Sweeten Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠤⠤⠤⠄⢀⣀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠉⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠉⠐⢤⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡁⢀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⠀⠀⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡁⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⢀⡤⠤⠴⠾⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⢿⣠⣇⣤⣄⡀
⠀⠀⠀⢀⠒⠀⠀⢠⠒⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢰⡇⣰⡖⠲⣶⣜⣿⠳⠙⠛⠛⠉⠛⠛⠛⠛⠉⠉⠉⠑⠃⢉⣿⣿⠇⠨⠛⠷⡄
⠀⠀⠀⠨⣄⠀⠈⠈⠉⢐⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡾⢇⠀⣄⠀⠘⠿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡿⠁⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠈⠑⠂⠓⠚⠉⠀⠀⠄⠀⠂⠄⠀⢠⠀⠀⠀⠀⠁⠀⠙⠶⢬⣄⣀⣒⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡗⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠇⠁⠀⣴⠞⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⠉⠙⢿⢿⣶⣄⠀⠀⠀⠀⢀⣠⡶⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠒⠊⠁⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠜⠋
⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠒⠛⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠐⠒⠛⢉⢻⡭⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⡠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠭⣥⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣹⣧⠆



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⣀⡁⠤⠀⠤⠐⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠉⠁⠈⠉⠒⠀⢀⠨⠄⠒⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠄⡪⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠈⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡈⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⢀⣀⠤⠔⠊⠊⠁⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠎⠀⠀⠀⢀⠀⠠⠐⠉⠈⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢀⠠⠐⠂⠒⢉⡉⠉⡉⢸⠉⠉⠉⠙⠒⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⢤⡄⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠲⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠠⠐⠈
⠀⠀⠀⠀⠁⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠁⠀⠀⠀⡀⠁⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠈⡔⠄
⠀⠀⠀⠀⠀⠀⡠⠄⠒⠒⠙⠒⠈⠉⠉⠉⠁⠛⠛⠲⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈
⠀⠀⠀⠀⢄⠰⠤⣬⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠈⠁⠒⠂⠠⠤⠤⠤⠤⠤⠤⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⢠⠀⠀⠂⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠈⠂⠀⠀⠈⠄⢀⡀⠂⠀⠉⡜⠀⡀⣓⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠄⠲⠶⠶⠶⠶⠤⠤⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠐⠂⠀⢀⡉⠛
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠃⠔⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠒⠒⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠋⠀⠙⡄⠀⠀⠀⠀⠀⢠⠮⢱⡀⠀⠀⠀⠀⠀⢀⠮⠧⡀⠀⠀⠀⠀⠀⣜⢁⡈⣣
⠀⠀⠀⠀⠀⠙⣶⡲⣶⠃⠀⠀⠀⠀⠀⡎⠉⠀⣧⠀⠀⠀⠀⡖⠁⠀⠀⠉⢢⠀⠀⠀⠀⡏⠁⠈⢸
⠀⠀⠀⠀⠀⠀⠸⣮⠃⠀⠀⠀⠀⠀⠀⢳⠒⠒⡞⠀⠀⠀⠀⠉⣎⡕⣊⡕⠁⠀⠀⠀⠀⠑⢢⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⡴⣢⠤⡀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⢀⡤⠒⠒⠒⠢⡀⠀⠀⠀⣀⣴⣚⣤⣷⡄⠀⠀⠀⢀⠊⠈⠈⢖⠁⠀⠀⠀⠀⣰⠒⠓⡄
⠀⠀⠀⠀⢯⠲⠤⠒⠤⠔⡬⠂⠀⠀⣧⠴⣶⠤⠶⡇⠀⠀⠀⠈⢓⣁⣩⠋⠀⠀⠀⠀⣤⠯⠬⠥⠽⡤
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠓⠒⠂⠈⠉⠁⠀⠀⠀⠀⠀⡏⡃⠀⠀⠀⠀⠀⠀⢩⣕⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢀⣠⡔⠒⢄⠀⠀⠀⠀⢠⢒⢭⢕⢢⠀⠀⠀⠀⢀⡴⠒⠒⢦⠀⠀⠀⠀⠀⣶⣶⣶⣶
⠀⠀⠀⠀⠰⠡⠒⢮⠱⡀⡇⠀⠀⠀⢸⢅⠣⠷⣱⠃⠀⠀⠀⡸⢤⢤⠤⡤⢇⠀⠀⠀⠀⣿⡿⠿⢿
⠀⠀⠀⠀⠘⣔⢜⠒⣠⠗⠁⠀⠀⠀⠀⠑⢻⠟⠁⠀⠀⠀⠀⢸⠸⢸⠀⠇⠇⠀⠀⠀⠀⡏⠙⠓⠾
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠓⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⢀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⢀⡞⡁⠀⢌⢱⠀⠀⠀⠀⠀⢀⢤⠪⣑⡄⠀⠀⠀⠀⣸⡤⢄⣇⠀⠀⠀⠀⣀⢤⠒⠒⣤⣀
⠀⠀⠀⠀⠸⠀⠁⠀⠄⢀⠇⠀⠀⢀⡤⡋⠆⢣⠏⠀⠀⠀⠀⠘⢤⠀⠀⡤⠃⠀⠀⠰⡓⣬⠦⠴⢥⢚
⠀⠀⠀⠀⠀⠑⠢⣤⠒⠊⠀⠀⠀⠀⠑⠮⠋⠁⠀⠀⠀⠀⠀⢠⡊⡠⢤⣙⡄⠀⠀⠀⠈⠛⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡷⣦⡀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⢀⣿⣿⣿⣎⢿⡜⠏⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢠⣬⡻⣿⣿⣿⣾⣿⢀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⢻⣷⣿⡗⣭⣶⣶⣾⣍⡈⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣮⢱⣾⣿⣿⣿⣿⣿⣷⡉⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⢰⣶⣶⣶⣶⣶⣶⣶⢴⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⡿⣵⣶⣹⣿⣿⢸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣭⣵⣿⣿⢾⣿⣿⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⢺⣿⣿⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⣽⣥⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣿⣿⣿⣿⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠄⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣾⣿⡿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣘⣭⣭⣭⣭⣭⣷⢟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠻⢭⣉⣿⣏⣿⣧⣽⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⡿⠿⢿⣟⠛⠯⣑⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣾⣷⣶⣶⣾⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡍⢛⣛⣛⣻⣿⣻⣿⣯⣭⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡹⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣾⣿⣝⣚⣛⣽⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡨⣦⣭⣽⣿⣿⣿⣿⣿⣿⣯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⣠⠀⣿⣿⣿⣦⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⢀⠈⡇⠀⠀⠀⠀⣿⡀⢘⣿⣿⣿⡃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⡿⠿⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡑⠄⣁⡀⢠⠟⠃⠈⠀⠌⢹⣧⠄⣌⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⣉⠈⣄⣀⣴⢫⣭⣭⢟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠈⠁⡖⠉⠌⠻⢈⣀⡁⡠⠀⠛⢿⣬⣿⣿⣿⣷⣾⣷⢶⣷⣶⡾⣷⣿⡿⠁⢈⣬⠀⠈⣿⣬⣿⣿⢾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠐⠄⠀⠀⠄⠻⣷⡇⠄⢘⣿⣿⣷⣀⠛⠘⢿⣿⣿⣿⣿⡟⠡⢬⣿⣿⣇⡤⠙⣿⣶⣿⣿⡿⠶⠿⠿⠟⣫⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⡀⠀⠀⣼⣄⢨⠈⣿⡿⣟⡉⢶⣽⣿⣝⣿⣿⡟⠀⠂⠤⢸⡟⠠⠒⠄⢸⡟⣻⣦⠷⠿⣛⡛⢻⣿⠟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⠷⠀⠀⠻⢿⣦⣴⣿⣟⡿⠁⣼⢿⡟⠁⠀⣻⣧⠈⠄⢢⠘⡇⠤⠀⡀⢠⣿⣙⡵⣿⡟⢩⣧⢜⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣥⡞⠉⢻⡟⠌⠀⠛⣿⣦⣀⣀⠀⠁⣀⣀⣤⢪⣿⣿⣉⣼⡿⣿⣽⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠘⣰⣿⣿⣿⣯⣿⢁⣰⠚⠓⠦⣤⣤⣾⣏⠉⠉⠤⠔⠉⠉⣹⣶⣿⣥⣍⣀⣶⣿⣿⣿⣿⣿⣿⣿⣉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⣻⠛⠡⢽⣿⣿⣿⠀⠀⠀⢛⠛⠍⠛⡷⠂⠒⡚⠁⠿⠛⠋⠛⣛⣿⣿⢿⡷⢿⡏⣿⠿⣿⣿⣭⣤⠂
⠀⠀⠀⠀⠀⠀⠐⡠⠄⠀⠀⠀⠃⢀⠀⣠⠀⡀⠀⣴⠌⢐⠊⣻⣿⣿⠷⠄⠀⠘⠣⣄⡀⠓⠀⠸⡷⠀⠃⢀⡀⠘⠁⠈⠓⢛⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢈⢶⡐⣆⠈⠀⢀⣀⡬⣤⢤⡄⠙⢧⡘⢛⣿⠇⡋⠀⢇⠻⢢⡀⠈⢿⣷⠂⠀⠀⠀⢻⡏⠀⠀⠶⠆⡋⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠐⡧⢬⡶⣸⣷⣶⠏⣽⣇⡀⣠⡽⠐⠾⡇⠈⣁⠀⢳⡄⠈⠑⠊⠁⡀⠸⠛⠛⣧⣠⡟⠛⠃⣀⠛⠀⠊⠀⣰⠇⠈⠙⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⣾⣴⠟⠛⣿⢂⡉⣁⣩⣭⣀⣀⣹⣦⠐⠈⠀⠀⠙⢦⣤⣤⣴⣦⡄⡁⠚⠁⠟⠙⠂⠃⣾⣤⣤⣤⡴⠋⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣟⠓
⠀⠀⠀⢀⡠⠐⢐⣃⡹⠋⠴⢂⠃⣬⣟⣿⣿⣿⣿⣿⡿⠇⠀⠛⠀⠆⠀⠀⢀⣿⠛⠂⠛⠐⣋⡅⠀⣍⡓⡰⠌⠻⣯⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣭⣀⡄
⠀⠀⠀⠉⠂⢀⣠⠖⢉⠀⠠⢼⠄⠈⣿⠛⡉⢹⣿⠫⠂⠳⠀⠀⠀⠀⠀⠀⢸⠏⠉⠑⢆⠀⠀⠤⢠⠄⠚⡴⠊⠉⠙⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠀⠀⡈⠀⠆⠘⠃⠄⢠⠀⠀⣿⠿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⣀⣀⠀⠊⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠉⠶⠁⠰⢾⡏⠀⠀⠀⣴⣤⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢸⡿⢿⣶⣾⠿⣷⠀⠀⠀⣀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠲⠮⠤⠞⠈⠀⠀⣾⣿⠟⠂⠀⠀⠀⠀⠀⠀⢸⣆⡀⠀⠛⠰⢿⠿⠐⠋⠀⢀⣼⠇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠄⢁⠀⠀⠈⠀⡐⠒⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢈⡐⢀⠀⠘⣧⡆⠴⠶⠚⠳⠶⠤⢠⡼⠃⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣧⡤
⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁⠀⣠⠊⠁⠀⢀⡀⠀⠈⡀⠀⠀⠀⠀⢀⣴⣯⠀⠈⠀⠀⠈⡇⣴⠞⠓⠛⠳⣦⢸⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣭⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄⠂⠀⠁⠀⠰⠀⣰⣤⣠⣴⣿⣵⣿⣆⢐⠀⠀⠀⠹⣏⠘⠓⢖⡛⢸⡟⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢃⣿⣿⣿⣿⣿⣿⣿⠋⠻⣆⠀⠀⠉⠛⠿⠿⠟⠃⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡈⠀⠸⠗⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡞⢻⣿⣿⣿⡏⠁⠀⠀⠀⣀⠀⠀⠀⣠⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡝⢧⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⠋⠀⠀⠀⡀⢀⣽⣃⠙⡀⠀⠀⠈⣿⣿⣿⣿⣿⡿⣿⡎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⢠⡀⢿⡞⠋⠰⠁⠀⠀⢸⣿⣿⣿⣿⡌⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⠉⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣿⣿⠿⣿⡿⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⢹⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⠶⡠⡟⠁⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⣏⣻⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⠓⢰⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢹⡤⠄⢀⠀⠀⢲⠖⣽⡃⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣬⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢻⣤⡶⠤⠄⠀⡀⢸⣿⣧⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡿⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⢤⠃⠀⠀⠀⠜⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⣰⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠖⠀⠀⠊⠀⠀⠠⡀⢀⠠⣀⡀
⠀⠀⠀⢀⣀⣰⠀⠠⠀⠀⠺⣷⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠻⣫⣼⠛⠯⠀⠀⢀⠠⠛⠡⢀⣾⡇⠈⢢⡀⠀⠀⠀⢤
⠀⠀⠀⠘⡿⠋⠴⠀⠈⠢⠐⣾⡄⠼⣴⣆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠁⠚⠀⠀⠔⠁⠀⠀⠀⣸⣿⣇⠀⠀⠇⠀⠀⢈⣀⢱⡀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡈⡗⢀⠃⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠈⡉⡇⢀⠔⠃⠀⠠⡀⠀⠀⠹⣿⣿⣶⣤⠂⠀⠀⠈⢛⠿⣿⡗⠈⠁
⠀⠀⠀⠀⠀⠀⢠⡈⢱⡄⠈⠻⠇⠈⠀⠀⠀⠀⣶⢠⠀⠀⠀⠈⠁⠀⠀⣵⡄⠀⠃⠀⠀⠈⣄⠀⠀⠀⠉⠉⠀⠀⠀⠂⢤⣤⡑⡿⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⣶⢀⠔⠈⠀⠠⣹⡆⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⢀⣘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠱⡟⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡷⠁⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠐⢀⠆⠀⠀⠀⠀⣴⣿⠁⠙⢦⠀⠀⠀⠀⢀⠀⠀⣤⣐⢻⣁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⢰⣶⣿⡆⠀⢈⡇⠀⠀⠀⠀⠈⠀⠙⢛⠏⠁⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡜⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠻⡛⠂⣾⡷⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠀⠀⠀⠀⠀⠀⣀⣴⣴⠁⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠟⢻⡤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⣓⠺
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢸⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣁⡀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠈⠀⠀⡄⠀⠀⠚⠋⠉⠀⠀⠉⠉⠛⠛⠓⠒⠲⡾⢯⣹⠿⠓⠻⠷⠶⠦⢤⡤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠠⢄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⡙⣿⣉⡛⣻⡛⣻⣟⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⣅⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣶⣶⣿⡇⣾⣷⣶⣶⣶⣶⣦⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠿⠛⠛⠋⠉⠉⠉⠀⠀⢀⣤⣴⡶⠶⠶⣿⡄⣿⣷⣶⣤⣤⣄⣀⠀⠀⠈⠉⠉⠉⠛⠛⠻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠙⠿⣶⣶⣶⣶⣿⣃⣿⣷⣤⣤⣄⣉⣙⣿⡆⠀⠀⠄⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠿⠶⢶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣄⣠⠞⠓⣦⠀⢀⣩⢽⣛⠛⠿⠶⠶⢶⣦⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠶⠶⠿⣛⣋⡉⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠶⠿⣶⣿⣤⣤⣹⣆⣀⣴⠞⠛⠶⡀⠀⣠⡴⠿⣭⠉⠉⢉⡭⢤⣄⠀⠀⣠⠴⠶⣄⠀⣾⣁⣤⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠻⠿⠷⠶⠿⣶⣶⣾⣷⣴⣿⣤⣤⣼⣦⣼⣧⣶⡶⠿⠟⠛⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⠹⠷⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⣠⡄⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠘⢷⣦⣀⣀⠀⠀⠀⠀⠀⣀⣠⣴⠟⠁⠀⠀⠀⠀⠀⣠⠞⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠛⠿⣷⣲⠤⢄⣀⡀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⣀⣠⠤⣖⣯⠵⠚⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠉⠛⠛⠿⠾⣿⣷⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⣒⣿⣭⠭⠶⠛⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢀⡞⠋⠁⠀⠙⠂⠀⠀⢠⡞⠃⢀⣶⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠘⢧⣄⣀⡤⠠⣶⣲⢤⣼⠓⢊⣽⠥⠴⠿⣤⣶⣄⡤⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠁⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠻⠶⢶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠶⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⣦⣬⣍⣀⣀⣐⣒⣒⣒⣒⣂⣀⣈⣭⣥⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠙⢦⣀⣀⣀⣀⣤⣤⣤⣶⣶⣶⡿⠿⠿⠿⠛⠛⠃
⠀⠀⠀⠀⠀⠈⠳⣀⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⣀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⣶⣶⣶⣶⠶⠶⠾⠿⠟⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣎⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⣿⣿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠈⠙⠛⠿⢿⣿⣶⡀⠀⠀⠀⠀⠀⠀⢰⣿⣦⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣀⠠⠤⠀⠀⠀⠈⣿⣿⠂⠀⠀⠀⠀⠀⠁⠈⠉⠻⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣼⣿⣦
⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⣀⡾⠃⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⢴⠟⠁⢠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢶⠛⠓⡋⠙⡒⠛⠲⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣈⣧⣤⣧⣤⣇⣠⡞⢀⡴⠉⠉⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢯⡽⠒⢣⡬⠷⢶⡛⠻⡛⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⡴⣋⡀⢠⠋⠙⣦⣀⠀⠀⠀⢀⡴⠚⢲⠀⠀⢠⣾⣽⠷⢚⣉⣉⣉⣉⣉⣙⠒⠿⣉⢓⣟⢦
⠀⠀⠀⢀⡴⠚⠉⠉⠲⡄⠴⣅⡈⢉⡗⠢⢜⡀⢈⣇⣠⠴⠋⢀⣠⠜⢦⠀⣏⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⠛⠛⢾⡄
⠀⠀⡴⠋⢀⣖⣻⡓⣆⢹⡀⢀⡽⠛⢦⣀⡴⠻⣌⢹⡤⠴⠚⠉⢀⣀⢸⡀⠈⠒⢤⡻⢿⣶⣿⣭⣭⣭⣿⣿⡿⠏⣀⣀⡼⠁
⠀⡼⠡⣄⠸⣜⠧⢟⡞⢸⡏⠛⢆⣀⡼⠙⢢⣀⠞⠻⡄⠖⠒⠋⠉⢀⣠⡇⠀⠀⠀⠉⠒⠦⠬⢭⣭⣭⣭⠥⠖⠚⠉
⠀⢳⢒⡎⡇⠈⠉⠉⠀⢼⢀⣀⠞⠉⢢⣀⡼⠉⢦⣴⢃⣚⣛⠉⠉⠉⠀⡇⠀⠀⠀⠀⠀⣀⣀⡤⢤⡤⠤⣤⠤⣄
⠀⢻⣀⡴⠃⠀⠀⠀⠀⣸⠹⢽⣂⣤⡊⠉⣧⣴⠋⢸⠦⣭⣉⡉⠉⠉⢸⠇⠀⠀⠀⠀⢸⡁⠰⡄⠀⡆⠀⡆⠀⡬⠉⢉⠲⠦⣄
⠀⠀⠈⠙⢦⣀⠀⠀⡰⣋⠓⠄⣸⠃⢱⣶⣧⠖⠊⡭⢤⣀⠀⠉⠓⣰⠋⠀⠀⠀⠀⠀⣠⡽⢶⣿⣟⠛⣏⣙⡾⣧⣴⡉⢀⣠⠞
⠀⠀⠀⠀⠀⠈⠉⠛⠒⠾⠷⠾⡗⠋⠹⣁⣨⠦⠴⠃⠀⠈⠑⠦⠤⠏⠀⠀⠀⣠⢾⣟⡭⠽⠷⠒⠾⠯⠤⣍⣦⠼⢗⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠉⣡⠔⠒⠛⠛⠛⠓⠒⠦⣌⠳⠼⢷⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣍⣓⠒⠒⠶⠶⠒⠒⣋⡤⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠰⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⡄⠀⠀⢀⣄⡱⢎⠈⠐⢄⠀⠀⠀⠀⠀⠀⠒⣢⢬⠀⡰⠓⠀⠀⠀⠞⠀⠈⠄⠀⠀⠀⠀⠀⠯⠦⢀⠠
⠀⠀⢀⠤⠐⠈⠀⠁⠀⠠⠄⠀⢂⡄⢀⠀⠀⠀⢀⠤⠻⣯⣛⣅⠀⢀⡀⢰⠎⢀⠣⣄⠀⠀⠀⠰⠦⢤⠤⠔⠃
⠀⠀⢸⠂⠀⠀⠀⠂⠂⠀⠓⢈⡡⠀⢀⠂⠀⠀⠏⡏⠉⡟⠛⣘⠄⠀⠀⢸⠀⡄⠀⢤⠀⠀⠁⠈⡜⠀⠀⠁
⠀⣊⡀⠉⠀⠀⠀⠈⠉⠁⠀⡀⠠⠐⠁⠀⠆⠀⠡⢘⣀⠀⠊⠞⠀⠀⠀⠈⠀⠂⠒⠈⠀⠀⠀⠎⡀⠀⢀⡂⠔
⠀⠁⠀⠀⠁⠉⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠂⡀⠀⡄⢀⡀⠀⢀⡀⠀⣀⢀⠀⢠⠀⠀⠀⠀⢀
⠀⣀⠀⢀⠔⡦⡁⢤⠒⢄⠀⠀⠀⢀⠖⢂⠈⠠⠈⠀⠀⠀⡄⠜⠈⠘⡄⠀⠀⠀⠀⣀⠅⠁⠀⡈⠀⠀⠀⠄
⠀⠀⠀⢤⠞⡱⠒⠚⣾⠚⠆⠀⡐⢁⠂⠁⠀⠀⠨⢰⠃⠀⠀⠁⢐⠂⠈⠀⠀⠀⠀⠢⡑⠢⠂⠀⠁⠀⢀
⠀⠀⠀⠛⠠⠃⡀⣠⢫⣧⠀⠐⡀⠁⡀⠀⠀⠀⠀⠢⠢⡀⠠⠀⠀⠂⢰⠁⠀⣤⠆⠀⠀⠉⡔⡡⠐⠒⠈⠍⣢
⠀⠀⠀⠈⠪⠉⠃⠶⠁⣊⠀⠀⠙⠢⣌⠂⠀⠀⠀⠀⠅⣠⠀⠀⠀⠀⠀⢓⠁⠶⠂⠀⠀⢰⡌⠀⡊⠈⠑⠀⡜
⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡀⠂⢊⡤⠚⠁⡀⠤⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⢄⠆⣀⠒⠊⠀⡶
⠀⠀⠀⠀⢀⡄⠲⣲⠉⢯⠈⠂⠀⠁⠀⠀⠙⠚⠉⠀⠐⢰⢀⡤⢏⡈⠀⠀⠠⠄⠀⠀⠁⠀⠀⠁⠀⠑⠂⠈
⠀⠀⢀⡔⡉⢠⠄⠈⠀⠀⢀⠂⠀⠀⠀⢀⡄⠒⠠⢀⠀⠀⠃⠠⠸⠉⢻⣈⢹⠉⠉⡉⢹⡍⠇⠀⠀⠀⡄⠀⠐
⠀⠀⠁⠀⠃⠀⠀⢀⠪⠐⠁⢠⠀⠀⠰⠵⠁⠦⠘⠦⢁⠀⠀⠀⠀⠀⡀⠄⠀⠄⡀⠀⠀⠀⠄⠀⠀⡘⢁
⠀⠀⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠪⡄⠄⠄⠠⠤⠠⢠⠈⠁⠀⢀⠊⠠⠐⠀⠂⠀⠠⠀⡀⠀⠀⠄⠀⠀⢢
⠀⠀⠀⠀⡠⠀⠠⡔⠀⠁⠖⠀⠂⠀⠈⠐⠐⠒⠒⠒⠁⠀⠀⠀⢈⠀⡁⠰⠀⠈⠀⢀⠀⠀⠀⢆⠀⠀⠀⠈⡄
⠀⡀⠀⢈⠀⠀⠀⠈⠄⠀⢀⠂⠀⢀⠄⣠⠀⢄⠀⠈⠁⠀⠀⠀⠀⢂⡐⠠⠠⠊⣀⠌⠠⠄⠀⠈⣟⡴⡶⡋
⠀⠁⠀⠌⠁⠢⢀⠀⡰⠊⠀⠀⣄⢴⠁⢉⢈⠑⠑⠀⣀⠀⠂⠀⢔⠕⠁⠁⠀⠉⡀⢀⠀⠀⠀⠀⠟⣒⢋⠁
⠀⠀⠀⡧⢀⡀⠔⠀⠰⠀⠀⠀⠶⡀⠌⠄⢂⢢⡄⠀⠀⠀⡠⡠⠀⠀⠰⡁⣁⠀⠥⠕⡀⠀⠀⠀⠰⡦⡏⠀⠉
⠀⠀⠀⠃⠂⠀⠀⢀⠀⠀⠀⠀⢔⠨⣊⣊⢪⠄⡑⠀⠀⡬⠊⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⡄⠀⠀⠘⠯⢃⡀
⠀⠀⠐⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠁⠂⠁⠐⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⣾⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⣩⣽⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣾⣿⣿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⣡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣤⡙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⣁⡛⢋⣉⣙⠛⣉⣉⠛⣋⣉⡙⢃⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⢻⡏⠉⠙⡿⠉⠉⢿⠋⠉⢹⡟⠉⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⣭⣿⣇⠀⠀⣸⣿⣭⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢢⡤⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢣⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠲⠎⠉⢳⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⡼⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢰⠲⠔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠁⠀⠀⠀⠀⠀⠀⠸⢠⣠⣤⠀⠀⠈⣇⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡉⠉⢳⣎⣠⣃⣀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⠃⠀⡁⠘⣉⠉⠳⡒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢋⠭⠒⠒⣏⠀⢀⡀⠀⡁⠀⣰⠯⠭⣍⣓⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢯⠀⠀⠀⠘⠢⠤⠤⠤⠖⠋⠀⢰⣂⣐⠆⣹⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⡀⠀⠀⠙⠒⠶⠶⠶⠶⠶⠦⠤⠤⠤⣤⠀⡰⠊⢁⣀⡠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣟⠑⠭⢭⣓⣒⣒⣒⡒⠲⠖⢒⣒⣒⣒⣒⣾⠁⢻⡭⠭⠴⢚⣿⠉⢻⢲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢸⢶⣤⣤⣀⣀⣀⣀⣀⣀⣈⣬⣭⣭⣤⣤⣬⡇⡿⠴⠖⢒⣩⡟⠀⡇⡼⢠⢷
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣿⣄⡉⠓⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿⠗⠒⠋⠉⣠⡗⠺⣥⠇⣞⣼
⠀⠀⠀⠀⠀⠀⠀⠹⣗⢾⣉⠻⢭⣟⣒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠖⢒⣒⣩⣭⣛⣀⣠⣼⠾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠺⣭⣭⣛⣛⣛⣛⣓⣒⣒⣒⣒⣒⣒⣒⣻⣿⠭⠭⠭⠿⠛⠛⠋⠉⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣹⡈⣇⢣⢹⠉⣏⠉⠉⡏⢻⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠾⠶⠶⠾⠾⠦⠶⠶⠒⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠿⣿⡿⢿⡟⢿⡟⢿⡿⠻⣿⡿⢻⣿⠟⣿⡿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⣠⡶⣿⡟⠻⡆⠹⡆⠹⡇⠘⠃⠸⠇⠸⠁⠀⠟⠀⡼⠁⢰⠏⢀⣿⠟⢉⣿⠿⣿⣿⣦⣄
⠀⠀⠀⠉⠳⡈⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣤⡄⠀⠀⠀⠀⠘⠁⠀⠊⠀⠘⠁⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡙⣿⡝⢧⡘⣷⡀
⠀⠀⠀⢠⠐⣆⠹⣏⠉⢷⣄⠀⠀⠀⠈⠛⢷⣌⣻⣮⡳⣼⡻⣶⢄
⠀⠀⠀⣨⣷⡈⠣⣌⢑⣦⡙⢷⡒⢦⣄⠀⠀⠈⠻⣿⣧⢻⣇⢹⡌⢄⡀⠀⣀⠤⣴⠤⣤⣤⣶⣆
⠀⠀⠀⠀⠉⠛⠶⣿⣿⣯⣻⣆⠳⣌⠻⣦⣀⣀⣀⣈⣻⣦⣍⠓⢯⠱⡄⢹⡵⠚⢉⡭⠯⠭⠤⠼⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣏⣽⣾⣿⡿⣍⠻⣝⢮⣧⡬⠿⣷⡈⢧⡙⠦⢰⣞⢩⡴⢋⣩⣻⣋⣩⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⣉⡽⠿⣿⣿⡷⡜⢷⡙⠷⠾⠿⠷⢶⡭⣻⣆⠹⣆⢳⡾⠿⠿⠛⠉
⠀⠀⠀⢰⡿⢛⣩⠽⠿⠭⠤⠞⢉⣠⠞⠋⠙⣷⡘⠦⠭⠭⠭⢍⡙⢦⡙⢮⣻⣦⡙⢮⣷⢄⣀
⠀⠀⠀⠰⢿⣿⣴⠞⠋⡉⣉⡓⠛⢁⣄⣼⡿⠿⠟⠻⢿⣿⣷⣦⣝⢦⡙⢦⣈⣽⡟⠲⣌⠓⢮⣍⣝⠖
⠀⠀⠀⠀⠀⠈⠻⣮⣿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣥⡿⠿⠛⠛⢾⣆⣸⡿⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣷⣶⣤⡀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡉⠉⠉⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠟⠛⠛⢁⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⡟⢻⣿⣿⣿⣦⠀⠘⠻⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠋⣿⣿⡿⠋⠀⢸⣿⣷⠀⠙⣿⣿⣿⣧⠀⠀⠀⠈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠀⠀⣿⡿⠀⠀⠀⠘⣿⣿⠀⠀⠈⢽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⢀⣿⣥⣄⣀⠀⠀⣿⣿⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣤⣤⣠⣾⣿⣿⣿⣿⡿⠿⣤⣿⣿⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏⠈⢿⣆⠙⠻⣿⣷⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⣿⡏⠀⠀⠘⣿⡆⠀⠈⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠻⠀⠀⠀⠀⢹⣿⡀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠂⣈⢁⣀⠠⠤⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠀⠀⠀⠒⠢⠀⠀⠀⢀⡀⠤⠔⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⡶⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠒⠀⠁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠞⠠⠀⠀⠀⠅⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠉⠀⠀⠀⠀⠀⣀⠤⠴⠜⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⣀⣠⠠⠖⠊⠉⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⢰⠐⡤⢤⣄⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠀⠠⡀⡀⠄⠂⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀
⠀⠀⠀⠀⡠⠔⠒⠋⠉⣁⠀⠠⠤⠬⠀⠀⠠⠀⠀⠀⡉⠉⠛⠶⠄⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠌⠄⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡄⠐⠤⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⡌⠄⠀⠀⠀⠠⠄⠀⠀⠠⣀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠀⠀⠀⢀⡼
⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠼⠁⠀⠀⢐⠠⠖⠈
⠀⠀⠀⠀⠈⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠂⠀⠀⠀⠀⠀⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡠⠐⡰⡀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠶⠶⠴⠖⠠⠖⠂⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠚⠉
⠀⠀⠀⠀⠀⠐⠊⠁⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠘⠂⠍⠙⣛⠓⠂⠠⠤⠤⠤⠤⠤⠀⠀⢀⡀⠂⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢠⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡄⠀⠀⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⢈⠑⠀⠠⠂⠇⡤⠀⢀⠀⠂⠈⢃⡅⠀⡀⠓⠀⠠⠌
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠤⠤⠤⠤⠤⢤⣀⣀⣀⣀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠆⠴⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠉⠠⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠀⠠⠤⢉⣉⠛⠶
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⡆⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⠓⠢⠤⠄⢠⣄⢤⠤⠀⠤⠤⠤⠤⣀⡀⠀⠀⠐⠊⠉⠀⡀
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠐⡦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠘⠁⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠑⠫⠐⠠⠄⣀⡀⠀⠀⠀⠀⣀⡀⠤⠐⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠈⠉⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠁⠀⠀⠀⢀⡠⠤⠖⠓⢒⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⣀⠴⠊⠁⣀⡠⠴⠚⠋⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡠⣪⣴⣂⠔⠋⠁⠀⠀⠀⢀⣠⠞⡛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⢄⡴⢞⠵⠋⠀⠀⢀⣠⠴⠒⠋⠉⠀⠀⠆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀⡴⡥⣫⠔⠁⢀⣠⠴⠚⢁⣠⡃⣜⡧⠟⠶⠄⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠊⠀⠀⠠⠊⢀⠞⠁⣠⠔⢋⢰⣬⢰⡸⢸⠷⠏⣁⡠⠴⠒⠛⠋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠐⢁⡴⠁⡠⠞⠡⣰⣯⣧⣿⠻⢃⡤⠖⠉⢀⣀⣤⠤⠴⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠈⢠⠎⢠⠞⣥⡛⢖⠓⠉⢀⡠⠚⠁⢀⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠠⠃⡰⠃⠀⠁⢀⠈⢁⡴⠋⢀⣴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠦⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠈⠀⡠⠁⡼⠁⢀⣤⡤⠀⡠⠊⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⢀⣠⠔⢋⡩⣴⣶⡿⠿⠿⢓⣮⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⡰⠁⢰⡧⠞⣻⡟⠁⠜⠁⢀⡔⠁⠀⠀⠀⠀⠀⠀⣠⢖⣩⣴⠌⠑⣊⠥⠒⢊⢉⣭⡉⠙⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡜⠀⠀⠉⠀⡰⠋⢠⠊⠀⡰⠊⠀⠀⠀⠀⠀⢀⣴⣫⡶⠟⠋⣠⠔⣋⡤⡀⢿⣿⣿⣷⠠⢀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⣠⠞⠁⡰⠁⢀⠞⠁⠀⠀⠀⠀⣠⣶⠏⡡⠊⢀⡤⢮⣥⣝⠿⠿⠿⠷⣏⣙⣛⠀⠀⠓⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠄⣀⣠⠖⠁⠀⠀⠀⠴⠋⠀⠔⠀⣠⠋⠀⠀⠀⠀⣠⡾⢿⡿⠋⣀⠔⣣⡝⠮⢟⣩⠵⠒⠊⠉⠉⣀⣈⣉⠉⢲⠎
⠀⠀⠀⣀⠀⣀⠀⡀⠈⠀⠀⠠⠀⠀⣼⠓⡽⠁⠀⠀⠀⢀⡰⠦⠊⢀⡔⠁⠀⠀⢀⢴⣿⣯⠔⠂⡠⠚⣁⠶⠈⡠⠖⠉⠀⢀⡠⠔⠒⠉⠉⠉⠉⠉⠋
⠀⠀⠸⣄⢀⡉⠈⡀⠀⠐⣠⣶⣿⠛⠁⠚⠀⠀⠀⢀⡴⠏⠔⠀⢀⠌⠀⠀⡀⢂⢕⡛⠚⣁⡤⠊⠀⠀⣈⠔⠈⠀⢀⡤⠚⠁
⠀⠀⠀⠁⠀⠀⠀⢚⠽⢷⣿⡉⠡⠔⠀⠀⠀⠀⣔⢂⠐⠁⠀⠔⠁⠀⠠⣪⠞⠕⠊⠠⠚⠁⠀⢀⠀⠊⡠⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠈⢦⣀⣀⠊⣹⠁⢸⡀⠀⢤⠶⡓⠂⠀⢀⠔⠀⠀⢀⡡⠊⠉⠀⠀⠀⠀⠀⠀⠔⠁⠤⣊⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠎⠀⡆⢀⢨⡙⠷⣯⣥⣴⣇⣾⣡⡀⠄⠚⠉⢀⠀⠀⠀⢀⡠⠔⠊⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡇⢸⣷⢼⣤⣤⡠⠄⢀⣀⡀⠤⠔⠒⠈⠁⠀⠀⢛⡥⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣦⣾⣿⣶⣦⣤⣐⣤⡲⠖⠛⠂⡀⠤⠚⠁⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡿⠟⠿⠋⠉⠉⠈⠀⠀⣈⣡⠤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢍⣃⣀⠠⠤⠒⠉⠁



⣿⡟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⢄⠒⠂⠉⣉⣉⡒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣥⢀⣀⣀⡀⠤⡇⣰⠉⠀⠀⠀⠉⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠚⠂⠦⠤⢤⡚⠠⣏⠀⠀⠁⠀⠐⠀⠀⠈⠢⡤⠤⢀⣀⡀
⠀⠀⠀⠀⣀⡠⠔⠒⠉⠉⠁⠀⡰⠁⠀⠀⠠⠀⠄⠀⠈⠙⣌⠢⡀⠀⠁⠄⠀⠠⠄⠀⢸⠀⠀⠀⠀⠉⠉⠒⠢⣀
⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢂⠄⢀⠀⠁⠀⢀⡿⠢⣈⠑⠒⠂⠠⠤⠤⠔⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⠀⠀⠀⠤⢔⡪⠕⠁⠀⠀⢉⠗⠒⠀⠠⡔⠚⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⢣⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣔⠤⢑⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⢆
⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠂⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡇
⠀⠀⡌⠙⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡆⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⣀⠔⠊⠱
⠀⠀⢰⡀⠀⠀⠈⠉⠓⠠⠭⢍⣈⡁⠉⠉⠑⠒⠣⠬⠬⢰⣁⣀⣀⣀⣀⡀⠠⠤⠤⠔⠒⠉⠁⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⡰
⠀⠀⡎⠈⠓⠤⢄⣀⣀⢀⠀⠀⠀⠀⠉⠉⠁⠚⠒⠐⠒⠒⠒⠂⠀⠤⠤⠤⠤⠤⠀⠤⠤⠤⠔⠒⠊⠉⠀⠀⠀⠀⡠⠒⠉⡇
⠀⠀⡱⣄⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠤⠔⠊⠀⠀⣴⠁
⠀⠀⢇⠈⠑⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠁⠒⠤⠤⢀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⣀⠤⢊⡎
⠀⠀⠀⣱⢤⡀⠀⠈⠑⠢⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠤⠤⠤⠤⠤⠤⠀⠒⠒⠈⠉⢀⡴⣋⠆
⠀⠀⡎⠸⡀⠙⢕⠢⢀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡶⠁⠀⠈⠆
⠀⠀⠱⢄⡙⠢⢄⠉⠒⠬⠕⣲⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⠒⠊⣁⠔⠁⠀⠀⠀⣤
⠀⠀⢠⠀⠈⠑⠂⠬⣑⡂⠤⢀⣈⠉⠉⠛⠃⠭⠄⠈⣀⣀⣀⣈⣀⣈⣉⣉⣉⠁⠤⠄⠐⠒⠒⢂⡠⠋⠀⠀⢀⡀⠔⠊⠀⡇
⠀⠀⠀⢦⠀⠀⠀⠀⠀⠈⠁⠒⠢⠬⣉⡁⠒⠒⠒⠐⠐⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠂⠈⠉⠉⠀⠀⠀⢀⠔⠁⠀⢀⡠⠞⡃
⠀⠀⠀⠈⠳⡢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠠⠤⠤⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡤⠔⣁⠤⡴⠚⠁⠀⡠⠁
⠀⠀⠀⠀⠀⠹⣑⠒⠩⢄⣒⡒⠠⠤⠤⢀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠒⠂⣁⠔⠉⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⠉⠉⠁⠒⠒⠠⠤⠭⠬⠩⣭⡍⠉⢉⣉⣉⣉⣉⣉⣉⣁⣀⠠⠤⠒⠉⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠐⠒⠂⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⠱⣆⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠻⠄⢠⠀⡀⠈⡼⣋⠄⠀⠂⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⣉⣉⣉⡛⠷⢦⡈⣵⠲⣮⢢⣞⡼⠋⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡒⢼⣋⣬⢿⣾⣲⠅⠒⣝⢾⣷⢳⣿⣋⣴⣆⣽⣥⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⡴⠄⣶⣤⣠⠆⡀⠀⠀⠉⠒⠾⣾⣷⢿⣭⣭⣍⡤⢀⣀⣈⠁⢀⠙⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠛⠓⠒⡯⠔⢊⢹⡇⠡⠀⡀⠈⠉⠒⠶⣤⡛⠘⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆⣠⡾⣫⣿⠁⠀⠏⠙⠑⠴⠄⣀⠀⠉⠢⡀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢁⡼⢋⡴⣇⡟⡧⠀⢸⠈⡓⠀⠀⠁⠘⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠰⢋⣴⡟⢰⢸⠇⣶⠀⠘⢀⡋⠀⠀⠀⠀⠀⠘⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣠⡟⠉⠀⠆⣿⠀⢹⡀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣤⠋⠀⠀⠀⠘⡿⠀⠋⠷⢀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢰⢰⣷⠀⠀⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠻⣷⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣄⠻⣿⡆⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⡇⢰⡿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠉⠉⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣭⣅⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣨⣭⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⣄⣉⠉⠛⠛⠻⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⠛⠉⣉⣡⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢁⣼⠋⡉⢻⡟⢿⠛⡿⠻⢿⡿⠿⣿⠻⣿⢻⡟⣛⡟⢉⣽⠉⡉⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡈⢻⡀⠟⢸⡇⡄⠀⡇⢸⠀⡇⢈⣿⢀⠈⢸⣧⠉⡷⠠⢿⠀⢃⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣿⣶⣿⣶⣷⣤⣾⣧⣤⣿⣴⣷⣾⣷⣶⣿⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣸⣿⣿⡀⢸⣿⣿⡆⠀⠀⣶⣶⠀⠀⣶⣶⠀⠀⠀⣿⣿⠀⣾⡿⠁⢸⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡿⢿⡇⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⣿⣿⣼⣿⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣿⡇⢸⣿⣿⠃⣿⣧⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⣿⣿⢿⣿⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢸⣿⡇⠈⣿⡿⠀⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⣤⡄⠀⣿⣿⠈⢿⣿⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠉⠁⠀⠘⠛⠀⠀⠛⠟⠀⠀⠛⠛⠛⠃⠀⠋⠉⠀⠀⠀⠀⢸⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣤⣈⣉⠛⠛⠻⠶⠶⠶⣶⣶⣤⣤⣤⣤⣶⣶⠶⠶⠾⠟⠛⠛⣉⣁⣤⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠶⠶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠶⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⢒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠐⠿⠿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠫⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⢬⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠖⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠾⣤⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣴⡚⠉⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡘⣷⣦⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠻⢿⣄⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠻⣦⣠⠆⢀⣼⣴⡏⠁⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠛⡷⢻⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢀⣏⠁⠀⠳⣶⡿⠷⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⢿⠀⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣂⡀⠀⠀⠨⠏⠀⠀⢈⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣏⠓⠒⠛⠛⠿⠶⠤⠴⠤⠤⠤⠭⢽⡦⠔⠶⠚⠛⠛⠋⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡀⠀⠀⢀⣀⣀⢀⡀⠀⢀⡀⣀⡀⠀⢀⠀⠠⣤⢴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢣⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣢⠤⠀⠀⠆⣀⣀⣀⣀⣀⣀⣀⡤⠤⣴⣗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠓⠒⠒⠒⠒⠒⠚⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣻⣿⣿⣿⡿⣻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⢿⣷⣶⣶⣶⣶⢶⡶⣶⣦⣤⣬⣽⡾⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠊⠙⠿⠿⢿⣿⡐⣿⠛⢛⣿⠛⣰⡟⡛⣻⡟⠻⣻⠋⣸⣿⡏⣉⣹⠛⢸⠇⣻⣿⣿⣿⣿⠿⠥⣶⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠿⠿⠿⠷⠾⠿⠿⠿⠿⠿⠿⠿⠿⠯⠼⠿⠿⡿⢿⣿⡿⣿⣿⣶⡷⡿⠿⠿⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⣖⣚⣛⣋⣉⣉⣉⣭⣭⣭⣭⡭⠭⠭⠭⠭⠭⠭⠭⠭⢭⢭⣭⣉⣉⣉⣛⣛⣓⣶
⠀⠀⣼⣉⠟⠉⣉⠟⢙⣟⡖⠒⢒⡶⢶⢲⡖⠒⡶⢲⢶⠖⠒⢒⡶⢒⣿⡟⠉⣹⢋⡟⢹⡆⠀⠀⠀⢀⣀⣀⡀
⠀⠀⣿⣻⠒⢚⡷⠶⣿⡿⠶⠾⠯⠦⠾⠯⠴⠾⠥⠾⠧⠶⠾⠿⠶⠾⢿⣶⠾⢷⡟⠺⣿⡇⣠⠶⢯⠉⠙⡏⢹⡟⢦⡀
⠀⠀⣿⠁⢠⠟⢿⠞⠁⢀⡴⠒⠚⠛⠒⠲⣄⠀⠀⢀⡴⠚⠉⠙⠓⢦⡀⠈⢳⡿⠻⡄⠈⡟⢻⣄⡼⠷⠾⠧⣼⡁⢀⡿⡆
⠀⠀⣿⠀⣾⢷⡏⠀⢰⠋⢀⣠⠤⢤⡀⠀⠈⣧⢀⡏⠀⠀⣠⠶⣄⠀⠹⡄⠀⢻⣋⣿⠀⣧⡞⠁⠀⠀⠀⠀⠀⠹⡟⢀⣿
⠀⠀⣿⠀⣿⣺⡇⠀⢸⠀⠈⢧⡀⢀⡟⠀⠀⣼⣼⡇⠀⢸⣁⠀⣸⠆⠀⣿⠀⢸⣏⣻⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢹⡞⢸
⠀⠀⢻⢀⣿⣺⣇⠀⠘⢧⠀⠀⠉⠉⠀⢀⡴⢛⣛⢷⡄⠀⠈⠉⠁⠀⢠⠏⠀⣼⡉⡿⡀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣾
⠀⠀⢸⢸⡈⢧⣼⣦⡀⠈⠙⠶⠤⠤⣴⢋⡜⣡⣌⠳⡝⣶⠤⠤⠤⠖⠋⢀⣴⣿⡽⠁⡇⡇⠀⠀⠀⠀⠀⠀⠀⣰⣋⣀⣸
⠀⠀⢸⡆⢧⡈⠻⣄⣿⠶⣤⣤⣤⠴⡏⢸⠈⢧⡸⠃⢷⢸⡦⣤⣤⣤⠶⣿⣤⠞⠁⡼⢡⡇⠀⠀⠀⠀⠀⠀⡴⠛⠛⢻⠏
⠀⠀⠈⡟⢦⡑⢦⣈⠙⡷⢋⡤⢭⡳⣷⠸⡄⠀⠀⢀⡟⣸⠟⡩⠥⣍⢳⡋⣁⡤⢊⡴⢻⠇⠀⠀⠀⠀⣠⠾⢧⡀⣰⠋
⠀⠀⠀⣇⠀⠉⠲⢬⣙⡇⢿⡀⢦⠗⠈⢳⡙⢆⣠⠎⡴⠃⠸⠴⠆⣽⢸⣏⡥⠖⠋⠀⢸⠀⠀⢀⣠⠞⣿⠀⣨⠟⠁
⠀⠀⠀⢻⠀⠀⣠⠴⢮⡻⣄⠓⢦⣀⠀⠀⢹⣌⢡⡾⠀⠀⢀⡤⠞⣡⠞⡵⠲⣄⠀⠀⣾⣠⡔⢻⡇⢀⡼⠟⠁
⠀⠀⠀⢸⣆⠀⠙⠦⠼⠃⠘⣗⣦⣈⠳⣄⠀⣿⢻⠃⣠⠞⣉⣤⣺⡇⠈⠧⠴⠚⠀⣠⡏⢸⣇⡤⠟⠉
⠀⠀⠀⠀⣿⣓⠦⠤⣤⣴⠊⠉⠀⠈⠑⢮⢳⣸⣟⡴⣡⠞⠁⠀⠈⠙⢦⣤⡤⠴⣚⣽⠗⠋⠁
⠀⠀⠀⠀⢹⡈⠉⠛⢺⡇⠀⠀⡴⠒⠦⣄⢳⣿⣿⡟⣡⠴⠒⢦⠀⠀⠈⡗⠛⠉⢁⡇
⠀⠀⠀⠀⠈⡇⠀⠀⠀⢧⡀⠀⠳⣄⡀⠈⢧⢿⣿⣰⠃⠀⣠⠞⠀⠀⡼⠁⠀⠀⣸⠁
⠀⠀⠀⠀⠀⢻⠶⠤⣤⣀⣙⣦⣄⣀⠉⠓⢮⣿⣿⡧⠖⠋⣀⣠⡴⢚⣀⣀⣤⠴⡏
⠀⠀⠀⠀⠀⠘⣇⡴⠋⠁⠈⠉⠉⣩⠿⠛⠛⠛⢻⠛⠛⠛⢯⡉⠉⠉⠉⠙⢦⣼⠃
⠀⠀⠀⠀⠀⠀⠙⠓⠲⠦⢤⣤⣾⣁⣀⣀⣀⣀⣾⣀⣀⣀⣀⣹⣤⠤⠴⠖⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⡘⠁⠁⣀⠤⠥⠖⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡴⠁⠀⢰⠁⠀⣀⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣋⠥⠚⢀⡤⠤⠊⣼⠟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⣪⠅⠒⠊⡡⠤⠴⠊⠀⣀⡤⣣⠀⣠⠤⣤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡔⢉⡁⠴⠒⢉⢔⠶⠆⠉⠛⠋⠈⢠⠒⢁⠊⠁⠀⣀⠭⠶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠞⢠⠃⠠⡦⠶⠛⠁⠀⠀⠀⠀⠀⠀⡄⢀⠞⠀⠀⡎⠀⠀⡰⡟
⠀⠀⠀⠀⢀⡠⠖⠒⠤⠤⠤⠠⠖⠁⣠⠎⢰⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⠖⣊⠤⠃⢀⡠⠔⠡⡶⠛⠊
⠀⢀⠔⠁⠈⢀⣀⣀⠤⠤⠠⠔⠊⣩⠔⠒⠁⠀⢠⣾⠇⢀⠀⠄⠒⡩⠭⠔⢊⡀⠠⠔⠁⠀⣠⡔⡇
⠀⠸⠀⠀⡜⠉⢠⠔⠒⣒⠒⣒⡉⠀⢀⣾⠍⠭⠛⠁⢀⠇⢠⠒⢉⡠⠔⠒⡡⡦⠶⠦⠉⠙⠓⠉
⠀⠸⠤⠼⠥⠴⣁⣀⣯⠖⠭⠕⠒⠛⠒⠁⠀⠀⠀⡔⠉⡠⠃⡜⠁⣔⠤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡤⠠⠤⢀⢄⣀⡤⠎⠀⡼⢡⡜⠀⠀⣺
⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⠉⢀⣀⣀⢤⣀⡠⠖⢊⡩⠔⠊⠀⠀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠊⠁⡠⠄⠠⠐⠒⡒⠉⠀⣰⡢⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠰⠇⡠⠮⠤⡞⠀⣸⡭⠭⠭⠒⠁⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡟⠛⠛⠒⠒⠒⠒⠶⠶⣶⣶⠶⠶⢶⣶⣶⣦⣤⣤⢤⠤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⣠⣶⡟⠛⠋⠉⠁⠛⠶⣤⡀⠀⠈⠃⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣰⠿⢃⣴⡿⠟⠉⣀⣤⣶⡄⠀⠀⠀⠘⢿⣦⡀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣼⣏⣴⡿⠋⢀⣤⣾⡿⠉⠈⠻⣦⣄⣀⣀⠀⠈⢻⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢠⣤⣤⣾⣿⡿⢋⣠⣾⠿⠛⠁⠀⠀⠀⠀⠈⠙⠙⠻⠿⢦⡀⣙⣧⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⣰⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤⣤⣶⣦⣄⠀⠀⠀⠀⣀⣀⣀⡀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⠟⡋⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⡿⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⢨⣿⣿⣿⡿⣿⡍⠙⠋⠉⠀⠠⢀⣀⣀⣠⣤⠐⠀⠈⠙⠛⠟⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣾⣿⣿⣿⣧⡸⣷⠀⠁⠀⠀⠀⠈⢉⣉⣁⣀⣀⡀⠀⠀⠀⠀⠀⣴⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣧⣿⣿⣾⣿⡇⠸⣧⡀⠁⢠⣶⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⢉⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠹⣷⡀⠈⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣠⣾⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣄⠈⢿⣆⠀⠉⠉⠉⠉⠁⠀⠀⠀⣀⣴⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡇⠈⠻⣷⣄⣀⣀⣀⣠⣴⠶⠛⠉⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣸⠋⠀⠀⠈⣿⠙⠋⠉⠁⠀⠄⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⢠⡟⠃⠀⣀⣴⣿⣤⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢀⣈⣥⡶⠟⠋⠀⠹⣿⣷⣄⣀⣀⣀⣀⣤⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣻⣿⣿⣿⣛⣋⣽⣿⣧⣴⠶⠶⠶⠾⠟⠛⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣰⡷⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣁⡴⠛⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣤⡦⢂⣥⡞⣻⡍⠀⢺⠟⠄⠀⢠⣤⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣀⣤⣠⣀⠀⢀⠒⠚⠉⢨⡥⠶⠟⠋⣀⡴⠋⣰⢟⢴⠒⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⠀⠠⠤⣤⣤⠄⠠⢀⡀⢀⢀⠐⢛⠁⠴⠚⢁⣾⡇⡀
⠀⠀⠀⠀⠀⢀⣤⠤⠀⠿⠛⠙⠛⠷⠶⠶⠂⠄⣊⣭⡾⢛⣥⢞⡩⢄⠀⣀⡤⠟⠈⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢸⣟⡙⠛⠓⠒⠒⠀⠂⠀⠀⠀⠉⠉⠤⠶⢛⠕⣹⠯⠗⠋⠀⠀⠀⠠⠤⠚⠁⠀⠀⣨⠞⣻⣦
⠀⠀⠀⠀⠀⢸⣿⠿⢯⣤⣀⣉⣉⣉⣀⡩⢹⣡⣬⠵⠖⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⠁⢸⡟⠛⢳
⠀⠀⠀⠀⠀⠸⠑⠀⠀⠀⠉⠙⠛⠋⢉⡀⠭⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠴⣊⣤⠟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢠⠠⢤⣀⣀⡄⠤⠄⠴⠶⠶⣚⠉⢀⠔⠁⠺⠛⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠶⠦⠴⠂⠈⠉⣿⠛⠋⡡⡬⠞⠛⠁⠒⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⡤⠤⠶⠚⠋⠉⠀⠀⠀⠀⠀⣀⠀⢀⡔⠈⣰⠃
⠀⠀⠀⠀⠀⠀⠠⠄⢀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠉⠉⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠘⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣽⣦⠀⣿⣡⣴⣿⡿⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣟⣥⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⢿⣿⣿⣿⣿⣟⠛⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣴⠏⠀⣀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠁⣼⣿⠿⡿⠙⣿⡆⠀⠈⠋
⠀⠀⢀⣤⣶⣾⣿⣿⣶⣿⣾⣿⣾⣟⠿⢿⣆⠀⠀⠀⠀⣿⠁⠀⣰⣿⠇⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⢀⣴⠶⠿⣿⣿⣿⡉⠛⢷⡀⠈⠀⠀⠀⠀⠃⢀⡼⢹⠏⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⢰⡟⠁⠀⣼⠛⢿⣿⣷⠀⠀⡇⠀⠀⠀⠀⠀⢀⡾⠁⠈
⠀⠀⠀⠀⠈⠁⠀⠀⢿⡆⠈⢷⡙⡇⠀⠁⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣾⣷⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣼⣿⣿⣿⣼⣿⣷⣯⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠿⠿⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢨⣿⣿⣿⣿⣿⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⣿⣿⠿⣿⣿⣿⣾⣿⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢻⣿⣷⣶⣴⣿⣷⣄⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⣴⣿⣷⣴⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢀⣀⣈⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣏⣀⣀
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣯⡿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⡿⣯⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣇⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠐⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣭⣿⣿⣿⣿⣿⣿⣿⣿⡉⣹⣿⣿⣿⣿⣿⣿⣿⣯⣵⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢯⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢹⣿⣿⣿⠿⠋⠀⢸⠀⠈⠻⣿⣿⣿⣿⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠋⠁⠀⠀⠀⢸⠀⠀⠀⠀⠉⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠪⡀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣗⢿⣽⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣣⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠤⡀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣶⣶⣶⡾⣛⢿⣿⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡸⠀⠀⠀⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣇⣿⣿
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢠⠧⠚⠁⠀⠀⠀⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡿⣘⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⣀⣀⠴⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣮⣻⢿⢿⣿⡿⣻⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠒⠉⠉⠉⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠂⣉⣠⡀⠀⠤⠖⡒⣀⠌⠉⠿⠛⢋⣰⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠉⠁⠀⠀⢠⠤⠦⠤⠤⠀⠀⠀⠒⠋⠁⠀⠀⢀⠀⠀⠀⢀⡤⢀⣴⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢀⠄⠀⠐⠛⠀⠀⠊⠀⠀⠀⣀⣤⣾⠁⠀⠙⠛⠷⣶⣿⣧⣭⣉⣟⣿⠟⠁
⠀⠀⠀⠀⠀⠊⢀⣰⡴⠚⠋⠁⣀⣤⣶⣶⡦⢰⣿⡿⠿⣿⣿⣿⣿⣿⠇⠀⠊⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢁⣴⣶⣾⠿⠿⣛⣥⣤⣶⣶⣶⣿⠶⠖⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡻⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢴⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠠⢶⣾⠿⢿⠉⠁
⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠖⠶⠦⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠑⠾⢟⠛⠛⠃
⠀⠀⠀⢀⣴⣞⡭⢴⣟⣠⠤⢤⡶⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⡜⠰⠿⡛⠛⠃
⠀⠀⠐⠋⠿⠋⠀⠛⠁⠀⡴⠟⠒⠒⣲⠶⠂⠀⠑⣄⠀⠀⢰⠳⢾⣟⠋⠀⢀⣠⠤⠐⠒⠒⠒⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⣉⣷⠶⠌⢣⡀⡆⢠⠞⢁⠴⠊⠁⠠⣤⣄⣀⠘⣶⠦⣄⣰⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣉⣉⡓⠦⡳⣵⢃⠞⠁⣐⣶⣖⠒⠺⢷⡀⠉⠺⠇⠀⠙⣧⠙⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢉⠀⢀⣤⠎⢉⣷⣼⡏⣹⢾⡿⠷⠮⠭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⣶⣰⢻⡴⢩⠗⠊⢹⠋⠻⣩⠩⡷⣅⢰⣧⢠⡆⢉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣤⡜⠸⠇⠘⠁⠀⠀⠀⠈⣻⠁⡟⠒⠁⠈⠻⢹⡼⢻⢸⡇⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡋⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠈⠀⢸⡎⣧⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣇⠀⠀⠀⠀⠀⠀⠈⠀⢸⢿⣾⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠈⠁⢀⣸⣆⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⢻⠀⠈⠉⠀⠘⣆⠀⡾⡔⣇⠔⠡⢝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⢆⣣⠈⡧⠴⠖⠚⠁⠈⢣⣧⠃⠻⢰⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠙⠀⣸⠄⠀⠀⠀⠀⣠⠭⠷⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣯⣀⣽⣭⣭⣭⣭⣿⣽⣭⣭⣯⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣯⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠃⢸⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢀⡇⠃⢸⣰⢙⡦⠖⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡌⠉⠑⢄⠈⠃⠊⠀⢈⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⢸⣉⠢⠀⠀⠁⣏⡠⠊⠄⡇⡇⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡑⠒⠢⣤⠏⢣⡀⢹⠇⠀⠑⠢⠀⠀⢀⡔⢸⠃⣠⢞⠙⠦⠔⠶⢒⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣻⡀⡇⠀⠀⠀⠙⠞⠀⠀⠄⠀⢀⡶⠋⠀⠈⠛⠁⠃⠀⠀⠠⣴⠏⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣃⢑⠙⢧⡀⠀⢰⠀⠰⡀⠀⡀⠈⢙⣇⠠⡁⠀⠀⢠⠀⣀⡀⠳⢂⣻⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠌⠉⠙⢙⣲⣬⠀⠀⡇⠀⡈⠢⣼⡟⣠⠁⠀⡀⠐⠄⡁⠉⡀⠀⠬⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣝⣤⣀⣉⠈⢀⣠⠽⢦⡇⠀⡇⠀⢀⣷⠃⠀⢠⣤⠚⠠⠀⠠⠤⡤⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⡄⠄⠀⣀⠤⡽⢶⡇⠀⢸⡷⠀⢂⡴⠁⠀⠀⣀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣤⣑⡀⠀⢈⠻⢷⣤⣼⠟⣠⢟⣿⡿⠚⠋⠉⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⢀⣀⠠⢬⠶⠚⣲⠶⢿⣿⣿⡷⢿⣅⣀⠄⠂⢀⡐⣛⣓⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⢷⣒⣟⡉⠀⢐⣁⠀⢐⣁⡴⢻⠿⢦⡈⠑⢦⡀⠉⠙⠯⣉⣩⠽⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠉⠁⠀⣾⠀⠀⠙⠒⠒⠽⣦⣾⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣽⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⢀⣠⣆⣬⣿⣿⣿⣦⠀⠀⠀⢀⢀⠀⠤⣤⣄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣓⣶⣤⣤⣖⣲⡄⠀⠈⠮⢻⠿⠟⠋⠉⠁⠀⠀⠀⠘⣜⢷⣾⣿⣿⣿⡄⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠
⠅⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⢀⢷⣾⡿⠟⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⡠⠀⠀⠐⠒⠒⠢⢤⠀⠙⠛⠿⠿⠟⠁⠀⠀⠀⠀⠀⠘⣦⣄⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸
⠀⠰⠠⠀⠀⠀⠀⠀⠀⠀⢀⣢⣾⠋⠀⠀⠀⠀⡐⢶⡄⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⢠⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠣⡀⠀⠀⠀⠀⠇⠀⠀⡌
⢠⠀⠀⠀⣈⠀⠀⠠⠀⠀⢀⢸⡏⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⢰⣻⣭⣭⣑⠒⠶⠤⠴⣷⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠙⣄⠀⠀⢰⠀⠀⠀⠇
⠸⣌⠀⢀⠜⠀⢠⠁⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⡇⢿⣦⣀⠈⠳⢤⣿⣿⡿⢷⣶⣶⣿⣿⠀⠀⠀⠀⠄⣸⣷⣦⠀⠀⠀⠀⠀⣸⢿⡇⠀⠘⡄⠀⢸⠀⠀⠸
⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⢈⣿⣇⠀⠀⠀⠀⠀⠙⢦⣽⣿⣧⡀⠀⠀⠀⠀⠀⠀⠉⢪⣿⡀⠀⢀⣴⣿⣿⡿⠏⠀⠀⠀⠀⣴⡿⣸⠁⠀⠀⢱
⠀⠀⡇⢁⠀⠀⡇⠀⠀⢀⣼⣿⣿⣷⣵⣄⠀⠀⠀⠀⠀⠈⠙⢮⠻⢿⣾⣿⣶⣄⣀⣀⣼⣿⣿⣾⣿⠟⠉⠁⠀⠀⠀⢀⡠⣺⡟⣡⡟⠀⠀⠀⢸⠀⠀⠀⢠⡇
⠀⠀⡇⢸⠀⠀⡇⠀⠀⢸⣿⣿⡟⢿⣿⠻⣕⡤⣀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠿⣭⡽⠟⠛⠳⣽⡇⠀⠀⠀⠀⣠⠔⣫⠾⢋⣴⣿⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⡇⠀⡆⠀⢳⠀⠀⠘⣦⣿⣿⡀⠹⣗⢄⡙⠮⣝⡒⠤⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣷⣤⠤⠒⣋⡴⠞⣡⢔⣵⣿⠇⠀⠀⠀⠀⡞⠀⠃⠀⢸
⠀⠀⡇⠀⢡⠀⠈⢧⠀⠀⠈⠳⢽⣾⣄⠈⠳⢭⣒⠤⣉⠛⠶⢤⣤⣈⣉⣙⠒⠒⠒⢒⣈⣩⣭⣭⣿⣷⠖⣋⡥⢖⣩⢶⢟⣽⠋⠀⠀⠀⠀⣼⠁⢸⠀⠀⡏
⠀⠀⠀⠀⢸⡀⠀⠈⠳⡄⠀⠀⠀⠹⣿⣷⣖⢤⣉⠛⠲⢯⣕⡒⠦⠤⢍⣉⣉⣉⣉⣉⣉⣉⠿⡿⣻⣿⣯⡴⢚⡫⢞⣡⠞⠁⠀⠀⠀⢀⡾⠁⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠢⡀⠀⠀⠈⠪⠿⠷⢮⣙⠢⢤⣀⡉⠛⠓⠲⠶⠶⠒⠒⠒⠒⠛⢻⣿⣷⣿⡯⠖⣋⣴⡿⠃⠀⠀⠀⢀⣴⠏⠀⠀⢠⠇⠀⢰⡇
⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠈⠙⠓⠶⠬⣭⣍⣑⣒⣒⣒⣒⣉⣉⣩⣭⣭⣵⣹⣷⣟⡿⠋⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⡸⠀⠀⢸
⠀⠀⢸⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠑⠈⢓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠗⠚⠛⠉⠁⣀⣠⢴⡺⠝⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⡀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠭⢒⠒⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠤⢖⡒⠍⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣠⣄⡏

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠘⠿⠷⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢁⣼⡇
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣠⣴⠟⠉
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⢀⣴⡾⠋⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣶⣿⠋⠀⣴⡟⠋
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠋⢹⡟⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣿⠀⠀⢸⣇⣤⣾⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣧⣤⣶⠶⠿⠛⠛⢉⣡⣾⠋⠀⠀⢸⣏⠁⠈⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⣠⣴⡶⠾⠟⠋⠀⠀⠀⠀⠈⠻⣶⣤⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠉⠛⠿⠷⠶⣶⣶⣶⣶⣶⣶⠶⠾⠿⠛⠉⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣤⣄⣀⣠⡾⠋⠉⠉⠉⠉⠛⢷⣄⣀⣠⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠉⠛⠋⠀⠀⣰⡿⢿⣆⠀⠈⠙⠛⠉⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢰⣟⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣤⣀⠀⠀⠀⠈⠿⣦⣴⠿⠁⠀⠀⠀⣀⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠉⠛⠿⢶⣦⣤⣤⣤⣤⣤⣤⣶⡶⠿⠛⠉⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⡀⠀⠀⠉⠉⠉⠉⠀⠀⢀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠶⠶⠶⠶⠿⠟⠛⠋⠁

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠶⠶⠒⠒⠒⠒⠶⠶⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣚⣭⠥⠶⠒⠒⠚⠛⠉⠉⠉⠉⠉⠉⠉⠛⠒⠲⠦⢭⣙⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠽⠛⠉⠀⠀⠀⣀⣀⣠⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣀⡀⠀⠈⠙⠲⢬⡑⣆
⠀⠀⠀⠀⠀⠀⡞⡿⠁⠀⣀⡤⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⣀⠀⠀⠀⠉⠙⠲⢤⣀⠀⣷⢸⡆
⠀⠀⠀⠀⠀⢸⣇⢷⣠⠞⠁⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⠴⠋⠀⣠⣄⠈⢧⠀⠀⠀⠀⠀⠀⠈⡷⢋⣼⡇⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⡍⠦⣙⠲⣄⡀⠀⠀⠀⠀⣏⠀⠀⠀⣤⠤⠤⠾⠁⠈⣧⢸⠀⠀⠀⢀⣠⠴⣋⡵⢋⡼⠗⠋⠉⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠈⡇⠀⠈⠓⠦⣍⣓⠲⠦⢤⣼⣦⣀⣀⣻⣄⣀⣀⣀⣠⣯⣾⠴⠖⢛⣉⡴⠚⠉⠀⣏⠀⢀⡤⠖⠛⠉⠙⠲⣄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠈⠙⠒⠶⠦⠤⣭⣉⣉⣉⣉⣉⣉⣥⠤⠴⠖⠚⠉⠁⠀⠀⠀⠀⠉⣻⠉⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠉⠛⠒⠒⠒⠒⠋⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⣀⡴⠚⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣶⠤⠤⠤⠤⠴⠒⠋⠁
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠈⠛⣆
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢧⡙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⣿⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣍⣙⡛⠒⠒⠒⠶⠶⠒⠚⠛⣉⣉⡤⠴⠋⠁⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠈⢿⡲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⣀⡤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠉⠳⠯⣝⣒⠶⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⣒⡯⠗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠬⠭⣭⣝⣛⣛⣛⣛⣛⣛⣛⡭⠭⠵⠖⠚⠉⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡲
⠀⠀⠀⠸⠽⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣀⣀⣀⡀⠀⡐⠛⠀⡟⠀⠀⠀⠀⠀⠀⠀⡏⠳⣴⠋⡇⠀⠀⠀⠀⠀⠀⠰⣦⣤⣤⣡⣬⣭⣭⡭⢭⣭⣉⣩⠇
⠀⠀⠀⠀⠀⠀⡖⢒⠒⠤⢀⡖⠀⠀⠉⠁⢀⡤⠀⡀⠈⣞⠀⠠⠀⠈⢳⠀⠀⠀⠀⠀⠀⠱⡄⢉⡴⠃⠀⠀⠀⢀⡀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢄
⠀⠀⠀⠀⠀⠀⢱⠄⠈⠀⢸⡇⠊⠀⠀⠄⠀⠀⠈⠀⠀⢻⠀⠠⡀⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠐⠦⠞⠁⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠷
⠀⠀⠀⠀⠀⠀⢏⡀⠰⠀⠸⡇⢠⠀⠀⠄⠀⠂⠀⠘⠁⣸⠲⢤⣀⡁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣰⠲⡄⠀⡀⢐⠉⡳⠀⢸
⠀⠀⠀⠀⠀⠀⢠⢋⡤⢀⡴⠛⢤⣄⣀⣀⣤⠤⠤⠶⠚⠁⠀⠀⠀⠉⠉⠁⠀⠦⠀⠀⠀⠀⠀⠔⢦⠀⠀⠀⠀⠀⠀⣼⠀⠀⣠⡈⢋⠥⡘⠚⣤⡉⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠳⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠲⡀⢇⠈⣇⠀⠀⠀⠀⢀⡏⠀⠀⣉⠁⠐⠴⠃⠸⡤⠇⡸⡇⠘⡇
⠀⠀⠀⠀⠾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠘⠞⠀⠉⠲⣆⠀⠀⣸⠁⠀⢎⣀⠇⢰⡱⠠⣲⠀⠘⠤⠋⠀⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⡀⠀⠀⠀⠠⣖⡁⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⣴⠢⢀⡀⠀⡄⠀⠀⠀⠀⢀⡟⠀⠀⠙⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠴⠦⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢻⠞⠒⠃⠀⠀⠀⠀⠀⢀⡤⠴⠁⠀⢀⡟⠀⠳⣤⣤⣤⠶⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠀⠀⠀⠋⠀⠀⠀⢰⡆
⠀⠀⢀⣀⠀⠀⠀⡴⠞⠛⢋⡩⠝⠛⠷⠶⢤⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠉⠀⠀⣄⠀⠀⠀⠀⢀⢔⡶⠀⠀⠀⠀⠀⠀⠀⠀⠺⠖⢉⣀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠨⠼⠀⡀⠀⠀⡠⢞⡁⠀⠀⠀⠃⠀⠀⠉⠳⣄⠀⠀⠠⠥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣠⠠⠠⠀⠤⠤⣀⠀⠜⠁⠀⢿
⠀⠀⠀⠀⡌⠰⢀⠞⠀⠈⢁⡴⠖⠶⢤⣀⠀⠢⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠉⠲⣄⠀⢠⠞⠉⡀⠀⠰⢀⠀⠀⠈⢻⣀⢌⠤⣏⠀⠀⠀⠀⠀⣠⡤
⠀⠀⠀⠸⠀⢀⡏⠀⠀⣰⠋⠀⠀⠠⣄⠈⢳⡀⢠⠆⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠄⠀⢈⢶⠃⡐⡰⠁⢀⠃⠄⠀⡠⢀⠂⣿⠀⠀⠈⠙⡧⠀⠀⢀⣽⠏
⠀⠀⠀⡇⠀⢸⠀⠀⠀⣏⠀⢠⠀⠀⢸⠆⠀⣧⠀⠀⠀⢹⠀⠀⠀⢠⣖⠗⠀⠀⠀⣞⠡⠀⠀⠌⣿⡔⡱⠁⢀⠎⡜⠀⡰⢁⠎⣰⣋⠀⠄⠰⡖⠁⠀⠀⠙⠼⠃
⠀⠀⠀⡇⠀⠸⡄⠀⣄⠈⠣⣄⣀⡤⠞⠀⠀⡿⠀⠆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⡠⢂⠌⣸⢷⣅⣠⢃⠎⢀⠔⣡⣯⠞⠁⠉⠳⣎⠀⣿
⠀⠀⠀⢳⠀⢀⠳⣄⠀⠀⠀⠀⠀⡀⠀⢀⡼⠁⡀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⠔⢁⡴⠁⠀⠈⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣤
⠀⠀⠀⠈⢣⡘⠀⠈⠳⢦⣄⣀⣀⣤⠴⠋⠀⠀⠁⢀⡾⠁⠀⠀⠀⠀⠘⠋⠀⠀⠀⠈⠓⠒⠋⠀⠀⢀⣄⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠙⠢⣄⣀⠀⠀⠁⠀⠀⠀⡠⠆⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⢤⠀⠀⠀⠀⠀⠓⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⣤⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⡶⣶⠶⠖⠛⠉⠀⠀⢀⡀⠀⠀⡤⠤⠞⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⢠⣤⣄⡀⠙⣧⡀
⠀⠀⠀⢀⠗⢒⠂⠀⠀⠰⡶⠀⡇⣿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠱⡀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⢠⡾⠁⢦⣴⠋⠀⠈⡷⠠⣹⡇
⠀⠀⠀⠓⢤⡬⠧⠀⠀⠀⠀⠀⣧⢸⠀⢀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠸⠒⠶⠶⠞⢣⡀⠀⠀⠀⠀⠀⠀⠠⠼⠂⠀⠀⠀⣴⡏⢰⣠⡞⠁⠀⢰⠟⠁⢂⣿⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⢸⡆⠘⢏⠀⠀⠀⠀⣸⡄⠀⢀⠆⠀⠀⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⣇⣰⠋⠀⠀⠀⠈⠓⠞⠛⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠿⠀⢸⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⣰⠧⠀⣤⠡⠋⠀⠀⠀⣀⠹⣆⠀⠀⠀⠀⠀⠀⣠⡞⢸⣀⡿⠁⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠾⠇
⠀⠀⠀⣀⣠⡋⢱⠀⠀⠀⠀⠀⢸⠈⡇⠀⠀⠀⠀⠀⠀⢀⡞⠹⢤⠀⣀⡗⢚⡁⡠⠀⠚⠀⣿⠀⠀⠀⠀⠀⣰⠃⢹⣴⠏⠀⠀⠀⠀⠀⠀⣯⣄⡀
⠀⠀⠀⠣⡈⠁⡸⠀⣤⣠⢄⠀⢸⡀⡇⠀⠀⠀⠀⠀⠀⢸⡀⠶⣎⠁⡄⠀⢰⠀⣔⡡⠃⣰⠏⠀⠀⠀⠀⣴⠈⣧⡼⠃⠀⠀⠀⠀⠀⠀⡼⢡⠲⡌⢦
⠀⠀⠀⠀⠀⠀⠁⠀⠡⣙⠞⠀⢸⡇⢷⠀⠀⠀⠀⢀⠀⠀⠙⠲⠦⣤⣬⣭⣥⣤⣤⠶⠞⠉⠀⠀⠀⠀⠀⠈⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⣷⣌⣠⠎⢸⠃⢀⢤
⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠈⠳⠏⠀⠀⠀⠀⠻⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠷⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠈⠒⠒⠚⠁⢠⣾⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠊⠉⠉⠉⠉⠉⢭⠍⣩⣽⡒⠲⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠉⠀⠀⠀⠀⢠⣶⣤⣤⣄⡀⠀⠀⠈⠁⠊⠁⠀⠀⠀⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠐⠒⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣾⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⣀⡀⠀⣠⡌⠘⠛⣦⣴⣦⣄⣀⡀⠀⠙⠉⣿⣶⣤⠶⠀⠘⠖⠚⠁⠀⠀⠀⠀⠀⣠⡮⠑⢄
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡾⢿⢿⣿⡟⠉⠉⠉⣉⡀⠺⢿⣻⡇⠤⠀⠀⠀⢹⡆⠠⠠⣤⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⠑⢄
⠀⠀⠀⠀⢀⠜⡀⠀⠀⠀⣰⡶⣟⣷⣶⣿⣾⣗⣾⢻⣁⡖⢈⣼⡇⣀⣀⡁⠐⣷⣶⣾⣇⠀⠀⣾⣶⣆⣀⡀⠀⠀⠀⠀⢀⣈⠁⠈⠢⡀
⠀⠀⠀⢠⢃⣾⣿⣧⣀⡤⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⢲⣿⣿⣿⣿⣤⣠⣴⣿⣿⣎⣹⣷⣤⡄⠀⠈⠛⠀⠀⠀⠑⡄
⠀⠀⠠⢡⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣤⣀⠀⠰⣧⠘⡄
⠀⢠⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⠻⣿⠋⠿⠟⢻⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣦⣤⣾⣷⡐⡀
⠀⡆⣼⢿⣿⣿⣿⣿⣿⣿⣿⠋⠉⢉⣉⣍⠀⠀⠀⠀⠈⠙⠁⠀⣠⡄⠀⢿⡟⠀⠈⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠁
⢨⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⢿⣷⣤⣴⣦⣀⣠⣾⣧⣄⠙⣩⣀⠀⣀⡀⣀⣈⡻⠿⠄⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⡀⢸⢿⣿⣿⠿⠿⠟⠛⠛⠛⢛⣫⡤⠖⠺⠟⠋⢻⡛⠛⢿⠙⠛⢿⡟⠛⣿⡿⠛⣿⣿⠻⣿⡾⢿⣭⣵⣿⠟⠻⣿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣧
⡇⣾⣶⠆⠀⠀⠀⣀⣀⣀⣴⣿⣁⡇⠀⠀⠀⠀⢀⠁⠀⠈⠀⠀⢸⠀⠀⢸⠁⠀⢸⠃⠀⡏⠀⠘⠛⠋⣻⣿⣿⢁⡀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠿⠏⠺⠶⢿⠿⠿⠋⢩⣿⣿⣿⣷⣄⣴⣦⣀⣼⣇⡀⣼⣆⣀⣾⣀⢀⣸⡀⢀⣿⠀⣴⠀⠀⡀⠀⠀⣹⣼⣿⣿⣷⣾⣿⣿⣿⣿⠿⠛⠻⠿⠟
⡇⢀⠀⠀⠀⠀⠀⠀⠩⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣼⣷⣴⣿⣿⣿⣿⣿⣿⠿⣋⠳⠟⢠⠤⢖⣴⣿⡇
⠀⠨⠄⠀⠀⠀⠀⠀⠀⠰⠿⠟⠻⠟⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⣀⠈⠅⢀⣥⡶⢛⣠⣤⡄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠀⠈⠁⠀⠉⠁⠀⠀⠈⠉⠉⠙⠋⢉⡀⣀⣀⣤⣮⣷⣦⣶⢟⣫⣾⣿⣿⣿⢣
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣶⣮⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡘
⠀⠀⠀⠀⠲⡛⠛⠋⠱⣤⠤⣤⡄⣠⡤⠀⢠⠀⡀⠀⠀⠀⠀⣰⣾⠿⠿⠿⠟⠉⠋⡉⠛⢻⣿⣿⣿⡿⠿⢛⣿⣿⣿⣿⣿⣿⡿⠛⠻⢿⡿⡰⠁
⠀⠀⠀⠄⠛⠀⠲⠤⣤⠛⠶⣾⣷⠾⠻⠿⠿⠛⣉⣠⡄⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠀⠀⠀⠀⠶⠖⠀⠳⣿⢿⠟⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠈⢂⠀⢀⣀⣀⣀⣐⣿⣿⡷⢶⣿⣓⢀⡙⠛⢚⠛⠀⢂⣤⣄⠠⠤⠤⣰⣶⣦⣠⣄⠀⠀⠀⠀⠀⠀⠀⣊⡀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠣⡈⠉⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠘⠉⠛⠋⠀⠺⠶⢾⣿⣿⣺⠟⠉⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣒⣂⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣀⣒⣊⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠶⠿⣿⡿⠟⠉⠉⠉⠉⠙⢛⣿⠿⠶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠉⠀⠀⠀⠀⣿⣅⣀⣀⣀⣀⣀⣴⡿⠋⠀⠀⠀⠀⠉⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠻⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡾⠟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣦⣄⣀⢹⣧⠀⠀⢠⣶⠿⠿⢷⣦⡀⠀⠀⣰⡿⠛⠉⠉⣀⣠⣴⣿⣁
⠀⠀⠀⠀⠀⠀⣠⣶⠿⠻⣧⡈⠉⠛⢻⣿⠀⠀⣿⣧⣤⣤⣤⣽⣷⣤⣴⣿⣷⠶⠾⠟⠛⠉⣁⣼⠟⠿⣶⣄
⠀⠀⠀⠀⢠⡾⠋⠀⠀⢸⣟⠛⠷⣶⣬⣿⣄⣸⣟⠀⠀⠉⠉⠉⠉⠈⠁⣀⣀⣀⣤⣤⣶⠾⠛⣻⡆⠀⠀⠙⢷⡄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠻⣶⣤⣀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⣀⣤⣶⠟⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠉⠛⠛⠿⠶⢶⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⡶⠶⠿⠛⠛⠉⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⠶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠶⠾⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣾⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣶⣾⣿⣆⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣇⠀⠀⠀⣰⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠙⢿⣿⣿⣿⣦⠀⣰⣿⣿⣿⠿⢁⣠⣤⣶⣶⣶⣶⣶⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡙⠋⠛⠛⠛⠻⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠀⠀⣡⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣿⣿⣿⣿⠋⢸⣿⣿⣿⣿⡇⠀⠙⣿⣿⣿⣟⠻⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠃⢀⣿⣿⣿⠉⠀⢸⣿⣿⣏⣿⣷⠀⠀⠙⢿⣿⣿⡄⠀⠉⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⢸⣿⣿⠃⣿⣿⠀⠀⠀⠀⠙⠟⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠋⠁⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⠀⠀⠤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢶⣥⣏⣀⣀⣀⣀⣀⣀⣀⣀⣤⣬⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡉⠙⠶⢤⣤⣀⣀⣀⡀⠀⠉⠉⣉⣀⣀⣀⣤⠿⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠙⠛⠒⠦⠤⠤⣭⣭⣭⣭⣭⣭⣭⣥⠤⠤⠴⠒⠛⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⡦⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣶⣶⣶⣯⡭⡭⠭⠭⠭⠭⠷⠦⠶⡟⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⣿⣿⣿⠻⢿⠛⠃⠀⠀⠀⠀⠀⠂⢀⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠉⠉⠛⠒⠒⠒⠲⠤⠤⠤⠤⠤⠶⠞⠒⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣦⢠⡌⢯⢳⡀⣀⠐⠢⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⣤⣀⣈⡘⠑⠋⠈⢧⠣⠃⠓⠤⠃⢀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣾⣿⣿⡟⢻⠑⠀⠙⠒⠋⠛⣽⡏⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⣿⣿⣿⣇⣿⠀⠀⠀⠀⠀⠰⢿⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣼⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠾⢿⣿⣿⣑⠂⠀⠀⠀⠀⣀⣿⡤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⢿⣿⣿⣖⡒⠒⠒⠒⠛⣿⠋⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠤⠤⠤⠤⠴⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⢠⡐⠉⠙⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⠀⠀⢺⠀⠀⠀⠘⠀⠐⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠘⠒⠊⡁⠀⠈⢹⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣭⣍⣛⣛⡻⠿⠷⠶⠶⠶⠒⠒⠀⠀⠐⠉⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣀⣈⡉⠉⠛⠛⠿⠿⠷⠒⠚⠛⠃⠀⠀⠀⢴⠿⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣮⣉⡙⠛⠛⠿⠷⠶⠆⠀⢒⠀⠀⠀⠀⠐⠈⠀⡴⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢤⣉⠛⠛⠻⠷⠶⠶⢶⣤⣤⡤⠀⠀⠀⠒⠀⣀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠙⠻⠶⣶⣶⣤⣤⣤⣤⡤⠤⠄⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠶⢶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠛⠋⠙⠛⠉⠉⠉⠉⠉⠙⠛⠉⠛⠋⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⠀⠀⢀⣐⣋⡉⠀⠀⠐⠿⠇⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢠⠎⠀⠀⠉⢢⢀⠄⠂⠠⡀⠀⠈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡎⠀⠀⠀⣀⠀⡏⢀⠀⠀⢱⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠀⢱⡀⠀⠀⠀⣰⢇⠀⠀⠀⣸⠀⠈⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⡀⠀⠙⠶⠤⠚⠁⠈⠳⠤⠖⠁⠀⣠⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠉⠀⠒⠒⠒⠂⠈⠁⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⣇⣀⣀⡀
⠀⠀⠀⠀⠀⣠⠖⠋⢁⣀⣀⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⠉⠓⢦⡀
⠀⠀⠀⠀⡜⠁⢠⠞⠉⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⢆⠀⠱⡄
⠀⢀⡤⠞⠁⠀⢮⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢨⠆⠀⠙⠦⣄
⣰⠋⢀⣤⡄⠀⢼⡀⠀⠀⠀⣼⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⣸⠄⢀⣤⡀⠈⢣
⢧⣠⡞⡸⠘⠦⣤⠇⠀⠀⠀⡟⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⡇⠀⠀⠘⢧⡤⠞⠸⡘⣦⣠
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠦⠤⠤⠦⣤⢤⣤⣤⢤⡤⠦⠤⠤⠦⠤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣿⠀⡇⡇⢸⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠉⠉⠉⠙⠓⠿⠀⡇⡇⠸⠖⠛⠉⠉⠉⠉⠉⠓⢤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣗⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣃⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⢘⢿⣿⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣫⡶⢃⣞⣼⡟⢮⣝⠦⡈⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⡩⡆⠀⢩⡴⠺⠋⠀⣷⢸⣷⠀⣽⡄⢻⡿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣄⣷⡶⠿⠥⣶⠶⢚⣁⣼⣟⣠⣿⠁⣸⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠉⢉⣡⠤⣶⠶⠚⠛⠿⠍⠁⣀⡀⠠⠚⡿⢂⣼⣷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠴⠚⠉⠀⢀⣀⣤⣴⣶⣶⣿⣧⠤⠤⣤⡴⢶⡿⢻⣿⡿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⣠⣶⠾⠟⠛⢻⢏⣁⣀⡠⠴⢶⠞⢉⣴⢟⠧⢸⣿⣿⡜⣿⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⣀⣴⠞⠉⢀⣠⠴⠒⠈⠁⣀⡠⠴⠒⠊⠉⠉⠠⠋⣠⡿⣿⣿⠃⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢰⡟⠀⣠⠴⠋⠀⢀⣤⠖⠋⠁⢀⣀⡤⠤⠒⠚⠛⢋⣡⡾⠛⣁⣼⣿⣹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠘⣿⣇⢰⡏⠀⠀⣰⠋⠀⢀⣠⠖⠉⠁⢀⣠⡤⠶⠲⢋⣩⠐⠋⠉⠈⣠⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⡀⠀⠈⠻⢌⣧⡀⠠⡇⠀⢠⡟⠀⣀⣴⠞⠋⠀⠀⣠⣶⣿⣥⣤⣤⣴⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣛⠳⠤⣄⣠⡉⠙⠲⠷⠤⣿⣥⣴⣿⣯⣤⣴⡶⣿⠛⣻⣷⣴⣾⣿⣿⢿⣟⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠄⠀⠀⠀⠉⠙⠙⠒⢶⠶⡶⠶⣦⠽⡶⣿⡿⢿⡟⢻⠙⡏⣿⢸⡇⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠀⠷⠀⡏⢰⡇⢸⡇⣼⣿⢀⡇⣼⢳⣿⣿⣿⠇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡄⠀⠀⠀⠄⠀⠀⠀⠰⠀⠀⢸⠀⣏⢸⡁⢸⢀⡇⣼⣸⣷⡏⡸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢠⠘⠀⠀⠀⠀⠀⠀⢸⡄⣟⢠⠀⣿⢸⡁⠦⡟⢸⢣⣷⣿⡿⣿⣶⡄⡀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢸⡄⣇⢝⢾⠐⡀⢠⠘⠀⡇⢟⢼⢰⡇⣼⣿⢲⡏⣿⣾⣿⣿⣿⣟⣿⣿⣿⠳⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⣯⠿⡜⣾⣇⠱⠘⡅⡇⣿⣿⣿⢸⣧⣯⣿⣿⢵⣿⣿⣿⣿⣿⣿⣿⣮⠛⢃⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠿⢿⣬⣧⣃⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠙⠻⠷⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣿⣿⣿⡿⣿⣿⣿⡿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠⠀⠁⠀⠀⠀⠀⠀⡞⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣢⣄⣀⣀⣐⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠔⠁⠀⠀⠀⠀⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠰⠤⠊⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⢠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠐
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⠒⠐⢆⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡠⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠁⢤⡀⠀⠀⠀⠃⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢡⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠘⡲
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠠⠃⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠪⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⢤⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠉⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⠀⢸⣇⡤⠂⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⠀⠀⠀⣈⣦⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣞⣀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠤⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠱⣆⠀⠀⢸⡇⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⣀⠠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢸⡇⠀⢀⡜⠁⠀⢀⣀⠤⠄⠲⠀⠀⠀⠀⠀⠎
⠀⠰⠶⠦⠉⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠤⠴⢤⣀⣙⣦⣇⣴⣯⠴⠒⠋⠉⠀⠀⠀⠀⠀⠐⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠹⣿⠉⡀⠀⠀⠀⠀⢀⠐⠀⠐⠢⠨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠄⠤⠀⠒⠂⡄⠀⠀⠀⡠⠔⠊⠀⠀⣿⠀⠀⠁⠒⠤⢀⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⡠⠔⠈⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠈⠢⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠻⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡋⠀⠀⣿⡏⠉⢻⣿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠑⠀⠀⣰⣿⡇⢀⣾⣋⣾⡏⢈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⣰⣟⡿⠀⣾⢣⣏⣿⠃⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⢀⣼⣿⡟⠁⣼⣏⣾⣿⣿⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⡀⠀⢀⣾⣿⠏⠀⣰⣿⣿⣻⣿⠇⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠰⠀⢀⣞⣿⠃⢀⣼⣿⡻⣫⣿⠋⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢀⣿⣼⠇⣴⡏⡵⢨⣿⡟⠁⠀⣰⡿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⢸⣿⡟⢸⣿⣝⡆⣺⡟⠀⢀⣾⢟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢠⣿⣿⠁⣾⣿⣿⣷⣿⠁⣰⡟⣡⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢀⣴⢿⡿⠃⣰⣿⢿⣿⣿⠏⣴⣫⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⢀⣴⢟⣡⡟⠁⣰⡿⣿⡟⣹⠏⣰⡟⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⢀⡴⠟⣡⡾⢋⣤⢾⡿⣿⢋⡽⠋⣠⣿⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢀⣴⣯⣾⡾⠋⣴⣟⣽⣋⠀⣡⠟⠁⠐⣱⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⣾⣿⣿⠫⠀⣼⢿⡟⡾⢃⣾⠋⢀⣴⠟⣱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⣸⣿⣿⠃⠀⢰⡟⣿⡟⢀⡾⠁⢀⣾⠃⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠈⠀⢠⣿⣿⠏⠀⢀⣾⣿⣿⠖⣼⠇⠀⣼⠇⣾⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⣴⣿⣿⠏⠀⢠⣾⢿⣿⡷⣻⡟⠀⢸⡿⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⣰⣿⡿⠋⠀⢀⡴⢛⠏⡂⣿⣿⠟⠀⠀⣼⢁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣼⣿⠏⠀⠀⣴⠏⡀⠤⢊⣾⠟⠁⠀⢀⡼⣳⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠰⢢⣾⣿⠋⠀⠀⣼⣿⣬⣻⣿⡟⠁⠀⠀⣴⢟⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠛⠑⢁⣠⣿⡿⠁⠀⣠⣾⡟⣿⣿⡿⠿⢂⢀⣢⣞⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⣠⣶⣿⡿⠋⣁⣴⣿⣿⣿⣿⣿⣥⡤⠶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⡀⣠⣾⢻⡾⠋⢀⣴⠟⢰⢿⣿⡿⢿⣿⣷⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣯⡉⢀⣴⣿⣿⠟⠁⣠⣾⣫⣴⣽⣿⣭⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣏⣀⣤⣾⡿⣯⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⣿⣻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⢤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣋⠥⠔⠒⠒⠒⠐⠒⠒⠒⠤⠭⣙⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⢿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣴⣶⣶⣚⣽⣿⡿⢣⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣮⣭⣛⣛⣛⠛⠿⠿⠿⠛⢛⣛⣛⣭⣵⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠤⠤⠤⠤⠤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢸⡛⠦⢤⣄⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⡤⠴⠚⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠚⠉⣀⣤⣤⣤⣶⣶⣶⣶⣤⣤⣤⣀⡉⠛⢦⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⣠⣤⡇⠘⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⢀⡞⠁⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠸⡇⠀⣇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠹⣄⡈⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠈⠙⠓⠚⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢻⠓⠦⢤⣀⣀⣀⣀⣀⣀⣀⡤⠤⠒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⢀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⣀⣀⣀⣀⣀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉





⣿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣷⣶⣦⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⣶⠛⠉⠉⠉⠈⣻⣟⣻⣿⣿⠿⠛⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⣟⠾⡅⠓⣶⡿⡯⠽⣏⣹⣟⣥⣬⣶⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉⠲⢭⣽⣼⣉⣉⣉⣩⣽⣿⣶⠿⠛⣩⢜⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢮⣅⣺⣾⡇⠤⠤⠤⠽⠾⠿⣾⣻⣽⣿⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠷⣌⠹⣿⣿⠭⠭⠭⠿⠿⠟⠋⣋⣊⡭⣛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠤⣈⢹⢿⡷⠬⣭⣭⣙⣭⣬⣭⣷⣶⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠳⢮⣍⣿⣧⣤⣤⣬⣵⣶⡿⠷⣿⣿⣿⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣶⠤⢤⡟⡟⠉⠩⠉⠉⢭⣒⡫⢵⣾⣿⣧⠈⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠐⠁⠀⢀⠴⢾⣿⣔⠝⠒⡿⣯⣭⣭⣉⣭⣥⣴⡾⣛⣭⣿⢿⡋⠒⢄⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢯⣰⢼⡄⠙⠿⣿⣾⡧⠤⠤⠬⢭⣯⣭⣵⣾⣿⣿⣿⣗⡰⣎
⠀⠀⠀⠀⢀⠀⠀⠀⠲⣷⣿⣿⣶⣤⡾⣹⣏⣉⣛⣛⣛⣭⠽⢶⣻⣽⣿⣿⣿⡶⠋⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠉⢻⢿⣟⠨⠿⢿⣿⣶⣶⢶⣾⡿⢿⢿⡿⠟⠋⠉⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⠉⠓⠛⠛⠛⠛⠛⠒⠒⠚⠉⠉⠁⠀⠀⠀⣀⡠⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠐⠒⠒⠒⠒⠒⠉⠁

⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣆⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⡆⠀⠰
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⡀⠀⠀⠀⠀⠰⠀⢻⢛⡛⣛⢛⡛⣛⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⡄
⠀⢠⣾⣿⡄⠀⠀⠀⢰⢤⣤⣀⣠⣤⣤⣤⣤⣤⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠀⠀⠀⠀⠀⠀⠛⠛⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⢿⠆⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⡄⠀⠀⡄
⠀⢠⡀⠀⠀⠀⠀⠀⢠⣤⣠⣀⣀⣀⣀⣀⣠⣠⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣴⠀⠀⠀⠀⣴⣿⡷⠄
⠀⠘⣧⠀⠀⠀⠀⠀⠈⠉⠙⠋⠛⠋⠋⠉⠉⠉⠀⠀⠀⡰⠾⠶⠶⠶⠶⢶⠏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢻⣆⣀⣀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣁⣁⣈⣀⣁⣀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⢻⡁⠉⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠉⠉⠹⠉⠉⠁
⠀⠰⠀⢸⣧⠀⠠⠄⠀⠀⠀⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡀
⠀⠀⣀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀⡀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⢿⣿⠶
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠶⠶⠶⠶⢶⠏⠄⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⢀⣀⣁⣉⣀⣁⣉⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠿⠟⠛⠀⠀⠀⠀⠀⠀⠈⠉⠩⠉⠉⠩⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⣿⣿⠗⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⣀⣀⡄
⠀⠀⢰⡏⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠒⠀⠺⠇⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠷⡿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤⣤⣤⢤⣴⡄
⠀⠻⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠚⢛⠛⡛⢿⢿⡟⠡⠀⠀⠀⠀⢷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠂⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣠⣀⣀⣀⣀⣠⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠈⢷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢠⣾⣿⣿⣦⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣴⣿⣷⣄⠀⣿⣿⣿⣿⣿⣿⣿⡏⢀⣴⣿⣷⣤⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛⠛⠛⠉⢻⠋⠙⠛⠛⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣄⣀⡀
⠀⣰⣾⠿⢟⣛⣫⡭⠭⠭⠭⠒⠒⠒⠒⠒⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠩⠉⢛⣛⡻⢿⣶⡄
⠀⣿⡟⠻⠧⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⠿⠟⣿⡇
⠀⣿⡇⠀⠀⠁⠀⠂⠉⠿⠭⠭⠽⠿⢿⣛⣛⣛⣛⣛⣛⣛⣻⠿⠿⠯⠭⠭⠿⠛⠚⠃⠀⠁⠀⣿⡇
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠛⠉⣉⣉⣉⡉⠛⢿⣶⡄
⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⣾⠿⠟⠛⠛⠿⣷⣄⠙⣿⡆
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⣾⡿⠁⠀⠀⠀⠀⠀⠈⣿⣇⢹⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢠⣿⡇
⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⣸⣿
⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⢰⣿⠇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⣰⣿⠋
⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣴⣿⠟⣡⣾⡿⠁
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣀⣠⣴⣾⠿⠋⣡⣾⡿⠋
⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠟⠛⠉⢀⣤⣾⠟⠋
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣤⣾⡿⠋⠁
⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⣿⣷⣶⣶⣶⣶⣾⣿⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠈

⠀⠀⠀⠀⣀⢰⣤⡀⣠⣴⠀⡀⠀⠀⠀⠀⠀⠀⢐⣳⣶⡄⢀⣤⡤⠀⠀⠀⠐⠲⣶⣠⣶⣖⠀⠀⢀⣀⣀⣻⣆⠀⣀⣄⣀
⠀⠀⠀⠀⠘⢻⣿⣷⣿⣿⣿⣥⡀⠀⢀⠀⠀⣰⣯⡿⠛⣿⠿⣿⡻⣆⠀⠠⠾⢿⣿⣿⣿⡿⠷⠄⢴⣿⣿⠟⣯⢼⡿⠿⣿⠂
⠀⠀⠀⠀⠒⠛⡿⠁⠙⣿⣍⠁⠀⢠⣾⡿⢶⠿⣟⠣⣼⠃⠀⣸⠇⠋⠀⠀⠐⠛⠁⣿⠙⠀⠀⠀⠘⠹⡏⠀⡇⠈⢿⡆⠈
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠉⠀⠀⠈⠸⡇⠀⢧⠈⢰⡟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠃
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⠀⠀⠀⠀⠀⢠⣤⣤⡀⣠⣶⣖⣀⠀⠀⢀⣤⣶⣦⣀⣠⣴⡦⠀⠀⠀⠀⠀⡀⢠⣶⣦⠄
⠀⠀⠀⠀⣀⠀⣴⢠⣬⣿⣿⣶⣞⣀⡀⠐⣻⣿⣿⣿⡿⠿⣯⡅⠀⠀⣴⢿⣿⣿⣿⣿⣿⣷⡄⠀⢠⢶⣿⣿⣿⣿⣿⠯
⠀⠀⠀⠤⣿⣷⣿⠛⠛⡠⢻⣏⠛⠿⡄⠀⣟⠰⡏⠀⢆⠀⠀⠙⠀⠀⠀⢸⡟⣸⠃⢻⡿⠛⠀⠀⠰⠛⣿⠀⠱⡀⠈⠃
⠀⠀⠀⠸⠋⣿⠀⢳⡾⠁⢰⠟⠀⠀⠀⠀⠈⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠁⡟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠈⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⠿⠸⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠂⡠⠤⠀⠀⠀⠄⠀⠀⠑⡄⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⣄⣀⡄⠠⠀⠀⠀⠀⠇⠀⠐⠁⠀⠠⠰⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⢀⠈⡁⠈⠛⠓⠛⠛⠁⠀⠓⠛⠛⠇⠈⠛⠁⠀⠀⣢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠐⠀⠀⠈⠛⠋⠤⢄⣀⡀⠀⣀⣀⣐⣂⣀⡀⠊⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠁⠒⠢⠤⣄⣀⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠐⣒⣀⠔⠊⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠅⣀⣀⡀⠀⠀⠀⠈⠉⠙⠀⠉⠉⠉⠉⠉⠉⠉⠀⣂⡴⠞⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠈⠉⠉⠛⠚⠒⠒⠆⠀⠲⠶⠶⠒⠚⠚⠉⠉⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣤⣀⣀⡇⠀⠀⣀⣀⣠⣤⣤⠤⠴⠒⠋⢁⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤⣀⡀⠀⠀⠈⠀⡀⢀⣉⣉⣁⣤⣤⡤⠶⠖⠛⠉⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⠲⠤⣄⣀⠈⠉⠉⠉⠉⠉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠃⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠤⣤⣀⣀⣀⠀⠈⠉⠛⠒⠒⠶⠶⠶⠶⠶⠶⠒⠒⠚⠛⠉⠉⣀⣀⣀⡤⠄⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠩⠭⢉⣛⡓⡒⢒⣒⣒⣒⡒⠒⣒⣻⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠘⢇⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠈⠙⠢⢤⡐⠌⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠈⠦⠀⠀⠀⢻⡿⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠤⣺⣠⠒⣠⠤⠤⣀⡝⠷⡀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠈⠁⠀⠀⠈⢩⠅⢠⠞⠁⠀⠀⠸⣄⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠞⠀⣸⠀⠀⠀⠀⢸⠏⢟⡋⢳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⡀⣀⣤⠖⠋⠀⣸⡇⢰⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠖⠢⠄⠒⠒⠒⠐⢀⡠⠔⠚⠉⠉⠁⢀⣀⣠⣴⣿⣿⣟⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠉⠲⣤⣀⣀⣀⣤⣾⣉⠀⢀⣠⣤⣴⣾⠿⢟⡽⢋⢙⢙⣅⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠂⣀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠉⡍⢋⠙⠁⠀⠠⣀⣤⣯⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⡍⣟⠲⣦⠤⣶⢤⣦⢶⣶⣶⣶⣽⣶⢶⠿⣿⢹⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢱⢸⡄⢿⠀⢸⠀⡇⢀⡇⢸⡇⣼⠀⣼⢠⣿⣾⣿⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢊⣧⢸⡇⢸⡇⣷⣈⣇⢸⠇⣿⢰⡿⢼⣿⣿⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣿⣸⣧⠀⣷⣾⢼⣷⣼⣳⣿⣾⣧⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡼⣿⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣃⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠿⠛⠛⠙⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠁⠀⠁


⣶⢶⣶⢷⢿⠷⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⠿⡶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠃⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠘
⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡠⠒⠐⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠢⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠄⠊⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁
⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠑⡆⠀⠀⠀⠀⠀⠀⠀⡗⠊⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠠⠤⠷⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⠾⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⡇⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠔⠁
⠀⠀⠀⠀⢀⠀⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⡇⠀⣠⠞⠀⢀⣀⠤⠔⠂⠉⠃⠀⠀⠀⠀⠐⠁
⠀⠈⠛⠃⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠶⢤⣬⣷⣧⣴⠷⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⣿⠡⠄⡀⠀⠀⠀⠀⡄⠀⠀⠀⠂⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠧⠀⠒⠂⠀⠠⠀⠀⠀⢀⠄⠂⠁⠀⠀⣿⠀⠀⠀⠁⠒⠀⢀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡀⠄⠂⠁⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠖⠛⠛⠉⠉⠉⠉⠐⡒⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⢀⢀⡴⣆⣈⠥⠭⠥⠦⠦⠤⣤⣼⣯⣿⣿⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠙⡀⠄⢐⣊⣩⣥⣤⡶⠶⠖⠖⠒⣖⢲⣶⣶⣤⣤⣉⠉⠛⠻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠛⣂⣿⣶⡾⠻⠉⡏⠑⠂⠂⠃⠈⠃⠉⠀⠀⠃⠶⠰⠉⠉⠹⣿⢶⣤⠁⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣢⣾⣿⡟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⢀⢈⢱⠘⣿⣟⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢸⡿⢏⡟⣁⣀⣶⣤⣤⣦⠴⠶⠞⠛⠓⠛⣛⣛⣛⠛⡛⠛⠒⠿⠷⢾⡶⠿⠛⢠⣿⡏⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠊⠈⠛⠟⠛⢛⣉⣭⠍⡶⠲⠒⠶⠚⠋⠛⠙⢿⠉⠟⠙⠿⡿⠟⢷⣶⡶⠀⠀⣨⣿⢏⣥⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢅⣦⡀⢰⣶⡟⠡⠈⠀⠀⠀⠀⠠⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⣄⢠⣔⡨⠋⣠⡴⡟⢁⣾⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠻⣇⡌⠻⠻⢔⡴⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢐⢠⡭⠅⣋⣠⣴⠟⣋⣥⢶⣿⣧⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡙⠻⠶⣤⣬⣙⡓⠒⠾⠶⢷⣦⠤⠄⠄⣐⣒⣒⣫⢥⠶⠟⠉⢁⣴⠿⢋⣡⣾⣿⣿⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣄⣉⣙⢙⡋⠙⠛⠙⠛⠛⠫⠙⠋⠉⣈⣁⣤⣤⡰⣶⣿⡥⠶⣛⡟⣻⣿⡏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⢉⣉⣩⣙⠛⠛⠶⠶⠶⠶⠾⠿⠟⣛⣙⣉⣁⡤⠞⢏⣡⡜⢩⣿⠟⠁⠁⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠌⢙⣓⣲⡤⢤⡴⠶⠚⠛⡉⣁⣉⣥⠆⢞⣋⣻⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠯⢭⣭⣭⣽⣿⠷⠶⠶⠞⠋⠁






⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⡿⠻⡿⠛⠿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡀⠀⡄⠒⠂⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣈⡥⠆⠀⠀⢄⣀⣀⣠⡀⠈⠀⢤⡄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠄⠀⡀⢀⠄⠚⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⢵⠀⢠⣄⣀⣹⠀⠀⣠⣄⣒⡂⠤⠭⠍⠐⠒⠊⠉⢢⢀⡠⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⢀⣀⣀⡀⡉⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠤⠒⢟⡈⢀⡨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⢀⣀⠁⠀⠀⠀⠈⢱⡀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⠀⠑⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢄⣉⠀⠀⠐⠒⠂⠶⡇⠠⠤⠤⠒⠒⠒⠋⠉⠉⠁⢀⣀⡗⠐⠈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣭⠈⠒⠒⠒⠒⠶⣇⠠⠤⠤⠤⠔⠒⠒⠊⠉⠉⠀⢀⡠⠔⠊⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⡓⠤⣀⡈⠑⠒⠒⠒⠂⠆⠀⠠⠤⠤⠤⠤⠔⠒⠚⠉⠽⣧⡀⠀⢀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⢃⠢⢄⡉⢁⠒⠶⠦⠄⣴⠀⣀⣀⣠⣤⠤⠤⠤⠴⡶⠋⠀⠁⠀⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠙⠲⢤⡄⠀⠋⠁⠀⠙⠶⣼⠁⠀⠀⠰⢤⡤⠤⠤⠤⠤⠤⠤⠴⠶⣶⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⢄⣀⠤⣉⠀⠀⣀⠠⢄⣀⣀⠠⠝⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣘⣴⣿⣿⣿⣿⣿⣿⠇⠚⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡶⢶⣃
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⢾⣿⣿⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠹⡆⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⡿⣻⣿⡷⢸⣿⡿⠁⣴⠿⣠⡶⣆⣼⣿⡿⠿⣷⣶⣿⣄⡀⠘⠀⠀⠀⠀⠀⣰⠁⠈⣾⣿⣥
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣿⣿⡿⢿⣿⣏⣷⣾⣿⣞⢅⠹⡄⡀⠂⠀⣿⣿⣿⣦⡿⠉⠉⠉⢛⡟⠳⢶⣤⣤⣤⣄⣀⢉⣿⣿⣿
⣧⡀⠀⠀⠄⠀⠀⠀⢠⣿⣿⣿⣿⣶⣾⣟⣬⣿⣿⣿⠈⠀⠻⠇⠀⢠⣿⣿⣿⣿⣇⠀⠀⠀⡾⠀⠀⠀⠠⣿⡍⠹⠛⣿⢿⣿⣿
⣉⢓⠂⢀⡄⣠⣬⣽⣛⢻⡿⢿⣿⣿⣿⣿⢷⣿⣻⣿⡇⠀⠀⠀⣼⢾⣿⣿⣿⣿⣿⣦⣄⠘⠃⠀⠀⠀⠀⠻⣇⣼⠰⠿⣾⣿⣿
⣿⣿⡇⣈⣼⣿⣿⣯⣹⣿⣿⣿⣎⠻⣿⣟⣿⣿⣿⣯⠛⠂⠀⢰⣇⣸⣿⣿⣿⣿⣿⣿⡿⣄⠀⠀⠀⣀⢀⣀⣹⠇⠀⢠⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡾⣮⠝⢿⣿⣿⠟⠿⠶⠀⠈⠛⠿⠿⣿⡿⠛⠙⠛⢣⡹⣶⣶⡟⠁⠈⠉⢙⣷⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣽⡻⣿⣽⣿⣿⣦⣤⣤⣀⡉⠊⠁⠦⣄⡈⠛⠻⣿⣷⣦⣤⣤⣬⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠯⣙⡻⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣙⡻⠿⢿⣿⣿⡶⠶⠏⢉⡉⢉⡉⢡⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣼⣿⣿⣿⣻⣷⣿⣿⣿⣿⡿⢛⣿⣿⣿⣿⣿⠿⣋⡉⢭⡿⣿⠹⠋⢉⣒⠂⠀⢠⣾⣿⣿⣿⣿⡿⢋⣽⣿⣿⣷⣾⣿
⣿⣿⣿⣿⣿⣿⡿⠇⠛⠛⢛⣉⣭⣽⣷⣿⣿⣿⡿⣿⣿⠛⣡⣴⣾⣿⣿⠆⣴⣾⣿⣄⡀⣾⣿⣿⣿⡿⠋⢱⣿⣿⣿⣿⣿⡿⣿
⣉⠉⠉⠉⠡⣽⣶⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠇⣹⣿⣿⣾⠟⢫⣽⣯⡿⠉⢀⣄⣾⣿⣿⣿⣿⣿⣤⣜
⠁⠰⠏⠀⠀⣨⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢠⣿⣿⣿⣿⣿⣿⣷⠆⡗⢀⣾⣷⣿⣿⣿⣿⣿⣿⣿⠈
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠇⢈⣭⣍⣭⣿⣷⣿⣿⣿⣿⣿⢿⣿⣿⣿⡏⠀⠃⣾⣿⣿⣿⣿⣿⡿⣷⣿⣿⣆
⠀⠀⠀⣀⣾⡿⢏⣽⣿⣿⠿⣿⣿⢿⠏⠁⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣭⣽⣛⣁⡀⢸⣛⣻⠗⠹⣿⠋⣰⣿⣿⣿⣿
⠀⠀⠀⠉⠁⢠⠌⠛⠉⠀⠀⠁⠠⢏⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣶⡀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠓⠦⠶⠚⠛⣁⣻⣄⣤⣻⣿⣩⣿⣿⣿⡟⠋⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⢿⣿⣿
⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣏⣟⢿⣿⠋⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭
⣿⠆⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⠻⡟⠃⠀⠸⠟⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣏⣴⣷⣦⡔⠀⠀⠈⠈⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣖⣤⣜⣿⣿⣿⣿⣿⣿⣿⣿⣩⠃⠀⢀⡀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢻⣿⣿⢿⠟⠉⠀⣹⣿⣿⣶⡿⠋⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⣿⣿⣿⣿⠀⢠⣾⣿⣿⣿⣿⠿⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣰⣿⣿⡿⡟⢉⠁⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠻⢿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣯⠘⢂⡛⢠⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣄⣶⣢⣽⣿⣿⡅⠀⠀⣼⣿
⣆⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠀⣀⠀⠟⣟⣿⡙⣿⡟⣿⣿⣼⣯⣤⣽⣟⣿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢿⣿⠇⠀⣾⣿⣿
⣿⣶⡀⠀⠀⠀⠀⠈⢿⡆⠀⠁⠀⠙⠻⠟⠛⣓⢚⡛⠂⠰⠿⠋⢈⠛⣿⣿⣿⣿⠦⣍⠙⠛⠻⢿⣿⣿⣿⣿⣿⡏⠀⠀⠸⠟⣿
⠀⠀⠀⠛⠛⠦⡀⢀⣺⡇⠀⠀⠀⠀⠈⠀⠀⠀⠸⢷⣦⣀⣀⣀⣷⣆⣼⣿⣿⣿⡄⠀⠀⠗⠾⣂⠀⠻⢿⣿⣿⣿⡆⠀⠀⠄⠋
⠀⠀⠀⠀⠀⠀⠸⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣾⣿⣿⣾⣿⣿⣝⡓⣾⣿⡇⠀⠀⠸⠟⣿⣿⣿⣧⣬⣉⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⢀⣠⣿⡿⠓⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠋⠓⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⣾⡿⠟⣉⡤⠞⣠⠟⠉⠛⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⣀⣤⣾⡿⠋⣸⠛⡶⢤⣀⡀⠀⠀⠀⡟⠀⢀⣿⡏⢀⡞⣡⠖⠋⠁⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣼⡿⠋⣡⠴⣚⣥⠞⠁⠀⠈⣹⠀⠀⡴⠃⠀⣸⣿⠁⣼⢰⡏⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢰⣿⠃⣰⢃⡞⠉⠀⠀⠀⣠⠖⠃⡤⠚⠁⢀⣴⣿⠋⢠⠇⣼⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⢀⣾⡟⢀⡏⣼⠁⠀⠀⠀⣰⠃⢀⠞⠁⢀⣴⡿⠋⢁⡴⢃⡼⠃⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⢀⣴⣿⠟⢀⡜⢡⠏⠀⠀⠀⠀⡇⠀⡟⠀⢀⣾⡟⠁⡴⢁⡴⠋⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⣰⣿⠏⢀⡔⢋⡴⠋⠀⠀⠀⢀⡼⠃⣰⠃⠀⣸⣿⠀⡼⢡⡟⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢰⣿⠃⣠⠋⡴⠋⠀⠀⠀⣠⠞⢋⡥⠞⠁⢀⣴⣿⠃⢠⠇⣾⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⢀⣾⡟⢀⡇⣼⠁⠀⠀⠀⡼⠁⢠⠏⠀⢀⣾⡿⠛⢁⡴⢋⡼⠃⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⣴⠚⠁⢀⣴⣿⠟⢀⡜⢠⠇⠀⠀⠀⢸⡇⠀⡞⠀⢀⣿⡟⢀⡜⢡⡶⠋⠀⠀⠀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⣼⠁⠀⣴⣿⠋⢡⠔⣋⡴⠋⠀⠀⠀⢀⡞⠀⡼⠁⠀⣼⣿⠁⡼⢠⡏⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⢰⠇⠀⢰⣿⠃⣰⢃⡼⠉⠀⠀⠀⣠⠞⠉⡴⠋⠀⣠⣼⣿⠃⣰⠃⡾⠀⠀⠀⢀⡏
⠀⠀⠀⢀⣠⠏⠀⢀⣿⡟⢠⡇⣼⠁⠀⠀⠀⡞⠁⠀⠀⠓⠦⣾⣿⠋⣡⠖⣃⡼⠁⠀⠀⠀⡼⠁
⠀⠀⢰⣏⡀⣠⣶⡿⠟⢀⠞⣰⠇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠈⠙⠳⠾⣅⡀⠀⠀⣠⠞⠁
⠀⠀⠀⠈⠙⠻⠿⣤⡞⣡⠖⠋⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠤⠤⢴⣾⣷⣤⣤⣤⣤⠤⣤⣽⣿⣽⣿⣿⣦⣀⣐⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣴⣾⣿⣿⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⣿⣟⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⣸⡟⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣟⢠⡟⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣧⡿⠋⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣀⡤⢰⣢⣴⣤⣶⣴⣶⣿⣷⣧⣤⣀⠀⠀⠀⠄⢠⣼⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠈⠀⠈⠀⠀⠂⢰⣝⡛⠯⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠆⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠤⣤⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣻⡿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠁⠀⣄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⢠⣿⡷⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣖⣿⣿⡿⣁⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢰⣶⡿⣸⣿⣿⣯⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⢌⣱⡞⣿⢛⢿⢻⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠒⠚⠛⣋⠩⣴⡗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠂⠀⠀⠀⠚⣷⣲⣴⡾⠖⣠⣼⢹⠇⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠂⢼⠣⢈⣭⡶⠷⢞⣛⣳⡿⠍⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⣠⡔⠊⠉⠀⠀⢘⣠⡞⢁⣀⣤⡾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠓⠀⠉⠉⠑⠲⠖⠚⠋⠉⠛⠛⠛⠩⢼⣶⣯⣽⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠆⠀⠀⠄⢠⠀⢀⡀⠀⡀⠀⡠⣦⠔⢶⠙⣯⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣄⠀⣆⠌⣂⢀⡇⠀⡇⣠⡆⠸⠳⡿⣾⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⢗⢻⣀⣄⠉⣿⠂⡿⢁⡇⣾⢧⣷⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣳⢾⡾⡏⣽⠃⢿⣬⣷⣾⡇⣼⣾⣿⣿⣿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣧⢿⢻⣦⣸⣍⣿⣸⣏⣿⣹⣿⣿⡿⣓⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⢼⣧⣽⣿⣿⣻⣿⣿⣿⣿⡿⠽⣋⠿⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣿⠟⠥⠏⡏⠉⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⣰⣿⣿⣿⣿⣆⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣿⣿⣿⣿⣿⣿⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣰⡆
⠀⠀⠀⠘⣷⣶⣤⣤⣴⣿⣿⣷⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⣾⣿⣿⣦⣤⣤⣶⣾⠁
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠄
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣯⣟⠿⢿⣿⣿⡇⢸⣿⣿⣿⠿⣻⣽⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣌⡙⠃⠘⢋⣡⣶⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠉⠁⠀⠀⢸⡇⠀⠀⠈⠉⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠐⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⣠⠆⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠈⠛⠤⣤⣤⣤⣤⣀⠀⠀⣠⣤⣶⠏⠁⢀⡤⠞⠁⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠻⢿⣧⣧⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠂⠈⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⡀⠉⠛⠷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠄⠀⢀⠀⠀⠀⠀⠐⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⣤⣦⣃⠀⠀⠀⢀⠐⠀⠀⢤⣄⣀⡀⠠⠤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣴⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣼⢄⡀⠉⠉⠛⠶⢤⣿⡀⠀⠀⡦⠭⠉⣝⣓⣦⣄⣀⡦⣆⣀⣄⣀⣠⣤⡶⣞⣻⣯⣵⠶⣫⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠛⠷⣧⣈⠀⠀⠀⠀⠀⠀⠋⠃⠘⠛⠒⠿⠿⣿⣿⣿⡗⣿⠿⠿⠿⠟⠛⠛⠉⢉⣁⡤⡾⢻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⡀⠈⠉⠛⠻⠷⣶⣀⣾⣤⣤⣀⠀⠀⠤⠀⠀⠀⢠⠀⡀⠀⠀⢀⣀⣠⣤⣼⣿⣽⠷⠾⠋⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡯⣙⠳⢤⣤⣄⣠⡀⣎⡴⢈⠉⠉⠛⠳⠶⠶⠶⠴⠞⢠⠟⠛⠋⠉⠉⠉⠀⠀⠀⠀⢀⣰⣶⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠻⠾⣷⡀⠀⠀⠈⡉⢉⡟⠛⠲⠶⠤⢄⣑⣀⡀⠀⢐⠊⡀⣀⣀⣀⣠⣦⡴⠴⠶⠚⠋⠉⣹⣾⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⠀⠀⠉⠑⠶⡆⢰⣤⣀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠉⠉⠉⠀⣀⣀⣀⣤⡤⠶⠛⠉⢐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡻⠿⢦⣀⡒⢀⣀⢃⠀⢡⠈⠉⠉⠛⠲⠲⠦⠤⠤⠄⠒⠶⠖⠒⠒⠋⠉⠁⠀⢀⡀⠀⢀⣄⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣤⣤⡉⠙⠻⠿⠾⢧⣼⣠⣈⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⢀⣉⣀⣠⣶⠶⢿⣛⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣤⣀⣈⡙⠛⢿⣟⢿⣾⢶⣶⣴⣤⣤⢤⣶⠶⠶⣾⡿⠿⢋⣡⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠉⠉⠛⠛⠿⠷⠶⢦⣭⣭⣭⣤⣬⣥⣭⣭⠤⠖⠓⠛⠉
⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠁
⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠾
⠀⠀⠀⠂⠘⠛⠋⣿⣿⣿⣿⣿⣿⡿⠿⠿⠤
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⡛⠹⣦⡀⠈⠀⠀⣀⣠⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⢿⣿⠿⠲⠇⡆⠀⠀⠙⠓⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠠⣞⡿⠟⠛⢩⠟⠉⠀⠀⢼⡇⢸⡌⠻⣿⡶⣠⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠙⠀⡑⢼⣿⡄⠈⠰⢽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢸⡇⠈⣿⣇⠀⠀⠈⠙⠉
⣀⣀⣀⣀⣀⣀⣀⣀⡼⠀⢀⣀⡀⣀⠘⣀⣈⢇⣀⣿⣷⠀⢀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀
⠉⠉⠉⠉⠉⠉⠙⠉⠇⠀⠺⠛⢋⠦⠉⠙⠷⠶⠇⠸⡿⠀⠈⠉⠉⠉⠛⠛⠛⠛⠟⠋⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⣀⣀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠐
⠒⠠⠤⠤⠤⠤⠀⠀⠀⠀⡔⠈⠈⠉⠀⠀⠘⠓⠒⠀⠀⠀⠀⠀⣀⣀⣉⣘⡋⠢⠤⠤⠄⠀⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠓⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣦⠈
⠀⠀⠀⠠⡀
⠀⠀⠉⠀⠴⠦⠤⠲⠒⠐⠂

⠀⠀⠀⣀⣀⣤⣤⣤⠶⠶⠶⠶⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠶⠶⠶⠶⢦⣤⣤⣄⣀⡀
⣴⣾⣿⣿⠋⢩⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣬⠉⢻⣿⣿⣶
⣌⡙⠻⠿⢷⣦⣤⣭⣉⣉⣙⣛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣉⣉⣩⣥⣤⣶⣿⣿⣿⣿
⠻⣿⣿⣶⣶⣤⣬⣭⣉⣉⣉⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣻⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⣦⣌⣉⡙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⢛⣩⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠘⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣾⣿⣿⡟⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣇⣴⣶⣶⣤⣼⣿⣿⣿⣿⣧⣀⣀⣀⣾⣿⣿⣿⣧⣤⣴⣶⣦⣼⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠙⢿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇
⢸⣿⠿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⢻⡇
⢸⣿⠀⠹⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣿⣿⣿⣿⣿⡟⠀⢸⡇
⢸⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢸⡇
⢸⣿⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⢸⡇
⢸⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢸⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⡇
⢸⣿⠂⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡄⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⣸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣸⡇
⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇
⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇
⢸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡇
⢀⡙⠛⠿⠿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧
⠻⢿⣿⣷⣶⣤⣬⣍⣉⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣋⣉⣩⣭⣿⣿⣿⣿⠿
⠀⠀⠀⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⡀⣀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⢀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠀⠠⡄⠀⠀⠀⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣒⣒⣒⣒⣆⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠟⠛⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢈⠈⠋⠋⠋⠙⠉⠓⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠂⠀⠀⠈⠉⠛⠻⠋⠙⠟⠙⠀⠀⠀⠰⠿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⡆⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢰⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⢠⣦⣶⣶⣶⣦⠀⠀⠀⠾⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠍⠩⠩⠉⠹⠃⠂⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠂⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠒⢒⠒⠒⡖⠒⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠄⢼⡄⠤⠄⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⣀⣠⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠩⠉⠁⠀⠀⠀⢀⣗⣒⣲⣛⣻⠀⠀⠀⠀⠹⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠚⠙⠈⠉⠩⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠠⠿⠿⠂⠀⠀⠀⠀⠈⠉⠏⠉⠩⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠖⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣿⠏⠀⠀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣤⣤⣤⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠠⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⡟⠀⠀⠀⠀⢿⣶⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⡶⠶⢶⠶⠖⠒⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡶⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⣀⡄⠀⠉⠉⠁⠀⠀⢠⣤⣤⣀⣀⣁⣀⣤⠀⠀⠀⠐⣆⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣦⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⣿⠳⣄⣤⡤⠀⠀⠀⠀⠀⠀⣦⠀⠀⢀⣠⣾⣷⠖⠚⢡⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠋⠀⠀⠀⣇⠀⣠⡞⠀⠀⢠⣿⢷⠞⠋⢉⡟⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⢀⠀⠀⠀⠙⠛⢹⡇⠀⢠⣿⣽⠈⠀⠀⣸⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠉⠻⡄⠀⢸⡠⠀⠀⠀⠀⠘⡇⢠⣿⡇⡏⠀⠀⠀⣯⠀⢀⡀⠐⠊⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢸⠁⠀⠀⢀⠀⠀⢻⣾⠁⠁⢻⠀⠀⡐⠛⠉⠉⠀⠀⢴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠠⡀⠀⠈⡆⠀⢀⠎⠀⠀⢸⡟⠀⠀⠘⡦⠊⠀⠀⠀⠀⢀⣪⣽⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠒⠤⣇⢀⠎⠀⠀⠀⠀⠀⠀⣠⠚⠓⠐⠂⠀⠀⠀⠙⢩⡿⠁
⠀⠀⠀⠐⣦⡤⠶⠋⠛⠻⠋⠀⠀⠀⠀⠀⢸⠋⠀⠀⢀⡄⠀⢀⠞⠁⠀⠀⠀⠀⠀⣀⡤⠞⣁⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠈⢷⡀⠢⣀⠀⠀⠠⡀⠀⢳⣦⡄⡇⠀⢀⡞⠀⡴⣃⠤⠀⠀⠀⠀⠀⠚⠛⠋⠉⠉⡉⠉⠙⡟⠉
⠀⠀⠀⠀⠀⠀⠹⣦⠈⠑⠦⣀⠑⠀⠀⠈⠻⣷⠀⣸⣡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⢙⣷⠶⢶⠄
⠀⠀⠀⠀⠀⠀⠘⠯⡀⠀⠐⠪⠽⣧⣔⣢⣄⠘⣧⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⣉⣁⣀⣠⣾⣷⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣉⣉⣉⣉⣹⣿⣿⣿⠶⠶⠶⠒⠒⠒⢶⣒⠒⠒⠊⠉⠛⢬⣀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣠⡿⢻⣬⠳⢄⡀⠀⠀⠀⠀⠉⠻⢧⣤⣄⣀⣀⡀⠈⣀⣠⣤⣀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠘⣿⡆⢸⠙⠶⣎⠀⠀⠀⠀⣠⠤⠤⢤⣀⡀⠘⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠹⣷⣌⠂⠀⡠⣱⡦⣄⣀⣹⣄⡀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣾⠋⠀⠀⠀⠀⠀⠀⠈⠙⠷⣶⡘⣾⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⢖⡫⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠉⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠻⢯⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⢳⣄⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠹⣆⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠼⠃⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠛⠛⠛⠛⠓⠲⠶⠶⠶⠶⠶⠶⠶⠶⢤⣤⡀
⠀⠀⠀⠀⣠⠶⠞⣋⣥⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⡀
⠀⠀⠀⠀⣷⣄⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⣀⣤⡿
⠀⠀⠀⠀⠻⣬⡉⠛⠶⢤⣤⣀⣀⣀⣠⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣈⣉⣉⣩⣥⡴⠞⠋⣩⣼⡃
⠀⠀⠀⠀⠀⢰⡟⠛⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣉⣉⠉⠛⠛⠛⠛⠉⢉⣉⣉⣀⣀⣀⣀⣀⣀⣠⣤⡴⠖⠋⠁⠉⠙⢳⡄
⠀⠀⠀⠀⠀⣸⡷⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⣿⠃
⠀⠀⠀⢀⣞⠉⠙⠶⣤⣉⠛⠳⠶⠶⠶⠶⠶⠶⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠚⠋⣁⣤⣾⠋
⠀⠀⠀⠀⣿⠳⠦⣤⣀⠉⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣄⣀⣀⣠⣤⣤⠴⠶⠶⠶⠶⠶⠶⠶⢖⣻⡿⠿⢻⡏
⠀⠀⠀⠀⢈⣿⢦⣄⣉⠛⠛⠒⠒⠒⠒⠒⠒⠲⠶⠶⠤⣤⣤⣤⣤⡤⠴⠶⠶⠒⠒⠒⠒⠒⠒⠒⠚⢋⣁⣤⠾⠿⣤⡀
⠀⣀⡤⠶⢻⣧⣀⡀⠉⠛⠛⠒⠒⠒⠒⠒⠒⠲⠶⠶⢤⣤⣤⣤⣤⣤⠴⠶⠶⠒⠒⠒⠒⠒⠒⠚⠛⠋⠁⠀⢀⣠⣼⡷⠦⣄⡀
⣾⠉⠀⠀⠈⢷⣌⡉⠛⠶⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⠶⠛⢉⣥⠟⠀⠀⠈⢹
⠙⠷⣄⣀⠀⠀⠈⠙⠳⠶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣉⣉⠉⠙⠛⠛⠋⠉⣉⣉⣀⣀⣠⣀⣀⣀⣠⣤⣤⠶⠚⠋⠁⠀⢀⣠⡴⠟
⠀⠀⠀⠉⠛⠓⠶⠦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠖⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣽⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣭⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⢾⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣷⣦⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⣿⣷⣿⣷⣿⣿⣿⣿⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣿⣿⣿⣿⣿⣿⠖⢿⡟⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠈⠻⠿⡛⠋⣿⣿⣿⢀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⠓⠀⠛⣿⡿⠙⢹⣿⠏⢹⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠆⠀⠀⠀⣿⡇⠀⠻⠋⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣼⣿⣿⣿⣿⣿⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢹⣿⡿⠻⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠘⣿⠀⠀⢸⡏⠁
⠀⠀⠀⠀⠀⢀⣤⣴⣷⣶⣶⣤⡀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢾⣯⡿⣿⣯⡿⠿⠃⠀⠀⠀⠰⡿⠛⢰⡟⠻⣿⠖⠀⢸⡇
⠀⠀⠀⠀⠀⠘⠿⠃⢸⡟⠀⠈⠛⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣿⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠿⠄⠀⠸⠧

⠀⠀⠀⢠⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⣦⣄
⠀⠀⠀⣿⠀⡉⠙⠛⠛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢠⠞⢡⠊⠳⣄
⠀⠀⠀⣿⠢⢄⠀⠉⠉⠀⠒⠒⠚⠷⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⢶⣦⠾⢋⠔⠁⡰⠁⢀⠔⠉⠳⢤⡀
⠀⢀⡴⢏⡀⠀⠈⠁⠀⠀⠈⠉⠉⠉⠁⠒⠫⢍⠋⠭⠹⢛⡛⢲⣦⠖⠋⠁⠀⢀⡴⠋⠄⠈⡠⠔⠊⠀⡠⠃⠀⣠⠴⠖⠛
⠀⠛⠷⢶⡈⠉⠒⣂⣨⠥⠤⢤⣀⡈⠑⠂⠤⣀⡈⢐⣒⣤⠶⠋⡁⠄⠤⠤⢠⣾⠕⠁⢀⡎⠀⢠⠔⠊⠀⡴⠋⠁
⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠉⠓⠲⣶⣤⠾⠋⠁⠀⠐⠁⠀⠀⣀⣠⡿⠃⠀⢀⡜⠀⢠⠃⣀⣴⠾⢅
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠉⢉⣉⣛⣛⣉⠭⠤⠀⠂⣀⡀⠤⠔⢒⣭⡿⠋⣠⠔⢒⡉⠤⠔⠁⡼⢥⣀⣠⠏
⠀⠀⠀⠀⢀⡴⠋⢀⡠⠒⠉⢁⣀⠀⠠⠤⠐⠒⠈⢁⣀⠠⠤⣲⡯⠊⠀⢠⠁⢰⠁⣀⣤⣴⠞⢧⣀⣰⠃
⠀⣀⣠⠴⠋⢀⠴⢋⠄⠊⠉⠀⠀⠀⡠⠒⢁⡤⠤⠤⠤⣤⠖⡫⠤⠒⠂⠁⠀⢤⠞⠁⠀⠈⠻⢦⣼⠏
⢻⡉⢀⡠⠊⢁⡔⢁⠄⠀⠀⠀⠀⣉⣤⣴⠏⢀⠤⠒⢉⡤⢊⠀⠐⠒⢀⣀⡠⠏
⠀⠻⣤⡤⠔⠉⡐⠁⣰⠞⠛⠛⠋⠁⣰⢋⠞⡠⠔⠈⠁⠀⠀⠀⡴⠋⠉
⠀⠀⠘⢷⣤⠞⢀⡾⠁⠀⠀⠀⣠⠞⢁⠎⡜⠀⠠⠂⢀⣀⣠⡼⠁
⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠚⠥⣤⣋⡐⠀⠀⡄⠀⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣴⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡤⢄⡀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢹⡟⢛⠓⢺⠖⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠐⠀⠀⠐⠀⣤⠃⣴⠀⠀⢠⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠎⠀⢠⡊⣁⠸⠃⢈⡉⠀⠀⠈⠆⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⠀⠀⠀⣁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠀⠄⠀⠀⠀⠀⠞⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣟⣿⣿⣷⣶⣶⣤⣠⣄⣠⣀⡀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣾⣿⠋⠉⡉⢛⠟⠒⡶⢶⣤⣀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣾⣿⣿⣶⣷⣶⣾⣿⣿⣯⣥⡽⣧
⠀⠀⠀⠀⠀⢠⣾⠋⠞⠋⠀⠀⠀⢾⡃⣈⣁⣤⣶⠁⠨⡽⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠆
⠀⠀⠀⠀⠀⢸⣿⣄⠃⡀⠀⠀⠀⠉⠉⠉⠁⠉⠀⣠⢔⣡⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣤⣤⣄⣀⣀⣀⣠⣤⣶⣿⣿⢿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡾⠏⣿⡞⣿⡿⢿⣿⣿⣿⣿⣿⢿⣟⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡛⣿⣟⠉⠛⣿⠛⠛⠉⠉⠚⣐⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠘⠀⠉⠉⠉⠉⠉⠉⢙⡋⠂⣉⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢯⣇⣉⠀⠀⣿⣤⣠⣤⡴⢾⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣓⣲⠤⠤⠤⠄⠠⠤⠤⢴⣶⣯⡗⣿
⠀⠀⠀⠀⠀⢸⣿⣟⠋⣿⡦⠌⠙⠀⠀⢹⡥⡤⢃⠀⣽⢿⣿⡄⠀⠀⠀⠀⢸⣿⣿⡎⠀⢈⡉⠁⠘⠛⢛⠿⢿⣍⣱⠐⠊⣽
⠀⠀⠀⠀⠀⢸⣿⣿⡋⣿⡿⠦⠄⣈⡉⠉⠛⠀⣉⠁⣨⡻⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠹⢃⠉⠃⠘⢩⢹
⠀⠀⠀⠀⠀⢸⡿⣯⣽⣿⡯⠥⠐⠛⠓⠂⠐⠀⣀⣴⡮⠷⢻⠇⠀⠀⠀⠀⢸⣿⣿⣿⣖⣦⡤⠬⠍⠐⠋⠱⠦⣖⣋⠀⠀⢾
⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣾⠍⡐⣂⢀⠤⠴⠠⢀⡐⢦⣨⣽⠀⠀⠀⠀⠀⢸⣿⣿⡿⠤⠈⠙⣁⡃⠈⠟⠗⣂⡠⠤⢀⡒⣺
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣧⡤⢬⣬⡭⠯⠴⣶⣾⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠐⠂⠀⣀⠐⠂⠤⠴⠖⠬⡛⢭⢿
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⠗⠒⠄⠶⠶⠢⢀⡀⣛⣩⠽⠟⠀⠀⠀⠀⠀⢸⡿⣿⣟⣳⡾⠻⠾⠏⢩⣿⣿⡶⣺⠿⠛⠃⢹
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡭⠍⢻⠋⠛⢻⣟⠉⠉⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⣿⣷⡄⡙⡉⠀⠀⠀⠀⠛⠉⠉⠁⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⣹⠀⠀⢸⡽⣶⡄⣤⣤⡶⠋⠀⠀⠀⠀⠀⠀⠙⠿⣿⣟⡀⠀⠀⠀⠀⣠⠀⠀⠀⢀⣰⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠦⠤⠼⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠦⠤⠤⠿⠶⠚⠛⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠁⠀⠃⠴⡟⡁⠀⠀⠐⠒⠢⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣾⣷⡀⠀⠀⡴⡠⠀⠀⠀⠀⠈⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠂⠉⠁⠀⠀⠒⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⢄⡀⠈⠙⠓⠲⠶⠶⠤⠤⣤⡤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠶⠒⠚⠋⠁⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠂⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠙⠒⠶⠤⠤⣬⣭⣉⣁⣀⣀⣠⡤⠤⠤⠖⠒⠊⠉⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠓⠤⣀⣀⠀⠀⠤⠀⠀⠀⠀⠀⠀⡀⢀⠀⠄⢀⣀⣀⣤⠤⠴⠒⠉⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠉⠉⠉⠛⠛⠓⠒⠒⠒⠒⠒⠒⠚⠋⠉⠉⠁⡀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠉⠓⠲⠶⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠤⠶⠖⠒⠛⠉⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⠀⠙⠦⣄⡀⠈⠉⠛⠓⠷⠶⢶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠶⠖⠛⠊⠉⠀⠀⢀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠛⠓⠶⢶⣤⣤⣤⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠂⠉
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠘⠷⣤⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠀⠉⠉⠁⠀⢀⣀⡤⠶⠛⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⣀⡀⠀⠀⠀⠀⠉⠛⠒⠶⢤⣤⣤⣤⣤⣄⣀⣤⣀⢤⣀⣀⣀⣤⣤⠴⠖⠒⠚⠉⠉⠀⠀⠀⠀⠀⣀⡀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢶⣶⣤⣤⣤⣤⣄⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣠⡤⠤⢶⣖⡖⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠚⠦⠤⠤⠭⠭⠭⠭⠭⠭⠭⠭⠭⠤⠤⠴⠀⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉
⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠊⠀⠂⠠⢶⠂
⠀⠀⠀⠀⠀⢀⠀⠀⢖⣿⡀⠀⠀⣰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠠⠀⠐⠚⠉⠁⠀⠂⠀⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠒⠚⠁⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠈⠳⣦⡀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⢻⣦⣄⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣇⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢿⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣀⠀⢸⡆⠀⠀⠀⢸⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠁⠉⠛⢻⠇⠀⠀⠀⠘⢧⡈⠉⠙⠛⠲⠶⢤⣤⣀⡀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠉⠉⠛⠓⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⣄⣀⣤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠦⣤⣀⢀⣠⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠙⠓⠶⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠟⠋⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢤⣀⡀⠀⠀⠀⢀⡶⠋⠉⠀⠀⠀⠀⠀⠉⠙⢶⡀⠀⠀⠀⢀⣠⡴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠙⠓⠶⠶⠛⠀⠀⠀⠀⣠⣦⣄⠀⠀⠀⠀⠛⠶⠶⠚⠋⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠉⠛⠓⠶⠶⢤⣤⣤⣤⣤⣤⡤⠴⠶⠚⠛⠉⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣷⣄⣀⠀⠀⠀⡀⢀⠀⠀⢴⣿⣿⡃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⡿⢿⣿⣿⠟⢡⣴⠦⣵⣤⡰⣿⣰⣿⣿⣿⠀⠀⠱⡀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠈⠉⠉⠰⢾⣿⣿⣷⣺⣿⣿⡿⣿⡿⠃⠀⠀⠀⠎⠀⢈⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⣤⣀⢀⡙⣿⣿⣿⣿⣿⣫⠿⢡⣾⣤⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠟⠻⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣃⣾⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣰⣶⣾⣿⣿⣿⣿⣿⣿⣵⣶⣠⣤⣶⣾⡉⠀⠒⠖⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⣿⣟⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⠻⠛⢁⡜⠙⣠⣿⣿⣿⣿⣿⣿⡟⠻⣿⡿⢿⣿⣿⢿⣿⠄⠀⢄⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠀⢀⣴⡯⢀⣾⢿⣿⠿⠛⣿⡿⣿⣿⣮⣽⣻⠈⠹⣿⣿⣿⣿⡢⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢰⡿⠇⠀⠀⢠⣾⠟⣠⣯⢿⣟⠏⠀⢰⣿⡇⠙⣿⣿⣿⢿⣿⡆⠸⠛⣿⡾⣿⣆
⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⡾⠃⠀⢿⡸⠈⠀⠀⠀⢸⣿⠀⠀⠈⠇⢻⣿⣳⢯⠀⠀⠁⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠸⠁⠀⠀⠀⠀⢸⣻⠀⠀⠀⠀⠈⠛⣾⣗⠄⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⢈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⢠⡄⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢⠤⠀⠂⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⠤⠔⡆⠀⢠⣦⡀⠀⠀⠀⢠⡔⠂⠒⠄⠀⠉⠀⠀⠀⠀⠀⣸⢢⡀
⠀⠀⠀⠀⠓⠀⠀⣀⠤⢚⠆⠊⣹⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣟⣛⢤⡠⠟⠀⠀⠀⠀⠀⠎⣠⠀⡄⠘⠀⠀⠀⠀⠀⠀⠔⠊⠉⠉⠐⡄⠃
⠀⠀⠀⣠⠤⠂⠭⠤⠤⠬⠅⠭⠐⠂⠐⢒⡈⡅⠠⢀⠀⠀⠀⠀⢀⠤⠚⠸⣯⣓⣲⣦⠀⠀⣀⠀⠀⠗⠠⢸⠱⢦⡇⠀⠀⢀⠀⠸⠂⠐⢤⠒⠞⠡
⠀⠀⠀⢰⠦⠒⣐⣒⡒⠦⠴⠆⠐⠉⠃⠒⢂⡀⠀⢈⠆⠀⠀⠀⡏⠽⠋⠙⠿⠽⠇⠸⡄⠀⠈⠀⠀⠀⠀⡘⠀⢸⡇⠀⠀⠉⠀⠘⡈⡄⠸⠀⠈⠈⢀
⠀⢀⠔⠈⠂⠠⠤⠤⠄⠤⠀⠒⠂⠒⠒⠈⠁⢀⡤⠊⠀⣀⠀⠀⢆⢏⢂⢸⠑⢀⠊⡳⠁⠀⠀⠀⠀⣆⣀⣁⣀⡀⠇⠀⠀⠀⢀⠔⠳⡀⠀⠀⣀⠄⡠
⠀⠰⠖⠤⠄⢀⣀⣀⣀⣀⡀⠀⠠⠤⠀⠒⠊⠁⠀⠀⠀⠈⠀⠀⠀⠑⠆⢠⠀⢀⠊⠁⠀⠀⠀⠀⠀⠀⠀⡂⢘⠀⠀⠀⠀⠀⠀⠑⠠⠤⠬⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⡀⠀⡴⡄⢠⡀⠀⠀⢇⡀⠀⠀⣀⠀⠔⠐⢢⠀⠀⠀⠀⠀⠀⣄
⠀⢀⠀⠀⠀⢀⠔⣂⠈⣉⡴⡢⢄⠀⠀⠀⠀⠀⢀⡶⠒⢈⡉⢀⠬⡑⢄⠀⠀⣀⣀⡭⢕⠀⠞⠀⠀⠀⠀⠀⠀⠀⣀⡆⠙⠐⢠⠞⠀⠀⠀⠀⣄
⠀⠈⠁⠀⡐⣡⠜⠀⣩⣌⣧⡱⣬⣆⠀⠀⢀⠔⠁⠤⠐⠁⠈⠀⢀⠃⢸⠀⠀⠈⠂⠤⢀⡠⠀⠓⢄⠀⠀⠀⠀⢎⠹⢬⣐⢠⠌⣑⠆
⠀⠀⠀⠀⣅⡛⢠⡊⠀⠀⠙⣟⢠⠈⠀⢀⠊⢠⠚⠀⠀⠀⠀⠀⠈⠆⢋⠀⠀⠀⠀⠀⠀⡆⠀⠀⡸⠀⠄⣀⠀⠈⠑⠤⣈⠇⢀⡠⠀⠀⠀⣀⢄⡀
⠀⠀⠀⠀⢃⠑⡌⢑⠠⢤⠾⢼⡻⡐⠀⠘⡀⠈⢄⠀⠀⠀⠀⠀⠀⠘⢤⠣⡀⠀⠶⠀⠀⠘⠂⢖⠁⠀⠠⣄⠙⠀⠀⠀⠀⡠⢋⡌⠈⠓⠁⠀⠥⣙⣆
⠀⠀⠀⠀⠈⠣⢕⠛⠳⢰⠜⡩⠜⠁⠀⠀⠙⠲⢤⡑⢄⠀⠀⠀⠀⠀⢀⡇⢈⡄⠀⠀⠀⠀⠀⠈⡆⡎⠐⠤⠃⠀⠀⠀⠀⣅⠁⠀⡞⠉⠉⠰⠀⢀⠟
⠀⠀⠖⠀⠀⠀⠀⠈⠉⠉⠉⠀⠙⠀⠀⠀⠀⠀⠀⠻⣄⠤⢀⣀⡠⠂⣁⡴⠛⠀⠀⠀⣀⣀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⡌⠢⠀⠣⣀⣀⠔⠀⢸⢠
⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡀⠀⠀⣠⠀⠀⠙⢦⣀⣀⣴⠞⠋⠀⠀⡠⠚⢝⣄⣀⡱⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠈⠦⡂⠀⢄⡀⡠⢒⡱⠁
⠀⠀⠀⠀⠀⠀⡠⣔⠚⢭⡃⠀⢻⠄⠈⠆⠀⠀⠀⠀⠀⠈⠛⠊⠁⠀⠀⠙⠰⡆⣠⡴⢏⣀⡀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠘⠂⠁
⠀⠀⠀⢀⡴⡉⠑⢈⡆⠀⠛⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⣀⠤⠦⠄⡀⠀⠀⠀⠑⠄⣀⣈⡇⠉⣯⠀⠙⡯⠉⠙⠹⠉⢿⡉⢱⠀⠀⠀⠀⠀⡀⠀⠀⠶
⠀⠀⠠⠋⠑⢺⡀⠈⠁⠀⢀⠠⣊⠔⠁⠀⠀⠀⠀⢠⢊⡷⠀⡦⠰⣌⠱⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣼⠁
⠀⠀⢀⠀⠀⠀⡁⠤⢄⡂⠅⠊⠀⠀⠀⠛⠀⠀⣀⠁⠋⠀⠀⠃⠀⠘⠃⠳⢄⣤⠀⠀⠀⠀⡠⠒⠉⠉⠉⠁⠢⡀⠀⠀⠀⠛⠀⠀⢀⠔⠁⠱
⠀⠀⠰⠗⠒⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡤⠐⠰⠠⠄⠤⠤⠔⢰⠊⠀⠀⠀⠀⡎⠀⡰⠂⠉⣉⠑⠀⠈⠆⠀⡄⠀⠀⠀⠌⠀⠈⠁⠑⠄
⠀⠀⠀⠀⠀⠀⡀⠄⠤⣠⠒⠈⠑⢤⠐⠐⢢⠀⠀⠈⠒⠂⠀⠤⠠⠄⠐⠁⠀⠀⠀⠀⢀⠀⠘⠀⠀⡋⡄⠙⡄⠀⢸⠀⠀⠀⠀⡉⠀⠒⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠎⠀⠀⠀⠈⢂⠀⠀⠀⢃⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢇⠈⢆⡀⠈⢀⡔⠀⢀⠎⠀⠀⠀⠀⢑⢢⡄⢤⢤⡤⠊
⠀⣤⠀⠀⢘⣄⠀⠀⠀⠀⠈⠆⠀⡠⠆⠀⠀⡠⢖⡩⡽⡀⢭⡐⢄⠀⠀⠀⠀⢀⠀⠀⠀⢀⢷⠄⣈⠉⢁⡀⠴⠋⠀⠘⠃⠀⠀⠘⡶⡞⢒⣋⡄
⠀⠀⠀⠀⡆⠀⠉⠂⢄⡀⢀⠜⠁⠀⠀⢀⢎⠴⡯⡀⠱⠑⡘⡘⡎⡆⠀⣤⠀⠈⠁⢀⠔⡱⠋⠀⠀⠀⠀⠀⡠⠄⣤⠀⠀⠀⠀⠀⠏⣓⣛⢠⠁
⠀⠀⠀⠀⠣⡀⠀⠀⡠⠀⠀⣤⠀⠀⠀⡌⣶⡀⠱⡐⡄⢣⠐⡁⡐⠐⠀⠀⠀⠀⡰⡡⠊⠀⠀⢐⠉⠑⠀⠰⣓⣪⠔⠀⠀⠀⠀⠀⢸⣬⣨⡎⠀⠘⠃
⠀⠀⠀⠀⡇⣀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢡⠡⡈⢢⠱⡘⠄⢣⢘⠇⠆⠀⠀⢠⢊⠜⠁⠀⠀⠀⠈⠓⠊⠑⠢⡀⠀⠘⠃⠀⠀⠀⠀⠘⣪⢬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠑⢌⡑⠷⠴⠄⠛⡡⠊⠀⠀⠀⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠙⠊⠰⠆
⠀⠀⠀⠙⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⢉⡛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⡗⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢀⡠⠔⠒⠛⠋⠙⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⣠⠞⠉⠀⠀⠀⠀⣠⠞⠛⢲⡄⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⢠⡾⠁⠀⠀⠀⠀⠀⠀⡏⠀⢠⣤⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢠⣿⠁⠀⣠⡶⠶⢦⡀⠀⠹⣄⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢸⣿⠀⢸⡏⡠⠒⣦⢹⡆⠀⠈⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠘⣿⡄⠸⡇⢧⡀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠙⣷⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠘⣧⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢹⡆⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠘⠁⠊⠀⠀⠀⠀⠀⠂⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⣶⠛⠉⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⢳⣄⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠙⠻⠄⠰⠞⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠾⠋⣠⡿⣿⣦⡤⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣉⣁⣠⡤⠴⠚⠉⠀⣼⡿⠞⠳⣦⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⣸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠋⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠤⠒⠋⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣀⡀⠀⠀⠈⠛⠲⠤⠤⠀⢀⣀⣀⣀⣀⣀⡠⠤⠤⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠤⣤⣄⣀⣀⣀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⢤⣤⣤⣀⣀⣀⣀⣀⡀⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠛⠋⠉⠉⠛⠓⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⢀⢰⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⣰⢾⣷⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⣦⡀
⠀⠀⠀⢰⠏⠁⠘⠻⠋⠀⢠⣾⣷⣦⣤⡀⠀⠀⠀⣤⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋
⠀⠀⠀⢹⡀⠐⣦⡀⠀⠀⠻⠋⣿⡟⢿⣿⣷⣦⣤⡿⠀⢸⣿⣤⣤⣶⣾⣿⣿⡧⠀⣸⣿⣿⠟⠋⠲⡄
⠀⠀⠀⢀⡇⠀⠈⠀⠀⠀⠀⠸⣿⡇⢸⣿⡏⢸⠏⡅⠀⢠⣿⣍⢻⣿⡿⢀⡆⠀⠀⠙⠛⢁⣴⠀⠀⡗
⠀⢀⣀⢸⠁⠀⢿⣆⠀⠀⠀⠀⠈⠻⣤⣭⠴⠋⢸⡇⠀⠀⡏⠙⢦⣉⣁⣼⠃⠀⠀⠀⠀⠈⡉⠀⠰⠃
⡴⠻⡅⠘⢧⡀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣠⠀⠛⠈⠃⠀⠐⠃⠶⠀⢯⡉⠁⠀⠀⠀⠀⠰⠿⠋⠀⡆
⢧⣀⣀⠀⡴⢻⣄⣠⡟⠀⠀⠀⠀⢺⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⡀⠀⢀⣼⣷⣄
⠀⠈⠉⢉⣷⣾⣿⣿⣿⠀⠀⠀⣰⠏⠀⠈⢳⣦⣀⣀⠀⣀⣠⠔⠋⠀⠀⠈⢷⡀⠀⢸⣧⣴⣿⣿⣿⢿⣷
⠀⢠⡖⣉⣼⣿⣿⣿⣿⡄⠀⢀⡿⠀⠀⠀⢀⣙⣻⣿⣿⣭⠥⠤⠤⣤⣤⡄⢸⡇⠀⢸⣿⣿⣿⣿⣿⣿⠷
⠀⠈⠉⣡⣿⣿⣿⣿⣿⣷⠀⠸⡇⣠⣶⣿⡿⠟⣩⠄⠀⠀⠀⠀⠀⠀⠈⠁⠘⠃⠀⢹⣿⣿⣿⣿⣯⡵⠞
⠀⠀⠘⣳⣯⣾⣿⣿⣿⣿⣇⢠⢿⣿⡟⣋⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⠾⠃
⠀⠀⠀⣩⣿⡿⠋⠹⣿⣿⣿⣏⡾⠻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣰⣿⣿⣿⣿⢿⣿⡵⠋
⠀⢀⣾⣷⣯⡀⠀⠀⠈⣷⡷⣦⢧⡀⡀⣀⢢⡄⠄⠀⠈⢁⠀⡀⠀⠀⠀⠀⣰⣯⣿⣿⣫⡵⢟⣯⠖⠁
⠀⠘⢷⣿⣿⣽⣤⣤⣀⢙⣹⣿⠀⠙⢧⡈⢻⠈⢂⠀⡄⠴⢂⠀⠀⠀⣠⡼⢋⣿⣿⢟⣫⠖⢋⡴⠂
⠀⠀⠀⠙⢿⣽⣟⣿⣿⣿⣿⣿⣇⠀⠀⢹⣶⣦⣀⠀⠚⢀⢀⣀⣦⣾⡋⢀⣾⣿⡵⢋⣥⠾⣫
⠀⠀⠀⠀⢩⡾⣽⢿⣿⣿⣿⣿⡿⠀⣰⠟⠉⢹⣿⣿⣿⣿⠛⠉⠉⠙⢿⣾⡿⣫⡶⣛⠵⠋
⠀⠀⠀⠀⠀⠸⢋⣾⡿⣿⢟⣿⣿⡀⠀⠀⠀⢠⣿⣿⣿⣇⠀⠀⠀⠀⢸⣿⣿⣽⢟⡡⠂
⠀⠀⠀⠀⠀⠀⠈⠾⣽⢧⡟⣽⠃⠁⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⣾⣿⢟⡵⢋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠰⠃⠀⠀⠀⠀⠀⠈⠛⠛⠛⠏⠀⠀⠀⢰⡟⠑⢋⠚⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠰⣆⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⢻⣦⢰⣿⣿⠏⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⣼⣿⡿⠀⣀
⠀⠰⠟⣻⣿⣷⣌⢻⣿⣿⣵⣾⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢛⣿⣿⣦⡘⣷⣿⣯⣴⣿⡃
⠀⠀⠀⣀⣤⣭⣿⣿⣿⣿⣿⣯⣭⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⣽⣿⣟⣿⣿⣯⣥⣤⣤⣤⣄⡀
⠀⢠⣾⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡻⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⢿⣟⡿⣿⣯⡟⠛⠉⠙
⠀⣼⣿⡿⠋⣿⣿⡿⠀⢿⣿⣷⡝⢿⣷⠀⠀⠀⢰⣄⠀⣠⠄⢀⣀⠀⠀⠀⢠⣿⡿⠟⢹⣿⣿⠇⠘⡿⣿⣦⠛⢿⡄⠀⠀⠀⣤⡀⢀⡄⠀⣀
⠸⠋⠁⠀⠀⣽⣿⠇⠀⠘⡆⢿⣿⣀⣨⣶⣶⣤⣄⣿⣾⣿⣿⡟⠁⠀⠀⠀⠟⠉⠀⠀⢨⣿⡻⠀⠀⢳⠘⣿⣆⣀⣵⣶⣤⣤⣸⣧⡟⣴⡟⠉
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⣷⢀⣻⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⢸⡄⢈⣿⣿⣿⣿⣿⣿⣿⠋⣼⣳⣿⠿⡿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠉⠉⠉⠉⣉⣿⣿⣿⣿⠻⣿⣿⣦⡝⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠈⠉⠉⣩⣿⣿⣿⡟⠻⣿⣶⣌⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⢴⣿⣿⠏⣿⣿⡇⠈⠉⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡠⣾⣿⡟⢸⣿⣿⠀⠁⠙⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡴⠃⣾⣿⠋⠀⠸⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠞⢠⣿⡏⠁⠀⢿⣿⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡞⠁⠀⠹⠃⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠃⠀⠀⠛⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⣯⣞⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿⣿⣵⣃⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢰⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠓⣆⠀⠻⣿⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⠆⠀⢸⣿⡇⠀⠀⠸⡆⠀⢻⡆⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⣷⠀⢸⣿⠁⠀⠀⠀⠀⢿⣧⠀⠀⢀⡇⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣧⠸⣿⡇⠀⠀⠀⠀⠈⠻⣦⣠⡾⠁⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡿⠋⣸⣄
⠀⠀⠀⠀⠀⣤⣴⠞⢷⡆⠀⠀⠀⠀⠀⠀⠀⢿⣧⡴⢿⡆⠹⣿⣄⠀⢀⣤⠀⠀⠈⠁⠀⠀⠀⠀⣀⣴⣾⣿⡿⠿⠛⢁⣤⣾⠛⠉⡆
⠀⠀⠀⠀⠀⢻⣇⠀⣸⣇⣠⣤⣄⣀⡀⠀⠀⢸⣯⠀⢸⣿⡀⠈⠛⠻⠟⠁⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠟⣁⠤⠤⠴⢞⣹⠁⠀⣀⡇
⠀⠀⠀⠀⠀⠈⣿⡾⠛⠉⠉⠉⠉⠻⣿⣷⡀⢸⣿⠀⠀⢿⡧⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣛⡽⠋⣠⠾⣗⡼⠤⠵⠋⢀⣠⣾⡿⢷
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⢹⣿⠇⢸⣿⠀⠀⠀⠀⠀⠀⠀⣠⣶⣟⠫⢿⠟⠛⠛⠋⢀⡼⣧⠤⠞⠀⢀⣠⣶⣿⠿⠃⣠⡟
⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⣠⠿⠋⠀⠈⠀⠀⣀⣤⣶⡶⡾⣟⢣⣼⣮⠟⠁⠀⣠⡄⣴⣿⡽⠃⣀⣴⣿⣿⠟⠋⠀⡠⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡴⠖⠋⠁⠀⠀⣀⣀⣴⣾⣿⠿⠛⠛⠛⠋⠉⠉⢀⣀⣀⣼⣿⠷⠋⣡⣶⣾⣷⡷⠛⠁⢀⠔⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⣀⣤⣴⡶⣿⣿⣿⣿⣿⠿⠁⠀⠀⢀⠔⠢⠤⢲⣻⠿⠛⢁⣠⣴⣿⣿⠟⠋⠁⢀⣤⡶⠕⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⣻⠻⣷⣼⠿⠃⠛⠋⠀⣀⡠⠔⢾⣩⣄⡤⠚⢉⣤⠤⢶⣿⡶⠟⠚⠋⣀⡴⠞⠉⠁
⠀⠀⠀⠀⠀⢀⣾⣿⣷⣿⣭⠜⠋⠀⠀⠀⣤⢶⢁⠁⣠⠴⠋⢀⣠⣴⣾⣿⡿⠿⠋⠁⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⣤⡖⣤⡉⠁⣤⡶⠍⢡⣥⠴⠛⠟⢛⢉⣠⣶⣿⣿⣿⠟⠋⠀⣀⣤⡶⠟⠊⠁
⠀⠀⠀⠀⠀⣿⣿⣜⣩⠿⡿⠖⠋⠉⢀⣠⢴⣾⣏⢾⡿⠓⠾⠋⣀⣤⡶⠞⠋
⠀⠀⠀⠀⠀⡄⡇⠀⢀⣀⣠⣤⣶⡞⣛⣳⠾⠷⠛⠋⢀⣠⡴⠞⠊⠁
⠀⠀⠀⠀⣼⣿⠋⢩⣿⣿⣿⠿⠋⠉⠉⠀⢀⣠⣴⠞⠋⠁
⠀⠀⠀⠀⢯⡏⠉⠉⠀⠀⠀⠀⠀⡀⠄⣂⠕⠊
⠀⠀⠀⠀⠘⡴⡤⠄⠠⢄⣐⠠⠁⠒⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢉⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⢣⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠖⠂⠹⣋⠁⠠⠀⣈⡋⠹⠉⠉⠉⡭⠽⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣉⠠⠄⠀⠁⡠⠔⠒⡄⠉⡩⠄⠲⢤⠀⠀⠀⢉⣉⠉⠉⣹⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⡁⣀⣀⠠⠯⠀⠀⡔⡠⢄⢣⠤⠄⠀⣇⠀⡜⠉⠈⢣⢸⢤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⡿⠍⣙⣗⠀⠠⠰⡀⡇⢸⠀⠆⠀⠒⢇⣴⡇⠌⠉⢸⠸⣀⡟
⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⢏⡒⠼⢨⣉⣉⣁⣳⡇⠚⢛⣂⣉⣉⠁⠸⠱⠀⠐⣚⠀⣯⣤⣀⣀⡀
⠀⣠⠴⠒⠋⠉⠁⠀⠘⡇⢰⡀⠱⠼⠒⠂⠠⠶⢹⠄⠄⠀⠀⠒⠂⡆⢸⠂⢁⡨⠟⢷⡄⠀⠈⠉⠉⠓⠢⢤⡀
⣟⣄⠀⠀⠀⠀⢀⠤⠴⢷⡸⠨⠍⠀⠒⠒⠒⣧⡔⠒⠒⣒⠒⠒⠒⠣⠜⠍⠁⢒⣁⠴⠣⠤⡀⠀⠀⠀⠀⢐⢝
⠈⠻⠵⣒⣢⠤⠬⢒⣂⡀⠙⠲⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠖⠒⠛⠛⠉⠁⣀⣀⣀⠡⠤⣤⣒⡪⠝⠊
⠀⠀⠀⠀⠉⠉⠓⠲⠦⠤⣄⣀⣀⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣀⣠⠥⠤⠶⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⢀⣠⡈⠗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⣷⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⠂⠠⠊⠛⠁⢌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠻⡧⠶⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠂⠁⢀⠃⠀⡆⠱⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⠁⠀⠠⠁⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⡠⠀⠊⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢄⣀⠀⠀⠀⢀⢠⠀⠉⠉⠉⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠄⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠒⠀⡀⠑⢤⠜⠀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⢠⠄⠼⣦⣀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⢠⢃⠈⠁⠀⠀⢉⠠⠄⠒⠉⣇⠀⠀⠰⡁
⠀⠀⠀⠀⠁⠀⠀⢀⠀⡇⠈⠉⠉⠀⠀⠀⠀⠤⠀⠀⠀⡠⠶⠢⢀⠔⡀⡠⠢⡠⠊⠙⠄⣿⠀⠀⢠⠊⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠉⠘⠤⠁⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⢧⠲⢄⡁⣀⠈⢀⡀⢀⡠⠄⢸⠓⠂⠤⠔⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⢸⠆⢦⡄⢀⣀⣀⣀⠀⢤⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢄⣀⠀⠀⠀⠀⠀⠇⠁⠀⠀⠈⠒⠂⠀⠒⠊⠁⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠃⠄⡀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡔⢉⠤⠒⠐⠠⢍⠢⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠒⠤⣤⣀⠀⠀⠂⠄⡀⣃⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢌⡔⡡⢊⠍⠊⠕⢄⠱⡌⢆
⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠈⠙⠳⢦⣄⠈⠸⠀⠀⠀⡠⠔⠚⠧⢤⡀⠀⠀⢸⠱⠁⡆⠀⠰⠎⠀⡆⢱⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⢉⠷⠔⠀⠀⢰⠄⠐⢁⠠⠂⢱⠀⠀⠸⡐⡀⠘⢄⠀⠀⢀⠃⡸⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⠒⠒⠈⠀⠀⠀⠀⠀⢰⠀⠂⠁⢀⠄⢘⡀⠀⠀⠳⡈⠢⣀⡉⠁⣁⠴⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠐⠈⡠⠐⠈⡇⠀⠀⠀⠈⠒⠤⣀⣉⣀⡠⠔⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠒⠈⠁⡠⠂⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⣮⣤⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡠⡀
⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠑⠊⠊⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⢀⣴⣿⣿⠟⢁⣴⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⡿⠛⣉⣤⣀⡉⠻⣿⣿⣿⣿⠿⠟⠁⠀⢸⣿⣿⠋⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢞⣋⣽⡟⣩⣖⣠⣾⣯⣿⣷⡀⢿⣿⣿⣥⣶⡶⠀⢰⣿⡿⠃⡰⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⣉⡀⢀⣾⣿⣿⢿⣿⣿⣿⣿⠇⣸⣿⠟⠉⠁⠀⠀⣸⣿⠁⡰⠟⠋⠁⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣶⣶⣟⠛⣵⣿⣿⣿⣟⣠⣽⣿⡿⠁⠀⣯⠴⠆⠀⠀⠀⠀⣿⠇⣰⣿⣶⣤⣀⣀⣀
⠀⠤⢀⠀⠀⠀⠄⢠⠶⠛⠉⠁⠈⣀⠈⠉⠁⣠⣿⣿⣿⣿⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⢦⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠒⠂⠠⡀⠀⠀⠀⢠⣤⣶⣾⣿⡿⠿⠋⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠻⠇⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠱⡀⠀⠀⠀⢀⠐⣡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡙⠂⠀⢀⡤⠞⠓⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣄⣀⣉⣈⣀⣠
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡷⣦⣬⣽⣿⠟⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡁
⠀⠀⣤⣶⣶⣶⣿⣿⣿⡿⣻⣿⣿⡟⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠛⠑⡄
⠀⠀⠛⢻⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠘⣆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⣿⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⢠
⠀⠀⣠⡾⣻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⠄⠙⠢⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⣠⠞
⣠⣾⣭⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢠⠞⣡⠞
⣻⣿⣿⣿⣿⣿⣿⠏⠉⠙⣿⣿⣿⣿⣿⣫⣧⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣤⣤⣤⣤⡀⠀⠀⣼⣯⣀⣀⣴⡯⠞⠁
⣿⣿⣿⣿⣿⣿⠟⣤⣀⠀⢹⣿⣿⣿⣏⠉⠉⠀⠀⠀⠀⠀⠠⣿⣿⣷⣿⣿⣿⣿⡿⠁⢀⣾⣿⣿⣿⣿⠞⠁
⣿⣿⣿⣿⣿⠟⠀⠀⠈⠙⢮⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠉⠉⠛⠛⠋⣀⣴⣿⣿⣿⣟⡼⠃
⣿⣿⣿⡿⠉⠀⠀⠀⠐⠄⠈⣻⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠞
⣿⣿⣟⠀⠀⠀⠀⠀⠀⠘⣦⢻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠋⠉⠻⣿⣿⠟⠁
⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⢷⣿⣿⣷⠈⠛⠳⠶⠶⠴⠶⠛⠁⠀⠀⠀⠀⠀⠀⢠⡿⠁
⣿⣿⣿⣿⡀⠀⠀⡀⢀⠀⠀⠈⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣾⠇
⣿⣿⣿⣿⣇⠀⠀⠉⠉⠙⠲⢦⣻⣿⣿⠷⣤⣀⠀⠀⢀⡀⡴⠋⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⠀⠀⢤⡀⠀⠀⠂⢹⢿⣿⣷⠀⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣤⣤⣄⡉⠉⠙⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣿⣶⣄⠀⣿⣧⠀⠀⠀⢀⣤⣶⡶⣶⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠂⠀⠀⣀⣀⣠⣤⣖⡢⢿⣿⣦⠈⣿⠀⢀⣴⡿⠿⠍⠉⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠉⢁⣴⣾⣿⣯⠛⠟⡻⡫⢿⣷⡝⢿⣇⣽⢠⣿⣟⠃⠀⣠⢄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣰⣿⡿⠛⠀⠀⣀⣀⣀⡀⠉⢴⣿⣆⢿⣿⣿⡟⢡⣴⣾⣿⣿⣿⣟⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⢸⣏⢠⣿⡿⢋⣴⣾⣿⣿⣿⣿⣿⢿⣶⣍⡻⣼⣻⠟⣡⣾⠿⠿⠿⡟⠛⠻⠃⠁⠀⠉⠳⡄
⠀⠀⠀⢠⡶⠛⠉⠛⠻⠿⣿⣦⣄⢿⣼⡟⣰⣿⣼⢻⣿⣿⣿⣶⣬⣦⣅⣭⣿⣶⡦⣬⣕⠿⣿⠿⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣰⣦⣬⣝⠻⣿⣿⣇⡿⢋⡯⢾⣟⣛⣻⡿⠏⢩⣿⣿⣿⡟⢿⡎⠻⣿⣮⡳⡀⠉⠙⣿⣿⣦⡀
⠀⠀⠀⢀⣴⣿⣿⡿⣿⣿⠿⣿⣷⡮⣛⣭⣴⣿⣏⡛⣻⠿⣿⣿⣋⣿⡟⢹⡿⠀⠘⣿⡀⠉⢿⣻⣿⣦⡀⠘⠁⡿⣿⡄
⠀⠀⢠⡟⠁⠈⠈⢀⣥⣾⠿⡻⣩⡾⢻⣾⠿⣿⣿⣿⣦⡀⠁⠊⢹⣿⢹⠈⠇⠀⠀⢻⡇⠀⠰⣿⣿⣿⣇⠀⠀⠀⢹⣷
⠀⠀⠈⠀⠀⠀⣰⣿⠏⠁⣠⣴⡿⠁⢸⡇⠀⠉⣷⠀⢻⢿⣆⠀⢸⣿⠆⠁⠀⠀⠀⢸⣷⠀⠀⠸⢹⣿⣯⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⢰⣿⠙⠀⢰⣿⣿⣇⠀⠘⣿⠀⠀⠉⠀⠈⢸⣿⡆⠈⣧⣃⠀⠀⠀⠀⣸⣿⠀⠀⠀⢱⣿⣿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢿⡏⠀⠀⢸⣿⣿⡃⠀⠀⢿⣇⠀⠀⠀⠀⠈⠋⣿⠀⠹⡾⠀⠀⠀⠀⡿⣿⠀⠀⠀⢉⣿⡟
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣿⣷⡁⠀⠀⠘⣿⡆⠀⠀⠀⠀⠸⢿⠀⠀⠱⠀⠀⠀⠀⡏⣿⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠹⣼⣄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢰⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢻⣜⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣎⢧⡀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠳⣄⠀⠀⠀⠀⠀⠀⡾⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣄⡄⠀⡀⠙⣶⡈⢶⣧⢴⢇⣎⣼⢃⣾⣿⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠦⣝⢾⣿⣄⠈⣻⣿⣷⣀⡙⢿⣿⣿⠃⢸⣿⣿⣿⣿⠷⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢙⠛⣿⣿⣿⣶⣾⣬⣻⣿⣧⣄⣽⠃⣀⣿⡿⣿⣿⣟⣉⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣧⣽⣿⣿⣝⣯⣅⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⢂⡀⠈⣣⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠑⢺⡧⠈⡀⠐⠈⠀⠀⠰⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠤⠐⠒⠀⢰⠛⠓⠆⠀⠄⣿⣧⠌⣤⣤⠄⣤⡷⢀⡠⠄⠀⡀⠐⠒⠠⠶⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⢉⠁⠀⠀⠀⠀⠀⢀⡘⡀⢀⡀⠂⠀⠀⠙⡿⠛⡁⠀⠉⠙⣯⠁⠀⠀⠘⠳⠶⢤⣀⠐⠂⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠈⠀⠀⣠⠴⠚⠋⠉⢙⣬⠉⠵⣐⠀⠀⣼⠃⠀⠄⡁⠀⠀⢸⣇⠴⠛⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⣀⠀⢀⡞⠁⠀⠀⠀⠐⠒⠺⣧⣐⢢⣨⣶⣛⣆⠀⠁⠀⠀⢔⣬⢯⠉⠁⠀⠀⠀⠀⠀⠀⡄⢳⡄⢀⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⢿⠑⢤⣀⠉⠀⡼⠀⡄⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠻⠿⠦⠌⠛⠛⠁⠀⠀⠀⠀⠀⠤⠤⢔⠃⠀⢸⠇⠉⠀⠀⣠⡴⠛⡇
⠀⠀⠀⠀⠀⠀⡸⢧⠀⠈⠻⣦⣱⠀⢈⡐⠒⠒⢈⡉⠤⠤⠤⠄⠀⠀⠀⠀⠀⣐⣢⣭⣭⣽⣦⣶⣤⣤⣄⡀⠀⠀⠀⢠⢏⣀⡤⠖⠋⠉⢀⣼⣅
⠀⠀⠀⠀⠀⢸⡇⠈⢷⣤⡀⠀⠉⢆⠀⠈⠀⠀⣠⣴⣶⣿⣶⣤⡁⠀⢀⣴⠛⠩⠅⠀⣂⣀⣒⠤⠤⠤⠭⢿⣦⠀⢰⠟⠋⠁⠀⢀⣠⡴⠟⢁⢸⣧
⠀⠀⠀⠀⠀⢹⡙⢦⣀⠈⠛⠷⣦⣬⣥⠀⠀⢰⠏⠀⠀⠀⠀⠀⢣⠀⣼⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣽⣤⣴⠶⠟⡋⠁⠀⣀⣤⠏⣿
⠀⠀⠀⠀⠀⠘⣧⠀⠙⠻⢦⡀⠀⠉⠙⡀⠀⡿⠿⠶⠶⣶⣶⣶⣾⠀⢻⣦⣤⣤⣶⣶⣶⠶⠶⠶⠾⠿⠟⠛⠻⣦⡿⠁⠀⢀⡠⠥⠒⠛⠉⢀⣼⣇
⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠙⠶⢦⣼⠀⢰⡁⠈⠀⠀⠀⠀⠀⢸⠄⠸⡇⠀⠤⠀⠀⠀⠀⠀⠠⠀⢀⣀⣤⣤⠤⠶⠶⠚⠁⠀⠀⢀⣠⣴⠟⠁⢸⣧
⠀⠀⠀⠀⠀⠀⢘⢿⡙⠻⢶⣤⣄⡀⢸⡆⠀⡟⠛⠒⠒⠒⠒⠲⣾⠀⠀⡷⠒⠒⠒⠲⠶⠶⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⣀⣤⡴⠾⠛⠉⠀⠀⣠⠎⣿
⠀⠀⠀⣀⡤⠒⢹⡆⠳⢦⣤⡈⠉⠛⠻⣷⠀⢹⣤⣤⣄⣀⣀⣀⣿⣀⣴⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣴⠶⠞⠛⠉⠁⠀⠀⢀⣠⡴⠛⢁⣼⡛⠳⢤⡀
⠀⣴⠞⠉⠀⢀⣾⠻⣄⠀⠀⠙⢦⣀⠁⢸⡇⣼⠁⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⡛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠁⣀⣠⡴⠟⠋⠁⢀⣤⠞⠉⣷⡀⠀⠙⢦
⣾⠁⠀⠀⠀⢨⢳⡀⠈⠳⢦⣀⠀⠈⠋⢻⠁⣿⠒⠶⣤⣄⣀⣤⣤⣤⣤⣄⣀⣀⣀⡁⣀⣀⣀⣤⡤⠶⠶⠖⠛⠛⠛⠋⠉⠁⣀⣠⣤⠾⠋⠁⢄⣰⢿⡇
⢿⡄⠀⠀⠀⠸⣆⠙⢦⣄⡀⠉⠛⠷⣶⡇⠀⢿⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⠟⠛⠉⠀⢦⣠⠴⠛⢁⣼⠇
⠈⢻⣄⠀⠀⠀⠙⢷⣄⡀⠉⠓⢦⣄⠸⣧⣤⡿⠛⠿⠿⠶⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⠶⠶⠿⠛⠛⠉⠉⠀⠀⠀⢀⣀⣤⠖⠋⠁⢀⣤⠾⠋⠀⠀⣠⡾
⠀⠀⠈⠻⢦⣀⠀⠀⠙⠻⣶⣤⣀⠈⠙⠢⢤⣀⣀⣘⣂⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣀⡤⠔⠚⠛⠛⠛⠛⠋⠉⢀⣠⣴⠾⠛⠁⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠉⠳⠶⣤⣀⡀⠉⠙⠻⠷⣶⣤⣤⣄⣀⣀⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣀⣀⣀⣤⣤⣴⠶⠟⠋⠉⢀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣦⣤⣤⣈⣉⡙⠛⠛⠛⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠛⠛⠛⠋⣉⣉⣠⣤⣤⣶⡶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⢭⣭⣛⣛⠛⠛⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠛⣛⣉⣩⣥⠴⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⠛⠛⠉⠉⠁