Chef Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣧⣶⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⠁⠀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡾⠁⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠖⢶⡀⠰⢖⡓⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠉⠉⠀⠀⠈⠁⠁⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣀⠙⠃⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢄⠀⠈⠙⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⣾⠐⣤⣀⣀⡠⠀⢸⡧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠤⠞⢿⣟⠢⡐⠉⠉⠀⠀⠈⠀⠀⠀⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠈⠛⠂⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠒⠉⠻⣇⢰⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⣖⣈⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣀⣻⡿⢿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠈⢿⣂⠀⠀⢸⣿⠀⠀⠀⠀⠀⣠⠟⠋⠁⠑⢆⢻⣷⡂
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⢻⡆⠀⠀⠀⠘⠁⠀⠀⠀⠀⢈⣠⡏⠉⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⡄⢀⣀⣀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠛⠉⠉⠉⠛⠶⣄⠄⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡿⣱⠞⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⣼⡏⠀⠀⠀⠀⠀⠀⢀⡠⠰⠿⣻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⠄⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠈⡐⠉⠁⠀⠀⠀⠀⠀⠱⡔
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢀⠄⠒⠀⠚⡆⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠤⠄⠀⠀⢀⡄⢠⠤⠄⠒⠀⠘⠛⠨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠐⠀⠀⡰⢋⡄⠃⠀⡐⠀⠀⢠⢀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⡀⠐⠈⠔⢣⡘⡀⠀⠸⠇⠀⠀⠘⢞⠃⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠀⠀⡇⠉⠀⢀⠀⠀⠀⠀⠀⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⠀⠀⠈⠲⣆⠀⠐⠂⠃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠄⡀⠀⠐⠡⠅⠠⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠑⠤⡀⠀⣠⢔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠈⠀⠀⠀⢠⣄⠀⠐⠤⢌⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⢀⠀⠀⡟⠀⡴⣇⣀⠘⡄⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⣀⠤⠒⠂⠉⠉⠀⠀⠉⠉⠙⠒⠠⢄⡃⠀⠀⠀⠃⠀⠈⠑⠢⡀⠀⠀⠀⠘⠀⠀⡐⠊⠀⡠⠚⡀
⠀⠀⠀⠀⠀⡠⠐⠂⢒⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⡾⠀⠀⠀⠐⠉⢀⣄⠠⠀⠀⠀⢐⠒⠈⣀⠤⢑
⠀⠀⠀⢀⠌⡀⠈⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡇⠀⠌⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⡊⣉⡀⠄⠊⠄
⠀⠀⠀⢀⠈⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣤⠀⠄⠊
⠀⠀⢀⡼⠞⢉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠚⢆⠀⠀⠀⠀⠁
⠀⠀⢎⠀⠀⢪⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠤⠐⠂⠀⡀⠔⣸⠃⠀⢠⠃
⠀⠀⠈⠳⢤⣀⠑⠢⠤⣀⡀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⣀⣐⣠⠥⠖⠋⣁⠤⠚⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠦⠭⢭⣽⣿⣟⣛⣛⣛⣛⣻⣿⡭⠭⠭⠐⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠒⠚⠛⠛⠛⠒⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠉⠉⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠠⠤⠒⠊⠉⠀⠀⠉⠉⠓⠒⠒⠒⠒⠘⢥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠉⠁⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠤⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⠀⠤⠤⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠄⠒⠒⠶⠦⢤⣄⣀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣔⣂⡀⠀⠀⠀⢀⠤⠈⠉⠑⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡡⠀⠀⡞⠁⠙⡀⠀⠀⠐⢋⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⣍⠤⡃⠀⠀⢣⡔⡄⠀⠀⠀⠀⠋⠂⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢀⢀⡇⠀⢀⠞⠁⠀⠀⠀⠉⠀⠀⠀⠀⠠⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⡇⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢣⠀⠀⣀⣀⠀⣀⡠⠀⠈⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠲⡀⢠⠓⠤⣀⠀⠣⡀⠣⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⣧⠱⡼⣄⠀⠈⠳⡄⠙⠦⡀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠤⠒⠓⠢⡈⢳⡀⢀⠋⠀⢀⡜⡖⠀⣀⡴⠊⢣⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⢀⣠⡤⠚⢦⡴⢀⡌⠀⠀⠈⢧⠘⢾⡉⢀⡠⠃⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⡠⠴⢮⠠⡜⠀⠀⠀⠀⠀⠉⠀⠉⢟⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠤⠤⢤⠴⠚⣄⢣⠀⠀⠀⠀⠀⢀⣀⣠⠦⢖⠴⠂⣸⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠊⠀⣸⠋⡆⢣⣀⡤⠖⠚⠉⢁⡠⠖⠉⠀⠀⢻⣇⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡴⠞⡄⠁⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⢸⡟⢦⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢁⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⡴⠒⣬⣇⡼⠋⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠎⠐⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠋⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠇⠀⠈⠳⡄⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠚⠋⠉⠀⠀⠀⠀⣠⣧⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡈⠙⠒⠶⠦⠤⠴⠿⠶⢶⣶⠶⢒⣀⣠⠄⣠⠞⠉⠉⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⣀⠀⠀⠀⠀⠀⠉⠉⣉⣉⡳⠿⠓⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⣷⣶⣶⣾⣶⡿⢿⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣾⣿⣿⣿⣾⣷⣿⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣿⣷⡾⢿⠿⠿⠿⠾⣷⡶⠶⠶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠒⠂⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢀⡾⢺⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢈⠒⠀⠀⠀⠀⠀⠀⠀⠙⢄⢠⣟⡾⣿⡀⠀⠀⠀⠀⠀⣠⡾⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑
⠀⠀⠀⠀⠀⣀⠄⠒⠈⠁⠀⠀⠀⠈⠁⠀⠠⢀⠀⠀⠀⠀⠈⢺⣯⠈⠌⠛⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⢿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⢂
⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸⠄⠀⡀⠀⠀⠀⠀⠐⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⡤
⠀⠛⠤⣄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠒⠉⠱⠀⠀⠀⠀⠉⠀⠀⣤⢺⠈⠲⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠐⠀⠀⠀⠰⠐⠋
⠀⠀⠀⠀⠈⠈⠙⢛⠲⣴⠢⣄⡤⢀⡀⠀⣀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠐⠀⠐⠘⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠄⢀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⠓⠈⡍⢀⠐⠟⡙⠛⠛⠈⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⢬⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠓⠀⢿⠗⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⢳⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠈⠄⠀⠀⢀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠁⠀⠒⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠁⠀⠀⠀⣓⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠄⠀⠀⠐⠉⣇⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢹⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⡐⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠊
⣿⣿⡟⠻⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⠠⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠉⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠻⢶⣤⠙⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣶⣬⣿⠟⠦⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⣼⠋⣻⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣤⠁⠀⠀⠀⠀⠀⠀⠀⢿⣆⠆⣠⡅⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠐⣋⣽⡇⠀⢾⣭⡉⢸⡿⢤⠖⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡷⠀⠀⠀⠉⠀⠀⠀⠀⠸⠿⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢀⠀⠓⠒⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠙⢓⠂⠛⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⡀⠀⠀⢀⣴⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⢿⣿⣿⣿⠋⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣍⡋⠁⠀⣠⠎⠀⠉⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⣌⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣤⣤⠴⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⣢⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⡰⡁⠀⠀⠀⡀⠐⠀⠀⢸⡀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⠁⢀⡸⠇⢀⠀⠀⠀⠸⡆
⠀⠀⡼⠁⠈⠀⠀⡌⠀⠀⠀⡀⣼⠁⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣤⣾⣿⣿⡟⢻⡗⣦⠙⢷⣌⠀⡀⠀⠀⣿
⠀⢠⡇⠤⠒⠘⣼⡃⠀⠀⠀⢥⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠋⣾⣿⣿⣿⣧⣸⣧⠀⠂⠀⠻⣷⡘⣆⠀⢸⡀
⠀⢸⡇⠀⠀⠀⠹⣁⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠃⠠⣿⣿⣿⣿⣿⣷⠟⠂⠀⠀⠀⠻⣖⠘⡆⢸⡇
⠀⣜⠁⠀⠀⠀⢀⣻⣴⣶⡞⠁⣇⠀⠀⠀⠀⠀⠀⢸⡇⢰⣴⣿⣿⣿⣿⣿⣿⣶⡶⠀⠀⠀⠀⠀⢹⡇⣷⢸⡇
⠀⠘⣦⡀⠀⠀⠸⣴⣿⡟⠀⠀⠏⠀⠀⠀⠀⠀⠀⠘⠣⣾⣿⣿⣿⣿⣿⣿⣿⡉⢻⠀⠀⠀⢔⣀⢴⠂⠀⠸
⠀⠀⠈⠻⣄⣀⣀⣻⠟⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠳⠦⠶⣄⡐⠀⠒⠀⢁⡾⠀⠀⠁
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠷⠲⠾⣽⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⢸⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⠜⠁⠁⠀⠀⠀⠀⠆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⣃⠠
⠀⠀⠀⢀⠀⠀⠀⠌⠀⢀⠔⠱⠋⠅⠀⠀⠀⠈⡁
⠀⠀⠀⠈⠢⢀⠀⠄⠠⢀⠘⠀⠰⠇⠀⠀⠰⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠈⠧⠠⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠤⡀⠁⠁⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠔⠀⢢⠑⠀⠀⢀⡀⠁⢈⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡔⠀⠸⠤⡄⠄⠀⠀⠈⢄⠀⠀⢀⣤⠄⠀⠐⠀⠢⣞⠆⠀⠀⡠⡄
⠀⡀⠀⠀⠀⠀⡀⠀⠀⣆⠄⢨⣣⠀⠈⡄⠀⠀⠀⢀⠈⠀⠨⣄⠈⡁⠀⠄⠐⠀⢀⡀⡄⠀⠀⠈⠡⣄⠀⡆⠈
⡌⡀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⢸⠂⢀⠃⠀⠐⠀⠾⣄⠀⠛⢁⠓⠀⠂⠀⠀⢀⡀⣄⠀⠂⡄⣊⠀⠐⠋⢁⣠
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠊⠁⠀⠀⠀⠀⠀⠾⡙⡶⡤⢤⣈⣉⣉⣉⡀⠉⣉⣉⣉⣀⣠⣤⣴⠾⠛⠉
⠘⠤⠀⠠⠀⣄⠀⠀⠀⠀⢀⢠⠉⠉⠉⠉⠉⠉⠈⠉⠀⠈⠉⠉⠉⠉⠉⠋⠙⠛⠻⠛⠟⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣉⣢⣤⣤⣔⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠙⠶⠦⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⢀⣰⡩⠂⠀⠀⠀⠀⢠⣞⣠⣴⡂⠀⠀⢸⣄⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⢀⡴⠊⠉⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠙⢧⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⣇⡀⠀⠀⠐⠒⣞⠉⠁⠀⠀⠀⠀⢀⣠⠤⠐⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⢹⠁⠀⠀⠀⠊⢈⢦⡀⠀⠀⣠⠖⢋⠀⠀⠀⣂⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠸⣄⠀⠀⠀⠀⠁⣠⣷⣄⣼⣅⠀⠀⠶⠀⠀⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠈⣷⣶⣦⣴⣶⣿⣿⣿⠟⠿⠂⠀⠀⠀⠄⠚⠀⣤⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣌⠓⠀⠀⠸⢆⡀⠨⠄⠺⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣟⣛⣛⣟⣿⣛⣿⣿⣛⣦⣤⣤⡀
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐⠀⣾⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠾⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠟
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡄⠀⠙⠲⠶⠶⠚⣹⣤⠼⠷⠛⢻⠈⠉⠙⠛⠿⣿⣿⣿⣶⣶⣶⡀⠐⠶⠶⢶⡶⠶⠒⠒⢒⡶⠒⠊
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠐⢧⣀⡀⠀⠀⣠⠶⢿⣁⠤⠀⣠⠟⣄⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠠⣴⣾⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈⠉⠉⢹⠉⠶⣄⣹⠆⠉⣀⠴⠊⠀⠀⠀⢀⠀⠀⠀⠙⢿⣿⣿⡏⠛⠷⣦⣤⣤⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠈⠑⠒⠲⢶⠶⣶⠺⡘⢆⢸⢹⠀⠀⠈⠢⡀⠀⠀⠀⣈⡆⠀⠀⢱⡀⠙⣿⠁⠀⠀⠀⠀⠀⠀⣸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢰⠙⣆⠰⡑⢷⡀⡸⣺⠀⠀⠀⢀⡜⠀⠀⠀⠘⢷⡆⠀⠀⠇⢠⡏⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⢻⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡄⠀⡜⠠⡿⢤⣀⠈⠛⣥⠇⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠸⣿⣿⣿⣿⡿⢁⡴⠀⠀⠀⠈⡿⠚⠧⠰⠧⡀⠈⠉⠒⣾⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃
⠀⢿⣿⣿⡿⣡⠏⢀⣤⠤⢶⡞⠉⠀⠀⠦⢀⢹⠀⠀⠀⣿⠊⣀⡤⠤⠀⠀⠀⣤⣄⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡏
⠀⠈⢿⣿⣷⠃⠀⠁⠀⣠⠾⠀⠁⠀⠠⢀⢈⡇⠀⠀⠀⡸⢿⡁⠰⡶⠀⠀⠀⠈⠡⠀⠀⠀⠀⢹⣄⡀⠀⠀⠀⢀⣴⣿⣿⣿⡿
⠀⠀⠈⢿⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠢⢤⠟⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠈⣷⠀⠀⠀⠘⡇⠀⠢⣼⣷⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⣴⡄⠀⠀⠀⠉⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠱⣄⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠳⠦⣤⣤⣬⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣶⡆⠀⠀⠈⠋⠀⠀⠀⣼⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣤⣶⣿⠿⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣻⣿⣷⣶⣶⣶⣾⣶⣶⣿⣛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠀⡰⠚⠉⠉⠩⢭⣉⡙⢦⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠉⠁⠈⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢰⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠑⠆⠀⠀⠰⠚⠉⠉⠁⠀⠀⠀⠀⠀⡎⠀⠠⠴⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠄⢰⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣇⠀⠐⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠐⠚⠉⠉⠁⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡤⠶⠒⠛⠛⠉⠙⠛⠛⢻⡇⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠁⢀⡤⠤⠀⠀⠠⠖⠲⠄⠘⡇⠀⠀⠀⠘⢦⠀⠀⠀⢰⠑⢮⠱⢄⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡀⠀⠀⠈⠁⣠⣤⣤⣷⡌⡦⢀⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⡄⢹⡀⣠⣴⣾⣿⣶⣿⣿⣿⣶⣾⣧⣴⡗⠀⠀⠀⣴⣋⣀⣄⠈⢡⠃⢰⠟⠀⣱
⠀⠀⠀⠀⠀⢰⠀⠀⠀⣾⠀⠀⠘⢷⣾⣿⣿⡿⠟⢉⣉⣉⠉⠙⠛⣿⠛⠋⠀⠀⠀⢰⠛⡇⠀⢸⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⢸⡄⠀⠀⢿⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠉⠁⠀⣀⡼⠃⠠⡄⠀⠀⠀⢸⡀⠙⠤⠎⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠸⡆⠀⠀⠀⠀⡎⠀⠈⠙⠓⠶⠚⠛⠉⣀⠀⢀⡇⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢳⡀⠀⠀⠀⣹⠖⢋⣩⣤⣷⠀⢸⣧⠤⠈⠻⣄⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠱⡄⠀⢰⠇⠀⠈⣈⠅⠛⠒⠋⠑⠢⢄⣀⣈⡷⢰⠚⢲⣀⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠈⠢⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡶⠃⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣤⣄⣀⣀⣀⣀⣀⣠⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣻⣉⣟⣻⣋⣿⣫⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣤⠤⠴⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠤⡀⠀⠒⠒⠒⠒⠢⢄⡀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⢹⡆⠠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠭⠍⠉⢩⠭⠑⢦⣸⡧⠯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠀⠀⠘⠃⠀⠸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣟⣀⡀⠀⠀⠀⠀⠀⡇⢜⣁⣀⠒⠄⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠉⢗⢦⡀⠀⠀⢣⠈⠻⠛⠋⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡝⡄⠀⢠⡗⠢⠄⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⡔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢗⡦⣟⠣⣄⣀⠤⠖⠋⢡⠷⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠙⠦⢭⣙⣒⣒⡒⠒⣲⣶⣶⣾⣿⡿⠟⠃⠘⢦⡈⡇⢀⣠⠔⠁⠀⠀⠀⠈⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⡏⢿⡉⣹⢹⠃⠀⠀⢰⠃⣬⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⠁⠘⢦⣄⠀⡞⢀⠁⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣦⣄⠤⠞⠉⠳⣅⠘⠃⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠶⠀⠒⠈⠉⠐⠲⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡊⢠⠄⠠⣄⠀⢰⡄⡄⢠⠂⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠣⣜⢀⢀⡙⠧⠀⢁⣷⣃⠀⠀⠴⣾⡧⡠⠶⠆⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡇⢀⠜⠋⠀⠀⠀⠀⠀⠀⠁⠘⠃⠾⠋⣨⣔⣥⣤⡄⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⠯⢷⣶⣦⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣬⣤⣻⠶⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡥⠴⠒⢛⣛⠩⠉⠉⠉⠛⠒⢦⣄⠀⠀⠀⠀⣼⣀⣒⣋⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⠛⣡⠶⠶⢦⠀⠀⠀⠀⣿⣿⣦⣀⣰⣏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⣽⠛⣛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠶⠄⠀⠀⡀⠠⢒⠤⠀⠐⠂⠀⠀⠀⠀⣷⢈⡇⠹⢻⡄
⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣇⠀⠒⡧⠀⠓⠐⠁⠂⠀⠈⠁⠀⠀⠀⠀⠸⡌⠰⣡⡏
⢀⡀⠀⡎⠠⠄⠄⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠸⠞⢻⠃⠀⠀⢀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢹⡶⣿
⢀⠹⡀⡇⠀⣠⠀⠐⣋⡔⠒⠶⢄⡀⠀⠀⠀⡜⠀⢸⣄⡰⣶⣬⣜⠩⠉⠀⡇⠀⠀⠀⠀⠀⣸⠁⣯⡀
⠈⠄⠡⣡⡀⠙⣆⡀⠛⠀⣰⠂⠀⣉⠀⠀⠀⢧⠠⡇⢾⠡⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⢃⠔⣵⠟⠉⢹⡆
⠀⠐⣄⠘⠓⠂⠀⠀⠁⠀⠘⡆⠀⣯⠀⢧⠀⠘⣆⠉⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠑⢱⠞⣩⡤⠀⠈⣿⠲⠖
⠀⠀⠀⠢⡀⠙⠄⡀⠉⠀⠀⠁⠀⣼⠀⡼⠀⠀⠈⢢⠸⡇⠀⠀⠀⡠⠈⠀⠀⣠⠎⢂⠄⠈⠀⠀⠀⠀⠰⠾⠿
⠀⠀⠀⠀⢧⠀⠀⠈⢦⠀⠀⠀⠀⠘⢦⠃⠀⠀⠀⠀⠑⣌⠐⠒⠈⠀⢀⣠⡞⡡⠂⠄⢒⡁
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢠⡀⠀⣠⡧⢄⡀⠀⠀⠀⠀⢘⣶⣤⣴⠚⣉⢂⣪⠔⠒⠚⠉
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡠⠚⠓⠒⠓⢌⠒⠃⠉⢩⡶⢚⠿⡠⣞⠻⠛⠄
⠀⠀⠀⡔⢿⠁⠀⠀⢀⠀⡌⠀⠀⠀⠀⠀⠀⠑⠦⠤⠗⠋⠁⠀⠀⠈
⠀⠀⢰⠀⠈⠀⠀⠈⠀⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠁⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠍⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⣏⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠄⠠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢸⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠉⠉⠀⠀⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⣄⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢤⣂⡤⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢤⠎⠭⢥⣴⠞⣣⠈⠢⡀⠀⠀⠀⠀⠀⢸⠋⠉⠀⠀⣀⡴⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡔⡊⠀⠧⠀⠈⢂⠀⠀⠀⠀⠈⠑⣶⠀⠈⠉⠀⠀⠀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠴⠿⠻⣷⡆⠀⢧⠀⠀⠀⠀⢠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠳⣀⠀⠐⢲⡍⠻⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠁⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠈⢣⡔⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠙⢶⣄⢀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠻⣧⠁⠀⠀⠀⠀⠀⠀⢠⡖⡾
⠀⠀⠀⠀⠀⠀⢀⣀⣄⣤⠐⢶⣶⠶⣎⠈⠛⠻⣿⣿⣿⣿⠿⠒⠛⣛⠃⠈⠀⠀⠀⢠⣿⣄⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⢀⣴⡟⣸⠃
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣂⣸⡿⠇⠿⠧⠀⠘⢛⣩⣥⣦⡀⠸⣿⡹⣶⡗⠀⠄⠀⣼⢸⣿⣦⡀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢀⣴⡿⠏⢠⡏
⠀⠀⠀⠀⠀⠀⢰⠘⣿⣿⣿⣿⣿⣿⣟⠁⠐⡆⢀⣿⡇⠘⠁⠀⠈⠻⣧⣞⣶⠀⠀⡿⠀⠉⣿⣿⣄⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠳⣌⠿⣿⣿⣿⣿⣿⡒⣠⠀⠸⠟⠃⠀⠀⠀⠀⠀⢸⣿⡇⣠⣴⡇⠀⠀⠸⣿⣿⣷⣆⣀⣠⣶⣿⣿⡿⠏⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣴⣿⣿⠁⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣛⠿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠆⠀⠐⠂⠀⠀⠁⠀⠄⠸⠂⠀⠀⠀⠶⠰⠀⠀⠀⠀⠠⠰⠀⠄⠈⠐⠀⠀⠀⠐⠢⠒⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠈⠙⠳⢶⡖⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡄⠀⠰⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⢀⡀⠀⢻⣷⣄⡀⠘⣷⣤⣀⠀⠀⢀⣀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⣯⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠘⢉⣛⡋⠀⠉⣛⣋⡛⠈⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢺⡇⠀⠙⠛⣷⠀⢸⡟⠛⠉⠈⢿⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡗⠀⠀⠀⡼⠀⠸⡄⠀⠀⠰⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢶⡶⣿⠶⢶⡷⡶⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢦⡀⠀⠀⣀⡐⠀⠀⣠⠞⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡉⠁⠀⠈⢲⣄⣀⣀⣤⠊⠁⠀⠈⣳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠉⠀⠉⠳⠤⣄⣭⡃⠀⠀⣸⣅⡤⠴⠋⠁⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⡞⠁⠀⠀⠀⠀⠀⠀⠀⡔⠁⠘⢁⡼⣄⠙⠀⠙⡶⢤⡀⠀⠀⠀⠀⠈⢹⡶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢹⡀⠀⠀⠀⠀⠀⢀⣜⣀⡤⠖⠋⠀⠈⠓⠤⣄⣘⣄⢈⡗⠀⠀⠀⠀⡼⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠲⡀⠀⠀⠀⠀⣀⣀⡞⠀⠀⠀⠀⢠⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣇⣀⠀⠀⢀⠉⡞⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠊⠀⠀⠈⠉⠳⣄⠋⡼⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⣐⣼⠿⠖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠸⠀⣿⣸⠃⠀⠀⣀⣠⡞⠉⠁⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⣀⠀⣠⣀⣠⡄⠀⢠⠿⠛⠲⠶⣿⣉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠎⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠉⠉⠓⠚⠋⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⢢⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢻⡀⠀⠀⠀⢀⡀⣀⡤⠞⠓⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⣹⣦⣤⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣄⣀⣀⣀⣈⣷⡶⠦⠤⠶⠿⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⠀⠁⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⢀⡠⡤⠈⠛⣦⡄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣒⣂⠀⠋⠔⠒⠂⠀⠸⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⠴⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠉⠀⠀⠀⠀⠀⡔⠁⠳⠀⢀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠳⢄⠀⣀⠀⡞⠀⠀⠀⠈⠰⠀⢡⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁⠈⠄⠀⠁⠀⠀⠀⠀⠀⠀⠐⢀⠀⠈⠀⠁⠆⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠃⠀⠀⠀⠀⠀⠲⠰⠆⠀⠂⠔⢆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠣⠀⢀⡄⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢰⣿⠇⠀⠀⠀⠰⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠠⠉⠀⠀⠀⠀⠀⠀⠈⠒
⠀⡠⠐⠀⣀⣤⡼⠅⡈⠁⠀⠀⠈⠀⠉⠀⠈⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢇⠀⠀⠉⠋⠀⠨⢓⡶⠀⣠⣄⠀⠀⠀⠀⠀⠀⢈⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠁⠢⠀⠘⢃⠁⠚⠡⣶⣿⡿⣀⠐⠀⠀⠀⠀⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠈⠣⠰⢦⡀⠶⠤⠤⠄⠤⠤⠐⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⢤⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⣀⡀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠈⠉⠳⠀⠈⠉⠉⠁⠈⣇⠀⠀⠀⠀⢠⡇⢸⠇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠀⢀⡏⠀⠀⠀⠀⢾⠁⠘⠀⣟
⠀⠀⠀⠀⠀⣿⠉⠁⠀⠀⡠⠀⣠⠤⠖⠒⠉⠀⠀⣿⣠⠞⠀⠀⠀⠀⠀⢀⣠⣼⡧⢤⣄⡀
⠀⠀⠀⠀⠀⠘⢦⣀⡀⢸⣇⠈⠁⠀⠀⠀⢀⣠⠀⢻⠁⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣇⠀⠀⠀⣀⣀⣤⣤⣼⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡴⠞⠋⠉⠀⠀⣀⡀⠀⠀⠀⠙⠶⢤⣤⣤⣤⣄⣀⣤⣤⣬⣭⡵⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠲⠆⠀⠀⠉⠀⢰⠀⢀⡄⠀⠀⠀⠙⣧⠀⢀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⣴⣶⣶⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⡟⠲⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⠿⠃⠈⠉⠏⠁⠀⠀⠀⠀⠀⡼⠁⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⠤⠎⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠓⠒⠀⠀⣀⠔⠀⠀⠀⢠⠷⣦⣄⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⣄⠀⠀⠀⠰⡎⠁⣀⠀⠀⠀⡇⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠘⣆⠀⠀⠀⠹⡄⠉⠀⠀⠀⠁⠀⡏⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⣿⠀⠀⠀⠀⢱⡐⠗⠀⠀⢀⡤⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⢈⠟⠀⠠⢜⡧⠗⣨⠴⠚⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠎⢀⠀⠀⢈⡲⢻⠠⡤⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢗⡠⠜⠁⠸⢀⡀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⠈⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠
⠀⠀⠀⢀⣤⣤⠖⠠⠀⠀⠀⠐⠁⠀⡠⠀⠀⠀⢀⣀⢤⠀⡀
⠀⠀⢠⣿⣿⡿⠦⠀⣤⡖⠞⠁⠀⠒⠀⠀⠀⠀⠈⠁⠀⠀⡀
⠀⠀⢈⣿⣿⡂⠆⠘⠁⠈⠃⢀⠀⠀⠀⠀⠀⠰⠚⠁⠀⠂
⠀⢀⣼⣿⡿⢁⠀⠀⠀⠀⠀⠸⡆⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢀⣼⣿⡟⠀⠘⠀⢀⡐⠂⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡄⠐⠀⢀⠠⠐⠒⠉⠉⣀⢠⡆
⢰⣿⡏⠀⠀⠀⠀⠀⠑⠤⠀⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣻⠃
⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠠⠄⠀⠉⠀⠂⠀⠠⡀⠀⠀⠀⣸⣿⣿
⠀⠙⢺⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢀⡄⠒⠆⠁⠀⠀⠀⢉⢽⣿
⠀⠀⠈⠀⢶⠦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠘⠒⠒⠒⠒⠲⠶⠶⢶⠦
⠀⠀⠀⠀⣁⣤⡿⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⣼⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠉⠁⠀⠀⠀⡀
⠀⠀⠀⠸⡻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠐⠀⠸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⢞⢉⣎⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠈⠁⠀⠘⠉⠀⠀⠲⠄⠄⠀⠀⠀⠀⠀⠐⠄⠴⢧
⠀⠀⠀⠀⠀⠁⣧⡠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⢖⠀⡄⠀⢀⠀⠀⢀⠀⢠⢀⣦⡀⡁⠀⠀⡔⠁⠀⠀⠀⠈⢑⠄⠠
⠀⠀⠀⠀⠀⠀⡧⢌⡉⠁⠀⠒⠒⣊⣉⣹⠤⡀⠀⠀⠀⠀⢢⣀⠀⠀⠂⠈⠀⠀⠘⠐⢀⣀⣤⣞⠀⠀⠀⢸⡶⣐⢢⣤⣒⠋⠄⠂⠄⠀⢤⡀
⠀⠀⠀⢐⠀⣾⡠⣤⣤⣤⣤⣤⣭⡭⣅⣋⡰⠈⡇⠀⠀⢀⢰⠓⠬⢁⠀⠀⠐⠒⣒⣉⡭⠴⢛⠆⡂⠀⠀⠀⢧⣵⣸⣸⣸⣤⠤⢄⣤⢴⣼⠁
⠀⠀⠀⠀⠑⠦⠭⣖⣀⡀⠀⠀⠲⣒⣋⠩⠀⠈⠀⠀⠀⠈⠓⠭⣄⣀⠀⠀⠀⠀⠀⣀⣀⣠⠥⠚⠁⠀⠀⠀⠀⠀⠉⠁⠀⣻⣹⢹⣹⣫⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠔⢡⡀⢀⠠⠀⠒⠒⠀⠀⠒⠒⠬⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠰⠄⡐⠅⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⡠⠖⠩⠙⢥⠒⣀⣤⣀⢀
⠀⠀⠀⠐⠎⠑⠚⠶⠤⠤⠤⠤⠤⠤⠀⠘⠒⠙⡆⠀⠀⠀⠀⠀⠀⠰⠁⡀⠀⠀⠐⠐⠈⠄⡸⠁⠀⠀⠀⠀⢠⠔⠋⠀⠀⢨⠁⣀⠰⠄⡤⠀⢀
⠀⠀⢠⠀⠶⣤⣤⣀⣀⣀⠀⠀⢀⣈⣀⣠⣤⡾⠀⠀⠀⠀⢠⣔⡊⠀⠡⠀⠀⠀⠡⠀⢐⡔⠁⠀⠀⠀⡠⠚⡀⠀⠐⡀⠉⠁⢎⡀⣀⠆⠀⡠⠁
⠀⠀⠸⠀⠀⠈⣉⠉⠛⠛⠛⠛⠋⠙⠛⢉⡼⠁⠀⠀⠀⠀⠈⢋⠻⢔⡂⠁⠀⠐⣀⠜⠉⠀⠀⠀⠈⠣⢕⡀⠁⠀⠂⠡⠊⠁⠐⡄⠀⠀⠄
⠀⠀⠀⠑⠲⠖⠁⢀⡂⠀⠤⠤⠴⠶⢚⠁⠀⠀⠐⠃⠀⠀⠀⠀⠑⠌⣳⣴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢄⡀⠑⠒⠊⡠⢀⠂
⠀⠀⠀⠀⠀⠀⣟⠉⢁⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠬⠭⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠀⢀⠵⠀⠀⠀⠀⠀⠉⠀⠱⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⢀⣁⠀⠤⠀⠀⠐⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⠎⡥⠤⠀⠴⠖⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⡐⠐⠀⠃⢀⣠⡶⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⣧⠀⢀⣼⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡅⣀⣴⠶⠛⠁⠐⠣⡘⢆⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠈⠀⢹⣷⣿⠏⠀⡇
⠀⠀⠀⠀⠀⢐⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠁⠀⠀⠀⠀⠀⠘⡮⣧⢦⣄⠀⠀⠀⠀⠀⠀⠼⠀⠀⠰⠦⣬⣉⣠⢠⡇
⠀⠀⠀⢴⢈⠜⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⡳⡌⠇⠀⠀⠀⠀⠘⢀⢠⡄⠰⠘⠛⣙⣉⡼
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⣄⣰⣿⣀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠞⠋⠉⠀⠀⠉⠙⠳⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠃⠀⠀⠀⣿⣿⠀⠀⠀⠘⣧⣴⠄⠀⠀⠀⢀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡇⠀⠀⢀⣤⣼⣧⣤⡀⠀⠀⢸⣇⡀⢀⣤⣀⣿⠷⠶⠿⠶⣾⣟⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣸⠋⣡⣀⡽⠋⠀⠀⣠⣤⡀⠀⠈⠻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢹⣆⠀⠸⠿⠿⠿⠿⠇⠀⣰⠟⠿⠉⡿⠁⠀⠀⠀⠻⡿⠃⠀⠀⠀⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⢻⣦⣤⣤⣠⣤⣴⠛⠿⠆⠘⠻⡇⠀⠀⢠⣾⣿⣿⣿⡆⠀⠀⢈⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠙⢳⣦⠿⣣⡈⠛⠀⠀⠀⠀⣶⣿⡀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡴⠟⠛⠛⠛⠛⠿⣇⣠⣄⠀⠀⠀⠀⣠⣿⣤⡈⠋⠛⠛⠛⠃⣠⣾⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣴⠋⠀⠀⢠⣶⣦⠀⠀⠈⢻⣅⣀⠀⠀⠀⠈⠀⢸⡿⠛⣶⡶⠺⣿⡁⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⣈⣛⣋⡀⠀⠀⠀⢻⡋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⣸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢷⡀⠀⢸⣿⣿⣿⣿⡇⠀⣠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢦⣈⡈⠉⠉⣈⣠⠾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠙⣿⠋⠹⠷⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠿⠿⠿⠷⠖⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠰⠶⠂⠀⣤⢦⠀⠀⠐⠒⠀⠀⢀⡤⢄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡖⠆⠀⠀⠀⠀⠀⠀⣠⢦⠀⠀⠐⠂⠀⢠⠲
⠀⢠⣦⠀⠀⠁⠈⠀⠀⡀⠀⠀⠀⠈⠁⠘⠀⠀⠈⠀⠀⠀⣠⠄⠀⣄⣤⠀⠘⠁⠀⠀⢹⠃⠀⠀⠠⠀⠀⠀⠻⡤⠀⠀⠃⠀⠀⠘⠿⠂⠀⣶⡆⠀⢰⡆
⠀⠈⠇⠀⠀⠀⠀⠀⢀⣛⣘⡀⠀⠀⠀⢀⠀⠀⠀⢠⣄⠀⠘⠂⠀⠙⠁⠀⠀⣁⠀⠀⠈⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡆⠀⣼⣧
⠀⠀⠀⠀⠀⠴⡶⠄⠸⡿⢹⡇⠀⠀⠀⠄⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠋⠙⠁⢸⡇⣿⠀⠀⠁⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⣐⠀⠆⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠃⠙⠀⣀⠀⠂⠐⣀⡀⠀⠀⠀⠘⠃⠙⠀⢀⠀⠠⠀⠘⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠸⠋⠀⠀⠀⡈⠉⠁⠀⠀⠐⠄⠀⠀⠸⠙⠂⠀⠀⠀⠀⠀⠉⠁⠀⠘⢏⡹⠀⠀⣴⠋⢹⠀⠀⠉⣷⠀⠸⠓⠀⠀⢛⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠾⠛⠗
⠀⣀⠀⠀⣿⣿⣿⣷⣀⠀⠀⠀⠀⠰⠿⠂⠀⠀⣘⣁⠀⠠⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠺⣿⠇⠀⢸⡇⠀⣾⣶⡂⠀⢀⣀⠀⠀⠀⠠
⠀⢀⡿⠀⢻⠟⠛⠛⠋⠀⣾⣶⣖⠀⠀⠀⠀⠀⣿⣿⠀⠈⠁⢰⠄⠀⣾⣿⠀⠀⣤⣤⣤⠀⠀⠒⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠰⢂
⠀⠀⠀⠀⠑⠋⠀⢀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠸⡧⠀⠀⠀⠈⠀⠀⠟⠸⠀⠀⠙⠋⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣺⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠋⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣿⠿⣿⢿⣷⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠀⠀⠀⢀⣤⣤⣶⣿⣿⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⣿⠟⠁⣠⣧⡀⠙⢿⣧⣀⠀⠀⠀⠀⠀⠀⠀⣀⢼⣭⠶⠷⢮⣷⣄⡀⢨⣿⣿⠟⠉⢉⡟⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠠⣤⣶⣿⣷⣶⣶⠸⢿⣿⡟⠛⣿⣿⡟⠛⣻⣿⠿⠁⣤⣀⣤⡄⠀⠘⣏⣟⠙⢦⣴⠋⢹⣼⢡⣾⣿⡏⠙⢲⣿⣄⣀⣿⣿⡿⠇
⠀⠀⠀⠀⠀⢀⣰⣿⠯⣤⣞⠹⣿⣦⢀⣿⣷⣦⣀⣇⣠⣶⣿⣇⢠⣾⡿⡛⣻⣿⣶⢄⣦⢷⣴⠋⠉⢢⣼⣹⡏⠹⣿⣿⣄⣼⠁⢀⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠙⣿⣦⣜⣉⣽⣿⠃⠈⣿⠋⠛⣿⣿⡟⠋⢻⡏⢸⣿⣤⠛⢣⣼⣿⠀⣤⠙⣭⡛⠟⣭⡟⣡⡄⠀⢾⡿⣿⣿⣿⣿⡿⠟⠟
⠀⠀⠀⠀⠀⠀⢶⡟⠛⢿⡿⠛⢹⡷⠀⠸⣇⠀⣨⣭⡀⢀⡾⠀⢶⡉⢿⠿⠿⠟⢀⡖⠹⣇⠈⣤⣤⡀⢀⡿⠀⠀⠐⣆⠀⠈⠛⠋⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠈⢧⢰⣿⣿⡄⡼⠁⠀⠀⠹⣆⠿⠿⢇⡾⠁⠀⠈⣧⢰⣿⣿⠄⡾⠁⠀⢹⡌⠿⠿⢃⡾⠁⠀⠀⠀⠹⣆⢼⣿⣷⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣭⣭⣼⠃⠀⠀⠀⠀⢻⣷⣶⣿⠃⠀⠀⠀⠘⣧⣭⣭⣼⠃⠀⠀⠀⢻⣷⣶⣿⠁⠀⠀⠀⠀⠀⢹⣬⣭⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠀⠸⣿⢿⡏⠀⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠸⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠶⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⢁⡴⠖⠲⣄⠀⠖⠒⢦⡀⠐⣶⠒⢶⡖⠲⣆⠰⣖⠀⢲⡖⠀⣲⠆⣠⠖⠲⢦⡀⢲⡶⠖⠀⡇⣰⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣟⠚⠒⠛⢰⡏⠉⣻⡇⠀⡇⠀⢸⠇⠀⡿⠀⠹⣤⡟⢹⣰⠏⠀⣧⠀⠀⣸⠇⢸⡇⠀⠀⡟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠉⠛⠛⠁⠈⠙⠛⠛⠃⠘⠛⠀⠛⠃⠘⠛⠀⠀⠙⠀⠀⠋⠀⠀⠈⠛⠛⠉⠀⠛⠃⠀⠘⠛⠀⠉⠃

⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠂⢂⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠞⡠⠂⠾⠰⡄⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠒⢉⠀⠄⠀⠀⢑⣤⡚⠳
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⢀⢴⣾⣙⡏⠀⠓⡄
⢂⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⢀⢴⣿⣭⠞⠒⡛⠉⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀
⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠓⠉⠉⠙⠋⠁⠀⠀⡴⠀⡇⠀⠁⠠⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠐
⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠈⠀⠀⠐⠀⠀⡂⠙⠧⠄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢂
⣖⠂⠤⠄⣀⣀⣀⠀⠀⠀⠀⢀⡌⠘⣥⠀⠀⢀⢄⠀⣀⣀⠂⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠻⡄⠀⣠⡀⠀⠀⠀⠀⠀⠈
⡇⠈⢩⣥⠴⠒⠒⠉⠁⠀⠐⠁⢴⡄⢈⣀⠮⠁⢈⢂⠐⠛⣦⣀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠄⠙⠁⠀⣐⣋⣂⣀⣁⣀
⣟⣋⣒⠀⠤⠤⠄⣀⡀⠑⢬⠕⠀⢥⠀⣃⠘⡇⠘⠗⡀⠀⠠⠉⠀⠀⠀⠀⠀⡰⠒⠒⠒⠛⠢⠀⡄⠄⠊⠉⠁
⣓⣒⣢⠤⠕⢛⠭⠀⠀⠀⢐⠈⠢⣰⡦⢴⡀⠛⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠄⠀⠀⠀⠐⡄⢀
⡋⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⡴
⠃⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⡄
⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠄
⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠐⠠⠀⢠⠄⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣤⣶⣦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣿⣿⣿⣯⡀⠀⠀⠀⢻⡗⠀⠀⠀⡀⠈⣻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣟⠀⠀⢾⡷⢾⣿⣶⣿⠚⠿⠚⢿⣿⣿⠟⠀⠀⠀⠀⣤⣶⣶⣄
⠀⠀⠀⣁⠀⠀⠀⣠⣴⣶⣤⣿⣿⡿⣿⠟⠻⠿⠃⠀⠀⠀⢴⣿⠿⣿⠆⠀⠐⢾⡟⠡⣿⣦⡀⠀⢸⠏⠉⠛⠻⠄
⠀⠀⠀⠙⣷⡀⢸⡿⠋⠉⠉⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠈⢷⣦⡻⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣴⣦⣀
⠀⠀⠀⠀⠀⠀⢻⣿⣧⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠻⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣴⡿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣤⡄
⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⣤⡉⠁
⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣥⣤⣤⣤⣤⣼⣿⣧⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣯⣬⣿⣿⣧
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⢰⣿⡧⠀⠀⠀⠀⠀⠀⢾⣿⡆⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠰⣿⣷⠀⠀⠀⠀⠀⠀⢰⣿⡇
⣠⣤⣶⣤⣠⣴⣿⣶⣄⣠⣴⣤⣄⣠⣶⣿⣦⣀⣤⣶⣤⣄⣤⣾⣷⣦⣀⣤⣶⣤⣀⣴⣾⣷⣄⣠⣴⣦⣤⣠⣴⣿⣶⣄⣤⣶⣦
⠉⣿⣿⡍⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⠉⠉⠉⢹⣿⣯⠉⠉⠉⠉⠉⠉⢹⣿⡏⠉⠉⠉⠉⠉⠉⣿⣿⡍⠉⠉⠉⠉⠉⠩⣿⣿
⣠⣼⣿⣤⠀⣠⣤⣤⡀⣠⣽⣯⣄⢀⣤⣥⣄⢀⣤⣿⣥⡀⢀⣤⣤⣀⢀⣬⣿⣤⡀⣠⣤⣤⡀⣠⣼⣯⣄⢀⣠⣤⣄⡀⣠⣽⣧
⠉⠛⠛⠋⠉⢹⣿⡏⠉⠉⠛⠋⠉⠉⣹⣿⡍⠉⠉⠛⠋⠉⠉⣽⣿⠉⠉⠙⠛⠉⠉⢉⣿⣯⠉⠉⠙⠛⠉⠉⢹⣿⡏⠉⠉⠛⠛
⢀⣀⣄⡀⠀⣨⣿⣅⡀⢀⣈⣀⡀⢀⣨⣿⣅⠀⣀⣁⣀⠀⢀⣽⣯⣀⠀⣀⣈⣀⠀⣠⣽⣯⡀⢀⣈⣈⡀⢀⣨⣿⣅⡀⢀⣀⣁
⠛⣻⣿⡛⠛⠛⠛⠛⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⢛⣿⣟⠓⠛⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⠛⠛⣻⣿⠛⠛⠛⠛⠛⠛⠛⣿⣿
⠀⣙⣟⡁⠀⢀⣐⡀⠀⢀⣻⣟⡀⠀⢀⣂⠀⠀⣈⣻⣋⠀⠀⣀⣁⠀⠀⣘⣿⣃⠀⢀⣐⣀⠀⢀⣹⣟⡁⠀⢀⣀⡀⠀⢀⣻⣟
⠿⠿⠿⠿⠖⢻⣿⡟⠓⠿⠿⠿⠿⠚⢻⣿⡛⠺⠿⠿⠿⠷⠚⣿⣿⠓⠺⠿⠿⠿⠗⠛⣿⡟⠓⠿⠿⠿⠿⠖⢻⣿⡟⠲⠿⠿⠿
⠀⠀⠀⠀⠀⠘⣿⡃⠀⠀⠐⠀⠀⠀⢹⣿⠃⠀⠀⠀⠂⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠘⣿⡃
⠶⢿⣿⠷⠴⠿⠿⠿⠧⠾⣿⣿⠷⠾⠿⠿⠿⠦⠿⣿⡿⠦⠾⠿⠿⠷⠴⢿⣿⠿⠶⠿⠿⠿⠷⠾⢿⣿⠷⠴⠿⠿⠿⠦⠾⣿⡿
⠀⠻⡿⠃⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠘⢿⠗⠀⠀⠠⠄⠀⠀⠸⣿⠇⠀⠀⠀⠄⠀⠀⢿⡿⠂⠀⠀⠠⠀⠀⠐⢿⡿
⣴⣿⣿⣷⣤⣶⣿⣶⣤⣾⣿⣿⣶⣴⢾⣿⣶⣤⣾⣿⣿⣦⣴⣾⣿⣦⣴⣿⣿⣷⣦⣶⣿⣷⣦⣴⣿⣿⣷⣤⣶⣿⣶⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠰⣿⡷⠀⠀⠀⠀⠀⠀⠸⣿⡇
⣠⣶⣶⣦⣠⣶⣿⣶⣄⣴⣶⣶⣤⣰⣶⣿⣶⣄⣴⣶⣶⣄⣴⣾⣷⣦⣠⣶⣶⣦⣠⣶⣾⣷⣦⣠⣶⣶⣦⣠⣶⣿⣶⣄⣴⣶⣶
⠀⣾⣿⡆⠀⠀⣁⠀⠀⠀⣿⣿⠀⠀⠀⡈⠀⠀⢰⣿⡧⠀⠀⠈⠁⠀⠀⢼⣿⡆⠀⠀⢁⡀⠀⠀⣾⣿⠄⠀⠀⠀⠀⠀⠠⣿⣿
⣠⣾⣿⣦⣀⣤⣶⣤⣀⣤⣾⣷⣄⣠⣤⣦⣤⣀⣴⣿⣶⣄⣠⣴⣦⣄⣠⣴⣿⣦⣀⣤⣴⣤⣄⣠⣾⣷⣦⣀⣤⣴⣤⣀⣴⣾⣷
⡿⠛⢻⠿⡿⠿⠿⢿⠿⠿⢿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡿⣿⡿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⣄⢀⡀⢀⠄⣤⡀⣀⡀⣀⢀⠀⣤⣀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠚⠑⠊⠚⠛⠛⠐⠃⠙⠒⠀⠛⠋⠒⠉⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠚⠉⠉⠳⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠴⠒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢴⠄⠀⠀⢄⣀⣠⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠣⢀⡞⠙⢆⡀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣎⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢅⠀⠘⠧⠒⢦⠉⢙⠎⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡀⠉⠐⠒⠒⠒⠉⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠑⠢⠤⣤⣬⠶⠻⡄
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡟⣄⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢰⠤⢄⡀
⠀⠀⠀⠀⠀⠀⢸⣠⡠⠀⠀⡇⠈⠑⠒⢤⡘⠄⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⣀⠴⠖⠁⢸⠀⠀⠠⢀⠂⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠹⡙⡆⢀⠁⠀⠀⠀⠀⠈⠉⠉⢒⠒⠋⠀⠀⠀⠀⠀⠀⠀⢐⠅⢒⡊⠉⠉⠀⠀⠂⠀⠈⠦⠤⠖⠾⣀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢟⣌⠊⠀⠀⠐⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⡄⠀⠀⠀⢀⠀⠀⠀⠀⣠⠊⠀⠞⠳⠴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⢵⣆⢄⣀⣈⢒⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠷⠄⣀⠐⠃⠀⡀⠄⣖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡁⣈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⡍⠒⠊⠁⠀⢹⠄
⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠄⠚⠀⠀⠈⡗⢄⡀⠀⠴⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⣚⠠⠁⠀⠀⠀⠐⠻⠀⠀⠀⠀⡼⠠⣰⠇
⠀⠀⠀⠀⠀⠀⢀⠴⣶⠀⣤⠀⠘⠀⠀⠁⠀⠁⠀⠀⠤⠀⠀⣧⡍⠲⣤⣤⣤⠄⢡⣀⣈⣀⠀⠀⠀⡤⠤⠁⠀⠀⠀⠤⠚⣆⢀
⠀⢀⠀⠀⠈⠀⠘⢦⣼⠖⠁⠀⠒⠒⠀⢰⠀⠀⣶⣈⠁⠀⡬⠃⠐⠃⠀⢀⡁⠀⠃⢈⡿⠉⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢧
⡀⢠⠀⠀⠀⠀⠐⠠⡉⠀⠀⠀⣠⠀⠀⠀⡄⠀⢱⡈⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃⠄⣤⠂⠀⢠⣀⠀⠀⠀⢀⣄⣤⣤⣺⣀⣀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠍⠠⠄⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⣀⠉⠉⠉⠉⣗⠀⠤⠤⠤⠀⣂
⣇⠀⠀⠀⠀⠀⠐⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⣄⣀⡀⠀⢀⣀⣀⡤⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠋
⡟⠶⠦⠤⣤⣄⣀⡀⠀⡐⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡤⠛⠛⠲⠶⠭⠭⠭⠵⠖⠒⠋⠀⠀⠀⠀⠀⠀⠰⠦⠤⠤⠶⠩⡀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠉⠓⠚⠛⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⡇
⡇
⠃
⡿⠿⢿⠿⠿⡻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠿⣿

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠀⠀⠈⠀⠀⢄⠀⠀⠀⢀⣠⡂⣶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⣀⠀⠀⠀⣁⡀⠀⠀⠀⣄⠀⢀⣤⡶⠋⠀⠀⢔⠃⠀⠈⠀⠘⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙⣷⠉⠉⠀⢙⡟⠁⠀⣼⠁⠀⠀⠀⡿⠀⠠⠤⠄⠈⣻⣧⠀⠀⠆⠀⠀⠀⠢⠀⠂
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢀⠀⠀⣠⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣥⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣇⠀⠀⢰⠀⠀⠀⡄⠀⣠⡗⠀⠀⠀⠀⠀⡀⠀⠂⠀⠃⢀⣀⡀⠀⠀⠀⠈⠰⠰⠄⠠
⠀⠀⠀⠀⢰⠏⠟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⣠⣿⡀⢀⣰⣇⣴⠟⠁⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠈⠉⡅
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⢰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⣴⠀⡀⣠⠀⠀⢀⠠⡂⢀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠆⠋
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠄⠀⠀⠈⠃⡏⠀⠅⠹⠆⠀⠈⣳⠀⠀⢤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⡄⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠈⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣾⠂⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠄⡐⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠉⠛⠃⠀⡠⠄⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠆⠀⠀⠀⣄⣠⣧⠈⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠠⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐
⠀⠀⠀⠀⢀⡀⢀⠀⣀⡌⠀⠀⠀⠀⠨⠟⠉⠊⠋⠉⢙⠈⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠘⠀⠀⠀⠀⢺⠁⠀⠁⠀⠀⠀⠀⠀⢀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠩⠀⢀⡀⠀⡀⠀⡠⢀⣠⠆⠀⠀⢠⠐⠄⢀⡠⠄⠀⡀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠠⡄⠀⠀⠀⠀⡄⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠂⠀⠋⠀⠀⠀⠀⠉⠒⠦⠴⠾⠓⠁
⠀⠀⠀⠀⠀⠀⢀⡄⣂⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠐⠂⠀⠀⠻⠛⠃⠀⠠⠶⠶⠥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠅⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠈⠉⠉⠱⠆⠀⠰⠄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠘⠶⠶⠦⠤⠀⠀⠀⠀⠂⠐⠂⠒⠂
⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣒⠠⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡂
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠁⠀⠀⣠⠀⡅
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⢠⡄⠀⠺⠷⠂⠀⠀⠀⠀⠀⠀⠀⠯⢀⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠀⠀⠀⢈⠴
⠀⠀⠀⢿⣿⣛⡛⡟⣟⣿⢻⢛⣿⢻⣛⣟⣿⣿⠀⠀⠈⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠍⠙⠉⠑⠛⠛⠻⠼⠇
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⡦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠀⠒⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠚⠋⠁

⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠿⣯⠉⢉⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⣨⢷⡋⠁⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣷⣷⣦⡴⢿⣿⣁⣀⣻⣾⣿⢡⣤⣼⣷⣴⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⣡⡜⢻⣿⡎⠉⠛⠻⠋⠋⢠⣶⡿⣫⣽⣭⡹⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣶⡄⣾⣿⣿⡠⢻⠃⣴⣿⠇⠀⠀⠀⠀⠀⢠⣿⡇⢿⣯⣽⡿⣹⣿⡁⣠⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⣱⡿⢋⣿⣿⣿⡿⡟⣯⣴⣦⣴⣀⡀⠀⠀⢹⣿⣦⣭⣯⣷⣿⣿⣇⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⡿⠋⠀⠀⣀⣿⢻⣧⡶⣾⣿⣷⡆⠀⠀⠀⠛⠉⠻⠻⣿⣬⣿⣷⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢼⡇⢾⣏⠶⣩⣷⣻⣟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⣆⠀⠀⠀⠀⠀⠐⢿⣾⡛⠻⣛⣷⣿⠉⢀⡀⠀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠈⣿⠃⠀⣀⣀⣶⣤⣼⡍⠛⠻⠏⣛⣸⣷⣾⠿⢿⣿⣶⡆⠀⠀⢰⡿⠋⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⢰⡿⢠⣤⣿⣿⣿⣿⣿⣿⡟⠀⣀⣿⡿⠻⣷⡀⠀⣨⣿⣿⣿⠀⢸⣿⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⣶⣿⣶⣿⣤⡄⢠⣿⣿⣿⠁⠀⣿⣿⣿⠟⣹⣿⠁⠀⣈⣷⡾⠋⠁⢹⣿⠗⠈⠛⣠⣀⣌⢛
⠀⠀⠀⠀⠀⠀⠀⠠⣴⡿⠋⠀⠀⠀⠈⢻⣷⡆⣹⣿⣿⣷⣾⣿⣿⠿⠃⢙⣿⣶⡿⠋⠁⢻⣦⣀⣾⡿⠦⣤⡿⠛⠛⠛⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣇⡀⠀⠀⠀⠀⠀⣿⡷⠀⠈⠿⠛⣿⠛⠛⠀⠀⠈⠙⣿⣷⣦⣤⣤⣿⣿⠟⠇⢺⣿⠀⠀⠀⠀⠀⣿⡗
⠀⠀⠀⠀⣴⣿⣿⣦⢿⣿⣇⠀⠀⠀⣀⣼⡿⠇⢻⣧⢀⣴⣶⣄⠀⣿⠇⠀⠀⠁⠈⠛⠙⣿⡏⢩⣴⣦⣝⣿⣄⢀⣄⠀⣠⣿⡋
⠀⠀⠀⠀⠻⢿⡿⢋⣼⣿⢻⣿⣿⡿⣿⠛⠋⠀⢸⣿⠸⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⣿⡿⠈⢿⣿⠿⡿⠛⠃
⠀⠀⠀⠀⢠⣶⣶⣾⣿⣥⣾⡿⠋⠀⠀⠀⠀⠀⠈⠻⣷⣬⣭⣵⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣭⣭⣤⣶⡿⠃
⠀⠀⠀⠀⢸⣿⣿⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢀⣿⠇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣿
⠀⠀⠀⣼⡿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿
⠀⠀⠀⠙⠁⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠘⠟

⣿⡟⠉⠈⢻⣷⡆⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⢀⣤⣤⠀⠀⠀⠀⣸⣿⠋⠁⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⢠⣤⣄
⣿⡄⠀⠀⢈⣿⡏⠀⠀⠀⠀⠀⠘⠿⣿⣿⡶⣾⣿⣿⣷⣶⠶⣤⣿⣿⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠸⢿⣿⣿⢶⣾⣿⣿⣶⣶⢶⣼
⣿⣷⠀⠀⢿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠈⠟⠙⠀⠀⠀⠹⢿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠸⠏⠁⠀⠀⠂⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⠈⠉⢱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤
⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⣷⠀⠀⠀⠀⠀⠀⠀⢀⡄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⡛⠛⠻⣧
⣾⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠘⣳⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣿⠀⠀⠀⠀⠘
⡏⠂⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠘⠟⠁⠀⠀⠀⠘⢿⠇⠀⠐⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠂⠀⠚⠛⠀⠀⠀⠀⠘
⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠈

⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⣾⡿⠟⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⣾⡿⠟⠻⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⣿⠃⠀⠀⠸⣿⡷⠀⠀⠀⠀⠀⠠⣾⣿⣦⣤⣼⣿⣿⣦⣤⣤⣀⣾⣿⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣦⣤⣼⣿⣿⣤⣤⣄⣰
⣿⣧⠀⠀⣴⣿⡁⠀⠀⠀⠀⠀⠀⠉⡿⠛⠉⠹⡿⠛⠉⠉⣉⢻⣿⣿⣄⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⢹⡟⠋⠉⢻⡟⠏⡉⠉⡉⣿
⠙⠋⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠛⠟⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⣿⣆
⣴⣾⣿⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡆⠀⠀⠀⠠⢤⣴⣾⣷⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠁⠀⠁⣿⡀⠀⠀⠀⠤
⡟⡋⠛⢻⣿⠉⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄⠀⠠⣿⠇⠀⠀⠀⠠⣿⡟⠉⠛⢻⡿⠉⠀⠀⠀⠀⠀⠀⠀⠹⣧⠄⠀⢴⡿⠀⠀⠀⠀⢰
⠇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠈⠁


⣤⣶⣶⣶⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣤⣶⣾⣶⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀
⣟⣻⣛⣛⣟⣙⣻⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠿⣿⣶⣷⠄⠀⣠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⠛⠛⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⣡⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡄⣿⣿⣦⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠆⠀⠀⠊
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⢻⣿⣿⣧⠀⢀⠘⢋⣿⣿⣿⣿⣿⠃⠀⢻⣧⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠐⠈⡃
⠀⠀⠀⠀⠀⣸⣿⠿⣿⣿⣿⢸⣿⣿⣿⣾⡿⡷⠟⠛⠋⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠸⠄⠀⢠⡃⠀⠠
⠀⠀⠀⠀⠀⣿⣿⣤⣿⣯⣤⠈⣿⣿⠙⠋⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠉⠙⣋⣿⣿⣿⠀⡿⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡯⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⡗⠿⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠸⡷⠾⠿⠿⢿⡇⠙⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠛⠛⣿⣷⣦⣤⣀⣈⠄⠇⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣤⣴⣶⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⢿⣿⡿⠁⠈⠈⣗⣀⣤⣴⣾⣧⣤⣤⠤⠤⠀⠐⢲⠒⠂⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣿⠟⠁⣼⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⡥⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣿⣿⠙⠿⣿⣻⡿⠟⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡃⠀⠀⣿⡉⠀⠀⠀⠀⠿⠿⠿⠿⢿⡟⠋⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣿⡆⠀⡸⣿⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠻⣿⣴⣧⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠇⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠠⠊⠁
⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢠⣀⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠙⠳⡄
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢠⠞⠃⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡴⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣴⠦⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢿⡆⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⠛⠲⢤⣀⣀⡀
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠸⣷⣴⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡿⠶⣶⡿⠃
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣅⣀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣲⡟⠁
⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣏⠁⠠⣄⠀⠀⠀⠀⠈⠙⠻⠂⠀⠀⠀⠀⠀⠀⣤⣤⣤⠬⠿⠶⡦⠤⢤⣤⣶⣶⣶⣶⡶⠾⠿⢷⣶⣤⡦
⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠈⠳⢦⣤⣄⣀⣀⣤⡷⠀⠀⠀⠀⠀⠀⠀⠉⠛⣳⠶⢶⣶⠾⠟⢛⣫⣭⣴⠾⠶⢖⣺⠾⠋⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠹⣧⠀⠀⣀⣀⣀⢸⣿⠉⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⢠⡟⠁⠀⠀⠀⠀⣀⣐⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠉⠉⠉⠉⠉⢻⣧⣾⣿⣷⣤⣦⠀⠀⠀⠀⠀⣼⡏⢠⡿⠀⠀⢀⡶⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⣦⣀⠒⠿⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣰⡿⠀⣾⠇⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠈⠉⠛⠿⣿⣦⣄⣰⡿⠁⣼⡏⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠙⠻⣿⡅⣠⡿⠀⠀⠀⣿⡇
⠀⠀⠠⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠁⠀⠀⢸⣿⠁
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣷⡀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⢀⣿⡏
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠋⠀⠀⢀⣽⠿⣷⣄⡀⠀⣴⡿⠁⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣀⠀⠀⣰⡿⠁⠀⠀⠙⠿⣾⠟⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢿⣿⣿⣾⣿⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣿⡟
⠀⢀⠐⠀⠀⠀⠀⠀⣼⠿⢷⣶⣤⣀⡀⠈⢿⣿⡄⠙⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⣠⣿⡟
⠀⠠⠀⠀⠀⠀⠁⣼⡇⠀⠀⠈⠙⠛⠿⣿⣶⣿⣿⡀⠀⠀⠈⠙⠻⠿⣶⣤⣄⣴⣿⠟
⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣧⠀⠀⠀⠀⠀⠀⣼⣿⠏⠛⠁
⠀⠀⡀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⣀⣀⣤⣤⣽⡿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠠⠀⠀⢸⣿⡄⢀⣠⣴⣾⣛⣭⡵⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠛⢿⠿⢿⡟⠛⠉⠁⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢿⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣄⢖⡶⠆⡰⢀⡄⠠⢰⠀⠇⠶⠀⠆⠀⣴⠆⢴⠦⢴⡄⢰⡢⠀⢄⣴⠀⡦⢠⠶⠀⠠⠀⠠⠆⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠋⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣰⣶⣦⠀⠀⠀⠀⠀⢀⣸⣿⡇⢠⠀⠀⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⠙⠿⢿⣷⣶⣴⣶⣿⣿⣿⣿⣧⣀⣠⣾⣭⣿⣤⣤⣀⡀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠻⣿⣿⡏⠀⠤⢾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⠿⣿⡿⠿⣿⣿⣿⣿⡆⢰⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣶⣿⡟⠁⠀⠀⠀⣾⡟⠹⣿⣿⣿⣿⣿⣿⡀⠀⢋⣴⣤⠀⣉⡉⢹⡟⣿⣿⣿⡟⣉⠉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣿⠿⠿⠿⢷⡄⠀⠉⠀⠀⠻⣿⣿⣿⣿⣿⣧⠀⣌⣛⣋⡘⠿⠿⢘⠁⢿⡿⠙⣧⣉⣤⡉
⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣿⣿⣿⣶⣋⣠⡄⠠⣶⣶⣷⣾⣿⣿⡿⠂⠈⠀⠀⠉⠉⠙⠀⢹⣿⣭⣿⣶⣶⣿⣿⣿⣷⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⣀⣢⣽⣿⡛⠛⢁⣤⡀⢀⠸⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢸⣿⡿⠉⠛⠿⢿⣇⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣶⣿⣿⡿⠛⢹⠀⢶⣿⣶⣿⠏⠀⠀⠀⠀⠀⠀⠤⠀⠀⣤⠀⣀⣸⣶⣤⣤⣴⣶⡅⠘⠃⠀⢲⣼⣿⠟⠛⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢥⠾⠿⣣⣾⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿⡿⣀⠀⠀⡀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣿⣣⣿⡿⠟⠉⠹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠀⠈⣿⣿⢀⣤⣤⣤⣀⣿⣷⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⢀⣀⣀⠀⠀⠀⢣⡀⠀⠒⠀⢼⢿⡿⢿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠹⢿⣿⠏⠀⠀⣀⣠⣿⣿⣧⠀⠀⣴⣿⡿⠿⢷⣀⣀⠀⠀⠀⡀⠀⠈⣼⢠⡀⠀⣀⢸⡟⣧⣀⣀⣠⡦
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠁⠘⢿⠀⢠⣿⣿⣿⡿⠛⠉⠶⣤⣿⠁⠀⠀⠀⢹⣿⣧⣾⣿⣿⣿⣷⣾⣀⡤⣤⢉⡼⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠀⠀⠀⠠⢶⣿⣿⣿⠇⠐⠀⠓⠶⠆⠒⠀⠀⠒⠀⣿⡏⠉⠉⣻⣿⣿⣿⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢻⣿⣿⡆⠀⠀⠛⠛⠛⣶⣟⣷⣶⡾⣿⣇⠀⠀⠋⣿⣿⠟⠃⠀⠀⠹⠀⠀⠀⠀⠀⠶⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⠀⢘⠛⠛⠛⣛⠛⠛⠋⣾⣿⣿⣷⣤⠄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣶⣿⣏⠉⠉⠀⠐⣶⠂⠈⠉⣉⣿⣿⣦⣤⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⢠⣿⡄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠸⣿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢣⠀⠀⠀⢠⠀⠀⠀⠀⣀⠄⠤⠐⠊⠀⠀⠀⠀⠀⡘⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠁⠀⢂⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠈⠁⠀⢀⡼⠙⢦⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠙⠦⡀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⠑⠄
⠀⠀⠀⠀⠀⠠⠀⠀⠈⠃⠀⠀⠀⠀⠁⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⡀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⣀⠀⠀⠁⠀⠀⠀⠁⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠄
⠀⠀⠀⠀⠘⠒⠒⠒⠋⠉⠁⠈⠀⠀⠀⠢⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢰⡠⠀⠀⠀⠠⠐⠀⠉⠀⠀⠀⠀⠑⠄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣴⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠹⢀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦
⠀⢀⠠⠂⠁⠀⠀⠀⢠⠁⠀⠀⠀⡇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠉⠉⠒⢤⡀⠀⠀⠈⣆⡀⠀⢠⣇⠀⠀⠀⠀⠲⢌⡀⠀⠀⠀⠄⠀⠀⠀⠀⠃⢀⠏⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⢀⣤⡦
⠀⠠⠤⠤⠤⣌⣷⣀⣠⣤⣤⣤⣤⠿⠤⠤⠤⠤⠤⠤⠐⡀⠀⠀⠠⡐⠀⠀⠠⠤⠘⣄⢀⠀⠄⠀⠊⠁⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠁⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⢠⣀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠓⠠⠀⠀⠀⣁⠀⠀⠀⠒⠀⣀⣀⣤⣴⣾⡿⠟⠉⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠐⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠈⠲⡄⠈⠱⠚⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠠⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠤⠄⠀⠒⠤⠤⠤⠖⠀⠘⣄⠀⠈⠑⠲⠄⠀⠀⠀⠀⢀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠰⢀⢀⣀⣀⡤⠀⠲⠤⠤⠤⠵⠆⠈⠑⠮⠥⢄⠀⠀⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⠀⠀⠀⠀⡀⠀⢆⣀⣀⣀⣀⣀⠀⠰⣈⣄⣀⣠⡄⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣉⣉⣭⣥⠀⠀⠈⠉⢩⣭⣤⣤⢀⠀⠶⠶⠶⡶⠶⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠤⠠⠊⠀⠀⠈⠀⠐⠀⠀⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⢀⣠⡀⠀⠀⣀⡀⠐⠂⠀⠄⠀⠚⠀⣀⡀⠀⠀⡈⠛⠙⠉⠃⠀⠀⠀⠀⠀⠁⠀⠀⠢⡀⠁⣠⣀⠀⠁⠘⢀⡀⢀⠑⠈⠉⠀⠀⠀⠀⠁⠀⠘⡟⠄
⠈⠉⠀⢀⣾⡁⣀⠀⠱⠀⠀⠀⡴⠋⠩⣽⣿⣿⣯⡽⣆⠀⠀⠀⢨⠿⠟⠷⢤⠀⠀⠀⠀⠀⢰⠃⠬⠽⣆⠀⠀⠘⠀⠀⡀⠁⠈⠲⠖⠀⠈⠀⠀⣀⠉⠀⡈⣷
⠀⠀⠀⠸⡄⠀⠈⠀⢸⠀⠀⠐⠃⢀⠀⠆⠀⠘⠉⠄⡋⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡞⠀⠀⠀⣼⠀⠀⢠⠀⠀⣧⠴⠄⠴⠤⣖⠒⠒⠒⢣⠂⠀⠀⠀⢷
⠀⠀⠀⠀⠁⠀⠥⠀⠀⠀⠀⢨⠀⠈⣀⠀⠦⠀⠀⢀⠁⠀⠀⠀⠁⠀⠤⠄⠀⠁⠀⠀⠀⠀⠈⠀⠤⠀⠁⠀⡄⣤⠄⠀⢻⡀⢠⠄⠐⣉⣉⣉⣉⣉⡀⠠⠤⠀⠊
⠀⠀⠀⠀⠀⣅⣀⣠⠀⠀⠀⢈⠀⠀⢨⠂⠀⢺⠀⠀⠀⡆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⡠⢄⠀⠙⠛⠋⠀⠀⠀⠈⠢⠇⡄⠀⠀⠈⣇⣤⠜⠁⢀⠀⠀⠒⠦⠒⠂⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⡤⣀⣀⣢⣄⠀⣀⡀⠐
⠀⠀⡯⠀⠀⠀⠀⠀⠀⠀⡙⠯⠯⡲⡄⠀⠀⠀⠀⠠⡀⣀⡀⠈⠀⠀⠀⠀⠀⠀⠚⠝⢀⣶⠀⠀⠈⠀⠀⢰⠋⣡⠉⡆⠠⠉⠂⠋⢀⠶⠝⠧⡠⣵
⠀⠀⠐⢃⡘⠀⠀⠀⠀⣦⠠⡤⠤⢴⡅⠀⠀⠀⠸⡀⠀⠀⠀⣀⢀⠀⠀⠀⠀⡌⠀⠀⠀⠀⡅⠀⠀⠀⠀⡟⠀⠀⠀⢻⡆⠀⠀⠀⠘⡆⠀⠀⠀⠸⠀⠀⠀⠀⠐
⠀⠠⠀⠈⠀⠠⠀⠀⠠⠐⠄⠴⠄⠐⠀⠀⠀⠀⠐⠳⠀⢂⠀⠊⠀⠀⠀⠀⠀⠁⠠⠔⠤⠀⠃⠀⠀⠀⠀⡇⡀⠀⠀⣸
⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⡖⠀⠐⠀⠀⠀⠀⠀⠀⠀⡄⡀⢀⡄⠀⠀⠀⠀⠀⣇⡗⠀⠘⡯⠀⠀⠀⠀⠀⢰⠁⠀⠈⡀
⡀⠀⣀⣤⣀⣤⠀⣀⡤⠈⠂⠠⠀⠄⠀⢀⠀⠀⠀⠐⠀⠄⠀⠀⢀⠀⠀⡀⠀⠀⠒⠠⠄⠓⢀⡄⠀⢠⡀⠋⠀⢀⠠⠉⠂⠀⠀⠀⠄⠀⠠⠤⠄⠀⠠
⠉⡛⠉⠉⠇⠉⠁⠀⢰⠂⠀⠀⠀⠀⠀⢺⠁⠀⠀⡀⠀⠀⠀⠀⠨⠏⠭⠟⢦⠀⠀⠀⠀⠀⠿⠤⣀⠀⢙⡄⢀⠀⣀⠀⠠⠁⡏⠁⠀⠀⠀⠀⠀⢀⡛⠣
⠀⠁⠀⠀⠏⠉⠀⠋⠸⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠃⢀⠀⠠⠀⠀⠀⠀⢻⠀⠀⠀⠀⠐⡂⠀⡀⠀⠸⠁⠢⡐⠂⠀⠀⠄⠁⠠⡁⡞⢀⠀⠀⣾⠀⠀⠀⠀⡬
⠀⠀⠀⠀⠀⡀⠋⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠘⣀⠸⡄⠀⠄⠀⡸⠈⠤⠀⠀⠈⠤⢾⠆⠠⠁
⠀⠀⠀⠀⢈⠀⠀⠀⠅⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰
⡤⣠⠀⠀⠀⠄⠒⠀⠀⠀⠢⢄⣀⠀⠀⠀⠀⠁⠀⣠⠀⠀⢄⠀⠁⠂⠀⠂⠁⠀⡄⡀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⠘⣫⡄⠀⠀⠀⢀⡀⢾⡀⠀⠈⠀⠀⠀⠀⠰⠂⠓⠀⠀⠀⡐⠀⠀⠀⠀⢀⣀⠀⢸⡗⠀⠀⠀⠀⠀⢠⡶⠪⠳⣶⡉⠀⠀⠀⠀⢈⠃⠈⠉⣷⠀⠰⠋⠐⢴⡳
⠀⠀⢰⡆⠀⠀⠀⢠⡇⠀⠈⠀⢲⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⣺⡿⡄⢤⣿⣄⠀⠀⠀⠀⢿⢔⠀⠐⢿⡇⠀⠆⠆⠆⢰⠂⠀⠈⠀⣿⠀⠀⠀⠘⡇
⠁⠀⠘⠀⠀⠀⠀⠀⢣⠀⠤⠀⣈⠂⠀⠀⠀⠀⢲⣀⠢⠄⣰⠃⠀⠀⠀⠀⡟⡀⠰⠀⢨⠅⠀⠀⠀⠀⣏⠁⠉⠉⠉⢃⠀⠀⠀⠀⠘⠆⢀⣢⠀⠟⠀⠀⠀⠀⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠈⢗⣀⣺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠇⡘⠀⠀⣧⠃⠀⠀⠀⠀⢻⠆⠘⠁⠀⠻
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠙⠓⠛⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⢀⡀⢀⠠⠤⠤⠤⠼⠠⠀⢀⡄⣀⠓⠃⠀⠀⢀⠋⠀⠀⠀⠀⠀⠚⠢⠔⠚⠀⢀⣀⣀⡤⡄
⡄⠀⠀⠀⠀⠄⢉⡁⠦⠀⠀⠀⠀⢠⠔⠇⠶⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⡝⠊⠉⠚⡆⢀⡠⠀⠀⠠⠔⠋⠢⣤⡀⠀⠀⣀⡀⢎⡁⢩⡀
⠉⠀⠀⠀⢪⠀⠀⠀⠀⣤⠀⠀⢲⣎⡉⠀⠀⠁⢁⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡔⠐⠒⢂⠂⠂⡇⠀⠀⠀⡇⠈⡇⠀⠀⢰⣶⠀⠈⣧⡐⠒⠒⠺⣗⣅⠀⠈⠉⠉
⠀⠀⠀⠀⠸⡄⢀⡀⢠⡧⠀⠀⠀⠈⠆⢀⣦⡄⠠⠁⠀⠀⠀⠀⠈⠣⠼⠆⠜⠁⠀⠀⠀⠀⠀⢰⠀⠄⢰⠀⠀⠀⠀⠀⢿⠋⢃⠙⢙⠇⠀⠀⠀⠘⠻⡄⠨⠀⠸
⠀⠀⠀⠀⠠⠸⠆⠠⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠀⢦⠀⠀⠀⠀⠀⠀⢈⠁⠀⢀⠀⠀⠀⠀⠀⡇⡄⠀⠀⡼⠐⠂⠒⠒⠒⠂⠈⠀⠈
⠀⠀⠀⠀⠁⡀⠠⡀⠁⡈⠠⠄⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠉⠒⠒⠈⠀⣠⠀⠀⣀⠐⠀⢀⣤⡀⠐⠀⣀⣀⠀⠉⠋⠀⠀⢛⠉⠀⠀⠁⠈⠉
⠀⠀⡌⠓⣄⠈⠒⣄⡐⠁⠀⠈⠑⣀⡄⠀⠀⠀⠀⠈⠀⠐⣶⣤⡀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠁⠀⠁⡐⠉⠄⠹⢻⠀⠘⠡⠈⠀⠀⢴⠀⠀⠀⠀⡀⣀
⠀⠀⠑⠄⢠⠈⠉⠉⠚⣆⠁⠀⠀⢀⠁⠀⠀⠀⢘⠀⠠⠀⠀⢹⡇⠀⠀⠀⠀⡆⠁⠀⠀⠀⡔⠀⠀⠀⠰⢖⠀⠀⠀⢘⡇⠀⠀⠀⠀⣠⠈⠀⠄⣠⠰⠀⠀⠀⡀
⠀⠀⠀⠰⠈⠀⠀⠀⠁⠛⠄⠲⠂⠋⠈⠈⠀⠀⠈⣤⠀⣤⠀⠸⠁⠀⠀⠀⠀⢠⡀⠂⠂⣐⠇⠀⠀⠀⠐⠘⠀⠠⠀⠠⠡⠀⠀⠀⠀⠇⠀⠄⠀⠘⠈⠈⠉⠀⠉
⠀⠚⡠⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⡄⠀⠀⡍⠀⢂⠀⠀⠀⠘⡄⠉⢰⡎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠙⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠿⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⢋⣴⡄⠀⠀⢀⠀⠀⢸⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⠀⡇⠘⠙⠀⠀⢰⠿⠇⣠⠏⢸⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠋⡞⠀⠀⠘⠤⣀⣀⣀⡠⠴⠊⠁⠀⡼⢀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⡇⠀⢧⠀⠀⣴⡦⠀⠀⠀⠀⠀⠀⢀⠜⢱⢿⠃⢠⠃⠑⡄
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠳⡀⠀⠑⠦⣅⣁⡀⢀⣀⣀⠤⠚⠁⠀⠀⢀⡴⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⢰⣶⠈⠲⠤⣀⣀⠀⠉⠉⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⢸⢹⡦⣄⡈⠀⠀⠀⠀⠈⠉⠉⠛⠒⢚⣻⠉⠉⠀⠀⠀⢀⣤⠞⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠈⡆⡇⢻⠙⡗⢲⢤⡤⣀⣀⣀⣀⣀⣀⣋⣠⠤⡴⢲⠋⡇⢸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢇⢱⢸⠀⡇⢸⠈⡇⢸⠀⡇⢸⢸⠀⡇⢸⠀⡇⣸⢸⠁⡏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡀⡇⢧⠘⡆⡇⢸⠀⡇⢸⢸⠀⡇⢸⢠⡇⡇⢸⢠⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⢇⢸⠀⡇⣇⢸⡄⡇⢸⢸⠀⡇⡏⢸⠀⡇⡏⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢹⢸⢸⠀⡇⢸⠘⡇⡇⢸⢸⢀⡇⡇⢸⢸⢁⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⡼⡌⡇⢹⢸⠀⡇⡇⣼⢸⢸⠀⡇⡞⢸⢸⢠⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣇⡇⡇⢸⠸⡀⡇⡇⡏⢸⢸⢰⠇⡇⡞⣼⣸⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠚⠶⠧⠧⣧⣧⣼⣼⠼⠤⠷⠓⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠢⠤⠂⠢⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⢠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣘⡍⠀⠀⠀⠀⠀⠀⠀⡦⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠤⠤⠄⠀⠤⠄⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣛⠀⢐⠦⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠘⠁⠑⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡠⠶⣄⡠⠏⣀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⠃⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⣀⠴⢉⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠸⠀⠀⠘⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠄⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠈⠒⡤⠀⢀⡉⠄⠂⠛⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢁⠀⠀⡆⠀⣀⡠⡚⠉⡉⠑⢆⡀⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠊⠀⠀⠀⡸⠀⢃⠀⠀⠀⠀⣆⢸⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠁⠀⠙⡆⠣⠤⠔⠁⢉⠋⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠳⢅⠀⠈⠁⠒⠒⠂⠁⠀⣷⠀⠀⠈⡆⠰⠲⠶⠶⠶⢌⠉⠉⠉⠉⠀⡸⠀⠀⢠⡀⠀⠀⠀⡘⠀⠀⢸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡀⠀⠀⠱⠤⡀⠀⠀⣀⡀⢤⠞⣧⡀⠀⠸⠀⠀⠀⣠⡶⠚⠛⠛⠛⠋⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠈⠉⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡀⠀⢦⣧⢔⡩⢄⠖⠃⣠⠟⠁⠀⢰⠀⠀⢘⡯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢹⢋⣼⠿⠓⠁⢀⡠⠋⠀⠀⡴⠃⠀⣭⠾⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠓⠤⣄⡀⠿⠁⠀⢀⡤⠊⠀⠀⠀⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠀⠀⠀⠀⠈⠀⠐⠂⠉⠀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⢁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡆⠇⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⢀⠏⠈⠈⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⠀⠀⠈⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⡇⠀⠀⠀⠀⢳⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠃⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣎⣀⡀⠀⠀⢀⣀⣠⠇⠀⠀⠀⠀⠸⠤⣤⣀⣀⠤⠤⠜⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠐⠒⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣊⣭⣽⣿⣿⣿⣛⡳⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠉⠁⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢀⢰⠚⢶⠚⠙⡶⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠟⠀⠀⠀⠙⢿⣶⣤⣤⡴⣦⠀⠀⠀⠀⠀⠀⢹⡄⢸⣇
⢴⣦⣤⣤⣤⣤⣤⡏⢹⠀⢸⠀⢀⡇⠀⣿⣭⣭⣧⣤⣤⣤⣶⣶⠀⠀⠀⠀⠀⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⣄⠀⠀⢸⡇⢸⡿
⢸⣿⠀⠀⠀⠀⠀⢣⡼⠧⠞⢦⠞⢧⠔⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠈⠛⠶⠟⢁⣾⠃
⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢠⠤⠄⠀⠈⠉⠛⢶⣤⣀⠀⠀⠀⠀⠀⠙⠻⢷⣶⡶⠾⠛⠁
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢾⠆⠀⠀⠀⠀⠀⠈⠙⢿⣶⣄⡀⠀⠀⠀⣼⠃
⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣠⣤⠤⠄⠀⠀⠀⠀⠀⠀⢀⠠⡄⠈⠻⣿⣦⣀⣼⠃
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠉⠀⠀⠈⠻⣿⠃
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⣿⣆⡀⠀⠀⠀⣠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡴⢸⡗
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠛⠿⠷⠾⢯⣁⡀⠀⢀⣠⡞⠃⠀⠀⠀⠀⠤⠶⠛⠁
⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠤⠬⠽⠿⠛⠉
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠔⠒⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠟⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢧⣀⣨⠗⠒⠒⣒⣛⣻⣿⣯⣭⣭⠤⠤⠀⠀⠀⠀⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠈⠙⠷⡶⠤⣜⣟⣠⠤⢶⠶⠋⠁⠀⠀⠀⠀⢸⡇⠀⠀⢰⣆⠈⣤⣀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠻⠿⠷⢶⣶⣦⣤⣌⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡟⠘⣇⠘⣟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣿⣄
⠀⠀⠀⠀⢀⡀⠠⡤⢤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠇⢀⡼⠁⠀⢿⡄⠸⡆⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⠇⠀⢸⣷⡀⢻⡀⠘⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⡴⠋⠀⠀⠀⠹⣷⣄⡙⠂⠈⠛⢿⣿⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡟⣀⡤⠎⠀⠀⠀⠈⠙⠻⠿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠂
⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⡀⠂⠀⠁⠖⠒⠛⠛⠛⠟⠁
⠀⠀⠠⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⠐⠻⢦⠀⠀⠀⠀⠀⠀⠑⠂⠄
⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⡤⠐⢤⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣼⡻⠀⢀⡞⠀⡠⣶⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢾⡄⡔⠀⢻⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡿⠃⠀⠀⠈⣧⢁⠀⠀⢻⡞⡆⠀⠀⣠⣤⣶⣦⣤⡤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢦⠀⠀⢀⡤⡞⠋⠀⠀⠀⠀⠀⠀⢸⡆⡆⠀⠀⣷⣷⣶⣾⠏⠙⢩⣿⠟⠟⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠸⣶⡺⠟⠉⠀⠀⠀⠀⠀⠀⣀⣴⣾⣷⣶⡶⣿⣿⡟⠿⠻⣷⠀⠈⠳⣌⢂⠀⠉⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⢸⣁⣀⢀⣀⠀⠀⠀⠀⣰⡿⠿⠊⠋⢻⣿⣿⢸⣿⢂⢀⢡⡀⠀⠀⠈⠈⢢⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠺⠛⠛⠉⠁⠀⢀⣤⣼⣯⡀⠀⠐⠀⡀⢻⣟⣾⠻⣧⠠⠀⠡⠡⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠈⠀⣆⠀⠀⠀⠀⢀⣼⣿⡿⣿⣷⣄⠀⠀⠀⠈⠉⢉⠣⡘⣧⠠⠠⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⠤⢤⣌⢳⠀⠀⠀⣾⣿⠉⠀⠉⢿⣝⣆⠀⠀⠀⠀⠀⠀⠁⠘⢧
⠀⠀⠀⠀⠀⠀⠀⢠⢟⡆⠀⢰⣏⡏⠀⢀⣴⣿⡉⢆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⢀⡟⣸⠀⢠⡟⣸⢀⣶⡿⡟⡩⢷⡱
⠀⠀⠀⠀⠀⠀⣸⢣⠃⠀⠈⣹⣷⣾⣿⡄⠘⡓⡀⠧
⠀⠀⠀⠀⢀⣤⣿⣶⣶⣶⣿⣿⡿⠛⣿⠙⡄
⠀⠀⠀⢀⡾⣿⣿⣿⢂⠙⢝⣿⠣⠨⡿⢀

⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠂⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠈⠀⠀⠁⠀⠐⠃⠈⠀⠠⠀⠀⠌
⠀⠄⡀⠀⠀⠀⠩⡀⠄⠀⢀⠌⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⢀⢠⠀⠀⠀⠀⡀⠀⠐⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠄⠂⠃⢀⡀⠘⠒⠠⠀⠀⠀⠠
⠀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠄⠐⠁⢀⠀⠀⠀⠀⠀⠀⡃⠈⢢⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠐⠈⠀⠀⠀⠀⠁⠄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⢀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢀⠈⠀⠀⠀⠐⠠⠐⠂⠄⠀⠀⠀⠀⠀⡀
⠀⢡⠀⠀⠀⠀⠠⠂⠀⠀⠀⢀⠀⠀⠀⢀⡐⠀⠀⢊⡀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⡀⢌⠀⠀⡱⢀⠀⠀⠀⠁⠀⠁⠉⠀⠂⠀⠀⠀⠀⡈
⠀⢁⠌⡄⠃⠀⠐⡀⠀⠀⠀⠄⠀⠐⠀⠀⠐⠠⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠁⠌⡑⠊⠁⠀⢀⠀⠀⠀⠀⡀⠀⠀⠘⠠⠁⡈
⠀⠈⠀⠀⠀⠐⠀⠐⠂⠐⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⢊⠀⠁⠀⠀⠀⠀⠁⠊⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠑⠀⠀⡀⠀⠀⠃⠀⠀⠘⠁⠀⠀⠀⠀⠠⠒⠐⠂⠄⠀⠀⠀⠃⠀⡰⢆⡀⠀⠐⠀⠀⠀⠐⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠢⠀⢀⠐
⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀⠖⠂⡀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠑⠄⡀⠀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡐
⠀⠀⡁⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢈
⢀⡀⠁⠀⠀⠀⠊⠀⢄⠀⠀⠐⠀⠑⠄⡀⢀⠀⢀⡀⢀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠢⠡⠀⠀⡄⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠡⣀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠠⢀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⣀⠀⠐⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠤⠀⠀⠀⠀⠂⠈⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠄⠑⡀⠀⠀⠀⠀⠠⠀⠠⡀⠀⠈⠁⠈⢀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠰
⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠁⠈⢉⡁
⠈⢠⠀⠀⠔⠀⠀⠀⠀⠀⠁⠀⡀⠠⠀⠠⠈⠉⡀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠈⠀⠀⠀⠊⠈⠁⠀⠂⠀⠄⠨⡀⠂⠀⠀⠰⢀⠁⠠⠀⠀⡄
⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠈⠢⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⡀⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⡈⠀⠀⠀⠈⢀
⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⡀⠐⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⡄⠀⠀⠀⠃⠁
⠀⡠⠁⠀⠀⠀⠀⠠⠀⠀⠄⣀⠀⠀⠄⠀⠡⠀⠠⠄⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠡⠀⠀⠈⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⠀⠀⠀⠈⢄
⠀⠐⠂⠀⡀⠀⠁⠀⠀⠀⠀⠨⠀⠀⠀⠤⠊⠀⠀⠑⠤⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠠⠀⠆⠀⠀⠀⠀⢀⡀⠰⢀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠆
⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢄⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣰⠉⣄⠀⠀⡠⠉⢆
⠀⡀⠀⠀⠀⠀⢤⠆⠀⠀⠰⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⢆⠀⠀⠰⢀⠀⡀⠀⠀⠀⢀⡀⡀⠀⠀⠀⡀⠤⠃⠀⠀⠘⠄⢈⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠜⠣⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠰⠨⡄⠀⠁⢠⠄⠂⠀⠀⠀⢂⠀⠀⣁⢐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠚⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⡄⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠡
⠀⢊⠀⠀⠀⠀⣄⠀⢀⡀⢀⣠⠀⠀⠀⢀⡕⠀⠀⢊⠀⠀⠀⢠⠀⠔⠑⠀⠀⠀⡀⠀⠀⠈⡅⠀⠀⣨⠀⠀⠀⠀⡂⠈⠀⠠⠂⢤⠀⠀⠀⠀⡑
⠀⠐⠁⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠐⠣⠜⠀⡐⠣⠜⠂⠀⠄⠠⠀⠀⠀⠀⡀⠀⠠⠐⠈⡁⢉⡅⠈⠁⠒⠄⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⢀⣀⣀⠀⠀⣼⣿⣿⣷
⠀⠀⠀⠀⠀⣀⡻⣿⣿⠟⣁⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣤⣬⣛⣛⣥⣤⠀⠀⠀⣠⣬⣛⣟⣫⣤⠰⣿⣿⣿⡆⣠⣬⣻⣟⣫⣄
⠀⠀⠀⠀⣾⣿⣿⣶⣶⣿⣿⣷⠀⠀⠀⣰⣶⣭⣿⣵⣶⡽⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⢿⣵⣽⣛⣯⣶⣝⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠿⠿⠿⠿⠛⠁⠀⠀⠻⠿⣿⣿⠿⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⠉⠛⠿⠿⠿⠟⠛⠉⠀⠀⠀⠙⠛⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣦⠀⠀⠀⣴⣿⣶⠀⠀⠀⢰⣾⣷
⠀⠀⠀⠀⠀⠀⣶⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣴⣽⣿⣵⡄⠀⢠⣮⣿⣯⣦⠀⢠⣬⣿⣯⣦⡀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⣼⣿⡄⢠⣼⣿⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣺⣿⡛⠛⣻⣿⡛⠛⣻⣿⡀⠀⠀⠀⠀⠉⠉⣵⣾⣽⣷⣮⠉⢹⣾⣽⣷⣾⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡗⠸⣿⣿⡿⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠀⠸⣿⣿⣿⡿

⣴⠈⠀⠀⠀⠀⠈⠙⣿⣿⣿⡏⠐⠀⢤⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⡇⠀⠀⠘⠃⢀⣠⣴⣿⣿⡿⠃⣀⠀⠀⠀⠈⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⠉⢻⡿⠛⠉⠛⠿⠿⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⣳⣶
⠁⠀⠀⠀⠀⣴⣿⣿⣿⠟⠁⠀⡀⠀⠀⠀⠀⣾⣿⠟⣹⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠸⢠⣶⣶⣄⠀⢀⣀⡀⠈⣿⡟⣿⣿⣿⣿⣿⡅⣤⠀⠀⢤⡎⠁
⠀⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣧⠀⠀⠀⠀⠙⠁⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⠘⠿⠿⠛⠐⣿⣿⣿⠀⡟⠀⣻⣿⣿⣉⣹⣇⣀⣀⣄⡀
⣴⣴⣦⠀⣾⡿⠋⠉⠁⠀⠈⢉⢻⣧⠀⠀⢀⣤⣤⣤⣤⣄⠈⠛⠛⠿⠿⠿⠿⣿⠀⠸⣷⣶⡶⢶⣄⡈⠙⢉⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠞⠁⢀⣀
⣿⣿⣿⣷⣏⣠⣤⣄⠀⢰⣿⣿⡿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠻⣿⣷⣶⣿⣿⡿⠿⢿⣿⣿⣿⠛⠻⢿⣿⣿⣿⣧⣶⣿⣿⣿
⢐⣍⣻⣿⡟⠿⠟⠛⣀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠸⣿⡿⢃⣀⠀⢀⣀⠈⠁⢻⣿⣿⠫⢄
⣿⣿⣷⠿⡇⢀⣀⣴⣟⣛⣷⣶⡞⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠸⡏⣢⣥⣥
⣟⡙⠑⢺⠇⠈⣻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠐⠶⠀⣠⣤⣶⣿⣿⣿⣶⣾⣿⣿⣿⣷⣦⣄⠀⠀⠀⠤⠀⠀⠀⠀⠀⢻⣿⣿⠟
⣿⠏⣂⣴⣿⣿⣿⣿⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠿⠷⠀⠀⣀⣼⣿⣷⣦⣄⣀⣣⡃⠞
⡤⣵⣿⣿⣿⡿⠿⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⢸⡿⠛⠛⠛⠋⠉⠉⢿⣿⣿⡏⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠲
⣾⣿⡿⠛⠁⠀⠀⠀⠹⠿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⢡⣶⣿⣿⣿⣶⣾⣿⠁⠀⠈⠉⠀⠈⣿⣿⣿⣿
⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣆⠀⠀⠓⣠⡀⠀⠐⠛⠀⠀⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣧⠀⠀⠤⣴⣾⣿⣿⣿⣿
⣿⣿⣿⡿⠀⠀⣀⠀⠀⠀⢻⣿⠟⠃⠆⠀⠀⢰⣿⣿⠟⠛⠛⠛⠻⣶⣶⣄⠉⠀⠀⠂⠀⠀⠀⠛⢸⠇⠀⠀⠉⠉⠉⠀⣿⡇⠀⠀⠀⠸⣿⠟⠋⠉⠙
⢹⣿⣿⠀⠀⠀⣁⣀⣤⣴⣿⣿⣿⣶⠀⠀⠀⠸⡿⠁⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⢺⠐⠖⠀⠀⠠⠦⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠙⣿⠀⠀⣴⣿⣿⣿⣿⣿⠿⠛⠋⠿⠶⠶⠶⠇⣤⡄⠀⠀⣤⡄⠀⣿⣿⣤⣤⣤⣤⣄⣀⣤⣴⣿⣆⣠⡴⢶⣄⢀⣼⠂⠀⠀⠀⠀⠀⢸⡔⠀⠈⠁
⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⡇⠀⠀⢀⣴⣦⣤⣤⣆⣀⣀⣤⣄⡀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⠧⠀⢠
⠀⣠⣴⣤⣭⣿⣿⣿⣿⡟⠃⠙⠁⠀⢠⣤⣤⣤⣼⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠛⠛⠛⠇
⠐⣿⣿⣿⠿⠿⠿⣿⣿⣿⠀⠀⠀⢢⣾⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠐
⠀⠻⠋⠀⠀⠀⠀⠀⠹⣿⣿⡆⠀⣿⣿⠿⠿⠿⢿⣿⣿⣄⣀⣀⠐⠚⠿⠿⢿⡀⠀⠀⠉⣿⣿⣿⣿⣿⠁⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠴⠀⠀⠀⣀⠀⠀⣿⡏⠀⠀⣿⣥⣥⣲⣴⣶⣽⣿⣩⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢻⣿⣿⡿⠀⣠⣴⣶⣶⣶⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⡀
⣀⠀⣀⢀⣀⡀⠈⠀⠐⢻⣿⣿⣿⣿⡿⠿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠃⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣤⣤⣤
⣿⣧⢻⣿⣯⣽⣶⣤⣶⣿⣿⣿⣿⣿⣷⡰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠛⢁⠀⠀⣿⠟⠁⠀⠀⠀⠈⠹⡟⠀⣠⣾⣿⣿⣿⣿
⠈⢿⡇⠙⠛⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠉⠉⠉⠉⡟⣠⣶⣿⣿⣿⣷⣄⣀⣀⣴⣄⡀⠃⢠⡤⠀⠀⠠⠦⠀⢰⣴⣿⣿⡟⠁⠉⠉
⣠⠈⠇⠀⠀⠀⠀⠘⠛⠿⢿⣿⣿⣿⣿⣿⣿⡟⠛⠀⠐⠆⠀⠠⡤⢠⣿⣿⣿⠿⠛⠉⠉⠉⢿⣿⣿⣿⣿⣷⣄⣀⣠⣤⣤⣀⣴⣦⣿⣿⣿⡇
⠉⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⢨⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⡀⠀⣠⣿⣿⡿⠁⠀⠀⠀⠀⠖⠈⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡿⢶⣿⣿⠟⠁⠛
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣆⠀⣾⡟⢿⣿⣿⣿⣿⣧⣤⣤⣤⣾⣿⣿⣿⣿⣿⡄⠶⠂⠀⢀⣠⣤⣼⣿⣿⣿⣿⣿⣿⠙⠛⠋⠁⠀⠀⢻⣿⣷
⠀⢀⠄⠀⠈⢿⣿⣿⣿⣿⣦⣙⡀⠀⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣴⣟⣡⣿⣿⣿⣿⣟⡋⠉⠉⠠⣤⡆⣠⣴⣶⣾⣿⣿⣇⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠚⠉⠒⡶⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠊⠲⢶⣦⣤⣤⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣠⣾⣷⡿⠂⠀⠀⠀⠉⠋⠛⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⣠⡴⣦⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⢸⡿⠾⠛⠀⠀⠐⠛⠛⠆⠀⠀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠓⠘⠀⠀⠀⠘⢋⡥⠀⠀⡤⠤⢬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⢀⡓⠂⠀⠀⠀⠀⠀⠀⠀⠇⠛⡄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡰⢊⡴⠋⢋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣕⡀⠙⣿⣷⡀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡇⠀⠀⠀⠀⡠⠋⠄⠚⠡⠀⡀⠀⠀⠘⣰⠋⠀⣀⠤⠛⠖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣴⣄⠈⣽⠆⠰⠋⠰⣄⠀⠀⠀⠀⢾⣿⠀⣼⠀⠀⠀⠀⡀⠀⢀⢈⠂⠀⠉⠜⣁⠠⠳⠁⠐⡛⢁⣠⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣛⣿⡏⢻⣷⣾⡇⠀⠙⠳⢦⣀⠀⠈⢁⣾⠟⣄⠀⠀⡰⠐⢀⠏⢂⠀⣰⣿⣿⣶⣶⣤⣄⣀⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⣰⣿⣿⣾⣿⣿⣾⣡⠇⡰⠁⡠⢎⢀⠂⣼⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠛⠻⣁⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣷⣿⠟⡀⡀⡰⠁⢀⡆⣼⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⠴⠋⣷⡄⠀⠈⠲⣄⠀⠀⠀⠀⠅⠀⠀⡘⢿⠉⠁⡘⡀⡐⠁⢀⡼⣽⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢸⡇⠉⠾⠋⢄⠁⡀⠈⢆⠈⢆⠀⠀⠀⠉⠒⣤⣄⣀⡀⢸⡆⢀⢀⡜⠐⠀⡰⠀⢜⣿⣿⣿⣿⣿⡟⠳⣤⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣼⣿⠿⣷⣤⠺⢗⠈⠀⠈⢢⢛⠳⣤⣀⣴⠂⠀⢠⣽⠟⠎⠀⢾⠞⠀⠀⠀⢀⢎⣾⣿⣿⣿⣟⠃⠑⣄⢹⣿⡿⠟
⠀⠀⠀⠀⠀⢹⣤⣟⢾⣿⡄⠀⠱⣿⡄⠀⢡⢢⣌⠉⠑⢀⣤⡀⠉⣤⣶⣄⠸⠙⠓⢶⡀⢮⣿⣿⣿⣿⣿⡏⠰⢂⣇⠜⠉
⠀⠀⠀⠀⠀⣼⢿⣿⠀⠈⢳⣶⠄⢨⠙⣆⠀⢳⡉⣠⠖⡋⠉⠁⠀⠐⠀⢤⣅⠄⣄⡸⠎⢟⡛⠛⣻⠛⡛⠃⠞⢈⠃⠀⠀⠀⠀⠀⣤⣠
⠀⠀⠀⠀⠀⢿⣮⡙⢠⣶⠁⠁⠪⣾⣄⠙⡆⠀⠣⠉⠀⠀⠀⠀⢀⡴⠋⠀⢱⣶⠀⡇⠀⠘⠿⣴⡏⠠⠀⠀⠀⠎⠀⠀⠀⠀⢀⣼⣫⡕⣡⠄
⠀⠀⠀⠀⠀⠀⠻⣿⠆⠉⠂⣠⣬⠟⠣⣠⣧⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠤⠄⠊⡉⢿⣂⣀⣽⠁⠸⠇⠀⡜⠀⠀⠀⠀⠀⠙⠻⢛⠄⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡖⠒⠴⣍⢻⣦⣄⠉⢻⠀⢀⡠⠖⠉⠁⠀⢀⣴⡧⠒⢲⠀⡀⢰⡏⠁⠀⢠⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⢋⡀⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⣿⣫⠴⠚⠋⠁⠀⠀⣠⠞⠉⠉⠁⠀⠀⠘⣶⡇⢸⣇⢀⡴⠋⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⢂⠉⠤
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢠⠐⡇⢸⠉⢻⣬⣤⣷⠄⢠⡏⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠚⣡⠖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⢠⠃⠀⢣⢬⠔⠁⢃⠀⠁⢠⠞⠀⠀⠀⠀⠀⢿⣤⡼⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⢀⣀⣤⠶⣿⠀⠀⣴⠃⠀⠀⠀⠀⠀⠸⡄⠀⠸⣀⡀⠀⣼⣀⣴⠋⠀⠀⠀⠀⠀⠀⡼⠀⢠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠋⣉⠀⠀⢿⡗⠐⠛⠓⠒⠲⠦⠂⠀⠂⠙⡶⠦⢭⡦⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢃⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠆⢸⣷⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡇⠀⣿⡻⡁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠓⠊⠁⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢧⣤⡴⠛⠁⢀⡼⠃⠀⠀⠀⠀⠉⠉⠉⠁⡀⢀⠐⢀⣇⣗⣱⡀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣶⣆⣀⣉⣀⡈⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠸⣧⣄⠀⣤⣤⡄⠤⠤⣤⣄⣀⢤⡤⠤⠤⠤⠤⢀⠆⠀⣿⡇⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠉⠉⠉⠒⠋⠀⠀⠀⠀⠀⠁⠀⠀⠰⣿⠀⠀⣿⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣿⡆⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⠀⠀⣿⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠘⣿⣄⠈⠻⠆⠀⠀⠀⠀⠀⠀⣠⡎⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣇⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠈⢻⡗⣲⣄⠒⠒⠒⠒⠲⣟⠋⠀⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠆

⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⣀⠀⠀⢀⠀⠀⢀⠀⣠⣤⣠⠀⠀⢠⠤⣀⠀⢀⣀⢤⠀⠀⣀⣀⠀⣀⣀⣀⡀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⢹⠀⠉⢢⠀⠀⠀⠀⠀⢸⠀⢠⠋⠀⣇⣀⡀⠀⠀⣧⠤⠚⠀⠺⣄⡀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠘⡦⠊
⠀⠀⠀⠀⢸⠀⠀⡜⠀⠀⠀⠀⠀⠈⣤⠃⠀⢠⠃⠀⠀⠀⢰⠏⠢⡀⠀⠀⢀⡽⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠸⠔⠊⠀⠀⠒⠒⠂⠀⠀⠉⠀⠀⠈⠀⠉⠁⠀⠈⠀⠀⠈⠀⠈⠁⠀⠀⠉⠉⠁⠀⠀⠀⠁⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣄
⠀⠀⢸⣶⣄⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶
⠀⠰⣾⣿⡿⠀⠀⣶⡿⣷⡄⠀⢹⣿⣿⡏⠀⠀⣶⣾⣶⡄⠀⢹⣅⣤⣇⠀⠀⠀⡀⠀⠀⠀⠀⢹⡉⢙⡖⠀⠀⣄⣦⣶⡆
⠀⠀⡝⡩⠛⡄⠀⠈⣷⣞⣳⠀⡰⠿⠛⢲⠀⠀⢻⣿⡏⠀⠀⠸⠒⠒⠚⡄⢻⣿⣿⣇⡀⠀⠀⡼⠿⠯⠽⢄⠀⢹⣿⣿⣧⡆
⠀⢸⡜⢁⠴⡇⠀⠀⡇⠀⠀⠃⠧⠀⠀⠸⡄⠀⢺⢿⡇⠀⠀⢠⠀⠀⠀⡇⠀⢻⣿⠋⠁⠀⢸⠀⠀⠀⠀⠸⠀⠀⣿⣿⣿⣤
⠀⢸⡴⠁⢠⠀⠀⠀⡇⠀⢸⠀⢸⠀⠀⠀⡇⠀⢸⣿⡇⠀⠀⢸⠀⠀⠀⡇⠀⢸⣇⠀⠀⠀⢀⠶⠤⠄⠲⡁⠀⠀⣿⠉⠀⢸
⠀⠸⢀⠔⢹⠀⠀⠀⡇⠀⠘⠀⢸⠀⠀⠀⡵⠀⣼⣿⡇⠀⠀⢸⠀⠀⠀⡇⠀⢸⡿⠀⠀⠀⢸⠀⠀⠀⠀⢱⠀⠀⣿⢰⢸⢸

⠀⠀⠀⠀⡀⣀⣀⣀⠀⠀⣀⣀⣀⠀⢀⠀⣀⣀⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠤⣄
⠀⠀⠀⠀⠀⠓⠓⠋⠐⠀⠛⠓⠛⠘⠃⠀⠛⠁⠊⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠊⠀⠀⠀⠀⠈⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡗⠄⠀⠀⠠⣀⣀⣀⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣞⠀⠀⠀⠀⠀⠀⠉⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠑⠢⠤⣄⣀⣠⠤⠜⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠖⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠑⡖⠒⠒⠦⠤⠒⠚⠓⠉⠀⡏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⣀⢠⣤⣤⠄⠀⠝⡂⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠄⠀⠀⡀⢀⢂⡀⠘
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡤⢯⠀⠀⠀⠀⠀⠀⠀⠪⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠳⠋⠱⡳⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠢⣀⠈⠲⠒⠁⠀⠀⠀⣀⠤⣺⠁
⠀⠐⠀⠀⠀⠀⠀⠈⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⣰⡢⢄⠀⠀⢀⡽⡶⠶⠶⣶⠊⠙⢊⠉
⡇⢹⠁⠀⠀⠀⠀⠈⡇⠀⠀⢨⣀⠀⠀⠀⠀⠀⣀⠀⠨⠁⠀⠀⠱⣻⡆⠙⠀⠟⠬⠈⠁⠐⡇⠀⠀⠀⠁
⡇⠸⡀⠀⠀⠀⠀⢀⠇⠀⢀⡘⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡤⠶⡊⠑⠀⠂⠀⠑⠒⠠⢀⡑⠠⢤⠜
⡇⠀⠙⠒⠒⠒⠒⠋⠂⠀⠀⠆⠀⠀⠀⢈⡀⠈⠁⠲⠄⠀⠈⢆⠀⠀⠀⠀⠀⡀⠀⠤⠒⠋⠁⡠⢷⠞⢣
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠸⠀⠳⠆⠈⠀⣄⠀⠀⠠⣛⠄⠀⡠⠔⠊⠀⢸⡫⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠁⠀⠀⠀⠀⠀⣧⡅
⡇⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⢱⠒⠒⠀⠐⠀⡆
⡇⠀⠀⠀⢀⠈⠉⠁⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡀⠀⠀⢸⠀⠀⠀⠀⠀⢹
⠓⠒⠒⠒⠂⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠀⠈⠉⠒⣏⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠈⠂⠀⠀⠰⡉⠉⠒⠂⠦⠤⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠐⠐⠒⠁
⣿⣿⣿⣿⣛⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣟⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠈⢳⠀⠀⡠⠄⢿⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠸⡆⠉⣹⠀⡜⠀⠀⠈⡦⠊⠈⠱⡀⠀⢠⠊⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠐⠻⠖⠁⠀⠈⢢⠀⢰⠀⠀⡀⠀⡇⠀⣊⠑⣠⣎
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⢢⠀⡀⠀⠀⠀⣄⠀⠳⡏⢀⠀⡇⢠⠃⡔⠞⠉⠀⣁⠀⢀⠐⠂⠲⡄
⠀⠀⠀⠀⡠⠤⢤⣸⠤⡴⠁⠇⠁⢄⠀⠀⡏⡧⡀⠱⡟⢠⠇⡼⠀⡧⠂⢀⠀⡏⠀⢘⢄⣀⡀⡇
⠀⠀⠀⢸⠀⠀⢠⢻⢢⢨⠏⠀⣠⣀⠑⠲⢇⡇⠱⣀⡅⡘⢀⡧⠊⣣⢔⣋⣰⠥⠆⠀⣉⠈⣨⠁
⠀⠀⡔⠉⢂⣤⠎⣤⣀⡉⠀⠠⣇⠀⠑⠢⠴⣑⣤⣶⡿⠟⢍⠉⠭⠤⠳⠤⢤⠤⠄⣼⡀⠀⠈⡆
⠀⢸⠁⠀⠀⠀⠀⢀⡨⠐⠒⠛⠊⠀⠀⠈⢉⠽⠫⡀⠕⢶⡰⣉⠒⠧⢀⣀⡎⠦⣀⣡⡭⠦⢼⠃⠈⢆
⠀⠈⢇⠀⠀⠀⢀⡎⠀⠀⣼⡤⠤⣔⡲⠚⣣⠊⢠⢃⠘⡆⠉⠊⠑⢦⡀⠈⠉⠉⣵⠋⠐⢄⠀⠀⠀⡸
⠀⠀⠈⢆⠀⢰⢡⢣⠖⢉⠭⠔⠈⠁⣠⡞⡡⢤⠃⠈⡄⠘⡄⠀⠘⣄⣙⣯⣐⠒⣿⡄⢀⣸⡄⡀⣰⠁
⠀⠀⠀⠘⡗⢶⢸⡇⠀⠀⠀⠀⣠⢊⠎⡜⠀⠀⡇⠀⢱⠀⠘⢄⢰⠋⠀⠈⡆⠀⠀⠀⠀⣇⡇⣧⠇
⠀⠀⠀⠀⡇⠈⠄⢷⣄⠀⠀⠀⣧⡇⠀⢧⣀⣠⠇⠀⠈⡆⠀⠈⠛⣦⣀⡴⡃⠀⠀⠀⠀⠉⢀⡏
⠀⠀⠀⠀⡇⠀⠰⡆⢹⠓⠂⠉⢸⡇⠀⠀⠉⠒⠈⠐⢄⢸⡀⠀⠀⠈⠉⠀⠑⢦⠀⠀⠀⢹⠋⠇
⠀⠀⠀⠀⠃⠀⠀⠀⡼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠸


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠘⠂⠀⠀⣰⣿⣿⡿⠋⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠏⡀⠀⠀⢹⣿⣀⣤⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢛⡁⣨⣽⠃⠀⠀⠀⡀⠀⠀⢀⣠⣶⣷⣿⣾⣶⣼⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠋⠉⠉⣠⣴⣿⣿⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠈⣿⣿⣿⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠂⠈⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠆⠀⠀⣠⣿⣾⡷⠶⡎⠉⠁⢹⣿⣿⣿⡟⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⣿⣿⠃⠀⠀⠈⠄⠀⠀⠀⠀⠹⠁⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⡟⠁⠀⠈⠹⡆
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣤⣼⣿⣿⡀⢀⠀⠈⠀⠀⠀⡀⠀⣤⢸⣿⡿⠿⠿⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠤⠀⢠⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠁⠀⠀⣼⣿⢷⣶⠄⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀
⠀⠀⠀⠀⢸⣿⡟⠛⠿⠿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢻⣿⣦⣤⣤⣤⡌⠀⠀⠀⣀⣀⣀⡀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⡿⠀⠀⠀⢀⣠⣴⣶⣦⣿⣿⣿⣾⡿⠀⠀⢠⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠘⢿⣦⡄⠀⠀⠙⠉⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠋⠻⣿⡉⠉⠀⠀⠀⣿⣿⣿⣿⠛⠻⠿⡟⠋⠉⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠙⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠠⣿⠇⠀⠀⠀⠀⢿⣿⠉⠁⡄⠀⣀⠀⠀⠀⠀⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⢀⡀⠀⠆⠀⠀⠀⣿⣿⡏⠙⠀⠐⠂⡿⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⣠⣿⠀⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠈⠁⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠲⠀⠀⠖⠀⠀⠀⠀⠀⣿⣿⡿⠀⢠⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠙⠻⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠣⠤⠔⢑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠒⠁⠀⢀⡀⠀⠈⠢⠤⣄⣀⠖⠤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡔⠁⠀⠱⠀⠀⡴⢪⡇⢸⡱⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉⡀⠀⠘⠤⠤⠊⠀⠀⠳⣄⣘⠖⢤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⢈⡆⠀⣀⡀⠀⡴⢊⡶⠝⠀⠁⠉⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠕⠈⠉⠁⠘⠵⢮⠀⠈⠆⠂⢀⠅⠀⡎⢉⠆⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⡐⠉⠑⠀⠀⡎⠀⠀⠀⠀⠈⠑⡄⣈⡁⡠⠬⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠤⠀⢠⠃⠀⠐⠤⠔⠁⢀⢉⠆⠀⠀⠀⢀⡀⣍⠀⠑⠁⠀⠀⠀⢠⠤⠋⠑⠊⡆
⠀⠀⠀⠀⠀⡌⠜⡠⡾⡍⢑⠀⡤⠤⣀⡅⢔⢄⠀⠀⠀⠘⣃⡬⠀⠀⠀⠀⠀⠀⡴⠀⢎⢉⠆⢨⠃
⠀⠀⠀⠀⠀⢣⠑⢴⠁⠟⠀⠉⠀⠀⠀⠀⠈⠪⡒⠀⠐⠂⢸⡆⠒⠤⣀⣀⢔⡲⠀⢇⠤⣀⡔⠚⢠⢖⣄
⠀⠀⠀⠀⠀⠀⠱⡀⢸⢈⡑⠠⢐⡩⠃⠀⠀⠀⠀⠀⠘⠀⢸⠁⠘⠐⠒⠒⠁⠀⠀⠀⢔⠤⡀⠀⠳⣰⡃
⠀⠀⠀⠀⠀⠀⠀⢃⡼⢠⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠆⡄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠨⠭⠥⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡳⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢐⠠⡈⢢⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⣉⡡⠂
⠀⠀⠀⠀⠀⠀⠀⡠⢃⠄⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⡄⢈⢹⠰⡄
⠀⠀⠀⠀⠀⠀⣎⠔⠁⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠘⡈⠢⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠃⠀⠈⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢸⠀⠀⠰⢿⠿⠿⠿⣿⣧
⠀⠀⠀⠀⠠⠔⠂⢠⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍
⠀⠀⢀⠀⠀⠄⡀⠀⠆⠀⠀⢳⠀⠀⠀⢀⣤⠄⣽⣿⡒⠒⢲⡄⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠤⣀⡀⠀⢄
⠀⠀⠀⠀⠀⠂⠀⠀⠇⠀⠀⢸⠀⠀⢀⡼⠋⠀⠛⠓⠶⣿⣾⣇⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⣙⢦⠈⠆
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⣻⡴⠆⠀⠘⠃⠙⢻⣦⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⣿⣿⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢤⣿⡀⠀⠤⠄⠀⢈⡼⣿⣿⡀⠈⠈⠀⠀⠀⠀⠀⢄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠛⠟⠁⠈⢂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢄⠀⠀⠀⠂⣳⣹⣆⠀⠀⠈⠀⠑⢿⣮⡿⠀⠀⠀⠀⢀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠦⠈⠃⠀⠀⠀⡿⣿⠏⠀⠀⠀⠀⢠⣿⣯⠁⠀⠱⠄⠀⠋⠁⠀⠀⠀⢱⣆⠀⠀⠀⠀⠀⣀⣀⠀⠈⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠊⠀⠂⠂⠀⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠙⠧⠀⠀⢀⣄⢀⠀⠀⠀⣀⣠⠎⠟⠀⢀⣀⣔⣛⣛⡛⡳
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⣿⣶⣤⣾⡿⠿⠛⠀⡠⠀⡿⠟⠿⠿⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⠉⠛⠋⠀⠀⠀⠀⠀⠀⠴⠀⠐⠓⠂⠈⢛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢀⠀⠀⠀⣠⣤⣄⣶⣶⣄⠑⠀⠀⠄⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡄⠀⠈⠀⠀⠀⠀⠱⠀⠀⠀⠈⡆⠀⣼⣿⡿⠟⠻⣿⣿⣄⠁⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣻⣻⣿⣟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⢠⡟⠁⠀⠀⠀⠈⠻⣿⣧⠀⠀⠀⠀⠀⠀⢱⠒⠀⠠⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣷⣼⣇⠀⠀⠀⡬⠄⠆⠀⠀⠀⡀⠀⢸⡇⠐⠀⠀⠀⠁⢀⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠉⠋⠉⠉⠉⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠠⠄⠀⣼⣟⢿⣿⡆⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠀⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠩⣿⣇⠀⠀⠘⣯⣿⡂⡿⠟⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⢉⡀⢀⡀⠀⢉⠀⠉⠀⠀⠀⠁⠀⠂⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⠄⠀⢀⡀⠀⠘⣀⢀⠀⣀⣠⡀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣍⠀⠀⠀⠀⠀⠉⣤⠀⣀⠀⠀⠄⡄⠉⠀⠉⠉⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⣀⠀⠀⡀⠀⢠⡄⠀⠀⠁⡀⠀⣰⠣⠀⠀⠀⠤⠀⠀⡠⠔⠁⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠆⠉⠀⠀⠀⣀⠔⠀⣀⠾⠭⠄⠁⠁⠉⠈⠠⠄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠓⠀⠀⠠⠀⠀⠂⠁⠀⠀⠚⢅⡀⠁⠠⡀⢀⠄⠀⢀⡀⠀⠐⠂⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠂⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠒⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠖⠚⠛⠉⠁⠉⠉⠉⠑⠒⠠⠄⣀⣀⣀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠶⠶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣇⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢹⣿⣄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣄⡈⣿⣇⠐⣄⠀⡀⠀⠀⠈⠀⢀⠀⢀⠀⠀⢰⠃⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣝⠿⣿⣿⣦⡈⠳⡌⢄⠀⠀⠀⠀⡇⠘⡀⠀⢧⠤⠤⠿⠦⠄⠀⠀⠀⠀⢀⡀⠃⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⣿⣿⣿⣆⠸⠈⢠⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⠀⠀⢀⣀⡤⠴⠒⠒⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⢆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⡴⠞⢋⠁⠀⠤⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⡔⢠⠿⠿⣷⣦⡀⠀⡀⠀⠀⠀⣠⣴⣶⡶⣦⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣿⠐⠋⠀⠀⠀⠈⠙⠛⣤⠓⠀⠁⠾⠟⠁⣀⠀⠈⠀⠀⣿⠛⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡙⢿⡿⢿⠃⠀⠀⢀⣄⠀⠀⣼⡆⠀⠘⠀⡄⣡⡈⢳⣄⠂⠀⠈⢴⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢁⣸⣧⡄⣩⠄⠀⢌⣉⠅⠈⣿⡁⠀⠀⠀⠤⠉⠉⠄⠈⠁⠀⢤⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⠀⠀⠀⠀⠀⠀⠀⠸⣧⢋⢻⣿⣿⣤⠰⠀⠀⣀⢀⡞⣿⠋⠀⠁⠀⠀⣀⣠⠀⠀⠀⠀⠻⠀⡸
⠀⠀⠀⠀⠀⠀⠀⡴⠳⠄⠞⠉⠁⣼⠆⢀⣀⡀⠀⠀⠀⠀⢻⠈⢸⣿⢿⣿⠖⢾⡿⢯⣌⣿⣿⠷⠆⠘⠈⢹⣿⣿⡇⠀⣆⠀⠈⠟⠁⠲⣄
⠀⠀⠀⠀⠀⡠⠊⠀⣸⠀⠀⡺⠉⢀⣀⣈⡨⠃⠀⠀⠀⠀⣸⣦⣼⡇⠈⡟⠀⠈⠧⠠⣬⣉⠻⠶⢰⣾⣛⣧⠄⠋⠁⠀⢸⠀⠀⢰⠀⠀⠘⢷⣤⣄
⠀⠀⠀⠀⢰⠁⠀⠞⠁⣠⡞⠀⢀⡘⠉⠀⠀⡔⠳⣀⣠⣾⣿⡿⢿⣇⠀⢿⠀⠀⠀⠀⠈⠻⣶⣤⣤⣴⠞⠁⠀⠀⠀⠀⡈⠀⠀⢸⠀⠀⠀⠀⠀⡟⢦
⣠⣄⡀⠀⡌⠀⢀⡄⠀⠁⢸⠀⣩⣅⣠⣤⣶⡀⠀⠙⡏⢸⣿⠁⠸⣿⣆⠘⢧⡀⠀⠀⠀⠳⣤⣉⣁⡅⠀⠀⠀⠀⠀⡰⠁⠀⢠⠎⠀⠀⠀⠀⠘
⢣⡀⠘⢾⡇⠉⠉⢱⣄⡠⠀⠉⠁⠸⢿⣿⡿⡿⠦⠀⣧⠸⣿⡆⠀⠻⣿⣄⠀⠙⢦⣀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣠⠞⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢫⡀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⢄⡀⠟⠀⠻⢧⠀⠀⠈⠻⣶⣦⡀⠉⠳⢦⣀⠀⠀⠀⣀⠔⠋⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠈⢳⡄⠀⢙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠉⡆⢠⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠦⣤⣙⢛⡛⢛⣁⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⠎
⠀⠀⠀⢳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠾⠿⠷⠶⠦⡤⢀⠀⠈⢫⡉⣉⠅⠈⠀⠀⠀⠀⠀⢀⡀⠀⢀⠔⠁⡴⠃
⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣌⣀⣀⣉⢀⣀⣀⣤⣤⣴⠶⠞⢋⣠⠖⢃⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠈⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠈⠛⠿⠶⢶⣶⣤⣤⠶⠶⠖⣿⡟⠙⠻⠿⣯⣭⣭⣤⣴⡶⣛⣩⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⢠⣤⠀⠀⠀⠀⠀⠀⢼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⡰⢉⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣬⡘⠇⢷⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢹⣞⢳⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠆⣎⣱⠟⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠸⣾⠟⠀⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡍⡉⠀⠀⣼⡰⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢰⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⢿⣁⠀⣀⠠⠛⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⢠⠇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠙⠫⡀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⡴⠃⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣠⠞⡄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠞⠁⠀⢰⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠈⣧⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣄⠀⠀⠀⠀⠀⣿⣿⣿⣦⡀⠀⣀⣤⡄⠀⠀⠀⠀⠀⢰⢰⢰⢰
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢻⣿⣿⣿⣧⣾⣿⣿⠇⠀⠀⠀⠀⠀⢸⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⣾⣸⣸
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠃⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⢼⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠁⠀⢀⣾⣿⣿⡿⠟⠋⠉⠉⠉⠛⠻⣿⣿⣿⣦⡀⠀⠈⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠹⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠃⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠙⣿⣿⣿⣦⣄⡀⠀⠀⠀⣀⣠⣶⣿⣿⡿⠃⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣻⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣃⣤⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⡿⢿⡿⠋⠛⠿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣁⠀⠀⠀⠀⠈⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠰⠖⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀⣠⣤⠴⣋⠜⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⣻⡿⠋⢹⣁⣰⣊⣁⣾⡿⠁⠀⠉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢿⡀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠻⡄⠀⠀⣾⠏⢙⣿⣿⣶⣶⣶⣿⣿⡇⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢠⣇⢀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢠⠴⣿⠴⠞⣿⠀⢐⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠹⣦⣤⠖⠚⠛⠛⣾
⣾⠀⠀⠀⠀⠸⣏⠻⣆⠀⠀⠀⠀⣸⠃⠀⠀⣰⠋⠀⠀⠀⠀⣿⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢰⡟⢳⠀⠀⠀⠀⢸
⣿⡇⠀⢀⠀⡴⠿⠀⠸⣿⠀⠀⢰⠃⠀⢀⡜⠁⠀⠀⠀⠀⠀⣿⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣀⡼⠓⠉⠀⠀⠀⠀⢸
⢿⣿⡄⢸⣆⢃⠀⠀⢀⡏⢀⡴⠃⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠁⠀⠀⠀⠀⠀⣠⡴⠋
⠸⡄⢻⣞⣸⣦⠑⢒⠋⠀⡏⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠲⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠹⣿⠙⠿⠹⣆⣾⠀⣀⢧⢠⣶⣠⠏⠀⠀⠀⠀⠀⠀⠀⣴⠃⣰⣾⣿⡿⠋⠁⠀⠀⠀⢀⣠⣴⣶⣏⠁
⢀⠀⢹⣆⠀⠀⢻⡟⣇⣿⣮⡏⠋⠁⣤⠀⠀⠀⠀⠀⠀⡼⠁⣸⣿⣿⡿⣶⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡀
⢸⣷⣨⣿⡄⠀⠈⠃⠘⠛⠘⠧⣤⡾⠋⢷⡀⠀⠀⢀⡞⣁⡿⠟⣋⣥⣴⣃⣜⣴⣤⣼⣿⣿⣿⣿⣿⣿⣇
⠀⣇⠙⠛⠃⠀⠀⣌⣧⣀⢠⡀⠀⠀⠀⢸⣿⣿⣷⣾⣛⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢹⡄⠀⠀⠀⢸⣾⣿⠟⠿⣿⣆⡀⠀⣸⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢹⣦⡀⠀⠘⢿⣧⣀⣀⣀⡿⢧⣐⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠈⢿⠿⢻⣶⣬⣭⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠳⢤⡈⠻⢿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⣶⣶⡶⣶⣶⣶⣾⣷⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠚⠛⠋⠿⠶⣤⡀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠐⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⡅⢠⢈⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⡇⠘⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⡇⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠘⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠠⠂⠀⠀⡇⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⡄⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠄⢈⠑⠢⠤⡄⢀⡀⠤⠤⠖⠻⠂⠀⠀⠀⠀⠀⠃⠀⠢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣇⣿⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠤⠔⠢⢤⣀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡤⣤⢤⠀⡠⢠⣄⢀⣄⢀⣀⠀⢠⣤⡀⣠⠄⡤
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠁⠉⠉⠉⠉⠈⠁⠁⠈⠉⠀⠉⠉⠁
⠀⠀⢀⡴⠊⠁⠀⠀⢀⡤⣄⠀⠀⠀⠀⡼
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠙⠒⠒⠊⢳
⠀⠀⠈⠓⠒⢷⠒⠂⠸⠤⠖⠉⠛⣄⢀⡮⣀⡀
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⡨⠋⠀⠀⠉⠓⢤⡀
⠀⠀⠀⢀⠎⠈⠙⠒⠒⠒⠒⠋⠁⢀⠀⠀⠀⠀⠀⠀⠘⢉⡆
⠀⠀⠀⡜⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠑⠄⠀⠐⠠⠀⣀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠃⠀⠀⠀⠀⠀⢳⠤⣀⣈⡠⢄⣀⠈⣀⣤⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠈⡄⢀⠀⠀⠀⠀⠀⢧⠀⠤⠀⠀⠈⠉⠁⠐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢐⠀⠀⠀⠀⡀⠠⠤⣌⠢⡀
⠀⠀⠀⡇⠠⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠉⠉⠉⢨⠀⡠⡖⠁⠀⠀⠀⠀⠀⠈⡖⢄
⠀⠀⠀⡇⢠⠀⠀⡆⠀⠀⡇⠀⠀⢀⡀⢀⡠⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠳⡌⠉⠭⠭⠭⠭⠍⠋⠥⠎
⠀⠀⢴⠃⠠⠀⠀⠀⠀⢀⣇⣀⣀⠀⠈⣉⣠⣴⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⡀⠀⠀⠉⠳⣄⣠⣄⡀⢸⠉⠙⢏⠙⠛⢡⡟⢦⡀⠀⣀⠄⢢⢧⠄⠀⠀⠀⡀⠀⠇⠀⠀⠀⠀⣤⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢘⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠘⡏⠉⠀⠰⠚⠂⠀⠀⢴⠋⠂⠄⡀⠀⠀⠐⠂⠤⠔⠒⣁⠀⣇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠶⣼⣄⣀⠤⢞⡥⠒⡂⠉⠁⠙⠁⠀⠊⠓⠲⠦⣅⠀⠀⠁⠀⠀⠀⠈⠀⠈⠗⠀⣤⠀⠀⣸⡀⠀⠀⠀⠀⠀⢸⠗
⠇⠀⠀⠀⠀⠀⠈⠂⡤⠏⠡⡀⠈⠑⠨⠑⣒⠀⠀⠀⠀⠒⠒⠂⢁⠅⠀⠀⢠⣀⣀⣀⣠⠀⠰⡀⠀⠇⠀⡀⠩⣷⣦⣤⣤⣤⣶⡧
⡆⠀⠀⠀⠀⠀⠀⡔⢸⠁⠀⠘⠢⠄⡀⠀⠀⠰⢁⠀⠀⠀⢀⡠⠂
⠁⠀⠀⠀⠀⠀⠀⣗⠃⣀⠀⠀⠀⠀⠈⠐⠒⠚⡳⣶⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⡬⠤⠭⢭⡭⠄⡾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠙⢄⠀⢀⡾⠄⡀⢥
⠁⠀⠀⠀⠀⢀⠞⠁⠉⢦⠤⠉⣽⠂⠠⠐⠚⡦⡀
⠁⠀⠀⠀⠀⠘⡀⠀⠀⠘⢄⣀⠼⡄⠀⠉⠈⠀⠈⠆
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⡳⠄⣀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠤⠚⠁
⣶⡶⢶⢶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⢶⣶⣶⠶⠶⢶

⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡤⣬⣀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠩⠀⠀⠂⠰⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠠⠀⠀⣀⠀⢐⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠠⠐⠈⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠠⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠰⠁⠀⠐⠊⠀⠈⠀⠈⠁⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⠠⠐⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⡠⠊
⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣽⣽⣽⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣷⣾⣿⣷⣶⣶⣶⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣴⣿⣿⢟⣫⣭⣭⣭⣛⢿⣿⣷⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⣼⣿⡿⣱⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣿⣧⢻⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⠁⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠈⢿⣿⣷⣝⣻⠿⠿⢟⣫⣵⣿⡿⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⣴⣷⣦⠀⠀⣶⣶⣶⠀⠀⢀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⢀⣼⣿⣶⠄
⠀⠀⠀⠀⠈⠻⣿⠟⠀⠀⠻⣿⠏⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣿⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣠⠟⠋⠁
⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⡄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠈⢿⣆
⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠸⣿⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃⠀⠀⠈⠿⠇

⣿⣿⣿⡿⠛⠋⠀⠙⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⡟⡠⠄⠀⠀⠤⡌⢿⣿⣿⣿⡀⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⣄⡀⠀⠀⠀⣰⣶⣾⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⣠
⣿⣿⡇⠀⠀⠀⠀⠈⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣿⣷⣶⣤⣶⣼⣿⣿⣿⣿⣷⣴⣶⣶⣴⣄⡀⢠⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⡿⠿⢿⣿⣿⡟⣻⣿⢿⢿⡿⣛⠿⠛⣿⣿⣿
⣿⣿⣿⣦⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠋⠀⠀⠀⠀⠹⣿⡏⢹⢁⣤⡄⠀⠀⣤⣄⠹⣿⣿
⢿⣿⣿⡿⠀⠀⠀⠀⢹⣿⣿⡿⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠐⠀⠀⠀⠐⠀⠀⠙⠿
⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠙⠛⠋⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⣀⣀⠀⠀⡠⠤⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣠⣶⢗⣀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠐⠂⠀⠀⠒⠊⢹⣿⡄
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠁⠀⠀⠈⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶
⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⣿⢋⣀⡀⠈⣩⣍⢻⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣶⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿
⡟⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣿⠀⠀⡀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐



⠀⠀⠀⣠⣤⣤⣤⣄⣀

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⣤⠐⠀⠂⡉⠀⠀⠤⠀⠀⠀⠀⠐⠁⠈⠉⠁⠄⠤⠀⡀⠀⠀⠀⠀⡎⠀⠀⠐⢀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⡀⠧⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⢂⡀⠀⠄⠀⠀⠀⠆⠀⠀⠀⣿⣻⡟⣿⣿⣿⡇⠀⠀⠀⡀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣿⣫⣵⣽⣿⣿⠇⠀⢠⠀⠀⡠⠀⠃⠀⠰⠀⠈⢀⡀⠀⠀⠠⠄⠀⠀⠀⠀⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠄⡀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠛⠋⢉⠀⠀⠘⠉⠀⠉⠀⠘⠅⠀⢀⠄⠀⢁⡀⠰⡂⠀⢠⠀⢈⡠
⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠁⠀⠀⠀⠀⠰⠞⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⡄⠀⠀⠀⠀⠀⠘⠀⡀⣰⠂⠉⠠⠀⡇
⠀⠀⠀⠄⠀⠀⠀⡾⠃⢈⣢⡤⠄⠂⠨⠁⠠⠐⢧⡠⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⢀⠜⣠⣇⠀⠀⠀⠀⠀⠉⠂⠠⠤⢀⠀⠀⠀⠀⠠⣴⠄
⠀⢆⠀⠀⠀⠀⠀⣄⠁⠈⠁⠀⡋⠈⠀⠀⠐⠀⠁⠀⠀⠀⠎⠀⡀⠀⠀⠀⡀⠰⡆⠀⠈⢲⠓⠑⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⢁⠀⠀⠂⠀⠀⠀⠀⠀⠃⠈⢁⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⣀⠀⠀⠀⠀⠠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠱⢀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⡀⠂⢀⠀⠁⠀⡀⠈⠀⠀⠀⠠⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠠⠠⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⠀⠁⠀⠀⣀⣀⣤⠠⡀
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⠀⠀⠀⠀⠀⢀⠀⠤⠃⠀⠀⠀⠀⢀⣲⡄⢛⢄⠣⡜⢡⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠒⠃⡤⠀⠀⠀⠀⠀⠀⠀⣼⡿⠏⠀⠀⠀⢌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠈⢣⠌⢱⡜⢣⠄
⠀⢀⣀⣠⠀⠀⠀⠀⣧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠊⠤⠐⠀⠃⠈⠀⠀⡀⠀⠀⠀⠀⢠⠤⡄⠀⠀⠀⠐⢀⡢⢠⡀⠛⣠⠚⢡⠘⠻
⠀⠀⠈⠀⠀⠀⠀⢹⢸⣞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠍⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⣄⠰⢆⡀⠦⣀⠶⣀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⢀⢱⡇⠀⠀⠀⠠⠶⠤⠀⠀⠀⠤⠀⡀⢀⠔⠊⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠤⠈⢡⡔⢁⠔⠉⡦
⠀⠀⠀⠇⠀⠀⠐⠺⣙⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠁⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⠹⣄⠘⣀⠜⢃⡐⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⢂⣀⣤⣄⡀⠂⡠⠀⠀⠀⢀⣄⡀⠀⠀⢈⡄⠀⠀⠀⢨⠉⠀⠀⠀⠀⡙⣈⠖⣀⠲⠀⠠⢴
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⣿⠁⠀⠀⠀⢐⡀⠃⠀⠀⠀⠂⠀⠠⠩⠀⠈⠁⣸⡧⠄
⠀⠀⠀⣀⠤⣒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣹⣯⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣠⡗⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠋⠉⠀⠀⠀⠀⠀⣇⠐⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⡀
⢰⢎⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⢲⠀⠀⠀⠈⠀⢀⠀⠀⠀⡀⠌⠀⠄
⢸⠀⠀⠰⠄⢀⡀⠠⢆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠂⢀⠠⠖⠁⠘
⢸⠀⠀⠀⠀⠈⡠⠑⠣⠀⠀⠀⠀⢨⠁⠀⣀⣀⣠⣤⣤⣤⣤⣴⡄⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠀⡀⠀⠈⠀⠀⠀⢰
⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⣿⣟⣿⣿⣟⣻⢿⣽⠻⣿⣧⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡄⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠉⢀⠀⠀⠀⠀⠀⣿⣧⣿⣝⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠰⠀⠀⠄⠀⡤⠤⠀⠀⠀⠀⠈⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠀⠀⠀⠀⠸⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⡇⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠈
⠀⢇⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⢓⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⢀⠀⠀⠂⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠑⢄⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇⠈⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⢳⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⣾⠀⠀⢸⡏⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⠀⠀⢸⡇⠀⢸⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⢸⡇⠀⢸⠁⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⠀⠀⢸⡇⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⢸⡇⠀⢸⠀⠀⢸⡇⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢳⣸⡇⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡤⡀⠀⠀⠀⠀⣧⠀⠈⣿⡇⢀⡤⠶⢤⣀⣤⠴⢤⣄⠀⢸⡇⠀⠀⠀⠀⠀⢠⠤⣄
⠀⠀⠀⣠⢤⡇⠀⣸⠚⢶⠀⠀⢹⡄⠀⠸⠃⡟⠀⠀⠀⠈⠀⠀⠀⢸⡆⢸⡇⠀⠀⠀⣠⢦⣧⠀⢸⠛⢳
⠀⠀⠀⡇⠀⡇⠀⢻⠀⢸⡤⣄⠈⣇⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⢀⡼⠁⢸⠁⠀⠀⠀⡇⠀⣿⠀⢸⠀⢸⡦⣤
⠀⠀⠀⡇⠀⡇⠀⢻⠀⢸⠀⢸⡆⢹⡄⠀⠀⠀⠈⠳⣄⠀⣀⡴⠋⠀⠀⣼⠀⠀⠀⠀⡇⠀⣿⠀⢸⠀⢸⠀⢸
⣠⣄⠀⡇⠀⡇⠀⢼⠀⢸⠀⢸⡇⠀⢻⡄⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⢀⡏⠀⣠⣤⡀⡇⠀⣿⠀⢸⠀⢸⠀⢸
⣿⠈⢷⡇⠀⠁⠀⠀⠀⠘⠀⢸⡇⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢿⠈⢳⣗⠀⠀⠀⠀⠀⠈⠀⠸
⢸⡆⠘⡇⣰⠞⠛⠶⠛⠳⣆⢸⡇⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆⠘⡿⢰⠛⠙⠲⠛⠛⣆
⠀⣧⠀⠀⠹⣄⠀⠀⠀⣠⠏⢸⡇⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣧⠀⠀⠘⣦⡀⠀⢀⣰⠏⢸
⠀⠸⣆⠀⠀⠈⠳⠶⠞⠁⠀⢸⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠘⣆⠀⠀⠀⠙⠶⠛⠁⠀⢸
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⣶⣶⠶⠶⢾⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣾⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⢀⡀⣀⢀⡄⠀⠀⣀⣀⠀⢀⣀⢠⡀⣀⣀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣗⡇⢿⣼⢹⣏⡁⣿⡿⠃⠻⢯⣸⡎⢹⡏⠉⢧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠘⠇⠘⠋⠸⠷⠆⠛⠛⠀⠷⠾⠛⠃⠘⠃⠘⠻⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⡄⠀⠀⠀⣤⣤⣦⣤
⠀⢠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠈⢿⣿⣿⠿⠁⠀⠁⣀
⠀⠘⡙⠛⢻⠇⢀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠸⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠚⠂⠀⠀⢀⣼⣿⣦⡀
⠀⠀⠀⠀⠘⠀⠀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⢷⡀⠀⠀⠀⠀⢹⣿⠛⠛⠀⠀⠀⠀⢠⡾⠛⢷⡀⠀⠀⠀⢠⣿⠁⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣶⣻⠇⠀⣠⣴⣶⣄⠀⠀⠙⠒⠝⠚⠃⠀⣠⣴⣦⡌⢳⣄⣢⠀⠀⠀⠀⢸⡀⠀⢸⡇⠀⠀⠀⡈⠉⠁⠀⠈⠉
⠀⠀⠀⣴⣶⣦⡄⠀⠀⠙⠋⠀⠀⠀⠿⠛⠻⠿⠀⠀⠀⠀⠀⠀⠀⢸⡿⠅⠈⡿⠀⠈⣁⣀⡀⠀⠀⠀⠑⠒⠊⠀⠀⠀⢀⣿⡿⠿⠗⠀⠀⠀⣠⣤⡀
⠀⠀⠸⠏⣮⣹⠟⠀⠀⣠⣤⣦⣄⠀⠘⢶⠶⠂⢠⣴⣶⣶⣤⠀⠀⢸⣿⡄⠀⠀⢠⣾⡿⠿⢿⣷⡀⠈⢀⣴⠶⢦⡄⠀⣼⣿⡀⠀⠀⠀⠚⠛⣿⡿⠃
⠀⠀⣰⡟⠙⠉⠀⠀⠀⢫⡬⢡⡍⠀⠀⠀⠀⠀⢿⡟⠛⠻⣿⠃⠀⠸⢿⠗⠀⠀⠸⣿⠀⠀⢀⣿⠇⠀⠈⠳⣂⠶⠁⠀⢿⣿⠇⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠶⠴⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠁⠀⠀⠀⠙⠿⠞⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡈⠉⠃
⠀⠀⠀⠀⠀⢀⢀⠁⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠸⠿⠿⣿⣼⠛⠿⠃⢸⣿⠟⠦⠾⠿⠃⠀⠀⢀⣿⣿⣧⠀⠀⠀⢸⣿⣿⡀⠀⣸⣿⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣤⣤⣤⣤⠀⠀⠀⣾⡿⠘⣿⣆⠀⠀⢸⣿⢻⣧⢀⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢀⡆⠀⣡⣿⠀⠀⠀⢸⣿⠉⠉⠉⠉⠀⠀⣼⣿⣷⣤⣿⣿⡄⠀⢸⣿⠘⣿⣼⡿⢸⣿⣿⣏⠘⠃
⠀⠀⠀⠀⠸⡿⠀⣿⣿⢀⡤⠂⢸⣿⣶⣶⡶⣶⡆⣰⣿⢏⢱⣤⠬⢻⣿⡀⢹⣿⠀⢿⣭⡇⢸⣿⡿⠇
⠀⠀⡀⠀⠀⠀⠀⠀⢠⠞⣀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠
⠀⠀⠀⢠⣦⠀⠀⢠⣤⠄⠀⠀⣠⣤⠀⠀⣠⣴⣶⣦⣤⡀⠀⠀⣤⣤⣤⣤⣤⣄⠀⠀⣠⠈⣇⡀⢀⣤⣤
⠀⠀⠀⠸⣿⡇⠀⡾⠁⢀⡀⢀⣿⡏⢀⣾⡿⠉⠁⠉⠋⠹⡄⢀⣿⡏⣩⣭⠙⣿⡇⠀⣿⣶⢉⣴⣿⠟⠁
⠀⠀⠀⠀⢿⣷⢸⣶⠀⢿⣧⢸⣿⠁⢸⣿⡇⠀⠀⠑⢄⣾⡇⠀⣿⣷⣾⣿⣿⠟⠃⠀⣿⣿⣿⢿⣷⡀
⠀⠀⠀⠀⠸⣿⣿⡏⠀⠸⣿⣿⡏⠀⠈⢿⣷⣀⣀⣀⣼⣿⠃⠀⣿⡇⠀⠙⢿⣷⡀⠀⣿⣿⠁⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠛⠛⡡⠄⠀⠛⠛⠁⠀⠀⠀⠙⠛⠿⠟⠛⠁⠀⠀⠛⠃⠀⠀⠈⠛⠛⠀⠛⠛⠀⠀⠀⠛⠛⠂
⠀⠀⠀⠀⠀⠐⠉⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⡅⣙⠿⢷⡦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡀⠘⠿⢻⡿⠁⠀⠻⠛⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣿⣇⣠⣾⣿⣿⡰⠶⠺⠾⢿⣿⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⢿⡟⢛⣶⡦⠤⠛⠁⠀⣪⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⠈⠁⠀⢀⣠⣶⣾⣿⣿⣿⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⠿⠿⠟⠛⠉⠉⠉⠉⠉⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠛⠶⠤⠤⣤⣠⢠⣷⠾⠿⠛⠉⠀⠀⠀⠀⠀⠈⢻⢂⢠⣿⠿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⣾⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠠⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⢶⡚⣻⠻⢿⣼⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡉⢋⣉⢽⣬⡀⢠⠐⠂⣤⢙⣿⣃⢿⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠐⣻⢒⡼⢖⠦⣰⡤⣄⣘⣥⡀⢠⣿⣾⣿⠿⢧⣿⣶⡏⠺⢿⣿⡆⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⣯⠁⠀⠈⠛⠁⡰⠼⠋⣸⠴⠋⠀⠻⠿⡄⠀⠙⢻⣿⠀⠀⣾⠇⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣻⠀⠀⡄⠀⠀⠀⠀⠀⣿⢀⡀⠀⠀⣤⣧⠀⠀⣾⣿⡀⢸⣿⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣿⠀⠀⢶⠀⠀⣿⢸⣧⡇⠀⣿⡿⡀⢀⣿⡿⠀⣸⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢹⡇⢀⣸⠀⢀⣿⣿⣷⡇⠀⣿⣿⡃⢸⣿⣿⠨⣿⢿⣿⣷⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⣾⣧⢘⣿⣿⢸⣿⢼⣿⣇⢸⣿⣿⠀⣼⣿⣧⢸⡿⣾⣿⣯⣔⣢⣶⡧⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⣿⣿⢩⣿⣿⡅⣿⣾⣻⣷⣾⣿⣻⠀⣿⣿⣟⢸⣾⣿⣿⣗⣮⣥⢴⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠤⠿⣿⣅⣉⣽⣿⣋⣹⣷⣿⣥⣿⡯⢽⣿⣿⡿⢿⣿⣿⢿⣧⠴⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⣿⠿⠷⠛⠓⠛⠛⠁⠤



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠉⠉⠉⠉⠉⠒⠂⠤⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⡠⠤⠒⠋⠁⠒⠒⠤⠤⠤⠤⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠓⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠠⠤⠤⠤⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠐⠚⠒⠒⠢⠤⣀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠘⢅⠀⠀⣠⠯⡂⠀⠈⢘⡇⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⢄⡸⠀⠀⠁⣀⠀⠀⠀⠀⡴⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⠇⠈⢩⠆⠀⠀⠙⠚⠀⠀⠀⢀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⢸⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢇⠀⢀⠀⠀⠀⠀⠀⠁⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⢢⠀⡟⠢⢄⠈⢦⠘⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⣇⡚⠣⡀⠀⠱⡄⠑⢤⡀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠀⣈⡆⢙⠀⡐⠀⠀⡎⢰⣀⠶⠋⢸⠃⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠉⣁⣼⢋⡰⠀⠀⠀⠘⠢⠙⢦⡔⠁⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⢀⣀⣡⠼⡈⢇⠀⠀⠀⠀⠀⠀⣠⡣⢄⠤⢰⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠔⠁⢰⠋⡌⢆⣠⠴⠖⠋⢉⡠⠖⠁⠀⢹⣇⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠴⠛⠈⢀⣠⠴⠚⠁⠀⠀⠀⠀⠈⡟⢆⢀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣉⡠⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⢰⠋⢷⣧⠞⠁⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣅⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠒⠉⠀⠀⢈⣇⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⢤⣀⣀⣰⡯⣤⣀⠤⠄⠀⡀⢀⡴⠛⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣀⠀⠀⠀⠀⠀⠙⠓⠛⣛⡲⠯⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠚⠛⠋⠉⠉⠙⠛⠓⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠉⡀⡀⡀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⡇⡇⡇⡇⠀⣀⢀⣿⣿⣀⡀⢀⣾⣿⣿⣿⣆⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢠⡇⡇⡇⡇⠉⠁⢸⣿⣿⠀⠀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⣠⠖⢸⡇⡇⡇⣷⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⢀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⢀⡞⠀⠀⠀⠀⠀⣠⠊⠁⠀⢸⣧⣇⣧⣿⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠱⣄⠀⠀⠀⠀⠀⢳⡀
⠀⢀⡟⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⡟⠀⠀⠈⢣⡀⠀⠀⠀⠀⢻⡀
⠀⡾⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠀⢀⣿⣿⣿⠀⠀⠘⠻⣿⣿⡿⠛⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢷
⢸⠃⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠘
⣾⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⡏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠻⣿⡿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢿⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⢸⡄⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢠
⠀⢷⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⡾
⠀⠈⣦⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⡼⠁
⠀⠀⠈⢧⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠙⠦⣀⠀⣿⣇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⡄⣀⠴⠋⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⢹⣿⣿⢄⣀⣀⢸⣿⣿⠀⣀⣀⡠⣾⣿⡏⠁⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⢸⣿⣿⠀⠀⠉⢹⣿⣿⠉⠁⠀⠀⣿⣿⡇⠀⠀⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠉⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⢤⣤⣤⣀⣀⣤⣤⡤⠴⠖⠚⠋⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢰⡇⣾⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣿
⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇⣿⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠛⠶⢦⣄⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠙⠈⠁⣿⠀⠀⣴⠟⠁⠀⢀⣠⣤⣤⣤⣄⡀⠀⠉⠳⣄⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣄⢀⣴⠏⠀⣾⠃⠀⣠⡾⠋⠁⠀⠀⠀⠈⠙⢷⡄⠀⠙⣧⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⣼⠃⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢸⡇⠘⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣷⠀⣷⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⢿⡀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢰⡏⠀⢸⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠘⣧⡀⠀⠻⣦⡀⠀⠀⠀⠀⠀⢀⣴⠏⠀⢀⡾⠁⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⡇⠀⠀⠈⢷⣄⠀⠈⠙⠳⠶⠶⠶⠞⠋⠁⠀⣠⠟⠁⠀⠀⢸⠃⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣿⠀⠀⠀⠀⠉⠳⢦⣄⣀⣀⣀⣀⣀⣤⡴⠟⠁⠀⠀⠀⠀⣿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡾⠃

⠀⠀⠀⠀⠀⢀⡠⠔⠒⠀⠀⠐⠒⠤⣀
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⢀⠎⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡄⠀⠀⠦⣀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀
⠀⠀⡘⠀⠀⠐⢤⢀⠆⠀⠀⣴⢀⠀⢤⢀⢀⠘⠀⠀⠀⠀⠀⢼⠃⢀⣀⡀⠻⣷⣄⠀⠀⠀⣠⡀⠘⣯⠀⠀⠀⠀⠀⠀⠀⠈⢀⡏⢮⡁⠄⢰⣣⡀⠰⣀⣠
⠀⠀⢃⠀⠀⢀⡎⢈⡶⠈⠆⠚⠒⠀⠈⢩⠃⢀⠀⠀⠀⠀⠀⢿⡀⠘⠛⠀⢀⠈⠻⣷⣄⠀⠛⠁⠀⡯⠀⠀⠀⠀⠀⠀⠀⠤⠎⠐⠊⠉⠉⠁⠉⠀⠀⠀⠃
⠀⠀⠘⡄⠘⠊⠀⠁⠀⠀⠠⠐⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠹⢇⠀⠀⠔⠃⠀⠀⠈⠙⠦⠀⠀⠨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠦
⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠑⠦⢄⣀⠀⠀⢀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⣠⣾⠟
⠀⠀⠀⠀⡀⠀⠀⠈⢿⣦⡀⣠⣾⠇⠀⠀⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣛⣸⣷
⠀⠀⠀⢨⡇⠀⠀⠀⢠⣿⣑⣿⡇⠀⠀⠀⠘⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⣷⣾⣿⣧
⠀⠀⠀⢾⡆⠀⠀⠠⢾⣿⣷⣿⣷⠄⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦
⠀⠀⠀⢘⣧⠀⠀⢀⣾⣿⣿⣿⣷⡄⠀⠀⢬⠃⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠿⣷⣄⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⢠⡤⣄⠸⠿⢿⠏⢹⡿⠿⠀⡀⢤
⠀⠀⠀⠀⠘⠁⠀⠘⠿⣿⠛⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⠻⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⣻⢳⢳⢻⡟⡟⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠿⠴⢦⢾⡼⠷⠷⠿⠿⠈⠁


⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢰⠂⢠⡄⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠘⠒⠚⠓⠸⠇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⡌⠀⠀⢠⣶⣿⣿⣷⣿⣿⣷⣦⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⡀⠀⣀⢀⡄⣀⠀⠀⡀⠀⢀⡀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⢿⡀⠀⠈⠉⠛⠛⠛⠛⠛⠋⡉⠀⠀⢠⠇⠀⠀⠀⠀⣤⣴⣶⣦⣤⣾⡟⣿⡇⠛⢸⡇⡆⣿⠈⢡⣿⠘⢻⣿⠸⢃⣿⡘⣸⣿⢛⣷⣦⣤⣶⣶⣤
⠀⠀⠀⠀⠈⠁⠀⠀⠃⠃⠁⠨⠬⠘⠇⠧⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠙⠿⠿⠇⠛⠸⠇⠶⠸⠸⠧⠻⠜⠋⠿⠰⠌⠿⠇⠿⠿⠿⠏⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠐⢧⣰⠒⣶⣲⣙⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠁⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡠⠖⢀⠀⢠⣌⡑⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠐⠘⠛⣩⣿⣿⣿⣿⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠉⠉⠉⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⡟⠁⠀⠀⠀⠀⠀⣔⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣄⢦⢾⣿⣷⡀⠀⠀⠀⠀⣀⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⠿⠿⠧⠀⠀⣴⣾⠉⠉⠉⠉⠉⠙⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣆⣻⣿⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠁⠀⠐⢦⡤⠀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠁⠀⠀⠉⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠐⠒⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣏⠣⠀⠀⠀⠉⠐⠊⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣽⣿⣦⡀⠀⠀⠀⣀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⢤⡙⢿⣿⣿⣿⣦⣄⡛⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠛⠉⠳⣄⠀⠙⢦⡙⠛⠿⠿⠿⣿⣿⣯⡹⡓⣄
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠈⢷⠀⠀⠙⣆⠀⠀⠀⣿⣿⣿⣷⣧⠈⡄
⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠘⣧⠀⠀⠻⣿⣿⣿⣿⣷⣵⠀⠀⠀⠀⠀⠈⣽⡂
⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠈⢧⡤⠤⢿⣿⣏⢻⣿⡇⣇⠀⠀⠀⠀⠈⣽⠃
⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠙⠃⠹⡆⠀⠀⢠⠌⠁⠀⣀
⠀⠀⠀⢀⠱⠀⠀⠀⠀⠀⠀⠀⠐⣿⠈⡆⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢹⠀⢠⠏⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡟⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⢦⡏⠀⠀⣄⢀⠰⣿⡗⠈
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣾⣇⣼⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠐⠚⢉⣁⠀⠀⠿⠶⠦⡞
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠔⠈⣀⣷⡟⠁⣀⣀⡠⠤⠔⠒⠚⢛⠀⠀⠀⠀⠀⢠⡆⣥⠀⠀⠂⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⢱⠀⠀⠀⢀⠔⣋⠔⡹⠋⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣧⡿⠄⠂⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠘⡆⠀⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠋⠁⠘⡇⠲⠄⣿⣟⣉⠉⡉⠉⣉⣉⣉⠁⣤
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠈⢧⣀⣀⣀⣠⣤⣤⣤⠴⠲⢖⠀⡀⠈⡿⣱⣄⣀⣄⣀⣷⣀⣀⣀⣟⣈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⡀
⠀⠀⠀⠀⠀⠀⢀⡉⠀⠉⠉⠃⠀⠒⠀⠈⠘⠀⠉⠀⣏⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠛⠳⢦⠶⠛⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠤⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣤⡆⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣷⡀⠀⠀⠀⠀⠀⠈⢷⣀⣤⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⣀⣴⠟⡉⠉⠙⣏⠉⠙⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠶⠶⠛⠉⢹⡜⠟⠀⢀⡿⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢀⣹⡶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣠⣤⡈⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠳⠄⠐⣿⣀⣀⣴⣿⡿⠿⣿⣷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⣀⡀⠀⠈⠙⠛⢩⡏⠷⣄⣈⣉⠉⠉⣿⣿⠛⣛⠒⠛⠛⠛⠉⣠⠊⠁⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠀⠀⠀⠀⠀⠘⣷⣄⣀⣹⠀⠀⠀⡟⠻⣤⡾⠀⠀⠀⠀⠀⢷⢫⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸⠉⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠷⠖⠚⠀⠀⠀⣻⡀⠀⠀⠀⠀⠀⠀⢶⡇⡇
⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⣶⡶⠶⠶⠶⠤⣤⣶⠶⠶⢒⣛⡿⠛⣿⠶⣤⣄⣀⣀⣀⣼⣷⣿⣷⠆
⠶⣤⣬⣷⣶⣄⣤⣤⡤⠶⠟⠋⣿⠀⠀⠉⠛⠒⣶⢾⡉⣉⣷⡛⠉⠀⠀⠀⢸⡆⠀⠈⠉⠉⠉⢻⢻⠿⠇
⠲⠶⣚⡿⠛⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⣁⣰⠏⢻⣄⣻⡀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠸⣸
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠉⠀⠤⢸⠈⠉⠀⠀⠀⠀⠀⢻⠀⠀⣏⢄⠤⡣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣤⣤⣤⣤⣤⣄⣤⣤⣼⣤⣤⣤⣤⣤⡤⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⠀⠀⠀⠀⢸⣇⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⣥⣉⡉⢙⡦⠀⢰⣋⣥⣄⣀⣀⣹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣤⠨⠁⠀⠀⠀⠀⠀⠀⠨⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠓⠐⠀⠀⠀⠀⠀⠀⠀⠈⢉⠦⠀⢠⠀⠀⡄⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⠊⢹⣿⠍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠈⠀⠀⠀⠀⠈⠋⠉⠉⠘⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⡀⡀⡀⢠⢀⠠⡀⠠⠀⠠⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠄⠄⢠⠄⡄⠀⡄⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠚⠂⠀⡄⢇⠀⠸⣀⠀⠀⢀⡸⠊⣄⡤⠼⠀⠠⠀⠀⡀⠈⠁⣃⡠⣀⠇⣀⠇⠰⣀
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⠀⠠⠀⠀⠰⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠦⠀⣀⠀⠀⠀⠀⠀⠂⠀⠀⢀
⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⣀⡀⠉⡍⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠆⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⢁
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⢃⠄⠃⢀⣀⠔⠊⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⣸⠄⠶⠀⠀⠀⠀⠀⠀⡀⠀⠠⢤⣾⣦⣤⠔⣂⡀⠒⠀⠀⢀⢀⡄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⢀⡠⠀⠀⠀⡀⠀⢀⠖⠂⠀⠀⠀⠀⠐⠀⢁⢟⡌⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠤⠤⠤⠤⠀⠀⠐⠓⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠙⢗⠂⠀⠀⠀⢐⠀⠀⠀⠂⢿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⢫⡨⠀⠀⠀⢥⡀⢡⣪⠞
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠠⠂⠃⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠀⢹⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⠀⢸⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⠀⢸⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⠀⣿⠀⢸⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢳⣿⠀⠀⠀⠀⠀⠘⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠚⣦⢤⡀⠘⡇⠀⠿⣰⠋⠉⠒⠋⠉⢢⠀⡇⠀⠀⢀⣰⠛⣦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡏⢸⠀⣿⢸⡧⣄⢳⠀⠀⠹⣄⠀⠀⠀⢀⡜⠀⡇⠀⠀⡏⢸⠀⣿⢸⡧⣄
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⢸⡇⢸⠈⣇⠀⠀⠈⠳⣤⠔⠋⠀⢸⠁⠀⠀⡇⢸⠀⣿⢸⡇⢸
⠀⠀⠀⠀⠀⡎⢳⡇⠘⠀⠉⠘⠃⢸⠀⠈⢢⡀⠀⠀⠀⠀⠀⢀⡞⠀⡏⢳⡇⠘⠀⠉⠘⠃⢸
⠀⠀⠀⠀⠀⢹⠀⠇⡔⠛⠒⠙⣦⢸⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸⠀⠀⢹⠀⠇⡞⠉⠚⠉⣦⢸
⠀⠀⠀⠀⠀⠈⣇⠀⠙⢦⣀⠔⠃⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠈⣇⠀⠘⠦⣠⠖⠁⣸
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢀⡏⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡀⠀⠀⠘⢦⡀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠾⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸⠅⣸⡀⣷⢼⠅⣿
⠀⠀⠀⠀⠀⠀⡏⠀⠙⠁⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⢉⡁⠈⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠙⣆⡀⢾⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡰⠟⠋
⠀⠀⠀⠀⢀⡠⣀⠀⠙⣀⡁⢀⡠
⠀⠀⠀⠀⢸⠀⡿⡁⣧⠏⣇⢸⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣄⣠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⢀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠲⠆⠙⠚⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⢹⠛⣽⡏⣿⢿⡟⣟⡿⢿⢟⢿⠉⣿⡿⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣼⣜⣛⣴⣜⣮⣤⣧⣤⣼⣮⣾⣰⣽⣦⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡰⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠉⠙⠛⠛⠛⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⠀⣿⡈⠉⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⢟⣳⠶⣞⣻⣿⣿⡿⠂
⠀⠀⠀⡄⣠⢀⣀⢩⡅⡄⣀⢩⡁⣀⡄⢠⠔⠖⢀⣄⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠻⣿⣼⣿⣿⣿⣷⣿⠟
⠀⠀⠀⠃⠉⠈⠛⠂⠙⠃⠋⠐⠙⠉⠃⠊⠃⠃⠋⠈⠁⠙⠀⠀⠀⠀⠀⣀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢉⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⡴⢰⡀⡆⢠⡶⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣏⣹⡉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠁⠈⠁⠁⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⡀⢄⠠⢄⣀⢠⡄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠰⠁⠏⠘⠉⠚⠸⠣⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠳⠶⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⣀⡮⠕⠀⠀⠀⠀⢀⣴⣁⣤⡂⠀⠀⠀⣧⣴⣶⣶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⠻⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⢠⠃⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⢸⣄⠄⠀⠀⠉⣻⡉⠉⠀⠀⠀⠀⠀⣀⠤⠄⠒⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⢀⣀⣀⡴⢦⡀
⠀⠀⡟⠀⠀⠀⠐⠁⡽⣄⠀⠀⢀⠴⢋⠀⠀⠀⣀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠿⠦⢤⡴⠋⠁⣀
⠀⠀⠳⡀⠀⠀⠀⠀⢀⣾⣦⣴⣧⡀⠀⠶⠂⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣠⢾⠆⠀⢉⣢⡴⣣⠷
⠀⠀⢀⣿⣷⣶⣶⣶⣿⣿⣿⡟⡛⠇⠀⠀⠀⠐⠘⣁⢶⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠠⡏⢰⠋⠀⡼⠉⠀⢀⡇
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⠂⠀⠘⠣⡄⠭⠀⠸⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠓⠈⠀⠀⠁⠀⠀⣸⠇
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⣸⡄⠹⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠭⠥⠬⠙⠳⠶⠦⠶⠮⠛⠿⠥⠤⠉⢹
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣧⠀⠈⠓⠒⠒⠊⢹⣥⠖⠓⠋⢹⠀⠈⠉⠛⠿⣿⣿⣿⣿⣶⣶⣒⣒⡒⠲⢶⠒⠒⠒⢚⡠⠛⠋⠉
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠇⠻⢤⣀⣀⢀⣠⠖⠻⣤⠤⢂⣠⠟⢦⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢰⣶⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⡀⠀⠀⠀⠀⠉⢹⠉⡙⢦⣸⠏⠉⢠⠒⠋⠀⠀⠀⢠⠀⠀⠀⠉⢿⣿⣿⠏⠙⠷⢶⣶⣾⡿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠈⠙⠒⢒⡶⢶⡚⣆⠳⠈⣽⠀⠀⠀⠳⢄⠀⠀⠀⢈⣇⠀⠀⢱⠀⠙⡟⠀⠀⠀⠀⠀⠀⢀⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⡏⢳⠀⠎⠻⣆⡸⣹⠀⠀⠀⢀⠜⠁⠀⠀⠈⢿⡇⠀⠀⠇⣸⠇⠀⠀⠀⠀⠀⠀⣸⣿⡇
⢹⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⣠⠀⣸⠀⢾⠲⢤⣀⠉⢣⡏⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠘⣿⣿⣿⣿⣿⠃⡰⠋⠀⠀⠀⣹⠋⠙⠀⠓⢤⠀⠈⠉⢻⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠈⠀⠀⠀⠀⠀⢠⣿⣿⡿
⠀⢹⣿⣿⣿⣇⡞⠁⡤⠖⠒⣾⠁⠀⠀⠑⠢⣨⠇⠀⠀⢸⡊⡠⠔⠒⠀⠀⠀⢴⡦⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⢻⣿⣿⡟⠀⠀⠀⢠⠞⠃⠉⠀⠀⠂⢄⡾⠀⠀⠀⠀⠻⡄⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⣀⣀⠀⢀⣠⣾⣿⣿⠏
⠀⠀⠀⠻⣿⡁⠀⠀⠀⣾⠀⠀⠀⣀⠉⢲⠞⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠙⣇⠀⠀⠀⢹⠀⠀⢺⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠾⠆⠀⠀⠀⠈⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢳⣤⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠲⣦⣶⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠾⠇⠀⠀⠀⠁⠀⠀⢠⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣶⣤⣤⣀⣀⠀⠀⠀⢸⠀⠀⣀⣠⣤⠶⠟⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣷⢀⣴⣿⡿⠿⣿⣦⡀⣾⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣻⣻⣟⣿⢸⣿⣿⡶⢃⣿⣿⡇⣿⣻⣟⣿⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣿⣿⣿⣿⣿⠘⢿⣿⣿⣽⣿⡿⠃⣿⣿⣿⣿⣿⣿⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⠀⠋⠉⠙⠛⠉⠁⠀⠀⠉⢻⡟⠉⠀⠀⠈⠙⠛⠋⠉⠙⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣿⣿⣿⣿⣿⡟⠻⣿⣷⣄⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⣠⣾⣿⠟⢹⣿⣿⣿⣿⣿⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠿⠟⠁⠀⣿⣿⣿⣿⣿⡇⠀⠈⠻⠿⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠛⠛⠁⠀⢸⣿⣿⣿⣿⣿⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⣴⣿⡿⠋⢹⣿⣿⣿⣿⡏⠙⢿⣿⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢹⣿⡇⠀⠀⠀⠙⠋⠀⠀⢹⣿⣿⣿⣿⡇⠀⠀⠙⠋⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠸⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠘⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⢤⣀
⠀⠀⠀⡀⠀⠚⠱⣏⣹⣫⡇⢀⢋⣸⣿⣎⣞⡡⢸⡢⡛⠀⣿⡷⣋⡶⣏⣰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠋⠀⠀⠀⠀⠈⠳⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣷⢤⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠀⠀⠀⠀⠀⠈⠙⠓⠒⢛⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠊⠀⠀⠀⠀⠈⠑⠢⠤⣀⣀⣀⣀⣠⠤⠞⠁⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠗⠤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣒⠒⠊⠀⣀⡠⠂⠀⢀⣀⣤⣀⡤⠞⠉⡏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠣⠠⠘⠉⠉⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⢉⠉⠓⠒⠋⠉⠀⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠩⣅⣀⣀⣀⣀⣀⣀⣀⣀⡽⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⢇⠀⢀⠞⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠉⢀⣼⠢⠬⠥⠤⠤⠤⠤⠤⠤⠤⢴⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠶⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠸⣄⡾⠚⣍⢧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⠤⠖⠚⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢠⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠈⠋⠁⠀⠀⠀⠀⠀⢀⣠⠖⣡⠟
⠀⠀⠰⠉⠀⠀⠀⠀⠀⠉⢁⡶⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⠀⠐⢰⠦⣄⡀⠀⠀⠀⠀⣩⢷⠶⢶⣴⢶⡶⠛⠛⠳⡍⠉⠁
⡖⠲⣶⠀⠀⠀⠀⢀⡀⠀⠰⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠒⠒⠒⠚⣏⢷⣄⠙⠶⣒⠒⣞⣡⠛⠂⠤⠤⣼⠃⠀⠀⠀⠸⠒⠒⠒⠒⠒⠒⠒
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⡗⠠⠤⠄⠀⠀⠠⠤⠄⠐⠒⠒⠉⠠⠀⠖⠀⠈⠲⢽⠀⠤⢀⠉⢁⡛⠀⠀⠀⠀⢼⠀⠀⠀⠀⢸
⡇⠀⢿⡀⠀⠀⠀⠀⠀⠀⢰⠇⠄⠀⢄⡀⠁⠠⠀⡠⠀⢠⣄⠀⠀⡄⠀⠠⠀⠀⠀⡴⠒⠛⣋⠈⠛⠀⠈⠉⠉⠉⠙⠓⠂⢦⣬⠋⠓⢦⠴⠋
⡇⠀⠀⠳⠖⢤⣤⣤⠴⠲⠛⠴⡀⠀⠀⣇⠀⠀⠀⢄⠀⠸⠈⠉⠰⠡⢤⣄⠀⠀⡀⢣⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠴⠚⠉⠀⢀⣿⣴⠟⢷
⡇⠀⠀⠤⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠄⠠⠀⢞⠋⠉⠀⠀⠰⡄⠤⢌⡡⠄⠷⠀⠀⠀⠀⠀⠀⢘⣭⠍⠁⠀⠀⢀⣠⠔⠋⠈⣧⡴⠾⠠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⡌⠀⠀⠈⠁⠀⠈⠀⠲⠄⣀⣀⣀⢳⣖⣠⣄⠤⠚⠉⠀⠀⠀⠀⣧⠹⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡇
⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢉⡏⠈⠃⠒⢲⠖⠂⠠⠤⠤⠜⠸⡄
⡇⠀⠀⠀⠀⠀⠈⠒⠤⢄⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢷
⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢤⣀⡀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠃⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠉⢻⡃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡄⠂⠉⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠤⠀⡀⠘⢦⣀⣀⣀⣀⣼⠉⠓⠢⠤⣤⣀⣀⣀⣀⡹
⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠦⠀⠀⠠⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡀⠀⠀⠀⠀⢀⣉⠽⠃
⠀⠀⣀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣿⡟⡉⣻⠟⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠛⠀⠀⠀⠈⠀⠀⠰⡆⢀⡀⣀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠐⠇⢠⠖⠒⢢⣀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣀⠀⢠⠚⠁⠀⠀⠉⠢⠀⢀⡀⠀⠰⡶⠀⠀⠀⠀⢀⣀⠠⠤⣌⣁⠀⠀⠀⠶⠀⠀⠀⠀⠀⣤⠀⢀⡼⢆⡀⠀⠀⠀⣽⡠⣀⠀⠀⠉
⠀⠀⢀⡴⠋⠉⠉⠑⠀⠀⠀⠀⠀⠔⠉⠉⠉⠱⣆⠀⠀⠀⢠⠶⠇⠄⠆⠀⠀⠚⢛⢦⡀⠀⠀⠀⠰⠂⠀⠀⡰⠋⢀⡀⣙⠢⡀⠀⠀⠀⠾⠤⡀
⠀⠀⣼⠁⠀⢫⠑⢄⠀⠀⠀⠀⠀⢀⠔⢩⠃⠀⢸⡆⠀⢰⠃⠆⠁⢢⣴⣐⢤⡀⡈⡀⢱⡀⠘⠃⠀⠀⡠⠊⠀⡴⢲⡯⠼⠃⠈⠳⢄⠀⢰⠚⠁
⠀⠀⠹⣦⡀⠀⠣⡀⠆⠁⠀⢀⠃⠂⡠⠂⠀⢀⡾⠁⠀⣿⡀⢃⠘⢯⡀⠀⢹⠇⠀⢄⢸⡇⠀⠀⣠⠊⠀⠀⠀⠉⢀⣀⣀⠤⡤⠶⠒⢻⢸⡄⠀⠺⠃
⠀⠀⠀⠈⠻⣆⠀⠀⠐⠘⠀⠈⠀⠀⠀⢀⡴⠋⠀⠀⠀⢫⠱⣄⠂⠀⡌⠉⣁⡆⠀⢀⣼⠁⢠⡾⠥⠴⢖⢒⣈⣿⣥⣶⣲⣶⡿⠷⣏⣻⡸⠃
⠀⠀⠀⠀⠀⠈⢳⠀⢀⣀⣄⣀⣀⠀⢰⠏⠀⠀⠀⠀⡦⠀⠳⣌⠑⠢⠶⠦⠤⠤⢚⡵⠃⠀⢸⡶⢶⡶⠛⣯⣛⠯⠿⠷⠚⠚⠉⠉⠉⠀⠀⠀⠶
⠀⠀⢶⠀⠀⠀⢸⣧⡬⡥⣥⡥⡬⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠤⠴⠒⠉⠀⠀⠀⠀⠓⠈⠉⠁⠀⠀⠀⠶⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠸⣥⣦⣧⣧⣧⣧⣾⡼⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⣀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢀⡤⠒⣭⠉⡍⠒⠄⠀⠶
⠀⠀⠀⠀⠀⠀⣤⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠶⠆⠀⠀⠀⠀⢀⠀⠉⠀⠀⠀⣀⠀⠀⠀⡄⠀⢠⠋⠙⠠⠄⠀⠀⣀⢸
⠀⠰⠆⠀⠀⠀⠁⢀⣴⠶⠶⢶⣦⠀⠀⠀⠀⠰⡦⠀⠀⡠⠾⠵⢄⡀⠀⠀⠀⢀⠴⠿⠳⢄⠀⠀⠀⠉⠀⠀⠀⠀⠀⢤⡄⠉⠀⠐⠂⠀⠀⣸
⠀⠀⢀⡀⠀⠀⠀⡟⠁⠀⠀⠀⢹⡇⠀⠀⢠⡄⠀⠀⢸⠙⠒⠒⠊⡇⠀⠀⠀⣏⠉⠉⠉⢉⡇⠀⠀⣤⠀⠀⢀⣀⠀⢀⠀⠰⠀⡄⠰⠀⠀⡅
⠀⠀⠈⠁⠀⠀⠀⡧⠤⠤⠤⠤⢼⡇⠀⠀⠀⠀⠀⠀⡰⡛⠛⠛⡛⢇⠀⠀⢀⠞⡛⠋⠛⠛⣄⠀⠀⠀⠀⠌⠘⠃⠑⣼⠀⡀⠀⠄⠀⠀⠤⡇⠀⠀⠘
⠀⠀⢀⡄⠀⢠⡄⣷⣆⠀⠀⡀⢰⡇⣤⠀⠀⠀⠀⢰⢁⠑⠒⠃⠁⢨⡇⠀⡼⠀⠓⢿⠛⠃⠘⡆⠀⠀⠀⡀⠀⠃⠀⠀⠈⠁⠠⠄⠈⠁⠀⡇
⠀⠀⠀⠁⠀⢸⡇⣷⡄⠀⠘⠁⠀⡇⣿⠀⠀⠀⠀⡘⣻⢺⡗⣿⢺⣻⣧⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢡⠘⠂⠠⠀⠁⠀⠀⠀⠀⠒⠀⠇
⠀⠀⠀⠀⢠⡾⢡⠇⠀⢤⡀⠴⠆⢳⠹⣦⡀⠀⠀⣏⣹⣹⣏⣿⣹⣿⣿⢠⠂⡀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢢⠚⠠⠄⠘⠁⠈⠀⠰⠆⠘⢰⠀⠸⠇
⢠⡤⠀⠀⣿⢁⡼⠀⠀⠀⠸⠃⡀⣈⣇⣹⡇⠀⠀⠳⠼⣄⣧⣸⣌⠿⠟⠘⠦⣀⣀⣀⣀⣀⡤⠎⠘⠇⠀⠀⠀⠣⡀⠠⠆⠈⣘⡧⠴⠒⢒⣾
⠀⠀⠀⡾⠉⠉⠀⠛⠒⠆⠀⠀⠛⠉⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣠⡞⢋⣀⣠⡴⠟⠁
⠀⠀⠀⡇⠻⢲⣾⡛⢉⠉⠍⠹⢉⣉⣿⡂⠰⣧⠀⠀⠘⠃⠀⢰⡏⢹⡄⠀⠀⠿⠀⠀⠀⠈⢠⣤⠀⣀⡀⠀⠀⠛⠀⣼⣩⠏⠉⠉⠀⠀⢠⣆⠀⠠⠦
⠀⠀⣀⡇⠀⠨⣿⡹⢀⠰⠘⠀⣠⢇⣿⠅⠀⢸⠀⠀⠀⣀⠀⢸⡇⢸⡇⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠉⣇⠠⠶⠹⠽⣶⣤⣬⣤⣶⠯⠓⠶⠀⡸⠀⢀⡀⠉⠀⠈⡇⢸⠀⠀⠀⠸⣏⣉⣉⣉⣉⣩⣭⣭⣭⣍⠱⠶⠶⠶⠶⠶⠶⠶⠶⢀⣥⠞⠁
⠀⠀⠀⠈⠢⣄⠶⠆⠀⢠⡍⠁⠶⠆⢀⣠⠞⠁⠀⠘⠃⠀⢀⣠⡇⢸⣄⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⣤⡤⠤⠶⠚⠋⠀⠀⣀⠀⠠
⠀⠀⠰⠆⠀⠀⠙⠒⠒⠲⠶⠒⠒⠚⠉⣀⡀⠀⠀⢤⡄⢰⠫⢅⢲⣶⠒⠽⡆⠀⠀⠻⠀⠀⢀⣀⣀⣘⡃⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⡯⡩⣩⢭⣭⣥⡇⠀⢰⠟⠶⢤⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣦⣤⠴⢦⡀
⠀⠀⣙⣡⠶⠶⠶⠦⠤⠤⢤⣤⣶⡿⠿⠿⠿⣛⣻⣦⠀⢸⠀⠀⠒⠀⠠⠀⡇⠀⠸⣤⡴⠶⣇⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠠⣼⡧⠦⣤⣴⠇
⠀⠀⣉⠙⠓⠶⠖⠒⠛⠛⠛⠛⠻⢿⣶⣶⣮⣭⠿⠟⠀⠘⠶⠶⠶⠷⠾⠾⠃⠀⠀⠀⢀⠀⠈⠉⠉⠉⢉⣉⠉⠉⠉⠉⢩⡍⠙⠛⠛⠁
⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠂⠁⠀⠀⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⡟⠀⠈⠀⠀⠀⠀⢠⠂⠷⠲⠒⠄⠑⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠒⠚⠋⠉⠁⠀⠀⠀⣐⣂⠂⠀⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠒⠉⠁⣸⠁⢰⠅⠐⠁⣠⣤⣶⢤⣿⣇⣤⣷⣾⡀⠀⢹⡄
⠀⠀⠀⡠⠔⠚⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠄⠀⠀⢠⠃⠀⠀⠀⢀⡀⠀⢀⣈⠀⡰⣶⡛⡶⣞⣻⡟⠀⡾⠀⠀⣼⡧⠙⢦⡇⢸⣯⣭⡵⠟⠀⢀⣾⡇
⠀⠈⠑⡀⠀⠈⠐⠡⠅⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠁⠀⠀⢤⠚⠀⠀⢀⠞⠁⢠⡞⢉⡯⠂⢹⠿⢯⣧⡆⢻⠁⠀⠇⠀⠀⣇⣉⡀⠤⠰⠋⠀⠀⠀⣀⣠⠞⡻⠁
⠀⠀⠀⠀⠑⠒⣦⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡠⠂⠁⠀⠀⢀⠎⠀⣠⣺⣇⢸⡇⢅⠞⣿⡶⠛⠦⠊⠀⡘⠀⠀⠀⠳⢽⡛⠛⠛⠓⠒⠒⠋⠉⠁⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠉⠀⠂⠉⠀⠊⠀⠀⠀⠀⠀⠀⠀⡾⠀⡠⢼⠥⣬⠑⠒⠛⠿⣿⠋⠄⠁⢀⡴⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣿⣟⢓⣋⡠⠶⠴⠋⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⣠⡤⠶⠚⠛⠛⠛⠛⣳⠦⢤⡀⠀⠀⠀⠀⠀⢀⠇⠀⢰⠿⠛⠉⠀⠀⣀⡤⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠄⠀⠀⠀⡀
⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⢠⠀⢰⠃⠀⠀⡈⠳⡀⠀⡠⠀⣺⠀⢀⠃⣀⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠰⡤⠴⠉⠀⠈⣏⠅⠀⠀⠤⠀⠉
⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠒⢀⡂⢸⡀⠀⠐⠌⠄⠁⠘⠤⠤⡇⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⡤⢾⠁⠀⠀⠀⠻⠆⠀⠀⡀⠀⠄⠐⠀⠂
⢠⠏⠀⣀⡀⠀⠀⠀⠀⠀⢰⠸⢁⡼⢿⡞⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣤⠤⠏⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠑⣀⠦⠚⠂⠁
⢸⠊⠉⠀⠉⠳⣄⠀⢀⣀⡤⠒⠉⠀⡞⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣴⠈⠀⠀⠀⠈⠶⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠒⠉⠁
⢸⠀⠀⠀⠀⣰⣿⣋⣉⡀⠀⠀⠀⠀⠓⠀⠈⠀⠀⠀⠀⠀⡀⢤⣀⡶⠉⠀⠀⠓⠼⠀⠀⠀⠀⠄⢀⠀⢠⠤⠖⠒
⠘⢧⠉⠀⠀⠀⠀⠀⠀⠉⠱⠀⠀⠀⠀⠀⠀⢀⡠⣄⠴⠁⠀⠀⠱⠽⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢀⣀
⠀⠈⠙⠦⠤⡀⣀⠤⠄⠂⠀⠀⠀⠀⠀⢀⠞⠁⠀⠈⠙⠃⠀⠀⢀⠂⠀⣀⡠⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⡀⠤⣦⣤⣤⠞⠁⠠⢊⡟⣩⠟⠉⠀⠈⢉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠐⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣸⣿⠗⠁⠀⠀⣾⡴⠋⠀⠀⠀⠀⣠⡾⠜⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠿⠃⠀⠀⠀⢸⠟⠀⠀⠀⠀⠀⣰⣯⠶⠚⠉⠀⠙
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠤⢀⣀⠀⠀⠀⠀⠀⣀⠤⠐⠒⢂⣉⠉⣉⠑⠒⠒⠢⢄⡀⠀⠀⡇⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⠔⠊⠁⠀⠛⠛⠶⠶⣶⣎⠙⡆⠀⣠⠊⠀⣠⠶⠶⠉⢩⠋⢽⠗⠶⠖⣦⡀⠉⢦⠀⠹⠦⣀⠈⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢴⠏
⠀⣀⡴⠋⠀⠀⠙⠛⠻⠶⢶⣤⠀⣈⣥⠞⠁⠰⠁⠀⠀⠛⠐⢿⠖⠒⠗⠂⢰⠶⠾⡫⠞⠠⠀⠂⠀⠃⠪⠉⠓⠒⠦⠤⠤⠤⠤⠤⠤⠶⠶⠖⠒⠋⠉⠁⢀⡞
⣰⡯⠠⣬⣙⣛⠛⠶⣦⠀⣀⡭⠞⠋⠀⠀⠀⢀⠀⠐⡠⠀⢀⠙⠓⠶⠷⠶⠶⢚⠩⠐⡈⠂⢀⠀⠀⠸⡆⠤⣈⠀⠀⠒⠀⠀⠤⠀⠀⠆⠀⠐⠂⠀⠈⢡⡼⠁
⡇⠶⢿⣷⣶⣶⡝⢦⣠⠖⠋⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠅⠭⠈⠌⠵⠐⠂⠁⢀⡴⠋⠀⠀⠀⠙⢶⣆⡁⠁⠐⠐⠂⠔⠰⠀⠆⠖⠐⠐⢀⣡⠾⠁
⠈⠙⠀⠀⠀⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⣀⣀⣀⣀⣀⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠶⠶⠒⠒⠚⠋⠉

⠀⠀⡀⠤⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢀⣤⣶⣦⣶⣦⡀
⠀⠌⠀⠀⢀⡠⠤⠚⡆⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⡠⣤⣶⣄⠀⠀⠀⠀⠀⠀⢌⣴⣾⣶⡐⠀⠀⠀⣼⣿⣿⠟⣿⣿⣿⡀
⠰⠀⢀⠞⠁⠠⠀⠀⡆⠀⠀⠀⠄⠀⠒⠒⢺⠀⠀⠀⠀⣼⠟⠙⢿⡆⠀⠀⠀⠀⢘⠿⠛⠉⠻⠇⡆⠀⢰⣿⡏⠀⠀⠀⠹⣿⣿
⠀⠢⠁⠂⠀⠀⢀⠐⠁⠀⠀⢨⡀⠀⠀⠀⢸⠁⠀⠀⠀⡄⠀⠀⠀⡆⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⡇⠀⢸⣿⢃⠀⠀⠀⠐⣿⣿
⠀⠀⢶⠟⣙⢻⠞⠀⠀⠀⠀⠀⠣⣀⣀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠇⠀⢀⠀⠀⠀⢣⠀⠀⠙⢛⠇⠀⢸⠛⠋⠁
⠀⠀⠀⠑⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠈⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⢿⣆
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣼⣿⠟⠁⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡗⠀⢸⣿⣇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠙⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠋⠁⠀⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠷⠧⠦⠧⠶⠤⠄
⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠄⣀⣀⣀⠠⠄⢂⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠂⠤⠤⠤⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⢀⠉⠄⢀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠁⢀⠄⠠⠀⠐⡒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠤⠖⠁⠀⣤⡌⠣⡀⠀⠀⠀⠀⠈⠀⠐⣊⡅⠀⠀⠀⠐⢈⡑⠤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡰⠐⠋⢁⠀⠀⠀⢀⣠⡄⠀⠀⢀⠀⠀⠀⢀⡀⠀⠑⣀⠀⠀⠀⠀⢀⠉⠈⠗⠁⠀⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⢀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠁⠑⠀⠀⠛⠀⡠⠀⠀⠛⠁⠀⠠⠒⠀⠀⠀⠀⠐⡀
⠀⡐⠐⠤⡈⠒⠀⠀⠀⠀⠠⠙⠀⠀⠀⠀⠀⣄⣀⣠⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⣠⣬⣆⣤⣤⢤⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀
⠀⣹⡜⣸⣆⣂⡼⠦⠀⠀⠀⠀⠀⠀⠀⢀⢁⡒⠀⢂⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⡐⠠⠠⠄⣀⣡⡄⠀⠒⠀⢀⡀⠀⠀⠀⠀⠀⢸⡀
⠂⠠⣤⣤⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠉⠉⠀⠀⠈⠀⠀⠀⠒⠀⠀⣒⠠⡀⠀⠁⠈⠰⠀⡄⠂⢰⠀⠀⡀⠀⠀⠀⠀⣧⠇
⠀⠀⠈⠉⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢀⠐⠉⠛⠀⠈⠁⠀⠚⠀⢶⠓⠋⠐⠀⠁⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠠⠄⠠⠈⠈⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠡⢀⡀⠀⠀⠀⠀⠀⠀⢰⠈⣅⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⣉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⣠⠀⠀⢊⣁⠀⠀⠀⠀⠀⠀⠰⡄⠠⠤⠦⠀⠋⠑⠉⠑⠉⠉⠁⠁⠀⠀⠀⡠⡆⠀⠀⠀⠀⠠⠀⠃
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠳⠤⠔⠃⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣧⣼⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⡿⠟⠿⠏⠉⠙⣉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⢀⡠⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣶⠟⢶⠛⡩⠔⣡⣾⠟⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⣾⠇⠀⠈⠉⠉⢱⣿⠇⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⡄⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠹⡄⠀⣰⡏⠈⣿⣿⣿⣿⣾⣿⡇⠀⠀⠰⣄⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣆⡄⠀⠀⠀⠀⢠⠏⠀⠀⣠⠞⠟⠚⢹⡇⢨⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢹⣶⠞⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣸⡎⠙⣤⠀⠀⢀⠏⠀⢀⡴⠁⠀⠀⠀⢸⡇⠠⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣾⡩⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢠⡀⠋⠁⠀⣹⠀⢀⠞⠀⢠⠞⠀⠀⠀⠀⠀⠸⡇⠈⣿⣿⣿⣿⣿⣿⣿⣷⡤⠞⠉⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⢻⣼⢳⡔⠠⠴⠃⡔⠁⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⡇⠨⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠙⠿⢻⣠⡇⢀⢣⢀⣄⣠⠃⠀⠀⠀⠀⠀⢀⡼⢁⣴⣿⡿⠋⠉⠀⠀⠀⣀⣤⣶⡎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⣇⠀⠀⣿⢻⣸⢧⡟⠋⢠⣄⠀⠀⠀⠀⢀⡞⣠⣿⣿⠿⡶⠀⠀⠀⢠⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⣿⡆⠀⠈⠀⠙⠈⠳⠴⠟⠙⣦⣀⡀⢠⣏⡼⢛⣩⣤⣶⣡⣦⣦⣶⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠉⠁⠀⣀⣾⣦⣤⣤⠀⠀⠀⣿⣿⣿⣿⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢿⣿⠃⠉⠛⣷⡀⢠⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣤⣬⣿⣿⣶⣮⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣈⠻⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠠⠂⠀⠂⠀⠀⠀⠂⠂⠀⠀⠉⠉⠁⠁⠈⠁⠁⠁⠉⠀⠈⠈⠉⠈⠉⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠀⠠⠤⠠⠀⠠⠀⠠⠠⠀⠀⠂⠀⠀⠂⠀⠀⠈⠉⠈⠁⠁⠁⠈⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢁⠐⠀⠀⡀⠈⠁⡄⢀⠀⠀⠀⠀⠉⠨⣆⠀⠀⠀⠀⠀⡃⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠀⠤⠠⠀⠀⠂⠀⡀⠀⠀⢀⣤⣿⡏⢇⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠰⠒⠂⠐⠀⠉⠄⠀⠐⠀⢀⠠⡤⠈⠁⠂⠀⣀⣰⠂⡘⡦⠍⠘⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣤⣤⣀⣀⢈⠁⠀⠀⠚⠁⠀⠂⠀⠐⢀⠴⢰⢢⠀⠃⡏⠀⠂⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠛⠛⠀⠁⠁⠉⠀⠈⠉⣖⠀⢤⣀⠀⠀⠋⠘⠉⠉⠀⣦⠀⠀⠀⠀⣠⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡌⡇⡀⢀⠀⠢⢅⠆⠀⠽⠃⠀⠀⣸⢭⣀⠄⠸⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢽⣇⣧⠀⠐⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠋⠃⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠠⠀⠀⠀⠠⠤⠤⠤⠄⠤⠤⠤⠀⠀⠒⡄⢀⠀⡔⢀⠀⠒⠐⠢⡄⠀⡶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⡄⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⡡⢀⣄⣠⠘⣁⣄⢄⡤⡅⢀⣆⣤⡤⡤⢤⠄
⠀⠀⠀⠀⠀⠤⠀⠒⠖⠒⠂⠰⢾⠱⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠈⠉⠉⠀⠨⣇⣭⣴⣱⢀⡀⠀⡅⢶⡥
⠀⠀⠀⠀⠀⠀⢰⠆⣀⠀⠀⠘⠉⠉⠁⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘