⠎⠀⠀⠀⠀⠀⠀⢀⠂⢸⣣⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⠃
⠀⠀⠀⣤⠚⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢐⡄⠘⢦⡀⠀⠀⡽⠁⠁⠉⠉⢉⠙⣄⡞⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁
⠀⠀⠀⠀⠉⠑⠒⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⢂⠀⠱⡄⣸⠁⠀⠀⠀⠀⠀⠀⠸⡄⣠⡴⠖⠚⠋⠉⠉⠉⠙⠛⠓⠶⣤⣀⠀⠀⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠐⠡⠀⢹⡇⠀⠀⠀⠀⠀⠀⢉⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠉⠳⢦⡀⠀⠀⠀⠀⠀⣹⠆
⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⠀⠙⣦⣀⡤⠒⠋
⠀⠀⠀⠀⠀⠈⠈⠓⠲⠆⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄⠀⢼⣿⣷⡆⠀⠈⢿⠑⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠘⠻⠿⠏⠄⠀⠘⣧⠌⠓
⠀⠀⠀⠀⠀⠀⠰⠛⠒⠒⠒⡓⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⣇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣀⡀⠐⠊⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠈⠑⠢⠤⠔⠊⠙⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢀⡤⠴⠶⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠀⢀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠠⠔⠊⠉⡆⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢀⠴⠋⠐⡀⠀⠀⡹⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠁⠀⠀⠀⠘⣤⠞⠀⠀⠉⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⠶⠿⠿⠿⠿⠿⠿⠿⠶⢶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⢀⣴⢖⠋⠀⠀⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⢀⡴⠊⣰⠁⡎⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⣴⠋⠀⠀⣿⠀⢧⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡉⠐⢄⠀⠀⠀⠀⠀⢹⡇
⣿⠀⠀⠀⠘⠳⠾⠃⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿
⢸⣇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠸⡏⠀⠀⠀⠒⠒⢲⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠁⠀⢀⠀⠀⠀⠀⠀⠀⢿
⠀⠈⠲⢤⣀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠁⠀⣀⡤⣀⠀⠀⣾
⠀⠀⠀⠀⠈⣿⣟⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣾⣿⠀⠈⠂⠀⣿
⠀⠀⠀⠀⠀⢸⡝⠧⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠈⠢⠀⠀⣾⣿⡟⠀⠀⠄⢠⣿
⠀⠀⠀⠀⠀⠀⡧⠤⠼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠀⠀⠀⠀⡿⠋⠀⠀⡐⠀⣼⡇
⠀⠀⠀⠀⢀⠜⢡⠂⠀⠤⣉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠋⠀⠀⠠⠀⠀⠳⠤⠤⠊⠀⣰⡟
⠀⠀⠀⢀⡞⢠⡇⠀⠀⠀⠈⢻⡆⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠓⠶⢤⡱⢄⣀⣠⠴⠚⠲⠦⣄⠀⠀⠀⠀⣰⡟
⠀⠀⠀⡾⠀⢸⡇⠀⠀⠀⢀⡾⠀⡏⠀⠀⠀⢀⣤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣹⠁⠀⠀⠀⠀⡿⡄⡼⠁⣴⠛⠛⠓⢮⡇⠀⢀⣼⠟
⠀⠀⠸⠁⠀⠀⠙⠶⢦⡤⠟⠁⣸⣿⣦⠀⣴⢋⠖⠒⠲⢤⡉⠳⣄⠀⠀⠀⠀⠀⣸⠁⠙⠖⠲⠤⣄⣸⠇⣶⠃⠀⢧⠀⠀⠀⣸⡇⣠⡾⠋
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣿⣿⠃⡎⠀⠀⠀⠀⢱⡄⢸⡇⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠳⠶⠶⢻⡿⠛
⠀⠀⢺⠁⠀⠀⠀⠀⠀⠀⢀⡞⠀⠠⣿⠃⠀⢇⠀⠀⠀⢠⡞⠀⣼⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠒⠉⠀⠀⢰⠏⠀⠀⠈⠙⠒⠒⠋⠀⢰⣇⠀⠀⠀⠀⡾⢰⠀⠀⠀⠀⠀⠀⠀⣿⠷⣄⠀⢀⣀⣤⣶⠃
⠀⠀⠀⠈⠳⢦⣤⣀⣠⣴⣖⣋⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⣿⠿⠶⣾⠇⠹⢤⡀⠀⠀⠀⠤⣾⡇⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⡴⠷⣄⠙⢄⠀⠀⠀⠀⠀⣰⣇⡀⠀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠋⠀⠀⠹⣦⡀⠀⠀⠠⠤⣶⠟⠉⠉⠛⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣒⣦⣤⣴⣖⣛⣉⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣛⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⢿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣦⡄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠟⠛⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣆
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣭⣭⣿⣽⣿⡟
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢠⣄⣴⣦⡾⢦⣶⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⢤⠴⠶⠤⣤⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠓⠲⢤⣀
⠀⠀⠀⣤⣼⠟⢋⡩⠄⢒⠀⠀⠂⠀⠈⠛⢷⣠⡀⠀⠀⠀⠀⠀⠀⢀⣤⡼⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⣤⣄⠀⠀⠀⠀⢠⠎⢠⢐⣪⣥⣐⢄⠉⢆
⠀⠀⣶⠞⠁⠀⢡⠀⠢⠞⠘⠳⠭⠰⠂⠀⠀⠙⢧⡆⠀⠀⠀⠀⣀⡼⠋⠀⠀⡄⠄⠰⠃⠚⠀⠀⠂⡄⠀⠈⠻⣤⡀⠀⠀⣽⠠⡄⡟⠙⠫⣟⡆⡄⢸
⠀⢶⠏⠀⢁⣘⠒⣡⣴⣾⣿⣿⣶⣤⡈⠰⡄⠐⠈⢷⠆⠀⠀⢀⡾⢁⠂⣀⡃⢀⣤⣶⣾⣾⣷⣦⣄⠠⢤⠈⠂⠹⣇⠀⠀⢺⡀⠠⡹⠿⠿⠟⡡⠀⡼
⠰⣿⠠⠁⢈⠉⣰⠟⡿⠛⡿⠛⠿⠻⣷⡀⠁⠀⢀⢨⡷⠀⠀⢻⠃⠀⠀⠉⢀⡾⠻⠛⢿⠟⠻⡟⢿⣆⠈⠁⠀⠀⢻⠃⠀⠀⠱⢦⣉⡑⠂⣁⣠⠞⠁
⠺⣗⠀⠨⠿⠃⣿⡆⡀⠗⣀⠿⢸⣧⣬⡇⠹⠏⢀⢈⡿⠂⠐⢿⠀⠁⠸⠇⢸⣧⢀⠸⢂⡸⠇⣾⣤⣽⠀⠶⠀⠇⢸⡗⠀⠀⠀⠀⠈⣉⣉⡉
⠘⢿⡈⠀⢀⣀⠘⣿⣿⣾⣿⣶⣿⣿⡟⢀⣀⠀⠀⢸⠟⠀⠀⣼⡆⠀⠀⡀⡀⠻⣾⣷⣿⣷⣾⣿⣿⠇⣀⡀⠀⠁⣿⡄⠀⠀⣠⣾⣿⣿⣿⣿⣷⣦⡀
⠀⠛⣧⠐⠌⠲⣂⡈⠙⠻⠿⠿⠛⢉⠠⣌⠈⠀⢀⡞⠃⠀⠀⠀⣿⡀⠀⠐⢂⠀⠉⠛⠿⠿⠟⠋⡁⢤⡑⢃⠂⣼⡇⠀⠀⣼⣿⣻⠋⠉⠉⠙⢿⣻⣿
⠀⠀⠛⣳⣌⠀⢈⠑⠋⠴⢶⢶⠘⠈⠃⠀⢂⣴⡟⠁⠀⠀⠀⠀⠈⢹⣆⡀⠠⠉⠒⠀⠲⡶⠎⠃⠙⡠⠐⣠⣾⠉⠁⠀⠀⣿⣯⡇⠸⠿⢿⠛⢪⣯⣿
⠀⠀⠀⠈⠹⠷⢦⣤⣀⣐⡒⢒⣀⣀⣤⠶⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠱⣦⣤⣀⡀⠒⠒⠂⣀⣡⣤⣾⠉⠁⠀⠀⠀⠀⠻⣿⣿⣦⣀⣀⣤⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠁⠙⠋⠻⠋⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠟⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣄⣄⡀
⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣗⡤⣖⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⡴⠟⡉⠤⠤⠉⠙⢶⡀
⠀⠀⢀⢴⣿⣯⣿⣟⡻⢙⣟⡹⢫⣽⠓⣦⣛⣗⢀⠀⠀⠀⠀⠀⠀⣠⡾⣩⣷⣿⣻⠏⣹⣋⠟⣭⡟⣳⣿⡻⣦⡀⠀⠀⠀⣼⠁⠘⡴⢿⡿⣶⡐⠀⢻
⠀⢠⡿⣿⣟⠳⣞⣾⠿⠟⠛⠛⠳⢿⣯⢨⣝⣷⡴⣦⠀⠀⠀⢀⣼⢏⣾⡛⢶⣷⡿⠿⠛⠛⠛⠾⣿⣅⣭⣻⣎⢿⣆⠀⠀⢿⠈⠁⢧⣥⣬⣿⠇⠁⣸
⢀⣿⣽⣏⡫⣸⠟⠁⠀⠀⠀⠀⠀⠀⠈⢷⡄⣘⣷⡽⣦⠀⠀⣼⢣⣿⣙⣥⡿⠋⠀⠀⠀⠀⠀⠀⠈⠻⣷⢂⣻⣿⣿⡆⠀⠈⢧⡈⠢⠭⠍⠑⠊⣰⠏
⢸⣿⡿⢟⠿⡏⠠⣶⣴⢶⣄⡶⣦⢦⣔⠀⣿⠟⡿⡟⣿⠀⢠⣿⣼⡿⢿⣿⠁⢰⣦⡶⣦⣰⢶⡴⣦⡂⢹⣿⠿⣿⣾⣷⠀⠀⠀⠉⠓⠶⠶⠞⠋⠁
⢹⣿⣿⣭⣸⡇⠀⢿⢻⣼⠟⣧⡿⠈⠙⠁⣿⣭⣵⡇⣿⠀⢸⣿⣿⣦⣴⣿⠀⢸⡟⣧⡿⢻⣼⠇⠁⠉⢸⣷⣤⣾⣿⣿⠀⠀⠀⣠⣤⣶⣶⣦⣄⡀
⠸⣿⢿⡿⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⡿⣿⢧⡧⠀⠀⣿⡜⣿⠿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⢿⣿⣿⡇⠀⢠⣾⣟⡿⠞⠓⠿⣿⢿⣆
⠀⠻⣿⢷⡭⣸⢿⢦⣄⣀⢀⣀⣀⣤⣾⡋⠾⣵⣿⠿⠃⠀⠀⠘⣿⡹⣮⢁⡽⣷⣤⣀⣀⣀⣀⣠⣴⣟⢵⢮⡼⣿⡟⠀⠀⣾⣿⡏⢠⣤⡴⡤⠸⣾⣿
⠀⠀⠙⢷⣝⢦⣄⢂⡂⠏⠛⠛⣥⢰⣄⣥⢚⣵⠟⠁⠀⠀⠀⠀⠈⢿⣮⡷⣤⣐⣹⠹⠛⠛⢻⡅⣦⣸⣷⣿⣾⠏⠀⠀⠀⢻⣿⢷⡈⠉⠉⢀⣼⣿⣿
⠀⠀⠀⠀⠑⠻⢿⢭⣛⣛⣖⣓⡛⣻⡿⠷⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣝⣛⣿⣷⣾⣿⣟⣿⣿⠾⠛⠁⠀⠀⠀⠀⠈⠻⣿⣽⣓⣺⣿⣷⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠂⠀⠈⠉⠉⠉⠁⠀⠀⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠈⠠⠄⠀⡀⢄⡂⠌⡂⠠⢐⠂⢠⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠄⢂⠀⠀⠉⠀⠀⠀⠍⡂⠅⡂⠅⡒⠍⡂⠀⡂⢡⠹⠖⠀⠑⢄
⠀⠀⠀⠀⠀⢀⠔⢁⠀⡡⠂⠁⠀⠀⠀⠀⠠⠀⠀⠀⢈⠀⢈⠀⠁⠂⠁⠊⠐⠘⣁⡀⠁⠢⠑⡀
⠀⠀⠀⠀⢀⠊⠔⡡⠊⡠⠐⡁⠀⠀⠐⡡⡊⠔⡁⠀⠁⠊⡡⠀⢊⡁⠀⠀⠀⢸⡂⢎⢧⠀⢰⡌⢄
⠀⠀⠀⢠⢃⢊⠜⡠⠊⡠⢊⠔⠀⠀⠪⠊⠄⢈⠔⢁⠔⢁⠄⠀⠁⠀⠀⠀⠀⠈⢯⡲⣿⡆⠀⠀⠈⠆
⠀⠀⠀⠄⢠⢂⠔⠠⢊⠔⡡⠊⡡⠀⠀⠈⠀⢀⠎⡡⠊⡀⠐⡡⠂⠁⠀⠀⠀⠀⠈⠓⠔⠁⢠⣤⡄⠘⡀
⠀⠀⠐⢀⢃⠊⡐⡡⠃⠔⠡⠊⡠⠂⠀⠀⠀⠀⢈⠄⢊⠄⡡⠂⠀⠀⠀⠀⠜⠜⡔⡀⠀⠀⢸⣋⣿⠀⠂
⠀⠀⠸⠀⠆⡔⡐⢁⠊⡐⡡⢊⠔⢁⠄⠀⠀⠀⠈⠔⠁⡪⢊⠀⠀⠀⠀⠪⢊⢌⠜⡐⡡⢃⠈⢷⡾⠀⢰
⠀⠀⢰⠀⡐⡰⢀⠂⠌⡐⠡⠃⡰⢁⠔⡠⠂⠀⠀⠀⠀⠀⢐⠅⠀⠀⠀⠀⢁⣀⠈⠐⡡⢂⢆⠆⠄⡀⢸
⠀⠀⢘⠀⢠⠁⠎⡌⡐⠠⠁⠜⠠⢃⠊⡠⠊⡀⠀⠀⠀⠀⠑⠡⡂⠀⢠⡮⢕⢽⣿⢂⠐⠑⡡⢊⢜⠄⡀
⠀⠀⠀⠆⠀⠘⠰⠰⢁⠃⠌⠠⠡⠁⡔⢀⠞⡠⠀⠀⠀⠀⠀⠈⠀⠀⢾⢿⠢⢊⡿⠸⠀⢊⢔⠕⡡⢀⠂
⠀⠀⠀⠐⡀⠀⢃⠃⠆⡌⢠⢣⠃⡘⢠⠃⠔⡠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠽⣁⡁⠔⠁⠀⢐⠥⡪⠃⠌
⠀⠀⠀⠀⠱⡀⠀⠘⠠⠀⠂⡄⠰⢁⠁⠎⡐⢡⢊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠜⠀⡜
⠀⠀⠀⠀⠀⠙⢄⠀⠀⠈⠰⢀⠃⠎⡌⡐⠠⢡⢃⢊⠄⠄⡀⠰⠠⠀⠀⠀⠀⢀⢀⠠⠈⠀⢠⠎⢀⠂
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠐⠰⢠⢁⠃⠂⠂⠌⡌⡐⡠⠡⢁⢃⢂⠎⠌⠀⠀⠀⢀⠔⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠕⠤⡀⠀⠀⠀⠈⠈⠈⠈⠐⠐⠐⠑⠁⠁⠂⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠔⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠈⠈⠙⠋⠝⠉⠉⢉⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠄

⠀⠀⠀⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤⡇⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⣇⢰⣾
⠀⠀⠀⠿⠿⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠁⠘⠁⠀⠀⠀⢀⠀⢀⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠⡄⣠⡇⠀⠀⠀⠈⢡
⠀⠀⠀⠀⠀⢱⡀⠀⠀⠇⠾⡇⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠸⠀⡿⠁⠀⠀⠀⠀⢀⡀⣀⡆⠀⠀⠀⠁⠹⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢧⠀⠀⠀⠆⢰⡞⠀⠀⠃⠀⠀⠀⠘⠳⣇⠀⠀⠀⠀⠘⠂⠿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀
⠀⠀⡀⠀⠀⣀⡇⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠱⡀⣤⡆⠀⠀⢠⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡇
⠀⠀⠁⠒⠒⢻⠇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢓⠚⠃⠀⠀⠀⠀⠛⢀⠀⠀⠇⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠷⠀⣀⣻⠃
⠀⠀⠀⠀⠀⣼⣤⡶⠇⠀⠐⣾⠇⠀⠀⠀⠀⠀⢸⣠⣠⡶⠰⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠄⠀⡒⡾⠀⠀⠀⠤⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣤⣤⠀⠀⣠⡼⠁⠁⠀⠀⠀⠀⠀⠀⠾⠛⠁⠐⡀⠀⣤⣴⠀⠀⠁⠀⠈⣡⣴⠞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⠈⠀⠈⠛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠃⢀⣠⠄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⢄⠀⠀⠀⠀⣠⡷⠖⠛⠁⢐⡀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⣤⣶⠖⠂
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠋⠁
⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠾⠿⠷⠶⠶⠶⠖⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣷⣺⣯⣛⣻⣏⣫⣿⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠌⠀⢀⠀⡀⠀⠢⠀⠀⠀⠀⠀⠀⢀⠁⠑⡖⢀⡃⠀⠀⠀⠀⠀⠀⠀⠓⣤⠈⠢⡀⠀⠀⠀⠀⠀⠀⢈⡌⡛⢰⠺⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢳⠊⠑
⠀⢠⠔⢂⡁⠀⢸⡐⢢⠀⠀⠀⠀⠀⢰⠃⠀⠠⠈⠚⡃⠀⠀⠀⠀⠀⠀⣼⠉⢂⠌⢂⠐⡀⠀⠀⠀⠀⠀⠘⠚⠧⠿⠓⡇⠀⠀⠀⠀⠀⠀⢠⣂⡥⠞⠕⠀⡀
⠀⠀⠢⡀⠀⠀⠀⢄⠔⠀⠀⠀⠀⠀⠈⢈⠈⠀⡀⡁⠁⠀⠀⠀⠀⠀⢀⣨⢧⠜⠊⠢⠃⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠈⣠⠛⠀⠀⣀⠔
⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉

⠀⠀⡿⠉⠉⣉⠉⠉⠇⠀⠀⠀⠀⠀⠀⢀⡄⠠⢀⣴⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠳⡄⠀⠀⠀⠀⠠⠊⠁⠀⠁⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢱⡀
⠀⠀⠀⠐⠥⠤⢬⠂⢀⠀⠀⠀⠀⠀⢀⠊⠀⠔⠙⠅⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⢦⢹⠁⠁⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠰⢎⠀⠀⠀⠀⠀⠀⣠⠪⢌⠽⡞⡆
⠀⠀⠀⠠⠁⠀⡈⠀⠈⠀⠀⠀⠀⠀⠘⢉⠉⠉⢉⠏⠀⠀⠀⠀⠀⠀⡀⢸⣌⣳⠥⠃⠀⠀⠀⠀⠀⠀⢸⠙⠋⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⡂⢀⡑⢁⡣⠊⠁
⠀⠀⠴⠀⠄⠉⠄⠀⠇⠀⠀⠀⠀⠀⠀⠠⠼⠯⠄⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠢

⠀⠀⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⡠⢀⣠⣄
⠀⠀⠈⡕⠁⠀⠈⢢⡀⠀⠀⠀⠀⠀⠐⣴⣉⣰⣀⣁⣄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠠⡠⠠⠀⠀⠀⠀⠩⡈⠒⠤⠤⠒⢁⠍⠁⠀⠀⠀⢠⠘⣟⠕⠊⠉⠉⠣
⠀⠀⠀⠈⡄⠄⠅⡅⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡐⣗⢺⢺⡊⠱⠁⠀⠀⠀⠀⠀⠈⠐⡗⢰⡇⠁⠀⠀⠀⠀⠀⠘⢔⠂⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠓⠖⠔⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡰⣇⠵⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⡠⠊
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⡀⡀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠰⡄⠃⠐⠀⠂⡼⠀⠀⠀⠀⠀⠀⠀⢀⡄⠰⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠐⡄⠀⠀⠀⠀⠀⢠⠧⠾⠿⠼⡆⠀⠀⠀⠀⠀⠀⡰⠴⡄⢠⣂⢣
⠀⠠⠧⠀⠼⠀⠾⠧⡠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⢀⠔⡎⢓⡠⠈⠁⠀⠀⠀⠀⠀⢰⠂⠤⠤⠀⡇⠀⠀⠀⠀⠀⠀⣇⠤⡀⠡⠀⠀⡅
⠀⠀⢘⣀⠒⢒⣚⠀⠁⠀⠀⠀⠀⠀⠀⠀⡅⠠⠈⠇⠀⠀⠀⠀⠀⠀⢔⠁⠈⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡀⢀⡀⡇⠀⠀⠀⠀⠀⠀⡄⠁⠀⠠⠤⠤⠄
⠀⠈⠙⠞⠁⠙⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄⠐⠁⠀⠀⠀⠀⠀⠀⠣⠤⠄⠠⠤⠤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣰⡦⠄⠤⡀⠉⠉⠉⠉⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠋⠉⣡⣞⡡⠥⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠶⠖⡾⠛⠉⠉⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⠰⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⡸⠁⠀⠀⣣⡀⠀⠒⠒⢀⡀⠀⣿⠀⠀⣞⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢋⡴⠂⠀⢠⠃⠀⠀⡼⠁⠉⠓⠒⣦⠎⠀⠀⡇⠀⣼⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⡴⢋⡤⠂⣠⠋⡀⠀⣰⠃⠀⠀⠀⢠⠃⠀⣠⡤⡇⣤⣄⣹⣒⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣮⡁⠊⠠⠾⠑⣮⣷⠞⠁⠀⠀⠀⠀⢸⠉⡟⠁⠀⠙⠃⠀⠀⠀⣇⢠⠉⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⣿⡷⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣠⠞⠰⡀⢀⣀⣀⣀⣀⣀⣀⣸⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠪⠽⠛⠁⠀⢀⣀⣀⣤⣤⠴⠶⠶⠚⢯⡤⠤⣄⡏⠉⢉⣉⣩⠭⠽⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⣡⣤⠶⠞⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣟⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⠷⢶⣤⣄⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⢱⡀⠈⠆⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣄⠀⢸⣦⠀⡆⠀⠀⠀⠀⠀⠀⢺⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⣄⠘⡦⠀⠱⢧⣿⣄⠀⠀⠀⢀⣠⠬⣣⣸⡄⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⢻⡿⠶⢄⡼⠀⠈⠑⣄⠀⠈⣇⣠⡿⠋⠉⠢⡀⠀⣠⠄⢠⠊⡠⠃⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠓⣄⠠⡀⠀⡇⠀⠀⠃⡀⠀⠀⠈⢧⣠⠟⢿⠁⠀⠀⠀⣿⣈⡤⠖⢻⡟⢁⠔⠒⠒⠒⣶⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡀⠘⣆⠈⠂⠡⡣⠀⢡⢳⡀⠀⠀⢸⡇⠀⠘⣆⠀⠀⢐⣿⠁⠀⢀⢀⡇⢹⣆⣀⡠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⣿⣮⣄⡀⠈⠢⣄⣱⡑⠄⣀⣸⠇⠀⢀⠜⠢⣶⣿⣟⠀⣠⣣⠞⠀⣀⠀⢀⣠⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⠀⣹⠁⠀⠙⠓⠦⡈⠁⠈⠛⣩⣽⣶⠖⠚⠤⣄⡈⠉⠈⣛⠭⠤⠤⠤⠤⣾⣿⣡⡤⠔⠁⢀⣀⣄⡀
⠀⠀⠀⠀⠀⠐⠢⢄⠀⠀⠒⠀⠫⣄⠀⠀⠀⠀⠙⡄⢐⣵⣿⣫⡗⣠⡄⠀⠈⠻⣠⠎⠁⠊⠁⣀⣠⠤⠾⠥⠄⢤⣤⣆⣁⣠⣴⠇
⠀⠀⢀⢲⣉⠉⠑⠢⢭⣲⣤⠤⠶⣿⡢⡀⠀⠀⠀⠸⣿⣿⣿⢳⡄⠉⠁⠀⠀⠀⢹⠀⠀⣴⡞⠁⠀⠀⠀⠀⠀⣠⣻⠍⣀
⠀⠀⠀⠀⠙⢛⣛⠓⠚⠻⢧⠀⠀⠀⠈⠲⢄⡀⠀⢀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⠗⠒⣇⠀⠀⠤⢄⣂⡤⢊⠬⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠶⠷⢤⣂⣀⣀⣀⣩⣞⣋⣯⢿⣝⢿⣿⣶⣤⣤⣬⣶⠟⠀⠈⠉⠉⠉⠉⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⢤⠤⣤⣤⣄⡀⠀⢨⣵⣯⣓⣾⣯⣿⣿⣿⡻⠒⠒⠒⢒⣒⣒⡲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠐⠲⠦⠍⠙⠛⠛⠓⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢰⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⣸⣿⣿⣿⣿⣇⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣆⠀⢸⣿⣿⣿⣿⣿⠀⠀⠻⠿⠿⠿⠿⠿⠇⠀⣿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⡿⠀⠀⢀⣄⣀⣀⣀⣄⠀⠀⢹⣿⣿⣿⣿⡏⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠻⠿⠿⠿⠃⠀⠀⠘⠿⠿⠿⠿⠿⠀⠀⠘⠿⠿⠿⠿⠃⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⢴⣶⣶⣶⠀⠀⠀⣠⣤⣤⣤⣤⣤⡀⠀⠀⣶⣶⣶⣶⠀⠀⠀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣉⡀⠀⠀⣈⣉⣀⣉⡀⠀⢠⣿⣿⣿⣿⣿⣿⣷⠀⠀⣈⣉⣈⣁⡀⠀⢀⣉⣉⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⠀⣸⣿⣿⣿⣿⣷⢀⣾⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣧⠀⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢿⣿⣿⣿⣿⡆⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⠁⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣿⠄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢼⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⢀⣀⡀⠀⠀⠸⡇⠀⠀⠀⠀⣸⠃⠀⠀⠀⢀⣀⣤⣀
⠀⠀⣠⠤⠦⣄⢠⡟⠁⠀⠈⢳⡴⠉⠀⠙⣧⠀⠀⣿⣶⣶⣶⣶⣿⠀⠀⠀⢠⡏⠀⠀⠈⢷⠤⢤⡀⢀⡤⠶⢤⡀
⠀⢸⡇⠀⠀⢸⡿⡇⠀⠀⠀⣸⣇⠀⠀⠀⡿⠀⠀⢸⡷⠶⠶⢾⡇⠀⠀⠀⠸⡇⠀⠀⠀⣾⠀⠀⢹⣿⠀⠀⠀⢹
⠀⠸⣇⣀⣀⣸⠁⢿⣿⣿⣿⡇⢹⣿⣿⣿⡇⠀⠀⣾⠿⠿⠿⠿⣧⠀⠀⠀⠀⣿⣿⣿⣿⣇⣀⣀⡾⢸⣄⣀⣀⣿
⠀⠀⣿⠶⣾⣿⠀⢸⣿⣿⣿⡇⢸⡷⠶⣿⡁⢀⣼⣃⣀⣀⣀⣀⣸⣧⠀⠀⠀⣿⣷⣶⣾⣿⢾⣿⡇⢀⣿⠶⢿⡇
⠀⢰⠟⠛⠛⠻⣄⡟⠀⠀⠀⠹⡾⠛⠛⠛⣷⣾⠟⢿⣿⣿⣿⣿⣿⣿⣷⣄⢰⠏⠀⠀⠀⠹⠛⠛⣇⡼⠛⠛⠛⢷⡀
⢠⡏⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠠⣼⠁⠀⠀⠀⠈⢷
⣾⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢠⣿⠃⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣾⡇⢰⣯⣀⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⢸⡄⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣿⠀⣿⡀⠉⣿⣿⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸
⠈⣧⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣀⣿⣿⣿⡇⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢀⡏
⠀⠈⢧⣀⣀⣠⡼⣆⠀⠀⠀⠀⣠⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣠⠟⠹⣦⣀⣀⣠⠟
⠀⠀⠀⠈⠉⠁⠀⠈⠓⠦⠴⠞⠋⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠳⠦⠴⠚⠁⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠠⠴⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠤

⠀⠀⠀⠀⠀⡀⠄⠤⡀⠀⢀⠄⠀⠐⢒⠒⠤⡀⢠⢄⢀⠒⠂⠐⠀⠐⢦⠀⣀⠀⢀⣤⡶⠶⣀⠄
⠀⠀⠀⠀⢰⠁⠀⠀⢸⡐⠁⠘⢀⠤⠨⡁⡄⠑⠘⠉⠀⠍⢣⠲⠢⢴⡞⠀⠓⠀⣺⡿⡯⠑⣶⡇⡁
⠀⠀⠀⠀⠀⠑⠀⠔⠃⢃⠀⡄⠈⢄⠀⡐⠀⡐⢠⠁⢉⠈⢐⠈⡼⢮⠤⡀⣄⠀⣿⣯⣚⣎⣿⣷⠃
⠀⠀⠀⠀⠀⢀⠄⠀⢠⠈⢆⠀⠠⠀⠀⢚⠐⠁⠸⣀⡒⢠⠏⠀⡇⠀⠀⠸⣸⠀⠙⠛⠓⡿⠟⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⢁⡀⠀⠈⠀⠀⠤⡄⠀⠀⡀⡀⠈⠮⠶⠴⠮⠟⠀⢖⠀⠀⡇⢰⠓⠇
⠀⠀⠀⠀⠀⡡⠴⠖⠒⠦⣤⠘⠀⠀⠀⠀⠀⠀⡗⠒⡌⠣⡀⠡⠀⣂⣆⠀⣤⠠⢠⡄⠀⠇⠈⠈
⠀⠀⠀⠀⠘⠆⠀⠀⠈⠀⠇⠸⡉⠥⢀⡁⠄⣀⡃⡰⢰⠀⠈⠁⣏⣀⣉⣉⣁⢈⣧⣶⠘⠄⢀⢄
⠀⠀⠀⠀⠀⢐⠀⠀⠀⠐⠁⠁⢣⠀⠀⠀⠀⡜⠁⠐⢁⠀⢀⡀⠙⠉⠋⡉⠉⠉⠛⠁⠐⠂⠀⠁
⠀⠀⠀⠀⠀⠈⢆⠀⠤⠠⠀⡀⡀⠉⠀⢀⡅⡆⣄⠀⡴⠤⠤⢤⣭⢀⠁⣦⠑⠂⡍⠑⠀⠀⡤⠤⢄
⠀⠀⠀⠀⠀⢴⡀⢰⠢⠀⠀⠒⠁⠀⠀⢥⠼⢿⣼⠀⠑⠢⠀⠐⠁⡈⡀⣄⠀⡘⠁⡜⠀⠀⣟⣻⠁
⠀⠀⠀⠀⠀⠑⠀⢐⠉⢄⠀⠶⢧⣈⠐⢌⠑⠉⢠⡂⠐⠒⠀⠐⣂⠀⠑⠄⠀⠡⢊⢄⠀⠀⡃⢰
⠀⠀⠀⠀⠀⠀⢱⠀⡁⢠⠠⠈⠢⣉⠛⣦⠆⠀⢀⠀⠀⠉⠉⠁⢸⡒⠢⠀⢠⠊⠉⡷⠃⠀⡇⢨
⠀⠀⠀⠀⠀⠂⠘⠀⠃⠁⢐⠍⠵⠀⡠⡀⠀⠀⠸⠀⠀⠀⠀⠀⣠⣬⡠⠠⡊⠀⢈⠁⠀⠀⠯⠽⠄
⠀⠀⠀⠀⠐⣧⣴⣦⣆⣇⣈⡒⠀⠊⠀⠈⠢⠠⠀⠑⠠⠤⠤⠐⠁⠀⠀⠀⡌⠲⡁⠀⠀⡠⡄⢤
⠀⠀⠀⠀⠀⠇⠀⠀⣀⡃⡆⡅⠀⢁⣈⡀⣨⠤⢄⠀⠈⠚⠋⠟⢋⠂⣰⠁⣋⢉⣋⡂⠀⠈⣿⡊
⠀⠀⠀⠀⠀⢈⠉⠘⠉⠟⠒⠀⢸⠀⠀⠀⡁⢐⣼⡇⠆⡞⡇⠷⢾⠀⠇⢀⠀⠀⡈⡑⢀⠎⠀⠱⡀
⠀⠀⠀⠀⠀⠈⠂⠀⠐⠀⠀⠀⠀⠉⠁⠉⠉⠈⠀⠀⠐⠙⠑⠛⠈⠀⠈⠒⠄⠀⠐⠁⠈⠂⠠⠬⠁

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

⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠹⣿⣿⣿⣿⣿⠏⢻⣿⣿⣿⣿⣿⠋⢻⣿⣿⣿⣿⣿⠋⢿⣿⣿⣿⣿⡿⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⢀⣼⣿⣧⡀⠀⠀⢀⣼⣿⣧⡀⠀⠀⢠⣿⣿⣄⠀⠀⠀⣠⣿⣿⣄⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣇⣠⣤⣈⣿⣿⣿⣃⣤⣤⣸⣿⣿⣿⣀⣤⣄⣹⣿⣿⣿⣠⣤⣄⣻⣿⣿⣏⣠⣤⣀⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠁⠀⣠⠤⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠐⠋⠀⠀⢠⡟⠁⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠴⠶⠦⢤⡀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⣠⠟⢀⡞⠁⠀⠀⠀⠀⠹⣆⠘⢧⡀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣌⡛⠲⠖⠛⣁⣤⢸⡇⠀⠀⠀⠀⠀⠀⣿⢠⣤⣉⠛⠒⢛⣉⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠈⠉⠛⠛⠉⠁⢹⡄⠳⣄⠀⠀⠀⢀⡴⠃⣼⠁⠈⠉⠉⠉⠁⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢠⡞⢋⡴⠦⣌⣙⠛⢛⣉⡤⠶⣌⠛⣦⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⠏⣴⠋⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⢳⡈⢷⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡟⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠘⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⠧⣤⣤⣤⣀⣀⣠⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⣤⣤⣤⣤⠤⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠦⠤⢤⣤⣤⣤⣤⣤⣤⠤⠤⠴⠾


⣶⣶⠶⡶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠋⠉⢙⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣴⢇⠀⠀⠀⠀⠀⠀⢀⣴⡶⠶⡀⠀⠀⠀⠀⠀⠀⢠⠛⢶⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠑⢄⠀⠀⠀⠀⣼⠋⠀⠀⠁⠀⠀⠀⢀⡠⠔⠙⠳⡞
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡆⠀⠀⠀⠀⠱⡀⠀⠀⢃⠀⠀⢀⠇⠀⣀⢔⡵⠊⠀⠀⠀⡇⠀⢀⣴⠿⠿⠶
⠀⠀⠀⣿⡇⠀⠉⠢⡀⠀⠀⠀⠀⠀⢰⢸⡇⠀⠀⠀⠀⠀⠘⡄⠀⢸⠛⠿⠏⢀⢜⣵⠟⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⡰
⠀⠀⠀⠻⣧⠀⠀⠀⣵⣄⠀⠀⠀⠀⢸⣺⡇⠀⠀⠀⠀⠀⠀⠸⢀⠜⠻⠿⣀⢦⣿⠋⠀⠀⠀⠀⠀⠀⣄⠼⣦⣀⡠⠖⠁
⠀⠀⠀⠀⠈⠑⠒⠾⣋⣼⠗⠂⠤⠤⠴⣼⣿⠀⠀⠀⠀⠀⠀⡠⣫⠀⠀⠀⠹⣿⠇⠀⠀⠀⠀⡀⠤⠔⠙⢷⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠀⡴⣽⠃⠀⠀⠀⠀⢻⠀⢀⡤⣲⡵⠚⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡴⡄⠀⠀⠀⠀⠀⠀⠀⠈⠳⡴⣼⡏⠀⠀⠀⠀⠀⠈⣖⣵⡿⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡹⣦⡀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠃⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢝⢿⣦⡀⠀⠀⠀⠀⠀⣿⣿⣤⣤⣤⣄⣀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⡻⣷⣤⣀⡴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠾⡀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢩⡟⣼⣿⣿⣿⣿⣿⣿⣿⣏⢹⣶⣿⣆⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢿⡖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣉⡛⠻⠿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣰⣆⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⢀⣿⣿⣦⣼⣿⣿⣦⣼⣿⣿⠀⣀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⢸⣿⣿⡿⠟⣛⣫⣭⣭⣭⣭⣝⣛⠻⢿⣿⣿⡇⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⣋⣵⣾⡟⢿⣀⣿⣈⣣⣍⣏⡿⣻⡷⣮⣙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣶⣾⡿⣫⣾⢫⣳⣧⠾⠛⠋⠉⠉⠉⠉⠙⠓⠿⣴⣟⡝⣷⣝⢿⣷⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠏⣼⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠻⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡏⣼⣿⣿⡿⠁⣼⡇⢀⡿⢿⡄⣾⢿⡆⢸⢻⡆⣸⠃⠈⢿⣿⣿⣧⢻⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⢰⣿⣿⣿⠁⠀⢸⡇⢸⡇⢸⡇⣿⠘⡇⢸⣸⣇⡏⣴⣄⠈⣿⣿⣿⡌⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⡿⢸⡏⢹⡏⠀⠀⢸⡇⢸⡇⢸⡇⣿⢠⡇⠈⠋⣸⢱⡇⣿⠀⢻⡋⢹⡇⣿⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⢸⣿⣿⣧⠀⠀⠿⠿⠆⠿⠿⠁⠻⡿⠃⠀⠀⠏⠈⠷⠟⠀⣼⣿⣿⡇⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⡘⣿⣿⣿⡄⠠⡆⣰⠰⢢⢴⢰⣂⣆⡔⠒⠆⡴⢢⣤⡄⢠⣿⣿⣿⢣⣿⣿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⠹⣿⣿⣿⣔⠛⠉⠀⠈⠒⠉⠈⠉⠑⠂⠁⠙⠊⠈⣡⣿⣿⣿⢏⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⢿⣷⡹⣿⣿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⣿⢋⣾⡿⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣮⡻⣮⡾⢫⣿⢶⣦⣤⣤⣤⣤⣴⡶⢟⣿⣳⣵⢟⣵⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⣿⣿⣦⣝⡻⠷⣿⣼⣬⣻⣁⣿⣁⣿⠾⢟⣫⣴⣿⣿⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠟⠛⣿⣿⣷⣶⣬⣭⣭⣭⣭⣵⣶⣾⣿⣿⠛⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠋⠘⣿⣿⠟⠻⣿⣿⠃⠙⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠘⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠛⠛⠛⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠁⠀⠀⠀⠈⠻⣦⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣰⠟⠉⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢀⣀⣠⣄⣀
⠀⠀⠀⠀⣰⡟⠉⠉⠈⠉⠻⣦⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣴⠟⠉⠁⠈⠉⠻⣆
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣹⡆⣿⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣼⠇⢰⡏⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⣿⠀⢻⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡷⢦⣤⣤⣤⣤⣴⠶⣿⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡏⠀⢸⡷⠶⢤⣤⣶⠶⢾⡇⠀⠀⣼⡷⣤⣤⣤⣄⣠⣤⡴⣿⠀⠀⠀⠀⣿⠶⢶⣤⣴⠶⠾⣿⠀⠀⢹⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⣿⢦⣤⣤⣴⣾⡇⠀⣸⡟⠶⠶⠶⠶⠾⢻⡇⠀⢀⣿⠀⠀⠀⠉⠁⠀⠀⠀⢸⡇⠀⠀⢰⡿⠷⠶⠶⠶⠶⠿⣿⠀⠀⢸⣷⣦⣤⣤⡴⣿⠃
⠀⠀⠀⠀⢰⡟⠻⠶⠶⠟⢻⡇⢠⡿⠀⠀⠀⠀⠀⠀⠈⣿⡀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⣼⠃⠀⠀⠀⠀⠀⠀⢻⡆⠀⢸⡟⠻⠶⠶⠟⢻⡆
⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⢻⣾⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡟⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡅⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⢀⣾⠃⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⣾⠃⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠘⣷
⢸⡇⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢿⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠
⢸⣇⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸
⠀⢻⡀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠈⢿⡄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠈⠻⣦⡀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠈⠛⠶⣦⣴⠶⠿⣦⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⣰⡟⠻⠶⣦⣶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⣀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣤⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢶⣤⣤⣤⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⢀⣠⡴⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣿⣶⣶⣿⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠋⢉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡠⠒⠉⠀⠈⠑⢄⡀⠀⠀⠀⠀⠀⠀⡖⠀⠠⢄⡀⢀⠀⠂⠙⡏⠀⠀⠀⠠⠫⠋⠀⠀⠀⠀⠀⢠⠛⠒⠀⠉⠁⠀⠛⠋⠇
⠀⢰⡥⠒⠂⠠⠤⣀⣀⠙⢆⠀⠀⠀⠀⠀⠈⠒⠤⣂⢀⣀⣀⡠⠀⠀⠀⠀⢠⢁⠄⡌⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⢀⢰
⠀⢸⢠⠀⣁⠀⠘⠁⠀⠢⡘⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡹⡅⠀⠀⠀⠀⠀⢁⠈⡰⠀⠀⠐⠁⠀⠐⠀⠂⡈⢀⠡⡀⠀⢠⡘
⠀⠈⠒⠠⠁⠅⠀⠉⠃⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠑⡅⡔⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠆⡂⠐⠀⠀⠁⠠⠙⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢀⣽
⠀⠀⠀⠀⣀⠞⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⡀⠀⠀⠀⠟⠀⠀⠀⠠⠬⠅⠊
⠀⠀⠀⢈⡒⠲⠿⣐⠀⠄⡀⠀⠀⠀⠀⠀⢰⠃⠀⢠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠐⢖⡈⣠⠀⠐⡌⠑⠠⣄⡱⡄⠀⠈⠀⠉⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠈
⠀⠀⠀⠈⢣⠑⢄⠈⠳⡀⠈⡢⠃⠀⠀⠀⠀⡀⠠⠤⠴⠀⠒⠂⠀⠈⠉⢉⡉⠀⠐⠒⢀⠀⠀⠀⠀⠀⠔⠈⠠⡈⠂
⠀⠀⠀⠈⡀⠱⡄⠑⡀⠈⠉⠀⠀⠀⠀⠎⠀⠀⣴⠂⠀⡴⢠⡖⣰⡆⠀⡜⣧⠀⠀⠀⢈⠀⠀⠀⠀⠀⡠⠄⠂⠐⠉⢄
⠀⠀⠀⠀⠹⡀⠸⢀⡠⠀⠀⠀⡀⠀⠀⠀⠀⠀⢿⣼⠟⠓⠈⠁⠈⠁⢀⡉⠉⠀⠀⠀⣨⠀⠀⠀⠐⠀⠀⢀⠀⠂⠀⠘⠮⠨⠃
⠀⠀⠀⠀⠀⠙⠀⠉⠀⠀⠀⠈⠁⠀⠀⡀⠀⡀⠀⡀⢰⡄⡤⣀⢠⡖⣺⣧⡆⢌⡶⢀⠇⠀⠀⡄⠀⠀⠀⢬⠀⠀⠀⠁
⠀⠀⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢡⠀⣹⠞⠛⢾⠜⠓⠿⠈⠉⠈⠋⠉⠀⡠⠊⠀⠀⠀⠀⠁⠁⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢔⠋⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⣀⣄
⠀⠀⠀⠀⠘⠀⠀⠀⢀⢄⠄⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂⠉⠁⠀⠀⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠌⡀⠀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⢤⡆⢄⣀⢄⢤⡎⡄
⠀⠀⡐⠁⠊⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⡀⠀⠡⠀⢐⠀⠀⠀⠀⠀⠠⠌⠏⠀⡄⠀⠊⠀⠈⠁⠀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⠖⠂⠀⠂⠀⠀⠀⠀⢈⡠⠁⠀⠐⠋⠁⠠⠀⠀⠀⠀⠀⠀⠪⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡠⠒⢒⠒⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢄
⠀⢘⢀⠎⣀⢨⠏⢲⠨⡈⢆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡌⢀⠂⠙⢓⡄⠀⡃⠢
⠀⠈⢌⠲⡄⠠⣖⠫⣀⠆⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠕⠀⠀⠀⠀⠐⡠⠈⠀⠀⠈⠁⠀⠀⠀⡆⢀⣤⡴⠟⠡⣠⠗⠀⡅
⠀⠀⠈⠣⢍⠢⠬⢓⡣⠞⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⢔⠊⢉⡠⠎
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡠⠔⠂⠉⠀⠀⠀⠙⠛⠶⣄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠠⠄⠤⠀⠀⠀⠀⡀⠀⠀⠀⠈⢾⡄
⠀⠀⠀⠀⠀⠀⠉⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢸⣱⠀⠀⠁⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⢠⠀⠀⠀⠀⠢⠀⠂⠀⠀⠀⠀⠀⠀⣸⢿⠀⠀⢀⠔⠁⢀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⢠⡟⠄⠀⠄⡌⠀⠀⠡⠂⠁⡂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠑⠤⢀⡀⠀⣀⠠⠔⠁⠀⣠⢯⠎⠀⠀⠀⡇⠀⠀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟⠁⠀⠀⠀⠀⢁⠀⢀⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠛⢳⡶⠶⠶⠶⠟⠃⠁⠀⠀⠀⠀⠀⠀⠀⢆⠀⠨⠀⠄⡂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠈⠀⠀⠀⠁⠀⢀⠉⠐⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠐⠈⠀⡀⡴⠐⠨⡐⠜⡈⠑⠡⢐⢈⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠧⠗⠒⢄⠀⠀⠀⠀⠀⠀⠠⠁⠀⠂⡁⠀⠁⠀⡠⢑⠀⠀⠌⡚⠉⠐⠲⢥⠊⡠⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡊⠒⠶⠀⣀⠙⠢⡀⠀⠀⢠⠁⡴⠄⡀⢀⠠⢀⠅⠂⠄⡈⢔⠤⠅⠀⠀⠀⠀⠑⠨⠔⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⡏⠑⡌⠳⢀⠑⡄⠠⠃⠠⠐⢸⠃⠠⠂⠔⠀⢔⠬⢂⠀⢴⡄⠀⠀⠀⠠⢁⠶⠑⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠐⢆⠳⢕⠈⡋⢀⠀⠜⠀⠀⡊⢄⠀⡂⠄⡀⠀⡀⠔⡀⠀⢀⠴⣁⠌⢂⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡉⠚⡡⠊⠀⠀⠀⠊⢄⠨⢐⠁⡀⢅⡠⢃⠐⠤⠗⢊⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢽⠊⠛⠉⠛⢙⡋⠛⠋⡵⢟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠂⢽⡫⠒⠛⠚⠍⠚⠓⠚⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠅⠊⠂⡙⠂⠉⠊⠙⠝⠫⢓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠑⠄⠈⠀⠃⢠⠀⠰⢀⠀⠂⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢀⠠⠀⠸⠀⠈⠀⢀⠄⡈⠔⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠁⠊⠀⡀⠀⠀⠄⠀⡨⢐⠁⠂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡇⠀⠀⠀⠀⢀⠄⡠⢀⠄⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠈⠀⠀⠃⠀⠀⠰⠀⠀⡨⢂⠀⠂⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠠⠀⠀⣰⠊⠀⠀⠀⠀⠒⢠⠄⡠⢀⠄⡀⠀⠀⠁⠌⠐⠀⠉⠖⡠⢀⠄⡀⣀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠂⠁⠊⠙⠒⠖⠲⠐⠖⠲⠋⠑⠈⠔⠈⠝⠲⠖⠶⠒⠖⠉⠕⠡⠊⠁⠊

⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣠⢤⠀⢠⠠⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡄⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⡃⢣⠰⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣄⡠⠂⡀⠀⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺
⠃⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠾⠀⠶⠀⠀⠀⢠⠆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠦⡄⠀⠀⠸⠀⠰⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⢇⠀⠀⠀⠇⠀⡆⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠸⠀⠠⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠀⢰⠀⠀⠀⢠
⠂⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠃⠀⠆⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⡄⠀⡄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⠀⠰⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢠⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠇⠀⡆⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⡄⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠸⠀⢰⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢁⠀⡀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠃⠀⠆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⢁⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠰⠀⠰⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠘⡀
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠃⠀⠀⠀⠰
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠁⢰⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠸⠀⠀⠀⠀⠄
⠀⠀⠀⢀⠁⠀⠀⠀⠘⠀⠆⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠇⠀⠀⠀⠰
⠀⠀⠀⡀⠀⠀⠀⢀⠃⠰⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠸⠀⢰⠀⠀⠀⠀⡆
⠀⠀⠀⠄⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠆⠀⡆⠀⠀⠀⠰
⠀⠀⢠⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⡀⠀⠀⠀⠀⣀⠀⠸⠀⠀⠀⢰⠀⢠
⠀⠀⠆⠀⠀⠀⠘⠀⠸⠀⠀⠀⠈⣿⠉⠉⠓⠲⠶⠤⣄⣀⣀⣀⣀⡤⢀⣀⡁⠀⠀⠑⠀⠀⠀⠰⠛⠆⠀⡇⠀⠀⠀⡄⠈⡀
⠀⠰⠀⠀⠀⠀⠃⠀⠆⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠤⡨⢁⠀⠀⠀⠀⠀⠀⠀⠓⢧⡀⠀⢀⡠⠀⠀⠀⢰⠀⠀⠀⢠⠀⠁
⠀⠇⠀⠀⠀⠘⠀⠸⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠷⠋⠀⠀⠀⠀⠘⡆⠀⠀⠈⡄⠘
⡴⠀⠀⠀⠀⠇⠀⠆⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⡤⠞⠷⡄⠀⠀⠀⠀⠀⠀⠙⠃⠀⠘⠁⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢁⠀⠁
⠇⠀⠀⠀⠘⠀⠸⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠘⠃⢂⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘
⠀⠀⠀⠀⠃⠀⠆⠀⠀⠀⢠⠁⠀⢀⣄⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠇⠐
⠀⠀⠀⠐⠀⠸⠀⠀⠀⠀⡜⢀⣀⠇⣸⣇⠀⠀⠊⠉⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠘⠀⠇
⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⠣⠉⣀⠄⢀⣀⠛⢂⡜⠀⠀⢀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠰⠀⠀⠀⠀⣸⠀⠀⠁⠀⠛⠉⠀⠀⠀⣠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡆
⠀⠀⠘⠀⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠐⠀⠰
⠀⠀⠂⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠄
⠀⠘⠀⠒⠀⠀⠀⠀⠛⠐⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⣆⠴⢶⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠦⣄⣠⢤⠄⣠⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⢀⠼⠀⠈⠀⣼⠀⠉⠁⠀⢸⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡀⠀⠀⠀⠀⠛⠒⣆⢀⡤⠬⠷⠀⠀⠀⠸⢶⣇⡔⠀⢠⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⢠⣼⠿⣆⠀⠈⣆⠀⠀⠘⠏⠀⡀⣠⣤⠀⢠⠟⢻⠈⠀⠙⡅⣀⡎
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠙⠿⣥⡞⠉⠙⠲⠶⢶⣦⣤⡀⠀⡇⠈⠉⣰⡏⠀⠀⡆⠀⠀⠉⠈⢁⣀⣀
⠀⠀⠀⠀⠀⠰⡞⠁⢓⠖⠀⠈⢳⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⡜⢸⠀⠀⠀⣿⠀⠔⣡⠞⠉⠀⠈⢳⡀
⠀⠀⠀⠀⠀⣀⠾⠚⠺⠆⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⢸⡀⠀⣰⠇⠀⡴⠃⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢸⡿⠏⠀⠒⠠⠤⣀⡀⢻⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠈⠓⠚⠋⠀⡼⠁⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢠⣧⠄⠹⢷⣦⣤⣀⢀⣀⣠⣾⢿⢿⢐⡋⠉⢀⢾⡷⣄⡀⠀⣠⠞⠁⢂
⠀⠀⠀⠀⠻⣄⣨⡷⢢⡤⠤⠤⢭⣤⣀⡀⠉⠛⠻⠿⣿⣿⣗⣯⠟⢀⣡⢶⡿⣄⠘⠻⣿⠟⠁⠀⢶⠉⣳⠆
⠀⠀⠀⠀⠀⠀⢉⣙⠻⢤⡀⠀⠀⠀⠈⠙⠳⣄⠂⠀⡄⠠⣀⡴⠞⠋⠀⠀⠙⢿⣿⡾⠑⠒⠠⠤⠉⠉⠓
⠀⠀⠀⠀⠀⣨⠁⠀⣀⠀⠙⢶⣤⣀⣀⠀⢀⣸⣷⡸⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠉⠀⠈⠛⠃⢀⣄⠈⠙⠛⠛⠛⠛⡱⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠢⢀⣀⠸⠷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠠⠂⠘⠦⣤⠀⠀⠀⠠⢱⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⢀⡜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢀⣀⢸⠁⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⡀⠀⢺⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠉⣡⣤⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠙⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⢿⣧⣀⣀⣀⣤⡶⠟⡋⠀⠠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡠⢲⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⣤⡀
⠀⠀⠀⠀⠀⠞⡄⠫⠐⠃⠀⠀⠀⠀⠀⣠⠾⠉⢁⣤⣌⢹⡷⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣤⣶⣶
⠀⠀⠀⢀⣦⣧⣵⡠⢲⠀⠀⠀⠀⠀⠀⢇⣀⣚⣉⣀⣉⣀⡻⠆⠀⠀⠀⠀⠀⢀⣴⣿⡺⡿⣽⡏
⠀⠀⠀⢸⣿⠛⠿⣿⣿⠂⠀⠀⠀⠀⠀⣿⠉⣉⣉⣉⡉⠉⡿⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⢿⣿⠓
⠀⠀⣰⣿⣿⣿⣶⣿⠁⠀⠀⠀⠀⠀⠀⢸⡆⠛⠛⠛⠋⢸⡇⠀⠀⠀⠀⠀⠀⢸⣯⣉⣙⣻⡏
⠀⠀⠉⠉⠻⠿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣤⣤⣤⡾⠁⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀
⠀⠀⠀⣠⠴⢶⠶⢤⡀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠘⢀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣒⣨⣵⣶⡄
⠀⠀⢰⡟⠀⡤⢀⣩⣷⠀⠀⠀⠀⠀⠘⠦⢤⣀⠀⠀⠀⣀⢼⠃⠀⠀⠀⠀⣠⣮⣵⡾⠿⠛⠉⠉⢿⠁
⠀⠀⠘⢿⡂⢴⡋⠦⠤⠀⠀⠀⠀⠀⠀⣘⣿⣮⣕⠶⣪⣷⣿⠇⠀⠀⠀⠘⢿⣿⠁⠀⠀⠀⣀⣠⠞
⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣟⡽⠋⠁⠀⠀⠀⠀⠀⠀⠘⠲⠴⠚⠋⠁

⠀⠀⠀⠰⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠶⣶⣬⡙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⡶⣄⣿⢸⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⣀⣼⣿⣿⡦⠙⠁⠀⠀⠀⠀⠀⠰⡚⡇⠇⠻⢸⡇⡏⠀⠀⠀⠀⠀⠀⢀⡶⠶⠾⠶⠶⡆
⠀⠀⢸⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣶⡆⠀⠀⠀⠀⠀⠀⠘⡆⢀⣤⡄⢀⡇
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠏⠙⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⣿⡿⢿⠃
⠀⠀⢘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠉⠁⢸
⣀⣀⣘⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿