Surprise birthday Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠈⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠀⠠⠤⠤⣀
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢢⠀⠀⢀⡠⠔⢂⣒⣉⣀⡀⠀⠀⠀⠀⠀⠀⠉⠓⢄⡀
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢀⠤⠤⠤⠤⠤⣀⡀⢀⣄⠀⠀⣀⡠⠀⡔⠁⠠⠊⠁⠀⠀⡠⠈⠓⣄⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⠀⠸⠀⠀⠀⢀⠀⢠⠞⠀⡇⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⢀⠴⠊⢩⠀⠏⡔⢆⢳⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⢇⠀⢠⠊⢉⡒⠊⠀⠀⠀⠀⢠⡆⠀⠀⠀⡀⢸⠀⢰⠊⠀⠃⠀⢣⡘⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢧⠀⡸⢠⠇⡇
⠀⠀⠀⠀⠀⠘⢄⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁⡜⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠀⠑⠢⣀⠀⠀⠀⣀⣈⣀⡱⠃⢰⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣎⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠇⠇⠀⠀⠀⢰⠄⠀⠀⠀⠉⠉⠹⠇⢈⠉⡑⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠒⠤⢄⣀⣈⣀⣀⣀⠤⠔⠉⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢾⢀⠇⠸
⠀⠀⠀⠀⠀⢀⠔⠊⠙⢶⣤⣀⣠⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠓⠄⣀⠙⠘⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⡆
⠀⠀⠀⠀⡰⠁⢠⠀⠀⠀⠀⠀⠈⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠥⣤⣀⣀⠽⠉⢉⡏⠉⢉⣧⠤⠋⠀⠀⡀⢠
⠀⠀⠀⣘⠀⠀⠀⠁⠀⠀⠊⠈⠀⠀⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⡀⠀⠀⣠⠃⠀⠉⢉⠬⡳⢤⠴⠮⠔⠁
⠀⠀⢠⠋⠒⣴⠁⠀⠀⠈⠀⠄⠀⡸⡇⠱⣀⢤⠀⠀⠀⠀⠀⢸⡏⠥⢉⡀⠠⠚⠛⠓⠲⢉⣂⠐⠐⠋⠀⡗⠈⣆
⠀⠀⣸⠀⢰⠁⡄⠀⠀⠀⠀⠀⠊⠀⠓⠠⠄⠀⠀⠀⠀⠀⠀⢸⠊⠁⢨⣃⠤⠀⣘⡂⢤⡧⠞⠃⣀⠤⠛⠁⠀⢸⡀
⠀⠸⣁⣀⢢⠀⡿⠤⢒⣒⣒⣒⠢⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠐⠠⠉⠀⠀⠀⠻⠄⢲⡞⠒⠒⠒⠒⠘⠅⢱⡀
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢠⡄⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡐⠠⠄⠀⠈⠀⣀⣀⡿⠀⠈⠀⠘⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⣸⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠒⠃⠉⠁⠀⡃⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠇⠀⠀⡠⠻⢄⣂⣠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠒⢲⠲⠒⠒⡏
⠀⠀⠀⠀⠀⠀⠑⠒⠮⠇⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⣩⠄⠦⠤⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠚

⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢶⡆⠀⠀⣠
⠀⠀⠀⠀⠀⠻⠟⣉⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠟⠹⠀⠀⣮⡅⠀⠘⠚⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠈⣉⣵⣾⣿⠏⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠀⠀⠐⠀⠀⡿⠂⠀⢠
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠇⠀⠀⠀⠈⠉⠀⠈⠽⠋⠉⡉⠛⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⢈⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⠀⠀⠀⠀⠀⠠⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢀⡀⠀⠀⠀⠠⠤⢤⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠀⠀⠀⠀⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣰⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢠⣿⣷⣟⣴⣿⣷
⠀⠀⣤⣤⡀⠀⠚⠀⠀⠑⠀⠀⠀⢹⡠⢤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠋⢀⡹⠀⠀⠀⠀⣠⣾⣦⡺⣿⡌⢻⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⣿⢻⡀⠀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⣿⣿⣿⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠶⠛⠊⠀⠀⢿⣿⣿⣿⣿⡿⢿
⠀⠀⠀⠀⠀⡆⠁⠀⠀⢸⣿⣿⣿⣿⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠺⠤⠄⠀⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢻⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⠛⡽⣧⣈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠾⣨⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⣆⣦⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⠉⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠤⠒⠒⠤⡀⠈⢢⡀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠈⠁⠢⡄⠑⢄⠀⠀⠀⢀⠴⠁⠀⠀⢠⠃⠀⠀⠀⠀⠈⢆⠀⠈⠽⢁⡈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠈⠄⠀⠨⠏⠤⠬⠂⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠂⢀⡀⣀
⠀⠀⡠⠒⠐⠢⢄⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣠⠊⠁⠀⠀⠉⢆
⠀⡌⠀⠀⠀⠀⠀⠑⡀⠣⡀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣰⠀⠐⣣⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠾⣁⣀⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⠇
⠀⠣⡀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠙⢦⠄⢀⡠⡴⠋
⠀⠀⠈⠒⢲⢀⣈⡷⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠆⠓
⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢁⣀⡀⠀⠊⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⡄⠀⠑⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠉⠉⢩⠋⠉⠉⠉⠉⢹⠒⠓⠒⠒⡆⠀⠀⠀⠰⡆⡜⢹
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⡎⠀⠀⠀⢀⠇⠀⠀⣄⢠⢣⡇⣸⢞⡅⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⠀⢠⠇⠀⠀⠀⢸⠀⠠⣄⡋⢾⢸⠇⡿⢸⡠⡺⣀⠈
⠀⠀⠀⠀⠀⡴⠒⠢⣀⠀⠀⠀⠑⢀⣠⢼⡂⠀⠀⡗⠀⠀⠀⠀⢸⠀⠀⠀⠀⡺⠶⠦⡵⢿⠼⠾⠤⢧⣿⣾⠟⠋
⠀⠀⠀⠀⡜⠀⠀⠀⠈⣢⠤⠤⢴⠋⠀⠀⡇⠀⠀⠧⠀⠀⠀⠀⢸⠀⠀⠀⠀⣇⢀⡴⠃⠀⠀⠀⠀⠀⠉⠉⠙⡟⠐⠒⠲
⠀⠀⠀⣸⠀⠀⠀⠈⣸⣥⣀⣀⣸⡃⠀⠀⣗⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡏⠐⠒⠀⡄⠀⠀⠀⠀⢀⢀⣠⢃⡖⠐⣄
⡠⠤⠤⠗⠒⠚⣻⠛⠁⡞⠁⠈⢣⠈⠻⡟⢻⡀⠀⠸⡀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠉⠉
⡇⠀⠀⠀⠀⡔⠁⠀⢰⠀⠀⠀⠈⠀⠀⠘⣎⡄⠀⠀⠇⠀⠀⠀⢒⠀⠀⠀⠀⡗⠠⠄⢠⠀⠀⠀⠀⠀⡃
⢁⢀⠀⠄⢼⠁⠀⠀⠸⠀⠀⠀⢀⠃⠀⠀⢸⢧⠀⠀⢺⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠉⠉⠉⡹⢹
⠈⠀⠀⠀⢸⡀⣴⠀⠀⠀⠀⠀⡸⠀⣶⠀⣸⠸⡄⠀⠈⡆⠀⠀⠸⡆⠀⠀⠀⢳⣀⣀⠴⠀⠀⠀⠀⢠⠀⠈⠒⠐⣚⣁⣼
⠠⡀⡀⠀⠀⠹⢽⡤⠖⠀⠀⠀⠙⠢⠟⠗⠃⠄⢗⠀⠀⢰⠀⠀⠀⡧⠀⠀⠀⠸⡀⢀⣤⡄⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⡄
⠀⠸⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢰⠀⠀⠀⠀⠘⡆⠀⠈⡇⠀⠀⢰⠄⠀⠀⠀⣇⡇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡇
⠀⠀⠱⣀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢸⡀⠀⠀⢧⠀⠀⠀⠻⣈⣒⣣⠀⠀⢀⣄⣀⣰⣀⣀⣤⡼
⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠂⠀⠒⠋⠈⠉⠉⠉⠉⠉⠓⢉⠉⠑⠒⠒⠊⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⢻⡆
⠀⠈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠞⠋
⠀⠀⠀⠀⠀⢹⡝⠛⠶⢶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⠶⠟⠛⢉⣿⠁
⠀⠀⠀⠀⠀⠈⣷⡀⠀⣠⣼⠿⣏⠛⠛⠛⠿⣿⣶⠶⣶⣶⣶⣶⣶⣿⠀⠀⣿⣶⠶⠶⠶⠾⢿⣿⣟⠛⠛⠋⢩⣿⢷⣄⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠘⣷⡾⠋⠀⠀⠹⣆⠀⢀⣴⠟⢻⡄⠀⠀⢀⡴⠛⡯⠀⠀⣿⠙⢦⡀⠀⢀⣿⠁⠙⢷⣄⣠⡿⠃⠀⠙⠷⣦⣼⠏
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠙⢷⠟⠁⠀⠀⢿⣤⡾⠋⢀⣠⣧⣤⣴⣿⣤⡈⠻⣦⡾⠃⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⢀⣴⠾⠛⠛⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⢼⣏⠀⠀⠀⠀⣸⡇⠀⠈⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠛⠷⣦⡀
⢠⡿⠁⠀⠀⠀⠀⠀⢈⣷⣤⣤⣄⡀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠈⢹⡟⢻⣟⢻⡏⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⢀⣠⣤⣤⣾⡁⠀⠀⠀⠀⠀⠈⢿
⣾⠁⠀⠀⠀⠀⠀⣴⠟⠉⠁⠀⠉⢻⣦⠀⠀⢶⣤⡀⠙⠀⠀⢀⣼⢇⡏⠹⣾⣷⡀⠀⠠⠎⠀⣀⣤⠀⠀⣴⡟⠉⠀⠈⠉⠻⣦⠀⠀⠀⠀⠀⠈
⣿⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠈⠁⠀⠀⣠⣾⠋⣼⠁⠀⠹⣎⠻⣦⡀⠀⠈⠉⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠘⣷
⢹⣆⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⢀⣾⠟⠁⢀⡏⢀⠀⡀⢿⠀⠈⢿⣦⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣰
⠀⠻⣦⡀⠀⠘⣷⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⣾⠇⠀⠀⢸⣷⠿⣤⠟⠻⠁⠀⠀⢿⡇⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⣾⠃⠀⢀⣴⠟
⠀⠀⠉⠻⠶⣶⡿⣧⡀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⢀⣼⢿⣶⠶⠟⠉
⠀⠀⠀⠀⠀⠈⣧⠙⠻⢶⣶⠿⠛⠁⠀⣠⠶⠋⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠰⢤⡀⠀⠈⠛⠿⣶⡶⠟⠋⣼⠁
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣿⠀⠀⠀⡾⠃⠀⠀⠀⠀⠀⠈⠻⢶⣤⣄⣀⣀⣠⣤⡶⠟⠉⠀⢀⡀⠀⠀⠙⢧⡀⠀⠀⣿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⢀⡿⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⢿⡄⠀⠀⠈⠁⠀⠀⢿⡄⠀⠀⠀⢿⡀
⠀⠀⠀⠀⢸⡇⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣴⠾⠛⠶⣄⠀⢀⣤⡶⠶⣤⣄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠲⠶⠦⠀⠀⠀⠀⠀⣀⡀⠀⣴⠟⣋⡉⠛⣦⠀⠘⣿⠀⣿⠇⣽⠇⢸⡇⢰⣶⠆⢹⡇⠀⣸⠏⠙⣷⠀⠀⢀⣀⣀⡀⠀⠀⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢸⡋⢿⡄⢿⡀⠻⢛⣤⡟⠀⠀⣿⠀⢠⡞⠋⠀⢸⡇⠀⠀⣶⠟⠁⠀⣿⠀⢸⡇⠀⣴⠋⠉⣉⠙⣦
⠀⠀⢀⣤⣤⡀⠀⢿⡉⢻⡜⣧⠘⣷⠘⣧⠀⢀⠙⢷⡄⠀⢻⠀⠸⣧⠀⠀⢸⡇⢰⣦⠸⣧⠀⢰⡇⠀⣿⠀⣀⠻⣦⡸⣯⡙⠁⠀⢀⣴⣤⡀
⠀⡼⠋⢀⡬⠇⠀⠘⢷⡀⠻⠿⠀⣿⠀⢹⣆⣸⠿⠶⠿⠀⠸⠷⠾⠟⠀⠀⠈⠳⠛⠉⠓⠛⠀⢺⣧⣼⠇⢸⡋⠓⠞⠁⠈⣿⠀⣠⡟⢁⣈⠙⣷
⠈⢧⣀⣘⡛⠛⢶⡄⠈⠻⣦⣤⡾⠋⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⣦⣤⣤⠞⢃⣴⠋⢀⣻⣿⡛⠋
⠀⠀⣩⣭⠟⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣧⣀⠻⣭⡉
⠀⠀⢷⣄⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠁
⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠐⠀⠍⠈⠈⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⢄⠠⠒⠀⠐⠀⠄⡀⠀⠀⡐⠢⠅⠰⠂⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⠐⠁⢀⠂⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢒⡚⢒⡈⡀
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢐⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⢀⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁⠉⠉⡄
⠀⠀⠀⠀⠀⡀⠀⠡⡁⠀⠀⠀⠁⠀⠀⠔⠸⠶⠄⢠⢤⡈⢢⠆⠤⠄⠀⠀⠬⠄
⠀⠀⠀⠀⡈⠨⠀⠀⠀⠀⠀⠑⠀⠀⠃⣀⡈⠂⢁⣄⠉⢠⠆⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⢇⡀⢠⢠⠀⠉⠀⠐⠂⠈⠉⠉⠣⡀⠈⠋⠀⡘⠀⠀⢂⠄⠘⢤⠂⠂⢣⠀⠀⠌⠀⢀⠠⠂⢑
⠀⠀⠀⠀⠘⠈⠀⠀⠀⠲⠀⠩⡁⠐⠀⠀⠀⠘⣫⠖⠀⠀⠈⠈⠁⠀⠀⠂⠁⠀⠀⠀⠲⢀⠀⠁⢀⠴⠁
⠀⠀⠀⠀⠀⠐⠬⠂⠀⠀⢄⠀⠘⠶⣤⣤⢄⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢦⡄⠀⡄⠈⠀⠀⠁⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢡⠀⠀⠀⢀⠁⠀⠀⠀⠀⠠⠀⠀⡀⠠⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠁⢶⢳⠄⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢭⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠠⣾⣛⡦
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠑⠠⡀⣶⡀⠒⠀⡘⠀⠀⠀⠀⠀⠈⠀⠀⠚⠒⠀⠀⠁⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠘⠀⠀⠀⣤⣶⣶⡇⠀⠀⣶⣶⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠰
⠀⢀⡀⠀⠲⠀⠀⠀⠀⠉⠉⠢⠀⣬⠉⡁⠀⠀⠀⠀⠐⠢⢌⢁⠀⣁⠠⢔⠂⣠⠚⠀⠀⠠⠂⠀⠀⠂
⠀⠀⠀⠀⢈⢲⠤⠀⠀⠈⡀⠁⠀⠈⢀⡃⠀⠀⠁⠀⠀⠀⡄⠰⠤⠧⣬⠴⡃⢀⡀
⠀⠀⠒⠠⣞⡿⠉⢀⠄⠈⠉⠀⠒⠂⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠄
⠀⠀⠀⠀⡶⣗⠀⠘⠀⠀⠀⠀⢀⠀⠀⢠
⠀⢀⠤⠐⡧⣗⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠠⠁⠀⠀⠉⠁⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢇⡀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠈⡈⠁⠈⠂⢐⢀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁
⠀⠁⠀⠨⠙⠉⠉⠁⠃⠠⠀⠀⢀⢀⠠⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇

⠀⠀⠀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠤⠤⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠘⢿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣼⣷⡄⠀⠀⠀⣀⠀⢀⡀⠾⠿⠂⣠⡇⠀⢙⣛⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣿⣿⡷⠀⠐⢟⡁⢀⠈⢠⣔⣦⢼⣿⡟⠀⢉⡉⠻⢟⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠀⣿⣿⡷⢀⣤⣾⠟⠁⠐⢦⣍⢈⣿⣿⣿⠚⠉⣠⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⣿⢸⠁⠆⠀⡶⠀⠐⠀⠾⣿⣷⣬⡟⠋⠀⠀⠋⢁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⡠⠀⠀⠀⠀⢀⣠⣤⣤⣐⡊⠀⠄⢤⣭⠎⣠⣼⣄⠘⢿⣇⠀⠀⠁⠉⠛⠀⠶⠀⠀⣠⣾⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠀⠀⠀⠀⠀⠁⡼⠷⠤⢾⣿⣾⣿⣿⣅⣀⡀⠀⢻⠁⠈⠉⠙⠓⠀⡼⠙⡷⠀⢤⣤⣤⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠊⠛⠻⣭⡍⣋⠈⠥⢄⣈⠀⠃⠞⢯⡁⠒⠀⠈⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠙⠳⠞⠳⠦⠀⢈⣭⣥⣤⣤⡄⠀⠤⠄⠀⢴⣶⢤⣬⣶⣶⡦⣤⣴⣄⣀⡀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡤⣿⣿⣿⣿⡿⣶⣦⡤⠄⠀⠀⠀⢤⣭⣍⠡⢨⣤⠀⣩⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣀⠀⠘⢿⠉⢉⣁⠤⠐⠒⢀⡀⠘⠋⠀⠀⠰⠤⢠⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠑⠀⠛⠓⠂⠉⠰⠯⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢦⡀⢄⠀⢤⠀⠀⠘⠀⢠⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⢀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠚⠀⠀⢂⠀⠈⠁⠰⠄⠀⡄⠉⠀⠀⢻⠤⠍⠀⠀⠘⠀⠀⠈⠁⢄⠄⠜⠀⠤⠛⢾⠁⠀⢀
⠀⠀⠀⠀⠀⢀⡄⠘⠂⠐⡀⠀⠀⠈⢀⠀⢂⠘⠚⢤⡀⠀⡀⠀⠀⢄⡄⡀⢠⠐⠂⠀⣤⠀⢀⠀⠀⡀⠀⡤⠒⣀⠀⠆⣠⠃⣀⠜⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠄⠀⠀⠈⠁⠀⢡⠀⢺⠁⢂⣀⠋⠁⢤⢰⡀⠀⠀⠄⠀⠀⠀⠉⡀⠀⠁⠀⠃⠚⢀⡀⠪⠀⢁⠘⠒⠆⠀⠄
⠀⠀⠀⠀⠀⠰⠀⠀⠖⠀⠀⠀⠢⠘⠂⠀⣄⡈⠄⠋⠀⠀⠀⠀⠼⠛⠀⢀⠀⠔⣉⢠⠀⢲⡟⢉⠁⡀⠢⠆⠈⢀⠀⢠⠀⡑⠣⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠆⠀⠃⢁⢄⠘⡄⠧⠈⠀⡀⠰⠆⠀⠀⠂⠀⠠⠧⡰⠀⡄⠰⠆⢄⠀⢀⠐⢤⠀⠀⠀⠀⡀⢀⣩⣤⡏⠇
⠀⠀⠀⠀⠀⢀⡕⠠⡀⠉⠋⠀⢠⡀⠀⡀⠀⠀⣀⡩⠀⢠⠀⠀⢀⠀⡂⠈⢤⠀⠁⠁⣤⠐⡤⠆⠀⠃⢃⡀⢃⡄⠐⢀⠀⠠⠀⠀⠔⠁⢠
⠀⠀⠀⠀⠀⠘⡇⢀⡈⠢⡐⠂⠀⠀⠤⠀⠘⠖⠋⢀⠡⠤⣴⠃⡈⠁⠀⠀⡄⠢⠈⣀⠀⠠⠒⢀⡄⠘⡠⠁⢠⡄⢀⢈⠚⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠀⢔⠄⠀⠶⠈⢌⠡⡊⡖⢀⠓⠀⠁⣀⠐⠂⠀⠂⠐⠈⢀⠚⠃⠀⠉⠔⠀⠤⠀⠀⠈⠨⢂⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⡤⢔⠢⠁⠐⠈⠄⠈⡀⢫⡀⢀⡰⠂⠠⣤⠀⡀⠗⢀⠜⠀⠆⢀⠤⡄⠂⠰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠊⠠⠠⠘⠀⠈⠄⠋⠐⠠⠈⠑⠂⣀⠎⠀⣵⡦⠂⠘⠀⣤⠀⢉⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠶⠶⠼⢭⣭⣏⣸⣏⡋⠉⠉⠃⠩⠉⡇⢸⡌⠁⢁⣓⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢠⡔⢒⠣⡜⠳⣘⣁⣨⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡖⠈⠅⠸⠧⣤⣾⠟⠿⠛⠛⠻⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⢹⠁⠀⠀⡘⢅⣿⠆⠶⠦⠿⡲⠾⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡂⠀⠀⠀⠠⡻⡶⠲⡶⠲⠺⠶⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢔⠁⡆⠀⠀⠸⠈⠀⣇⠀⣃⠀⠀⢈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠤⠀⠀⠀⠀⠀⡈⠠⠁⠀⢼⠀⠀⠀⠀⢊⢸⠀⢼⠄⠁⢁⠐⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠈⠁⠒⠂⠀⠠⠤⠶⠮⠦⠤⢼⠐⣀⠀⠆⢘⡨⠄⠜⠂⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⡀⢀⠉⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠁⠂⠀⠀⠀⠁⠠⠂⠄⠠⠈⠔⠠⠢⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠀⠃⠈⠁⠉⠀⠂⠀⠀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⠈⠁⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢠⠁⠐⠀⢒⠀⠀⠀⠀⠄⠀⠒⠀⠃⠀⢉⠀⠒⣀⡈⡓⠀⠀⠀⠐⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠠⠄⠘⠀⠘⢀⡀⢀⡀⠐⠂⠁⠃⡈⠁⠈⡠⣤⠈⣁⠀⠨⡆⠀⠠⠀⠀⠃⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⠀⡈⠠⠄⢉⡆⠀⠀⠀⢱⠀⣀⠈⢁⡀⠑⢡⠀⠁⠘⠀⠆⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢉⠀⢡⠀⠠⠆⢤⠄⠀⠀⠀⠙⠋⠀⠑⠜⠆⠠⠀⠉⠐⠂⠀⠉⠁
⠀⠀⠀⠀⠀⢀⢤⠠⠄⠈⠀⢀⡄⠀⠆⠠⠄⠀⠀⠠⣔⡆⠸⠵⠐⠂⠤⢀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠃⢀⡀⠰⠀⠐⠈⠀⠀⠀⠄⠠⣶⠄⡐⠂⠠⢄⡀⡈
⠀⠀⠀⠀⠀⡀⡂⠀⠀⠠⠒⢀⡀⠤⠀⠒⠀⠀⠀⠀⠉⠀⢀⣀⠃⡁⠐⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣂⠄⡁⢀⠶⠈⡁⠁⠐⣠⠀⠀⠀⠠⠅⠀⠠⠆⠙
⠀⠀⠀⠀⠀⠀⠀⢆⠂⠀⢠⠰⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠊⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⣀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⣠⠅⣀⠁⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⠀⠉⠠⠈⠠⠀⠂⢐⢋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⢤⡤⠐⠆⠀⠈⠉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠠⠢⢣⠔⠠⢃⣀⠈⢎⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡬⠀⡁⢀⢂⠀⡑⠂⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⢂⠀⠈⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⡀⠀⠀⠐⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⠀⠀⣀⡀⠀⣈⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⣠⣤⣄⢠⣶⠀⢠⣶⣶⡞⠳⣶⣴⣿⠙⢿⣾⣿⠙⣿⣾⣿⣶⣿⠹⡿⣾⡏⢹⣿⠀⣿⡇⢠⡄
⠀⠀⠀⠀⢸⣿⡀⠛⢻⣿⠀⠘⣿⣿⣧⣴⡛⢸⣿⡤⠞⢹⣿⣴⣏⠁⣿⡟⠿⣦⣄⣿⣧⣸⡟⠀⣿⡇⠈⠁
⠀⠀⢀⣄⠀⠙⠻⢶⣬⣿⡆⠀⣿⣿⣿⢻⣷⡀⣿⡇⠀⢸⣿⡙⢿⣆⣿⣧⣤⡀⢹⣿⣿⣟⢰⣿⣛⡁
⠀⠀⠀⠉⠀⢶⣆⣸⡿⠻⠿⠼⠟⠸⠟⠀⠹⠟⠛⠃⠀⠀⠛⠃⠈⠋⠉⠉⠉⠙⠋⠁⠈⠙⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡀⠀⠈⢢⠀⠀⠐⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⢦
⠀⠀⠀⠀⠙⡀⠀⠈⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡀⠀⠈
⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⣉⣝⡀⠀⡀⠀⠀⠀⢤⠄⠀⠀⠀⠁⠠⡄
⠀⠀⠀⠸⠀⠀⠀⠀⠐⡂⠐⡖⠀⠀⠀⡀⠀⠀⠀⠀⢠⢉⡇⠀⣰⣏⣹⡆⠀⠻⠄⠀⠠⠋⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠘⢣⠀⠀⠀⠂⠀⠀⠀⠀⢼⢀⠛⡆⠠⠧⢵⠖⠀⠀⠀⣀⡰⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠈⠀⢠⠀⠀⠀⢠⣔⠀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠐⠀⠀⠀⠿
⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠔⠄⠠⠤⠀⠀⠀⢀
⠀⠀⠀⠀⠴⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣶⠄
⠀⠀⠀⢀⣟⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡷⣾⠃
⠀⠀⠀⠠⠎⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢲⠆⠶⠀⠀⠈⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣮⣦⣮⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠷⡦⡀
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠀⠀⠐⠂⢄⣀⠀⢀
⠀⠀⠀⢀⠠⠤⡎⠠⠁⠀⠀⠀⠀⠐⢢⠤⠒⠒⣦⣄⠀⠀⢰⠁⠀⠀⠀⠀⠀⡔⠨⠢⠔⠀⠈⢂
⢀⠔⠈⠠⠄⢰⡇⠀⠀⠀⠀⡄⢀⠏⡁⠀⠀⠀⠈⠉⡂⠀⠘⠄⠀⠀⠀⠀⠀⢆⠀⠀⡀⠀⠀⠀⠑⡀
⢸⠀⠠⠄⠀⢸⠀⡀⠀⠀⠰⠀⠎⠈⠀⠀⡠⠂⠀⢠⠁⠀⠀⡠⠓⠒⠈⠀⠀⠬⢦⠀⠁⠀⠀⠀⠘⠌⠄
⠈⠤⠀⠤⠄⠸⡀⠀⠀⠀⠀⠼⠀⠀⠀⡀⠤⠀⡠⠃⠀⠀⠘⠀⠀⢀⠀⠀⠈⠁⠀⠡⡀⠀⠀⠀⠀⠈⠸
⠀⠀⠀⠀⠀⠀⠑⡀⠀⣀⠌⠀⠑⠠⠄⠠⠄⠊⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⡕⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⠀⠀⠀⠐⠉⡲⠈⠂⠄⢀⡀⠜
⠀⠀⠀⠀⠔⠁⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⠄⠜⠊
⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡐⠂⠈
⡇⠀⠐⠀⠀⠀⠀⡠⡊⠀⠀⠀⠀⠀⠀⠀⠂⠔⢀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠑⢀⠀⠀⠀⡠⠊⠀⠀⠁⠂⠠⠀⠀⠀⠀⠠⠐⠁
⠀⠀⠀⠉

⠀⢐⡀⠀⣂⠀⠀⠀⠄⠀⠀⠀⢜⠀⠀⠆⠀⠀⠀⢤⢀⠠⡀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣶⣤⠀⠀⠀⠠⠄⠄⣠⢀⠄⡀
⠠⠈⠂⠐⣈⠂⠀⡄⠂⠄⠀⠁⠉⠀⠂⡥⠆⠑⣈⢰⣇⡠⠋⠢⡀⣿⣰⣧⣝⣀⣇⣷⣅⣿⢀⠔⠀⠀⡤⠀⠠⡄⠀⢠⡔
⠀⡀⢀⠀⠄⠄⡀⢔⢊⠌⠀⡨⣀⠀⠀⠰⠀⠰⠢⢓⠀⢳⡄⠐⠀⠙⠛⠛⢻⡿⢛⡛⠛⠁⢨⠀⡳⠆⠀⣼⡄⢡⣶⡄⠁⢳⡇
⠀⠁⠁⠂⠀⠸⠅⠃⠀⡂⢀⠦⢀⠀⠁⢉⠀⠀⠾⠂⢠⠂⠀⡠⠊⠖⠀⡀⠈⠀⢠⡄⠀⠈⠀⠁⠠⠘⠠⣀⠔⠢⣀⠄⠁⠤
⠀⡠⢂⠅⠰⢈⢐⠀⡨⠇⣀⠪⠂⠁⢀⠰⢁⠀⠀⠠⠂⢰⡁⠄⠀⠀⠀⢧⢀⠀⡇⢩⠀⡞⢰
⠠⠠⠠⢐⠀⠂⠂⠂⠰⠃⠀⠄⠡⠂⠂⠀⠠⠃⠠⠄⠠⠊⠧⠤⠤⠀⠂⠈⠄⠀⠀⢸⠀⠇⢸⠀⡀⡤⠀⠠⡢⠀⠀⢰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⡀⠐⢖⠂⠡⠀⠁⠀⠀⠔⠀⠈⠀⠀⠈⣹⡄⠀⠀⠐⠤⠘⠀⡄⢼⠀⠀⠈⢀⠀⠀⠀⢀⠅⠘⢂
⢀⠄⠀⢠⢀⠀⠀⣉⣂⠐⠻⠲⡶⠲⡶⠆⠀⠰⢆⠂⢒⠶⡀⠁⡁⠈⠁⡀⠀⡐⠈⠀⠁⢀⠀⠀⠀⣄⠀⢠⠀⠀⠌⠈⡀
⢀⣤⣤⣄⡏⢄⢾⣿⣿⠀⠀⠀⠹⠜⡀⠈⠤⠂⠈⠳⠎⠀⠈⠒⠀⠀⠂⠂⠀⠀⠀⠀⠀⢀⠀⠀⣄⠈⠘⠈⠑⠠⠆⠀⠀⠠⠘
⠀⢿⣿⡟⠀⠀⠈⠋⠙⢆⠀⠀⠀⢀⢆⠀⠀⠠⠀⢀⠀⠲⠲⠒⠒⠒⠖⡤⢄⠀⠀⠀⠀⠞⠉⠀⠐⠄⠀⠀⠆⠀⠀⠀⣀⡀
⠀⠀⢖⠀⠀⠀⡠⠒⡀⠘⠣⢀⠠⠼⠸⠄⢀⠀⠀⡞⠀⢀⠀⢀⠐⠀⡆⠀⢘⠆⠘⠋⠁⠀⠀⠀⡀⠈⢆⠈⠀⠀⠃⡀⠉
⠀⠂⠛⠑⠀⠀⠅⠀⠀⢂⠀⠔⡀⠀⠀⠀⢰⡀⠀⡁⢈⠁⠄⠁⠈⠀⠁⠀⠀⠀⠀⠀⢢⡀⠀⠀⡀⡀⠀⠁⠀⠈⠂⠁⠀⠐
⠀⠀⠀⢰⡄⠸⡀⠀⠀⡘⢸⠠⠄⣁⠀⡀⠡⢸⠀⢃⢄⠀⠐⠐⠢⠀⠠⠄⢀⠇⠀⠀⠀⠀⠀⠤⠀⠸⠄⠀⠀⡀⠈⠁⠈⠂
⠈⡐⢠⠃⠱⡀⢡⠀⢀⠀⡸⣀⠀⠀⠀⠀⢐⠸⠀⠀⠘⠀⠤⠤⠤⠤⠄⠐⠁⠀⠰⠆⢰⣶⠞⠉⠁⠀⠙⢶⣶⠀⠀⠐⠈⠄⠁
⠀⠀⠎⠀⠀⠰⠀⠉⠈⠀⣈⠂⢩⠄⠀⠬⠀⠊⠀⣄⠀⠀⠀⢀⠀⠢⠉⠁⢐⠀⣀⠀⠀⠙⠀⠀⠀⠀⠀⢸⠁⢀⠠⠀⠒⢐⠢
⠀⢸⢀⡀⣀⠀⢇⠀⣸⠀⠉⠀⠙⠁⠀⠀⣠⠊⡀⠊⡀⣀⡂⠘⡬⢄⣲⠅⠈⡀⠓⠁⠀⠈⢀⡀⠀⠀⣠⠊⠀⢀⠘⠂⢒⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⠀⢀⢐⠀⠀⡰⠁⠀⠀⢠⡅⠃⠀⠀⠀⠀⠀⡀⢀⠅⠀⠴⠢⠀⣤⣬⣷⣯⣤⣇⠀⠀⠑⠀⠐⢂⠁
⠀⢀⡀⠙⢋⢄⠀⠈⠢⠈⠈⠈⠀⠀⠐⠧⠐⠂⠉⢀⡀⡀⠀⠀⠀⠉⠐⠒⠁⠐⡿⠠⡤⠤⡄⡉⠉⣉⣉⣀⣀⢀⠀⠠⠄⠺⠃
⢀⢮⣈⠻⠧⣔⠢⡄⠐⠠⠀⠀⠀⢀⠆⢀⠀⢠⠞⠁⠀⠰⠞⠉⠁⠀⠀⠀⠂⡀⠀⢠⣀⣀⡃⠀⠀⠀⠀⣰⠀⠀⠀⡀
⠘⠉⠀⠤⠀⠐⠀⠅⠐⠀⠊⠀⠈⢂⣀⡈⠀⠐⠀⠁⠂⢐⠆⠼⠀⠷⠀⠀⠀⠁⠀⠀⠀⠈⠁⠁⠈⠁⠈⠁⠉⠀⠁
⠈⠢⡀⠀⠀⣀⠤⠀⠚⠒⠀⠀⠂⠀⠙⠀⠀⠃⠒⠀⠀⠌⢸⠀⣇⠀⠍⠀⠀⠀⠘⠄⢱⠀⢉⠀⠀⠎⠂⠰⡰⢢⠚⠁
⠀⡠⠬⠉⠀⠀⢄⠀⢀⢆⠀⠀⠀⠀⠀⠀⠀⠄⠘⠀⠌⠀⠀⠲⠀⢀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠂⠐⠃⠀⠊⠀⠐⠂⠀⠈
⠀⠀⠀⡖⠲⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣰⣰⡌⢄⡸⠀⠀⡂⢀⠀⢀⠂⡂⠀⠚⠈⠇⢹⠈⠐⠄⠀⠡⠀⢡⠋
⠀⠒⠀⠀⠀⠀⠚⠀⠁⠂⠤⠁⠁⠀⠈⠉⠈⠒⠒⠒⠒⠒⠀⠂⠀⠁⠀⠑⠀⠈⠂⠘⠑⠃⠐⠂⠐⠁⠊⠐⠒⠀⠀⠀⠂⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡊⠀⠀⠀⠠⡀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⠈⠹⣧⡀⠀⠀⠘⣆⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⡠⠤⠦⢤⣌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠷⠶⢤⣌⣻⢦⣀⠀⠈⢧⡸⡄⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⠔⠉⠀⠀⠀⣸⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠈⠉⠛⠿⣶⣄⣀⠉⢃⣀⣀⣀⡀⠀⡠⠊⢀⠔⠁⠀⠀⠀⣀⡔⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠈⠛⠳⠶⣤⣤⠬⣍⠛⡏⡀⠘⠋⣿⠉⠴⠞⠓⠚⠋⠉⠉⠉⠉⠉⠛⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠒⠈⠉⢀⠀⠈⠁⠒⠤⣄⣀⣀⣀⣀⣀⣤⣿⣧⠀⢸⡇⠈⠒⢄⡛⠛⠛⠛⠛⠛⠒⠒⠤⠀⣹⠀⣉⣠⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠛⠲⠦⡭⠛⠛⠛⠀⠀⢀⠀⠄⠐⠀⠁⠙⠻⠷⠾⠳⣄⠀⡈⣷⡈⠛⠲⠶⣦⣤⡀⠀⣼⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠀⢷⠠⠤⠤⠤⣤⣤⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠙⠒⠒⠢⠤⠤⣀⣀⡤⠖⢻⠀⢸⣧⣄⠀⠀⠀⠀⢹⡿⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠀⢀⠉⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⢀⡏⠀⠘⡇⠈⠛⠷⠶⠾⠛⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠐⠦⠤⠖⢻⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⡾⠁⠀⢠⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡼⠁⠀⠀⢸⡇⠀⠀⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⣸⠧⡀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⡀⠀⠀⡟⠀⠈⢳⡇⠀⠀⢠⠂⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠋⠉⠹⡄⢸⠃⠀⠀⠀⡇⠀⠀⢸⢣⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⡇⠀⠀⢸⠀⠱⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⠀⠀⠰⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⠀⠀⠀⢣⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡜⠀⠆⠀⠀⢸⠀⠀⠀⠀⢇⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡜⠀⢸⠃⠀⠀⢸⡀⠀⠀⠀⠀⠃⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠞⠀⠀⢸⠀⠀⠀⠘⡇⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣄⣸⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⡷⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢤⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣸⡆⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣄⣀⠀⠀⠀⠀⡇⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣤⣯⠤⠒⠉
⣤⣤⣤⣤⣀⣤⣤⣄⣀⣄⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⠄⣟⣋⣛⣛⣋⣛⣙⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⣁⣀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢙⣦⡀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠘⡄⠙⢦⠀⠀⠸⡄⠀⣀⣀⣀⣀⣀⣀⢀⠀⠀⣠⠏⠀⠀⠀⠀⢀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢇⣀⣤⠷⡀⠀⢹⠉⠉⠉⠀⠀⠀⠀⠀⢈⡞⠁⠀⢀⣤⠶⠒⠛⡆⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠷⣄⠘⣇⠀⠀⠹⡄⠀⣇⠀⠀⠀⠀⠀⠀⣠⠋⠀⢀⡴⠻⣄⠀⠀⠀⢹⡄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠚⠉⠀⠀⠀⠘⢷⣌⠳⣄⠀⠹⡄⠸⣄⠀⠀⠀⠀⡴⠃⣀⠴⠋⣀⡤⠾⠓⢦⡀⠀⢷⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣌⡳⣄⢹⡄⢻⣦⣄⣀⠞⣡⣾⡥⠖⠋⠁⠀⢀⣀⣤⠿⢦⣸⡀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠈⢟⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣷⣷⠏⠁⢠⡿⠛⠋⠁⣀⣠⡤⠖⠛⢋⣉⣀⣀⣤⡿⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⢻⡏⠀⠀⢸⠛⠒⠒⠿⠛⠛⠛⠛⠋⢉⣉⣉⣭⠤⠖⠋⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠉⠲⣄⠀⢀⣀⣤⠴⠖⠛⠉⠀⠀⠀⢉⡿⢷⡛⠁⠉⠙⠛⠻⣟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⠀⣀⣤⠶⠚⠉⠀⠀⠙⢦⡀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⢀⣠⠤⠖⠊⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠢⣀⠀⠀⠀⢸⠀⠀⠀⣰⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⢀⣩⣷⣤⠔⠊⠉⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠑⠦⡀⣸⠀⠀⠀⣿⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢷⡶⠛⠉⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢹⡶⣄⠀⣿⠀⠀⠉⠲⣄⠀⠀⠀⢀⡠⠔⠂⠉⠀⠈⣇⠀⠀⠀⢸⡇⠀⠀⣀⣠⡤⠶⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃⠙⠷⡇⠀⠀⠀⠀⠀⠑⡄⠊⠁⠀⠀⠀⠀⠀⠀⣿⢀⣠⣤⡾⠟⠚⠉⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣾⠀⠀⢰⡏⠢⢄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣠⢴⡏⠉⠁⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡏⠀⠀⣾⠁⠀⠀⠉⠓⢄⠀⡇⣀⣀⠤⠖⠋⠉⠀⠈⣇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣄⠀⠀⠀⢸⠇⠀⣸⠇⠀⠀⠀⠀⠀⠀⢙⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⣾⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣇⠀⠀⠀⠀⣀⣀⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⢧⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣀⣿⡶⠖⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠶⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⡇⠀⠀⢀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡇⣠⠖⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⠒⠒⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⡠⣖⣒⢤⠀⠀⠀⠀⠀⡠⣒⣒⢄⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠐⡷⡁⢀⣇⡇⠀⠀⠀⠸⡸⡀⢈⢏⠆⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠙⠚⠓⠊⠀⠀⠀⠀⠀⠑⠚⠓⠋⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣔⣒⣒⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⣎⠞⠁⠀⠀⠙⢎⡆⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠘⡼⡀⠀⠀⠀⠀⡸⡜⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠱⣕⠤⣄⡤⢔⠵⠁⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⣀⠀⠀⠀⢀⣀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⢿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣇⣷⣾⣿⣿⣿⣻⣣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈
⠀⠀⠀⠄⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠡⠀⡀⠀⠀⠀⠀⠠⣘⡂⢾⣒⠠⠤⣀⠀⠀⠀⠄⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃
⠀⠂⠀⣀⠙⢄⠔⠂⠀⠀⠀⠀⠈⠘⡀⠐⠁⠀⠀⠀⠀⠀⠈⠤⢆⡛⠢⠄⠀⠀⠀⠀⠀⠐⠄⡀⠀⣀⠄⠀⠀⠀⠀⠀⠘
⠀⠃⠀⡀⠉⡀⠈⠀⠀⠀⠀⡀⠀⡁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⠀⠂⠀⠀⠀⠀⠀⠣⠀⠀⠀⠄⠂
⠀⠀⠀⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢆⠄⢀⠐⠀⠀⠀⠀⠀⠐⠓⠶⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠆⠘⢍⡀⣂
⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀
⠀⠀⠈⢶⠀⣠⡄⠀⠀⠀⠀⠀⡀⠀⠀⠓⠘⢃⠀⠀⠀⠀⠂⠀⢀⠀⣀⠈⡂⠀⠀⠀⠀⠀⢀⢀⠀⡀
⠐⠢⠔⠉⠁⠉⠀⠀⠀⠀⠀⠀⠁⠈⠂⠀⠘⡰⠀⠀⠀⠀⣆⠀⢨⠲⢉⡑⠠⠄⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⡈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠖⠋⠁⠠
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠆⠀⠀⢀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠴⠠⣆⡿⢒⣃⠤⠀⠀⠀⠀⠀⠀⢀⣀⢄⡀⠀⠀⠀⠀⠰⡀⢸⡦⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⡛⡸⠖⠃⠀⠀⠀⠀⠀⠀⠐⠉⠈⡀⠀⠀⠀⠀⠀⠀⠓⡀⠀⣁⠁⢲⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢄⠀⠀⠔⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣦⣴⣴⣤
⣿⣿⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁⣀⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⢠⣶⣶⣤⣤⣤⣀⠀⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠉⠉⠉⠉⠉⠉⠁⠀⢸⡇⠀⠀⠈⠉⢉⠉⠉⠉⠁⠀⠀⠙⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢿⡟⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⣸⠁⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢹⡁⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠋⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡄⠀⠀⠠⠄⠀⠰⠄⠀⠀⠀⠀⣀⡄⠀⠀⡀⠀⠀⣀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠉⠀⠀⠈⠁⠀⠀⠉⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠉⠀⠀⠃⠀⠀⠆⠀⠀⠀⠀⠒⠀⠀⠀⠒⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣀⠀⢠⠄⠀⠀⠐⠂⠀⠀⠀⠠⠄⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⠀⠀⠀⠶⠀⠀⠈⣃⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠉⣬⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠁⠛⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⢴⠄⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠁⠀⠈⣣⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠔⡴⠀⠈⡑⠀⠀⡠⠖⠒⢤⠀⠀⠀⣏⣁⠀⡂⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠴⠁⠃⡠⠤⠁⠀⠀⡇⠰⠇⠀⡇⠀⠀⠀⡤⢉⢔⠡⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⢠⡎⠻⠄⠀⠀⣠⡤⠈⠒⡆⠊⠀⠀⠀⠀⠰⢋⠂⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠁⠀⠀⠀⠰⡄⣰⠀⠀⡇⠀⠀⠀⠀⡠⡇⠀⠀⠀⠄⡠⠀⢳
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡾⠀⠀⠳⡀⠀⢅⡰⢡⠃⠀⠀⣀⠇⠀⠀⠀⡰⠀⠛⠇⠀⠌⣼⠔⠃⢸⡆
⠀⠀⠀⠀⠸⡟⡄⠀⠀⢰⠇⠀⠀⣠⠽⢦⡀⠡⡎⢀⣠⣘⠀⠀⡞⠀⡔⢁⠰⠒⢂⢈⠞⡁⠀⠀⠸⡇⠀⠀⠀⡰⣷
⠀⠀⠀⠀⠀⣷⠸⡀⠀⣾⣀⣀⠞⠀⠀⠀⠙⢦⡀⡌⣻⣭⣿⡁⢴⠣⢷⡈⢦⠀⠾⠋⠀⠘⠲⠄⠀⡇⠀⠀⡜⢱⡇
⠀⠀⠀⠀⠀⢹⡆⢡⢀⡏⠀⠈⢆⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⢱⠐⣄⠀⢱⠀⡰⢁⣿⠁
⠀⠀⠀⠀⠀⠘⣿⡄⣾⡥⣶⢾⡽⠐⠒⠋⠉⠉⠉⠉⠁⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢇⠈⣦⠘⣴⢁⣾⠇
⠀⠀⠀⠀⠀⠀⢹⣷⡏⢸⢫⡟⠀⠀⢀⡠⢖⣩⠔⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠰⠸⡀⠀⠀⣇⣼⡟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⣄⣤⣴⣾⣛⣩⡵⡇⢀⠇⠀⢸⠀⢣⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⢣⡴⣻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡿⢋⡾⢁⡎⠀⠀⠀⠇⠈⢧⠀⠀⠀⠀⠀⢀⡇⠀⠀⢰⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⣿⠟⠉⣠⡟⠀⡜⠀⠀⠀⠀⠨⠀⠈⢦⢀⠀⠀⠀⢸⠀⠀⠀⠘⡄⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⡹⢁⡠⠞⢁⡀⠀⠛⠀⣜⣀⠀⠀⠀⠀⠐⡄⠀⣈⡴⠃⠀⠤⠤⢤⠤⠶⢖⠇⠀⣟⠛⣟⡀
⠀⠀⠀⠀⠀⢀⣯⠖⠉⠀⢰⠋⠙⡆⠀⠀⢿⠋⠑⢄⠀⠀⠀⠁⢏⣁⣀⠀⠀⠀⠀⠀⠀⠀⢯⣤⡤⠼⠀⢸⡇
⠀⠀⠀⡠⣖⣩⣀⡀⠀⠀⠘⡄⠀⠸⡀⠀⠀⢑⡄⠈⠳⣄⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠘⠁⠀⠈⠙⠷⣤⣤⡞⠀⠀⠈⠙⠉⠉⠀⠀⠀⠀⠙⠲⠤⠤⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⢀⣶⠖
⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣶
⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠃⠀⠀⠀⣀⣀⣀⣴⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠿⠟⠋⠀⠀⠀⠀⢠⣖
⠀⠀⠀⣈⣿⣁⠀⠀⠰⣤⠉⠀⠰⣿⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⠿⠿⠿⣿⣇⠀⠀⠘⠀⣆⣀⠠⠀⣤⡄⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠰⠟⠉⢹⡇⠀⠀⠀⠀⠀⣼⠁⢸⡏⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠉⠃⡈⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⢀
⠀⠀⠀⠀⢀⠀⠀⣴⣤⣤⡀⠀⣿⣠⣤⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠛⠂⠀⠀⠀⠀⢙⠇⠀⠀⠀⠀⠀⠀⢳⡆⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⣠⣷⣦⣄⡘⢿⠿⢷⠀⣿⣿⣿⠏⡀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⣿⡇⠀⠀⠀⠀⠀⠀⣬⣁⣴⡿⠋⠁⠀⠀⠀⠀⠀⠸⣿⠄⣿
⠀⢠⣴⢛⡛⠛⠛⠁⠀⠀⢀⡀⠉⣶⣤⣶⣷⣤⡀⠙⢿⣧⡀⠀⠀⠿⠀⢸⣦⠀⣴⣦⣼⣃⠀⠀⠀⠀⠀⠀⠛⣙⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠛
⠘⣿⣿⣾⡏⢀⠀⠀⠀⠀⠀⠃⠀⠉⠘⠛⠉⠁⠀⠀⠀⠈⠻⠞⢿⡗⠀⠀⠋⣀⣿⣿⠟⠛⠻⣦⠀⠀⠀⠀⠘⣿⡦⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢰
⠠⣿⣿⣿⡿⠛⠛⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⡄⢷⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠛⢿⡗⠀⠀⠀⠿⣿⣆⣰⣜
⠀⠀⠉⠻⠀⠀⠀⠀⠀⠰⣾⣿⡇⠀⠀⠀⠀⠀⠀⡀⣧⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠠⠬⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠙⠃⠀⠀⠶⠂⠀⣠⣷⣼⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢸⣶⢦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣠⣾⣿⠟⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢶⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘
⢸⡁⠀⠀⠁⠀⣀⣠⣤⣤⣀⠀⠀⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⠀⣼⠃⠀⠀⣤⠀⠐⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠛⠃⠀⢶⡄⠀⠀⠀⠀⠀⣰⣆
⢸⣷⡀⠀⠀⠀⣽⣿⣿⣗⠀⠀⠀⠀⠸⠛⣿⡟⠀⠀⠀⠀⠀⠀⠈⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠶⠆⠈⠁⠀⠀⠀⠀⠸⣿⣿⡆
⠈⢹⣷⡆⠀⠀⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡇
⢀⣾⡁⠀⠀⠀⠀⠘⢻⣿⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒
⢸⣟⢁⣌⣅⠀⠀⠀⠈⠻⢷⣦⢴⣿⣸⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⠀⠶⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢟⠁⠇⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠋⠁
⢈⣉⠉⠻⢿⡇⠀⠀⠀⠀⠀⠀⠘⠋⣩⡄⠀⣀⡠⠘⠉⠀⣹⣿⠿⠿⠗⣀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠐⣷⠀⠘⢠⠄⠀⢈⣿⣂⠀⠀⠀⠀⠀⠀⠐
⠸⠿⣿⣦⣀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⢠⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠃⠀⠀⠀⠉⠆⢩⡄⠀⠀⠀⠀⠙⠉⠛⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠹⣿⢻⣧⠀⠀⠀⠀⠠⣤⣄⠀⢾⠁⠀⠀⠀⠀⠈⣥⡀⢀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡇⢰⣶⣶⣷⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡿⢸⡇⠀⠀⠀⠀⠀⠀⢹⣷⣾⠇⠀⠀⠀⠀⠀⣿⣷⡈⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣍⣛⡛⣧⠟⠉⠛⠿⢿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠹
⠠⠀⠀⢀⡄⠀⢀⡀⢀⣀⣀⣀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣴⠏⠉⠈⠉⠻⣿⡿⠛⠿⠃⠀⠀⠀⠀⢹⣏⣽⡆
⠀⠀⠀⣼⣣⣶⣾⣿⣿⣿⣿⣿⣷⡶⠈⡅⠀⠀⠀⠀⠀⠀⠀⠈⢁⣄⣤⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⣿⣠⣤⠀⠀⠀⠀⠀⠀⣁⣀⡴⣶
⠀⠠⢾⡿⠿⠿⠿⠿⡿⠿⠟⠋⠁⠙⠰⠀⠀⠀⠀⠀⠀⠀⢾⢿⠿⠿⠿⢶⡦⡤⠏⠀⠀⠀⠀⠀⠸⠹⡿⣿⠀⠀⠀⠀⠀⠈⠉⠻⣧⢷⠀⠀⠀⠀⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⣛⣛⢋⠉⣉⠙⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⣶⣶⣴⣦⢸⣿⠃⢹⡇⠘⢾⡟⢠⡟⠉⠉⠉⢛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁⢸⡄⣟⠁⠛⠹⠆⠉⠓⢀⣁⣑⣛⣁⣸⣇⣀⣀⣀⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠛⣻⠂⢀⣸⣧⣤⠤⠶⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠉⠉⢁⣀⣴⠊⠀⢀⠴
⠀⠀⠀⠀⠀⠀⢰⢸⡇⢴⠀⢰⣧⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⢺⡇⠀⠀⠀⣿⡷⢲⣒⡶⣶⣻⠛⢆⣿⠀⣀⢸⣯
⠀⠀⠀⠀⢀⣀⣸⣸⣇⣿⠀⠈⠁⠀⠀⡶⣶⠲⡖⢲⢺⣿⣿⡇⠸⣮⢿⠀⠀⠀⠛⢀⡘⠟⠀⠈⠛⠀⠀⠁⠀⣿⢸⣿⢀⣶⣆
⠀⢀⡞⠩⠁⣀⡈⣉⣁⠉⠍⠙⢦⡀⠀⠯⡏⠐⡷⢾⠀⠿⠛⠇⠀⠁⣀⠀⣀⠀⢀⣼⡅⠀⢾⡆⠀⠰⣿⡆⠀⣿⡜⠋⠀⠈⠀⣀⡀
⠀⠀⡇⣤⣤⣴⢶⢠⡶⣄⣶⣦⢈⡇⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⢷⢴⠻⠔⣇⠀⣼⠽⡇⠀⣟⢳⡀⢀⣯⡿⣄⠙⢁⢠⣿⣆⡆⠘⠁
⠀⠀⢻⣇⠻⠿⢾⣼⠧⠿⠃⣿⣼⠃⠀⠀⢀⣸⣿⣿⣽⣃⠀⠀⠘⠄⠑⣒⠋⠘⠳⠛⠃⢸⣯⢈⣧⠸⠷⠿⠛⠀⠈⢳⣻⠯⠂
⠀⢠⠼⢯⣔⣆⣈⣩⣄⣀⣺⠴⠛⠑⣆⢏⣉⣩⣿⣧⠤⢼⡇⠈⣏⣧⣼⣹⠀⠀⢠⢒⠤⢠⡬⠉⡁⠀⣀⣀⠀⢀⡤⠠⡁⠀⢴⣤⠄
⠀⢸⠈⣀⠀⣠⡀⣰⣦⠀⣠⡀⢠⣄⢹⠈⡇⠀⣿⡇⠀⣸⠀⢺⡿⣿⣿⠿⡿⠀⠜⠬⠗⠠⠽⠖⠁⢸⡁⠈⠳⠏⠀⢀⡇⣀⠈⠁
⠀⠘⡟⢹⣾⠇⣿⣿⢸⣸⡏⢧⣾⠘⡾⠃⣇⠀⣿⣷⣀⡏⠀⠘⣷⠀⢐⢿⡇⠰⠀⠀⣠⡴⠲⣄⠀⠘⠧⣄⠀⠀⢀⡼⠁⣿⣿⠟
⠀⠀⣧⣠⢉⠠⡙⠇⡀⣙⣂⢬⡗⢔⡿⠀⢈⡅⢠⡀⢀⣀⡀⠀⠻⠼⠿⠾⠃⠀⢀⣠⡟⠀⠀⢸⣇⡤⢄⠀⠉⢷⡟⠁⠀⠈⣁⡀
⠀⠀⠛⠬⣏⣙⣙⣋⣹⡥⠽⠖⠚⠉⠀⠀⠘⢦⣾⣷⣯⡴⠃⢀⣤⢀⣀⠀⢀⣰⣋⠸⡇⠀⠀⢸⡏⠀⢸⡇⠀⠸⣅⠀⢠⠞⠁⠀⠉
⣀⠞⢦⡄⣀⠀⣀⣤⡤⡄⠀⡀⠀⢿⡷⠀⡏⠉⣿⠀⣿⠁⢹⣾⣿⣿⠟⣰⠏⠀⠉⣷⢳⣀⣠⣾⡞⠋⠉⢳⡄⠀⢸⡆⡏
⢨⡶⠼⠆⢹⡽⠛⠛⠉⠃⣸⣽⢠⣏⣷⠀⢻⠀⢿⢀⡯⠀⣼⢸⣿⢹⠇⣿⠀⠀⠀⣿⡏⠉⣷⣼⠁⠀⠀⢸⠇⢀⡾⠁⢹⣄⣀⣠⠼
⢠⣄⡄⠀⠀⠀⠴⣿⡆⠀⠧⠿⢀⠌⡄⠀⠘⠲⠺⠞⠓⠒⠋⠘⠛⠚⠀⠘⠦⣤⠞⣿⠀⠀⣿⡿⢄⣀⣠⠟⠀⠀⠀⠀⡾⠉⢠⣤⣤
⠚⢶⠛⠠⠴⣱⠴⢋⡀⣰⣆⢻⡿⠀⡗⣠⡀⠀⠀⠀⣀⢤⡎⣳⣀⠀⠀⠀⠀⢩⠀⡻⣄⣠⠏⡀⠘⢋⡴⠊⠉⠑⢦⠀⢇⠀⠮⣿⠿
⠴⠆⢀⣀⡉⠋⠀⠈⠙⠾⠋⠁⠀⢀⣔⣛⣓⣄⠀⢴⡙⠦⠟⠗⣚⣳⠀⠤⠀⠀⠄⢃⠘⠉⠠⠁⠌⢸⠀⠀⠀⠀⠈⡇⠈⡆⢀⣁
⠀⠀⣿⠁⡭⠉⠉⣉⣷⡆⠀⠀⢠⡗⠢⠗⡒⢚⡇⢸⣸⣿⢻⢫⣿⠇⠀⡿⡄⠀⠀⠱⣄⢀⢀⠊⠀⠸⣄⠀⠀⣀⡾⠁⣠⠇⠺⠿⠂
⠀⠀⣿⣀⣠⣴⠞⠛⣩⡿⡵⠀⢪⣤⣑⣭⣤⡭⣼⠀⠛⠛⠛⠛⠋⡤⠀⠛⠳⠤⡀⠀⠸⡶⠁⠀⠀⣠⠟⠛⠉⠁⣤⡀⠁⣤⣤⡄
⠀⠀⠈⠙⣯⣄⡴⢞⡥⠚⣁⢀⠈⢿⣿⣿⣶⣧⠏⢀⢀⣴⡄⠀⢺⡁⠀⠀⠀⠀⣈⠇⢰⣧⠀⠀⠀⡇⠀⣴⣶⠀⠉⢁⡄⠘⠋
⠀⠀⠀⠸⣯⠵⠚⠉⠀⠠⣻⣟⠄⠀⠀⠀⠀⠀⠀⢹⠾⡿⣿⠃⠀⠹⡟⡟⣿⣿⣿⠀⠸⠁⠀⠀⠀⢸⠀⠈⠈⢀⣽⡁⡷⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣧⣿⣟⣛⣛⣀⣀⣀⣀⣀⣴⣃⣀⣀⣀⣈⣉⣉⣃⣀⣀⣀⣀
⣿⣿⠛⡿⢛⠻⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⣉⠉⠑⠢⣀⠀⠀⠀⠀⣀⡀⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠉⠙⠒⣒⣳⣖⣲⡭⠤⠴⠒⠋⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⣀⡠⠤⠘⠃⠐⠒⢊⡹⢋⡾⠙⡹⡆⣤⣀⡀⠀⢀⣿⡒⠂⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⢸⡄⠉⠈⠓⠒⠶⠊⡝⠀⢠⣆⠈⠀⠘⢦⣀⠈⠛⠯⢄⣈⡏⠑⢷
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⠀⢇⠀⠘⣇⠉⠙⠒⠲⢬⠋⠙⢲⢦⠀⠙⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⠈⠻⡀⠘⢦⡀⠀⠀⢹⠀⠀⠈⡆⣀⣼⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢤⣈⠉⠒⠢⣀⠀⠀⠀⠀⣀⣀⢼⡇⠀⠀⠀⠀⠸⡇⠀⢼⣀⣀⡴⠃⠀⠀⢸⡀⠀⠀⢧⠀⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠉⠑⢒⣚⣷⡖⢲⡭⠤⠴⠺⡇⠀⠀⠀⠀⠀⡇⠀⠸⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⢹⠀⢸⡇
⠀⠀⠀⠀⢀⣀⠤⠄⠳⠤⠔⠒⢚⡽⢫⡮⠛⣹⡢⢄⣀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⢸⠀⢸⡇
⠀⠀⠀⢰⡏⠉⠐⠒⠒⠦⠖⢊⡏⠀⢸⡄⠊⠀⠙⢄⡀⠉⡇⠀⠀⠀⠀⠀⡇⠀⠀⡟⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⠀⢸⠀⢸⡇
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⡇⢸⡀⠀⢸⡍⠙⠒⠲⠤⣭⠗⣧⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠠⢸⠀⠸⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠑⢆⠀⠻⣄⠀⠀⠀⢸⠁⣿⠀⠀⠀⠀⠀⡇⠀⠀⣧⠀⠀⠀⠀⠀⠀⡇⠂⠀⢸⠀⢸⠀⠀⡇
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣿⠀⠀⣼⠀⠀⣸⠆⠀⠀⢸⠀⢸⡀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠀⢸
⠀⠀⠀⠀⠈⠉⠓⠒⠲⠤⠿⣤⣄⣸⣉⡉⠁⠀⠀⠀⠸⡆⢸⡇⢠⠄⠀⠀⢹⠀⠀⢹⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⢸⢋⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠲⠤⠤⠧⠚⠧⡀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⢀⡇⠀⠀⢻⢀⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣼⣈⡀⢸⠀⠀⠀⠀⠀⢸⡇⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠦⢤⣄⣨⠷⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠉⠲⢄⡀⢀⣀⣄⣀⣀⡴⠊⠁⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢹⠋⠉⠉⠉⣧⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⢲⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣏⠓⠒⠦⠤⣀
⠀⠀⠀⠀⢰⠯⣍⣀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⣟⠀⠀⠀⠀⠀⠀⠀⢸⣀⠤⠖⠊⢹
⠀⠀⠀⠀⣿⠀⠀⠀⠉⠙⢇⠀⠀⣀⣀⠤⣶⠋⠁⡸⠁⠀⠙⡄⠉⠓⢶⣤⣄⣀⣀⡼⠀⠀⠀⠀⢸⠓⠲⠤⢄⡀
⠀⠀⠀⠀⠻⣀⡀⠀⠀⠀⠀⠉⠉⠁⠈⡽⠁⠀⢰⢇⣀⣀⡤⠼⡄⠀⠈⢣⡀⠀⠀⠀⢀⣀⠤⠖⣊⡠⠴⠒⠉⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⡤⢤⣀⣀⠞⠀⠀⢠⠇⠀⢸⠁⠀⠀⠹⡄⠀⠀⠱⡤⢒⣊⣭⠴⠚⢹⡅⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⣯⠤⡄⢀⠮⣄⣀⣸⠀⣀⡤⠴⣷⡀⢰⠤⣹⡍⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢧⠏⠀⠀⢸⠉⠉⠀⠀⠀⢸⠑⠏⠀⠈⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠇⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠦⢄⣀⢸⡃⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⠀⢀⣸⠧⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠲⠤⣼⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⡤⠞⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⢼⡤⠴⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢀⡠⠍⢉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠖⠓⠠⠍⢢⣀⡠⠤⠤⢄⣀⠀⠀⢠⡇⠀⠀⠀⠀⢀⣾⣛⡛⡞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣰⢲⢶⣤⡀⠀⠀⠀⠀⠙⢶⠺⠂⢄⡈⠑⢦⡼⠀⠀⠀⠀⢀⡾⠥⣞⠃⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡹⣊⠥⢒⡹⢆⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⣰⢃⡀⠀⠀⠀⡼⢍⣉⠂⡼⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣖⣳⣝⠮⣡⠖⡩⢷⣄⠀⣄⠐⢄⢈⣳⡤⣴⣃⢞⡀⠀⢀⡴⠯⢗⣢⠞⡵⠥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠊⠉⠁⠰⢣⢎⢝⡾⣽⣶⡥⠾⣷⣤⡙⢲⠏⠀⠀⠀⢊⡻⣖⡠⣾⣭⡵⣊⡵⣞⣩⡴⠖⠿⢦⣀
⠀⠀⠀⠀⠀⣠⢴⠇⠀⠀⠀⠀⣀⣀⣠⡤⣬⠬⣥⠬⣭⣙⣃⡖⡖⣻⣬⡆⡄⣤⣶⢶⣺⠿⢿⣿⣿⣿⡛⠉⠀⠀⠀⠀⠐⠠⣉⠓⠢⢄⣀
⠀⠀⠀⠀⠈⡏⢻⠒⣡⣴⠾⣫⠟⡱⠋⠔⠁⠞⠁⠚⣱⡮⠿⠛⡟⠩⢻⣝⣷⣷⣬⣓⠊⠉⠀⠀⠀⠈⠉⡳⢄⡀⠀⠀⠀⠀⠀⠁⠒⣆⡬⠙⣲⡄
⠀⠀⠀⠀⠀⢣⠘⢞⠋⣟⠶⢥⣀⠀⠀⢀⣠⠴⠒⠉⠁⠀⠀⠀⠀⠀⢁⣫⡿⠟⢯⣟⠩⠶⣦⣄⡀⠀⠀⣓⠄⠑⣄⠀⠀⠀⠀⠀⣈⣡⡔⡺⢫⡇
⠀⠀⠀⠀⠀⢸⠀⠘⠘⠉⠀⠀⠈⢱⠖⠉⠐⠡⠀⠀⠀⠀⠀⣠⡴⠾⠉⠁⠀⠀⠀⠈⠳⢮⣁⠖⢋⡱⢦⡐⢋⡠⢾⣀⣀⡤⡴⢪⢇⢞⡜⡵⣻
⠀⠀⠀⠀⠀⠀⢳⢄⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡴⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡐⠉⠭⠟⣪⠍⠈⠙⣞⡵⡱⢃⢎⠞⡰⡟
⠀⠀⠀⠀⠀⠀⢸⡎⠙⢲⢤⡀⠀⡇⠀⠀⠀⠠⠅⢰⡟⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣈⣷⣤⠀⠀⠀⣖⡆⢸⡜⡱⣣⢋⣞⡼⠃
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠙⠳⡇⠀⠀⠀⢐⠂⢸⠅⠀⠀⠈⠙⠲⢄⡀⠤⣀⠤⠘⠂⢁⡤⢲⠏⡽⡡⣻⡇⠀⠠⠵⡂⢸⣼⡱⢷⢏⢿⠃
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡇⠀⠀⠀⢨⠄⢸⡀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡡⣴⢎⠝⡵⢡⢎⣜⡝⡵⡇⠀⠐⣋⠅⢸⡽⡡⣫⢫⢾
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⣧⠀⠀⠀⠠⠁⢸⢟⠦⣄⠀⠀⠀⠀⠀⢸⠀⡷⡡⣣⠊⡴⡡⢫⢮⢞⣾⡇⠀⣩⠔⡂⢹⡜⡱⡱⢃⡎
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢿⠀⠀⠀⢈⡁⢸⡂⠀⠉⠛⠦⣄⡀⢠⣶⠀⡟⡱⢡⢎⣼⡱⡷⢻⢏⣼⠁⠠⢖⡩⠁⣟⣜⡵⣡⢋⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⠀⠀⠀⣐⠀⢸⠆⠀⠀⠀⠀⠀⠋⠳⢦⣥⣿⢴⡟⣩⢋⠞⡴⢣⢋⣾⠀⢘⡡⠖⠁⣟⡞⡴⢡⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡆⡀⠀⠀⢸⡆⠀⠀⣺⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢊⠜⡱⢫⠞⡴⣳⢫⡟⢀⡥⢖⣩⢴⠋⡜⡱⡳⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡈⠃⡄⢸⡅⠀⠀⠨⠁⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⣡⢋⠞⣠⢋⢞⡜⡱⢫⡇⠀⢚⣉⠴⢺⢏⣜⡟⣱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⡇⠀⠀⠰⠄⢸⡀⠀⠀⠀⠀⠀⠀⠠⠀⡿⣡⢋⡴⣡⢫⢎⡜⡱⣫⡇⢐⣫⠴⢊⣸⢎⢎⣜⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⢌⡂⢸⡆⠀⠀⠀⠀⠀⠀⠨⡄⡟⣡⢊⡔⣡⢣⠎⡜⡵⣩⡇⢩⠴⢊⡁⣾⡯⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡒⠅⠸⡇⠀⠀⠀⠀⠀⠀⠐⠄⡟⣡⠊⡴⣡⢫⢞⡜⡵⣹⠀⠠⢚⡭⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⡷⢤⡀⠀⠀⠀⠠⢍⡀⢸⢁⠞⡴⣡⢫⢎⣞⣼⠿⠀⣘⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣒⠯⣕⠂⢸⢋⠞⡴⣱⠧⠛⠉⠈⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⣀⠀⢀⡤⠀⣀⠀⢀⡀⠀⢈⠙⠲⠥⣸⣯⠜⠋⠁
⠀⠀⠀⠀⠀⠀⡠⢞⡥⢚⡥⠒⡩⠔⢋⠴⠚⣁⠴⠊⡡⠔⢉⡠⠒⣁⠔⠃⡠⠆⢀⠴⢋⠴⢊⡤⠂⣠⠔⢃⡠⠂⢀⠴⣀⠄⣠⠆⢀⡄⢀⠄⣠⠆⣠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⠔⢋⡠⠚⣁⠔⠊⣠⠔⠋⣠⠔⢉⡠⠚⢁⠴⠋⡠⠖⣡⠚⣁⠔⢋⡠⠚⢁⠔⠉⡠⠖⣡⢞⡥⠚⣡⠖⣩⠞⣡⠞⣡⠞⡡⠊⠐⠋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠊⠀⠰⠋⢠⠖⠁⡠⠊⡠⠊⣠⠞⢁⡴⠋⣠⠞⢁⠔⠋⡠⠊⡰⠋⠠⠊⠀⠚⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠈⠀⠀⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⡬⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⣁⡃⠀⠀⠀⠀⠀⢱⠚⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠺⡀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠂⠀⠀⠁⠀⡀⠀⠀⣀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⢀⠠⠀⠀⠐⠀⠑⢄⠈⠂
⠀⠀⠀⠀⠀⢀⠀⢤⠤⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠔⢁⠄⠀⠈⠀⠀⠄⠀⠀⠠⠀⠔⠀⠀⢸⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢫⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡠⢂⡄⠀⢈⣨⡄⠉⠛⢄⠀⠀⠀⠀⢀⡀⠀⠀⠰⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣐⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠀⠈⠀⣀⡣⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠈⠀⠀⠀⠠⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠐⠞⠉
⠀⠀⢀⠂⠁⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠁⠈⡆⠀⡠
⠀⠀⡀⠁⠈⠀⠐⠀⠀⢳⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠄⠀⠀⠀⡯⡊⢀⠴⠁⠀⡀
⢸⡂⠧⠀⠀⠀⠀⠀⠀⢄⠁⠠⡄⠀⠀⠀⠗⢄⠀⢀⡀⠀⠀⠊⢀⡌⡀⠀⠀⠀⠀⢀⠇⠀⠁⠀⠈⠀⠀⡀
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢀⠀⠀⡀⠔⠑⠐⣦⣤⠀⠘⡀⠐⠁⠀⠀⠀⡠⠊⠉⠀⠐⠀⠀⠒⡤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⢑⠶⠀⡀⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠈⠛⢦⡶⠾⠿⠿⢶⡞⠉⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠉⠙⠓⠲⢦⣄
⠀⠀⠀⠀⠀⣼⠯⣍⣀⡀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⡀⠀⠀⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡤⠖⠚⠉⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠛⠒⢧⠀⠀⠀⠀⠀⣀⣤⡶⠚⠉⢸⠏⠉⠉⢯⡉⠙⠲⢦⣤⣀⡀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⡷⠶⠖⠛⢻⢻⡞⠁⠀⢠⡏⠀⠀⠀⠀⣳⡀⠀⠀⠻⡏⠉⠉⠛⣿⠀⠀⠀⠀⣀⣠⠟
⠀⠀⠀⠀⠀⠘⡗⠲⠤⣤⣀⡀⠀⡇⠀⠀⠀⣸⠏⠀⠀⢀⡞⠙⠓⡶⠚⠉⠉⢷⠀⠀⠀⠹⣆⠀⢀⣿⡤⠶⠚⠋⠁⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠙⡷⠶⢤⡴⠃⠀⠀⠀⡼⠁⠀⠀⡇⠀⠀⠀⠈⢧⠀⠀⠀⠘⢿⠉⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢼⠥⢶⡆⠀⡼⠷⠤⣄⣀⣇⣀⡤⠴⠚⠛⢦⠀⡞⠲⣌⡇⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣧⠞⠁⠀⠀⠀⠈⡏⠁⠀⠀⠀⠀⠈⢷⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠈⠧⣤⣀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⢀⣠⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣧⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢇⣠⡤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠻⠤⣄⣀⠀⠀⠀⠀⡇⠀⠀⠀⣀⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠤⣧⠴⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠉⠈⠙⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠀⠀⠀⠀⠀⠠⢦⣤⣀⡀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⣀⠸⣿⣦⣤⣤⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡙⣿⣋⣉⣉⠁⠈⠉⠉⠙⠛⠛⢿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣴⣤⣤⣿⣿⣿⣿⠿⠿⣿⣿⣦⣄⣤⣶⠿⠟⠋⠁⠈⠉⠛⠻⢿⣶⣤⣄⣀
⢀⣀⣤⣤⣶⣶⣾⣿⡿⠿⠟⠛⠛⠉⠉⠀⠉⠛⠛⠛⢿⣿⣿⡙⠛⢿⣿⣿⠋⠀⠀⠀⠀⠀⠙⢿⣿⡀⠀⠀⠀⣀⣀⣀⣀⡀⢀⣨⣽⡿⠿⢿⣶⣦⣄⡀
⣿⣿⣿⣿⣋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣬⣿⣿⢷⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⡾⠟⠛⠛⠛⠛⠻⠿⠿⣿⣿⣶⣤⡀⠀⠉⠛⠿⣷⣦⣄
⣿⣿⠀⠙⠿⣿⣦⣄⡀⠀⠀⠀⣀⣠⣤⣶⣾⠿⠟⠛⠋⠉⠀⠀⠀⠈⣹⣿⣿⣦⣀⡀⠀⠀⠀⣀⣼⣿⣧⣴⣶⣶⣦⣤⣀⠀⠀⠀⠉⠙⠻⣿⣦⡀⢀⣀⣤⣽⣿
⣿⣿⠀⠀⠀⠈⠻⢿⣿⣿⣿⠿⠛⢛⣿⣤⣀⣀⠀⠀⠀⠀⠀⣀⣠⣾⡿⠃⣽⡿⠿⠿⣿⣿⣿⣿⣿⡿⠛⠋⠉⠉⠉⠛⢿⣷⣄⠀⠀⠀⠀⠸⣿⣿⠿⠛⠋⠁⢸
⢻⣿⠀⠀⠀⠀⠀⠀⣿⡿⣿⣷⣄⣸⣿⠟⠿⠿⣿⣿⣿⡿⠿⠿⠛⠉⢠⣾⠋⠀⠀⣰⣿⣯⣀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⠈⣿⡇⠈⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠃⠀⠀⢰⣿⠏⠀⠙⢶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠃⠀⠀⠀⣿⣿⣴⣶⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⠀⠀⠀⠀⠀⢸
⢸⣿⡆⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣦⣤⣴⣶⣾⡿⠟⢋⣁⣿⡏⠀⠀⠀⢸⣿⡿⠛⠋⠁⠀⣿⡟⣿⣷⣤⣀⠀⠀⠀⢀⣠⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠿⠿⠛⠛⢿⣷⣾⣿⢿⣿⡇⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⣿⡇⠈⠛⠿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠸⣿⣇⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⢿⣿⡄⠀⠀⠀⠀⠙⣿⣦⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⢻⣿⣶⣶⣾⣿⡿⠟⠛⠋⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣟⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⣿⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢻⣿⡄⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠹⣿⣆⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿
⠀⠀⠀⠀⠙⣿⣦⠀⣿⡇⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠈⢻⣷⣿⡇⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⣤⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⢺⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣀⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⢼⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣀⣠⣴⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣷⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⣿⣿⣀⣠⣴⣾⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠟⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⡠⠐⠒⠒⠒⢦⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠛⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⡿⠹⠁⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠘⢿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⡇⠀⠀⠀⢀⠁⠀⠀⠀⠀⢀⣼⣿⣏⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠙⠛⣿⣿⣿⡇
⠀⠀⠀⠀⡇⠀⠀⠀⡈⠀⠀⠀⠀⠀⠸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠂⠉⠉⠁⠀⠀⣈⣴⣿⣿⣿⡇
⠀⠀⠀⠀⢠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡄⠢⠄⠀⠀⠀⠀⠀⠠⠂⢄⠠⣶⣿⣿⣿⡿⠟
⠀⠀⠀⠀⢸⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⡀⠈⠁⡀⠀⠀⠑⠁⠀⠉
⠀⠀⠀⠀⠈⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢠⠊⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠒
⠀⠀⠀⠀⠘⠢⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⣖⣰⣄⢰⢰⣴⣶⣦⡆⠀⠰⡆⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣻⣿⣟⣟⣛⣿⣿⣿⣛⣛⣿
⠀⠀⠀⠀⠀⠈⠀⠈⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣞⣙⣆⣀⣜⣉⣦⣀⣴⣋⣳⣀⣰⣋⣳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠈⢩⠏⠉⠉⢩⠟⠉⠁⢉⡽⠉⠉⠉⡽⠃⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠀⢠⠀⠀⡄⠀⢠⠀⠀⡄⠀⢠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⢀⠀⠀⡀⠀⢀⠀⠀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠈⠀⠀⠁⠀⠈⠀⠀⢁⠴⠚⢉⡉⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠘⠀⠀⠃⠀⠘⠀⣰⠃⠀⠀⢀⡄⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠆⠀⠰⠀⠀⠆⠀⠰⠀⡇⠰⠀⠀⣸⠇⠀⠀⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⡄⠀⠀⠁⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣈⣁⡠⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠈⣤⡄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡄⠀⠀⠀⠀⠀⣀
⠖⠠⠀⠀⠀⠀⠀⠀⣿⠁⠀⡀⠄⠀⣤⢠⡀⠀⠀⠀⠉⠁⠀⢀⣤⣤⣾⠟⠃⠄⡀⠀⠀⡀⢰⠏⠛⠉⠀⠀⠀⣼⣾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠟⠛⠇⠀⠉⠀⢠⠃⠟⠁⠀⠀⠆⠀⠀⢰⡇⠀⠀⠙⣧⠀⠀⠉⠓⠘⠏⠀⠀⠀⠀⠀⠀⢰⣏⣠⣶⣿
⢺⡏⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⢸⣶⣦⡀⢀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣍⠀⠀⠀⠀⡹⠀⣤⠶⠀⠀⠀⠀⣿⢻⡟
⠘⠁⠀⠀⠀⠀⣠⢾⡿⠛⠋⠁⢙⣿⣭⣥⣿⡿⠿⢷⡀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⡿⠇⠀⠀⠀⠀⠀⠻⠘⠃
⠀⠀⠀⠀⢀⣀⡛⠚⠃⠀⠀⠀⠈⠉⢨⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠘⠟⣀⣤⣶⣶⣦⣴⣄⡀⣦
⢀⡄⠀⠀⢀⣇⣿⠛⠂⠀⠀⠀⠀⣿⣿⢻⠀⠀⠀⠀⢸⣠⣤⣼⣷⣷⣶⡄⠀⠀⠀⠀⠀⣇⠝⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⡀
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⡠⠀⠀⠀⠀⢀⡦⢄
⠀⠀⠀⢸⠀⡸⠁⠙⢉⠇⠠⠀⣠⢐⠚⣓⠢⢀⠴⠂⠀⣴⢿⡭⠭⡻⣧⠀⡀⠀⠀⠇⢰⠙⠒⡉⡂
⠀⠀⠀⢸⣰⠁⠀⠀⣌⠀⠐⠄⡁⠬⣶⡟⠁⣇⡄⠀⠸⢟⡸⡀⠀⡀⣸⠀⠀⠒⠊⢊⠓⠘⠉⢏⡿
⠀⠀⠀⢀⠏⠀⠀⠀⠘⡄⠀⠀⠉⢄⡀⢀⡜⠁⡂⠁⡄⠑⢯⢮⣯⠾⠋⡀⡀⠀⠀⠈⠦⠀⠠⠼⠃
⠀⠀⠀⠈⠉⠉⠉⠉⡉⢁⠀⡀⢠⣤⣄⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⡤⡌⣳⡁⢸⠂⡒⠄⠀⠀⠀⠒⡱
⠀⠀⠀⠀⠀⡲⢠⡀⠂⠠⠃⢱⡟⣷⡎⡆⠐⠂⢰⠀⢰⠅⠀⠀⠀⠈⠁⠉⠁⢸⠀⡃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠄⢵⠐⠃⠀⣀⢀⢀⢉⡒⠊⠀⠀⠀⠈⠀⠀⡀⠀⡀⢀⠀⡀⠀⠀⠀⠰⠥⠤⠤⠤⠤⠤⠱
⠀⠀⠀⠀⠀⠀⠤⣸⣆⠹⢯⣟⡿⠃⠀⠀⠀⠀⠀⠀⠈⠀⠇⠑⠈⠠⠥⠀⠀⠀⠀⠀⢀⠰⠄⡀
⠀⠀⠀⠀⠀⢀⣀⡈⣁⡀⠀⠀⠀⡀⠀⠀⠙⠰⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⡍⣔⠁⠀⠀⠈⡴⡆
⠀⠀⠀⠀⡊⠁⠀⠀⠀⣈⠄⢀⠙⢋⡇⠀⠳⠀⣉⣀⣀⠈⢸⢓⣲⢀⢨⣿⣔⡁⠀⢿⠀⣴⣶⡀⠃⣤
⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁⠀⠸⠤⠼⠆⠀⠀⠂⠡⣀⡠⠒⠉⠉⡀⠘⠿⠟⠉⠀⢁⠀⠈⠛⠞⠁⡀⠽
⠀⠀⠀⢸⡑⢒⠒⠠⢂⡄⠀⠰⢪⢍⢢⠀⠀⢶⠀⡀⠀⠀⠀⡼⣁⣥⠀⠀⠐⠉⠂⠉⢢⣀⣀⡜⠁
⠀⠀⠀⠘⠁⠀⠒⠒⠂⠝⠀⠰⢌⡈⠜⠐⠟⠀⠐⠘⠀⠀⠀⠀⠃⠔⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡖⢿⣿⣯⣾⣻⣿⣥⣌⡙⠷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣤⡀⠩⣿⣿⣿⣿⣿⣷⣄⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣯⠈⣿⣿⣿⣷⣦⡭⣯⣭⠹⣿⠿⠟⢾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣾⣿⡟⣠⡿⠻⠿⠏⠁⠀⠀⠀⠀⠀⠀⠘⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⡿⢟⣶⡿⠁⣠⠤⢤⣄⡀⠀⠀⠀⢀⡴⠒⠸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢠⣧⣼⣿⠃⠘⠁⠀⠀⠈⠁⠀⠀⠀⢉⠀⢀⠀⢿⠃⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣡⣾⡟⡹⡟⠀⠀⣰⡖⢲⠂⠀⠀⠀⢀⡵⢛⢉⣟⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣯⣰⣿⣿⡇⢳⠀⡀⠀⠈⠁⠈⠀⠀⠀⠀⠈⡁⠈⠉⠀⠸⣿⣿⣿⣷⣄⠀⡖⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⣀⠀⠀⣿⣿⣿⣿⣷⣿⣄⡠⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢠⣿⣿⣷⣷⣿⠜⣠⠃⠀⠀⠀⡠⢎⠔⠋⢀⡀
⠀⠀⠀⠠⠑⠁⣀⣿⡍⠉⣻⣿⣿⡿⣷⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣼⣿⡟⣿⡿⠋⢀⠁⢀⠴⠒⢁⣴⠇⠒⣡⠔⡋
⠀⠉⡐⢄⠁⠙⢄⠙⣿⣿⣿⡃⠸⣿⣟⠻⣆⠀⠀⠀⠈⢲⣤⣤⣤⡶⠀⠀⢠⣿⣿⣿⠊⠀⠀⡄⢀⠀⢴⡆⠉⡠⠔⢊⠕⠊
⠀⠀⠈⠌⠓⣦⡈⠑⠄⠙⠻⠿⠀⠩⣷⣿⣿⣆⠀⠀⠀⠠⡄⠀⢈⠆⠀⢠⣿⣿⣿⠃⠀⢀⣼⣃⠈⠀⠀⠛⢁⡤⠒⠁⠤⣠
⢀⡀⠠⡈⣢⡈⠑⠄⣠⡆⣀⢰⡄⠀⠈⢻⡿⢻⣷⣄⠀⠀⠑⣤⠤⠎⠀⣾⣿⢼⣿⡇⠀⠜⠉⠁⠀⠀⠀⣀⡩⠤⠐⠒⠁
⠀⠀⠀⠀⠀⢉⡀⠘⠉⠘⠓⠒⠂⠀⠀⠘⠁⠀⢻⡏⠳⣦⣀⠀⠀⢀⣼⣿⣿⠸⣿⢹⠐⢦⣀⢠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⣀⠀⠀⣰⢦⢠⠂⠀⠈⢿⠀⠈⠙⠻⢿⣿⠟⢻⣿⠀⠉⠸⠀⠀⠉⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡻⡇⠘⠀⢧⠀⠀⠘⡄⠉⠙⠒⢦⡀⢠⡞⣹⠀⢀⠀⠘⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡀⠀⠀⠸⣶⣦⢄⠐⢄⠀⠀⠀⠁⠀⣠⠿⠀⠸⠢⠀⡀⠀⠀⠘⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣤⠸⣤⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⡿


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣛⣛⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⣒⡒⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠶⠖⢾⢳⠟⠓⠚⠓⠛⠒⠒⠒⠒⠒⠒⢶⡏⣴⠿⠿⠿⠦⠽⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣛⣛⣛⣛⣛⣛⣛⣛⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠦⠤⠤⣤⣤⣤⣄⣀⣠⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡧⠤⠴⠶⠶⠤⠬⠭⠽⠿⠿⠿⠒⠶⠶⠾⠿⢿⣿⣛⣛⣓⣲⣶⣤⣤⣠⠞⣿⡿⣦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣿⣇⡇⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⢣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣼⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⣇⡟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣸⢻⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⡏⡿⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⢱⢇⡟
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⣿⡏⣸⠁
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡼⣽⢁⡏
⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣸⢳⠇⣸⠁
⠀⠀⠀⢠⣞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⠏⡟⢀⡏
⠀⠀⠀⠀⠈⠉⠉⢛⡷⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡟⢸⠃⡾
⠀⠀⠀⠀⠀⠀⠀⣾⣤⣤⣄⣈⣉⠓⠶⠤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣼⢁⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠦⠭⣭⣿⣿⣶⣶⣶⣶⣾⡿⠿⠿⠯⢽⣿⣶⣶⣤⣤⣠⠞⠁⢠⡇⣼⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠿⢭⣽⣛⣛⣿⣿⣿⣒⣶⣶⠾⢡⠇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⠿⠿⠴⠃

⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⡃⢄⠀⠸⣧⠯⠃⠀⠀⢀⣀⣐⠀⠉⠁⠀⠀⠐⠀⠠⢀⠀⠀⠐⠻⡟⠂⠀⠀⠀⠀⠐
⠀⠐⢤⠃⠀⢀⠔⡅⠀⠀⠐⣋⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⣀⣂⣀⠀⣴⠆⠾⠇
⠀⠀⠀⢀⠔⢁⠢⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠽⠒⠅
⠀⢀⡔⠁⢌⢢⢳⡇⠀⢹⡃⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠤⠑⡄⠀⠀⠀⠀⣤⡞⠀⠹⠆⠬
⠀⠫⣉⠉⠚⢦⣻⠁⠀⡈⠳⠎⢸⣀⠄⡟⢃⡠⢧⣀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡠⠒⠀⠋⠀⠑⢲⠊
⠀⠀⠈⠉⠓⠒⠛⠀⠀⡷⠴⢶⣺⢅⡜⠀⠀⠹⢪⠋⣹⠒⠶⠤⢤⠴⠶⡊⠟⣄⠴⠄⠈⠁⠲⠉⠀⢠⠄
⠀⠀⢀⣀⣀⡀⢀⣀⠀⠫⠻⣷⣮⡟⢠⢒⡒⢄⠀⠉⠀⠃⠰⠢⠌⠦⠆⠈⠃⠈⠀⠀⠀⢀⡈⠀⠀⠈⠂
⠀⠀⠙⢄⠀⢠⡋⠶⠥⡀⠀⠀⣀⠀⢧⡇⠈⢪⡄⣰⠞⠉⢢⠀⠀⢤⠀⠀⢰⠠⠂⠀⡠⢊⢁⠝⠑⠒
⠀⠀⣀⠀⠑⢼⢢⣔⠀⠉⠳⣄⣈⣯⠚⠿⣄⠌⣷⡟⠀⣀⠎⠀⠒⠈⣁⣠⠤⠒⠂⠀⠀⠁⠃⠔
⠀⡰⢤⡂⠐⣢⠉⠈⢳⡉⣉⠽⡏⠉⡑⠲⢬⡟⢙⣶⣚⣉⠙⢧⠀⣸⠑⠀⡗⠀⢀⠀⠀⡠⠀⠀⠀⠠⡀
⠀⠰⠔⠄⢀⡽⡄⠀⠀⠻⠴⡞⠁⠀⢦⠀⢠⠀⢀⡀⢀⡰⠀⠀⠀⢇⢀⡔⠱⢄⡜⠀⢠⠁⠀⠀⠀⠀⠄
⠀⠀⢀⣀⣸⢀⠇⢀⠀⠀⠀⢹⠀⠀⡆⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡨⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢘⢒⠢⡿⠤⠒⡃⠀⠀⢸⠀⠀⡇⠀⢸⠀⠃⠀⡆⠀⠀⠀⠴⡇⠀⠀⠀⢠⡼⠤⢀⡆⠄⠤⠒⠁
⠀⢀⣸⠂⠈⠑⠎⢸⠀⠀⢿⡛⣏⠀⠉⠉⠉⠘⠒⠒⠃⠀⢠⡤⠐⠀⠤⠶⠂⠀⠀⠀⠙⠂
⠀⠘⠤⣑⣦⣤⣔⡃⠴⠀⡰⠉⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇⠃⡰⡟⣭⣕⡄⢯⡟⠀⠀⠀⢀⣸⠙⣳⠆
⠀⠀⠀⠈⣒⣒⡉⠀⠀⠀⠀⠀⠴⡀⠀⠀⠢⡁⢀⡀⢀⣀⢧⣆⠘⠷⢝⠿⠃⢀⠀⠂⠀⡠⠢⡀⢙
⠀⠀⢠⠊⡉⢳⣹⣀⠠⠰⣷⠄⠣⡁⠠⠀⢠⣌⠀⠲⠟⠉⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⢀⡫⠒⣠⠏
⠀⠀⠈⠦⣑⣊⣁⣀⠤⠒⠉⠀⠀⠈⠢⠘⠋⠈⠀⠈⠀⠈⠀⢠⡀⢢⠘⢑⡀⠀⠀⠀⢏⣋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠁⠊⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⣀⣀⡀⠀⠈⠙⢦⡀⠀⠀⠀⠀⢀⣴⠞⠁⠀⣀⣠⣤⣄⡀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢠⡞⠉⠉⠉⠓⢦⡀⠀⠙⣆⠀⠀⣠⠞⠀⢀⡴⠚⠉⠀⠀⠀⠹⡆⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠹⣄⠀⠈⢧⣰⠏⠀⣴⠋⠀⠀⠀⠀⠀⠀⢰⠇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢷⡀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠋⠀⡼⠁⠀⠀⠀⠀⠀⢀⣴⠋⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠻⢦⣀⠀⠀⠀⠀⠘⡆⠀⠀⢸⠇⠀⠀⠀⢀⣠⠶⠋⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠉⠛⠲⠦⣤⣄⣿⠀⠀⣿⣠⡤⠶⠚⠋⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠿⠶⠦⣤⣄⣀⠀⠁⠀⠀⠉⠀⠀⢀⣠⡤⠶⠯⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢀⡴⠛⢳⡀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⢠⢿⡇⠀⢘⣷⡀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢠⠏⢸⡇⠀⠠⡇⢻⡀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢠⠏⠀⢸⡇⠀⠀⡇⠀⢷⡀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡴⢶⣠⠏⠀⠀⢸⡇⠀⠀⡇⠀⠈⣧⢠⠷⣌⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠸⡟⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⢶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⠀⠀⠀⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⠶⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⡤⢤⡤⠷⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠼⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡗

⠀⠀⠀⠀⣠⡄⠐⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⣤⠀⠀⠀⣶⠀⠀⠀⡀
⠀⠀⠀⠀⠉⠀⠀⠀⠈⠂⢀⡄⠀⠀⠀⢠⠀⠀⠀⠿⠀⠀⠀⠀⠈⠓⠀⠀⠘⣿⣦⣾⡏⠀⠀⣠⣡⣄
⠀⠀⠀⠀⣠⣄⠀⠀⢡⠀⣾⣇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢻⣿⣿⠃⠀⠀⠙⠿⠁⠀⠀⡒⠄
⠀⠀⠀⣰⣿⣿⡀⠀⠀⠀⠈⠉⠀⣴⣶⣶⠀⠀⢸⣿⡇⠀⠀⠰⢿⣿⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡆
⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⣿⡏⣿⡇⠀⠈⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣿⡇
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢹⡇
⠀⠀⠀⠀⠀⢄⠀⠀⠤⠀⠀⢀⠀⢻⠀⢹⡇⠀⠻⠛⣇⠀⠀⠀⡀⡀⠀⠀⢀⠀⠀⠀⢀⡀⠀⣀⠀⡀⠀⠟⠀⠸⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⢄⠈⠀⠀⢀⠀⠀⠅⠑⢶⣦⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣁⣀⡴⠖⢾⡟⠀⢀⠀⠀⢀⢀⡄
⠀⠀⠀⠃⠦⠈⢱⠚⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠶⠈⠡⠈⢲⣶⠄⣠⠠⣤⡀⢠⡤⣀⣀⣀⢀⠄⣀⣤⠀⠄⠰⡶⠒⠄⠉⠏⠁⢀⠀⠀⠀⠈⠀⠐⠲⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠀⠛⠀⠀⠀⠈⠟⠁⠁⠀⠙⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⠻⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡸⠟⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠈⠓⠖⠒⡀⠤⢤⠀⠀⢀⣀⠀⠀⢌⡀⢀⠀⠀⣠⠶⠤⡤⢀⠀⠰⠛⠀⠸⠁⠉⠉⠀⠅⠀⠀⠀⠈⠁⠀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠠⡖⢆⠀⠰⣴⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣀⠈⠀⠀⠈⠀⠀⠀⢱⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠽⠁⠀⠀⠀⠁⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠘⢶⡄⠀⠀⢀⣀⠀⢀⡈⠰⠆⠀⠀⠀⠁
⠀⠀⢤⣴⠾⠶⠶⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠘⣛⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠂⠄⠈⠁⡀⠈⠈⠒⢀⣀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠈⠁⠀⠀⠆⠁⠀⠐⠦⡈⠉⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⡰⠒⠇⠀⠰⠂⠀⡀⠀⢀⠀⠈⡧⢀⠀⠀⣀⠈⠉⠁⠀⠀⠀⣃⡦⠀⢀⠇⠐⠀⠀⠀⠺⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠋⠀⠀⠀⠀⣀⣷⣄⢈⡧⠀⣧⠈⡷⡀⢨⣦⣀⠀⠀⠀⣠⡝⠀⠀⠀⠀⢀⠤⠈⠀⢠⡦⠀⢀⠀⠘⠛⠅
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠣⡾⢀⡨⠉⡍⢛⡍⣯⠉⡍⡈⢺⠀⠀⠁⠛⠀⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⣇⡘⡄⢀⢁⠉⡍⣀⢠⢀⡁⣸⠀⠀⠀⣀⡀⠀⢀⠟⠀⠀⠀⠰⠆⠘⠇⡀⢀⠀⡀⠌
⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣘⣋⡶⣀⠏⡇⠀⢿⡜⠙⠇⠘⢿⣟⠃⢙⠛⡆⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⣁⠘⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡟⠘⠃⢀⠻⠃⢀⠓⠀⠘⠀⠐⠎⠁⠆⠉⠰⡧⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢆⠂⠀⠀⡀⠀⠘⠋⠀⢀⠘⠁⠀⠘⠁⠀⠀⠖⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢢⣶⠱⠀⢿⠎⠀⣀⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⢥⠀⡛⠩⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣣⣾⠃⠀⠀⠀⠸⢛⠄⠀⠀⠶⠃⠚⠛⢠⣤⣿⠂⠁
⠀⠀⠀⠀⡀⠀⠄⠀⢀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠘⣞⠀⣟⠢⠤⡉⣀⠒⠀⠀⠀⣀⠤⠤⠤⠐⠒⠀⠀⣈⠔⠀⡀⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠀⠘⠛⠁⠀⠀⠀⡀⠀⠉⠀⢨⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠠⠄⠒⠋⣀⠄⠨⢁⣀⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⢠⣊⣅⡰⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠑⠢⠤⠄⢈⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⠀⠀⠉⠀⡀⠀⠀⠀⠄⠀⡄⠠
⠀⠀⠀⠘⠉⠞⠀⠙⢍⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠄⠀⣰⣆⠀⡀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠶⢎⡛⣠⣀⢠⡄⠀⠀⢈⠀⠈⠱⠀⠀⠠⡄⢸⠀⡇⢀⠆
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠌⡆⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠘⠳⠦⠴⠁⣁⣀⣴⣄⠀⠐⠀⠀⢬⣻⢨⣻⣯⢹⣶⢃⡢⠀⠠⠀⠁⢨⠀⠀⠀⡇⢸⠀⣇⢸
⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠾⠿⡧⣥⢿⣼⡇⠀⠐⠐⠒⠐⠀⠀⠀⠁⠈⠀⠁⠈
⠀⠀⠀⠀⠠⠤⠀⠛⠠⠤⠤⠀⠀⠈⠑⣄⡀⠐⠀⠉⠁⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁⢻⠶⠜⠃⠀⠰⣤⠖⡩⡀⠀⢴⣤⠀⠀⠠⠈⠓
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢑⠑⡇⠀⠀⠁⣀⠀⢀⡤
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠾⢗⠀⠀⢤⡀⠀⠀⠀⠀⠄⡀⢸⣦⠀⠀⠈⠀⠀⡅⠀⠀⢀⠀⠑⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠉⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠀⠀⠀⡐⠿⠁⠈⠋⠀⠀⠠⠀⠰⠀⠀⠀⠈⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠠⠠⠄⠤⠀⠄⠀⠠⠀⠴⠤⠀⠐⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⢂⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡢⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⡉⡅⠀⠀⠀⠀⠈⠗⠑
⠀⠀⠀⠀⠀⠀⠀⠸⢂⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠐⠤⠀⠀⠀⠁⠀⠑⠀⠐
⠀⠀⢀⠀⢀⠀⠐⠖⠒⠂⠀⡄⠀⡀⠀⠀⠀⠀⠀⠁⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢴
⠀⠀⠀⠀⠀⠀⠐⠁⠄⠀⠀⢀⠀⡀⠀⠀⠀⠀⠊⠠⠀⣾⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠄⠀⠖⠀⠠⠒⠓⠀⠈⠢⡀⠀⠀⠀⠠⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⣀⡴⠇⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠀⠄
⠀⠀⠐⠀⠈⠂⢀⠀⠰⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠐⢁⡄
⢠⠠⣂⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠼⢁⠈⠀⠀⠀⢨⠀⠀⠉⠀⠀⠂
⠈⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⠀⠀⠀⠀⠢⠶⣥⠀⠀⠄⢺⠛⠋⠀⠀⠐⠀⠻⢯⣚⠏⠈⣡⣖⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠕⠀⠀⠉⠉⠀⠀⠑⣬⣭⠤⡠⠈⠁⠉⢛⠓⠤⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢺⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠲⣉⠁⠀⠈⠑⢤⡴⠲⠦⢤⣀⣀⡴⠋⠀⠀⠀⠈⢉⠵⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⣄⠈⠳⢄⠀⠀⠀⠙⢆⠀⠀⢠⠋⠀⠀⠀⠀⡠⠞⠁⣠⢞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⢋⡽⠯⠤⣈⡙⠲⠤⣍⣒⢤⣀⣀⣑⣶⠃⣀⣀⡤⣒⡩⠴⠒⣉⡠⠤⠝⠶⣤⣀⡀
⠀⠀⠀⠀⣀⡤⠴⠖⠚⠉⠁⠀⠀⠀⠀⢪⡀⠀⠀⠀⠈⠁⠒⠦⠌⣉⣹⣿⣁⣀⣹⣟⣋⣉⡤⠴⠒⠉⠀⠀⠀⠀⢀⡔⠀⠉⠉⠒⠲⠤⢤⣀
⠀⠀⠀⠀⡏⠉⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⣉⡴⠖⠒⣒⣠⠴⠞⣛⡭⠵⠊⢩⢇⠉⠒⠮⠙⠛⠯⠭⣉⡛⠲⢦⣈⠉⠀⠀⠀⢀⣠⠴⠖⠋⢹
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠓⠲⡶⣞⣉⡤⠶⠒⠉⠁⠀⠀⠀⠀⠀⣀⡴⠃⠀⠑⠦⣄⡀⠀⠀⠀⠀⠉⠑⠲⢄⣉⡷⣶⠚⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡷⠊⠁⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠙⢦⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠓⣄⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢹⡗⠦⢤⡠⠋⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠉⠓⠲⢤⠴⠚⠉⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢀⢇⣠⠤⢲⡋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⣠⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⣄⠀⠀⠀⢀⠞⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⡄⠀⠀⠀⠀⢀⡼⠧⢼⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠞⢢⡀⣠⣾⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢆⠀⠀⣠⠋⠀⠀⢸⡏⠉⠒⠦⢤⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⡤⠖⠋⢹⠀⠀⠀⠙⠁⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠳⡞⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠉⠒⠦⢄⣸⣠⠴⠒⠉⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠒⠦⣄⡀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢧⣀⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡤⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣸⢀⡠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠒⠋⠉⠉⠉⠀⠀⠀⠉⠉⠑⢢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⢀⠂⠀⠀⠀⢀⣀⣀⣠⡤⣶⠦⠛⡻⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡀⣀⣀⠀⠠⡶⠶⠞⠛⠋⠙⢉⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣀⠀⣒⠛⠚⠋⠑⠻⠉⠁⠀⠀⠀⠀⢠⡤⠀⠀⠀⠂⠳⢄⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠍⠉⠓⠀⠀⡀⠀⠈⠂⢀⡤⠐⠉⠀⠀⠀⠀⠀⠀⠀⢀⣈⠦⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠌⠳⣒⠒⠓⠂⠈⠂⢠⡏⣀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡠⠤⢄⡀⢤⣌⠑⢀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⡀⠊⠀⠀⠀⠀⠈⠀⠈⠛⢦⢌⠣⢀⡐⠆⠀⠀⠀⠀⠀⣠⠴⠩⠀⠐⠂⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⣴⠀⣀⠀⣈⠳⡴⢢⡑⢤⡤⠤⡔⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⣠⢊⠽⠋⠀⠀⠀⠈⠰⡄⠈⠂⠐⣤⠠⡀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠕⠉⠀⠀⠀⠀⠀⠀⠠⢻⠀⠀⠀⡈⠣⣀⢀⣀⣀⠠⠤⠒⠁⠀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⡠⠈⡠⠁⠀⠀⠀⠀⠀⠀⢀⠴⢷⣘⠗⠁⠈⠛⠑⢌⠃⠀⢀⡤⠐⠂⠉⠀⠁⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⠀⠀⠀⠀⠀⠀⢀⠔⠓⠚⢲⠏⠀⠀⠀⢠⠀⠀⠣⡠⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⡀⠔⠈⠀⠀⠀⣠⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠬⡛⢤⠤⣤⡀⠠⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠈⠀⠀⠔⠒⠈⠒⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡄⠀⢠⠂⠀⠀⠀⠀⡆⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢀⠔⠀⠰⠁⠀⠀⠀⠀⢐⠁⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡁⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⠀⣀⠀⠀⠀⠀⠀⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢸⡇⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡀⠀⢠⢞⠤⣀⠀⢲⡟⠂⡠⠄⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠐⠁⠧⠀⠇⠀⠻⠀⠘⠁⠐⠲⠏⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣴⣶⣶⣶⣠⣤⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⠛⠻⠿⢻⣿⣿⣿⠿⢿⣿⣿⡿⠿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣴⠃⠀⠀⠉⣴⣦⠀⠀⠀⠀⠀⠀⠀⣠⡄⠃⠀⠀⣇⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠦⣼⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣯⣴⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⡠⠖⠦⡄⠀⠀⠀⠀⠀⠀⠀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣼⣷⣾⣶⣾⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡟⠛⠿⠛⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢣⣀⢀⣠⠇⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠤⠤⠴⠚⠉

⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⡀⢻⡅⠈⢇⡴⠛⠻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⡀
⠀⠀⠀⠈⣛⣓⣚⣓⡰⢦⣴⣚⣵⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠟⠻⢿⠿⠛⠿⠷
⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢹⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡀⣀⢀⣀⡄
⠀⠀⠀⠀⠘⣿⣿⣿⡇⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⢠⣶⣶⣄
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⣠⡝⠒⠶⠶⠶⠶⠶⠖⠒⣄⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣌⠙⠟⠁
⠀⠀⠀⠘⠿⢿⣷⣶⣦⣤⣤⣴⣶⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇
⠀⠀⠀⠈⠉⠙⠗⠲⠶⠶⠶⠶⠲⠖⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉
⠀⠀⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠘⠀⠇⡇
⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡓⠂⠀⢠⡿⡀
⠀⠀⠀⠰⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠠⢦⣉⣉⣠⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣴⣤⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠻⢦⣬⣭⣭⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠈⠳⠦⢤⣤⣤⠾⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠓⢶⢛⡲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣛⣛⣿⣿⡛⠉⠉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⢤⣤⣬⡶⠶⠎⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡃⠀⠠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⢠⣤⡘⣦⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠿⠙⠁⠀⢘⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠓⠀⠘⡻⣼⣯⡏⣿⢿⡄
⢰⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣍⢷⣤⠀⠀⠀⠟⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠄⠀⠀⢙⡿⣋⣴⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤
⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠛⢧⡀⢻⢆⠀⡑⠀⣀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡠⠂⡀⠀⠈⠟⣠⠏⠐⠤⣀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⢸⣿⣿⣿⡟⠀⠀⠀⣠⠞⠁⢀⠀⠀⠒⠒⠦⣷⠈⣬⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⢀⣎⡔⠁⠀⠀⠀⠈⠳⣄⠀⠀⢸⣿⣿⣿⣿
⢸⣿⣿⣿⠇⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠂⠢⠀⠀⢸⠟⠀⠉⠈⢲⡀⠀⠀⠈⠳⡀⢸⣿⣿⣿⣿
⢸⣿⣿⡿⠀⣰⠏⠀⠀⠀⠐⠶⠾⠿⠿⢿⣂⣀⣈⠹⢝⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠴⠀⢇⠀⡼⠰⠒⠚⣋⠄⢣⠀⠀⠀⠀⠘⢆⢻⣿⣿⣿
⢸⣿⣿⠁⡴⠃⠀⠀⢀⡢⣀⠀⠀⠒⠛⠛⠛⠛⠂⠀⠀⣷⣡⣱⣢⠰⠒⠆⠀⠀⠀⠀⠀⠀⠀⢀⡘⠈⠒⢄⢈⢆⡷⠶⠛⠉⠀⢀⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿
⢸⣿⣿⠘⠁⠤⠶⠞⣻⣿⣿⣟⣶⢤⣀⠀⠀⢀⣤⡤⠴⠀⠉⠉⠁⣂⢀⣄⠈⠁⠀⠀⠀⠀⠀⡇⢢⣀⢰⠈⢧⡼⠁⠀⠀⣠⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠉⠁⣶⠿⠋⢁⠞⠉⣿⢶⣄⡀⠀⠀⠀⠀⠀⠀⢿⢀⡇⢰⠀⠀⠀⠀⠀⠀⢳⣳⠀⡎⢠⠎⠀⠀⠀⣰⠛⠲⠶⣖⣒⣠⠀⠀⠀⠀⠀⢸⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠸⣿⠎⠀⠉⠓⠢⢤⣄⣀⣠⠟⢸⣷⠸⠀⠀⠀⠀⠀⠀⢸⠿⠀⠃⢸⠀⠀⢀⡴⠋⠀⠀⠀⢷⠀⠀⢔⡀⠀⠀⠀⢸⣿⣿
⢸⣿⣿⣀⣀⣀⣤⣤⣀⣄⡭⠖⢁⣤⠞⠁⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⣼⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿
⢸⣿⣿⠉⠉⠉⠉⠉⠉⢀⣠⣠⢞⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠺⣦⣝⢲⠦⢤⣤⣤⣤⣼⣿⣿
⢸⣿⣿⠀⢀⣀⣤⣴⡾⠛⠛⣫⢾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣻⠿⣆⣟⠛⢷⣍⠛⣶⠶⠄⢸⣿⣿
⢸⣿⣿⡀⢛⣛⣉⣤⣴⠂⠘⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⣿⡇⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣟⢿⣦⣍⣶⣤⣽⣦⣼⣿⣿
⢸⣿⣿⣿⣿⠿⣻⠿⠋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣽⣿⣿⡏⠙⣧⠀⣿⡇⣼⡿⠛⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣷⣶⣾⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠡⠾⠟⢹⡿⠟⠀⠀⠈⠀⠛⠁⠈⠀⠀⠈⠛⠿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⠉⠉⢑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⣠⣶⣿⣿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣤⣶⣿⣿⣿⠟⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⡟⠿⠟⠋⠀⠀⠀⣠⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⠀⠀⠀⣽⠉⠀⠀⠀⠀⢀⣠⣾⣿⣿⠶⣶⣶⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢾⡿⠋⠉⣲⣄⠀⠀⠀⣰⠃⠀⠀⣀⣴⣿⣿⣧⢀⣀⣤⣾⣿⣿⠟⠁⣾⣀⠀⠀⠀⠐⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⣀⠀⠀⠀⠈⠛⠳⡄⠀⣷⣶⣾⣿⣿⣿⠟⠋⠀⠙⢿⡿⠟⠉⠀⢀⡜⢻⡿⠂⠀⣠⣄⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢽⣿⠆⠀⢰⣿⣦⠀⢻⠀⣿⡿⠛⠋⠁⠀⠀⠀⢀⣴⣿⠀⠀⠦⣶⣯⡀⠀⢀⡀⠀⢿⣿⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀⠀⠀⢠⡄⠈⠛⠋⠀⢈⡆⢡⠀⠀⠀⠀⢀⣤⣾⣿⡿⡟⠀⣠⠀⠙⣿⡇⠀⠈⠁⠀⠀⠀⢀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠈⠁⠀⠀⠀⠐⣿⡇⠀⠣⣤⣶⣾⣿⣿⠟⠋⣰⠁⠀⠀⠈⠀⠉⡀⠀⠀⣶⣦⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢏⠀⠀⠐⣿⡇⠀⠀⣠⠇⠀⠀⠙⠻⠛⠉⠀⢀⡴⠃⠀⠀⢀⠃⠀⠀⠑⠀⠀⠙⠋⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⣀⣀⡀⢀⠟⠀⠀⠀⠀⠀⠀⠙⢚⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠙⢷⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠀⠀⠀⠀⠀⠀⠠⠢⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣠⣚⢢⠀⠀⠀⠀⠀⣷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⠀⠀⠀⠀⢀⠼⣁⣪⠉⠀⠀⠀⠀⠶⠿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡛⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠃⠀⠀⠀⠀⠀⠀⣠⣜⠣⠙⠋⠀⠀⠠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⡅⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠐⠄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣱⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⠄⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⣀⡀⠀⠀⠠⢄⠚⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠈⠒⡤⠂⢄⠀⠀⠀⠀⠀⠀⣶⢑⣈⡄⠛⠁⠀⠀⠀⠀⣥⠐⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠉⠢⡀⠀⢄⣀⣴⠀⠀⠀⢴⡙⣄⣨⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⢙⣄⠀⠀⠀⠑⢶⡑⣤⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣾⠉⠨⠿⠁⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣰⣿⣿⣷⣄⠀⢠⣶⡷⡉⠀⠀⠀⠀⡊⠀⢐⠀⠀⠀⠀⠤⣾⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣾⣿⣿⣿⠿⠋⠉⠢⡀⠀⠈⢆⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣴⣿⣿⡿⠟⠁⠀⠀⣀⣤⣾⣷⣦⣤⠇
⠀⠀⠀⠀⠀⣸⡟⠀⣴⣿⣿⠟⠋⠀⢀⣠⣴⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⢰⠟⣠⣾⡿⠛⢁⣠⣴⣾⡿⠟⠛⠉
⠀⠀⠀⢠⢏⣼⠟⣋⣤⡶⠟⠛⠉
⠀⠀⢀⣿⡿⠷⠛⠉
⠀⠀⠈⠁

⡿⢿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⢿⣿⣿⡿⣿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣆⣳⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠏⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⢠⣂⣀⣤⣿⣻⣍⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠁⠀⠀⠐⣦⢿⣿⣶⡦⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠃⠀⠀⢱⣯⡵⣻⡏⢻⣿⡚⠁
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠳⡆⠀⠈⠌⡼⠯⢶⣒⡚⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣄⡀⠀⠀⢠⣦⣬⢿⣣⣨⠛⠃
⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⡀⠐⠐⠌⠉⠀⠀⠀⣉⡀⠀⠀⠀⢀⣴⣿⡳⡀⠀⠀⣾⣻⣷⠀⠀⡠⢚⣩⣿⡿⣛⠳⣄
⠀⠀⠀⠀⠀⠑⠀⠀⢀⠀⠙⢬⠋⠀⠀⠀⢘⣓⡒⠀⠀⠀⢠⠞⠁⠛⠃⠀⠀⠀⢹⣿⡟⠀⠀⡉⢭⡤⢁⣿⣦⡉⠁
⠀⠀⠀⠀⠠⠁⠂⠀⢀⠹⣮⣗⣲⠤⠴⠂⠀⠀⠀⠀⠀⢠⠏⢀⢀⡀⣀⣀⠀⠀⠀⠙⠁⠀⠀⠉⠛⢣⣾⡶⢵⡛
⠀⠀⠀⠀⠀⢤⣀⠀⠀⠈⣾⡍⠁⠀⠀⠀⠀⠉⠀⠀⣠⢏⠔⠣⠺⣿⠂⠸⡒⣠⠀⠀⠀⠀⠀⢠⠀⣡⢛⡼⢦⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠁⢀⣠⡇⠒⠀⠐⢦⣀⡀⠀⠀⢐⡦⡦⠖⡇⡇⣻⡥⣣⢦⡮⢿⠄⠀⠀⢀⣿⣛⠽⣻⡾⣖⠬⠇
⠀⠀⠀⠀⠀⠀⠀⠐⣲⣾⣄⠐⠲⠤⠤⠬⠍⠀⠀⣸⣷⡭⡾⡏⣧⣮⡶⣽⡻⣶⣿⣄⡄⠀⠀⠈⢀⠊⣿⣿⣮⣳⡄
⠀⠀⠀⠀⠀⠐⠀⠀⢻⡏⠀⠀⠀⠠⠀⠂⠀⢀⢠⣿⢺⢰⣻⡞⣕⡾⡻⣵⢶⢪⡟⡃⣦⠀⠀⠀⣆⡼⠹⡆⡆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠪⢕⠄⡀⢄⠂⠀⠀⠀⡴⣻⣵⣙⣾⡈⡇⡯⡝⡯⣺⣟⣼⣇⢷⡄⠀⠀⠉⠁⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⢛⠒⠀⠀⠀⢸⢧⡞⢛⣞⢮⡓⡆⣵⣧⣆⠿⡽⣲⢭⢽⡵⡄⠀⣰⣞⢳⡀⣰⣶⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⠟⢓⣸⣽⡚⡇⠏⡽⣆⣿⣜⡻⢤⠋⠑⠃⠀⣿⣿⣿⠃⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠂⠓⠁⠀⠁⠉⠋⠉⠀⠀⠀⠀⠀⠀⠸⢿⡿⠀⠹⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⡂⠀⢀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⡿⠂⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠰⣶
⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠆⠀⠀⠠
⠀⠀⠅⠀⠀⠀⠀⣶⡶⠸⡆⠀⣸⡀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢀⠀⠀⢦⡀⠀⢻⠟⠀⠙⢦⣽⣥⠾⠁⠀⣦⠀⢀⣇⠀⢰⠆⠀⣦⠀⢀⣆⠀⠀⠀⢠⡄⠀⣠⣇⠀⣾⠀⠀⢰⣀⠀⠀⣀⠀⡀⠀⠀⢀⣄⠀⢀⡆
⠀⠈⠀⣀⣻⣋⣠⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠘⢦⣙⣟⣤⠟⠀⠀⠘⢦⣙⣟⣠⣄⡀⠀⠹⣦⣙⣯⡴⠃⠀⠀⣙⣿⣤⠴⠋⠀⠙⠶⣤⣘⣟⣠⠞⠁
⠀⠀⢸⡍⣿⣿⠀⠀⠀⠀⢀⣼⠋⠻⣄⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣿⣿⡇⠀⠙⠂⠀⠀⢹⣿⡇⠀⠀⣞⠹⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⡇⢀⡀
⠀⠀⠀⠙⣿⣿⠀⠀⠀⢠⡾⠁⠀⠀⠙⣧⠀⠀⠀⣿⢻⡅⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⡟⡇⠀⠀⠙⠀⣿⠻⣆⠀⠀⠀⠀⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⢸⠇⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡿⡏
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠁⠀⠀⠀⠀⠀⠛⠘⠃⠀⠀⠀⠀⠀⠘⠃⠃⠀⠀⠀⠘⠃⠘⠋⠀⠀⠀⠀⠘⠋⠃

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠻⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣆⠀⠐⠀⠀⠐⠠⠌
⠀⠀⠀⠀⢠⡾⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣵⣄⣀⣠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⣿⢋⡿⠋⠙⠁⠀⠀⠀⠀⠀⢐⣯⣽⡛⣿⡏⠛⠁⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠳⠀⠘⠀⢰⡃
⠀⠀⠀⠀⠀⠀⣿⡌⢿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⠛⢷⣄⠀⠠⠄⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⣿⡄⠄⠀⣠⣤⡀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⣿⠁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢠⡟⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣼⡀⠀⢈⣻⡆⠘⢿⣿⣿⣿⣷⡿⠁⠀⠂⠉
⠀⠀⠀⠀⠀⠘⠋⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠟⠁⠀⠀⠀⠀⣾⣿⡄⠀⠀⣾⣿⣇⠀⠀⡌⠻⣷⣶⣸⣿⣿⣏⣀⡀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠿⣦⣤⡻⢿⣋⣠⡄⢟⢆⢸⡿⣿⣻⡟⣿⣿⣿⠀⠀⢀⣠⣾⠇
⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠐⢿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣯⡿⣧⣬⣭⣥⣴⡶⠟⠋
⠀⠀⠀⣠⣧⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣴⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⣶⣿⣿⣟⣯⣴⣮⣥⣼⣿⢿⣿⠀⣿⣿⣿⣿⡀
⠀⠀⠀⢙⣛⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢛⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢸⣿⣻⣷⣸⣿⣿⣿⣿⢸⣿⢸⣿⣿⣿⣿⣧
⠀⠀⠀⢸⣿⡷⠶⠶⠾⠀⠀⠸⠶⠶⢶⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⢸⣿⠙⣿⣹⣿⣿⣿⣿⢸⣿⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⡿⠿⣿⣿⣿⢸⣿⢸⣿⢹⣿⢻⣿⠟⠘⠿⠘⢛⣿⠟⣿⡏⠁
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⡿⠿⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢸⣿⠁⠉⢸⣿⢸⣿⢸⣿⢸⣿⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠐⠿⠋⠀⠀⠀⢸⣿⠀⠀⠈⠉⠀⠉⢸⣿⢸⣿⠀⠀⠀⠀⠐⠿⠃⠿⠃
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠸⠿⠘⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣷⣴⣾⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠟⠛⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡏⠉⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⠁⠀⠀⢀⣙⠆⠀⠀⠀⠀⠀⠴⢚⡍⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⢿⠿⠁⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠀⠀⠀⠀⣿⣿⡿⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢠⠧⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣖⠒⠈⠐⠤⡠⠤⢄⣀⡀⣠⠔⠉⠉⠐⢒⡦⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣦⡀⠑⢄⠀⠀⠈⠢⡀⠀⡜⠁⠀⠀⢀⠔⠁⣠⢆
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠒⠊⢉⠟⠓⠠⠬⢁⠒⠬⢶⣤⣤⠹⠯⣤⣤⠴⠮⠔⣊⠥⠄⠒⠛⠢⠤⣄⣀
⠀⠀⠀⠰⠶⣉⡁⠀⠀⠀⠀⠀⠘⠤⣀⣀⣀⡀⢈⣱⡶⢶⣻⢦⠰⢟⡲⠶⣦⡭⠤⢀⣀⡀⠤⠊⠀⠀⠀⠉⣙⡶⠆
⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢄⣤⠔⣊⡩⠤⠒⠊⠉⠀⠊⠉⣀⠞⠑⢄⡀⠀⠀⠈⠁⠒⠤⢍⠒⢦⣤⠴⠒⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠔⠋⠁⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠉⠣⡸⠀⠀⠀⠀⠀⡁
⠀⠀⠀⢰⣀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⠔⠛⠲⠤⣀⡀⠀⠀⠀⣀⡠⠔⠚⠢⡀⠀⠀⠀⠀⠙⢄⠀⠀⠀⣀⡄
⠀⠀⠀⠀⢸⠉⠓⡶⠋⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠉⠑⡖⠉⠁⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⣰⠤⠒⠉⡇
⠀⠀⠀⠀⢸⠀⠀⠸⡄⠀⠀⠀⠀⢠⣎⣠⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⢸⣈⣢⡀⠀⣰⠁⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠹⣄⠀⢀⡔⠁⠀⠙⠒⠢⢄⣀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡠⠔⢺⠁⠀⠑⠊⢹⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢢⠎⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢄⣭⠤⠖⠊⠁⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⢄⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡼⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣔⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⠀⠀⠀⡇⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⡧⠔⠊⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠒⠒⠛⢲⣤⠞⠉⢳⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣀⡀⣿⠀⠀⢤⣤⡀⠀⣾⣁⣀⡠⠞⠋⠉⠓⡆⢀⡤⠶⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡞⠁⠀⠀⠀⠉⠳⣤⣀⠀⠛⠛⠀⣯⡴⢯⡀⠀⠀⠀⢠⣿⠏⠀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⣏⣉⠉⠉⠀⢸⣠⠄⠀⠀⠀⣀⣤⣬⣳⣞⣒⡀⣰⣿⠵⣺⠃⢀⠀⢄⣾⣯⣾⣭⣭⣿⣦⠇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⢦⣉⠽⢯⣿⣾⢰⢠⢰⣻⣿⠿⣟⣿⠭⢥⣟⡂⢄⣏⣼⠿⠞⠛⠛⠛⢾⣌⣧⣷⡯⠖⠛⠓⢆
⠀⠀⠀⠀⠀⠀⠀⠈⡿⠚⠁⠀⠀⣨⣿⣿⣾⣌⢿⡻⣿⣿⡻⡯⡝⡡⢚⣽⣿⠶⣄⠀⠀⠀⠀⠀⢹⣿⠟⠞⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⣼⠷⣄⡰⡭⡾⠹⣏⣎⡞⡏⣟⣷⣿⣽⣖⢿⣿⣷⣿⣿⣯⡟⡿⠀⣠⢀⣐⠀⢠⣗⣿⢽⡆⢀⡴⠃
⠀⠀⠀⠀⠀⠀⣀⣹⢾⣮⣽⣲⠃⢀⣠⣭⣍⣍⣩⣠⣶⣶⣽⣿⣿⣷⣿⣿⡿⠾⠥⠶⣭⣧⣶⣾⣿⣿⡵⠿⠿⠧⢤⣀⣀
⠀⣀⡤⠴⠒⠋⡥⠾⣿⣯⣭⣿⠀⣞⡏⡟⠉⠁⠀⠀⠀⠀⣴⣿⣿⣷⣿⠿⢿⡄⠀⠀⠀⠈⣿⠷⠶⠖⠂⠀⢀⣠⢴⣾⢋⡇
⣸⠙⠒⠦⢤⣀⠀⠀⠀⢛⣻⣿⣦⣬⣥⡇⡆⡄⢀⠀⠀⢰⣟⡟⢈⣼⣻⠽⣿⢃⢖⡔⣰⣰⣏⡙⠲⠦⣴⡾⡩⡡⡟⣡⣻⠇
⣿⠀⠀⠀⠀⠈⠉⠓⡶⠋⠀⠀⠀⠒⠨⢽⣷⣷⣼⣜⣄⣸⣿⣯⣾⣍⠻⠛⠛⠛⠛⢿⣟⠛⠂⠀⠀⡀⠈⣿⢞⢌⠜⠕⣽⠃
⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣴⡯⣅⣈⠉⠉⠙⠋⠙⠋⠁⠀⢀⣠⡤⡶⡺⣫⢾⡆⠀⢀⢎⢃⠀⢻⣣⡪⡪⣮⣾
⢻⣀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣿⡁⠀⠈⠉⠓⠲⠦⢄⡤⡶⠊⡋⡫⣪⢏⢔⣕⣿⡇⠀⠰⡱⠁⠄⢸⣷⣵⣷⣿⠃
⠀⢸⣿⢷⣦⣤⣀⣐⡇⠀⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⠀⢸⣯⢖⢔⢜⢝⡵⡡⡪⣪⣺⣇⠀⢀⣀⣠⠄⣸⢏⢌⢬⣿
⠀⢸⠄⠀⠈⠙⠛⠿⡇⠀⠤⣤⡀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⡗⡵⡱⡡⢊⢊⢠⢞⣽⣿⡟⡠⠚⠉⠁⠀⣿⡣⡣⡵⡿
⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠀⠀⠹⣷⣦⣤⣀⡀⠀⠀⠀⣿⢊⢆⣔⣴⡵⡻⢫⢎⣿⠃⠀⠀⠀⡠⠀⣿⣜⢋⢜⡇
⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠈⠉⠙⠙⠙⠶⣦⣷⢗⣿⢯⢟⢞⡕⡱⡡⣿⠀⠀⠀⢄⠄⠀⢻⣧⢣⣺⣇
⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠕⢕⢕⠝⡡⡫⡪⢊⢔⣿⠀⠀⠆⡦⠊⠀⢸⣱⠟⣼⣿⡟⠀⠀⠀⡀
⠀⢸⡆⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡮⡢⢊⢾⢎⢞⡔⡱⣪⣿⠀⠀⢴⠔⠁⠀⢸⢏⢮⢚⣟⣓⡀⣒⠂⠀⠐⠂
⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣗⡕⡵⡵⡣⣫⢞⢞⡵⣿⠀⠀⢘⡔⡡⠀⢸⣷⣷⠟⠧⠄⣀⡀⠒⣒⣂⣀
⠀⢸⣄⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣯⢪⢊⢞⡕⡕⡡⣫⣾⣿⠀⠀⠀⠈⢀⣄⣾⣛⣉⣀⡒⠀⠀⠀⠀⢀⠴⠄
⠀⠀⠀⠉⠻⠲⣤⣄⡇⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢸⣗⡕⡡⣪⢊⢞⠕⣵⣿⢿⠀⢠⣠⣾⠾⠭⡩⠁⠁⠀⠀⠀⣐⡒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠹⣬⣶⣆⣄⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣫⢞⢜⡕⡡⣣⣾⣻⡷⠿⢖⢛⣉⣀⡐⣀⡀⠐⠀⠒⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⢧⣹⣦⣄⡀⠀⠀⠀⢸⡵⡷⣫⣊⣼⣿⣉⠉⠉⠉⠁⠠⡍⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢶⣤⣸⣯⡖⡛⠀⠐⠒⠖⠒⠐⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⣤⣶⣿⣿⣷⡄⠀⠀⠀⠀⠀⢠⣾⣿⣿⣶⣦⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣠⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢤⣄⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣠⠋⣠⣶⣶⣶⠎⠱⡄⠀⠀⠀⡰⣫⣶⣶⣶⠆⠈⠳⡄⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⢰⣿⣿⣿⣯⠀⠀⣳⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠧⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠡⠘⣿⢿⣿⣿⣿⣿⠇⠀⠀⠀⠻⣿⢿⣿⣿⣷⡾⢀⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠄⠹⠿⠟⠁⠀⠀⠀⠀⠀⠁⠀⠽⠟⠫⠔⠁⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⣤⣤⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣄⣀⣀⣀⣀⣀⡤⠤⠖⠚⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⢐⠀⠉⢹⠰⡁⠀⠈⠉⡟⠀⠘⠒⢂⠇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠈⠺⡄⠀⢸⣰⣷⣄⣀⠎⠀⠀⢀⡤⠛⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣉⣢⣄⣠⠞⠈⠙⢻⡿⢿⠋⣀⡤⠖⠡⢤⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢦⡀⠈⠹⡀⠀⠀⢠⣃⣨⣿⣷⣤⣤⣄⡀⠈⠙⠪⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡞⡠⠔⡎⠉⠙⡏⠁⠀⡇⠀⠀⠀⡇⠀⢸⠢⡀⣠⢾
⠀⠀⠀⠀⠀⠀⢠⠠⢄⠀⠀⡇⢀⠟⢦⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣇⠀⢸⠀⠉⢀⠼⣐⡀⢄
⠀⠀⠀⠀⢀⣠⡇⠑⢺⣶⡖⣯⠟⠚⢁⣧⣇⡀⠀⡇⠀⠀⡇⠀⠀⣰⡩⢭⣝⢶⣲⠱⠻⣤⣌⡇⣃
⠀⠀⠀⠀⠨⠉⡟⢲⠾⣛⣃⣙⡷⣖⠺⡖⠒⠋⡇⣇⣴⣶⣓⠉⠉⢣⢻⣰⠶⢻⣍⣻⣿⠭⣭⣚⡉⠉⠉⣒⣲⠤
⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⡇⠀⠀⢸⢀⣇⣀⠀⣧⠟⢉⣽⡆⠉⡟⠛⡿⠾⡿⣷⠉⠀⣈⡱⢆⠙⡞⢛⠋⠁
⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⡧⠤⠤⠾⢾⣤⢤⡏⠉⣿⣉⣻⡖⠶⠧⠤⢇⡀⡇⠋⠈⡏⠀⠀⢸⠀⣇⠨
⠀⠀⠀⠀⣰⠤⡇⢸⠀⢸⢑⠶⣶⠶⠬⢽⡴⠯⠽⢦⠽⢦⠥⠶⠒⠒⠚⣧⣇⡆⠀⡇⠀⠀⠈⢢⣨⣾⠖
⠀⢀⣀⣴⡍⠓⠾⣾⠤⣼⠿⠒⢩⠦⠤⠌⠀⠀⣠⣼⠀⢸⡤⠤⠤⠴⡖⠉⠞⠀⠀⡇⠀⠀⠀⢸⠀⢹⠃
⠀⡏⠉⠛⠓⠲⡾⣿⣶⡿⡷⢶⢻⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⢸⡀⡠⠔⠊
⠀⡇⠀⠀⠀⠀⡇⡇⢸⠀⡇⢸⢸⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⣧⡀⠀⠀⠀⡇⠀⠀⠀⣸⠄⠊⠁
⠀⠉⠒⠢⠤⣀⡇⡇⢸⠀⣇⡼⠜⠒⠒⠒⠂⠀⠤⠘⠒⠚⠓⠂⠁⠁⠀⠈⠑⠠⢀⣇⠤⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡚⣧⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠇⠀⠀⠀⠀⠀⠀⣀⠤⣶⠂⠛⣀⠒⢤⡀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⡬⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⡀⠉⠠⠀⠉⠀⠀⠋⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡔⠂⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠛⠀⠀⠈⠀⠐⠗⠀⠀⠂⣠⣷
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢸⠑⠊⠁⠀⠀⠀⠀⣴⠄⠀⠀⢸⠀⠂⠆⠀⠀⢴⠄⠐⠀⡀⢀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠘⡜⠃⠀⠀⠄⠀⠀⠀⠈⢁⢮⠐⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠘⠑⠒⡢⢄⡀⠀⠀⠀⠀⠀⠈⢺⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⣔⠷⠀⠀⣶⠀⠀⠒⣥⢄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡁⠀⠀⠸⠇⠀⠈⠀⠉⢦⡀⠀⠀⠀⠀⠀⠢⡀⠀⢈⢈⠇⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀⣀⣀⡥⠖⠋⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠘⠟⠀⠀⠀⠀⠀⣀⠀⠠⣤⢧⠀⠀⠀⠀⠀⠀⠈⠀⠚⠣⠄⠀⠤⠆⠀⠀⠀⢠⠖⡸⠻⡽⡀⠀⠀⠀⠀⠀⠀⠎⡨⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠐⠀⠀⢠⡄⠀⠀⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠙⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣇⢄⠀⠁⠀⠀⠀⠠⠀⢠⣄⠀⡸⠀⠀⠀⠀⡀⠑⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡰⠆⠀⠀⣴⡆⠀⠈⠁⡴⠃⠀⠀⠀⠘⠄⠉⠀⠀⠀⠐⠟⡸⠀⠀⠀⠀⠧⣹⠴⠂⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢤⣆⣀⣀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠿⠀⠀⠀⠠⢤⡜⢳⠖
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣸⠟⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠒⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠙⠄
⠀⠀⠀⠀⠀⠀⢴⢁⡦⠀⠀⠀⠀⠈⣓⡛⠁⠀⠀⠀⠀⠀⠀⠀⡰⠋⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠣⡎⣀⡽⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⡃⡇⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣜⢀⡸⠀⠀⠀⠀⠀⠀⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐⠿⡗
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠛⠀⠀⠘⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢣⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀⠀⠀⢀⣠⠔⠁⠀⠀⠀⣠⠲⡀⠀⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢉⣿⡿⡟⠒⣄⠀⠀⠀⠀⠑⣄⠃⠀⠀⠀⠀⠐⠠⢀⣰⡂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠻⣤⠖⠀⠀⠀⠀⠘⠲⠔⠚⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠉⠒⠒⠁
⠀⠀⠀⠀⠀⢮⠈⠆⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣸⠚⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠰⠇⡸⠆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠷

⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠈⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠒⠠⡀
⠀⠀⠀⠀⠀⣀⢀⡮⣤⠄⣀⣠⣄⠰⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⢰⡂⠀⠀⠀⠀⠘⠀⠠⠤⠤⠠⠄⠤⢀
⠀⠀⡔⠊⠉⠉⠈⡇⠀⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⠀⠈⠀⠀⠈⠉⠉⠐⡀⠀⠸
⠀⠀⡇⠀⠀⠀⠀⠈⡗⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠥⠄⡃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡄⠀⠈⠉⠉⠁⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠔⠒⠢⡀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡀⠀⠀⠀⠀⢠⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡅
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⡰⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠐⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠐⠂⠀⠉⠁⠀⠀⠀⠀⠐⠒⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢀⠀⠀⠀⠘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠸⠀⠀⠀⠀⡆
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡠⠜
⠀⠀⠈⠦⠤⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠐⠒⠒⠒⠒⠒⠂⠐⠃⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣾⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠢⠀⠀⠀⠀⠐⡍⠀⠀⠋⠃⠀⠀⠀⡤⠠⠀⠀⠀⠀⠨⠑⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⢤⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠦⢤⡄⠀⠀⠀⡀⠀⠀⠄⠉⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠀⣄⠀⠀⠀⣠⠐⡀⠀⠀⢀⠔⠁⠐⠀⠀⠀⠀⢀⠄⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢄⠁⢂⡤⠐⠀⡆⠀⠘⠉⡐⠂⠤⡰⢶⣀⠈⠒⠃⠀⠂⠀⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠚⡌⠀⠀⠀⠐⢡⠀⠀⠀⠁⠀⠐⠚⠀⠐⠐⠃⠀⠊⣀⠀⠀⠘⠁⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠄⣀⠠⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢁⣀⠃⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⡀⠀⢠⠀⠀⠀⠑⣀⡀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⠉⠀⡀⠈⠈⠀⠀⠀⢀⡤⠤⠀⠉⠀⡀⠙⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣦⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⢠⣶⣿⡿⠇⠀⠀⠀⠻⣿⣷⣦⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠿⠀⠀⠀⠀⢀⣀⠀⠐⢳⠀⠀⢀⣀⣀⠀⠀⣛⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠂⠀⠀⠀⠞⠿⠟⠁⠀⠀⡀⠈⠉⠛⠋⠀⣴⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣧⠀⠀⠀⠀⠀⢀⡄⠀⢧⡀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠘⠳⠤⠜⠃⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⣤⣶⣿⠿⠿⢿⣷⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡁⢐⣒⣀⢸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠙⠛⠻⠿⠛⠋⠁⠰⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⣆⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⡲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠙⢄⡀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⣠⠾⡁⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⠞⠉⢢⠄⢀⡀⠀⠉⠉⠉⠉⠉⠉⣀⣠⠴⡎⠁⠀⠱⡄⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⠎⠀⢠⠃⠀⠀⢸⠋⠉⠁⢰⠋⠉⠉⢹⠀⠀⠘⡀⠀⠀⠙⡄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠎⠀⢠⠃⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡆⠀⠀⢡⠀⠀⠀⠘⡄⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡜⠀⢀⠇⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠘⡄⠀⠀⠀⣘⡄⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠴⠧⢄⣬⣀⣀⣀⣀⡆⠀⠀⠀⠀⢸⣀⣀⣀⣀⣯⠠⠤⠤⠓⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣷⣶⣄⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠿⣿⣇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣆
⠀⠒⢛⣿⣿⣿⣷⣶⣤⣀⠸⣿⣿⣇⠀⠀⠹⣿⡀⠀⢸⣿⡟⠁⠈⢻⣿⣿⣿
⠀⠀⠉⠉⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣷⣤⣀⣻⣶⡆⣾⡿⣁⣀⣠⣾⣿⣿⡟⠀⠀⠀⠀⠀⢀⣀⣴⣇⣴
⠀⠀⠀⠀⠀⠀⣤⣰⡶⢾⣿⣿⣷⣉⣭⣭⣭⣽⠿⣳⡿⢶⣿⣿⣿⣿⣿⣻⣶⣶⣶⣶⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣿⡟⢻⣿⣶⣾⣯⣝⣯⣻⣿⢯⣿⣿⣭⠿⣶⣢⣝⣤⣾⣿⣽⣿⣿⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⣿⣿⣿⡋⣹⣿⣷⢰⣿⣿⠛⣿⣿⣿⣿⣿⣿⣯⣨⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠛⣿⣿⣿⣿⣟⣿⣿⣿⢿⣿⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⢿⣾⣿⢸⣿⣾⣿⣯⣿⣿⣿⣿⣿⣿⣧⣼⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢸⣿⡟⠛⣿⣿⣿⣿⣿⣿⣯⣽⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⣿⢸⣿⣿⣯⣽⣿⣿⣿⣿⣿⣇⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠟⢻⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣼⣿⢸⣿⣷⡿⠿⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠻⠿⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠻⢧⡀⠈⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠀⠀⠀⠈⠀⠀⠀⠀⠑⢦⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⡀⠀⠀⠉⠉⠉⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⣤⣤⣤⣤⣀⣀⡀⠀⠀⠲⣤⡈⠳⡄⠀⠀⠀⡼⠁⢀⣴⠋⠀⠀⠀⠀⠀⠀⢀⣀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠋⠀⠀⠀⠉⠙⠲⣤⣈⠻⣧⡘⡦⠒⢰⠃⣴⠏⠁⢀⣠⣤⣤⣶⣶⠶⣶⣾⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣄⣀⣀⠀⠀⠀⠀⠀⢸⣿⣷⣾⡟⠛⠒⠻⣼⣡⣶⣿⠟⠋⠉⠉⠁⠀⠀⠉⠙⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡖⠛⢛⠿⠛⠛⠛⠛⠛⠛⢛⣿⣿⣿⣿⣿⣶⣶⣿⣿⣧⣤⣤⣤⣤⠤⠤⠴⠶⠖⣛⠳⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈⠉⡿⠋⠀⠀⠀⢀⣠⠶⠚⠉⢀⠼⠋⠀⠉⠉⠛⢯⣉⠛⠓⠲⠬⣄⡀⠀⠀⠀⠀⠀⠉⢲⡚⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣠⠞⠁⠀⠀⠀⣰⣿⣷⣶⣦⣴⣅⡀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⢀⣀⣩⣦⣠⠀⡼⡸⠀⣧⠸⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⡣⠀⠀⠀⢀⣼⡇⣿⣿⣿⣿⠁⠀⠉⠛⢲⢤⣄⡀⢀⣀⣤⠴⠿⡾⠿⠟⠿⢿⣿⡇⠀⠁⠀⢸⡄⡇
⠀⠀⠀⠀⠀⣠⣖⣋⣁⣸⡁⠀⠀⢠⣾⣿⡇⣿⣿⣿⣿⠀⠀⠀⠀⢂⡄⢈⣿⠉⠀⠀⠀⠀⣇⠀⢰⡅⠸⢹⡇⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⢀⣴⡿⠛⠾⣇⡏⣿⣿⣿⠀⠀⠀⠀⠐⣤⣷⣿⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣋⣴⡿⠻⠀⠀⠀⡟⠆⢹⡿⣿⢶⣤⣀⠀⠐⠠⢍⣿⠀⠀⠀⠀⢀⣻⠀⠀⠀⠀⠀⣿⣆⠀⠀⣦⣄⡘⢦
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠁⢸⡇⠀⠀⠀⠀⣳⠀⠧⡙⣿⠀⠈⠉⠛⢷⣾⣿⣿⣀⣤⡴⠞⠛⢿⠀⠀⠀⠀⢀⠉⢿⣧⡀⡛⡇⠉⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟⠀⢼⢃⣿⠀⠀⠀⡤⠂⠈⢻⡏⠋⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠉⠿⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡇⠀⠀⠀⢻⠀⠀⠌⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⡇⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠈⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢄⣿⣦⡂⠀⢸⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣸⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣆⣸⠀⠀⠀⠀⠀⠀⢴⡇⠀⠀⠀⠀⠀⢸⠀⢠⣾⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⢤⣀⠀⠀⠰⡇⠀⠀⢀⣀⣤⠾⠶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣷⠴⠚⠋⠁
⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠙⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠧⣤⣿⠶⡀⠙⢆⣀⠖⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⡴⢤⣤⡄⠋⠀⠀⠈⢿⢀⡞⠀⠀⢀⣤⣀⣀⠴⠖⠋⠁⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢡⠎⢀⡞⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡴⠋⠀⣴⠛⢦⡀⠀⠀⠀⠈⠛⢻⡏⠀⠀⠀⠀⣀⠤⠚⣶⡄⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⣀⣠⡞⠉⠻⣦⡙⠷⢄⣀⣀⠴⢾⠀⠀⣀⠴⠛⠁⠀⠀⠈⡇⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠉⢣⠈⠳⣄⢀⡈⠙⢶⣄⠀⠀⠀⠘⠳⠛⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣿
⠀⢀⣀⡀⣀⣀⣠⡞⠉⠘⢦⡀⠛⠄⠀⠉⠀⠀⠀⠙⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿
⣾⡋⠁⠀⠀⣨⣿⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⣠⠞⠛⢳⡀⠀⠀⢀⠀⠀⠀⠀⠀⣿
⠈⠁⠀⢀⡾⠁⣿⠱⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠏⣠⡄⠀⣿⣀⡀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢾⡀⠀⣿⠀⠀⠂⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠿⢀⡿⠀⠀⡿⠿⠟⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠁⠀⣿⠀⠀⠀⠈⢏⠻⣄⠀⠀⠀⠀⣠⣴⣿⠟⠋⠀⠀⠀⠀⠀⢘⣦⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⢠⣤⣄⠀⠈⠻⣿⣦⣀⣴⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠚⠻⣿⣦⣦⣾⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢀⣀⣀⣿⣿⠉⠻⣷⣄⣀⣼⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⠀⠀⠀⢠⣿
⠀⠀⢰⣿⠿⣿⣿⢻⣧⡀⠘⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢻⣦⡈⣿⠀⠙⠻⣶⡿⠃⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠸⡆⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠉⠛⣿⠀⠀⠀⢀⣤⠞⠉⠓⢤⣀⠀⢰⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⣠⠔⠋⠀⠀⠀⠀⠀⠉⠳⣤⡈⠺⣽⣶⣄⠀⠀⠀⠀⠀⠀⢠⠃⠀⣇⠀⠀⠀⢸⣿⣄
⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣌⠛⠛⠁⢠⡄⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⢸⡟⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠘⠁⠀⠀⢠⡎⠀⠀⣿⠀⠀⠀⠘⠇⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⣼⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⣿⣆
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡷⣦⣼⣷⣄⠀⠀⢹⣦⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⡰⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⢠⣿⣿⣿⣿⠓⢦⣀⣼⠃⠀⠀⠀⠀⠀⠈⣿⣿⡄
⢠⡶⠋⠁⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⢸⡸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡄
⠈⣧⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⡹⠁⢸⣿⡟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠈⣦⠀⠀⠀⠀⠘⢦⣀⣠⣄⠀⠀⠀⠀⠀⠰⣎⠉⠳⣄⠀⠀⣠⠞⠁⠀⣼⣿⣇⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇
⠀⠀⠘⢧⡀⠀⠀⠀⠀⣟⠀⠈⠳⠶⠖⠉⠱⠞⠙⢦⡀⠙⣤⡔⠋⠀⠀⠀⣿⣿⣿⡄⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠳⣄⠀⠀⠀⠘⢷⡤⣤⠀⠀⠀⠀⠀⠀⡎⠛⠂⠁⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⣶⠁⠈⠳⠆⠀⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⢠⠋⠀⠛⢉⡿⢄⡀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠋
⠀⠀⠀⠀⠀⠈⠳⠤⠚⠁⠀⡎⠉⢛⡟⠁⠿⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⢠⠎⣸⠛⠳⢤⣀⣀⣀⠠⠤⠈⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣠⠎⠀⠀⢀⡂⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⠛⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣻⠀⠀⠀⠀⠀⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠉⠉⠉⠉⠙⠛⠛⠦⣤⣀⣈⡗⠀⠀⠀⢀⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠒⠒⢲⡆⠀⠀⠀⠀⠐⢿⣿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⣶⡞⢿⣧⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠻⣷⡌⠛⠛⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⢀⣠⠔⣿⣸⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣀⡀⣀⢀⣤⡤⠖⠋⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢀⣀⡀⠀⠀⠀⠀⣄⣠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣤⡴⢶⣶⣶⣤⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣠⣴⠶⠛⠛⠛⠂⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠈⠙⢿⣧⡀⠀⠀⠹⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⢀⡾⠁⠀⠀⢀⣤⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⣀⣴⠾⠟⠛⠻⠷⣦⣄⠀⠀⠀⠘⢷⡀⠀⠀⣸⠁⠀⢀⣴⠟⠋⠁⠀⠀⠀⠈⠉⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠘⣇⠀⠁⠇⠀⢠⠞⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠳⡀⠀⠸⡀⠀⠀⠀⠊⠀⢀⣴⡾⠿⠿⠿⢷⣦⣀⠀⠀⠀⠀⠀⠀⣀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⠷⠶⣤⡀⠀⠘⠄⠀⠃⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠐⠂⠀⠀⠉⠲⠄⠀⠀⠀⣼⣾⣀⠞⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⣰⣶⡀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠴⠚⠉⠙⠛⢿⣦⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠃⠀⠀⠀⡼⠁⠀⠀⠀⠀⣠⣿⠟⠉⠀⠀⠉⠀⠀⠀⠸⠧⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠹⣧⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠸⠁⢠⣤⠀⠀⢰⡿⠁⠀⠀⠀⢠⣶⠶⠆⠀⠀⠀⠀⠈⠁⠈⠙⣷⡀⠀⠀⠀⠀⢹⡆⠀⢠⣿⣿⡄⠀⠀⠆
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⠁⠀⠀⠀⢠⡟⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠁⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠼⡿⠆⠀⠀⠸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠀⠀⢀⡀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠁⠀⠀⠀⠀⠁⠀⠀⣠⡄⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠀⠀⠀⠀⠀⠀⣎⢑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠛⡗⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡀
⠀⠀⠀⣖⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣏⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣒⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⠯⠬⠭⠍⣵⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢴⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⠀⠀⠀⣀⣠⣋⡛⠒⠒⠖⠴⡇⠀⠀⠀⠀⠈⠁⠛⠃⠀⠀⠀⠀⠀⠉⠲⠤⠼⠾⣀⣀⡄⠀⠀⠀⠀⢠⠞⠉⠀⠉⠙⠢⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⢀⡉⠉⡉⠐⠒⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣀⣠⠒⠒⠹⡄
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠒⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠞⠁⠡⢀⡤⠤⢷
⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢾⡏⠉⠙⠲⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠤⠞⠛⠋⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⣸⠀⠀⢠⣶⡄⠀⠀⢸⣧⠀⠀⣀⡗⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⣰⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⣰⠃
⠀⠀⠀⠀⡤⠠⢤⣠⠔⠒⢢⠀⠀⠀⠀⣾⡷⠤⠤⠤⡏⠀⠀⣾⠇⢷⠀⠀⠘⣿⠒⠋⠁⢠⡶⠒⠢⣄⠀⠀⢿⡇⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢙⠾⡅
⠀⠀⠀⠀⢇⠀⠀⠁⠀⠀⣸⠀⠀⠀⢐⣿⠇⠀⠀⢰⡇⠀⣼⣯⣤⣼⡇⠀⠀⣿⠀⠀⠀⢸⡟⠀⠀⠈⡇⠀⠈⣿⡞⠁⠀⠀⠀⣠⢄⣠⢤⠀⠀⠀⠳⣏
⠀⠀⠀⠀⠈⠳⣀⠀⠀⡰⠃⠀⠀⠀⠸⣿⠀⠀⠀⢸⠀⢰⣿⠀⠀⠀⢹⠀⠀⢿⡆⠀⠀⢸⣇⣀⣠⡴⠃⠀⠀⣾⡇⠀⠀⠀⠀⠙⢄⣡⠞⠀⠀⠀⠀⢸⡄
⠀⠀⢠⠀⠀⠀⠈⠑⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⣼⡏⠉⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠻⠃⠀⢀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⣴⡞⠉⠓⢦⠀⠀⠀⢸⣧⠀⠀⣰⡞⠛⠲⡄⠀⠉⠉⣿⡏⠉⠀⣀⠀⠀⠀⠀⢸⡇⠀⠙⢦⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⣠⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣸⠀⠀⠀⣼⡇⠀⠀⣿⡇⢀⣠⠇⠀⠀⠀⣿⠇⠀⢸⣿⠀⠀⢸⠇⢸⣷⠀⠀⠘⡇⠀⠀⠀⠀⠈⣿⡄⠀⢠⠇⠀⠀⢸
⠀⠀⢀⡀⠀⠀⠀⢸⣷⠶⠚⠧⣄⠀⠀⣿⠇⠀⠀⣿⡿⣍⠁⠀⠀⠀⠀⣿⠀⠀⢸⣧⣀⣀⣼⠀⠈⣿⠀⠀⢠⡇⠀⣰⣿⡀⠀⠘⣿⣀⡞⠀⠀⠀⢸
⠀⠀⠁⠑⠙⠆⠀⠈⣿⠀⠀⠀⢸⠀⠠⠿⠄⠀⠀⣿⡇⠈⠳⠀⠀⠀⠀⠿⠀⠀⣿⡇⠀⠀⡿⠀⠀⣿⣀⡠⠏⠀⣰⡟⠀⡇⠀⠀⠸⣿⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣠⠞⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢀⡇⠀⠀⠙⠉⠀⠀⣰⡟⠉⠙⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠈⠃⢀⡶⡄⠀⠀⠰⠟⠀⠀⠀⠾⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣓⡆
⠀⠀⠀⠀⠀⢀⣠⡀⠀⣀⡉⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢹⠄⠀⠀⠀⠀⠀⠀⣞⠑⠞⢱
⠀⠀⠀⠀⠀⠋⠀⠀⠀⢹⠉⢳⠦⣄⡀⠀⠞⠀⠀⠠⠖⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣇⣠⠤⠴⢆⠀⠀⠘⢦⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠘⡏⠑⣲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⣋⣉⣩⣤⡤⠖⠒⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡠
⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⠀⠸⡆⠁⡷⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣉⣁⡨⠤⠔⠒⠋⠉⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠙⠁⠀⠀⠀⠀⢸⡀⢀⡷⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⠉⠀⠀⠀⣀⣠⢤⡤⢴⣛⡦
⠀⢀⡤⢤⡀⠀⠀⠀⠀⢠⠗⠋⠀⠀⠀⠀⠀⠖⢦⣀⣀⡀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⢸⣓⣮⣽⠯⠽⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤
⠚⠉⠀⠀⠙⠢⠤⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⢦⠀⠀⠀⡴⠖⠲⣄⠀⠀⠀⣠⠤⢤⡀⠀⠀⢀⡤⠶⢤⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣿⠶⠶⠾⠿⠿⢿⡿⠿⠿⠿⠿⠟⢿⡟⠿⠿⠿⠟⠿⣿⠿⠶⠾⠷⠶⢶⣿⣶⣦⣿⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠙⠶⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠳⠦⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢻⡧⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠤⢤⣤⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡔⠒⠒⢲⢠⠒⠒⠒⠂⡔⠒⠒⠒⠀⡖⠒⠒⢢⡔⠒⠒⠒⡴⠒⠒⠒⡆⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀⣀⡀⢣⣤⣤⣼⠘⣤⣤⣤⡄⢣⣤⣤⣤⠀⣧⣤⣤⡜⢣⣤⣤⣤⠳⣤⣤⣤⠇⠀⢸⡏⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠁⠀⠀⢹⡇⠀⠀⠨⢸⠀⠀⠀⠁⡉⠀⠀⠈⠀⡇⠀⠀⢸⡇⠀⠀⠈⡏⠀⠀⠀⡇⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠥⠼⠛⢅⡩⠭⠤⢬⢈⠥⠬⠽⠅⡨⠭⠥⢬⠀⡭⠭⠤⢍⡨⠭⠤⢬⣨⠤⠤⠾⡁⠀⢸⣧⣿
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠰⣀⣀⣀⡸⣇⠀⠀⢸⠠⡀⠀⠀⡀⠇⠀⠀⢀⠀⡇⠀⠀⣸⢇⠀⠀⢸⢟⠀⠀⢀⠇⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠰⠉⠉⠙⣶⠉⠉⠉⡇⡎⠉⠉⢹⣴⠚⠛⠛⣧⣶⠛⠛⢻⣦⡞⠛⠛⣧⡞⠛⠛⣿⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⡿⠀⢀⣷⡤⢤⠔⣛⣤⣤⡤⢃⣣⣤⣤⢼⣃⣀⣀⡀⢉⣰⣀⣶⠾⠻⠴⠦⠴⠟⠧⠤⠤⠏⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⣰⡟⠁⠀⣇⣀⣀⢀⢳⠀⠀⠀⡟⡇⠀⠀⢀⣿⠀⠀⠁⣉⡉⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣿⣿⡿
⠀⠀⠀⠀⠺⠿⠤⣤⣤⣄⣀⣀⣀⣈⠉⠉⠉⠁⠉⠉⠁⠈⠈⠉⠙⠛⠉⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠈⠉⠉⠉⠉⠛⠛⠛⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⣾⣟⣛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣠⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠿⠛⠁⠙⠻⠿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠁⢀⣤⠀⠀⠀⢠⣤⡀⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠏⠐⠛⠁⠀⠀⠀⠀⣙⠃⠀⠈⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠺⠟⠀⠀⠀⠈⠻⠁⠀⢀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠐⢄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣤⣁⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⠀⠀⠀⠰⢄⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠲⠂⠀⠀⠀⠀⠀⣠⠏⡐⠶⠃⠀⠀⠀⢀⡐⠂⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⢀⣞⠉⠳⠞⠀⠀⠀⠀⠀⠀⠀⠡⡌⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⡂⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢦⡵⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⠸⠃⠀⠀⠀⠀⠀⠀⠀⣾⠤⣌⠆⠀⠀⠀⢰⠁⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢥⠊⠀⠀⢔⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣤⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⡔⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⢠⡏⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢢⣀⡀⠀⠀⠀⢤⡐⠀⠁⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⡀⠀⠈⠢⡤⠀⢄⠀⠀⠀⠀⠀⠀⠠⣎⢂⢈⡰⠂⠁⠀⠀⠀⠀⢈⡤⠂⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡐⠉⢆⠀⠀⢄⡀⣠⠀⠀⠀⠀⡖⢢⠀⣩⠜⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡐⠀⠀⠀⠑⡄⠀⠀⠀⠑⢤⡈⢆⢙⡸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡠⠒⠣⣰⣁⠁⠑⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡐⠁⠀⠀⡠⠊⠈⠢⡀⠀⠀⡠⣅⡜⠉⠀⠀⠀⢀⠊⠈⠲⠀⠀⠀⠀⢀⣠⢉⠀⠐⠗⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠃⡜⠀⠀⢀⠔⠁⠀⠀⢀⠜⠓⢄⠑⠉⠈⢢⠀⠀⠀⠀⠢⠄⠘⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠃⡜⠀⠀⡰⠃⠀⠀⡠⠖⠁⠀⠀⠀⣁⠦⢀⡀⠡
⠀⠀⠀⠀⠀⠀⡌⠌⠀⡠⠊⠀⢀⠤⠊⠀⠀⢀⠠⠒⠉⠀⣀⠠⠘⠋
⠀⠀⠀⠀⠀⡐⠌⢀⠔⠁⡠⠐⠁⢀⡠⠐⠈⣀⠠⠐⠊⠉
⠀⠀⠀⠀⢰⠏⡠⢃⠴⠊⡀⠄⢂⠡⠔⠒⠉
⠀⠀⠀⢠⣏⣨⢔⣡⠔⠍⠒⠉
⠀⠀⢠⡿⠟⠛⠉


⣿⢿⠿⡿⡿⠛⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣟⣛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿

⠀⠀⡈⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠒⠈⠳⡀
⠀⢰⠀⠀⠀⠀⠈⠁⠐⠒⠒⠤⠤⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠠⠤⠤⠤⠒⠒⠒⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠪⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠈⢫⠒⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⢀⣀⣤⠔⠈⢡⠃
⠀⠀⠀⠀⡆⠀⠀⢩⠙⠒⠠⠄⢀⡠⠀⣀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⢁⣀⠠⡀⠦⠐⠈⠁⠸⠀⠀⢸
⠀⠀⠀⠀⡇⢰⡄⢸⠀⠀⠀⠀⠀⠀⠈⠁⠐⠊⠠⡄⠀⠀⠰⢄⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⠛⠀⡁⠀⠀⠁⠀⠀⠰⣤⡄⢸
⠀⠀⠀⢰⡵⠃⠸⡰⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⢠⠀⠀⠀⠀⠈⢡⡀⠀⠹⡄⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡌⠻⡄
⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠠⠡⣀⠀⠀⠀⢠⠀⠀⠀⣀⠀⣀⠙⣄⠀⠜⢢⡤⠖⠋⠀⠀⢀⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠑⢄⠀⢉⢠⠴⠁⣉⡀⠉⣀⠛⠳⡄⠙⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢂⠾⢀⠔⠋⠉⠀⠀⠉⠙⠢⡉⣤⢀⠔⠘⠀⠘⠓⢦⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⠀⠀⠀⠀⢱⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠓⠄⡀⠀⠀⠀⢀⡄⡼⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠓⠄⠘⣶⡶⠶⠶⠶⢾⠇⢠⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠸⠀⣤⣀⠊⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠲⢄⡀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠴⠛⣫⠟⠣⢖⡀
⠀⠀⠀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠫⡀⠀⠚⠃⠶⠭⠬⡍⠉⢁⡱⠤⡀⠀⠀⠁⠈⠀⠀⢡
⠀⠈⠁⠀⠀⠉⠢⣄⠐⠀⠀⠀⠀⠱⠇⠀⠀⠀⠀⢀⡤⠐⠉⠁⠀⠀⠀⠈⠓⠢⢄⠹⠒⠛⠊⠐⠋⠀⠀⢸⠃⠀⠀⠀⠀⠀⢸
⠀⠄⠐⠀⠀⠀⠀⠀⡆⠀⡰⠒⠖⠒⠶⢧⡱⡖⣶⠀⠄⠀⢀⠘⠃⡄⠀⢤⠃⠄⠀⡳⠃⠀⠀⠠⡦⠄⡴⢈⡀⠀⡠⠐⠒⡄⢘
⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⠃⠀⡄⠀⠀⠀⠀⠐⡋⣴⠂⠀⠁⢌⠀⠀⠀⣤⠀⠀⡄⠀⠛⠙⠺⣥⠓⢦⠶⡮⢷⡬⢶⣦⣄⠛⠌⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡀⠬⡖⠀⠂⠡⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠹⠒⢶⠙⠚⣆⠀⢈⠇⢀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠭⣄⠁⠀⠀⠀⠀⠀⣠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠠⠀⠀⣸⠀⠀⠹⣦⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆⠀⢈⣁⠤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠅⢄⠉⠘⢄⡄
⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠃⠀⠀⡎⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⡓⠀⠀⠈⠐⠂⠄⢄⠠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢸⠀⡀⠣⡀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⢀⠔⠉⠓⠦⣄⡀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠈⠂⠤⡤⣉⣐⠂⠀⠠⠤⠤⠔⠚⠋⠁⣠⡤⠔⠁⠀⠀⠀⠀⠀⠉⠳⠀⠂⢰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣅⡼⠀⠀⠀⠀⠀⠩⠈⠉⠛⠛⠛⠙⠒⠋⠈⠭⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠰⠖⠓⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡠⠄⠐⠒⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⢠⡄⠀⠀⢀⡆⠀⠀⠈⠠⠀⠀⠀⠄⠀⠀⠀⠐⢤⠴⠃
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣇⠀⠀⠀⠈⠢⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣀⠜⠀⠈⠳⣄⠀⠀⢀⡗⠀⠀⠀⠀⠀⠠⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣄⣀
⣿⢻⡟⢿⢛⣟⣛⡛⢿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣛⣿⣿⣛⣻⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⠂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠝⠉⣁⣨⠄⠂⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠎⠁⣠⡄⠒⠊⠉⠉⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⣙⠉⡒⢤⣀⡀⠀⠀⢰⢋⣥⡐⡉⠀⠈⠁⠀⠀⠀⠐⢁⡀⠤⢶⣶⠶⢶⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⣾⢽⣿⣷⣿⣄⡤⠙⠈⠐⠀⣀⣤⣔⣉⠀⢡⠀⣠⠈⠁⠊⠉⠉⠡⠄⣍⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠹⠄⠈⢻⡟⠋⡙⢦⡉⢉⣡⠤⢿⠦⢼⣦⣬⣵⠂⣀⠄⠀⠐⠐⠂⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⠀⠀⣞⡅⡺⠜⠂⣿⠇⠀⣐⡂⠐⢀⡀⢽⣭⣿⢿⠦⣤⡺⠗⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⠀⢠⣇⠀⠃⠸⠦⣈⣧⠀⡙⣧⡈⠛⣀⠀⠶⠒⠦⣫⣙⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⢸⠉⠹⠀⠁⠠⡜⢿⣷⣽⡀⣀⣤⡄⠀⠁⠈⢀⢀⠉⣈⡋⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢼⡀⡀⠀⡀⠐⢘⡰⣽⢏⠛⣍⠀⠀⠀⠀⠀⡤⢆⠐⠠⠤⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠸⣿⠁⠐⡅⠂⠈⠣⣿⠀⣾⠉⠙⢶⣦⣹⣆⣀⣀⡄⠤⠾⠿⠤⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⠲⡄⠁⢀⡁⢈⡿⠀⣏⡄⣻⠃⠁⠘⠘⣧⣸⡗⠀⣼⠃⡉⣀⠀⢿⣽⡿⡶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠓⠶⠥⠚⠁⢸⡇⠀⠛⠈⠻⠂⢤⡻⢿⠳⡢⡈⠦⠈⢉⡀⠀⠖⠲⢬⣍⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢄⠙⠦⣀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠳⣦⠈⠁⠠⣬⠻⢿⣧⠙⢦⣀⡤⣶⠀⠈⢀⠀⡀⠈⣁⣙⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⢻⠃⡀⠂⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡀⠈⠐⠀⢃⡲⣽⡀⡤⠛⢤⡀⠀⢀⡀⠱⠖⠀⠀⠤⠆⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠈⠛⠀⠹⠃⠰⣮⣛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣢⡆⢀⣁⠐⡀⠘⢷⣿⣿⠄⢠⡕⠋⡳⢾⣿⣀⣤⠤⠰⠶⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠟⠶⡶⠦⣟⠴⣤⣮⡙⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣁⠀⠸⠀⡄⠣⠌⣿⢿⡘⠰⡄⢲⠄⢀⡻⢧
⠀⠀⠀⠀⢀⠔⠻⠿⠁⠀⠠⠾⠁⢠⠮⣡⡅⠈⢳⡌⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣄⣈⡓⢠⠏⠘⡧⢤⣀⠒⠀⢤⡙⠿⣧
⠀⠀⠀⠠⠋⢀⣰⠆⠬⠀⡀⠀⠂⣤⠆⠈⠞⣠⠎⡙⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠃⠀⠁⠀⠈⠀⠈⣦⢻⣧
⠀⠀⢰⡄⡀⢀⡀⠐⠀⠐⡰⢆⣤⡂⣀⣠⠾⠁⢌⠳⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⢈⡐⠀⢈⠙⣿⡄
⠀⠀⠸⣀⣀⡊⣥⣴⣛⣶⣤⣿⡵⠟⠛⠓⠤⣤⣨⣋⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣀⣀⠑⠀⣉⠂⣻⠃
⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⠾⠤⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡏⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⢸⠀⠀⡿⢰⠀⠆⡿⢠⠀⡆⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠁⠃⠸⠀⠇⠸⠀⠀⠀⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⣩⣽⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⣿⣆⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⣠⣾⠟⠋⠁⠘⢯⡻⣆⠀⠀⠀⢀⡀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⣀⣿⣿⠿⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⣼⠟⠁⠀⠀⠀⠀⠈⢷⡹⡄⠀⠐⣿⡇⠀⠀⣠⣾⠋⠀⠀⢀⣠⡾⠟⠉⠹⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠈⣧⢱⡀⠀⠀⠀⢀⣴⠟⠁⢀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢈⣥⣶⣿⣿⣿⣶⣤⡀⠀⠀⠘⣧⣷⢀⣠⣶⣿⡿⢿⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣆⢠⡿⠋⠉⠀⠀⠀⠈⠙⠻⣷⣄⠀⠈⢻⡟⣫⣥⣴⣶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠃
⠀⠀⠀⢀⣠⣤⣴⣶⣾⡿⠿⠛⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⣟⠀⠛⡛⢉⣩⣿⠀⠀⣀⣾⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⠿⢷⣦⣄⣀
⠀⠀⠀⢸⡟⠛⠛⠿⠿⠿⣿⣶⣶⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣶⣶⣿⣷⣿⣿⣿⠟⣿⠛⠁⠀⠸⠿⠿⠟⠛⠛⢿⣿⣯⣉⣭⣭⣥⣶⣶⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢉⣽⡿⠛⢉⣼⠟⣩⡿⠋⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠛⠛⠛⠛⠛⠉⠉⠉⠉⢹⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⣠⡿⠃⣰⠏⠀⠀⠀⠀⠀⢰⡌⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⣰⠟⠀⢰⠏⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠘⢿⣿⣿⣶⣶⣶⣦⣤⣤⣤⣄⣠⡿⠁⠀⠀⢠⡏⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣠⣤⣤⣴⣶⣶⣾
⠀⠀⠀⣠⡀⠀⢸⡏⠉⠙⠛⠛⠛⠻⣿⠏⠀⠀⡀⠀⠘⠀⠀⣾⣀⣀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠛⠛⠛⢻⡛⠋
⠀⠀⠀⢸⡇⠀⢸⠇⠀⠀⠀⠀⠀⣸⠋⣀⣴⣿⡇⠀⠀⠀⣸⠋⠛⠿⢿⣿⣿⣷⣶⣦⣼⣇⣤⣶⣾⣿⡿⣇⠀⠀⠀⢠⣀⠀⠀⠀⣸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣯⣾⠟⠉⠀⡀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⡿⠛⠛⠉⠀⠀⢸⡀⠀⢠⡟⠻⣆⠀⢀⡏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⢹⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⢠⡟⠀⠀⠹⣄⡾⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢡⡼⠁⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⠿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢿⣷⣶⣶⣤⣤⣀⣸⣇⢀⣀⣠⣴⣶⣿⣿⣿⠿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⢿⠿⠿⠛⠛⠉⠉