Granddaughter Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠀⢉⠉⠉⠉⠉⠉⠉⠁⠐⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠤⠄⣠⠀⠒⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡌⠀⠀⠀⠈⠉⠀⠀⠀⠈⣁⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⠁⠀⠀⠐⠛⠁⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⢠⡀
⠀⠀⡠⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠇⠘⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢃⠀⠃⠈⠄
⢀⠾⣁⠶⠘⠂⠀⠀⠀⠀⠀⠀⠀⢆⠀⡹⠃⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠈⠀⠀⠀⠀⠈⢺⠀⣰
⠣⣱⠊⠀⠀⠉⠢⢄⠀⠀⣀⡀⠀⠀⠑⠇⠀⠀⠀⠀⠔⠀⢄⡀⠀⠀⣀⠄⠑⠀⠀⠀⠀⠸⠊⠁⠈⢂
⠀⠀⠀⠐⠢⣀⠀⢠⠼⢉⠠⠁⠣⠄⡀⠰⡀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠁⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠑⡞⠀⠂⠀⠀⠀⠀⠈⠑⢵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⢀⡀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢂⠀⠀⠀⠀⠀⠠⠋⠀⠈⠁⠖⡄⠠⣀⣀⣀⠤⠀⡶⢒⠁⠀⠐⠄⠀⠈⣺⠄⠤⠤⠀⡀⠠⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⢠⠁⠀⠀⠀⢸⠀⠈⠢⡀⠀⠀⣀⠜⠁⠈⠀⠀⠀⠈⡤⠊⠀⡠⠀⣒⠀⠀⠐⢒⠠⢀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠈⠢⣀⡀⠬⠒⠪⠄⣀⣀⠔⠀⠀⠀⠀⡆⢠⠞⠈⠀⠀⠀⠀⠀⠀⠈⠐⠑⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⢀⠦⡀⠀⠀⠀⠀⠀⠈⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠆⠀⠀⠀⠀⠀⠀⠀⠀⡅⠇⡠⠐⠁⠤⠌⠢⢄⠀⠀⠀⣠⠷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⣄⠆⠀⡧⢸⠁⠀⠀⠠⠆⠄⡐⠄⠉⠀⠸⢀⡄⠠⣄⠔⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠐⠒⠂⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠸⠦⣀⠀⠳⢀⡀⠹⠀⠀⠀⠰⢤⣤⡤⠂⠀⠀⠰⠉⣇⡸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⠦⣀⠛⠁⠳⣄⡀⠀⠀⠀⠀⠀⢀⣴⠓⠚⣁⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠘⣄⠀⡌⠀⠀⠉⠁⠒⠸⠁⠀⠈⡄⢀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⢤⢑⠂⠀⠀⠀⠀⠆⠀⠀⠀⠀⣗⣡⠨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡄⠀⠀⠀⠀⠓⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠀⠃⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠜⠀⠁⠀⠒⠒⠒⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠁⠂⢲⡄⢠⣤⡄⠐⣆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡖⠢⠔⠒⠤⢺⢄⣼⠂⠤⠋⠂⢤⡌⠑⠃⠀⠀⠠⠚⠳⠼⠁⢠⠁⢏⠜⠓⢄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣴⣶⣭⣭⣤⣬⣭⣶⣶⣶⣶⣥⣤⣤⣤⣤⣴⣶⣶⣶⣮⣤⣵⣶⣦⣶⣬⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠀⠉⠉⠉⠉⠉⠉⠁⠒⠢⠤⣀
⠀⠀⠀⠀⡠⠒⠉⠁⢨⠛⠑⡶⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⡜⠀⠀⠀⠀⡃⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⡘⠀⠀⠀⠀⠀⢑⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⢠⠁⠀⠀⠀⠀⣠⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⣴⣃⡀⠤⡾⠤⠊⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⣀⣀⣀⠀⠄⠒⣉⡀⠀⠀⠀⠀⢈⣷⣤⠄⣀⣀⣀⡀⠀⣐
⠀⠀⠀⠀⠀⠀⠀⠞⠁⠃⡘⠁⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠱⢸⠉⠉⢆
⠀⠀⠀⠀⠀⠀⠐⡀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠡⡀⢸⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠞⠑⠄⠑⠊⠀⠀⠀⠀⠀⠀⢸⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠄⣀⡀⠀⠀⣀⡀⠤⠀⠀⠀⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠷⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠙⢶⠤⠄⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠴⠚⣦⠀⠒⠒⠀⠀⠰⡖⠀⠈⠉⠀⠀⠀⠥
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⡆⠀⠈⠅⠉⠐⠒⠒⠊⢩⠁⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢸⣀⡀⠤⠀⠒⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⢠⠁⠀⠀⠈⠲⢤⠄⠠⠔⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣐⠁
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⢹⠛⠀⢰⠊⠀⠀⠀⠀⠀⠀⠀⠀⣖⡦⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⢞⠆⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠘⢦⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⠼⡜⠉⠐⠒⠀⠤⠤⠀⠀⠠⠀⠀⠠⠤⠤⠐⠒⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠑⢊⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤
⠀⠀⠀⠀⠀⠈⠐⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡏⠉⠉⠉⡋⠉⡋⠉⠉⢈⡅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣿⣄⠉⢶⠂⠉⠅⠀⠉⠑⠲⠎⢁⣼⡧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠉⠈⠢⠜⠀⠀⠂⠐⠀⠀⠘⠤⠊⠈⠁⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⣀⣀⣀⣀⡀⠤⠄⠃⠀⠃⠤⠄⣀⣀⣀⣀⣀⠠⠂
⣿⣿⡿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⣠⠔⠚⠉⠙⢦⣀⣀⣀⡤⠚⠉⠙⢦⡀
⠀⠀⠀⢠⠞⠁⢀⢴⡆⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⢳
⠀⠀⢠⠏⠀⣰⠃⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⡇
⠀⠀⡾⠀⢰⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⢠⡇⠀⡾⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⢤⡀
⠀⢸⠁⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢿⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣃
⠀⢸⠀⠀⡇⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⢀⣠⠏
⠀⢸⣄⣀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⢀⡤⠖⠒⠚⠛⠛⠒⠒⢦⡀
⡼⢁⡤⣄⠙⡆⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢀⣸⣇⣀⣠⠴⠚⠁⣀⣴⠀⠀⠀⠀⠀⢀⡀⠹⡄
⠓⠚⠀⢸⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣏⠉⠉⢁⣀⣠⠴⠚⠁⣼⠀⠀⠀⠀⠀⠘⣇⠀⢳
⠀⠀⠀⢸⠀⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠉⠉⠁⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⡿⡄⠘⡇
⠀⠀⠀⢸⠀⡇⠈⠙⡟⠉⠋⠙⣿⡏⠉⠉⢻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⢳⠀⢹
⠀⠀⠀⢸⠀⡇⠀⠀⡇⠀⠀⠀⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣿⠘⡆⠈
⠀⠀⠀⢸⠀⡇⠀⠀⡇⠀⠀⠀⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢹⠀⠹⣤
⠀⠀⠀⢸⠀⡇⠀⠀⡇⠀⠀⠀⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡖⠒⣶⡖⠒⢲⠛
⠀⠀⠀⢸⠀⡇⠀⠀⡇⠀⠀⠀⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⢸
⠀⠀⠀⢸⠀⡇⠀⠀⡇⠀⠀⠀⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⢸
⠀⠀⠀⢸⠀⡇⠀⠀⣇⠀⠀⢀⡿⡇⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⢸
⣤⣤⣤⣬⣶⣥⣤⣤⣬⣷⣶⣯⣤⣽⣶⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣾⣯⣽⣶⣭⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⢦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣁⣀⣀⣀⣀⣈⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⣀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⡴⠙⣆⢀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⡞⠠⠀⠀⠀⣠⠞⠁⠀⠈⢦⡑⠦⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢰⣴⡁⠄⢢⠵⠋⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢸⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⢸⢀⠇⠀⠀⠀⣀⠀⠀⡁⠀⠀⠀⢀⡀⠀⠀⠘⡇⠸
⠀⠀⠀⠀⠀⢰⠚⡟⠀⠀⠀⠁⠋⠀⢠⠗⠀⠀⠀⠘⠋⠀⠀⠀⠘⡟⠲⡄
⠀⠀⠀⠀⠀⠱⣄⢱⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡰⠆⠀⠀⠀⠀⠀⠀⠀⣀⠴⠖⠚⠛⠛⠓⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠘⠀⠀⠄⠀⠐⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⡠⠊⣡⠔⠊⠁⠀⠀⠀⠀⠒⠂⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⠀⢀⣀⣀⠀⣀⣤⡖⠻⢤⣀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⣀⣀⠤⠦⠰⠖⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⣿⡉⠉⠉⠉⠉⢀⣼⠃⠀⠀⠈⠑⢄⡀⠀⠀⡠⠾⠤⠒⠉⠁⠀⠀⢀⡎⢀⣤⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢀⡔⠋⢸⠀⠀⠀⢸⠈⠑⠒⠒⠊⠁⣬⠀⠀⠀⠀⠀⣰⠛⣄⠘⣇⠀⠀⠀⢰⣿⠇⠀⢀⡷⠃⠸⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡰⠋⠀⠀⢸⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢰⠃⠀⠈⢣⡜⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡀⢠⠀⠀⠀⡆
⠀⠀⠀⢀⡜⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢾⠀⠀⠀⠈⣧⢻⡀⠀⠀⠀⡀⠀⠀⠀⠀⠿⠂⠱⣌⠀⠀⡜
⠀⠀⠀⡘⠀⠀⠀⡀⠀⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠘⡄⠀⠀⠀⢈⡾⠳⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠒⢎
⠀⠀⠀⡇⠀⠀⠀⢼⠋⡇⠀⠀⠀⢠⠋⠉⠩⠴⠄⠀⢹⡀⠀⠀⠀⠀⢳⠀⠀⠀⢈⡧⣤⡬⠓⠦⢄⡀⠀⠀⠀⢀⣀⡴⠤⡤⠊
⠀⠀⠀⢣⠀⣀⣤⣶⠀⡇⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⠒⠀⠀⢀⠇⠘⡳⢄⣀⣸⣅⠉⠉⠉⠁⢸⠁⠀⡄
⠀⠀⠀⠀⢻⡏⠁⠈⢣⡇⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣠⠟⠂⢀⣀⡞⢸⡄⣷⠀⠠⡀⠈⣣⠀⠀⠀⡎⠀⡰⠁
⠀⠀⠀⠀⠀⠑⠒⣶⠋⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡞⠁⣾⠀⢦⡘⣧⠘⣄⣴⠖⠉⢙⠀⠀⠀⠗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠁⢻⣦⡀⠉⠛⠶⣿⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⢀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠴⡖⢛⠀⠉⠒⠤⢤⣟⣆⣤⣾⠁
⠀⠀⠀⠀⠀⠀⢸⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣀⣀⠀⠀⠀⠀⠙⢦⡙⢦⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⠙⠥⠬⢙⣛⡒⠲⠆⠨⣇⠉⢦
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡟⠒⠲⠤⠤⣤⣀⣉⣉⠁⣿⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢃⠀⠀⠀⠀⠀⠉⠓⠤⢀⣉⢛⣰⠶⠴⠖⠋⣻⠸⣇⣙⠓⠒⠲⠦⠤⠤⣉⣉⣻⡀⠀⣡
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠋⠁⠀⠀⠀⠂⠀⠀⠉⠉⠀⠀⣀⡤⠔⠋⡟⠀⠀⡏⠉⠉⢛⠓⠒⢲⡆⠀⢻⠁⠉
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠈⡟⠓⠒⠢⠤⢶⠄⠐⠒⠛⠈⡏⠀⠀⠀⢠⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⡷⢤⡜
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢱⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⢻⠀⠀⠀⢼⠀⠀⠀⡟⠀⢠⡎⠀⠀⠀⠣⠞⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠣⣄⡀⢻⡄⠀⠀⣶⠴⠈⢱⠄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⠚⠲⢤⡀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠓⠤⠤⠜⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠂⠀⠀⠈⠁⠀⠀⠐⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡣⠠⢀⡠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⡄⡆
⠀⠀⠀⠀⠀⠀⢠⡶⠞⢆⠀⠀⠀⠀⠸⡀⠱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠀⠀⠀⠀⢀⠐⢶⣆
⠀⠀⠀⠀⠀⠀⠻⣂⣠⠚⠠⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠠⠚⢢⣀⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⣀⠃⡠⠉⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠐⠊⠁⡈⢃⡀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡐⠁⠀⠀⠀⠁⠁⠂⡤⠀⠀⣀⠤⡀⠐⠈⠳⡀⠀⠀⠀⣀⠄
⠠⣔⡂⠀⣀⡠⠐⠈⢩⠁⠉⠒⠄⣅⠀⠉⢢⡤⠀⠀⡇⢀⠲⢄⠀⡇⠀⠀⠀⠐⡀⠐⠁
⠀⠈⠁⠈⢨⠁⠀⠀⡠⠀⠀⠀⠀⠁⠉⠈⠁⠀⠀⠀⠀⠀⠰⠀⠉⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡸⡤⠒⠊⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠣⠁⠀⠀⠀⠀⠀⠀⢸⠞⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⢀⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⡄⠘⠀⠀⠀⡘⠁⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣎⠙⢂⠀⠀⠀⠄⠀⠀⠐⠊⡱⠀⠀⠁
⠀⠀⠀⠀⠁⠙⡄⠀⠀⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡐⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣼⠀⠀⠀⠠
⠀⠀⠀⠠⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⣿⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠈⣁⡂⠖⠖⡒⡈⠁⠀⠀⠀⠀⠀⢀⡐⠀⠐⠛⠖⠂⢊⡀
⣀⣀⣀⣀⣀⣊⣉⣡⣀⣠⣭⣭⣆⣀⣀⣀⣀⣐⣧⣤⣤⣔⣀⣀⣤⣤⣼⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠉⠢⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣁⠥⠊⠈⠁⠣⠀⣨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⠈⣠⡤⡴⠚⢧⠈
⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⢢⠮⠛⠿⠃⠀⠠⠃
⡀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠘⠬⠛⠓⠦⠖⠀⠀⢠⠀⡄⠉⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠐
⠀⠀⠰⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠑⠀⠀⡄⢳⣀
⣶⠤⠠⢀⣀⠀⠀⠀⠀⠀⢀⡀⢒⣌⠀⠀⠠⡀⣀⢀⡆⠀⢀⠂⠀⠀⠀⡀⠀⠀⠾⢄⠀⣀⠀⠀⠀⠀⠀⠐
⣃⠈⣭⠤⠔⠒⠋⠁⠀⠐⠁⢦⡀⣉⡐⠏⠈⣃⠀⠚⢦⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠀⢀⣚⣑⣀⣀⣀
⠏⠑⠒⣂⠀⠀⣠⠄⠁⠨⠗⠀⠦⠨⡄⢻⣀⠟⡀⠀⠠⠁⠀⠀⠀⠀⠀⠆⠒⠒⠉⠐⠀⠖⠀⠁
⣭⡤⠴⠒⠈⠥⠂⠀⠀⠠⠈⠁⠴⠐⠣⠈⡁⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠂⠀⠀⠀⠐⡄⠄
⡇⠀⠈⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⠃
⡅⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘
⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠈
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈
⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀⣠⣴⡴⠶⠶⠖⠰⠶⠦⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠰⡩⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠳⣤⡤⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⡁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠉⢳⡒⠢⠤⢠⠤⠖⡲⠚⠉⠀⠀⠳⡄⠀⠈⠲⣄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⢤⡴⠅⢀⣀⠈⠀⠀⠀⠤⣀⠙⢤⡀⢀⣠⠞⠁⠀⡀⠀⠀⠀⠹⡄⠀⠀⠈⢧⢀
⠀⢶⣋⡁⠀⢀⣰⣢⠎⠁⠀⢰⡇⠀⠀⠈⠳⡜⣆⡀⠀⢈⣛⠶⠛⡙⠓⠠⠴⠚⠀⠀⠀⠀⠀⣃⠀⠀⠀⣸⠠⡆
⠀⠀⠉⠉⠉⠁⢸⠃⠀⠀⠀⢀⡅⠀⠀⠀⠀⠘⠃⠉⠈⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢶⠦⠚⣉⣻⢇⣀⡀⠀⠀⣠⠴⣶
⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⡤⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠒⠉⠁⠀⠉⠉⠙⠛⠁⠀⢽
⠀⠀⠀⠀⠀⢰⢙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣦⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠣⣠⣿
⠀⠀⠀⠀⠀⠈⠳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠚⠓⠒⠲⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠒⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠒⠒⠀⢳
⠀⠀⠀⠀⠀⠀⢀⢰⠏⠙⠒⠀⠤⣶⠒⠋⠣⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⣴⡁⢸⠀⠀⠀⠀⣬⠁⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢜⠀⠙⢻⠀⠀⠀⠀⡤⠀⠀⠀⠀⡓⠋⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠙⠒⠚⡆⠀⠀⠀⠦⠀⠀⠀⠀⠛⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠒⠲⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢶⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠒⢤⠤⣤⣤⡤⡤⠴⠰⣦⢿⣀⣀⣴⣁⣠⣆⢀⣄⡀⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠶⢿⡆⠙⠘⢠⣿⠶⣄⠀⠀⢿⣥⣤⣬⡤⠼⠋⠣⠤⣤⣤⣴⡿
⣀⣀⣀⣀⣀⣀⣀⣳⣤⣶⣶⣂⣀⣲⣶⣶⣞⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢴⡞⠉⠑⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣠⣴⣿⣷⣶⣦⣤⣤⣉⢡⠀⠀⠀⠀⠀⢀⣀⡤⠶⠖⠒⠒⠶⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⢏⡝⢃⡀⠠⢰⣆⣨⣿⣿⠀⠀⢀⡠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣣⡎⠁⠀⣀⢀⡶⢿⣡⣿⣿⣇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡲⠖⠒⠂⠑⠲⡖⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡧⠶⠝⣦⣿⣵⣂⣤⠃⠚⠁⠀⠁⡏⢻⣿⣿⣷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡠⠒⠒⢲⡄⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢸⣦⣤⠃⠀⠀⣩⣀⣶⠽⠃⠀⡸⠁⣾⣿⣿⣿⡟⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡔⠀⢀⠤⣘⣁⠐⣁⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⣚⣻⣿⡄⠖⠀⣀⠀⢿⣦⠾⠛⣣⣾⣿⣿⡏⠀⠀⢹⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠰⠁⡘⠀⠀⡌⠀⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢩⣿⡝⣷⠀⠈⠋⢀⣿⡟⠂⣴⣿⣿⣿⠟⠀⢠⣴⣾⣿⣟⣽⣿⣿⣶⣄⠀⠀⠀⠀⠅⢀⠃⠀⡐⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢸⣯⡀⣜⡆⢉⠉⣿⣿⡇⣾⣿⣿⠿⠃⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢸⠀⠀⠀⢀⠁⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠉⠀⡌⠮⠙⠈⢿⠉⠏⠁⠀⢀⣤⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠂⠀⠺⢠⠀⠀⡌⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠀⢠⡞⠀⠀⠀⠄⠘⡄⠀⠀⠠⠖⠟⠴⠶⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠇⢸⠁⠀⠃⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠁⠀⠀⠀⠀⠸⠧⡐⠁⢛⣷⣀⡑⢦⣤⡜⢡⣆⠀⢰⡄⣿⣹⢯⣿⣿⣿⣿⡿⠿⠿⡛⠛⠛⠛⠋⠉⠉⠁⠀⠉⠆⢸
⠀⠀⠀⣠⠴⠊⠁⡀⠄⠂⠈⠁⠀⠈⠀⠀⠠⢏⠓⡷⢻⣿⣿⣿⣿⣿⡟⢿⣿⣦⣟⣯⣿⣼⣁⣿⠏⠁⢀⣀⡀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠸⡌
⠀⡠⣾⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⢀⣴⣾⣿⣿⡿⢿⣿⠧⢭⢿⣿⣿⡿⣿⣿⣿⠇⠀⠀⠋⠀⢸⠀⠠⠃⠀⣀⣀⣀⡠⠤⠔⠁⡇
⡾⡇⠃⠀⠀⠀⠀⡀⠀⠊⠀⠀⠀⠀⢀⣠⣴⣶⣻⡟⡻⢻⠁⣆⠉⢃⡀⣫⡾⣿⣿⣿⣿⣿⡿⠀⠀⡀⠀⢰⠗⠚⠁⠈⠉⠉⠀⠀⠀⠀⠀⢠⠁
⡟⣾⠢⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⠟⠫⡟⢽⠏⠐⠃⠉⠀⣀⠠⠤⠄⡀⠀⠻⣿⣿⣿⣿⡇⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠑⡌⡄⢓⠀⣠⠎⠀⠀⠀⠀⢀⣾⠅⠀⡡⠀⠁⠀⠀⢀⠔⠊⠀⠀⣠⠊⠀⠀⠀⠛⣿⣬⠛⣧⠀⠀⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣇⣺⡄⠙⠡⠀⠀⠀⠀⡴⠋⠀⠠⠊⡐⠁⠀⢀⡔⠁⠀⠀⣀⠔⠁⠀⠀⠀⠀⢸⠇⢈⠇⢨⠀⠈⡇⠀⢿
⠀⠀⢹⡁⠘⠛⣦⠀⢀⠞⠀⡀⠐⢀⠀⠀⠀⠠⠋⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠋⡠⠁⣠⣾⠀⠀⡇⠀⢸
⠀⠀⠀⠸⢄⡁⠂⢶⣶⣀⡈⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⡠⠐⠈⡄⢀⣾⣿⠀⠀⠃⠀⢸
⠀⠀⠀⠀⢀⠟⢲⣿⣿⢟⣿⣶⣤⣤⣀⠀⠀⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⡠⢀⠎⢠⢀⣾⣿⡉⡀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡄⢠⣿⣿⣻⣿⣿⣿⣿⣿⣿⣦⣤⣴⣥⣀⠀⠀⢀⡀⠄⣀⣩⣀⣤⣤⣴⣿⣿⣿⡇⠀⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣷⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠠⣾⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢸⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⠀⠀⡃⠀⠀⠀⠀⠀⣀⣀⡠⢤⠄⢺
⠀⡴⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⣼⠀⠀⡧⡤⠒⠚⠋⠉⠉⠀⡅⠀⠀⢸
⠀⠻⠶⠞⠿⢿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣟⣿⣃⠀⢨⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⠀⠀⡇⠃⠀⠀⠀⠀⠀⠀⠙⠶⠖⠊
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣾⣿⡿⠟⠛⠛⠿⠛⣻⣛⣿⣛⡻⠛⣿⣽⠿⠿⠿⠛⠿⠟⠛⠋⠀⠀⢸⡀⠀⣇⡰
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣄⡀
⠀⠀⠀⠀⢀⡴⠋⠀⢀⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠑⢦⡙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⢸⡇⠀⣠⣤⣤⣤⡈⠳⣌⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⠀⠹⣦⢻⠀⣀⣀⣀⣄⣠⣤⣄⠀⢀⣀⠸⠄
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠀⠈⠃⠈⣿⢧⠉⠃⠁⠈⣧⠴⢶⡄
⢀⡤⠞⠋⠀⠀⠐⠒⠛⠁⠀⠀⠀⣶⠛⠒⠀⡴⠚⣾⡗⠀⠁⡄⢶⡄⢀⣤⠾⠗⠋⢻⡄
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⠃⢀⢠⡄⠚⠉⠛⠛⠀⢙⡛⢡⠀⢀⣄⠐⠚⠋⢀
⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠷⠶⠆⡈⠋⡀⠀⣰⠀⠐⠤⠞⠷⡼⠃⠘⠾⠇⢠⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⣶⣀⣷⣀⣿⠋⠹⣄⡀⠳⠾⢤⣀⣀⡦⣈⣷⣄⣹⠇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⢰⡏⢀⣾⢦⣄⣽⠋⠀⠉⠁⠉⠁⠀⠀⠀⠀⢣
⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢄⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣇
⠈⠉⠛⠛⠛⠲⣦⡀⠀⠀⠀⠀⠀⠳⢤⣤⣀⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢀⠀⣹⢂⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠖⠛⠋⠉⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠘⢦⠿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢨⡇⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⠞⠋⠀⣀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠇⠀⠀⣼⠉
⣿⣿⠟⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣍⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⣀⣄⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⠋⠁⠀⠀⠻⣄⡄⠊⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⢀⠼⠤⠿⠿⡀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⢘⢺⡶⠃⠀⠀⠀⠀⠀⠀⢹⡀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠴⠂⠀⠀⠀⠀⠈⡇⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⡖⠣⠖⢦⠤⡴⠺⣜⢲⠟⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⣧⣒⡶⢠⢠⡔⣖⣢⣧⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡳⠤⠴⠊⠀⣙⠦⠤⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⡏⠡⠄⣎⠀⡂⠈⠙⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠉⣀⣤⣀⠀⠂⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠀⠠⠤⠀⠒⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣧⠉⠉⠀⠀⠀⠉⠉⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡤⣀⡀⠀⠐⣇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠒⢄⠀⠀⣀⣠⠞⠉⠈⠙⢳⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⢀⢘⠛⠳⠦
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠀⠈


⣶⣶⡶⣶⠶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢚⠛⠛⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⢀⣄⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠶⠄⠐⠁⠀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣆⠀⠈⣀⡠⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠗⢄⣀⣀⠔⠉⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣦⣾⣿⣿⣦⡄⠀⠘⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠏⠀⠘⣿⣿⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠻⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⡠⠒⠠⠤⢾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⢿⠛⠟⠁⠀⡿⠿⠟⠛⠀⠀⠀⠀⠀⠀⠻⠿⠿⡇⢈⣤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⣾⡆⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢹⡜
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡗⠛⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣠⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⢇⠀⠀⠀⠀⠀⠀⡠⢴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡂⠒⠈⡇⠀⠈⣄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢿⣷⣤⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣼⡇⠀⠀⠀⠙⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠲⠄⢲⡀⠀⠀⠀⠀⠀⠐⠃⠚⠗⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠈⠀⠀⠈⠛⠧⡀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠾⠀⠀⠀⡁⠘⢂⡀⢀⠉⠀⠀⠡⠀⠀⠀⠀⢰⡀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠐⠀⠀⠱⣄⠙⠶⠾⠟⢯⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⡆⠀⠀⠀⢸⠃⢳⠠⣾⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠁⠀⣈⣀⣘⢷⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⣒⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠒⢂⡼⠉⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠂⡴⠋⠀⠀⠀⢀⡤⠤⠤⠐⠶⡉⢠⣽⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠻⡃⠲⠀⠀⠔⠁⠀⠀⠀⠀⣀⠀⢺⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠲⠀⠀⠀⠀⠀⠔⠋⣀⠈⠸⣿⣿⣷⣴⣷⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠶⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣆⠀⠀⠀⠄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡄⠀⢸⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠁⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠃⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣉⡟⠻⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⢸⣿⣿⣿⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣷⣤⣤⣤⣤⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀⣤⡴⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⣉⣁⣀⣀⣀⣀⡀⠉⠙⠻⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣿⣾⣿⠿⠿⠟⠻⠿⠿⢿⣿⣶⣦⣄⡈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠿⠿⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁⠀⠙⢿⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⢀⣴⣿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⢿⣿⣷⣶⣦⣤⡀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣦⡀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣿⠏⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠘⣿⣷⣴⣷⡄
⠀⠀⠀⠀⠀⠀⣼⣏⠉⠉⠁⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠒⠛⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠂⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⣿⣿⣤⣤⣶⡾⠿⠿⠿⠿⠿⣶⣦⡀⠀⠀⠀⣠⣶⠿⠿⠿⠿⠿⠿⣶⣤⣤⣿⣿⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⡀⣀⣴⣿⠋⠙⣿⠃⠀⠀⢰⣾⠿⠷⣆⠹⣿⣶⣶⣾⡟⢠⡾⠿⢷⣦⠀⠀⠈⣿⡏⠙⢿⣧⣀⢀⣀⣤⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣿⠁⠀⠀⢻⣦⠀⠀⠈⠁⠀⠀⠈⢠⣿⠁⠀⢿⣇⠈⠀⠀⠀⠉⠀⠀⢠⣿⠁⠀⠀⣹⣿⠿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠻⣷⡀⠀⠀⠹⢷⣦⣤⣤⣤⣤⣶⠟⠁⠀⠀⠈⠻⢷⣤⣤⣤⣤⣤⣾⠿⠃⠀⠀⣴⡿⠁⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣜⢿⣦⠀⠀⠀⠀⠉⠉⠉⢀⠠⣆⠀⠀⠀⠀⢀⡶⣄⠉⠉⠉⠁⠀⡄⠀⣠⣾⣏⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣤⣿⣷⡐⢧⣀⣀⣀⡴⠋⠀⠙⠷⠶⠶⠶⠛⠁⠈⠳⣄⣀⣀⣴⠇⣴⣿⣧⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣹⡇⠀⠉⠉⠁⠀⢠⣀⣀⣀⣀⣀⣀⣀⣀⣠⠀⠀⠉⠉⠀⢸⣿⠀⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣧⡀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⣸⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠛⠘⢷⣄⣀⠀⠀⢀⣀⣴⠟⠈⠉⠉⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡟⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⡄⠻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⠃⣼⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⢿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⠿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣈⣉⣉⣉⣉⣉⣀⠀⠀⣀⣈⣉⣉⣉⣉⣉⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣦⣴⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠑⣶⣤⡀⢰⣤⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣻⣿⣷⣼⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣛⣛⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢁⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⣀⣤⣤⣴⣦⣤⣤⣄⣀
⣿⣿⣿⣿⣿⡇⠀⢀⣀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⣿⣿⣿⣿⣿⡇⢀⢸⣿⠃⠆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠈⠹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣿⣿⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣦⣴⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠀⠠⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⡀⢠⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠸⠀⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⡸⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⠀⢸⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⡇⠀⢸⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⡏⠀⢸⣿⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋
⣿⣿⣿⣿⣧⡀⢸⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⣿⡇⠀⠈⣿⣿⣿⣿⣿⣿⣭⣤⣤⣄
⣿⣿⣿⣿⡿⣾⣿⣿⣿⣏⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢀⠀⠀⠀⢸⡇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁
⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⠀⠀⠀⢸⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣦⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⡀⠆⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⡼⠃⠀⠀⠀⠀⣾⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣌⡙⠿⢿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠠⠈⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣷⣦⣤⣄⣀⡀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠻⠿⢿⣿⣿⣿⣾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠛⠀⠀⢙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣤⠤⣤⣿⠗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠈⡷⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠐⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⠠⠀⠀⣀⢼⣟⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⣾⡻⢤⣀⡀⠊⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⠠⠤⠌⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣼⢳⣎⢀⠑⠂⢈⢱⡞⣢⠀⠀⠀⠀⠀⠀⠤⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢏⡷⣭⢣⡜⢀⡄⠀⣮⣿⣼⢣⡄⠀⠀⠀⠀⠀⠀⠀⣀⠠⡦⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⣰⢎⡳⣭⢿⡹⢣⡍⠁⡄⠥⠙⠾⠛⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠈⠀⠈⠁⠉⠉⠉⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⢳⢎⡹⣵⠮⣽⠷⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠈⠀⠀⠀⠨⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠛⣮⡽⣯⣛⢶⣴⠐⢶⡽⣇⠀⠀⠐⢖⣶⣤⣤⡖⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⣠⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡝⢱⠾⣡⠟⡶⢱⡟⣷⢤⣀⠀⠈⠿⠟⠁⠀⠀⠰⢿⡷⠾⠟⢀⣠⣾⣏⣸⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣜⢃⡤⠀⠛⠁⠀⠀⠀⠀⠋⠛⠳⠶⢄⣤⣤⣤⣤⠴⠶⠖⠚⠛⢹⣁⣥⢼⡧⠘⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢿⢙⣇⣥⢴⡦⢼⣶⠄⣀⣤⡶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢮⣷⣄⣀⡀⠀⠀⣀⠀⠀⢄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠃⣸⡤⣷⠤⣷⠄⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠽⣟⡿⠿⠛⣹⡄⠀⠀⠈⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⠖⣿⠋⠉⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢰⡆⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⢸⡇⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠛⠋⠈⠁⠀⠀⠀⠀⠊⣾⣿⢦⣤⠖⠐⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠆⠀⠀⠀⠀⠈⠙⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢸⠱⠶⠆⠀⠀⠀⠀⠀⠸⠿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⣀⣀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⡿⠋⠉⠉⠉⠛⠿⠿⠛⠋⠉⠉⠙⢿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⢰⣿⡿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⣆
⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀
⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣬⣽⡿⠷⢦⣄⣀⣀⣀⣀⣠⡴⠾⢿⣭⣥⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄
⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁
⠀⠀⢀⣾⣿⣧⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⣾⣿⣷⡀
⠀⢀⣾⣿⠏⢹⣧⡀⠀⠀⠀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⢀⣼⠋⠹⣿⣷
⠀⢸⣿⡏⠀⠀⠈⠛⠿⠿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠿⠋⠁⠀⠀⢹⣿⡇
⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠋⢉⣩⣭⣽⣿⣦⣄⠀⠀⠀⠀⣠⣾⣟⣭⣭⣉⠉⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⢠⣿⣿⠏⠀⠀⠀⡲⢤⣤⣾⡟⠀⢀⠴⢻⣿⣿⣿⡌⠁⢻⣷⢒⡶⣾⠏⠁⣻⣿⣿⣿⡟⢦⠀⠈⣿⣷⣤⠤⣤⠀⠀⠀⠻⣿⣿⡀
⠀⢸⣿⣿⠀⠀⠀⠠⡝⢶⣿⡏⢳⡀⠀⠀⠀⢈⣉⡉⠉⠀⣼⠇⠀⠀⢻⡄⠈⠉⢉⣉⣉⠀⠀⠀⢠⠟⣿⣧⠶⠋⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠑⢄⠀⠀⠉⠉⠀⣀⡼⠃⠀⠀⠀⠀⠻⣄⡀⠈⠉⠉⠀⢀⡴⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠈⠻⣿⣿⣯⢿⣿⣿⣷⠀⠀⠀⠉⠀⠀⠐⠛⣩⠀⠀⠀⠀⠀⠀⠀⠀⣍⠛⠀⠐⠛⠉⠀⠀⠀⣿⣿⣿⡟⣽⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠈⠙⠿⣯⣿⣿⣿⡄⠀⠀⠀⠀⠀⣠⡾⠛⠲⣦⣄⣀⣀⣠⡴⠞⠉⢦⣄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⢻⣷⠀⠀⠰⠶⠛⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠛⠶⠆⠀⢀⣿⡏⠉⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⣿⣿⡆⠀⠀⠀⠀⢸⠀⠈⠚⠿⠿⠿⠿⠿⠿⠓⠀⠀⡇⠀⠀⠀⠀⢸⣿⣿⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣤⣾⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠇⣿⣿⣿⣿⣿⣶⣶⣦⣴⣶⣶⣿⣿⣿⣿⣿⠹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣤⣾⣿⡿⠟⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⠻⢿⣿⣶⣤⣀⣀
⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣯⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣤⣾⣿⣿⣿⠉⣹⣿⣿⣿⣷⣄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠉⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣠⣤⣀⣀⠀⠠⢀⣄⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⣐⣒⡂⠐⠢⣐⣢⣵⣦⣶⣀⣬⡙⣷⣤⡢⢄⡑⠢⣥⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠖⠀⠂⠀⠈⠓⠢⡀⠙⠿⣯⣻⢿⣿⣾⣿⣿⣷⣭⡲⣴⡙⢳⣟⢿⣷⣆⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠈⠛⠛⠿⠿⣿⣯⡻⣟⢾⣝⢿⠻⣿⣧⣿⣷⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠘⠢⠙⢎⢷⢩⠻⣏⣿⣧⢌⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠈⣿⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠠⡀⠀⣀⠀⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠄⠰⢀⣡⢰⣿⣼⣼⣆⣿⣷⣿⣿⣿⣷⣶⣦⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣌⢇⣳⣿⢷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢳⣷⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠿⣿⣿⣿⣿⣻⠿⣿⣿⠛⠻⠿⢿⣿⣿⣿⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣹⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⡋⠹⠤⠚⣿⣽⣿⣿⣾⠀⠰⣿⠀⠀⠀⠀⠈⠉⠛⢿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣭⣿⣿⣿⣿⠿⠟⠁⠀⠘⣆⠀⠀⠀⠀⠉⠙⠓⠁⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠂⠒⠂⣤⠄⠀⠐⠘⠦⣄⡀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠠⠂⠔⠂⡀⠀⠈⠉⠛⠓⠒⠛⠓⠒⠚⠛⠉⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠙⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠐⠒⠒⠒⠂⠸⠦⣀
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣰⢆⢀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⢀⡼⠁⠈⠣⡑⢄⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⢨⡀⢀⢠⡶⠋⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠌⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡄⠀⡇
⠀⠀⠀⠀⠀⣲⡸⠀⠀⡀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠸⣀⡃
⠀⠀⠀⠀⡜⠹⡁⠀⠀⠀⠁⠀⠘⠁⠀⠀⠀⠁⠀⠀⠀⢹⠉⠂
⠀⠀⠀⠀⠘⠤⢧⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠔⠃⠀⠀⠀⠀⠠⢖⣦⣠⠤⠤⠤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠁⠀⠀⠀⠀⠁⠀⣠⣔⠁⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⡀⠀⠀⠀⠀⠈⠂⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⠑⣖⠂⠒⠒⠐⠂⣫⠇⠀⠉⠲⢄⠀⠀⠀⠀⡜⡡⠒⠁⢣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢀⠔⠋⡇⠀⠀⢹⠑⠂⠀⠒⠊⠹⠀⠀⠀⠀⢀⠗⢄⡠⠲⠋⠀⠀⣀⠀⠑⢄⢸⡆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⡴⠁⠀⠀⡇⠀⠀⠘⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡌⠀⠈⢣⡀⠀⠀⠸⠟⠀⠀⠀⠉⠙⢆⠀⠀⠀⠀⢸
⠀⠀⢀⠎⠀⠀⠀⠀⠇⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢲⡇⠀⠀⠀⢧⠀⠀⠀⠀⠐⠀⠀⢀⣶⡆⠑⠤⣀⢀⠆
⠀⠀⡌⠀⠀⠀⣀⣼⠀⠀⠀⠀⡇⠀⠀⠀⠀⢃⠀⠀⠀⠀⢡⠀⠀⠀⠸⣆⠀⠀⢈⠀⠠⠀⠀⠁⠀⠀⠰⣥⠊
⠀⠀⡇⠀⠀⠀⠏⢸⠀⠀⠀⢰⠀⠀⠒⠒⠀⠘⡄⠀⠀⠀⠨⡆⠀⠀⢨⣄⣑⠤⣀⠁⠀⠀⠀⠀⠀⣀⡀⡸
⠀⠀⠘⣤⡶⠚⠧⣸⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⢀⡁⠀⠀⡸⠈⢧⣀⠈⣟⠐⠒⠒⠊⠉
⠀⠀⠀⠘⢤⣄⡤⠻⠦⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣀⠁⢀⡰⠁⠀⠀⠀⠉⠀⢳
⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠲⡇⠣⡀⠀⠀⠀⡠⠊⠉⡇
⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡶⠻⢆⡈⠁⢲⠉⠁⠀⡰⠁
⠀⠀⠀⠀⠀⢸⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠉⠓⠚⠦⠔⢺⠁
⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣷⣠⡑⢤⣀⡀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠢⣄⠀⠀⠀⡀⠀⠀⠀⢀⣠⡇⠇⠀⠉⠉⠛⠛⠛⢻
⠀⠀⠀⠀⠀⣼⠀⠘⡄⣀⣀⣀⣀⣀⣉⡀⠶⠤⠘⠊⠉⣀⣴⠀⠃⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠧⢄⣀⣀⣀⣀⣀⣀⡠⠤⡔⠂⠉⠀⠸⠀⠀⠀⠀⠸⢳⠀⠀⢸
⠀⠀⠀⠀⠀⡏⠀⠀⢸⡀⠀⠀⠀⡇⠀⠀⠀⠀⢣⠀⠀⠀⡇⢈⠄⠀⠀⠼⠸⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠈⢧⠀⠀⠀⠳⠀⠀⠀⠀⠈⢆⡀⠀⢇⠈⠇⠀⢀⡇⠀⢦⣤⣼⡄
⠀⠀⠀⠀⢀⡏⠀⠀⠀⡞⠲⢄⠀⡆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠿⢉⠹⠇
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠑⠢⠤⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⡰⠈⠉⠱⠀⠀⠀⠀⠀⣀⠤⠔⠊⠉⠉⠒⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⣠⠂⠀⠀⣠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⣄⠀⠀⠀⠀⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠊⠁⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠐⠀⠀⠘⠃⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⠀⠁⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⡀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁
⠀⠀⠀⠀⠀⠀⠠⠔⠒⠂⠀⠀⡀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠶⠥⠤⡄⢠⠤⠬⠶⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠐⠄⣀⠀⣀⠔⠀⠀⡕⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡗⢺⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⢀⡀⣀⣼⠀⠀⠀⠀⠀⠁⡇⢸⠈⠀⠀⠀⠀⠀⣧⡀⢀⠀⠀⠀⢀⡼⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣎⡠⠋⠹⣤⢤⣤⡤⢤⣠⣇⣸⣤⡤⣤⣤⣤⣤⠇⠙⢌⣱⠀⢀⡜⠀⡄⠀⠉⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⠞⠁⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠐⠢⡀⠀⠀⠀⠀⠈⠳⡔⠋⠀⠀⠈⠂⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⣀⢀⡀⡠⢄⡀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠈⠠⣄⠀⠀⠀⠀⠑⠢⠤⠤⠤⠤⠔⠊⠀⠀⠀⠀⡠⠈⠁⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢀⡠⣄⢀⡀⡀
⢀⠤⢧⡀⢸⠁⠀⡅⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠈⠉⠒⠠⠄⣀⢀⡎⠑⠊⢱⡀⣀⠠⠄⠒⠉⠁⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⣈⠀⠈⡇⢀⡼⠤
⢰⡄⠀⡘⠈⠀⠀⠨⣄⣀⠀⠀⡞⠁⢣⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⡜⠀⢳⠀⠀⣀⣠⠅⠀⠀⠇⢃⡀⢠
⢧⣀⣀⣁⠀⢀⣀⡀⠀⠈⠙⠐⢧⣀⣈⢦⡀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⢀⡼⣁⣀⡼⠃⠋⠁⠀⢀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠈⠁⠉⠀⠉⠓⠦⣤⣀⡰⠉⡝⠀⠙⢦⣄⡀⠀⠀⠑⠦⣄⣀⠀⠀⢀⠀⠀⠀⢀⣠⠴⠊⠀⠀⢀⣠⠴⠋⠀⠫⠉⢆⣀⣤⠶⠚⠉⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠁⠀⠘⠦⣀⡠⡎⠉⠛⠲⢦⠤⣤⣤⣭⣭⣭⣭⣭⣭⣤⡤⠤⡴⠖⠛⠉⢱⢄⣄⠴⠃⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠙⠦⣀⠤⠻⡄⠀⠀⡰⡄⠀⠀⣠⢆⠀⠀⢠⠟⠤⡀⠴⠋⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠉⠉⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⢤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡤⠄⠞
⢀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠟⠛⢛⣛⠒⠒⠒⠒⠒⠒⠒⠒⠒⣒⠛⢛⡛⠿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠠⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠊⠀⠀⠀⡄⠀⠀⠀⠙⡔⠀⠀⠈⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⠀⡠⠀⡁⠀⠀⠀⠀⠀⠦⠀⢀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⠀⠃⠀⠧⣀⠀⠘⠂⠃⠀⠀⠀⠀⢀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⠆⠀⠀⠐⠠⢄⡀⢈⠀⠰⠂⣀⠀⠠⠆⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠂⠒⠤⢤⠤⠂⠘⠃⡀⠀⠀⠀⠄⠀⠀⠀⢌
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠄⡀⠀⡈⡁⠔⠌⠤⠇
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡀⠀⠀⠀⠒⠊⠀⠀⠀⠘⠢
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠄⣀⠀⠀⠀⠀⠀⠠⠚⡀⠱⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢡
⠀⠀⠀⠀⠀⠀⢸⠀⣀⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠁⠈⠂⠀⠀⠀⠀⠤⠐⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠊⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡏⡅⠀⠀⠀⡈⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢐⠀⠇⠀⠀⠀⠇⠀⠀⠀⠀⠈⠑⠂⠒⠒⠒⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠅⢰⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠀⠀⠐⠤⠀⠔⠀⠀⠀⠀⠀⠀⠀⠸⠰⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠚⠄⠀⠀⠌⠒⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢁⣀⣠⣰⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡀⠈⠁⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣿⡿⠿⠛⠛⣿⣇⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡿⠿⠓⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠐⢧⡀⢈⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠅⣀⣓⠁⠀⠀⠀⠀⢀⠀⠀⠀⣬⣷⠟⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡉⠓⠂⠂⠁⠊⠀⠀⠀⢀⠀⢸⣷⡂⠀⠀⠈⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡀⠀⠀⠀⠀⠀⠀⢼⣧⠀⠀⢸⠀⢘⣿⡇⠀⠀⢺⠀⠀⠀⣾⣿⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡏⠀⠀⠀⠀⠀⠀⢸⣿⣦⣄⣿⢴⡶⢻⡟⠀⠀⠀⠀⠀⢸⣿⠹⢷⢤⡘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣾⣷⡘⢫⠚⠁⠀⠀⠇⠀⠀⢹⡇⠀⠈⣿⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⣿⠻⠑⠢⣤⣤⣤⣤⣤⣶⣶⣾⡿⠟⢿⣷⡈⠀⠀⠀⠀⢀⠀⠀⢸⡇⠀⢐⠙⢳⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠿⠃⢠⡀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣇⠀⢸⣿⣧⡸⡄⠀⠀⣼⠀⠀⠀⣿⠐⣿⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢾⡦⠀⡀⠈⠐⠶⠴⠾⠿⠶⠶⠶⢊⣃⠀⢸⣿⡟⠀⠹⣶⣶⣿⣶⣶⣾⣿⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠰⡶⣶⣶⡶⡶⣶⣿⣿⣿⠀⣾⣿⣷⣤⣔⣿⣿⣿⣿⣿⣿⡏⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⡟⠁⠛⠛⠛⣿⠀⢹⣿⡿⣫⣾⢿⣿⣿⣿⣿⣿⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢠⡏⠀⢸⣿⣷⣿⣿⠈⣿⣿⣿⣿⣿⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣤⠀⢰⣿⣿⡄⠀⠀⠀⣾⠃⠀⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⣿⡄⢸⣿⠘⣧⠀⠀⢸⡿⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣿⣧⠀⣿⡁⣿⡄⠀⢸⡇⠀⠀⢸⣿⣿⣿⡇⠀⠈⣿⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣼⣿⠁⡿⣷⡀⣿⡇⠀⠀⢸⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠁⠉⢿⣿⠀⡇⠁⠀⣿⡇⠀⠀⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⠀⢸⣿⡆⡇⠀⠀⢻⠀⠀⠀⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⡏⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢨⡅⣼⣵⠃⢡⠀⠠⡌⠀⠀⢀⠙⠛⢿⣿⡃⠀⠸⣿⡿⣿⡇⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⠳⠖⠞⠁⠀⠸⠦⠴⠇⠀⠀⠷⠾⠾⠛⠋⠀⠀⠀⣄⣴⣿⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⢋⣉⣉⣉⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣰⠞⢋⣩⣭⣍⠛⢦⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣧⡤⠷⠶⠞⠓⠒⠚⠒⠶⠶⠶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⢉⣡⠶⠁⣠⡴⠂⢀⡾⢤⣀⠙⠶⣤⡈⠻⣯⣝⡳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣁⣤⡶⠛⣁⡴⠞⠉⢀⣴⠟⣠⢤⡉⠛⢦⣄⠉⠓⠦⣍⡛⠷⢮⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⠚⠛⣉⣡⠴⠚⠉⠀⣠⣴⠟⣡⡞⠁⠀⠙⠷⣤⣈⠛⠦⣄⡀⠉⠛⠲⠶⠤⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠰⠶⠚⠛⠉⢀⣀⣤⡶⠟⣩⡴⠟⠁⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⣉⠛⠶⠦⣤⣤⣀⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣤⣤⣤⠶⠶⠞⢛⣩⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠙⠛⠶⢦⣤⣄⣉⡀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⣠⣤⣤⠶⠶⠛⠋⢁⣀⣤⡴⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⡀⠀⠀⠀⠀⠉⠉⢹⣥⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠓⠦⢠⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠀⠀⠀⠀⠠⠟⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣸⡆⠀⠀⠀⠀⠀⠀⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⡀⠀⠀⠀⠀⠀⢀⡿⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠉⠁⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡛⠳⠶⢚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⣤⣤⣄⣀⣀⣀⣤⣤⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣾⣿⣿⣿⣭⡭⢝⣛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣟⢷⡮⡳⢔⠭⣊⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⣰⣿⣿⣿⣿⣕⡭⣊⠕⠊⠔⠪⣦⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠙⠀⠀⠀⢸⣿⣿⡿⣿⣝⡻⣮⢉⠓⠦⣤⡐⢿⣿⣾⣯⣉⣉⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⡷⣝⢪⡲⣕⢦⡀⡳⣝⣦⣽⣿⡯⠓⣤⣄⡑⣷
⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣮⡳⡱⡌⢳⡑⢈⢻⣿⣿⡟⡠⡺⠛⢽⣿⣾
⠀⠀⠀⠀⢠⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠙⠋⠉⠻⣗⠔⡙⣄⠙⢦⡹⣿⣿⣿⠈⢀⠀⣿⣿⡃
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦⠘⣎⢪⣳⣽⣿⣿⣿⢸⢨⢦⢫⢸⠃
⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⡳⣌⠿⣾⣿⣿⣿⣿⣿⣿⣊⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⡝⣝⢮⣓⣚⣫⠵⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡻⣝⠮⣕⣒⣲⡴⠞⡽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣕⠮⣟⣒⣒⣦⠴⢋⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡫⠖⢒⣒⣬⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠈⠀⢄⠀⠄⠀⠠⠤⡄⠀⠀⠀⡀⠀⠀⠀⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠂⠀⠀⠀⠒⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⣀⣀⣐⣀⣀⣀⣀⣁⡉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠙⠀⠀⠠⠠⢄⠀⠀⠀⠀⠀⡇
⠀⠀⡇⢀⠤⠀⠀⢄⠀⢠⠔⠢⡀⡔⠒⢄⠁⠀⠀⠀⠀⠀⡎⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⡸⠂⠀⠀⠀⠀⠁
⠀⠀⡇⡎⠀⠀⠀⠀⢡⠘⢄⠤⠃⠣⣠⠎⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠶⠲⡄
⠀⠀⡇⠇⠀⠀⠀⠀⠸⢰⠁⠉⡄⡋⠈⠁⠀⠀⠀⠀⠀⠀⡉⡈⣁⠀⠀⣀⣀⡀⠀⢀⠀⢠⠀⠀⠀⠘⢄⡀⡀
⠀⠀⡇⠈⠢⢄⡠⠔⠁⠈⠒⠊⠀⠁⠒⠁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠉⠈⠉⠉⡉⡉⠁⠀⠀⢀⠀⠠⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠁⠀⠀⠀⠡⠀⡀⠀⠀⠀⠀⠀⠀⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢩⡋⠽⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⡀⠇⡄
⠀⠀⢰⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠔⠂⠐⢀⠀⠀⣇⠆⠀⢹⢠⠎⡗⠀⡔⠉⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢀⠀⠀⠀⠘⢤⣠⣤⣎⡫⠃⠀⢇⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠉⠢⠈⠁⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⢀⠀⢀⡀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠸⠀⠀⠆⠀⢡⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⡀⣨⠑⠒⠀⠁⠀⠀⠀⠀⠈⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣡⠶⠚⣿⠋⢁⢘⣧⡀⠀⠀⠀⠂⠀⠀⠰⣾⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡵⠃⠀⠀⠈⠙⠶⢾⠿⠛⠛⣶⣆⣀⠐⢉⠲⣄⠋⢿⠀⠀⠙⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡉⠀⠀⠺⣿⣷⣾⣿⡏⠀⠀⠀⢂⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠿⠛⠛⠿⠻⣿⣿⣯⣭⡒⠀⠀⠀⠀⣿⢣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠘⣷⡀⠀⢄⠀⠀⠀⢂⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⢶⡄⣠⠏⡙⢿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡐⣿⣿⣦⣀⣀⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢷⡙⠂⠑⠄⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⢰⣿⣿⣿⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⢆⠘⢠⠏⢸⣿⠏⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣶⡎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠘⠂⠁⠀⢺⡇⢠⣰⣶⣾⠷⢶⣶⣤⣄⣠⠀⠰⢀⣤⢴⣶⠶⠾⢶⣶⣄⠘⣷⡀⢈⠼⣿⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡑⠄⠀⠀⢀⣿⣵⡟⠁⢀⣀⣷⣶⣅⡄⢼⣷⠶⠶⣿⢥⣄⣡⣾⣇⣀⠀⠉⣷⣼⣷⢀⢲⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⣀⣠⡴⢿⡟⣿⠿⠛⢯⠀⠀⠀⠀⢸⡏⠀⠀⣿⠀⠀⠀⠀⣨⠿⠻⢿⡿⣹⠿⣼⣸⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢥⣨⣿⣿⡇⠙⠦⣄⣀⣀⣀⣤⠶⠛⠀⠀⠀⠈⠳⢦⣄⣀⣀⣀⡤⠞⠃⣿⣶⣾⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⢷⡌⠀⠀⠉⠉⠀⢠⣾⣃⣀⣀⣀⣀⡱⣆⠀⠀⠉⠉⠀⠐⣰⡿⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢿⡶⠀⣀⢤⣤⠀⠀⠁⠈⠉⠉⠉⠀⠈⠠⣄⣀⣐⢢⣶⡟⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣇⢼⠉⠀⠈⠒⠲⠶⠶⠶⠶⠶⠶⠒⠀⠘⠛⢸⠂⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣄⣠⡀⠀⠀⠀⠀⢐⡀⢥⠴⢛⢽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡎⢻⣿⣷⡉⠉⠙⠿⢶⣶⡖⠊⠉⠀⠀⠀⠀⣼⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠒⠉⠁⢠⠀⢻⣧⣌⠲⡅⠀⠀⠀⠀⠐⠂⠀⠀⣠⢞⡟⢸⡇⣟⡗⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠋⣉⠀⠀⢠⡄⠀⠀⠈⡄⠀⢣⡙⢧⡀⠀⠀⠀⠀⠀⠀⣠⠞⣁⠟⠀⡟⠀⠈⠺⠻⢳⢉⡓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⠀⠀⠀⣠⣾⠵⠛⠉⢳⣤⠃⠀⠀⠀⠀⠁⠈⠜⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⢀⡀⠀⠀⠁⠀⢠⠄⠀⠉⠳⣄⣾⠟⠁⠀⠠⠄⠀⠉⠀⠐⠛⠀⠀⠲⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠃⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠲⠀⠀⠘⠀⠀⠈⠁⠀⠀⠀⠀⠀⠆⠀⠀⠉⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠖⠒⠒⠒⠒⠲⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⢀⣀⣠⡤⣤⣀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣀⣤⢤⠖⠋⡉⠉⠉⠉⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⣠⠞⠉⠈⠀⠀⠀⠈⠳⣤⣀⠀⠀⣀⠔⠀⠹⡄⠀⠀⠀⠀⡾⠁⣠⠶⠛⠋⠉⠀⠀⠀⠛⠓⠶⣌⡳⣄
⠀⠀⠀⠀⠀⡼⠁⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⡁⢀⡠⢺⡁⠀⠀⢀⡼⣡⠞⠁⠀⠀⠀⠀⣠⠶⣄⣀⠀⠀⠈⠳⣌⢳⡀
⠀⠀⠀⠀⠀⠹⣄⢀⡞⠀⠠⠀⠀⢠⠀⠀⠀⠀⢀⠐⠀⠀⢄⠹⣅⣠⠞⠁⠀⣠⠟⢰⠏⠀⠀⠀⢀⡴⠞⠁⠀⠀⠉⠳⣄⠀⠀⢹⡆⢷
⠀⠀⠀⠀⠀⠀⠈⢿⠀⠈⣀⡤⠤⣀⡀⠀⠀⠀⣠⠤⠢⠤⡈⠀⣟⠁⠀⠀⢸⠁⠀⢿⣤⣤⠴⢚⡉⠀⠀⠀⠀⠀⠀⣀⠈⠳⢦⣤⡿⠈⠓⣦
⠀⠀⠀⠀⠀⠀⡤⢼⡇⡜⠁⠤⠄⠀⢹⣄⡤⢼⠁⠐⠲⠄⢸⠂⣿⠓⢦⠀⢈⡇⢀⣠⡇⠀⠀⠉⢀⡀⠀⠀⠀⠀⢀⡈⠑⠀⠀⣿⣲⠚⢠⠏
⠀⠀⠀⠀⠀⠸⡅⠀⡇⠹⢤⣀⣀⡤⠞⡄⠀⢀⠓⠒⠒⠒⠁⠀⢿⢀⡜⠀⣼⠀⣾⠀⣿⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⠋⠀⠀⠀⡏⠈⣷⠈⣷
⠀⠀⠀⠀⠀⠀⠙⠶⡇⠀⠀⠀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⠀⠀⣽⠉⠀⠀⠘⢦⠘⠦⣿⠀⠀⠀⠀⠀⠀⠣⠜⠀⠀⠀⠀⠀⠀⣇⡴⠋⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠹⣖⠒⢒⡿⠁⠀⠀⠀⣠⠇⠀⠀⠀⠀⠘⣆⠀⢸⡀⠀⠀⠀⠀⢰⠒⠒⡒⠀⠀⠀⠀⢰⡇⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⡀⠈⠉⠉⠀⣀⡤⠔⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠲⢷⣄⡀⠀⠀⠈⢦⡤⠃⠀⠀⠀⣠⣞⡤⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⢹⠉⠉⠉⢹⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⢦⢤⣤⢤⠖⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⢹⡄⠀⠘⠲⡤⠞⠉⠀⢠⠏⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣶⠞⠹⣫⠽⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠹⣦⣠⠞⠉⠳⣄⣴⠋⠀⠀⠀⠪⣳⡄⠀⠀⠀⠀⠀⢀⡴⠋⠀⢸⣿⠀⠀⠀⠚⣹⠂⣷⠈⢳⡀
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢻⣳⡀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠐⢹⣆⠀⠀⠀⢀⡞⠀⠀⠀⢸⣿⠀⠀⠀⡼⠁⠀⣿⠀⠀⠹⡄
⠀⠀⠀⠀⢠⡟⠆⣀⠀⠀⠀⠀⠀⠀⢳⣷⣼⡿⠁⠀⠀⠀⠀⠀⢀⡀⢳⢻⡄⠀⠀⡞⠀⠀⠀⣀⡼⠋⠙⠲⠴⡧⠤⠴⢿⡄⠀⠀⢹⡀
⠀⠀⠀⠀⣿⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢏⣷⠀⢸⠃⠀⢀⡼⠉⠀⠀⠀⠀⢸⠃⠀⠀⠀⠳⡄⠀⠈⡇
⠀⠀⠀⢠⡏⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⢹⡀⣾⠀⢠⠏⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⢹⡄⠀⡗
⠀⠀⠀⢸⡇⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⣧⠀⡇
⠀⠀⠀⠀⣷⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠀⣇⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸⡄⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠙⣦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢀⣴⠃⠀⢹⣄⠀⠀⠀⣀⣤⣾⠁⠀⠈⢧⡀⠀⠀⠀⠀⠀⢾⡆
⠀⠀⠀⠀⠀⠈⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠋⠁⠀⠀⢸⠏⠙⠛⠉⠉⠀⣿⠀⠀⠀⢨⡗⠶⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⡿⠛⠋⠉⠉⠉⠙⠛⢿⣿⣷⣶⣶⣶⣶⣤⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠛⠛⠻⢿⣿⣿⠿⠿⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀
⠀⠀⠀⠀⢀⣾⣿⡿⠟⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆
⠀⠀⣠⣾⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣾⡃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀
⠀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠰⣿⣇⠀⣿⡄⠀⠀⠀⠀⠀⠀⢀⣼⠃⣠⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⠈⢿⣿⣷⣦
⣸⣿⡿⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣿⣦⣿⣿⣦⣀⠀⣀⣠⣤⣾⣷⣾⣿⠟⠁⠀⠀⠀⣀⣀⣤⡶⠃⠀⠀⠈⠻⣿⣷
⣿⣿⡇⠀⠀⠰⣇⠀⠀⠀⠻⣦⣄⣀⣀⣀⣀⣠⣤⠆⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠉⠙⠛⠛⢿⣿⣯⣉⣁⣀⠀⠀⢸⡆⠀⢹⣿
⣿⣿⡇⠀⠀⠀⠙⢿⣶⣶⣶⣾⣿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣉⡉⠉⢀⣠⣿⠃⠀⢸⣿
⠸⣿⣿⣄⠀⠀⠀⠀⠀⠉⢁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⠿⠛⠁⠀⢀⣼⣿
⠀⢻⣿⡟⠃⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡙⠳⠄⠀⠀⠴⢿⣿⣿
⠀⢸⣿⣧⠀⠀⠀⠀⠉⢹⣿⠁⠀⠀⣠⣶⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⢿⣦⡄⠀⠀⢿⡏⠁⠀⠀⠀⠀⣸⣿⡟
⠀⠈⢿⣿⣧⡀⣿⣶⣤⣼⣷⣶⣶⣾⣿⠿⠶⢶⠶⠿⣿⣧⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢷⣶⠶⠶⢾⣿⣶⣶⣾⣧⣠⣴⣿⠀⣴⣿⣿⠃
⠀⠀⠀⠻⣿⣿⣾⣿⣿⣿⣿⣿⠟⠉⠀⣰⡾⢿⣿⣿⣦⡙⢻⣷⣤⣤⣤⣤⣴⡿⢋⣵⣾⣿⣿⠿⣶⡀⠈⠙⢿⣿⣿⣿⣿⣿⣾⣿⡿⠁
⠀⠀⠀⠀⠸⣿⣿⡝⠻⢿⣿⡟⠀⠀⠈⠁⠀⢿⣧⣿⠿⣿⡆⢿⣿⡿⢿⣿⣿⠁⣾⡟⢿⣼⣿⠆⠀⠉⠀⠀⢸⣿⣿⠟⢩⣿⣿⠏
⠀⠀⠀⠀⠀⠹⣿⣿⡄⠈⣿⣧⠀⠀⠀⠀⢦⣀⣉⣠⠄⠀⠀⣸⣿⠁⠀⢹⣿⠀⠉⠀⢨⣍⣀⣠⠆⠀⠀⠀⢸⣿⡇⢀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⢹⣿⣄⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⣿⠃⠀⠀⠀⢿⣇⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⣿⡿⢀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⢻⣿⣄⡀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⣀⣾⡿⠃⠘⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⣀⣼⣯⠛⠿⠷⢶⣶⣶⠾⠟⠋⠰⢾⣶⣤⣠⣴⣾⠷⠈⠛⠿⠶⣶⣶⡶⠿⠟⣻⣧⣀⣠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣆⠠⣤⣤⣤⣤⣶⠏⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⢷⣦⣤⣤⣤⠴⢠⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⠀⠉⠉⠉⠀⠒⠶⣤⣤⣤⣤⣤⣤⣴⣤⡶⠖⠂⠈⠉⠉⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠛⠿⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠟⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣷⣦⣤⣀⣀⣠⣤⣾⣿⠃⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⡇⠹⠀⠀⠈⢿⡏⠉⠛⠛⠋⠉⣿⠃⠀⠀⠘⢀⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⡀
⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠘⢇⠀⠀⠀⠀⢸⠇⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠉⠉⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠲⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢂⣴⠶⣿⣿⣿⣅⡀⠀⢀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⡰⠋⠀⣠⣿⢉⣧⡙⡙⢷⡀⢀⠢⡀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⣀⡴⢡⣿⠘⣿⣧⢹⡌⢻⣄⢣⠘⢦⣀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⣀⣤⣤⠶⠛⣁⣴⠟⢹⡆⠙⠛⠦⣷⡀⢻⣎⢧⠀⠙⠷⣤⣌⣙⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⣾⡿⠛⢉⣠⣶⠿⠛⠁⠀⠀⢿⡀⠀⠀⠸⣷⠄⠹⣷⡀⠀⠀⠀⠀⠉⣉⠬⠙⢧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠙⣧⡀⢉⠻⢦⣤⡤⠖⡋⠁⠀⠀⢀⣤⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡀⠀⣴⣿⣿⣿⠒⠶⣄⠀⠀⠀⣠⡴⠞⢿⣄⠀⠀⠈⠻⢦⣿⣶⡶⢖⡉⠄⠀⢖⣤⣿⣗⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣾⠁⣸⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢀⡴⢾⣿⣿⣮⢳⣦⣤⣤⣤⣴⣶⠿⣷⣶⣾⣿⣎⢿⣏⠳⠄⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣿⣿⣿⣿⡏⠉⠉⠀⣰⠀⠀⠀⠈⠈⠉⠉⠘⢿⣽⣿⣿⣿⣿⣿⣷⡘⢆⠹⣷⡹⣟⢿⣷⣶⣤⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⣿⡿⢹⡇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⡜⣿⡿⢹⡟⡞⡄⢹⣷⢹⣶⣽⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠟⢀⣿⣿⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⡿⠸⢧⡿⠁⣸⠧⠀⣿⢸⣿⡇⣿⡆⢿⣆⢙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣾⣿⣿⣷⡄⢠⣶⣶⢶⣤⡄⠀⠀⠀⠀⠀⡸⢁⣴⠟⠁⣰⠏⡀⡰⡏⢸⣿⣷⠘⣿⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣾⢫⣶⣿⣿⣿⣷⡈⠛⠷⠖⠉⠀⠀⠀⢀⣤⣎⣴⠟⠁⣠⡞⠁⣰⠟⣸⡇⣼⣿⣿⠀⡿⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣸⠁⢠⣿⠙⣿⣿⣿⣷⡀⠀⠀⠀⠀⣠⣴⣿⡿⠋⢀⣠⠾⢋⣠⠞⠁⢰⣿⠁⢿⣿⣿⡾⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⡏⢸⣌⢻⣿⣿⣿⣷⣶⡾⣿⣿⡿⠋⣀⣴⠟⣡⠶⢋⣤⠶⠖⠛⣸⣢⠸⣿⣿⣱⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠘⣷⠀⠻⣷⣝⣿⣿⣿⣿⠇⣿⠟⢠⣶⢟⡵⠞⠁⣠⡯⠔⣲⣯⡴⠛⠻⣦⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⢳⡀⠘⣦⠑⠸⡏⢻⣿⣿⠏⣰⡏⣰⣿⢣⡞⠁⣤⠞⠁⣠⠞⣽⡟⠀⠀⢀⣼⣦⣿⣿⣯⡳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣸⠇⢸⣷⡀⠘⣧⠀⢰⣼⠟⠁⣴⣿⣀⣿⢣⡟⢀⣾⠁⠀⣴⠏⠘⣿⠀⠀⢠⠟⠁⠀⢿⡄⢧⠻⣮⡙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡏⢠⣿⣿⣷⠀⠘⣇⠈⠁⣠⣾⢻⣿⣿⣿⠸⡇⣼⡇⠀⢸⡿⠀⢸⡟⠀⠀⠰⠄⠀⠀⠈⢧⠈⢧⡈⠻⣦⡙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⢸⡇⢹⣿⠀⡀⢿⣠⡾⢿⡿⢸⣿⣿⣿⣇⢸⣿⠁⠀⣿⡇⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⢻⣦⡈⠛⢦⡈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡇⠀⣿⣠⣷⠿⠋⣠⢪⠃⣼⣿⣿⣿⣿⡌⣿⠀⠀⣿⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣿⢦⣄⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⢀⣷⡴⠟⠋⠀⢀⡴⢡⠎⣰⣿⣿⡟⢸⣿⣷⣿⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠈⢳⡍⠳⢦⡻⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⠟⠁⣀⣀⡤⠔⢁⣴⢋⣴⣿⣿⣿⡇⡄⣿⣿⣿⠀⠀⣿⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡻⣄⠀⠙⢶⣄⡉⠻⠄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠗⠛⣉⣉⣡⣤⡶⢟⣵⣿⣿⠏⣼⣿⢰⡇⣿⣿⣿⠀⠀⣿⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣍⠳⣦⣀⠉⠻⢶⣤⡿
⠀⠀⠀⠀⠀⠀⢠⠞⣁⣤⣶⣿⣿⡿⡿⠿⠿⣿⣿⣿⠋⣼⣿⠇⣼⢡⣿⠏⣿⠀⠀⣿⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣿⣿⣷⣌⠙⠿⣿⠛⢹⠁
⠀⠀⠀⠀⠀⠀⣿⣿⡿⢛⣭⣥⣴⠾⡷⢶⣶⣞⣋⣡⣾⣿⠟⠠⢣⣾⡟⢰⣿⠀⠀⣿⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⣻⡿⣿⠈⣷⣀⡿⢀⠆
⠀⠀⠀⠀⢠⡾⢋⠏⢰⣿⣿⡿⣡⢫⡆⢾⣿⠉⣿⣿⡿⢋⣀⣴⠿⡿⢁⣿⡇⠀⢀⡇⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⣀⣴⣾⠟⠀⠿⣳⡿⢋⡴⠃
⠀⠀⠀⠀⢻⣅⠈⢧⠘⣿⣿⡇⣿⢸⣧⡈⠉⢶⣮⣛⣻⣛⣋⣥⡞⢁⡾⣿⠁⠀⣸⣷⠟⢧⣀⣀⢀⣀⣠⠴⠒⠋⠉⢉⠾⠋⠁⢀⡰⣶⣯⠖⠋
⠀⠀⠀⠀⠀⠙⠳⣤⣙⠺⣿⣿⣼⣆⠻⠦⣭⣜⠛⢿⣟⣿⡿⠋⢀⠞⣽⠃⠀⣰⣿⡇⠘⣷⣬⣉⣉⣁⣠⣤⣤⣀⣤⣤⡤⠶⢚⣩⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣍⡙⠛⠿⠶⠦⠽⠷⠾⠛⠉⢀⠴⣋⡾⠁⢀⣴⡿⣡⣯⡾⠛⠛⠛⠋⠉⠀⠀⣽⣿⣿⠙⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⣭⣙⠲⠤⣀⡀⠀⠀⠀⣈⣤⠞⢋⣠⡴⠻⣿⠀⣿⠁⠀⣀⠔⠒⠒⢒⣒⣚⡋⠭⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⡋⠛⠻⢿⡿⣷⣶⠞⠋⠹⡇⠰⠙⠦⠬⢉⣉⠴⠖⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠛⠛⠉⠀⠀⠀⠀⠈⠑⠒⠻⠛⠋⠁

⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠛⠛⠛⠓⠶⣤⣀
⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⣀
⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⢣⣤
⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢴⣿⣿⣿⡥
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢞⣛⢿⣿⣶⣀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠉⣣⡔⢲⣿⣿⣿⣻⠞⠁
⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢉⠽⣷⡞⣟⣏⣯⣮⡤
⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣶⣦⣿⣻⣤⣾⣯⢘⣿⡿⣿⠛
⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣃⡷⣾⣿⢿⣏⢹⡿⢿⣋⣴⣚⠇
⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⡾⢒⣾⡭⢷⡤⣿⠛⢿⢧⠜⠛⣿⣯
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢃⡿⣥⡋⡟⣌⣦⣧⢸⠙⢄⡟⠍⣡⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡿⣍⣿⡏⠳⡬⠿⢯⡭⢊⡜⡼⢀⠞⣧⠞⢿⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⡠⠄⢀⢐⠛⢏⣇⣰⡇⠸⡦⠬⣯⣉⡩⠝⣩⠗⢁⡠⣿⠒⢂⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠐⢷⣾⣾⣸⣶⣊⣉⣹⡟⢢⣍⣒⣦⠦⠔⠚⢓⡪⢿⢭⠈⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣌⠙⣿⣿⣯⣉⣡⣿⣿⢀⡞⠙⠒⢋⣱⢶⡲⢭⠥⠤⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠋⡽⠁⣻⡇⣯⠤⢴⣒⣪⠟⠓⢤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠟⠁⠛⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡥⠄⠆⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠀⠀⠀⠀⠀⠤⠄⣀⠀⠀⠙⣧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠑⠀⠀⠀⢀⠠⠂⠀⢀⠀⠀⠈⡱⣄⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠡⠄⠒⠀⠈⠀⠀⠀⠀⠀⠑⢄⠠⠈⢪⢳⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠘⠢⣄⣀⣀⣀⠤⠊⠀⠀⠀⣠⠃⠀⠀⠠⡅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠖⠒⠠⠄⡀⠀⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠲
⠀⠀⠀⠀⠀⠀⢈⡵⠋⠀⠀⠀⡀⠀⠀⠈⠢⠐⡡⣿⡉⣑⣶⠒⢲⣶⢒⠍⠢⠀⠀⠀⢢⠀⡀⠂⠛⢀⡀
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡟⠀⡏⠣⣀⣄⣉⣰⠃⡎⠀⢀⡦⠖⠂⠁⠀⠈⠑⡄⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢠⡇⠰⠁⠀⠀⠈⠉⠀⠐⢇⠀⡈⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠈⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⢉⡰⠃⠇⠀⠀⠀⠁⠀⠀⠀⠀⡀⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠤⠒⢹⡦⣀⠀⠀⠑⠀⠀⠀⣄⠀⠀⠁⠀⠘⠣⡐⠄⡀⠀⡀⠜⠀⠀⢀⡴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠂⠋⠀⢨⠠⠀⠀⠠⠀⠀⠀⠀⠠⡄⠀⠋⣀⠈⢢⠠⢀⢀⡀⠄⠐⠉⠀⠃
⠀⠀⠀⠀⢀⡀⠀⠀⠸⣀⢀⠃⠀⠀⠀⢘⣸⠦⠀⢀⠀⠄⢀⠒⠀⠀⠀⢀⣠⠇⠉⡀⠘⠀⠀⡃⠀⠀⠀⠤⣷⠭
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠤⠀⠐⠒⠉⢈⠆⠀⠀⠀⢐⡆⠀⠒⠚⠉⠁⠉⡢⠀⣄⠀⠀⠀⢣⠀⠀⠀⠀⠘⠃⠀⠀⠀⠈
⠀⠀⠐⠂⠄⠙⡆⠄⠠⠀⡈⠀⠀⣀⣴⣠⢞⠀⠀⠀⠀⠈⣀⢀⣀⡠⠠⠆⠊⠁⢘⡉⠉⠒⠒⠈⠀⡄⠠⠀⠀⠀⠀⠜⠁⠪⢀
⠀⠀⠈⠢⠄⡀⠉⣠⠀⠋⠉⠉⠉⠉⠀⠀⣈⠈⠀⠀⠀⠈⠀⠀⠀⢠⠤⠠⡀⠀⠀⠀⠈⠈⠁⠨⣀⠀⠀⢀⠀⣀⡠⠤⠀⠄⠂
⠀⡄⠀⠀⢂⠀⠀⠜⠀⣆⠘⠈⠑⠀⠈⠃⠁⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⢀⠐⠁⠉⠒⢖⠃⠀⠐⠋⠰⠄⠀⠐⠹⠃⠆⠢⠄
⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣠⡔⠋⠈⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⡀⠀⠁⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡈⠭⠭⢁⠢⢒⠩⠭⠭⡐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⠂⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠡⠐
⠀⠀⠀⠀⢀⡀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠇
⠀⢀⠄⠐⡡⠂⠉⠁⠢⡑⠄⠀⠀⠀⢀⡀⠐⡌⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡐
⡐⡡⠊⠀⠀⠀⢀⠠⠄⠓⠀⠀⠀⠀⠒⠠⠅⡚⢦⡑⢄⠀⠀⠀⠀⠀⢀⠔⣡⢊⣀⠄⣂⠭⠀⠀⠀⠈⠭⢐⠂⠄⡀
⡇⡅⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⢢⡑⠤⡀⡀⠔⡁⢀⠅⠐⠢⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⡂⠄
⠐⡌⡲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠹⡑⠤⠔⡘⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⢢
⠰⢰⠀⠀⠀⠀⠀⣀⣀⠀⠤⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⢡⠡⠀⠰⠰⠀⠀⠀⠀⠀⠄⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢃
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠒⠁⠐⢄⡀⠀⠀⠀⠀⠄⡀⡄⢀⣀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤⠣⣠⠀⠀⠀⠀⠀⠸⠈
⡰⠸⠄⣤⡀⠬⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⣄⠀⠔⣐⠢⡙⢰⠁⠘⠛⢤⠋⢀⣀⣀⠂⠀⠀⠀⠀⠀⠀⠑⠇⠀⠀⠀⢀⢀
⡇⡁⠀⠀⠈⠇⠠⠋⠉⣑⠄⠀⠀⠀⢔⡉⠉⠳⠀⠛⠀⠀⠀⠐⠸⠀⠀⢀⠈⠀⠊⠀⣨⡁⠀⠀⠀⢀⠔⠒⢦⠀⢮⠌⠥⠈⢂
⠡⠰⡀⠀⠄⠀⠀⠀⠘⠛⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠢⠀⠠⢁⢆⠣⡀⡁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠷⠀⠈⠀⠀⠀⠀⠀⡰
⠀⠑⠀⠂⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢒⠡⠂⠀⠑⢢⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⢀⠔⡡
⠀⠀⠀⠈⢆⢂⠀⠀⠀⠀⠐⠂⠀⠐⠂⠀⠀⠀⢀⠎⠌⠀⠀⠀⠀⠀⠀⠆⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡭⠅⠊
⠀⠀⠀⠀⠀⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠣⠊⠀⠀⠀⠀⠀⠀⠀⠈⢢⡣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠎
⠀⠀⠀⠀⠀⠀⠈⠂⢅⡂⠤⠤⠤⠤⠄⣂⠡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢑⠂⠤⠄⠀⠤⠄⢒⠡⠀⠁
⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣤⣤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⡀⢈⡹⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⣒⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⣀⡤⢤⠾⠟⡀⡠⠲⢭⣱⣤⢄⠀⠈⡓⠚⠃⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⠛⠁⠀⠀⠈⠀⠀⠀⠉⠟⠶⣄
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢇⡁⢀⠀⠀⠙⠻⠾⠵⢵⡧⠴⡄⠸⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠡⠐⠠⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡄
⠀⠀⣼⠃⠀⡄⢸⠁⢠⠀⠀⠉⠁⠀⠀⠀⠀⠀⠙⠉⠀⠈⡁⠀⠧⠜⣧⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⣿⠀⠀⢝⣢⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠒⡴⡿⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄
⠀⠀⡿⢢⡠⠤⣼⡃⠀⠀⠀⠠⠤⠤⠒⢒⡀⠔⣂⣉⣭⣁⠀⣄⢠⣾⡅⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠄⡀⠀⠀⠀⡸⡇
⠀⠀⢷⠀⡙⠚⠋⣾⠀⢀⣤⠴⠶⠦⣀⣀⣠⣴⣩⣵⣤⣈⣳⣪⣿⡾⠀⠀⠀⠀⠀⠀⠀⢸⣦⡼⢿⠿⠾⣤⣀⢀⠀⠀⠐⢶⣤⣀⡀⠀⠈⠩⠤⠇
⠀⠀⠘⣧⠙⡶⣭⡇⠀⣿⢀⣶⠿⠗⢺⣿⠀⠿⣀⣉⣉⡨⠏⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠛⢶⡶⠦⡿⠛⣿⠟⠿⢭⣝⡻⣦⡤⠶⠞⢒⣷⡄
⠀⠀⠀⠘⢷⡘⢻⠓⠉⠛⠯⣄⣠⠤⠊⠸⠇⠀⠀⡉⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠣⠌⣁⣠⠜⠁⠀⣿⡄⠀⠊⠩⢉⡏⠀⠀⢰⣿⡿⠁
⠀⠀⠀⠀⠀⠉⠻⣆⠀⠀⠀⠀⠀⢀⢰⡻⢧⠀⡤⠏⠶⣄⡀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣰⠊⠐⠀⠈⠓⠸⡖⠂⠒⠊⠀⠀⠀⢙⠟
⠀⠀⠀⠀⠀⠀⢾⡉⡇⠀⡀⢀⡴⢋⣀⣠⣤⡤⠴⠶⠂⢸⠃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡾⠃⠀⠙⠰⠾⠓⠃⠻⣄⠀⠀⠀⢠⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠉⢷⠀⠱⣜⢇⠈⠛⠗⣒⣂⣨⠂⠀⡎⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢇⠈⠙⠣⢀⡀⠰⠶⠂⡈⢀⠔⡰⡞
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡗⢤⡈⠑⢄⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣇⢸⠀⠀⠐⠒⠂⠐⠁⢀⣃⡞⣰⣹⣿⣆
⠀⠀⠀⠀⢀⣠⡞⠹⣿⠄⠀⠈⠓⠊⠳⢤⣀⣀⣠⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠋⠨⠀⠀⠀⠀⠀⠀⢀⣼⣛⣵⣿⣿⡿⠟⠛⠂
⠀⠀⠀⠀⠈⠉⠀⠀⣿⣇⡀⣀⢀⠀⣀⣀⡀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠉⠙⢫⣀⢦⣌⡉⠲⠦⣤⣴⣾⠿⢛⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣁⣀⣉⣙⣶⣰⢷⡓⠚⠉⠉⠛⠛⠓⢲⢲⡦⠀⣠⠴⠎⣀⡴⣉⡈⠙⣶⣻⠀⠈⢿⣀⣀⣤⡀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⡠⢋⠉⠉⠉⠉⠻⣿⣦⡣⡀⠀⡠⢒⣬⡿⠟⠛⠁⠀⡟⠀⠠⠀⣠⣄⡈⠑⠸⣿⢀⠀⠀⢀⡠⠄⣸⣆⣹⡄
⠀⠀⠀⠀⠀⣞⠕⠉⠉⠙⠓⡍⠉⠛⠻⣷⣄⣭⡾⠛⠁⠀⠀⠀⠀⣸⠁⠀⠀⢀⣸⣀⡈⠑⢤⣈⣦⣭⣿⠷⠾⠛⠉⠉⠉
⠀⠀⠀⠀⢴⠻⠋⢀⠙⠛⡶⡙⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⣤⣤⣄⣉⣓⣶⠚⠉⠁
⠀⠀⠀⠀⠈⢧⠘⠉⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢀⡎⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣶⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⢛⡻⢿⣿⠇⠀⠀⠀⠘⣿⣿⠿⠿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣿⣿⣿⡿⠀⠀⠀⠀⠀⢿⣷⣿⣿⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠛⠀⠀⣄⣠⡄⠀⠈⠻⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⣉⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠉⠉⠐⠢⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣄⡠⠊⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠂⠐⠂⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⠀⠀⠀⠐⠈⠂⡁⠒⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢂⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⡀⠀⠀⠀⠀⠂⢀⣂⠀⡀⠀⠀⠈⡒⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡌⠀⢀⠊⠴⠂⠀⠀⠊⠀⠀⠀⢀⠀⠁⠀⢀⠠⠠⠌⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢨⠀⠠⠀⡠⠆⠀⠀⠀⠀⠀⠀⣈⡈⠀⡄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢈⣤⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⡀⠀⠀⡐⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡁⠀⠨⠛⠒⠂⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⡗⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢱⠆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡠⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⠁⠔⠱⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢈⠃⡀⠒⠡⠒⠆⠀⠐⠊⢉⠀⠐⠆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⢄⠀⠀⠄⠀⢠⠂⠔⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠈⠖⢃⡀⠠⣌⠙⠀⠀⠀⠀⠀⠑⠤⠀⠀⢀⠄⠦⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠔⠈⣄⠀⠀⠀⠉⠀⠁⠁⠀⠀⠀⠫⡢⣀⡀⠀⠀⠀⡀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡰⠁⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠁⠀⠘⠃⡤⠊⠑⠠⢀⣀⣀⡀⠠⠤⠀⠀⠤⠤⠐⠋⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠠⠴⠀⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡰⠉⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡰⢁⠴⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠤⠒⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⠃⠙⡤⠂⠀⢀⣀⠀⠀⠀⠀⢀⣀⣀⡀⠠⠤⠐⠒⠈⢁⡄⠀⠛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⠃⠀⠀⠑⠒⠢⡤⠀⠤⢀⣠⣄⣀⣀⠤⣤⠄⠴⠒⠒⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠁⡇⠀⠀⠀⠀⠀⠸⠤⠐⠊⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⠢⠐⠂⠐⠀⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡁⠀⡀⠀⠀⡀⠀⣃⠁⠀⠀⠀⠀⠀⢀⠄⠀⠐⠒⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠁⢀⠀⠀⠀⡇⡁⠀⠀⠀⠀⠀⠆⠠⠀⠠⡀⠀⠀⠡
⠀⠀⠀⠜⣏⢂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠡⠐⠰⠒⠲⠑⠠⠁⠀⠀⠀⠀⠀⡈⠀⡄⠀⠀⠐⠴⡄⠀⠡⡀
⠀⠀⠀⠉⠪⡃⠙⢠⠀⠀⠀⠀⠀⠀⢀⣰⣷⡒⠂⢒⣺⣷⣀⠀⠀⠀⠀⢰⠀⢴⠦⢒⢤⡔⠓⠜⣖⠂⡀
⠀⠀⠀⠀⠡⡀⠃⠀⡀⠀⠀⣠⠔⠈⠀⠹⣿⣿⣦⣿⣿⣿⣅⡨⠐⢄⠀⠀⠱⡈⢄⠘⠢⠴⠀⠠⣁⠏
⠀⠀⠀⠀⠀⠀⣶⣾⣿⡄⢠⣿⡀⠀⠀⠀⠈⠻⣿⡿⠛⡝⠅⡁⠂⣼⣷⠀⠀⠀⣁⣓⠂⠂⢻⣉
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣼⣿⡇⠀⠀⠀⠀⠀⠉⠁⠀⠈⠢⣀⡴⣿⣿⢀⣴⣿⣟⠂⠭⠯⠐⢺⣽⣶⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠽⢾⠷⠿⠇⠀⠀⠀⠀⠸⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⠿⠛⢛⣉⣉⠉⠉⠙⠛⠉⣁⣤⣤⣤⣤⣤⣄⡈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠉⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣤⠈⠻⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠙⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣀⣤⣤⣾⣿⣇⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⣾⡿⠋⠉⠙⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠁⠈⠹⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⢀⣿⡇
⠀⢀⣼⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣤⣀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣼⣿⣷⡀
⣼⣿⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⠿⣶⣤⣀⣀⣤⣴⡾⠟⠋⠉⠀⠀⠀⠉⠻⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣿
⣿⣇⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣤⣶⠿⠋⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠹⣿⣦⡀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠃⠀⠀⠀⠀⢀⣴⣿
⢀⣿⣿⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⣧
⢸⣿⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢹⣿
⠸⣿⣆⠀⠀⠀⠀⢀⣤⣤⣴⣶⠶⠾⠿⠿⣛⣿⡿⢷⣶⣤⠀⠀⠀⢠⣴⡾⠿⣿⣿⣿⠿⠿⠿⢶⣶⣤⣤⣄⠀⠀⠀⠀⠀⣼⣿
⠀⠙⣿⣷⣄⠀⠀⠸⣿⣾⣿⠁⠀⠀⡴⢿⣿⣿⣷⡀⠈⢻⣷⣦⣤⣿⠋⠀⣼⣿⣿⣿⡷⡄⠀⠀⠘⣿⣾⣿⠇⠀⢀⣤⣾⡿⠃
⠀⠀⠘⢿⣿⣧⡀⠀⠀⠸⣿⠀⠀⠂⠀⠘⠿⠿⢮⡇⠀⣸⡏⠉⠉⣿⡀⠀⠣⠾⠿⠿⠃⢨⡄⠀⢠⣿⠁⠀⠀⣠⣿⣿⠟⠁
⠀⠀⠀⠈⣿⣿⡁⠀⠀⠀⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⢹⣿⡟
⠀⠀⠀⠀⠈⠻⣿⣶⡆⠀⠀⠻⢷⣤⣄⣀⣀⣤⣴⡾⠋⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⣀⣀⣠⣴⡾⠋⠀⠀⣴⣶⣿⠟
⠀⠀⠀⠀⠀⠀⣨⣿⣧⡀⠀⠀⠀⠈⠉⠉⠉⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠉⠉⠉⠉⠁⠀⠀⠀⢀⣿⣿⣀
⠀⠀⠀⠀⠀⣾⣿⠋⠙⣿⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠲⣤⣀⣀⣀⣀⣤⠞⠉⠢⡀⠀⠀⠀⠀⠀⠀⢰⣿⠋⢻⣿⡆
⠀⠀⠀⠀⠀⠹⣿⣷⣾⣿⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⢃⠶⠀⠀⠀⠀⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⣿⡄⠀⠀⠀⠀⠃⠙⠢⠄⢀⡀⠀⠀⠀⠀⠀⢀⡠⠤⠂⠈⠀⠀⠀⠀⣸⣿⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⣿⣿⣿⣿⣿⣷⡀⣄⡀⠀⠀⠀⠀⠀⠀⢀⡠⢠⣿⣿⣿⣿⣿⣿⣷⣄⣀
⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠙⠶⢶⣦⣶⠶⠞⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⡠⠔⠋⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⡀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠙⠢⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣷⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠛⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠿⠛⠛⠿⢶⣤⣶⠾⠟⠛⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣷⣠⣴⠾⠟⠛⠉⠉⠉⠉⠉⠉⠉⠙⠻⢷⣦⣄⣰⣿⣶⣶⣶⣤⣄
⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠿⠛⠉⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⣿⠇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⡶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⣿
⢿⣆⣴⠃⠀⠀⠀⣼⡟⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⢳⣀⣿
⢸⣿⠃⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⢻⣿
⣼⡇⠀⠀⠀⣰⡿⠁⠀⠀⠀⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠃⠀⠀⠀⠀⢻⣦⠀⠀⠀⠈⣿
⣿⡇⠀⠀⣼⣿⣀⣀⣀⣤⡾⢟⣛⢛⣛⢛⡻⣦⡀⠀⠀⠀⠀⠀⢀⣴⠟⢛⣛⢛⣛⠻⣶⣤⣀⣀⣀⣻⣷⡀⠀⠀⣿
⢻⣇⢠⠀⠈⠻⣿⡍⢻⡟⠀⠈⠀⣿⣿⡇⠈⢸⣷⣴⠾⠿⠷⣦⣾⠃⠀⣿⣿⣧⠀⠁⠈⣿⠟⢩⣿⠛⠁⠀⣦⢠⣿
⠈⢻⣾⡄⠀⠀⣿⠃⠈⣿⡀⠀⠀⠙⠛⠁⠀⢸⣿⠁⠀⠀⠀⠈⣿⡄⠀⠙⠛⠃⠀⠀⢠⣿⠀⠸⣷⠀⠀⢀⣿⡿⠃
⠀⠀⠙⢿⣄⡀⠹⠄⠀⠙⢷⣤⣀⣀⣀⣠⣴⠟⠻⣦⣄⣀⣠⣴⠟⠻⣦⣄⣀⣀⣀⣴⡿⠃⠀⡰⠃⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠙⠻⣦⣄⣀⠀⠀⠉⠉⠉⠉⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠉⠉⠉⠁⠀⠀⣀⣠⣴⡿⠋⠁
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⡨⠛⠛⠶⠶⠶⠛⠛⠅⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⠿⣿⡿⠏⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠙⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡛⠿⢷⣶⣶⣶⣶⣶⣾⠿⢛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣇⣀⠀⠀⠉⠉⠉⠀⠀⣀⣰⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣇⣀⣤⠀⢠⣄⣸⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⢾⣿⣿⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠐⠠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡼⠒⠊⠉⠉⠑⢦⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁⠀⠀⠀⠀⠀⠀⠈⢅⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠾⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣤⠖⠒⠂⠀⠀⠀⠀⢠⡄⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠀⠠⡶⠛⠂⠀⠀⠀⠨⡅⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢦⣀⠀⠀⣸⡛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣤⠾⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣺⠽⣯⠒⠂⠀⠀⠀⠀⠀⠀⠀⠴⠤⠤⠐⠀⠀⢀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠳⢌⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠳⡌⠑⠲⣤⣤⣤⣤⣤⣤⠶⠶⡶⠺⣧⡴⢲⡆⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠹⣆⠀⠀⠀⢀⡠⢤⣼⡞⢁⡴⠋⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⢠⣄⣤⠶⠶⠦⣄⣹⡄⢀⡴⠾⠀⠀⠉⠳⠛⢿⢦⣀⣀⣀⡀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣿⠛⠁⢼⣿⣿⡶⠙⢳⡾⠃⠀⠀⠀⠀⠀⠀⢸⡆⠡⠀⠀⠉⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⢧⡀⠀⠈⢿⡋⠀⣀⣨⡷⢤⣀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠙⢿⣍⠀⢀⡞⢻⣟⠛⢲⠛⠋⠀⣀⠀⠀⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠘⠓⠶⡀⠀⠉⣛⣺⣧⡼⢻⡆⠈⠳⠶⢏⠉⠿⠉⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣄⠀⠀⠀⠀⠀⠀⣠⠴⠷⠶⠛⠉⢉⣩⣷⣾⢧⡤⠤⠖⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠳⣤⠆⠀⣰⠚⠁⠀⠀⠀⠀⠘⠛⢋⣩⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠻⣀⣈⣱⣦⣀⠀⠀⠀⠀⢀⣨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠳⢤⣉⠉⠉⠀⠉⠓⣦⡤⠴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡟⠛⠳⠶⠞⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠛⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠉⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠦⠤⠶⠒⠛⠛⠉⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⢹⡟⠛⠛⠒⠲⠖⣾⠛⠛⠋⠉⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⢷⡄⠀⠀⠀⣾⡏⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⢀⡟⢸⠇⠀⠀⠸⣆⢳⡀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⡇⠀⠀⣠⠟⡀⢻⠀⠀⠀⠀⢻⠀⠙⢦⣀⡀⠀⣬⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠙⠒⠛⣡⠞⠳⠊⠀⠀⠀⠀⠘⠶⠛⠶⣄⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⡠⠐⢄⠀⠀⠸⣧
⢀⣶⠟⠛⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢀⠊⠀⠀⠀⠆⠀⠀⣿
⣾⠃⠀⠀⡀⢀⠈⠻⢷⡄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠶⠶⢶⠶⠶⠿⢤⣤⣁⠀⠀⠀⠀⢠⠀⠀⣿
⣿⠀⠀⢰⠀⠀⠑⢄⠈⠻⣦⡀⣀⣤⡶⠟⠋⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠘⠀⠀⣿
⣿⡀⠀⢸⠀⠀⠀⠀⠑⢀⣸⠟⠋⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⢰⡟
⢸⡇⠀⠈⡀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⠻⠿⠦⠀⠀⠘⣦⣾⠃
⠈⣿⡀⠀⢡⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢠⠀⡀⠀⡀⠀⠀⠈⢿⡀
⠀⠸⣧⠀⢀⡾⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠔⠁⠀⠀⠀⣠⣖⡊⠙⠓⢴⠁⠀⠀⠀⠘⣷⡀
⠀⠀⢻⡆⢸⠁⠀⢠⡿⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣠⡆⠀⢇⠀⠀⠀⠀⠸⣧
⠀⠀⠀⢿⣾⠀⠀⠈⠀⠀⢀⣱⢄⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⢿⣿⠿⠃⢀⡎⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠈⣿⠀⠀⠀⠐⠢⡞⣿⣿⣄⣠⡄⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠠⠴⠊⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⡇⢿⣿⣿⡿⠃⢀⠇⠀⠀⠀⣀⡠⣤⣄⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠘⢦⣈⣁⣀⡠⠊⠀⠀⣀⠾⣷⣶⣿⠏⠀⠀⢠⠀⠀⠉⠑⠢⡀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠈⣧⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠘⢤⡀⢀⡴⠃⠈⠉⠉⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠈⠁⠀⠄⠀⠠⠂⠀⠀⣀⡤⠞⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣤⣄⣀⣁⡀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠶⠚⠋⠙⡄⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⡏⡙⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠸⣧⣀⡀⢀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡆⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⡏⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣴⠟⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣇⣴⠟⠁⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⢷⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡟⠋⠁⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠿⠛⠒⠒⣾⣤⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢰⣃⣀⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⡀⠀⠀⣰⢋⠁⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣠⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠶⣿⣦⣤⣅⣀⣠⣤⣇⠀⠀⡄⠀⠠⠀⢀⣇⣀⣀⣤⣴⠾⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣛⣿⣶⣷⣶⣿⣾⣟⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀
⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⢍⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠔⠂⣀⠒⢂⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢈⠀⡀⠀⠙⠂
⠀⠀⠀⠀⠰⠀⣔⠀⠒⠂⠀⠈⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠂⠠⡀⠀⠣⠚⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠂⠑⠀⠨⠄⡑
⠀⠀⠀⠀⠀⠀⠀⢈⣱⡀⢐⣥⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢼⢓⡠⠐⠰⢓⡆⠀⠄⡀
⠀⠀⠀⠀⢀⠔⠊⠁⠈⠉⠉⠀⠀⠈⠑⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⡀⢴⠷⠬⢀⡀⠙⠀⠢
⠀⠀⠀⠀⡄⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⡄⠀⠰⡄
⠀⠀⠀⠀⠃⢂⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠆⠀⠀⢸⠀⠀⠁
⠀⠀⠀⠀⡀⠈⢢⠀⠀⠀⠀⠀⠀⠀⢣⠘⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⡘⠀⠀⣤
⠀⠀⠀⠀⡇⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢣⠃⠀⠀⠀⠀⠀⢀⠔⠂⢄⠄⠀⠀⠀⠐⠤⢀⡧⣀⣀⣆
⠀⠀⠀⠀⠑⠠⠗⠤⠤⢤⣀⣀⣀⡀⠠⢮⠇⠀⠀⠀⠀⠰⣁⡀⠑⠌⠀⠀⠀⠀⠀⠀⠀⠐⠀⣰⠇
⠀⠀⠀⠀⠐⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢢⠀⠰⠅
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢉⡒⠒⡖⠲⠒⠒⡋⠉⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⢀⠊⠀⠈⣀⡽⠰⣇⠈⠀⠙⣆⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠈⣀⠀⠈⠛⠀⠀⠰
⠀⠀⠀⠀⠀⠈⠑⠊⠀⠉⠀⠀⠈⠓⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⢀⣀⣀⣤⡤⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠖⠉⡐⠒⢲⣿⣿⣮⣍⠉⠛⠶⣬⣣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢠⣫⣠⠞⠛⠉⠭⢭⣿⣿⣶⣬⣿⣿⣄⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣵⢿⡀⢰⣏⢄⣢⡴⠶⠛⠉⠉⢹⣿⣿⣿⣿⣿⣷⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡮⣪⢏⣼⡇⢸⣛⠉⠀⠀⠀⠀⠀⠀⠸⣿⣿⣝⢿⡻⡀⠑⠝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡾⣷⢿⣿⣇⡌⣉⠁⠀⠐⠚⠒⠲⢤⠀⠹⣿⢿⡀⠹⣌⠳⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⡞⣵⣿⣿⠟⠁⠙⠁⠀⠀⠀⠾⠳⠄⠀⠀⢻⣷⣝⢦⡈⢗⢌⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⡜⡜⣟⣿⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡼⢸⡧⡠⡙⠢⢭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⢸⢸⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡋⢇⣹⣌⡛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠈⢀⢂⣿⠀⠁⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡆⡆⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡏⣰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢥⣙⡻⣿⣿⣦⣤⣤⡤⠠⠀⠂⠀⠀⠀⠀⢸⠀⢧⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣺⠟⣿⣿⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣜⠌⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⣡⢀⠜⠋⠻⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣦⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠛⠁⠀⠠⣺⢿⠈⣿⣹⢿⡆⠀⡐⠁⠀⠀⠀⠀⠀⠀⠉⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⣡⡞⡱⣾⠀⣼⢋⡞⣠⢣⢟⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠻⣧⡁⢳⡀⣯⠞⡰⡱⣣⣣⣿⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⣻⣿⡇⠁⡸⠐⣱⠿⡫⢾⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣇⡇⡇⡇⡏⠞⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⢻⣌⣄⠡⡀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⣪⠿⣿⡴⠃⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠉⠉⠒⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⢀⣠⣒⡈⢁⣒⣄⡀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠀⠀⠀⠀⠀⡎⠉⠉⠀⠀⡷⢾⠀⠀⠉⠉⢱⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⡠⣠⡇⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⢸⣤⢄⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣸⠔⠁⠓⠒⠒⣒⠲⠇⠸⠖⣒⠒⠒⠚⠈⠢⡃⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠁⠠⡀⠀⠀⠉⠒⠂⠐⠒⠉⠀⠀⢀⠄⠈⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢿⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠀⠀⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⡿⠢⡀
⠀⠀⠀⠀⡠⠖⠉⠙⣄⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣠⠋⠉⠲⢄
⣀⠔⠒⠚⠀⡠⠒⢲⠎⢉⠀⠑⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠊⠀⡍⠱⡖⠒⢄⠀⠓⠒⠢
⠣⡔⡚⠂⢀⠞⠀⡌⠀⠀⠑⠒⠻⡀⠉⢹⡗⠒⠒⡖⠒⠒⢲⠒⠒⢚⡏⠉⢀⠟⠒⠊⠀⠀⢡⠀⠳⠀⢐⢗⢢
⠀⠀⠀⠙⠒⠁⢰⠀⠀⠀⠀⠀⠀⠈⠀⠁⠑⠒⠒⠙⠂⠐⠋⠒⠒⠊⠈⠁⠁⠀⠀⠀⠀⠀⠀⡆⠈⠒⠋
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰

⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠒⠠⠦⢠⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡄⠴⠄⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⠄⢀⣠⠎⠉⢷⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡰⡞⠈⠀⠀⠀⢀⠈⠵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢧⢱⠀⠒⠂⡀⠈⠀⢀⠉⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⡀⠈⠓⠂⠀⠀⠀⡀⠀⠉⠀⠙⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡀⠐⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⣀⣹⠀⠀⠀⠀⠀⡀⠤⠀⡂⠱⠌⡰
⠀⠀⠀⠀⢀⠔⠊⠒⠒⠀⢄⡀⠀⠰⠈⠈⠉⠀⠀⠀⠁⠀⠈⠈⠀⠉⠁⠠⠀⠀⠀⠀⠘⠰⠈
⠀⠀⠀⠔⠃⠈⠀⠀⠀⢀⠀⠀⠁⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡄⣆
⠀⠀⡎⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠡⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡔⠈⠠⠀⠀⠀⠑⢢
⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠨⠀⢀⡴⠒⠒⠒⠲⢄⡀⢠⠀⢸⠀⠀⠈⠀⠀⠀⠂⠀⡇
⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢸⡧⠀⠀⣿⠆⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠘⡀⠀⣤⠀⠀⠀⠀⠹⠃⠀⡇⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠙⡒⠒⠠⠤⠤⠤⠴⢤⠂
⠀⠀⠀⢱⠀⠉⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⠈⠲⠤⠤⠤⠴⠚⢡⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣇⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠤⠄⠒⠒⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣽⣿⣿⣶⣶⣾⣭⣝⡲⣬⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⣝⣛⡻⠿⣷⣯⡛⢦⡙⢎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⠀⠉⠛⠿⣷⣮⣻⣿⡴⡍⡊⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⢸⣦⡙⣿⣿⣷⢻⡜⣆⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢷⡘⣿⣿⢸⣿⢸⣿⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⣿⡏⣾⣿⢸⣿⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣾⣿⣯⣿⢋⣾⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣵⠟⠡⠞⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢫⣾⣻⠇⣼⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠸⡏⣿⣼⣯⠁⠐⣃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡸⣯⡘⢿⣿⣄⢠⣞⣚⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣮⡳⠌⠻⣶⣝⢻⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠠⣾⣿⣦⡈⣷⡝⠧⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢉⣵⣿⠿⠃⠐⢵⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢠⢤⠤⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠯⠖⠾⠶⠿⠼⠶⠮⠽⡶⠀⠰⠿⠴⠾⠾⠯⠀⢰⠵⠾⠧⠗⠾⠷⠦⠾⠶⠾
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⢰⣿⣿⣿⣿⡟⠿⠋
⠀⠀⣰⣿⣿⣿⣿⣿⣷⣄
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣤⣀
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣿⣿⠛⣿⣿⣿⣿⣿⠈⠙⠿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣯⣴⣿⣿⣿⣿⣿⡀⠀⠀⠀⠉⠙⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣷⣄⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷
⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣷⡀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡇
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠉⠛⠛⣿⣿⣿⣿⣿⣿⣿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⡿⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣼⣿⣿⠃⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢰⣿⣿⡇⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢸⣿⣿⡇⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⠈⠉⠉⠛⠛⠛⠻⣿⣿⣿⣿⣿⡏
⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢿⣿⡇⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⡇⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀⣠⣴⣶⣶
⠀⠀⠀⠀⢠⣿⡇⠀⠀⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⣿⣁⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣿⣿⣿⡀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣧⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿
⠀⠀⠀⠀⠃⠹⣿⣧⡀⠀⠘⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⢿⣿⣿⡇⠀⠀⠘⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠁⠀⠀⠀⡿⠟

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣴⣶⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⣾⣷⣤⣤⣤
⣠⡄⠈⠃⠉⠘⠁⣦⡄⠀⠀⠀⠀⠀⠀⠀⡸⠏⠀⠻⠙⠉⢧⠀⠀⠀⠀⠀⠀⠀⠸⢉⣤⡀⢀⣤⡙⠃⠀⠀⠀⠀⠀⠀⢻⡿⠋⠁⠀⠀⠈⠻⢿⣤
⢸⣏⠁⢐⣀⡀⢈⣹⠁⠀⠀⠀⠀⠀⠀⠠⣯⣁⣤⣤⣤⣄⣽⡆⠀⠀⠀⠀⠀⠀⠇⣾⠟⠃⠈⠻⣿⠸⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⣀⣀⠀⠀⢸⡟
⡀⠲⠂⠀⠉⠁⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣿⣿⣿⣾⠟⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠙⠛⠀⡠⠃⠀⠀⠀⠀⠀⠰⣿⣷⣀⠀⠈⠉⠀⣠⣾⠿
⢛⠀⣘⠀⠐⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⡿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣷⣶⣶⣿⠛⠃
⠘⣴⡇⢠⠄⡤⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⠭⢭⢭⠽⡄⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⣶⢰⡆⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⠠⠃
⠈⠛⠆⣸⡀⣇⠰⠛⠃⠀⠀⠀⠀⠀⠀⠀⣾⢉⣻⣛⣼⡉⠷⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⢿⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⢿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢶⣆⢀⣶⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠟⠈⠁⠀⠉⠙⢿⠟⠀⠀⠀⠀⠀⢀⣠⣄⠀⣀⡀
⠀⢀⠀⠐⠒⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣠⣤⣤⣤⣀⣈⠀⠀⠀⠀⠀⢠⠋⣸⣾⠿⠿⠿⣷⣄
⠣⠁⠀⠀⠀⠀⠀⠡⠇⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢰⣿⢻⡟⠁⠠⠀⠀⠀⢻⡛⣷
⠠⠀⠀⢿⣿⠿⠀⢀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⣿⢩⠉⠀⠀⠀⠀⠀⠀⠀⠹⣏⠳⢄⠀⠀⠀⡀⠈⢛⡯
⠀⠑⠤⠀⣀⠠⡔⠁⠀⠀⠀⠀⠀⠸⢿⣿⣷⣄⠀⠀⠀⣠⣾⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠐⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡏⠉⠹⣷⣄⢿⣞
⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠈⣬⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⢀⣿⣜⡞
⢠⠂⠀⠀⠀⠀⠀⡓⡄⠀⠀⠀⠀⠀⠀⠀⢸⡿⢃⣴⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢘⠀⡄⣤⢚⠎⠀⠀⠀⠀⠀⠀⠐⢿⣿⣷⣤⣾⣿⣿⣿
⠈⠉⠺⠭⠶⠭⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠖⠲⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠓⠻⠆⠀⠉

⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⣠⡀⠀⣠⡄⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⣀
⠀⠀⠀⢠⣶⣿⡍⠀⠀⠀⠀⠀⢀⣽⣥⣤⣼⣧⡀⠀⠀⠀⠀⠀⣈⣿⣆⠀⠀⠀⠀⠀⠀⠈⣹⣿⣦⡀⠀⠀⠀⠀⣾⣧⠀⣠⣾⡟⠁
⠀⠀⠀⣿⣿⡇⠹⠄⠀⠀⠀⠀⢾⣿⣿⠆⣿⡿⠙⠀⠀⠀⠀⠘⢉⣿⣿⣴⠀⠀⠀⠀⠀⠰⠇⢹⣿⡇⠀⠀⠀⢸⣿⡏⢻⣿⡏⠳
⠀⠀⠀⢻⣿⢃⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⣿⡇⠀⠀⠀⠀⠀⠀⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⠛⡇⢸⣿⠁
⠀⠀⠀⠈⠈⠈⠀⠀⠁⠀⠀⠀⠀⠁⠁⠀⠁⠁⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠀⠀⠀⠈⠀⠀⠁⠁⠁⠀⠀⠀⠈⠀⠁⠈⠉
⠀⠀⠀⠀⢤⣤⡀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⢀⣦⠀⢠⣤⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⣾⣯⣄⠀⠀⠀⠀⢀⣾⣧⡀⣠⣶⣟⠁⠀⠀⠀⠀⢀⣾⣷⣄⠀⠀⠀⠀⠀⣠⣿⡤⢶⣿⣦⡀⠀⠀⠀⢛⣷⣦⣀⣼⣷⡀
⠀⠀⠀⣼⣿⣷⡘⠁⠀⠀⠀⢸⣿⡏⢹⣿⣏⠹⠆⠀⠀⠀⠐⠋⣿⣿⠈⠃⠀⠀⠀⠘⣿⣿⡟⢾⣿⡧⠙⠀⠀⠀⠞⠙⣿⡏⢹⣿⡧
⠀⠀⠀⣿⣿⡏⣷⠀⡀⠀⠀⢸⠁⡇⠘⣿⡏⠀⠀⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢹⣿⠃⢸⣿⡇⠀⠀⠀⡀⠀⠀⣿⡇⢸⠃⣇

⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢠⣦⠀⠀⠀⠀⢰⡦⠀⠀⢠⣄⡀
⠀⠀⠀⢀⣾⣿⢯⠀⠀⠀⠀⠀⠀⢀⣺⠇⢀⣼⣿⡄⠀⠀⠀⢀⣴⣿⣦⡀⠀⠀⠀⣠⣾⣷⣄⢀⣾⣷⣄⠀⠀⢀⣾⣷⣀⣴⣿⡏
⠀⠀⠀⢸⣿⡇⠈⠃⠀⠀⠀⠀⣺⠽⠿⡗⠉⢘⡿⡇⠀⠀⠀⠋⣼⣿⡇⠉⠀⠀⠘⠁⣿⣿⠈⠉⣿⣿⡌⠃⠀⢸⣿⡇⣽⣿⡇⠹
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠾⠣⠀⡀⠀⠀⢸⡇⡧⠀⠀⠀⠀⢹⣿⠃⠀⠀⠀⠀⠀⣿⣿⠀⠀⢹⣿⠁⠀⠀⢸⠘⡇⢹⣿⠃

⠀⠀⠀⢰⣶⠄⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⢶⡆⠀⠀⠀⠀⠀⢴⣞⡀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⢴⡆
⠀⠀⠀⣴⣿⡶⣄⡀⠀⠀⠀⠀⣠⣾⣶⠚⣿⣷⣄⠀⠀⠀⠀⢠⠿⣿⣦⡀⠀⠀⠀⣿⣷⣄⡐⣿⣄⡀⠀⠀⢀⣴⣿⣷⣄⣠⣾⣷⣄
⠀⠀⠀⠟⠉⠑⣮⡁⠀⠀⠀⠀⠋⣿⡏⠛⣿⡿⠈⠁⠀⠀⠀⠚⠀⣿⣿⡇⠀⠀⠐⡿⣿⠀⠈⡿⠛⢭⣅⠀⠈⠀⣿⡇⠈⢁⣿⣿⠈⠁
⠀⠀⠀⠠⠀⠠⠎⠿⠆⠀⠀⠀⠀⠿⠇⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⢹⢿⠀⠀⠀⠀⠇⢹⠀⠀⠀⠀⠴⠻⠂⠀⠀⠿⡇⠀⠀⠿⠯

⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠟⠛⠉⠉⠉⠛⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠛⠁⠀⠀⠀⣀⣠⣤⣤⣴⣾⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⣀⣴⣾⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣴⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣶⣦⣄⡀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⡉⠛⢿⣦⡀⠀⠉⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠿⠿⠿⣿⣿⣶⡀⠙⠧⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⣰⣿⡟⠁⠀⠀⠀⠀⠈⠙⠿⡄⠀⠀⠀⢤⣀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠠⣤⣤⣴⣶⡿⠟⠋⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣶⣄⠙⢷⡄⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢀⡀⠀⠈⠁⠀⠀⣀⣤⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠘⣿⡄⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠙⣿⡿⠿⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠘⣷⡀⠀⢹⣿
⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠰⢶⣶⣾⣿⠇⠀⠀⠀⠀⠀⠀⠦⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠀⠀⠘⣿⣧⠈⠿⠆⢸⣿
⠀⠀⠀⢠⣿⡟⠀⠀⠀⣀⣀⣀⣤⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣷⡀⠀⠀⢀⣾⡿⠋⠉⠀⠀⠀⠀⢹⣿⣷⣄⡀⠈⢿⣷⡀
⠀⠀⠀⢸⣿⠁⠀⠀⠀⣿⡟⠛⢻⡿⠻⣦⣀⠀⠀⠀⠶⠶⣷⣶⣤⡀⠀⢻⣷⠀⠀⢸⡏⠀⣠⣶⣶⡤⠀⠀⠘⣿⡉⢻⡇⠀⠀⣿⣷
⠀⠀⠀⢹⣿⡀⠀⠀⠀⢸⣿⣄⣿⠇⠀⠀⠙⢷⣦⣀⣤⠤⠿⠿⠿⠷⠦⣼⣿⣀⣀⣸⣀⣨⡭⠿⠿⠷⠶⠶⢤⣿⣧⣼⡇⠀⠀⣼⣿
⠀⠀⠀⠘⣿⣧⡀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠹⣏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠉⠉⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠁⠀⣰⣿⡟
⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠦⢶⣶⡶⠚⢋⠀⠀⠀⠘⣿⠲⠶⠶⠴⠶⠶⠚⢻⡿⢀⣴⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣶⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠛⠻⠷⡶⠿⠛⢿⣷⡀⠀⠀⠀⠀⠀⣾⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠙⣿⣆⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⣠⣴⣶⣶⣶⣦⣤⠀⢻⣧⠀⠀⠀⠀⣼⡟⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣾⣿⣦⡀⠀⠀⠀⠀⠸⣿⠀⠀⣴⣿⣋⣉⣉⣉⣹⡟⠀⢸⠏⠀⠀⢀⣾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣷⣄⠀⠀⠀⠀⠙⠂⠀⠹⣿⣍⣉⣉⣉⣻⣇⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠋⠀⢀⣀⣼⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡟⠉⠀⢻⣿⣿⣷⣦⣴⣤⠀⠀⠀⠐⠶⠶⠶⠂⠀⣠⣾⡿⢻⣿⠇⠀⢹⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⡇⠀⠀⠘⣿⡇⠉⠛⠻⢿⣷⣤⣀⣀⣀⣀⣀⣤⣾⡿⠋⠀⣼⡿⠀⠀⢸⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡟⣿⣷⡀⠀⠀⠙⣿⣦⡀⠀⠀⠈⠛⠻⠿⠿⠿⠟⠛⠁⠀⢀⣼⡿⠃⠀⠀⣼⡇⠸⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⣿⣷⡀⠀⠀⠈⠻⢿⣶⣤⣀⣀⠀⠀⠀⢀⣀⣀⣤⣶⡿⠛⠀⠀⠀⢠⣿⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢼⣿⣷⡀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⢿⣿⠿⠟⠛⠁⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⢀⣿⣿⣷⣄⠀⠀⢀⣠⣴⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⣀⣀⣀⣉⣉⡛⠛⢛⠛⡛⠛⠛⣀⣀⣀⣀⣀⣈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣁⣀⣀⣀⣀⣀⡛⠛⢛⣛⣋⣉⣁⣀⣀⣀
⣿⣿⡿⠿⢿⡿⠿⠿⢿⢿⠿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣶⣾⣷⣾⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⢀⠲⡄⠀⠀⡠⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠢⡜⢱⠀⠣⠤⠧⣤⠎⠙⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⣧⠈⢧⠀⠀⡴⠙⢄⣀⡎⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢫⢣⡀⠑⢌⡲⢤⣀⡼⠀⠀⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠳⣝⠢⢄⣉⠑⠚⢒⣚⡩⠥⢺⡤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣀⣠⣀⠀⠀⠀⠀⠉⠒⠂⠬⠝⠓⠚⠉⠉⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡠⢤⣀⠀⠀⠀⠀⡇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠼⠶⠠⠤⡀⠀⠀⠀⠑⠐⠷⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⡇⠀⡘
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠈⠳⠀⠀⣀⠄⠒⠒⠢⢄⠀⠀⠀⠀⠀⠀⡼⠀⢠⠃
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⡠⢾⠀⡰⠃
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡷⡖⡏⠀⢸⠀⡧⣂⠢⣄
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⣇⠀⢸⠀⠳⠽⠃⠈⡆
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠉⠒⠚⠀⠀⠘⡆⠈⢧⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠵⢖⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠲⠤⠄⠀⠀⠀⢸⠤⢄⣀⣋⣻⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⣠⡃⠀⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡦⠤⠤⠖⠚⡗⠒⢶⠒⠋⠀⠐⢲⡏⠣⣌⡉⠓⠚⠉⢉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡸⠁⠀⢸⡀⠀⠀⢀⡜⠙⢦⣀⠈⠉⠉⣉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠙⠲⠖⠋⠀⠀⠀⠈⠉⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠈⣿⣶⣦⣴⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⢀⣠⣴⡿⠟⠀⠰⣿⣿⢻⣿⡿⣿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠛⣿⣾⣿⣿⡜⣦⡹⣿⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣷⣽⡇⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣷⠀⠀⠀⠀⠀⢸⣿⠙⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣷⣄⣀⣀⡀⣀⠘⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣀⡀⢀⣠⣄⣀⣀⣀⣀⣐⣦⡄⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣿⠟⣼⣿⣻⣧⣄⣙⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠋⠀⢚⣿⣿⣿⣿⣿⡿⢟⠋⠙⠂⠀⣀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢰⣦⣾⣿⣿⣿⣿⣿⣟⣀⡄⢀⡀⠀⠀⡹⣷⣶⣶⣦⣄⣀⠀⢀⢀⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣷⣿⣴⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⢿⠅⠀⠈⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢂⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣂⣤⢀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⡆⠈⠁
⣗⠀⠀⠀⠀⠀⣄⣐⣀⣀⢀⡈⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⢡⣸⡇⣀⣴⣿⣄
⣿⣿⣦⣄⣠⣾⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠋⠀⠙⠟⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣋⣧⣿⣶⣦⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠉⠉⠉⠉⠁⠐⠒⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⣤⠤⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡤⡇⠀⠀⠀⢠⠖⠉⠀⠀⠒⣂⢰⠀⠦⠄⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⣐⠝⡜⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠐⠸⠉⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠊⠑⠢⠤⠭⠭⠭⠭⠟⢻⠀⠀⢰⠀⠀⠀⠈⠀⠀⡠⠤⠤⠄⠤⠄⠤⠤⠤⠤⢤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⠀⠇⠀⠐⠛⠀⠀⠀⠈⠓⠀⠐⡆⢀⢼⠀⠀⠀⢰⠀⠀⡇⠀⠠⠖⠂⠀⠀⠐⠦⠀⠀⡆⢀⣇
⠀⡤⣲⢢⠀⠀⠀⠀⠀⡁⢸⣸⠀⠀⠀⠠⠆⠀⠀⠢⠀⠀⠀⢣⡄⠀⡇⠀⠀⡋⢹⣹⠓⠢⢤⡊⠑⡤⣴⠁⢢⠤⠒⢻⡹⠀⡇⠀⠀⠀⠀⠀⡔⢄⣄
⢞⣵⠋⣡⠧⣀⠀⣀⣀⠑⢼⠀⠀⠀⡠⣀⠀⠉⠁⢀⡠⠀⠀⠀⡧⠜⠀⠀⠀⠣⣨⠁⠀⠀⢀⠑⠚⠃⠚⠒⢅⡀⠀⠀⣃⡴⠣⡀⠀⠀⢀⡸⠉⠻⣮
⠀⠑⠚⠧⡀⠀⡱⠁⠨⠃⠌⡄⠀⠀⠀⠈⠋⠭⠝⠋⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠑⠖⠶⠁⠀⠀⠀⢀⡇⢀⠴⠍⡑⠲⡁⢀⡠⠖⠚
⠀⠀⠀⠀⠈⡱⠁⡰⠀⠀⠀⡨⠢⣄⣀⡀⠀⠒⠀⠀⣀⣠⠴⠓⢤⠤⠤⠄⡤⢠⡤⣼⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⠉⠀⠀⠀⠀⡄⣳⠁
⠀⠀⠀⠀⠀⠈⠒⠧⣀⠀⡜⠀⢸⢠⢡⠉⢹⡶⣟⠉⢱⠘⡀⡂⠈⢇⡠⡶⠋⠁⡰⠁⠀⠀⡏⢫⠐⠒⠒⢉⠟⢱⠀⠀⢣⠀⠀⢀⠜⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⡄⠘⠈⠦⠜⣋⡙⠤⠊⠀⡇⠇⠀⠈⠁⠁⠀⢀⠃⠀⠀⠀⠧⣀⡉⢦⡶⣁⣀⠜⠀⠀⠀⡖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⡆⠀⠀⠀⣈⡀⠀⠀⠀⠃⢠⠀⠀⣧⣑⠤⢴⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡇⡇⠀⠀⠀⢈⡀⠀⠀⠀⠀⢸⠀⢀⠃⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡇⠇⠀⠀⠀⢈⡁⠀⠀⠀⡄⠘⠀⡘⠀⠀⠀⠀⡗⠂⠠⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠐⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢁⢰⠀⠀⠀⠈⠁⠀⠀⠀⡇⡆⢠⠃⠀⠀⠀⠀⡑⠂⠤⠤⠤⠀⣀⣀⣀⡀⠠⠤⠤⠄⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡼⠆⠤⠤⠤⠤⠤⠤⠰⡤⢻⠁⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⢸⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢸⡄⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠒⠒⣞⡓⠀⠚⠢⢄⠀⠀⠀⠀⠀⠀⠀⠈⠥⡉⣉⠉⡉⢉⡉⢉⡉⣩⡉⢉⣌⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣋⣶⣖⣚⣋⣙⣓⣲⣶⣙⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣵⣽⣾⣿⣋⣻⣿⣷⣼⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⢀⠖⠋⠉⠳⡄
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⣀⡤⠼⠶⠖⠒⠒⠛⠲⢤⡀
⠀⠀⡴⠋⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠳⡄⠀⣸⠀⠀⠀⢀⡄⠀⢀⢶⡀⠀⠀⣧⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⣇⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⡇⢀⣿⠀⣠⣶⣋⡤⠖⠉⠀⠳⢤⡀⣿⠀⠀⠀⠀⠀⣠⠶⣄⠀⠀⠀⠀⡇
⠀⠀⢻⠉⢿⠀⣠⣤⣠⠤⣄⣠⣄⢸⠏⢙⠇⢺⡈⢻⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣟⡇⡴⠒⠒⠋⠁⠀⠈⠓⠲⢦⣀⣇
⠀⠀⠈⢑⡏⠀⠙⠛⠁⠀⠀⠛⠛⠈⣷⠉⠀⠀⡟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠘⡧⠀⣴⣤⠤⠤⣤⣤⡄⢸⠏⣸
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢸⡘⢆⠀⠀⠀⠀⠀⠀⠀⢀⡞⡼⠙⡇⠀⠙⠋⠀⠀⠈⠛⠁⠈⡟⠁
⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⣠⡤⢚⡁⠀⠀⠀⠀⠀⠣⣌⠓⠦⣄⣀⣀⡤⠖⢋⡰⠃⠀⠘⣄⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⢠⠴⡞⠉⠉⠹⠦⠏⠙⢯⣬⠧⡀⠀⠀⠀⠀⣀⡽⠖⠿⠦⠿⠒⢾⣍⠀⠀⠀⠀⢈⣑⢦⡤⢤⡤⠖⠋
⠀⠀⡴⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠙⢆⠀⣠⠞⠁⣹⠀⠀⠀⠀⠀⣇⠈⠳⡀⠀⣠⠯⣬⡟⠓⠚⢻⠛⠲⣄
⢀⡞⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢿⣧⣠⣼⠁⠀⠀⠀⠀⠀⠘⢦⢀⣽⣰⠃⠀⠀⡇⠀⠀⢸⠀⠀⠘⠆
⢾⡳⠤⣄⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣠⠴⢚⡧⠀⠹⡀⠀⠀⠀⠀⠀⠀⡾⠙⢾⠃⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠘⡄
⠀⣿⢳⢶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⠚⣯⠀⠀⠀⢱⡀⠀⠀⠀⠀⡼⠁⠀⡟⠀⡄⠀⠀⡇⠀⠀⢸⠀⠀⢠⠀⢱
⠀⢿⢸⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡇⢸⠀⠀⠀⠀⡇⣀⡀⣀⣠⣇⠀⢸⠁⣸⡇⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇⠈⡇
⠀⠘⡄⢇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⢸⡇⠀⢀⡞⠀⠀⠀⠀⠀⠸⣄⡏⢀⡏⡇⠀⠀⡷⠒⠖⢾⠀⠀⢸⢹⠀⢻
⠀⠀⠹⡼⢾⡗⠲⠦⠤⣤⠤⠴⠶⢺⠀⠀⢸⡜⡇⠀⡾⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⢰⠃⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇⡇⠘
⠀⠀⢰⡗⠚⡿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣸⠃⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⣾⠀⡏⠈⢳⢤⣀⡇⠀⠀⢸⣀⣤⡞⠁⢹
⠀⠀⢸⡇⠀⡇⠀⠀⠀⣶⠀⠀⠀⢸⠀⠀⠈⠛⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⣷⢶⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡶
⠀⠀⢸⡇⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢧⣼⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠺⣄
⠀⠀⢸⡇⠀⡇⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣅⡀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⡇⠀⣧⠀⠀⠀⣿⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠙⡖⢤⣤⡤⠴⡞⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⡇⠀⢹⠀⠀⢠⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡧⡇⢰⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⡇⠀⢸⠀⠀⢸⣿⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇⢸⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⢸⡇⠀⢸⠀⠀⢸⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡇⡇⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⢸⡇⠀⢸⠀⠀⢸⠏⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣷⡇⣼⠀⠀⠀⠀⠀⠀⠓⠒⡖⢶⣶⢲⡖⠚
⠀⠀⢸⡇⠀⣸⣄⣀⣸⠀⣇⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⣏⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡟⢸⠃
⠀⠀⢸⡇⣾⣡⣤⣤⡽⠀⢯⣤⣀⣉⣷⠀⠀⠀⠀⠀⢀⡴⣯⣾⠟⢷⣼⠷⡄⠀⠀⠀⠀⢀⣴⣣⣾⢷⣌⣷⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⣤⣤⣤⣬⣯⣭⣤⣤⣤⣬⣭⣧⣤⣤⣤⣤⣾⣾⣭⣥⣤⣭⣷⣾⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠙⠳⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⡾⠶⠟⢿⣿⠟⠛⠻⠤⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡂⠒⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢿⣷⣀⢐⣶⡁⠢⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠄⠀⠀⠀⠉⠛⡟⠉⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⣤⡄⠀⠀⠀⠈⠋⠸⠟⠋⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⡀⠀⠀⠿⣿⣆⠀⠀⣦⣄⠀⠀⠀⠀⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠙⣿⢿⡇⠀⠀⠀⢿⣿⣦⣤⣬⡿⠞⠁⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡴⠆⠠⠉⠉⢄⠀⠀⠈⠼⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⢋⣷⡄⠀⠀⠳⠂⠓⠀⠀⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⠀⢋⣿⠁⠀⠀⠀⠀⠀⠀⠺⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣾⠇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢠⣿⢹⡿⠻⡟⠀⠐⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠸⣿⠊⠀⠀⠀⠀⠀⣻⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⠅⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠠⠂⠈⡉⠈⠈⢉⢢
⠀⠀⠀⠀⠀⡐⠀⢊⡤⠈⢀⠄⠄⠂⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠈⠁⠈⠲⠄
⠀⠀⠀⠀⠀⡄⠀⡠⠄⢠⡥⠀⠐⣇⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠈⠉⠐⣀⠀⠄
⠀⠀⠀⠀⠀⢀⠘⠃⠀⠠⣄⡤⠄⠘⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⡠⡀⠀⣛⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⡈⣀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣁⣨⡀⠀⣈⢁⣆⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠔⠾⣤⡤⠗⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠁⠊⢀⠌⠠⠏⠁⢀⡂⢄
⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠒⠛⡟⠚⠀⢄⠀⠀⠀⠃
⠀⠀⠀⠀⢸⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠰⠠⠀⠀⢳⠀⠀⢳
⠀⠀⠀⠀⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⡰⠃⠀⠀⠀⡇⠀⠀⠆
⠀⠀⠀⠀⢸⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠘⡄⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⢀⠌⠀⠀⠈⠀⢀⠃⠀⠀⡆
⠀⠀⠀⠀⠸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⠀⠀⠀⠀⠀⠀⠀⡠⠂⢀⡠⠀⠀⠀⠀⠰⢤⠀⢣⣀⣀⣰⢁
⠀⠀⠀⠀⢐⠄⣰⢇⣀⣀⣀⡀⠀⠀⠀⣀⣠⣣⠃⠀⠀⠀⠀⠀⠊⠀⠒⢄⠋⠀⠀⠀⠀⠀⠀⠉⠃⠆⣀⣽⠁
⠀⠀⠀⠀⠀⡀⡼⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⡷⠀⠀⠀⠀⠀⠀⠁⠛⠆⠂⠀⠀⠀⠀⠀⠀⠸⠀⠀⠰⠞⠉
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⡸
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢹
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠆⢀
⠀⠀⠀⠀⠘⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠞⠁⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣈⡉⠙⡓⠚⡋⠉⣉⡀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⣉⡽⠀⣏⡉⠀⠀⠑⡄⠀⠀⠀⠀⠀⠑⠘⠀⠀⠀⠀⠀⢆⣀⠂⠈⠙⠂⠠⠐⡎
⠀⠀⠀⠀⠀⠀⠙⠒⠛⠐⠉⠀⠀⠈⠛⠒⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠒⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠜⠁⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠔⠢⡀⠀⢀⠔⠉⠀⠀⠀⠉⠢⠤⠊⠉⠒⠨⠕⠢⡀
⠀⠀⠀⠀⠀⠘⢄⠀⠀⠇⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠉⠘⣀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢠⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⡆⡇
⠀⠀⠀⠀⠀⠀⠸⡸⡀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠑⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢎⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠔⠊⠁⠀⠆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠋⠑⠒⡄⠈⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡒⠤⠤⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⠜⠀⠀⣆⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠂⠉⠀⠈⠑⠂⠤⠐⠂⠁⠀⠀⠀⠈




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

⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠦⠬⠉⠉⠁⠐⠂⠤⡀
⠀⠀⠀⠀⠀⠀⢠⠊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⢀⠟⠸⠙⠻⢖⡒⠀⠀⠀⠻⡒⠠⠄⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⢄⡀
⠀⠀⠀⠀⠀⢸⢀⠆⠀⠀⠀⠀⠉⠐⠒⠀⠘⠂⠀⠀⠃⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣀⠀⠀⢀⡈⣆
⠀⠀⠀⠀⠀⢠⠎⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⢸⡅⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⢠⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⢀⣸⠀⢀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠎⠀⠀⠀⢀⡼⠁⠣⡀⢄⠀⠀⠀⠈⢆
⠀⠀⠀⠐⠁⢸⠀⠀⠀⠹⠿⠀⠀⠀⠀⠀⠨⠿⠀⠀⠀⠀⠀⢱⠀⠀⢘⠀⠀⠀⡴⠋⠀⠀⠀⠙⢆⡁⠀⠀⠀⠈⡄
⠀⠀⠀⠈⢄⠀⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠴⢀⠌⠀⠀⡏⢁⠤⠊⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄⡀⠀⡇
⠀⠀⠀⠀⠀⠉⢳⠀⠀⠀⠀⡆⠈⠉⠀⠰⠀⠀⠀⠀⢀⠋⠁⠀⠀⠀⠇⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠁⠀⠉⠉⠀⠀⠀⠀⣠⡂⠀⠀⠀⠀⡔⠻⠁⠀⠀⠙⠁⠀⠔⠀⠀⠀⠙⠁⠀⠀⢙⠓⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢶⠤⠤⠤⠀⠀⠤⠄⠐⣾⡻⠉⠒⢄⠀⠀⠣⠤⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡬⠤⠞
⠀⠀⠀⠀⠀⠀⠀⣴⠁⠘⡄⠀⠀⠘⠀⠀⠀⡪⡷⠁⠀⠀⣠⠓⢄⠀⠀⠈⠢⣀⠀⠀⠁⠀⠈⠀⠀⠀⢀⣠⠊
⠀⠀⠀⠀⠀⠀⡔⢠⠀⠈⠜⣄⠀⠀⠀⢀⣴⠟⠀⠀⠀⢀⠁⠀⠀⠣⡀⠀⠀⡠⠟⢲⡦⠄⠲⠆⢐⢖⡛⠀⠉⠢⢄
⠀⠀⠀⠀⠀⡰⠀⠘⠀⠀⠈⠢⡙⠒⠫⠑⠁⠀⠀⠀⠀⠸⠃⠀⠀⠀⢱⡔⢛⠀⠀⠀⡗⠩⠀⠨⠔⠊⡇⠀⠀⠀⠀⡱⢄
⠀⠀⠀⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢥⢸⠆⠀⠀⢁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠀⠀⠣
⠀⠀⠀⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠒⣦⣂⠀⠐⡈⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠑⡇⠀⠀⠀⠇
⠀⠀⠀⢀⠜⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠝⠀⠀⢠⡇⠀⠀⠀⢸⠀⠀⠀⠀⠈⡀⠀⠀⠀⠰⠀⠀⠀⢸⠄
⠀⠀⢀⡏⠒⢢⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠂⠒⠤⣠⠃⠀⠀⠀⠀⡜⠈⠉⠁⠀⠀⢃⠀⠀⡀⣀⣇⡀⠀⢠
⠀⠀⠮⢤⡂⡎⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⠀⠉⡷⠖⠁⠀⡁⠀⠀⡔⠀⠀⠀⠀⠀⠀⠈⢢⣾⠄⠈⢻⡄⠀⡜
⠀⠀⠀⠀⠉⠁⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⢶⡃⠀⠀⠀⣴⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⢽⡟⢹
⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠁⣿
⠀⠀⠀⠀⠀⠘⢰⢸⠂⠤⠤⠀⠉⣉⣉⣙⣉⡉⠉⠁⠤⠤⠀⠚⡇⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡴⠟
⠀⠀⠀⠀⠀⠀⡎⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹
⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⢀⠄⠺⡆⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠇⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡁⠘⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠂⠤⣆⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢈⢻⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠃
⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠄⠀⢤⠀⠀⠀⠀⢲⠈⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠄⠀⠀⠀⠈⡆⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣇⡠⠤⠤⣁⠀⡆⠀⠀⠀⠀⠀⢶⠚⠛⠓⢶⠄⠀⠀⠀⠈⣇⠀⠀⠈⡁⠀⠀⠀⠀⠑⢤⣤⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣈⣉⣈⣉⡻⠇⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠰⡉⠑⢦⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀⢸⣿⣿⣹⣿⣧⠀⣀⣠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⢻⣿⣿⢻⣿⣿⢸⣿⡿⣻⣿⣿⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⢿⣿⣧⢿⣿⡟⣿⣿⣻⣿⠇⣿⣿⢣⣿⣿⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⣠⡿⣿⠮⠟⠁⡘⠉⠈⠋⠀⠻⢣⣿⠿⣁⢺⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣫⣿⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣟⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠴⠛⠙⠛⠿⣶⣦⠀⠀⠀⠀⣴⣶⠿⠛⠋⠛⠦⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢯⡀⠀⣀⣤⣶⣶⣄⡄⠁⠀⠀⠀⠀⠈⢠⣠⣶⣶⣤⣀⠀⢠⡹⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⢺⡇⠀⠙⠣⠴⣿⠭⠿⠆⠀⡁⠈⠀⠰⠿⣭⣿⡦⠜⠋⠀⢸⡗⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣂⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡾⢐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢂⠀⠀⠀⠀⠀⠠⡵⠦⢄⣠⠴⢮⠄⠀⠀⠀⠀⠀⠰⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡀⠀⣨⠀⠀⠀⠀⠀⠀⢀⣠⣤⣠⣄⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣆⠀⠀⠀⠰⠶⠫⠤⠤⠤⠤⠽⠲⠆⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⠈⠦⣄⣀⣀⣠⠴⠁⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⠦⣄⣀⣀⣀⣀⣠⡴⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⣀⣤⣤⣶⡾⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢷⣶⣤⣤⣀⡀
⢶⡿⠿⠟⢛⡿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣼⣿⢿⡙⠻⠿⢿
⠘⣷⡘⢃⡞⠺⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⡟⠻⢳⡤⢠⡶⢶⡄⢤⡾⠛⢻⡍⣿⢾⣿⣿⣿⣿⣿⣿⣿⠛⠗⢳⡘⢃⣾
⠀⠈⢿⣌⢡⡍⢉⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠳⣤⣼⢧⣽⡦⢠⣿⡼⣧⣤⠞⠉⠁⢸⣿⣿⣿⣿⣿⣿⣿⡉⢩⡌⣡⡿⠁
⠀⠀⠀⠻⣦⣀⡮⠔⠚⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠈⠑⠚⠁⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⡗⠂⢵⢀⣴⠟
⠀⠀⠀⠀⠈⠻⣄⠛⢀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣀⣠⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇⠛⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣦⣿⣿⣿⣧⣿⡇⠀⢸⣟⡙⢋⢻⣿⡻⢟⣿⡿⡙⠫⣹⣇⡈⢘⣿⣿⣿⣿⣿⣷⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣧⣿⡏⠠⠤⠃⣤⠈⣌⡕⢁⡈⢪⣡⠁⣤⠈⠲⠅⠙⣿⣯⣽⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠛⠷⠶⣤⣤⣄⣙⣀⣈⣁⣀⣉⣠⣤⣤⠶⠾⠛⠋⠁⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣋⣉⣿⣀⣬⣉⣭⣉⣸⣟⣩⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠓⡀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠔⠈⠑⠂⠁⠀⠈⠂⠀⢀
⠀⠀⠀⣄⠠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠐⠈⠂⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⠦⠀⠀⠈⣐⠁
⠀⠀⠀⣏⠀⠟⠀⢠⢨⡁⠀⠀⠀⠈⠒⠁⠀⠀⠈⡈⠀⠀⠘⡐⢂⠀⠀⠐⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢎
⠀⠀⠀⠀⠓⠀⠀⠀⠓⠉⠢⡀⠀⠀⠈⠀⠀⠀⠠⠦⠄⢀⣨⠠⠴⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⡆⠀⠀⠀⠀⠸⠆⣄⠀⠀⡀⠀⢀⠀⠀⠤⠀⠉⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠠⣶⡄⢀⣒⣼⡓
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢙⡁⠀⠀⠰⠿⠀⢈⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣠⡁⠄⠀⠀⠀⠀⢅⠁⠢⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⣄⠉⠡⠋⠘⠌
⠀⠀⠀⠀⠉⠂⠂⠚⠀⠀⠀⠀⠊⠀⠰⠈⡄⠀⠀⠀⠀⠉⠃⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠔⠀⠂⠈⠀⠀⠀⠀⠈⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠠⢀⠁⠀⠀⠀⠀⠀⡅⠀⠠⣶⠆⠀⠢⠤⠀⠀⠀⠀⡐⠁⠀⠀⠐⠂⠠⠀⢀⠁⠀⢀⡀⠀⠀⠐⠄⠐⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠂⢀⠂⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠂⠀⠀⠠⠚⡔⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠀⠊⠁⠀⠀⡀⠀⠀⠀⠀⠇⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⢆⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⡐⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⠐⠢⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠢⡈⠢⡡⠀⠀⠀⠠⣹⠓⠂⠀⠃⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠱⠄⠀⠀⠀⠀⡅⠁⠁⠐⠀⠀⠀⠀⠀⢀⠀⢠⠀⠠⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⢖⣈⠶⢌⠀⠀⠀⠈⠂
⠀⠀⠀⣂⡠⠐⠀⠃⢀⠚⠀⠀⠀⠀⠀⢈⠂⠒⠁⠀⠀⠀⢈⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⣀
⠀⠀⠈⠚⠢⢀⣀⠠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠄⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠢⢀⡀⠄⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠠⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⡀⠚⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡃⠀⠀⠀⢤⡆⠀⠘⠠⠄⠁⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⠀⣠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱⠃⠀⢤⠊⠈⠁⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠠⠌⠁⠈⠄⠤⡀⡹⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⢧⠄⡰⠓⠀⠀⠀⣄⣀⠼⠦⣀⡉⢀⠀⠐⠂⠴⠆⣀⠀⠀⠀⠀⠀⢸⠁⠠⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠀⠀⢹⠃⠀⠀⠁⠀⡞⠁⠀⠀⠁⠀⠁⢤⠀⣄⠀⠀⠁⠙⢏⠀⠀⠀⠈⢇⡐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣼⠀⠀⠀⠀⢸⠁⠀⠀⠀⢠⠈⠚⠐⠂⠈⠂⠀⠀⠐⢾⠀⠀⠀⠀⠺⡄⠀⠀⢀⠈⠈⠁⠠
⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⠋⠀⠀⠀⢸⡇⠀⠀⠂⢸⡇⠠⠠⡄⠀⠀⠀⠀⠈⠐⢀⠀⠀⠀⣼⢂⡀⠡⠀⠀⡷⠀⠀⠠⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠒⠀⠠⠨⣈⣷⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣤⡏⠀⠀⠀⠀⠄⣿⡤⠀⡐⠆⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠺⡇⠀⠀⠰⠀⠈⢿⠖⠀⠀⠀⠈⢳⣤⠄⠀⢴⠀⠀⠀⠀⢁⠄⣠⠾⠀⠀⠀⠀⠈⢡⡟⠀⠀⢹⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠦⢾⡆⠀⠀⠀⠠⠀⠙⠳⣤⡟⠀⠀⠀⠀⣤⠾⠋⠀⠀⡀⠀⠀⠀⢼⡗⠃⢀⣼⠃⠀⠠⠤
⠀⠀⠀⠀⠀⢀⠐⠀⡀⠉⠉⠖⠲⣤⡀⣼⠃⠀⢀⠀⠂⠄⠈⠇⠉⠛⠷⣤⣽⡞⢡⠀⠀⠀⢠⡈⠠⠃⠀⣼⠃⣰⠾⠁⡶⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠻⣿⠀⠀⢠⠀⠀⠀⡀⠀⣰⣴⠾⠋⠉⠛⠿⢶⣤⣩⡊⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠢⡤⠀⢀⣵⣾⣋⡁⡐⢂⡄⠠⠀⠀⠛⠙⠻⣶⣮⣀⢸⣿⡇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⡀⠈⢹⣧⠀⠀⣠⣴⠟⠁⠀⠈⠀⠁⠈⠀⠀⠀⠀⢐⠄⢺⠉⠙⢿⣿⣿⣧⠀⠀⠲⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢃⠀⠀⠀⠀⣿⣆⣹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠈⠐⠂⡀
⠀⠀⠀⠀⠐⠐⣄⡀⠠⠄⠀⠀⠀⣄⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢁⠀⣾⣿⣿⡀⠀⠀⢰⡏⠉
⠀⠀⠀⠀⠀⠀⠀⢈⣳⡆⠀⠀⣰⠅⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏⠀⣲⡴⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣹⣇⣠⣀⠀⢘⣿⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠻⠦⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⣔⠓⠀⠀⠉⠉⠉⠛⢷⣬⣿⣿⡇⠀⠀⠀⠀⠀⣤⣤⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⢰⣿⡿⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⣧⠀⣽⣿⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⡿⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣼⣿⣿⣠⣿⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢀⣿⣿⣿⠛⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠈⢻⣿⡟⠁⢹⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⠈⢿⣷⡀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀⠀⠀⢠⣟⢹⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣷
⣤⣤⣤⣤⣤⣄⣀⣀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣦⣶⣿⣿⣷⣤⣤⣦⣶⣿⣿⡿⠿⠿⠷⠶⠶⠶⠶⠶⣤⣤⣄⣀⣀
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⢿⡾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⡃⡼⠃⠀⠀⠀⠀⣴⣶⣶⣼⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢤⠀⠀⠀⠀⠈⢃⣀⣀⡸⢱⣀⠤⠙⠛⠛⠛⠿⣿⣿⡿⠃⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⢠⣄⣀⣠⠊⠁⠀⠐⢁⠜⢣⠀⢸⣏⣶⣤⡀⠘⣿⣿⣷⡄⠐⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠈⠒⢤⠤⠀⠠⠀⠘⠀⢀⠀⠆⢠⣉⡙⠻⠛⢹⡏⠉⠉⠁⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⢏⠤⠒⠚⢄⠀⠢⠴⢶⡊⠁⠀⢰⠀⡏⢙⡀⠀⠠⡀⠀⠀⣴⠟⣻⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠱⣁⣀⢶⣾⠂⢣⡀⠀⠸⣨⠗⠈⠙⠚⠀⠛⠳⡦⡁⠇⠀⠠⠵⣾⣿⣝⢿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠀⠐⡇⠛⠿⣤⣽⣿⣿⣇⣀⣰⠁⠀⣀⣀⠀⠠⢆⡀⠀⠙⣄⠀⠀⠀⠈⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⢠⣀⡇⠀⠒⢃⠜⠋⢡⣂⡄⠀⣉⠲⠄⠋⢀⣠⣴⠀⣀⠤⡀⠉⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡡⠨⢳⡀⠀⡏⠀⠀⢨⡽⢗⠻⠀⠀⡎⠁⡪⠽⠋⠰⠥⡀⡨⢰⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠰⡅⡀⡇⠙⠂⠖⠂⠘⠁⢀⣎⠀⣀⠔⠋⠉⠑⠢⣾⠼⣀⣈⢣⠸⣌⠆⡀⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠂⠳⢌⣘⡦⡄⣠⡀⠀⣤⠈⠈⢶⠁⠠⡄⣠⣶⣴⠈⡞⠀⡏⡆⣠⠃⠊⠐⠂⠐⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠕⠠⢊⠼⢿⠀⠉⣀⣀⣄⠀⠀⠸⠀⠀⢀⠁⢀⡀⠀⢰⠊⠀⠀⢹⠍⠉⡓⣎⠔⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠲⠡⡐⠆⠆⠀⠻⠽⠟⠀⠀⠀⡆⠧⡸⠀⠿⠃⠀⠸⠀⠀⠇⢸⠈⢑⠈⢘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⣿⠖⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⣶⣶⣶⣿⡖⠒⠒⠚⠧⠁⣂⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠋⠀⠀⠀⠀⠀⣴⣾⣷⣦⡀⠀⠀⠈⠛⠟⠛⠁⠀⠀⣠⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣶⣤⣀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⠀⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢀⣀⡀⠀⠀⠀⣰⣶⣶⣦⡀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⣿⣿⣷⡶⠆⠀⠀⠀⠀⠀⣀⣠⣤⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⢀⣼⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠈⠁⠚⠛⠛⠓
⠀⠀⠀⠀⠀⠘⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀
⠀⠀⠀⠀⠀⠴⣆⠀⠀⠀⠁⣠⣶⡀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⢀⣶⣄⢀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣶⣤⣾⣿⠋⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣶⣿⡟⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⡿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢐⡅⠀⢀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⢀⡤⢴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠒⢤⡀
⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⡏⠀⡼⠀⢀⣠⣄⠀⠀⠀⣠⣄⠀⠘⡄⠀⢣
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⡴⠛⠒⠒⠒⠒⠒⠚⠙⠢⡀⠀⠀⠸⡄⠀⠀⠀⡠⢧⣴⣥⣤⣜⡩⣻⡤⠤⣴⣉⡫⣥⣤⣽⣤⡦⣄
⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀⠏⠀⠴⠶⠦⠀⠀⠶⠲⠄⠀⢱⠀⠀⠀⢧⠀⠀⠀⡅⢸⢫⠀⠙⠓⠄⠾⠀⠀⠹⠤⠜⠛⠀⢸⠙⠀⢸
⠀⠀⠀⠀⠀⢸⠂⠀⡔⣼⠀⡤⠒⠁⢒⠀⠐⠊⠒⢦⠀⠘⣤⠲⡀⢾⠀⠀⠀⠙⢻⠀⠳⠤⣤⠶⣾⡶⢶⡶⢶⡦⠤⣤⠞⢸⠓⠁
⠀⠀⠀⠀⠀⠈⡇⠈⠧⢇⠀⠀⠀⠀⢁⣀⡀⠀⠀⠀⠀⠀⡸⠴⠀⡏⠀⠀⠀⠀⠘⡄⠀⠀⠸⣄⡈⢷⠾⣀⡠⠃⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠈⠢⣠⣈⣦⡀⠀⠀⠀⣤⡄⠀⠀⠀⠀⣴⣡⡠⠊⠀⠀⠀⠀⠀⠀⣈⣦⣀⠀⠀⠉⠉⠉⠁⠀⠀⢀⣤⠮⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠛⣶⠢⣀⣀⣀⡠⠖⣶⠛⠢⢄⡀⠀⠀⠀⠀⢀⠔⠉⠁⡏⠀⢹⡃⠒⢲⣶⠒⠚⡹⠁⠀⡇⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⣿⠀⠈⢦⡴⠁⠀⣿⠀⠀⠀⠙⢦⡀⠀⡰⠁⠀⠀⠀⡇⠀⠀⠑⠊⠁⠀⠉⠒⠁⠀⠀⡇⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⡿⠖⠚⠉⠉⠓⠶⢿⠀⠀⠀⠀⠀⢳⡼⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢇⠀⠀⢠⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⡆⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⢀⢟⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⢧⠀⠀⠀⢇
⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄⠀⠈⣗⠋⠸⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢸⠁⠳⡖⠒⡇
⠀⠀⠀⠀⢀⠁⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣇⠀⠀⡟⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⢧⠀⢰
⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠀⠀⠘⡄⢸
⠀⠀⠀⠀⠈⡀⢠⠻⣄⠀⠀⠀⣃⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣠⣿⡆⠀⡇⠀⠀⣯⢓⣞⣦⠤⠤⠤⠤⠤⠤⠤⠤⢤⢾⣖⡪⢿⠀⠀⠀⡣⠘⡇
⠀⠀⠀⠀⠀⡏⣹⠀⢸⠉⠒⠢⠞⠀⠀⠀⠀⠀⠀⠷⠔⠒⠉⣷⣁⡷⠲⡇⠀⠀⠇⠀⠀⠉⠉⠐⠒⣒⣒⣒⠒⠈⠉⠉⠀⠀⡎⠀⢀⢮⠀⠀⢘
⠀⠀⠀⠀⠀⢧⡿⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣼⣿⣇⠃⠀⠀⢸⠀⠀⠀⠀⠀⢸⠁⠀⠈⡆⠀⠀⠀⠀⠀⡇⠀⢸⡞⠓⠒⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⡆⠀⠀⠀⢃⠀⠀⠀⠀⢠⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠀⠈⡄⠀⠀⠀⡌⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠀⣏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡇⠀⠀⠀⠀⠀⢡⠀⠀⢰⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢒⣊⣀⢴⠀⠀⠀⡦⣀⣙⡒⣤⣀⠀⠀⠀⢀⡤⠐⠒⠒⠧⠤⢠⡇⠀⠀⠀⠀⠀⢸⡄⠤⠾⠐⠒⠂⢾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠤⠀⠒⠊⠀⠀⠀⠑⠒⠀⠤⠤⠜⠀⠀⠀⠚⠤⠤⠤⠤⠤⠒⠚⠁⠀⠀⠀⠀⠀⠈⠓⠒⠤⠤⠤⠤⢼⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢤
⠀⢨⣿⣿⢟⢿⣿⣧⠀⠀⠀⣰⣞⡛⣳⣦⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠶⢤⠁⠆⠰⠆
⠀⠸⣿⣯⠈⠠⣿⡗⠀⠀⢠⡟⠛⠁⠙⠛⡇⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢿⣤⣬⣇⠁⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⢶⢆⣶⣶⣇⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄
⠀⠀⠀⠤⠤⠤⠆⠀⠀⠂⠀⠉⠉⠈⠉⠈⠀⠀⠀⠠⠄⠀⠤⠀⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠻⡿⠿⠃
⠀⠀⠀⠀⢀⢀⠀⠀⣈⠥⠀⠀⠰⣠⡆⣴⢰⣴⢰⠀⣤⣤⢀⡆⣦⢰⠀⣆⣤⡄⠀⠀⠀⠀⠀⠀⠀⠓⠂
⠀⠀⠀⠐⠻⠳⠀⠀⠉⠁⠀⠀⠈⠁⠀⠈⠈⠈⡈⡀⣁⡈⡀⡅⠀⠈⠀⠉⠁⠁⠀⠀⠀⠀⠀⢠⡀⠀⠀⣠⡄
⠀⠀⢠⡀⣀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣤⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣿⡶⣾⣏
⠀⠀⠀⠁⠈⠁⠁⠀⠀⣶⡄⠀⠀⢠⣶⠀⠀⣀⣻⠟⢉⠙⢿⣃⡀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⣿⠙⢹⣿
⠀⠀⠀⠏⠀⢒⣛⠀⠀⠘⡇⠀⠀⢸⠇⠀⠀⣭⣿⠀⠀⠀⢈⣯⡅⠀⠀⠀⠈⢹⣿⠀⠀⠀⠀⠀⣿⠀⢸⡟
⠀⠀⠀⣿⣿⣿⡗⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⢹⡷⣶⣶⣿⡍⠁⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⢸⣶⣾
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣍⠀⠀⠀⠀⠶⠖⠀⣬⣿⡄⠘⠿⠗⠀⢸⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠺⢥⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠁⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⡀⠒⠢⠔⢛⡧⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣔⣍⣦⡀⠀⠀⣠⡿⠀⠀⠀⠀⣂⢀⡄⢸⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⣿⠟⠟⠛⠿⣯⡀⠂⠀⠀⠀⢀⣈⣉⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⣦⠀⠀⠀⠈⠻⣶⣿⣿⠿⢿⣵⣿⠯⠽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣴⣾⣦⣀⡆⠀⠀⠀⠀⠀⣿⠉⣀⠞⠹⣇⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠈⠻⣇⣆⡀⢻⠀⠀⠉⠊⠀⣩⠿⣙⠀⢀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡙⢿⡄⠀⠀⠹⣿⣷⠸⡆⠀⠀⢰⠞⠥⠤⣼⡗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠁⠙⣿⣄⠻⡆⠀⠀⠘⢿⣷⡝⣌⣠⠈⠀⠀⠈⢻⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠻⣝⢷⡝⠄⠀⠀⠀⠻⣞⢷⣭⣷⣶⣶⣿⣿⡀⢸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠘⠈⢻⣎⢆⠀⠀⣠⠟⢿⡘⣿⠛⣿⣿⣿⣯⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢰⡆⠀⠻⣯⣧⡾⠛⠲⣄⡙⢿⠀⢸⣿⣿⣿⠀⣿⡞⣇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠻⣆⠀⢻⡀⠀⢹⣯⠀⠀⠀⠀⠉⠻⣧⠀⢿⣿⣿⡆⢹⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠹⣦⠘⣧⠀⠀⢻⣦⠀⠀⠀⠀⣤⣹⡄⢸⣿⣿⡇⢸⣧⣿⣆⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢿⡆⢹⡇⠀⢸⣿⠁⠀⠀⠈⠻⢻⡇⠀⣿⣿⣿⠀⣿⣿⠾⠖⠒⠒⣄⣴⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⢿⣸⡇⠀⠸⣿⠃⠀⠀⠀⠀⠀⣧⠀⢿⣿⡿⠖⠉⠀⢀⣨⣦⣴⣿⣿⡟⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠻⣦⡀⢸⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⢠⡾⠋⠀⢀⡴⠾⠛⠛⠉⠉⠀⢈⣿⣎⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠈⠛⠶⣿⣇⠀⢾⣿⣀⣰⠟⣷⠦⣤⣿⠋⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠛⢫⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⡤⣽⢷⣾⣿⣿⡏⠀⠸⡇⢨⠅⠀⠀⡄⠐⢄⠘⠋⠁⣀⡤⢴⣾⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⠀⠉

⣶⢶⢶⢶⠷⠷⣶⠶⢶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⢶⣶⣶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣾⣿⣿⣷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣿⠿⢿⣿⡟⠿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠞⠁⠀⠀⠙⢿⣷⣦⢭⣑⣒⡾⠤⣬⣙⠮⡻⡿⣿⣿⣿⣿⣿⣿⣭⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠃⠀⠀⠀⠀⠀⠀⠈⠛⠳⣶⣤⣍⣙⡓⢮⣿⣾⣾⣾⣿⣿⣿⣿⣿⣿⣿⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠀⠀⠉⠛⠛⢿⣿⣿⣿⣿⡛⠛⢻⣿⣿⣿⣿⣟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⣴⡿⠛⠁⠀⠀⠀⠀⠀⣀⠤⠐⠛⣿⣿⣿⣷⡀⡀⣿⣿⣿⣿⣿⡣⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣤⠶⠆⠀⡤⠂⠀⠒⢿⡿⠛⠚⠁⠀⠀⠀⠀⠸⡏⠹⣿⢻⣶⣿⣿⣿⣿⣿⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡤⢤⣀⢤⣠⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣟⡧⠔⠏⢺⠉⢢⡀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢿⢹⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣇⡇⠀⠠⠘⠁⠀⠉⠒⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡱⠢⠤⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠘⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠢⠀⢀⣠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢟⢆⠀⠀⠀⠑⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠸⢻⣿⣿⣿⣿⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⡈⢧⡀⠀⠀⠀⠠⡀⠀⠀⠀⣠⠐⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀⢘⠻⣿⣿⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢥⠈⠳⣄⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⠋⠉⠀⠁⠉⠉⠙⢳⣄
⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⣠⡶⢶⣿⠋⣻⣿⣄⣤⣀
⠀⠀⠀⠀⢀⣾⣿⠓⠦⢤⣄⡤⠺⣥⠒⢦⡤⠒⠋⣿⣧⠇⠀⢀⡟⠽⢣⢿⣛⣵⡿⡷⠽⢻⣆⣀⡀
⠀⠀⠀⠀⠈⠳⢿⡄⠀⢣⡈⠣⠴⠋⠂⠘⡀⠃⢠⡿⠉⢀⣾⣻⡷⠒⠋⠀⠀⠀⠙⠛⠛⠛⢿⣻⡗⣀
⠀⠀⠀⠀⠀⠀⢈⣿⣦⣀⠑⠢⠤⠤⠀⠊⣀⣴⣟⠁⠀⣾⢽⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡃
⠀⠀⠀⠀⢀⡴⠋⣹⠉⣽⠛⢲⡖⣶⠒⢻⡟⠙⣿⠳⣤⣟⣿⠛⠢⣄⡀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣸⣧⠟
⠀⠀⠀⣠⠞⠀⠀⡏⠀⠹⡶⣾⡋⠙⠓⣾⠁⠀⢹⡄⠈⠻⣿⡀⠀⢀⡍⠻⡋⢳⣶⣉⣱⢶⡋⠉⢸⣻⡇
⠀⠀⣴⠋⠀⠀⢸⡇⠀⠀⠀⠈⣧⠀⢸⠃⠀⠀⠈⣧⠀⠀⣸⣧⡀⠀⢢⡀⠈⠉⠀⠉⠁⡠⠁⢠⡟⠉
⠀⠸⣧⡀⠀⠀⣾⠁⠀⠀⠀⠀⠸⣇⡟⠀⠀⠀⠀⢹⣦⡞⢁⡾⣿⠶⣤⣌⣐⣒⣒⢒⣉⣠⣶⢿⡀
⠀⠀⢸⡿⠲⢶⡏⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠈⣿⣷⡾⠁⣿⠀⠀⠈⠻⣛⠿⠋⠉⠁⢹⡄⠻⣄
⠀⠀⢸⠇⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠹⣆
⠀⠀⣾⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⢈⣧⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢹⣆
⠀⠀⣿⠀⠀⣿⠧⣤⣀⣀⣀⡀⠀⣸⡆⣀⣀⣀⣤⠴⠛⢻⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⠻⣇
⠀⠀⣿⠀⠀⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢹⠟⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢻⡄
⠀⠀⢿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⡇
⠀⠀⠸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⢰⡇
⠀⠀⠀⢻⡄⣿⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⢸⡇⣸⣦⠤⣤⣤⣤⣤⣤⣠⣤⠴⠶⠞⢋⡹⡇⣸⠃
⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢀⡴⣄⡀⠀⠀⡰⢢⡀⠀⠈⡤⠤⢿⠏
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⡇⣿⠰⡍⠷⢠⠇⠀⢈⠱⡦⠰⠀⢀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠲⠚⠉⠀⠀⠀⠤⠂⠊⠔⡊⡐⠄⠀⢿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠆⢉⡊⢡⠀⠀⠀⠁⣑⡁⠅⠀⢸⡆
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠠⣋⡑⠚⣀⠇⠀⠀⠀⠀⠀⠀⣀⡼⠇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⠙⠓⠦⣤⣤⣵⣶⣤⣤⡤⠴⠶⠶⣿⠋⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢹⡀⠀⠀⠀⣿⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠙⣷⣦⠤⠤⠤⣤⣤⣤⠾⣷⠤⠶⠖⠒⠛⠛⢯⡀⠀⢀⣼⡗⠓⢲⣄⣻⣀⣤⠖⢚⡳⢦⡀
⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠹⣶⠁⠀⠀⠀⠀⠀⠀⠀⣿⢠⡟⠀⠈⠉⠁⠘⣿⠋⠀⠁⠁⠀⠈⣷
⠀⠀⠀⠀⠀⣿⣤⣀⣀⣀⣀⣀⣠⡴⣿⠷⢤⣄⣀⣠⣀⡠⠴⣿⢸⡟⠛⠛⠛⣛⣛⣿⣛⠒⢒⢒⣚⣩⡿
⠀⠀⠀⠀⠀⠙⠷⠬⠭⠭⠭⠭⠤⠴⠟⠓⠒⠒⠒⠒⠒⠛⠛⠋⠀⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠹⠡

⠀⠀⠀⠈⠆⠀⠀⠃⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠈⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣦⣤⣤⣴⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⡀⣢⣉⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢹⣿⣿⣿⣿⢿⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⢠⣆⢢⣲⣶⣠⣀⡀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠘⠋⠀⠉
⠀⠀⠀⠀⠀⠀⣜⣶⣿⢿⠎⠑⡛⢽⣿⣟⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⡔⠃
⡀⠀⠀⠀⠀⢠⣟⡫⠇⠈⡀⠀⡇⠈⠈⡿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠳⡀
⢿⠆⠀⠀⠀⣞⣛⣒⠀⠂⡆⠀⡇⠀⢈⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⢰⣿⠷⠤⠤⠤
⣾⡆⠀⠀⠀⠴⣖⣚⠀⠀⠹⡿⠇⠀⠈⣙⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠈⢂⢀⡀⠀⠀⠀⠀⠀⠀⣾⣤⣦⣄⠀⢠
⣿⡇⠀⠀⠀⠈⠟⣶⣧⠀⡀⣐⣀⣀⢠⣻⡦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠒⠻
⡻⠃⠀⠀⠀⠀⠘⠻⣻⣿⣿⣿⣴⣷⣿⠏⠛⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢻⠀⠒⠒⠐⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣤⣤⣶⣦⠠⣶⣶⣶⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⢻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣄⣄⡀⡀⡄
⠀⠀⠰⠖⠂⠀⠀⠀⠒⠚⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣀⣀⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠼⠍⠉⢼⢗⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⣀⡿⣿⣿⡂
⠀⠀⠀⠀⠈⠡⠻⠓⡈⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠀⠀⠀⠤⣯⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣤⣤⣤⣼⣿⢚⣛⡁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⢤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⢲⡞⠂
⠀⠀⠀⠘⡀⢸⠀⠐⠀⡰⠁⠈⠀⠀⠀⠀⠀⡴⢧⡖⠀⠀⠐⠊⠀⠈⠹⣮⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢦⠠⠀⢠⣥⡀⠄⢰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⣿⡷⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣦⣿⣿⣤⣾⣆
⠀⠀⠀⢀⣀⡀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⡿⢿⠿⡿⢿⣿⣧⣶⣦
⠀⠀⠀⠈⠁⠙⣿⣿⡅⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⡇⠀⡆⠀⢸⣿⣿⡿⠃
⠀⠀⠀⢀⠠⠀⠛⣿⠃⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢻⣀⣿⠀⠀⡀⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⡇⠀⠃⠆⠸⣿⣿⣿⠗
⠀⠀⠀⢸⠛⠰⣶⣶⡆⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠂⠈⠟⠛⠃⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⢷⠇⠀⣿⣿⣿⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠋⣉⣷⣖⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠘⠟⠀⢬⣾⡋⠁⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠔⣰⠇⣰⠒⠛⠀⠀⠽⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡖⠀⠀⠀⠈⠉⠠⡄⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠎⡻⠁⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⠻⣾⣳⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢰⣷⣖⠃⠀⠰⡿⣏⣊⠀⠀⠌⢻⡧⠨⢻⣷⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣭⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣴⠎⢻⣧⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢽⣿⢰⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠶⠓⡦⠀⠀⠀⠀⠀⠀⠀⠉⣺⡙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢢⢰⡇⠀⠀⠉⠀⠀⠀⠀⢠⡖⣦⢔⣿⡟⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⣼⣇⣤⠀⠀⠀⡀⠀⠀⠀⡠⠎⡸⠀⠀⠈⢑⡄⠀⠄⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⣽⣿⡆⠀⠰⠟⢛⠂⠉⢀⡔⠁⠀⠤⠚⠉⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⢛⣻⡟⠁⠀⠀⠠⠇⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠏⡻⠁⠀⠀⠀⠼⠀⠀⠀⠀⠀⢀⡆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣿⣶⢏⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⡾⠟⠀⠀⠐⠛⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⣿⣿⡘⠀⠀⠀⢰⡋⠀⠀⠀⠀⠀⠸⠟⠉⠀⠀⠀⠀⣀⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⢸⠃⠀⠀⠠⡀⠰⡀⠀⠀⢓⣶⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⣄⠀⠀⠀⠁⠀⠀⠀⠂⢈⣙⠻⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣠⡜⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⠟⢠⠀⠀⠀⠉⠙⢯⣀⢦⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠄⡀⢁⣄⠲⣽⣷⠦⣿⢳⣔⠏
⠀⠀⠀⠀⠉⠉⠙⠛⠋⠙⠋⠉⠉⠋⠐⠀⠀⣀⣇⠀⠀⠀⠀⠀⡙⣦⠙⢷⠀⠹⡷⢻⠆⠐⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡀⠀⠀⠀⠀⠁⣈⣠⣬⣇⣀⠀⠀⣧⢦⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣷⡿⣟⣿⣿⣄⠰⣻⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠹⣯⠙⢿⣿⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣴⠀⢰⠂⢰⠀⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢀⠇⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠀⠀⠀⠀⠀⠀⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠛⠶⢶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡛⠁⠀⡀⠀⠄⠀⠀⠘⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⣶⢀⢼⣿⠀⡐⠀⣠⡀⣹⣿⡆⠀⠀⠉⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⡼⣿⣾⣿⣿⠟⠹⠈⣿⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⡏⠙⠃⠉⠁⠀⢠⡀⢁⠀⠀⠒⡄⠀⠠⢦⡀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⠉⠙⠛⠁⠃⠀⠀⠀⠀⠈⠂⠉⣈⡀⠀⠀⠐⢤⣄⡀⠑⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⣆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣨⡇⠰⣀⢧⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡄⠀⠀⠐⠅⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡙⡐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⢼⠂⠹⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣾⡿⠳⠾⣿⣠⠘⠮⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡰⠃⠀⠀⠀⠈⠙⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣧⡼⠃⠀⠀⠈⡌⢹⠋⣄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣴⠁⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⢠⠜⢻⠟⠀⠀⠀⠀⠀⢧⢸⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿⠀⠀⠀⠀⠀⠀⠀⢀⠈⢿⡀⠀⠀⠀⠀⠀⢨⠟⠁⠀⠀⠀⠀⠀⠀⠈⢠⡄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⢻⠀⠀⠀⠀⠀⠀⠀⢀⠱⡸⣧⠀⠀⠀⣤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣧⡐⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⡆⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⢀⣴⠟⢹⠳⡽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢾⠿⣤⠀⢀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣿⡏⢠⡄⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠾⠁⢠⡇⠀⠀⣀⠤⠖⠈⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠉⠀⠀⠀⠀⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⡇⠀⠀⠠⠄⢀⡒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⣿⣿⣏⠁⠀⠀⡶⠤⠤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⣷⣴⣦⣤⣦⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡿⣿⠛⠟⠛⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡧⠀⠠⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣦⣶⣶⣶⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⡿⠏⠀⠉⠙⣿⡏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⡋⠁⠀⠀⠀⠀⠠⣿⡃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣿⡟⠃⠀⠀⠀⠀⠀⢀⢀⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠤⢀⣿⣿⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡀⠀⠀⠀⠀⠀⠘⣿⣿⡄⣿⡇⠀⠀⠀⠀⠀⠀⠀⡄⢸⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢛⠄⠀⠀⠀⠀⠀⠀⡏⣿⡇⢻⡇⠀⠀⠐⠀⢀⠀⠀⠀⣾⠇⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⠀⢠⠘⡄⠀⠀⠟⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠘⠃⠀⠀⠀⠀⠀⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡙⣉⡉⣿⠁⣽⠉⣩⡍⡋⣭⡟⢩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠶⠛⠶⠂⠀⠀⠈⠁⠂⢄
⠀⠀⠀⠀⠀⢠⢂⣄⣀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⡌⡜⠀⠈⠙⠡⠄⣀⠀⢡⡈⠑⠲⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠙⠒⠢
⠀⠀⠀⠀⠀⡠⠁⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠆⢰⠀⠀⠀⠀⠀⢀⠤⠚⠈⢁⠉⠀⠁⠓⠠⡀
⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣎⠀⠀⠀⠀⡔⠁⠀⠀⢀⠎⢡⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠐⠉⡇⠈⠀⢘⡧⠀⠀⠀⠀⠀⡷⠀⠀⠀⠉⠁⡄⠀⢐⠀⠀⢀⡴⠋⠀⠀⠑⢌⠂⠀⠀⠀⢃
⠀⠀⠀⠠⡀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠇⡠⠁⠀⡏⢁⠔⠉⠀⠀⠀⠀⠀⠀⠑⠢⢀⡀⢸
⠀⠀⠀⠀⠈⠙⡀⠀⠀⠀⡄⠈⣀⠀⠆⠀⠀⠀⡘⠉⠀⠀⠀⢃⠂⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⢃⢸
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀⠀⠀⠀⡎⠹⠀⠀⠀⠉⠀⠜⠀⠀⠀⠉⠀⠀⠀⡏⢱
⠀⠀⠀⠀⠀⠀⠀⡔⢻⠂⠒⠒⡐⠒⠒⢈⣿⠇⠀⠑⢤⡀⠈⠒⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠈
⠀⠀⠀⠀⠀⢀⠺⠀⠸⣇⠀⠀⠈⠀⢀⡼⠋⠀⠀⠰⠁⠑⢄⠀⠀⠑⠢⣄⣀⠀⣀⣀⢀⣠⡔⠫⢄
⠀⠀⠀⠀⠀⡌⠸⠀⠀⠡⡁⠤⠤⡒⠝⠁⠀⠀⠀⢰⠀⠀⠈⢂⡠⢖⠁⠀⢻⢕⡀⢐⡂⢴⠁⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠰⠀⡀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠈⡇⢰⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠈⢄
⠀⠀⠀⠀⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⣦⣀⠀⠀⠘⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠈⠆
⠀⠀⠀⡠⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡄⠘⠀⠀⣰⡇⠀⠀⠀⠀⠀⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣸⠍⠒⣬⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⢀⡐⠢⡤⠁⡇⠀⠀⠸⠁⠉⠁⠀⠈⡄⠀⡀⠀⣴⡀⠀⢸
⠀⠘⠒⠮⣼⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⣹⠋⠀⠀⡇⢀⠰⠁⠀⠀⠀⠀⠀⠈⠺⣇⣀⣸⣇⣠⠂
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠇
⠀⠀⠀⠀⠀⠇⠸⡀⠀⠀⠀⠄⠦⠤⠷⠦⠤⠀⠀⠀⣀⡅⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⢸⢠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠉
⠀⠀⠀⠀⠀⢸⣼⠀⠀⠀⠀⠀⡀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠀⠘⠈⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠠⢴⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠇
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠠⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠘⡒⠒⠀⠠⡄⠐⠒⠀⠏
⠀⠀⠀⠀⠀⠀⣞⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⢀⣀⣀⣀⡁⠀⠀⠀⢡⠀⠀⠀⡇⠀⠀⠀⠸⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠙⢳⠪⠍⠡⠦⣤⠀⠀⠀⠀⠈⠛⠠⠬⠙⠃⠀⠀⠀⢨⠦⣀⠀⡇⠀⠀⠀⠀⠈⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠬⠛