Active Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠁⠀⠁⠀⠀⠀⢰⣠⣄⠖⣀⣀⣴⠀⠒⠀⢠⢂⡠⠒⣶⡂⢆⡀⠀⠒⢀⠐⠀⠀⣲⠀⣆⠐⠂⣲⡐⢀⡆⢰⠀⡆⠀⡴⠀⡰
⠀⠠⣄⠘⣃⡄⠀⠀⠀⠁⠈⠀⠀⠈⠉⠁⠁⠀⠈⠂⠈⠂⠁⠂⠀⠁⠀⠀⠈⠐⠁⠀⠊⠁⠉⠈⠂⠉⠈⠀⠀⠉⠀⠑⠀⠉⠀⠀⠀⠀⠀⠀⠋⠀⠶
⠤⠤⠤⠿⠴⠤⠤⠤⠤⠤⠤⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠦⠤⠤⠤⠤⠤⠶⠤⠦⠤⠤⠴⠤⠤⠤⠤
⠀⠀⠀⠀⠀⡀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⠀⠀⠀⠀⣠⠤⣀⠀⠀⠀⢀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⣄
⠀⠀⢀⡄⠉⠀⠀⠀⠀⠀⠁⢆⠀⠀⢀⠤⠀⢄⠀⢠⣏⠀⠀⠀⠱⠀⠀⠸⠿⠀⠈⠑⠂⠒⠁⠀⠀⠁⠀⠀⠀⠀⢠⠂⠁⠀⠂⠀⠈⠀⠀⠀⠀⠑⡀
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⠀⠀⠀⠀⠈⠉⡠⠀⠒⠁⠀⢬⡷⠀⠀⢼⣦⠄⠀⠀⢀⡠⠂⠠⠤⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⢂⡀⡰⠃⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⠁⠀⠸⠀⠀⠇⠰⢀⡀⠀⠀⣀⠐⠀⢀⠂
⠀⠀⠑⡄⠀⠘⠃⠰⠇⠀⠀⢂⡀⠀⠀⣀⠜⢣⡀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠑⠆⠑⠀⡼⠒⠋⠀⠀⠉⠀⠰⠀⠉
⠀⠀⠀⠘⡀⠀⠀⠀⠀⣶⡦⢄⠈⠀⠈⠀⢀⡰⠃⠔⠀⠀⠀⠀⠀⠄⠀⣀⣠⣄⣐⣀⠀⠄⠀⠀⢀⠔⠀⡰⠀⣀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠜⠀⡤⠊⢉⠠⠀⢸⠀⠀⠀⡄⠑⡄⠀⠀⠀⠀⡀⠠⠄⣀⠴⠂⠉⡤⠒⠉⠰⠁⠀⠈⠒⠻⠟⣀⠐⠀⠐⢦⡀⢀⠄
⠀⠠⠚⠂⣆⠀⠀⢀⠀⠀⡈⠀⢘⠀⠀⠃⠀⠈⠁⠢⣀⢀⡀⠀⠱⢄⣀⡠⠊⠀⠀⢠⠉⠲⠄⢸⠀⠀⠀⠐⣄⠀⠀⡀⠔⠘⠋⠒⡄⠀⠀⠁⠁
⠀⠁⠤⠔⢸⠀⠀⠀⠀⠀⠡⣀⠘⢄⠀⢣⣀⡔⠁⠀⠀⢀⠀⢀⡴⠖⠒⠂⡄⠀⠀⢀⠀⠀⠸⠀⠉⠐⠀⠀⢢⠉⠁⠀⠀⠀⠢⠤⠂⣠⡄⡏⠉⠓⡄
⠀⠀⠀⠀⢨⠀⢠⠖⠀⠑⠀⠈⢣⠀⠀⣀⡉⠀⠀⡀⢀⠉⠀⡎⠀⠀⢱⠀⠈⢦⡀⠈⠓⠚⣁⠔⠂⠒⠒⢄⠀⠄⠀⠀⡔⠀⠂⠀⢏⣙⣁⠈⢢⠀⠘
⠀⡠⠂⠒⡄⢉⠖⠉⠑⠄⠘⠀⠎⠀⠊⠀⡀⠀⡰⠀⠀⡷⠀⠐⣀⡰⠂⠠⠒⠒⢏⠐⠒⠊⠁⡠⠈⠉⠢⡀⠱⠈⠢⠤⠃⠀⡈⠀⠀⠀⠀⠉⠁⠀⠀⡆
⠀⠑⢄⣀⡟⠁⢀⠀⢀⠈⠢⡀⠀⠈⣥⠔⠁⡠⠃⠀⢀⠓⡐⠀⠉⠀⠁⠀⠀⠀⠀⠁⢤⡀⣠⠁⠀⠀⢀⠇⠀⠗⠀⢀⡀⠚⣁⠀⠀⠀⠀⠁⠀⠀⢠⠁
⠀⠀⠀⡰⠁⠀⠀⠀⣌⠉⠂⠀⢈⡡⠤⣌⠉⠀⠀⡠⠎⡀⠃⠀⠀⠀⠐⠂⠐⠂⠀⠄⡈⠁⢹⠀⠒⠈⠁⠀⡴⠂⡔⢁⠄⠉⠻⠁⠀⠄⠀⠀⠀⠔⠁
⠀⠀⡜⠁⠀⠀⠀⠀⠿⠁⠙⠀⡏⠠⣶⠄⠀⠒⠈⢡⠊⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠠⠀⠚⡀⠀⢀⠊⠁⠀⠀⠀⡄⠀⠀⠀⠀⠘⠦⠀⣀⣀⠈⠂
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠃⡀⠈⠂⡀⠀⠀⠀⢱⠀⠀⠆⠀⠀⠙⠋⠘⠛⠀⠀⠠⠀⠀⠈⠑⠢⡕⠢⡀⠀⢣⠈⠒⠀⡄⠀⠀⠁⠐⠄⠀⠽⠆
⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠆⠀⢰⠀⢀⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠀⠀⠀⠀⠀⡸⠀⠀⠀⢀⠤⠄⠠⠌
⠀⠀⠀⢱⠀⠠⠤⠒⠂⠀⠀⢠⠀⠀⠀⠀⠸⠀⠁⠀⠀⠀⠀⠀⠛⠀⠀⢀⠀⠀⠀⠛⠀⠀⠀⠰⠿⠗⠈⢠⠃⠀⠀⢰⠀⠨⡁⢀⠄⠴⠓⠂⢄
⠀⠀⠀⠈⡄⠀⠀⠀⠀⣀⢤⡠⠤⠄⣀⠠⠃⢀⡀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⣀⣀⠔⢋⠁⠐⠂⠸⠀⣀⠤⠤⠤⢀⢣⠀⠀⠡
⠀⠀⠀⠀⠈⠁⠒⠀⠉⠀⢈⠀⠀⡠⠃⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢰⠀⡃⡁⠘⠢⠀⠀⠀⠀⠈⠛⠁⠀⠈⠒⠒⠰⠁⠰⠀⠀⠀⠀⠀⢰⠉⠈⠁
⠀⠀⢠⠔⠂⠀⢈⡈⣀⢀⡠⢖⠋⠀⠎⠀⠲⣀⠜⢀⠀⠀⠀⠀⠀⢸⠀⡥⡄⢘⡁⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠀⡖⠢⡄⠓⠀⡁⠀⠁⠀⢄⣀⣀
⠀⠀⠈⢄⠀⠀⠸⠀⠠⠀⠥⠈⠤⡀⠈⠉⠉⡃⠠⠿⠃⠀⠠⠀⠀⢨⠀⡆⠀⢨⣁⠀⠀⠄⠀⡰⠃⢀⠆⠉⠁⠀⣡⣤⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠰⠀⠀⠀⠀⣰⣦⠀⠀⢀⡀⢰⣦⠀⡠⠄⠶⢄⡀⡇⠀⠀⠀⠀⢉⠉⠀⠀⠀⣀⣤⡀
⠀⠀⢀⠚⠀⠈⠀⠀⠀⠀⠀⠀⢰⣶⠔⠀⡸⠀⠀⡇⠀⡤⠂⠉⠀⠀⠀⠹⠏⠀⡹⠸⠀⢠⡄⠀⠐⢵⢄⡀⠀⠀⢸⠀⠀⠀⠘⠈⠡⠠⠄⠀⠀⠀⠸
⠀⠠⠁⠀⠀⠀⠀⠙⠃⠚⠁⠀⠀⠀⡠⠘⢀⡠⠖⠀⠘⠀⠀⠀⡠⠄⠊⠁⢀⠜⠁⠀⠇⠈⡇⠀⠀⠀⡆⠙⠄⠀⠸⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡰⠁
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⡜⠀⠀⡤⠀⠀⢯⡀⠀⡁⠀⠀⠀⠘⣄⠀⠀⡀⠀⠱⡀⠀⢠⡅⠀⡀⠀⠀⠣⡀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⣎⠀⠀⢀⡴⠉⡆⠱⣄⠀⠀⠀⠀⠑⠀⣀⠀⢀⣉⠐⠁⠀⡐⠀⠀⢀⣀⣈⠒⠠⠤⠄⠒⣀⠄⠒⠠⡀
⠀⠀⠑⠒⠒⠒⠂⠠⣀⠀⠀⠀⠀⡆⠀⠐⢄⡈⠁⠈⠉⠠⢀⢃⠀⠈⠑⠒⠒⠒⠈⣡⠐⢠⡀⠀⢣⠀⢰⠀⠀⡐⠁⠀⠈⡆⠀⠔⠉⠙⠋⠰⡀⠀⢰
⠀⠀⢀⠒⠉⠈⠁⠢⡈⠑⠠⠄⠰⢁⣀⡀⠀⠈⠢⠀⠇⠀⠈⢋⠃⣰⣆⠐⠆⠀⢰⠁⠀⢀⡅⠀⡘⠀⢠⠀⢠⠁⠀⠀⢀⠇⠠⡀⠀⠠⡀⠀⡠⠀⡘
⠀⠀⣄⡄⠠⡀⠀⠀⠈⠑⠒⠂⡉⢀⡀⠈⠢⡀⠀⡇⠈⠑⠢⡈⢆⠉⣹⠀⠐⠄⠀⣦⡄⠋⠀⢠⠁⠀⠈⠦⢘⣀⢀⠔⠃⠀⠀⠿⠃⠀⠈⠉⠀⡐⠁
⠀⠀⠛⣄⠀⣨⠀⠀⠀⠀⣶⠎⠀⠀⠈⠰⡀⠂⠀⡇⢀⠤⠀⢾⢄⠁⠀⠀⠀⠀⠀⠀⠀⢢⡀⢸⠀⢀⠤⠤⡀⠋⡀⠀⢀⣲⡖⣖⣲⣒⣖⠀⡐
⠀⠀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠰⠠⠄⠠⠌⠀⡗⠀⠀⣤⠀⣤⠀⠀⠀⠈⠆⠱⣇⠀⢀⡐⠀⢠⠀⠀⠉⠁⠉⠉⠉⠉⠀⠇
⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠶⣦⠀⠀⠀⠸⠀⡇⠀⠑⣄⠀⡤⠊⠀⠀⡀⢉⠁⡉⠀⠀⡀⠀⠘⠀⠩⡈⠁⠀⢀⡘⠀⠀⠀⠓⠛⠒⠺⠀⠀⠰
⠀⠀⡠⠃⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⠀⠀⠈⢂⠑⠀⠒⢂⠀⠈⠁⠁⢠⣤⠀⠠⠚⠀⠀⠈⠉⣉⡁⠀⠀⠀⠀⠘⠗⠺⠂⠀⠀⠀⠦
⠀⠘⡁⠀⠒⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⡰⠀⠀⠠⠀⠀⠀⠸⠀⠀⡴⠃⠀⠀⠀⠀⠈⠹⠀⠣⠀⠀⠤⠐⠉⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖
⠀⠀⠈⠒⠈⠉⠉⠒⠤⠄⠀⠀⠀⠴⠊⠀⠀⠀⠀⠁⠒⠊⠀⠀⠀⠑⠂⠤⠤⠤⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠀⠀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣉⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠐⣊⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠒⠉⠉⠁⢇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢍⡑⠒⠒⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⣭⠭⠭⠙⠀⠀⠀⠀⠀⠀⣀⢤⣒⠬⣛⡏⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⢀⣀⣣⠃⠀⠀⠀⠀⣀⡠⣔⡪⠝⠊⠁⣠⣾⠗⢹⣃⣀⣀⡀
⠀⠀⠀⠀⡠⢒⠭⢒⠈⠉⠉⢑⣶⠭⡲⢄⢤⡲⠍⠒⠉⣿⠀⠀⡠⡪⢚⠥⣒⠭⡵⡖⠒⠒⢭⣕⠦⡀
⠀⠀⢀⢎⠜⠁⡀⠀⠀⠀⢀⣾⠏⠀⠈⠢⡣⡀⠀⣀⠤⢿⡠⡪⠊⡴⡡⠊⠀⠀⢧⢣⠀⠀⠂⠀⠙⢮⢢
⠀⠀⡎⠞⠀⠀⡈⠢⠀⢀⣾⡏⠀⠀⠀⠄⢳⢡⠎⠀⣀⡀⠈⠆⡘⡼⠀⠀⠁⢄⠘⡾⡀⠀⠀⠈⠀⡀⢧⢡
⠀⢸⢸⠄⠀⠀⠀⠁⢰⡞⢻⣴⣾⠶⠦⠬⢽⣼⠓⢴⡷⠃⠀⡼⡃⡁⠀⠀⠁⠀⠀⡶⢷⡀⠐⠀⠁⠀⢀⡎⡆
⠀⠈⡜⣄⠀⠀⠀⠀⠀⠉⠋⠠⡀⠀⠀⠀⢰⢰⠉⠢⡸⠖⠊⠀⡇⡅⠀⠀⠀⠀⡈⠛⠚⢃⠂⠀⠀⠀⢀⢇⠃
⠀⠀⠱⡱⡀⠀⠀⠀⠈⠀⠀⠂⠈⠢⠀⣠⢣⠃⠀⠀⠀⠀⠀⠀⠱⡹⡀⠀⡀⠈⠐⠀⠀⠀⠁⠀⠀⢀⠞⡜
⠀⠀⠀⠘⢮⡣⢄⡈⠀⠀⠀⠀⣀⡠⢞⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⠦⡀⠠⠀⠀⠀⠀⠀⢀⡱⣪⠎
⠀⠀⠀⠀⠀⠈⠐⠪⠭⠬⠥⠭⠕⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⠕⣒⣒⣒⣒⡪⠕⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠛⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠛⠛⠛⠛⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠉⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁⠀⠘⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏⠀⠀⠀⠈⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠈⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠂
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣘⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣙⣃⣀⣀⣀⣀⣀⣀⣀⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠋

⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢠⣦⣦⠀⠀⠀⠀⣀⢠⣤⢀⡀⠀⠀⠀⠀⣤⣠⠆⠀⠀⠀⢤⣠⣄⡤
⠀⠀⣼⣿⣷⠀⠀⠀⠀⠰⣿⣿⠆⠀⠀⠀⠐⣾⣿⡷⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠉⢻⣿⠉⠀⠀⠀⢀⡴⣾⡏⠀⠀⠀⠀⠀⣿⡟
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⢿⡀⠀⠀⠀⠀⢠⠏⢧⠀⠀⠀⠀⠀⡾⠹⡄⠀⠀⠀⠀⢰⠟⢧⠀⠀⠀⠀⢀⡟⢻⡀
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠈⠀⠈⠁
⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⢠⡄⠀⠀⠀⢀⣀⣰⠆⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣀⣶⢀⡀
⠀⠀⢛⣿⠑⠀⠀⠀⠀⠐⣿⡛⠆⠀⠀⠀⠀⣼⣿⡶⠇⠀⠀⠀⢨⣿⠿⠖⠀⠀⠀⠘⢿⡟⠋⠀⠀⠀⠐⢻⣿⠳⠂⠀⠀⢀⠚⣹⣟⠋
⠀⠀⠞⠙⠧⠀⠀⠀⠀⠠⠟⠹⠄⠀⠀⠀⠴⠻⠏⠙⠂⠀⠀⠠⠞⠽⠁⠀⠀⠀⠀⠴⠋⠹⠆⠀⠀⠀⠠⠞⠸⠃⠀⠀⠀⠈⠙⠋⠉⠧
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣠⣿⣄⠀⠀⠀⠀⢀⣼⣧⡴⠀⠀⠀⠀⠀⣰⣦⡖⠀⠀⢠⣄⣠⣿⣀⡄⠀⠀⠈⢷⣷⠞⠀⠀⠀⠀⣠⣿⠞⠁⠀⠀⠀⠀⣼⣥⠆
⠀⠐⢫⠿⡟⠂⠀⠀⠀⢋⡟⢷⠀⠀⠀⠀⠀⢋⣿⣧⡀⠀⢀⣠⠴⠿⣇⠀⠀⠀⠀⠀⣸⢿⣄⠀⠀⠀⠈⢹⣷⡄⠀⠀⠀⠀⠸⡿⣦
⠀⠀⠛⠀⠙⠀⠀⠀⠀⠘⠁⠋⠀⠀⠀⠀⠚⠋⠀⠚⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠐⠃⠘⠁⠀⠀⠀⠀⠚⠘⠁⠀⠀⠀⠀⠘⠃⠁
⠀⠀⠀⣤⠀⠀⠀⠀⠀⢰⣠⣄⡆⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢦⣴⣤⠆⠀⠀⠀⢰⣠⣦⡰⠀⠀⠀⠀⠠⣦⡴⠀⠀⠀⠀⢢⣰⣆⡖
⠀⠠⣴⣿⡦⠄⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⣹⣾⠃⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢾⣿⡁⠀⠀⠀⠀⣀⣻⡏
⠀⠀⢰⠟⡇⠀⠀⠀⠀⠸⠏⡽⠃⠀⠀⠀⢀⡼⠻⣿⠇⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠰⠿⠃⠀⠀⠀⠀⠀⢠⠇⠟⠀⠀⠀⠀⠉⠛⠃
⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉
⠀⠀⣀⣰⠆⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣠⡦⠀⠀⠀⠀⢀⣀⣴⠀⠀⠀⠀⠀⣀⡶⠀⠀⠀⠀⠀⣀⣀⡶⠀⠀⠀⠀⢀⣀⢀⣄
⠀⠀⢡⣿⡙⠁⠀⠀⠀⢈⣿⢷⣺⠂⠀⠀⠘⢹⣟⠷⠂⠀⠀⠀⢫⣿⡛⠃⠀⠀⠀⠟⣻⣛⠛⠀⠀⠀⠘⢫⣿⡷⠆⠀⠀⠀⢻⣿⡿⠿⠃
⠀⠰⠋⠸⠇⠀⠀⠀⠠⠞⠽⠉⠉⠀⠀⠀⠤⠞⠽⠀⠀⠀⠀⠀⠾⠁⠻⠀⠀⠀⠀⠀⡝⡍⠀⠀⠀⠀⢠⠟⠽⠃⠀⠀⠀⠰⠾⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⣤⢀⡄⠀⠀⠀⠀⢠⡄
⠀⡤⢤⣼⡧⠤⠀⠀⠀⣤⣿⣶⡦⠀⠀⠀⠀⣤⣿⠃⠀⠀⠀⠀⠤⣾⡷⠆⠀⠀⠀⠀⣨⣿⠃⠀⠀⠀⢠⢾⣿⠉⠀⠀⠀⠀⠠⣾⣷⡄
⠀⠙⢻⡏⠀⠀⠀⠀⠀⠙⡟⠁⠀⠀⠀⠀⠀⢹⣟⡦⠀⠀⠀⠀⠐⠛⡿⠀⠀⠀⠀⠘⣹⢛⡦⠀⠀⠀⠀⢈⣿⠆⠀⠀⠀⠀⠘⠛⠿⠃
⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠈
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⡄⣠⠀⠀⠀⣠⣀⢠⡄⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢶⠀⡀
⠀⠠⢴⣾⠧⠆⠀⠀⠀⡶⣾⠏⠁⠀⠀⠀⢷⣾⠟⠛⠋⠀⠀⠀⣴⡟⠻⠓⠀⠀⠀⢀⣤⣯⣤⠆⠀⠀⠀⡴⣿⣆⡀⠀⠀⠀⠺⣿⣏⠁
⠀⠘⠻⠛⡆⠀⠀⠀⠀⢠⣟⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡿⢻⡄⠀⠀⠀⠀⣨⡼⠻⢦⡀⠀⠀⠀⣀⡿⣄⠀⠀⠀⠀⠀⣿⢸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠀⠐⣾⠊⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⢿⢆⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢅⠀⠀⠀⠀⢠⠃⢸⠘⣆⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠃⠀⢸⠀⠘⡆⢠⣊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢠⠃⠀⠀⢸⠀⠀⠘⡄⠀⢉⡝⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⢀⡠⠤⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⣀⣠⠃⠀⠀⠀⢸⠇⠀⠀⠸⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠈⠢⠁⠒⢲⠠⡆⠐⠒⡡⠔⠁⢠⠋⠀⠀⠀⠀⠸⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⢘⠄⠀⠀⠀⠀⠀⠀⠀⢅⠀⠘⠀⡇⠀⡎⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠰⣦⠤⠤⠤⠼⠆
⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⢨⡇⢀⠎⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠑⠒⢄⠀⠀⠀⣀⢠⠊⠀⠸⠈⠇⠈⣄⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠳⡱⡀⠀⠀⢀⠜⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⠃⠰⢏⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠠⢤⠠⠤⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⣀⣀⠤⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠉⠀⣰⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠙⠢⢄⡀
⠀⠀⠀⣀⡀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡴⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢈⣑⠢⠤⣀
⠀⠀⠸⣀⡨⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⣀⠤⠊⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣀⠇⠀⠀⣉⡱⠂
⠀⠀⠀⣸⣊⣁⣀⣀⣀⡀⠀⠀⠀⠀⣀⣠⣴⡮⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣓⣧⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠑⠒⠒⠒⠒⠒⠒⠒⠒⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡽⡥⢤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⣀⣾⡑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠻⠀⡔⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠗⠟⡠⠐⠉⢁⣀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠱⢆⡅⠀⠀⢐⡁⢊⣉⡴⠒⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣁⢺⡏⠀⠀⠘⠁⠀⡨⣧⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠑⠊⠀⠁⠀⠐⠐⠉⠀⠀⠈⠑⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠉⢀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⠠⡀⠤⠤⠚⠟⠉⠉⠀⠈⠀⡄⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢎⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⠀⣠⡀⠀⠀⠀⢀⠳⣄⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⣿⢠⡅⣠⠈⠲⠄⠄⢸⠀⠈⠉⠐⠒⠈⠀⢀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠘⠿⠘⠂⠀⠀⠀⠀⠀⠀⠉⢂⡀⠀⢀⠀⠠⠒⠲⡦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⡆⠀⠀⠀⠈⢌⢆⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠈⠁⠀⡷⠷⠘⠲⠉⠢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠌⠈⠸⢢⡀⠀⠀⠀⠙⠃⠀⢀⠱⠀⠀⣀⡀⠀⠤⠀⠖⠈⠉⠒⠄⡀⠠⢴⣷⠦⣀⠰⠂⢀
⠀⣠⠔⠀⠒⡒⠒⠒⠄⢄⣺⣂⡗⠂⠧⢄⡀⠑⠄⠀⠀⠠⠀⠀⠰⡀⣳⡜⣣⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠁
⠀⢏⠀⠀⠀⠑⢘⣿⣹⠁⣿⠃⠻⡀⠀⠀⠀⠑⢦⢢⠀⠃⠀⠀⠀⣇⣰⡍⢠⣀⣀⣤⣄⣀⣀⡀
⠀⠀⠂⣀⣤⣥⣜⣻⣷⣷⣿⣧⣀⠈⠀⠀⠀⠀⢘⣮⣼⣄⠀⣀⣀⣬⡷⠶⠒⠻⢍⠋⠳⠶⠿⢪
⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⡟⠛⠚⠚⠋⡀⠉⡆⢈⣾⣶⣶⣤⣤⣤⣤⣤⣤⣼⣿⣶⠄
⠀⠈⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣿⠄⠀⠀⢰⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⡄⠀⠀⠀⠰⣷⡟⠀⢠⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣭⣿⣧⠀⠀⠀⣀⠀⠀⠀⠀⠀⣻⣿⣿⠟⠀⠀⠈⣻⣿⣛⣂⣠⣽⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⡶⠖⢀⣼⣿⣇⠀⠀⠀⢨⣿⣿⣿⣿⠄⠀⢠⣿⣿⣿⣭⢶⣿⣿⣿⣿⡶⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡷⠟⢈⣿⣿⣟⡃⠀⢀⣘⣾⣿⣿⢏⠀⠀⢤⣽⣿⣿⣥⣌⣻⣿⣿⣷⣟⡤⢀⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⣿⣿⣿⡷⠮⣿⣿⣿⡇⠀⠰⢿⣿⣿⣿⣿⠆⠀⢚⣿⣿⣿⣽⣷⣿⣿⣿⣿⣿⡷⢪⣿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣶⣿⣿⣿⣿⢀⣾⣿⣶⣤⣙⡛⠿⠀⠈⠛⣿⣿⣿⠯⠞⠻⣿⣿⣯⣷⡤⢻⣿⣿⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣉⣹⣿⣿⣿⣿⣿⣿⡿⢡⢋⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣷⣌⠻⡿⠿⠿⢿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⣿⣿⣿⣿⣿⠟⡰⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠳⡘⠿⠿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿⢋⡜⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢦⡙⢿⣿⣿⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⢣⢎⡜⣰⣿⣿⠇⣿⣿⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠙⢎⠻⣿⣿⡿⠖
⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⡟⡱⢃⡞⣡⣿⣿⣿⠀⣿⣿⣿⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢦⡱⣌⠻⡿⠓
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⡴⣡⠏⣴⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢟⣿⣿⣿⣧⡙⢎⠳⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢃⠞⠐⣡⣾⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣦⠁⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢠⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣴⠶⡤⢈⡉⠉⠉⠙⠉⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⢠⡴⢶⣄
⠀⠀⠀⠀⠀⠀⢼⣇⠀⠀⣾⠁⢠⡗⣴⡄⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠘⠇⠀⠀⠀⠀⢀⡄⠀⢰⣶⡀⣿⣇⢻⡀⠙⣧
⠀⠀⠀⠀⠀⠲⣤⣽⡇⣼⠇⠀⡾⢡⡿⣧⣾⡿⣸⣿⢠⣿⣇⣿⠒⠂⣴⠶⣆⠀⠀⠀⢠⡟⠙⠁⠀⢸⡿⣄⠀⣿⡻⡏⢹⡎⢷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠾⠃⣾⠁⠋⣸⡇⣿⢹⡟⣿⢸⣏⣀⠀⣿⠀⢹⡆⠀⠀⢸⡇⠀⠀⠀⢸⣧⣿⣦⠸⣧⠀⠀⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣶⣦⣤⣀⠀⠁⠀⠀⠛⠸⠇⠀⠀⡿⢸⣏⣉⡁⣿⠓⣿⡁⠀⠀⠈⠳⠦⠶⠂⠻⠁⠀⠉⠁⣀⣤⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⣿⣿⣿⣷⣯⣿⣿⣿⣿⣿⣾⣻⣻⣿⣿⣟⣍⣷⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⠾⠿⠿⠿⠿⠿⠿⢷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄
⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆
⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠉⣼⣷⡀⠀⣀⠙⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠿⠛⠋⠁⠀⠀⠀⢿⡿⠃⣸⣿⡇⠀⠈⠛⢿⣶⣄⡀⠀⠀⠀⠀⠀⠸⣿⡀
⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣀⣀⡀⣻⣇
⢸⣿⣀⣀⣠⣤⣤⣤⣶⣶⡶⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣿
⣿⡟⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡾⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠿⠿⠶⣶⡆⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⣿
⣿⡇⠀⠀⢀⣿⣷⠀⠀⠀⠀⠈⠁⣀⣠⣤⣶⡾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⣿
⣿⡇⠀⠀⣸⡟⣿⡇⠀⠀⠀⠀⣾⡟⠋⠉⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠉⠙⣿⠆⠀⠀⠀⠀⠀⢿⣷⣿⠀⠀⣿
⢸⣷⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠘⢿⣦⣀⣀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⣀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢰⣿
⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⢠⣶⠀⠀⠀⢰⡆⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣼⡏
⠀⢻⣷⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⢰⣿⠁
⠀⠈⢿⣆⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠃⠀⢀⣿⠇
⠀⠀⠈⢿⣆⠀⠀⠀⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣷⣦⣾⠿⣷⣦⣴⡿⣷⣤⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠈⠉⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠷⣶⣶⣶⣶⣶⣶⣶⡶⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠴⢶⣶⣶⣶⡶⠶⣤⣄⡀⠀⠀⣴⢻⡀⠀⠈⠉⠉⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⡾⠋⠉⠉⠙⢿⣦⠀⠈⠙⢦⣀⣿⣈⣧⣀⣀⣀⣀⠀⢸⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠹⣧⢀⡤⠚⠉⠁⠀⠀⠀⠀⠉⠉⠉⣉⣉⣩⣭⣍⣻⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⡠⠔⣚⡭⠵⠒⠋⠉⠁⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠁⣾⠁⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⢀⠔⢊⡡⠖⠋⠁⠀⠀⠀⠀⢀⣠⠴⠒⠒⠉⠉⠉⠉⠙⠓⠶⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⣠⠞⣡⠔⠉⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⡴⠁⡴⠁⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣷⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⡼⠁⡼⠁⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣿⠀⠀⠀⢰⡏⠀⠀⠀⠀⢰⠁⡸⠁⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⠀⣾⠁⠀⠀⠀⠀⡏⢰⠃⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⠀⠀⢰⡏⠀⠀⠀⠀⢰⠁⡏⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣠⡄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡀⠀⣾⠃⠀⠀⠀⠀⡌⢰⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠾⠭⠭⠭⠭⠽⠶⠿⠟⠛⠛⠛⠛⠒⠛⢻⢿⠃⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠃⡸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⡇⢰⡏⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡜⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣾⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⡇⢸⡇⠀⠀⠀⠀⡿⠀⡇⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⣧⠸⡇⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠙⠷⣇⠀⠀⠀⠀⡆⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⠤⠴⣶⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⡏⢹⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣧⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⠀⠈⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣤⠾⢤⣄⣀⠀⢀⣀⣠⣤⠤⠤⢤⣤⣤⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠐⠀⠘⠀⠀⠂⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⢀⢀⣆⠀⡀⢠⠀⠀⠀⠀⣀⠀⠈⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼⣦⢀⣄⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⣿⣿⣶⣷⣾⡇⠀⠀⠀⣿⣸⣆⡆⢸⣧⣿⡄
⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠠⠰⠿⠷⠼⠿⠿⠋⠼⠿⠦⡄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠉⢿⣿⣿⣿⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⣀⣿⣿⣷⡄⣤⠀⠀⠀⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⢰⣶⣹⣖⡄⣤⣠⣦⣀⡀⠀⠀⠀⠀⠀⠀⠸⢛⣿⢻⠿⠖⠖⠺⠿⠿⢛⣻⠀⠀⠀⠀⣀⡀⣀
⠀⠀⡼⠉⡍⣉⢠⣠⡀⡴⠛⡿⣟⡇⠀⠀⠀⠀⠀⠀⠀⢀⢈⠉⢀⡀⣈⡉⠁⠉⠀⠀⡄⠀⠀⠀⠀⠠⠌⠧⠬⠤⠤⠴⠶⠤⠤⠬⠼⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠦⠐⠓⠛⠛⠛⠁⠇⠀⠛⠓⠧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠂⠀⠐⠐⠂⠀⠀⠀⢀⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢻⣿⠉⠁
⠀⠀⠀⠀⠠⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠘⠛⠀⢠⠄⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⡀⣰⣿⡿⢻⣷⢭⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⠀⠀⠠⠤⠀⠤⠄⠀⢠⠶⠟⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⣽⣿⣤⣿⣿⣩⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣠⣿⣿⣿⣿⠆⢴⡇⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂⠄⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡹⣧⠀⠘⠃⠘⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠻⡛⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠛⡛⠛⠋⠀⠀⠀⠀⠀⠀⣀⣀⣿⡟
⠀⠀⠀⠻⣟⡮⢿⣿⣿⣻⣴⣻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⠆⣄⡤⣀⡄⠴⡮⠑⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⡁⢀⠀⠀⠀⠠⠀⠀⢸⣿⣿⣿⣿⣄⡤⠆
⠀⠀⠀⠀⠀⠑⠚⠂⠙⠛⠉⠁⠀⠀⠀⣤⣶⣄⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⢸⡄⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⣴⠀⠀⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⣤⣰⣾⡇⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⡇⣤⣀⣷⣿⣇⣤⠀⠀⠈⠉⠉⠉⠙⠁⠉
⠀⠀⠀⢠⡶⣶⣾⣶⢶⣷⣦⢶⡆⠀⠀⠀⠀⠀⠀⢠⡂⣠⣪⡭⡭⣭⣝⣇⣤⢵⡤⡆⠀⠀⠀⠀⠀⢰⣿⡟⠉⠉⢹⣿⠟⢻⣿⢿⣷
⠀⠀⠀⢸⣷⣿⡿⠁⠈⢿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⢀⠀⠉⠁⠈⠀⠀⠛⠀⠂⠀⠀⠀⢀⢪⠉⣍⣏⣙⡟⢫⠈⠇⢉⢉⣙⡆⠀⠀⠀⠀⢠⣷⡀
⠀⠀⠀⢸⣿⡟⠀⣰⣷⡄⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠴⠶⡴⡾⠇⠀⠁⠁⠀⠀⠀⠀⠀⠋⠃⠉⠉⠉⠃⠉⠀⠓⠘⠉⠉⠁⠀⠀⠀⣰⣿⣿⣿⡄
⠀⠀⠀⠈⠛⠋⣉⢉⣉⣉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⣸⣿⠟⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⣆⣦⢰⣧⣿⡀
⠀⠀⠀⠼⠻⠟⠻⠇⠘⠀⣷⣿⣶⣾⣷⡄⠀⠀⣠⢤⠤⠤⣤⣠⠀⣠⠒⢂⣀⢠⣄⢀⡄⣤⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢠⣤⣤⣤⡀
⠀⠀⢰⢶⠀⠃⣰⣄⠀⠀⢰⡆⢰⡄⡶⡆⠀⠀⠻⠿⡿⠿⠿⣷⠐⢧⠰⠧⠿⠏⠟⢹⡿⠔⠁⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡧⠂
⠀⠀⢼⣀⡀⣼⡟⠻⣆⠀⢸⣷⣿⡇⡗⠃⠀⠀⠈⠙⠻⢏⣙⢋⡽⠿⠛⠋⠉⠉⠈⠋⠀⠀⠀⠀⠀⠘⠿⠟⠿⠿⠿⠛⠀⠀⠻⠿⠇
⠀⠀⠈⠉⠀⠈⠀⠀⠀⠀⠈⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠘⠂⠀⠀⠀⠀⠀⠀⣀⣴⣾⣆
⠀⠀⣀⣀⣀⣀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⡆
⠀⠀⣿⣸⣿⣉⣉⣿⡛⢛⡄⢠⣶⣒⣒⣒⣖⣂⣀⣀⡀⠀⠀⣾⣏⣹⣏⣉⣉⣩⡿⢒⠛⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠐⠒⠛⢻⣿⣿⣿⣿
⠀⠀⠿⠿⢿⢿⣿⠿⠿⠿⠿⢈⣿⣿⣿⠿⠿⠿⣿⣿⠇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⢿⣿⠆⠀⠀⢀⣿⣇⠀⠀⠀⣀⠋⡉⣁⡀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠻⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠦⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠈⠉⠀⠈⠉⠉⠉⠉⣠⠁⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡏⠀⠠⣤⣜⡓⠘⠰⣶⣂⣀⣀⠀⢠⠋⡄⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⠃⠀⠈⠉⠉⠀⠀⠀⠉⠉⠛⠋⠁⢸⠀⣇⣠⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣀⣀⠀⠀⢀⠀⢀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⠒⠋⠉⠒⠶⠮⠜⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠢⠤⠚⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡦⡠⠖⠋⠙⠶⠊⡇⠀⠀⠀⠀⠀⠀⠀⡶⢄⠜⠲⢄⡉⠙⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠒⠂⠤⠤⠤⠤⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡼⠉⠉⠉⢹⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⢤⡇⠀⠀⠀⠀⣇⡸⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠀⠀⠀⠀⠀⠓⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⣀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣶⣿⠉⣷⣄⠀⠀⠀⠀⡧⣿⢯⠷⡾⡿⢭⠂⠀⠀⠀⢾⡟⠉⠉⠉⣯⡆⠀⠀⠀⢀⣂⣂⣉⠈
⠀⠀⣿⣤⣿⣻⣏⣸⡆⠀⠀⠀⢈⣿⣯⣤⣼⣿⣁⠀⠀⠀⠀⡱⠆⢸⡷⠀⠶⡇⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠹⢧⣼⣏⣨⠟⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠻⠀⠀⠀⠀⢾⣧⣀⣀⣠⣟⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⣰⠁⠀⢘⡄⠀⠀⠀⢸⠙⠋⠋⢳⠃⠋⠋⢻⡇⠀⠀⣼⣛⣛⣛⣻⡄⠀⠀⠀⡠⠴⠟⠙⠋⠹⢶⣄⡀
⠀⠀⡼⡌⠀⠀⠀⡿⣆⠀⠀⢸⢀⣀⡀⣰⡀⣀⣀⣸⡇⠀⠀⠀⠙⣤⡜⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠘⠚⠒⠒⠒⠒⠚⠚⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠁⠀⠀⠠⠊⠀⠈⠢⠀⠀⠀⠀⠃⠀⠀⠀⠐⠀⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣀⣀⢀⡀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⣀⡠⠤⣲⣷⠀⠀⠀⣴⠤⣤⢦⡤⢶⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠻⡇⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣿⣭⣶⣾⣿⣿⣿⠀⠀⠀⢹⠀⣿⢸⡇⣾⠀⠀⠀⠛⢷⣾⢫⡙⣶⠟⠃
⠀⣴⡇⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠋⡟⠻⢿⣿⣿⣿⠀⠀⠀⢸⡆⣿⢸⡇⡿⠀⠀⠀⠀⢀⣿⡿⢿⣧
⠀⠉⠉⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠁⠀⠈⠉⠀⠀⠀⠘⠧⠿⠼⠶⠇⠀⠀⠀⠀⠚⠁⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠠⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⠀⡠⠄⠀⠀⠀⠀⠠⢀⠀⠈⠑⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⡔⠉⠀⠀⠀⡀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠌⠀⠀⢀⠔⡩⠀⠑⣄⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡠⢚⠒⢤⠀⠀⢰⣀⠠⠔⠁⡐⠀⠀⠀⠀⠉⠒⠄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠊⠀⡀⠀⠀⠑⠠⠎⡀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠚⢄⠀⢀⠴⡒⢄⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡔⠉⢱⡀⠀⠀⢀⣀⠨⠜⠁⠀⠠⡦⠀⠀⠀⡠⠒⠂⢤⡀⣰⡈⠀⢀⣸⡆⠈⠒⠤⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣞⠀⠀⣺⡃⢀⠔⣵⠀⠀⠀⠀⠀⡸⡯⠓⡤⠊⠀⣺⡂⠀⠈⣻⡓⠉⠀⠨⡏⠀⠀⢼⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⣟⠀⢐⢻⡏⠀⠀⣻⣀⡀⢴⡖⠉⠸⡯⠀⠈⣾⣯⡉⠉⠀⠀⠙⠉⠚⢫⣮⡧⠀⠀⢽⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⡯⠀⢀⣸⡁⠀⢘⢽⠁⠀⢸⡇⠀⠩⡯⣁⠜⠹⢿⡙⢦⣀⠀⠀⠀⠀⠀⠹⡧⠁⠀⢹⡆⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠨⡏⡀⠀⠘⢠⠀⢐⣽⢑⠀⢼⡏⠀⠀⡡⢂⠆⠁⡀⢷⠀⠑⠅⡂⠄⠀⠀⠀⠈⢆⡀⠼⠤⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡀⠀⢀⣤⠛⢶⣠⣘⣀⢠⣭⡯⣡⠔⢠⠊⢨⠀⠇⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠄⡑⣄⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢀⣬⠩⡸⢉⢻⡄⠀⠀⠀⠛⠠⠆⠁⢀⡆⠤⠄⠀⠈⣵⠄⠀⠀⠀⠀⠀⠀⠠⡖⠋⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⠘⠭⠦⡄⣥⣺⡹⢠⠀⠙⠃⠀⠉⠀⠈⠀⠒⠒⠀⠤⠥⠤⠤⠀⠐⠂⠀⠀⠈⠁⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠉⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⢀⠠⠔⠒⠒⠂⠤⢀⠀⠀⠀⠀⠀⣀⠔⢉⣀⠀⠀⠀⠈⠂⠄⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡒⠀⠀⠒⠠⢄⡈⠐⠒⠒⢉⡠⠂⠁⠀⠀⠀⠀⠈⢑⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢄⡀⠀⠀⠁⠐⠊⠁⠀⠀⠀⠀⢀⡠⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠀⠉

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

⠀⠀⣀⣤⠤⣄⡀⠀⠀⠀⠀⢠⣺⣭⠷⣠⣚⣯⡷⡄⠀⠀⠀⢀⡤⡤⠤⠤⢤⢤⡀⠀⠀⠀⠀⡀⡀⠄⠄⠀⠂
⠀⣼⠛⣿⣤⡾⠻⡆⠀⠀⠀⠘⠯⣯⠝⠉⢯⣿⠽⠁⠀⠀⠀⠠⣛⠀⢠⣀⠀⣛⡇⠀⠀⠀⢰⣶⣶⣦⣦⣦⣤
⠀⣿⠟⢻⡾⠷⣶⡇⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣴⠀⠀⠀⠀⠐⣭⠀⠘⠉⢨⣭⡇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⠀⠈⠳⠾⠷⠴⠋⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠉⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠃⠀⠀⠀⠛⠛⠿⠿⠿⢿⡟

⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⣬⠀⠀⠀⣣⠀⠀⠀⠀⡇⠁⠁⠁⠑⠁⠁⠁⠁⡇⠀⠀⠠⣯⣭⣭⣭⣽⠄⠀⠀⠀⡤⣶⣿⣁⠍⠠⠿⠷
⢀⣞⡇⠀⠀⠀⢨⢳⡀⠀⠀⡇⡀⡀⡀⡠⡀⡀⡀⡠⡇⠀⠀⠀⠀⢑⣴⡋⠀⠀⠀⠀⠀⢺⣿⣿⣿
⠘⠚⠒⠒⠒⠒⠚⠓⠃⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠔⠁⠀⠉⠢⠀⠀⠀⠀⠓⠒⠒⠂⠉⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣶⠀⠀⠀⢠⡬⣥⡤⣤⡬⣦⠀⠀⠀⠀⠀⠀⢰⣧
⢿⣿⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⢰⣖⣫⣥⣶⣾⣿⣿⠀⠀⠀⠘⡇⢸⡇⣸⠀⣿⠀⠀⠀⠲⣶⣶⠟⡟⢶⣶⠶
⣠⣿⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠘⠻⡿⢿⣿⣿⣿⣿⠀⠀⠀⠀⣷⢸⡇⣿⠀⡏⠀⠀⠀⠀⠀⣽⣾⣿⣾⡁
⠛⠛⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠈⠓⠒⠋⠉⠙⠻⠀⠀⠀⠀⣿⣸⣇⣿⣰⡇⠀⠀⠀⠀⠰⠟⠁⠈⠙⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠟⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠅⠀⢀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⠃⠂⢀⣿⣿⣿⣷⣄⠀⠘⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠾⠿⠿⠿⢿⣟⣛⡓⠚⠿⠭⠭⣭⣽⡀⠀⡟⠙⠛⠻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀⠀⠀⠉⠙⠳⢤⣀⠀⠀⠈⠙⢷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠠⠤⠈⠙⢦⡀⠀⠀⠀⠀⠉⢳⡀⠀⠀⠈⢷⠀⠍⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢠⡤⠤⠶⠶⠶⠾⠛⠋⠁⠉⠑⠒⠒⠒⠀⠈⠉⠃⠀⠀⠙⢦⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⢇⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣿⣟⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠠⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⣿⣯⠅⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠤⠼⢙⡛⠲⠖⣤⠄⣀⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣿⣟⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⠀⠈⠁⠊⢷⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⠀⣿⣯⠅⠠⠀⠀⠀⡤⢤⠠⠄⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⣇⠀⠀⠀⠀⡤⢀⠀⠀⣿⡇⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⠀⠀⠀⣯⣟⠂⠀⠀⠀⢀⡀⢈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢀⠜⠀⠐⠉⡂⠾⣿⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡠⠛⣯⠅⠀⠀⢀⠉⠀⠀⢉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⠊⠀⠀⠀⠀⠀⢠⣿⡆⠘⣷
⠀⠀⠀⡈⢐⠤⡀⢸⣿⠀⢸⠀⠔⠑⠀⢿⠂⠀⡐⠉⠀⡀⠉⠉⠁⡄⠀⠰⢀⠀⣰⠉⡍⠐⢲⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠁⠠⠀⠠⠀⣿⣇⠀⢿⡄
⠀⠀⠀⡆⠀⢀⠐⢿⠻⠀⠀⡌⠁⠀⠀⣸⡅⠀⡇⠀⠀⠀⠐⠀⠀⡇⠀⡆⡠⠦⠬⡤⠇⠉⠑⠠⡀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡐⠁⠀⢹⣿⠀⠸⣧
⢀⡴⢄⠱⠀⠁⠀⠈⢠⠀⠀⠁⠀⠀⢠⠋⠳⡀⡇⡾⢘⡄⠀⠀⠀⡇⢰⡇⠀⠀⢀⠀⢀⠀⠀⠀⠐⠀⠀⡜⠁⠀⠀⠀⢀⢀⠜⠀⠀⠀⢸⣿⡄⠀⣿
⠋⢄⠀⠉⢣⡀⠀⠀⠈⢃⡠⠀⠀⠀⢨⡇⠀⢣⡄⠅⠀⠇⠀⠀⠀⣃⠻⠄⠀⠀⠈⠀⠀⠀⠀⢀⠆⠀⡞⠀⢀⠴⣂⡠⣀⠆⠀⠀⠀⠀⠈⣿⡇⠀⢸
⠀⠀⠑⢠⣊⣈⣒⣄⣀⣀⣡⣀⣀⣄⣤⣥⣤⠾⠷⠶⠶⠶⠶⠖⠒⠓⠚⠚⠒⠶⠶⠶⠦⢤⣤⣼⣰⡆⡅⠀⣸⠁⠀⡡⠈⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸
⠀⠀⢀⣾⠿⠛⠟⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠙⢳⣄⠈⠢⣤⡀⠀⠀⠀⠀⠀⠄⠒⠻⡇
⠀⢠⡿⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠉⢳⡔⠀⣇⡼⠢⡤⠂⡁⠀⠀⢸⡀⢀
⠀⣾⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻⡞⠁⢡⢮⠔⠁⠈⠑⠒⢺⠀⠀⢸
⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣸⠁⠀⠀⠀⠀⢻⣿⡇⠘⠀⠀⠀⠀⠀⠘⡆⢀⣾
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡟⠀⠀⠀⠀⠀⠈⣿⢠⢠⠀⠀⠀⠀⠀⠀⣧⡾⠃
⠘⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⣸⠃⠀⠀⠀⠀⠀⠀⣿⠈⢠⠀⠀⠀⠀⠀⠀⣿
⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢹⠀⠉⠀⠀⠀⠀⠀⢠⣿
⠀⠸⣷⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠹⣧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠄⢀⣾⠋
⠀⠀⠀⠹⣧⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢠⣿⡾⠟⠁
⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣦⣤⣤⣤⣄⣤⣤⣤⣤⣴⣶⣶⣏⣩⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠶⠾⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⢹⠙⠛⠛⠓⠒⠒⠒⠲⠶⠦⠤⠤⠤⠤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿⡆⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣿⣧⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⣿⢸⡀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣠⠏⠀⢀⣟⡿⡏⣇⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⡔⢣⠃⠀⣠⠃⡅⠀⡇⠀⠀⠀⠀⠀⡰⠃⠀⠀⡼⢸⢃⢿⢹⡀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⣸⠃⠈⠀⣰⠏⠀⠀⠀⠳⠄⠀⡄⢀⡜⠁⠀⠀⡸⠁⣟⣈⣈⣇⢧⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⣿⠀⢴⠚⢁⡄⠄⠀⠂⠀⠄⠀⠸⣮⣤⣤⣄⡀⠁⢠⡏⠉⠁⠹⡈⢇⠀⠀⠀⠙⢦⠀⠀⢀⡀⠀⠀⠀⣀⣀⣀⣀⣤
⠀⠀⠀⠀⠀⢻⠀⢀⢸⠉⠀⢀⡔⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠓⠀⠘⠓⠒⠲⠦⢽⣤⡰⠷⠒⠋⠉⠁
⡀⢤⢤⡤⣾⣿⣧⣀⣴⣤⣾⣯⡤⣎⣿⣷⣦⡖⢦⡀
⠳⣬⣿⣷⣦⣤⣑⣾⣿⣻⣯⣟⠋⡿⠿⠿⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠁⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣴⣦⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⣧⣄⠀⡀⠀⢀⣶⣿
⠀⠀⠀⢠⣴⣾⣿⠿⠛⠿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⣾⣿⣯⠋⠉⡁⠈⠹⣿⣿⠃⠀⠀⢀⠔⠛⠻⣆⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃⠀⠀⣿
⠀⠀⠀⠘⢿⣿⡄⠐⠛⠆⢦⡿⠋⠀⠀⠀⠊⢠⡀⢠⣻⠀⠀⢠⣴⡞⠉⠙⠛⠙⠃⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⠟⠁
⠀⠀⠀⠀⠀⠉⣷⢉⢦⣼⣥⠀⠀⠀⠀⠘⠀⠀⢤⠿⣸⠀⠀⢺⣿⡇⠀⠀⠀⠀⢀⣿⣿⠆⠀⣀⡤⠤⠤⢦⣄⠀⠀⠀⠀⠠⠖⣨⡿⠉⠟⠃⠀⠈
⠀⠀⠀⠀⢠⣿⣿⣿⣆⠺⣴⣶⣶⣦⡀⠀⠀⠐⠒⠁⠇⠀⠀⠀⠻⠇⢀⡀⣀⠀⠸⠿⠋⠀⣼⠏⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣬⣥⡤⠚⣞⣶
⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠝⣻⣿⣹⣵⢲⡀⠲⣶⡆⠀⢀⡀⠀⡃⠰⡀⣒⠀⡄⠀⠈⠀⡀⣿⠢⢤⠀⢠⠤⠀⠟⠁⠀⣿⡇⣿⣿⢷⡶⣿⣿
⠀⢸⣿⠟⢋⢓⣾⣿⣿⣧⣤⣿⠟⠋⠁⠸⣿⣦⣽⣏⡀⠘⣷⡀⠀⢀⠀⠭⠀⠀⠖⠀⡠⠻⣜⣷⣾⣤⡄⠀⠀⠀⠀⢰⢁⣼⣋⠿⣤⡽⠿⢾⡇
⠀⠈⠛⠳⣤⣿⣿⣿⣻⣿⡿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⢷⡤⣀⡙⢸⣋⣡⣀⡔⠁⠀⠈⢿⣿⣟⡛⠀⠰⠀⠀⢸⣾⣿⣿⡒⢂⣠⣐⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠁⠀⣄⣄⣰⢿⣿⣽⣿⡿⠿⠋⠉⠀⠀⠀⠙⠁⠁⠀⠀⠀⠀⠺⢿⣻⡿⠿⠀⠀⠐⠈⠁⠀⢹⠿⣿⣿⣿⣿⣿⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⡽⠖⠚⡎⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠹⣿⣭⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⢀⡆⠀⠀⢀⣀⡠⠤⠤⠤⠄⠄⠠⠀⠤⠀⠄⣀⡀⠐⠏⠒⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠐⠊⠉⢀⠀⠠⠄⠀⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠉⠀⠀⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⡇⠉⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⠀⠀⢈⡆⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⢀⠎⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠱⡀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⢀⠄⠀⠀⠀⠀⡠⠀⢔⠁⠀⠀⠈⠡⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠠⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⡄⡀⡄⠀⠀⢢
⠀⠀⠀⠠⠀⠀⢠⠀⢄⠁⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠤⢘⠁⢄⠏⠀⡍⠀⠀⠱⡀
⠀⠀⠀⠅⠀⠀⢘⠀⠈⠓⢡⡌⣁⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⡄⢠⠒⠊⡁⢨⢀⣀⠑
⠀⠀⠀⠀⠀⠀⠘⠂⠀⠰⠒⠃⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⠀⠀⠀⠈⢸⠀⠀⠀⠐
⠀⠀⠀⠀⠁⠀⠀⠄⠀⡆⠀⠀⠀⠁⠀⠀⢀⠄⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄
⠀⠀⠀⠀⠀⠀⡒⠈⠀⡆⠐⢰⠀⠀⠀⠠⠀⠀⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠐⡄⢀⡇⠀⠑⠄
⠀⠀⠀⠀⠀⠀⡏⠂⠂⠐⠒⣸⠀⠀⠈⠀⠀⠀⠈⠶⠅⠀⠀⠒⠂⠀⠀⠀⠀⠀⠐⠒⠂⣁⣀⠀⠘⢎⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢢⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠠⠂⢠⠊⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠇⠈⠴⠁⠀⠀⠀⠀⡀
⠀⠀⡀⠀⠀⠀⠹⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢁⠤⡐⠈⡤⠀⠀⡴⠋
⠀⠀⠀⠑⢄⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠱⠐⠀⠐⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠰⠀⠐⠀⠆⠄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠁⠀⠀⠄⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠘⠁
⠀⠀⡈⠀⠀⠀⠀⠀⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠁⠀⡠⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀⠀⠀⠀⠀⠠⠂
⠀⠀⠐⠂⡀⣄⣤⡄⢠⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢴⡄⠀⠀⠀⣲⠂
⠀⠀⠀⠀⡀⠈⠐⠀⠀⠈⠉⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠉⡔⡁
⠀⠀⠀⠀⠁⠖⣀⠶⢄⡀⠦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⠆⢸⡆⠀⡼⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⢀⠉⠐⠒
⠀⠀⠀⠀⠀⠂⠀⠙⠉⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⡆⢰⠁⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠐⠒⠠⢄⠄⠀⠈
⠀⢰⠋⠃⡇⣔⠀⡀⡀⢤⢠⠀⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠘⠆⠀⠙⠂⠀⠉⠁⠀⠻⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣄⠀⡀⠀⠛⠛⣤⣀⠀⣀⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠠⠀⢀⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⢀⡀⢆⠀⢀⠈⠉⡀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣼⣦⡀⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠈⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⢿⠀⡀⡞⠀⠉⢠⠸⢃⠗⢠⣀⡴⠀⠀⠀⠀⠀⢸⣷⠒⠋⢸⣧⠈⠒⣲⣿⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠘⢀⡀⠈⠀⠁⠀⠠⠄⠀⠀⠀⠀⠀⢐⣞⠠⠯⠬⠅⠡⠌⠋⢅⣲⠚⣄⡠⡴⠃⠀⠀⠀⠀⠀⢻⠙⠀⠀⠀⠀⠀⠀⠈⠀⢣⡀
⣧⣄⣀⠀⡀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠠⠠⠤⠀⠀⠀⠀⢀⠈⠉⠚⠁⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠄⠠⠠⠀⠀⠄⠀⠱
⠙⢿⣿⣤⢿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⢀⠁⢂⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣌⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⣄⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡖⠛⠉⢿⣦⣶⣿⣿⣿⣿⣿⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⢀⣖⣾⣧⠀⢸⣿⣟⡿⠟⠻⣿⣿⢿⣿⣿⣿⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⢀⠀⡀⢾⣿⣿⠄⣠⣾⠏⠻⣷⣷⣶⣂⠙⠲⠄⠘⠻⣗⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠁⢀⣠⡴⠋⠀⣘⠲⢀⣴⣾⡋⢽⠂⠀⢰⣿⣿⡇⠑⠂⠀⠀⠁⡻⣆⠀⠀⣄⢄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⣁⣤⣤⠴⠾⠛⠁⠀⠀⢰⡏⠀⢸⢉⣉⣵⣿⠀⠀⠈⢿⣉⡝⠀⠀⢇⢲⠀⠀⢘⣶⣦⡙⠾⠇
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢻⡏⠁⠀⢰⣧⣤⣤⠾⠛⢿⡏⠓⠻⣿⢿⣿⠇⠀⠀⠀⢸⡟⣍⠁⠀⠈⠉⠀⠀⠿⣧⣻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣗⠛⠉⢷⡀⠀⠈⢳⣭⠅⠀⢀⡀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⣧⠈⠑⠀⠀⠀⠀⠀⠀⣄⣙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⣠⣷⠄⠀⣸⡇⠀⢰⣯⠗⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⠉⠈⢷⣬⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠙⢿⣟⠳⣤⡀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⣠⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠈⠻⣆⠀⠉⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠞⠋⠁⣠⡞⠁⠀⠈⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠉⢙⡷⢄⠀⣼⢿⡛⠋⠉⠁⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⣴⣏⣦⡘⣾⡁⢈⣿⣄⠀⠀⢀⣴⠞⠉⠀⠀⠀⢠⣆⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠃⠈⠛⡿⢺⣿⡭⠔⠻⣧⣰⠟⠁⠀⠀⠀⠀⠀⠀⢹⡄⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣾⡏⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠶⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠴⡾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡞⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⠞⢿⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⡇⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠶⠻⡏⠁⠐⣦⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⣿⣇⡏⠀⠀⠀⠀⠀⠀⢠⣾⣿⣅⣀⣤⣴⡇⠀⠀⢹⡀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⣦⣄⣀⠀⠀⠀⠀⢸⣇⡿⠙⠛⠃⠀⠀⠀⠀⠀⠀⠘⠛⠛⣿⠟⡏⠀⣿⠀⠀⢸⡅⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣶⣤⣬⣭⣁⣠⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣾⠿⢻⠇⠀⢿⠀⠀⠘⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠘⣦⠀⣾⠀⠀⠀⡇⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠘⠛⠓⠏⠀⠀⠀⡿⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠈⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠒⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢸⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⡒⠩⠅⠒⠛⠁⠘⠛⠛⠻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠤⠒⠠⡁⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡷⣮⡻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠔⣉⠳⣜
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇⣿⡿⢮⡻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⠀⣿⠀⢨⠳⣄
⣿⣿⣿⣿⣿⡿⣿⣿⣿⣇⣿⣧⣿⣯⣼⣿⣮⡻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⠔⠁⠀⠗⠀⠚⠀⠸⠂⠈⠳⣄
⣿⣿⣿⣿⣿⣽⣿⣿⡟⣹⣿⣿⣯⡿⣿⣽⣿⡿⣺⣿⣿⣿⣿⣿⠀⠀⠀⠀⠐⢅⠠⠀⣰⣇⠐⢐⣒⣒⠐⡀⠀⡨⠃
⣿⣿⣿⣿⣿⣿⣿⡟⠟⢸⣿⣿⣿⣾⣿⣿⣫⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠑⣴⣽⡐⡡⠀⠈⠁⠀⢡⠚
⣿⣿⣿⣿⣿⣿⣛⣽⣄⣘⢿⡻⠟⢋⣙⣛⣙⣛⣻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⠴⠛⠿⢧⣡⣤⣤⡶⠦⠶⠦⠤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠊
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠛⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣀⠀⢻⡇⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⣼⡟⣁⣀⣠⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠉⠛⠻⢿⣿⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠘⠿⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⣾⠟⠛⠻⣷⣄⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠻⣧⣤⣤⣤⣿⠿⠂⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠐⢿⣿⣍⡀⠀⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⣀⡀⠀⠀⠀⣀⣠⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠂⠀⠀⠉⠉⠉⠁⠈⠉⠁⠀⠀⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣴⣀⡁⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣽⣿⣿⠿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⢟⣿⣟⣹⣿⣿⣻⣿⡤⠂⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⣿⣯⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣳⣿⣿⣿⣿⣯⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠔⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣏⡛⣛⣽⢿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠈⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠤⠀⠄⠀⢀⠀⢠⡄⠤⠄⠒⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⢠⣄⣀⣠⣴⣶⣷⣞⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠋⠁⠀⠀⠀⠁⠀⠀⣠⠊
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡄⠤⠛⠉⠀⠀⠀⠀⠀⠈⢧⣄⣀⡀⠀⢀⣀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠶⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠁⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠀⠀⠀⠀⠀⠠⠄⠤⠤⠤⠭⠀⠀⠀⠶⠶⠶⠶⠴⠦⠀⠀⠀⠀⠀⣤⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣦⣞⣠
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠐⠒⠒⠒⠒⠒⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⣀⣀⣰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣖⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⢀⣤⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⠀⣾⣿⣿⣿⣿⡟⣠⣾⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⣼⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⢠⣿⡿⠿⠛⠛⠿⢃⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠈⣿⠿⠟⢋⣉⣤⣤⣶⣾⣿⣷⡀⢺⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿⣿⣿⣿⠇⠹⣿⣿⣿⣿⣿⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣈⣉⣙⢋⣠⣄⡙⠛⠻⠿⠟⠀⠹⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣦⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣧⠀⣉⣉⣉⣙⣛⠁⣼⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⠛⠛⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠻⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡛⣛⣛⡛⢛⣻⣿⣿⠿⠿⢿⡿⠿⣿⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⡑⠀⠘⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⠀⠀⡠⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⠤⢶⠊⣠⢼
⠀⠀⠀⠀⠀⠀⢠⠖⡉⠂⠄⡻⣀⠀⠀⠀⠀⠀⠀⢠⣮⡤⢤⡈⣙⣿⣯⡫⠠⣌⣷⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⡟⠘⣧⠀⠉⠀⢸⠄⠀⠀⠀⠀⠀⣿⣿⡇⢀⢼⠋⢩⡇⠳⣸⣃⢀⣼⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠙⣦⡙⡒⠤⠤⣤⠇⠀⠀⠀⣀⣼⣿⣿⢷⢃⡏⡄⠀⠙⡲⣿⡻⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢀⠠⢤⣸⠙⠈⣦⣄⣚⣻⣷⣴⣿⣿⣿⡿⠋⢸⡾⠀⠀⠀⢠⠗⣻⣷⣿⣿⠟⠛⠋⠉⠁
⠀⠀⠀⢀⣏⠿⠼⠃⠀⠀⣿⣿⣟⣿⣿⣿⣿⣿⣿⡄⠀⢸⡇⠐⠀⠀⡞⣼⣿⣿⣿⠋
⠀⠀⠀⠘⢧⣤⠀⢀⣠⠞⠙⢿⣾⣿⣿⣿⣿⣿⣯⢀⣤⣿⣿⡀⠀⠀⣷⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣻⡿⠿⡿⢷⣤⡾⠿⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⣛⡿⡟⠁⠀⠀⠀⢸⡛⠂⠀⢀⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⣿⣿⣿⡇⠀⠀⠀⠀⢸⣧⣿⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⣿⣿⣿⣿⡿⠀⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡛⠕⠉⠈⢳⡀⠀⢀⣿⣿⣿⣿⠃⠀⠀⠈⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢿⣷⣄⠀⠀⠀⢳⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠏⠀⢀⡟⠉⠙⢆⠀⠈⢧⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⢀⡤⠊⠀⠀⠀⠈⣇⠀⠈⢻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆
⠀⢀⣤⠤⠴⠊⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡄
⣤⣾⠏⠉⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠛⢆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠙⣶
⢸⣧⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⣏⠀⠀⠀⢣⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⣆⣿
⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣤⠞⠁⠀⠀⠀⢿⣧⡀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠋⠁⠀⠀⠀⠀⠀⠀⠻⠳⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⢠⣶⢀⠄⢶⡆⣴⡦⢄⢠⣶⠒⠀⠀⠀⢠⣶⡄⠰⢠⢲⢦⣶⣦⡀⣴⢲⠄⠀⠀⠀⣤⢀⡄⢠⡄⣶⢠⢦⡀⡤⠦⡀
⠀⠀⠀⠀⢸⣿⡎⠀⣿⡇⣿⡇⢀⠘⢿⣧⠀⠀⠀⠘⣿⣤⠃⢸⠙⢸⣿⣿⣿⣙⢳⢦⠀⠀⠀⣿⠛⣦⢸⡇⣿⡇⢸⡟⢷⡘⠃
⠀⠀⠀⠀⠸⡿⠘⠄⠿⠇⠻⡧⠊⠀⠄⠿⠀⠀⠀⠀⠿⠇⠱⠼⠽⠜⠿⠟⠃⠫⠾⠃⠀⠀⠀⣿⠀⢿⢸⡇⢿⣠⠞⠀⣀⢹⠆
⠀⠀⠀⠀⢰⠂⡄⣤⣴⣆⡄⣇⣠⢘⢰⡳⠀⠀⠀⠀⡇⢤⢰⣤⡰⣦⣀⢶⡄⠀⣆⠀⠀⠀⢠⡄⣤⣠⣤⣄⢠⣤⣄⣀⣤⣥⡄
⠀⠀⠀⠀⣀⡱⠷⠧⠇⡏⠷⢻⠏⢺⡘⢧⠄⠀⠀⠀⣑⢾⡸⡏⠁⣿⠁⢸⣇⡀⠿⠀⠀⠀⠐⠟⠻⠻⠟⠉⠻⠃⠋⠛⠇⠛⠁
⠀⠀⠀⠀⢀⡤⠀⠀⣴⡅⢠⣦⢀⠀⣶⠀⠀⠀⠀⠀⡈⢀⢀⣀⣀⢁⡀⡈⠀⢀⣤⣁⠀⠀⢀⡶⠢⡔⢳⡄⣾⣀⡼⣧⣶⠖⡆
⠀⠀⠀⠀⢸⡇⠀⢸⡿⠆⣸⡏⠀⠀⣿⠄⠀⠀⠀⢰⣿⠀⣿⣹⡿⣿⡇⠘⠁⢼⣧⠼⠀⠀⠸⡇⡄⣞⢽⡇⣿⠹⠇⡏⢻⠘⠁
⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠈⠁⠃⠙⠁⠐⠀⠘⠃⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠁⠈

⠐⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠒⠈⠛⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀⠸⠿⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣤
⠀⠀⠀⠀⠀⡠⠊⠉⠉⠉⠉⠉⠒⢵⡀⠀⠀⠀⠀⠀⠀⣼⡿⢀⠖⠈⠁⣈⣛⡲⣤⡀⠀⠀⠀⠀⠀⢀⡔⠚⠛⠉⠀⠀⠩⣯⣤
⠀⠀⠀⠀⠀⠁⢀⠄⠒⠒⠒⠂⠄⡀⡇⠀⠀⠀⠀⠀⣼⡿⠃⠘⠀⢠⠏⠁⠠⣤⡆⢹⠀⠀⠀⠀⠀⢸⠇⡠⠂⠐⠂⠀⠶⣷⣿⡆
⠀⠀⠀⢠⣶⢤⡅⠀⠤⠤⠤⠀⠀⠷⣿⠀⠀⠀⠀⢸⠊⢀⠀⡆⠀⣼⠀⠀⢀⡈⣠⠼⠀⠀⠀⠀⠀⣘⠀⠀⡏⠉⠉⠉⠁⠹⠹⢧
⠀⠀⠀⠈⢸⡆⠀⠀⠄⠤⠂⠀⠀⠀⣿⠂⠀⠀⠀⣿⡆⡆⠉⠓⡂⡇⠠⠀⠰⣴⠂⢀⢼⠀⠀⠀⢀⠛⢠⠀⣤⠀⠀⣰⠀⡅⠀⢸⡇
⠀⠀⢸⢑⣿⠃⠀⠑⢄⡀⢀⠀⠀⠀⢻⠈⠂⠀⠀⠉⢳⠇⠀⣠⡇⠀⣘⣰⠦⠀⠐⠙⢼⠀⠀⠀⠈⠀⠈⡆⡉⠐⠂⣠⠀⢇⢰⢻⠇
⠀⠀⠈⠊⠻⣄⠀⠀⠀⣈⣁⣀⣀⠠⠛⠂⠀⠀⠀⠀⠘⢦⣀⣠⠃⠀⣻⠃⠀⠊⢀⣀⡼⠀⠀⠀⠈⠾⠒⠃⠃⠁⠀⠀⠒⠋⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠓⠋⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣀⣤⡴⠛⠛⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠭⠭⢵⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢦⣤⣦⡤⠄
⠀⠀⠀⠀⢸⠁⣠⠖⠛⠫⠿⡿⠽⣷⡄⠀⠀⠀⠀⠀⠀⡾⡠⢒⣚⣛⣓⡒⢝⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⢀⠀⠀⠀⠃
⠀⠀⠀⠀⢸⠀⡇⢠⠀⣀⠠⠣⠀⢄⣡⠀⠀⠀⠀⠀⢸⠁⣻⠋⠀⠀⠀⢹⢎⢺⡄⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠇⠀⠀⠈⠁⢀
⠀⠀⠀⠀⢸⠀⡇⢸⡎⠀⠀⠀⠁⠀⣮⠀⠀⠀⠀⠀⣎⣰⠛⠀⠀⠀⠀⠈⠸⣦⡇⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⡆⣦⠀⠉⠉
⠀⠀⠀⠀⢸⠀⡇⠈⠁⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣉⡏⠀⠀⠀⠀⠀⠀⠀⢧⣼⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡇⡥⠐
⠀⠀⠀⠀⣾⢤⡇⡗⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⡏⠀⣰⠀⠀⠀⠀⠀⣷⠀⡜⡄⠀⠀⠀⠀⣾⠀⠀⠀⠀⢰⠇⡇⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠙⠸⠗⠧⣄⡀⠀⠀⠀⡀⢀⡇⠀⠀⢀⡞⠁⠀⠙⠀⠀⠀⠀⠀⡜⠀⠁⢹⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠷⠀⠒⠉⢹⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠟⠃⢰⣦⣴⢓⣻⠇⠀⠀⠀⠇⣰⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠼⠁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⣀⡠⠚⠁
⠀⠀⠀⠘⠢⢤⣾⡀⠀⠀⠀⠬⠓⠋⠁⠀⠀⠀⠀⠈⠛⠲⠶⠦⠤⠶⠶⠶⠶⠚⠋⠀⠀⠀⠀⠀⠐⠛⠀⠀⠠⠾⠗⠚⠉⠁

⣿⣟⢻⡿⢿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠕⣄⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠉⠁⠉⠉⠉⠉⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⣀⡄⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡀⢀⡀⡄⠰⠇⠸⡀⠰⣠⡀⢠⣤⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⠀⠀⣝⢤⠑⡌⠹⠘⠃⢴⠃⠱⠀⠋⠓⠁⢡⠘⢃
⠀⠀⠀⠀⠈⠄⠊⠉⠀⠀⠀⠀⡆⠀⠰⠀⠀⢫⠀⠀⢀⣀⣀⡴⠀⡤⠀⠠⣄⠀⠀⢸⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠓⢤⡀⠀⡌⠰⣠⡎⠥⠒⠉⠱⡄⠧⠀⡠⠊⠻⢍⣮⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠔⠒⠵⣤⡃⠀⡇⠀⠀⠀⠀⣷⡁⠀⠸⠁⠀⠀⣾⠈⣿⡓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠐⢊⡇⠀⠀⢠⠀⠐⢄⡀⠀⢀⠵⠁⠀⠈⡣⢄⣀⡠⠚⡝⢫⡃⠀⠀⠛⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣐⡃⠀⠀⠉⠀⠀⠀⠈⠉⢡⠀⠀⠈⠉⠃⠀⠀⠀⢀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⢀⠀⠔⠀⠀⠀⠀⠈⡄⠀⠀⠋⠀⠀⡄⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⡇
⠀⠀⠀⠀⠐⠕⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢠⠃⠀⠀⠛⢀⠔⠊⠉⢢⠀⠀⠀⠐⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠼⠦⣄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡀⠀⠀⠠⡏⠒⠢⠤⠤⣄⣀⣀⣀⣀⣀⡤⢤⡒⠉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⡴⠀⢀⠀⠀⠙⢄⠀⡠⠒⠲⣥⠟⢦⠀⢀⠇⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⡀⠈⠁⢀⠃⠀⠀⠂⠀⠉⠀⠀⠀⠁⠀⢀⠖⢱⠀⠀⠀⢸⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠈⠉⠒⠖⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣧⠠⠤⠚⡇⠀⠀⠨⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢡⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣽⣿⣿⣿⣿

⠀⠀⣄⡀⢠⡀⠀⣠⡄⠀⠀⠀⠀⠀⣶⡦⢀⠀⠀⠀⠀⠀⢀⣤⣾⣦⣄⠀⠀⠀⠀⠠⠐⠒⣄⡒⠀⠄
⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⣼⣿⡗⣲⡄⠀⠀⠀⠀⠛⠛⢻⠛⠛⠃⠀⠀⠀⢃⢠⡦⢐⣴⡆⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣯⠀⠀⠀⠀⠀⣐⣀⣸⣀⣐⡀⠀⠀⠀⠨⢑⡨⠤⢂⡰⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⣾⡟⠛⠛⠛⣿⡆⠀⠀⠀⢸⡏⠉⠉⢹⡇⠀⠀⠀⠀⣴⡺⡽⣖⢄⠠⠀⠀⠀⢀⣤⣤⡄⣤⡤⣄
⠀⠀⡟⠀⠰⡇⠀⠙⡇⠀⠀⠀⢸⡷⠤⠤⢼⡇⠀⠀⠀⢸⢞⢎⢏⣎⣾⠀⠀⠀⠀⢸⣿⣿⣷⣿⣤⣿
⠀⠀⣿⣲⣴⢳⣴⣾⡇⠀⠀⠀⢸⣇⠀⠀⢸⡇⠀⠀⠀⠈⣯⢗⣷⣮⣎⠀⠀⠀⠀⠻⠟⠃⠀⠈⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉

⠀⠀⣦⣬⣀⣀⣥⣴⡄⠀⠀⠀⢠⣶⣾⣷⠂⠀⠀⠀⠀⠀⠀⢀⣒⣒⡀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⡄
⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠱⢆⣾⣿⠀⠀⠀⠀⢀⣿⡿⣿⣷⠄⠀⠀⠀⢸⣿⡏⣄⡌⣿⣿
⠀⠀⠿⠿⢿⡿⡿⠿⠃⠀⠀⠀⠈⠀⣾⣿⠿⠃⠀⠀⠀⠀⣼⣿⣷⣶⣶⣆⠀⠀⠀⠀⠉⢡⣟⣣⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠉⠃⠀⠀⠀⠀⢠⠖⢻⠛⠲⡄⠀⠀⠀⠀⠾⠮⠴⠧⠷⠇
⠀⠀⠀⢸⡁⠀⠀⢀⡀⠀⠀⠀⠀⠀⣹⠶⠀⠀⠀⠀⠀⠀⡟⠑⣾⣷⠊⣻⠀⠀⠀⢠⠀⢴⠄⢴⠄⢠
⠀⠀⣀⣀⣙⣲⣒⣋⡀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠘⠾⣅⣨⠶⠃⠀⠀⠀⠈⢶⣾⣶⣾⣶⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣀⣤⣴⡶⠶⠿⠟⠛⠛⠻⠿⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠾⠛⠉⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⢀⣴⣾⡏⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠋⣽⣿⠁⠀⠀⠀⠀⠀⠀⠘⢁⡴⠀⣠⠂⠀⠈⢳⣄⠀⠘⠋⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⢠⣾⠙⡆⠀⠀⡏⢹⣧⡘⡀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⢀⡆⠀⠀⠸⣷⣿⣿⠀⡇⠀⠀⡇⢸⣿⣿⣷⠀⠀⠀⠀⠘⡇⢀⣠⡀
⠀⠀⠀⠀⠀⢸⣿⣇⠉⠀⠀⣼⡇⠀⢠⣾⠛⠟⠃⢀⡇⠀⠀⢇⠀⠛⠟⢻⠷⠀⠀⢀⠀⣿⡏⣠⢼⡆
⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⢸⢻⡇⠀⠀⠘⣧⠀⠀⣼⠃⠀⢸⡜⣦⡀⢀⡜⠀⠀⠀⣾⠀⢿⣷⣏⢸⠃
⠀⠀⠀⠀⣾⠋⢀⠙⢧⡀⠙⠛⠀⠀⠀⠳⣌⣙⠋⠁⠀⠀⠀⠹⣮⠙⢩⡴⠃⠀⠀⢻⠇⠸⣧⠉⡟⠀⠀⠀⢀⡤⠴
⠀⠀⠀⠀⢿⣠⡎⢳⡄⠀⠀⠀⢸⡇⠀⠰⠬⠉⠀⠐⠋⠀⠀⠀⠈⠳⣤⡘⠁⠀⠀⢸⡆⠀⢻⣷⠇⠀⠀⠀⠉⠠⠒⠋⣉⣉⡁
⠀⠀⠀⠀⠈⢿⡁⠸⣿⡄⠀⠀⣼⣿⠀⠀⠀⠀⡤⠀⠛⠛⠷⠶⣤⣀⠈⠙⢦⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⡴⠚⠉⠉⠉⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠙⢦⡈⠁⡀⠀⡆⡟⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⠉⠐⢋⠀⠀⠀⠀⠀⢸⣿⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠸⣴⠇⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⣾⡏⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⢾⠃⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠸⡇⠀⠀⣼⡿⢁⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣷⡀⠀⠀⠀⢸⣤⠞⠋⠉⠁⣤⣤⣤⠀⠉⠙⢶⣄⡇⢀⣾⡿⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠙⢿⣄⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣵⡿⠋⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠙⠷⣦⣄⣘⣦⣄⡀⠀⠀⠀⠀⠀⣀⣠⡾⠋⠁⠀⠀⠀⠀⠈⠳⢤⡤⡀⠀⠀⠀⠀⣠⠾⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠰⣿⠓⣄⠀⣉⠽⠛⢿⠿⢷⣶⣾⣿⣿⠋⠀⠀⠀⠀⠀⣠⠤⣄⣠⣎⡼⣿⠦⠤⠖⠋⠁⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡴⠛⠉⣀⡤⣶⢿⣦⣠⣴⣮⠀⡾⠛⣦⡀⠀⠀⣰⢧⡀⠈⠻⣏⢰⡏⠀⠀⠠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣾⣧⠀⠀⠀⠀⢫⡴⢾⠟⣿⠶⣌⠡⠞⠁⠘⡎⢷⡀⠀⣷⣄⢙⡏⠁⠈⠹⣷⡀⠀⠐⠚⠉
⠀⠀⠀⠀⠀⠀⢀⣴⣯⡤⠴⠾⠃⠀⠀⠀⠀⠈⢀⡿⠀⣷⠀⠀⠀⠀⠀⣠⣿⡇⣹⠶⢿⠆⢺⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⣠⣤⣤⣤⣶⡏⠀⣼⣿⡆⠀⠀⠀⠀⠈⢷⡾⠁⣠⠼⢷⡀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⢀⣠⠞⠁⠀⠀⠀⢺⣤⣴⣿⣿⣿⣀⣀⣀⣀⣀⣸⣇⠀⡏⠀⠀⠉⢠⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⣀⣄⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣻⣤⣤⣴⣾⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣏⣀⣠⣤⣤⣤⣤⣤⣄⠻⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⢿⣿⣿⡇⢹⣿⠁⠀⠀⠈⣿⡇⠀⢿⡿⣿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠖⠚⠯⠤⠝⢻⡄⠸⡿⠀⠀⠀⠀⣿⡇⣠⠾⠤⠼⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄⣉⣀⡤⠖⠲⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣤⣤⣀⠈⠿⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄
⠀⠀⠀⠀⠀⡰⢿⡿⠤⢿⡿⢄⠀⠀⠀⠀⠀⠀⠀⠖⠐⠀⠉⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⢿⣿⣿⣶
⠀⠀⠀⢸⠭⠤⠔⢀⣀⡌⠢⠤⠯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⠁⠈⠉⠙⣿⡇
⠀⠀⠀⢸⠀⠀⢻⣿⣿⣿⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠛⠋⢸⣿⠠⠁⠀⠀⠀⠀⠀⠀⢸⣿⣤⣶⣶⣷⣶⣦⣿⡇
⠀⠀⠀⢸⢤⣠⣾⣿⡿⠿⠟⠛⠀⢃⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⢿⣿⣿⣿⠟⣿⡇
⠀⠀⠀⢉⠀⠉⠉⠀⠐⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠾⠆⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠿⣄⢤⣥⣌⣤⣤⠿⡇
⠀⠀⠀⠀⠀⠣⡀⢠⠒⡀⠀⠊⠀⠀⠀⠀⢀⣀⣰⣶⡶⠄⣀⠄⢄⣤⣶⢄⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠛⠋⠛⠉⠁⠀⠀⠙⠛⠋⠉⠉⠀⠀⠀⠀⠉⠻⠿⠿⠿⠿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣤⡤⠤⣾⣿⡧⠤⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢞⣷⣿⡳⣤⣄
⠀⠀⠀⠀⣿⣿⣿⡏⠉⢹⣿⣿⣿⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⣿⣶⣬⠛⠛⣵⣶⣿
⠀⠀⠀⠀⡿⣿⡿⢀⣴⠛⢿⣿⠿⠀⠀⠀⠀⠀⠀⡠⡔⢶⢾⡣⡣⡦⢄⡇⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⣿⡟⠁⣾⣯⣶⡈⣿⣿⠀⠀⠀⠀⢠⡶⣭⣵⣷⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣤⣿⣷⢠⣯⠈⠁⠀⣿⣿⣤⠀⠀⠀⠀⠁⠘⣿⠿⢿⣯⣿⠏⠀⠀⠀⠀⠀⠀⢹⣿⠻⣟⣻⣷⡿⠃⣿⡏
⠀⠀⠈⢿⣯⣥⣤⣶⡴⢶⣦⣭⣿⡟⠀⠀⠀⠀⠀⢀⣈⣻⣟⣉⣁⣀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣬⣥⣴⡾⠿⣿⠇
⠀⠀⠀⠀⢅⡐⠋⠛⠙⠀⠐⣀⠍⠀⠀⠀⠀⠨⠿⠛⡛⠛⠃⠑⠘⡛⠻⠯⠀⠀⠀⠀⠀⠙⠛⣏⠉⠀⣰⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠈⠇⣠⣴⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣤⣴⣶⣭⣤⣁⣀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠇⡂⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠢⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⠠⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠉⠀⠉⠉⠁⠶⠶⢤⣄⣀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠈⣄⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢧⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠤⠤⠴⠶⠶⢿⣍⡻⢿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇⠀⠀⠀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇⠀⠀⠀⠀⠀⠀⠘⡗⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⡏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡾⠁⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⡀⠀⠀⠀⠈⠉⠁⠒⠒⠒⠒⠲⠛⠛⠉⠃⠀⠀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠀⣈⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣶⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣿⣛⣥⣤⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠿⠿⠿⠿⠛⠻⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⢰⣿⠄⠛⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⡀⠀⠀⣴⣿⣿⠋⣀⠀⠰⡄⠈⠲⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⢠⡿⣻
⠀⠴⣾⡟⡶⠄⠀⠀⠀⠀⣼⡇⠀⠀⠀⣠⡄⠀⠀⢀⣾⣷⡄⠀⠀⠀⢰⡇⠀⠈⢹⣿⣿⣆⠀⠁⢤⠈⠂⠀⢀⡀⠀⠀⠀⠀⣼⡇⢀⣴⠂⠀⠀⠀⣼⣿⣿
⠀⣪⣿⠃⢄⡐⠀⠃⠀⢠⣟⠠⠀⣠⣾⡋⠀⠀⠀⠀⠘⠿⢁⡄⠀⠠⣿⡟⠀⠀⠀⠉⠙⠻⠃⡄⠀⡐⠀⠀⣾⡀⠀⠈⠀⢠⣿⣭⠄⠈⠁⡀⢀⣈⣿⣿⡎
⣴⣿⣿⣠⣄⡄⠀⠀⠀⣚⡆⠀⠆⠈⡉⠿⣠⠀⠂⠀⠀⠀⣼⣇⠀⢴⣟⣙⡁⢰⡆⣧⣤⡆⢸⡇⢠⡇⣠⢸⣿⣣⢀⢰⡄⣸⣿⠋⠀⢠⢸⣇⢨⣽⣿⣿⣿
⣿⣿⠻⢫⡄⡀⠀⠄⠀⣷⡇⠀⠠⢴⣧⠀⠈⠀⠀⠀⠘⠃⣿⣿⣒⣿⣿⡇⠐⣾⣿⠛⠻⠷⣿⣧⣿⠿⣿⣿⣿⣖⣿⣿⣿⣿⣷⣶⢠⣾⣶⣿⣾⣷⣿⣷⣯
⣿⠉⠁⠈⡌⠀⡀⡀⢨⣾⣿⢿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢿⣿⣿⣶⡶⣾⣩⠁⠀⠀⠘⠛⠋⡏⠀⠀⠹⣿⣿⡷⠈⠇⠈⠉⠛⠛⠻⠿⠻⣿⣿⡿⢉⣿
⣡⠺⠀⡄⠀⠈⠛⠀⠀⠙⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠉⠙⠉⠛⠋⠁⠀⠀⠄⠀⠀⢠⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢶⣾⡿
⡔⢦⠄⣱⠏⠀⠀⢠⡀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠸⠿⠀⠀⠂⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⢸
⠀⠁⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠠⠤⠀⠀⠀⠀⠀⠆⠀⢤⡄⠤⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⢰⠀⠀⣀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠉⠙⠂
⠀⠘⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠠⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⠀⣀⣤
⢿⠇⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⡿⠿⠿⠻⠿⠶⠶⣶⣤⡤⠤⣤⣤⣤⣶⣶⣤⡀⢀⣈⣉⢀⡀⠀⠀⠀⠠⣤⣀⡀⠀⢸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⢈⣿⣿⣿⣿⡛⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣉⠛⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣀⣀⣬⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣼⣿⣿⣿⣶⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢠⣤⣀⠀⠉⠉⠉⠉⠛⠋⠛⠛⠛⠻⠛⠿⠿⠿⠿⣿⣿
⣾⣷⣿⣿⣷⣶⣷⣿⠿⣿⣿⣟⡋⠙⠛⠋⠉⠉⠁⣀⣠⣶⡶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⣿⣿⣾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣉⣁⣈⣿⣿⣻⣿⣿⣿⣟⣁⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡡⠊⢑⡄⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⢐⢑⠔⠁⠈⢐⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠣⠔⠀⠀⠀⢠⣿⣿⣿⣿⠛⠛⠛⠛⠉⠁⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠅⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢅⠕⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠠⠄⠀⠀⠀⠀⢸⡇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁
⠠⢁⠓⢄
⠱⣆⠀⢎⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣤⣶⣶⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢨⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠻⠿⠿⠉⢙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡨⠖⠋⠑⢌⣳
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠠⠈⠒⢿⡷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣩⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢄⡽⠚⠈⠀⠀⠀⠂⠀⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⢄⠀⠈⠀⠠⣿⡄⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠻⢆⠀⠀⠀⠀⠀⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠈⠙⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣤⣤⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠛⠛⠿⡇⠀⠀⢸⣿⡿⠿⠛⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⠀⠠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠔⠊⠉⠁⠉⠉⠑⠢⡙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠁⠉⠉⠀⠀⠒⠒⠂⠀⠠⠤⠷⠆⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠋⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡔⢡⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⡀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⠊⠀⡌⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢣⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⠔⠁⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⠤⣉⢢⠤⢤⠀⠤⠐⣊⠤⠒⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⡠⠤⠄⠤⠭⢓⣋⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢄⡀⠀⠀⠀⠀⢸⠀⢠⢂⡢⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢸⠀⢧⠃⠸⢸
⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢠⡎⠀⠠⢸
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢨⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠇⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠐⡌⢦⡑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡇⢸⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⣁⡒⠠⠍⣁⡒⠒⠠⠄⣀⣀⠀⣀⠎⢸⠀⢀⠸⠀⠀⠀⡜⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠉⠩⠤⠒⣀⣀⠤⠐⠁⠀⠸⡀⠳⠤⢊⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠱⠆⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⡄⣤⢠⢶⡀⡄⡄⢴⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠌⠀⢸⠿⢻⢸⢸⡇⣿⠇⢸⢸⡇⠸⠓⡀⠰⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠢⠔⠀⢁⠀⠀⠆⠈⠠⠀⠠⠉⠁⠒⢠
⠀⠠⣐⣦⣤⣄⡢⡌⠁⡀⠀⠘⠀⠘⠈⠛⠁⠘⠀⠚⠈⠓⠀⠀⠀⠁⠀⠀⠈⠀⠀⠃⠐⢤⠀⠈⠂⠀⠀⡠⠃⠈⠇⠀⠀⢐⡀⣥⠀⣀⠴
⠀⠌⠛⠛⠋⠉⠀⠃⠀⠈⡆⠀⠈⠄⠀⢀⠄⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠆⡠⠃⠀⣬⠆⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠈⢀⢐⡀⣀
⠀⠄⠀⢿⣿⡗⠀⡬⠀⢂⣀⠀⠀⡼⠰⠄⠀⡀⣀⠀⠀⠃⠀⠀⠠⠞⠀⠀⠀⢀⠑⠐⡂⠡⠄⣀⠀⠀⠣⠀⡔⠉⠢⡀⠀⡰⠊⠁⠀⠀⣄⠑⡄
⠀⠈⠂⠠⠭⠀⠊⠀⡈⣤⣴⣶⣶⣶⣶⣦⣤⠀⠀⠈⠀⠃⠀⠀⢀⢀⡀⠀⠀⠀⡄⠙⠤⢄⠀⣻⠈⢈⠁⠸⠀⠀⠀⠓⡀⣄⠀⢁⠀⠌⠠⢀⠇
⠀⢀⠄⠠⠠⡀⠀⠀⡇⣿⣿⣷⣿⣟⣿⣿⣿⠀⠀⠀⡆⢰⠀⠀⠙⠺⠹⠈⠠⠀⠰⠀⠀⢛⠒⠚⠀⠂⡃⠘⡐⠀⠀⣈⠃⠈⠐⠀⠀⠀⠀⠁
⠀⠊⠀⠀⠀⠈⢂⠀⡇⣿⣿⡟⡿⣿⡿⢿⣿⠀⠀⠀⠆⠀⣤⠆⠂⠐⠒⠒⠒⠒⡖⠀⠀⠸⠀⠀⠀⠨⠁⠀⢱⠀⠀⡄⠀⠀⢠⢴⣯⡀⠀⠈⠢
⠰⣅⣀⣀⢀⣀⠃⠀⠇⠛⠿⠿⠿⠿⠿⠟⠣⠀⠀⢀⠀⠀⠉⠀⢀⡠⠄⠐⠀⠤⡀⠀⠀⠀⠆⠀⠀⡠⠀⠀⠈⠒⠒⠁⠀⠀⠁⠀⠈⢿⣄⠐⡰
⠀⢽⠔⠛⢸⡇⠀⠀⠸⠋⠉⠉⢉⣉⡉⠉⡈⠉⠻⠅⡠⠐⡠⢤⣌⡰⢄⠀⠀⡠⠄⢹⠒⠄⡀⠀⠀⡴⠒⠒⢠⡀⠀⠐⠄⠀⠐⢀⠀⠸⠿⠋
⠀⢺⢤⡄⢾⠀⠀⠀⠀⢸⠉⠀⠀⣠⡤⠀⠃⠀⢀⡐⠀⠠⡀⠀⠀⠀⢀⠀⢸⠀⠀⣘⠢⠱⠈⢄⠀⠀⠀⠒⡆⣶⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠙⠲⠧⠟⠂⠀⠀⠀⠈⡄⠀⠠⡤⠛⠀⠸⠀⠑⢄⠀⠖⡱⠆⢩⠀⢊⠐⢚⡁⢎⠠⢊⠜⢀⠒⠀⠢⠄⠤⠬⠤⠤⠤⠊
⠀⠀⠀⢀⣤⣶⣶⣦⣀⠀⢃⠀⠀⠶⠐⠂⠀⡆⠀⠀⠑⠠⢀⠀⠀⠉⠉⠀⠀⠀⠀⢁⠠⠐⠁⠀⠀⠀⠀⢀⠠⢀⠀⠀⣿⣻⠏⣿⡟⣿⡏⡉⣿
⠀⠀⣴⢟⣭⡶⣶⣭⡻⣆⠘⠒⠒⠈⠉⠉⠀⠀⠀⠀⢀⣀⡤⡀⠀⠀⠀⠀⠀⢀⢠⠶⠤⠤⠀⢄⠀⠀⢠⠁⠀⠀⢑⠀⠟⠛⠛⠃⡀⠈⠛⠛⠛
⠀⠰⠿⠟⠁⠀⠈⠙⢳⢿⠀⢠⣴⣶⣶⣶⣶⣶⣶⡄⠈⢥⠜⠁⣀⡤⡄⠀⠼⢁⣄⡀⠀⠘⠃⠃⠀⠀⢸⠀⠀⠐⠘⡄⠀⠀⢘⠀⠉⠈⠈
⠀⢃⣀⡀⠀⠀⠀⠀⣄⣠⠀⢸⡏⠁⠸⠀⡏⠠⢸⡇⢠⣀⢶⡄⢈⣀⠀⠀⠄⠈⠛⢀⠔⠒⠤⣀⡸⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄
⠀⠀⢹⣷⣄⠀⢀⣼⣿⠀⠀⢸⣇⠀⡀⡄⠃⠰⢺⡇⠈⠁⠀⢠⣼⣙⠃⠀⠈⠒⠒⠁⠀⣀⠀⠀⢀⡀⠀⠑⢰⡶⠒⠁⢐⠀⠀⠀⠐⠈⠀⠐
⠐⠂⠘⠋⡻⠀⠘⢋⡙⠀⠀⢸⣿⢾⣷⣿⣾⣤⣾⣿⠀⠀⢀⠀⠈⡉⠀⠀⠀⠀⠀⠀⢺⠁⠋⠙⠛⠛⣿⣆⠘⠷⠇⠀⠺⣶⣿⣛⣓⣚⣻⣷⣾
⠀⠈⣠⣉⠤⡀⠀⠀⣈⡄⠀⣰⣿⠀⡇⡆⢀⠘⢸⣿⠀⢰⠓⠲⠾⠶⢾⠄⢄⠀⠀⠀⠇⠀⣘⢀⡀⠀⠈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢀⠿⠏⢀⡠⣿⣿⠿⠷⠀⠉⢿⣶⡷⠶⠿⠶⠾⠟⠀⢸⠀⠀⠄⠠⠀⠀⠀⠀⢀⡞⠋⡧⠰⠎⠀⡤⠢⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⢈⠅⡇
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⡀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠀⢠⠆⠌⠀⠸⡄⠀⡀⠀⠀⠀⠃⠀⢸⠀⠂⠀⠞⣀⣀⡀⠙⠀⠈⠀⠁
⠀⢀⡀⠤⠤⠤⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⠀⠀⢀⣀⡀⠀⠑⠀⠀⠀⠊⠀⠀⠰⠀⠘⠒⠁⠀⠀⠀⠧⠴⠃⠐⠀⠀⢼⣿⣿⣿⡗⠀⡠⠀⠐⠄
⠀⠂⠀⠀⠀⠀⠀⣳⠀⠀⡖⠃⠀⣀⠀⠀⠑⡄⢸⠀⠀⠉⠐⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠄⠀⠀⡄⠀⠀⠙⡍⠉⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠇⠀⠈⠿⢿⠀⠀⠃⠈⡀⠀⠀⠀⠀⠀⡥⠐⠭⠂⠀⠀⠀⢀⠄⠠⣀⠀⠀⣀⠀⢀⠇⠀⠀⢠⠇
⠀⠀⠀⠀⢶⠂⠀⢸⡀⠀⠀⠀⣀⠀⢀⣀⡀⠀⠀⠇⠀⠀⢀⠔⡨⢔⠁⠀⢀⠀⠀⠐⣵⡿⢷⡌⡆⠀⠜⠀⠣⡀⠀⢔⠺⠂⡄⠀⢸⠀⠘⠃
⠀⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠰⠟⠣⠞⠃⠈⠀⢀⠀⠀⠈⠁⠀⠸⠀⠀⠀⠠⡊⠀⠙⠾⠋⠈⠀⢸⡀⠀⠀⡄⠀⠀⠀⣤⡆⠀⠈⡄
⠀⠑⠚⠚⠛⠛⠛⠋⠀⠀⠑⠆⠀⠀⠀⠀⠀⠐⠃⠀⠉⠑⠒⠒⠢⠤⠆⠀⠀⠀⠈⠢⠠⠊⠀⠀⠀⠈⠑⠀⠊⠁⠀⠀⠀⠈⠁⠀⠀⠐⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢠⣶⣦⡄⠀⠀⢀⣀
⠀⠀⠀⣀⠀⢠⣶⣿⣿⣿⣷⣿⣿⣿⣷⣿⡿⣿⡿⠋⣰⣾⣽⣾⣷⣦⠀⠀⠀⠀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⣼⣿⠙⠋⠉⢉⠛⠛⠛⠋⠀⠀⠀⠁⠀⢿⣿⠉⠁⠀⢈⠁⠀⣶⠋⠀⠀⢀⠁
⠀⠀⠀⠀⠀⢩⡉⠀⠉⠀⠈⠠⠀⠀⠀⠀⠤⠄⠀⠀⣠⡅⠀⠁⠤⠄⠀⠀⡄⠀⠈⠤⡌
⠀⠀⠀⠀⠀⡄⠈⠂⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⣡⡙⠂⠀⠐⣆⣤⠜⠁⢰⡄⠤⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠈⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠀⠀⠰⠄⠂⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠠⡤
⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢆

⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣽⣯⣿⢷⡄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣄⢀⣀⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⣿⣿⡏⢘⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣿⣿⣧⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣻⣿⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢿⡿⢿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣾⣿⣿⣿⡆⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠾⡿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠙⠿⡿⠟⠁⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠒⡗⠂⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣼⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠀⠀⠰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠸⠐⢠⢆⡠
⠀⠀⠀⠀⠀⠀⠀⡈⠂⠀⡠⠤⠦⢄⡀⠉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⡜⠀⠀⠀⠀⢱⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠈⢁⠀⠱⡀⠀⠀⢀⡜⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⢡⠔⢂⡈⠉⠉⠉⠀⢄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢠⠃⠸⠄⠈⠀⠀⠀⠀⢠⠞⠉⠉⢢⠀⠀⠀⠠⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢦⠀⠀⠀⢃⡜⠛⠲⣄⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣲⡿⣗⡇⠀⢀⠞⠍⠳⢤⠈⢹⡆⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠣⡀⠉⠉⣏⢰⠄⠩⣈⡇⣸⠃⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠑⠤⠬⣉⠁⠈⠓⠒⠒⢟⠲⠋⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⠀⠀⡁⠈⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅⠈⠉⠉⠉⠉⠉⢉⣉⡏⠉⠙⢄⡀⠐⢀⡜⢱⠄⡇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠲⠤⠝⠒⠒⠊⠉⠉⡁⠀⠈⢢⠀⠀⠉⢲⠎⠀⠀⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠬⠤⢄⣀⣀⣀⣁⣀⣀⣀⣀⠤⠧⠤⠔⠋⠀⠀⠀⠱⢌⣻⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠂⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⠁⠀⠀⢀⣠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠤⠍⠁⡤⡄⠀⢀⠀⠀⠀⠁⠈⣀⣀⠀⢠⠃⡎⠀⠀⠀⠰⠆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⣀⠔⠊⠢⣀⠀⠀⢀⠘⡄⠣⣀⠀⢀⡀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠰⠆⠀⠀⢁⡠⠤⠔⠮⡀⠀⠀⠒⠄⠑⠢⣀⠀⠈⠒⠤⠝⠋⠀⠀⠀⠀⠈⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠂⠀⣀⡠⠒⠁⠐⠒⡀⠀⠈⠓⢤⡀⠀⠀⠀⠈⠑⠤⢀⣀⠐⠂⠀⠀⠠⠄⠀⠀⠀⠀⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⡠⠒⠛⠤⣀⠀⠀⢠⠀⠀⠀⠀⠐⠀⠜⢉⠁⠑⢄⠀⠐⣀⡀⠭⠝⠒⢤⡀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢀⡴⠋⢀⠀⠀⢀⣌⠱⠀⣿⡇⠀⠄⢀⠒⠂⠒⢄⠀⠀⠀⠙⢯⡀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⣠⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⣿⡄⠀⣰⣗⠀⠀⢿⣇⠀⢠⠁⠀⠀⠀⢀⠳⣄⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⠁⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠂⠀⠀⢠⣿⡇⠀⣺⣟⡂⢕⣺⡒⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠤⠤⠤⠤⠤⠴⡆⠀⠀⢀⠿⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠬⠤⠤⠭⢼⠬⠤⠤⡧⠤⠬⢼⠬⠼⢤⡤⠬⣥⠤⠤⠤⠤⠬⠥⡴⠉⢂⠀⠀⠀⠀⠁⠘⡤⠤⠤⠼⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣸⡆⠀⠤⠚⠤⠀⠊⠏⠀⠤⠚⠂⠤⠀⠙⠆⠂⠉⠒⢌⠀⠀⡰⠁⣘⠀⢢⢎⡡⣀⠀⠀⠘⣄⠐⠊⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣠⠀⠔⠊⠉⠒⠀⠀⠀⢀⡠⠄⠂⠀⠀⡀⠀⠑⡄⠡⠌⠉⠉⢙⣼⡑⠬⢯⠁⠀⠉⠠⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣺⡓⠄⠀⣿⠅⠀⠀⠀⣀⠄⠒⠉⠁⠀⠀⠀⠀⠀⠠⠄⢀⠃⠀⢀⠈⠀⣸⣵⡥⣞⣾⠀⢀⣷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢺⠢⠅⠨⣿⠅⠀⢠⠊⠀⠀⠀⠈⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⣿⡇⠀⠸⠥⠚⠽⠏⠀⢸⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡪⢿⢕⡀⢠⠀⠐⠂⠀⠀⠀⠀⠂⠀⠃⠀⠈⠉⡀⠀⠠⣿⡧⠀⠐⠀⠀⣦⠀⠀⢕⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⢹⠐⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢀⠀⠀⠨⣺⡪⠀⠀⠀⢨⣿⠀⠀⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠂⠈⠢⡀⠀⠠⠀⠀⠀⠐⠀⠈⠲⢄⠀⠀⠈⢹⠉⠀⠀⠀⠜⡟⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⡈⠓⢄⠀⠀⠀⠀⠀⠁⠀⠀⠙⠢⠄⠸⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠁⠀⠀⠀⡀⠀⢀⣽⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠴⣿⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⣿⣿⡇⢙⣿⣿⣿⡓⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡄⠀⢠⡀⠀⢐⣿⣷⡀⣦⠨⣿⣿⣿⣿⣿⣿⣿⣟⠃⣿⣿⠄⢠
⠀⠀⠀⠀⠀⠀⣰⣷⣄⠀⣴⣿⣿⣿⡴⠟⣹⣿⣼⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣿⣿⣗⣿⡧
⠀⠀⠀⠀⠤⠾⠿⠿⠿⠿⠿⠿⠿⠿⠤⠾⠿⣽⣿⣿⣿⣿⣿⣿⣿⡿⠃⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣗⡀⢀⣨⣿⣿⣿⣿⣿⣿⡿⠁⢀⠀⠹⣿⣿⣿⡿⠛⣿⣿⣿⣿⣅
⠀⠀⠘⠿⣿⣿⡿⣿⣷⣿⣿⣿⣿⣿⣿⣋⣀⣾⣿⣿⣿⣿⣿⣿⣀⣠⣿⣧⣀⣸⣿⣿⣳⢶⣿⣿⣿⣿⣶⣄⣀
⠀⠀⠀⣉⣉⡉⢉⡉⣉⢉⣉⠉⣉⡉⣉⡉⢉⣉⢉⣉⣉⢉⣉⣉⡉⠉⠉⣉⣉⡉⠉⣉⡉⠉⣉⠉⢉⡉⣉⣉⣉
⠀⠀⠀⣿⣈⡋⢸⡇⣿⢸⣿⣀⣿⡇⣿⡇⣸⣿⢸⣇⣀⢸⡏⢸⡇⠀⠀⡏⠘⠃⢠⡟⣇⠀⣿⡆⣿⡇⢹⠀⣿
⠀⠀⠀⣤⣉⡿⢸⣇⣿⢸⡏⣿⢹⡇⣿⢹⡏⣿⢸⣏⣁⢸⡟⢿⡁⠀⠀⣇⣠⡄⣸⠧⣿⠀⡏⣿⢻⡇⣽⠛⠋
⠀⠀⠀⣈⣉⣁⣈⣉⣉⣈⣁⣉⣈⣁⣉⣈⣁⣉⣈⣉⣉⣈⣁⣈⣁⣀⣀⣉⣉⣀⣉⣀⣈⣁⣁⣈⣈⣁⣉⣀⣀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⡉⢉⣉⢉⢩⠍⠉⠉⡉⢉⣍⡉⣉⣩⠉⠉⢉⣍⢉⡉⣉⢉⢉⣍⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⠈⡞⢸⡂⡀⠀⣇⢸⣸⢹⠇⣗⡀⠀⢸⣀⡸⢧⡿⣾⢸⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⢉⡹⠟⠓⠲⠦⣀⠀⠀⠀⣀⡤⠤⠤⠤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠒⠛⠓⠒⠒⠛⠓⠚⠛⠛⠲⠾⠷⠤⢮⣅⣀⢀⡤⠞⣉⡥⠤⢬⣙⠲⣄
⠀⠀⠀⠀⠀⢀⡠⢖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡬⠭⠿⣦⠚⠉⠙⢦⡀⠈⠳⣌⠳⡄
⠀⠀⠀⠀⣠⠋⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠊⠳⢗⢦⠀⠈⠳⡄⠀⠀⠳⡄⠀⠘⢦⠹⡄
⠀⠀⠀⢰⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢧⠀⠀⢳⣳⡀⠀⢻⠀⠀⠀⠹⡄⠀⠈⣧⠹⡄
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠸⡆⠀⠀⢯⢷⠀⠘⡇⠀⠀⠀⢻⡀⠀⠸⡆⢳
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⢸⠀⠀⡇⠀⠀⠀⠈⡇⠀⠀⣷⢸⡆
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⢺⠀⠀⡇⠀⠀⠀⠀⢹⠀⠀⢸⠀⡇
⠀⠀⣸⠀⠀⠀⠀⠀⠀⢀⠴⠂⠠⠤⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⣼⠀⠀⣇⠀⠀⠀⠀⢸⡄⠀⢸⡆⣧
⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⠀⠀⢈⣯⡄⠀⢻⠀⠀⠀⠀⢸⡇⠀⢸⡇⣿
⠀⠀⠹⣄⠀⠀⠀⠀⢀⠱⠤⣤⠤⣤⣄⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢻⠀⠀⢀⠀⠐⠀⠸⡆⠀⠀⠀⢸⡇⠀⠀⡇⢹
⠀⠀⠀⢸⡟⠒⠲⠶⠤⠤⠤⡇⠀⠀⣯⣀⣀⣀⣀⣀⣀⣀⣀⣤⠴⠃⠀⠀⢸⡄⠀⠀⠮⠵⠀⠀⢷⠀⠀⠀⢸⡇⠀⠀⡇⣸
⠀⠀⣠⠼⠓⠒⠒⠒⠒⢒⠶⠇⢀⣨⡗⠒⠒⠶⠶⠶⠤⠤⢤⣄⣀⠀⠀⠀⠀⣇⢀⡴⡶⠒⠛⠛⠉⠛⠲⢦⢸⠁⠀⢸⡇⣿
⠀⢰⠇⠀⠀⠀⠀⠀⠀⡗⢎⣉⣁⡶⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢻⢸⣿⣿⡄⠀⠀⠀⠀⠀⠈⣿⠀⠀⢸⠃⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠛⠓⢒⠖⠋⠀⠀⠀⠀⠀⠀⠀⠘⢠⡟⣷⠀⠀⠀⢸⡜⡏⠉⣧⠀⠀⡄⢀⠀⠀⢹⠀⠀⣼⢠⡇
⠀⣿⠀⠀⡀⡀⠀⠀⠀⠀⠑⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿⠀⠀⠀⠘⡇⣧⠀⢹⠤⠤⠤⡤⠶⠒⣯⠀⢠⡇⣼
⠀⢻⡀⠘⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⣱⠄⠀⠀⣼⠀⣿⠀⠀⠀⠀⡇⢻⠀⠸⠀⠀⠀⣵⠀⠀⢿⠀⡞⢠⠇
⠀⢸⡟⠒⠶⠤⠤⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠞⡇⠀⣿⠀⠀⠀⠀⡇⢸⠀⠀⡄⠀⠀⢸⠀⠀⢸⡞⣠⠏
⠀⠸⡇⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⢀⡇⢸⠀⠀⡇⠀⠀⢸⠀⢀⣟⡴⠃
⠀⠀⢷⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⡇⢀⡏⠀⠀⠀⣸⠁⠸⣤⣀⣇⣀⣀⣽⠴⠛⠉
⠀⠀⠀⠉⠻⣗⡒⠶⠦⠤⠤⣤⣤⣤⣀⣀⣀⣀⣸⣀⣀⣀⣸⣠⠞⠁⠀⠀⣰⣃⣠⡤⠤⠤⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠶⠦⠤⠤⢤⣤⣤⣄⣀⣈⣩⣥⡤⠤⠴⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠠⣄⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡄⢠⠀⡂⠀⠀⠀⣀⣤⣠⡈⡆⠀⠀⠀⠀⢰⠀⡄⠀⠀⢀⣴⠿⢿⣦⠀⠀⠀⠀⠀⡜⢃⠀⠀⠀⠀⠀⠀⣀
⠀⢸⣿⠻⠏⣿⡇⠀⠀⠀⣈⠀⠶⠀⠀⠀⣿⣿⠛⣿⡆⠀⠀⠐⣻⡏⠁⠁⠀⠀⢰⡇⢸⢀⡇⠀⠀⠘⣿⣤⣴⡿⠀⠀⠀⠀⠋⠠⠋⠀⠀⠀⣿⣫⣿⣻⡇
⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠈⠀⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣒⢴⣲⠀⠀⠀⣠⣴⡞⢿⡆⠀⠀⠀⠉⠸⢃⡀⠀⠀⣾⢷⣶⡶⣶⠀⠀⢠⡤⡤⢠⢤⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣠⣄⣄⠀⠀⠀⠀⠴⣷⡄
⠀⠀⢶⣿⣾⣿⠀⠀⠀⣷⣴⠿⠟⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⡆⡇⢠⣾⠀⠀⠰⣷⠀⠀⣾⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⢰⣿⡇



⠀⠰⠶⠆⠰⠆⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⣀⣠⣞⣟⡷⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢠⡄⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⢀⠀⠀⡀⠀⠀⢠⠤⠤⠤⠤⢤
⠀⢀⡀⠰⢆⡰⠆⠀⠀⢠⣄⢠⣤⠀⠀⠈⢿⡾⠾⠁⠁⠀⠀⣶⣦⣶⣤⣤⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⢺⣧⣼⠇⠀⠀⠀⢸⢡⡆⡟⠀⠀⢸⠹⠀⠇⠳⢸
⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠁

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⣐⡿⢣⣤⡄⠀⠀⠀⠄⠛⣿⡆⠀⠀⣀⣀⣤⣄⡀⠀⠀⠠⠀⠀⢂⠀⠀⠀⠀⠀⢠⣮⠛⠀⠀⢠⠊⡉⣳⣀⡀⠀⠀⠀⢰⣦⡤⠀⠀⠀⣄⣰⡀⡄
⠀⠘⠿⢣⣾⠍⠀⠀⠘⣿⣦⡄⠈⠀⠀⠀⣿⡏⣶⢎⣿⠀⠀⠀⠱⣖⣻⡆⠀⠀⠀⡴⠟⠁⠀⠀⠀⠘⢄⣀⠜⠉⠁⠀⠀⠛⢻⣿⠟⠀⠀⠘⡘⢤⢼⡘
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠈


⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⣼⣿⣿⣧⠀⠀⢸⢾⡯⡭⡿⠇⠀⢀⣾⣶⣶⣾⣷⠀⠀⠠⠤⢤⠤⡄⠀⠀⢠⣼⣟⡷⣤⠀⠀⠀⠀⡖⠿⠀⠀⠀⠀⠀⡠⠚⠁⡀⠀⠀⣤⣶⠀⠀⡀
⠀⠀⠘⠾⠷⠃⠀⠀⠈⠀⠀⠁⠁⠀⠀⠈⠻⠟⠉⠻⠟⠀⠀⠠⠤⠦⠹⠄⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠶⠀⠚⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠛⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠐⠠⠢⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡄⠀⠀⠀⠀⣿⠁⣿⠀⠀⠀⠀⠘⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄
⠀⢸⣿⡇⠈⠁⠀⠀⢰⣋⣟⣹⣏⡇⠀⠀⠘⠁⠀⠁⠀⠀⠀⠀⡏⠀⢹⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠦⢀⢀⠂
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠈⠁


⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣿⣿⣼⣷⣶⣧⣼⣧⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣷⣶⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠎⠀⠀⠉⠁⠈⠓⠒⠓⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⢲⡀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣵⡾⣿⣶⣄⢷⠀⣰⠇⣠⠞⠙⢦⡲⣄⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠫⢎⡥⠖⢺⡽⡏⠁⢧⠀⠀⠀⠀⠀⢩⡄⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠈⢠⣞⣉⡤⣇⠀⠀⣶⠀⢤⣀⡀⠈⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢀⡟⠉⠈⢉⡁⠀⠀⠀⠀⣀⠈⠀⠀⠛⠻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⡴⢿⠉⠀⠀⠀⠘⠋⠀⢀⡀⠈⠙⠀⠀⠀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⡞⠁⠈⠙⢲⡀⣦⠀⢀⡤⣤⡤⢤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸⠁⠀⠀⠀⠈⢳⡀⠀⠈⠓⠚⠓⠚⠁⠸⢆⡞⠀⠀⠀⡐⡆
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠙⢲⡤⣤⡄⠀⠀⠤⢶⠋⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠙⡾⠲⢤⣀⡤⢼⡁⠀⠀⠀⠀⠀⢈⡧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⡼⠁⠀⠀⢻⣀⣠⡽⣆⠀⠀⢀⡴⣫⣀⣀⡟⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡞⠁⠀⠀⠀⠈⠁⠀⠓⠮⣷⣤⡮⠴⠏⠀⠉⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠉⠁⠀⣇⣀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠬⠀⠀⠀⣀⡀
⠀⠀⢠⣄⣾⣦⡀⣠⠤⡹⠻⣆
⠀⢀⣠⣼⣿⢿⠄⠂⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠖⠓⠘⠛⣿⢰⡄⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢈⣧⣿⣞⣋⠉⠈⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠐⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⠾⢻⡙⣿⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⣿⡃⠉⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸
⠁⠀⠸⢳⣿⢳⡦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠚⢙⢹⣧⡄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠑⠒⢰⣾⣖
⠀⣀⣴⣶⣿⠻⠇⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢾⠯⠅⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢈⣽⡟⠙⠢
⠛⠁⢘⠹⣿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠰⢸⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠘⠀⠉⠨⣿⣿⣤⡀
⠀⠀⡝⢂⣿⠄⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⢴⣾⣟⠇⠀⠙⠆⠀⠀⡠⠖⡀⣠⣤⠀⠀⠀⠀⢀⣤⢤⣾⡶⠀⠈
⠀⠀⡄⠘⣿⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⢹⡏⠃⠀⠀⠀⠀⠀⠀⣀⡀⣹⡿⠲⢄⡀⠀⠁⠀⢘⣽⡿⠖⡤
⠀⠰⠈⢤⣿⠃⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡇⠃⠀⠀⠀⠀⠀⠘⠉⠉⢿⣿⣤⡀⠀⠀⠀⠀⢀⢸⡇⠂
⠀⠐⠀⢸⣿⡂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡇⠤⠀⠀⠀⠀⠀⠀⡠⠀⢺⡏⠀⠈⠀⠀⠀⠀⡀⢼⡇
⠀⠁⠈⣺⢿⠉⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣸⡇⢈⡀⠀⠀⠀⠀⠀⠀⠀⢾⡟⡉⠂⠀⠀⠀⠀⠁⣸⡇⠰
⢸⣳⣄⢼⣾⡧⣀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⣭⣧⡀⠂⠀⠀⠀⠀⠀⠀⠀⢊⡇⠀⠀⠀⠀⠀⠠⠁⣼⡇⢀⡄
⣻⣀⣈⣻⣻⣃⣉⣹⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣏⣈⣿⣗⣉⣵⣀⣀⣀⣀⣀⣀⣀⣠⣇⣀⣀⣀⣀⣀⣸⣉⣻⣟⣉⣇
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⢀⡴⠛⢉⡻⣟⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠞⠛⠛⣯⠉⠉⠳⣏⠀⣴⠋⠀⠸⡆⠀⠸⣆
⠀⠀⠀⠀⠀⠀⢀⡴⠞⣩⠞⠉⠀⠀⠀⠀⢸⣆⣤⣤⡽⠶⠷⠶⠖⠒⠓⠒⠲⠾⢦⣄⣀
⠀⠀⠀⠀⢀⡴⠋⢀⡼⠃⠀⠀⢀⣤⠶⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⢠⠞⠀⣠⠟⠀⠀⠀⣴⣋⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⣰⠏⠀⣸⠏⠀⠀⠀⣼⡟⠉⠁⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⢠⡏⠀⣰⡏⠀⠀⠀⣸⡏⠀⠄⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⡾⠀⠀⣿⠀⠀⠀⢀⡿⠀⠈⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢰⡇⠀⣸⡇⠀⠀⠀⢸⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⣧⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⠀⠀⣿⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡏⠀⠀⣿⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣇
⡇⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠠⡀⠀⠀⠀⢠⠞⠋⠙⣧⠀⠀⠀⠀⣼⠃⢻⡄
⣷⠀⠀⢹⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠹⣄⠀⠀⠁⠀⠀⢸⡄⠀⠀⢹⠚⠛⠛⠋⠁⠀⠈⣧
⣿⠀⠀⢸⡆⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⠦⠤⠴⠞⠛⣷⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢹⡄
⢸⡀⠀⠸⣇⠀⢀⡟⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠛⠛⠀⢀⣀⣤⡤⠶⢤⡈⣧
⠘⡇⠀⠀⣿⠀⢸⠇⠈⠇⠈⠉⠉⠉⠁⠀⠀⠀⢸⠃⣿⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⡴⠶⠛⠋⠉⠀⠀⠀⠀⢳⢹
⠀⢻⠀⠀⢻⡆⣾⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⣿⠀⠀⢹⡏⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⡇
⠀⠘⣇⠀⢘⣷⡇⠸⣶⠶⠶⠶⠦⢤⣤⣄⣀⣀⢀⡷⣿⠀⠀⢸⣇⠀⠀⠀⠀⠄⠀⢀⣀⣈⣤⣤⣤⡤⠶⠺⡇⠈⡇
⠀⠀⢻⡄⠀⢸⠃⠀⡟⠀⠀⠀⠀⠀⠀⠈⠉⢹⢻⠃⣿⠀⠀⣾⢧⣀⣤⡤⠶⠞⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⢻⡀⡇
⠀⠀⠀⣧⠀⡿⠀⢠⡇⢀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠀⣿⠀⠀⣿⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢧⡇
⠀⠀⠀⠘⣦⡇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⠀⣿⠀⣸⠃⠀⠀⢀⣀⣠⣤⡤⠶⠶⠶⠚⠛⠋⣡⡞
⠀⠀⠀⠀⢸⡇⠀⠘⢷⣄⣀⣀⡀⠀⠀⠀⢀⡿⢸⡇⡟⠀⠀⠉⠓⠿⠶⠶⠛⠋⠉⠀⠀⠀⣀⣀⣠⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠓⠋⣸⠇⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠦⣤⣤⣄⣀⣀⣠⣴⣯⣤⡤⠶⠖⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠭⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠈⠈⢳⡀
⠀⣀⡤⠤⣤⣀⠀⠀⠀⠉⠓⠦⣄⠈⣇
⡞⠁⠒⠒⠂⠉⠓⢦⡀⠀⠀⠀⠈⢧⣿
⢧⡀⠀⠀⠀⣀⡀⠀⠙⣆⠀⠀⠀⠸⡏⠀⠀⠀⢀⣠⣀⣠⣤⣤⣀⡀
⠀⠉⠙⠋⠉⠉⠉⠙⠲⣌⣧⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⢠⡏⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠙⣆
⠀⠀⣠⢚⣛⠛⠒⠶⢤⡀⠀⠀⠀⠀⠀⢸⡇⣠⠴⠛⠉⠀⠈⠳⣤⣀⠀⠀⣹
⠀⠘⣇⠁⠀⢀⣤⠴⠦⢭⣷⡄⠀⠀⠀⣼⠛⠃⠀⠈⠓⠚⡁⠀⠀⢉⠛⢶⠏
⠀⠀⠈⠓⠛⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠹⣦⠀⠀⠀⠉⠉⠀⠀⠘⠬⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠶⠦⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣟⠻⢦⣀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⠉⠁⠀⠉⠙⠛⠛⠛⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⡛⠶⠦⢤⣤⣀⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠉⣻⠋⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠘⣧⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⡀⣠⡏⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⢸⠿⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⡿⣋⡴⠳⠦⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⣀⠀⠀⠀⠈⢷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⢀⣤⣄⣉⣙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢲⡏⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠓⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠉⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢶⣿⣿⣿⠿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠏⠄⠀⠘⠋⠃⠀⠀⠿⡿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢉⠩⠉⠍⠙⠭⠉⠂⠀⠀⠐⠀⡰⠁⠸⠿⣿⠿⡿⠓⠒⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣾⣷⣤⠑⡦⠾⣹⠄⠈⠁⢈⡀⠚⡼⠃⠂⠀⠀⠘⢗⡙⣛⢛⢂⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢾⣿⣿⣿⣿⣷⠓⡲⢔⡂⣐⠃⣀⠀⡜⠀⠀⡀⠀⠀⠀⠈⢆⠀⠐⠚⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠿⣣⢤⡔⣽⡙⠃⠛⠠⣄⣀⣀⣀⠀⠀⠀⠈⠶⣻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣽⣿⣿⡏⡹⣇⣜⣿⣿⣧⠀⠀⠀⠀⢠⠞⡍⠂⣄⠀⠀⢀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡟⣿⣿⣷⢳⣾⣿⣿⢿⣿⣆⠀⠀⣰⡇⠀⠘⡄⠈⢦⣦⣿⢻⣏⣿⣿⠶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣛⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣦⣾⣿⡇⠀⠀⢰⡀⠀⢻⣵⣾⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣹⣿⣿⣿⣿⣿⣿⣇⣐⣿⣟⣻⣿⣿⡿⣯⣀⣀⣠⣧⣼⣦⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣟⣋⣽⣿⣿⣭⣥⣶⣷⣿⣿⡿⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠛⠿⠿⠿⣿⡿⢿⣿⣿⣿⣿⠿⠿⣿⠿⠿⠿⠟⠋⠁







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣤⣴⣶⡟⠉⠹⣿⠀⢨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢊⣉⣩⡍⠙⠛⠻⣆⠸⣿⣧⣼⣿⣗⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢁⡒⠛⠛⠿⢿⣧⢠⣆⣿⡛⣯⡭⠏⠛⠉⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡈⣽⣵⣁⡃⢀⣸⣿⣦⡿⣿⣿⣷⣶⣶⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣿⣿⣿⣿⢿⢺⡇⣿⣯⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⣸⣘⣿⣿⣿⣿⣿⣿⣿⣿⠻⠀⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠈⢁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡟⡕⡻⢻⣿⣿⣿⣿⠙⢿⣍⠢⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣹⢏⠌⠀⠉⣟⢅⠙⣆⠻⣢⣽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡎⣀⡀⢀⣬⣺⣶⣌⣷⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣷⣶⣾⣿⣿⣿⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⡿⣿⢸⣿⡟⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣷⣿⣽⡿⣧⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⠹⡇⣿⣿⢹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢿⣿⠀⢘⡇⠻⠋⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣾⡇⠄⢸⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⠇⠀⢸⡄⡄⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⢸⠀⠀⠈⡇⡇⠀⣿⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⣷⡇⠀⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⢸⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠘⣿⠀⠘⣷⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⢹⠀⠀⢸⡼⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣃⣀⣀⣀⣀⣀⣈⣀⣀⣀⣓⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠗⣠⡂⠶⠀⡏⢤⠸⠰⠇⠰⢲⣦⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⡀⢘⣥⠬⠗⠚⠛⠛⠛⠛⠛⠛⠳⠦⣮⣸⡕⢃⢗⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⢄⣬⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣾⣇⠁⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠤⣹⠞⠁⠰⣆⣀⠀⠀⠀⠐⣿⣆⣀⣤⠰⣶⡀⠀⠀⠀⠀⠀⠉⠳⣤⡋⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⠏⠀⠀⠀⢇⠉⠁⠀⠀⠀⠀⠉⣁⡀⠀⠹⣷⣄⠀⢀⠀⣀⠀⠀⠈⠻⣄⡴⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣛⡇⠀⠀⢀⣤⣾⡆⠀⠀⠀⠀⠀⠀⠙⠿⠦⠐⠋⠉⠀⢘⣧⡿⣁⠀⠀⠀⠈⣧⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡥⢸⠀⠀⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⡀⠈⢳⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣠⣤⣀⠀⣀⣠⡄⢰⣾⣿⣿⣶⡶⠂⠀⠀⠻⠿⠃⠀⠈⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⣧⣶⣿⣿⣿⣿⣷⠌⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⡿⣆⠀⠘⣟⠛⠉⢻⠁⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢺⡟⠹⠀⠀⢰
⠀⠀⠀⠀⠀⠀⣀⣀⡀⣠⡄⠀⠒⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠈⢧⠀⠈⠳⠶⠟⢠⡟⠁⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⣨⡅⢴⠖⠀⠸
⠀⠀⠀⣠⡴⠶⠿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⡾⠀⠀⠀⠸⡇⠀⠀⢸⣦⡀⠀⣀⡀⡈⠉⠀⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠻⣤⡀⠀⣴⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠤⠞⠁⠀⠀⠀⠀⢷⠀⢀⠸⠆⠀⠀⠉⠻⠉⠀⠀⠀⠀⠀⠀⢰⠁
⠀⢴⡞⠛⠋⠀⠀⠈⣳⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣤⣶⣄⠀⠀⠸⣄⠿⠗⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⣠⠏
⠀⠀⠙⠳⣦⠀⠀⢼⡁⠀⠀⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⡴⠃
⠀⢸⣿⠉⠉⠀⠀⠀⠉⠉⠒⢦⡙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠙⢶⣄⣀⠀⠀⠀⢀⣀⣤⠶⠋
⠻⣏⣉⠀⠀⠀⠀⠀⢀⣔⠒⠉⠀⠀⠙⢿⣿⣷⣶⣤⣀⡀⠀⠀⠈⠉⢻⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠉
⢠⡏⢍⠙⠛⠲⢲⡆⠀⠉⠛⠓⠒⠠⣤⡀⠈⠙⠻⢿⣿⣿⣿⣷⣦⣤⣾⣿⣿⣿⣶⣦⣤⣀
⡼⠃⠉⠀⠘⠆⣾⠁⠀⠀⢈⣿⡏⠉⠁⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⡀
⢧⠺⣤⣂⣠⣶⠛⠚⠛⠉⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠉⠉⠛⠛⠿⢿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣀
⠘⢧⣈⣉⡴⢻⣶⠏⠀⠶⠆⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⢿⣷⣦⣄
⠀⠀⠉⠉⠀⠈⣇⠀⠧⠄⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠈
⠀⠀⠀⠀⠀⠀⠻⣌⡓⣋⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧⡀⣀⣀⣤⣤⣴⣶⣶⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣤⣶⣶⣶⣶⣶⣶⣶⣿⣿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠟⠛⠉⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣒⢒⡒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⠎⡔⠁⠀⠀⠈⢆⢱⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⢀⣉⢦⡀⠉⢢⠀⠀⠸⡀⡇⠀⠀⠀⠀⡜⢸⠀⠀⢀⠔⠊⢡⠔⠯⡁⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⡠⠚⠁⠀⠀⢱⠀⠀⣇⣀⠤⠷⠷⠤⠤⠤⠴⣥⡃⠀⡰⠃⠀⡰⠃⠀⠀⠈⢆⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠞⠀⠀⠀⠀⠀⡨⠗⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠣⣄⢠⠃⠀⠀⠀⠀⠈⢆⠀⠀⢳
⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⠎⠀⠀⠀⠀⡰⠋⠀⠂⢀⠤⠒⠊⠉⠉⠉⠉⠒⠢⢄⠀⠀⠂⡈⠻⡀⠀⠀⠀⠀⠀⠸⡄⠀⠈⡆
⠀⠀⠀⠀⠀⡏⠀⠀⠀⡎⠀⠀⠀⠀⡜⡀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠈⠂⠱⡀⠀⠀⠀⠀⠀⡇⠀⠀⢳
⠀⠀⠀⠀⢸⠀⠀⠀⣸⠁⠀⠀⠀⣸⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠁⣇⠀⠀⠀⠀⠀⢰⠀⠀⢸
⠀⠀⠀⠀⡟⠀⠀⢀⡇⠀⠀⠀⢠⡏⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⢹⡄⠀⠀⠀⠀⢸⡄⠀⠸
⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢇⠀⠀⠀⠀⠀⠃⠀⠀⡅
⠀⠀⠀⢠⠃⠀⠀⡌⠀⠀⠀⢀⡇⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠈⠒⠢⠤⠤⠤⢀⣀⣤⣤⣤⣤⣄⡠⠤⠤⠤⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⢠⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢠⠀⠀⠀⠁
⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⣀⡇⠀⢠⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠑⡄⠀⢸⡤⡀⠀⢸⠀⠀⢸
⠀⠀⠀⢸⠀⠀⢸⠀⠀⡏⢸⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⡆⢸⠀⢸⠀⠀⢸
⠀⠀⠀⠈⡄⠀⢸⠀⢸⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⠀⡆⠀⠀⡟
⠀⠀⠀⠀⡇⠀⠈⡆⢸⠀⠸⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡇⢸⢠⠇⠀⢸⠇
⠀⠀⠀⠀⢷⠀⠀⢣⢸⠀⠀⠀⠀⡏⠓⠒⠒⠒⠂⠀⠤⠤⠤⡤⠤⠤⢤⠄⠀⠐⠒⠒⠚⠛⠓⠉⢹⠀⠀⡇⢸⡼⠀⠀⡞
⠀⠀⠀⠀⠘⡀⠀⠘⣾⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡇⡸⠁⠀⡸⠁
⠀⠀⠀⠀⠀⢳⡀⠀⠘⣄⢸⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡏⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠈⢻⠀⠀⠈⠓⠢⠤⠤⠤⠄⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠤⠤⠚⠀⠀⢠⡥⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠒⠒⠒⠒⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣴⣾⠛⠃⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢪⣾⣿⣿⡁⠀⡄⢉⠸⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⣿⣿⣿⣧⠀⠁⡠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⢿⣿⣿⡟⢀⣴⡅⠰⢐⠖⠁⣹⣖⠈⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠑⢾⣻⣧⠀⡚⠚⠉⡽⢌⣁⣁⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠛⠟⠀⠈⠲⠊⠀⠀⠀⠉⠀⠀⠀⠀⠐⠒⠒⠒⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠋⠁⠀⠀⡄⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢩⠀⠀⠀⠀⠄⠀⢢⣄⠀⠀⠀⠈⠀⢰⠀⠀⠀⠀⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⠀⠀⠀⠀⠠⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠁⠠⠄⠀⠃⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⠀⠀⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡰⠁⠔⠚⠈⠈⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠧⣀⠀⢆⣀⡠⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠉⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠖⠒⠚⢱⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠹⠉⡗⠒⠂⠀⠀⠀⠀⠀⠀⠀⠤⣴⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⠃⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠄⡀⡌⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠚⠁⠘⠧⢔⡒⠠⠤⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⡀⠀⢠⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣁⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⡚⣃⡭⠳⡤⠋⡆⢹⠀⠀⠒⡖⠆⠀⢀⡠⠊⠓⢄⡠⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⢐⡇⠑⢄⠀⠀⡇⠀⢣⠀⢀⠀⠀⢠⠁⠀⢥⠂⡄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢸⠁⠹⣯⡧⢠⠇⠀⢸⡆⠉⣀⠀⣸⡀⣀⠈⠘⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢒⠞⠀⠀⢀⣙⣞⣡⣴⠲⣻⠙⣯⣽⣼⢛⠋⣫⢀⣉⢠⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠹⠀⠀⠙⠛⠿⠿⠿⣯⠴⣷⣾⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⠿⠿⠿⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠍⠁

⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⢀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀⡀⠀⢀⠤⠒⡤⡀⠀⢀⡠⠤⣀⠀⢀⡀
⠀⠀⢸⠏⢳⣝⣿⣦⣙⣷⣌⠻⣿⣕⡄⠈⣁⠤⠐⠒⠉⠁⢡⡁⡇⠀⠘⠻⠅⠀⡓⢓⣈⣅⢡⠔⠡⡄⢖⡄⠱⡈⠉
⠀⠀⡾⢄⣀⣦⣤⢀⣀⠀⠈⠉⠉⠉⠁⢂⢀⠀⡀⠀⠀⠀⠘⠞⡇⢸⠲⠤⣔⣒⠽⣼⣧⠔⠉⡇⠉⡥⡠⣒⢂⠇
⠀⢠⠁⠉⠙⠃⠶⠾⣛⣩⣵⡶⠦⣤⡄⠀⠉⡏⡝⠉⣀⡒⠒⠓⠁⠘⠚⡲⠢⢌⡉⠑⠺⣿⠒⠸⢦⣭⡡⠴⠊⢀⡀
⠀⡌⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠛⠫⠃⠀⢀⠑⠃⠀⠉⣁⢀⡀⠀⠀⢠⠃⠀⠀⠀⠉⠡⠀⠀⠀⠀⠀⠈⢳⠀⠛⠃
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠽⢿⠃⡤⠂⠉⠀⢀⠈⠉⠢⣌⠒⡦⠤⣀⡀⠀⠀⢀⠀⠀⠀⢀⠎⠉⠁⡹
⠀⢀⣍⠁⠒⠠⠤⢁⣀⠀⠀⠀⠀⡇⠀⠀⢀⠎⠀⠀⢆⠀⠸⠁⢀⡠⠈⢣⠈⠁⡒⢩⡝⡒⠤⢌⡀⠀⣜⡀⠀⢰⠁
⠀⠘⠉⣀⡤⠔⠊⠠⣀⣈⠉⠒⠒⠀⠀⣠⡿⠀⣀⣀⠠⠆⣘⠂⠁⢀⡀⠈⡆⠹⣛⣀⣉⠐⠚⠓⢡⣌⠀⠈⠲⠃⣤
⠀⢠⠊⡀⠀⠐⠑⠂⠀⠌⢳⠀⠲⠖⠛⠁⢠⡀⠀⢀⣠⢴⡀⠺⡁⡡⠁⢀⠇⡴⠇⠤⢈⠓⡄⠀⡠⡔⠂⠀⢲⣄⠉
⠀⢹⣦⠤⣀⣁⣀⣀⣀⣀⣼⠀⢠⠿⠴⣄⢄⢢⡀⠉⠒⠰⠇⠀⠑⠁⢀⠜⠀⣇⢐⡒⢡⡅⡇⢔⠑⠒⠒⣂⡭⡊
⠀⠸⣿⠀⣿⠀⣿⠇⣿⢸⠇⢀⡖⢔⣒⠨⡉⠃⠙⠢⢄⣀⣀⣀⡠⠔⠃⠰⠀⣿⢨⠬⢸⡇⢱⠀⢷⢋⣴⢟⣴⢷
⠀⠀⣿⡄⣿⠀⣿⢰⡏⡞⠀⡾⢁⡨⠙⠃⡉⠀⡀⠖⠠⢀⠶⠰⣖⠀⠉⣿⠀⢫⠃⠫⠄⡧⢸⠀⠘⣟⣵⠟⣡⠾
⠀⣤⠙⠓⠻⠶⠿⠾⠓⠁⣸⠐⢌⠛⣣⡐⠀⡾⠂⡖⠄⡘⠄⠀⣿⠆⠀⢻⡆⠘⢄⡀⣓⣠⠚⢠⡄⠘⣥⢞⣥⡾
⠀⣉⣤⣀⣠⣄⣠⣒⣂⣄⡈⠐⠀⠀⡰⡠⠊⠀⠠⠛⠂⠌⣃⣄⢩⣇⣀⣸⣇⡀⠀⣍⠉⠀⠀⢠⡀⠀⠀⠉⠀⢤
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣠⠀⠘⢠⣤⢄⠀⠀⠀⠀⠉⢉⠾⣿⣴⣟⣻⢆⢀⣸⣧⠰⠆⢠⣀⣀⡀⠀⠀⣠
⠀⠉⠢⠠⠠⠂⠒⠤⠤⠔⠊⠘⠛⠁⠘⠠⠀⠀⠈⠑⡇⢀⠔⠁⢦⢆⠀⣤⠈⣿⠀⠀⢿⡄⢰⣽⡃⠀⠀⠀⡀⣹⠢
⠀⠐⠂⠒⠂⡆⠀⠀⠀⢀⠀⠀⠀⠐⠈⠐⠐⠉⠁⠀⠉⠁⡀⢴⠟⢢⠠⠁⠀⠶⡇⠀⢹⡇⢰⣏⢁⣀⠀⢤⣠⠀⣽
⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠐⠦⡀⠀⠀⠀⠀⠐⢿⢶⠶⠥⢐⠡⠀⠠⠱⠑⡀⢸⣥⡀⠸⠗⣀⡙⢦⠑⣷⡺⢃⡴⠁
⠀⢰⠀⠀⠐⠀⣸⣿⣷⣿⣬⣀⣵⠄⠃⠀⠀⡄⠈⢢⠣⠀⠀⠰⢐⡂⡂⠱⠝⠀⠀⣠⠚⠁⠐⡵⡌⢻⣟⢻⣟⠐⠗
⠀⢸⣄⡠⠂⠀⠸⣄⡜⢛⣻⣿⠏⠀⠈⢀⣀⡇⠀⠀⢣⠑⡈⠚⣉⠡⠌⣁⠂⠀⢸⡠⠉⢀⢀⡤⣧⠋⡹⢧⡑⢷⡀
⠀⢸⠻⣄⠀⠀⠀⠈⠉⠸⠋⠁⠀⠀⠀⣠⠟⡇⠰⣿⡇⠛⠊⠀⢈⢦⣤⣤⠚⢥⠺⠓⠭⠔⡉⠀⢡⡜⠁⠀⠳⣾
⠀⠈⠓⠚⠓⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠒⠂⠁⠀⠈⠙⠂⠀⠘⠋⠀⠊⠉⠉⠉⠁⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢀⢻⣴⣖⣸⠦⡀⠀⠀⠀⠀⢀⠠⠤⠀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⡞⠸⢻⡀⠀⡇⢰⠀⠀⡠⠊⠀⢀⠔⠫⡀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⡤⠄⠘⠃⠒⠐⠃⠠⢥⡀⣆⠞⠀⠀⡰⠁⠀⠀⠘⡄⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⢉⡈⠓⠒⠒⠀⠦⠀⡀⠀⠀⠀⠉⠢⣀⡰⠀⠀⠀⠀⠀⠰⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠒⠉⠁⠀⠀⠀⠉⠐⠢⢄⠀⠀⠀⠠⠄⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠇⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⠀⠀⠀⢵⡆⠀⠀⠈⢄⠀⠀⠀⠀⠸⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠘⠁⠀⠀⠈⠁⠐⠂⠄⢀⠀⠀⠀⠀⠑⡀⠘⡛⢻⣇⠀⠀⠈⡄⠀⠀⠀⠀⡆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣠⣄⠀⢠⡀⠠⠄⣀⠀⠀⠀⠀⠉⠶⣆⠀⠐⡀⠀⠘⢯⡇⠀⠀⢘⠀⠀⠀⠀⢁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠈⠀⠒⠨⢐⡢⠄⠄⠀⢋⠇⠀⠁⠀⠀⠀⢤⠀⠀⠀⡆⠀⠀⠀⠈⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠄⠀⡀⠀⠘⠀⠀⠀⠐⠀⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⡆⠀⠀⠀⣇⠀⠀⠈⠀⠀⠀⢸⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠸⠐⠂⠖⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠁⠀⠀⠀⣠⠀⣀⠀⡇⠀⠀⠀⢄⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠀⢸⠀⠀⠀⣗⠒⢹⡇⢀⠀⠀⠀⢎⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠀⣀⠀⠀⠀⠀⠁⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⣻⡄⢸⠀⠀⡠⠉⠉⠈⠣⣸⠀⠀⠠⠉⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⠀⠒⠀⠨⠁⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⡇⠈⠀⠀⡇⠀⠀⠐⠁⢹⠀⠀⡌⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠠⢁⡐⠂⠄⠀⢀⣀⣈⡤⠂⠀⡆⠀⠃⠀⠀⠀⠀⣿⣷⡼⠁⣼⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠉⠒⠠⠄⣀⠀⠀⠀⠀⣷⡷⠦⠄⠀⢀⠀⠼⠿⠇⢰⠿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠓⠀⠀⢿⡛⣧⡖⠺⣼⠁⠺⣷⢇⠞⠀⠳⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠊⠁⠀⠀⠀⠘⠷⡏⢑⠴⡉⠲⠧⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⠠⡆⡟⠣⢀⡘⠃⠖⢸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣥⢧⡔⠁⢈⠟⢂⡾⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠭⠅⠒⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠠⢀⠀⢸⠀⠹⢦⡗⠋⣣⠴⢷⢷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠨⠄⡂⠤⠄⠠⠶⠊⠀⢘⠀⠠⠤⠛⠛⠛⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⢾⡷⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣡⢾⡷⠌⠉⠁⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⢺⡷⣼⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣠⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣸⡷⢾⣿⣥⠀⠀⣀⠀⠀⡴⠿⠃⠀⠀⠀⠀⢈⣽⣿⣷⣍⡀
⠀⠀⠀⠀⣠⡄⠀⢀⠀⠀⠀⠰⠾⢛⣽⡇⠀⠀⠀⢀⣼⣉⠹⣄⠀⠀⠈⠀⠀⠀⠀⢉⣡⣾⣷⣍⠁⠀⣰⡄
⠀⠀⠀⣾⣿⣯⣠⣿⣦⡀⠀⠀⢀⡉⢸⣧⣼⣧⣀⣾⣿⣿⣥⣽⣷⣄⠀⠀⠀⠀⠐⠛⣫⣿⣯⡛⠷⢾⣿⢿⣆
⠀⠀⢉⣩⢼⣷⢞⣿⣿⣇⡀⢠⣾⠷⢻⡇⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠴⠾⢟⣿⣟⠿⣷⢟⣿⡙⠿⠤
⠀⠀⠉⠁⢸⣷⢟⣿⣆⠉⠁⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⣶⠿⣿⡿⢷⣾⡏⣿⣻⣶⣆
⠀⠀⣠⣿⢾⣧⣾⣿⡿⣷⣆⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣦⣾⢟⣿⣤⡈⠉
⠀⠀⣩⣵⢿⣯⣾⣿⡿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡻⢿⣏⣿⠿⣿⡄
⠀⠀⠀⣠⣼⣯⣾⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡝⣆⠈⠉⠛⠛⠛⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣷⣤⡀
⠀⠀⠀⣀⢸⣯⣵⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⠟⢰⣇⠹⡷⡀⠀⢀⣀⣀⣀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣭⡁
⠀⠀⣴⡿⢾⡇⠶⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣾⣿⠀⠘⣟⢆⠈⢿⣿⣿⣧⠘⣿⣿⣿⣿⣿⣿⡇⣿⠳⣤⡀
⠀⠀⠀⠀⢸⡇⡀⣼⡜⣿⣟⣿⣿⣿⣿⡿⠃⠀⣼⣿⣿⣧⠀⠈⢻⣷⣮⡉⣉⠉⠁⢈⣻⣿⣿⣿⣿⣇⣿
⠂⢀⣰⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣾⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣿⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣍⠁⠀⠀⠀⠀⠀⠀⠤⠄⢀⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⢹⣿⣿⣏⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣟⣰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡏⣴⣶⣮⡙⢃⣩⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣼⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡿⢿⣿⣿⣿⣿⣿⣷⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠹⣌⡿⠟⠻⠿⠿⠿⠅⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⣹⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⢀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⣠⣴⡟⠁⠀⣀⢠⣦⣞⣿⣶⡴⣷⣞⣿⣴⣿⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⣾⡏⢰⣾⠿⣻⡄⣼⣆⣿⣿⡿⣸⣏⣿⠱⡿⢸⣿⠿⠧⢺⣿⢁⣷
⠀⠀⠀⠀⠀⠀⠀⢿⣄⡤⣤⡴⣿⣷⠻⣟⣟⡘⠃⠻⠏⠀⠀⠀⠀⣉⠀⢠⡜⠷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣹⡇⠉⣡⡾⢋⣽⠇⠀⡀⢠⣄⣼⣿⣾⣿⢃⣿⡷⠛⣷
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣴⡿⠁⣼⡟⠲⠟⣵⡟⣿⡟⣾⡿⢸⡟⣼⢇⣾⣿⣡⣴⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣸⣿⠃⠀⣸⣿⡿⢿⠟⣉⣥⣴⣶⡝⢫⣿⠇⠉⠉⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣀⣼⠏⣈⣴⠶⠛⠛⠉⠉⠉⠁⢼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣀⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⠏⠀⠀⠀⠀⠈⣿⠇⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡄⢠⣿⣿⣿⣿⣿⡀⢸⣿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢋⣤⠀⡟⠀⠀⠀⠀⠀⠀⢻⠆⣤⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⠞⣿⠁⣿⣿⣿⣿⣿⣿⣧⢸⡟⢶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⡄⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣆⢹⣿⣿⣿⣿⣿⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⣀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠿⠿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠛⠀⠀⠉⠉⠀⠀⠛⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⢛⠉⢡⣿⣿⣿⣶⣿⣿⣧⡍⠙⡛⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⡞⠉⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠉⢳⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢋⣴⣿⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣷⣎⢻⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠁⠀⠀⢸⣿⣿⣦⣀⣀⣀⣀⣤⣿⣿⡇⠀⠀⠀⣷⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⢸⣿⣿⣿⠀⠀⠙⠿⠿⠿⠿⠟⠁⠀⢸⣿⣿⣿⡆⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀
⠀⠀⠀⠘⠛⠛⠛⠻⠗⣀⡀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣤⣾⣿⣷⡀
⠀⠀⠀⠀⠀⣤⣀⣀⣰⣿⠷⠶⠿⠿⢿⡏⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣀⣤⣴⡆
⠀⠀⠀⠀⠀⣿⠿⠿⢿⣿⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠈⠙⣻⣿⣿⣿⣿⣿⢶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⢿⣿⠏⠉⠈⠁⠉⠉⠛⠿⠇
⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠖⠀⣤⣽⣿⣧⡄
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⣯
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢴⠆⢶⠆⢠⣾⣷⣶⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿⡀
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⣾⠇⠀⢹⡇
⠀⠀⠀⠀⠀⠈⠉⢻⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠰⣦⠈⠻⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡯⢀⣤⠄⣠⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⢤⣶⣾⣿⣿⣿⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⢀⠔⢹⠑⢄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠻⣘⡿⠿⠿⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠂
⠀⠀⠀⠀⢀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣶⣶⣶⣶⡆⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⣀⣀⣀⡀
⠀⠀⠀⠀⢹⡿⣿⣿⢿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠯⠿⠿⠽⠽⠽⠿⠿⠯⠯⠯⠿⠿⠽⠽⠽⠽⠿⠇
⠀⠀⠀⠀⢸⡁⢀⣀⢈⣿⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣼⣿⣼⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣶⣶⣶⢲⢲⢲⣶⣶⡖⡖⣶⣶⣶⣶⢲⣲⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣒⡼⠿⡿⠿⢿⣒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣡⠞⠁⠊⠉⠉⠙⠲⡝⣦⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢰⡏⠀⠀⠀⠀⠀⠀⠀⠹⣾⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣞⣀⣀⢀⣀⣀⣀⣀⣀⡀⣹⠀⢸⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⠁⠀⠀⠈⢉⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⣌⠀⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠀⠀⠀⢠⠟⠉⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⡁⠲⢈⠇⠀⠀⠀⠀⠀⠀⠀⠀⡖⣀⠐⢺⠀⠀⢠⡏⡶⠂⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠞⠤⢀⠀⠀⠉⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠰⠏⠀⠀⡼⢰⡇⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠰⠀⠀⠙⠉⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⠁⠳⠤⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⠀⠀⠀⣠⡞⠁⠀⠀⢀⠒⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣦⡤⠶⠚⢻⠁⠀⠒⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⣠⠄⠀⠀⠀⠐⠚⠋⡄⠀⠀⠀⠈⠉⠁⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⡏⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠶⡇⠀⠀⠀⠀⠀⣄⠀⠈⣆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢟⠀⠀⢀⡇⠀⢸⠃⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣄⡀⠀⠀⠀⣿⠙⠛⠁⠀⠉⠉⠉⠉⠉⠉⠑⠒⠒⢻⡄⢿⠀⠀⢸⡇⠀⠸⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⠿⡇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣼⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠱⠄⣏⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠊⣧⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡆⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠉⢉⡽⣛⣲⡄⠀⠀⠉⠙⡐⠀⠀⣠⡿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⣾⡿⡿⠛⠁⠀⠀⠀⠀⠈⢹⡛⣿⠀⠀⣻⠦⣤⣀⣀⣀⡤⠖⠋⢡⠅⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⡀⠀⠀⠀⠀⢿⠁⡇⠐⠂⠒⠠⠤⠄⣀⣠⣧⢾⠀⠀⣿⠀⠀⠈⣉⡉⠄⠀⠀⢸⠁⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠙⡋⡟⠒⠛⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢹⠀⠀⠀⠘⠀⠀⠀⠀⢘⡀⠀⠐⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⡇⠀⠀⠨⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⣠⣴⡭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⢠⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠓⢺⢸⡄⣄⣀⣸⡴⢃⠀⠀⠀⠀⠀⠀⢸⡇⡼⠀⠀⢸⠀⠀⢀⡀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⣈⣀⣀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠒⠚⠋⠀⠀⠀⣾⠀⡤⠞⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠶⠶⠤⣤⣤⣄⣀⣀⣀⣀⣠⣿⣦⠶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠋⠉⠀⠀⠀⠀⠀⠈⠉⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠻⠦⠄⠀⣤⣞⣧⡀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⢠⡿⠁⡀⠀⠁⡀⣴⣤⠀⠀⠀⡠⠊⣼⣿⡿⣿⣦⡀⠀⠀⢠⣀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⢀⡿⢁⠀⠀⠀⠀⠀⠠⣄⢠⣤⠎⠀⣸⢋⠿⣿⣿⠟⣷⣦⣀⣈⠉⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⣼⢇⣿⡀⠀⣠⡖⠛⢪⡿⠃⢠⣤⠞⠁⢀⠄⠈⠛⣶⣿⣿⣿⣿⡷⣄⣀⣧⣼⣧
⠀⠀⠀⠀⠀⣿⣸⣿⣧⢮⣿⣯⣴⢿⣇⢚⣻⣇⡀⠀⣸⡀⢰⢠⣿⠌⠛⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣌⣿⣿⣧⣼⣿⣼⡿⠿⠃⠛⡟⠳⠾⣿⣿⣗⠀⣻⣿⡿⢘⣿⣿⡏⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣥⣿⣿⣿⣿⠟⠁⠀⠀⠀⢸⣿⠀⠀⠈⠙⢿⡫⣽⣿⣿⣭⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠃⣼⣿⣿⡇⠀⣾⣿⡆⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⢁⢀⣛⠛⢛⣣⣾⣿⣿⣿⡄⠀⠀⢀⠀⠻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣄⣤⣼⣿⣶⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢠⠖⢫⠗⠋⠉⠉⠻⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⡀⠙⢆⣠⠃⢀⠏⠀⠀⠀⠀⠀⠘⡎⢣⠀⣠⠞⠉⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢰⠙⡆⢠⣏⠠⠞⠀⠂⠀⠒⠂⠐⠠⣹⠈⡷⠁⢠⠞⡇⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠀⡸⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣀⠋⠀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢸⠎⠀⠀⠀⠀⢀⣀⠤⠤⠀⠀⠤⢀⡀⠀⠀⠈⢀⠀⠰⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠋⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠡⡤⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣠⠂⠀⠀⣰⠒⠒⠒⠒⠒⠒⠒⢦⠀⠀⠈⢆⠀⠱⡐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⣰⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⠈⢂⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡸⠀⠀⠀⠀⠀⣇⠐⠀⠀⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠰⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⢻⠷⠲⠶⢦⢤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣤⣤⣤⡤⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠆⢡⢤⣀⣀⣀⠀⠀⠈⠈⠉⠈⠉⠉⠁⠉⠈⠁⠀⠁⢈⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠓⠒⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠛⠛⣻⠟⠀⠀⠐⣟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⢳⡄⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⢷⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣾⠆⠀⠀⣶⠚⠇⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⢀⣠⠞⠃⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠈⢧⡠⠿⠴⠶⢚⡿⠃⠀⠀⠀⠀⢠⡶⠦⢴⣿⠄⠀⠀⠀⢴⡿⠷⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢳⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣁⣀⣤⡄⠀⠀⠀⠀⢀⡀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡴⠋⠀⠀⠀⠀⠀⢿⠋⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⡈⢿⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠹⣟⠒⠢⠿⢇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣰⠟⢷⣄⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣠⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣩⡿⠋⠀⠀⠀⢀⣼⠏⠀⠀⠙⠳⣿⣄⡀⠀⠀⠀⠀⠀⣀⣠⣶⣿⡿⢻⠇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠈⢿⡿⠛⠛⠒⠛⠉⠉⢁⣴⡟⠁⣿⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⣀⣀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⣠⡾⠋⠀⢰⡏⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠉⣉⡭⠟⠉⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣼⡇⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠠⣶⣿⡯⠤⠤⠴⠖⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣸⣿⡄⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠠⡀⢸⡄⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⢀⣠⠴⢻⡷⣦⣀⠀⠀⠀⠀⠀⠙⣾⡇⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢈⣷⣄⡀
⢚⢃⣀⡤⠖⠉⠀⠀⢸⣇⠀⠉⠛⠲⠦⣤⣶⣶⣿⣃⣤⠤⢶⡿⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⡇⠀⠉⠓⠶⣤⣀⠀⠀⠶
⠚⠻⣇⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠤⢤⣄⣀⣀⣀⠀⢀⡴⠋⠀⠀⣿⠀⠀⠀⢀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣷⠶⠶⠶⠶⠶⠶⠇⠀⠀⠀⠀⠀⠈⢙⣿⠖
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⡀⠉⢉⡽⠛⠛⠛⠛⠛⠛⠒⠛⠛⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣴⣶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⡀⠀⠀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠂⠐⠴⠐⠲⠀⡀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⠀⠀⠞⠁⢀⠀⠀⠀⠡⣤⠄⠤⠠⠔⢰⠀⢈⠀⠈⠳⠀⠀⢢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡞⠘⣁⠤⣀⠉⠓⠀⠁⠰⠜⢀⡄⡀⣀⢀⡁⠀⠉⠁⠀⠘⠭⣂⠤⣈⠃⢳⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠵⡊⠕⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠪⢕⠮⠇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⢎⠔⠋⠀⠀⠀⠀⠠⠴⠂⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠤⠄⠠⠀⠀⠀⠀⠀⠀⠑⠢⡰⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡄⠁⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢊⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⢀⢢⠌⠀⡐⡄⠀⠈⠁⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡑
⠀⠀⠀⠀⠀⠀⠀⠸⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠊⠐⠒⠀⠊⠋⠐⠐⠙⠐⠀⠄⠆⠆⠀⠂⠀⠀⠀⠀⠰⠂⠀⠀⠀⠱⠅
⠀⠀⠀⠀⠀⠀⢀⡄⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⢀⡆⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⡁⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⡀⠀⢀⡀⠀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⢈
⠀⠀⠀⠀⠀⠀⣀⠀⣠⢀⠀⠀⠀⠀⠀⢀⣄⣄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠁⠀⠀⠀⠀⢀⠀⣠⠄⠀⠀⠀⡀⣄⠀⢀
⠀⠀⠀⠀⠀⠤⢈⠙⠡⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠋⠁⠔
⠀⠀⠀⠀⠐⠀⠂⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐⢄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣹⡇⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⡄⡠⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⣿⣿⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠖⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠑⠤⡁⠀⡀⠀⠀⠀⠀⠀⡠⠀⠀⢀⢠⣿⣿⡄⠀⠀⠘⠀⡀⡀⠀⠀⠀⢀⠀⢈⠴⠊⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⣀⠀⡀⠀⠁⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠐⠈⠀⠂⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠐⠒⠠⠍⠐⠀⡀⠀⡀⢀⠀⢀⠀⠂⠩⠄⠒⠀⢌⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠒⠒⠀⠒⠐⠓⠐⠒⠒⠒⠀⠐⠒

⢠⣶⣶⣶⣶⣶⣶⣶⣖⣴⣶⣴⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦
⠀⣿⣿⣿⣽⣿⠿⢿⣿⣻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣽⣻⡿⢯⣬⣛⣿⣻⣿⣿
⠀⠙⠙⢻⣿⠿⣦⣀⡿⢿⠿⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠁⠀⠀⠀⠁⠀⠀⠀⠀⠈⠙⢿⣿⣇
⠀⠀⠀⠈⠁⠠⣿⡟⠀⠀⠀⠀⠀⣀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣲⣿⣿⣾⣿⣿⣠
⠘⣶⣄⣤⢤⣄⣿⣧⣤⣴⣦⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠀⠸⠿⠿⣷⣶⣶⣿⠶⢿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣯⣉⣉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣾⣿⠿⣿⣿⣿⣿⣿
⠐⠟⠉⠉⠉⠉⣿⡟⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢰⡇⡀⠀⠀⠘⠿⢿⠟⡿⠿⣿⣧⣿⣿⣿⣿⣿
⠀⢈⠋⠀⠀⣀⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣾⣷⡀⠀⠀⠀⠀⠉⠙⣋⠀⠀⢸⣿⣿⣿⣿⣿
⠀⢰⢲⢧⣤⢶⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣿⢿⡄⡄⠀⠀⠀⠀⠀⢀⠀⡄⢨⠝⠩⣿⣿⣿
⠀⠸⢽⣺⣽⢸⠥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡂⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣟⠛⣄⡀⠀⡙⣿⣿
⠀⠀⠽⢺⡼⠜⠰⢤⣀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢘⢛⡻⢿⣷⠀⢀⣀⡤⠤⠠⠐⠒⠰⠛⠛⢛⠑⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠷⠶⠦⠐⣀⣀⣠⡤⠤⠖⣠⣤⡄⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⢂⣈⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡤⠼⠛⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⡱⡀⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠠⠏⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡤⠀⡀⠐⡈⡉⠤⠄⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠭⢑⡑⠧⠘⢀⠞⠁⠀⢀⣤⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣁⣀⡀⠉⠀⠀⠀⠀⠀⠀⢀⣀⠀⢡⠔⠂⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠔⠊⢉⡀⠀⢀⠠⠐⢈⡁⠄⠤⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢔⢞⡉⢠⠖⢁⠤⠊⠁⠠⠐⢀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⠁⠀⢀⠷⢅⡐⠀⢸⡇⠴⠊⠙⠒⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠨⠑⠀⠀⠀⠁⠤⡴⠙⠂⠉⠉⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠘⠀⡇⠀⠀⠀⡀⣈⠀⢀⣀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠠⡀⡃⠀⢀⡾⠉⠉⢀⠎⠀⠈⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡤⣾⡿⡀⠨⠃⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠰⠉⣴⠋⠀⠙⠢⠄⠊⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⠉⠈⠉⢀⡄⢀⠄⠀⠐⠒⠃⠀⠘⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⢀⠄⠂⠉⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡘⢷⣤⣀⠌⠀⠀⠒⠂⢤⠆
⠀⠀⠀⠀⠀⠀⢠⠤⣀⡱⡆⠹⣉⠒⠠⠤⠄⣤⠜
⠀⠀⠀⠀⠀⠀⠎⢠⠏⢩⢧⡀⠘⠝⠁⣎⠉⢽⠋
⠀⠀⠀⠀⠀⣸⢦⣴⣀⠔⠋⠳⠄⢈⠀⠈⠁⠁
⠀⠀⠀⠀⣴⠉⢈⡾⠍⡲⡶⢶⣤⢷⣄
⠀⠀⢀⣤⣴⣿⣷⣷⣶⣿⣶⣚⣋⣅⡀⢙⣿⣿⣷⣶⣶⣤⣤
⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠁⠄⠀⠠⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡇⠘⠓⠊⠚⠀⠂⠀⠸⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠎⠀⢀⣩⡀⢂⠀⠀⠅⠖⠒⠒⠐⡁⢇⣀⠄⢀⣍⡀⠀⠱⡀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⡏⡀⠔⡢⠒⠈⠀⠈⠈⠊⠐⠆⠄⠄⠀⠀⠀⠀⠀⠀⠁⠒⢄⠢⣀⠹⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠘⡫⠂⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠈⠐⢝⡃⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢔⠱⠋⠀⠀⠀⠀⠀⢘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠉⠀⡁⠀⠀⠀⠀⠀⠀⠑⢊⡢⡁
⠀⠀⠀⠀⠀⠀⠀⢠⢨⠊⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⡀⡀⠀⢀⡀⡇⠴⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠑⡕⡄
⠀⠀⠀⠀⠀⠀⠀⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠒⠤⠄⠜⠜⠡⠸⠦⢃⢇⣀⢤⠀⠥⠀⠀⠀⠀⢀⠄⠀⠀⠘⣶
⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠁⡁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⢠⠄⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⢀⠃⡀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⡀⠀⠀⡀⠈⢈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⢘⡀
⠀⠀⠀⠀⠀⢁⡀⣀⣊⡀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠉⠁⠀⠀⠀⠁⠁⠀⠀⠁⠈⠀⠀⠀⠀⢀⡀⣠⠄⠀⠀⢀⣑⡀⢀⡈
⠀⠀⠀⠀⡀⠤⠉⠉⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠙⠉⠤⢃
⠀⠀⠀⠀⠐⠂⠁⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠐⠊
⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠂⠀⠀⠀⠂⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠡⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⣀⢴⡦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠌⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠡⡁⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣾⣇⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⢊⠌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⡁⠄⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⡀⣿⣿⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠠⢉⠡⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡂⠈⡀⢁⠂⠄⢀⠀⠀⠒⠁⠀⠀⠐⠛⠛⠓⠀⠐⠀⠒⠐⠀⡀⠠⠐⡀⠀⠁⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠃⠀⠀⢀⠈⠀⡐⠠⠄⠀⠀⠀⠀⠀⠀⠠⠄⢂⢀⠁⡠⠀⠀⠘⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠁⠀⠈⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠈⠐⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠄⠲⠂⠢⠰⠖⠒⠲⠶⠆⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⣟⠛⠛⠛⠒⠒⠒⠒⠲⠶⠦⠤⠤⠤⢤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡟⡇⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠰⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣸⣧⢷⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⣿⣿⡸⡆⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⢀⢴⠠⠃⠀⢠⠒⡂⠀⡇⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⡞⣼⠹⣇⢧⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⡾⠀⠃⠀⣠⠇⠀⠀⠀⣇⠀⠀⠀⠀⢀⡞⠀⠀⠀⡼⠁⡇⠀⢹⡘⡆⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⡇⠀⣀⡴⠋⢀⠀⠀⡀⠈⠀⠀⢡⣠⠏⠀⠀⠀⠰⠁⢸⠷⠷⠚⢧⠹⡄⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⢸⡄⠈⠃⣠⠶⠀⠀⢀⠀⢠⠁⠀⠈⠛⠛⠛⠛⠒⢀⣀⡾⠀⠀⠀⠘⣆⠑⣀⣀⣀⣀⠙⢦⡀⠀⢲⢀⣀⡤⠤⠖⠒⠒⠒⠛⠂
⠀⠀⠀⠀⢀⣼⣇⠀⢂⠃⢀⣀⡴⠁⣠⣦⣄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠶⠋⠉
⡔⠰⣭⣴⣛⠿⢿⠶⣴⡷⣞⣿⣶⣿⢷⣿⣿⣿⣏⣱⡆
⠙⠶⠾⠿⠿⠶⠦⠽⠷⢯⣿⣾⣧⣠⠟⠛⠋⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣀⠛⠿⢿⣷⣤⡀⠀⠀⣿⡏⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⣉⣉⣉⠉⠉⠛⠛⠿⢿⣶⣤⡈⠻⣷⡄⠀⣿⣧⣼⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⠿⠿⠿⠿⠿⠿⠿⣷⣶⣦⣄⡉⠻⣿⣦⣼⣿⣿⡿⠟⠛⠛⠿⢿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠉⠙⣿⣷⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣤⡀
⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⣿⡿⠛⠛⠻⢿⣷⣾⡿⠟⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣦⡀
⠀⠀⢰⣿⣋⣩⣽⣿⡿⠿⠃⠀⣠⣴⡿⠛⠉⢀⣴⣿⠟⠁⠀⠀⢀⣿⡿⠉⠀⠀⠀⠀⠀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧
⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⢀⣾⠟⠁⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⣤⡀⣤⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⣶⠶⠾⣿⣿⣄⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⢹⣿⠃⠀⠀⠲⣮⣿⣿⣿⣷⣄⠀⠀⠀⠀⠘⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡿⠛⠛⠋⠀⠀⠀⢀⣴⣾⣷⠘⣿⣆⠀⠀⠀⠀⠈⠀⠀⣶⣶⣌⠻⣯⡙⢿⣿⣷⣄⠀⠀⠀⠙⠿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⢰⣟⣡⣾⠇⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠙⢷⣄⠻⣿⣿⣷⣄⠀⠀⠀⠀⠉⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠀⣼⣿⠃⢀⣠⣄⡀⠀⠀⠘⠻⠿⠋⠀⠀⠀⠀⠻⣿⣦⣄⣀⠀⠀⠀⠹⣿⣿⣦⠈⠻⣧⠈⠋⢹⣿⡆⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⣠⣤⡶⢾⣿⣿⡿⠁⢠⣿⡿⠀⠉⠉⠙⠿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣄⠘⣿⣿⠃⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣾⣿⠃
⠀⠀⣾⣿⣁⣤⣾⡿⠋⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⡶⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⣶⡀⠈⢹⣿⠀⠘⣿⡀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣸⣿⡏
⠀⠀⠻⠿⠿⠟⠋⠀⠀⠀⠀⣼⣿⠁⣀⣀⡀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⣿⠀⠀⢹⣷⠀⠀⠀⠀⢸⣿⣧⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠻⣿⡿⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣽⡀⠀⠈⣿⡄⠀⠀⠀⢸⣿⣿⣀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢸⣿⣇⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⠀⠀⠀⠀⠉⢫⣥⣄⡀⣹⣯⡀⠀⠀⢸⣿⣿⣿⠟⠋
⠀⠀⠀⠀⣠⣴⣾⣿⡿⠀⠀⠀⣿⣿⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣇⠀⠀⠈⣿⣿⠁
⠀⠀⣰⣿⣋⣰⣿⠟⠀⠀⠀⠀⠘⣿⣧⠀⣰⣿⠏⠀⠀⣀⣤⣶⣾⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡹⣿⡄⠀⠀⢻⣿⡀
⠀⠀⠙⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣏⣠⣶⡿⠟⠛⠉⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣿⣿⣆⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠟⠋⠁⠀⢴⣤⣤⣀⡀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⡿⠛⣿⣦⠀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣄⠀⠀⠉⠙⠻⢷⣶⡄⠘⢿⠄⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠋⠀⠀⠈⢿⣧⠀⠀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣤⣀⣀⠀⠈⠁⠀⠀⠀⠀⢀⣠⣤⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣶⣶⣶⣿⣿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠁⡇⣿⠀⠀⠀⠀⢾⣿⣿⣦⡄⠀⠀⢰⣾⣷⣿⣶⠀⠀⠐⡀⠐⢆⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣴⣤⣤⣄⣤⡀⠀⠀⡀⠠⠤⢀⡀⠀⠀⠀⠀⣠⣶⡄⠀⠀⢰⡖⣒⣒⢺⣶
⠀⠀⠀⠀⠀⠀⢹⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠆⠀⠀⣠⣾⣿⣿⠿⠀⠀⢸⡇⠧⡼⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠀⠀⠓⠀⠀⠘⠃⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠋⠉⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⡀⣄
⠀⠀⠀⠀⠀⠀⢸⣼⣶⣦⣦⡆⠀⠀⠀⠶⠾⠄⠀⠀⠀⣔⣻⢾⣳⡀⠀⠀⠀⢸⡛⠟⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣻⠞⠁⠀⠀⠀⣰⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠀⠈⢦⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠗⣦⣠⠟⠙⠋⠁⠀⠀⠀⠈⢳⡟⠑⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡲⢛⡔⠀⠈⠈⠂⠀⠀⠀⠀⠀⠐⠀⠀⠉⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠄⠐⡫⠂⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠈⠓⢄⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⢨⣄⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⢠⣿⣿⡆⠀⣀⠀⠀⠀⠀⣴⠤⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⣿⣿⡄⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧⣸⣿⡀⠀⠀⢠⣿⠃⡐⠠⡀⠀⠀⠀⠀⢠⣿⣧⣽⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢾⣿⡆⠑⢄⠀⡁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⠆⣸⣿⣧⠀⠀⣢⡀⠀⠢⣠⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠿⠯⠭⠽⠟⠛⠓⠺⠣⠾⠥⠴⠿⠿⠿⠷⠶⠞⠲⠶⠿⠿⠷⠼⠷⣚⡛⡻⠯⠭⠼⠿⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠒⠒⠀⠉⠉⠉⢉⣉⣉⣉⠁⠀⠉⠉⠛⠃⠀⠀⠀⠀⣠⣶⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡦⠰⣿⣿⣾⠃⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣼⢏⡀⠀⣤⠀⣤⣠⣶⣤⣶⡦⢠⣾⡿⣿⣶⢿⡇⣸⣿⠟⣡⡾⠋⣹⣇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢻⣴⣿⠃⣼⠇⣼⡟⣼⣿⠏⣿⠏⣼⣷⣿⠟⢸⡟⠁⣾⠿⠻⠷⠟⠉⠀⠀⠛⠋⠀⠘⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣴⣿⣿⠟⠐⠿⠟⠻⠿⠛⠋⠀⠁⠀⠛⢁⡤⠤⠴⠶⠖⠒⠚⠛⠛⠛⠛⠛⢛⡛⠛⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⠙⠉⠁⠀⠀⠀⢀⣴⠾⠛⠛⣷⠀⠀⣀⣤⡀⠀⣠⣤⣴⣤⣴⣶⡄⢀⣾⡿⢛⣿⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣥⡤⠶⠟⣡⡴⠟⢁⣿⠁⣴⣿⢋⣿⠏⢠⣿⣴⣿⣿⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⣯⠀⠀⢀⣤⢾⣟⣁⣴⣿⣧⡞⣿⠃⠸⠃⠀⠿⠋⣼⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠃⠈⠛⢛⣉⣥⣤⣽⠿⠷⢾⣥⣤⣤⣄⣀⣀⡤⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⢀⠀⠐⣆⠐⠀⠀⠈⠁⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠔⠒⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠁⠀⠚⠓⠀⠤⠄⣀
⠀⠀⠀⠀⠀⣀⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⢄⡀
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠲⢀
⠀⢀⡞⠁⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠙⠄
⢀⡟⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢀⠀⠀⠀⠀⠀⡔
⡸⠃⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⡆⠀⠀⠀⠀⢠⠃⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⣀⠃
⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣿⠈
⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡀⠲⣄⠀⠰⣀⠀⠀⢄⠀⠀⠀⠀⠀⠈⡀⠀⠀⠛⡀⠉
⢸⡄⠀⠀⠀⡀⠀⡇⠀⠀⡁⢰⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡈⠃⣀⠈⢳⡀⠈⢷⡀⠀⠀⠀⠀⢳⡄⠀⠀⢣
⠈⣧⠀⠀⠀⣧⠀⡇⠀⠀⠇⠈⠖⠈⣠⣤⠗⠒⠒⠀⠉⠉⠉⢉⣩⣥⣬⣴⡉⠉⠉⢱⡄⢈⡿⢄⢠⠷⣤⣀⠑⢤⡈⠉
⠀⠘⡆⠀⠀⠏⠘⠳⠄⠂⠚⠛⡾⠛⣿⣶⢶⣄⠀⠀⠀⢀⡶⠉⣾⣿⣿⠉⢹⣆⠀⠀⠀⠈⠀⠀⠙⠘⠋⠙⣷⠂⠉⠛
⠀⠀⠸⣀⢀⡸⠂⠀⠀⠀⠀⢼⠀⠸⣿⣧⣴⣿⡆⠀⠀⢸⠀⠀⠹⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣶⣀⣠⠟
⠀⠀⢰⠛⠈⢁⠀⠀⠀⠀⠀⠘⣆⠀⠙⠛⠛⣻⣿⠀⠀⠀⠻⠦⣄⣀⡠⠾⠋⠀⡀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠈⠲⠖⠋⠲⡀⠀⠀⠀⠀⠈⠙⠒⠒⠛⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠊⡴⢃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⠀⠀⠀⠀⠓⠦⣤⢤⣤⠖⠋⠀⠀⠀⠀⢀⣠⣴⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⠶⠤⠤⢄⣀⣀⣀⣀⡠⡤⠤⠤⠒⠛⠉⠁⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠉⠓⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠲⢤⣀⠀⠀⠉⠳⢆
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣠⠶⠋⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⢉⠃⠀⢠⡜
⠀⠀⠀⠀⠀⡜⠁⠀⢠⡞⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣠⠔⠁⢀⡴⠋
⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠁⠐⢲⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢏⠁⠀⡴⠋
⠀⠀⠀⠀⠈⠛⠒⠶⠶⠶⠤⢤⠇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡀⠈⢡⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢀⣈⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣀⣀⠀⠀⠤⠤⠤⠤⠤⠄⠀⠙⠛⠛⠛⠒⠀⠠⠤⢤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣄⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⡀⣿⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⠂⠀⠉⠒⠾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣛⣉⣀⣀⡀⠀⠀⠀⢈⡇⠀⠀⠀⠘⣆⡠⠤⠤⠀⠀⠐⠒⠒⠂⠒⠘⠣⠤⠤⢀⣀
⠀⠀⠀⠀⠀⣀⠴⠚⠋⠉⠉⠉⠀⠉⠀⠉⠉⠁⠐⠛⣾⠃⠀⠀⠀⠀⠯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣳⡄
⠀⠀⠀⠀⢰⣍⣀⣀⣀⣀⣀⣀⡠⠤⠶⠶⠶⠒⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⠠⠄⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡀⢴⡠⢖⠒⢦⣄⡆⡄⠉⠐⠠⠄⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠃⠒⢸⠀⠀⠉⠀⠀⠙⠀⢀⠀⡀⠀⠀⠀⠀⠀⢸
⠤⠄⠒⠒⠂⠀⠀⠒⠒⠬⡢⣀⠀⡼⠐⠒⠠⡀⠀⠜⠀⠓⠒⢲⠄⠴⠭⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠹⣀⠤⠀⠤⠜⡼⠤⠤⣀⠠⠊⠀⠀⠀⠀⠀⠋⠒⢄⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⢀⡔⠀⠡⡀⠇⠑⠂⠤⠄⡲⠀⢀⠆⠀⠀⠈⠑⠒⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢡⠈⠢⣀⡠⠔⢡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⣀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠈⣾⠃⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣰⢻⢧⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⢰⠃⢸⠈⢧⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢠⠃⠀⢸⠀⠈⢆⠰⠁⣀⣀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠬⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠀⠀⢠⠃⠀⠀⢸⡄⠀⠈⢆⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⡀⠠⠒⠒⠤⠄⣈⢦⡀⠀⠀⠀⠀⠀⣠⠇⣀⠤⢤⠃⠀⠀⠀⢸⡇⠀⠀⠈⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡏⠙⠈⠉⡤⠚⠀⢠⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡇⠀⠀⢸⡀⠀⣠⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠙⢎⠂⠀⠂⠉
⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⡘⠀⡇⡀⠀⡎⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠈⠢⠄⠀⠄⠲⡪⡀⠀⡇⡿⠀⡼⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢕⢤⠁⣧⠞⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⢉⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠈⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠉⠒⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢠⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠎⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠲⢖⡊⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⢀⠞⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠑⠢⢄⡀
⠀⠀⢰⠉⠠⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢀⡴⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢐⠉⢰⠀⠉⠒⢤⡄
⠀⠀⠀⢹⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣚⡡⠔⠚⠁⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣽⡁⠤⠔⠊⠁
⠀⠀⠀⠉⠉⠉⠉⠁⠀⠒⠒⠒⠒⠒⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠐⠃⠤⠤⠤⠤⠤⠤⠄⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣻⣫⣻⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣯⣭⣭⣽⣿⣿⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡇⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⡀⠀⠇⠀⢀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡖⠒⢢⡀⠉⠓⠤⠀⡡⠔⠒⠢⣄⢀⠴⠊⢁⡤⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠣⠀⠀⠀⠊⠉⠑⠀⠀⡼⢀⣄⡀⠀⠈⡆⠀⠈⠉⠁⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⢆⠀⠀⠀⡱⠋⠈⢫⢣⡠⠃⠀⠀⢀⠔⠉⡏⠱⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠞⠉⠀⢹⠘⣄⢀⠜⠁⠀⠀⠀⣇⠱⡄⣠⠔⠁⠀⠀⠈⡆⠘⣄⣠⣀
⠀⠀⠀⠀⠀⣠⠞⠉⠑⠚⠁⠀⠀⠀⢸⡀⠈⢯⠀⠀⠀⠀⠀⠸⡄⢸⠁⠀⠀⠀⠀⠀⠣⡀⠈⢇⠈⠓⢄⡀
⠀⠀⠀⠀⠘⢷⡶⠶⠤⠤⢄⣀⠀⠀⠀⠈⢢⡠⠃⠀⠀⠀⠀⠀⣙⠞⠀⠀⠀⠀⠀⠀⢀⣈⣉⣙⣠⣀⣀⠝
⠀⠀⠀⠀⠀⠀⠑⢄⣀⣀⣀⠤⠭⢲⣄⣀⣈⡁⠀⠠⣤⣀⣀⣀⣹⡀⠀⣀⡠⠤⠖⠛⠛⠒⠒⠒⠲⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠠⠀⠀⠀⣉⡭⠚⠁⠀⣠⠔⢹⡟⢮⡁⠀⠀⠘⠛⠀⠐⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡉⣀⠀⠀⠀⣸⣉⣉⣉⣉⣹⣁⣠⣏⣱⣀⣸⣂⡀⠀⢀⣀⣉⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢭⠤⠄⠄⠁⠭⠏⠉⠉⠁⢩⡀⠉⠉⠨⠍⠈⠄⠠⢬⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣖⠂⠀⠒⠂⠀⠀⠀⢿⠁⠀⠀⠀⠂⠀⠒⡲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡭⠬⠀⣀⠤⣿⢟⣷⠢⢥⠀⠤⣭⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠓⣿⡵⠋⠲⣿⠏⣀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣆⣀⡀⠂⢀⣀⡲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡤⠤⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⢿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡀
⣿⣿⣿⣿⣿⠟⠛⢻⣧⣤⣿⣿⣟⣾⣿⡿⣟⣿⣦⣤⣽⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣠⣦⡄⠚⠛⠀⠀⠠⠁⠀⢄⢠⡄⠛⠛⠓
⣿⣿⣿⣿⣷⣶⣶⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠉⠉⠁⠀⠔⠙⠀⠉⠀⡠⠂⠈⠄⠂⡀⡀
⣿⣿⣿⣿⣿⡿⠹⢻⡏⢿⢍⣿⣿⣿⣿⡿⡿⡿⣿⣿⣿⣻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣦⡄⢱⡁⣲⠀⠘⠀⠄⣀⢀⢠⠀⠀⠀⢄
⣿⣿⣿⣿⣿⣷⠆⠈⠁⠙⠀⢹⣿⣟⣿⣇⣠⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⣽⣿⣾⣷⣿⡆⠀⢢⠀⠼⠟⠻⠀⠀⠀⠀⠄⠂
⣿⣿⣿⣿⣿⣿⣿⣵⣤⣤⣤⣶⣯⣶⡾⠿⠾⣋⣛⣿⣿⣿⣧⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠉⢘⡛⢓⣀⣁⣶⡦⠁⠒⠂⠚⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿