Feeble Emojis & Symbols

⠀⡠⠒⠊⠉⠉⠙⠲⣄⠀⠀⠀⡠⠔⠂⠀⠀⠒⠢⡀⠀⠀⠀⠀⡠⠔⠒⠒⠒⢤⡀⠀⠀⠀⢀⠔⠊⠉⠉⠙⠢⣄
⡜⠀⢀⠀⠀⠀⣀⠀⠈⡆⠀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢀⠎⠀⠀⠀⠀⢀⡀⠙⡄⠀⢠⠃⠀⠀⠀⠀⠀⠀⠈⢧
⡇⢀⠉⠀⠀⠀⠉⠀⠀⡇⢸⠀⠀⠀⠻⠀⠋⠀⣠⠀⢸⠀⠘⠀⠙⠁⠀⠀⠈⣁⡀⢸⠀⢸⠀⢤⢀⣀⠠⡦⢸⡄⢸
⢡⠀⠢⢄⣀⣀⠴⠃⢠⠃⠈⡄⠘⠤⠄⠤⠠⠆⠚⠀⡌⠀⠸⡀⠻⣯⡯⣯⢽⡽⠃⡸⠀⠘⣄⠘⢦⣁⣀⣠⠞⠀⡼
⠀⠑⠤⢄⣀⣀⠠⠔⠁⠀⠀⠈⠢⢄⡀⠀⠀⣀⡠⠊⠀⠀⠀⠙⠦⣈⣉⠉⣉⡠⠔⠁⠀⠀⠈⠒⠤⣀⣀⡠⠴⠊
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡠⠚⠉⠉⠉⠉⠓⢄⠀⠀⠀⣠⠔⠊⠉⠉⠙⠲⣄⠀⠀⠀⢀⠔⠂⠉⠉⠉⠲⢄⠀⠀⠀⢀⠴⠊⠉⠉⠉⠑⢦
⡘⠁⠀⢀⠀⢠⠀⠀⠈⢇⠀⡰⠁⢀⠀⠀⠀⣀⠀⠈⣇⠀⢰⠁⠀⠠⠀⠠⠀⠀⠈⢧⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⢣
⡇⡄⠀⠀⠀⠀⠀⠀⢸⢸⠀⡇⠀⠈⠁⢀⣀⣀⣀⠀⣸⠀⠆⠀⠀⢀⣀⠤⠴⣲⠀⢸⠀⢸⠀⠰⠆⠀⠀⠠⡶⠀⢸
⠣⡑⢄⠀⠀⠀⠀⡠⢣⠇⠀⢣⠈⢯⡙⠚⢒⡲⠃⢠⠇⠀⠸⡄⠐⠡⠃⠀⠀⠀⠁⡜⠀⠘⣄⠀⠀⠋⠙⠀⠀⢠⠎
⠀⠈⠢⠬⠭⠭⠭⠒⠁⠀⠀⠀⠑⠦⢌⣉⣡⠤⠖⠁⠀⠀⠀⠈⠢⠤⣀⣀⡠⠴⠊⠀⠀⠀⠈⠓⠦⠤⠤⠤⠚⠁
⠀⠀⢀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄⡀⠀⠀⠀⠀⠀⢀⣀⡤⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀
⢀⡔⠉⠀⠀⠀⠀⠉⠢⡀⠀⠀⣠⠊⠁⠀⠀⠀⠉⢢⡀⠀⢀⡔⠉⠀⠀⠀⠀⠙⢦⠀⠀⠀⡠⠊⠁⠀⠀⠀⠈⠙⢦
⡎⠀⢰⣄⠀⢰⡆⠀⠀⢱⠀⢰⠃⠀⣀⡀⠆⣀⡀⠀⢣⠀⡘⠀⠀⠀⠀⠀⠂⠀⠈⡇⠀⢰⠁
⡇⠀⠀⠀⠀⢀⡠⡤⠀⢸⠀⢸⠀⠈⠀⢀⣀⠀⠀⠀⡼⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠠⣤⣤⠤⠔⠒⠂
⠘⢄⡀⠀⠀⠀⠉⠁⡠⠃⠀⠈⢦⡀⠀⠈⠋⠀⢀⡴⠃⠀⠈⢢⡀⠀⠖⠉⠂⢀⡴⠁⠀⠈⠣⣀⠘⠜⠀⠀⠀⢀⡴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠁⠀⠀⠀⠀⠉⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⢻⠀⢠⣶⡄⠀⠀⢀⣤⡀⠘⢿⢿⣿⣷⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⣿⣿⠃⠀⢀⣾⣿⠇⠀⠈⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠉⠉⠀⠀⠘⠿⠏⠀⢠⣾⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠐⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠹⣿⣦⡀⠀⢀⣠⣴⣿⡿⠋⠀⢉⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡀⠘⢿⣿⣿⣿⡿⠟⠋⠁⠀⣠⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⣈⣉⠁⣀⣀⣀⣤⣾⣿⡿⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⡠⠒⠂⠐⠢⡀⠀⠀⠀⡠⠒⠀⠐⠢⡀⠀⠀⠀⡠⠒⠀⠒⠦⡀⠀⠀⠀⡠⠒⠀⠒⠢⡀⠀⠀⠀⡠⠒⠀⠐⠢⡀
⠀⠀⢰⠀⠶⠀⠰⠆⠰⠀⠀⢰⠁⠶⠀⠰⠆⢱⠀⠀⢰⠀⠖⠀⠰⠆⢰⠀⠀⢰⠀⠖⠀⠰⠆⢱⠀⠀⢸⠀⠞⠀⠸⠆⢱
⠀⠀⠘⡄⠠⢀⠠⠀⡜⠀⠀⠘⣄⠄⠒⠢⢄⡜⠀⠀⠘⡄⠀⡀⠴⠂⡜⠀⠀⠘⡄⢆⡀⣀⠆⡜⠀⠀⠘⡄⠔⠀⠐⢄⡜
⠀⠀⠀⠈⠑⠒⠒⠉⠀⠀⠀⠀⠈⠑⠒⠒⠉⠀⠀⠀⠀⠈⠑⠒⠒⠉⠀⠀⠀⠀⠈⠑⠂⠒⠉⠀⠀⠀⠀⠈⠑⠒⠒⠉
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⢀⠎⣁⠀⢀⡈⢆⠀⠀⢀⠊⣄⠀⢀⠌⢆⠀⠀⢀⠊⣀⠀⢀⡈⢆⠀⠀⢀⠊⡀⠀⠀⡉⢆⠀⠀⢀⢊⣀⠀⢀⡉⢆
⠀⠀⢸⠀⠉⠀⠈⠁⢨⠀⠀⢸⠀⠈⣀⡈⠸⢺⠀⠀⢸⠀⠉⠀⠀⡀⢸⠀⠀⢨⠀⢉⠀⣈⠁⢸⠀⠀⢸⠀⣉⣀⣈⡁⢸
⠀⠀⠀⠣⢄⣀⣀⠤⠃⠀⠀⠀⠣⢌⣀⣈⠤⠃⠀⠀⠀⠣⢔⣒⣊⠤⠃⠀⠀⠀⠣⢌⣒⣊⠤⠃⠀⠀⠀⠢⢕⣒⣒⠥⠃

⠀⠀⠀⡠⠔⠒⠒⠤⡀⠀⠀⠀⡠⠔⠒⠒⠤⡀⠀⠀⠀⡠⠔⠒⠒⠤⡀⠀⠀⠀⡠⠔⠒⠒⠤⡀⠀⠀⠀⡠⠔⠒⠒⠤⡀
⠀⠀⢰⠁⠶⠀⠰⠆⠱⠀⠀⢰⠁⠶⠀⠰⠆⢱⠀⠀⢰⡁⠒⠀⠐⠂⣱⠀⠀⢰⠁⡦⠀⠠⠆⢱⠀⠀⢰⠈⠰⠀⠰⠀⢱
⠀⠀⠘⡄⠀⠰⠀⠀⡜⠀⠀⠘⡀⠤⠤⠤⠤⡘⠀⠀⠸⡆⢶⡒⣲⠆⡞⠀⠀⠘⡄⠤⠤⠤⠄⡜⠀⠀⠘⡀⢔⠉⢑⠀⡜
⠀⠀⠀⠈⠐⠒⠒⠈⠀⠀⠀⠀⠈⠐⠒⠒⠈⠀⠀⠀⠀⠈⠐⠒⠒⠈⠀⠀⠀⠀⠈⠐⠒⠒⠈⠀⠀⠀⠀⠈⠐⠛⠓⠉
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⢀⠎⣁⠀⢀⡉⢆⠀⠀⢀⠎⢁⠀⢀⠉⢆⠀⠀⢀⠎⣁⠀⢀⡉⢆⠀⠀⢀⠎⡁⠀⢀⡙⢆⠀⠀⢀⠎⣡⠀⢠⡉⢆
⠀⠀⢸⠀⠉⠀⠈⠁⢨⠀⠀⢨⣜⠄⠀⠀⣔⣼⠀⠀⢸⠀⠀⣤⡄⡤⢸⠀⠀⢸⠀⣉⠀⢈⡁⢸⠀⠀⢸⠀⣉⢀⢈⡁⢸
⠀⠀⠀⠢⣀⠂⢀⡠⠂⠀⠀⠀⠢⣀⠁⢁⡠⠃⠀⠀⠀⠢⣀⠀⢈⡠⠂⠀⠀⠈⠢⣀⠛⢃⡠⠃⠀⠀⠈⠢⣈⠒⢊⡠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⣄⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣋⣿⣟⣻⣷⠀⠀⠀⠀⠀⣾⣟⣿⣿⣻⣷⠀⠀⠀⠀⠀⣾⣟⣻⣿⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⡟⠛⠛⢻⣿⠀⠀⠀⠀⠀⣿⣿⠛⠛⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠿⠛⠁⠀⠀⠀⠀⠀⠈⠻⠷⠾⠟⠁⠀⠀⠀⠀⠀⠈⠛⠷⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⣾⡿⣿⣿⢿⣷⠀⠀⠀⠀⠀⣴⣿⣿⣿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⣿⣿⣿⡿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠙⢿⣿⣷⡿⠋⠀⠀⠀⠀⠀⠙⢧⣶⣾⣿⠟⠁


⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣷⣾⣿⠀⠀⠀⠀⢰⣿⣷⣿⣿⣾⣿⡆⠀⠀⠀⠀⣿⣷⣾⣿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣯⣿⣿⠟⠀⠀⠀⠀⠀⢿⣿⣿⣽⣿⡿⠁⠀⠀⠀⠀⠻⣿⣤⣦⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣭⣿⣯⣽⣿⠀⠀⠀⠀⢠⣿⡉⢉⡉⢉⣿⡄⠀⠀⠀⠀⣿⣯⣽⣿⣭⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⣿⡿⠀⠀⠀⠀⠘⣿⣿⡿⢿⣿⣿⠃⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢀⣤⠴⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⢦⣄⡀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣠⠞⠋⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠋⣿⣿⣦⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣏⠀⠀⠿⣿⣿⡆⠀⠀⠀⠀⢀⡀⠀⠀⠀⣿⠀⠀⠿⣿⣿⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⣿⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⣴⣿⣿⡇⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣯⣀⣀⣿⣿⣿⣇⠀⠀⣴⠟⠋⠀⠀⠀⢀⣿⣀⣀⣿⣿⣿⣇⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠻⣦⣤⣤⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢸⣿⡿⠟⠛⠛⠛⠛⠿⢿⣿⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⣀⣀⣀⢀⣀⣀⣀⣀⢀⡀⢀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠙⢿⣿⣿⣿⣿⡿⠋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣉⣉⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⡠⠚⠉⠉⠉⠲⢄⠀⠀⠀⠀⢀⠔⠊⠉⠉⠑⠢⡀⠀⠀⠀⠀⢠⠔⠋⠉⠉⠑⢦⡀⠀⠀⠀⠀⡠⠖⠉⠉⠉⠓⢤
⡞⠀⡠⣄⠀⡠⣄⠈⢇⠀⠀⢀⠏⠀⣤⠀⠀⣤⠀⠘⡄⠀⠀⢰⠃⢀⣤⠀⢀⣄⠀⢹⠀⠀⠀⡸⠁⢠⡄⠀⢠⡄
⡇⠀⢀⡀⠀⢀⠀⠀⡼⠀⠀⠸⡀⠀⣈⠀⠀⣀⠀⢀⡇⠀⠀⢸⠀⠀⣁⡤⣄⡁⠀⢠⠃⠀⠀⢇⠀⢠⠤⠤⠤⡄
⠘⢆⡀⠉⠉⠁⢀⡴⠁⠀⠀⠀⠳⣄⠀⠉⠉⠁⣠⠞⠀⠀⠀⠀⠣⣀⠁⠀⠀⠉⣠⠎⠀⠀⠀⠈⢦⡀⠓⠤⠒⢁⡰
⠀⠀⠉⠑⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠊⠉
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⢀⡔⠋⠁⠀⠉⠑⢦⠀⠀⠀⠀⢠⠖⠉⠁⠈⠉⠳⣄⠀⠀⠀⠀⡠⠚⠉⠀⠈⡉⠢⡀⠀⠀⠀⠀⡴⠊⠉⠀⠈⠙⢢
⡎⠀⢠⡄⠀⢠⡄⠀⢧⠀⠀⢠⠃⠠⣀⠄⠠⣀⠄⠘⡆⠀⠀⢰⠁⠀⠘⠀⠐⠃⠀⠸⡀⠀⠀⡼⠀⢒⡤⠀⢠⡒
⣇⠀⠠⠤⠤⠤⠄⠀⡜⠀⠀⠸⡄⠀⣀⠤⠤⣄⠀⢀⠇⠀⠀⢸⡀⣏⠉⠉⠉⠉⡝⢰⠁⠀⠀⢣⠀⢀⡠⠤⢤⡈
⠈⠦⣀⠀⠀⠀⣀⠜⠁⠀⠀⠀⠑⢄⡀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠳⣌⡓⠲⠒⢊⡤⠋⠀⠀⠀⠈⠣⣀⠀⠀⠀⣀⠴
⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⢀⠖⢉⠤⠀⠤⠉⢢⡀⠀⠀⠀⡠⠊⠁⠀⠀⠈⠑⢄⠀⠀⠀⠀⡴⠋⠁⠀⠀⠈⠓⣄⠀⠀⠀⢀⡔⠉⠀⠀⠀⠉⠲
⡏⠀⠰⠆⠀⢰⠆⠀⢣⠀⠀⢰⠃⢀⡲⠄⠠⢖⡁⠈⡆⠀⠀⢸⠁⢘⢶⡁⢘⢶⡀⠸⡀⠀⠀⡜⠀⢘⡦⠀⠠⡦
⢇⠀⢀⠴⠒⠬⡀⠀⡞⠀⠀⠘⡄⠀⠤⣀⣀⠤⠀⢠⠇⠀⠀⠸⡀⠀⠤⠤⠤⠤⠀⢰⠁⠀⠀⢳⠀⠀⢄⣀⡠⠄
⠈⠣⣄⡀⠀⣀⡠⠞⠀⠀⠀⠀⠘⠦⣀⡀⢀⣀⠴⠋⠀⠀⠀⠀⠑⢤⣀⠀⢀⣀⠔⠁⠀⠀⠀⠀⠳⢄⣀⠀⢀⣠⠞
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⣀⡠⠤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢄⣀
⢠⠎⢁⡤⠀⢄⠈⠳⡀⠀⠀⠀⡴⠉⠠⠄⠠⠄⠈⢢⠀⠀⠀⠀⡔⠉⣄⠀⢀⡄⠙⢆⠀⠀⠀⢀⠞⠁⡀⠀⢀⡈⠱
⡇⠀⢤⠴⠀⢦⠴⠀⢳⠀⠀⢰⠁⠀⡴⠋⠙⢢⠀⠀⡇⠀⠀⢸⠀⠠⡦⠀⠠⡦⠀⠘⡄⠀⠀⡞⠀⠰⠏⠀⠹⠆
⢧⠀⠐⠤⡠⠴⠂⢀⡎⠀⠀⠈⣆⠀⢣⣒⣒⡾⠀⣠⠃⠀⠀⠸⡄⠀⠴⠒⠒⠄⠀⡸⠀⠀⠀⢱⡀⠐⠒⠒⠒⠂
⠀⠓⠤⣀⣀⣀⠴⠊⠀⠀⠀⠀⠈⠳⠤⣈⣁⡤⠖⠁⠀⠀⠀⠀⠘⠢⣄⣀⣀⡠⠞⠁⠀⠀⠀⠀⠑⠦⣀⣀⣀⠤⠊

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡴⠒⠀⣠⠄⠀⠀⠀⠀⢀⣤⠶⠒⠒⠢⠤⡀
⠀⠀⠀⡞⠀⣴⡄⢴⣦⠐⡄⠀⠀⠀⠀⣠⠤⠤⠤⢄⡀⠀⠀⠈⣯⠤⠜⠁⢩⠠⠐⣏⠀⠀⢠⡟⠁⣴⡆⠀⢰⡆⠈⠢⡀
⠀⠀⠀⢣⠀⠉⠰⠀⠀⡠⠃⠀⠀⡰⠋⢠⡄⠀⣤⠀⠈⠆⠀⠀⢃⠠⠂⠀⠀⣰⠀⡿⠀⠀⡿⠀⠀⣘⣃⣀⣘⣃⡀⠀⢻
⠀⠀⠀⠀⠉⢀⣀⡼⠉⠀⠀⠀⠠⠀⡀⠈⠁⠀⠉⢀⠀⠸⠀⠀⠈⠳⠄⢲⠾⠅⠚⠁⠀⠀⢧⠀⢸⠀⠀⠀⠀⢠⠏⠀⣼
⠀⠀⢰⠽⣆⣉⣉⣡⣤⣦⠀⠀⠀⢆⠙⠦⠀⠀⠠⠋⠀⠂⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠈⠳⣄⠑⠤⠤⠴⣋⡠⠾⠃
⠀⠀⠘⠴⠒⠛⠀⠉⠱⠞⠀⠀⠀⠀⠑⠦⠤⠤⠴⠊⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠉⠳⣦⠀⠀⠉⠙⠛⠉⠁
⠀⠀⠀⣀⠴⠒⠒⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⠟⣿⣿⣷⣾⣧
⠀⠀⡴⠁⣀⣀⠀⣀⣀⣀⢳⡀⠀⠀⠀⠀⠀⠀⠘⠀⣧⠀⠀⠀⠀⡇⢠⡉⠉⠁⠀⠉⢛⠛⢁⣿
⠀⢰⠁⠀⠀⠉⠀⠀⠈⠉⠀⡇⠀⠀⠀⢠⡀⠀⠀⠔⠁⢹⠀⠀⠀⠳⡀⠙⠦⠀⣠⠴⠋⣀⡾⠃
⠀⠸⣇⠀⠀⠀⢀⣀⡴⠀⣠⠁⠀⠀⡄⠀⠉⠀⠀⡠⠒⠁⢰⠀⠀⠀⠈⠀⠠⠤⠤⠴⠚⠉
⠀⠀⠙⠲⢤⣀⣉⣡⣤⠞⠁⠀⠀⠠⠓⠤⠄⠀⠈⠀⡀⠔⠋⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⡀⠴⠃⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⡔⠀⠀⣾⠆⠀⢰⠀⠀⠣
⠀⠀⠀⢀⡴⠋⡁⠀⢀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⣤⣠⠴⠖⠛⠓⠲⢮⣸⠇⠀⠀⡇⠀⠀⠙⠀⠀⠙
⠀⠀⠀⡞⠀⠘⠗⠀⠻⠇⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠞⠁⠀⠀⠀⢀⡤⠄⠹⣦⠀⠀⣇⠈⢧⡀⠀⠀⠀⣰⠃
⠀⠀⠀⡇⠀⡴⢄⢀⡤⢄⣠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠒⠛⠷⠀⠿⠀⠀⠀⣿⠀⠀⠘⢄⡀⠙⠒⠒⠋⠁⢀⠄
⠀⠀⠀⠻⣄⠁⠀⠉⠀⠀⣁⠜⠀⠀⣀⣤⠤⢄⣀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⣰⠃⣰⠏⠀⠀⠀⠀⠉⠙⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠉⠑⠒⠚⠋⠁⠀⣰⠟⠁⣄⠀⠀⣤⠁⠢⡀⠈⠢⠀⠈⠐⠒⠚⡡⠞⠋⠀⠀⣠⠖⠉⠉⠉⠓⢦⡀
⠀⠀⢀⠄⠒⠀⠀⠢⡀⠀⠀⢰⡏⠀⠈⠿⠀⠀⠿⠀⠀⠹⣄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢰⠃⠀⠿⠀⠸⠀⠀⠑
⠀⠀⠀⠀⠋⠀⠃⠀⠀⠀⠀⢸⡀⠀⡖⠒⠛⠛⠛⣻⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣠⠔⡂⡂⢲⢢⠀⠆
⠀⠀⠀⡑⠀⠀⢈⠀⠀⠀⠀⠈⢷⣀⠳⣀⣀⣀⡴⠃⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠁⠒⠂⠒⠚⢊⡔
⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠓⠦⠬⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠴⠶⠒⠚⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣋⣡⣤⠤⠤⠴⠒⠒⠒⠦⠄
⣀⣤⣴⣶⣾⣿⣿⣿⣗⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⢟⡻⠟⠉
⠋⠉⠉⠈⠀⠀⠀⠉⠉⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⣡⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢁⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠛⠛⠂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠁⠈⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠨⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⡀⠀⢠⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⣰⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠔⠛⠳⣶⣿⣿⣷⠀⠀⠀⠀⠀⣤⢄⡐⢶⣤⣾⣿⣿⣇⣼⡇⠀⠀⠀⠀⣠⢺⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠈⢻⡟⠻⣿⡿⢿⡿⢿⡿⠛⠛⠛⣻⣿⣿⠆⠻⢿⣿⣦⡀⠀⡜⢱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⢸⣿⢆⠀⣀⣾⠀⠀⢣⣀⣴⡾⠛⠻⣿⣇⠀⠀⠉⣿⣿⡈⠁⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⣠⣿⠃⠀⣹⠟⠉⣳⠶⠉⠨⣿⡁⠀⡀⠹⣿⣷⣤⣴⣿⣿⠉⠙⢻⣦⡀⠀⠀⠀⠀⠀⠀⡠⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠋⠀⢠⠇⠀⣼⡷⠀⠉⠘⣿⡅⠀⠘⠀⠈⠛⣯⣍⣽⢧⠀⣀⣼⡏⢳⡀⠀⠀⠐⢤⣜⣣⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢯⡿⣤⣄⡆⠀⢸⣧⣿⠀⠀⠀⢸⣿⣤⠁⠀⢠⠞⠋⢻⣻⣧⣴⣿⠟⠁⠈⠀⠀⠀⠀⣐⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⢛⠋⠀⠘⠛⣷⠀⠈⡏⣹⡷⠚⠷⣟⠟⢻⡀⡰⠂⠀⠀⠸⣿⣿⠋⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠻⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠈⠀⠀⡠⠸⣿⡟⠿⢻⣿⣆⠀⠀⢰⠦⣾⣷⡖⠀⠀⠀⠀⣼⠿⣧⣀⣀⣀⣠⣤⠟⣿⣿⣿⡄⠀⣿⡇
⠀⠀⠀⠀⠀⢠⣿⡟⠻⣆⠀⠀⠀⢀⡄⠰⣿⣀⠀⢉⣿⣯⡤⠀⣠⡟⣠⠟⠱⢀⣤⡤⢾⡟⠀⠄⠉⠋⠩⡝⣣⣾⣿⣿⣿⠁⠀⣾⣿
⠀⠀⠀⠀⢀⣿⠏⠀⠀⠈⢻⣦⣴⡟⠀⠀⠹⡧⣼⣿⡟⢿⣿⡿⠋⠁⠘⣦⠶⠫⠁⠀⢸⡇⠘⠀⣴⣷⣷⣾⠟⠉⣰⣿⣿⠄⢀⣿⣿⠂
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⣿⡏⠀⠀⠟⣧⡀⢀⣼⠇⠀⠀⠀⠀⢸⡇⠀⠘⢋⣿⠋⠑⠖⠚⣹⡿⠙⠀⣸⣿⡟
⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣾⣿⣿⡇⠀⠀⠀⠀⣿⠀⡀⠀⠀⢸⣿⢌⡁⢈⡤⠀⣀⡴⠻⣧⠀⣼⠋⠀⠀⠀⣠⡾⠋⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⢸⣷⣤⠶⠛⠉⠉⢻⣿⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⡇⠀⠀⡞⠻⣧⡿⠀⠀⢿⣌⠁⠀⠀⢰⡿⠋⠀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⢻⠀⠀⠀⠀⢻⣅⠀⠀⣧⡴⠪⡟⠲⡄⠈⣿⠀⠀⢠⡟⠀⣠⡾⠛⠉⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⣰⣿⣿⡄⠀⠀⠀⠀⢀⣿⠿⢿⣆⠀⠀⢸⠀⠀⠀⠀⠀⠙⠛⣬⣿⣆⣀⣾⡀⢹⡄⠘⣆⠀⢸⠇⣼⣿⣶⣶⣶⣦⣤⣞⣛⣻⣿⣽⣿
⠀⠀⣼⣿⡿⠋⢻⡆⠀⢀⣤⠞⠁⠀⠀⠹⣧⡀⠀⡆⠀⠀⠀⠀⢀⣴⠟⠁⣿⠛⠉⢉⣿⡇⠀⠙⡄⠈⠀⣿⣿⡿⣶⣤⣾⡉⠙⠛⠛⠛⠛⠉
⢀⡼⢿⣿⠁⠀⡘⢧⣶⣿⠃⠀⠀⠀⠀⠀⠈⢿⣄⣿⠀⠀⠀⠀⣼⠛⠳⣄⡿⠀⠀⢸⣿⡇⠀⠀⠘⣆⣀⠘⣿⣀⡼⠋⠙⢯⣷⣄⠀⠀⠀⢀
⣼⠁⠀⡇⠀⠀⢀⡿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢧⠀⠀⢰⣿⠀⠀⠀⠉⠳⣤⣿⣿⣿⡀⠀⠀⢿⣿⢀⣿⣯⣴⣶⣶⣦⠉⠛⠿⠿⠿⠋
⡇⠀⠀⠃⠀⠀⢿⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⠀⠈⣿⡀⠀⠀⠀⢀⣿⠟⢙⢻⣷⣤⣄⣤⡿⠿⠿⡿⣟⠀⠈⣿
⠀⠀⠀⢸⠀⠀⠸⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⢿⡇⠀⠀⢀⣾⠏⠀⠀⠑⠿⠷⢿⣧⡄⠀⠀⢈⣿⠀⠀⠏
⠀⠀⠀⠀⠁⠀⠀⣽⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⣇⢀⣠⣿⡿⠀⠀⠀⠀⠀⣠⣾⣿⣁⠀⣀⣼⡇
⠀⠀⠀⠀⠘⣤⡾⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⢻⡇⠀⠀⠀⠀⡼⠏⢀⣼⢏⣸⣿⠋
⠀⢀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡇⠻⣆⣀⣀⣤⣼⠷⠞⠋⠑⣦⡿⠁
⠺⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣶⣶⠾⠛⠛⠛⠋⠉⠀⣷⠄⡀⢹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠿⠋⠑⠀⠀⠀⠀⠀⠀⠀⢠⠤⠐⠋⠀⠙⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⠿⡋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣼⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠠⠅⠰⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⠃⠠⠄⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⢿⣿⡇⠀⠀⠀⠰⠂⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠐⢂⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡸⠁⠤⠄⠙⠋⠀⠀⠀⢢⡈⠁⡸⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢣⡈⠁⡸⠀⠀⡿⢏⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠉⠁⠀⠀⠘⣧⡼⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡄⢤⡀⣭⠱⣄⠀⣀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⠊⣭⠀⡤
⠑⢂⡈⠻⠆⠈⢶⠉⣨⠃⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠘⡅⠈⡶⠃⡰⠟⢁⡀
⠷⢄⣉⠒⠈⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠬⠽⠿⠿⠿⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠁⠒⣋⡠
⠀⠀⠉⠙⠢⠄⣀⣀⡸⠛⠦⢀⡀⠀⠀⠀⠀⠙⠳⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡀⠴⠚⢇⣀⣀⠠⠔⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢆⡀⠀⠈⠃⠦⣀⠀⢠⡤⣤⣈⣙⣛⠲⠶⣦⡤⠤⠤⢤⣤⠶⠖⣛⣉⣁⣤⢤⡄⠀⣀⠴⠚⠉⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠉⣻⠀⠀⠀⠀⠀⠉⠋⠛⠀⣀⣀⠀⠛⠉⠉⠀⠀⠀⠀⠀⣟⠉⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠤⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣤⣤⣤⣤⣤⣤⣤⣤⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⡴⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢳⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢸⢿⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣾⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠈⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡇⠀⣷⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣣⢀⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢹⢸⡇⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣾⢸⣇⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣹⠋⠀⠀⣍⣹⣼⡇⡅⠀⠈⢙⣏⠻⣦
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣶⣶⣶⣾⣭⣷⣶⣶⣶⣶⣶⣶⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠛⠿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⡿⠿⠿⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣭⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠙⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⡟⠋⠁⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⢸⣿⣿⠀⠀⠀⢘⡍⢉⠀⠀⠀⠀⠀⠀⠶⠄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⡴⠁⠀⠀⢐⠀⠀⠀⠀⣴⡒⢢⠘⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⢳⠂⠀⠉⡠⠀⠀⠀⠀⠑⠬⠋⠀⣇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠿⠀⠀⠀⠐⠈⠀⠀⠀⠀⢳⡀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠴⣆⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⠇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠈⠙⠚⠉⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡴⣿⣧⡄⠀⠀⠀⡴⠋⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠓⠙⠿⠁⠀⡠⠊⠀⠀⠐⠃⠌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⢶⣉⣳⣄⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠂⠀⠁⠢⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠔⠂⠐⠁⠠⢄⣼⢩⣿⡟⠹⣖⡾⠋⣿⣷⣀⣀⣠⣔⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢨⠃⠄⡠⠚⠳⡤⢿⣾⡷⠁⡜⠉⡇⠀⢻⡇⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡰⠁⠀⠀⠘⢿⡃⠀⠘⠁⠀⡥⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣦⣊⠉⠉⠓⠚⠣⠤⣄⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡕⢠⡦⣄⠀⠘⣫⠈⣢⣄⡀⠀⠀⠢⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠫⠊⣤⠞⠁⠈⠦⠟⠁⠉⠀⠀⠉⠙⠓⠛⠻⣽⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣉⡛⠛⠋⢉⠗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠾⠦⠯⣭⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⢶⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠛⢶⠃⠀⠸⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠳⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⡇⠀⠀⣷⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣄⠀⢶⣄⣀⡸⠛⢢⣤⣠⣴⣛⡦⣤⣤⣼⢳⣦⣤⠟⠲⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡷⢀⠀⠀⠀⠀⢸⠙⠋⢠⡤⠔⠊⢁⣀⠀⠀⠈⠁⠀⠀⣤⠀⠀⠉⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢸⡉⠭⢦⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠂⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠉⠉⠉⢱⠖⠚⠁⠀⠐⠮⡠⢄⡀⢀⡀⠔⠀⠀⠢⠤⠀⠀⠀⣠⠖⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡤⣄⡀⠀⠀⠈⠁⠁⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⠄⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠰⣥⣈⣵⡤⠇⠀⠀⠀⠀⠀⣀⡼⠋⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠁⣄⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⣀⡴⠛⣇⠀⠀⠀⣰⣧⠞⠀⠙⠒⠦⢬⣭⣉⣉⣻⠉⠉⠉⠉⣉⣀⣀⣀⣠⡼⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠈⠛⠛⠛⢡⡟⠀⠀⠀⠀⠀⢀⣀⡀⠀⢸⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠐⢰⡖⠋⠉⠉⠇⢚⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢧⠀⠀⠀⢸⡼⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⢇⠀⠀⠈⣇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢹⠀⠸⣤⣀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣨⣦⡀⠘⠓⢦⠤⠼⠷⢤⡾⢻⠞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡤⠚⠉⠀⠀⠀⠙⠛⠛⠋⠙⠛⠒⠚⠁⡞⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠓⠒⠶⠶⠶⠶⠶⠦⢤⣤⣤⣤⣤⣤⠟⠋

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⠛⠙⠛⠻⠛⠛⠛⠿⠿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⢀⠤⠤⠀⠀⡤⠔⠠⠤⣤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⣀⣀⡠⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠶⠶⠶⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⠠⠴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠤⠤⠠⢄⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠁⠀⡶⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠤⣤⡔⠋⠉⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⠸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠹⣿⣿⣿
⣿⣿⣿⣿⡇⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⢹⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠿⠟⠛⠛⠉⠉⠛⠛⠿⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠐⠒⠒⠀⠀⠤⠴⠶⠄⠀⠐⠒⠛⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠛⠒⠒⠂⠀⠀⣀⣀⣀⣀⣀⣠⣀⠀⠀⠐⠒⠚⣏⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠣⣄⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⢀⡨⢿⣿⣿⣿⣿
⣿⣿⡿⠋⠁⠀⠀⠈⠑⢦⣀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣠⠶⠋⠀⠀⠙⠻⣿⣿
⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠜⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻
⣿⣶⡾⠀⢀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⣿⣿⣄⣼⠟⠀⣠⡆⠀⣠⣄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣀⡀⠀⢠⡀⠈⢷⡄⠈⣿
⣿⣿⣿⣿⣄⣰⣿⣀⣼⣿⣿⣿⣶⣶⡦⠤⠤⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠤⠤⠤⠀⢸⣏⣁⣴⣿⣿⣄⠈⣷⡀⢈⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠲⠂⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡀⣸⣀⣄⣠⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠙⢿⡴⣤⣤⠴⠦⠤⠤⣬⡉⠀⣿⠃⠈⢫⣤⣤⣤⣤⣤⣤⣬⠙⠛⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠐⠛⠁⠉⠁⠐⠒⠛⠛⠛⠋⢀⡹⢆⠘⠛⠉⠁⠀⠀⠀⠀⠀⠃⠒⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡐⠂⣀⠀⠀⠀⠀⠀⣀⡀⠒⢉⣿⣄⠑⠂⣀⠀⠀⠀⠀⠀⠀⣀⠐⢢⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠙⡦⠄
⡞⠁⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠁⠀⠀⠀⠀⢀⠊⡠⠖⠚⠉⠉⣉⠂⠀⡥⢤⡤⠤⠤⡄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡭
⡂⣷⠤⠀⠀⣤⠘⡥⠀⠂⣥⠖⠁⡠⢾⣤⠂⢂⠄⢀⠈⣄⣶⢰⡟⢸⠁⠀⠄⠀⠀⠀⣠⢠⡀⠀⠂⠀⠀⠂⡈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰
⠛⠀⠀⠀⢀⣿⢢⠑⢀⡴⠋⣠⠊⠠⣄⠈⠦⢓⠀⣀⣿⠸⣿⢸⡇⣼⢾⣅⣀⠠⠊⠀⠀⠀⠑⠀⠀⢈⡉⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⣼⠁⣼⣷⠎⠠⡾⢏⠂⠀⡍⠀⣞⠶⠎⠀⠈⢀⠛⠘⢫⣹⣿⣲⢗⠾⠿⣿⡦⠀⠘⠀⣇⠈⠀⠀⠀⡘⢠⠀⠀⠀⠂⠘⠙⠎⠑⣦
⢀⣦⡀⢰⣿⢸⣿⡇⢠⠁⢱⡦⢉⠂⡍⠉⠁⡀⠀⠀⠀⡀⢥⠰⠙⠓⠘⠗⢩⣿⠿⠿⡇⢤⡠⡄⢸⣕⡋⢤⣡⠄⠈⡌⣀⠀⠀⠲⢀⠀⢸⣟⣷
⡉⠀⠀⢸⣿⡜⣿⠀⡆⠀⣄⣹⡾⠍⠀⠀⢀⣴⠝⠈⡐⢏⡌⠄⠸⠇⣴⢳⣤⠯⡄⣄⠚⡆⠳⣀⠋⡳⣴⠆⠑⠓⠀⠀⡭⠐⠂⠀⡈⠀⢀⠹⣾⡇
⠧⠄⠀⠈⢿⣷⡘⡆⠘⢄⠘⠁⠀⠀⢀⣴⢩⣱⣦⠀⠀⢄⠁⢄⠀⠀⡙⢮⣏⢑⠻⡃⢚⢡⡆⠻⠆⢹⣝⣿⣷⠟⡀⠠⠀⠀⠀⠀⠀⣠⣾⣷⣿⢻⡀
⡀⠀⠀⠀⠀⠙⢿⣬⣦⣀⠉⠁⢀⠀⢮⣶⣿⣿⣿⣦⠐⡈⢲⣦⡑⡀⡘⠉⠻⡬⠷⣀⣿⠇⣿⣿⣣⢘⡮⢈⡟⢎⢀⠀⠀⠀⢀⠀⡐⢿⣏⠂⣽⣿⠇
⠷⠀⠀⠀⠀⠀⣸⣿⡏⣭⠀⠀⡪⠶⢀⡉⠁⠋⠟⠟⢡⢨⠮⣧⣷⠋⠀⣾⠛⡆⣿⢮⣵⠀⢿⣌⣉⠀⢐⠟⢘⢄⡼⢀⣾⣿⡏⠀⠌⣠⣿⣤⣿⠃⣆
⠀⠀⠀⠀⠀⣰⣿⠛⠋⢓⡆⠀⠀⡷⢆⠉⢉⠛⡔⠒⠴⠟⠋⢠⠃⠀⣾⣫⣶⡧⣝⢶⣎⣄⢽⣟⣿⠀⢘⠓⣰⠃⠀⠘⠋⠉⠀⣡⡆⠈⡷⣸⡏⠀⠿
⠀⠀⠀⠀⢰⣿⣿⣆⠘⠿⢷⠀⠀⣏⣀⡈⠙⡳⠼⢷⣶⣯⠀⠎⠀⠀⣿⣽⣞⣎⠁⣐⢹⡄⣍⠁⢿⡆⢩⡜⢪⡀⢘⠆⠀⠀⣰⠿⠇⣌⢳⣟⠁
⠀⠀⣠⠀⠀⣿⣿⣿⡆⠀⢸⣇⢠⣬⣩⣵⣒⡀⠙⠛⠍⣿⡿⠀⠀⠘⢤⣿⣿⡇⢣⡉⠸⠇⢹⡗⢿⡷⠈⠖⠉⣠⣇⠴⢄⣴⣯⠴⢦⠤⠘⠈⠁
⣁⣴⠏⠀⠀⠹⢿⣿⣿⡄⠈⠋⣿⣇⣱⣥⡛⣥⠀⡗⣀⣩⣦⣼⣷⡦⢌⡻⣿⣵⢾⠉⠛⠶⢤⣤⣤⣤⡴⢶⠛⠁⠉⠳⣤⣈⣁⣀⣤⡔⠁
⣿⣃⣠⡾⢁⣼⠃⠈⣻⣷⢀⠀⠉⠁⠀⠀⣿⠦⣍⠛⠻⠯⠀⠚⠋⢰⣟⣛⡻⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠒⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀
⣎⡿⣿⢛⣾⣃⣃⢠⡟⣽⣧⣁⠩⠔⢚⣭⣶⡿⠛⢿⣶⣤⣴⡶⠗⠁⠀⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⣿⣿⣿⣿⠃⢉⢱⡿⠳⠿⣿⣷⣾⣿⠟⠋⣁⠠⠐⠁
⣿⣿⣿⣷⣵⢾⠬⣧⣀⣀⣹⣿⡭⠴⠚⢋⡡⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣞
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⣠⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣽⣿⣯⠛⣻⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠉⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡀⠠⠶⠍⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠁⢴⣤⣄⠀⢙⣇⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡆⠀⠀⠀⠀⠀⣨⣟⡍⢀⠇⣠⢆⣴⣶⣖⣠⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠈⠉⠉⠁⠀⠀⢰⡏⢿⠀⠀⠀⢠⠀⠀⢉⡀⣼⣴⣱⣷⣾⣿⣿⣿⣿⣿⣿⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠽⣦⣤⣴⡯⣤⠴⠋⣸⣿⠟⢉⣥⡀⠀⠉⠉⣿⣶⣾⣿⣯⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠟⠋⠀⠀⠀⣼⠏⠁⣠⣤⣤⠀⠀⠀⠀⠀⣽⣿⣿⣟⣿⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠘⠳⣄⡀⠀⠀⠀⠀⡋⢀⢿⣿⠇⠀⡆⠀⠀⠠⡄⠙⠛⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠐⣠⡿⠁⠀⢠⡟⠙⢢⣠⠂⠸⠂⣿⡀⠀⢀⡰⠀⠀⣿⣷⢰⡀⠀⡷⣶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡶⠞⠛⠁⠀⢀⣴⠟⠁⠀⠀⢹⡀⡀⠀⠈⠙⠛⠁⠀⠀⠘⣍⠁⢨⡇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⠛⢻⣷⣦⣤⡴⠞⠛⠁⠀⠀⠀⠀⣸⣿⠄⠀⠀⠀⠀⢷⣄⠀⠀⡙⠻⠃⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢸⡇⠀⠀⠀⢻⣷⡈⠛⠒⠀⠀⠀⢀⣤⣾⣿⠁⣠⣶⡶⠠⡄⢈⠛⠿⠛⠁⠀⠀⠀⠀⢄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠘⣿⡀⠀⠀⠀⠹⣮⡁⠀⠀⠀⠀⠻⢿⣿⣷⠐⢿⣽⣳⣶⣀⡜⠑⠦⠀⠀⠀⢀⣒⡐⠂⠀⠸⣦⣌⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡽⡆⠀⠀⠘⣷⡀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠈⣹⣷⣤⠄⠀⠀⠉⠻⣟⣷⡇⢠⡴⠋⢁⣿⠶⠠⠠⠧⣹⡧⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣹⣄⠀⠀⠈⠻⣄⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠘⠻⣿⣿⣄⡀⠀⠀⠀⣉⣉⡠⠊⠀⢀⣾⠟⢶⣶⣶⣦⡀⠀⣰⡤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡛⠿⣳⣼⣦⣄⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠉⠛⠷⣦⣄⣀⠉⠉⠁⠙⢿⠟⠉⠁⣀⣠⣴⡿⠃⣎⣀⣀⣀⠀⠀⣰⣿⡇⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠣⠜⠂⠉⠙⢳⣬⣟⣲⢤⣀⣀⠉⠛⠳⢶⣤⣀⣀⠀⠀⠈⠉⠛⠛⢶⠶⠖⠚⠛⠛⠛⠛⢥⣄⠀⢈⡟⠛⠉⠁⣰⣯⣽⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⠔⢋⡠⢀⠠⠄⡒⠬⠳⠙⡫⢝⡳⣿⢽⣏⣿⣒⣶⣤⢭⣙⣛⣻⣶⣶⣶⣮⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢏⣹⡿⣿⡿⠞⠋⠀⠀⠀⣠⠞
⠀⠀⠀⠀⢀⣴⡻⠝⡂⢥⡐⠬⢓⠨⢐⢈⡁⣄⢢⣥⣬⣦⣍⣭⣿⣿⣿⣿⣿⣿⠿⣍⠷⢉⣣⣤⡾⠿⢓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠦⠤⠤⠒⠋⠁
⠀⠀⠀⣰⡿⢅⡲⠍⢂⠡⣐⣨⡴⢞⠯⢒⠉⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⠿⡿⠶⣳⣶⣾⣉⠫⠕⢢⡑⢮⡳⠬⢛⣄
⠀⠀⡴⠏⠐⢡⠀⢌⡠⠕⠋⠡⢚⣥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣟⡋⢀⣡⣴⠶⣛⢃⣍⠦⣔⠶⠟⠛⣛⣦⡀
⠀⠀⣷⠐⢈⡀⠈⣠⠴⣲⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣦⡷⠛⠋⠈⡑⠬⡑⢊⣉⣤⡽⢿⡀
⠀⢀⣿⢂⣡⡲⣝⣶⣿⠻⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠺⣷⣼⣷⠿⢟⡻⢍⡳⢸⡇
⠀⠸⣯⢿⣿⠿⢛⢩⠲⣵⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣽⣧⣙⣦⣽⣶⢿⣿⠃
⠀⢸⡿⣻⡷⠞⣺⣿⢾⠮⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣭⡹⢌⣣⣴⣟⣿⠋
⠀⣷⠞⣿⣤⠞⢽⣿⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣭⣷⢾⣻⣿⣿⣼⡇
⠘⣷⢺⣯⢔⡪⢼⡟⣧⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⠿⣿⢟⣿⠁
⠀⢹⣯⢲⠟⡩⣴⠟⠺⢑⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⡻⢝⢢⣽⣟⣧⡿
⠀⢨⣿⡾⢓⠭⢂⣵⠞⢪⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢜⢣⣝⣾⣿⡿⢛⣸⡇
⢀⣼⣷⣻⣷⣾⠿⢶⣶⣿⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⡿⣟⣣⣼⣿⣿
⠸⡀⢉⣿⣿⣧⣤⣤⣸⣿⠞⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠛⠻⣧⣿⡾⣿⣵⣾⣿⡄
⠀⠈⣿⣾⣟⣛⡛⠿⣟⠻⢢⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠙⢿⡟⣉⣀⣼⣁
⠀⣼⣿⣿⣭⣭⣿⣷⡚⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣦⣤⣤⡶⣻⡶⣋⣉⣽⡇⢀
⢰⡿⠟⠉⠀⠀⠀⠀⠈⠙⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣯⣄⣡⣿⣷⣏⣠⡴⠶⣧⣄
⡏⠑⠶⢤⣤⡀⢀⠀⠀⣀⡸⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠞⠛⠉⠑⢦⡀
⠘⠢⠤⠤⢀⣀⣀⣀⠀⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⣶⣶⣤⣤⣬⣍⣙⠛⠻⠿⣡⣀⠀⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣤⢭⡭⠽⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠰⡇⢸⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣤⣠⣶⣼⣧⣤⣄⣄
⠀⠀⠀⠀⠀⠀⣠⣴⠿⠟⠻⠿⢿⣆⡀⠀⠀⠀⠀⠀⠀⣼⡟⡁⠁⠀⠈⠉⠉⣙⠻⣷⣯⡀⠀⠀⠀⠀⠀⠀⣤⠀⣄⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⢸⣿⠙⠋⠀⠀⠀⠀⠀⠙⠓⠘⢻⣇⠀⠀⠀⠀⠀⠀⠻⣿⣷⡏⠀⠀⠀⠀⠠⣷⣿⣾⠃
⠀⠀⠀⠀⢸⣧⣤⣤⣤⣀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⡇⠰⠆⢠⣾⠀⠀⠀⣴⡄⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢼⣿⡆⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠷⣦⣤⣠⣿⠀⠀⠀⣶⣾⡇⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠈⣟⠁⠀⠀⠀⠀⠐⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣤⣤⣬⣿⣿⣆⠀⠀⠻⣿⡇⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⢐⡿⣷⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡄⠀⢹⡧⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢘⣣⣿⠀⠀⠀⠀⠀⠀⢠⣟⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⠃⠀⠈⣿⠀⠀⣤⣶⣶⣤⣄⠀⠀⢠⡿⠋⠁⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⠀⠀⢨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣦⢰⣾⣷⣆⣶⣄⠀⠀⠀⠀⠀⠀⠙⢧⣀⠻⠟⠛⠿⠿⠃⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣦⣿⡾⠟⠛⠛⠛⠙⠛⢿⣷⡄⠀⠀⠀⠈⠙⠛⢶⣶⡶⢶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣦⣀⠀⠀⣰⣿⡅
⠀⠀⠀⠀⠀⠀⣸⣟⣡⣄⠀⠀⠀⢀⣀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡁⠀⣀⣀⠀⠀⣠⣴⣶⠿⢿⡟⢋⣤⠚⠻⣧⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡟⢁⡈⢡⡄⠀⠀⠉⠃⠀⢸⣿⣄⠀⠀⠀⣾⣷⣴⡟⠀⠘⣷⣾⣿⣿⣧⣾⠿⠟⣁⣰⠶⠀⠿⢿⣤⣀⠀⣬⣽⣿⠁
⠀⠀⠀⠀⠀⢀⣿⠀⠈⠁⣸⠇⠀⠛⠁⠀⠀⠸⣿⢿⣧⠀⣸⣿⣿⣿⣦⣤⣶⣿⣿⣿⣿⣿⣿⡶⠛⠋⠁⠀⠀⠀⠀⠀⣉⠓⠦⢭⣧⢀⡀
⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⢿⣶⡦⠀⠀⠀⠀⠀⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠰⣟⢻⠄⣷⠀⠀⢰⢞⣉⠃⠀⠀⠹⣿⣻⠆
⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⢀⣤⡀⠀⠀⠀⣸⣿⣿⣿⣿⢻⢟⣿⣿⣿⣿⣿⣿⢹⣿⢻⣏⠀⠀⠉⠀⢠⣿⠃⠀⠀⠈⠉⠀⠀⠀⠀⣹⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠛⠃⠀⢀⣴⣿⣿⣿⣿⠇⢎⡎⣽⣿⣿⣿⢿⣿⡇⣿⡄⢿⣆⠀⠀⠀⣸⣷⣾⡟⠀⠀⠀⠀⠀⠀⣠⣿⡿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣿⣦⣤⣀⣤⣿⠟⠛⠻⢿⣿⣆⠘⣰⣿⣿⣿⣿⣿⢿⠇⠈⠀⣿⡏⡟⢹⢃⡟⠉⢸⠟⢻⣟⣛⣿⣷⠞⢻⡟⠁⠀⠈⣳⣄
⠀⠀⠀⠀⠀⠀⣰⡟⠉⠈⠉⢿⡏⠁⢨⡿⠀⠀⠀⠀⠻⣿⣤⠟⣡⣿⣿⣿⣿⡄⠀⠀⢠⣿⣇⣀⡉⢸⡇⠀⠉⠀⠀⢨⣿⠋⠀⢠⡟⠁⠀⠀⠀⠟⠹⣆
⠀⠀⠀⠀⠀⢰⡟⠀⡄⠀⠀⠘⢿⣶⠟⠁⠀⣤⢀⠀⠀⠙⣧⣰⣿⣿⣿⣿⣿⣿⣤⣤⣴⣿⣿⣟⠿⣿⣿⣿⣿⠖⢷⣾⡟⠀⣰⡿⠇⠀⠀⠀⢀⡄⠀⢻
⠀⠀⠀⠀⠀⣿⣧⣠⡇⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣷⡶⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣾⣿⡏⠻⣧⠈⢿⣿⢀⣿⣇⣀⡀⠀⠀⢿⡇⠀⠈
⠀⠀⠀⠀⣴⠏⢠⣿⠀⠀⠀⠀⠀⠀⠠⢀⠰⠆⠸⣿⣷⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢠⣿⠿⣇⠀⣹⣧⠈⢻⣿⣿⣿⡿⢿⡆⣴⣾⠃
⠀⠀⠀⣰⡟⢀⡾⢻⠀⠀⢠⠀⠀⠀⡄⠘⡄⢀⠀⣿⣿⣇⠀⢿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⠁⠀⠚⠁⣠⣿⣶⣿⣿⣧⠀⢻⣿⠏⠀⢸⣧⣾⣿⣾⣷⣶
⠀⠀⢰⡏⢀⣾⠁⣼⠀⠀⠚⠀⠰⠚⠇⠀⠛⠘⢂⣸⣿⣿⠀⠘⣿⡟⣽⣿⣿⣿⣽⣿⣿⡇⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡇⠀⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠁⣾⠃⢠⣿⣿⠿⣿⡿⠿⣿⣿⣷⣶⣾⣿⠻⣿⣿⣧⠀⢿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣾⣿⣿⣿⣿⣟⣛⣿⣿⣿⣷⡀⢀⣴⣿⣿⢿⣿⣿⣿⣿⣿
⠀⢰⣿⢀⡇⠀⢸⣿⣫⣾⣿⣿⢶⣿⣿⣿⡝⣿⣿⠀⣿⣿⣿⣆⠈⣿⣿⣿⡿⣣⡿⣫⣿⣿⣿⣿⣿⠻⣿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣯⣿⠷⣿⣿⣿⣿⣿
⠀⢸⡇⢸⡁⠀⣾⣿⣿⣿⣿⣿⡺⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⡆⠘⣿⣿⣿⣿⡾⠻⣟⣿⣿⣿⣿⠿⠛⠻⣦⢹⣿⢟⠉⢻⣷⣦⡝⢿⣿⣿⣿⣿⣿⡿
⢠⡟⠀⢈⣻⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡿⠋⡁⠀⠘⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⠀⠀⣰⣿⡆⣿⣿⣇⠀⠸⣿⣿⡤⣿⣿⣿⣿⠏
⢠⣴⣦⣾⠏⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣷⣿⡇⣀⢴⣼⣿⣿⣷⢟⣿⣿⣿⣿⡏⠀⣴⣿⣿⠿⣿⣿⣿⣆⠀⢻⣿⣷⣿⠿⠛⠁
⠋⡿⠻⡿⠀⠀⣿⣿⣿⣿⣿⣿⡿⠘⣿⣿⣿⣿⣿⣷⢸⣿⣷⢿⣷⣿⣼⣿⣿⣿⢲⣿⣿⣿⣿⣿⡇⢠⡏⠀⠀⠀⠀⠉⠙⢿⡆⠸⣿⡋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣶⣿⣿⣶⣿⣿⣿⣿⣿⢃⣩⡾⣿⣿⣿⣿⠀⣾⠁⠀⠀⠀⠀⠀⠀⠈⢿⡀⢿⡇
⠀⠀⠀⠀⠀⢼⣿⣏⣭⣿⣿⣿⣿⠐⣿⣿⣿⣿⣿⣿⣿⣿⣟⡳⣿⣿⣿⣿⣿⡿⣋⣴⣿⣿⣿⡿⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⡇
⠀⠀⠀⠀⠀⠈⠉⢹⣿⡆⢻⡏⠀⠀⠀⢹⡧⠀⢻⡏⢸⣿⣿⣿⢿⣿⣿⣿⣋⡾⢋⣿⣿⣿⣿⠃⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⣿⠀⠀⠀⢸⣧⠀⣻⡇⢸⣿⢟⠅⣸⣿⣿⣿⣿⡴⢛⣵⣿⣿⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸⡏⠀⠀⠀⢸⣿⠀⣿⣇⣸⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⢟⣛⡙⣧⣼⣿⡆⠀⠀⠀⠀⠀⠀⣀⣀⣹⣿⣧⣼⠟⠳⣦
⠀⠀⠀⠀⠀⢀⣀⣸⣿⣷⣾⣷⣤⣴⣶⣿⣿⣶⣿⡟⠛⠋⠁⠀⢹⣿⣿⡿⠿⢛⣿⡎⢿⣷⣿⣿⣿⠀⠀⠀⠀⠀⢀⣾⡿⢿⣿⣿⣄⣩⣤⣾⣿
⠀⠀⠀⣠⡾⠛⠛⣻⡟⢹⣿⡿⠋⠀⢉⣙⠃⢹⣿⣇⢀⣀⣀⢀⣾⣿⣿⣄⠛⠻⠿⠷⠾⠛⠛⣿⣿⠀⠀⠀⠀⠀⢸⣿⣒⠚⠛⠟⠛⠛⣛⣉⣿
⣀⣀⣀⣙⣷⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣉⣉⣀⣉⣀⣀⣈⣉⣉⣁⣀⣀⣀
⣿⣿⣿⢿⠻⢿⡿⠻⡿⢿⡿⢿⠻⢿⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣶⣼⣷⣴⣧⣦⣾⣤⣼⣿⣤⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠉⠉⠉⠛⠻⠿⣿⣿⡿⠿⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠠⣏⠁⠀⠀⠀⠀⠀⠀⠂⢄⡀⠈⠀⢄⠀⠉⠠⢀⢀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠃⠀⢀⣴⣿⠟⠁⠀⠙⣄⠀⠀⠄⡀⠀⠀⠀⠀⠉⠓⠄⡀⠀⠀⠀⡀⠉⠻⣷⡄⠀⠈⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⠏⠀⠀⠈⠀⠈⢢⣀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⣂⡬⢷⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⣼⠏⠀⠀⠀⠀⠀⠔⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡴⠖⢀⣠⠄⣛⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⢀⡀⠉⠀⠔⠈⠀⠄⠋⠀⠀⠀⠠⠀⠀⠀⠐⠤⠀⠀⠀⠀⠄⠤⠀⠀⠛⠁⠛⣛⠄⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠁⠀⠈⣿⡶⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣿⣿⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠸⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⢿⣿⣿⣿
⣿⠿⠿⠋⠀⢀⣾⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠛⠛⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣆⠀⠈⠻⢿⣿
⣧⣀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⠁⠀⠀⠀⠀⠀⠀⠒⠀⠙⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠂⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡖⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣧⣾⣷⣤⠄⠀⠠⠀⠀⢀⣀⣀⣀⡀⠀⠠⠄⠀⠀⣦⣴⣿⣄⣧⣸⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣀⠀⠀⢀⡀⣀⢀⣀⣀⣄⣀⣀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠍⠛⠛⠛⠻⠇⡿⠠⠟⠒⠒⠒⠩⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀⣀⣤⣿⣤⣀⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⡀⠀⣀⣀⠀⢀⣀⣀⠀⢀⣀⡀⠀⣀⣀⡀⠀⣀⣀⠀⢀⣀⣀⠀⢀⣀⡀⠀⣀⣀⡀⠀⣀⣀⠀⢀⣀⣀⠀⢀⣀⡀⠀⣀⣀⡀⠀⣀⣀⠀⢀⣀⣀⠀⢀⡀⢸
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸
⠀⡇⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⣀⠀⢀⡀⠀⢠⠀⠀⢠⠀⠀⠀⠀⣠⣴⠶⠶⠶⣤⣀⠀⠀⠀⢠⣤⠀⠀⠀⢠⡀⠀⠀⣠⣤⡄⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⡂⢸
⠀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⣼⣇⠀⢸⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠙⢷⡀⠀⢸⡇⠀⠀⠀⣿⡇⠀⢸⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠇⠀⠀⠀⠀⠀⠀⠸⣇⡀⠀⣿⠀⠀⡇⠀⣿⢹⡄⢸⠀⠀⣿⠀⠠⡶⠀⠀⠀⣶⠀⠈⣷⠀⢸⣇⡀⠀⢸⠃⣧⠀⢸⠃⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠂⢸
⠀⡄⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣿⠀⠀⡇⠀⢿⠈⣧⢸⠀⠈⣷⠀⠀⠀⠐⠒⠃⠀⠀⠀⣿⠀⠸⡏⠀⠀⣾⠀⣿⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠃⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢻⡄⠀⡇⠀⢸⠀⠸⣿⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⡇⠀⢀⡟⠛⣿⠀⢸⡆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠳⠴⠃⠀⠸⠀⠀⠻⠀⠀⠀⠈⠛⠶⢦⣤⡴⠾⠋⠁⠀⠀⠀⠋⠀⠘⠁⠀⠛⠀⠀⠛⠶⠟⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⡂⢸
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠓⠂⠀⠒⠒⠀⠐⠒⠒⠀⠐⠒⠂⠀⠒⠒⠂⠀⠒⠒⠀⠐⠒⠒⠀⠐⠒⠂⠀⠒⠒⠂⠀⠒⠒⠀⠐⠒⠒⠀⠐⠒⠂⠀⠒⠒⠂⠀⠒⠒⠀⠐⠒⠒⠀⠐⠃⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣀⣒⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠙⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣶⣾⣭⣽⣛⡇
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃⠙⠋⠐⠟⡿⠿⠂
⠀⠀⠀⣴⠟⠹⣯⡀⡇⠠⡞⠙⡷⣧⣶⣰⠆⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠⠟
⠀⠀⠀⠻⣦⡲⠾⠃⠛⠓⠙⠛⠁⠈⠀⠉⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠊⠁⠀⢦⡀
⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠞⠁⢻⣀⠀⠀⣈⡆⠒⠦⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠃⠀⠀⠸⡆⠈⠉⢨⠃⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡦⢦⠀⠀⢣⠀⢀⡞⠀⠀⠂⠈⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⣷⢰⣞⣉⣉⡈⢇⡞⠀⠀⢀⠁⠂⢠⠇⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠌⠁⠀⠀⢸⠂⠀⠀⡌⢀⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣿⠀⣿⠀⠀⠀⠈⠀⠀⢰⠀⡄⠀⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⣠⣽⣆⣁⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⢿⠀⢸⣤⡄⠴⠀⠀⠀⠀⠀⡀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠛⠉⠉⠛⠛⠓⠶⠆⠈⠙⠛⠃⡜⠀⠈⠂⡀⣠⢠⣀⡀⠀⡀⣁⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠚⠉⠠⠡⡽⢸⠅⠀⠀⠥⠬⠭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠐⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠓⢒⡒⢚⣫⣍⣹⡏⠟⠒⠔⣦⣬⢇⢀
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⡠⣄⣘⡋⢯⣄⠀⢀⣁⣀⡀⣀⣸⡿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠛⠉⠉⢈⡳⠿⢷⠎⠉⣭⠵⠟⢻⡇⠐⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⠻⠭⣖⡨⣡⣿⡁⠤⡈⠉⠲⣼⡇⠘⠖⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠦⠤⣤⡄⣩⠟⢸⡇⠠⠖⠚⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⡀⢹⠃⠀⠘⣧⠐⠲⠦⠄⣸⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⢿⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠋⠀⠀⠀⠀⠠⣤⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠉⠈⠳⣄⠀⠀⠀⣠⠞⠛⠻⢿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠙⣷⣠⡾⠃⠀⠀⠀⠀⠈⠙⠛⠿⣷⣦⣄⣀
⠀⠀⠀⢀⣠⣤⣶⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣶⠶⠤
⠀⠀⠋⠉⠉⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⠤⣿⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡷⠖⠒⠛⠉⠉⠹⣏⠉⣹⣍⣿⠟⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⣤⣼⣭⠿⠁⠀⠉⠉⠉⠉⠒⠒⣿
⠀⠀⠀⠀⠀⠸⣿⠀⢀⡠⠄⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⢹⣷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣿⣷⠀⠁⠀⠀⢠⠀⠀⠀⠀⠐⠲⣄⡀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⢀⣹⠀⠀⠀⠀⠀⠀⠀⢽⣠⡾⠁⠘⠷⣄⡀⠀⣮⠁⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣬⠇⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠈⠙⠛⠧⠤⠤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠋⠉⠙⠻⣿⣿⣓⠾⣯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣈⣷⣮⣝⣷⢯⣝⡳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠉⠛⠻⢧⣍⣷⠮⣝⣓⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣯⣙⢲⣭⣛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣮⣿⡒⢬⣝⡲⢤⣄⣀⣤⠶⠶⠶⢤⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢧⣽⣗⠮⣭⣀⣤⣶⠦⡄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠀⢿⣿⣷⢤⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⠻⠿⠿⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠒⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠛⠙⠳⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢻⡀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⡀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⡞⠁⠀⠀⠀⠀⠀⠀⢠⣿⠻⣦⣀⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⣠⣤⠉⠛⠛⠿⠿⠿⠿⠷⠶⠶⠶⢶⣿⣟⡋⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠉⢀⣤⣤⣄⠀⠀⠀⠀⣸⣶⣦⣤⡈⢿⣍⠛⢷⡄⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⣠⣴⣿⡿⠁⠀⣠⡾⠋⠉⢛⣛⠳⣄⠀⣼⠋⠀⠀⠈⠙⢾⣿⣦⡀⠳⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⢠⣤⠶⠟⣫⣾⡟⠀⠀⣼⡏⠀⠀⠀⣿⣿⡷⢸⣦⣿⡀⣾⣿⡆⠀⠈⣿⣿⢳⣄⠀⠀⠀⠀⠈⠻⣦
⠠⠤⠤⠤⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠀⠀⢀⠹⣇⠀⠀⠀⠀⢉⣠⡟⠁⠻⣷⣌⡉⠀⠀⣰⡟⢹⡆⠙⣦⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⠛⠀⠀⠈⠻⠾⣿⣿⡿⠿⠛⡁⠀⠀⠀⠈⠻⣿⢿⣿⣿⡾⠛⣷⣤⡜⣧⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⠀⢀⣠⣿⠂⠀⠀⠀⠀⠹⢩⡿⠘⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⣦⡀⠀⠀⠀⠀⢸⣇⢸⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢶⣾⠃⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣈⠻⣦⣄⠀⠀⠀⠻⣤⠙⠻⣿⣦⡶⠀⠀⣠⣶⠾⣿⠟⠻⣿⠟⠛⠻⣿⠛⠛⣿⠿⢷⣦⡀⠙⣧⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠷⠶⠶⠶⠶⠿⣿⠋⢀⣤⡾⣿⠁⠀⢿⡀⠀⣹⡆⠀⢰⣿⠀⢀⣿⣀⣀⣾⣿⠀⣿⡇⠀⠀⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⢤⡀⢰⡏⠀⣿⣿⣧⣼⣷⣴⡿⣿⡟⢿⠟⢷⣾⣿⣾⠛⠿⣇⠉⠻⣿⢷⣿⣧⣤⣤⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠻⣾⣧⣀⣈⣽⠟⢁⡾⠃⠀⣿⠁⣾⠆⢸⣿⠀⠙⣷⡀⠙⣷⠀⢻⡆⢻⡏⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠙⢻⡟⠀⣾⠃⠀⢰⡇⠀⠛⠀⠈⣿⢧⡀⠈⠓⠀⠸⠇⠘⠿⠘⣿⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡤⣄⢻⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⢻⣆⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣸⡇⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⣻⣶⠆⠀⠀⠀⠀⠀⠀⢰⡿⠀⢸⣯⠀⠀⠀⠀⠀⠸⣿⣤⠶⠶⠶⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⣤⣤⣼⣿⣡⡾⠛⠁⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠙⢷⡄⠀⠀⠀⠀⠈⠻⣤⣤⣄⡀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠞⠋⠉⠀⠀⠀⣹⡿⠋⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠈⠻⣭⡙⢷⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⣀⣤⡶⠾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠈⠿⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⣾⠏⠁⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣄⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠙⢿⣷⣶⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣠⣤⠶⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠑⡄⠀⠀⠀⠄⠀⠀⠀⠀⠑⠀⠀⠀⢀⠚⡁⠀⠀⠀⠑⠀⠀⠀⢠⠊⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢸⢤⣀⡀⠀⢈⣠⢤⠀⠀⢸⣶⣶⠖⠲⢶⡶⡷⠀⠀⢸⠀⠐⠀⠀⠀⠀⠀⠀⠀⢨⠀⠰⠀⠀⠆⠀⠀⠀⠀⠀⠀⠁⡀⡀⠁⠀⡁
⠀⠀⠀⠯⠮⡘⠉⠉⠝⡩⠦⠇⠀⠀⢆⠀⠠⠤⠒⡠⠂⠀⠀⠈⢦⡈⠂⠐⢁⡔⠁⠀⠀⠈⢆⡀⠠⠄⢂⠔⠀⠀⠀⠘⢄⠈⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⢀⡀⡀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠔⢁⠀⠀⡀⠑⠄⠀⠀⠀⠎⠀⠀⠀⠀⠑⡄⠀⠀⢀⠀⠁⠀⠀⠈⠑⡀⠀⠀⢀⠊⡀⠀⠀⢈⠱⡀⠀⠀⠀⠊⠀⠀⠀⠈⠣⡀
⠀⠀⠀⢸⠀⠹⠂⠀⠗⠀⢠⠀⠀⢀⠀⠈⠂⠀⠆⡀⠰⠀⠀⠀⠀⢮⠆⠰⡿⠀⢰⠀⠀⠀⠀⣤⠆⢰⣦⠀⡁⠀⠀⠀⠀⠰⠀⠀⠂⠀⡅
⠀⠀⠀⠀⠣⡰⠋⡿⠃⡠⠃⠀⠀⠀⠢⡀⠀⠒⢈⡠⠃⠀⠀⠈⠀⠠⠒⠒⠆⡠⠁⠀⠀⠈⢆⡀⠀⠀⢀⠔⠁⠀⠀⠈⠢⡚⠛⠛⢃⠜
⠀⠀⠀⠀⠀⠰⠴⠃⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉
⠀⠀⠀⣠⡄⣀⡀⢀⡀⡰⡀⠀⠀⠀⠀⣀⠤⠤⢄⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⡀⠀⠀⠀⠀⠀⢀⣠⠠⣀⠀⠀⠀⠠⡐⠈⠀⠁⠀⠉⠓⢆
⠀⠀⠀⠳⠊⠀⠀⠀⠈⠳⠃⠀⠀⠀⠎⠀⠀⠀⠀⠙⠄⠲⡒⢄⠎⠑⠀⠀⠈⠑⠤⠔⠖⠀⡔⠁⠀⠀⠀⠉⢄⠀⠠⠍⣛⣞⠀⢐⣖⠚⠣
⠀⠀⠀⡄⠐⠦⠀⠠⠖⠀⢳⠇⠀⠀⠁⠐⠂⠀⠂⠀⠇⠀⠁⢸⠀⠐⠂⠀⠒⠀⠠⠘⠀⠀⠀⠀⠁⠀⠁⠀⡸⠀⠀⠼⡔⢚⠃⠚⡳⠓⠃
⠀⠀⠀⠙⢄⡀⠒⠂⢀⠔⠁⠀⠀⠀⠢⣀⠈⠁⢀⠴⠁⠀⠀⠀⠣⣀⠀⠀⢁⡤⠁⠀⠀⠀⠢⡀⢿⣿⠇⡠⠃⠀⠀⠘⢄⠈⢴⡆⠁⡠⠃
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢈⡁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⢁⠀⣄⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠐⠤⠀⠉⠻⢕⣀⣀⠀⠀⠂⢠⠆
⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣴⠞⣡⠆⠀⣁⠠⢖⠀⠠⠄⡛⢿⣷⣇⠀⠦⠀⠀⢀⠞⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣧⠋⢤⢖⡠⢄⠦⠠⣤⡶⠦⡉⠚⢻⣬⣳⣧⠔⠒⠌⢀⠀⠀⣠⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⢀⡾⢃⣠⠖⠀⠉⠛⠂⠁⠀⠀⠈⠀⡩⢴⣿⣮⣿⣦⣐⣀⡘⡀⠀⢀⡀⢀⡠⠰⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⢸⢋⠀⠁⠀⢈⣤⣶⣶⣶⣃⠀⠀⡘⡴⠃⠙⣿⣷⣝⣿⠖⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠿⠉⠀⠈⢦⠀⠀⢀⢴⡿⣊⣤⣡⣾⣿⡟⢹⠉⠉⠉⠛⠀⠴⠁⢄⠀⡞⣿⣜⣧⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⣸⡟⢛⣋⣈⢸⠟⡉⠀⠀⡀⠀⠀⢀⠂⢰⠆⠀⣠⢹⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂⠈⠁⠀⣠⣿⠛⠟⠛⠙⠉⠀⠃⠀⠀⠀⠀⠠⠁⠀⠋⠰⣰⢧⣿⣿⣳⣤⡄⡀⠀⠀⠀⠀⡊⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣡⣿⢿⢳⠀⠀⠀⠉⠀⠠⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠠⠀⠀⠀⠁⠀⠢⢀⠀⠀⣤⣷⡆⠀⠀⠀⠀⠀⣠⢾⠟⠀⣈⡸⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⡿⣦⣧⠀⠀⠀⠀⢾⡟⢠⡇⣿⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⡀⠠⠄⠀⢀⠀⠀⠀⠙⢿⡟⣧⠀⠀⠀⠘⠆⣾⡏⢸⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠑⣎⠆⠀⠀⠄⢲⣽⡿⠘⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⣠⣜⠞⠀⠀⠀⠈⢿⠇⠀⠀⠁⢀⠀⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠘⢿⠝⠻⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠙⢄⡃⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠚⠶⢄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠠⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⣰⣦⠀⠀⣰⣶⠀⠀⠀⠀⠀⣠⡦⠀⠀⣴⣤⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢤⣀⠀⠀⣄
⠀⠘⠁⣤⡤⡌⠑⠀⠀⠀⠀⠀⠊⠁⢀⡀⠉⠑⠀⠀⠀⠀⠠⠁⢈⣀⡈⠀⠠⠀⠀⠀⠀⠠⠌⢁⣀⡀⠈
⠀⠀⠀⠘⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋



⢸⣿⠄⠀⢸⣿⠀⠀⠀⠀⠀⢠⣀⠄⠀⢄⣀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠐⠂⠀⠀⠀⠀⠀⠔⠢⠀⠠⠒⠲
⠁⠀⠠⣀⠄⠀⠉⠀⠀⠀⠀⠐⠀⠀⠆⠀⠐⠀⠀⠀⠀⠀⠁⠀⣤⢤⠀⠈⠀⠀⠀⠀⠈⠁⠀⢰⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈




⠀⣾⡇⠐⠁⣾⡆⠀⠀⠀⠀⠀⣶⡆⠀⠀⣲⡆⠀⠀⠀⠀⢸⣶⠀⠀⠰⣿⠆⠀⠀⠀⠀⠰⡴⠄⠀⠤⡴
⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠉⠡⢤⡤⠉⠑⠀⠀⠀⠀⠐⠀⢀⠤⡀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠂⡄⠰⢄⣴⣯⣷⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⢻⣿⣿⣿⣯⣿⣿⣿⣿⣦⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⣿⣿⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣮⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠍⠀⠛⠙⠯⠿⠽⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁

⠀⠀⠀⣠⣤⣾⠃⠀⠀⢀⣰⠟⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠂⠀⢀⣶⣖⣶⣆⢠⡀⠀⠀⠀⠀⢀⡀
⠀⠀⣼⣿⡿⠁⢀⣤⠶⠛⡁⠄⠒⠛⣿⡿⠃⣀⡤⡶⠦⣀⣺⠶⠚⢉⠁⠀⠀⣉⣯⣟⡻⠶⣄⡰⡄
⢀⣼⡿⠋⢀⡶⠋⠀⠀⠁⠁⠀⠀⠈⠙⢀⣢⣭⣿⡐⢛⣋⡄⢐⢎⡴⢒⡽⣚⡵⢋⡥⣒⠾⢛⣻⡄
⣾⡟⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣫⡵⢋⣴⢾⡫⠅⡰⠃⡾⣣⡟⢱⡷⡐⢉⡞⣡⣾⡿⡋⣧⡸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠏⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠣⣢⠽⣾⣁⡐⣚⠁⡸⢥⣏⢰⡏⣰⠣⢸⢰⠟⢋⠄⢠⡼⣿⣂
⠀⢀⣿⡇⠀⣤⠀⠀⠀⠀⠀⠀⢀⠟⠀⠐⣱⢞⡟⡱⣱⡇⢰⠃⠀⣡⠸⢰⢧⣤⢰⢀⠄⡀⠀⢀⡴⣿⢽⡰
⠀⣼⣿⠁⠀⣿⣆⡀⠀⠀⠀⡄⡛⣀⢂⣰⠷⠊⣴⣣⠟⣡⢇⡀⢰⢇⠀⣎⣟⡿⡿⠈⣴⠃⠂⢠⣆⣿⣷⠹
⢠⣿⠃⠀⢠⣿⣿⠛⣦⡀⠀⠀⢚⣦⠌⡋⠠⡠⠓⣫⣰⢫⠎⠜⣠⡎⢸⣹⠻⣱⠇⠀⣿⣷⠁⢊⢎⣯⣤⠐⠆
⣼⠃⠀⣠⣿⡿⢁⣾⡿⠋⢓⣶⠈⠀⢨⠁⢨⣰⢰⢗⢣⡍⡀⠀⠟⡄⠠⢧⠄⡃⡌⠀⢻⠋⣤⠆⠈⠛
⡏⠀⣴⣿⡟⢀⣾⠏⢀⣴⣿⠏⢠⡰⢸⡇⢎⡏⣼⢊⡴⢳⢡⡄⠾⢁⢖⠄⡐⠸⣵⡴⡼⢚⡄
⢇⣼⣿⣿⠠⠾⡋⠀⣼⡟⠁⠀⠀⠀⠸⢏⡜⠰⢃⣼⢡⢋⢫⠔⠰⠁⢀⡾⢿⢹⢏⠍⠀⠀⡀
⠀⣿⣿⣿⡄⠀⠀⣾⡏⡠⠀⠀⠀⠀⠀⠰⡆⠀⠘⠇⢣⡏⠀⠀⡰⢠⣼⡇⠾⠀⡌⠀⠀⠀⠂
⡀⠈⢻⣿⣷⣀⠀⡿⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢫⡀⠇⠁⣿⠰⡁⢸⠀⠀⠁⡀
⣷⡀⠀⠙⠿⣿⣷⣞⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⠋⠇⠘⢰⡜⢾⠁⠀⠀⠀⠀⠉
⣿⣿⣦⡀⠀⠀⠉⠏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⠀⠃
⡘⣿⣿⣿⣦⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣌⡛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄
⠟⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠖⢻⡿⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⣶⠆
⠀⠀⠀⢀⣀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⢀⣤⠶⠛⠉⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣁⣤⣤⣤⡄⠀⠀⠀⠰⠿⠿⠶⣶⣶
⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠉⠁
⠀⠀⠀⢀⣿⡏⠀⢀⣠⣴⣶⣿⡷⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⢸⣿⣷⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠙⠟⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⣠⣤⣶⠾⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠐⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⠀⠀⠀⠀⠀⠀⠙⣿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀
⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇
⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⢻⣿
⠀⠀⢰⣿⣿⠹⠇⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢶⣤⣶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⣤⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿
⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟
⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣾⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠉⠉

⠀⠀⠀⠀⠀⣀⠐⣪⠄⠀⠀⠀⠈⠀⢀⠀⠀⠀⢠⠄⠂⠀⡀⠀⡄⠂⠐⠀⢰⡈⢀⠀⠀⠀⠀⠀⣴⠀⠀⣤
⠀⠀⠀⠀⢂⡭⠞⠠⠀⠄⠀⠀⠀⠑⠛⠁⠘⠀⠈⠂⠠⠋⠀⠀⠀⠂⠀⠀⣝⣤⡏⠀⠀⠀⢠⡄⣿⢰⡆⡇
⠀⠀⠀⠀⠈⡰⠈⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⠀⡘⠮⡱⢉⠆⠀⠀⠈⠁⠈⠘⠀⠁
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢸⡏⣦⢠⡄⣿⠀⠀⠀⠟⠒⠚⠁⠠⢈⠢⡑⢂⠐⠀⠄
⠀⠀⠀⠀⠈⠀⠠⠤⠀⠠⠀⠀⠀⠘⠧⠙⠈⠃⠙⠂⠀⡀⠀⠀⠀⠀⠀⠊⡰⣀⠒⠀⠀⠀⠆⠀⠠⠀⠀⠂
⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠘⠁⠀⠏⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⡖⠀⠀⠂⠀⠀⠀⠈⠆
⠀⠀⠀⠀⡓⠀⠀⠐⠀⠀⠂⠀⠀⠀⠤⠀⠊⠂⠀⠀⠀⠈⠀⠀⠀⠀⣿⠸⣷⡿⠸⣷⡀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⡏⠉⠀⠈⠁⠀⠀⣁⠄⢶⠄
⠀⠀⠀⠀⠀⠀⢠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⢠⣀⡀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⣤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠈⣀⠀⢠⠚⢁⠾⠋⠛⠧⠀⢠⠀⠀⠀⠰⠃⠀⠀⢀⠄
⠀⠀⠀⠀⢀⠌⠁⠶⠉⠀⠈⠐⠀⠒⠂⠊⠀⠀⠀⣈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠲⣠
⠀⠀⠀⠀⢌⢠⡄⠀⠈⠢⠀⠀⠀⠀⠀⠠⠤⢐⠂⢻⣴⠐⣷⠸⡇⣿⠀⠀⡠⠤⠄⠀⠐⠂⠀⢈⠣⡙⠁
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠀⠀⢠⠀⠜⢄⠀⠀⢀⠀⡐⠁⠀⠀⢶⡆⠀⠀⠀⢨⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣷⠀⠀⢡⡀⠀⠀⡔⠉⠉⠳⡄⠀⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡀⠁⢠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠃⠀⠀⣆⠀⠀⢠⠇⠀⢀⠀⠁⠐
⠀⠀⠀⠀⠀⠠⠸⠆⠷⢺⠿⢹⠾⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠀⠄⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⡄⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠑⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⢸⠃⠀⣴⡄⠀⠠⡤⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣴⡄⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣴⡄⠀⢠⡤⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢸⡀⢀⠀⠀⠀⠀⣀⡴⠀⡆⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢸⡀⠀⢀⡠⠤⢀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠉⠛⠛⠋⠉⢀⡼⠃⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠈⢷⡀⠉⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣤⡴⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠤⠴⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠁⠙⢷⣄⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⡀⠀⠀⡀⠀⠈⣧⠀⠀⠀⠀⢠⡟⠀⠀⡀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⡞⠀⠰⡶⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢋⣀⣈⠃⠀⠀⣼⠀⠀⠀⠀⢸⡇⠀⠀⠉⠀⠈⠂⠀⠀⡿⠀⠀⠀⠀⠸⡇⠀⢀⣀⣀⣀⣀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠿⣿⡿⠇⠀⣠⠏⠀⠀⠀⠀⠈⢧⡀⠠⠚⠓⠊⠑⠀⣸⠇⠀⠀⠀⠀⠀⢳⡀⠈⠻⠿⠿⠿⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣀⣠⣤⠞⠃⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣄⣠⣤⠶⠋


⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⠀⠐⠲⣤⡀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠉⠛⠳⣤⡀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠉
⠀⠀⠀⠀⠀⠀⢰⠃⠀⣀⠀⠀⢀⣀⠀⢻⡄⠀⠀⠀⠀⣴⠋⠀⣤⠀⠀⢀⡀⠈⢳⡀⠀⠀⠀⠀⢀⡞⠀⠠⠄⠀⢀⠤
⠀⠀⠀⠀⠀⠀⣯⠀⠀⠉⠁⠀⠈⠁⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠉⣠⣦⡌⠁⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠐⠋⠉⠉⠀⠀⢠⡟⠀⠀⠀⠀⢻⡄⠀⠐⣿⣿⡷⠀⠀⢰⠇⠀⠀⠀⠀⠸⣆⠐⠢⠤⣤⠤⠶⠋⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⣀⡴⠏⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠉⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⢀⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉



⡿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣛⣿⣿⣿⣿⣿

⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⣠⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⣀⠀⠀⠀⢀⣠⣴⣶⣶⣦⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣦⣠⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣷⣆⣰⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣠⣶⣿⣷⣦⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡄⠀⠀⠀⣴⣾⣿⣿⣷⣶⣤⡀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣶⡄
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣄⢀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣯⣿⣿⡿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣻⣿⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⢏⣴⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣌⠻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣧⡘⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡾⢰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⢂⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠃⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣤⣄⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠿⣟⢛⣙⣟⡟⠛⢶⣤⣖⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⣨⣾⢿⣟⣻⣿⢿⣿⣆⡈⢿⣿⣶⡶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢇⣾⣿⡟⣫⣯⣿⣟⠻⣮⢿⣷⣆⠩⣿⣷⣄⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⢼⣿⣏⢸⣯⣰⣾⣿⡧⢻⣿⣿⡟⠈⢹⣿⣦⠩⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢨⣿⣿⣾⣽⣻⣵⣿⡇⣿⣿⣯⣶⠀⢈⣿⣿⡷⠨⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡉⠻⠿⢿⢿⠿⢋⣰⣿⣿⣿⣿⣭⣶⣾⣿⢿⣷⣿⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⠿⣿⣿⣟⣻⣹⣩⣵⣾⢳⠟⣟⣷⡶⣴⣐⠲⣎⠀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠤⠄⢀⣴⣮⣣⣋⡹⢿⣛⣛⣛⣛⣛⡋⢓⣛⣓⣛⣙⡛⣉⣚⣿⡮⣛⣺⢿⣭⡾⣿⣷⠄⢲⣯
⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠟⠛⠛⠉⠉⠉⠁⠀⠁⠉⠉⠉⠉⠀⠂⠉⠉⠉⠉⠉⠙⠓⠂⠤⠤⠴⠆⠚⠚⠛⠙⠃







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢘⢰⠀⢰⠁⡆⠀⡇⢘⡦⡇⠀⡇⠦⢸⠀⢸⢰⡐⡂⡆⠀⡿⢺⢖⠅⢶⢀⠸⢰⣶
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁
⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⢠⠋⣤⠀⣄⠑⡄⠀⠀⡰⠁⣤⢠⡄⠱⡀⠀⠀⡔⢁⡄⢠⡌⠳⠀⠀⠀⡜⢁⡀⢠⡌⢣
⠀⠀⠀⢠⠀⡀⠀⢀⠀⡇⠀⠀⢇⠀⣀⣀⡀⢀⠇⠀⠀⡇⢠⠀⠀⣠⢀⠀⠀⠀⡆⢀⠀⠀⡀⢨
⠀⠀⠀⠀⠓⠭⠄⠥⠊⠀⠀⠀⠈⠓⠤⠤⠔⠊⠀⠀⠀⠈⠢⠥⠬⠕⠃⠀⠀⠀⠈⠢⠥⠬⠔⠁

⠀⠀⠀⢀⡴⠒⠀⠒⢄⠀⠀⠀⢀⣴⠆⠀⠲⣄⠀⠀⠀⢀⠔⠂⠐⠲⢄⠀⠀⠀⡠⠒⠉⠉⠒⢄
⠀⠀⠀⠘⠀⠛⠀⠛⠀⡇⠀⠀⡎⠀⠛⠘⠃⠈⡆⠀⠀⡇⠐⠃⠘⠃⠈⡀⠀⠀⡁⠐⠃⠐⠂⢸
⠀⠀⠀⠘⢄⠂⠀⢀⡰⠃⠀⠀⠱⣄⠀⠂⢀⡴⠁⠀⠀⠳⣘⠿⠿⢋⠜⠀⠀⠀⠳⣔⣤⣤⣣⠎
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⢀⡀⠤⣀⠀⠀⠀⠀⠀⢀⡠⠤⣀⠀⠀⠀⠀⠀⢀⡠⠄⣀⠀⠀⠀⠀⢀⡀⠤⢀
⠀⠀⠀⠰⠋⣤⠀⣤⠙⡄⠀⠀⡰⠡⣦⢠⡦⠑⡄⠀⠀⡜⢡⡦⢰⡎⠳⠀⠀⢰⠋⣤⠀⣤⠙⡄
⠀⠀⠀⢠⠀⣤⣶⣄⢀⠇⠀⠀⢇⠀⣠⣤⡀⢀⠇⠀⠀⢇⠀⣠⣄⡀⢠⠂⠀⢸⠀⣤⣤⣤⠀⡆
⠀⠀⠀⠀⠙⠥⠤⠝⠋⠀⠀⠀⠈⠓⠥⠤⠕⠋⠀⠀⠀⠈⠒⠤⠤⠔⠁⠀⠀⠀⠑⠪⠭⠕⠊

⠀⠀⠀⢀⠄⠈⠁⠒⢄⠀⠀⠀⡠⠊⠉⠉⠒⢦⡀⠀⠀⢠⠔⠊⠐⠒⢤⡀⠀⠀⡠⠀⠀⠈⠑⢦
⠀⠀⠀⣼⠀⠃⠐⠁⠈⡇⠀⠀⡇⠀⠃⠐⠃⡀⣧⠀⢰⠃⠈⠃⠐⠂⠀⡇⠀⢰⠃⠀⠂⠀⠂⠈⡇
⠀⠀⠀⠹⣄⠲⠄⠋⡠⠃⠀⠀⠳⣈⠳⡐⡹⡠⠃⠀⠈⢧⣘⢇⠀⢚⠰⠃⠀⠈⢧⡀⠂⠈⠃⠜
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⢀⠠⠤⣀⠀⠀⠀⠀⠀⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣰⢡⢤⡄⡤⡷⡄⠀⠀⡜⠘⡄⢀⠁⠙⡄⠀⠀⡔⠀⡄⢠⡄⠱⡄⠀⠀⡔⢠⡀⢀⠈⢳
⠀⠀⠀⢧⠈⢉⡀⡈⢀⡏⠀⠀⣇⠀⡠⠤⡄⢀⡇⠀⠘⣇⠀⠄⠀⠀⢠⠇⠀⠀⣇⠀⠀⢤⠀⡸
⠀⠀⠀⠈⠓⠠⠄⠀⠈⠀⠀⠀⠈⠒⠀⠀⠀⠈⠀⠀⠀⠈⠓⠀⠀⠀⠁⠀⠀⠀⠈⠓⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠒⠒⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⡉⠀⠀⠀⠀⠀⠀⠠⢝⠢⣄
⠀⠀⠀⠀⠀⠀⢀⠔⠩⠐⠊⠁⠁⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠄⠃⠀⠀⠀⠀⠀⠀⠀⡐⠦⡙⢎⠣⡀
⠀⠀⠀⠀⢀⠔⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢠⠎⠤⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⢄⠱⡙⡄⢣⠱⡄
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄⠀⠀⣠⡀⢠⠏⠌⠁⠀⠀⠀⠀⠀⠀⢀⠐⠁⠐⡄⢢⠡⠘⡀⡆⢧
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠳⣌⠋⠀⠀⠀⠀⠀⠀⠀⢀⠐⠄⠁⠀⠱⠰⠀⡆⠃⢁⢰⢸
⠀⠀⠀⡆⡡⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣳⠃⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢌⢂⠱⠀⢠⠘⡀⠃⡆⢡⢰⠈⢸⢨
⠀⠀⠀⡇⠔⡡⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠈⠱⡢⣄⠀⠄⡐⠘⠈⡌⠀⠈⡴⠀⡇⠀⠁⢸⢸⠀⠼⡀
⠀⠀⠀⠀⠐⡡⢂⠀⠉⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡆⠀⠑⠈⠀⣹⡆⢀⠀⠃⡀⢀⠁⡇⠀⠇⠀⠀⢸⠘⢠⢀⠃
⠀⠀⠀⠰⠐⡱⢁⠔⡀⠈⠀⠊⡔⢁⠔⣠⢂⠔⡠⢀⠆⡄⠀⠀⢞⡟⠁⠀⠐⢣⡾⠋⠀⠀⠀⣄⠣⣽⠀⠟⠀⠆⡄⡆⡈⠀⡈⡌
⠀⠀⠀⠀⠡⠀⢁⠎⢠⠂⡠⠀⡀⠀⠀⠀⢀⠄⢀⠀⡀⢠⢀⢂⠌⣱⡄⠀⠀⠀⢱⣄⠱⡱⠰⠘⢠⠇⡘⡏⠰⢠⢀⠁⠁⡄⠷
⠀⠀⠀⠀⠀⢣⠀⠔⢁⠎⠠⠊⡠⠊⡰⢣⠊⠔⢁⠌⡐⡡⢣⠊⡴⣿⢃⡄⠀⣰⡟⠁⡰⢡⢃⢆⣾⢀⠿⢀⠃⡆⠘⠰⡰⢠⠃
⠀⠀⠀⠀⠀⠀⠡⡀⠈⠰⢁⠜⢀⠌⡐⢁⠊⠠⢃⠌⡐⠠⣡⢞⠞⠁⠈⠁⣴⡟⠀⡴⢡⢊⠎⣼⢃⠎⠇⡜⢠⢠⢃⢃⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠎⢠⠊⡜⠠⠃⡐⢡⠂⠌⠠⢁⠻⣏⠲⠀⠶⠀⢻⡄⠊⣠⢃⠎⡐⡌⡘⡜⢰⢁⠇⡆⡜⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠁⡌⠰⠁⡔⠠⢁⠎⠠⢡⠃⣮⡿⠀⠀⠀⠠⡟⠁⡔⢁⠎⡰⡱⡑⢰⢀⠇⡜⡰⡰⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⡔⠠⢁⠊⢠⢃⠂⡄⡈⠑⣦⡀⠙⠀⠉⣢⡀⠃⠴⠵⠳⢡⠇⠞⡴⣰⢣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢁⠃⡌⠀⠂⡌⠠⢠⢁⠮⡻⠡⠰⠀⣼⠟⢁⠞⡴⠱⢃⠏⠞⠰⠑⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠑⠤⡈⢰⢠⠁⠃⢸⣾⠁⠀⠀⠀⢹⠰⢃⠞⡰⢁⠏⡜⢸⡡⠋⠔⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⣐⠰⢀⢿⡆⣠⡀⠀⠀⢇⠉⡴⡡⠊⢎⠔⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠼⠇⠀⠀⠀⠀⠘⣌⣐⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⢜⠆⠀⣬⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⢀⣀⠀⠰⡦⠈
⠀⠀⠀⠀⢀⠄⡠⠀⡠⢀⠄⡠⢀⠄⢀⠄⡀⣀⠀⡀⠀⡠⢀⠀⡀⠈⡉⠀⠀⡀⢀⠀⢠⠀⡀⢀⠀⡀⢀⢀⠀⢀⠀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠉⠐⠉⠒⠑⠊⠐⠁⠒⠁⠊⠐⠡⠊⠴⠋⠔⠡⠊⠠⠊⠔⠠⠊⠔⠡⠊⠠⠊⠔⠁⠊⠐⠑⠃⠂⠁⠊⠐⠁⠊⠐⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣻⣟⣿⣿⣿⡿⣿⣿⢟⣛⣦⢾⡀⢀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⣟⣿⣷⣿⣶⣦⣿⢻⣿⣼⣿⣶⣶⣶⣦⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣿⡟⠙⣿⣿⣿⣿⣿⣟⠃⠛⠃⠛⠉⠋⠙⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢋⡞⣻⣯⣿⣿⣯⣭⣭⣷⣿⣶⣧⣄⣤⡬⢵⣄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡙⣿⢻⣾⣿⣿⠟⣿⣿⣿⣶⣿⣾⠿⣿⣾⣟⣛⣹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⢞⣭⣽⣾⣿⣾⡿⣻⣿⣿⣩⣿⣤⣿⠿⣿⡿⣽⣿⣍⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣨⣿⣿⠟⣻⣿⠻⣿⣿⣿⣿⣿⢷⣿⣟⣻⣿⣿⣿⣤⣿⣏⣿⣟⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⣙⣏⣿⣿⣿⣯⢻⣿⣿⣿⣬⣟⣿⣏⣻⣧⣹⣿⣿⣿⣿⢿⣶⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣟⣹⣿⣻⣿⣿⣿⡿⣿⣡⣿⣿⣧⣽⣿⣿⣿⢧⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⣼⢟⣿⠿⣿⢿⣯⣽⠿⣿⠟⠉⠛⠋⠿⠿⠿⠿⠷⠟⠻⣷⣾⡟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣾⣷⣿⣿⡾⡟⠉⠉⠁⠀⠀⠠⣄⠀⠀⠀⡠⠤⠀⠀⢻⣿⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢷⣿⣿⠛⠀⠀⠀⠀⠀⠀⣤⠈⠀⠀⠀⣤⡆⠀⠀⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠿⣏⠀⣶⡉⠁⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠛⠁⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢃⡝⢷⡭⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣯⣿⠁⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢨⣷⡻⣆⠀⠀⠀⠀⠋⠈⠉⠃⠒⠒⠒⠚⠋⠉⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⣹⣟⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢲⡵⣿⣿⡯⣟⣷⣶⣤⣤⣄⣠⣤⣤⣤⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣶⣾⣭⣿⡿⠻⣿⡶⣶⢿⡽⠷⣶⠂⠀⡒⠀⡁⠂⠄⢠⣬⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠏⣿⣷⣷⣶⣿⣯⣞⣭⣎⢷⢥⣐⡢⣀⣤⢠⣴⣿⢷⣽⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡻⢿⣾⣿⡿⣾⣿⣿⣿⣿⣿⣿⡻⣿⣟⡃⣿⣭⡜⣿⣿⣮⣿⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢛⢶⣻⣿⣿⡳⣾⣿⣿⣸⣿⡋⢩⣻⣿⣨⣽⣿⢹⠷⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣿⣿⣿⣾⣿⣯⣿⣿⣝⣛⣛⢛⡛⠋⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣹⣿⣿⣿⣏⣹⣿⣿⣿⣿⣭⣭⣤⣥⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣸⠞⢼⣿⣮⡓⢀⠀⠀⢰⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⡞⣿⠻⡏⠙⢦⣉⣲⣿⣍⢻⣿⣿⣗⣹⣿⣿⣿⣿⣿⡟⠉⠙⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣇⠀⣀⢻⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡄⡀
⠀⠤⡤⢤⣄⠀⡀⠀⠀⠈⠁⣘⣽⣿⣟⣯⣿⣿⣿⣿⡿⢿⣷⣿⣯⣹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠈⠃⠤⣶⡿⣿⣿⣿⣿⣧⢽⣿⣿⣯⣿⣿⣿⣟⣿⣿⡀
⠀⠀⠀⠈⢳⣠⢠⣤⡈⠀⣤⡀⠀⢀⡐⢀⣈⡽⠛⠛⣿⣾⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠈⠔⠁⠀⢊⠽⠿⢦⣴⣏⣙⣿⣧⡇⠴⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠈⠀⠀⠀⡠⠉⠀⠠⡀⢼⡷⠿⣩⣑⣠⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⣠⠤⠇⢠⡄⠀⠟⠋⠠⣠⣿⣿⣧⣿⣟⣿⣿⣿⣿⣿⣿⣿⣧⣿⣯⣿⠆
⠀⠀⠀⠀⠀⠘⠁⢸⣴⡥⠊⢁⣠⣤⣄⣤⣤⡵⣿⣿⣿⣿⢻⣿⡿⣿⡿⣿⣿⣿⣿⡏
⠀⠀⠀⠐⠀⠀⠴⡾⠋⠤⠌⠠⣶⣿⣭⡷⣬⢻⣿⠿⣿⣿⣿⣿⣲⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠣⡤⠀⠀⢶⣶⡾⡉⠐⠀⣌⣈⣰⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⣈⣉⣙⣩⡷⠶⢭⡋⢰⠐⣻⣿⡿⠻⠿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠳⡄⠀⣠⡿⠚⠛⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣴⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠠⠐⠀⡉⠐⠠⡀⠀⠀⡠⠒⠈⠁⠒⢄⠀⠀⢀⠔⠂⠉⠐⠢⡀⠀⠀⡠⠒⠈⠁⠐⢄⠀⠀⠀⡤⠂⠈⠁⠒⢄
⠀⠀⢠⠁⠀⠈⠉⠀⠀⢱⠀⡘⠀⠘⠀⠀⠃⠀⠆⠀⠇⠀⠀⠉⠀⠀⠘⠀⢰⠀⠀⠀⠀⠃⡀⢁⠀⡌⠀⠀⠰⠆⠀⠈⡆
⠀⠀⠘⡄⠀⠀⠀⠀⠀⡘⠀⠠⠀⠀⠀⠀⠀⢀⠃⠀⢄⠀⠀⠀⠀⠀⡰⠀⠐⡀⠳⠦⠰⠶⠃⡌⠀⠁⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠈⠂⠤⠤⠄⠊⠀⠀⠀⠑⠢⠤⠤⠐⠁⠀⠀⠈⠑⠤⠤⠤⠊⠁⠀⠀⠈⠂⠤⠤⠔⠊⠀⠀⠀⠑⠢⠤⠤⠐⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡐⠊⢀⠀⡈⠑⡄⠀⢀⠔⢁⠀⠀⠉⠢⠀⠀⡠⠊⠀⣀⠀⠑⢄⠀⠀⠔⠉⠀⣀⠈⠲⡀⠀⢀⠂⠁⠀⠀⠉⢢
⠀⠀⢰⠀⣤⣈⣀⣠⡄⠘⠀⠌⠀⠈⠁⠈⠁⠀⡇⠀⠃⣴⣶⣶⣶⣶⠈⠀⠸⠀⡀⠀⠀⠀⡀⠠⠀⡆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⢆⠈⠋⠉⠛⢀⠄⠀⠐⡀⠀⠀⠀⠀⡠⠃⠀⠣⡈⠛⠛⠛⢁⠜⠀⠀⢄⠉⠈⠉⠀⢁⠆⠀⠰⡀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠁⠒⠒⠂⠁⠀⠀⠀⠈⠑⠂⠒⠈⠀⠀⠀⠀⠈⠒⠒⠒⠁⠀⠀⠀⠀⠁⠒⠒⠊⠁⠀⠀⠀⠈⠑⠂⠒⠈
⠀⠀⠀⢀⠤⠒⠐⠢⢄⠀⠀⠀⣠⠔⠒⠒⠠⡀⠀⠀⠀⡠⠐⠒⠂⢄⠀⠀⠀⢀⠤⠒⠒⠢⢄⠀⠀⠀⡠⠔⠒⠒⠤⡀
⠀⠀⢠⠃⠀⠀⠀⠀⠀⢁⠀⠐⠁⠀⢠⡄⠀⠈⡆⠀⡎⠀⠀⣤⠀⠀⠱⠀⠠⠁⠀⢀⣀⠀⠀⢃⠀⡘⠀⠀⢠⡄⠀⠈⡄
⠀⠀⠘⡀⠀⠀⠛⠀⠀⠘⠀⢡⠀⡾⠟⠻⢷⠀⠄⠀⡄⢠⠄⠀⠀⢰⢠⠀⠨⠀⠀⠀⠀⠀⠀⡈⠀⢁⠀⠀⠠⠄⠀⠀⠇
⠀⠀⠀⠐⠤⢀⣀⠠⠔⠁⠀⠀⠑⠄⢀⡀⠠⠊⠀⠀⠈⠢⠀⣀⠠⠔⠁⠀⠀⠑⠤⢀⣀⠠⠘⠀⠀⠀⠓⠄⢀⠀⠤⠊

⠀⠀⠀⡠⠂⠉⠈⠁⠢⡀⠀⠀⠔⠈⠉⠉⠑⢄⠀⠀⢠⠒⠉⠉⠉⠒⡄⠀⠀⡠⠒⠉⣉⠑⠢⡀⠀⢀⠔⠈⢉⡉⠐⢄
⠀⠀⢰⠀⡀⠀⠓⠀⡀⢐⠀⠈⠀⠀⢠⠄⠀⠀⡇⠀⠃⢀⣠⣭⣤⣤⠘⠀⠘⠀⣶⣤⣭⣥⡐⢡⠀⡌⠀⠀⠈⠁⠀⠀⡆
⠀⠀⠈⢄⠙⠖⠒⠶⠃⡄⠀⠰⡀⠀⠀⠀⠀⢀⠃⠀⢇⠻⠿⠿⠿⠟⡐⠀⠐⡄⠙⠛⠻⡿⢃⠄⠀⠡⡀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠈⠑⠀⠄⠐⠈⠀⠀⠀⠈⠒⠠⠄⠒⠁⠀⠀⠀⠑⠂⠤⠐⠊⠀⠀⠀⠈⠐⠀⠄⠒⠁⠀⠀⠀⠈⠂⠠⠄⠂⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠂⠉⠉⠀⠀⠀⠀⠉⠉⠐⠒⠠⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠉⠀⠀⠀⠈⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⣴⣾⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠚⠋⠉⠀⠀⠀⠀⠈⠙⠻⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⢀⣠⣎⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⢑⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠸⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠏
⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣶⣦⣤⣤⣀⣀⣀⣀⣤⣤⣴⣶⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠄⣀⣀⠀⠀⠀⠀⣀⣀⠠⠤⠐⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣁⣀⣀⣀⡀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢚⣉⠤⡤⠤⣄⣈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢸⣷⠀⠀⠀⠈⡦⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⡞⠀⣎⣀⡜⠁⠀⠀⠈⢣⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠛⠒⠒⠒⠚⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢆⠈⣿⠉⡉⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠦⠆⠀⢣⠀⠀⠀⠀⠀⠀⠘⡇⠀⠁⠀⠀⠈⠀⠀⠤⠶⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠐⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠐⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣴⠲⠤⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⢰⠲⠤⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠸⣄⡀
⠀⠀⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⠢⠄⠀⠀⠀⠀⠀⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⣛⡛⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⣛⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⣤⣉⣵⣯⣿⣿⣧⠌⢳⡀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣠⢤⡘⢩⡷⠠⣤⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⣻⡾⣿⢿⣿⡟⠺⠿⠆⠀⡇⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢺⠀⢸⠀⢰⠋⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠸⣹⠞⠛⠛⠑⠀⠀⡀⡀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠸⡄⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⣀⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⣀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⣀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠈⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠿⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠟⡿⢋⡿⣀⠀⠀⠀⠀⣰⠏⠁⠀⡄⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠐⢋⠔⠁⠀⠀⠀⢠⣟⡀⠀⠀⠃⠀⠠⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠋⣀⡰⠄⠀⠀⠈⣿⠙⠛⠶⠶⣶⣶⣶⣦⣼⠷⠀⠀⢀⡴⠤⠤⣦⣤⣤⣤⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⠶⣦⣤⠶⠞⠛⠁⠀⠀⠀⠀⠐⡃⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢀⡴⠋⠀⣠⡶⠋⠀⠈⢦⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⡸⢋⡴⠋⠀⣠⡾⠋⠀⠀⠀⠀⠀⠙⢦⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠇⢀⡀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⣴⣿⡋⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡾⠟⠉⠃⠀⠀⣀⣽⣿⣧⠞⢁⣶⠖⠀⡶⠀⢀⣴⣦⠀⡀⠀⣀⢸⡷⣄⠀⣿
⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢸⣧⣤⡤⠶⠛⢻⡏⡇⠈⠀⠀⢀⣿⠸⣄⠀⠰⢊⣽⡏⣿⡇⠀⠀⠻⣦⠀⣇⠀⢸⡇⣿⠆⢸⣼⣿⣾⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⡇⢸⠀⠀⣿⣿⠀⠙⢷⡾⠋⣽⢩⣿⠉⢦⣄⠶⡿⠃⢿⡄⠘⠷⠟⠀⠘⠿⠘⠟⢀⡴⢻⣿
⠀⠀⠀⠀⠀⠀⠸⣆⡀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⡇⡇⢸⠀⠰⡏⣿⠀⠀⠀⠀⢰⠏⠘⣿⣄⠀⠙⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⡿⢿⣧⠀⠀⠀⠀⢸⡇⡇⠘⢸⠀⡇⢿⡀⠀⠀⠀⡿⠀⠀⠠⠙⢷⣄⠀⠙⢷⣄⠀⠀⠀⠀⠀⢀⣴⠋⠀⢠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠟⠛⠛⠶⣆⣸⣇⣇⠀⢸⣄⠇⢸⡇⠀⠀⣼⠃⠀⠀⢸⠀⠀⠻⣦⡀⠀⠙⢷⣄⠚⠃⣠⠞⠁⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟⠙⡛⣻⠀⠈⣿⣀⣼⠋⠀⠀⣷⠸⠀⠀⠀⡏⢹⡶⠶⢦⣽⡷⠾⢷⡦⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⡇⣿⠀⠀⠈⣿⠃⠀⠀⠀⡟⢸⠀⢸⠀⡇⢾⡇⠀⠘⣿⣙⠃⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢸⡆⡇⣿⠀⠀⢰⣿⠀⠀⠀⠀⡟⢸⠀⠈⣴⢁⣾⣧⠀⠀⣿⣩⣤⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⣟⣘⣁⣠⣿⠀⠀⢸⣿⠀⠀⠀⢰⠁⡆⠀⢸⣿⣿⣽⣻⣷⣄⡼⠿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢈⣻⠍⠀⢀⡀⠀⢘⣿⠀⠀⠀⢸⠀⡇⠀⢸⣏⢀⣀⢨⣧⠈⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣰⣿⠆⠉⢈⡇⠀⠈⣿⠀⠀⠀⡈⣶⠁⠀⡼⣿⣭⡭⠼⡟⠀⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁⣶⠀⣼⢰⡄⠀⠹⣧⣀⠀⠇⣿⠀⠀⣷⠃⣿⠀⠀⣧⠀⠿⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢻⡆⠈⣨⣥⡤⠴⠟⠙⠛⠛⠿⠶⠴⠿⠾⠋⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠻⣦⠶⢿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠈⠛⠷⢶⣦⣤⣤⣤⣶⠶⠾⠛⠋⠉⠁⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠲⠦⣤⣤⣤⣶⣶⠒⢲⡿⠁⠀⠀⠀⢸⣿⠈⠓⠒⠒⠓⠶⢺⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠐⠒⠒⠒⠀⠀⠉⠀⢸⡇⠀⠀⠀⠀⢸⣿⡉⠉⠉⠙⠓⠒⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠄⠐⠒⠒⣒⣂⣀⢸⡇⠀⠀⠀⠀⠀⣿⣯⣤⣉⡛⠒⠒⠒⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠛⠛⠛⠉⠉⠉⢹⠇⠀⠀⠀⠀⠀⣸⣿⠀⠉⠛⠛⠛⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠉⠲⠶⠦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⡀⠀⠀⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⣄⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢀⠖⠈⠑⣤⠀⠀⠀⠀⠈⠓⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⡀⠀⠀⢀⡘⠈⠙⢦⣀⠀⠀⠀⠀⠈⠃⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣩⡷⣦⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⡾⠋⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠆⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠈⡇
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠂⠀⠀⠀⣠⡟⠀⠀⣠⡶⠿⠄⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⣀⠀⠰⠿⢶⡄⠀⢀⣠⣾⠋⠀⠀⣿
⡄⠒⢤⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⠶⠷⠶⠋⠁⢸⡇⠀⠀⣿⠀⢀⠐⠒⡀
⠁⠰⠆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢸⠇⠀⠀⣿⠀⠘⠃⠐⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⠶⠶⣦⣸⡟⠀⠀⠀⢰⠏⢀⡀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣀⠀⠿⠄⠀⠀⡟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢠⠟⠁⣤⠀⠀⠛⠁⠀⠀⠀⠘⣄⠈⠁⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠉⢀⡶⠀⠀⢸⣿⡄⠀⠀⡟
⡏⢀⠜⠀⠀⠀⢸⡄⢸⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⢿⣀⡿⠀⢸⡇⠀⠘⢃⠈⡁
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠻⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣶⣶⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣄⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠄⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⠻⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⠇⢠⡝⠃
⠀⠀⠀⢀⣀⣰⠳⢆⠀⣠⢄⠙⢶⡠⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⣠⡾⠁⠘⣾⠁⠀⠀⡠⣄⠀⣠⠞⣦⣀
⠀⠀⠀⠻⣍⡙⠳⠈⠛⠁⢸⠁⠈⠻⣆⠀⠀⠙⠻⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠋⠀⢂⡿⠁⠀⠀⠈⠣⠈⠟⠁⠾⢛⣉
⠀⠀⠀⠰⣯⣁⡀⠀⠀⠀⠘⠧⣤⣄⠉⢳⣄⠀⠀⠀⠈⠉⠙⠓⠒⠶⠒⢶⡤⠀⠠⡤⠶⠶⠶⠞⠋⠉⠀⠀⠰⣹⠞⠁⠀⣀⣤⡸⠇⠀⠀⠀⠀⣉⣽
⠀⠀⠀⠀⠀⠀⠈⠛⠃⠚⠲⠦⣄⡀⠙⠋⠻⠷⠌⣡⣄⣀⣠⣶⡶⠶⠶⠾⣄⠀⢀⡴⢶⠶⠶⢶⣆⣀⣀⣀⣴⠿⠔⠛⠋⢀⣀⠴⠖⠁⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢤⡄⣀⡀⠀⠁⣹⡇⠀⠀⠀⠂⠀⠤⠌⠀⠀⠀⠀⢸⡏⠈⠀⢀⣀⣠⡤⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠻⢷⣴⡆⠀⠀⠀⠀⠀⠀⠀⢸⣦⣼⡿⠟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠛⠛⠛⠛⠛⠛⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⡀⠀⡀⠀⣀⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠋⠻⠟⠻⠟⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣿⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢰⣿⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠹⡇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣾⠀⣧⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⣿⠀⣿⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣿⠀⢸⡃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠾⣟⠛⣿⠀⣸⡛⣿⠿⠶⣦⣀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣷⣶⣶⣦⣤⣟⣀⣻⣤⣤⣶⣶⣾⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠂⡠⢔⡂⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠄⠀⠀⠀⠀⠠⠀⡄⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀
⠀⠀⠀⠀⢰⠰⢓⣰⡑⠁⡀⠀⠀⠀⠠⢦⡾⠆⠠⠄⠀⢪⠀⢀⡄⠃⠃⠁⢀⠌⠀⠐⠼⠃⡔⠀⠀⠀⠀⢀⠀⣿⢠⡄⣿⠁
⠀⠀⠀⠀⠀⠖⢉⠠⢫⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠⠻⡛⡧⡀⠀⠀⠀⢻⡄⠿⢸⡇⢻
⠀⠀⠀⠀⠀⢀⠏⠀⠈⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⢠⡆⠀⠀⠀⣤⡀⢀⣀⠆⠐⡤⠁⡌⠱⢁
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠰⣦⢸⡇⠀⠀⠀⠉⠈⠉⠀⠀⠚⣄⠙⡌⠁⢌⠀⠘
⠀⠀⠀⠀⠀⠁⠀⠦⠒⠂⠒⠂⠄⡀⠀⠙⠷⠈⠁⠙⠀⠙⢀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠔⣠⠉⠆⠀⠀⠘⠀⠀⠠⠀⠀⠈
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⡈⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠊⠀⠀⠑⠀⠀⠀⠀⠙
⠀⠀⠀⠀⢘⢆⠀⠀⠠⠀⠀⠰⠀⠀⠀⠀⣠⠀⠢⠣⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⢺⣦⣿⠐⣿⡄⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠄⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢸⡏⠛⠀⠘⠃⠀⠀⠀⠃⡀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠀⠀⠄⢀⠆⠀⠀⠀⡀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈⠉⠘⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠑⠞⠀⠈⠙⠃⠒⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⣄⠄⣻⠴⣦⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠄⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⢠⡄⠄⠤⠀⢠⡏⠀⠌⠛⠁⢀⠝⠁⠈⠦⢀⠀⠀⡀⠃⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀
⠀⠀⠀⠀⠀⣴⠁⠀⠛⠀⠀⠀⠀⠈⠐⠀⠉⠁⠀⠀⠀⢀⣤⠉⠉⠀⠀⣉⠉⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠳⡝⣵
⠀⠀⠀⠀⠀⡄⢤⡀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠐⠲⠄⡀⠀⠿⣾⠈⣿⠀⠿⢼⡃⠀⢀⠠⠶⠄⠀⠀⠛⠀⠀⢈⠳⡜⠁
⠀⠀⠀⠀⠀⠡⡄⠀⠀⢈⣁⡀⠰⣦⠀⠀⠀⠀⠀⠀⠀⡀⡀⠙⢄⠀⢀⠀⠀⠀⠎⠀⠀⠀⢶⣶⠀⠀⠀⠀⢠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⣣⠀⠀⠑⣀⠀⠀⠀⡴⠚⠙⠲⣄⠀⠀⠘⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠘⠁⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠛⠀⠀⠀⠸⠀⠀⠀⡅⠀⠀⠀⣹⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⡀⢠⡄⣤⢰⣷⡴⣆⡿⠠⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠙⠢⠤⠖⠁⠀⠀⠈⠀⢠
⠀⠀⠀⠀⠀⠀⡈⠈⠁⠙⠸⡏⠁⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠂⠀⠂⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠑⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⠤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢺⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠘⣇⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠌⢀⠞⠉⢧⡀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⡎⠉⠑⢦⡀⠀⡿⠀⠘⠀⠈⠀⠀⣴⠛⠶⣄⡀⠀⠁⠒⠠⠄⣀⣀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠈⡇⠀⣠⠒⠉⠙⡆
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢸⡇⠀⠀⠀⠀⠀⣼⠃⠀⠀⢨⣿⠿⠿⠶⢤⣤⣄⣀⣁⣀⡀⠀⠀⠀⠀⠈⢢⠀⣿⠈⠉⠉⠉⠉
⠀⠀⠀⠀⢴⠉⣉⡵⠂⠀⢸⡇⠀⠀⠀⠀⣼⠃⠀⠦⣀⣈⣥⣤⣀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠻⠿⣦⣀⠀⠂⡏⠀⠀⠶⣉⠉⢳
⠀⠀⠀⠀⠈⠊⠁⠀⠀⠀⠘⣇⠀⠀⠀⢸⠇⠀⠤⣤⠟⠁⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⣠⠤⢤⣠⠝⣧⣼⡇⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⡿⠀⠀⢠⡏⠀⠀⢀⣀⣀⣤⡷⠀⠀⠀⠀⠀⠰⡧⢤⣀⣹⠆⢹⡿⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢰⡇⠀⠀⠀⢿⠛⠉⠉⠈⠋⣠⠇⠀⠀⠀⣶⠀⠀⠳⢤⠴⠋⠀⢸⡇⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣤⣼⡇⠀⠀⠀⠈⠳⢤⣤⡤⠞⠃⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢸⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢫⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣤⠟⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣉⣧⠐⡄⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡀⢀⠀⢿⡀⠀⠀⠀⠀⢠⣾⣿⣷⣭⣿⠀⠀⠀⠀⠀⢀⡾⠃⣿⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣤⣅⣄⣀⣘⣧⠀⠀⠀⠀⣿⣘⣻⣿⣿⠏⠀⠀⠀⢀⡴⠋⠀⠀⢿⣐⠏⠉⡷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠉⠉⠉⢻⡄⠀⠀⠀⠀⠉⠉⠛⠿⠀⠀⠀⣰⣟⣡⣅⣀⣠⡟⠉⠛⢺⠀⢠⠛⣦
⠀⠀⠀⠀⠀⣀⣀⣀⡤⠖⢋⣩⠽⠚⠛⠻⣶⠒⠺⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠉⠉⣷⠀⠚⡟⠲⣇⡠⠏
⠀⠀⣠⠖⠋⠉⠉⠁⠀⣴⠋⠀⠀⠀⠀⠀⠛⠶⢶⡞⠻⢦⣄⣀⣀⣀⣠⡴⠛⢧⡀⠀⠀⠀⠀⠀⢀⡼⠃⣠⠄⠐⠉
⠀⠀⣯⣠⠄⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠈⠉⠉⠉⠀⠀⣷⣼⣿⣄⣀⣀⣠⡴⣋⡴⠞⠁
⠀⠀⠀⢿⣠⡔⠀⣴⠤⠟⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⠈⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠘⠷⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣆⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣴⠏⡔⠙⣇⠀⠀⠀⠀⢸⡿⠛⠛⠲⠶⠤⣤⣄⣀⡀⠀⠀⣸⠃
⠀⠀⠀⠀⣠⡟⣡⠊⢠⡴⢻⡆⠀⠀⣠⡼⢷⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⠏
⠀⠀⢀⣾⢋⡴⠁⣼⣿⢿⡅⢻⣄⣸⣿⡀⠈⢻⣦⣄⡀⠀⠀⠀⠑⠲⢶⣯
⠀⣰⠟⣡⠛⡄⢠⡞⠃⠀⢻⣄⣉⡽⠋⠙⠞⠋⠀⠈⠙⠳⠶⣤⡴⠾⣿⣻⠶⢦⣄⡀
⣾⣋⣞⣁⣠⠿⠋⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠆⣸⡇
⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠶⠾⣿⣾⡷⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⢤⠤⠤⠍⣁⣀⣀⣰⠋⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⠶⠶⢦⣤⣯⣭⣉⣗⣚⣧
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⢈⣉⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⢟⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠿⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡀
⠀⠀⠀⠀⠀⡤⠖⠒⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢳
⠀⠀⠀⠀⢸⠃⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⢸⡀⠀⠀⠀⠀⠀⣠⠶⢴⣦⡀
⠀⠀⠀⠀⢸⡄⠐⠀⠀⠀⡀⠤⠀⠄⠠⠄⠄⠀⠀⢀⠁⢀⠀⠀⠀⠘⠀⠀⠀⠀⢸⡇⠀⠀⠀⡴⢛⡥⠤⠼⠋⠁
⠀⠀⠀⠀⠸⡇⠀⠀⠀⡈⠀⠠⢤⠂⠀⠀⠀⠀⡀⠈⠁⢀⠀⠒⢀⠒⢄⠀⠀⠀⠘⡇⠀⠀⢰⡇⡞
⠀⠀⠀⠀⠀⡇⠀⡆⠀⠑⠀⢰⠀⠈⡄⠀⠃⠀⠌⠐⠂⠻⠉⠀⡀⢀⠀⠀⠀⠁⠀⡇⠀⠀⠸⡇⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⢐⠀⠀⠀⠄⠠⠀⡀⣀⠀⠀⠀⠀⠢⠐⠀⠃⠀⠀⠀⠀⠀⣇⠀⠀⠀⢳⢹⡀
⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⠉⠈⠀⠀⡀⠀⠀⠀⠀⠀⡄⠀⢀⡀⠀⠀⠶⠀⠀⠀⠀⣿⠀⠀⠀⠈⣏⢧
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠐⠀⠀⠀⠀⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⢸⠈⡆
⠀⠀⠀⠀⠀⠸⡆⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠄⠊⠀⣿⠀⠀⠀⠀⢸⢀⠇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡴⠋⠀⠀⢀⣴⢫⠎
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠛⠛⠛⠛⠉⠹⣍⣀⠀⢀⣀⡼⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⣠⣴⡫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠗⠀⠀⠻⣆⣀⣠⠶⢒⣿⣿⠾⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠎⠉⠉⣁⠤⣀⣈⢁⡭⠇⣾⣯⡤⠤⣤⣤⣄⣀⣤⣀⣀
⠀⠀⠀⠀⠀⣀⡤⠔⠚⣛⣙⠠⠖⢰⠂⠐⠒⡎⠰⠂⠛⠹⢃⡛⠁⠐⠃⠉⠀⠀⠁⠀⠀⠤⡟
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠤⠴⠶⠖⠒⠃
⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠚⠛⠉⠉⠉⠉⠉⠉⠉⠁



⠀⡉⡉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠈⠉⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡤⠒⣛⣓⣦⣤⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣡⠞⠋⢉⡟⢉⣩⡽⡿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠓⠒⣶⠟⠋⠉⢁⡴⠁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣰⠃⣀⣠⠴⠋⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣉⣀⡀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣭⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡆⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣠⡤⠔⠒⠒⠒⠒⠁⠐⠚⠁⠀⠻⠹⠇⢤⠤⠤⠤⠖⠒⢂⡽⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⣀⠿⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠶⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡖⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⣀⠴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⢀⡤⠶⠒⠉⠑⢶⠃⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠞⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢬⣦⡄
⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣠⠞⠁⢀⡤⠚⠁⠀⠀⠀⠳⣀⠇⣄
⠀⡴⠁⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡴⡏⠀⠀⠀⠀⣠⠞⢁⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠘⠻⠘⣆⠀⢀⠀⠀⡀⠀⠀⢀
⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⣠⠶⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢸⠄⠀⢹⡄⠀⠘⡇
⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⣠⠔⠒⠊⢹⠀⣠⠞⠁⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠀⠀⠀⢰⣘⣇⠀⠈⢧⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⡰⠃⠀⠀⢠⠇⠊⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠘⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡼⠃⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⢇⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾⠁⠀⠀⠀⠀⢀⡜⠁⡀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⡀⠀⠀⠃⠀⠀⢸⣤⠁
⢃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⡞⠀⠀⡇⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⢧⡄⠀⠁⠀⠀⢸⡇
⠘⣇⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⣼⡤⠆⠀⠇⢰⠏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡞
⠀⠘⢆⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠠⡿⠁⠀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠈⢷⠈
⠀⠀⠈⢣⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢦⡀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠋⠁⠉⠙⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠙⢦⠀⠻⣄⠀⠀⠀⠀⣠⠞⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣤⣤⣤⣤⣤⡤⠤⠤⠘⠇⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⣦⡀⠀⡼⠁⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡜⠀⠙⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠚⠻⠧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠉⠛⠻⣍⠉⠑⠶⣴⠲⠶⠖⠒⢫⣉⠁⠉⠑⢦⡀⠀⠀⠉⢲⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠈⠳⡄⠀⠈⢳⣤⠤⠤⠒⠉⠳⣄⠀⠀⠀⠀⠀⠀⡟⠀⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠙⠆⠀⠀⠙⢦⡀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠙⠚⠁⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠘⡆⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⡆⠀⠀⠀⠀⠙⢛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣠⠞⠉⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⢿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⢿⣯⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⡟⠻⢿⣿⣿⣿⣿⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⣿⡿⠟⠋⠀⠀⠀⠀⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣥⡀⠀⠀⠀⠀⢀⣬⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠌⠁⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠄⠀⠀⠠⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⢰⡆⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠒⠒⠒⠒⠒⠶⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠖⣀⣤⣴⣶⣖⣀⣤⢦⡀⠀⠀⠀⠀⠀⢰⡾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡴⠚⠁⣼⠷⠋⠁⠈⠉⠉⠉⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢠⢾⡀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢠⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⣿⡈⠿⢁⡤⢤⡀⠀⠀⠀⠀⢀⡴⠒⠲⣄⠀⠀⠀⠀⣿⠞⢻⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⡇⢠⡏⠀⠀⢳⠀⠀⠀⠀⢸⣁⣀⠀⠈⢧⠀⠀⠀⠀⠀⣼⡤⢤⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠸⣯⣷⠀⢸⠀⠀⠀⠀⢸⣷⣼⡆⠀⢸⠀⠀⠀⠀⠀⢩⣤⡀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠛⠋⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣸⣠⠇⣼⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣁⡼⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢤⣖⠒⠒⠓⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⣼⢉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢸⣤⠤⠤⣤⣤⣈⣷⠀⠀⠀⠀⠀⣠⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⣦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣴⢮⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⡾⠀⠀⠀⠉⠉⠉⠉⠀⠀⣰⠃⠀⠈⡿⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⢉⡟⢹⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⡞⠁⢀⣠⣤⢽⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡞⠀⢸⡄⠀⠸⣄⠀⠀⠀⠀⣀⡴⠋⠀⢀⡰⢋⡴⠚⠉⠀⠀⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⠁⠀⠀⠹⣄⡀⠈⠙⠓⠚⠋⠁⠀⣀⡴⠋⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣼⣀⣀⣀⠀⠀⠉⠓⠲⠶⠶⠖⠚⠋⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠒⠒⠲⠶⠶⠤⠤⢤⣽⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠓⠒⠛

⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀
⠀⠀⡴⠉⠀⠀⠀⠑⢀⠀⢀⠔⠁⠀⠀⠀⠑⡀⠀⠠⠊⠀⠀⠀⠀⠢⠀⠀⠠⠈⢀⠀⠀⡀⠐⡀⠀⢀⠔⢁⡄⢠
⠀⢰⠀⠀⠂⠀⠐⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠁⠀⢃⠀⢈⠃⠀⡇⢐⡉⣠⠀⠀⠀⠁⠐⡃⡀⠀⠀⠘⠀⠚⠀⠀⡄
⠀⠸⡀⠈⠇⢀⠌⢀⡜⠀⠠⡐⠁⠀⠀⠀⢀⠆⠀⢇⠀⠀⠒⠂⠀⡰⠁⠀⠡⡈⠻⠿⠿⠋⡰⠋⠀⢂⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠈⠲⠤⠤⠖⠉⠀⠀⠀⠉⠂⠀⠀⠒⠁⠀⠀⠀⠑⠢⠤⠖⠋⠀⠀⠀⢀⣤⡏⠒⠒⠉⠀⠀⠀⠀⠑⠒⠤⠔⠊⠁
⠀⠀⢠⠔⠀⠉⠃⠀⠀⠀⢀⡔⠊⠁⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⡑⢄⠀⠀⠀⡭⢒⠙⠛⠠⠛⠃⠀⢀⠔⠋⣁⠀⠠⡀
⠀⠰⠁⠀⢰⠀⡆⠀⠈⠀⡎⠀⡀⠀⠀⡠⠀⠁⠀⡌⠑⠀⠀⠸⣀⠸⠆⠀⡎⢀⡀⠀⠀⢄⠈⡄⠀⠆⠀⠊⠀⠙⠀⠈
⠀⠘⡀⠈⠦⠀⠄⠀⠰⠀⢣⠀⠙⡡⢉⠃⠀⡆⠀⡆⠀⠈⠀⠀⠀⢀⠇⠀⢇⠀⠀⠂⠀⠀⠀⡄⠀⢆⠀⠸⠀⠛⠀⢠
⠀⠀⠈⠢⠄⠀⠠⠐⠁⠀⠀⠑⠤⣉⣉⠤⠊⠀⠀⠈⠢⢄⣀⡀⠴⠋⠀⠀⠈⠢⠄⣀⢀⠠⠊⠀⠀⠈⠓⠤⠀⠤⠒⠁
⠀⠀⠀⣀⠤⠤⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⢀⡠⠄⠤⠄⠀⠀⠀⠀⢀⡠⠤⠤⠄⡀⠀⠀⠀⢀⠀⠄
⠀⢠⠊⢀⡀⠀⣠⠀⢂⠀⡐⠁⠐⠀⠀⠀⠀⠄⠀⠰⠋⣀⠀⠀⡶⠈⢂⠀⣰⠋⢀⠀⠀⡀⡙⠄⠀⡴⠁⢠⢠⠀⠀⢦
⠀⣏⠀⠀⠁⢀⠉⣀⢨⠀⡇⠀⠐⠀⠐⠂⠀⠀⠀⡇⠀⣿⠃⠀⢀⠀⢸⠀⡇⠀⠈⠁⠈⠀⠀⡸⠀⠓⠤⠞⠀⠓⠔⠁⡇
⠀⠘⣄⠀⠀⠚⠀⣉⠎⠀⠱⣄⠘⠂⠐⠀⡠⠃⠀⠐⢄⡉⠻⠿⢉⡴⠃⠀⠳⡀⠙⠷⠶⠀⡠⠃⠀⢢⠘⠿⠷⠶⢆⠜
⠀⠀⠀⠉⠒⠒⠊⠁⠀⠀⠀⠀⠉⠒⠂⠉⠀⠀⠀⠀⠀⡈⠉⠈⢁⠀⠀⠀⠀⠈⠓⠒⠒⠉⠀⠀⠀⠀⠁⠚⠂⠊⠁
⠀⠀⢠⠖⠋⠉⢉⠠⡀⠀⠀⡠⠒⠊⠁⠀⠄⠀⠀⠐⣿⠅⠒⠀⢬⡿⠀⠀⢠⠊⠩⠁⠦⠐⢄⠀⠀⢀⠔⠌⠁⠀⠠⡀
⠀⢰⠉⠂⠐⠀⢇⣀⢿⠀⡸⠁⣴⡆⢠⣶⡀⠡⠀⡜⢀⣄⠀⣠⡄⠑⡄⢠⠁⠀⠒⠀⠒⠄⠀⡆⢠⠃⢀⠀⠀⠀⡀⠘⡀
⠀⢸⠀⠀⠑⠰⠀⠀⡸⠀⢇⠀⢹⣁⢀⡟⠀⡜⠸⡀⠀⢀⣀⡀⠀⢀⡇⠸⡀⢠⣾⣿⣶⡄⠀⠇⠘⡄⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠑⠤⣀⣀⠤⠖⠁⠀⠈⠢⣀⣉⣉⡤⠚⠀⠀⠳⢄⣀⣀⣀⡠⠞⠀⠀⠑⠬⣀⣀⣀⠤⠊⠀⠀⠈⠢⠄⣀⠀⠄⠂
⠀⠀⢀⡠⠄⠤⢄⡀⢠⡀⠀⢀⠠⠤⠤⠄⡀⠀⠀⠀⣀⠬⠉⠁⠀⠀⠀⠀⠀⣀⠤⠤⠄⢀⠀⠀⠀⠀⡀⠤⠀⠀⡀
⠀⡰⠃⠀⠀⠀⠀⠈⡢⠀⡰⠁⢀⠀⠀⡀⠈⢆⠀⡜⠁⠀⠀⠀⠀⠐⡀⢀⠎⣀⠂⠀⢀⡀⢡⠀⢠⠊⠀⣀⠀⡄⠈⢦
⠀⡇⠈⠋⢀⡈⢡⡀⡁⠀⡇⠀⠈⠀⠀⠁⠀⣸⠀⡀⠐⠒⠀⠐⠂⠀⡇⢘⠀⡀⢀⣄⡀⠈⢰⠀⢯⠀⠀⢀⣀⡃⠀⢸
⠀⠱⡀⠀⠈⠁⢀⠔⠁⠀⠹⣄⠀⠀⠀⢀⡰⠃⠀⠱⣄⠀⠀⠀⢀⠜⠀⠘⢤⡀⠈⠛⠁⡠⠊⠀⠈⢦⣀⠀⠀⢀⡰⠃
⠀⠀⠀⠁⠈⠉⠁⠀⠀⠀⠀⠀⠉⠙⠉⠉⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⢠⣶⣶⣄⠀⣶⡄⠀⠀⣠⣶⣶⣄⢠⣶⠀⣴⡆⠀⣶⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⢿⣇⠈⠛⠀⣿⡇⠀⠀⣿⡇⢸⣿⠀⣿⡀⣿⣿⢠⣿⠀⠀⠀⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⣿⡇⠀⠀⣿⡇⢸⣿⠀⣿⡇⣿⣿⢸⡏⠀⠀⠀⠀⠀⠀⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⣴⡆⢘⣿⠀⣿⡇⠀⠀⣿⡇⢸⣿⠀⢹⣿⡏⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠘⠿⠿⠋⠀⠿⠿⠿⠇⠙⠿⠿⠋⠀⠸⠿⠇⠸⠿⠃⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⡿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⡿⣟⣛⡀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣯⣿⣿⣿⣿⣿⣷⣿⣾⢾⣷⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣯⣽⣿⣾⣦⣤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣯⣿⡿⢿⡛⣯⣿⣿⣿⡾⣿⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⣿⢿⣾⣿⣿⣿⣿⣿⣾⣿⣯⣽⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣿⣿⣿⣿⣽⣿⡿⣾⣿⣯⣿⣼⡿⣿⣿⣷⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣻⣿⣴⣷⣿⢻⣿⡽⠶⠿⠿⣾⡿⣯⠽⣯⣹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣼⢴⣦⠞⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢛⡺⠟⠁⠀⠀⠀⢠⡆⠀⠀⢰⡇⠀⠘⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡻⣬⡓⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠀⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⠀⢀⢔⡀⠀⠀⠈⣉⡀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⢤⣤⣀⣀⣀⣤⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢔⣾⣿⣿⣿⣿⣿⣿⣾⡗⡆⣄⠀⠀⢀⣤⣶⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣯⣽⣿⣿⣿⣿⣿⣷⡷⣶⠎⣾⣚⣿⣿⣻⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣮⣽⣿⣿⣭⣿⣿⣿⣷⣿⣿⣩⣿⡿⠏⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣭⠏⡉⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣛⣻⣻⣿⣿⣻⣻⣟⣛⣃⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠉⢅⠢⡀⠀⢆⠑⢄⢹⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡑⠢⣄⠳⡀⠑⢄⢣⠈⢢⡆⠱⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡢⢄⠀⠈⢢⠀⠑⠬⣆⠀⠳⣇⠀⢻⠀⢣⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠈⠢⡉⠒⢄⡱⡄⠀⠈⠳⠀⠈⠂⠈⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠒⢄⡲⣬⣢⣄⠉⠺⢆⠀⠀⠀⢀⣀⠀⢤⠂⢀⣴⠆⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡌⠊⠻⢍⡓⠄⠀⠀⣴⠉⠀⠀⠀⠠⠃⣈⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠆⣐⠒⠒⠷⢄⠀⠁⠀⠀⠀⠁⠀⠀⠀⢀⠗⠀⠀⠈⠀⣀⠔⠒⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⣛⣁⡀⠀⠀⠀⠀⠀⠀⡴⠀⠀⢠⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠂⠠⠀⠈⠉⠂⠀⠀⠀⠀⠀⠈⡀⠀⡆⠀⠀⠀⠀⠀⠘⢣⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⠶⠄⣀⠃⠀⣠⠔⠒⠒⠢⢄⡇⠀⡀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠁⠠⠀⢠⡄⠀⠁⠀⠙⠢⠤⠤⠒⠊⠙⠢⠤⢤⠔⠊⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡴⠋⠁⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠄⣀⡀⠁⠺⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣴⣿⣟⣓⣒⣂⠒⠋⠑⠢⢌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡎⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡜⠉⣠⣤⡈⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣄⡈⢟⡄⢇⣀⠀⢀⣀⠀⢠⡀⠀⢀⡀⠀⢀⠔⠉⣀⠀⠀⠀⡜⢀⡊⠁⢀⣈⠢⠄⠠⠜⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⠑⠻⠆⠀⠀⠈⣉⠁⢤⡈⢑⠀⠉⠁⠸⡀⠁⡰⠎⠀⢂⠀⠀⣏⠉⣉⠝⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠂⠜⠁⠀⢀⣯⠋⠀⠀⠀⠀⠈⠉⠁⠘⠢⣈⣦⡀⠈⣉⡤⠖⡡⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢨⠆⡀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⢙⡛⢄⠀⡰⢁⠞⠐⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠜⠀⠀⠈⣂⠉⠁⢓⡀⠤⢐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠑⠒⠢⢤⢤⠄⣆⠀⠀⠀⠀⣀⡧⢠⠀⠀⠉⢉⠡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠢⠄⠑⠒⢶⠀⠀⠀⠀⠠⠦⣼⣘⠀⠈⠈⢑⠀⠘⠠⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⣖⠁⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣢⣔⠈⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⠀⠉⠉⡠⡶⠖⢶⠘⢻⠉⠀⣈⢽⡈⠸⢒⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠈⠐⠊⢥⣍⣬⢷⣒⣒⡒⠈⠅⠰⠖⠛⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣳⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡀⠀⠠⡀⠀⠀⢀⣀⠀⣀⣀⠀⢠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠄⠈⠁⠐⠲⠌⠑⠒⠊⠉⠀⠁⠉⠀⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⣄⠀⢲⢀⣠⡶⢶⣿⠣⣽⣆⣮⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢰⠿⣄⡰⢣⢞⣽⣿⣿⣴⣿⣿⣡⡴⣣⠶⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢔⠫⣹⣮⡿⣷⣿⣿⣿⣤⣷⣿⣿⣿⣿⢟⣱⣦⡴⢤⣴⢟⣾⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣨⣿⣿⣻⣽⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡭⢷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢨⡙⢮⣻⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⡵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢱⣼⡼⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣯⣶⡟⢁⡔⣠⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠋⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣭⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⠶⣾⣿⣿⣿⣿⡿⠻⢡⠛⠙⠟⢿⣿⣿⣿⣽⠿⠟⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⡴⠛⢻⣿⡽⣯⠁⠀⠀⠜⡀⠈⠈⣿⡟⣿⢿⣻⣦⡀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢒⡀⢹⣟⡇⠉⠀⠀⠀⢠⡀⠀⠀⠘⠃⡛⠒⠈⠙⠓⠵⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠭⠄⠱⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡌⢀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢤⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠂⠄⠀⡴⠂⠉⠀⠀⢣⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⡆⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠊⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⣀⣠⡴⠶⠞⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⣠⣴⠛⠉⠁⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣀⣠⡴⠾⠟⠛⠛⠛⠛⠳⠶⢿⣋⠉⠛⠛⠛⠳⠶⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⠀⠀⠀⠉⠙⠃
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⣼⠋⣠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⢀⡾⠃⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⣼⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⢸⡇⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⣿⠁⠀⢠⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⡴⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣀⣀⣀
⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠉⠉⠉⠉⠻⠶⣦⣀
⠀⠈⢷⡄⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠟⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢻⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣧⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣤⣶⣶⢾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠋⢹⡟⠋⠉⢻⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⣆⠀⠀⠀⣾⠃⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣍⠻⢦⣴⡟⠀⠀⠀⢀⣼⠟⣷⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠹⢦⣀⣿⠀⠀⠀⠀⣾⣿⠀⠘⢷⣄⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢹⡏⠀⠀⠀⢰⡏⠹⣆⡀⠀⠙⠛⠛⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠐⢷⡀⠉⠳⠶⠆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⣀⣀⣀⣤⣀⣀⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⣀⣠⣤⠾⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⢉⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣶⣤⣤⣤⣶⣶⠶⠾⠛⠛⠋⠉⠀⠠⣿⡁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣴⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠿⠿⠿⠛⠛⠛⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠟⠛⢿⡟⠋⠁⠀⠀⠀⠀⠈⠸⣟⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⣀⣀⣀⣌⣥⣤⣿⠆⣠⣾⣿⠴⠤⠿⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠈⠻⣦⣤⣶⣴⣤⣤⣶⣶⣶⣤⡄⠈⠒⣦⣬⣷⣶⠶⠶⣤⠤⣤⣤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠈⢦⣾⣿⠁⠈⠉⢧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⢸⡂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⣿⣟⡄⠀⠀⠈⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⣿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣾⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢄⡀⠀⠀⠀⠀⡟⡔⠀⠀⠀⠀⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢞⣡⠀⠀⢠⡀⡷⢄⣠⠄⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⡅⠠⣤⠀⣿⠁⡾⠴⠀⠰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⠓⠀⠛⠛⠓⠊⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠒⠶⣒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡟⠈⠀⢽⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣾⣿⠉⠉⠈⣹⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡝⣿⣿⠐⠚⣳⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⣿⡟⠀⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠄⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡀⣸⣿⡇⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡯⠁⠀⢀⡠⠐⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣼⣋⣉⣿⣿⢱⣶⣿⣟⣿⠿⠟⠿⢿⠛⢿⣿⣼⣃⠉⠒⠉⠀⠀⢀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣤⣻⣿⣿⡋⠀⡀⢀⠀⠀⠄⠜⠋⠁⠀⠀⣀⣤⡾⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠿⠓⠴⠐⠴⠆⠲⣚⠉⠀⠪⠥⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠛⣈⣉⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠋⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⡀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠭⠇⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠗⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠆⠀⠾⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠲⢾⠂⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋⠻⣿⣿⣿⣯⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢀⣿⣿⣿⣿⣿⠿⠛⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⣫⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿
⣿⣿⠀⠀⠀⠀⠀⠳⠢⠄⠤⣀⠠⠤⠤⠤⠴⠾⠿⠿⠟⠁⠀⠛⠋⠙⠃⠖⠋⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠠⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣆⠀⠀⢀⠤⡄
⠀⠀⠀⠀⠀⣠⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠈⠻⣿⣿⡄⢰⡣⠔⠃
⠀⠀⠀⠀⢠⡿⢻⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⢠⣄⠘⢿⣧⠀⢀⡤⠤⣄
⠀⠀⠀⠀⣾⣧⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣿⡄⠈⠓⠲⠊
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣆⠘⠫⢳⡄
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠐⣾⠁⠀⠀⠀⡤⠦⣄⣀
⢎⣉⡭⠖⠈⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢹⣦⡧⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠸⣦⡴⠶⣼⠁⢠⠏⠉⣿⡀
⠀⠀⢀⠠⡄⠈⢿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⣿⢧⠀⠙⡖⢧⣠⠜⠉⣹
⠀⠰⣅⡼⠁⠀⠀⢹⡿⠛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣄⠀⠀⠀⣿⠘⣧⠀⠀⠀⢁⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢾⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡄⠀⢠⣷⠞⠁⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣩⠕⠋⠉⠀⣠⣟⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢿⡷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣤⡞⢁⠈⢳⡴⠋
⠀⠀⠀⠀⠀⠀⣞⠉⠛⠻⡄⠈⢷⡈⠙⢲⣦⣤⣤⣤⣤⣤⣴⣖⣿⠽⠛⠀⠀⠙⢯⡀
⠀⠀⠀⠀⠀⣰⠛⠑⠒⢲⠓⠀⠈⠉⠉⠉⠁⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠘⣶⠒⠒⠛⠀⣀⣤⠤⠤⠤⠤⠾⠷⢲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠙⠶⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢶⡟⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢀⣀⣤⠴⠚⠛⠉⠁⠀⠀⣇⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣅⠀⠀⠀⠀⣀⣠⡴⠖⢻⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⢤⣄⣀⣈⣹⣟⣛⣉⣿⠀⠀⠀⠀⣧⣤⣤⡿⡠⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡄⠀⣿⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣇⢀⠀⠉⠉⠭⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠇⣦⣻⠶⠤⢤⣤⠴⠞⠀⠀⠀⠀⠀⠳⠖⠒⠒⠒⠚⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⠙⢷⠎
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣼⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣴⡿⣿⡿⢷⣄⠀⠀⠀⣴⡿⣿⡿⢷⣆⠀⠀⠀⣰⡿⣿⣿⢷⣆⠀⠀⠀⣠⣾⣿⣿⣿⣦⠀⠀⠀⣠⡾⢿⡿⢿⣦
⠀⠀⠀⠐⣿⡟⠻⠟⢿⣿⠀⠀⠀⣿⡟⠿⠛⢿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⠿⠿⢿⣿⠀⠀⠀⣿⡿⠛⠻⢻⣿⠆
⠀⠀⠀⠀⠙⢿⣶⣶⠿⠃⠀⠀⠀⠙⠿⣶⣶⠿⠃⠀⠀⠀⠙⠿⣷⣾⠿⠋⠀⠀⠀⠘⠿⣿⣿⠿⠋⠀⠀⠀⠘⠿⣶⣶⡿⠋

⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⢀⣿⣯⣽⣯⣽⣷⠀⠀⠀⣿⣯⣽⣯⣽⣷⠀⠀⠀⣾⣑⣿⠖⢀⣷⠀⠀⠀⣾⣯⣿⣿⣽⣷⠀⠀⠀⣾⣯⣽⣿⣭⣿⡀
⠀⠀⠀⠈⢿⣿⡀⣨⣿⡟⠀⠀⠀⢿⣿⣟⣻⣿⡟⠀⠀⠀⢿⣿⣭⣭⣿⡿⠀⠀⠀⢻⣿⣭⣭⣿⡿⠀⠀⠀⢻⣿⣟⣻⣿⡿⠁
⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠉⠛⠛⠋

⠀⠀⠀⠀⣴⣾⣿⣿⣷⣄⠀⠀⠀⣰⡾⣿⣿⢷⣄⠀⠀⠀⣠⡾⣿⣿⢷⣄⠀⠀⠀⣠⣾⣿⣿⣷⣆⠀⠀⠀⣠⣾⣿⣿⣷⣦
⠀⠀⠀⠰⣿⣷⡿⢷⣿⣿⠀⠀⠀⣿⣿⡿⢿⣿⣿⠀⠀⠀⣿⣷⣿⣿⣾⣿⠀⠀⠀⣿⣷⣿⣿⣾⣿⠀⠀⠀⣿⣷⡿⠻⣾⣿⠆
⠀⠀⠀⠀⠹⢿⣿⣿⠿⠃⠀⠀⠀⠙⢿⣿⣿⡿⠃⠀⠀⠀⠙⢿⣿⣿⡿⠋⠀⠀⠀⠘⢿⣿⣿⡿⠋⠀⠀⠀⠘⠿⣦⣤⡿⠏

⠀⠀⠀⠀⢀⣠⡤⣤⣄⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⣠⣤⣤⣄⡀
⠀⠀⠀⢀⣿⡼⣷⡿⢾⣧⠀⠀⠀⣾⣏⣿⣿⣿⣷⠀⠀⠀⣾⣏⣿⣿⣹⣷⠀⠀⠀⣼⣯⣽⣿⣽⣷⠀⠀⠀⣼⣯⣽⣿⣽⣿⡀
⠀⠀⠀⠈⢿⣿⣛⣛⣿⡟⠀⠀⠀⢿⣏⡉⢉⣽⡿⠀⠀⠀⢿⣿⣯⣽⣿⡿⠀⠀⠀⢿⣿⣭⣭⣿⡿⠀⠀⠀⢻⣿⣅⣀⣿⡿⠁
⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁

⠀⠀⠀⠀⡰⠚⢿⡿⠳⡄⠀⠀⠀⣠⣞⢿⣟⣷⡄⠀⠀⠀⣠⡾⢿⡿⢷⣄⠀⠀⠀⣠⣞⢿⣿⣳⣄⠀⠀⠀⢠⡶⢿⣿⢷⣄
⠀⠀⠀⠠⣿⠀⡿⣿⢸⣿⠀⠀⠀⣿⣷⡿⢿⣾⣿⠀⠀⠀⣿⣯⣵⣮⣽⣿⠀⠀⠀⣿⣷⣿⠿⢼⣿⠀⠀⠀⣿⡽⠿⠿⢿⣽⠆
⠀⠀⠀⠀⠻⠀⣷⣾⠸⠃⠀⠀⠀⠹⢿⣿⣿⡿⠋⠀⠀⠀⠙⢿⣿⣿⡿⠋⠀⠀⠀⠙⢿⣧⣀⡰⠏⠀⠀⠀⠘⢿⣦⣤⡾⠟


⣿⣟⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣠⣄⡀⢀⣠⡄⠀⠀⠀⢀⣂⠈⠀⣀⡀⠀⠀⠀⠠⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⡀⠀⠀⢠⠠
⠟⠿⠟⠹⠿⠓⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⠀⠀⠁⠀⠄⠁⠀⠀⠀⠰⠀⠂⠠⠜⠁⠂⠀⠀⠀⠀⠂⠤⠀⠂
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠄⠁⠀⠀⠀⠁⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠔

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠂⠤⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠃⢀⠀⠀⠀⠀⢀⠈⠓⠁⢀⠄⠀⠀⠀⠀⠀⠙⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁

⠠⢒⡠⢀⡒⠄⠀⠀⠀⠀⡊⠀⠀⡐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢈⡀
⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠐⠀⠀⠀⠓⠀⢄⠄⠑⠂⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠑⠄⠀⠀⠐⠀⠀⠀⠀⠐⠄

⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⡀⡀⠒⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠄⠀⠀⠀⠂⠀⠁⠁⢀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⢄⠀⠋
⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠈⠀⠀⠁⠀⠀⠀⠀⡀⠉⠠⠀⠁⠀⠀⠀⠀⡀⠉⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠀⠉⢑
⣈⣂⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣢⣤⣔⣊
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⢀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠤⠐⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⡈⠠⠀⠄⠡⠀⠀⢠⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⡂⠂⠀⠐⢐⠒⢒⠄⠀⠀⠄⡰⢂⠀⠀⠀⠂⡀⢁⡠⠁⠀⠀⢀⠀⡀⠈⠀⠐⠄⠈⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡈⠂⠀⠠⢀⠁⢀⠀⠀⠀⠈⠀⠀⠑⠀⠀⠌⠀⠀⠀⠄⠀⠀⢁⠀⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠐⠀⡀⠀⠀⠐⠶⠶⠀⠀⠀⢐⢶⣶⠆⠀⠀⠁⠎⠀⢰⠐⠀⠠⠠⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠠⠂⠉⠀⢀⠀⠀⠔⠀⠐⠀⠀⠀⠀⠂⠐⠠⠀⠀⡠⠂⠀⠂⡀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⢁⢈⣀⣁⠨⠀⠸⠀⢀⡐⠀⠀⠀⠀⣀⣘⠀⠀⠀⡀⣀⣀⡃⡀⠀⠒⠙⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠈⠁⠊⠀⠀⠀⠂⠀⠂⠀⠀⠀⠁⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣄⠂⠀⠀⠀⠀⠀⠀⠸⠀⠐⠀⡂⣐⠀⠀⠀⠀⣂⣀⠂⠀⠀⠂⣘⠀⣃⠐⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠐⠀⠰⠊⠀⠀⠐⠉⠍⠃⠀⠀⠀⠈⠉⠁⠀⠀⠈⠂⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠐⢀⠀⡀⠁⠀⢀⠀⠀⠀⠀⠀⠀⢢⣤⢤⡴⠆⠀⡀⣴⠀⣦⠠⠀⠐⠠⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⠄⠀⠀⠀⠒⢀⠔⠀⠀⢄⠐⠂⡠⠀⠀⠀⠀⠐⡀⠃⠀⠀⠐⠶⢂⠌⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⡑⠀⠀⢠⢈⠀⣁⢂⠀⢀⠀⠁⠈⠂⠀⠀⠐⠁⠚⠛⡁⠀⠀⠈⠀⠁⢀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢡⣄⠀⠀⠨⠤⠌⠠⠀⠀⡰⡄⣠⠆⠀⠀⡀⢠⣠⠀⠃⠀⢁⠁⠀⠀⡐⠀⠀⠈⠥⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠁⠈⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⢤⣶⣶⣶⣶⣄⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠴⢒⠒⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣿⡄⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠈⠙⠳⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠋⠁⠹⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠐⠀⡀⠀⠑⠠⠀⠈⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠑⣄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠐⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⠀⢀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⢒⠹⣿⣿⣿⣿
⣿⣿⣿⠋⠀⠀⡠⠀⠀⢀⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⡀⠭⠚⠁⣉⠃⠘⠿⠿⠿
⣿⠟⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠀⡄⠈⠀⠀⠀⢀
⣿⣿⡗⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣠⣴⣿
⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⠛⠛⠻⠿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⣾⡄⠀⠀⠘⠇⠐⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⠛⢉⣉⣁⣤⣤⣤⣤⣶⣤⣤⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣶⣿⡟⢛⡉⠉⠉⢀⣀⣀⠙⣿⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣏⠀⠿⠿⠦⠀⠺⠿⠿⠄⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⢀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠈⠉⠁⢀⣴⣿⣤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⣶⣾⣿⡟⡦⠤⠴⠊⢹⣿⣯⡀⠀⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣿⣿⣿⡇⠀⠀⠀⠀⠘⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⠀⠀⠀⠀⢐⣿⣿⣿⠁⠀⠀⠀⠀⠀⣸⣿⣥⠀⠀⠠⠐⠈⠀⡀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⢷⣶⣦⣄⢸⠉⠈⣿⡦⢀⠀⡀⠤⠂⢹⣿⣟⠀⠀⠀⠀⠀⠀⣀⣶⠟⠉⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⠈⢹⣟⣿⣿⠀⠀⢿⡄⠉⠉⠁⠀⠀⣸⣿⡏⠀⠠⠀⣤⣶⣾⡿⠋⠀⠀⠀⠀⠉⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡈⣿⣿⣿⣆⠀⢸⡇⠀⠀⣀⡔⠺⣿⣿⠁⠀⠄⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⣤⢹⣿⣿⣿⡿⣧⡘⣿⢠⡞⠁⢀⣤⣿⡇⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⣀⠈⠛⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢰⣶⣿⣼⣿⣿⣟⡇⠈⢷⣿⣿⠁⠀⣾⣿⣿⡇⣠⡾⢋⣴⠶⣶⡄⣾⡆⢀⣾⠛⣷⡘⣇⢸⣿⠀⣿⠀⠙⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠈⢿⣿⣹⣿⣿⡇⠇⠀⠈⣿⡏⠀⢸⣿⣿⣿⣿⠟⠀⠘⣿⣤⣉⠁⣿⡇⢸⣿⠀⢹⡇⣿⣿⢻⣶⡿⠀⠀⠀⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢠⣶⣶⣿⢿⣿⣿⡇⡀⠀⠀⠙⠃⠀⠸⣿⣿⣿⣿⣦⡀⢀⣀⠉⣿⡆⢹⣇⠈⣿⣀⣼⠇⢿⡇⢸⣿⠃⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⣀⣬⣿⣿⣾⣿⣿⣧⠁⠀⠀⠀⠀⠀⣠⣿⣿⠏⠈⢻⣿⣾⡿⠿⠟⠁⠘⠛⠛⠃⠉⠁⠀⠈⠁⠈⠉⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠛⠻⣿⣿⣏⣿⣾⣿⠃⠀⠀⠀⠀⠀⠴⠟⠈⠀⠀⠀⢸⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣃⢀⣀⠀⠲⣿⣿⡍⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣷⣄⠀⠀⠰⠆⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠈⠀⠀⠀⠀⢸⣷⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣭⡛⣿⣿⡟⠻⣷⣦⣀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⢀⣤⠶⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣨⣤⣿⣿⡇⠀⣨⠙⢿⣀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⠀⠀⢸⣷⣿⢿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠆⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣛⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⠉⣉⣩⣟⠀⠀⢸⡇⢸⣿⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣩⡎⠀⠀⠀⠀⢀⠀⠠⣤⣤⣦⡤⡀⢠⣦⡀⠴⢾⣏⠀⠀⢸⡇⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡷⣶⣶⣾⡇⢸⢿⡿⠿⣿⣿⣿⡿⢿⣿⡿⢿⡛⠉⣶⡆⡘⠷⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⢸⣷⣼⣿⣷⣾⠷⠶⠿⠿⣿⣄⣼⣿⢠⠞⠁⣨⣿⠟⠀⣄⡢⠄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠈⠙⠻⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠁⠀⠀⠀⠀⢴⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⠴⣦⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠘⠓⠶⠾⠿⠛⠋⠉⠉⠁⠈⠉⠉⠉⠉⠛⠛⠛⠛⠉⠁⠀⢻⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣠⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⠛⣿⡟⣿⡟⣿⡟⣿⡟⣿⡟⢻⡟⢻⡟⢻⡟⢻⡟⢻⡟⢻⡟⢻⣿⢻⣿⢻⣿⢛⣿⠛⣿⠛⣿⠛⣿⠛⣿⠛⣿⠛⣿⡛⣿⡟⣿⡟⣿⡟⣿⡟⢻⡟
⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⣿⠀⣿⢀⣿⣶⠿⠶⠿⣤⣿⠆⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⣿⢸⣿⢸⣿⢸⣿⠀⣿⠀⣿⡿⠋⠀⠀⠀⠀⠀⠉⠂⣿⣧⣿⣧⣿⡇⢿⡇⢸⡇
⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⣿⢸⣿⢸⣿⠸⣿⣤⠿⠶⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⡇⢸⡇
⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⣿⢸⣿⢸⣿⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⡇
⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⣿⢸⣿⢸⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⢸⡇
⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⣿⢸⣿⢸⣿⢰⣿⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇
⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⣿⢸⣿⢸⣿⢸⣿⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⢸⡇
⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇⢸⡇⢸⣇⢸⣇⣸⣇⣸⡇⢸⡿⢸⣿⢸⣿⠨⣏⢥⣤⠶⣾⣄⡀⠀⠀⠀⢀⡠⣶⣤⣤⣤⣴⡶⣿⡇⣾⡇⢸⡇
⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡧⠜⠋⠕⠊⠛⠛⠻⢿⣿⣿⣾⣿⣭⣩⣛⡿⣾⣿⢀⣿⠀⣿⠀⣿⠙⣿⠋⣿⠀⣿⡆⣿⡆⣿⡇⣿⡇⣿⡇⢸⡇
⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣻⠗⠊⠀⠀⣀⣒⣬⣽⣻⣿⣿⣿⣿⡷⡶⠤⣉⠑⠂⠉⠉⠙⠻⣤⣿⠀⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⠀⣿⡇⣿⡇⣿⡇⢿⡟⠁⠀⢀⡴⠚⠉⠁⠀⠀⠈⠉⠛⢷⣮⡍⠊⠐⢄⠑⠄⠀⠀⠀⠀⠀⠙⢦⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⠀⣿⡇⣿⡇⣿⣧⠏⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡝⢦⡀⣄⠡⠀⠁⢄⠀⠠⡀⠀⠀⠹⣆⣿⠀⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⠀⣿⡇⣿⡇⡻⠁⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠳⡌⢢⠀⠀⠀⠀⠀⠀⠄⠀⢁⢻⣿⠀⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⠀⣿⡇⣿⡗⠁⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⠳⡀⠙⢦⡳⡄⠀⠀⠀⠀⠘⠀⠀⣆⣿⠀⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⠀⣿⡇⠏⠀⠀⠐⢰⠇⣀⣠⣄⡀⠀⠀⠀⠀⠾⠟⠛⠉⠉⠉⠉⠓⠉⢢⣌⡳⣿⣆⠀⢀⠀⠀⠂⠀⢻⣿⠀⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⠀⣿⡇⠀⠀⠀⢀⣾⡋⠉⣉⣉⠙⠆⠀⠀⠁⠀⣠⠶⢛⡛⢷⣦⠀⠀⠀⠹⣝⠛⣿⡆⢸⡄⠀⡀⠀⢸⣿⠀⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⠀⣿⠀⠀⠀⠀⣾⣿⣷⡾⠋⠭⡛⠀⠀⠀⠀⠀⠁⠀⠶⠷⠚⠋⠀⠀⠀⠀⠹⡽⢩⢳⢸⠇⠀⡇⠀⢀⢻⡄⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⠀⣿⡀⠀⠀⢸⣿⣽⠇⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣿⠀⠀⡇⢀⢸⠀⠙⣿⠀⣿⠇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⠀⣿⡗⣄⠀⠸⡇⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⡄⠙⠦⣀⣿⡜⡘⢧⡀⢌⠳⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⠀⣿⡇⣿⡦⡄⣿⢿⠀⠀⠀⠀⠀⠓⠆⠈⡓⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠈⠀⠀⢀⣿⣦⣀⠀⠈⢉⡙⠢⠙⣄⢀⣹⡄⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⡆⣿⡇⣿⡇⢻⡿⠸⣇⠀⠀⠀⢀⠂⢀⠆⠈⠀⢒⢒⡀⢀⣀⡝⠂⠀⠀⠀⠀⠀⢸⡏⣼⢿⣿⣶⣄⠙⢦⡀⢻⡆⣿⡷⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⡇⣿⡇⣿⡇⡜⠁⡰⢹⣦⠀⠰⠃⣤⣥⣒⠈⡁⠠⠤⠀⠊⠉⠀⠀⠀⠀⠀⠀⢠⣾⠱⡏⠈⣏⠙⠀⠡⠈⡇⢈⣵⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⡇⣿⡇⣿⡇⠀⠰⠁⡾⠀⣷⣄⠀⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣴⡟⢻⡄⡇⢄⠙⢦⣀⣀⣧⣇⣼⠟⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⡇⣿⡇⣿⡇⠄⠃⡈⢠⣇⣿⣸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⢸⣷⣜⢦⣵⣶⣴⣿⣿⠟⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⡇⣿⡇⣿⡇⣻⣆⠁⠸⡿⡙⣿⡦⢬⣿⣦⡀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⢸⡇⠉⠛⣾⡟⣿⠉⣿⠀⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⡇⣿⡇⣿⡇⣿⡏⣳⡤⣝⣬⣼⣷⣾⡇⢸⡇⢹⡆⢹⡇⢹⣧⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣴⣿⣿⣿⠀⣿⠀⣿⠀⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇⢸⡇
⣿⣧⣿⣧⣿⣧⣿⣧⣿⣧⣾⣧⣼⣧⣼⣧⣼⣧⣼⣧⣼⣧⣼⣿⣧⠀⠀⠀⠀⠀⢀⣿⣶⣿⣿⣿⣿⣿⣦⣿⣤⣿⣤⣿⣧⣿⣧⣿⣧⣿⣧⣿⣧⣼⣧

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣄⡀⠀⢀⣆⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢲⡆
⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⣠⣾⠟⠀⠀⠀⢀⡀⠀⠀⠁⠢⡄⠀⠀⠀⠀⠀⣀⠤⠐⠚⠻⣇⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠁⣀⡤⠤⢶⠟⠃⠘⣷⠠⠀⢸⡇⠀⠉⠁⠚⠛⣱⣄⠤⠒⠉⠀⠀⠀⢀⡤⠛⠀⠀⣴⡾⠛⠛⠋⠂
⠀⠀⠀⠀⡐⢀⣸⡇⠀⡀⢘⡀⠀⠀⣹⡇⠀⠈⠁⠀⠀⠀⣠⡾⠋⠀⠠⣄⡀⢀⣤⠔⠉⠀⣤⠤⠤⢿⠇⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠘⡻⠋⣿⠡⣾⢀⣠⣴⢾⣿⡿⠁⠀⠀⠀⠀⢠⡾⠋⠀⠀⣐⡀⠀⠉⠉⠀⠀⢀⣄⣸⣧⠀⠀⠀⢠⣆⠀⢻⠶⣴
⠀⠀⠀⢰⠁⠈⠁⠐⢻⡿⠋⠀⣾⡿⠁⠀⠀⢠⣾⣴⡟⠀⠀⠀⢀⡠⠘⠀⠀⢠⢀⡴⠋⠉⣿⠇⠀⣀⣠⣾⣿⣶⠟⢀⠞
⠀⠀⠀⢸⡲⠁⠀⡴⠋⢀⣀⠠⠿⢟⣿⣿⣿⠏⠀⢻⠀⠀⣠⡖⠁⠀⠀⠀⠀⠘⠉⠀⡀⣸⣿⣶⣿⠿⠟⠛⠁⠉⢠⠎
⠀⠀⠀⢸⡀⠀⠚⢶⣴⡏⠀⠀⠀⣿⣿⡏⠀⠀⠀⠈⠇⠊⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⡍⠉⠀⠀⠀⠀⢠⡴⠋
⠀⠀⠀⠀⠳⡄⠀⠀⣿⣏⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣀⣴⡿⠉⠷⣄⡀⠀⠀⢀⣸⣿⣿⡉⠀⠀⠀⠑⠒⣤⠞⠁
⠀⠀⠀⠀⠀⠹⣆⢲⣿⡏⠀⢠⣴⠿⠋⠀⡠⠒⠶⠞⢱⣿⣇⠀⠀⠈⠻⠿⠿⠟⠋⠉⠻⢿⣦⡀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠘⢾⡿⡅⡀⠘⣿⡄⢀⡜⠁⠀⠀⠀⣴⠋⠘⣷⣄⠀⠀⠀⠀⠀⠀⢀⣰⠞⠛⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣧⠀⠈⢻⣾⠀⠀⠀⢠⡤⠽⠆⠀⣼⡿⠀⠀⣦⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⢸⡀⠀⣾⠁⢀⣀⣠⠟⠀⠀⠀⢰⣿⡇⢀⣠⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠷⡾⠛⠀⣿⠈⠉⠀⢿⣇⢀⣸⠿⠿⠛⠋⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⣆⣀⡤⠞⠉⠛⠁⠀⠀⠀⠴⠟⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠾⠋⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁

⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⣰⠋⡡⠬⢽⣛⠛⠛⠛⠶⢤⡀
⠀⠀⡸⢁⠀⠔⣉⢽⣮⣟⠭⠍⡢⢤⣙⢦⡀⠴⠶⣦⡀⠀⠀⠀⠀⢀⡠⢦⠀⠀⠀⠀⠏⠸⠚
⠀⠀⠁⠇⣄⠀⠁⠐⣀⣄⣅⠒⠶⣤⣊⠠⠙⣦⡀⠘⡽⠀⠀⢠⣮⠟⠓⠈⠀⠀⠀⠀⡤⡀⢀⣄⣀⡀⡄
⠀⠰⠰⠀⡇⡆⠀⠐⢉⠈⠳⠀⠀⠈⣾⠀⠘⡈⢧⣠⠞⠢⠔⡟⠁⠀⠀⠀⠀⠀⠀⠈⠩⠖⠚⠊⠈⠉
⠀⣾⠀⠐⠁⢇⠺⣒⣢⠾⠚⠁⡅⢀⡯⠐⠀⠠⢸⡇⠀⠀⠀⣇
⠀⢹⡀⠀⠀⠁⠀⠀⡤⠤⠂⠁⠀⠈⠀⠀⠠⢁⣾⠁⠀⠀⢀⠌⡆
⠀⢸⣧⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠔⣂⣴⠏⠁⠀⠀⠀⠃⠈⡇
⠀⠀⠙⠿⠷⢶⣶⠶⠾⢾⠿⠿⠟⠛⡛⣍⠠⡤⡠⠀⠀⠄⢀⡴⠃
⠀⠀⠰⠖⠛⠓⠐⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣿⠋⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⡆⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⡜⠀⠀⠀⠀⠀⢀⠂⠀⠋⠀⢆⠀⡄⠀⠀⠀⠀⠀⠀⠀⠶⠸⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⡄⠀⠀⠀⠐⠀⠀⠀⠠⠄⣀⡐⠀⠀⠀⠀⠀⠀⡀⠤⡀
⠀⠀⠀⠀⠀⢀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⢡⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠸⠀⠤⠷⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⢀⠐⠄⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠆⠁
⠀⠀⠀⠀⠀⠁⠈⢁⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠄⠀⠀⠀⣅⠠⠃⠀⠀⠀⠀⠀⠀⠈⣠⣂⠠⠄⠀⠀⠀⠀⠀⠀⢈⠐⠂⠀⠐⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠤
⠀⠀⢀⠀⢀⡤⣄⠀⡄⠀⡀⠀⠀⢀⠐⠄⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠤⠄⠔⠠⠄⠄⠀⠀⠀⠄⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠂
⠀⠐⠃⠀⢀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠰⡶⡆⠀⠀⠈
⠀⠀⠀⠀⠀⣈⠩⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠂⠀⠀⢉⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠩⣂⣀⠄⠂⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠒⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠄⠀⠀⠀⠀⢀⠀⠘⠀⠃⠁⠀⠀⠀⠀⠐⠀⠔⠀⢴⠆⠀⠂⠐⠄
⠀⠀⠇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⣄⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠄⢠⡀
⠀⠀⠀⠈⠱⠀⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠩⠤⠰⠅⠀⠀⠀⠀⠀⠀⠀⠈⠠⠶
⠀⠀⡄⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠀⠊⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠐⠂⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠎⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⠉⠲⠾⡀
⢰⠤⠤⣀⣀⠤⠒⠊⠁⡠⠤⢄⡈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢃
⠀⠳⡄⠀⡠⠀⠀⠂⡄⠀⣤⠀⠱⠀⠀⠳⣀⡀⡄⠀⠀⠀⠀⡀⠀⠀⠀⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠸⢤
⠀⡰⠁⢸⠀⢠⣦⠀⢀⠀⠉⠀⠐⠀⠀⠀⠀⠐⢳⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠘
⢀⠃⠀⠀⢄⡀⠀⡠⡎⠀⠀⣁⣀⡀⠀⠀⠀⠀⢉⡇⠀⠀⠀⠀⠀⣠⠴⡄⣠⠔⡆⠀⠀⠀⠀⠀⠀⠀⢡
⣸⠀⠀⠀⠀⠀⠀⣠⠄⠀⢊⠉⠀⠘⡄⠀⠀⠀⡠⢋⢤⣠⠊⠱⠜⠁⠀⠛⠁⠀⢡⠔⠒⠀⠀⠀⠀⠀⢸
⠉⡲⠀⠀⠀⠀⢀⠉⢐⠡⠁⠘⠀⠊⠀⡀⠰⣋⡠⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⡘⠀⠰
⠀⠉⣟⢠⡄⠀⠀⠉⠁⠀⠀⠀⣀⠔⣎⠷⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠃⢀⡔
⠀⠀⠀⠈⠓⠒⠚⠉⠉⠉⢒⠯⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠖⠁⣀⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡗⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠆⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⡦⣀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠇⢰⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢱⡀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠘⡄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠎⡇⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣸⠀⠀⠱⡄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⡴⠋⠀⡇⡰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠻⢿⣿⣿⡟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡏⠀⣰⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣀⣀⠀⢠⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⣿⢿⣿⡟⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃⣰⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠃⠘⠛⢻⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣼⣯⣤⡶⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣡⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣾⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢾⣿⣿⣿⣿⣽⣻⡟⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠷⣿⣿⣿⣿⣿⣿⣿⣷⠺⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠏⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠃⠀⠀⠀⠻⣿⣿⣯⡿⠋⠀⠀⠀⠙⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣤⣄⣀⡀⣰⣿⣿⣆⢀⣀⣤⣤⣾⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣯⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠙⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⢰⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠾⠛⠛⠻⠷⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⠀⠀⢀⣤⡶⠟⠛⠉⠉⠉⠙⠛⠷⣦⣄
⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⣹⣿⡆⠀⠀⠀⠈⣷⠀⢰⣇⢈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢉⣿⣿⠇⠀⠀⠀⢰⡏⠀⠘⢧⣸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⡇⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⢀
⠀⠀⢣⡀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠈⠻⢷⣤⣀⡀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠀⠀⢀⡜
⠀⠀⠀⠳⣄⠀⠀⠉⠻⢶⣤⣀⣀⠀⠀⢀⣀⣀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠛⠛⠉⠁⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠾⠿⢿⠶⢶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⢻⠀⠀⠀⠀⢸⠀⠀⠀⠀⡟⠙⠛⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣟⣁⡀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⣗⠀⣀⣀⣠⠼⠟⢻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠈⠉⠉⠛⠛⠓⠒⠒⠒⠛⠓⠒⠚⠛⠉⠉⠉⠀⠀⠀⠀⣿⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠘⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⢈⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠛⠳⣦⠶⠛⠛⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⡤⣤⡴⠶⠶⠶⠶⢦⠤⢤⣽⣇⣀⡀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⢀⠀⢸⣿⠀⠀⠀⠀⢀⡴⠞⣿⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⣿⠛⠶⣄⠀⠀⠀⠀⠘⢿⣄
⠀⠸⠀⢸⣟⠀⠀⣀⣀⣸⣇⣀⣿⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣸⣄⣀⣀⣠⣤⣿⣤⣤⣼⣦⣤⣤⣤⣤⣴⣿⣷⣤
⠀⢠⠀⠘⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣴⠛⢳⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⠶⠃⠀⠘⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡆⠀⠀⣇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠐⣦⡀⣳⢄⣀⠼⠉⣲⣤⣤⣴⣋⡷⠤⠤⠾⢽⣶⣴⠋⠒⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⣀⣀⠀⠀⠀⢸⠉⠉⢰⡤⠒⠉⢠⡄⠀⠀⠀⡂⠀⠀⣤⠀⠀⠈⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠸⣁⠺⢧⡀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠠⠃⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⠉⠉⢹⠒⠋⠀⠀⠀⠢⠐⠤⠀⠀⠤⠒⠀⠀⠒⠠⠤⠄⢂⡠⠊⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣠⠄⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠸⠥⣤⠽⠶⠃⠀⠀⠀⠀⠀⣠⠞⠁⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠓⠦⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠋⠁⣦⣀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣠⡾⠋⣇⠀⠀⢀⣴⣧⣀⡤⠖⢉⣀⡀⢈⣉⣩⢭⣭⣍⡙⢅⣠⣀⣀⣠⣤⠞⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠛⠉⠁⠀⡾⢡⡞⣉⠔⠋⠉⠀⠀⣀⡬⠗⠛⠂⢹⡌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠉⠀⣀⣠⠴⠒⠉⠙⠢⣤⣀⣀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⠃⠀⠀⠀⠀⠀⠀⢿⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣶⠶⢶⡶⠶⠶⣶⠶⠶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠒⣾⠁⠀⠀⠘⣶⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡇⠀⠀⠀⠀⠘⣇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣡⡼⠀⠀⠀⠀⠀⠀⠸⣄⣹⡆

⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡀
⠀⠉⠀⠘⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠘⢃⡀⠛⠀⠀⠀⠠⣶⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠂⠁⠀⠀⠁⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠘⠁⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠀⠀⠀⠀⠛⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠰⠀⠀⠀⠀⡂⡀⡀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠂⠀⠒⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⢠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃⠀⠀⠀⠙⠿⠿⠟⠀⠀⠀⠀⠐⠑⠀⠀⠀⠀⠀⠀⠀⠀⠃

⡖⠂⠤⠐⠒⠀⠀⠀⢠⠀⢠⠀⠂⠀⠀⢄⡦⢠⣶⠀⠀⢰⣶⣤⢤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠁⠀⠀⠈⠈⣥⣠⠍⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠰⠀⠀⠀⠀⠤⠀⣤⠀⠀⠈⢻⣭⣽⣯⣭⣷⠀⠐⠐⠂⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠒
⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿