Marine biology Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠚⠢⠌⣁⣒⠒⣀⠀⠀⠀⠀⠀⡄⢌⠳⢤⣀⠀⠀⢀⣠⠞⢡⠎⡼⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⣾⠛⠁⠐⠄⠀⠐⠠⡀
⠀⠀⣷⠀⣄⠀⠀⠤⠌⣅⠠⠀⠀⠀⠀⠀⠸⡌⠳⣄⠀⠭⠭⠙⠁⣠⠏⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣯⠀⠦⡤⠀⠐⢂⠀⠁⢄
⠀⠀⡏⢸⡇⠀⠀⠀⠀⠀⡀⠠⡀⠀⠀⠀⠀⢻⢄⠘⢷⣄⠀⠀⣴⠋⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⣄⠁⠀⠀⠀⠉⠀⠀⠀⠒⠀⢈⣧⠀⠀⢃
⠀⠀⢁⠸⠀⠰⠁⠀⣾⣯⣷⠄⠀⢀⠀⠀⠀⣼⠀⠀⠀⠙⣷⣾⣃⠀⠀⠀⠀⠉⠲⠤⣀⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠂⠀⠘⢄
⠀⠀⠈⡀⠀⡀⣴⣿⣿⣿⡏⠀⡀⠀⠀⢀⡴⠃⢀⣠⠶⠚⠉⣿⠉⠙⠓⠒⠒⠶⠦⠤⠤⠌⠁⠂⠀⠈⠢⣄⡐⠀⠀⠀⠀⢀⣠⣾⠿⠀⠀⠀⢸
⠀⠀⠀⠁⠀⠀⣿⣿⡟⠋⢀⠔⠀⡀⠔⠉⠔⠚⠉⣀⠀⠀⠀⢹⡀⠠⢊⣴⠬⠖⠒⠛⠉⠁⠀⠀⠀⠀⠀⢀⣩⠉⡽⠷⢾⡿⠋⠉⠀⠀⠀⠀⠸
⠀⠀⠀⠐⠀⢰⡿⠉⣀⠴⠋⠀⠀⠁⠔⠐⠒⠉⠉⠉⠲⡄⠄⠸⡇⢈⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠉⢀⠇⠀⠀⠱⡄⠀⠀⠀⠀⠠⠞
⠀⠀⠀⠄⢀⡼⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠘⢆⠀⠀⣀
⠀⠀⠀⣴⠏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠢⠘⠁
⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠂
⠀⠀⠿⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡄⢦⠑⢀⠄⠈⠀⠰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡰⠁⠊⠰⠐⠃⠀⠠⡀
⠀⠀⢐⠂⠀⣦⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⠠⠤⢤⡄⠁⠈⣁⡀⠌⢃⣄⣘⢀⡄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠄⠀⠀⠀⠀⡀
⠀⠀⡈⠉⠐⠀⠄⡀⠄⢀⣠⠀⠃⠀⠀⠀⢐⠀⢀⣢⠀⠈⠈⠁⡲⠈⠉⢨⡈⣁⠤⠄⠐⠀⠀⠀⠀⠀⠀⠘⣄⢀⠀⠀⠀⠀⠀⡀⠀⠈⢀
⠀⠀⠀⠀⠌⠁⠀⠃⠮⠀⡀⠑⢂⠀⠂⢠⡌⠀⠒⠊⠀⠀⠀⠀⢡⠀⠡⠐⠚⠉⠂⢲⠐⠆⠀⠀⢀⠀⠀⠀⠏⠢⡀⠀⠇⠀⢈⠀⠀⠀⠃
⠀⠀⢂⡀⡀⠀⠈⠀⠀⠈⠁⠀⠐⠂⠀⠀⠀⠀⠀⠒⢠⠘⢃⠀⠋⠀⠒⠠⡌⠀⠈⠀⠌⠀⠀⠐⠈⠀⡘⠀⠐⠀⠐⡇⠨⠀⡀⠀⠀⠀⣄
⠀⠠⠀⠀⠀⠀⠀⠐⠉⠰⠠⠀⠁⠀⠠⠀⠀⠀⠀⡀⠱⠲⡃⡎⠀⡇⠀⠐⠀⠀⠳⢀⠀⠀⠚⠛⠀⠀⠀⠂⠈⠀⠀⠽⠬⢐⠁⠀⠀⠐⠘⠀⡀
⠀⠀⠀⢀⣘⠀⠈⠀⠀⠠⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠡⠀⠀⠀⠁⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⡇⠀⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠒⠀⠎⠂⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⢛⠀⠀⠁⠀⢰⠁⠀⠀⠀⣠⠀⠠
⠀⠀⠀⠀⠀⠆⠠⠐⠀⠃⠀⠭⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠐⣈⠀⠀⠀⠐⡘⠁⡄⠄⠚⢡⠁
⠀⠀⠀⠀⠀⠐⡀⢀⠀⠛⠀⠀⠣⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢀⣾⣼⠂⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡛⡿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛

⠀⠀⠀⠀⠀⠀⢀⣀⠤⠰⠤⠔⠂⠒⠉⠉⢀⡦⠋⠁⠀⠀⢰⠆⠀⢿⠃⣤⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣾⡿⣿⠉⠁⠉⠀⠃
⠀⠀⠀⠀⡰⠊⣹⠁⠀⠀⠁⢀⠀⢰⣀⡟⠈⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄⢀⠀⠀⡄
⠀⠀⠀⢸⣃⠴⠏⠀⠀⠀⠀⣸⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠄⠀⠀⠀⠀⠘⠁⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⠀⠁⠀⠈⠀⠈⢹⡏⠁
⡀⠀⠀⣈⡁⣤⣀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠀⠀⠀⠉⠞⠁⠀⠀⠀⠀⠀⠀⢈⠄⠉⠁⠀⢴⡗⠘⢒⣴⠸⠃
⠄⠊⠁⠀⠀⠀⠈⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡆⠀⠀⢀⡴⠋⡰⠋⠀⠶⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡸⠇⠀⠀⠀⠀⠀⠱⢀⠾⠁⠀⠀⠀⠀⠉⠀⠀⣤⡿⠰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⢿⡏⣻⠀⣿
⠀⠀⠀⠀⡀⠀⠀⡸⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠂⠀⠄⠀⣰⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣶⣬⢻⠟⢼⠁⡿⢀⡟⠀⠰
⠀⠀⠀⠀⠁⠀⢠⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠒⠋⢹⣿⡄⣸⢀⡇⣸⠃
⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠁⠀⠀⠀⠀⢆⠀⢸⡀⠀⡄⠀⠀⠀⠑⢂⠠⠜⠀⢀⠄⠀⠀⠀⠀⢠⣶⣶⢿⠙⡇⢹⠀⢤⠀⠀⠀⠈⠙⠓⠿⠼⠃⠏
⠀⠀⡀⠀⢰⡇⠀⠀⣤⠂⠀⠀⠀⢐⡄⠘⠀⢠⡇⣸⢟⠀⠀⠀⠀⠀⢠⣀⣀⡼⠀⠀⠀⠀⢆⣤⣽⣻⢸⡄⠇⢼
⠀⠀⢃⠀⢀⠀⡀⠀⢣⠀⠀⠀⠀⢀⣙⠣⢡⣦⣹⢏⣿⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠩⣿⣦⣇⡤⠞⠀⠀⠀⠀⠀⠀⠰⠆⠠⠆⠀⠒
⠀⠀⠘⡄⠸⠀⢘⠆⢠⠀⠀⠀⠀⡀⠤⢄⣻⢿⣿⣟⠗⣲⢾⠠⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⡀⣧⡾⠀⡘⠀⠀⠀⠀⠀⢀⡔⣡⢎⣿⡟⠓⢬⡳⠀⠀⠀⠀⠀⠀⣤⡿⣿⡆⠀⢀⣠⣤⣴⢶⣶⣶⠆⠀⠀⠀⠀⡆⠚⢹⡈⠀⠀⣆⠀⠄⢰⣒⠢⠄
⠀⠀⠀⠀⠨⣸⣷⡿⠁⠀⠀⠀⠀⠀⠛⡔⠫⠂⠸⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠙⣡⠿⡗⢰⠿⣋⣁⡵⠛⠋⠁⠀⠀⠀⠀⠀⠙⠆⡈⢻⠃⠀⠉⠁⠀⠘
⢰⡇⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠪⡘⠈⠉⡉⠀⠣⡞⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⠀⠂⠀⠀⢐
⠘⣷⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢟⣴⠞⡊⠀⠉⠀⠀⠱⠈⠈⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡍⠀⢠⠀⠀⢠⡄
⠀⠻⣦⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡀⠀⠀⠀⠀⠀⠀⣟⣼⠇⢘⣹⠈⠁⠂⣀⠐⢆⠀⢻⣥⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⡀⠀⢠⡾⠃
⡄⠀⣿⠃⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⡀⠀⠀⠀⢸⣷⡟⠀⠈⢄⠤⢐⠉⡇⠀⠀⡅⠱⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠀⢸
⡇⢈⣿⡄⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢄⠈⠋⠀⠀⠀⠀⠳⠍⢲⠌⢑⡀⠙⡒⢾⣄⠀⠸⠀⠀⠀⠀⠀⠚⠻⠄⠀⠁
⡇⢼⠏⣰⡟⠀⠀⠀⠀⢠⡌⢀⠠⠤⢄⠀⠀⠀⠀⠀⠀⢸⠀⠈⠂⠀⠀⠀⠀⠀⠀⠠⣿⣲⠭⠟⣯⠹⠛⠛⠂⢰⡀⠀⠀⠀⠀⠀⠐⢀
⠹⣾⠾⠏⠁⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⡄⣆⠀⠀⠀⠘⢦⡀
⠀⠉⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣏⠀⢨⡇⡇⠀⠀⠀⠀⠀⠳⡀
⣄⡴⠖⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⢹⣎⠀⠹⡇⠀⠀⠀⠀⠀⠀⢱⠀⠀⡄
⠋⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⠈⢀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢀⠁⠀⠀⠀⠀⠀⠀⠈⣧⠃⠣⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠓⠦⠄⣀⠀⠀⠀⠀⢢⡎⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⡼⣰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣆⠀⣴⠄⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡰⢀⠇⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⣿⢱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⠟⠢⠦⣀⣀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠓⠀⠐⠀⠀⠃⠄⠒⠃⠀⠚⠃⠁⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⡤⠆⠀⠀⠀⠀⠀⠀⠀⢀⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡈⣀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⢀⡄⠐⠠⡄⠚⡆⡞⠀⠀⠀⠀⠀⠀⠀⠠⠀⢄⡀
⠀⠀⠀⠀⠲⠶⠶⣒⣋⣩⠤⣽⠲⠶⢶⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠈⠄⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠡⡀⠳⠲⣟⠂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⡈⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠽⠆⠀⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣸⠜⡤⠀⢏⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢠⠊⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠸⠛⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⡌⡀
⠀⠀⠀⠀⢀⣶⢁⠄⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠸⢇
⠀⠀⠀⠀⢈⣏⠁⠀⠀⠀⠀⠀⠀⠙⠾⠆⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈
⠀⠀⠀⢀⠨⡝⠀⠀⠀⠀⠀⠀⠀⠀⠾⡇⡀⠀⠀⠤⡆⠈⠀⠀⠀⠉⠒⠀⠀⠁⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢠⠈⣦⠑⣷⣶⠟⠺⡮⡿⢯⣐⢾⢃⠇⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⡄⠽⠀⡐
⠀⠀⠈⠧⣈⣛⠟⠉⠳⠤⣀⡤⠞⠃⡼⣥⠔⠂⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠂
⠀⠀⠀⣠⠴⠛⢓⣒⡶⠦⠯⠥⣴⠊⠍⠒⠢⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠐⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠌⠀⠁⠈⠑⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⢠⣄⡤⣤⠔⠒⠋⠀⠀⠀⠀⠀⠀⠀⠉⢉⣠⣭⡿⣧⣀⡀⠀⣀⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⢰⠠⠂⠀⠀⠀⠀⠀⠀⢀⣠⡶⣫⡿⣬⡻⠃⢨⡙⣛⡆⠀⠈⠉⠻⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⡖⢃⠉⠀⣠⡙⢆⣿⡗⡟⣵⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠠⠄⠀⠀⠀⣰⣯⡿⡙⢣⣾⣧⠀⢿⠿⢀⣩⣮⠥⠀⡙⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⢰⣻⠽⢽⢿⣪⣚⣋⣣⠰⠶⠙⠈⠀⠀⠀⣾⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣼⣿⢿⠋⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠈⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠤⠈⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣾⣄⣀⡀⠀⠀⢀⡀⠀⠀⠄⠀⢀⡀
⠀⠀⠀⠀⠈⠁⠀⠀⣤⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠓⢿⠀⠀⠀⠀⣤⡀⠀⡙⠀⠀⠀⠠⠶⣾⣿⡶⠶⠶
⠀⠀⠀⠺⠆⠐⠚⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢀⣄⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⢀⡠⣴⣖⢻⣿⠁⠀⢀⣤⠄⠀⠀⠒⠀⢀⣀⣤⣶⣦⣄⣀⣀⡀⠀⠀⠀⠀⣁⠀⠋⠀⠉⠀⠐⠂
⠀⠀⠀⣠⣶⢹⡇⣿⡟⣾⣿⠀⣴⣿⠏⠀⠀⠰⠆⠀⠉⠙⢻⢟⠛⠛⠉⠉⠉⠀⠀⠀⠀⠉⠀⢀⣤⣤⣤⣤⠄
⠀⠀⢸⣿⡇⢸⡇⣿⡇⣽⣿⣾⣿⣿⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣶⣦⣤⣤⣶⠶⠖
⠀⠀⠀⠻⣿⢸⣷⢻⡇⣿⡿⠙⣿⣿⠀⢀⡄⢀⠀⠀⠀⠀⣀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠈⠙⢿⣿⠿⠿⠿⠿⠿⠛⠉⠈⠙⠓⠂
⠀⠀⠀⠀⠈⠻⣟⣸⣿⣿⣇⠀⠈⢿⣷⡄⠀⠘⡀⠐⠀⠈⠁⢀⠂⢐⣀⠎⠀⠃⢀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⡀⠀⠈⠉⣁⠃⠂⡁⣬⠂⢀⠀⠈⣠⠁⡀⠏⠀⠛⡁⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠠⠄⠒⢤⡆⢪⠤⢠⠑⠀⠹⡔⢀⠃⠄⡆⢨⠄⠰⠫⠀⠀⠂⠀⣀⠀⢀⠈⠃⠀⠀⠀⠀⠁⠂⠀⠀⠈
⠀⠀⠀⠀⠀⡄⠠⠦⣆⠠⣰⡁⢈⢢⠃⠀⠁⡈⠀⡖⠀⡘⠆⡂⠐⠀⠚⠂⢸⠀⠀⡄⠀⠀⠐⠄⠁⠃⣀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⢐⡁⢐⠠⡑⠠⠬⠦⢰⡠⠣⠀⠱⠆⠈⡖⠠⢀⠇⠐⠄⣠⠐⠘⡄⠀⠈⠀⡆⠀⠀⠁⠰⡏⠐⢐⠒⡐⠁⠇⠉⠒
⠀⠀⠀⠀⠀⠈⡀⠄⡀⠑⠤⣘⣃⠘⣄⠀⡤⠊⠰⠘⡠⠒⠡⡂⠀⢂⡅⠀⠀⣤⢉⢀⡌⠑⣀⠃⣀⠀⠠⠈⡅⠂⠃⡀⠒⠉⠇⠂
⠀⠀⠀⠀⠀⠀⠤⠉⡉⠱⡄⠤⢀⣠⠀⡘⢧⡀⠢⡼⠒⢂⠐⠒⠨⢄⠁⠆⠂⠘⠈⠠⢑⠅⡑⢀⠡⣀⠀⢹⠃⠥⢀⠰⠀⠀⠒
⠀⠀⠀⠀⠀⠀⢄⠀⠒⠀⠤⡀⠰⠲⠈⠡⠔⠉⠒⢧⣤⡀⢈⠢⢑⣄⣄⠁⢼⠀⠀⢐⠸⠆⠸⣆⠀⡨⠆⠡⡀⠍⠁⡔⠚⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠋⢤⡼⠣⠛⠉⠃⣁⣀⣁⣠⠄⠀⣀⠀⡐⠀⠀⠠⡄⠀⡈⠀⣉⢡⠖⠊⠁⢈⠁⠁⢀⡠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠐⢤⡟⠀⠀⠂⠐⢄⠀⡁⡤⠔⠋⠰⠀⢹⢀⢄⡉⠃⠍⡈⠐⢧⠊⢰⡦⠘⠛⠉⠛⠉⣉⣰⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠎⣌⡋⠀⠀⠀⢀⠈⣢⡜⠁⠈⠀⠠⠈⡀⢸⠃⢸⠀⠰⠀⠰⠀⣸⡀⢮⣤⡴⠂⠀⠀⠘⠟⠁⢀⣵⡄
⠀⠀⠀⠀⠀⠀⢘⣒⡛⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢀⠦⠇⡾⢀⡻⠶⡏⠣⡀⣇⡘⣡⡍⠁⠁⢠⣤⠀⠈⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠤⠭⠀⢀⡀⣀⣀⣇⣀⠀⣀⡀⠀⢼⠶⡚⣠⠈⠱⣎⠩⠵⡖⣨⠿⢢⡾⠧⣴⡶⠅⠠⣤⡾⠓⢀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢠⡔⠃⢠⡞⠁⠍⠉⣲⠂⠐⠚⠍⠆⠿⠄⣘⡀⠠⠋⠈⠢⠩⡁⠋⡄⢛⣳⠜⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠰⠏⠐⠀⠝⠀⠈

⠀⠀⠀⠀⠀⡀⠀⡀⢱⡀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠤⠂
⠶⠦⢄⡀⠠⠇⠄⠀⠀⠁⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⢠⠀⢀⠀⣰⠂⠀⢠
⢶⢤⣼⠇⣠⣄⡀⠀⠀⡢⠀⡳⢆⠈⣁⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠈⠀⠀⠈⢹⣉⠀⠀⠀⠸⠆⠀⠀⠀⢀⠄
⠉⠉⠁⠀⠹⣆⣾⡿⠟⠀⠐⠇⠈⠋⠁⠀⠀⠀⠒⠀⠀⠀⠀⠀⠉⠀⠂⠄⠀⠀⠄⠀⠁⠀⠀⠤⣤⡀⠀⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⢀⠀⠀⠀⠀⡄⠠⠤⠈⠈⠀⠀⠀⠀⠀⠀⠁⠀⠈⠓⡀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠋⠁⠀⠀⠀⠀⠉⠆⡂⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁
⠉⠈⠉⠁⠒⠶⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠁⣬⠝⠆⠀⠀⠀⠀⠀⠀⢠⢠⢤⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠰⢗⡖⠴⠂⠤⣭⣄⡂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠢⡶⠀⠀⠀⠀⠀⠀⠀⠐⢠⡾⠙⡗⢄⢀⣴⠦⠔⠒⠁⠀⠴⡈⠁
⠤⠠⡤⢴⣶⢟⠌⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⢧⢂⠈⠀⠀⠀⠀⠀⠀⠀⠔⠀⠛⠙⠶⠾⣞⣛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⠵⣆⠀⠀⢸⣦⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠁⠀⠀⡱⠾⠄⠀⠀⠀⠀⠀⠈
⠀⠀⡀⡀⢠⠀⠠⠀⠄⠀⠌⠀⠁⠀⠀⣠⣠⠞⡜⠀⠀⠀⠀⠀⡀⠀⠰⠄⡀⢁⠀⠄⠀⠂
⠻⠛⠗⠒⣶⣀⠀⠀⢀⡤⠂⠀⠀⠀⢸⡟⠃⣊⠀⠀⠀⠀⡀⡤⢧⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠢⠀⠀⠀⠀⠀⠉
⢀⡰⠀⠀⠀⢨⠀⠉⠈⠁⠀⠀⠀⢀⠀⠈⠀⡀⢀⠀⠀⠦⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⠀⠀⠀⣶⡆⠀⠀⠓
⠶⠀⠂⠀⠔⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠂⠀⠀⠀⠐⡔⠀⢘⠠⠀⠀⠀⠈⣄⠁⠀⠁⠀⠀⠀⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠇⠰⠋⠀⠀⠀⠄⠀⠀⠁⠶⠶⠀⢀⣠⢤⣄⢲⣦⣄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢪⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠂⠀⠀⢀⡟⣥⣊⠹⣅⣇⣬⡷
⠀⠀⠀⠀⡀⠀⠀⠀⢼⠄⣀⣠⡄⠀⠀⠀⠀⢠⠁⡈⡀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠹⠏⠀⠀⠀⠀⠈⠇⠀⠀⠼⡿⡃⠁
⠀⠀⠀⡠⠁⢂⠀⠀⠀⠀⠈⠉⠀⢀⠀⠀⠀⠀⠂⠃⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠰⡀⠂⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠈⠈⠠⠂⠀⡀⠀⠠⢏⠀⢀⡠⠀⠀⠀⢦⣸⠆⠀⠀⠀⠀⠀⠀⠀⢀⠖⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚
⠀⠀⢀⠀⠀⠉⠀⠀⠀⠑⠦⠡⠞⠐⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤
⣤⠀⠘⠃⠄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠄⡀⠀⠤⠚⢤⠄⠀⡀⣤⣏⠄⠠
⠄⠄⠀⠁⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⠀⠀⠀⠀⠀⠀⠞⠀⠀⠉⠁⠻⠷⡶
⠀⠀⠀⠀⠀⠀⠈⢆

⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡤⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠹⢿⣷⣎⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⢀⣀⢻⣴⣶⣶⣦⡝⣀⡀⠀⠀⠀⠀⠀⣄⠸⣿⣿⢃⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠐⠿⠋⠛⠿⠻⣿⡇⠀⠀⠀⠀⠀⢀⠎⣻⣿⣿⣟⡱⡃⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⢿⠔
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⡶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣼⣇⣬⠀⢀⡀⠈⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣶⣄⠀⠀⠀⠀⠀⢠⡿⠱⡶⡅⡬⠎⣿⡆⠀⠀⠀⠀⣀⢰⣿⠏⣿⣿⢂⣀
⠀⠀⠀⠀⠀⠾⢋⣿⢻⡿⡇⢹⡀⠀⠀⠀⠀⠀⠤⢾⢿⡿⢿⣿⣿⠿⠟⠀⠀⠀⠀⠈⠛⣶⣿⣷⣯⣼⠛⠁⠀⠀⠀⠈⠿⣼⣿⡂⣹⣿⠺⠿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠉


⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⡄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⢀⡼⠛⠛⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡤⠀⠀⣀⣤⣶⠞⠉
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠈⠻⠈⠙⢧⠀⠀⠀⠀⠀⠀⠀⡞⣽⣏⢏⠂⠀⠀⠀⠀⠉⠉⠙⢿⡿⣿⣿⢠⡄
⠀⠀⠀⠀⠀⠈⠙⠛⠛⢿⡁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⡄⠀⠀⠀⠀⠀⠁⠘⠈⠙⠳⡄⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂


⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⠶⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⣤⢤⡤⠄⡀⠀⠀⠀⢰⣿⡆⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢈⣹⡎⡓⢠⡜⠂⠀⠀⠀⠀⠰⢶⣤⢼⣽⡧⣤⡴⠆⠀⠀⠀⠀⠀⡀⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣾⣀⠝⠁
⠀⠀⠀⠀⠀⠀⠉⢀⠀⢹⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣿⣻⡍⠀⠀⠀⠀⠀⠀⠈⠁⣴⠋⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡂
⠀⠀⠀⠀⠀⠀⣴⣾⣷⣿⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠠⣤⢤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄
⠀⠀⠀⢀⣊⡥⠸⠃⠀⠀⠀⠃⡄⢀⠀⡀⠀⢀⠀⠃⠀⠀⠀⠀⠠⠀⠀⢠⡆
⠀⠀⠀⢟⣱⠂⠀⠀⠀⠀⠀⠐⠀⠙⢁⢊⠌⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢢⠀⠀⠀⠀⠉⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠰⠀⠃
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠑⡦⠦⠀⠘⠂⠀⠀⠋⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠠⢴⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⡄⠀⠀⢢⠛⠂⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠈⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⡬⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢀⠀⣀⢠⡄⠀⠀⠀⠀⠀⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠂⠃⠀⠀⠀⠀⠀⠀⠀⠱⣤⡀⠀⠀⠀⠲⠀⢠⠅⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠸⠇⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⣤
⠀⠀⠀⠄⠀⠀⠀⡀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠘⡡⠴⠄⠀⢰⠆⠤⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⡀⠀⠀⠈
⠀⠀⠀⠘⠃⠀⢀⢣⠙⠀⠈⠉⢠⠜⠂⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠋⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠟⠐⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢣⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠈⠏
⠀⠀⠀⠀⠀⠀⠙⠄⠀⠑⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠐⠊⢀⡴⠚
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠠⠊⠀⠁⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢆⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡐⠀⠀⠀⠀⠀⢗⡀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⡀⠄⢀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠹⣎⠀⣀⢠⠠⠀⠀⠀⠀⢀⣀⢴⣷⡇⠀⠀⠞⠁⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣣⠖⠀⢰⠛
⠀⠀⠀⠀⠀⠀⠀⠸⢧⡀⠉⠀⣠⣈⣣⡀⣀⠀⠤⣼⣵⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡄⢀⣀⡀⠀⠀⠀⠠⠃⡄⠀⠀⡏⢀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠌⢄⣀⠀⠤⢾⠀⡈⣾⣧⠀⠛⡆⠈⠂⠀⠀⠀⠀⠀⢃⣞⡇⠋⠈⠀⠔⠁⠀⠀⠀⡄⢀⠀⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢻⡀⡘⠀⠀⠀⠀⠀⢑⠈⠛⠄⡀⠈⠳⣴⡄⠀⢀⠀⠐⠈⠙⢿⢦⠀⠀⠀⠀⠀⠀⡄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠡⠀⠀⠀⠀⠀⠚⢣⡈⠒⠀⠉⠓⠋⠒⠒⠁⠁⠀⠂⠀⠘⣿⠷⠴⠤⠀⠂⠁⡆⡀
⠀⠀⠀⢰⠀⠀⠀⠀⣀⠀⠐⠉⢢⡀⠀⠀⠀⠀⢠⠂⣀⠓⠄⡀⠀⠀⠀⢀⡀⠠⠀⠀⠀⢰⡿⠒⠒⠀⠀⠐⠂⠀⠈⠀⠀⡠⠤⣄⡀⣀
⠀⠀⠀⠀⠀⢀⣤⡎⠁⠀⠀⠀⠀⠁⠢⣀⠀⠀⠈⢢⡍⠳⡄⠀⠀⠀⠀⠀⠈⠁⠀⠒⡠⢄⣤⣤⠶⣏⠀⠀⠀⠀⠀⠀⢰⡴⠾⣭⠀⠌
⠀⠀⠀⠀⠰⡁⠏⠁⠀⢴⣄⠒⠄⠀⠀⠠⡝⠲⢤⡶⢇⠀⠀⢀⠤⣄⠀⢀⣀⣠⠖⣡⠖⠈⠶⡇⠈⠪⣉⠶⣤⠤⠤⠳⠀⠂⠀⠹⠆⡄
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣀⠳⢶⡠⠄⠀⠙⢄⡀⣤⣌⣳⡬⢄⣀⡘⠩⢄⣡⠖⠨⠧⢀⠸⣢⡑⠀⠀⢀⡀⠤⠔⠋⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠒⠢⢤⣿⠟⠉⢦⠤⠴⢂⡉⠋⠁⡨⠢⠞⠑⠋⠕⢂⠟⢀⣀⡀⠉⠉⡠⠼⢿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⣿⠃⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠒⠿⠁⠀⠐⠋⠍⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠰⠊⠁⠀⠀⢨⣇⠀⠀⠸⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷
⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠘⠏⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠗⠀⠀⢰⣷⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⠉⠐⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠤⠂⢀⢄⡠⠁
⠀⠀⡀⠀⠀⠀⡀⠄⡴⠂⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠘⠋⡴⠋⠀⠀⠀⢴⠟⢋⠁
⠀⣠⠑⢄⠀⠀⠀⢠⠁⠀⠈⣠⢠⣕⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠄⠂⠀⠀⠀⠀⠀⡇
⠀⠠⣄⠀⠑⠢⣤⣀⣠⢖⣡⠞⢛⠋⡀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡲⠀⠀⣤⢀⠄⠁⠀⠀⠠⠁⠀⠀⠈⠀⠀⠀⢀⡴⠁
⠀⠀⠈⠑⠀⠀⢀⡈⣿⡟⠁⠀⠈⣠⠁⠉⠑⣂⠁⠀⠀⠀⠀⠀⠙⠆⠄⠉⠂⠀⠈⠁⠀⠀⠀⠕⡀⠀⢀⠄⠂⢐⠀⢀⡞
⠀⠀⠀⠈⠀⠂⠈⠙⣿⠔⠀⠀⡀⠥⠀⠀⠀⠀⠹⣤⠀⠀⠀⠀⠀⠀⡀⠀⠄⠊⠀⠀⠀⢠⠎⠠⠀⠀⣬⡶⠓⠁⢠⠞⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀⡁⠴⠀⠀⠀⠀⠀⠀⠉⡓⠦⠤⡔⢒⠭⠜⠀⢀⠠⠂⠀⠐⠁⡴⠁⠀⠀⡟⠀⣠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢮⠃⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠄⠀⠀⠀⠀⠀⠀⠀⡁⠀⠁⢸⠥⠚⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢘⠀⡎⠀⢠
⠀⠀⠀⠀⣠⣈⠙⠆⠀⠀⠈⢢⡀⢠⣆⠀⣀⠀⠀⢄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣁⠀⠀⠀⠁⢠⠆
⠀⠀⢠⠈⠟⠉⠨⣦⡐⠤⠀⠈⠛⠀⢩⣓⡘⠯⠖⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⡟⠛⠀⠀⠀⠀⠚
⠀⠀⠀⠉⠀⠀⠀⠁⡘⢶⡄⠤⠀⠀⠠⠀⠈⠐⣉⡭⢀⣬⠟⠒⠒⠚⠛⠛⢉⠝⠡⢠⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠤⠰⡤⠀⠢⠄⠖⠉⠁⠀⠀⠒⠊⠉⠉⠀⠁⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠤⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⢐⡁⡤⠲⠞⠃⠠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡤⢤⠤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣻⢦
⠀⠀⠀⠀⢀⠀⠛⠙⠇⠀⠁⡄⠀⠀⠀⠀⠀⠀⢼⣱⣫⣭⣿⣧⣴⡶⠀⠀⠀⠀⠀⠀⢀⡤⣦⣿⠏⡀⠁
⠀⠀⠀⠀⠀⢐⠀⠇⢣⣈⠀⠇⠀⠀⠀⠀⠀⠀⠘⣯⣷⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠛⠈⡇⠀⠀⡀
⠀⠀⠀⠀⣄⠀⠠⣶⣾⣿⣿⡆⠀⠀⠀⠀⠀⣠⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠘⠁⠰⣭⣝⣿⣿⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⣼⡛⠷⠏⡕⠃⠀⠀⠀⠀⠀⠀⢠⣖⡀⣈⡽⠀⡀⠒⢋⣠⡄⡂⠀⠀⠀⠀⠹⣄⠆⠀⠀⢀
⠀⠀⠀⠀⠈⠓⠂⠛⠁⠀⠈⠀⠀⠀⠀⠀⠀⣀⣐⣮⣍⣥⣽⣮⣧⡇⣠⡤⠀⠷⠃⠀⠀⠀⠀⠀⠹⠇⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠩⠼⢿⡿⡉⠛⠛⠻⢟⠉⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣸⣴⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠈⠀⢴⣶⣤⡀⠀⠀⠀⠀⠐⢶⣿⣿⣿⠟⢛⣷⡆⠀⠀⠀⠀⠀⠈⠦⠀⢀⠀⠠⠀⠀⣀⡤⠤⠴⠦⠆
⠀⠀⠀⢀⣄⣠⣶⣿⣿⣿⣿⡟⠋⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣏⡀⠀⣤⣄⡀⠀⠀⠈⠁⣁⠀⠀⣈⠈⠀⠀⢀⣀⣰
⠀⠀⠀⠀⢿⣿⡿⠿⠛⢛⣿⠟⠀⠀⠀⠀⠀⠝⢿⡿⠟⠙⠀⠀⠀⠀⠈⠙⠛⠛⠒⠚⠉⠉⠀⠀⠙⠮⠉⣉⣙⠛⠋
⠀⠀⠀⠀⠈⠉⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⢀⠰⡀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣀⡀⠀⢀⡴⠂⠆
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠙⣿⢎⡁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐
⠓⠄⠐⢦⠀⠀⠀⠀⠀⠈⠃⣐⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠸⢃⠂⠀⠀⠀⠀⠀⠀⠂⡉⢣⣾⠦⠄
⠀⠈⠓⠬⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤⠀⠀⠂⠰⠂⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⣁⠠⠺⠖⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡓⠀⠀⠀⡀⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢸⠀⠀⠀⠨⠔⠀⠀⠀⠀⠁⢆⠀⠈⠀⠀⠀⠀⠨⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣀⣀⣠⠆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢀⠢⡀⠀⠀⢀⢴⠿⠀⠀⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠋
⠀⠀⣀⠐⢀⠀⠀⢼⠁⠁⠀⠀⢠⠀⠀⠉⣑⣿⠁⠀⠀⠀⢰⠃⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⢋⠔⠃
⠀⡈⠏⠉⢦⠀⠄⠀⢆⠀⠀⠀⠠⠒⠀⠀⡀⠙⠀⣄⣤⣀⠊⡠⠔⢢⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠘⢶⡀⠀⠑⢦⠱⠊⡉⢛⠤⠁⢨⠷⣌⠉⠉⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠠⠘⢃⠜⢉⠁⠤⠂⠈⠋⠁⡌⠀⠈⠣⡠⢤⣀⡀⣀⠀⠀⣀⣀⢠⡄⢂
⠀⠀⠀⠑⠚⠷⠐⠀⠀⠈⠁⠀⠀⠂⠘⠒⠀⠀⠀⠀⠀⠁⠐⠂⠄⠥⠭⡍⣯⠿⠃⠈⠛⠠
⠀⠀⠀⠀⢰⠎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠟⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡾⠐⠀⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣿⣿⣿⣛⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠁⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣒⠒⠂⣤⢤⣀
⠀⠀⠀⠀⠀⢠⠜⠀⡄⠀⠀⠰⠀⠁⠀⠓⡄⠀⠀⠀⣰⠀⠃⠀⠀⠀⠈⠁⡐⠀⠀⠀⠀⠀⠀⢀⣴⠅⢠⠉⠁⠀⠈⠉⠁⣬⡖⢄
⠀⠀⠀⠠⠼⡐⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢁⠀⠀⢨⠋⠘⠋⠀⠑⠀⠀⡀⡨⠑⠀⠀⠀⠀⣸⢱⠁⠀⠃⠀⠀⠀⡠⠂⠀⠀⠡⣼⡀
⠀⠀⠀⠘⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠘⡀⠰⠀⠀⠀⠀⢀⡝⣢⠀⢀⠀⠀⡠⡇⠘⠀⡌⠀⠀⡠⠂⠀⠀⠂⠀⠀⣈⡱
⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠇⠀⠀⠂⠀⠀⡈⠠⢣⠀⠠⡇⠀⠇⡆⡄⢰⠁⠀⡔⢁⠔⣁⠄⠂⠈⣀⣈⠍
⠀⠀⠀⠀⠀⢹⠈⠀⠀⠀⠘⠀⠀⠀⠀⠠⡁⠀⠀⠀⢲⡐⡌⠀⠀⢀⡟⠀⠀⢀⠜⠀⠀⡄⠇⡃⡾⢀⢌⢔⠵⣊⠄⠂⠁⢀⠽⠂
⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⢡⡌⠀⠀⠀⠀⠀⡇⠈⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢳⡀⢀⡇⣾⡶⣫⡮⠖⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⠀⢳⠀⠀⡎⠀⠀⡦⠁⠀⠀⠀⠀⠀⢳⣸⣿⡿⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⢇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⢰⡘⢂⠄⠀⠀⠐⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠎⠀⠱⠄⠣
⠀⠐⠂⠀⠤⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠤⠠⡀⠈⠀⢁⣀⠀⠃⣄⢸⢠⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠉⠐⠠⠀⡀⠀⠄⢤⡈⠀⠀⠀⠀⡀⠀⣠⡀⠈⠀⠉⡰⠀⠉⠠⠀⣀⠤⠄⠀⠀⠀⠀⠀⠀⠀⢰⡀⡁⠀⠀⠀⠀⡀⠀⠀⢁
⠀⠀⠀⠨⠁⠀⠁⠦⠀⡀⠓⠆⠐⠀⠴⠤⠀⠚⠀⠀⠀⠂⢢⠀⠀⠀⢈⠉⠀⠀⠐⠀⠈⠀⡀⠀⠀⠀⠑⠄⠀⠺⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⡐⠁⠁⠀⠀⠀⠘⠁⠌⢋⠀⠋⣀⠈⠠⠄⠀⠈⠀⠈⠀⢀⠁⠀⠀⠁⠀⡁⠀⢱⠀⡀
⠁⠀⠀⠠⠀⠀⠉⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠇⢸⠀⡽⠀⠐⠀⠀⠑⢀⠀⠀⠙⠁⠀⠠⠈⢠⠁⠀⠰⠑⢄⠅
⠀⠀⠠⠬⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⠀⠙
⠀⠀⠀⠀⠄⠀⠀⠰⠁⢐⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⣁⠀⠀⠀⠠⡅⠀⠀⢀⣰⡀
⠀⠀⠀⠀⠂⠐⠌⠂⡀⠐⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠄⠀⠀⠀⢀⠃⡄⠀⠈⠰⠁
⠀⠀⠀⠀⠈⠤⠠⠀⠁⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢼⣧⠃⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠹⢸⠁
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣷⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠉⠉⠉⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⡠⠖⠋⠉⠉⠙⠲⢄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡞⠀⡴⠊⠉⠉⠑⢄⠀⢣⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢰⠀⠀⠀⠀⠀⠈⡆⠈⡆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠘⡆⠀⢆⡔⠊⠐⢲⠉⠁⠲⡁⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠈⠦⣀⠀⠀⠀⠈⡆⠀⠀⢣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡈⠀⠀⠀⠀⣏⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡏⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⢀⣸⣆⠀⡏⠀⠀⠰⠋⠉⠉⠉⠉⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⢹⠉⠀⠀⢰⡀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠐⠂⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡤⠓⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠇⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣠⠛⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠎⠩⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠜⠀⢀⡀⠀⠧⠤⢄
⠀⠀⠀⠀⡠⠚⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠰⣾⣷⠄⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⢄
⠀⠀⢀⠞⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⢤⠀⢸⡇⠀⡤⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠣⡀
⠀⢀⠎⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢪⠀⢸⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠱⡀
⠀⡜⠀⣰⡆⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢸⠀⣼⣧⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⣆⠀⢣
⠸⠀⢀⣿⣿⡄⠀⠣⡠⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢰⢠⣿⣿⡄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠜⠀⢠⣿⣿⡄
⡇⠀⣸⣿⣿⣿⡆⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡌⣾⣿⣿⣧⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⢰⣿⣿⣿⡇
⡇⠀⠈⠉⠻⣿⠀⠀⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⢰⣹⣿⣿⣿⣿⣆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⠀⢀⣿⠟⠉⠁
⠰⣀⠔⠢⡀⠈⠁⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⣸⣿⣿⣿⣿⣿⣿⣆⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎⠀⠀⠁⢀⠔⠢⣀
⠀⠀⠀⠀⠈⠦⡀⠀⠀⠈⠑⠢⠤⠄⢀⣀⠀⠤⠤⠒⠙⢄⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠑⠒⠤⠤⠀⣀⡀⠠⠤⠔⠊⠁⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣶⣿⣄⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠄⣀⠀⠉⠉⠛⠻⠿⣿⣿⣿⣿⣄⡀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⣀⠠⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠠⠤⣀⡀⠈⠙⠻⢿⣧⣄⠀⠀⠀⠈⠹⢿⣿⠟⠋⠁⢀⡀⠤⠔⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠉⠻⢿⣆⡀⠀⠀⠀⠀⠋⡱⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠙⠋⠐⡢⠄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡰⠋⢠⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢈⣉⠘⠀⡰⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⢡⠀⠀⠀⠀⠀⠀⡎⠰⠁⠐⠐⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡠⠀⢀⡜⣆⣰⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⡐⠁⢀⠊⢇⠳⡈⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠐⠀⠀⠀⠀⢀⡜⠀⢠⠃⡰⢈⠜⡸⠆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡰⠁⠀⡇⢸⠘⣀⠇⠀⢀⠔⠁⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢠⠁⠀⠀⠀⠀⠀⠀⡠⠃⠀⡔⠁⠀⢰⡉⡞⢀⠇⢀⠴⠃⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⡐⠁⢀⠜⠀⠀⠀⡎⢸⣠⡎⠠⠁⠀⠀⠈⠀⡌
⠀⠀⠀⠀⠀⢠⠀⢸⠀⠀⠀⠀⢀⠜⠀⢀⠎⠀⠀⠀⠀⢇⠸⠉⢃⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠸⡀⠈⡄⠀⠀⢀⠊⠀⠠⠃⠀⠀⠀⠀⠀⢨⢜⠇⢸⢀
⠀⠀⠀⠀⠀⠀⠣⠀⠈⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⣀⡳⠟⣁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠒⠒⢈⣁⠴⢺⣍⢦⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠤⠤⠤⠤⠄⠒⠊⠁⠀⠀⠀⠛⠛⠲⣶⢝⢲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠁⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠠⠈⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠈⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠶⠀⠀⢨⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢡⠀⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⠁⠀⡀⠀⠀⠀⠈⠉⠀⠀⠈⠑⠀⠀⠀⠀⠀⠙⠒⠀⢻⠂⠀⠸⢠⡀
⠀⠀⠀⠀⠀⡒⠠⠀⠈⡍⠹⣤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⣹⡏⠀⠠⠖⢠⠇
⠀⠀⠀⠘⠀⠘⣆⡠⢀⠑⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⢀⡔⠀⢀
⠀⠀⠀⠐⠂⠀⠈⠢⢤⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠴⠋⠠⢀⠉
⠀⠀⠀⠀⠀⣀⠀⢀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣂⣂⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠈⠀⠤⠵⠆⠀⠴⠶⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣵⣖⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠠⠔⠚⠋⢱⣄⣀⠀⠀⠀⠀⣀⣠⣤⢾⡑⠀⠀⠉⠉⠛⠓⠦⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠰⠊⠀⠀⠀⡶⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠙⠓⠞⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠆⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣷⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⣿⡿⠟⠻⢿⣿⣮⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡏⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢻⡷⠔⠒⢖⢲⣞⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⣤⡗⠛⣯⢹⣿⣵⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢳⠛⡇⢰⣿⣾⢻⡅⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⡗⠺⣴⡇⠀⣿⣿⣦⡇⢸⠁⢀⡤⠤⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⣹⠶⠶⠶⢶⣒⣋⣹⣄⡹⢧⣀⣿⣿⣦⣏⢻⣷⣿⣿⣷⠐⣷⣏⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⢳⣮⣉⠛⣿⣟⡧⢽⡶⢳⣼⣿⣼⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢙⣿⣶⡏⠻⡧⠶⠞⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣿⡇⣿⡆⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠨⣿⡇⣿⣱⠞
⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠀⢘⣿⡇⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀
⠀⠀⠀⠀⣠⢋⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣴⡟⠀⢸⣿⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣄
⠀⠀⢀⣼⠇⣼⡟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣐⣳⣇⡀⠸⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢳⣜⢧⡀
⠀⢀⣾⠏⢰⣿⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣄⡋⢙⠿⢿⣼⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣎⢷⡀
⠀⣼⡏⠀⣼⣿⣿⡄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠻⣷⣿⢦⣞⡉⣣⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣼⣿⣿⣿⣧
⢰⡟⠀⢀⣿⣿⣿⣷⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢨⣿⣿⠀⣏⣿⣹⠆⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⣿⣿⣿⣿⣿
⡼⢀⣴⠿⡿⣿⣿⡛⠷⢽⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠠⣿⣿⡆⣿⣡⠏⠀⠀⠀⠀⠀⠀⢠⡾⠷⢛⡿⣹⢻⠿⣟⠻
⡷⠋⠁⠀⢳⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡏⠸⠦⠐⣿⣿⣿⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣰⡿⡟⠀⠈⠙
⠀⠀⠀⠀⠘⣆⢈⡻⢿⣿⣶⣤⣄⣀⣀⣠⠞⣧⡞⠀⠀⠀⡼⢿⣿⣿⣧⡑⠦⢤⣀⣀⣀⣤⠤⠖⠉⣠⣾⡿⣱⠃
⠀⠀⠀⠀⠀⠘⣞⣿⣷⣍⡛⠿⠿⣿⣿⡟⢤⣿⣖⣀⠤⠚⠀⠀⠙⠿⣿⣿⣷⣶⣤⣤⣤⣴⣶⣷⣿⣿⢟⣴⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣝⡛⠛⠁⠀⠀⠀⣷⢬⡏⠉⠙⠲⡄⠀⠀⢠⠴⠛⠛⣻⣿⣿⣿⣿⣿⡿⢟⣫⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣄⣀⣹⡦⣽⡀⢀⣠⣾⣆⢰⠃⠀⠀⠐⠛⣛⣫⣭⣿⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠼⡗⠿⢿⣿⣿⡇⣀⡤⠶⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠢⡇⠀⠀⠈⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠈⡉⠉⠉⠉⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠐⠂⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⣀⣀⡀⣀⣀⣀⣀⣈⣁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⡂⠉⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⠤⣄⣀⣈⣠⣤⠤⠀⠌⠥⠴⣺⠁⠀⠀⠀⠀⢀⠀⠔⠁⠀⠀⠀⢀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠀⠀⠀⠀⠐⠂⠀⠣⠁⠎⠀⠀⠀⡸⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠰⠤⣀⠀⠀⢀⡴⠁⠀⠀⠀⣀⠤⠊⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⡠⠖⠉⣀⡤⠒⠈⢁⡠⢀⠔⠀⠀⠉⠉⡁⠀⠀⠁⢉⠁⠀⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡴⠃⠀⡠⠊⠁⠀⢀⠔⠉⣠⠃⠀⠀⠀⠄⢠⠀⠸⠀⠀⠸⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⠞⡑⠀⣠⠊⢀⠂⠀⡰⠃⠀⡴⠁⠀⡜⠀⠀⠀⡎⢠⠃⡇⠀⢠⡙⡄⠈⣆
⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⡔⠁⠀⠀⢀⠜⠁⡠⠊⠀⢰⢰⠁⠀⠀⡜⠀⡜⠀⣻⠀⢸⢡⠸⡄⠘⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠎⠀⢀⡠⠚⠁⠀⠀⡠⠨⠀⠸⢸⠀⠀⡜⠀⡜⠀⢈⠉⡆⢸⢸⠀⢣⠀⠸⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠐⠁⡰⠁⢀⡰⠋⡰⠁⠀⡇⡆⠀⢈⠀⡜⠀⠀⢸⠀⡇⠸⢸⠀⠸⡀⢠⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠜⣁⠔⠉⢀⠜⠁⢀⠜⢰⠁⠄⣡⠞⠀⡀⢀⣾⠀⡇⡇⢸⠀⠀⡇⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⢀⠔⢃⠠⠔⠁⣠⠃⡠⠊⠁⠀⡜⠀⡜⢸⢠⢡⠁⡌⠀⠀⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⡇⢰⠁⠀⠀⠌⠀⡼⠀⢸⠀⠆⢠⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠃⠀⠐⠀⠘⠀⠀⡆⠈⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⡀⠀⠀⠀⠀⠰⠁⢀⠊



⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣉⡉⡛⢿⣙⣠⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢀⣠⠶⠛⠉⠀⠙⢀⣁⣒⣤⡀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣿⣽⣿⡆
⠀⠀⠀⠀⠀⠀⣴⣿⠀⢠⠾⠋⠨⠛⣤⠀⠀⠀⠁⣸⠛⢻⢿⢿⣎⢥⢴⣶⣶⣦⣤⣀⠈⠛⢿⣿⢛⡟⠂⠔⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢰⣿⣉⣴⣿⠀⠀⢀⣤⠛⣷⠀⢻⡀⠛⣶⣿⣫⣬⠘⢾⡌⠉⠀⠀⠈⠙⢻⠒⣤⣽⣿⣿⡆⠀⠀⠀⠀⢰⣧
⠀⠀⠀⠀⠀⣿⡇⢿⣿⣿⣤⠴⣫⣀⣶⣼⠻⢮⠇⢈⣽⣿⣿⣷⡀⣨⡷⡀⠀⠀⠀⠓⢀⢤⡟⠈⠻⣿⣿⡄⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⢸⣿⡬⣿⣿⠇⣄⣽⣿⣿⣿⢆⣈⣰⣦⠙⣿⣿⣿⠗⠎⠀⢨⣿⣧⡤⠀⠐⠮⡿⣴⠃⠈⠿⡧⠀⢀⡾⢘⠇
⠀⡤⠀⠀⠀⠀⢹⣧⡘⠷⢶⣾⣿⣿⣿⣧⣶⣿⠛⢀⣴⠟⠛⢿⣿⣄⣴⣿⣿⣿⡿⠀⠦⠀⠈⢫⠙⡋⠀⡰⢂⣾⢣⠈
⠀⢳⣾⣦⣤⣄⠀⠈⠻⡖⢶⣿⣿⣿⣿⣟⠛⠧⠸⠟⠁⡄⢠⣿⣿⠛⠙⣿⢿⣟⠛⢷⡀⠀⠀⠀⠀⠀⠀⣠⠾⠡⠎⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠹⣿⣿⣿⡷⠤⢠⣦⣀⣿⣿⣿⣿⣿⣞⠀⠀⠀⠀⠀⢸⣿⠘⠀⣾⢻⣼⡿⠀⠈⠀⠀⠀⠀⠁⢠⣄⠘⢂⣠⡔⠒⢈⣩⠄⠀⠀⠀⠁
⠀⠀⠀⠈⠛⠾⠿⣶⣤⣼⣿⠻⣶⣿⣛⠙⠇⢗⡤⠀⠀⠀⠛⢠⡄⠀⢸⠂⠙⡿⢶⡀⠀⠀⠀⢈⣡⢶⢿⡟⡋⠉⠄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣭⣿⡟⠻⠿⠿⣿⣿⣦⡋⠁⠁⠀⠀⡀⣸⡇⠀⠾⠀⠀⠀⠁⠈⣀⣴⡾⠯⠄⡍⣌
⠀⠀⠀⠀⢠⠴⢋⣩⣤⡞⢡⣴⣦⣤⣄⣹⣿⣿⣄⣀⣠⠤⠉⠀⠀⠀⠄⢀⣀⣀⣠⣞⣋⣀⣴⡖⠚⠓⠛⢶⣌⠒⣄⡀
⠀⠀⠀⣴⠟⠚⠋⠁⠀⠀⣠⠰⠙⠉⠉⠛⢛⣏⣯⣻⠽⣿⣿⢿⣭⣿⣽⢿⣿⣾⣿⡀⠉⣩⣉⣛⣷⢆⠀⠀⠈⠓⠪⣱⣄
⠀⠀⠀⡷⠀⠀⠀⢀⣤⠾⠋⠛⠀⠀⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠓⠌⠉⠉⠛⢧⠱⠄⠀⠀⠀⠀⢬⣃
⠀⠀⠀⠀⠀⢀⢤⡿⠉⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣠⠿⠟⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢛⣛⣛⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣡⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⣿⣿⡏⠁⠀⠉⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣄⠀⢀⣤⢾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟⣡⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⣿⡇
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣠⣾⣿⣷⣶⣖⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃⢙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣲⣿⣿⣷⣄
⠀⠰⣿⣯⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣿⣿⠆
⠀⠀⠈⠻⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⡇⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡇⣿⣿⡄
⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣧⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⣿⡇⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿
⣿⣧⣿⣮⡻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣾⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⢻⣿⣿⣿⣿
⢹⡟⠉⢻⣿⣿⣦⡙⢷⣤⡀⠀⠀⠀⠀⠀⣿⠇⣼⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣤⠾⢋⣴⣿⣿⡟⠉⠻
⠀⠀⠀⠀⠻⣿⣿⣿⣶⣌⡙⠳⠶⣤⣶⡾⠟⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣟⣋⣥⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣦⣤⣀⣀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣭⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣷⣬⣙⠻⠟⣡⣴⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣷⣾⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⢲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠞⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠗⠠⠞⠿⣀
⠀⠀⠀⠀⠀⠀⠀⠒⠷⠧⠤⣤⡶⠶⠷⠗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠷⢧⣤⣤⣶⠶⠷⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⣇
⠀⠀⠀⠀⠀⡏⣄⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣄⠀⠀⢀⠀⠉⡏⠙⢶⡀⢀⡔⡇
⠀⠀⠀⠀⠀⠳⡉⠀⠀⠀⠀⠀⠳⠀⠀⠀⢈⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡉⠀⠰⡏⠀⠀⠻⠀⢸⠇⢈⡽⠃
⠀⠀⠀⠀⠀⠀⠙⢦⣥⣀⣀⠀⢀⣁⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣬⢋⣀⠀⢀⣉⣯⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢤⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⢴⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⣷⣟⡿
⠀⠀⠀⠀⠀⠀⠰⠾⠶⠶⢌⠉⣹⠦⠶⢦⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⣌⠉⣹⠷⠶⢦⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⡇
⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⣿⣟⠳⣤⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠛⣏⠀⠀⠀⠐⠈⢿⠀⠀⠀⠈⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣏⠀⢰⠞⠀⠘⠿⠀⢸⡇⠈⡿⠃
⠀⠀⠀⠀⠀⠀⠙⢧⣤⣀⣤⠤⡦⣐⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣜⣃⣠⠤⣦⣐⣛⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢰⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢰⡿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣇⡀
⠀⠀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⣶⣶⣀⣀⣀⣠⣴⣾⣿⡿⠛⠛⠻⣿⣿⣿⣄
⠀⠀⢿⣿⣿⣿⣻⣟⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠉⠻⠿⣿⣷⣿⣿⣿⣿⣶⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⢻⣛⣚⠛⠽⢻⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠀⠀⠀⠉⠻⠿⣦⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠎
⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣻⠋
⣀⣤⢤⣶⣾⡿⠿⢻⡿⢋⣭⠉⠁⣀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠛⠛⠻⠿⣶⣦⣤⣤⣤⣴⣿⣿⡽⠃
⠻⣿⠀⠫⢥⣀⠒⠛⠳⣿⣿⡀⠀⠀⠀⠌⠉⠁⠃⠀⠁⠀⠁⠁⠉⠉⠁⠀⠄⠀⠠⠀⠤⠰⠨⠭⠭⠭⣼⣿⣏
⠀⠈⠓⠦⣠⠒⠒⠾⡶⠠⠒⠂⠀⠠⡄⠀⡀⠀⢂⠉⢀⣀⣀⡄⠀⣀⣀⠀⠀⠈⠀⢀⣀⡤⣶⠟⠛⠓⢾⣿⣿⣦
⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣴⣶⣶⣤⣤⣤⣤⣤⣄⣤⣤⣤⠤⠤⠤⠤⠤⠤⣶⣖⣚⣩⣵⠞⠁⠀⠀⠀⠀⠙⢮⣋⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡏⠀⠀⠀⠀⠀⠈⠻⢿⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⢈⠉⠙⣳⣶⡤⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠰⢿⣶⠶⠂⠀⠀⠀⠈⠓⠦⣄⡉⢣⡀⡤⠋⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠑⢐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠉⠁⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⡦⠊⠀⠀⠀⠀⠀⢀⡀⡀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⣀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢰⠀⠶⠏⠀⠀⠀⠀⠀⠂⠀⢠⠄⠃⠘⠈⢳⣬⡆⠀⠀⠀⠀⠀⠀⡄⠈⠉⠁⠻⣧
⠀⠀⠀⠀⣄⠀⢀⠀⠀⠀⢠⣶⡆⠁⠀⠀⠀⠈⠀⠀⠀⠀⠐⣿⡧⠠⣄⡀⠁⢠⢶⠆⠀⠀⠀⣠⡇⢀
⠀⠀⠀⠀⢻⡄⠈⢆⡀⠀⠀⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠄⠈⡷⢀⣼⣿⠀⠀⣠⡾⠏⠀⡞
⠀⠀⠀⠀⠀⠡⣄⠀⠈⠀⠐⠯⡙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠛⠉⠉⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠢⠤⠤⣤⣀⡈⠩⢷⣄⣀⡀⣦⣄⣀⣦⣀⣤⣾⡿⠏⢀⣀⣐⡦⠀⠀⠋
⠀⠀⣠⣤⣤⣄⠀⣐⡀⠠⢧⠤⠤⠤⠄⢾⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣧⣤⠤⠄⢠⡬⠁⢐⠂⠀⢠⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠰⡠⠴⡂⢀⣀⠀⠗⠁⠀⠀⠈⠁⠀⠲⠀⣀⣀⢁⠶⠴⠆⠀⠲⠄
⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠼⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠦⠀⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣤⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢌⣆⠤⢆⡄⣠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠰⠀⠀⠀⠌⠢⠈⠢⡈⠂⠂⠚⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡵⡇⠀⠄⣀⠂⢄⠢⠀⢈⠀⣀⠠⠄⠤⢱⣄⣣⡀
⠚⠁⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠀⠀⠀⣠⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠳⢄⡀
⠀⠀⠀⠀⠓⠤⢄⡀⠈⠙⠢⢤⣀⠀⠀⠀⠀⠘⣠⠴⠊⠁⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⠀⠀⠁⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣁⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⠀⠀⠀⠀⠿⠘⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⡠⠖⠀⡠⠒⠀⠀⣏⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡠⢹⠀⠀⠀⠀⠀⠨⠭⠄⡠⠾⢑⠟
⠀⠀⠀⠀⠀⠀⢸⠤⠒⡉⡀⢀⠌⠀⠀⠀⠀⢣⠄⠀⡀⠀⠄⠈⠙⢁⡘⠤⠐⠈⠁⠀⠀⠑⢌⠠⠀⠐⠄⡴⠊⣠⠔⠁
⠀⠀⠀⠀⠀⠀⣁⡠⢊⠊⢠⠊⠀⠀⠀⠀⠀⢹⡒⠁⢀⣀⣀⠴⠈⣡⢀⠄⠠⠠⠴⠄⢀⣀⢀⡉⠀⡄⠐⠒⠉
⠀⠀⠀⠀⠀⠀⡝⡰⢁⡴⠁⠀⠀⠀⠀⠀⠐⠓⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⣠⣟⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠹⡈⠀⢀⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡴⠂⠀⠀⠀⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠊⠀⠀⠀⠀⡠⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣶⣄⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠋⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠈⢻⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠃⠀⢀⣠⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣀⠀⠀⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣯⠉⠁⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠉⢩⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⣿⣿⣿⣟⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⢘⣿⣿⣿⡟⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠹⢿⣿⣧⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢠⣿⣿⠿⠃⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠁⠀⠀⠀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⢿⣿⣷⣶⣶⣶⢞⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⢶⣶⣶⣶⣿⡿⠟⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣤⣄⣀⣉⣉⣭⣶⡿⠃⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠻⣷⣮⣭⣉⣁⣀⣠⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⠿⠿⣫⣴⣶⣶⣶⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣬⡻⠿⢿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢛⣿⡿⠋⠁⠀⠙⠻⠿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠈⠻⣿⣟⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠈⠻⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠟



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠐⠒⢶⣦⣤⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢰⣤⣤⣽⣿⣿⣿⢖⣦⣾⣿⡿⣿⣿⡟⢸⢞⢲⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠠⣤⣭⣽⣿⡿⢋⣵⣾⣿⣿⣿⢳⣿⡿⡇⠹⣙⡢⠋⠁⡬⠛⠁
⠀⠀⠀⠀⠀⠤⠶⣾⡿⢋⣴⣿⣿⣿⣿⣿⠃⠙⡛⠀⠀⠠⡀⠀⠀⠈⠉⢙⠇
⠀⠀⠀⠀⠄⠲⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠁⡀⠀⠀⠨⡀⠀⢁⠤⠃
⠀⠀⠀⠠⠶⢾⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⡀⠀⠂⠈⠳⠤⠝⠁
⠀⠀⠀⢀⣴⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠋
⠀⠀⠀⢠⣼⠣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⢀⡠⠄
⠀⠀⠀⢀⣼⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠀⣴⣤⣶⡿⠋
⠀⠀⠀⠁⣽⠨⣿⣿⣿⣿⣿⣿⣝⡛⣿⢕⣣⢴⠟⠛⠉
⠀⠀⠀⠀⢰⡆⣿⣿⣿⣿⣿⣿⡀⠅⠈⠉⠉⠁⠀⠀⠀⢀⣀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠐⠀⠱⠹⣿⣿⣿⣿⣯⢢⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠻⠿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠸⣄⢣⢽⣿⣿⣿⣿⣫⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢯⠙⢿⣧⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠹⡆⣬⡻⣿⣿⣿⣷⣷⣄⢀⣠⠾⠟⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠉⣿⠑⣮⠿⣿⣿⣿⣿⣀⠉⠋⠉⠙⠶⠆⠀⠀⠀⠀⠀⠀⠐⣘⣧⣤⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠜⡓⢮⡛⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠀⢤⣄⣉⡛⠿⠿⣿⣶⣶⣶⣶⣾⣷⡶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣦
⠀⠀⢻⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣬⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀
⠀⠀⠘⣿⣿⣿⣿⣆⣀⣀⣀⣰⣶⣶⣶⣶⣾⡿⠿⢿⡟⠛⣿⠛⢛⠉⠻⠏⠁⠉⠙⠟⠛⠛⠿⠻⢿⢿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⣿⣿⣿⣿⠿⠿⠿⠿⠿⠭⠿⠀⠠⠾⠀⠀⠒⠒⠒⠂⠀⠙⠀⠀⠀⠉⠀⠀⠴⠀⠐⠇⠚⠛⠛⠿⠻⠿⠿⠩⣿⣿⣿⣶⣤⡀
⠀⠀⠀⣿⣿⣿⣷⡤⠄⠀⠀⠀⢀⣀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣦⠠⠄⠀⠤⣤⠀⣈⠉⠹⠯⠄
⠀⠀⣸⣿⡿⠟⠋⠀⠀⠀⠀⠀⠈⠿⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠛⠋⠉
⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡎⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡀⠈⡆⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠄⠼⠤⠚⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡙⠢⠼⠤⠼⡄
⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⡜⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠀⠀⠘⡄⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⢰⠁⠀⢀⠞⠀⠀⠀⠀⢶⠆⠀⣀⣀⣀⡀⠀⢶⠆⠀⠀⠀⠘⢆⠀⠀⢱⠀⠀⠀⠀⠀⣏⡇
⠀⠀⠀⠀⢻⡉⢢⡀⠀⠀⣏⢉⡶⠯⣄⠀⢀⡠⠔⡊⠉⠉⠀⠀⠀⠈⠉⠉⠒⠤⣀⠀⢀⡬⠷⣎⢉⡇⠀⠀⣠⠊⡹⠃
⠀⠀⠀⠀⠀⠙⢄⡱⢄⡀⠈⠫⣀⡁⢊⠗⠩⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢏⠀⣁⡨⠋⠀⣀⠴⣁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠤⣹⠒⠒⠢⠽⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠅⠒⠒⢺⡡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣓⣒⣒⣛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣓⣒⣒⣋⣉⣉⣀⡀
⠀⠀⠀⠀⠀⢀⠴⢺⡥⠼⠦⠤⠤⠤⣤⡤⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⣤⡤⠤⠤⠤⠾⠤⣽⠢⢄
⠀⠀⠀⠀⠀⣷⠞⠁⠀⣀⠴⣋⣉⣉⣁⣠⠤⢜⣲⣄⣀⠀⠀⠀⠀⠀⢄⣂⣴⣛⡤⢤⣀⣉⣉⣉⡳⢄⡀⠀⠙⢶⡇
⠀⠀⠀⠀⠀⢿⠀⠀⢰⢧⠞⠁⠀⠀⡶⢿⡭⠥⠬⠷⠚⠋⠉⠉⠉⠉⠛⠲⠯⠤⠬⣭⢷⡄⠀⠀⠑⢦⢷⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠘⠇⠀⣼⢸⠀⠀⠀⠀⢱⠔⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠓⢴⠁⠀⠀⠀⢸⢸⡄⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢭⣢⣄⠀⠀⠀⠑⠮⠤⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠬⠖⠁⠀⠀⢀⣤⣪⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⡀
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣰⣶⣦⡀⠠⠄
⠰⠶⠲⠤⠤⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⡖⠐⠒⠂⠈⠉⠉
⠀⠤⠀⠀⠀⠀⠀⠠⠤⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠢⠽⠿⠟⠁⠀⠀⠀⠀⠀⠠⠤⠮⠤⠄⠀⠀⠀⠀⠀⠀⠤⡤⢤
⣤⣤⣤⣤⣄⣀⣀⡀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⣀⡀⢀⠀⠀⣤⣤⡄⠀⣤⣤⣤⣤
⣿⣿⣿⣿⣶⣿⣿⣷⣶⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⠄⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⣤⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠹⣷⣤⣾⢿⠯⠿⠛⣿⣯⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⠀⡒⡺⠟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠴⣚⠶⠿⢦⣶⣮⣽⣿⣿⣿⣿⣿⣛⢛⣿⣶⣦⣌⣛⢷⠶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⢤⣯⣛⣻⠿⣿⣟⣛⡛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⠒⠶⠒⠶⢶⣦⣴⡶⢦⣍⠉⢿⣿⣿⣿⡿⣛⢛⠉⣙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣛⣫⣾⣻⣿⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠤⠥⢤⣄⣿⣷⣮⡥⢶⣶⣤⣬⣹⣿⣿⣿⠷⠖⠂⠙⢿⣿⣷⣶⣮⣝⡛⠿⢿⣿⣧⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣀⣀⣀⣤⣤⣍⣙⣛⣛⡛⠛⠀⠀⠀⣊⣉⣙⡛⠛⢛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠚⠛⠛⠳⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣠⣴⡶⠷⠶⠦⣄⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣿⠏⠀⠀⠀⠀⠈⣧⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣀⣠⠤⠤⢤⣀⠟⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠛⢰⡇⠀⠀⢸⣿⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢽⠿⡇⠀⠀⢸⣿⠟⠋⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠄⢲⣳⡟⠊⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠶⢰⣯⣀⡀⢀⣀⣀⣀⡀⢀⠀⠀⠀⡀⣄⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣴⣾⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢻⡜⣿⠛⢻⠻⢿⣿⣿⡛⠋⠉⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣹⣦⣸⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⣿⡟⠳⣟⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⣿⡇⠀⠈⢧⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠸⢯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⣲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣷⡀
⠀⢸⠟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⣠⢳⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅⢹⣧
⠀⣾⣇⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⢻⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣽⢁⣎⣿
⠀⣿⣿⡈⢻⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⣄⣷⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣾⠃⣸⣯⣿
⠀⣿⣿⣷⡌⢿⣧⡵⠶⠀⠀⠀⠀⠀⠀⠀⢀⡠⣣⠴⠊⣵⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠃⣰⣿⣹⡿
⠀⠸⡿⣿⣿⣆⠳⣄⠀⠀⠀⠀⠀⠀⢀⣔⣣⠿⣇⡤⠆⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠙⢿⠃
⠀⠀⠹⠏⠘⢿⣧⡀⠑⠦⣀⠀⢀⡔⣣⠟⢁⡜⠉⠀⠀⢸⣧⣿⣿⣦⡀⠀⠀⢀⡀⠄⠂⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠉⠉⠙⠓⠊⠁⠀⠀⠀⠀⢸⣿⡟⠝⠛⠉⠀⠉⠁⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⠀⠀⢀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣷⣶⣤⣀⠀⠀⠀⠀⢸⣿⡫⣴⣦⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣿⠀⠀⠉⠛⠿⣿⣶⣤⡀⣸⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠷⠀⠀⠀⠀⠀⠉⠻⣿⣷⣿⡿⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢄⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠉⣆
⠀⠀⠀⠀⠀⠀⢀⡏⠿⠹⣽⣅⠓⠶⠆⠤⣀⡠⠄
⠀⠀⠀⠀⠀⠀⢀⡆⠀⠈⠙⠟⠤⢄⣀⠸⠏⢠⣾⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣠⢤⣤⡄
⠀⠀⠀⠀⠀⠀⣸⠐⠀⠀⠀⠀⠀⠈⠙⡀⢰⡄⠹⣿⢯⣟⡵⠟⠻⣷⡖⣶⣤⣶⢆⣉⠫⠻⠋⣹⣆⣭⣼⠾⣟⡏⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢽⣇⣌⣙⡇⣨⣞⢙⣻⣟⣿⣿⡟⠋⢀⣤⣸⡾⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠨⠿⢿⠟⠝⠱⢾⣿⣽⡿⣱⡼⣻⡿⠟⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡁⠤⡴⠢⠔⡾⢟⣻⠟⡯⠁⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣤⣤⣤⡌⠒⠀⠤⢀⣴⣿⣿⠟⣤⣴⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⢋⣠⣼⣿⡿⠏⠀⠐⠉⠀⠀⠀⢀⣼⣿⡿⠇⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣩⣟⣭⠋⢀⣴⠂⠀⠀⠀⠀⣠⣶⣟⠟⠋⠁⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠂⣠⡟⠙⠁⠀⠠⠜⠋⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢇⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢏⠟⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣾⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⡄⣻⣶⣤⣀⣀⡀⠀⠀⠀⣀⣤⠞⢉⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣦⣤⡎⢀⣀⣤⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢡⠇
⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣻⣿⣿⣟⣻⣯⣿⣿⣟⣾⣻⡿⣻⡟⣻⡿⣻⡿⣿⡿⣿⢿⣿⣿⣿⣿⣷⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⡴⠋⢠⠏
⢀⣠⡴⠾⠟⠛⣿⣶⡛⠀⠙⠻⣮⣉⣠⣤⣤⣥⣿⡬⢥⣬⣧⣭⣠⣟⣘⠛⠁⠋⠘⠋⠾⠇⠸⠟⠹⡿⢻⣿⣿⣿⡿⣶⣦⣤⣤⣾⣷⣶⡟
⠀⠙⢣⡀⣀⠻⠿⠟⠀⠀⢀⠀⣻⡟⡟⠓⠉⠉⠈⠉⠉⣁⣀⣨⠭⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠾⠿⢿⣿⣿⣻⡅
⠙⠻⢦⣥⣞⡃⠀⠀⠀⠀⣴⡀⣿⣓⣳⣤⣤⠖⠈⠉⠉⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⠉⠁⠉⠀⠀⠀⠀⠻⣝⠛⢻⡀
⠀⠀⠀⠀⠉⠛⠻⠶⣶⣮⣿⣿⡟⣛⣛⣿⣿⣿⣭⣭⣭⣵⣶⣶⠶⠶⢷⣴⣤⣶⣴⣶⣶⣾⠿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⣿⣿⣯⣭⣭⣭⡭⠽⠿⠿⠿⠟⠛⠛⠛⠻⣿⡻⠏⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⣷⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢩⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣀⣄⡀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⠤⠤⠴⠿⡿⠷⠤⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠷⠶⠤⣶⣶⣶⣶⡶⠴⠶⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣶⣶⣶⣶⣶⣴⣦⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⢦⠀⠀⠀⠀⠀⠀⠀⡇⣸⡇⠀⠀⠀⠀⠀⠀⢠⢦⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⣸⠀⢧⡀⠀⠀⠀⠀⠀⡇⢹⣷⠀⠀⠀⠀⠀⢠⠏⢸⡆⠀⠀⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡟⠈⣿
⠀⣿⡄⠉⣳⡄⠀⠀⠀⢀⡇⢸⣿⠀⠀⠀⠀⣰⡋⠀⣼⡇⠀⠀⣿⣿⣿⡆⠀⣹⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡋⠀⣸⣿
⠀⠻⠻⣄⠈⠦⣀⠀⢀⣸⠃⣼⣿⣄⠀⢀⣠⠞⢁⣾⠻⠃⠀⠀⣿⣿⣿⣿⣄⠈⠻⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⠟⢁⣴⣿⣿
⠀⠀⠀⠙⢷⣄⡈⠛⠛⠁⠀⣿⣿⣿⣭⣭⣤⣶⠟⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⡈⠛⠛⠁⠀⣿⣿⣿⣭⣩⣤⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠷⣶⣶⣴⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠢⣄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠱⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠐⣇⢀⡎⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣄⣠⣸⣶⠞⡵
⠀⠀⠀⠀⢠⠃⠀⠈⠙⠓⣅⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠈⠀⠫⠇
⠀⠀⠀⢀⡇⠀⠀⠀⠀⣀⡌⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢈⣰
⠀⠀⠄⢸⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣺⣻⠿⠓⠐⠻⠚⠻⢟⣻⡀⠀⠀⠀⠀⠀⠹⠇⠀⢄⣘⣿
⠀⢠⠀⢿⣶⠔⢀⡛⠟⠁⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⢠⡇⠀⠐⠀⣬⡇⠀⠀⢰
⠀⢸⠆⢸⡇⠀⠀⠀⡴⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢡⠀⠸⡅⠀⠀⠀⢰⠀⠀⠀⠈
⠀⠘⡄⢻⡅⠀⠀⠉⡇⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡄⣷⠐⠀⢤⡞⠀⠀⠰⡞
⡇⠀⠘⣼⢿⣄⡀⢀⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣯⣴⣟⠀⠀⠀⣰⠃
⢧⡀⠀⠘⢜⢿⣿⣿⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣭⣭⠎⠀⢤⡴⠃
⠈⢇⠀⠀⠈⠢⡙⣿⡿⠿⣧⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⣿⡿⠟⡁⠈⣠⠞
⠀⠈⢳⡴⠤⠀⠉⠻⢶⣶⣾⣷⢴⣶⡆⣄⡀⠀⣢⠀⣴⡤⢦⡀⠀⠀⡀⢀⣠⡂⣼⠖⢲⣮⠴⠋
⠀⠀⠀⠙⠒⠒⠒⠒⠛⠓⠶⠾⣿⣿⣿⣧⣦⡟⠁⠀⢻⠇⠀⠑⣰⣾⣿⢻⣷⠞⠛⠛⠋⣡⠀⠀⠀⠀⣠⣤⠎
⠀⠀⠀⠀⠀⠀⠀⣠⣧⣬⣄⣀⣀⣬⣿⣿⣿⡄⣀⣁⣈⣁⠀⣠⣾⡿⣳⣿⡛⠉⠉⠉⣛⡍⠛⠒⠛⠋⠁
⠀⠀⠀⠀⣰⣦⡴⠋⠀⣠⣄⠈⠉⣉⣙⣻⣿⣻⢿⣿⣿⡿⢿⣿⣻⣽⣿⠛⠋⠉⣙⣿⠛⠷⡀
⠀⠀⠀⢐⣨⡟⠀⢀⡞⢹⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠛⠉⠉⠉⠻⣆⠀⠈⣶⠆
⠀⠀⠐⣿⡋⠀⠀⢸⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢶⠀⠹⣄⡀
⠀⠀⠀⠈⠳⡀⠀⠈⣇⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠙⠀⠀⠘⠿⠶⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠒⠒⠒⠒⠶⠤⢴⣿⠠⠤⣄⣻⠀⠀⠈⠁⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢷⣄⢀⣸⣄⣤⣤⣶⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠰⢶⣶⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀
⠀⠀⠄⠁⠀⠛⣒⣲⣦⣤⣬⣿⣛⣿⣿⣽⣿⠿⣿⣿⣿⣿⣿⡿⣛⣉⣭⣤⣶⣿⣿⣾⣿⢶⡄⠀⠀⠥⠤⠤⠭⠄
⠀⠀⣠⣏⣉⡉⢙⢻⣯⣵⣮⣭⣿⣿⣿⡿⠯⠼⣿⣟⣁⣍⣿⣿⣿⣿⣿⣿⣯⠿⣿⣿⣿⣯⣭⣥⣤⠀⠐⠂
⠈⠉⣉⡙⠛⠛⠛⠛⠿⣉⣉⣁⣚⡉⠛⢿⣿⣟⠋⢽⣿⣿⡿⠛⠒⠒⠉⠉⠭⠯⠭⠿⢿⣿⡇
⠀⠀⠈⠉⠉⠉⠀⠐⠲⠦⢭⣉⣉⡀⠀⠈⠉⠉⠥⠬⠽⣿⠧⠀⠈⠙⠛⠚⢛⣋⣉⡭⠭⠉⠁⠀⠚⠛⠛⠓⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⠀⠀⠛⠃⠀⠀⠉⠉⠒⠒⠀⠀⠀⠈⠉⠙⠓⠚⠛⠋⠁

⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣠⣤⣤⠤⠤⠤⣤⠤⣤⣀⡀⣀⣤⣰⣦⣴⠆
⠀⠀⠀⠀⠀⠀⠀⣰⠿⡿⣍⠻⣗⡒⠰⣟⡳⢉⣽⣟⢿⣿⡿⣶⣭⣟⠯⣫⡿⢿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⢿⢸⠐⣽⡧⠹⡿⠑⠺⡻⡿⠻⠍⡣⢛⢿⣿⡿⣿⣿⣮⣔⢩⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢘⣏⣶⣘⣿⡀⠇⢠⢀⡇⠀⠀⠐⠀⢸⣆⡠⢃⣿⣿⣿⣿⣷⣻⣽⣿⡀
⠀⠀⠀⠀⠀⠀⣴⢛⣭⣍⣉⡉⠘⠒⠋⢙⣷⠦⠤⠤⢔⣘⣠⠔⣨⠀⢛⣹⣿⣿⣿⣞⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠦⣦⡐⠀⠀⠉⠉⣛⣯⣤⣤⣄⣤⣾⣉⣁⠚⣠⣿⣿⣾⣽⣿⡿⣿⡾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⣴⡖⠮⢿⣿⡿⢿⡛⠷⠊⠁⠀⠙⢿⣿⣿⡿⢛⣭⣬⣿⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⣴⣤⣄⣀⡀⠀⠈⠁⠀⠙⢛⣫⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣤⡀⠀⠙⣿⣯⡹⣿⠀⠀⠀⠀⠐⣿⢿⢿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡿⠓⠦⢼⣾⣿⣿⠀⠀⠀⠀⠀⣴⣿⣿⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⠀⠘⢿⣿⡄⣤⣄⡀⢰⣾⢷⡏⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⢀⣀⣀⣀⣀⣀⡼⠃⣽⡿⣷⣝⣯⣿⡾⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠿⣿⣿⠀⠀⢹⣤⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡯⢿⣿⣤⡐⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣟⣽⢿⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⣡⢿⡿⠁⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠸⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣝⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣶⣾⣿⣷⣶⡶⣦⣠⣄⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠟⣛⣹⣿⣿⣿⡿⣿⣶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡸⢣⠟⠛⡟⠛⠓⠰⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣾⢠⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⡀
⠀⢀⠀⢠⡄⠀⠀⢠⣿⣿⡿⠁⣶⡿⣆⠀⠀⠀⠀⣶⡶⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣧⠐⠟⢼⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣥⠠⡄⠠⠚⢫⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⢸⠷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡶⣣⣴⣧⠈⡟⣿⡿⠟⢹⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⣿⣿⣿⣸⣀⠀⠀⠑⠘⠉⣛⠽⢟⣵⣦⠀⡀
⠘⠻⠯⢭⣷⡶⠿⠿⠿⠿⣿⠿⣿⣿⣿⣷⣿⢻⣤⣤⣶⣿⣧⣦⣿⣿⣷⣿⣷⣿⡷⢦
⠀⠀⣀⠀⠉⠙⠳⠷⠶⣶⣤⣤⣤⣭⡟⠟⣿⣿⣿⣿⡇⣿⣿⢟⣿⠯⣽⣿⡻⢷⣶⣿⣷⣄
⠀⠀⠙⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣧⣶⡬⠿⣿⣿⣿⡿⣟⣿⣿⣿⣯⣽⣟⠓⢬⣙⣛⠿⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣥⣶⣄⣬⠤⠈⠉⠀⠀⠉⠛⢿⣿⣿⣿⠷⣮⣟⣛⠧
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡫⠭⠷⢶⠀⠤⠄⠒⣀⠄⠈⠈⠷⡏⣿⣐⣈⣽⠳⠇
⠀⠀⠀⠀⠀⠀⠈⠙⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣬⣥⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠁⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⠉⠉⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠄⠈⠀⠁⠀⠉

⠀⠀⣴⡒⠶⢤⣀
⠀⢸⠁⣌⠑⠢⡉⠓⢦⡀
⠀⢸⣿⣸⡠⠄⠘⢶⡀⠉⠳⣄
⠀⠀⢻⣿⣿⡜⣥⠀⠈⢳⣠⡈⠳⣄
⠀⠀⠀⢻⡜⢷⠜⢷⣄⠘⣿⡻⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠻⣦⡣⢡⣻⢷⣎⠋⣦⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆
⠀⠀⠀⠀⠀⠙⢷⣸⣿⣷⠊⠳⡮⢻⠂⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇⠳⣜⠢⡁⣇⠀⠀⠀⠈⠛⠳⣤⡀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠻⣿⣯⠻⡄⠂⢹⠀⠀⢱⡄⠀⠀⠀⢉⠓⠶⠤⣄⣀⣀⣀⣨⠇⠀⠀⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⠻⣝⢮⣂⢼⡇⠀⠀⢹⡄⠀⠀⠀⠈⠑⠒⠀⠬⠉⠉⠀⣀⣠⣤⣀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢰⡏⠳⢿⣽⡇⠀⠀⠀⢷⢠⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠈⠙⠲⠤⠤⠼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⡇⠀⠀⠈⣧⠀⠀⠀⢸⡆⠈⠳⠦⠤⢴⣒⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠈⡇⠀⠀⠀⠸⡆⠀⠀⠀⡷⠶⠶⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢻⡀⠀⠀⠀⠹⣆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠙⣇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠾⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⢚⣳⠶⠂⠀⠀⠀⠀⠐⠶⣟⡓⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⢀⡴⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⢦⡀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠞⠁⠀⠀⢀⡴⠋⢀⡞⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⢳⡀⠙⢦⡀⠀⠀⠈⠳⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢻⡁⠀⠀⠀⢀⡞⠀⣠⠟⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠻⣄⠀⢳⡀⠀⠀⠀⢀⠟⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠈⠁⠀⠀⠘⠷⠞⠁⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⠳⠾⠃⠀⠀⠈⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⢀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢯⡖⠒⢦⡀⠀⠀⠀⠀⢀⡴⠒⢲⡽⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡀
⠀⠀⠀⠀⠀⠀⡿⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢸⠀⢀⣼⠃⠀⠀⠀⠀⠘⣧⡀⠀⡇⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢻
⠀⠀⠀⠀⠀⠀⡇⠀⠈⠒⠀⠀⠀⠀⣀⣤⠶⠋⠀⠀⢸⡆⣸⠃⠀⠀⠀⠀⠀⠀⠘⣇⢰⡇⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠐⠁⠀⢸
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣾⠉⠁⠀⠀⠀⠀⣀⣀⣷⡼⠤⠶⠶⠶⠶⠶⠶⠦⠯⣾⣀⣀⠀⠀⠀⠀⠈⠉⣷⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⠀⠀⡽⣄⠀⣀⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣀⠀⣠⠏⠀⠀⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠈⢻⠏⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠹⡟⠁⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠚⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠓⠒⠦⣤⣀
⠀⠀⠀⠀⣠⠞⢉⡴⠷⠶⢶⣶⣶⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣶⣶⡶⠶⠾⢦⡉⠳⣄
⠀⠀⢀⡞⢁⡴⠋⣀⡴⠚⠉⠁⣀⣠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⣀⠈⠉⠓⢦⣀⠙⢦⡈⢳⡀
⠀⠀⣿⣹⠋⢀⡾⢋⡾⠓⠛⣩⡽⠋⠉⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠉⠙⢯⣍⠛⠒⢷⡙⢷⡀⠙⣏⣻
⠀⠀⢿⠏⢠⠟⢀⡞⠁⢠⠞⣁⣠⢴⡾⠋⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡋⠙⢷⡦⣄⣈⠳⡄⠈⢳⡄⢻⡄⠹⡿
⠀⠀⠀⠀⡾⣴⠏⠀⣴⠋⢠⡏⢠⡞⣥⠖⠋⠈⠳⣶⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⠞⠁⠙⠲⣬⠳⡄⢹⡄⠙⣦⠀⠹⣦⢷
⠀⠀⠀⠀⢷⣿⠀⡼⣃⡴⠋⢠⡏⢀⡟⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠒⠒⠒⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⢻⡀⢹⡄⠙⢦⡘⢧⠀⣿⡾
⠀⠀⠀⠀⠈⠃⠀⣧⣸⠀⠀⡾⢤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡤⢷⠀⠀⣇⣼⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠻⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠟⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠐⢂⡄⠉⠉⠉⠁⠐⠒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣊⣥⣄⣀⠀⠀⠺⣿⢶⣿⣿⠄⣠⣶⣶⠀⠀⠉⠑⠢⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣼⣿⣿⣿⣿⣧⠀⢀⣠⣀⡠⢋⣾⣿⣿⡿⣣⣼⣿⠟⢠⣄⣤⣬⡉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡴⠀⣼⠟⡿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠙⠉⠉⠋⣾⣿⣿⣿⠙⠿⠿⠜⠛⠛⢂⠤⣜⡂⠤
⠀⠀⠀⠀⠀⠀⠀⣰⣫⣾⣿⢃⣼⡟⡌⠁⠋⢻⣿⣿⣿⠏⢰⠰⣴⡀⠀⠰⠀⠁⢀⠴⠒⠒⠒⠒⠀⠀⠀⢴⣿⣿⣶⡶⠖⠒⢷
⠀⠀⠀⠀⠀⠀⡼⣱⣿⣿⣷⣾⠛⢡⠀⠀⠀⠸⣻⣿⠃⢀⡟⣧⣿⣧⡀⠀⠀⠠⠃⠀⠀⡀⠊⠐⠀⣠⣶⣿⣿⣿⣿⠿⣆
⠀⠀⠀⠀⣠⣴⣷⣿⣿⣿⣿⡏⠀⢠⠀⠀⡀⡀⠈⠃⢰⣼⡇⠸⣿⣿⣷⢀⠀⠀⠀⡠⠊⠉⠀⢢⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰
⠀⢀⡴⢋⠝⣫⣿⣿⣿⣿⣿⠇⠀⢸⡀⠀⠱⡐⣄⠀⢸⣿⣿⣄⠈⠛⠿⣿⣦⠀⢸⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢠⡏
⣰⠋⠀⠀⠊⣡⣿⣿⣿⣿⡵⠂⢀⣾⡇⠐⢄⠘⠮⡳⢎⢿⣿⣿⣷⣦⣄⣀⠈⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⣿⣿⣿⡃⡜⠀⡞
⢻⣀⠀⠀⠀⠺⣛⣭⠞⢁⣠⣶⡿⠃⠘⡄⠈⠓⢤⣘⣆⠳⢍⡻⠿⠿⠿⢿⣿⡦⠀⠀⠀⡈⠉⠒⠄⡈⠙⠻⠿⣿⠞⠀⡸
⠀⠈⠙⣶⡾⠟⠋⠀⠀⣿⡟⠉⠀⠀⠀⠈⠦⡐⢤⣈⠉⢅⣀⠈⠉⠉⠉⠐⠢⣄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠤⣉⡒⠇
⠀⠀⢸⡇⠀⢀⣴⣾⣷⠋⣽⣗⡀⡀⢀⣠⣤⣌⡒⠤⣉⠉⠩⠕⠲⣄⡀⠀⠀⠀⠉⠒⠦⠤⠤⣤⣀⣀⣀⣀⠀⠀⠀⠉⣢⠁
⠀⠀⠘⣇⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣷⣤⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠭⠀⠀⠐⠊⠁
⠀⡴⠋⠁⠩⠾⠟⣻⣿⣎⣝⣛⣷⣆⣀⣈⣉⠉⠛⣿⣿⣿⣿⣿⠀⣶⣤⣄⣀⣤⡤⠤⠄⠒⠈⠁
⠾⠒⠒⢶⡠⡄⣍⠹⣿⠟⢿⣿⣿⣿⣫⡞⠁⠀⢸⣿⣿⣿⣿⡏⢸⣿⡟⢿⡿⠙⠀⢀⠁
⠀⠀⠀⠀⠳⡱⣌⢷⡅⠀⠸⢟⢭⣻⡿⣷⣄⠀⠀⠈⠙⠿⠟⢀⣿⣿⣿⢸⣷⡦⢔⡀⢀⣠⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⣠⣤⠄
⠀⠀⠀⠀⠀⠙⢾⡶⠇⡠⠶⣶⣦⣬⣉⡛⠻⢷⡀⣤⣀⠀⠠⣿⣿⣿⡟⣼⣿⣷⢰⣍⣾⣿⣿⣿⣿⣿⣿⠧⣀⡴⣨⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠂⢬⣛⣶⣮⠉⠛⠛⠛⠚⢻⣦⣝⢷⣤⡀⠈⠉⠰⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⡇⢰⢋⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣶⣦⣽⣿⣿⣿⣿⣶⣄⣀⠈⠁⠀⠹⠟⠙⠿⢿⣿⣿⣿⢧⣓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠁⠙⠋⠁⣠⠞⠉⠻⢿⣿⣿⣿⣷⣦⣄⡀⠀⣠⣴⣿⣿⣿⠛⡗⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⡼⠁⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠹⡙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣯⣻⣿⣷⣿⡏⠒⠄⢹⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣝⢷⡝⢟⢮⡋⠓⠤⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠠⣀⠙⠦⡀⢨⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢔⡀⠑⢄⣸⠿⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⠅⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠉⠉⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠔⠈⠁⠢⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⡐⠂⠒⠓⠀⠬⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⠒⠵⠤⠄⠀⣀⡐⠁⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠒⢁⠶⢀⢀⡠⠁⠈⢀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠄⠂⠀⠐⠐⠀⢦⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⢅⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠄⠊⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠋⠀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠁⠀⠀⡄⠂⠁
⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠀⡠⠐⠉⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⡔⠀⢷⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⡘⠀⠀⠈⠕⠄⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⢢⠂⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠌
⠰⠁⠀⠀⠀⠈⢆⠂⡀⠀⡇⠀⠰⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡰⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠜⠀⢀⠌
⡀⠀⡀⠀⠀⢔⢦⠓⠁⠀⢁⠀⠀⠂⢀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠊⠤⡤⠀⠀⠀⠀⡃⠀⠀⠀⠀⡠⠂⠀⠠⠂
⠗⠈⠱⠀⠀⠀⠢⡑⢄⠀⠀⠄⠀⢈⡎⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⡠⢀⠔⠁⠀⠀⠉⠐⠂⠀⠀⡠⠊⠀⢀⠔⠁
⠀⠀⠀⠱⡀⠀⠀⠈⠐⠬⠀⠈⠗⠋⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⣀⠀⠥⠒⠁⠀⠀⠀⠌⠀⠀⢀⠠⠊⠀⢀⠀⠁
⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠠⠄⠊⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⡀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠊⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⠙⢿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣠⠶⠒⠒⠲⢤⡀⠀⠀⣀⣤⣄⡀⢘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⠒⠋⠁⠀⠀⠉⠛⠋⠙⠻⣿⣿⣿⣄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢤⣀⡞⠙⢦⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⡴⠋⠉⠙⠳⣄⡼⠁⠀⠀⠀⠙⠓⠲⡄⠉⠙⠲⣤⠀⠀⠰⠿⣿⣿⡿⠟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠐⢦⠴⠃⠀⠀⡴⠋⠁⣀⣠⣤⣤⣤⠄⠀⠀⣿⠀⠀⠀⠈⢳⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⡀⠀⠀⣠⠞⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⢧⣠⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿
⠿⣍⣛⣉⣁⡴⠊⠀⣀⣀⣀⣀⣀⣤⠖⠋⠁⠀⡴⠚⠲⣄⠀⢀⣽⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⡿⠟
⠀⢀⣨⣭⣤⣤⠴⠚⠉⠀⠀⠀⠀⠈⠙⢦⡀⠈⢧⣀⡀⣈⣩⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣀⡴⠒⠒⠦⣀⠀⠈⠳⢤⣀⣀
⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠙⢄⣀⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⠀⠀⠈⠳⠦⣤⣤⠤⣴
⠀⢀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠿⠟⠛⠛⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣻⣿⣿⣿⣿⣷⣶⣤⣤⣶⣿⣿
⠀⢸⣷⣶⣶⣶⠾⠟⣋⣡⣴⣶⣶⣿⣷⣶⣶⣤⣄⡈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣿⣿⠿⠛⢉⣡⣤⣤⣤⣤⣤⣌⣉⠙
⠀⠀⠉⣉⣥⣤⣶⣿⣿⣿⣿⣿⣿⣿⢯⣭⣭⣤⣬⣥⣤⣤⣄⡉⠛⠻⠿⠿⠿⠟⠛⠉⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠘⠟⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⡿⠿⠿⠿⠿⢿⣿⠷⠶⠛⣣⣴⣿⣿⣿⣿⣿⣿⣦⠤⠤⠶⠿⠿⠟
⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠙⠷⣶⣶⣶⣶⣶⣶⠶⠾⠿⠿⠿⠿⣿⠿⠛⣉⣤⣤⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠙⢿⣿⣿⡿⠟⠛⠛⠻⣿⣿⣿⣿⣷⣦⣄⠀⠰⣶⣶⣶⣶⣶⣶⣶⠶⠟⣁⣤⣶⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢳⣴⠶⠶⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣬⣭⣭⣭⣭⣤⣶⣿⣿⣿⣿⣿⣿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣦⣤⣤⣀⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠛⠳⠦⠄⠉⠛⠻⠿⠿⢿⣿⣿⠿⠿⠟⢛⣉⣤⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣶⣾⣿⣷⣦⣀⠛⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣭⡛⢶⣄⣠⠾⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⢿⣿⣿⠶⠋⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠁⠀⠀⡾⠃⣠⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀⠀⠀⢷⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣿⡿⠛⠋⠀⣠⣏⣀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⣱⣾⣯⣭⣭⣤⣴⠟⠉⠀⣠⣤⣶⣿⣯⣭⣭⣭⣭⣛⣶⣿⣿⣌⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠿⢿⣿⣿⡿⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⠁⢠⡞⢹⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠰⡇⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣿⣿⣿⣿⣿⡷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠙⢲⡄⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠙⣇⠀⢷⡀⠀⠀⠀⠀⢀⣤⠞⣻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡄⠈⢓⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠸⡆⠀⣷⠀⣀⣤⣾⡛⢁⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⢿⣿⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⡧⠀⢸⠀⠈⠉⢉⣽⣿⡿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⡷⠀⢸⠂⢀⣴⣾⣿⠟⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡙⢦⣀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⡇⠀⠀⠀⣰⠃⠀⣼⠖⣻⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣌⠳⢤⣠⠞⠋⠉⡉⠉⠙⠛⠿⠷⢤⠴⠞⠁⢀⣼⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣉⠓⠾⣏⣻⣿⣷⣶⣤⣤⣠⣤⣤⣶⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢿⣿⣿⣷⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠲⠛⠁⣠⡼⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠦⠶⠛⠉⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣤⣤⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡟⠀⠀⢹⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠈⠲⠖⠋⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡶⢶⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⣀⡀⣀⣸⡇⢸⣇⣀⢀⣀⣴⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⣀⣀⣠⣭⣭⣍⣹⡇⢸⡏⣉⣭⣭⣅⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⢹⡇⢸⡏⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣿
⠀⠀⠀⠀⠀⠀⢀⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡄
⠀⠀⠀⠀⠀⢀⡎⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠸⡄
⠀⠀⠀⠀⠀⣸⠁⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢷
⠀⠀⠀⠀⢀⡟⠀⠀⢄⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⣠⠀⠀⢸⡆
⠀⠀⠀⠀⢸⡇⢀⡄⠘⣧⡴⠾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠶⣴⠏⠀⠀⠘⡇
⠀⠀⠀⠀⢸⣧⠞⣧⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⣸⠓⢦⡇
⠀⠀⠀⠀⠈⠀⠀⠘⣧⡀⠀⠈⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣠⡴⠟⠉⠀⠀⣰⠏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠈⠙⠛⠷⠶⢶⣤⣤⠁⠀⠀⠀⣧⣤⡴⠶⠾⠛⠋⠁⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⡀⠀⠀⠀⡞⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣭⣥⣬⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣷⣶⣾⣿⣿⣶⣶⣶⣶⣾⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⣿⡿⠷⢦⣕⠀⣀⣿⣿⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣼⠇⠈⡿⠀⠀⠘⠆⢰⠟⢡⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠰⡀⠀⠀⠈⠀⢸⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣤⣿⡾⠦⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢸⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠟⠉⠓⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣼⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⢀⣀⣄⣀⣻⣤⣴⣶⣿⣿⡿⣿⣷⣯⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣲⣦⣶⣿⣷⣾⣿⣾⣿⣽⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠁⠉⠲⠿⠿⠦⠬⠭⡍⢉⣉⣉⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠔⠒⠀⠀⢀⣠⣤⡴⠂⠄⠀⠀⢀⣀⡠⠭⠄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡀⠁⠠⠤⠀⣀⣐⣲⣦⣤⣤⣤⣤⣄⣀⣤⣤⣤⣤⣤⣶⠶⠿⠛⠁⣠⢄⠀⠀⠀⠪⠹⠭
⠀⠀⠀⠀⣀⣀⢀⣀⣀⡠⠭⠵⠶⠶⢚⢛⡛⠛⠽⠟⠛⠋⠉⠉⠉⠉⠈⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣳⣷⠶⢦⠀⠀⠐⠄
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⣤⣖⣲⡶⡺⠿⠿⠟⠛⠋⠀⢀⡞⠀⠀⠀⠀⠄⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠛⢛⣉⣁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠞⢫⡀⠀⠀⠀⢀⡠⠔⠋⡀
⠀⠀⠐⠒⢀⣉⣉⣉⣙⣉⣉⣭⠤⠤⠶⠟⠛⠛⠋⠉⠉⠈⠃⠛⣉⣀⢀⣙⣩⠽⠛⠛⠛⠛⢛⣫⠭⠉⠓⠋⠉⠒⠻⠧⠒⣒⠉⠁⠀⢀⣀⣱⠄
⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠴⠖⠒⡋⠉⠅⠐⠊⠉⠀⠀⢉⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢸⣶⣶⢷⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠒⠀⠀⠀⢀⣠⡤⠤⠒⢈⠄⠙⠂⠀⠀⠀⠀⠀⠀⢀⡤⢮⣱⣎⡿⢟⣓⣿⠉⢀⣉⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠒⠒⠒⠒⠛⠛⠉⠁⠀⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠐⠀⠂⠉⠰⠏⣥⠼⠃⠀⠉⠉⠁⠘⢋⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⣿⡿⢿⡿⣿⣿⣿⡟⠟⢻⣷⣿⠃⠀⠀⣀⣀⣀⣀⣤⣤⣤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡉⠉⠉⠉⠁⠀⠀⠀⠺⠋⠉⠀⠀⠀⠀⣴⣶⣶⣶⠞⠁⠀⠙⠻⣋⣍⠁⢸⠿⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣿⣿⠓⠶⠂⠀⠀⠀⠀⠔⠀⠀⣠⣿⢀⡄⠀⣰⠟⠛⠉⢄⠀⠀⠀⠀⠁⠀⠶⠦⢷⡄⡀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠯⠽⠿⠿⢭⣟⣿⣿⣿⣿⢻⣷⣤⣤⣤⣤⣤⣊⣀⣴⣿⠿⣷⡿⠓⠉⢀⣀⣠⡴⠿⠤⠤⠤⢄⡀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⢀⣔⣾⣁⠇⠀⠀⠀⠀⢀⣾⣿⣿⠿⢣⠿⠟⠈⠓⠂⣩⠇⣀⣀⣴⣇⣤⣠⣦⣠⠾⠋⠉⠁⠐⠒⠒⠛⠓⢺⣿
⠀⠀⠀⠀⢠⡟⢁⣿⡏⠀⠀⠀⠀⠀⢈⡿⠛⢹⣿⣷⣾⡻⢋⡽⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇
⠀⠀⠀⠀⠈⠀⠸⣿⠁⠀⠀⠀⠀⣠⣿⡄⢰⣺⣿⣿⡤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣰⣿⠛⣧⣸⠛⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠃⠀⠈⣻⡀⠀⡟⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠟⢆⠘⢦⡙⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡄⠀⠁⠒⠚⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣙⡛⣉⣙⢛⣟⣋⡿⢻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⣀
⠀⠀⠀⢻⡿⠛⠟⠛⠻⠿⠿⣶⣶⣶⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⣿⣿⣯⠟⠚⠓⠒⠶⠦⡄
⠀⠀⠀⠀⠻⣶⣀⠀⠀⠀⠀⠀⠀⠹⣯⣽⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⠀⠀⠀⠀⠈⡉⠙⠻⢿⡿⠟⣿⣿⣿⣿⣿⡶⣶⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡻⣷⠶⡠⠤⠴⠤⣀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣯⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣖⡀⣀⠚⠊⠀⠲⠄⠸⣿⣿⣿⣿⡟⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⣿⡷⠮⢶⣦⣤⣄⣀⣨⣿⡛⢻⣷⣷⡷⡆⠀⠉⠉⠉⣉⠩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡿⠃⠀⠀⠈⠉⠛⠛⠿⠿⠿⠷⡌⠻⣿⣾⣿⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣌⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣲⠽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠚⣟⣵⣋⣁⣀⡷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⢬⣙⠲⢤⣈⡩⠤⠶⠒⠒⠒⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⡀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠈⣹⢂⡇⣀⣀⣠⠤⢤⡄⡖⠒⢮⣳⡄⠀⠀⠀⠀⠀⣿⠤⠶⣈⣨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠾⣍⣓⠦⣤⠇⣼⢰⠁⢸⣿⠀⠀⢷⠹⣄⣤⠷⠙⢦⠀⠀⠀⠀⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢲⣟⣉⣀⣀⣀⣤⡬⠽⠶⣤⣇⡘⠒⠚⠉⣉⣉⣉⣠⣤⠤⠤⠤⠘⠳⣄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠋⣹⣋⠔⠊⠁⣸⠉⠉⡏⠀⠀⢱⡀⠀⠘⢦⢻⡉⠉⣉⣉⠉⣀⣀⠀⢠⠤⡀⢰⠒⡌⠳⣤⢼⣧⣤⡀
⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠻⣄⡀⠀⠀⠇⠀⠀⢃⠀⠀⢀⣇⣀⡠⠜⠊⢧⣀⣓⣚⠀⠓⠚⠀⠘⠒⠁⢈⣉⣀⣀⣬⠧⠔⠋⢻⡄
⠀⠀⢠⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣻⡿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣉⡉⣉⣭⣍⢩⠵⢦⠀⠀⠀⠀⢀⡹⣤⣀
⠀⠀⠈⠙⢿⣟⡛⠛⠛⠛⠛⠛⠋⠉⣩⣭⠤⠴⠖⠒⣦⠀⣀⣀⣀⢀⣤⣤⡄⣴⠶⠖⠒⠛⠛⡇⣏⣉⡇⢧⣤⠼⠘⠒⡚⡁⠀⣠⣤⣼⣷⠟⣸
⠀⠀⠀⠀⠀⠈⠙⠓⢦⣄⣀⣀⠀⠀⠀⠙⠲⣄⠀⠀⢘⡆⢳⣀⣸⡏⣧⡤⠇⠸⠶⠒⠒⠛⠋⠁⠈⠁⠀⢀⣰⣲⣿⣾⡶⠛⣟⢉⡤⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠉⠙⠛⠿⢶⣤⡀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣀⣠⣶⡶⠶⢶⢿⣿⣿⠿⠧⠨⠥⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡃⠀⠀⠀⠀⠀⠉⠓⠶⣦⣴⣿⠶⠶⠶⠶⠶⣿⡟⢿⣭⣽⣿⣾⡷⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠳⠶⠞⠛⠿⠟⠂⠸⠦⠔⠶⠖⠲⠿⠝⠛⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠈⠛⢿⣶⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⢸⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⠟⠻⢷⣦⣄⡀
⠀⠀⠀⠀⢻⣧⠈⠉⠻⢷⣦⡀⢀⣠⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠈⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢀⣴⣦⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⢈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠛⠛⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⢀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⣴⡟⠁⣀⣤⣾⠟⠋⠈⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⢸⣿⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣦⣴⡾⠟⠋
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢀⣠⣴⡿⠻⠿⠿⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣾⡿⣿⡿⣻⠟⢿⣿⣽⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣶⡿⠿⢟⠛⢉⢥⣴⣻⡿⠻⢽⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢡⣿⡟⢡⣲⣾⣿⣮⣿⠟⠉⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣠⣾⣟⡥⢙⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⣀⢻⣿⣿⣿⣿⣀⣤⡶⡾⠒⠄⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⡄
⠀⠀⠀⠀⠀⠀⡠⠂⣀⣤⣿⣿⠿⠀⢐⣾⣿⣿⡿⠟⠟⠉⢁⣀⣴⣀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠐⠉⠀⠀⠛⠉⢋⡀⣄⣶⣾⣿⣿⢯⠙⠐⠊⠉⠛⠿⢿⣿⣦⣀⣀⣠⣴⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢺⣿⣧⣧⣾⣿⣿⣿⣿⣿⡻⠆⠀⠀⣠⣴⣄⣻⣿⣿⣿⣿⠟⠋⠀⠋⠙⢻⠟⡁
⠀⠀⠀⠀⠀⣠⣶⣾⣿⣛⣛⣿⣯⣭⣿⠉⢂⠁⠀⣀⣸⣿⣿⣿⣿⣿⡿⠛⠂⠀⠀⢀⡤⣴⣾⠿
⠀⠀⠀⠀⠀⢻⡿⠿⢿⣿⣿⣿⡿⣯⡉⠉⠀⠀⢺⣿⣿⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠀⢠⣶⣦⣄⡀⠀⠅⠑⠀⠀⢀⣠⡀⣀⣦⣾⣿⣿⡟⠎⠙⠻⠷⠀⠀⠀⠀⢀⣤⣶⡿⠃
⠀⠀⠀⣿⣿⣿⣿⣷⣢⣦⣿⣼⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⣀⣚⣫⠭⠁
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠊⠈⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠐⡋⠐⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢪⠖⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣁⣈⣀⡰⢁⡎⠀⣀⠀⠀⠀⠘⠂⣀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡞⠛⠙⠽⡯⠙⠙⣿⣿⠿⢷⣤⣀⡠⠀⠀⠀⠠⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⣙⣾⢄⠀⠀⠠⠁⠚⡉⠨⢢⡙⣐⠒⢹⠕⠶⣤⠴⠀⠀⠀⣠⢏⡔⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠐⠒⠀⢀⣴⡿⣙⣍⣵⠋⢠⡔⠄⢢⠀⠀⠀⠀⠝⠀⠀⠀⠱⠌⠊⠀⢤⣀⣴⣷⡿⢀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⣤⡤⠀⢀⣴⣯⣿⣟⠛⠿⠷⠤⠟⠛⠦⠀⠀⣄⢀⠤⠀⢧⢪⠔⠁⠀⠀⠀⠸⢏⢻⣿⣿⡀⠀⠀⠀⠀⡠⠖
⢀⠏⠀⢔⡕⣊⠒⣍⠿⠀⢀⡶⠀⠀⠈⠀⠠⠄⠋⠅⠰⠈⠊⠼⠇⡠⠀⠀⠠⠠⣀⣾⣿⡋⠁⠀⠀⠀⢸
⡋⢣⠀⠸⡘⠿⢀⡾⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠨⠡⠁⠡⢠⢁⣠⣀⡶⠖⠒⣶⣾⣿⣯⡁⠀⠀⠀⠀⠈⢣
⠑⠈⠓⠀⢈⢙⡉⠄⠀⠀⠀⠀⢀⠘⡠⡀⠂⢀⡤⠰⣢⣌⣋⣰⣽⠏⠱⠈⠀⠸⣿⣿⠎⠓⠄⠀⠀⠀⠀⠐⢆
⠀⠀⠲⢄⠀⢠⠀⡀⠀⠠⠠⠐⢁⣼⡇⡀⠠⠀⡰⠀⣶⣿⣿⣿⡟⠀⠀⠀⠒⠒⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠉⠒⠄⣀⣀⣤⣖⣄⢺⣿⣧⣈⢀⣀⣠⡾⣿⣿⣿⠏⠀⠀⠀⠀⠀⣸⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⠳⡿⣿⣿⣿⣷⣻⣿⣿⡿⠟⠀⠀⠀⡐⠀⢠⠇⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠙⢿⣛⠛⠛⠛⠿⠛⠦⡀⠠⠀⠈⠀⠀⢸⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠡⠀⠀⠀⠁⠙⠲⠤⡀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠡⡀⠀⠀⠀⠁⠀⠀⠀⠰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⢆

⠀⠀⠀⠀⠀⢀⣠⠤⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢼⣏⠀⠀⠀⠀⢀⣠⣾⣾⣿⣿⣶⡴⣖⣠⣴⣾⢧⣍⣩⣭⣿⡿⣿⣿⣦⣄⡀⠀⠈⠳⣦
⠀⠀⠀⣠⡼⢿⡆⠀⢠⢰⣿⣿⣿⣿⣿⣿⣿⣝⢿⣷⣿⣿⣿⡿⠉⣹⣿⡟⣻⣿⣿⣿⣿⣷⡤⡔⣾⢤⡀
⠀⠀⠀⢹⣧⢞⢿⣴⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣾⢿⣿⡞⣿⣿⣾⣿⣼⡅⣼⡟
⠀⠀⠀⠀⣽⣧⣃⡹⣿⣿⣿⣼⣿⣾⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢄⣽⣿⣿⣇⣴⣯
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⢿⣿⣾⣟⣿⣞⠁
⠀⠀⠀⠀⢀⣀⣤⣤⣼⣯⣿⡧⣭⡲⣝⡻⣿⣷⡙⢟⣿⣟⡛⣣⣿⣿⡿⣋⢻⣏⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⣴⣿⣿⡿⠻⠿⠛⠁⠀⠻⠿⢾⢶⣾⢿⣿⣿⣿⣿⣿⣿⣿⢟⣴⣽⠿⠟⠀⠀⠀⠀⠀⠀⠈⠙⠛⠟⠟
⠀⠀⠰⠟⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠤⡀⠀⠀⣖⣚⣻⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣰⣿⡿⣿⣿⣿⠟⠁⢀⣀⣀⣀⣀⠀⢀⠄⠀⠙⠿⣿⣯⣷⣷⡄⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⢿⣿⣳⣭⣥⢀⣤⣶⣾⣿⣿⣯⣿⣿⣿⣾⣦⣀⣤⣴⣶⡿⡿⠃⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⡹⣿⣿⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣠⣶⠿⠋
⠀⢀⡴⠾⠿⠿⠒⠾⠷⠶⣭⣟⣷⣿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⣟⣻⣥⣷⣶⠾⠗⠿⠷⠶⢦⡄
⠀⠈⠀⠀⠀⣀⣤⣤⣴⣶⠴⣦⣭⣽⣟⣛⡿⣿⣿⣻⣼⣿⣏⣿⣿⣿⣿⣯⣾⣿⣿⣿⡿⠶⠶⣦⣤⣄⡀⠀⠀⠙
⠀⠀⠀⠀⢸⠋⠉⠀⠀⣀⣤⢰⣶⣦⣿⣯⣥⣅⠾⠟⣿⡿⣿⡟⠩⢶⣿⣿⣿⡿⢿⣠⣴⣤⣀⠀⠙⠻⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⡿⠷⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠛⠿⣿⣿⣇⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⠻⣾⣿⣿⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡏⣽⣟⣷⢀⣤⣶⣶⣦⣤⣴⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠿⣦⣤⣄⠤⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⢀⠀⣠⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢟⣿⡿⠿⠿⠯⠭⣽⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢦⣤⣬⣭⢶⣶⣾⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠛⡋⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⣿⣶⣶⣂⠀⡀⣠⣴⣿⣿
⠀⠀⠀⠀⢶⣮⣥⣤⣔⣶⣲⠒⠶⠶⣦⡤⢄⣤⣤⣤⣄⣀⣀⣀⠀⢀⣀⡀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⣤⣥⣄⣩⣉⣙⣛⣛⣻⣷⣶⣿⣾⣷⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⠟⠱⠃
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡟⠛⠻⣿⣿⣿⣿⣿⡫⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠏⠀⠀⠀⠉⠉⠛⠻⠿⠿⣿⣷⣮⣍⡻⣿⡆⠒⠀⢘⡿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⣶⣶⣿⡿⠟⠛⠃⢹⣿⣶⡿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠟⠛⠋⠁⠀⠀⠀⠀⠈⢿⣷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣾⠗

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣿⣶⣾⣿⣿⣿⣿⣿⣽⣿⣿⣯⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠀⠈⠁⠀⠛⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⢚⠉⠉⠛⠋⠀⠉⣩⡭⢉⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠈⠀⠀⠀⠀⢐⣊⣠⣭⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣀⣂⣰⠇⡬⠐⠨⠉⣠⣴⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠞⠛⠻⠿⡿⠡⣤⣀⣴⣶⣾⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠞⣡⣤⣶⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⣡⣾⣿⠟⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡄⣸⣿⠋⠘⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡷⣇⠆⢂⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠸⣜⢏⠀⠈⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡁⠀⠀⢻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠙⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣮⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠉⠈⠙⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠆⠰⠀⠄⠰⠐⠶⠴⠀⠴⠀⠜⠄⠄⠄⠆⠘⠋⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢋⠝⢋⡹⠿⣦⡀⠀⠀⠀⠀⠀⣠⡶⡿⢛⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⡁⡰⢁⠔⣁⠔⢊⡭⠟⢶⣦⠀⢠⡾⢫⠊⢀⠼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠾⠿⠿⠷⠷⢧⣾⣡⣞⡡⠖⣻⡟⠁⢠⡟⣰⠃⢠⠏⢈⠝⣻⣷⠆
⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠀⠐⠲⢤⡀⠀⠉⠙⠒⠭⣛⢷⣯⣽⠃⢀⡿⣰⠇⣠⠏⢠⠏⢈⡾⠃
⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠈⠑⢌⠳⣄⣸⡷⢋⡴⢋⣴⠏⢠⡟⠁
⠀⠀⢀⣴⡏⠀⠀⠀⢰⣿⣳⠀⢠⠀⠀⢹⡆⠀⠀⠀⠀⠀⠈⠀⠈⠻⠿⠿⢞⣋⠵⢢⡟
⠀⠀⠸⣏⠳⠆⠀⠀⠈⠛⠋⠀⢸⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣒⠂⢸⠃
⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⡼⠀⠀⣸⣧⣀⠀⠀⠀⠀⠀⣠⠶⢶⣦⣍⠲⢤⡀⠸⣇
⠀⠀⠀⠀⠀⠈⠛⢶⣤⣤⠄⠈⠀⣀⡼⢿⣗⢭⡳⣆⢀⣠⡾⣟⢦⣄⠹⣿⣝⢦⣍⠢⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠿⢯⣤⣤⣽⣿⠾⣿⣿⡧⡙⢮⣱⣿⠇⠹⣟⢧⡈⢷⡀⠈⡻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣾⣟⢦⡑⢌⠪⣳⡾⠋⠉⠀⠀⠀⢻⡆⠻⡄⠹⣄⠪⣑⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⣝⣧⡙⢦⡑⢽⣧⠀⠀⠀⠀⠀⠀⢻⣄⠹⡄⠘⢦⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣺⣿⣦⣝⢎⣿⠀⠀⠀⠀⠀⠀⠀⠙⢦⣘⢦⡸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠙⢯⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⣠⠀⢀⡀
⠀⠀⠀⠀⠀⠁⠽⡜⠙⢛⠒⠤⣄⣀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠓⢤⣀⠀⠠⠔⠐⠚⠛⠋⠉⠉⠉⠋⠙⠻⠶⣤⡀
⠀⠀⠀⠀⠀⣠⡴⠉⠀⠄⠀⠒⠂⢀⣀⣀⣤⣤⣤⣤⣶⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡠⠀⠠
⠀⠀⠀⠀⠈⠙⠛⠛⣷⣂⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⡶⠦⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠰⠞⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠓⠒⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠐⠂⠈⢥⢈⡄
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⣶⣶⣶⣶⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂⠁⠉⠉⠀⠀⠀⠀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⠟⠋
⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁⠀⠀⠀⢀⣀⠀⣀⣀⣀⣀⣀⣠⣠⣤⣽⣭⣭⣿⣿⣿⠟⠁
⠀⠙⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠈⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⢿⡿⠷⢦⡄⠀⣨⣿⡿⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠇⠀⡿⠀⠀⠘⠀⠸⠋⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⠀⠠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⣶⣿⡟⠂⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠞⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣧⠀⣆⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢦⣄⢠⣤⣤⣄⣿⢶⣴⠟⣽⣿⣿⣿⣿⣶⣿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣽⣦⠾⠿⠿⠿⡿⣿⠷⣿⣿⠿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠒⠒⠂⠀⠤⠄⠀⢉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠒⠃⠁⠀⣀⣤⡤⠒⠚⠐⠂⠀⣀⣤⠤⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣤⣤⣤⣭⣽⣶⣶⠶⠶⠶⠤⠶⠶⠶⠶⠾⠖⠛⠋⠁⠀⠰⢀⣀⡀⠀⠑⠘⠂
⠀⠀⠀⠀⠀⠀⠤⠤⠴⠖⠒⠚⠛⠉⠭⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣄⣤⡽⠟⠉⡧⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠶⠶⠒⠒⠚⠒⠒⠛⠛⠛⠉⠤⠰⠖⠉⠉⠉⠉⠁⠀⠀⣀⡤⠃⠀⠀⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠤⢤⣤⣀⣠⠤⠶⠶⠛⣛⣩⣤⣤⣶⠶⠶⡶⢶⠿⠶⠖⢾⡶⣦⣤⣤⣤⣠⣤⣠⡤⠴⢶⡾⣏⣁⣲⡄⠀⣀⠠⠄⠂⠁⠠
⠀⠠⠀⡤⠠⠤⠤⠤⠴⠖⠒⠒⠒⠂⠉⠉⠁⠁⠀⠀⠀⣀⣀⡬⠤⠄⣒⡤⠖⠒⠉⠀⠀⣀⠠⠒⠋⠁⠀⠀⠀⠈⠉⠉⠀⣶⡀⣀⡀⣤⡤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠒⠚⢋⡉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠠⡖⠋⠁⠀⠀⠀⠀⠀⠀⡠⢀⢠⣠⣀⠸⣿⠏⠿⠷⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⣀⣀⣀⣀⣠⡤⠶⠛⠉⠀⢀⣠⡤⠖⠐⠉⠁⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠤⠊⡴⠾⢷⡛⠒⠾⡧⠀⣲⡾⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠙⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⠛⠛⠷⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⣀⣀⣀⡀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⠂⢠⡞⠉⠉⠉⠛⣦⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⢸⠀⣿⡀⢀⣀⣀⡀⣸⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⣤⡾⢿⣋⣿⣉⣿⠟⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠋⣷⡾⢟⠉⠉⠀⠀⣀⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣁⣿⣿⠀⠧⠀⠀⠀⣹⣿⣯⠽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢇⣼⣃⣿⣀⣋⣠⣀⣀⣹⣇⣹⣿⢿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣾⡟⢤⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣧⠀⢹⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡟⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⣷⣿⣇⢺⣇⠀⢀⣀⣀⣀⣤⣄⣀⠀⠀⣿⠀⢸⡇⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣍⠽⣧⣼⣿⣿⣿⣿⣿⣭⣤⣤⡟⠀⣼⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡞⣻⣿⡟⠋⠙⠟⢹⠅⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣴⠛⣿⣦⣄⣺⡅
⠀⠀⠀⢀⡾⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⡿⣷⣴⣋⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠳⣆
⠀⠀⢀⣾⠃⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡅⢈⠉⠻⣶⡊⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢹⣆
⠀⠀⣼⠃⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⠀⠀⠀⢸⣿⡎⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡀⡇⠀⢻⡄
⠀⣸⣿⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣿⠀⠈⣷
⢠⣿⠋⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠇⢀⠀⠀⢺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⢿⠀⠀⠸⣇
⣼⠃⠀⠀⠀⠀⠀⡀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⡄⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡀⠀⠀⠀⠀⠀⢻
⣿⡀⠀⠀⠀⠀⢠⠇⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣿⠀⠄⠰⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢃⠀⣨⠀⠀⠀⣸
⠈⢷⣆⠀⠀⡄⢸⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢉⣷⣿⢀⣄⠘⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⠀⡇⠀⢠⣶⠋
⠀⠀⣿⡀⠀⢳⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠈⣿⠃⣿⢀⡄⢀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢰⠁⠀⣸⠇
⠀⠀⠸⣧⠀⠈⣆⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢻⡇⢠⡟⠀⠀⠠⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⠏⠠⢠⡟
⠀⠀⠀⠹⣧⠀⠘⢆⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣹⣷⣼⣃⣘⣆⣀⣀⣀⣠⣿⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⢀⣰⠋⢀⠎⠀⢠⡟⠁
⠀⠀⠀⠀⠘⢷⡀⠈⠣⡀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠛⢷⣦⣋⣉⣟⣉⡟⣉⣟⣉⣼⣭⣠⣿⣤⣟⣉⣹⠿⣦⠀⢀⣴⠟⠁⡰⠋⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠛⣦⡀⠈⠢⣄⠉⠛⠶⣤⣄⣀⣀⣀⣤⠾⢋⣽⡿⠋⢛⡿⣋⠉⠉⠙⠉⠻⢦⣄⣉⣉⣻⣦⡾⠟⠋⢀⡤⠊⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠉⠒⣤⢀⡉⠉⠉⠉⠀⠐⠋⠻⠛⢈⡼⠁⠸⣆⡑⠀⠀⠀⠀⠈⠉⠉⢉⣀⣠⠔⠊⢁⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣌⣉⠒⠒⠒⠒⠶⡶⠶⢂⣁⠀⠀⠀⢈⣉⠲⠶⠶⠖⠂⠚⠒⢈⣁⣤⣴⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢦⣄⡀⠀⠀⠲⣬⠳⠄⡴⢋⡴⠂⠀⠀⣠⣤⠶⠟⠋⢩⣿⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⠈⠁⣾⠁⠈⢀⣤⠶⠛⠉⠀⠀⠀⢀⡾⣇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣆⣘⣠⡾⠋⠁⠀⠀⠀⠀⠀⢠⡿⣇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⢠⡿⣀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣅⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿
⣿⣿⣿⡿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣾⠉⡧⢀⡴⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢧⣤⡇⠀⡿⢻⠃⠀⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢠⢃⡞⠁⣧⠀⡿⣿⠀⣰⠏⢿⡏⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡶⠦⠤⣀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⠉⠉⠀⠀⠀⠈⠉⠛⠛⠿⠶⠶⠿⠾⠷⣯⣼⠃⠀⣿⢦⣤⣤⡽⢦
⠀⠀⠀⠀⠀⠀⢀⣴⣋⣽⠦⠶⠚⠛⠲⢶⡏⠳⣄⢀⣠⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⣧⢀⡞⠉⣷⣀⠙⢳⣄⣧
⠀⠀⠀⠀⣀⣤⡿⠭⣶⠶⠶⠦⠤⢤⣀⠀⢳⡀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⡏⢀⣰⢁⠈⠳⣟⠉⢻
⠀⠀⣠⠾⣇⣀⣀⣠⠼⠲⠶⠶⠶⢶⡏⠛⢦⡙⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⡿⠋⠉⠉⠉⢻⣿⣄
⣠⢎⡴⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠉⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡇⠀⠀⠀⢀⡞⠀⠙⣿
⠿⠋⠀⠀⠀⠀⠀⣀⣤⢤⡶⠒⠒⠶⠶⠶⠮⠿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⡈⠓⡾⠳⠶⠶⣶⠋⠀⠀⠀⣸
⠀⠀⠀⠀⢀⡴⠋⠁⣠⠜⠓⠒⠒⣶⣶⣶⣶⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢜⣭⣿⣿⣤⡠⣄⣉⡿⠁⠀⠀⢠⡟⠲⠶⠒⢺⡏
⠀⠀⣀⡴⠚⢓⡶⠋⠁⣠⡴⠚⠋⠉⣉⣉⣭⠏⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣽⣿⣥⣼⢿⣿⡿⠃⣷⡼⠋⠀⠀⠀⠀⡟⠀⠀⠀⠀⣸⠃
⠀⣴⠁⣠⡾⠋⠀⠀⣰⠋⠉⡟⠉⠉⠉⠀⣸⠷⣴⠋⠀⡴⠀⢀⠀⢀⠀⠀⡀⢠⡿⣿⣿⠟⢹⡧⣄⣠⡴⠋⠀⢀⣠⠴⠒⡻⠛⢤⣀⠀⣰⠏
⠀⡟⢹⡇⠀⠀⠀⢠⡇⢀⡼⠀⠀⠀⠀⢰⠃⠀⠈⠻⢾⣥⣴⣏⣀⣸⡀⣀⣇⣈⣳⣤⣀⣠⣾⣷⠶⢿⡷⣀⠴⣫⡤⠖⣾⠇⠀⠀⢩⠟⠁
⠀⡇⢸⡅⠀⠀⠀⢸⠛⠉⡇⠀⠀⠀⠀⢸⠀⠀⢀⣀⣼⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠙⠟⠀⠀⠴⠟⠋⠁⢀⡾⠃⠀⠀⣠⠏
⠀⠹⣜⡇⠀⠀⠀⠘⣆⢰⡇⠀⠀⠀⠀⠘⡟⠋⠉⠀⠈⢳⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣛⣠⠤⠖⠋⠁
⠀⠀⠘⠧⠀⠀⠀⠀⠘⣏⠙⢆⠀⠀⠀⠀⢿⡀⠀⠀⢀⡼⠉⠉⠉⠛⠲⠦⣤⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣙⣦⣀⠀⠀⠛⠒⢾⠉⠀⠀⠀⠀⠀⠀⠴⣶⣦⡤⣶⠾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠓⠤⣤⣄⣀⣀⣀⣀⣠⡤⠤⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣈⣉⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡇⢻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⠈⡏⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡇⠀⢹⠈⢟⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡇⠀⢸⡄⠸⡜⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡇⠀⠘⡇⠀⢧⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡇⠀⢀⡇⠀⢸⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢀⡇⠀⢸⠇⠀⢸⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢸⡇⠀⣸⠀⠀⢸⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢸⡇⢀⡇⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⣾⡇⡼⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣀⡀⠀⢀⣿⣿⡥⠤⣤⣀⡀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢿⣿⡶⠿⠿⠛⠛⠟⠛⠉⢉⣉⣉⣽⣿⣿⡿⠟⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣒⣩⣭⣤⣤⣤⣶⣶⣶⡾⠿⠿⠿⠟⠛⠋⠉⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠉⠉⣉⣉⣉⣉⣁⣠⣤⠤⠤⠤⠖⠒⠒⠚⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⡆⠀⢀⡤⡀
⠀⠀⠀⢀⠞⠳⣄⠀⢰⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⡠⠖⠂⠉⠐⠲⢄⣠⠖⢉⣀⡀⠉⠓⣄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢰⠀⡴⠋⠀⢱
⠀⠀⠀⢸⠀⠀⠈⠣⣸⠀⠀⠀⠈⢆⠀⠀⠀⠀⡘⠀⢀⣾⣿⣿⣦⢀⠃⢰⣿⡿⣿⣆⠀⠈⡆⠀⠀⠀⠀⡰⠁⠀⠀⠀⠸⠊⠀⠀⠀⡸
⠀⠀⠀⠘⡄⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⢣⠀⠘⣿⣧⣸⣿⠀⣆⢸⣿⣧⣼⡿⠀⢠⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠳⢄⡙⠻⠟⣋⠔⠉⠲⢝⡛⣛⡡⠔⠁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⣀⠀⠈⠣⣀⠀⠀⠀⠀⠀⢀⣈⠤⠏⠈⠒⠒⠒⠋⠀⠑⠒⠂⠤⢄⡀⠀⠀⡠⠃⠀⢀⠖⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠈⠑⠦⣀⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠁⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠀⠀⠀⠀⠀⢎⠀⠀⠀⢀⡤⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣈⠙⠲⠤⣀⡏⠀⠀⠀⠀⠀⢀⣀⢤⡤⡴⢶⠒⡏⢩⣈⡖⣯⠃⠀⠀⠀⠀⠀⠈⠦⠔⠊⣁⠤⢦⠀⠙⢄
⠀⠀⠀⠀⠀⢠⠎⠀⢠⠊⠀⠉⢲⣤⣄⠀⠀⠀⠀⠀⠀⠈⠻⡖⠧⢶⠴⡶⢺⠈⣇⡷⠁⠀⠀⠀⠀⠀⠀⢀⣤⠶⢯⡀⠀⠀⢣⡀⠈⢣
⠀⠀⠀⠀⢠⠃⠀⡰⠃⠀⢀⡔⠁⣠⣉⡛⠀⠀⠀⠀⠀⠀⠀⠈⠛⠬⢦⣥⠴⠗⠉⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠢⡀⠱⡄⠀⠀⢱⡀⣀⠇
⠀⠀⠀⠀⠈⠂⠜⠁⠀⢀⠎⠀⡸⠁⠀⢩⠵⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢧⠀⠀⠀⢣⠀⠘⡄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡀⡴⠁⠀⠀⡏⠀⡆⠐⠒⠒⠒⠂⠤⠤⠤⠄⠀⠤⠤⠤⠔⠒⠈⠉⠉⠉⡇⠈⡀⠀⠀⠀⠧⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⣶⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣏⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⢹⠟⢿⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣴⣟⣃⣾⣿⢃⠀⢀⣄⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠯⠛⠛⣛⣈⣧⣤⣷⡶⠀⠀⠀⠀⠀⠁⠈⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣋⣤⣿⣦⣶⣶⣾⣿⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠠⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣭⡛⠉⢿⣿⠜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡥⠀⢸⣿⣇⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠊⣿⣿⠄⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄
⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⢀⡀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠀⠶⠀⠐⠃⠀⠀⠀⠀⠀⠀⠿⣻⣷⡾⣝⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⡀
⠀⠀⠀⠀⠀⠠⠄⠀⠡⠒⠀⠀⣿⣇⠀⠠⠀⠀⠄⠀⠀⠀⠀⢸⡟⢣⡔⢀⣤⠀⠀⠀⠀⠀⠀⠀⠺⠓⠻⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣝⢣⡀⠀⠀⠰⠆⠀⠀⠀⠐⠠⠡⣄⣴⣷⡠⠆⠒⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⢹⢀⠀⠀⠀⠀⠀⠀⣀⣷⣴⣿⡁⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⢻⡾⠷⠀⠀⠀⢀⣰⡟⠉⣿⣿⣧⠀⢹⡄⠀⠀⠀⠀⠀⠀⢠⢯⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠿⣆⠀⠀⠀⠀⠀⢬⡗⠀⠀⠹⣿⣿⡆⠀⢣⡀⠀⠀⠀⠀⠠⣠⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠁⠀⠹⣧⡀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣷⡀⣀⢡⣀⢀⡠⢈⣠⣼⠏
⠀⠀⠀⠀⠀⢤⣤⢀⠞⣏⢹⡅⠀⠀⠀⠀⠙⢿⣤⡀⠀⢹⣿⠀⠀⠀⣿⣿⣿⡇⠛⠀⣹⢍⣂⣴⣿⠏
⠀⠀⠀⠀⠀⠘⠛⠈⢃⣹⠘⢳⡄⠀⠀⠀⠀⠈⠹⢿⣶⣄⣉⠒⠀⠒⢿⣴⡿⣿⡄⣴⣴⣿⣿⡿⠁⠀⠀⠴⠶⠖⠀⠀⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠹⣿⣿⢷⣶⣶⡿⣿⣿⣷⣾⣿⣼⠋⠀⠀⠀⠀⡠⠄⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⣿⡿⠷⠿⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⢶⡶⣠⣤⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉⠐⢛⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⢼⣤⣤⣮⣤⠾⠤⠴⠒⠐⠒⣀⡐
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠿⠇⠀⠀⠉⠉⠉⠈⠁⠀⠀⣤⣴⣶⠶⠟⠛⠉⠑⠦⠄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣉⣉⣷⣶⣒⣂⣀⣤⣀⣀⣤⣄⠄⠠⣤⣤⣤⣤⣤⣤⣤⡄⣤⣤⣤⣄⡀⢐⠲⠶⢶⣶⡶⣶⣶⣶⡶⢶⠖⠁
⠀⠀⠀⠀⠀⣠⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠓⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⠛⢛⣓⡛⠛⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣠⠞⣛⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠞⠋
⠀⠀⠀⠀⠻⡏⢹⣯⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁





⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⣒⣚⡭⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⡴⠞⠋⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⣠⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⢀⡞⠉
⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⢨⡼⠃⠀⠀⠀⠀⠀⢀
⡄⠈⠑⢤⡀⠀⠀⠀⡀⣀⡤⠼⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡴⠋⠀⣻⣶⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠊
⢁⠀⠀⠀⠉⡓⠴⡎⠉⠁⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠌⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄
⠀⠢⠀⠀⠀⠸⠃⠁⡤⠀⣀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⣴⡟⢹⡇
⠀⠀⠀⠂⠄⣾⣄⠀⢺⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀⢞⡆⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⣴⡿⠊⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠙⡄⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠀⠁⠀⠀⢀⡤⢋⣿⠀⠀⠀⠀⠀⡜⠀⠀⠀⠐⠀⢸⡇⣀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡐⠤⢀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⢀⣤⠞⣠⢶⣏⡏⠀⠀⠀⠀⠀⡅⠠⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠙⠛⠀⠀⠀⠛⠉⠀⠀⠁⠀⢀⠄⠴⠞⣩⠔⠋⠀⠥⣾⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠠⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠤⢄⣒⠂⠀⠀⠀⠠⠤⠄⠀⠀⣀⣀⠠⠄⠈⢹⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⡆⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠠⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⡓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢸⡇⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠸⠁⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⢀⡼⢃⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇⢀⠀⠀⣠⣾⡿⠋⠁⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⢠⡞⢀⡼⠋⠁⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣿⡶⠋⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⡟⣹⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣰⡟⢀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⡼⡿⠁⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⡼⠁⠘⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⡼⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⡼⠁⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⡼⠁⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⡾⠁⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⣡⣾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠁⠀⠀⠛⠉⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⡀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⢀⣴⠟⣁⣠⠀⠀⠀⠀⠀⣸⣷⠏⠀⠀⠀⠀⠀⠀⠸⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⢀⣴⣿⣿⣿⣯⣥⡤⠶⠶⢻⣿⣯⣤⣤⣄⣀⣀⣀⣀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠃
⠀⠀⠴⠿⠟⢋⣉⣉⣀⣀⣀⣀⣴⣋⣀⣀⣭⣭⡭⣭⠭⠤⡌⣿⠗⠒⠠⠤⠤⠀⠀⢀⣀⣀⣠⣀⣠⣤⣤⣴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⡻⡟⠻⠿⣯⣭⣀⣀⠘⠂⠸⢧⣌⣙⣀⣀⣀⠀⠀⠀⠀⠤⢼⠆⠈⠉⣯⠉⠉⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢰⡶⠛⠛⠛⠿⣿⣿⡷⠶⠶⢤⣤⣬⣭⣭⣭⣭⣭⣥⡴⠶⠖⠋⢁⣠⡴⠁⡀⠴⠒⣒⣩⣥⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣾⠁⠀⠀⠀⠙⢿⣷⣄⡀⠀⠚⠉⠉⠙⢻⣿⣿⣿⣿⠿⠿⠛⠉⣀⣠⣴⠾⠛⠉⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠤⠀⠀⠀⠀⠙⠻⣿⣛⡒⠚⠀⢀⣠⡿⠈⠛⠻⠿⠿⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢩⠞⠁⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠈⠳⡍⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⠔⠋⡠⠶⢺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠕⠶⢄⠉⠢⡈⠓⣄
⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠞⡰⠚⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠓⢆⠳⡄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠓⠂⠀⠐⢖⠋⠁⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠈⠙⡳⠂⠀⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣣⣀⡠⠒⠜⠳⠞⠡⠐⢄⣀⣠⠛
⠀⠀⠀⠀⠀⠢⠀⠀⠂⠉⠙⠋⠢⠤⣀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⣀⠤⠔⠙⠋⠉⠐⠀⠀⠔
⠀⠀⠀⠁⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⢀
⠀⠀⠉⠇⠀⠀⠀⠙⠢⢀⣰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣆⡀⠔⠋⠀⠀⠀⠘⠉
⠀⠀⠀⣠⠦⠀⠀⠀⣀⡎⠁⠀⠀⠀⠀⢀⡠⠐⠒⢦⡀⠀⠀⠀⠀⠀⢀⡴⠒⠂⢄⠀⠀⠀⠀⠈⢱⣀⠀⠀⠀⠴⣄
⠀⠀⠀⠈⠁⠐⠠⢰⡅⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⢨⡆⠄⠀⠀⠁
⠀⠀⠀⢀⣀⣀⣠⠔⠀⠀⠢⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠔⠀⠢⢄⣀⣀⡀
⠀⠀⠀⠉⠛⠳⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⠛⠉
⠀⠀⠑⠀⠀⠀⠠⠚⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠃⠄⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⣨⡓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⢚⣅⠀⠀⠀⢀⠔⠀⠀⠀⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠉⠀⠂⠈⠉⠉⠉⠁⠀⠐⠂⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠐⠀⠉⠀⡄⠀⠀⠀⢀⠀⡀
⠀⠀⠀⢨⣠⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢉⡝⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠎⠀⠀⠈⣿⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡁⠀⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡖⡚⣿⡙⢧⣢⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢙⣧⣿⣿⣿⣜⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣄⣇⣀⣠⣀⣠⣽⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣄⣤⣤⣿⣿⣧⣤⣤⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣏⢋⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠛⠛⣿⣽⡏⠉⠉⠙⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⢣⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢪⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠖⡀⠀⠀⠀⠀⠀⡰⡱⢻⠃⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⡇⠙⠦⠀⠀⠀⡐⡱⠁⢸⠀⡟⣿⠀⠀⠀⠀⠀⢀⣤⡞⢸⡇⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⢷⣰⣄⠚⡁⠰⣰⠁⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⣹⣾⣗⣾⠁⠀⠀⠈⣿⣮⡲⠄⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢴⣾⣿⡏
⠀⠀⠀⠈⠋⠈⠻⣦⣙⠛⠤⣤⡾⢠⡇⠘⢤⠤⠀⣲⣿⠿⠋⠘⠃⠀⠀⠀⠀⠹⠟⢿⣦⣄⡀⠀⠀⠀⣸⣿⣿⣿⡀⠀⠀⠀⣀⣤⣾⠟⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣦⣄⡤⣾⣿⣤⣦⣶⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣟⣿⣶⣿⣿⣿⣿⣿⣽⣶⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡙⠻⣮⣵⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣀⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣍⣭⣫⣤⣍⣽⣉⣼⣏⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿