Examinations Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡄⠿⠨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⠀⠀⠈⢆⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠂⠈⠉⠉⠭⣁⡐⠆⠀⠀⠀⠉⠏⠀⠀⠀⠀⠘⠓⠒⠒⠿⠇⠘⡵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢴⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⠀⠈⠑⠋⠱⡀⠀⠀⠀⠀⠀⠀⠑⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⡠⠒⠒⠂⢄⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡁⠀⠐⡢⡀⠱⡀⡰⠁⠀⠀⣠⠄⠄⠀⣠⠖⠈⠑⠠⢈⡗⢄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡁⢀⠀⠈⠁⠀⠁⠀⠀⠀⢰⠁⠀⠘⠈⠃⠀⠀⠀⠀⠀⠀⠀⠑⢄⠇
⣠⡤⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣁⠰⠀⠁⠀⠀⠀⢀⠆⠀⠀⠘⠄⠴⢳⣤⡔
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠃⠀⡦⢠⠐⠆⠀⠴⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⡂
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠸⠤⢀⣇⣠⠀⠀⠀⠀⢀⠴⢄⠈⠀⠀⠀⠰⠀⠀⠀⠂⠁
⢰⡄⠀⠀⠀⠀⠀⠀⢠⠴⠊⠀⠀⠀⠀⠀⡸⢏⠈⠀⡀⡖⣄⠀⠚⠀⢇⡠⠄⡲⠂⠓
⠀⣇⠀⠀⠀⠀⠀⡠⠊⠀⢀⠋⠀⠀⠀⡐⠁⠀⠀⠀⠀⠘⢯⢂⠀⢀⡀⠔⠊
⠀⢳⣶⠒⢺⣇⡐⠁⠀⠀⠈⢘⣷⣶⠁⢳⣶⡃⠀⠀⠀⠀⠀⠀⡩⢄⡀⠀⠀⠀⠀⣀⢄⣀⡀⠀⣀
⠀⢸⠹⠀⠀⢸⠶⠒⠂⠒⠀⣼⠉⢹⠮⣀⡍⠣⠀⠀⠀⢀⠄⠂⠀⠤⣏⠡⣀⡴⠉⣠⣈⡅⠀⡋⠀⡠⠂
⠀⠀⡄⠀⢀⡏⠀⠀⠀⠀⠀⠸⠄⣈⣒⣘⣛⣒⣀⣈⣉⣀⠀⠀⠀⠀⡬⢇⡁⣀⣈⣀⣀⣨⠴⠂⠁⢀⡤⠄⠀⣀⣀
⠀⠀⣇⡀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠈⠓⠒⠚⠑⠚⠓⠚⠛⠛⠋⠉⠛⠛⠛⠛⠒⠒⠘⢉⡽
⠀⠀⡇⠀⠀⠀⠈⠉⠉⢉⣉⣉⣉⣁⣇⠀⠀⢩⠉⡏⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡏⢻⣧
⠀⢀⠾⠟⢩⣿⠏⠀⠀⠀⠀⠀⠻⢃⡜⠀⠀⠈⠀⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠
⠀⠌⢠⠀⡌⠈⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢨⢤⠾⠼⠤⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠈⢀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣑⡶⢶⣒⣚⠀⢰⠠⠀⠀⠀⠀⣱⣰⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⠀⢦⣆⡰
⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠘⠛⠿⣀⡼⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠐⠀⠄⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⣀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣶⣶⡶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⠀⠤⠤⠤⠤⢄⠀⣠⠀⠀⢠⡀⢀⡀⠤⠤⠤⡄⠀⢀⡤⠤⠤⢤⠀⠀⢀⡤⠤⠤⠤⢤⠀⠀⢀⡤⠤⠤⢤⠀⠀⠀⠀⣀⡤⠤⠐⠒⠂⠒⢢
⠀⠀⠐⠈⡇⠀⠀⠀⠀⠀⢸⠀⠵⠀⠀⠀⣆⠹⠀⠀⠀⠀⠘⡰⠈⡇⠀⠀⢸⠀⢠⢩⠇⠀⠀⠀⠈⡆⠀⡎⡇⠀⠀⢸⠀⠀⣠⢾⠃⠀⠀⣀⣀⡤⣤⡼
⠀⠀⢸⠀⡇⠀⠀⣖⣚⡲⠋⡠⢸⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⢿⢈⡇⠀⠀⢸⠀⡏⢸⠀⠀⣶⠀⠀⣇⠀⡇⡏⠀⠀⢸⠀⢰⠁⣺⠀⠀⢾⠂⠒⠐⠁
⠀⠀⢨⠀⡇⠀⠀⠓⠀⠀⢰⡆⢸⠀⠀⢀⡧⠘⡇⠀⠀⠀⠀⠸⡄⠆⠀⠀⢸⢀⡃⢾⠀⠀⠚⡆⠀⢻⠀⡏⡷⠀⠀⢸⠀⢸⠂⠱⡀⠀⠀⠉⠁⠒⠒⢤
⠀⠀⢰⠀⡇⠀⠀⣠⢤⣤⣬⡇⢸⠀⠀⢸⡗⢈⡇⠀⠀⠀⠀⠀⢿⡄⠀⠀⢸⢸⠀⡯⠀⠀⠧⠇⠀⢸⡄⡇⣷⠀⠀⢠⠃⠀⠱⢄⡑⠢⠤⠤⢤⡀⠀⠀⢳
⠀⠀⠰⠀⡇⠀⠀⡟⠈⠉⠀⡇⣹⢀⢀⣸⡷⣨⣇⡀⠀⣺⡀⠀⠘⠀⠀⠀⢸⣾⢐⡇⠀⠀⣀⠀⠀⠈⡇⡇⣿⠀⠀⢸⡀⢀⡀⠀⡀⢉⡉⢲⣄⠅⠀⠀⡘
⠀⠀⠰⠀⡇⠀⠀⡇⠀⠀⡠⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠄⠀⠀⢸⣿⢸⠃⠀⢰⡇⡆⠀⠀⣿⡇⣻⠀⠀⠀⠀⠀⣥⢞⠀⠉⠉⠀⠀⠀⠀⣰⠃
⠀⠀⠰⠀⡷⢤⢤⣇⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁⣰⠖⢖⡾⣇⠺⠤⠤⠼⠇⡳⠤⠤⡼⣧⠼⠤⠄⠤⢤⡼⢻⡸⠤⠤⠤⠤⣲⡶⠛⠁
⠀⠀⠘⠛⠁⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢐⣡⡴⠒⠒⠛⣏⠋⠀⠈⠈⠈⠉⠀⠛⠘⠙⠉⠀⠉⠈⠁⠊⠘⠉⠀⠈⠁⠁⠈⠉⠁
⠀⠀⠀⣔⠁⠀⠀⣀⣀⣀⡀⣀⠀⠀⠀⣠⠄⣠⣾⡾⠋⠼⠀⠀⠀⠀⡌
⠀⠀⠐⠋⠀⠀⣀⡠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣖⠀⣀⠁⠐⠓⠀⠀⡀
⠀⠀⠄⡀⠀⠀⠀⠐⠒⠒⠂⠀⠀⢀⣀⣀⠀⠀⠀⠀⠦⠛⠈⠁⣀⠠⣄⠀⠈
⠀⠀⠒⢀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠀⠁⠀⠒⠀⠀⠀⣀⠰
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠩⠛⠛⠂⠀⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⠤⠀⣒
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣉⠀⠀⢤⠀⠀⣀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠰⠁⠄⠀⠀⠠⠙⠛⡋⠅⠚⠋⠁⠀⠉⠙⠳⢮⣄⡈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠤⢀⡀⠀⠀⠀⠈⠉⠉⢿⢰
⠀⠀⠀⠀⣠⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣀⣬⡆⠀⠀⠀⢙⡄⠄⠀⠀⣀⣀⣌⣈⣲⣤⣀⡤⠊⣀⣇
⠀⠀⢰⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠠⠍⡉⠑⠒⠀⠐⠀⠀⠐⠲⠆⠀⡈⠉⠈⠁⠈
⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢿⡙⠠⠍⠀⠐⠀⠠⢄⣀⠀⠀⡘⢀⠔
⠀⠀⠀⠀⠌⢀⠁⠖⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠄⠁⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠁⠈⠀⠂⠨⣍⣳⣟⢰
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠒⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠉⠀⠀⠀⠀⢀⣤⣞⠋⠀⠀⠀⠀⠀⠀⢀⠋⠢⢀⠀⢀⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠁⠐⠀⠲⠒⠚⠛⠛⠉⣉⡓⠦⠄⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡴⠞⠋⠤⢀⡈⠉⠘⠒⠶⠤⣠⣀⢀⣤⣤⠕⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠠⠍⠶⠦⠀⠀⠀⠀⣽⣧⠶⠞⠁⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⣀⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⠶⡶⠶⢶⣶⡶⠶⠶⠶⣶

⠀⠀⠀⠀⠀⠀⢀⣐⣦⠤⠄
⠀⠀⠀⠀⠀⢤⡟⠉
⠀⠀⠀⠀⠀⣖⠀⡀⠀⠀⠀⠴⠒
⠀⠀⠀⠀⠀⠾⢿⣿⣧⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣒
⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠈⠙⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢸⢻⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣧⣼⠿⣤⠀⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⠂⣿⠏⠁⢹⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡥⢴⣿⡥⠀⢬⡀⠤⠤⠤⠠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠿⠗⠂⠨⠇⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣦⣤⣤⣤⣤⣤⣤⣤⣀⣁⡉⢳⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⣀⣠⣰⣿⡻⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⠾⠿⠿⠿⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⡤⠤⠤⣄⣀⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣤⠀⠀⠀⠀⢀⣠⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡟⠛⢻⡇⠀⠀⠀⠀⠀⣿⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣤⣤⣼⡇⠀⢸⣧⣤⣤⡄⠀⠀⣿⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⣿⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠛⠛⢻⡇⠀⢸⡟⠛⠛⠁⠀⠀⣿⠀⣿⠀⠀⣴⠟⠉⠙⠷⣦⣤⣈⣿⡛⠛⠶⠶⠶⠶⠚⣷⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣧⣤⣾⠇⠀⠀⠀⠀⠀⣿⠀⢻⣄⣼⠃⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢹⣇⣼⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠏⠁⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠉⢻⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⣿⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⣤⣤⡤⠸⣇⠀⠀⠀⠀⠀⢦⣄⣀⣤⠆⠀⠀⠀⠀⣸⠇
⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠹⣦⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣴⠏
⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⠀⢈⣿⠛⠛⠳⠖⠛⠛⢹⣇
⣿⠀⠀⠀⠀⠀⠀⠀⣠⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⠶⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠳⠶⠶⠶⢦⣤⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠙⠳⣤⡾⠋⠉⠉⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⣶⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⢀⣀⣴⠞⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣧⡴⠶⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣛⣒⣶⣶⣶⣶⣶⣶⣶⣶⣶⣚⣛⣛⣉⣉⣉⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡤⠒⠚⠛⠛⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢉⠀⠁⠀⠀⠀⠀⠙⢦⡀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡀⠀⠀⢦⣄⠀⠀⠀⠹⣆⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢶⡖⢻⣦⡀⠀⠈⠳⣄⠀⠀⢹⡄⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⣷⡞⠀⠘⣧⠀⠀⡇⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡏⠈⠉⡅⠹⣆⠀⢻⠀⠀⡇⠀⢸⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⠻⡿⠐⠀⠀⢹⡀⢸⡄⢠⡇⠠⡎⠀⡉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡀⠀⠀⠀⠀⢸⡇⢸⠁⠘⣧⠀⠑⠚⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⠃⠀⢸⡇⠘⠂⡔⠉⠑⣤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠔⣿⢿⠀⠘⢧⠖⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠈⠳⣄⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⢠⢍⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢶⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⣠⠖⣥⠞⠛⢧⡀⠀⠀⠀⠀⠀⠀⡜⡱⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣡⣠⠖⠀⠀⠒⠒⢾⡏⠷⡅⣴⡂⣿⣡⠞⠁⠀⠀⠈⠳⣄⠀⠀⠀⢀⣼⡾⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡿⣅⠀⠀⢰⡶⣦⢾⡟⣷⠒⠺⣡⠷⡀⠈⢷⡀⠀⠀⠀⠘⣆⠀⢠⡾⢿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠉⢀⡴⣟⠻⢉⠓⢺⠁⠹⣼⢧⡘⠦⠀⠙⣦⡹⣄⠀⠳⣄⠀⠀⠀⠘⣇⠀⡇⢸⠀⠀⡎
⠀⠀⠀⠀⠀⣀⡴⠚⠁⡠⠊⠹⠟⣿⠿⠂⠘⠛⠛⠔⣳⠆⣨⠟⣲⣄⠈⠛⠥⠃⠀⠈⠳⣤⠀⠀⠙⠀⣷⢸⠀⢠⠇
⠀⠀⢀⡤⠞⠉⠀⠠⠾⠤⠤⠀⠀⠀⠂⢐⣒⣒⣀⣈⣩⠞⢁⠞⠁⠈⢂⠀⠀⠀⠀⢀⡴⠁⠀⠀⣴⠀⣿⣿⣴⣿
⠠⠾⠯⠤⠤⠶⠶⠒⠒⠚⣛⣛⣉⣉⣉⣭⣭⣭⣭⣭⣽⣶⠋⠀⠀⠀⠀⣧⠀⠀⢠⠏⠀⢀⣠⡞⠁⠀⢿⡟⠉⡟
⠤⠤⠶⠶⢶⣶⡏⠉⠉⠉⠉⠉⣿⡇⠀⠀⠀⣿⡇⠀⠀⣿⡅⠀⠀⠀⠀⠘⡇⢀⠋⣠⡴⠋⢸⣇⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⣰⠏⠉⠛⣿⠶⠛⠋⢻⣆⠀⢸⡇⢸⠃
⠀⠀⠀⠀⡆⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⢿⠇⣀⣠⠔⠀⠙⢳⠦⣤⠟⠀⠀⠀⠀⠹⡇⢸⡇⢸
⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣿⡇⢀⣤⣾⠋⣿⢿⡶⢢⠔⠉⣀⣤⠄⠀⠀⠀⠀⠀⣏⣸⠃⣼
⠀⠀⠀⠀⢧⢸⣧⣠⣤⣤⣤⢤⡴⠦⠄⠴⠐⣿⡗⠉⠀⠙⣿⣁⡼⠓⢦⡴⣏⢡⠞⣀⠀⠀⠀⢀⡴⠁⣹⠠⢿
⠀⠀⠀⠀⢸⠈⡷⠴⢒⡂⢯⣸⠧⢷⡀⠀⠈⡏⡇⠀⠀⣀⣴⡻⠐⠇⠀⢠⣸⡇⢀⡅⠀⣏⠀⢸⣥⣤⣿⣴⠋
⠀⠀⠀⠀⢸⠀⡇⢠⡎⠀⠀⠀⠀⠀⢻⡄⠀⡅⡇⢠⡞⠁⠈⠁⡰⠀⠂⠈⠛⢷⣼⡇⠀⣿⠀⢸⣆⣠⢿⣿
⠀⠀⠀⠀⢸⡇⡇⣸⠇⠀⠀⠀⠀⠀⠀⣿⠀⡇⣿⡟⠀⠀⠀⠀⣉⣀⠠⠔⠊⢹⣿⠿⠦⣿⣷⠿⠛⡏⢸⣿
⠀⠀⠀⠀⠈⡇⡇⣿⣶⣟⣭⣭⣭⣽⣯⣿⡄⡇⡏⠻⣄⠀⠀⠀⠀⠙⡄⠀⠀⢸⡟⠀⠀⢸⡇⠀⠀⣷⢸⡿
⠀⠀⠀⠀⠀⣇⣧⣻⢸⡄⠀⠀⠀⠀⠘⠞⣧⣯⣇⣀⠈⠳⣄⠀⠀⠀⠘⣄⠀⠈⣿⠀⠀⢸⡅⠀⠀⡿⢸⡇
⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⣻⢩⡇⠀⠈⠳⣦⡀⠀⠘⢿⢦⣟⠂⠐⣾⠒⠒⠲⡇⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣼⢸⡇⠀⠀⠀⠀⠙⢦⡀⠈⢿⣿⣧⡉⣿⠈⠉⢠⡇⢠⡇
⠀⠀⠀⠀⠀⣿⣆⠀⣿⡉⣩⣭⣍⢩⡉⠉⣿⠀⡿⢸⠃⠀⠀⠀⠀⠀⣀⣽⣆⠀⠻⣀⡁⣿⠀⠀⢸⠇⣸⣧⡄
⠀⠀⠀⠀⠀⡇⢹⣆⠘⣷⠹⣻⠿⣽⠗⣰⠇⣰⠇⡜⢀⡀⣀⡀⠀⠀⠉⣽⣿⣧⠀⣿⣷⡌⠴⠶⠾⣠⣿⡿⠇
⠀⣤⣶⣶⣦⠥⠤⠿⣶⣀⠑⠒⠚⠛⢛⣥⢞⣡⠞⠃⣺⣴⣾⡷⠀⠀⣤⣼⡿⣼⣇⣼⡿⢷⡖⠒⢺⣋⣽⡇
⠀⠸⢷⡶⠿⠒⠚⠓⠚⠛⠛⠳⠶⠞⠛⠛⠉⠁⠉⠉⠉⠉⠉⠀⠀⠀⢷⣿⣿⣤⣿⡿⠓⠒⠛⠓⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣀⣀⡀⣀⠀⢀⡀⠀⢀⠀⠀⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡿⠤⠄⣻⡏⠀⣾⣻⡄⣿⣧⣰⣿⠀⠀⠀⢸⡇⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠋⠻⠸⠃⠈⠳⠻⠘⠏⠸⠀⠀⠀⢸⡇⠀⣿⣥⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⠀⢸⡇⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿⠉⠉⢹⡇⢸⡇⣿⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠄⠀⠿⠶⠶⠾⠃⢸⣷⡏⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⣀⣀⣀⣀⠀⢸⣿⠃⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⢹⡇⢸⡟⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠿⠶⠶⠾⠃⣸⠇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⣀⣀⣀⣀⠀⡿⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⢹⡇⣷⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠿⠤⠤⡾⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠇




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢀⠉⡆⠏⠘⠈⠀⠃⡄

⣀⠀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⢀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣄⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⡆⣀⣀⣀⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⢀⡾⠿⣿⡿⣿⢿⡿⣿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣠⣤⣀⣀⠀⠀⢀⣿⡇⠠⠆⠐⠃⡈⠁⠀⢸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣷⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠒⠒⠚⠓⠒⠚⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠓⣿⣿⠏⢈⡉⠉⣀⠉⡟⠒⢿⡗⠚⠓⠚⠛⠓⠚⠛⢟⠒⠒⠒⠒⠒⠒⠚⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠈⠁⠀⠉⢠⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⣿⣿⣷⣤⠀⠀⢰⣿⡧
⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣾⣿⣿⣿⡿⢅⠀⠈⠉⠀⠀⠀⠀⠀⠉⢀⣀⣀⡀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⡂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣟⣃⣚⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣵⣔⢀⣀⣀⣀⣤⣄⣀⣀⣀⣠⡤⣽⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠴⠤⠶⠶⠦⠴⠦⠤⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠶⠶⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠧⠜⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠻⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⡟⢀⣖⠀⣶⣀⣻⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠶⠆⠀⠀⠀⠀⠀⠀⡇⣸⣿⠀⣿⣿⣿⣿⣿⡿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠻⠟⠀⠻⠉⠉⠉⠀⠀⢙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠘⠊⠛⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⠆⠒⠀⠀⠀⠀⢠⡀⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⡂⢨⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣅⠙⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠛⠋⠉⠀⠚⠣⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡷⠶⢶⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⣿⣿⡏⠉⠉⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⢀⣠⣶⣦⣤⠀⠀⠀⢀⣰⣿⣟⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠁

⠀⠀⠀⢀⡠⠤⠤⠀⠠⠤⡄⢀⡄⠄⠤⡄⢀⡤⠤⠤⠤⢄⠀⢀⡤⠤⠤⢤⠀⠀⢀⡴⠶⠶⠶⠆⠀⠀⢠⠄⠴⠦⡄⠀⠀⠀⢀⡤⠀⠀⠒⠀⠒⠂
⠀⠀⠐⠨⡄⠀⠀⠀⠀⠀⣳⠀⡇⠀⠀⣀⡎⡇⠀⠀⠀⠈⣄⠎⡇⠀⠀⢸⠀⣐⢹⠀⠀⠀⠀⢸⠀⢰⢰⡇⠀⠀⡇⠀⢀⡔⠏⠀⠀⣀⡤⡤⣤⡴
⠀⠀⢸⠐⡇⠀⢰⡯⠭⠟⠁⠄⡇⠀⠀⣿⠆⡇⠀⠀⠀⠀⢹⠀⡇⠀⠀⢸⠀⡏⢸⠀⠀⢷⠀⠈⡆⢸⢸⡇⠀⠀⡇⠀⡎⢸⠀⠀⠸⡒⠐⠒⠁
⠀⠀⢸⢀⡇⠀⠈⠁⠀⠈⢹⠀⡇⠀⠀⣿⠆⡷⠀⠀⠀⠀⠈⣧⡇⠀⠀⢸⢰⡅⡿⠀⠀⣿⠀⠀⣇⢸⢸⡇⠀⠀⣣⠀⢇⠀⣆⠀⠀⠈⠉⠁⠐⠢⡀
⠀⠀⢸⠀⡇⠀⠀⡤⠤⣤⢾⠀⡇⠀⠀⣿⠂⡇⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⣾⠀⡇⠀⠀⠻⠃⠀⢹⢸⢸⡇⠀⠀⡝⠀⠈⠲⢤⣓⣖⡶⠶⢄⠀⠀⢹
⠀⠀⢼⠠⡇⠀⢀⡏⠉⠁⢸⢀⣇⠄⠀⠿⠇⠷⠠⠤⠼⣇⣀⠀⠁⠀⠀⢸⣿⢨⡇⠀⢀⡤⠀⠀⢸⣼⢸⡇⠀⠀⠧⠤⣤⠀⡤⠤⠠⠭⠧⠊⠀⠀⡸
⠀⠀⢚⠠⡇⠀⢸⡇⠀⡠⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠂⠀⠀⢸⡿⢸⠀⠀⢸⠂⡇⠀⠈⣿⠠⡇⠀⠀⠀⠀⣸⡎⡇⠀⠀⠀⠀⠀⣀⡴⠁
⠀⠀⢸⣠⠓⠒⣺⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢀⢀⣰⣏⣫⠟⢷⣘⣒⡚⠋⣦⢛⢒⡲⠋⣖⣛⣚⣘⣋⠟⠙⣧⣓⣂⡒⡺⠭⠟⠋
⠀⠀⠀⢉⠍⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣨⠴⠞⠉⠉⠁⠶
⠀⠀⢐⡦⠀⠀⠀⠤⠤⠶⠂⠤⠀⠀⠐⠊⠒⠛⠛⠡⠜⠁⠀⡄⠀⣹
⠀⠀⡁⠠⠀⠀⠒⠒⢒⣂⣀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣻⡠⠔⠂⠀⠉⡁⠀⠄
⠀⠀⣁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⠤⣤⠄⠉⠉⡈⠀⠀⠔⠊⢑
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠀⢀⣀⣀⣀⡀⠰⠁⠀⠀⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠘⠀⠀⠀⠈⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⢠⢂⡀⠠⠤⠤⡤⠴⠶⡠⠴⠛⠉⠓⠲⢤⣴⣄⠄⠊⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠁⢀⣀⡤⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠼⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠀⠀⠤⢾⢱
⠀⠀⠀⠀⢀⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢄⣀⠀⣰⠀⠁⠀⠁⢆⣀⠀⠀⠀⠁⠐⠢⢤⣀⡀⣀⠸⠀⣇
⠀⠀⢠⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂⣑⠃⠉⠤⠄⠠⠍⠭⠉⠭⣍⡉⠁⠉⠒⠋⠉
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠉⠀⠀⠠⡇⠓⠤⠁⠂⠤⠠⠄⣀⠀⠀⢰⠁⡰
⠀⠀⠀⠠⠌⡉⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⢀⠐⠉⠀⠀⠀⠀⣀⡤⠛⠀⠀⠀⠈⠀⠀⠒⠀⠨⢝⣛⣾⠠⡄
⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠉⠓⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠁⠀⠀⠀⠀⢀⣴⣞⡉⠀⠀⠀⠀⠀⠀⡔⠂⢄⡀⠀⠌⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠖⠛⠛⠛⠋⢉⣙⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠎⡇⠀⠀⣠⡴⠚⠃⠠⢀⡈⠉⠑⠲⠦⢤⣄⣀⡠⣤⠌⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠮⠕⠂⠄⣀⠀⠀⢽⣧⠶⠊⠁⠀⠀⠀⠀⠀⠈⠁⠒⠢⠤⣀⡀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣦⡀⠀⢠⣟⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⢸⡇⢠⡟⢉⣿⢶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣤⣤⣤⡄⣠⣤⡤⠀⣤⣤⡄⢠⣤⣤⡤⠀⣿⠀⢸⣧⡟⢀⣾⠃⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇⠀⣿⡷⠆⠀⠿⢶⡆⠀⢸⡇⠀⠀⣿⠀⢸⡿⢁⣾⠃⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠘⠁⠀⠙⠛⠋⠀⠛⠛⠃⠀⠘⠃⠀⠀⣿⢠⡿⠁⣾⠃⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢫⡿⠁⣼⠃⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⢾⣇⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⢸⣇⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⢰⡟⠉⠉⠉⢉⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⢀⠀⡀⣀⡀⠀⣀⣀⣀⠀⠀⢠⠒⠀⣆⣀⠀⢀⣀⠀⣀⡀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⢀⠿⡁⣔⣺⢸⡇⢹⠀⡇⠀⢈⣙⡆⡏⢸⡇⣗⢚⠰⣗⡚⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈⠁

⠀⠀⢀⡤⠂⠉⠉⠉⠉⠉⡆⡇⠈⠉⠈⠁⠂⣰⠂⠐⡆⠀⡤⠤⢤⢀⡤⢀⣀
⠀⠰⢹⠀⠀⠮⠭⠭⣍⠋⠷⣉⢶⠀⢐⡒⢚⢸⠀⠀⡇⡀⡇⠀⣸⢺⠀⠀⠀⠑⣄⡔⠊⠠⡀⠀⣀⠤⠀⠀⢀
⠀⢫⠘⢄⣀⣀⣀⡀⠀⠑⡄⢐⢸⠀⢸⠀⢸⢸⠀⢀⠁⡇⡇⠀⣿⡘⠀⣈⢣⠀⢹⡈⠣⡀⠈⠉⠀⡀⠼⠃⠃⠀⢨
⠀⠀⠉⠒⠒⠒⢢⣸⠆⣀⠤⠼⠼⠀⠼⠀⣨⣸⠀⠸⣀⢣⠁⢠⡇⡇⠀⣿⡌⠀⡸⠈⡆⡸⠀⢰⠉⠀⡘⠸⠀⠀⡎
⠀⠀⠊⡍⠉⠉⡩⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⢀⡼⢵⠁⠀⠉⠀⡠⠃⢰⢀⠇⠀⡏⠀⢠⠃⡇⠀⡘
⠀⠘⠦⣩⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣤⣾⣶⠴⠋⢰⡸⢤⣶⠶⠋⠀⠀⡎⡼⠀⢰⠁⠀⢜⣘⠠⡤⠁
⠀⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⢀⣤⡶⠚⡏⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣓⡲⠊⠀⢀⠸⣅⣈⠆
⠐⠊⠀⠀⠠⠤⠄⠀⠈⠀⠀⠀⠀⠀⠀⠰⡆⣀⠁⠒⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠁
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣄⡀⠀⠉⠁⠀⠤⠐⠆
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠄⠀⠉⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡠⠔⠤⣄⣀⡀⢀⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠈⠀⠀⢀⣀⠤⠐⠀⠀⠀⠀⠀⠀⠉⠙⠒⠠⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡖
⠀⠀⣀⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠶⠤⣀⣀⡂⠀⠀⠠⡀⠀⠀⠈⢀⣐⣠⣄⣀⣀⢃⣃
⠀⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠭⢉⠐⠂⠐⠀⠀⠐⠂⠀⠀⠉⠉⠈
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠉⠀⢼⠤⠠⠀⠀⠀⡀⠀⠀⠀⡘
⠀⠀⠌⠓⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠄⠈⠀⠀⠀⣀⠼⠂⠀⠀⠀⠂⠀⠠⠐⣒⢣⠆⡀
⠀⠀⠀⠀⠁⠀⠀⠉⠒⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠉⠀⠀⠀⣀⡴⣞⣁⠀⠀⠀⠀⢀⠒⠠⡀⠀⠋⠀⠉⠁
⠀⠀⠀⠀⠀⠉⠙⠉⠉⠉⠁⠈⠑⡒⠤⣀⠀⠀⠀⠀⠀⣿⠀⣀⠴⠋⠁⠀⠀⠀⡈⠁⠒⠢⠦⠤⠰⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠄⣀⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠤⠄⣴⣾⣿⣷⣾⣿⣷⣦⡠⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣦⣭⣭⣭⣭⣭⣭⣭⣭⣵⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⢰⣶⣶⠶⣩⣾⣿⠿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⢀⡀⣫⡘⣿⣿⢸⠷⡵⡷⣿⣿⣿⠰⢊⡹⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⢸⡇⣿⡇⣮⢻⣿⣿⣿⣿⣿⣿⣿⡜⣏⢷⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⢸⢰⣶⣶⠶⣩⣾⣿⣿⣹⡞⣧⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡟⣼⢸⣷⣵⡷⣿⣿⣿⣿⣧⢻⡸⡎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⡿⠛⢿⣟⢫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢷⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢡⣾⠾⣷⠍⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣧⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡘⢿⢿⡿⣣⢋⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⡹⡎⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠶⠾⠿⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓




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

⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣔⡆⠀⠀⠀⠀⠀⠀⢴⣆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣡⠤⣄⠀⠀⠀⠀⣠⠤⣅⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢻⡀⠙⢀⡿⠀⠀⠀⠀⢿⠀⠛⢈⡟⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠛⠉⠀⠟⠛⠛⠛⠀⠉⠛⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⢸⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⢸⠀⠀⠀⠀⢸⣧⠤⠤⠤⠤⠤⣾
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⣆⣀⣀⣀⡼⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢧⣠⠟
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠛⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢀⣔⣉⠑⣤
⠀⠀⠀⠀⠀⠀⢀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⡸⢫⢽⢯⢺
⠀⠀⠀⠀⠀⠀⡃⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢾⠀⢠⡊⠔⢢⣀⣼⢊⠒⢂⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢢
⠀⠀⠀⠀⡠⠞⢿⡖⡶⢅⣠⢄⠀⣀⡀⠀⠀⠀⠀⠈⡶⢡⣆⡸⢒⡈⢆⣿⣁⡷⡤⣤⡆⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢠⣦⡀
⠀⠀⠀⠀⣇⠈⠻⠷⠄⢧⠘⠚⣾⡿⢃⢖⠒⡄⠀⠀⠐⣺⡏⠃⠘⡗⠈⢯⠍⠇⡤⠄⢰⠀⢀⠀⠀⠀⠀⠀⢠⠵⠒⠟⠇⣼⠋⠑⡄
⠀⠀⠀⢀⠿⠳⣤⣤⡄⢀⣵⣾⣏⡷⠬⠅⡊⠤⠤⠄⠒⠻⣁⡾⢿⣷⡒⡶⢲⡂⣷⡤⠼⠀⠀⠉⢢⡄⠀⢠⠃⡄⢃⠙⠀⢫⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠂⢹⡉⠙⡞⢩⣷⠾⠧⡔⣡⡆⢲⡼⠦⠟⠋⠒⣎⣁⡀⠿⠹⡁⠉⡳⣇⢀⠤⠤⣠⣿⣤⣖⣛⠶⠻⢴⠶⠦⠴⢤⡀⠸⣿
⠀⠀⠀⢳⡶⠖⢞⠧⢠⢫⡟⠁⢰⡟⠱⠠⢵⠆⠀⡔⠀⢺⠀⠏⠁⠙⣌⣠⠿⡌⢝⠂⣹⡴⣖⠛⠫⣇⣠⣓⡶⣶⠃⠀⠀⠀⠀⠀⢠⢿⡄
⠀⠀⠀⠀⣿⠆⣌⠀⡖⣾⠀⠀⠀⠉⠑⠛⠼⢴⣼⣣⣲⣉⣬⣎⣼⢣⣿⣤⣦⣴⣮⠼⠽⠿⠏⠀⠀⣨⢿⡉⠀⠈⡏⠩⠯⢋⠉⠉⠉⠂⠰⡀
⠀⠀⠀⢠⠏⠀⠈⠙⣹⠇⠈⠢⠄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⢻⢾⣷⠀⠀⠲⠀⠸⡀⠀⠐⠀⠁⠀⠱⡀
⠀⠀⠀⢸⣦⠤⢤⣿⡟⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠤⠀⢀⣀⢀⣀⡀⣀⢤⢤⠤⠖⠊⢁⠁⣋⠷⡋⣼⡇⠀⡆⠔⠂⡾⠒⣄⡀⠀⠀⠀⣘⠄
⠀⠀⢀⠎⢺⡄⣾⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠡⠀⢂⠀⠀⠈⠀⠂⡀⣘⣺⠥⣣⣿⠀⠙⠀⢠⣿⡄⢹⠍⢹⠫⢹⠇
⠀⠀⠈⢢⡃⢰⣖⣿⣾⡟⠀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠌⠀⠐⠀⢥⠞⣺⢴⣿⠀⠐⠂⣸⢿⣣⠤⠆⠀⠀⡞
⠀⠀⠀⡄⠉⠏⣿⣟⡟⠀⢰⠀⠀⢩⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠉⢎⠌⠀⢀⠂⡚⣡⢦⣫⡏⠀⢂⢀⡿⠀⣿⡀⠘⡀⠀⠱⡀
⠀⠀⢀⠓⠺⢱⣮⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡒⡇⢸⠆⠀⠊⡰⠡⠓⠪⣟⠃⠐⠀⢸⡇⠀⢹⣟⠋⢿⠀⢀⡷
⠀⠀⣄⣀⣯⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡟⢡⠏⠀⠀⠀⣁⢋⣙⣾⡟⢀⡂⠛⣿⠁⠀⠸⣿⡴⠼⣾⠿⢷
⠀⠰⠈⠹⠒⠻⣿⠟⠒⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠋⠀⠀⠀⠐⢛⣊⢿⠎⠀⠨⣤⢨⡏⠀⠀⠀⣿⡇⠀⢃⣀⣠⠆
⠀⠑⢢⠇⢙⣟⣺⠀⠰⣥⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠁⠀⠀⠀⠀⡩⣥⣶⠊⠀⠀⠀⠀⣼⠁⠀⠀⠀⠸⣿⠉⠈⡄
⠰⠀⠄⠀⣾⣩⣇⡀⠠⡎⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠟⠁⠀⠀⢠⠑⠂⣯⢦⡀⠀⠀⢀⢿⡆⠦⢁
⠦⠼⠶⡖⠲⠍⠉⠁⠀⢈⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⢐⡷⠁⠀⠀⠀⠀⢆⡀⢸⡇⠀⡘⣕⢮⡀⠹⣧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠐⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢼⠁⠀⠀⠀⠀⠘⠀⠀⣿⠗⠚⠉⠀⠈⠙⠛⣿⡈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⣠⡥⠴⠒⠉⢈⡧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠩⡏⠀⠀⠀⠀⠀⡌⠙⢸⡏⠀⠀⠀⠀⠀⠀⠀⢹⡇⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠡⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠬⡇⠀⠀⠀⠀⠰⢠⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠤⠇⠀⠀⠀⣀⡄⠀⢰⣷⣤⣤⡤⠤⣤⣄⣀⣀⣀⣿⡎⠁⢃⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡰⠊⢩⠇⡰⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢸⠁⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠒⡄⠀⣾⠟⣋⡿⡿⠯⢿⠟⠥⣍⡙⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣷⡾⠁⠀⡁⢀⠀⡁⠀⠘⠿⢷⡷⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠉⡞⠀⠁⠀⠀⢀⠴⡍⠲⣤⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⢀⣤⠀⣠⡇⠀⠙⢦⣨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠑⢦⡤⠤⠔⡾⠁⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠑⠒⠊⠁⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢔⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⡔⠋⠻⣄⣀⣀⣀⣀⣀⠀⠀⡠⠞⠁⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡏⣤⠈⢇⠨⣷⣿⠛⠛⠛⠛⣳⡾⠛⠓⠒⢄⠀⣠⠖⠉⠉⠉⢁⡶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠫⠽⠋⠀⢀⠝⣉⣛⢀⡀⢀⠀⢹⣬⣔⣖⡤⠞⠋⠀⠀⠀⢀⡴⣫⠞
⠀⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣀⣀⣀⡐⠧⠤⠤⠤⠤⠤⠄⣄⡠⠋⠀⠁⠀⠀⠀⠀⠀⡴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠶⠦⠤⠤⠬⠭⠭⢭⣭⣭⣉⣛⣚⣒⣒⣒⣒⠒⠒⠲⠀⣾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⣤⠔⠒⠒⠚⠉⠉⢉⣩⣍⣉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣈⠁⡤⠆
⠀⠀⠀⠀⠘⣆⠀⠀⠀⣀⡴⢫⠚⢲⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠯⣷⣂⣀⣀⣤⣤⡟⢷⣄⠈⢡⣤⠀⠴⠆
⠀⠀⠀⠀⠀⢸⡀⠀⠩⣼⠀⢻⣀⠜⠀⠀⠸⡄⠀⠀⠀⠀⠀⢸⣵⡾⠫⡭⣽⣼⡊⠀⠀⡟⡆⣤⡄⠲⠆
⠀⠀⠀⠀⠀⠀⣇⠀⠀⣻⣄⣀⣀⡤⢊⡀⠀⡇⠀⠀⠀⠀⢰⡽⠃⠑⠤⠒⠏⠙⠯⣇⠀⠧⣇⡀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⢸⡀⠀⢉⡄⠸⢂⠐⢛⠀⣖⣛⣲⠀⠀⠀⠀⢳⠀⣎⠙⠋⠷⡄⠀⢸⢑⢦⣀⠈⠉⢲
⠀⠀⠀⠀⠀⠀⠀⢧⠀⢨⠖⠘⣃⠀⣋⡠⠀⢸⣯⠀⢀⡤⠤⣈⣦⡈⠉⠓⢒⡡⠤⢾⠛⠉⣀⡥⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⠁⠀⠁⠀⠀⢸⡿⠒⡏⠀⠀⢴⣋⢯⡼⢻⡉⠀⠀⠸⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⡠⠤⠤⠤⠤⠤⠴⠶⠴⣧⠞⠃⠀⢉⠏⠉⠒⠃⣄⣀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣜⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢀⣸⠀⠀⠀⠀⣰⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⣀⢀⡀⢀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡆⠟⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠤⠄⢀⡀⠀⠀⠀⢦⠟⠀⠀⠀⢳⡄⠀⣠⣤⠰⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠀⠀⠀⠀⠀⠈⠀⠐⠂⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢴⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⠀⢠⠀⠀⠀⠀⠀⠀⠲⡄⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠁⠀⠈⠉⠑⢄⠀⠀⠀⠀⠀⠈⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢠⠊⠁⠈⠐⠄⠀⢀⠀⠀⠀⠀⠀⠄⠀⣑⣤⠀⠀⣄⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠀⠀⠩⢦⠈⠆⠊⠀⠀⡰⠊⢱⡀⡜⠁⠀⠈⠀⠘⠑⢄⠸
⠀⠀⢀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠓⣈⠢⠀⠀⠀⠀⡀⠀⠀⡁⢠⣼⠀⠸⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⢀⣉⣀⢤⠀⢠⠀⠘⠀⢀⠀⠈⠁⠈⠛⠓⠂⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢰⠤⢠⠁⡃⠈⡀⠀⢀⢤⡀⠉⠀⠀⠠⠄⠀⠠⠐⠁
⠀⠀⠢⠀⠀⠀⠀⠀⠀⠠⠔⠁⠀⠀⠀⠀⡸⡇⠃⠀⢱⢀⠈⠒⠘⣄⡠⢐⠖⠓
⠀⠀⢸⠀⠀⠀⡀⠀⡴⠃⠀⡘⠀⠀⡐⡐⠀⡇⠀⠀⠀⠳⠢⣀⠀⠤⠂⠁
⠀⠀⠈⣷⠋⠹⢧⠌⠀⠀⠀⠀⣿⣿⢡⠹⣿⠁⠀⠀⠀⠀⢐⠕⢠⡀⠀⠀⢀⠤⠠⡄⠀⡀⠄
⠀⠀⠀⠇⡄⠀⣸⠋⠁⠀⠀⠼⡇⠰⣓⠤⣄⠁⠠⢤⠠⠀⠀⠀⠐⢦⡠⠔⠁⠤⠂⠇⠈⠄⠤⠂
⠀⠀⠀⠀⠁⠠⠇⠀⠀⠀⠀⠀⠑⠒⠀⠒⢲⠶⠂⣒⣒⣀⣠⣄⡰⠈⣷⣰⣤⣤⣤⣾⣮⣤⡤⠞⠋⢠⡴⢶
⠀⠀⠀⠐⠒⠈⠒⠒⠂⠀⠀⠀⠀⠀⡤⠀⠐⢲⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⢴⡇
⠀⠀⠀⢸⣶⠒⣶⠖⠂⠀⠀⠀⢻⠿⡹⠀⠀⡄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢘⠉
⠀⠀⠀⠂⠂⠘⠉⠀⠀⠀⠀⠀⠀⡬⠁⠀⠀⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠧⠴⠴⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇
⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣬⣖⢲⣬⡭⠀⣆⡀⠀⣀⣀⣥⣴⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠀⢰⡆⠄
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠘⠉⠓⠤⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠐

⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⣞⣇⣢⣗⠄⠀⠀⠀⠀⠀⠰⡫⠤⣿⠀⠀⠀⠀⠀⠀⠀⢰⣾⣶⠀⠀⠀⠀⠀⠀⠀⠐⣷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤
⠀⠀⠀⢸⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡄⠀⠀⠀⠀⠀⠀⠘⡛⡛⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠘⠀⠀⠀⠀⠀⠀⢠⣵⣾⣤
⠀⠀⠀⣏⡻⠜⠉⠀⠀⠀⠀⠀⠀⢸⣙⠶⠋⡇⠀⠀⠀⠀⠀⣰⣿⡿⣿⣷⡀⠀⠀⠀⠀⠀⣀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣤
⠀⠀⠀⠓⠒⠒⠒⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠃

⠀⠀⠀⠀⣴⣶⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⣴⠇⢀⡤⠀⠀⠀⠀⠀⢀⠴⠋⡤⢱⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⣀⣼⠗⠀⠀⠀⠀⠀⠈⠄⠓⠒⠚⠴⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣤⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⣠⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣟⠰⢉⡏⠀⠀⠀⠀⠀⠀⢸⡏⠛⢻⡄⠀⠀⠀⠀⠀⣴⣶⣄⢸⣿⡇⠀⠀⠀⠀⢰⣿⣧⣀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣦⣀⣀
⠀⠀⠀⠓⣆⡶⢳⠀⠀⠀⠀⠀⠀⢸⠱⣶⠋⡇⠀⠀⠀⠀⠀⣻⣿⣃⠸⣿⡇⠀⠀⠀⠀⠻⣿⣿⣿⠿⠗⠀⠀⠀⠀⠈⠛⠛⠛⠛⠓
⠀⠀⠀⠤⠼⠾⠞⠀⠀⠀⠀⠀⠀⠘⠒⠛⠛⠃⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡜⠁⠁⠀⠀⠀⠀⠀⠀⢀⠼⡽⠃⠀⠀⠀⠀⠀⢸⡇⠀⠈⡇⠀⠀⠀⠀⠀⠀⣈⣯⣀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠈⣿⡱
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠞⠀⠀⠀⠀⠀⠀⠀⢸⠗⠓⠚⡇⠀⠀⠀⠀⠀⣼⠁⠙⢻⢷⡤⠀⠀⠀⠀⢸⢿⣷⣀⣾⣧
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡈⡇⢹⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠒⠒⣒⠂⠀⠀⠀⠀⠀⠈⠚⠀⠀⠀⠀⠹⠂⠒⢰⣷⠌⠛⡤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⣴⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⢄⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠊⠢⡀⠀⠀⠀⠀⠀⠀⠣⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⠀⢀⠔⠒⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠢⡀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡇⠀⠐⡦⡀⠁⠀⠀⠀⠀⠀⢀⠤⢄⠀⣠⠖⠈⠑⠄⡈⡇⢄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡁⠄⠀⠀⠉⠀⠀⠀⠀⠀⢠⠁⠀⠸⠈⠃⠀⠀⠀⠀⠀⠀⠀⠣⣸
⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⢈⠨⠆⠉⠀⠀⠀⠀⠄⠀⠀⠘⠀⠐⢧⣠⡔
⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡇⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠐
⠀⠀⠀⢰⠀⠀⠀⠰⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠐⢀⠀⠈⠀⢣⡀⠔⡀⠂⠂
⠀⠀⠀⠸⡀⠀⠀⠀⠀⢀⠞⠀⠀⡈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠳⠣⡀⣀⡠⠔⠋
⠀⠀⠀⠈⣶⡖⠈⣿⡠⠃⠀⠀⠀⠉⣿⣿⠃⠛⣾⡏⠀⠀⠀⠀⠀⠀⡽⠤⣀⠀⠀⠀⢀⡠⢄⡀⠀⢀⡀⢀
⠀⠀⠀⠀⠋⠁⠀⠀⡑⠒⠒⠂⠀⣰⡆⠀⡶⠄⣠⠐⠀⢀⣀⡀⠀⠈⠀⠀⢥⡑⣀⠔⢁⣠⠠⡄⠀⠁⣀⡠
⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠆⠀⢉⡁⠉⣉⣀⠉⠁⣀⠀⠀⠀⠀⡄⢣⡀⠀⣀⣀⣀⣠⠀⠀⣀⣠⠄⠀⢀⣠⣄
⠀⠀⠀⠀⠀⣸⡀⢇⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣍⣉⡁⠀⢀⠀⠈⠉⠉⠀⠈⠉⠀⠉⠉⠉⠉⠁⠉⠉⢁⠀⠀⠉⣠⡏
⠀⠀⠀⠀⠠⣀⡀⠀⣀⡀⣀⡀⠀⠀⣄⣠⢿⡀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠇⠻⠇
⠀⠀⠀⠀⡘⠻⠁⠾⡟⠀⠀⠀⠀⠀⠘⠛⡜⠀⠀⠸⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢠
⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⡶⠁⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⢀⡀⠀⠶⠺⠚⠋⠅⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠆⡄
⠀⠀⢀⠁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠛⣯⣬⠁⢸⣀⣀⣀⣀⣀⣬⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡇⠀⣿⠧⠄
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠙⠢⠜⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠠⠤⠀⠄⠀⠠⠠⠠⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠆⠆⠀⠀⠆⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣡⠴⠚⠚⠛⠛⠛⠳⠶⣧⣌⠄⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⢀⡤⢿⢀⡤⡀⣰⢢⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠸⡴⢿⢸⡇⡵⢿⣸⢇⣖⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡂⠀⠀⠀⠀⢸⠈⠛⠁⠈⠉⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣷⠀⠐⡞⠀⠀⠰⠐⠆⠀⠀⠀⣸⠀⠀⢸⢣⠀⠀⢠⣤⠟⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣿⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⣯⠞⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⡠⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀⠀⠈⠓⠤⢄⠤⠤⠚⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⠀⠘⡜⠀⡟⢄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣶⣤⣤⣤⣤⣤⣤⠴⣞⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡤⠃⣇⠀⡷⠈⠢⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠯⣈⣸⠿⠗⠶⠶⠃⠀⠹⣆⣨⠷⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠓⠀⠀⠀⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢠⡯⣄⣀⣀⣀⣀⡠⠤⢿⡗⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠎⠑⠢⢤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠴⠊⠁⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠿⠦⣄⣠⡞⠢⠤⠤⠤⠤⠤⠤⠤⠤⠒⠚⣇⣀⣤⠜⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠘⣆⢸⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣼⠓⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠒⠊⢻⡄⠀⣀⣀⢈⡿⢯⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣀⣀⣴⣇⣞⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⢟⡛⢭⣽⣾⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣤⠿⣷⣦⣄⣒⣛⣋⣉⣽⣿⣿⣿⣿⣿⣭⣥⣴⠶⠾⠶⠞⠓⠺⡿⣄⡀⣠⠃⠀⣀⣀⣀⣀
⠀⠀⠀⡶⠒⠖⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠒⠛⠛⠛⠛⠙⠛⠛⠉⠛⠉⠋⠉⠉⠙⠙⠉⠉⠋⠉⠉⠉⠙⠋⢻⡇
⠀⠀⠀⠷⠤⠤⣤⡤⠤⠤⣤⠤⠤⢤⣤⣤⠤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⠶⡶⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⢰⠃⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠶⠶⠚⠛⠛⠓⠲⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀⣉⡻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣴⡾⣿⣿⣿⣷⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢰⣿⣿⣗⡖⠒⠒⠒⠒⣾⠀⠀⠀⠀⣀⣸⡄⠀⠀⠀⠀⢀⢀⡇⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢸⣿⠿⠏⡿⠀⠀⠤⠀⣿⠀⣰⣄⡴⢋⠉⣿⠟⡞⠀⢠⣯⣽⠁⠀⠈⠋⠌⣿⣿⣿⡿⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⢿⡘⠃⠀⠀⠀⠀⢀⣽⠀⢻⣿⢀⠀⠙⣿⡀⠳⡶⠾⠿⠿⡀⠀⠀⠀⠼⠛⠻⢿⡇⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟⠀⣀⣄⠀⡀⢀⡰⣿⢠⣿⣧⠀⠀⠀⢩⠁⠈⠀⠀⠀⠀⢇⣀⣀⡄⠀⠀⠀⣀⣧⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣠⣤⣾⣷⣄⡛⠃⢐⡶⠶⠖⢿⣼⣯⠛⠳⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣼⠒⠂⠉⠁⠀⠀⠈⢦⡀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠉⣥⣜⠀⠀⠀⣿⣿⣿⣶⣶⣷⠞⠉⢣⠀⠀⠀⠀⠀⢀⡴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⢆⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⢶⣿⣿⣷⡄⠀⣻⣹⣉⣁⡀⣿⠀⠀⠀⢧⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠀⣿⢿⠷⠀⠂⣹⡄⠀⠀⠈⢧⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⡇⣿⠿⠷⠢⠤⠤⢧⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡋⠁⠙⢿⣿⣿⣇⣿⢷⡶⠄⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄⠀⠈⢻⣿⣿⣿⣿⡇⠰⣶⣶⡖⠘⡄⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠹⣿⣿⣿⣿⣾⣿⣿⣧⣤⣼⣦⣤⣤⣶⣿⣿⠀⠀⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉






⣿⣿⣷⣾⣶⡷⢿⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣿⢶⠾⡷⢿⡿⠿⣾⡿⠿⠿⠶⠶⠶⠶⠾⡶⡷⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢀⡴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢴⡆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠈⢳
⠀⠀⠐⠐⠀⠀⠀⠠⠀⠒⠀⠀⠀⠀⠒⠄⠀⠀⢣⠀⠀⠀⠀⠀⠀⠄⠂⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⠐⢠⡔⠘
⠀⠀⠀⠀⠀⠀⠀⢐⠂⡉⢲⠀⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⢸⣟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⡠⢂⠶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⢭⠀⡁
⠀⠀⠀⠀⠀⠀⠄⣤⣤⣤⣀⡄⠀⠀⠀⠀⣄⡠⡉⣲⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡒⠒⠢⠀⡄⣇⠀⠂⠐⠊⡱⠄
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣷⣘⠆⠀⠀⢠⡔⣾⠇⢐⡀⠀⠀⠀⠀⠀⡰⠚⡥⠀⠀⠁⠀⠀⠰⣄⠁⠀⠚⢃⣀⡤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠟⠉⠃⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠛⣀⠀⠀⠀⠀⠀⠈⠉⠒⠚⠉
⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠐⠀⠠⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⢄⣤⢠⣄⠠⡄⠀⢐⡇⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⢬⡵⠄⠚⠀⠔⠰⠊⠑⡅⠀⠁⢈⠔⠁⠀⡁⠂⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠈⠛⠀⡠⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⢀⠎⣁⢀⠴⠀⠀⠀⠈⡐⢤⢄
⠀⠀⠀⠀⠀⠀⠀⠈⡁⣨⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠈⠀⠀⠀⠀⢠⢰⡎⢀⣽⠄
⠀⠀⠀⠀⢀⠤⡕⠀⠀⠉⠀⣀⣄⠀⠀⠀⠀⠊⠒⠊⠒⠓⠚⠒⠀⠀⠀⠀⠀⠀⠀⠀⣙⡀⠀⠀⠘⠀⠀⣔
⠀⠀⠀⠀⠳⠤⠄⠩⠿⠆⠐⠄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⠠⠐⠤⠤⠤⠀⠀⠠⠀⢮⣀⡀⠉⠳⠆⠀⠀⢄⣀⠤⢤⠄
⠀⠀⠀⠀⠀⠰⠚⠓⠲⠶⠛⠚⠲⠆⠀⠀⠀⠀⠄⠒⠒⢄⠀⠀⠀⠀⠀⠀⠂⢄⣀⠀⢰⠶⠶⠶⣤⠶⠶⠶⡆⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⡿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢟⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣯⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣯⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠹⣿⣿⣿⣿⣻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⣠⣽⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣟⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣾⣦
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⡇⠀⢸⣿⠟⠿⠿⠿⠿⠿⠿⠻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣥⣦
⠀⠀⠀⠀⠀⠀⠀⣿⡇⡀⠠⡀⠀⠤⢠⠄⢸⡇⠀⢸⣿⠀⠀⠄⠀⠠⡄⡤⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣸⡷⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠛⠱⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠛⠁⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠁



⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠦⠶⠶⠦⣤⣄⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⡠⢾⢠⢄⢠⢢⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠒⢻⢸⡼⠹⠜⠰⠽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡇⠀⡠⠀⠀⡀⡀⠀⠀⢰⠂⠀⣇⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡇⠀⠰⡀⠀⠀⠀⠀⢀⠎⠀⣸⠼⠀⠀⠉⠀⡆⠀⠀⠀⠀⢹⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆⡀⠈⠒⠠⠄⠒⠁⣠⠖⠁⠀⠀⠀⠀⠀⡇⢸⡆⡀⡆⣸⢢⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢳⣶⣒⣲⠞⢻⡉⢣⡀⠀⠀⠀⠀⠀⠧⠀⠓⠀⢧⡙⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠈⢉⣏⣁⣀⠀⣀⣀⢿⡏⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡌⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠏⠒⠤⡜⠀⠀⠀⠀⠀⠀⠀⣷⠒⠉⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠑⢦⡞⠑⠒⠒⠒⠒⠒⠒⠈⠙⣦⠖⠊⠱⡄⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡸⢅⣀⣀⣀⣀⣀⣀⣀⣀⠠⢽⡄⠀⠀⠘⣼⣾
⠀⠀⠀⠀⠀⠀⠀⡸⢀⣀⣴⣳⠂⣀⣀⠀⠀⠀⠀⠀⣀⣠⡴⠿⠚⣻⣷⠃⠈⢣
⠀⠀⢀⠀⠀⣀⣠⣃⣼⣿⣶⣤⣬⣭⣭⣿⣿⣿⣿⣯⣵⣶⣖⣒⣛⣛⣿⣧⣠⣎⣀⣀⣀⣠⡀
⠀⠀⢾⣉⣉⠉⢉⣉⣉⠉⠉⠉⢉⣉⣉⣉⣁⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⡇
⠀⠀⠈⠉⠉⢿⠉⠉⡏⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⢧⢉⠃⠶⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣤⣀⡀
⣾⣿⣿⣡⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣷⡄⡀⠀⠀⠀⢈⡟⠛⠿⣿⣷⣦⡀
⣿⣿⡏⠁⠀⡾⢯⠄⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠸⣾⠾⠿⠛⠛⢻⡻⠿⣶⣤⣀
⢹⣿⣧⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠈⠋⢀⡿⠛⢿⣷⣦⣀⣀
⢨⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠚⠉⠠⠀⡼⠋⠈⠙⣿⣿⣶⣄
⠈⠻⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣄⠀⣤⣤⣤⣤⣤⣤⣀⠀⠀⣠⣀⠀⠀⠀⣤⣤⣀⠀⠀⣠⣤⣤⣀⣤⣤⣤⣤⡄⠀⣇⠀⠀⠀⠸⣿⠿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⣽⠟⠀⠈⠻⣿⣿⡿⠋⠀⠀⢀⣹⣿⣇⠀⠀⢹⣿⣿⣆⣸⣿⣿⡏⠘⢻⣿⣜⠿⠇⠀⠙⠦⣀⡀⠀⢻⣇⡠⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣀⡀⢀⣴⣿⣿⣧⡀⠀⢠⣿⣿⣿⣿⣆⠀⢼⣿⣿⣿⣿⣿⣿⡇⢰⣦⡙⣿⣿⡆⠀⠀⠀⠈⠉⠉⠉⠛
⠀⠀⠀⠀⣴⡖⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠓⠿⠿⠯⠻⠿⠿⠶⠿⠿⠧⠘⠿⠿⠶⠻⠿⠷⠻⠟⠻⠿⠷⠞⠛⠻⠿⠟⠁⠀⠀⣀⣀
⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⡾⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⢡⣮⠿⠁⢁⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠈⠳⠴⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠈⠃



⣶⣶⡶⠶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠉⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⠿⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣀⠀⠀⣀⣹⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⠿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠛⠃⣀⣴⠟⠹⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⠉⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⠏⠻⣦⣀⠘⠛⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣥⡤⠶⠛⠉⣀⡴⠂⠘⣿⡄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⣿⠃⠐⢦⣀⠉⠛⠶⢤⣬⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠈⢉⡠⠖⠀⠘⣿⡄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣿⠃⠀⠲⢤⡉⠁⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⢠⠖⠋⣀⠘⣿⡄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣿⠃⣀⠙⠲⡄⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠰⠛⣁⡄⠘⣿⣆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⣿⠃⢠⣈⠙⠆⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠚⣁⡤⠆⠘⣿⣆⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⣰⣿⠃⠰⢤⣈⠓⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠉⠀⠀⠀⢨⣿⠂⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠐⣿⡅⠀⠀⠀⠉⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⣀⣤⣾⣿⠃⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿⣷⣤⣀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⡶⣿⡟⢿⡌⢿⣦⠀⠈⠻⠿⣶⣶⣶⣶⠿⠟⠁⠀⣴⡿⢩⡿⢻⣿⢶⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⢻⡌⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢡⡟⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠘⣿⣆⢻⡌⢻⣧⠀⣀⡀⠀⠀⢀⣀⠀⣼⡿⢡⡟⣰⣿⠃⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡈⠉⠛⠛⠿⣶⣦⠀⠀⠘⢿⣆⢻⡄⢻⣿⢛⣿⡇⢸⣿⡛⣿⡟⢠⡟⣰⡿⠃⠀⠀⣴⣶⠿⠟⠛⠉⢁⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡷⠀⠀⢀⡴⢻⣿⠀⠀⢠⣾⣿⡆⢻⣠⣵⡿⠋⠀⠀⠙⢿⣮⣄⡟⢰⣿⣷⡄⠀⠀⣿⡟⢦⡀⠀⠀⢾⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣏⡀⢀⣴⣯⣀⣸⣿⠀⠀⢿⣷⣶⡾⠟⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣶⣾⡿⠀⠀⣿⡇⣀⣽⣦⡀⢀⣹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⣿⠉⠛⠻⠿⣶⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⠿⠟⠛⠉⣿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣇⢠⡆⣰⡄⠉⣩⣍⠛⢛⣛⠛⠛⠛⠛⠛⠛⠛⡛⢛⣛⣛⣉⠉⡥⢤⡄⣶⡄⢾⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢻⡾⢠⣧⣽⣸⡇⠈⢣⡏⠉⠓⣼⢸⢷⣸⡇⣼⢳⡈⢹⠁⢸⢸⡇⢀⡟⣿⠙⡿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢷⣶⣬⣄⣈⡋⠓⠶⠋⠳⠤⠗⠿⠸⠀⠻⠧⠟⠛⠇⠸⠀⠘⠈⢛⣋⣠⣤⣶⡾⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠿⠿⠛⠛⠋⠉

⠀⠀⠴⠒⠒⠂⠐⠒⡆⡰⡂⠒⠒⣆⢴⠒⠒⠒⠲⡀⢠⡖⠒⠒⢲⠀⢀⢴⠂⠈⠉⠑⠀⠀⠰⡖⠀⠐⡆⠀⠀⢀⡴⠒⠈⠉⠉⠉⢹
⠐⡅⣿⠀⠀⣠⣀⣠⠿⡆⡇⠀⠀⣿⢸⠀⠀⠀⠀⢱⡃⡇⠀⠀⢸⠀⡇⡚⠀⢀⡀⠀⡇⢠⢚⡇⠀⠀⡇⠀⡔⣹⠀⢀⣴⣖⣚⡻⠛
⠀⡂⣍⠀⠀⠏⠚⠧⢄⡃⡇⠀⠀⣿⢸⡀⠀⠀⠀⠘⡇⡇⠀⠀⢸⢰⢂⡇⠀⢸⡇⠀⢣⢸⡅⡇⠀⠀⠇⠀⡀⢺⠀⠀⠓⠤⠤⢀⣀
⠀⡇⣿⠀⠀⢀⣀⣀⣸⠀⡇⠀⠀⣿⢸⡃⠀⠀⠀⠀⢹⡅⠀⠀⢸⢘⢠⡇⠀⢸⣇⠀⢸⢸⢚⡇⠀⠀⡇⠀⠳⡈⠢⢄⣀⣀⡀⠀⠈
⠀⡄⣹⠀⠀⡟⠛⠛⢹⠀⡇⡀⣀⣿⣸⣇⠀⠀⣆⠀⠈⠃⠀⠀⢸⣏⢸⠁⠀⢀⡀⠀⠈⣿⡥⡇⠀⠀⡇⢀⡀⠈⠁⠚⠚⣤⣩
⠈⠆⣽⠀⠀⡇⠀⢀⡸⠞⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⡄⠀⠀⢸⡗⣽⠀⠀⡾⢗⠀⠀⣿⡖⡇⠀⠀⠉⠀⣧⢾⠉⠉⠉⠁⠀⠀⣠
⠀⡇⡸⠤⢤⡧⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⢠⣖⢶⠞⣧⢳⡶⡶⢫⡸⠶⢲⠾⣧⢓⠖⡲⢲⡶⢻⡼⡶⢒⢶⢖⡶⠟⠁
⠀⠉⢉⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣭⡶⠛⠉⠉⢯⡁⠀⠀⠉⠁⠀⠈⠉⠉⠉⠀⠀⠉⠈⠉⠉⠀⠀⠁⠉⠉
⠀⣲⠄⠀⠀⠤⠤⠤⠤⠤⠀⠤⠴⠃⠶⠛⠫⣴⠃⠀⡠⢀⣿
⢠⠁⠄⠐⠒⣒⠒⠉⠉⠉⠉⠁⠉⠉⢉⣀⣀⣘⣧⠔⠂⠈⢀⠁⠐⠄
⠈⠔⡠⠀⠀⠀⠀⠠⠄⠀⠶⠶⠶⢶⣤⠀⠀⠄⣀⡀⠖⡈⠛
⠀⡐⠇⠀⠀⠒⠒⠒⠒⠂⠀⠠⠤⠤⡤⠴⠦⠠⠅⠀⠀⠘⢉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠀⠀⠀⠀⠀⠐⠒⠒⠲⠆⠀⠀⠀⠀⠀⠠⠀⣀⠤⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡍⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠠⢀⠀⠀⠀⠀⠀⡀⢀⠀⣀⣤⢤⣉⠀⠁⡆⠀⣀⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠐⠃⠢⠀⠀⠈⢉⡩⠍⠚⠉⠀⠀⠈⠉⠛⠿⠬⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠄⣀⣀⣘⣷
⠀⠀⠀⡇⣀⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠤⣀⣀⡠⠠⠤⣀⠄⠠⠠⣀⠤⠔⢒⣲⠤⢠⣀⠀⠀⠀⠨⡏⣇
⠀⡀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠺⢧⡤⠴⠤⠜⡀⡀⠀⠓⣓⣛⣚⡛⠷⠶⠯⠵⠿
⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣢⠈⠀⠈⠀⠁⠀⠉⠑⠐⠠⢀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⡏⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸
⠀⠣⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠔⡇⠲⢴⠒⠤⠤⣠⣀⡀⠀⠀⡏⢐⠄
⠀⠀⠴⢋⠑⡶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠉⠀⠁⠀⣀⡥⠟⠀⠀⠀⠁⠐⠂⠒⠨⠝⣚⣷⡧⡄
⠀⠀⠀⠀⡘⠀⠀⠀⠉⠓⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡞⠉⠀⠀⠀⢀⣤⣾⣋⠀⠀⠀⠀⠀⠀⠞⠢⢄⠀⠰⠁⠀⠉⠁
⠀⠀⠀⠀⠁⠸⠩⠛⠛⠉⠙⠫⠭⣙⢒⠤⣀⡀⠀⠀⠀⠀⠀⣸⡇⠀⢠⡴⠟⠛⠂⠤⣀⠉⠉⠓⠲⠤⣌⢤⠤⡤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠓⠢⠄⣀⠀⢼⡷⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠄⣀⠠⠊
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢠⠀⠀⠃⠈⡁⠀⠀⣠⠀⠠⠀⠚⠘⠀⠀⠀⠀⠀⠀⡁⢄⡀⠀⢀⠈⠀⠀⠀⠀⡀⢫⠏⠀⢀⠄⠀⠀⠤⠐⠀⠁⠂⢊⠀⠀⡀⠀⠀⣧⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠄⢀⠊⠀⠀⢲⠀⠘⠈⠐⠠⢭⠄⠈⠀⢀⠓⠂⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠄⠂⠉⠀⠀⠐⠁⠀⠀⠘⠀⢠⡤⠀⠁
⠀⠁⠂⠐⠀⠀⠀⠀⠑⠄⠠⠂⠀⠘⡀⠀⠀⠀⠀⠠⠐⡀⡀⠈⠀⠀⠸⠁⣀⣁⠀⠀⠀⡆⠀⠁⡄⠄⠀⠀⠀⠀⠀⡀⠀⠘⠀⠁
⠈⡠⡀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡈⡰⡀⠀⡗⠒⢩⡀⠜⠀⠀⠇⠠⠀⢀⠀⡈⢀⠀⠀⠀⠀⠀⢀⠀⠀⠠⡀⠈⠀⠘⠂⠁⠀⠀⠀⠐⣄⡁⠉
⠐⠒⠉⠀⠀⠀⠀⠐⠀⠀⢀⠀⠠⡀⠀⠀⠈⠱⠎⡄⠫⢀⠟⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠈⠀⠀⠀⠁⠀⠀⡤⠂⠎⠂⠀⠀⠀⠠
⣀⠀⠢⠀⠠⠀⠀⠀⠀⠀⢠⠃⣁⠃⠈⠀⠀⠁⢀⢡⠘⠱⠂⠐⠀⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠢⠀⠀⠐⠀⠀⠀⠒⣫⠗⠒⠁⠀⠀⠀⢄⢲⣃⠄
⠴⣀⠙⠀⠀⠀⠀⠀⠀⠀⠃⠠⠆⠊⠂⠀⢀⢀⣀⡀⠀⠂⠀⣀⡤⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠈⡡⢬⠀⠀⠠⡀⠵⠤
⠐⠃⠐⠂⡠⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⡖⠈⠁⠀⡉⠓⢦⠀⠘⢒⡄⢀⠄⠀⠀⠁⠀⢈⠁⠀⠒⠄⠀⠘⡌⠄⠐⠀⠠⠨⠀⠀⠋⠉⠔⠀⠀⡀⠁
⠀⠀⢤⡨⢀⠀⠀⠀⠀⠀⠠⠈⣥⠀⣸⠀⠀⡣⠱⠁⠆⠀⠷⠀⠡⡑⠃⠀⠐⠑⠄⠀⠀⢀⠀⢀⠀⠀⠄⠈⢀⡀⡀⠀⠀⠀⠀⠀⠀⠐⠠⠈⢁
⠐⠂⠀⠈⠁⠈⠀⠀⠀⠀⢲⠀⠛⠀⠛⠉⠈⠀⠀⠈⡀⣀⡖⠀⢀⣤⣴⡤⠀⠀⠀⠀⠀⠂⠓⡀⠀⠀⢂⢠⠥⠢⡈⡈⠀⠰⠀⠀⠃⠀⠀⢀
⠀⠠⠀⠀⢠⣀⠀⠤⠤⠒⠚⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠙⠀⣠⠸⣄⠜⠀⠀⠀⠀⠋⠀⠀⠠⠅⠀⠀⠤⣁⢡⢐⠒⠀⠀⠀⠤⠀⢠⠀⢠⠌
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠐⠀⠀⠉⠀⠀⠀⠶⠆⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⢂⠀⠤⢀⡀⠀⠀⠀⠁⠒⠈⠀⢀⡀⠀⠰⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠂⠡⡀⠀⠁⠀⡀⠀⠀⠄⢠⠤⠔⠉⠀⠀⠀⠈⡂⢀⠑⡀⠀⠀⡀⠶⠆⢀⠄⠈⡉⠒⠂⠠⠄⠀⢲⠘⡅⠀⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠄⠋⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⣤⢠⡀⠀⠀⠈⠐⠈⢃⠀⠁⠀⠶⠀⠀⠀⢀⠁⠈⠁⠀⡈⠀⢸⠁⠈⢆⠈⠀⠈⡀
⡄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠋⠀⠀⠀⠁⠂⡀⠀⠘⠑⠥⣊⡖⢤⣄⠀⠀⠁⠐⠈⠄⢀⠀⠈⢁⡀⠀⢠⠁⠀⠀⠀⠀⠠⠆⠀⠙⠂⠁
⠁⠀⠀⠢⠄⠤⠀⠁⠀⠀⣀⢤⠐⠄⠀⠀⠢⠀⠐⢁⠀⡀⠀⠀⠀⠈⠙⠂⠄⠀⠀⠠⠒⠀⠤⠨⠀⠀⠀⠉⠈⠀⠀⢀⡆⠀⠀⠠⡀⠀⠀⡄⠀⠦⢀
⡄⢀⠀⠀⠠⠀⠠⠐⢈⠁⠀⠀⠀⢊⠀⠠⠀⠀⠀⠠⠈⣑⠠⠀⠀⣄⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠒⠀⠠⠀⡓⠄⠈⡇⠀⢆⠀⠙⠀⠀⠁⠐⠃
⠉⠀⠁⠀⠀⠄⠀⠀⢎⠄⢊⠀⠀⠈⠄⠀⠣⡈⠀⠀⠀⢠⠂⠀⢰⠇⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⣰⠄⡀⠀⠀⠠⠀⠀⠁⡀⡄⠀⠁⠀⠀⠀⠄⠒
⡀⠀⣄⠀⠈⠄⠀⠀⢀⠀⠄⠂⠀⠀⠀⠀⠀⠀⢄⠔⠀⠉⠀⠀⣸⠃⠈⠀⠀⠀⠀⠀⠀⠄⠀⠠⡈⠂⠀⠁⠀⠐⠀⠈⠄⠈⠀⠀⠐⡀⠀⠀⠁
⠁⠓⠘⠀⠀⠀⠀⠈⠀⠀⣀⠀⠤⠀⠀⠦⠂⠀⠠⠀⠀⣀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠔⠀⠀⠸⡀⢠⠀⠀⠀⡂⠀⠀⠄⠀⠀⢀⠀⠳⠀⡀
⠀⡀⠁⠀⠀⠀⠀⠀⠃⠘⠃⠀⠉⠀⠀⠀⠀⠠⢠⠜⣃⠠⠣⠀⠁⠀⠀⠀⠈⠀⠠⠀⠐⡀⣌⠀⠂⠀⠄⠀⡘⠀⠀⠀⠐⠀⠀⢉⡁⣁⠀⠠⠉⡄⠁
⠀⠀⠀⠀⠠⠀⠐⠒⠂⠄⠀⢲⠧⡀⠀⠒⢀⡰⠁⠀⢈⡟⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠠⠈⠈⢃⠒⠀⡄⠃⠀⠀⡷⡈⠘⣴⠂⠀⠡⣄⠀⠂⠘⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠻⢥⣜⠀⠀⠈⠀⢓⡰⠊⡆⠐⠀⠊⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠈⠉⠀⠁⢀⣀⠀⠁⢁⣀⡀
⠀⠓⠁⠀⣀⠀⠀⠠⠁⠀⠀⠀⠚⠈⠂⠀⠁⠀⠈⠁⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⢠⢦⡙⠋⠄⠀⣀⠀⠽⠀⠀⠀⠀⠰⢎⡠⠂
⠁⠰⢄⡀⠐⢄⢉⡀⠀⠀⠀⠀⢀⡀⠀⠠⠂⢀⡀⠀⠀⠾⠑⠀⠑⠈⠀⠀⠀⠁⠀⠀⠀⢁⠂⠁⠊⠀⠀⠀⠀⠐⠊⠀⢀⡀⡄⠀⠠⡆⡦⠀⠀⢅
⠀⢀⡀⠙⠰⠀⠠⡐⠀⠄⡁⠔⠀⠀⠀⠀⠘⠒⠃⠁⠀⠂⠁⠄⣤⠀⠀⠀⠀⡔⡤⠀⠀⡏⠝⠀⠀⣀⠀⠄⠰⠄⠂⢈⠡⠀⢠⠀⠀⣀⡀⠀⠤⠀⠱
⢷⠀⠂⠀⠀⠀⠀⠀⠄⠈⠠⠀⠀⠀⠀⢀⡀⠀⠀⠀⢈⠄⠀⠀⠂⠭⠀⢠⠀⠈⠀⠐⠂⠀⠀⠈⠉⢠⡄⠠⠐⠀⠉⡀⢀⡄⢀⠂⢸⠀⡀⠀⠀⢋
⢀⣃⡀⣰⣀⡀⠀⠀⢀⡐⡀⣀⣀⢂⡄⠀⡈⣀⠀⣀⣈⣀⣀⣆⠀⢀⣂⡂⣀⣠⣀⣀⣀⡐⠀⠀⣀⣀⣠⡀⠀⣀⣈⣁⣀⣄⠒⠈⢈⣇⣀⠀⣀⢸⡀
⣿⣿⠿⠻⡿⠛⡿⢿⡿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⢀⣤⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠐⠒⠊⠉⠊⢀⠒⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡀⠀⠠⠤⠓⠂⢒⣸⠓⠂⠀⠠⣭⣥⣤⠀⠀⠀⢄
⠀⠀⠀⠀⠀⡠⠒⠀⢈⣩⡭⠭⠤⢈⡁⠈⠛⢍⠭⠉⠷⢻⠁⠀⠒⠂⠀⠀⠀⠐⢒⠂⣭
⠀⠀⠀⠀⠀⠛⠂⠀⠠⠬⠬⠉⠭⠭⠤⠀⠐⣁⡄⠀⣨⢾⡀⠀⠀⠀⠀⣜⢜⢕⠥⠪⠞
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠔⢔⠔⠐⠀⠀⢺⣠⣆⠠⠀⣹⠷⠲⠤⠴⠤⠷⠚⢶⡕⠄
⠀⠀⠀⠀⠀⠀⠃⣤⣀⡀⠐⠐⠅⠔⡕⣑⢹⣛⡙⠂⣠⢸⠁⠀⠈⠉⠉⠉⠉⠁⡀⢀⠀⡄
⠀⠀⠀⠀⠀⢀⠜⠉⣁⡈⠉⠉⠋⠉⠁⢋⠓⠷⡀⡄⠀⠈⠀⠀⠀⠀⠀⠠⠪⠦⠨⠘⣀⡇
⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠉⢉⢉⢉⠄⢀⠤⠂⣇⡂⠀⡂⠀⢨⡉⠀⢒⠒⡒⢂⡉⣉⢰⡇
⠀⠀⠀⠀⠀⡟⠢⠤⢀⡀⠀⠁⠀⠀⠐⠉⣉⠁⡇⣀⠀⡇⠀⢸⡈⡄⠽⣘⠀⢀⠀⡏⢻⡅
⠀⠀⠀⠀⠀⣷⣴⡆⡄⠄⣉⠉⠉⡉⢉⠡⠂⠀⡇⠀⠎⡇⠀⢸⣹⣶⡠⠭⠂⠨⠄⠒⠚⠄
⠀⠀⠀⠀⠀⠯⣈⠁⢆⠸⣀⠠⠀⠑⣌⢈⡁⠀⡇⠀⡀⠅⠀⢸⠀⡫⠑⣋⣉⡀⠀⣒⠀⡆
⠀⠀⠀⠀⠀⣾⠶⣭⡒⢒⠂⠤⠤⡤⠄⢒⠊⠁⡖⠒⢛⡀⠀⠀⠲⣦⠍⡀⠂⠀⠀⣀⠤⡇
⠀⠀⠀⠀⠀⢲⣅⣒⠀⠈⠝⠉⠀⠇⢸⡗⠀⡆⡏⠉⠉⢙⠀⠀⠀⠀⢉⣒⣒⣎⠩⠀⠀⡏⠀⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⡙⠪⢕⡀⠸⠓⠂⠀⠀⠈⠀⣀⠃⡧⠀⠀⠈⠢⡙⢤⣐⠛⡬⣍⣸⣶⠟⣸⣒⢒⣒⡂⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠁⡀⠀⠀⡉⠐⠒⢒⠂⢀⡊⣠⠂⣗⣒⡂⠀⠀⠈⠑⠪⢥⣬⣬⣬⣴⡛⠠⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠘⢜⡦⣀⠨⠲⠃⠔⣤⢦⣴⠟⡾⠀⠀⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠪⢍⣊⠗⠒⠚⢉⢠⠚⣁⠀⠀⡀⠠⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠉⠠⠤⠠⠀⠐⠂

⠀⠀⠀⠀⠀⠀⢠⢾⣯⣷⣼⡇⠀⠀⠀⠀⣤⣤⣤⣀⣀⣀⣀⡀⠈⠉⠀⣠⣴⣲⢿⡄⠀⠀⠀⠊⠀⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⡖⣾⡏⡿⣿⣽⣽⡇⠀⠀⠀⠘⣟⡮⡷⣷⠀⠀⣠⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⣀⡤⣆⢄⠀⠀⠀⠐⠢⢄⠀⡐
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣇⠀⠀⠀⢻⣻⣟⡧⣏⡄⣽⡇⠀⠀⠀⠀⢹⡾⣿⠟⠀⠀⠀⠉⠒⠂⠀⢌⠀⠁⠀⣿⣹⡳⣻⣹⣯⠷⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠠⠀⢹⣿⠿⠿⠛⠋⢀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠁⠀⠀⠸⣧⣳⣷⠷⠟⠚
⣶⠀⠀⢁⠀⠀⠠⠄⠀⠀⠀⢠⣿⣿⣯⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⢄⣀⠀⠀⢸⣿⣿⣷⠀⠀⠀⢀⣀⣀⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⣿⡟⡗
⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⠈⠀⣘⢻⠃⠈⣿⣷⡿⠀⢤⣾⣿⣿⣿⣗⡀⠀⢀⣤⠶⣴⢄⠀⠀⠀⠘⣿⣯⡇⣷
⠋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣤⣴⡶⢧⠀⠀⠀⠀⠀⠀⠀⠀⡟⠐⡂⠍⢀⡞⠀⠀⠁⠁⠀⠀⠀⢻⣿⠿⠛⠁⠀⠐⣟⢙⢖⣿⠗⠀⠀⠀⠀⠀⠙⠙⠛
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢻⡞⢭⣿⡆⠀⠀⠀⠀⠀⠀⡸⠉⠷⣦⠤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠈⡿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠈⠀⠀⠀⢠⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣳⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣶⡄⠀⠈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠻⠯⣷⣿⠿
⢠⠀⠀⠀⠀⠀⣀⡀⢠⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢀⣴⣯⣷⣄⠀⢀⡠⢞⠷⣗⠀⠀⠀⢀⣬⣍⡛⠃⠀⠀⠀⠠⣾⣶⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣆
⢹⠀⠀⢀⣴⣾⣿⣷⣌⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⡘⢿⣅⡷⠋⠁⠀⠀⣼⣿⣿⣿⣷⣶⠀⠀⠈⢾⣿⣿⣿⣿⡿⠀⠀⠀⠀⡀⠀⠀⠀⢸⣁⣉
⢸⠀⠀⠙⢿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠃⠀⠉⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠃⢀⣄⠀⠀⠉⠻⣷⠟⠀⠀⢀⠀⠀⡀⠀⠀⠀⢸
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⢀⣌⡛⠁⠀⠀⢰⣾⣿⣿⠀⢰⣏⡿⣿⣿⣿⠏⢰⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢷⣜⣥⣶⡀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡄⠀⠈⣯⣿⣿⠄⠀⠉⠙⠛⠛⠁⠀⠀⠙⢿⡿⠿⠀⠀⠀⠀⠀⠀⢀⣤⢿⣤⠀⠀⠀⠀⣀⣀
⡇⠀⠀⠀⠀⠀⣠⣾⢯⠣⡱⣝⢿⡶⢋⣢⠀⠀⠀⠈⠛⠿⣿⣿⠟⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣀⣤⡤⡶⣶⡀⢸⠧⠆⣸⠀⠀⢸⣿⣿⣿
⠋⢀⠀⠀⠀⠸⣿⢮⡳⡑⣝⢮⣢⡽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣤⣶⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣺⡟⡼⢱⢣⣇⠈⠑⠚⠃⠀⠀⠈⠿⠿⠛
⢠⣾⣿⣦⣄⠀⠈⠳⣽⣌⣮⠿⠋⣀⣄⣤⡀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡄⠀⣀⣴⢦⠀⠀⠀⠀⠙⠻⠟⠛⠋⠉⠀⠀⠀⠀⠀⣤⣤⣶⣶⣧
⠈⢿⣿⣿⣿⠏⠀⠀⠙⠟⠁⠀⠀⢿⣿⣼⡇⠀⠀⡀⡀⠀⡁⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡜⣏⡧⡪⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿
⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠘⠿⠟⠛⠉⠀⠈⢢⠞⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣢⣤⡀⠀⠀⠀⠀⠀⢀⡟⠳⠒⣢⣤⠀⠀⠀⠀⣤⣶⡞⣿⢸⡇⠀⠀⣾⣿⣿⣷⣆⠀⢀⣶⣿⣷⡄⠀⠁⠀⠀⠀⢀⣠⣴⣶⣦⡄⠀⠀⠤⠀⠀⠐⠀⠀⠀⠀⢀
⡽⣿⡵⠀⠀⠀⠀⠀⣸⠐⠉⠐⢠⡇⠀⠀⠀⢰⣷⣿⣿⣿⣿⣷⠀⠀⠙⠛⠿⣿⠏⠀⣾⣿⣿⡟⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣛⡄⠀⠀⠄⠀⠀⠀⠀⢀⣴⣿
⡾⠋⠀⠀⠀⠀⠀⢰⠷⣦⣄⣂⡜⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣻
⠀⢀⣀⣀⣀⣀⠀⠈⠐⠲⠇⠈⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣀⣠⠤⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⡀⠀⠐⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢶⡗⢸⠇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠷⠛⠛⠃⠀⣴⡞⠉⠄⠀⠀⢳⡞⢳⠀⠀⠀⣶⣾⣿⣿⣿⠀⠀⠰⠀⠀⠐⣾⣿⣻⣷⣦⣄⡀
⠀⠀⠛⠚⠓⠛⠀⢰⣶⣾⣿⣷⡄⠀⣀⣲⢤⣄⣀⣀⣾⣷⣄⠀⠀⠀⠈⡀⠈⠀⠈⠀⠘⣏⣀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⣰⣷⣻⣷⣳⣻⢯⡟⣄
⠀⠀⢀⣀⣀⡀⠀⠈⣿⣿⠿⠿⠿⠀⢹⣏⠣⡈⡟⠻⣿⣿⣿⡇⠀⠀⠀⢱⣦⣠⠤⠖⠋⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠛⠷⣿⣣⣯⢧⣿⢻⡗
⣿⣿⣿⡟⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠀⠀⠙⠍⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠷⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣭⡙
⠿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⡳⢤⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠃⠀⠀⠀⠀⡾⢲⡖⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣠⡿⡇⠀⠀⠀⠀⠀⠈⠻⢿⣿⡁⠀⠀⠀⠀⠀⠿⠼⠥⠿⠇⠀⠀⠀⠀⠀⠀⠀⣠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠒⠒⠷⣿⡖⠐⠢⠀⠀⠀⢀⠀⢀⣀⣀⠀⠀⠀⠀⡠⠒⢒⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⠙⠲⣶⣶⣦⣀⢈⡿⠶⠴⠁⠀⠀⢻⣦⣤⣤⣄⣍⣑⠖⠀⣠⣤⡿⠋⠁⣠⡑⣀⢀⡀
⠀⠀⠀⠀⠀⡠⠊⠉⠉⠛⠲⢶⣿⣀⠀⠈⢻⣿⢿⠟⠁⠀⠀⠀⠀⠀⠘⠿⢿⠿⠟⠻⡇⠀⣸⡿⠋⢀⣼⣿⡟⠋⢉⣁⣌⠃
⠀⠀⠀⢀⠊⢀⣼⣿⣶⣤⣤⣾⣿⣿⣷⣶⠾⢱⢺⠀⠀⠀⠀⠀⠀⠀⣴⣤⣾⣦⣼⠘⠀⣠⣿⠇⠀⣸⣿⠏⠀⣰⣿⣿⣿⡀⠆
⠀⠀⠀⢘⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⡧⣼⡦⠤⣄⠀⠀⠀⠀⡖⠒⠀⠀⢺⠘⠶⠋⢰⠀⣰⣿⣿⣤⣾⣿⣿⣿⣟⣳⠈
⠀⠀⠀⢸⠈⣿⣿⣿⣿⣿⣿⣿⣿⣏⣁⠄⣰⠊⠀⠀⣠⠞⠀⠀⠀⠀⡆⠀⠁⠀⠙⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢡
⠀⠀⠀⢘⠈⠋⠹⢛⣿⣿⣿⣿⣿⢿⣿⣶⣯⣤⣴⡿⠁⠀⠀⠀⠀⠀⢷⣶⣶⣶⣾⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⡆
⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⡿⣳⣿⣿⣿⣿⣿⢻⠃⠀⠀⠀⠀⠀⠀⢾⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢱
⠀⠀⠀⠠⠀⢿⣿⣿⣿⡟⣿⠁⣿⣿⣿⣿⡟⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠚⠛⠛⠛⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠸⡆
⠀⠄⠀⠘⣣⣉⣿⣿⣿⣇⢻⡄⠻⡟⠋⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡘⡻⣿⣿⣿⣿⣿⣿⣯⣤⡈⡇
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣷⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⢾⣿⣿⣿⣿⣿⣿⣷⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠂
⠀⠀⠀⠀⠰⡏⠉⠁⠀⠀⠀⠀⣠⢔⡒⠲⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠖⠒⠒⠒⠦⣤⣀⠀⢠⡶⠀⠚⠛
⠀⠀⠀⠀⠀⠹⡀⠀⠀⣀⢄⠞⣱⠉⢙⡆⠀⠀⢿⠀⠀⠀⠀⠀⠀⠰⡗⢾⣯⣤⣤⡤⣶⣾⠏⠻⣧⠀⠠⠶⠀⠐⠛⠃
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠒⡎⠀⠹⢤⠞⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣷⣿⠋⠶⢖⣧⣟⣁⠀⠀⡇⡇⡤⠦⠘⣛⡀
⠀⠀⠀⠀⠀⠀⠸⡄⠀⢀⣳⢤⣀⣀⠤⢚⡁⠀⢸⠀⠀⠀⠀⠀⢸⡽⠃⠘⢒⣒⠻⠀⠙⠬⣇⠀⠣⢧⣀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣉⡄⠸⠆⠀⡛⠂⢰⣚⡓⡆⠀⠀⠀⠀⢱⠀⢸⠉⠛⠹⢦⠀⠀⢾⢑⣦⣀⠀⠉⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢩⡕⠸⣋⡀⢛⣃⠀⠈⡇⡇⠀⢀⣀⣀⣈⢣⡈⠉⠓⠢⠜⡠⠤⣾⠟⠋⠁⣁⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠐⠋⠀⠚⠉⠀⠀⣇⡧⠰⡎⠀⠀⣨⠟⢻⣒⡶⡞⠉⠀⠀⢹⣤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣀⣀⣀⣀⣀⣀⣽⣁⣸⢃⡴⠇⠈⠙⣶⠓⠦⢽⢀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡸⠋⠀⠀⠀⢠⠯⠁⠀⠀⢸⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢠⢏⠂⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠤⠤⠤⠤⠾⠤⠤⠤⠤⠤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠸⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠰⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠐⢄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠢⡀⠀⠀⠀⠀⠀⢂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠎⠁⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢⠀⠀⢤⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠆⠀⠈⠅⠀⠀⠀⠀⠀⠠⠊⠉⡄⢔⠁⠀⠀⠁⠚⠈⠢⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠘⢢⠰⠀⠀⠀⠀⠀⠀⠀⢃⠀⠐⠄⡠⠂⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠐⠄
⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈
⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⢣⡠⢂
⠀⠀⠀⠀⡇⠀⠀⣠⠀⡠⠀⠀⣀⣀⡀⢀⣀⠀⡀⠀⠀⠀⠑⠓⣄⠠⠄⠊⠁
⠀⠀⠀⠀⢹⡏⠀⠿⡔⠀⠀⠀⠀⢸⠿⠇⠉⠿⠇⠀⠀⠀⠀⠈⠞⠒⡄⡀⠀⢀⠤⠂⡄⠠⡠⠄⢀
⠀⠀⠀⠀⠈⠀⠀⠀⠏⠉⠉⠈⠀⠃⠀⢍⠢⠧⠄⠀⠤⠄⠂⠀⠀⠀⢲⡴⠒⠁⠔⠂⠢⢀⠀⠀⠂
⠀⠀⠀⠀⠀⢐⠀⡄⠀⠀⠀⠀⠀⠈⠀⠐⠐⠒⠒⠒⠒⠒⠂⠀⠀⠈⠋⠽⠬⠒⠒⠒⠚⠃⠀⠀⠊⠭⠀⠀⣲
⠀⠀⠀⠀⢠⠋⠈⠁⠀⠂⠀⠐⠂⠀⠀⡀⠀⠀⢓⠂⠀⠒⠂⠐⠒⠂⠀⠐⠂⠀⠀⠀⠀⠐⠀⠀⠒⢲⡄⢰⣿
⠀⠀⠀⠀⠀⠛⠛⠠⡞⠁⠀⠈⠁⠀⡿⡁⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⢡⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠀⠰⠴⠗⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⢈⣀⣀⣀⣀⢀⣀⣀⣀⣀⣝⠙⣋⡉⠀⠘⠀⠀⠀⠀⠈⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠀⠘⠃⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⢉⣓⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⡇⠀⢀⡤⠂⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡇⠐⠣⣀⣘⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡉⠙⠇⡈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⢀⡴⠛⡆⣠⠁⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⢀⡔⣁⠤⠼⠒⠲⠤⣄
⠀⠀⠀⢀⣞⣀⣠⡇⠘⢦⣀⣠⠞⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⢰⣀⠞⠊⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⣠⠊⠉⢰⠃⠀⠀⠀⣀⠯⠂⠀⠀⠈⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⡴⠁⠀⠀⢀⡱⠖⠂⠉⠀⣀⠀⣀⡄⠀⠀⠈⢆⠀⠈⠀⡗⠂⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⡼⠳⣶⠖⠈⠀⠀⠀⣀⠔⠊⠁⠙⠳⠇⠀⣏⠛⠢⢦⠀⠀⠛⠀⠀⠐⡎
⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⣦⠐⠀⠀⠈⠣⣀⠀⠀⠀⢀⠇⠀⠀⠂⠀⢸⡎
⣀⣀⡠⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⡌⠂⢄⣀⡼⠓⠢⣀⡸⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠈⡦⡀⠀⠨⡃⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⠀⢠⠃⠈⠢⢄⠘⠤⣀⢀⡎⠈⠒⢄⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⠀⠀⡼⠀⠀⠀⠀⠑⠒⠈⠉⠀⠀⠀⠀⠈⠢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣦⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⢉⣨⢿⣿⣿⣿⣿⣿⣏⡙⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣠⡞⠁⠀⠒⠙⠛⠀⠙⠈⠉⢿⣷⣤⣿⡻⢿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠁⠀⠠⡶⣠⡆⠸⠆⠀⠀⠈⠁⢹⣷⡙⢦⣌⣉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⡄⠙⢯⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠦⠤⠆⠀⠀⣠⡞⠛⠿⠿⠛⠛⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠑⢦⠤⠠⠤⢴⡎⠉⠉⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠈⠦⢤⠤⠚⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣯⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠖⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣳⣤⠞⠉⠙⢦⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡫⣅⡀⢹⠀⢈⣶⢼⡿⠛⠿⠿⠿⢿⣿⡽⠛⠛⠾⢿⣦⡀⢀⣠⠞⠛⠛⠚⠓⢒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠡⡛⢲⠟⠁⠀⠙⣲⠞⢼⡇⠀⠀⣀⡀⢺⡟⡔⡠⣠⠄⣀⡿⠛⠉⠀⠀⠀⠀⢀⡴⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠁⠀⠀⣠⠞⠡⠤⠀⠐⠒⠒⠒⠂⠀⢙⡷⠳⠷⠋⠁⠀⠀⠀⠀⠀⢀⡴⢋⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠶⠶⠤⠴⠤⠤⠤⢤⣭⣍⣉⣉⣙⣓⣒⣒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠒⠒⠒⠲⠖⠶⠶⠶⠤⠤⠤⠤⢤⣭⣭⣍⣉⣉⣉⣉⣙⣛⣚⠓⠚⠒⣻⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⡀
⠀⠀⠀⠀⠀⣤⡄⠀⢠⣄⠀⢸⡿⠂⠀⠀⠀⠀⠀⠀⠀⣢⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⣿⢿⣄
⠀⠀⠀⠀⠀⠙⣧⡤⢞⣛⣒⣚⣷⣤⣶⠆⠀⢀⠀⠀⠠⣿⣿⣿⡿⠄⠀⠀⠀⠀⢴⣄⢀⣞⣵⣿⠟⠁
⠀⠀⠀⠸⣷⠞⣵⣾⣿⣿⣀⣹⣿⣿⣿⣦⣤⠿⠗⠀⠀⠉⠟⠛⠀⠀⠀⠀⢀⡴⠋⠙⢷⣝⠟⠁
⠀⣶⣦⣠⢏⡾⢿⣧⣾⣿⣿⣿⣧⡿⠛⢻⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠐⢦⡀⠀⣹⢷⠄
⠀⠀⠀⢸⢸⣧⣼⣿⣁⣨⣿⣋⣿⣷⣤⣼⣿⡟⠃⠀⠀⠀⠀⠀⢀⡴⢋⠀⠳⡄⠀⣠⠞⠁
⠀⠀⠻⠻⣼⣿⣏⣽⡟⢻⣿⣿⡿⢻⣿⣏⣿⠷⣶⣦⠀⠀⢀⡴⢋⡀⠈⠳⠀⣠⠞⠁
⠀⠀⣠⣤⠿⣿⣿⡟⣻⣿⡏⠀⢹⣿⣿⣿⣧⡀⠀⠀⢀⡴⠋⢤⡀⠙⠂⣠⠞⠁⢠⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠙⠃⠀⣼⡿⢿⣿⣿⣿⣿⣿⠿⣏⠀⠉⠀⢀⡴⢋⣄⠻⣿⣾⣿⠟⠁⠀⠀⣿⢸⡇⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠀⢀⣾⠉⠉⠿⠆⠀⠿⡷⢀⡴⢋⣴⣝⢿⣿⣿⠟⡁⠀⠀⠀⠀⠈⠛⡟⠛⠋⠉⠉⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⢀⡴⣯⣴⣦⣙⣿⣿⠟⠁⡜⡉⢆⠀⠀⠀⡠⠞⠁⠈⠉⠉⠁⠙⠲⡄
⠀⠐⢦⣰⣼⣡⡄⡀⠀⠀⠀⠀⠀⠀⡟⣵⣿⣿⣿⣿⠟⠁⠀⣸⢸⠁⠸⡄⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠤⣽⢾⡿⣾⣟⢻⡅⠀⠀⠀⠀⠀⢤⣝⢿⣿⣿⠟⠁⠀⠀⠀⢯⠄⠀⢀⡇⠀⡇⠐⠀⠒⠒⠒⠒⠒⠒⠒⠂⡇
⢐⣻⣿⣿⡿⣿⣿⠲⠀⠀⠀⣀⣵⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⢫⡀⠀⡇⣿⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠈⠀⠛⡟⠻⠛⠌⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⢃⣀⣀⡙⡄⡇⣿⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣿⣿⡇⡇⡇⢻⠀⢿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣿⣿⡇⡇⡇⣾⠀⣾⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣼⣿⣿⡇⡇⡇⣿⣦⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⠿⠿⠿⢇⠇⢧⡙⠿⠿⠿⠿⠿⠿⠿⠟⣣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠠⠀⠭⠉⠉⠍⠩⠭⠭⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢹⣇⢰⡟⢠⡿⣷⠀⣼⠟⠛⠃⣾⠟⠛⠃⣿⠄⣿⣧⣸⡇⠀⣾⢿⡄⠛⣿⡟⠃⣿⢠⡾⠛⢻⣆⣿⣷⣄⣿
⠀⣻⣿⢀⣾⠛⢻⣧⠻⣷⣶⠆⠻⣶⣶⠆⣿⣂⣿⡈⢻⣇⣼⡟⠛⣿⡀⣿⡇⢀⣿⠈⢿⣦⣾⠃⣻⡇⢻⣿⡀
⣿⣿⠛⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠶⠿⠿⠿⠿⠿⠷⣶⣶⣤⣤⣴⣶⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠛⠉⠉⠉⠉⠉⠙⠛⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠟⠁⠀⣀⣤⣤⣴⣶⣶⣤⣤⣀⠀⠀⠉⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⢀⣤⡾⠟⠋⠁⠀⣀⣀⣈⠉⠛⢷⣦⡀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⠋⠀⢠⣾⠏⠀⠀⠀⠀⠀⠉⠙⠛⢿⣦⡀⠙⣿⣄⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣴⡶⠷⠶⠶⠶⠿⠿⠿⠛⠛⠛⠉⠉⠀⣼⡏⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠈⣿⡄⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣼⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⣧⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠙⠻⣷⣤⣄⣀⣀⣀⣀⣤⣶⠿⠋⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⡀⠀⠉⠉⠛⠛⠛⠋⠉⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠛⠿⢶⣶⡶⠾⠟⠁⠀⠈⠙⠻⣷⣦⣤⣤⣤⣤⣤⣴⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠈⢉⣿⡏⠁⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣦⣄⣀⠀⠀⠀⠀⠀⣀⣠⣤⣾⠟⠋⠀⠀⣰⡿⠛⢻⣧⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣏⠉⠙⠛⠛⠿⠿⠿⠛⠛⢋⣻⣷⣄⠀⠀⠀⣽⡇⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠛⠿⣶⣤⡀⠀⠀⣀⣤⣾⠿⠋⠉⠻⣧⡀⠀⣿⣷⣤⣼⣿⣀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣭⣿⣷⣦⣤⣤⣴⣿⠃⠀⠀⠀⠀⠀⠉⠻⢶⡞⠛⠉⠀⠀⠀⠀⠀⢙⣷⣾⣟⠀⠈⠉⠙⠛⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⢷⣦⡀⠀⠀⠀⢀⣶⣿⣿⣦⠀⠀⠀⠀⢀⣴⡿⠋⢹⡷⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⢠⣾⠋⢹⡏⠹⣷⡀⣠⣶⠟⠋⠀⠀⠸⣿⣀⠀⠀⠀⠀⠀⢸⣿⠙⢿⣦
⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠃⠀⢸⡇⠀⠙⠿⠟⠁⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⢸⣿⠀⠀⢻⣧
⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣸⣧⣤⣶⡶⠿⠿⢿⣆⠀⣿⡆
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⡍⠁⠀⠀⠀⠀⠈⣿⡆⢸⣷
⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⡇⠀⠀⠀⠀⠀⠀⠘⣿⡀⣿⡆
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣧
⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⣠⣿
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠙⠻⠷⣶⣶⡾⠟⠁
⠀⠀⢸⣿⡀⢀⣀⣀⣀⣀⣀⣀⣿⣇⣸⣿⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⢀⣼⣇⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣀⡀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣧⣤⣤⣤⣤
⣿⣿⣿⡛⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣡⠞⠻⠉⠉⠏⠁⢲⣬⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠰⣠⠄⠆⠀⠀⠉⣹⢍⣑⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠈⢀⠀⠀⢀⠞⠙⢦⠙⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠗⢤⣈⣉⣠⡔⠋⠲⢤⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠑⠶⠊⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣖⡆⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠷⡞⢁⠙⣤⣤⣤⣤⣤⣄⣠⡔⠉⠀⠀⠀⣠⣃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⠧⡴⠀⠣⢔⢯⡟⠉⠉⠉⢼⢟⢁⡉⠉⣦⠤⠚⠁⠀⠀⡠⣻⠃
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠀⢀⣔⣁⠀⠐⠀⠀⠂⠈⡺⠓⠗⠋⠁⠀⠀⠀⡠⣪⠞⠁
⠀⠀⠀⠀⠀⠀⢹⣍⣉⣐⣒⣒⣒⣒⡒⠛⠬⠭⠭⠭⠭⠭⠤⠄⣀⣀⣀⣀⣠⣾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠙⠛⠒⠒⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠋⠉⠉⠉⠉⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠉⠉⠉⠉⠉⠉⠲⢄⡀⠀⠀⠀⠀⠀⣇⢠⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⡟⠋⠹⠬⡑⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⡀⢠⠀⢦⡀⢰⡄⢰⣄⠀⠀⠀⢣⠀⠀⠀⠀⠇⠀⣀⠀⠀⠀⠀⢣⠤⢤⠀⠈⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠦⠼⠙⠚⠈⠉⠉⠉⠉⠉⢤⡉⠉⠀⢈⠀⠀⠀⠀⣠⡀⢿⡇⣀⠤⢶⠁⠀⢸⠃⠀⢰⠀⠀⢰⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠐⠉⠁⠀⠀⠀⠀⠀⣀⠀⠁⠀⢰⡼⢄⠀⠀⠸⡀⠈⠉⠉⣀⠄⠸⠀⢰⡇⠀⠀⠘⠀⠀⢸⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢾⣄⠀⠀⢾⣧⠀⠀⠀⠀⣿⡆⢀⡠⠚⠃⢨⠀⠀⠀⠑⡄⠉⠉⢀⡠⠀⢳⠆⠀⢸⠀⢸⠀⣠⢃⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣈⡀⡗⠒⠚⣓⣊⡉⢉⣉⡉⠍⠀⢡⡰⠒⠁⠀⠀⠀⠀⠱⣈⠉⠁⢀⠴⠃⠀⢿⠈⣠⢆⡰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠤⠤⠤⡤⠤⠔⠒⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⡲⠀⣠⠚⢳⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠶⢖⣶⣖⣲⡂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⠟⢁⣠⠃⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⡏⠉⠈⠙⡀⡠⢖⡒⣀⡶⠴⠤⡀⠀⠀⠀⠀⡜⣨⠃⢒⠁⠞⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠖⣍⣭⣵⡀⣀⠷⣛⣩⣽⠋⠑⠢⠀⠀⡏⠑⠒⢴⡱⢁⢔⡥⠐⢦⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠐⣤⠁⠀⠈⠻⡓⢻⠁⠙⠯⠿⠖⠒⠉⠰⣅⡀⠀⣾⠔⡡⠊⠀⠀⠈⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠢⡀⠀⠀⠱⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣧⠎⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⢤⣤⡖⠘⠦⣀⠀⠀⢀⣀⡤⠤⠖⠒⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⣯⢕⣂⠀⠀⠉⠁⢙⡀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⢦⠙⠒⠒⢎⠉⠧⠄⠒⠒⠂⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠑⡄
⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡀⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡯⠀⠇⠀⠀⠀⠀⠄⣿⠈⠄⠁⠀⠀⠀⠀⠀⠠⠰⡀⠀⠀⠀⠀⠀⢠⡐⠂⠀⠬⠆⠀⢀⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠁⠀⠀⢰⠗⠊⠀⠀⠉⠂⠀⠀⠀⠀⠀⠄⠀⠇⠀⠀⠀⠀⠀⠀⠉⠂⠤⠤⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⡆⢉⠃⠁⠀⢀⠼⢇⠀⠀⠀⠀⠠⠆⠒
⠀⠀⠀⠀⠸⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠤⠀⡠⠀⠈⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠅⠠⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠁⠀⠀⠀⠀⡀⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠃⠀⢺⠉
⠀⠀⠀⠀⣐⠈⠁⠤⣄⡀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠠⣄⡀⠀⠀⠘
⠀⠀⠀⠀⠈⢉⣀⣀⣀⠁⣀⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠑⠒⠆⠀⠀⠁⠀⠀⠀⠐⢂⠩⠛
⠀⠀⠀⠀⠀⠈⠉⠀⠈⠙⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⡸⠋⠉⠉⠻⠋⠉⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⡞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠳⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣋⣤⣤⣤⣤⣄⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣿⣿⣿⣶⣦⡝⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢁⣾⣿⣯⣍⡩⠭⠉⣩⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⠉⢹⣿⣿⣿⣿⣿⣿⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢸⣿⣿⡟⡶⠀⢀⡀⢸⡇⢀⣄⢠⣞⠛⣧⡧⡤⠀⢠⣗⡾⠀⠈⠿⡛⣿⣿⣿⣿⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠈⣿⠰⡇⡇⠀⠀⠀⣘⡇⢸⣿⢏⠈⠲⣿⠐⢧⣤⣿⣿⣧⠀⠀⠀⢠⠿⠿⢿⡇⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⡾⠁⣆⡇⠀⠈⠀⣿⡇⣼⣿⠀⠀⠀⢛⠃⠈⠁⠀⠀⢸⠀⠀⣠⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⣀⣤⣾⣷⣄⠛⠃⢲⠴⠶⠸⣧⣿⠟⠓⡄⠀⢸⠀⠀⠀⠀⠀⠀⠉⢉⡕⠒⠒⠈⠉⠁⠘⢄⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠉⣤⡊⠀⠀⢀⣿⣗⣷⢶⣷⠔⠉⢃⠀⠀⠀⠀⠀⡠⠞⠛⠉⠀⠀⠀⠀⠀⠀⠑⠄⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢶⣿⣿⣧⡀⢀⣭⣧⡉⠀⣽⠀⠀⠈⢧⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣧⢠⡿⡟⠒⢀⠨⡇⠀⠀⠈⢧⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⠀⡟⠛⠛⠒⠒⢻⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡅⠀⠘⢿⣿⣿⣦⣟⣿⠓⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠻⣿⣿⣿⣿⣈⣿⣿⡏⠈⢣⠀⠀⠀⢀⣠⡴⠛⠁⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡙⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣶⣿⣿⣿⡇⠀⢀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠋⣹⣿⣿⣿⣿⣿⣿⡟⠻⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣠⡖⠋⠡⠌⠻⠿⠉⠛⠙⠛⣷⣦⣨⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡾⠟⠁⠀⢠⡄⢠⠀⣴⡀⠀⠀⠘⠋⢻⣯⠳⣌⡙⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠈⠱⣏⠀⠈⠀⠀⠀⠀⣠⣾⣿⣧⠈⠻⣗⠲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣄⠀⠀⠀⣀⣀⡀⠀⠀⠀⡼⢿⣿⣿⣿⠷⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠷⢄⣀⠈⠁⢀⣀⡴⠚⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⢻⡉⠉⢉⡽⠁⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡨⠧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠦⢷⡎⠀⣄⠘⣶⡤⢤⢤⣤⡤⠤⠤⢤⣤⡴⠯⣀⠀⠀⠀⠀⢀⣟⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⡵⢒⣦⠈⢧⡀⠾⡏⣷⠋⠉⠉⠉⠉⣹⣿⢏⠉⠉⠉⠛⣿⣀⣠⠖⠋⠀⠉⠉⠉⣩⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠯⡩⠟⠁⠀⠀⣨⠟⠙⠛⠀⠀⠀⠤⠄⢹⣃⡏⣞⡔⣀⠤⠟⠉⠀⠀⠀⠀⠀⣠⠞⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠠⣞⣁⣒⣒⣀⡈⠉⠉⠈⠁⠀⣩⠟⠙⠛⠁⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⡛⠛⠓⠒⠒⠒⠒⠲⠶⠤⠤⠤⢬⢭⣭⣭⣍⣉⣉⣙⣁⣀⣀⡀⠀⠀⠀⠀⠀⢀⡼⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠓⠒⠒⠲⠶⠶⠶⠦⠤⠤⠤⡤⣤⣭⣭⣉⣉⣉⣉⣉⣹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣎⡲⣴⢟⢦⣂⠄⠀⠀⣀⢤⣶⠃⣶⠀⡀⠀⠀⠀⢢⣴⣶⣾⣤⠂⣀
⠀⠀⠀⠀⢰⠷⠊⠃⠀⠀⠀⠀⠀⠀⠀⣀⣱⣾⣷⠐⢀⡫⣷⣃⠀⢰⣿⣿⣿⣍⣙⣿⣅⠀⢠⢨⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠠⠼⠀⠀⠀⠀⠀⠀⢀⠀⠀⠖⣯⣿⣿⣷⣶⣶⣻⡀⠀⠸⣿⣿⣾⣝⣶⣿⡟⠀⠠⢀⠹⣿⡅⠁⠀⠈⠀⠁
⠀⠀⠀⠀⠀⡌⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠊⡝⠿⡿⠷⡛⠃⠀⠀⠀⠘⠻⠿⠿⠿⠋⠀⠀⠀⠈⢠⠋⠀⠐⡌⠂
⠀⠀⠀⠀⢠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢀⡩⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃
⠀⠀⠀⠀⢸⣾⣷⡆⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠁⠠⠂⣃⢁⡀⠁⢁
⠀⠀⠀⣤⢽⣿⣿⣯⣄⠀⠀⢐⣟⠂⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⢸⠛⣿⣿⣿⡇⠀⠠⠀⢠⡚⠡⢟⠈⠃⠀⡆
⠀⠀⠀⠀⠸⢹⣟⠇⠀⠀⠀⢀⣿⠇⠀⠀⠀⣾⡥⢿⣿⣭⡄⠀⠀⠀⠛⡶⠻⡿⠟⠃⠀⠀⠀⠀⢂⡀⠢⠖⢠⠙⠁
⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠆⣿⡄⠀⠀⠀⠸⠗⠾⠿⠟⠁⠀⠀⠀⡠⠋⠀⠹⠀⠀⠀⠀⠀⠂⠀⠄⠋⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⢸⡟⠃⠀⠀⠀⡊⣿⡆⠀⠀⠀⣂⣀⣀⣀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⢸⣧⠄⠀⠀⠀⡇⣿⡅⠀⠀⠸⡟⠛⠛⠿⠿⣟⠀⠀⢠⣶⣦⣖⣾⣷⣶⠀⠀⠀⢀⣀⣦⣴⣢⣶⠀⡀
⠀⠀⠀⠀⠀⢸⡋⠆⠀⠀⠀⠏⣯⡄⠀⠀⠰⠆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠉⠀⠈⠀⠀⠠⢤⣿⣿⡿⢏⢛⣎⡁
⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⣀⣦⣿⣇⣀⠀⠀⡄⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢒⣯⣿⢕⡡⢁⣛⡒⠂
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠙⣿⠀⠉⠀⠀⣇⣀⣀⣀⣼⣿⠀⠀⠀⣄⣀⠀⣀⣁⡀⠀⠀⠈⢉⢿⣿⣮⣥⡣⣟⠉
⠀⠀⠀⠀⠠⠟⠓⠀⠀⠀⠀⠾⠛⠦⠀⠀⠀⠋⠛⠻⠟⠛⠋⠀⠀⠀⠛⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠃⠘⠀⠁

⣶⣾⠿⣶⠶⢶⣶⡶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠋⠙⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⣿⣿⠋⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⣽⠀⡏⢸⣆⠀⠀⡠⠒⣋⠭⠽⠿⠯⣍⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⠁⡇⢀⡇⢸⡏⡇⢰⠃⠸⡕⠂⠨⠍⠀⢺⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣼⢰⡇⢸⠁⢸⠇⡇⢸⡄⠀⠈⠉⠀⠀⠈⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⡇⠸⠇⢸⠀⢸⠀⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⣤⡼⠆⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⠀⠐⠃⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⢤⡤⠤⠴⠚⠉⢰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⡟⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⣀⢀⡄⠀⠀⡤⠀⡀⢠⣢⠀⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣰⣆⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⡿⠿⢿⣿⡿⣿⣦⡤⠀⠀⡴⣄⠀⢀⡽⣮⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣧⣀⣼⣿⣷⣿⣿⣇⠀⢀⡝⢮⡳⣏⠔⠁⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⣿⡏⠉⢹⠟⢁⠔⢁⠐⠀⠙⢮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⠖⢁⠴⡉⠢⠀⠀⢀⡴⢋⣉⣁⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⢁⠔⠁⠄⠈⠀⢀⡴⠋⢀⣾⡿⣿⣁⣹⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠣⡀⠁⠀⢀⡴⢋⠀⠀⠉⠹⣷⣾⣿⡿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢦⡀⢀⡴⠋⠀⢻⣷⡒⠀⠈⠁⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠓⠊⣉⠉⣴⣷⣤⣾⣿⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣸⣿⣇⣀⣸⣿⣏⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡻⠿⣿⣿⠿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠠⣄⣼⣤⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⡾⠛⠻⡆⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠚⠓⠂
⠀⠀⢺⢿⡿⠋⠀⠀⢿⣿⣿⠗⠀⠀⠀⠁⠀⠀⠁⠀⠀⡰⠋⠀⣭⠀⠀⢠⣶⣷⡆⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠠⠤⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄
⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠄⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⡀⠑⠄⠀⠀⠀⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠒⠐⢢⢀⡈⡄⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢠⢆⡀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢱⣁⣀⡀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⠔⠒⠒⢄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⠢⠎⠀⠀⡃⠀⠀⠀⢿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⠀⠀⠀⡠⠰⠁⠀⣀⣠⠂⡮⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠮⢀⠀⢀⡼⠒⠀⢲⠁⠀⠀⠀⢸⣿⣿⣿⡇
⠀⣿⣿⣿⣿⠀⠀⠠⠀⠀⢤⠊⠀⠀⠐⢀⡠⠚⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠉⠀⠀⡀⠶⠁⡧⠤⢀⡜⠀⠀⠀⠀⠸⣿⣿⣿⣗
⠀⣿⣿⣿⡏⠀⢀⠒⠢⠄⣨⡔⠁⠀⢡⡈⠓⠄⠀⠵⠦⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄⠐⠚⠅⠀⠀⡰⠄⠚⢀⡀⠰⣀⡀⠀⡁⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⣿⣿⣿⠃⠀⢠⡀⠀⠀⢀⠣⡀⠐⠁⠀⠉⠐⠄⠒⠐⢲⠲⠉⠉⠉⠉⡏⠉⠀⠀⢡⣀⡀⠤⠒⠒⡁⠀⠀⠘⠁⡀⢸⠀⠀⠙⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⣿⣿⡟⠀⠀⠀⠀⠁⠉⠉⠠⠉⢃⠀⠀⠀⠀⠀⠀⠀⠀⣌⡀⠀⣀⣠⣶⡂⠀⡉⠀⠰⠁⠀⠀⠀⠇⠀⠀⠘⠀⢁⡌⠁⠒⠺⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⣿⣿⠃⠀⠀⠣⠐⠐⠢⠼⢣⠄⠒⠛⠓⡂⠙⠗⡂⠒⠈⠁⢈⣁⡴⠞⠁⠀⠀⠰⡌⠀⠀⠀⠀⠾⠒⠢⡀⠠⠄⠺⠘⡤⠀⢀⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⣿⡟⠀⠀⠀⢢⠐⠒⠠⠜⠂⠀⠀⠀⠀⠇⠀⢀⡠⠙⠢⡜⠋⠁⠀⡠⠚⠀⠀⢀⡓⢀⣀⣀⣀⠀⠀⠀⢰⠀⠀⢀⡇⠃⡀⠀⢰⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⣿⠇⠀⠀⠀⡆⠀⠀⠀⠈⡰⠒⠒⠒⢲⠀⠊⠁⠀⠀⡰⠀⠀⠙⠒⠤⡀⠠⠐⠈⠉⠁⠴⠉⠉⠀⠀⠀⡈⠈⠃⠀⠘⠀⡅⠀⠉⠂⠀⠀⠀⠀⠀⠈⣿⣿
⠀⣿⠀⠀⠀⠀⠰⠒⠀⠒⡃⠀⠀⠀⣀⣀⠀⢡⠀⠄⠀⠥⢦⡀⡀⣀⠀⠈⢃⠀⠀⡀⠀⠐⢄⠀⠀⠀⠚⢄⠀⡆⠀⠙⡄⠸⠂⠀⠘⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⡅⠖⠉⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⡏⠀⠀⠘⣀⠈⠀⠀⢸⡄⠀⠀⠈⠱⠄⠀⢸⠀⡄⠀⢀⠆⢠⣀⣀⢀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠢⠤⠄⠤⢹⠀⠀⡀⠠⠤⠀⣀⡠⡞⠀⠀⠔⠁⢀⡀⠀⠁⠚⠂⠀⠉⡦⠑⠀⠀⠠⠤⠤⠊⣠⠋⠀⠸⡀⠁⠀⠀⡁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣇⣀⣀⣠⠈⡖⠁⠀⠀⠀⠀⠈⠑⡥⢀⣀⣀⠴⠋⠤⠊⠠⢀⠀⠀⠀⠱⠚⢄⠀⠐⠀⣀⠁⠀⠈⠖⡑⠉⠑⢢⠔⠁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⡆⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠁⠀⠐⠘⠀⢂⠀⠐⢤⣴⠚⠖⠁⠠⣀⠠⡞⠊⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠇⠀⡀⠄⠣⡈⠢⢀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⣠⠈⠉⣀⣸⣂⣀⠇⠠⠐⠡⠴⡀⠀⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠖⠁⠀⠀⠈⠖⠒⠳⠶⢒⡚⠣⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢄⢠⡐⠉⠎⠑⡎⠁⠙⠄⠀⢀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⢸⠀⠀⠀⢀⣜⠠⠷⡁⠤⠀⡀⠀⠘⡄⠀⠀⠀⠀⢀⣿⠄⣀⣠⣀⣠⣀⡀⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠒⠒⠀⠈⠀⠀⠀⠀⠙⠒⠁⠀⠀⠘⡄⠀⠀⠀⠈⠀⠤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⢰⣆⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢋⠈⠀⠁⠉⠀⠉⢻⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⢂⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠋

⠀⠀⠀⢀⡤⠐⠂⠐⠒⠒⣤⡆⠀⠐⠒⠠⠀⡤⠠⢄⠀⣀⠀⣀⢀⣀
⠀⠀⡰⡏⠀⢰⣦⣬⣭⠞⢣⢓⠦⠀⢰⡰⠓⡇⠀⡈⢸⡟⠀⣼⠼⠀⠀⠙⢆⣠⠐⠢⡀⠀⢀⠤⡀⠀⡀
⠀⠀⢇⠱⣄⣀⣀⣀⠀⠑⠄⢘⢸⠀⠈⠀⡇⡇⠀⡇⡜⡇⠀⡿⡇⠀⡟⡆⠸⣄⠳⣄⠈⠉⠀⡀⠜⠸⠀⠉⡗
⠀⠀⠀⢉⣒⣓⣒⣬⠇⢀⠴⠼⠼⠄⠤⢠⣇⡇⠀⣄⣣⠃⢰⣇⠃⢠⣧⠃⢠⠀⡱⠸⠀⢰⠉⠀⠆⠇⠀⡸
⠀⠀⢰⢸⡀⠀⡀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⡈⠁⢀⠞⢿⠀⠈⠀⣠⠎⢀⠃⡇⠀⡇⠀⣸⢸⠀⢠⠃
⠀⠀⠘⠢⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡤⠖⠾⣿⠛⠋⠸⣼⠲⠞⠋⠀⠀⡸⣼⠀⢸⠁⠀⣧⡼⣲⠂
⠀⠀⡤⠀⠀⠤⠤⠤⠤⠀⠀⠄⠠⠾⠏⠙⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠜⠁⠀⠨⣘⡤⠜
⠀⢨⠀⠀⠐⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⢽⠔⠂⠀⡀⠀⠒
⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠀⠀⠀⠄⠈⢀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠋⠉⠁⠈⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⡀⠀⠀⠀⠐⠀⠈⠁⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡤⣀⡀⢀⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠁⠐⠀⢀⣈⠭⠑⠈⠁⠀⠀⠉⠙⠳⠤⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀
⠀⠀⠀⡨⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⢔⣂⣠⠀⠀⠐⣄⢀⣀⣈⢐⣀⣤⣀⣀⡘⣰⡀
⠀⠀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⠀⡑⠂⠐⠂⠀⠐⠂⠍⠉⠉⠉
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⢰⡖⠢⠒⠀⠀⠀⢀⠀⢠⠁⠄
⠀⠀⠀⠈⢉⠓⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠈⠀⠀⢀⣠⠖⠁⠀⠀⠀⡈⠀⠀⠈⡭⠿⡶
⠀⠀⠀⠀⠀⠀⠀⣤⠤⢽⠶⠦⢄⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⡔⠛⠂⠤⠄⣀⠀⠐⠀⠑⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠵⠢⠤⢀⠀⠀⣿⠤⠖⠉⠀⠀⠀⠀⠒⠠⠄⣈⠁⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⢰⣻⣯⣧⣿⡀⠀⠀⠀⢠⣤⣤⣤⣀⣀⣀⣀⠀⠉⠀⢠⣴⣲⢿⡄⠀⠀⠀⠋⠀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣇⠀⠀⠀⢰⣲⣿⣿⣿⣿⣿⡿⠀⠀⠀⠘⣏⡺⣟⣷⠀⠀⣤⠀⠀⠀⠀⡼⢤⠀⠀⢀⣀⣤⢴⡤⣀⠀⠀⠠⠲⢤⡀⡐⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡄⠀⠀⠹⣟⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⡞⡿⠏⠀⠀⠀⠉⠀⠀⠴⢉⠈⠀⠸⣯⣿⢯⢯⣿⠽⠄⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠐⠀⠻⠿⠟⠛⠋⢁⣀⠀⠀⠀⠉⠉⠉⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⢻⣾⡾⠾⠚⠋⠁
⣿⠀⠀⡢⠀⢀⠐⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡴⣶⠤⢄⣀⠀⢹⣿⡿⣇⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡭⣿⣿⣟
⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠼⣿⣿⣿⠃⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⢰⠧⠀⣀⠀⣻⠃⠸⣿⣿⠏⠀⢶⣿⣿⣿⣿⡧⠀⢀⣤⡞⢷⣶⠀⠀⠀⠈⠻⣿⣏⣿
⠉⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠉⢳⣶⣿⣻⡄⠀⠀⠀⠀⠀⠀⢀⣞⡀⠀⠂⢠⠇⠀⠀⠀⠀⠀⠀⠀⢿⠿⠛⠁⠀⠀⠻⣗⢟⡵⠟⠁⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠐⠄⠀⠠⠀⠀⠀⠀⠀⠈⣿⣓⣯⣷⠀⠀⠀⠀⠀⠀⠼⢀⣋⡿⠒⠞⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣄⡀⠀⠀⠸⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣴⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣳⣽⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠘⠿⣿⣿⡟⠀⠀⠀⠀⣔⢦⣀⠀⠀⠀⠀⠀⠉⠛⠛⠿⡟⠀⠀⡀
⢼⠀⠀⠀⠀⣠⣶⡀⢾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡄⢠⡶⡝⣝⣿⠄⠀⠀⢰⣷⣦⣭⣀⠀⠀⠀⠸⣿⣯⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠧
⢸⠀⠀⢴⣿⣿⣿⡿⠆⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡦⠙⠶⠛⠁⠀⠀⢠⣿⣿⣿⣿⣿⠏⠀⠀⠈⠳⣿⣿⣿⡟⠁⠀⠀⠀⠂⠀⠀⠀⢸⠀⠂
⣼⠀⠀⠈⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⢿⠿⠋⠀⢀⣀⣀⣄⠀⢀⣾⣿⣿⣿⣿⡟⢀⣴⣦⣀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⢀⡀⠀⠀⠀⠈⠀⢠⣶⣦⣄⠀⠀⠈⣿⣯⣿⠀⠘⠵⢯⣿⣿⠟⠀⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⡀
⡄⠀⠀⠀⠀⠀⠀⣠⢾⡻⡙⢶⣟⣳⢄⠀⠀⠀⠀⢿⣿⣿⣿⣧⠀⠀⠿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠝⠁⠀⠀⢀⣀⡀⠀⣟⣛⢻⠃⠀⢠⣤⣶⣶
⡇⠀⠀⠀⠀⣴⣾⡫⡱⣝⢎⢮⡻⣥⠶⠃⠀⠀⠀⠀⠈⠙⠿⠃⠀⠀⠀⠀⠀⢀⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣿⡟⡽⣽⣣⠰⠧⣹⣾⠀⠀⠀⣿⣿⣿
⠀⣰⣤⣀⠀⠈⠻⡕⢝⢮⣢⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣾⣧⠀⠀⠀⣀⡀⠀⠀⠀⠻⢿⣾⣵⡧⠿⠀⠀⠀⠀⠀⠀⠀⣉⣡⣄
⠸⣿⣿⣿⣷⡆⠀⠘⢿⡷⠋⠀⢰⣿⣟⣿⠀⠀⠁⠀⠀⡔⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⢴⡮⣫⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿
⠀⠈⠛⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠿⠛⠀⠻⣤⡶⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉
⣀⡀⠀⠀⠀⠀⠀⠀⢀⣶⡤⠄⣀⠈⠉⠁⠀⠀⣠⣤⣤⠴⣦⠀⠀⣰⣿⣶⣦⡄⠉⠀⣤⣤⣄⡀⠙⠈⠁⠀⠀⠀⣀⣤⣤⡀⠀⠀⢀
⣻⣽⣢⠀⠀⠀⠀⠀⣼⠁⠀⠨⢹⠃⠀⠀⢀⣾⣽⣿⣾⣷⣿⠀⠀⠻⠿⣿⣿⡟⠀⣼⣿⣿⡿⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⡿⢆⠀⠀⢀⠀⠀⠀⠀⠀⣠⣾
⣿⠟⠁⠀⠀⠀⠀⢰⣧⣀⠠⢈⡏⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠉⠀⠀⠙⠻⠿⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿
⠁⣂⠀⠀⠀⠀⠀⠃⠨⢽⠏⠙⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⣿⡏⢹⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠿⠿⠿⠇⠀⣠⡔⠒⡉⠉⠀⢧⡴⢆⠀⠀⢠⣶⣶⣶⣿⡇⠀⢈⡄⠀⠀⢤⣿⣷⣦⣤⣀
⠀⠛⠓⠚⠒⠓⠀⢰⣶⣶⣶⣦⠀⠀⢠⣤⣀⡀⠀⣠⣶⣄⠀⠀⠀⠘⡑⠀⠐⠀⠀⠘⡗⢈⡀⠀⠀⣿⣿⣿⣿⣿⠀⠈⠀⠀⢀⣞⣞⣞⣞⣞⣟⡿⣄
⠀⠀⠀⢀⣀⠀⠀⢸⣿⣿⣿⣿⠇⠘⣿⡳⡬⢽⠛⢿⣿⣻⣗⠀⠀⠀⢱⣤⣀⡠⠶⠒⠉⠁⠀⠀⠀⢻⣿⣿⣿⣿⠂⠀⠀⠀⠺⢿⣾⣟⣞⣟⣾⠿⡶
⣻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠀⠙⠯⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣠⣤⡀⠀⠸⠯⠷⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠾⣯⡹⠃
⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠲⢤⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠀⠀⠀⠀⢰⡖⣶⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣮⡾⡗⠀⠀⠀⠀⠀⠙⠻⣿⣟⠀⠀⠀⠀⠀⠸⠧⠽⠼⠟⠀⠀⠀⠀⠀⠀⠀⢠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣄⠀⣠⡶⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠹⣿⡋⠀⢀⣤⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢋⣤⡶⢷⣄⠈⢻⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⣁⣴⠟⠙⠷⠀⣹⡇⠀⣹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⣠⡾⠋⠻⠆⢀⣠⡾⠋⣠⡾⠋⠁⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢉⣴⠟⠙⠷⠀⢀⣴⠟⢉⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣡⡾⠛⠻⠆⠀⣠⡾⠋⣡⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⣾⠋⠀⣾⠛⠷⠀⢀⣴⠟⢋⣴⠾⠋
⠀⠀⠀⠀⠀⣀⣄⡀⠀⢰⣿⣷⠀⢴⣷⠄⢸⣿⣿⡿⠀⠀⠀⠀⢹⣧⡀⠙⢷⣤⡶⠟⣡⡶⠟⠁
⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠙⣿⠀⠘⡟⠀⢸⣿⠏⢰⣾⡆⢀⣶⠛⣉⣿⣦⡀⢉⣤⠾⠋
⠀⠀⠀⠀⢀⡀⠉⠙⢷⣄⣠⣼⣷⣶⣿⣶⣾⣧⣠⠟⠉⢀⡵⠛⠞⠋⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠿⢿⣦⣀⣼⣿⡿⠿⣿⣿⣏⣹⣿⣿⡿⢣⠔⠉⢀⣤⣤
⠀⢠⣤⣤⣄⠀⠀⣿⣿⣿⡀⠀⢸⣿⡟⠉⠈⠉⢠⣴⣿⡶⠿⠿⠟⠁
⠀⠘⠿⠿⠟⠻⢾⣿⣿⣿⣿⣾⣿⣿⣧⡀⠀⣀⣿⡟⢻⣿⠤⠴⢾⣿⠇
⢀⣶⣿⣶⣶⣦⣾⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣶⣶⣶⣆
⠈⠻⠿⢟⣋⣁⣨⣿⣿⣿⣙⣿⣿⡟⠉⢹⣿⣧⣠⣿⣿⣇⡀⠉⠛⢿⡿⠏
⠀⠀⠐⢿⠿⠋⢀⣼⠿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡿⠋⠈⠻⣿⣿⡆
⠀⠀⠀⢠⣾⣿⡿⠁⠀⢈⡿⠻⣿⣿⣿⡿⠿⡟⠉⠻⣦⣀⠀⠈⠉
⠀⠀⠀⠘⠛⠛⠁⢀⣴⣿⡇⣴⣇⢸⣿⡀⠀⢳⡀⠀⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠻⢛⣾⣿⣧⠀⢸⣿⡆⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠟

⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⡿⠋
⣿⣿⣿⣿⠟⠃⣐⣤⡿⠋
⣿⡟⠿⠩⠆⣥⡿⠋
⣿⣿⣄⣰⡾⠋
⠉⢾⠝⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣬⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣴⠶⠶⠶⣶⠀⠰⠶⠶⠶⠶⠶⠶⠈⠉⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⠰⠶⠶⠶⠶⠶⠶⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⣿⡀⠰⠶⠶⠶⠶⠶⠶⠀⢰⡟⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠻⠶⠶⠶⠿⠂⠰⠶⠶⠶⠶⠶⠶⠀⣼⣧⣤⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⢿⣶⣶⣶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⢸⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⢸⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠲⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⢸⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠛⢛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠠⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠠⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡴⠡⠖⠒⠒⠒⠼⠭⠁⠉⠐⠀⢀⠄⠂⠀⠰⠺⠝⠶⠆⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢄⠀⡐⠁⠐⠀⡠⠂⢀⠈⠀⠀⠀⠀⠀⢠⠊⠀⢅⠀⢠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢈⠂⠀⠀⠔⠀⠀⠀⠐⠲⠢⠄⠔⠖⠒⠆⠀⠢⡀⠀⡠⠒⠒⠂⠂⠂⠂⠂⢀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣱⣗⠦⣙⢤⡀⠀⡸⠄⠀⣀⠀⠀⠀⠀⠠⠀⡀⠆⢠⠁⠀⣀⠀⠀⠀⠀⢄⣀⡠⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠪⠀⠀⠀⠑⠚⠂⡆⠐⠂⠀⠠⢨⡍⠁⠀⠀⡇⠀⢸⠀⠀⡀⠠⠄⠩⢅⡁⠀⠐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡶⠡⠁⠀⠀⠀⠀⠀⠀⡇⠐⠀⢀⡀⠠⠄⠀⠈⠀⡇⠀⢈⠀⠀⠀⠠⡀⠢⠔⠰⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢐⠞⠔⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠛⠺⠀⠀⠀⠀⠀⠀⡇⢰⢸⠀⠀⠃⠀⡀⡅⠵⢈⠀⠀⠲
⠀⠀⠀⠀⣀⠀⠀⢀⠮⠰⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠣⢒⠭⠀⠀⠀⠀⠀⠀⡇⢸⢸⠀⠀⠀⠀⠁⠀⢀⡀⢀⡀⠬
⠀⠀⠀⠀⠐⠁⠢⢮⡒⠅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠒⠀⠠⠤⠠⠄⠀⠀⠃⢺⢸⠀⠀⢄⡀⠀⠁⠀⠐⠒⣒⠤
⠀⠀⠀⠀⠀⠀⠈⠒⢍⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠠⡠⠂⠈⢸⠀⠀⠀⠀⠀⠈⠉⠉⠩⡉⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠐⠀⠀⠂⠐⠂⠀⠀⠒⠈⠈⠗⠤⣀⡐⠀⠒⠀⣊⢁⡑⠋⠘⢤⣀⠀⠀⢀⢄⠀⡨⠒⢠⠞⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⠁⠂⠐⠀⠀⠁⠊⠝⠒⠒⠴⠢⠖⠀⠀⠐⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⡖⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣵⣶⣶⣶⣶⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣮⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⣿⣿⣷⡂⠈⠀⢰⡇⠀⠀⢠⣬⣇⣀⠀⠀⢠⢠⠀⠸⣿⢿⣿⣿⣿⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣿⢹⠘⠁⠀⠃⠸⡇⢸⣷⠋⢢⣿⢩⣁⣰⣷⣾⠀⠀⠉⣸⣿⣿⡟⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣼⠂⢀⠀⠀⠂⣼⡇⣼⡇⠁⠀⢸⠂⠙⠉⠉⠉⠀⠀⡀⠁⠀⠈⣧⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢸⡇⣀⣠⣴⣷⣄⠛⠑⡶⠶⠼⣧⣿⠛⢆⠀⢸⠀⠀⠀⠀⠀⠉⢨⠗⠒⠀⠉⠉⠑⢄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⢹⣼⡀⠀⠀⣿⣿⡶⢾⡞⠉⢣⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⠀⠱⡀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⢻⣿⣿⣆⠀⣾⣷⠒⠰⡇⠀⠀⢣⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣸⣿⣿⣿⠐⣿⣧⣄⣀⣡⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡋⠉⢻⣿⣿⣆⡷⣦⠄⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⡄⠀⠹⣿⣿⣿⣿⢰⣶⣶⠉⢆⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣘⣿⣿⣿⣿⣿⣿⣶⣾⣷⣶⣿⣿⡇⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠁

⠂⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⡀
⠀⠌⠁⠄⠀⠀⠀⠀⠲⠄⢀⣈⣁⡀⠠⡶⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⢉⣷⡄⠇⠴
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣹⡿⢿⣯⠓⡄⠀⠀⠀⠀⣰⣿⡆⣇⣸⣷⡄⠀⠀⠀⢸⡀⠀⠀⢸⣿⡆⠚⡛
⠄⠀⠀⠀⠀⠀⠀⠀⠈⠓⠼⢿⡿⠧⠚⠁⠀⠀⠀⠀⣿⣿⡷⠛⣿⣿⡇⠀⠀⠀⠀⠳⠦⣤⠴⠟⠀⠇⠘



⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⣤⣄⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠰⣄⣠⣾⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⣇⣀⠀⠀⠀⢰⣶⡦⠀⣴⣶
⠀⠀⠈⠉⣿⣿⠀⠀⠀⠀⠀⣀⣉⣛⣋⣁⠀⠀⠀⠀⢀⣿⣋⡿⢿⢿⡿⢟⣹⣯⠀⠀⠀⢀⣿⠀⠀⠛⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣼⣿⣿⣯⣿⣿⣿⡄⠀⠀⠘⣯⣨⣷⡾⠘⣷⣾⣤⡿⠀⠀⠀⢸⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠟⠟⠀⠀⠀⠀⠻⠿⠟⠟⠻⠷⠿⠃⠀⠀⠀⠀⠿⠋⠁⠀⠈⠛⠟⠀⠀⠀⠀⠘⠿⠀⠀⠀⠿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀
⠀⠀⣠⣤⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠀⠀⣠⣴⣦⣾
⣠⡾⠃⠈⠀⡀⠈⠀⠙⢦⡀⠀⠀⢀⣴⣶⣶⣤⢀⣴⣖⣲⣦⠀⠀⠀⢀⣼⣿⠿⠟⠁⠀⠀⠀⣿⣿⣿⣿
⣟⠉⠉⠉⣿⠉⡟⠉⠉⠉⡏⠀⠀⢸⣿⣿⣿⣿⣤⣿⣿⣿⣿⠀⠀⠀⠚⠛⠁⠀⠀⠀⠀⠀⢾⣿⣿⠿⠃
⠙⠒⠒⠚⠃⠀⠙⠒⠒⠚⠁⠀⠀⠀⠙⠛⠛⠋⠉⠙⠛⠛⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠘⠋⠙⠇⣲⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⣀⠈⠀⢃⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠂⠁⠲⣄⠀⠀⣀⠘⢀⠄⠀⠀⠀⠀⠀⠒⠖⠀⣁⠀⠀⠀⠲⢄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡰⠁⠢⢄⡀⠀⠀⢈⠙⠂⡴⠀⠀⠀⠀⠀⠀⠸⠀⢀⢀⡀⠀⠀⢂⠀⠀⠀⠀⠀⢀⠤⢀⠌
⠀⠀⢀⠎⠀⠒⠲⠄⣙⠀⢀⠛⠀⠚⠈⠁⠀⠀⠀⠀⠀⠸⠀⠀⠈⠁⠀⠀⠈⠃⠀⠠⠀⠀⠀⠀⣨
⠀⠀⠀⠈⠁⠐⠂⠄⣀⠐⠛⠂⡄⠀⠰⡁⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⣀⠀⠐⣀⡤⠾⠀⠀⢀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡨⠂⠁⠐⡀⠀⠀⠀⠀⠈⢢⠀⠀⠻⣿⣿⠃⡀⠃⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣠⠂⠡⡀⠀⠀⠀⠀⠀⠙⠢⠤⠀⢉⢡⡤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠐⠀⠤⠤⠀⠄⠖⠀⠾⣂⡓⠋⠀⠀⠀⠀⣀⠀⢀⡀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠆⠀⠀⠈⠀⠁⠀⢠⠀⠉⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡆⢰⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⢸⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠠⢀⠇⠀⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡐⠀⠀⠀⠒⠀⠉⠀⠐⠠⢀⠀⠀⠀⠀⡀⠀⠀⠂⠁⠀⠈⢢⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⢀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠁⠀⠀⠀⢀⡠⠔⠊⠁⠀⠑⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⢀⡀⠀⠀⡀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠰⠤⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⢤⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠈⠒⠤⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣠⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣞⣯⣷⣼⡄⠀⠀⠀⠤⢤⣤⣤⣤⣄⣀⠀⠉⠀⢠⢴⡺⣧⠀⠀⠀⠈⠀⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧⠀⠀⠀⣾⣺⣿⣿⣿⡿⣿⠀⠀⠀⠘⣷⢽⣪⡇⠀⠠⡀⠀⠀⠀⡘⠠⠀⢀⣀⡤⡴⣦⣤⠀⠀⠀⠓⢤⢀⠆
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡆⠀⠀⠙⢽⣿⣿⣿⣳⣿⠀⠀⠀⠀⠸⠻⠟⠀⠀⠀⠀⠈⠁⠐⠄⠀⠀⢹⡷⡱⡹⣹⣟⣃⠀⠀⠀⠁
⣀⠀⠀⠀⠂⠁⠘⠛⠛⠉⢁⣤⣤⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡴⣆⠀⠀⠀⠀⠀⠀⠿⠷⠛⠋⠉⠀⠀⠀⠀⢀⣤⣀⣀⣀
⣿⠀⠀⠂⠀⠀⠁⠀⠀⢀⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡞⠛⠒⣢⣤⠀⣯⣿⣿⠀⢀⣠⣴⣶⣦⠀⠀⠀⢀⣀⣀⠀⠀⠀⢰⣶⣿⣸⣽
⠿⠀⠀⠀⠀⠐⠀⠀⠀⠘⠛⠿⣟⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢀⡇⠀⠘⠛⠋⠀⠙⣿⣿⣿⠗⠃⢠⡔⠋⢝⣿⡃⠀⠀⠀⠙⠿⠾⠾
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⢿⡄⠀⠀⠀⠀⠀⢠⠗⣦⣥⣀⡾⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢻⢾⣿⠃⠀⠀⠀⠀⠀⠈⠐⠒⠇⠂⠀⠀⠀⣴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣤⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠂⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠸⠾⣿⣾⠿
⣠⠀⠀⠀⠀⣠⡄⢠⣿⣿⣿⠏⠀⠀⠀⠀⠀⢠⣾⣾⣧⡀⢀⣠⢞⣻⣗⡀⠀⠀⣤⣭⣛⠃⠀⠀⠀⢼⣵⣟⣷⣄⡀⠀⠀⠀⠀⠀⠈⠁⢀⢰⣂
⢹⠀⠀⣴⣾⣿⣿⡦⠙⠛⠉⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣄⠩⣤⠞⠁⠀⠀⣰⣿⣿⣿⣿⡗⠀⠀⠸⢯⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠸
⣬⠀⠀⠙⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠟⠁⠀⣀⣀⡀⠀⢠⣿⣟⣿⣿⡿⠁⣰⣦⡀⠀⠉⠻⠃⠀⠈⠀⠀⠠⠀⠀⠀⢠⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⢀⠀⠀⠀⠀⠂⠀⣴⣦⣄⡀⠀⠘⣯⣿⣷⠀⠛⢼⣽⣿⠿⠁⠺⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠁
⡄⠀⠀⠀⠀⠀⢀⡴⣛⢝⢶⣟⡳⠄⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⢻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⢀⣀⣀⠀⢼⢩⢷⠇⠀⢠⣤⣶⣶
⠇⠀⠀⠀⠠⣾⣟⢞⢮⢮⡳⣻⣷⠞⠀⠀⠀⠀⠀⠉⠻⠋⠀⠀⠀⠀⠀⣀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⢏⣏⣿⡀⠿⠤⠼⠀⠀⠨⣿⣿⣿
⢀⣴⣦⣄⠀⠈⢯⣳⡱⣵⡽⠋⠀⣀⡀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⣠⡄⠀⠀⠀⠻⠿⡮⠾⠟⠃⠀⠀⠀⠀⢀⣀⣤⣤⡄
⠈⣿⣿⣿⡿⠀⠀⠙⠟⠋⠀⠸⣯⣿⣇⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡔⣏⡣⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡧
⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⡿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠹⠿⠛⠋⠁⠈⢯⢿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣢⣤⠀⠀⠀⠀⠀⢀⡟⡫⠒⣦⡄⠀⠀⠀⣤⣾⣟⣏⣿⠀⠀⣼⣿⣿⣷⡄⠀⣴⣾⣷⡄⠀⠀⠀⠀⢀⣠⣴⣶⣦⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠
⣟⡷⠆⠀⠀⠀⠀⡸⡁⠌⠀⡼⠀⠀⠀⢸⣟⣿⣿⣿⣿⡄⠀⠈⠙⠻⠿⠁⢰⣿⣿⡟⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠄⠀⠐⠀⠀⠀⠀⣠⣾⣿
⠟⠁⠀⠀⠀⠀⢠⢛⠶⡶⠴⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠘⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿
⠀⣠⡤⠤⢴⠄⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⢀⣀⡤⠔⠊⢣⣀⣀⠀⠀⠀⣀⣠⣤⣤⡄⠀⠀⠀⠈⢀⣰⣦⣄⡀
⠀⠛⢧⢬⢼⠄⠀⣀⣤⣤⣄⠀⠀⢀⡀⠀⠉⠉⣩⡉⠀⠀⠀⠹⠃⠀⠀⠀⠀⢯⠌⠂⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠂⣸⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡃⢰⣾⠯⣙⣶⢾⣿⣿⣦⠀⠀⠀⢃⡄⢀⣀⠤⠜⠒⠁⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠠⣿⣿⣿⣿⡿⣽⣧⡇
⣶⣶⣶⣿⣷⠀⠀⠉⠉⠁⠀⠀⠀⠉⠓⠺⠇⠀⠙⠟⠟⠀⠀⠀⠈⠋⠁⠀⠀⠀⢀⣀⡀⠀⠸⡏⠿⠟⠁⠀⠀⠀⠀⠀⠉⠛⠻⢿⣟⠺⠁
⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠲⢄⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠂⠀⠀⠀⠰⣶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⠸⡇⠀⠀⠀⠀⠀⠛⢿⣿⡁⠀⠀⠀⠀⠰⠾⠦⠟⠀⠀⠀⠀⠀⠀⠀⢠⣄
⣿⢿⡿⠿⡿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢈⡉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠪⠘⢂⠯
⠀⠀⠀⠀⠀⠀⢀⢴⢃⣲⠤⡤⡤⡠⢀⠖⠦⡀
⠀⠀⠀⠀⠀⣴⣿⠟⣠⣄⣁⡘⠘⠁⠣⠂⠁⠈⠂
⠀⠀⠀⠀⢀⠙⠋⠀⢏⠂⠀⠈⠑⢤⡔⡀⢠⣀⣀⣄
⠀⠀⠀⠀⠨⡁⠀⠀⠀⠀⠀⠀⠀⠈⠢⠅⠎⠀⢠⠟
⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⢀⣈⡱⠀⠀⠀⠀⠀⠀⣀⣀⡠⠄⠐⠠⢤⣺⢋⠜⠠
⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠈⠢⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⢘⠆⠁⠀⢀⡄⠀⠀⠉⢷⣞⢥
⠀⠀⠀⠀⠀⠀⠀⡠⣆⠑⣄⣠⠂⠈⠁⠀⠀⠀⠀⠀⠰⠀⢀⡀⠀⠫⠄⡀⠀⠀⠀⠀⠀⠀⠀⠹⣇⡅
⠀⠀⠀⠀⠀⠀⡜⢁⢨⡧⠈⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠆⠀⠀⠀⠀⢀⠀⠀⢸⠁
⠀⠀⠀⠀⢀⠎⠀⠇⠀⠁⠩⣀⢸⠀⠀⠀⠀⠀⣀⠀⣀⣀⠀⡀⢀⣩⠆⠰⠃⠀⠀⠀⠀⠬⠇⢀⠆
⠀⠀⠀⢠⠃⡄⠀⠈⠲⣄⡀⠀⠙⠃⠀⠉⠁⠀⣠⣼⡿⠟⠁⡠⢊⠕⠣⢀⡀⠀⠀⠀⣀⡠⠔⠋
⠀⠀⠀⠈⢠⠁⡆⠀⢀⡼⡉⢲⠶⠶⠒⠃⠈⠁⠈⠁⠀⠀⠀⠉⠀⠀⠀⢀⡞⠐⣏⠁
⠀⠀⠀⠀⡄⠀⢀⡀⢏⠨⠀⠈⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡑⠀⠀⢽⢀
⠀⠀⠀⣨⠀⢠⠂⠚⠠⡍⠀⠃⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⡠⡀⠒⠂⠠⣼⠈⢦
⠀⠀⡈⠁⠀⠐⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣔⢦⣀⠔⡡⣿⡀⣀⢀⡀⠂⠀⠀⠡⠦⠀⢄
⠀⠀⠀⠤⣀⣀⡈⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠎⡌⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢣⠀⠀⠉⠉⠳⣖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠭⠝⠀⠢⣦⠀⠀⠀⠐⠀⠀⠀⠀⠐⢶⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢴

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

⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⡿⢏⠐⢌⣿⡿⠋
⣿⣿⣿⡿⢛⠐⢂⣷⡿⠋
⣿⣧⡘⠀⣉⣼⡿⠋
⣿⡿⣻⣶⡿⠋
⠀⢾⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠛⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣶⠶⡶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣈⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠶⠶⠶⠶⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣧⣰⡿⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠾⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⢀⣶⣦⣉⠻⢿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⣄⠀⠀⢀⣴⣿⣿⣿⣿⣷⣦⡙⠻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⣦⣈⠻⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣷⣤⡉⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⢀⣶⣦⡈⠛⢿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋⣰⣿⣿⣿⣿⣷⣤⡉⠻⢿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠙⠻⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡀⠀⠀⠙⠻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⢃⡐⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⢃⣼⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⣰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠋⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣾⣿⠟⠁⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⣠⣿⣋⣉⣉⣉⣉⣉⣉⣉⣉⣙⠏⣰⣿⣿⠋⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⢃⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⣸⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣠⣾⣿⠏⠀⠀⢀⣶⣶⣦⠀⠀⣀⣼⣿⣷⣿⣿⣶⣿⣧⣀⠀⠀⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣴⣿⡿⠁⠀⠀⠀⠘⢿⣿⣿⣦⣾⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣷⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⣰⣿⣿⣿⣿⣿⣿⣿⡟⢁⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢙⣛⣛⣻⣢⣾⣿⡿⢿⣿⣿⣿⣿⣧⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠛⠻⠿⠋⣰⣿⡿⠋⠀⠀⠀⠀⠀⣀⣄⡀⢀⣿⣿⣿⣻⣿⣿⣿⣟⣟⡀⢀⣹⣿⣿⣿⣿⣦⣤⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⠏⠀⠿⢷⣾⣿⠿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⢿⣿⣿⣿⣷⣤⣼⣿⣿⡟⠁⠈⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⡿⣷⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡿⠋⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠙⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡏⠀⠀⠀⠀⢀⣹⣿⠛⠋⠉⠹⣿⡄⠀⠀⠀⠀⠙⠿⠿⠋
⠀⠀⠀⠀⠀⢠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⣿⣿⣦
⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠈⢿⣿⡿⠁
⠀⠀⠀⠀⠈⠉