Loosen up Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠉⠉⠉⠐⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠯⠭⠥⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠱⢄⠀⢀⡀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠖⠿⠓⠒⠲⠄⠄⠒⠚⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡀⠀⠀⡴⠚⠑⢦⡠⠚⠐⢦⠀⠉⢲⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⢸⠀⠀⠀⡀⠇⠀⣀⠀⢣⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠈⢦⡀⢀⡠⢧⡀⠀⠀⡸⠀⠀⠙⠢⠞⣉⡉⢳⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠈⠉⠲⠅⠉⠀⠀⠉⠒⠊⠁⡀⠀⠀⠀⠘⠁⠀⡼⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡽⠆⢠⠖⠒⠒⠋⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠴⡀⠀⠀⠀⣀⠴⠛⠉⠀⡠⠋⠀⠀⠀⠀⢀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⣠⠞⠁⠀⢀⡠⠀⡠⠿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠄⣀⣀⣀⣀⣀⡤⠔⠋⠑⠒⠒⠊⠙⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⠤⠤⢸⣛⣻⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣉⣹⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠁⣶⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡠⠊⠀⠈⠒⢺⡁⢀⠃⠀⠀⢀⢀⡀⢰⡗⠀⠀⠸⡀⠀⣀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃⠞⣀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢚⠀⠀⢉⡀⠠⠂⠀⠙⠉⠤⠞⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠽⡀⠀⠰⣼⣦⠀⠀⠾⠀⠀⠀⠀⠀⠀⠈⣄⡀⠘⢄⣁⠔⠂⠤⠀⠒⣂⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⠠⢻⢸⠀⢱⡀⢫⡭⠉⠉⠁⠐⠗⠈⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⡦⠞⠁⠪⣆⢢⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠈⡋⢧⠀⠀⠀⠀⠳⣀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡁⡇⠀⠀⠐⠆⠀⠀⠁⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⣇⠘⡄⠀⠀⠀⠀⠀⠀⠈⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣯⡀⠀⠀⠀⠙⡳⡿⠒⠂⠈⣉⠀⠀⠀⠼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢡⠿⠽⠛⠒⠦⣌⡀⠀⡽⠈⠐⠂⠀⠀⣄⣠⣴⡾⢧⠉⠉⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠭⠭⠭⠤⠬⠤⠤⠬⠭⠝⠻⠓⠲⠾⠿⠿⠥⠒⠙⠻⠿⠭⠭⠭⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⢁⠀⠀⠀⠠⢄⡈⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠐⢤⡀⠀⠀⣠⠞⠁⠀⠀⡜⠀⠀⡄⠀⠀⠈⠙⠓⠉⠁
⠀⠀⠀⠀⠀⠠⠂⣀⣠⡾⠛⡃⠈⠁⢉⣉⡛⠓⠸⠀⠀⡇⠀⠀⠀⣀⡷⠚⠹⣇⠀⠀⡔⣾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠛⠋⡀⢀⣠⣿⣷⣶⠶⠖⢬⣧⠀⡆⠐⣧⣠⣴⠏⠁⠀⠀⠀⠈⠓⢒⠁⡘⠶⠤
⠀⠀⠀⠀⠀⢰⣤⣠⠿⠛⠉⠀⠀⠀⠀⢀⣀⠈⡶⠁⠀⢻⡣⠘⠷⠀⠀⠀⠀⠀⠀⠈⣉⣁⡀
⠀⠀⠀⠀⠀⠐⢽⣇⠀⣠⣤⣄⠀⠠⢶⣿⠃⠀⠀⠂⠀⠀⠻⣶⣦⣤⡀⠀⠀⠀⠀⠈⠁⠐⠀⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠻⠄⠀⠉⠙⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠹⣿⣿⣿⣦⣀⠀⠀⠀⠀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⣤⣁⡀⠀⠀⠰⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠾⠟⠋⠁⠀⢀⣶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢰⡞⠽⣟⢉⣤⣶⠃⡀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⡀⣼⠟⠋⠀⠀⠀⠀⠤⣄⣀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠳⢾⣷⣦⣿⣿⡿⠁⠀⠈⠃⠒⠠⢀⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⢠⡇⢿⢀⣠⠀⠀⠀⠀⠀⠀⠉⢹⠄⠀⠑⠄⡀
⠀⠀⠀⣠⠴⠚⢉⡠⠀⠀⠀⡄⠈⠉⠙⠉⠁⠀⠀⠀⡀⠀⠀⠀⠈⠒⠦⣄⡀⠀⠀⠀⠀⠀⢨⠃⢠⠈⠁⢀⠀⠀⠀⠉⠁⢠⠏
⠀⠀⠀⠀⠰⠚⠉⠀⢀⠀⠀⠈⠒⠤⣀⣀⣀⠠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠄⠀⠨⠀⠀⡄⠀⠻⠀⠀⠀⠀⢸⠃⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠁⠀⢠⣷⡀⠈⠀⠀⠀⠀⣸⡀⠀⠘⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⠐⠀⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠰⣿⣿⣿⣦⣄⠀⠐⢈⣿⣧⡠⡃⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠚⠉⠁⠀⠀⠀⠀⠉⠐⠣⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠂⠀⠠⡀⠉⠉⠀⠀⠀⠀⠀⢸⠿⠟⠻⠛⠂
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⣠⠤⠖⠒⠒⠂⠀⠤⡀⠸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠢⠄⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡀
⠀⠀⠀⠀⠀⠀⢸⡇⣠⠖⠉⠉⣩⣭⡅⠀⢰⣿⣓⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣷⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⢿⡁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠟⠿⠿⠛⠻⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠤⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢀⡀⠀⠙⣿⣿
⠀⠀⠀⠀⠈⠀⠀⠀⠘⣷⣆⠀⠁⠀⠠⢤⠔⠂⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠴⠶⡤⠀⠀⠉⠉⠁⠀⢸⣿⡀
⠀⠀⣠⠴⠀⠀⠀⠀⠀⠸⣿⣷⣦⣄⣀⣀⡠⠎⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠁⠀⠛⠀⠀⠐⠤⠛⠁⠈⠈⣿⠃
⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠃⠀⠀⠠⠀⠀⠀⠙⡗⢢⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠡⠀⠀⠀⠘⣄⡠⠤⠐⠂⢡⡀⠀⠀⢉⡁⠀⠀⠀⠀⢰⠀⠀⠉⠉⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠈⠦⢄⣈⠁⢀⣀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⠛⠉⠀⠀⠘⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠋⠀⠁⠀⠀⠙⠒⠤⢄⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠙⠿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⠔⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠗⠊⠙⠀⢀⠔⠒⠒⢄⠀⠀⠀⠀⠀⢀⡦⠤⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠆⠀⠀⠀⡴⠊⠀⠈⠄⠑⡄⠀⡆⠀⠀⠀⢀⠎⠀⠀⠙⢦⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡄⠀⠀⣀⡰⠁⠀⠀⠙⢷⡄⠀⠀⠀⠀⠐⠀⢀⣄⠀⢀⣸⠉⠓⢲⠀⡜⠀⠘⡀⠰⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡶⠊⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠐⠉⠀⡇⣉⢀⣠⣄⠀⢻⠦⢀⠀⠱⠀⡇⢀⠂⠀⠀⠀⠀⠀⠈⡄⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠈⣧⣬⢄⣨⠁⠀⢀⠄⠾⠄⣀⡀⠀⠀⠀⠀⣆⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⢀⡴⠢⠤⠄⡀⢀⠖⠂⠀⠙⠁⢰⡞⠀⠀⠀⠁⠈⡆⠀⠀⢀⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠋⠀⠀⠀⠀⠈⡎⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿
⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠑⠂⢄⠀⠀⠀⡐⠀⠀⠀⠀⢀⣅⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿
⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠐⠀⠀⠀⠀⢀⠏⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⣿⣿⣿⣿⣿⣿⠛⠉⠑⠢⠤⠊⠉⠒⠤⢄⣀⣀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⡀⢾⣀⠀⠀⠀⠀⠀⡰⠈⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⣼⠀⠉⠐⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⢂⢡⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿
⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠳⣧⡀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿
⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⢄⣀⣠⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠍⠒⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣠⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣶⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣙⣟⣻⣹⣏⣩⣉⣻⣋⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣿⣿⣿⣿⣭⣭⣽⣭⣭⣭⣭⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠁⠀⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠉⠉⠉⠙⠛⠳⢦⣄⡀⠀⣸⡷⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠐⠲⠶⢤⣬⡿⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢶⣒⠲⠤⢤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣶⣾⠿⠛⠷⠦⣤⣀⠀⠀⠀⠀⠀⠙⠓⠺⠛⠛⠛⠋⠉⠀⠀⢀⡤⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⣠⣶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⣠⣴⠟⠁
⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⣾⣿⣭⣤⣶⡶⠟⠋
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠛⠋⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⣿⣇⠀⠀⠀⣀⣤⣶⠿⠛⠉
⠀⠀⠀⠈⠻⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠆⠲⠀⠒⠤⠄⠰⠰⠀⠸⠀⠐⡈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣰⣿⣿⡶⣇⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣩⡅⣈⣙⠛⠓⣛⣛⣿⠻⠸⣦⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠲⣀⠙⠻⠟⠛⠛⣩⠶⢀⠀⠀⠈⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣂⡽⠃⠀⢠⡀⠀⢹⣇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢘⡉⣡⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣾⡀⠀⢀⣀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⠁⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⢿⣶⣤⣠⣤⣦⡤⠾⠓⠶⢤⣠⣤⠤⠞⠛⠁⠀⠀⠀⠀⠀⠐⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⣾⠁⠀⡄⠙⣇⡠⠋⠁⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⢀⡋⠀⠀⣃⠘⣿⠏⠀⠶⠁⠀⠀⣄⢀⣀⢐⠀⡀⠀⠀⠈⢿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠃⠀⠀⠀⢠⡞⠛⠀⠀⡽⠠⣟⠆⠘⢠⡄⠀⠛⠁⢀⡞⢰⣿⣅⠄⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠇⠀⠀⠀⢀⣾⡄⠀⠀⢠⣷⣘⠪⣟⡀⢸⠁⠀⠀⢰⡾⠃⠚⣿⣟⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢰⣽⠛⢳⡄⠀⢠⣹⢯⡑⢮⡛⠾⠀⠀⠀⠟⠁⠀⠀⠻⠛⣤⡄⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠄⡁⠁⠀⠀⢧⠀⠀⢸⠆⠉⠂⠁⡴⠀⠀⡜⠀⠀⠀⢀⣾⠀⠈⠫⣴⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⣼⡆⠀⠀⠀⠈⠀⠐⠢⠀⠀⠚⢁⠀⢀⣾⠁⠀⠀⠀⣀⣹⠀⠀⣀⣘⣾⡏⠁⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣷⣄⠀⠀⠀⠐⠀⢸⠀⣀⡴⢸⢧⣾⣿⣦⡔⠋⠈⠉⠙⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡔⠚⠀⠀⠀⠀⢆⠁⠀⠀⢀⠘⣏⡄⠀⠀⠀⠀⠈⠁⠀⣆⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣀⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠷⢄⠀⠀⣿⣽⣿⣶⣀⠀⠀⣀⣴⣦⣾⣶⡾⠡⠤⠄⠄⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠳⢾⡿⡿⢿⣟⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⣴⢷⠄⠀⠀⢠⣶⢦⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⡡⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠢⣀⡠⠤⢄⠀⠀⠀⠀⠀⢀⠤⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠒⢦⣀⠀⠀⠀⠀⠀⠀⡐⠈⠁⠀⠀⠀⠑⠤⡀⠄⠐⠁⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠔⠂⠊⠉⠢⡀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠋⠀⡈⠀⠎⢁⠈⢀⠀⠀⠀⠀⡰⠊⠢⡀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠐⠐⠢⣄⠀⠀⠀⠀⠀⠊⠀⠀⠀⢠⠇⠤⡀⢘⠀⢣⠀⡀⠀⠀⡐⠀⠀⠀⠈⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠢⡴⠈⠁⠀⠀⠀⠈⠳⡀⠀⠀⠀⠠⠚⠰⠐⠊⢀⡀⠱⣦⡀⠀⠂⢠⠀⠈⠀⠀⠀⠀⠐⡀⠀⠀⠌
⠀⠀⠀⢀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠄⠈⠻⠁⠙⣤⣁⣈⠀⠀⡀⠴⠤⢀⡀⠀⠀⠀⣂⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⣠⠓⠢⠠⡀⠰⠃⠀⠈⠉⢲⠁⠀⠀⠁⠈⠀⠀⢀⠇⠐⠄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⡠⠂⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠄⠀⢀⠀⠀⠀⠀⡴⠓⢎
⠀⠀⠀⠀⠀⠈⠦⠄⠀⡠⠂⠄⣀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠄⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⢠⠁⠀⠀⠀⡀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠖⢁⠉⠐⠢⠤⠂⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠈⡄⠀⠀⠀⠀⠀⡠⠤⠄⠊⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⣀⠀⠀⠀⠀⣀⠊⠀⠀⠀⠀⠈⠳⣄⡀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡲⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢖⠒⠀⠠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡠⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣟⣛⣛⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⡿⢻⡇⣏⡷⢻⡾⡾⠁⠀⢸⠇⢾⣽⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⣿⣀⢴⣳⢴⣞⡦⣾⣧⢶⣷⢴⡞⡆⠀⢿⣀⡇⣶⣳⠄
⠀⠀⠀⠀⠁⠀⠈⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠛⠀⠀⠀⠀⠀⣀
⠀⢻⡞⢱⢲⣤⣆⣦⣶⠂⠈⣷⣿⣾⢰⢲⣤⡶⢸⣶⡂⠀⠀⠄⠀⠁⠀⠀⠐
⠀⠈⠁⠈⠉⠀⠉⠁⠉⠀⠀⠉⠀⠉⠈⠉⠀⠁⠈⠈⠁⠀⠀⠲⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘
⠀⠸⠤⠾⠖⠶⠶⠺⠰⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠄
⠀⢠⣤⢀⡀⢀⡀⣀⢀⢀⣀⢀⡀⣀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠘⠙⠙⠊⠘⠃⠛⠘⠹⠿⠉⠋⠓⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⠀⢀⣸⡇⣸⣷⡄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣾⢡⣿⣧⣷⣟⣹⣧⣿⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢿⣾⣿⣿⣿⣿⣿⢟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⠀⠀⢀⡀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡿⠀⣄⣤⡅⠀⠀⠈⠷⢿⠟⢻⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠘⣙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣬⣷⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⡾⣿⠀⠛⠀⠀⠀⠀⠀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡟⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠸⡇⡁⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠋⠀⠀⠀⠀⣶⣾⣿⡙⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢰⡟⢿⡿⠇⣸⠿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣧⢤⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠞⠉⠋⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢥⣿⠿⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢸⡅⠀⠀⠀⠀⠀⠀⠀⡙⠛⠿⢿⡄⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⣾⣇⠀⠀⠀⢀⠛⠛⠛⠛⠀⠀⠈⣅⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣶⣤⣾⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠂⠀⠀⢿⣿⡆⠀⣟⠛⠛⠛⠛⠀⠠⠞⠁⠈⠁⠀⣛⣿⣏⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⣿⣿⡿⠾⠿⠷⠶⠶⠶⠂⢀⡀⠠⡦⠞⢹⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣾⣿⡀⢀⡀⢠⣿⣿⣿⣶⣶⣶⣶⠶⠄⠀⢈⣀⣀⣀⣀⣤⡤⠤⠄⠘⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠟⠿⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣧⣀⣿⣿⣦⣤⣼⣶⣤⣤⡄⠀⠈⠿⢛⡋⣉⠀⣾⡇⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣯⣀⠆⢰⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢰⡆⠸⣯⣹⡏⢹⣿⣿⣿⣶⣦⡤⠄⠀⠀⠀⠀⠀⣿⣿⣿⢠⣿⠇⣼⡇⠀⢈⢇⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⢰⣿⡆⢹⠄
⠀⠀⠀⠀⠀⢸⡇⠀⣿⣹⣿⣼⣿⣿⣿⣿⣿⣦⣤⣤⡀⠀⠀⠀⢹⢿⣿⣿⠟⣰⣿⠃⢠⠘⠋⡿⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⢸⣿⡇⠈⡆
⠀⠀⠀⠀⠀⢸⡇⠀⣿⢿⣿⣽⣿⣿⣿⣏⣉⣩⣭⣄⣀⡀⠀⣶⣾⣿⡿⠁⣰⡿⠃⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠛⣿⣿⣿⣿⣿⠀⠹⣿⣿⡇⠀⡇
⠀⠀⠀⠀⠀⢸⣧⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠻⣿⡿⠁⣼⠏⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣄⣀⣿⣿⠃⠀⠃
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣴⢶⡾⢿⣿⣿⣿⣿⣿⠷⠶⠆⠀⠀⠀⢠⡞⠁⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠛⣿⠀⢀
⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣷⣼⣿⣅⣀⣀⣀⡀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠫⠙⠃⠀⢀⡿⠃⠀⠀⠀⠀⠀⢺⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣤⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠟⢋⣀⠠⠴⠚⠂⠈⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⣀⣉⠀⣀⡐⠶⠂⣠⣤⣀⣀⣀⣀⣠⣤⣥⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠁⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠛⠋⠉⠉⠁⠀⠀⠘⠛⠛⠛⠛⠉⠉⠉⠙⠿⠿⠿⠿⠿⠿⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠒⠐⠐⠀⠐⠒⠀⠘⠀⠒⠒⠒⠀⠗⠂⠀⠘⠒⠒⠒⠂⠐⠒⠀⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣦⣤⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠿⠛⢉⡍⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠹⣿⣿⡟⠛⠋⠀⠀⢀⣀⣀⣴⣾⠛⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠺⠛⠛⠉⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡿⠂⠠⠄
⠀⠀⠀⠀⠂⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⣠⣴⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣉⠟⠁⣸⣿⣿⣿⣿⣿⣿⣿⡇
⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠰⡄
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠈⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡶⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧⣈⣿⣿⣿⣿⣿⡏⠙⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣆⠘⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠈⢻⣿⣿⣿⣿⣿⣿⡄⠹⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠻⣤⡈⣿⣿⣿⣿⣿⣿⣷⣤⡙⢿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⠀⠀⠀⠄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠂⢤⣄⠡⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⡸⠉⢿⣿⣿⣶⣆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣹⣿⣿⣿⣿⣿⡿⠃⠉⢀⠀⢘⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢻⡏⣿⣿⣿⣿⡇⠀⠀⠀⣶⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡁⠀⠀⠀⠹⣯⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡗⣀⣴⣾⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⢉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⣝⣿⣿⣿⣿⣿⣡⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣭⠯⠙⠩⡶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣛⣿⠻⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢴⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣴⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣴⡾⠻⢦⡀⠀⢀⣴⣿⠉⠹⣄⡄⠀⠀⠀⡠⠄⠀⠀⠀⠀⢠⠊⠠⠀⠀⠀⠀⠀⠀⠀⢰⡉⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⡇⠀⠀⣀
⠀⢸⣿⣷⡀⣾⣇⡜⠉⣠⣿⢀⣸⠋⢻⡴⠆⢸⣤⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⡖⠙⠄⣀⣿⣿⣇⠀⠀⣠⣦⠀⠀⠀⢻⡁⣤⣤⣾⡿
⠀⠀⣿⢾⣿⣿⢿⣁⣾⢿⡿⠏⠉⡄⣾⠀⠛⢹⠃⣠⢐⠀⠀⠀⠀⢰⠓⠀⢸⠙⡀⠸⣤⣿⢟⣷⣸⣿⠀⣰⡏⢸⣆⣀⠀⢸⣧⣿⣿⣿⣿⡆
⠀⢠⣬⣻⣿⠏⢹⣿⡇⢰⣿⠄⠠⢧⢹⣎⠀⠈⠰⠿⠉⠹⡀⠀⠀⠀⠀⠀⠬⠀⠗⡄⡿⣿⣼⣻⣿⠈⣿⣯⣿⣻⡋⣿⣧⣬⣾⣿⡾⣿⣧
⠀⠈⢛⣿⣿⣰⣻⡇⢤⠈⢸⢦⠀⠷⠀⠈⣀⠃⠈⠀⠀⠀⠀⠒⠘⠀⠀⠑⢄⠀⠀⢠⠄⢿⢫⣿⢿⠀⠿⡟⣧⢼⣿⣽⣝⣿⣿⡿⢷⣞⡋
⠀⠀⢸⣧⣿⡿⠋⢇⢳⡀⣼⣹⣷⠀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠀⠰⢽⠀⠀⠉⢥⠀⢷⡄⢹⣿⡿⢦⡸⢧⣹⣅⢹⢦⣻⡇
⠀⠀⠀⠛⠀⠀⠀⠘⠋⠛⠉⠁⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⠀⠀⠈⠓⠛⠷⠋⠁⠀⠈⠱⠤⠇⠘⠛⠀⠙⠁

⠀⠄⢠⢠⠀⣆⣄⠰⢦⣴⠀⡾⠀⡇⡇⢺⠀⠀⢘⣅⢿⢸⠷⠾⠔⡆⢸⠀⠇⢶⠰⡆⠾⠺⠎⢴⡡
⠀⠘⠚⠈⠂⠃⠉⠘⠁⠈⠀⠁⠀⠁⠀⠈⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⠑⠄⠀⣠⠄⢀⢼⣣⠆⠀⠀⠀⠀⠀⠀⠠⠈⣄⠐⡄
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠘⠁⢀⠀⠀⠀⠀⠁⣂⢉⡟⡰⠃⡾⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠍⠀⠀⡀⢀
⠀⠈⠁⠰⠀⠀⢄⠀⠀⠀⠀⠀⠈⠀⠛⠋⠀⠀⠘⠀⠃⣸⠟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠃⠈⡆⠀⠀⠄
⠀⠀⠀⠀⢀⠀⠙⠦⢄⠀⠀⢀⣤⣠⡦⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠓⡄⠰⠌⠛⠛⢏⣛⣋⠀⢀⠀⠀⠀⠀⠦⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐
⠀⠀⠀⣠⢄⠐⡕⢠⠖⠁⠀⠀⠀⢌⠳⡌⠀⠀⠀⠀⠀⣀⡰⠚⢣⡀
⠀⣴⠞⣿⠇⡄⠃⠇⠈⠀⢡⡮⢧⠀⡇⠐⠀⠀⠀⠀⠀⣾⠃⠀⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤
⠀⠀⠈⠁⠀⡁⢀⠡⢘⠀⠲⠤⠀⢠⠁⠀⢠⠄⠀⠀⢠⢸⠀⠀⠀⠀⢀⠄⠀⠀⢰⡀⠀⠀⠀⢿⣽⣿⡀⣀
⠀⠀⠀⠀⠐⠰⣈⢦⠀⠀⠠⠀⣠⠆⠀⠘⠌⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠈⣷⠿⣷⣄⡄⣽⠻⠉⠉
⠀⠀⠀⠀⠀⢄⠳⠀⠙⠦⢠⡀⣀⢠⡀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣷⣇⣨⣿⠧⡟⠀⢡⡄⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠄⠸⡐⢂⡀⣀⠀⣀⣠⠀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠿⠟⠉⠀⠁⣼⠅⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠠⠶⠤⠤⠒
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⡀
⠀⠠⠬⠽⠷⠼⠿⠏⠩⠽⠿⠿⠟⠛⠿⠤⠀⠀⠀⠁⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠠⠰⢒⠀⠒⠄⢈⣦⡄⣠⣤⠞
⠀⢘⣒⣒⣚⣛⣉⣉⣉⣉⣉⣉⠉⣁⣉⣁⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⡆⣀⠃⠀⠀⠀⡈⣿⠟⠟⠁
⠀⢼⣿⣧⣤⣤⣤⣤⠴⠖⠂⠛⠚⠓⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠀⢀⡠⡀⢶⡀
⠀⡉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠋⠉⢀⠠⠾⡟⠇⠀⠁
⠀⠀⠸⣶⣷⣶⣶⣿⣶⡆⠸⣿⣿⠇⢏⠀⠀⠀⠀⢀⡜⣀⣄⡀⣀⢀⡀⠦⠀⠁⠀⠀⠀⠀⠂⠰⠁⣠⣤⣼
⠀⣠⠤⠤⠄⠉⢉⡀⠉⠁⠀⠈⠁⠀⠈⠀⠀⠀⠠⡿⠛⠏⠐⠟⠡⣈⠹⠲⡄⠀⠀⠈⠂⠀⠀⢠⠀⠁⠀⠈⠂
⠀⠀⢠⣄⡀⠀⠈⠙⠃⠤⠂⠀⠀⠀⠀⠀⠀⠀⣾⢀⠁⡀⡰⣿⣶⠀⠃⢘⣴⠀⠀⠀⠀⠀⠰⡌⠀⠐⢼⡆
⠑⠪⠑⠂⠸⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⠀⠐⡄⣯⠸⡀⡆⢉⣉⣉⡄⡂⠸⢯⠀⠀⠀⠀⠂⠀⢀⠘⡆⠀⠟
⠀⢀⠀⢀⠀⠀⠀⠀⠹⠄⠀⠄⠀⠀⠀⠀⠀⢸⡏⠳⡍⠚⠶⣤⣭⣥⠇⠀⣬⠴⠤⠆⠰⠀⢠⠾⠀⠀⠀⠣⠄⠋⠁⡀
⠀⠈⠃⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠸⣤⠀⠀⢀⣀⣠⣴⠟⠀⢰⠆⣹⣄⠢⢄⠐⢀⡄⢤⠆⠀⠀⠐
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠈⠑⠛⠓⠓⠋⠀⠀⣀⠚⠁⠉⢁⠀⠀⢀⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠐⠢⢤⠀⠀⡠⠊⣈⠛⢠⣄⠉⢠⡄⠛⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣿⠆⠀⠀⢹⡿⠛⠳⠶⣦⣙⣻⣿⣿⣿⣋⣉⣀⣀⣴⣾⢃⣀⣤⣄⣀⣤⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣤⣤⡄⠛⣡⣴⣆⠀⢠⡯⠙⠋⣉⣉⣿⣿⣿⣿⣿⠛⣈⡉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢰⣿⣿⣿⣁⣀⣤⣶⡿⠛⠛⠋⢹⡟⠻⣿⣿⡟⠛⠛⠿⠛⠻⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡶⠶⠋⢉⣩⣭⣿⠿⠿⠿⢿⣦⡴⠶⠞⠷⣿⣿⣿⣿⣿⣦⣤⣴⡾⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡋⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣶⣶⣶⢶⡾⠦⠤⠤⠤⢀⣤⣤⣽⣿⣅⣤⣤⣀⠠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠟⠉⠉⠉⠙⠻⠛⠁⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⠉⢉⣩⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣤⣤⣤⡀⠀⠀⠀⣀⣄⡴⠶⠐⢛⣿⣿⣿⣿⣿⣿⠶⠆⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠀⠉⠉⠉⠙⠻⢿⠿⠻⠿⠿⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣧⡤⠤⣤⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣶⣶⣤⣤⣤⣄⣀⣤⣀⣤⣀⣀⠀⢀⡀⠀⠉⢛⠁⠀⠀⣀⣤⡤⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡉⠙⠛⠛⠿⠛⠋⠉⠛⠛⠛⠋⠛⠛⣻⣿⣛⣛⣻⠿⣥⣤⢤⣤⣴⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣶⣄⣀⣀⣀⡀⣀⡀⠀⠀⣀⣽⣿⣿⣿⣭⣝⠛⠳⠖⠒⠛⠿⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢿⣿⡿⠿⠿⠿⠛⠛⠛⠻⣦⣬⣿⣿⣿⣟⣛⠂⠀⠀⣤⣤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠈⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣶⣶⣾⣛⠀⠀⠀⠉⠉⠀⠀⣉⣉⡁⠈⣛⣿⣿⣿⣉⣙⣳⣤⠦⠶⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣡⣤⣌⣽⣿⣶⣶⣶⣿⡿⢿⣿⣿⣿⡀⢉⣿⣿⣿⡏⠉⠁⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⠷⣡⣶⣾⣽⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣽⣿⣿⣶⣶⣦⣤⡴⣦⣤⣤⠤⠀⠀⢠⣤⣿⣿⠿⠇⠉⠉⣩⣽⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠿⠿⠋⠉⠋⠀⠀⢀⣀⣀⣠⣤⠴⠦⢴⣶⡶⠬⠛⢿⣧⣤⣶⣌⠛⠉⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣴⣶⢶⣾⠟⠛⠛⠛⠉⠁⠀⠀⢸⣅⣀⣤⣤⣶⣾⣿⡟⠛⠛⢷⡆⠀⠶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣹⣟⠉⠙⢿⣿⣷⣤⡀⣸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠟⠃⠉⠉⠉⠉⣛⣛⠛⣷⣾⣧⡈⠙⣧⠈⢀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⠶⠆⠀⠙⢿⡟⠛⠛⠛⢻⣿⣿⣿⣴⣿⠶⠛⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠛⠛⠃⣤⣀⣀⣤⣈⣷⣶⣶⡶⣶⣟⣿⣿⣥⠈⠿⣦⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡾⠿⠛⠛⢻⣉⠉⠉⠉⠉⠳⠀⠹⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠻⠿⣷⣧⣀⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠠⣾⣿⣿⠿⠛⠻⢦⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⡠⢤⣽⢿⣄⡀⢀⣼⣄⣠⣤⡶⠞⠛⠀⠻⣿⣿⡇⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⣀⣿⣿⣯⣷⣿⣿⣶⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣄⠀⠈⣩⣄⣤⡞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠿⢿⣿⣿⣿⠧⠜⣿⣿⠙⢿⠿⠿⠟⠛⠛⠛⠛⠉⣩⠿⠛⠻⣿⣿⣿⡿⠶⠿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠶⣾⣿⣿⠟⠓⠘⠛⠋⠀⠀⢀⣀⣀⠀⣤⣴⠾⠿⠋
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣟⠛⠛⠛⠋⠀⠀⠁⠉
⢸⣷⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⢿⡟⢿⠛⠻⠿⠿⠶⢦⣴⡾⠿⠿⠿⠷⢶⣶⣶⣶⣤⡄
⣿⡏⠈⠉⢻⡀⠀⠀⣠⣶⣶⣿⣧⣾⣿⠀⠰⣿⣿⣿⣿⣷⣤⣤⠶⣶⣤⣀⣀⣀⣄⣀⣾⣧⡀⢹⣿⠃⠘⢿⡶
⣿⣧⠀⠀⣴⡇⠀⢠⡿⠈⢿⣿⠿⣿⣿⠓⣾⣿⠋⠉⠉⣿⡟⠛⢷⣤⡈⠉⠉⠉⠉⢻⣿⣿⠛⠛⠛⠿⣷⡾⠇
⠙⢿⡇⠀⣿⠁⠀⣸⣿⣶⣤⣄⣀⣽⣿⣿⣿⣿⣶⣤⣴⣿⡀⠀⠀⠈⠳⣦⡀⠀⠀⠈⠛⣿⣤⣀⣀⣤⡿⠁
⠀⠈⡇⠀⣿⠀⠠⣿⠟⢿⣇⣭⠉⠛⢿⣿⣧⣄⣹⣿⣿⣋⣴⣿⣷⣶⣦⣼⣿⣷⣦⣤⡀⠈⠛⠛⠋⠋
⠀⠿⠀⣠⡟⠀⢠⡿⠀⠀⠙⣿⠷⣤⣤⣿⣯⣉⣹⣿⣿⣟⠋⠁⠀⠀⠀⢀⣈⡙⣷⣿⣷⣦⣤⣤⣤⣤⣀⣤⣤
⠀⢰⣄⣿⠃⠀⣿⠇⠀⠀⠀⣿⠀⠈⠉⢹⣿⣿⣿⠿⢿⣿⠿⢿⠿⠿⠿⠿⣿⣿⣿⠿⠛⠉⠉⠙⠿⣿⣿⣯⣉
⣠⣼⣿⡇⠀⢀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⡏⠀⠘⣿⣆⣀⠀⠀⠀⠀⠈⣽⣿⣷⣤⣄⣠⣤⣤⣿⠟⠋⠉
⠁⠉⣿⣧⠀⢰⡟⠀⠀⠀⣴⠀⢠⡄⠀⢶⣿⣿⡄⠀⠀⢿⣿⠉⠻⣦⡀⠀⠀⠉⠉⠀⠉⠛⠛⠉⠁
⠀⢠⣿⣿⣦⠀⣿⠀⠀⠸⠃⣀⠼⣷⣤⣾⠿⠻⣷⡀⠀⢸⣿⠀⠀⠘⠃
⠀⣼⡟⠁⢹⣧⣿⣁⣼⣷⠀⣿⠀⠛⣿⡟⠀⠀⣿⡇⠀⢸⣿⠀⠀⣰⡆
⣿⣿⠃⠀⠀⢸⣿⣿⠋⠀⠀⣿⠀⠀⢿⡏⠀⢠⣿⠃⠀⢸⣿⣧⣶⡿⠃
⣿⠋⠀⠀⠀⠀⣿⣏⣲⠀⠀⢟⣀⣀⠈⢿⣶⣾⠇⠀⢀⣿⣿⣿⣿⣥⡀
⠉⠀⠀⠀⠀⠀⣿⣿⣿⣧⣰⣿⠋⢻⡆⠀⠁⠀⠀⠀⣼⣿⠉⢹⣿⡟⠻⣷
⠀⠀⠀⠀⠀⠀⣽⣿⠀⠙⣿⣿⣀⣾⠃⠀⠀⠀⠀⣸⣿⡟⠀⢸⣿⡇⠀⢻⣆
⠀⠀⠀⠀⠀⣠⣿⣧⠀⠀⢻⣿⡿⠃⠀⠀⠀⠀⢠⣿⣿⠀⠀⢸⣿⠃⠀⢸⣿
⠀⠀⠀⠀⣴⣿⣿⠇⠀⠀⣿⣿⣷⡀⠀⠀⠀⢀⣿⣿⣿⠇⠀⢸⣿⣦⠀⢠⣿
⠀⠀⠂⢀⣿⢿⡿⠀⢰⣾⣿⣿⣿⣧⠀⠀⠀⣾⡟⢻⣿⠀⠀⢸⣿⣿⡇⢨⣿
⠀⠿⠄⠈⢻⣿⣷⡄⠸⣿⡿⠀⣿⣯⠀⠀⣼⣿⠁⣸⡏⠀⠀⢸⣿⣿⣧⢸⣿
⠀⢂⣴⣿⣿⣿⠁⠀⡧⠸⣷⣾⠟⠁⠀⢰⣿⡏⠀⠘⡇⠀⠀⢀⣿⣿⣿⢸⣿
⢠⣿⠏⠸⣿⣿⠀⣴⡿⠀⠈⠁⠀⠀⠀⣿⣿⣿⣤⣼⣷⠀⠀⢸⣿⣿⣿⢸⣿⠇
⣿⠃⠀⢠⣿⣟⣰⠟⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⣿⣿⠁⠀⠀⠘⣿⣿⡟⠸⣿
⣿⡆⢀⣿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⢸⣿⠀⠀⠀⠘⣿⣿⡇⢸⣿
⣿⣱⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣼⣧⣸⣿⠀⠀⠀⠀⠈⠻⣿⣼⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⡿⠟⠛⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⡀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣦⠀⠀⠀⠘⣿⣿⣿⢋⡔⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣷⡄⠀⢸⣿⢿⢿⣿⠿⣿⡏⠀⠀⠀⠀⠀⣹⣿⠋⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠂⠀⠀⠈⠙⠶⠟⠛⠋⠁⠀⠀⠀⠀⠀⣼⣿⣿⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠾⠋⢻⣿⣿⣶⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⣿⠟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠋⠉⡉⠋⠁⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠁⠀⠀⠀⣠⣴⣿⣿⠟⠙⠋⣩⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣶⣾⣿⠿⠿⡟⠀⠀⠀⢹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⠟⣿⣻⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠌⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⣠⠰⠹⢿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⢀⡴⠲⣶⣶⢶⣄⡀⢀⣿⡶⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁⣠⢤⡀⠂⠈⠹⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⢏⡈⡔⢡⣁⠊⠑⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠄⣸⣥⣤⣤⣤⣭⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣷⣾⣿⣿⡿⢃⠀⠁⢹⠛⡁⢁⣤⡄⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣠⡾⠁⢈⡼⢣⡈⠀⠁⠈⣌⠁⠀⠀⢿⣿⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢢⢄⡄⠀⠀⢸⠁⠉⡀⠉⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⡠⠈⠹⠧⠖⠋⠀⢀⠑⠀⠀⡔⠙⠀⠀⣤⠈⣿
⠀⠀⠀⠀⠀⠀⠐⠀⠸⣿⣿⣿⡏⢡⢇⡃⠃⠀⠀⠈⠀⠘⣧⠀⣀⣠⠤⠚⠁⠀⠀⠀⠠⣎⡀⢀⣀⠠⠤⠔⡚⢿⢃⡤⠊⠀⢀⠃⠀⠰⠀⡉
⠀⠀⠀⢄⡶⡾⣆⠀⠀⠹⣿⡟⡅⣵⡎⠙⢓⢠⠄⠀⠀⡀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡸⠀⠀⠀⠀⡇
⠀⠀⠀⠾⣿⢾⣧⠀⠀⠀⠙⢷⣴⡹⡇⠂⠈⠁⠀⠀⠀⣿⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢰⠁
⠀⠀⠀⠰⣿⣷⠁⠀⠀⠀⠀⠀⣿⣇⠈⣺⠀⠀⠀⠀⠀⠈⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠁⠀⢸⡆⠇⠀⠀⠀⣀⠔⠁⠀⠀⠀⢀⡌
⠀⠀⢀⣄⠘⣻⡅⠀⠀⠀⢀⣉⠁⠹⢆⣿⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡑⢌⣣⠸⢸⠀⠀⢰⠀⠐⠃⠀⠀⢀⡠⠖⣹⣷⡄⠀⠀⠐⣉
⠀⠀⠀⠏⠳⠹⣗⠀⠀⠈⠙⠛⠋⠀⡜⠛⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠑⡄⣏⠦⡘⠀⠀⠸⣄⢸⣤⡠⠖⠁⢀⣼⣿⣿⣷⠀⠀⠀⠈⠐
⠀⢠⣶⡆⠀⠀⢋⣄⠀⠀⠀⠀⢰⣾⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⣠⠀⠠⢸⠄⠃⠀⠀⢠⣿⣿⣯⠀⠀⢠⣾⣿⣿⣿⣿⡄⠀⠀⢠⠂⠠
⠀⠀⠻⠇⠀⠶⡀⠉⠀⢀⡠⠴⢏⡈⠙⠳⢷⢶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⢀⡧⠔⢻⡄⠀⠘⢃⣿⠻⠀⣰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⢸⠁⠀⣿⠙⠓⣿⠌⠉⣲⣤⣀⣀⣀⣹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠾⢿⡧⠶⠀⠸⣶⣾⣿⣷⡀⠀⣸⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⣸⡀⢼⡷⠟⠀⣈⢻⢫⣮⢿⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⢻⣿⣿⣿⣷⣠⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⢠⠟⣡⣤⠀⢰⣯⢓⠾⡀⠡⠊⢀⣀⣀⠀⠀⠀⢀⡴⠴⠦⠼⢄⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢠⡄⢀⡎⠀⠀⠀⡎⢸⠯⠜⡲⠆⠞⠁⠀⣀⠤⣊⠖⠲⣭⠻⢆⠀⠈⣞⣙⣋⠱⠂⠀⡝⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣠⢿⠀⡜⠀⠀⠀⣸⢁⣤⣦⠞⠀⣴⣶⣶⣿⣶⣶⣶⣾⣷⣿⣶⣾⣶⣶⣿⣷⣶⣶⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣰⣷⡾⣷⠁⠀⠀⠀⠧⣛⣃⣍⠀⠀⠈⠉⠹⠋⠉⠉⣉⣉⣭⣭⠭⢭⣿⡉⠉⠉⠉⠉⢩⣿⣿⠉⠉⠀⠀⢀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠉⢉⣿⣸⣄⣀⣀⣀⣀⣀⢠⡭⣍⡁⠚⠻⢞⣁⢾⡏⢩⣟⠷⠀⢠⣞⠛⣿⡀⠀⠄⠀⢸⡇⡀⠀⠀⠀⣠⠊⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠛⠛⠛
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⡠⢀⢽⢏⡀⣤⠀⢀⣐⡄⠀⠀⢹⣯⠆⢤⣴⣾⣿⡇⠀⠀⣀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠓⢾⡎⣧⡻⠃⠀⠺⠙⠁⠀⣾⠎⢹⣯⣱⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣦⣸⡆⢹⢁⣶⡿⠂⢠⣶⡇⠀⣴⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣇⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣷⣾⣿⣷⣇⡈⢉⣤⠤⠽⡿⠒⠋⠙⣇⠁⠀⠉⢻⣿⣿⣿⣿⣿⣏⠙⣷⢀⠤⠀⠰⠄⠂⢀⣩⣽⡮⠁⠈⠈⠁⠚
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⠉⠀⠀⡰⠉⢦⠀⠀⠹⣆⠀⠀⣾⠉⢿⣿⣿⣿⣿⣦⢹⡄⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⠛⠉⠀⡆⠀⢠⡅⠀⠀⠳⡀⠀⠹⣦⣀⣿⠀⠀⠙⣿⣿⣿⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⣇⣀⣀⡈⢻⡇⠀⠀⠀⠈⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢠⣁⠀⠘⣆⠀⠀⠀⠙⣿⣿⢿⣿⣿⣶⣤⣤⣀⢀⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡞⠀⠀⠀⢸⣿⠻⢿⣿⣶⣄⣀⠀⠈⣧⠀⣼⠀⠉⠛⣻⣿⣿⡿⠋⠈⠋⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠓⠚⠁⠀⠀⠀⠀⠃⠀⠀⠈⠙⠛⠿⠿⠗⠛⠚⠋⠠⠶⠟⠛⠋⠁⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⠚⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⣶⣿⣶⣯⣭⣿⣧⣿⣿⡿⣿⣿⡧⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣿⣿⠟⠹⠿⠟⢻⡟⢻⣿⡶⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡷⡌⣿⣿⡆⠀⠀⠀⢈⠉⠉⠈⢳⡦⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣿⣿⠀⠀⢺⣿⣿⣿⠀⣸⣇⡇⠀⠀⠀⠀⠀⣶⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢨⣿⣿⣿⣿⡟⠀⠀⠀⠉⠉⠛⢀⣿⠿⠁⠀⠀⠀⠀⠀⠹⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠗⠀⢤⣾⣿⣻⢋⣿⣿⣶⣤⣀⡀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⢰⣿⡽⣱⣿⣼⣿⡇⠙⠋⠉⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠙⢶⣤⣿⣿⣷⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⢁⠀⢀⣠⣤⣴⣶⠿⣿⡿⣇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⠀⠀⠘⡟⢜⢿⣿⣿⣷⡻⣧⠄⠀⠀⣾⣧⣭⣿⣿⣿⣿⡟⣼⣿⢟⣿⣿⣿⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⣰⡿⢸⠘⠿⣿⣿⢿⣟⠛⢦⣠⠏⠀⢺⣿⣿⣿⠟⣱⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣼⣍⠀⠀⠀⢠⣿⢣⣯⣴⡇⠨⣿⣿⡿⠃⠀⠘⠂⢀⠀⢹⡿⢻⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢀⣠⣾⣿⣿⣿⣿⣿⠀⠚⠁⠀⠀⠀⠀⠁⠘⡀⠀⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠘⢿⣇⣀⣩⣿⠟⠉⢻⣿⣿⡏⢇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢿⣿⣇⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢣⣿⣿⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣭⡤⠀⠀⠀⠰⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠶⠿⡥⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠉⠉⠉⠉⠉⠉⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡰⠳⣄⠀⠀⠀⠀⠀⣠⠞⠑⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠖⠉⠙⠦⣀⠀⠀⠀⢸⠁⠀⠈⠓⠦⣄⡀⢰⠃⠀⠀⠈⢧⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⠙⠲⠤⣬⡦⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡰⠓⢶⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣠⠶⠚⡇
⠀⠀⠀⠀⠀⠇⠀⠀⠘⠦⣀⡇⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠉⠁⠀⠀⠇
⠀⠀⠀⠀⠀⢢⠀⣀⠀⠀⠈⠁⠀⡜⠉⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠈⢳⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⢹⡹⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡿
⠀⣴⣤⣶⢀⠀⠈⢇⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠃⠀⡀⣶⣤⣦
⣻⣽⠞⠻⢯⠇⠀⠘⢎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠃⠀⠸⡽⠟⠳⣿
⠙⢯⡀⠀⠸⢤⡀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⠶⠶⠶⠶⢦⠀⠀⠀⠀⠀⠀⠀⢀⡜⡱⠃⠀⢀⡤⠃⠀⢀⡽
⠀⠀⠈⠳⡀⠀⠙⠦⣀⠀⠙⢿⣄⡀⠀⠀⠀⠀⠀⠹⡍⠉⠉⠉⠉⢉⠏⠀⠀⠀⠀⠀⢀⣠⣯⠞⠀⣀⠴⠋⠀⢀⠞⠁
⠀⠀⠀⠀⠙⢦⠀⠀⠈⠳⢄⡀⠙⠻⣦⣄⠀⠀⠀⠀⠈⠓⠒⠒⠚⠁⠀⠀⠀⠀⣠⣴⠿⠋⢀⡠⠞⠁⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠉⠳⢤⡀⠉⠛⠷⢦⣤⣤⣀⣀⣀⣀⣤⣤⡴⠾⠛⠋⢁⡤⠞⠉⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠉⠳⡶⣄⠀⠀⢘⡇⠀⠀⢸⡃⠀⠀⣠⢶⠞⠉⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⡰⠃⠀⠙⠒⠋⠱⢤⡤⠞⠙⠒⠋⠀⠈⢆⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠋⠉⣇⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡹⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣍⠀⠀⠀⣿⡈⠀⠀⡀⠀⠀⠀⠀⠀⡀⠆⠀⠔⠈⠀⡇⠀⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⡏⠐⢤⣴⣿⣿⡆⠀⠀⠀⣀⠀⠀⠀⠀⠘⣦⠀⠀⢀⠃⠠⠀⢀⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠋⠉⠁⠀⠈⠙⢻⣿⣿⠆⠀⠰⠂⠈⠀⠄⡀⠀⠁⠐⠠⠊⠀⠔⠀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⡀⠀⠀⠀⢹⠀⠘⣿⡷⠀⠀⠀⠀⠄⠀⠀⠁⠄⠀⠀⠀⠙⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠠⠁⠀⠀⠀⡆⠀⠀⠡⠀⠀⠀⠁⠂⠀⡔⠀⠀⠀⠀⠀⠀⠀⢡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢜⠀⠀⠀⣰⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠏⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⣼⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠖⠀⠈⠒⠇⠀⠀⠀⢠⣤⣤⣀⣀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠿⠿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⣿⣿⣿⣿⣿⠿⠛⠉⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢉⠙⠉⠀⠈⠁⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⢀⠆⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠰⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣊
⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠁⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠴⢚⡿⠀⠀⠀⢀⣶⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⢳⡄⠀⠀⢰⡏⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⢿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠒⠺⡏⠙⠹⠓⠗⠛⠉⡷⠦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠇⠀⠀⢀⣠⡴
⠀⣀⡀⠀⠀⠀⠀⠈⢳⡝⢦⡀⠀⠀⠀⠀⠀⠀⢀⡟⢠⡄⠀⣧⠀⠀⠀⠀⠀⣸⠃⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣾⠤⠶⣾⢕⡠⠶⠚
⢀⣬⡿⣷⣶⡖⠶⠒⠉⠀⠀⠙⡆⠀⠀⠀⠀⢀⡾⠀⡾⠀⠀⢸⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠠⡿⠟⡒⠒
⠀⢬⣭⣻⣯⠶⣠⡤⠦⢤⡀⠀⣷⠀⠀⠀⠀⣼⠃⢸⠇⠀⠀⠈⡷⠶⠶⢶⡞⠀⠀⠀⢹⠀⠀⢻⡄⠀⠀⠀⠀⠀⣸⠃⠀⡴⠛⠉⠉⠉⠉⠙⠛
⠀⠀⠀⠾⠖⠋⠁⠀⠀⠀⣧⠀⢸⡄⠀⠀⢰⡏⠀⢾⠀⠀⠀⠀⢷⠀⠀⣸⠁⠀⠀⠀⣿⠀⠀⠈⣷⠀⠀⠀⠀⢠⡏⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣇⠀⠀⣿⢄⡀⠘⣆⠀⠀⠀⢸⡄⣰⠃⠀⠀⠀⢀⣿⠀⠀⠀⠸⡆⠀⠀⠀⡞⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢹⠀⢰⠇⠀⠉⣱⠘⢧⣤⠤⢬⣧⣏⣀⣀⡀⢀⡾⠙⣧⣤⠴⠞⣧⠀⠀⣸⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣇⡏⠀⠀⣰⠏⠀⣎⣵⣤⡴⠷⠾⢤⣀⡿⡾⠁⠀⠘⣇⠀⠀⠸⡆⢠⠏⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡿⠀⠀⢠⡏⠀⠀⠀⢈⣆⣀⣀⣀⠀⠈⣹⠇⠀⠀⠀⠘⣆⠀⠀⠙⠞⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠀⠀⣠⡴⠋⠁⠀⠉⠉⠉⠉⠛⢦⡀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⡞⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠈⢦⡀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⡾⠁⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⣤⣤⠀⠉⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢹⣿⡇⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢿⣿⡄⠀⠁⠀⠶⢤⣴⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⢠⠎⠁⠈⠑⣄⠀⠀⣶⡆⢰⠃⠀⣼⣿⠁⠀⠀⠀⣠⣿⠏⠀⠉⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⢸⣿⡇⡀⠀⠀⠀⣿⡗⠀⠀⢸⠀⢠⣤⣤⡸⠀⢠⢿⣷⡘⠀⠀⣿⣿⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢸⣿⠟⠁⠀⠀⢠⣿⡇⠀⠀⠈⠳⣿⣿⣿⣠⠖⠁⠈⠻⠇⠀⠀⣿⡇⠀⠀⠀⣾⠃⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣻⡟⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⣿⣿⠁⡸⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⢿⡀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⡿⠁⠀⣀⣴⠿⠋⢀⣴⣿⠟⠓⡄⠻⣿⡶⠃⠀⠀⠀⠀⢀⣀⠀⠘⠇⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠛⠿⠿⣿⣆⠀⠘⠓⠛⠋⠉⠀⠀⣰⣿⡿⠁⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⣾⠉⠈⢳⠀⠀⠀⠀⢠⣴⡶⢦⡀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⡠⠋⠀⢀⠔⠒⠒⢤⠀⠀⢿⣆⣀⠞⠀⠀⠀⢰⣿⡏⠀⣠⠇⠀⢠⠾⠿⠛⠋
⠀⠀⠀⢀⣴⣶⣶⣶⣦⡄⠀⠀⡀⠀⠀⠀⠀⠀⣼⣿⣯⠴⠋⠀⠀⠀⢺⠀⠀⠀⠈⡇⠀⢈⠿⣿⣦⣔⠒⠉⢻⣿⡓⠋⠁
⠀⠀⠀⠘⠛⠛⠋⠁⠀⠀⠀⠀⠓⠂⠀⠒⠐⠒⣿⣿⡇⠀⠀⠀⠀⢀⣠⣵⡶⠤⢴⠗⠚⠁⢠⠄⠙⢿⣷⡄⠈⠻⠷⡤⠖⠀⠀⠲⢶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⡰⠋⠘⣿⣇⣠⠎⠀⠀⠀⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⠀⠀⣀⣤⠶⠦⣄⠀⠘⣿⣧⠀⢀⡜⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠳⢤⣠⣴⠿⠃⠀⠀⠀⠀⠀⠀⠈⢷⣦⡀
⠀⠀⠀⠀⠀⠸⠿⠋⠀⠀⢀⣾⠟⠁⠀⠀⠈⡆⠀⠈⠻⠖⠋⠠⣤⠀⢀⠄⠀⢠⡤⣄⠀⢠⣦⠀⢀⠀⠀⣀⡤⣀⠀⠀⠀⠀⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠘⣷⠏⠀⠀⣿⠀⢸⠀⢸⣇⠀⢸⠁⠀⣿⡤⠊⠀⠀⣤⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⡼⠀⠀⠀⠘⠃⠀⠀⠀⢿⠀⠀⠀⠻⢦⠜⠀⠈⠻⠤⠎⠀⠀⠹⠟⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⡄⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢠⣾⡟⠳⠄⠀⠘⣿⣿⠀⢠⠞⠀⠀⠀⣀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⡔⠉⢸⣿⡇⠀⠀⣀⣤⣦⣄⠀⠀⢀⣴⣿⡿⠛⣧
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢿⣿⡶⠁⠀⠀⡴⠛⢻⣿⡀⠀⠀⣿⣿⠁⠀⢠⣶⠄⠀⣾⡆⡜⠀⠀⣾⡿⠀⠀⣴⣿⠋⠀⠘⠀⠀⣾⣿⠟⢀⡴⠃
⠀⠀⠈⣿⣿⠀⠀⠀⠀⢀⡸⣿⣿⡀⠀⡜⠀⠀⢸⣿⡇⠀⠀⣿⣿⣄⡴⢻⡿⠀⢰⢻⣿⠀⠀⣸⣿⠇⠀⠀⣿⡏⠀⠀⠀⢀⣼⣿⡟⠋⠁
⠀⠀⠀⠹⣿⣧⡀⢀⡠⠊⠀⠹⣿⣷⢠⠃⠀⠀⢸⣿⡇⠀⠀⠈⡟⠉⠀⢸⡇⢀⡏⠀⠁⠀⠀⣿⡿⠀⢀⡜⢻⣧⠀⣀⠔⠁⢸⣿⡇⠀⠀⢀⠄
⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⡸⠀⠀⠀⠈⠷⠞⠀⠀⠀⠀⠀⢿⣇⢀⠎⠀⠀⠉⠉⠁⠀⠀⠀⠻⣷⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠈⢿⣇⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣥⣭⣯⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠀⢀⡀⠀⠀⠀⠀⠀⣰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⣹⠇⠀⠀⠀⠀⠀⠀⠀⢦⣤⣾⢃⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠸⣿⠟⠋⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠛⢋⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢱⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⢀⡸⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠚⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⢠⣿⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣤⣴⣾⣿⡄
⠀⢀⣼⠏⡀⠁⡈⠁⡀⢀⡀⢀⡈⢓⡦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣸⠋⡄⠁⡄⠁⡄⠁⡄⠁⡄⠁⡄⢁⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢀⠃⠄⠀⠄⠀⠄⠠⠄⠠⠄⠠⠄⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿
⢸⠂⠀⠂⠀⠂⠀⠂⠐⠂⠐⠂⠐⠂⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠚⣷⡃⡀⠃⣀⣧⣄⠃⡐⠃⡀⠃⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠈⠙⠷⣶⠟⠁⠉⠿⣅⡀⢁⡀⢁⣈⣥⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠁⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠇⠈⠿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠛⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠻⡿⠰⢀⠐⠸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⣤⣤⣀⣾⠋⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠙⠻⢷⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣆⣀⣀⣀⣀⣀⣠⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣦⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣰⣾⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣎⠻⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢡⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣤⠀⣀⣀⡀⠀⠀⣼⣿⣿⠋⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⣰⣿⣿⡟⠀⠀⣿⣿⣿⠀⢸⣿⣇⠁⠀⢀⣴⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃⢠⣿⣿⣿⠃⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⣠⣿⣿⡿⢋⣿⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⢠⣿⣿⣿⣿⠀⠀⠀⠀⠹⣿⣿⣦⠀⠀⢠⣿⣿⣿⡁⠘⠿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⣿⠇⢠⣾⣿⣿⣿⣿⠀⣴⣶⣦⠀⢸⣿⣿⡇⠀⣿⣿⣿⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣴⣿⣿⠟⣿⣿⣿⠀⣼⣿⣷⡀⣸⣿⣿⠇⠀⢿⣿⣏⠀⢀⣠⣴⣾⠿⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠛⠁⠀⢿⣿⣿⠀⠈⠙⠻⠿⠿⠛⠉⠀⠀⠈⠻⠿⠿⡿⠿⠟⠛⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡄⠀⠀⠉⣁⣀⣀⡀⣠⠖⡄⣶⠖⠀⣀⣀⣀⡀⠀⠀⠀⣰⣿⣿⡿⢿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⡠⢊⣴⡾⣛⣁⠙⠒⠃⠋⣰⣿⣿⠿⣿⣿⣦⠀⢰⣿⣿⡟⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⠀⣼⣷⣿⠏⠀⣿⣿⣿⠀⠀⢸⣿⣿⡇⠀⣿⣿⢹⠀⣿⣿⣿⠁⠀⠻⣿⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⢰⣿⣿⡟⠀⠀⢸⣿⣿⡆⠀⢿⣿⣿⡀⠀⣿⣿⣠⠀⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠘⣿⣿⣧⠀⠈⠉⠁⢀⣿⣿⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠈⢿⣿⣷⡄⠀⠀⢸⣿⣿⡇⣠⣿⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠻⣿⣿⣆⠀⠸⣿⣿⣷⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠸⣿⣿⣇⠀⠀⣼⣿⡿⠇⠀⠔⣶⣶⠀⠀⢻⣿⣿⡄⠀⠻⠿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣄⣼⡿⢋⠎⠀⢰⣸⣿⡇⠀⠀⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠒⠁⠀⠀⠘⣿⣿⣧⣀⣠⣿⣿⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣬⣯⣬⣽⣯⣾⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⢀⣤⣶⣾⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣯⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⢿⣦⣠⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⣼⣿⣿⣿⡿⠟⣋⠀⣿⣿⠟⠁⣩⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣴⣾⣿⣿⣭⠿⠛⠉⢁⣴⣿⡿⠗⠀⠁⠀⠀⠱⠚⠋⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⠏⠁⠀⠀⣰⠟⠉⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⡿⣿⣿⠀⠀⢀⣰⢋⣴⡷⠀⠀⠀⠘⠿⣿⠿⠁⠀⠀⠀⠀⣠⣤⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠙⣿⣿⣶⣿⡷⠶⠟⠛⡡⠞⠉⠀⠀⠀⢠⣦⣀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢛⣩⣵⣶⣯⣿⡿⡅⢰⣾⣿⣷⣶⣬⡛⠛⢿⣶⣶⣿⡿⠋⢿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⠟⠁⠀⠘⠂⠉⠉⢉⡉⠉⠉⠁⠀⠈⠉⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣉⣠⣀⣠⣾⣿⡇⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠏⠀⠀⠀⠸⣿⣿⣿⡿⠉⠉⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣄⣀
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣶⣄⠈⣧⡀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⣶⣤⣀⣤⣤⣄⡀⠀⠀⠀⠀⠈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠃⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠋⠀⠀⠀⠙⠛⠛⠋⠉⠙⠿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠀⠀⣀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠠⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠄⠐⠂⠀⠀⠂⠀⢠⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠈⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠤⡀⠀⠀⡟⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄
⠀⠀⠀⠙⠛⠈⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠸⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠰⠆⠀⠀⢄⣠⡤⠄⠀⠀⠀⠀⠰⢄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⣠⠆⠀⠀⣴⠂⠈⠀⠐
⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠁⠀⣤⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⠀⠀⠀⠈⠁⠀⠈⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠶⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡰⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠤⠀⠀⠠⠂⠀⠀⠀⢤⣀⠟⠀⢀⠀⠀⠠⠰⠃⠆
⠀⠀⠀⠀⠀⠀⠠⡀⠠⠞⠀⠔
⠀⠀⠀⠖⢠⡰⠖⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⣀⡤⠄
⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⣀⣯⠁⠀⢠⣬⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⠀⡄
⠀⠀⠆⠀⢠⡀⠀⠀⠀⠘⠀⡆⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡠⢆⠀⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠉⠂⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⣀⠄
⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠊⣧⠀⠀⠀⠀⠀⠊⠁⠶⠀⠀⠂⠀⠐⠋⠑⠀⠀⠀⠀⠈⢰⠒⠦⠀⠀⠀⠀⠀⠶⠊⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡄⠀⠀⣀⣤⣄⣀
⠀⠀⠀⣴⣿⡟⣛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢫⡭⣹⣿⣿⣀⣾⢟⣿⣿⣻⣿⣆⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⣿⣷
⠀⠀⠀⠹⣏⠓⣸⢟⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢡⣿⢱⣷⣾⣿⠟⣛⢶⡿⣻⣭⢟⢿⣤⣾⢿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣶⡿⣄⡃⣿⣿⠁
⠀⠀⠀⠀⠈⢻⣇⠻⢋⡽⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿⢇⣿⢇⡿⣛⣿⣿⣜⣻⣼⢳⣿⢛⣩⣾⡟⣧⣌⣡⣬⣿⣷⠀⠀⣀⣴⣿⡛⢓⣦⣿⡿⠃
⠀⠀⠀⠀⠀⠈⢿⣿⡋⢐⠗⣹⣿⣶⣄⠀⠀⠀⠀⢠⣿⡟⣾⡟⢼⣭⣭⣽⣿⣭⣭⣿⢸⢿⣘⣛⢿⣿⢋⣵⣮⡝⣾⣟⣠⣾⣿⣯⡍⡙⣶⣟⡿⠁
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣼⢟⣁⣶⣽⣿⣿⣦⣤⣴⣿⣿⢱⣿⣿⣖⣖⡖⣾⡇⡿⣏⣿⡏⡿⡟⣛⣸⡏⠾⣿⢛⣋⣼⣿⣿⡏⢼⣿⣧⡵⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⢿⡿⠞⠻⢿⣿⣿⣿⣶⣤⣬⣭⣭⣥⣿⣧⣛⣋⡿⣧⢩⡇⣿⣫⣷⡙⠿⣶⠶⣹⣟⣉⡿⢶⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡏⣴⢞⡛⢦⡈⣯⣿⠻⣿⣧⣿⠿⠳⣿⣭⢮⣿⠿⠿⣷⣾⡋⣉⡿⢿⣳⣖⣿⣻⣿⣟⣿⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣽⡇⣧⡾⣛⢾⣧⡿⣟⣿⣿⡇⣿⣮⡝⣿⣵⣳⣿⠞⣓⢸⣿⣿⣻⡶⢮⡻⠿⢿⣫⠶⣝⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣫⣾⣯⣿⣶⣿⣿⣿⣿⣷⣿⣶⠶⢿⠿⡿⠿⣮⠽⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣛⢿⡭⢷⡟⢲⡗⠒⣧⡏⠭⡝⣇⣿⢡⢹⡉⢛⠟⠒⣿⠶⠶⠶⠶⣼⡏⢹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡹⣅⢦⠳⡟⠙⢾⢱⢠⣿⠀⠸⠜⡜⣿⡜⡎⢧⡞⡼⣱⡏⣞⠩⠬⣤⠿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣎⠳⣀⠛⢦⡼⢸⡏⡟⣈⣉⢳⢹⣷⡘⣌⡜⣱⢿⡇⡏⢩⣤⣿⡐⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣌⣰⢳⣤⣿⣇⣀⣇⣸⣌⣡⣏⣷⣌⣴⣋⣸⣇⣓⣲⠆⢸⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⡿⣟⣶⡶⣿⣟⣳⣻⡾⣲⡶⣶⢶⣶⣶⣰⡾⢭⣭⣭⣭⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣿⣳⣽⣿⣻⣿⣹⣿⡯⣿⢹⣿⣿⢿⣿⢸⣿⢧⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣨⡍⣿⣾⣿⡸⣿⡇⣶⢸⣿⣿⣿⣿⢼⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣾⣻⣿⣿⣿⢿⡿⣷⢾⡷⡿⢾⢿⡿⣿⣿⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣬⣭⣭⣭⣭⣭⣩⡭⣿⣛⣛⣛⢛⣛⠛⠛⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠓⠟⠿⠽⠿⠿⠽⠿⠿⠽⠟⡿⠿⢿⡟⣿⡿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⡄⣿⡀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣬⡻⠟⠋⠁⡱⠒⡑⠊⣁⣠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⡠⠚⣡⣾⣿⣿⡿⢃⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠰⢯⣟⡣⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠗⣠⡅⣰⠟⠋⠀⢀⣤⠀⠈⣹⠶⡀⠀⠀⠀⠀⢈⡎⠱⡄⠉⡿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢡⡴⠀⠀⠋⢀⠀⠀⢄⢰⠜⢆⠀⠀⠀⢀⣷⢠⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡘⠇⠀⠀⠀⠙⠁⠀⣤⣉⠆⠈⣇⠀⢀⣾⣷⣙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡀⠀⡄⡰⢊⡁⠀⣀⡠⠤⠀⣿⣠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄⠁⠁⠸⣶⣶⣶⣾⠏⢀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⡐⢄⠈⠛⠛⢉⢄⣾⣿⣿⣿⣿⠃⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣦⣥⡒⢂⣵⣿⣿⣿⣿⣿⢵⣾⢫⣯⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⡇⠈⠑⠼⡏⠙⠉⠙⢟⠛⠛⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠁⢃⡀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠛⠿⠿⣿⣿⣿⣶⠶⢤⣤⣀⠀⠀⠀⠒⠒⠰⠀⠓⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢁⠇⠀⠈⠸⣩⢻⢦⡀⠀⣠⡼⢤⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣼⡀⢀⢃⢇⠀⠄⢋⢞⠃⢀⠀⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⢩⠇⡸⠌⠈⡄⣴⠟⡌⣄⣴⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⡏⠈⠊⡤⢇⣄⣠⠟⢁⣤⡾⡿⡱⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⢰⢰⠀⢡⣠⣟⡥⢶⣏⠁⠁⠁⠃⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣤⡆⠆⢾⣠⡾⠛⠁⠀⠀⠙⣶⡰⡆⠀⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣶⡞⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⣥⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠃⠀⣠⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⣰⣿⣫⣶⠿⠛⠂⠀⠀⠀⠀⠐⠂⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⠟⠛⠀⠀⠒⠂⠀⠶⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⠟⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡶⠀⠀⣠⡤⡀⠀⠀⠀⠀⠀⠒⠫⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠐⢿⡿⢋⣵⠄⣦⣄⠀⠈⣧⠀⠀⠀⠀⠀⠀⣀⣀⡠⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠈⣇⠀⡀⠀⠀⠳⣈⢀⡴⠛⠉⠱⡄⠘⡑⠒⣒⣛⣋⡩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣽⠋⢀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡆⢹⣿⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠁⠘⠃⠁⢐⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠿⠓⡿⠀⠁⠀⠀⠀⠀⠀⢠⠊⠉⠳⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠄⠀⠀⡉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠚⠁⠀⡄⠀⠀⠀⠀⠀⣸⠄⠲⠖⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠑⢄⡠⠒⠀⠀⠀⡀⠀⡇⠀⢀⠔⠊⣀⠀⠀⠠⠂⢀⠴⠒⢀⡠⠆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣌⠁⠢⣀⣰⠉⡒⠀⣜⣡⣀⠈⠀⢃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⠎⠀⠀⠉⠙⠀⣡⣬⣌⢣⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⣗⢷⡙⠲⠦⠀⠀⠀⠖⢹⣻⠇⠀⠁⣴⠅⢀⡅⠀⠀⠀⣀⠤⠒⣶⠀⠀⠀⠀⠀⡠⠚⠉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣻⣆⣹⣟⠳⣤⠤⢤⡔⣊⢟⡀⠀⠀⠀⠑⠦⣄⣀⡠⠊⠁⠀⠀⣿⠀⠀⠀⡠⠊⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠹⣉⡙⡿⠿⠛⠉⠛⠃⠀⣀⡤⠖⠚⠛⠁⢀⣠⡄⠀⠀⠘⡦⢚⠾⠁⡨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡭⠍⢉⡀⠀⠀⣀⠠⠐⠚⠁⠀⠀⠀⠀⠔⠁⠘⠀⢀⡴⣊⠔⣁⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣠⠞⠁⠀⠀⠁⢀⡥⠚⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡠⠀⠀⠤⠀⠘⠈⠀⠀⠀⠀⠒⢦⡁⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠎⢸⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼⢸⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣠⠇⣼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡛⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⢀⣀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⣀⡀⠀⠀⣀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠉⠓⠶⠿⠟⠾⠿⠛⠲⠮⠛⠛⠻⠛⠛⠳⠂⠈⠛⠒⠲⠦⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠉⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠔⣐⣴⢀⡤⢊⠅⠀⠀⠀⢀⡀⠀⠈⠁⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⢹⠃⣀⣿⣿⡏⡴⢃⡴⠋⠁⢤⣀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⡾⢿⣸⣆⣿⣻⣿⢹⣷⣿⠀⡼⣱⣋⣤⡐⢆⠀⠀⠀⠀⢈⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣇⠠⠾⣿⡙⣿⣻⣽⣿⣿⣾⣧⣿⠟⣻⣿⡌⡆⢸⣧⣰⠏⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡽⢻⠀⢰⣾⣷⣬⠙⠛⠉⠉⠀⠀⠀⠀⠙⣿⡇⢹⠀⢿⡇⠀⡀⠘⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣗⡃⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡆⠸⢧⣞⣠⡼⠁⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢬⣤⡤⠴⢃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⡇⢄⠑⠻⠶⠦⢤⠾⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢲⣿⢀⣤⣀⣀⡀⠀⠀⠀⣤⠶⢛⣭⣽⣦⠣⣑⠑⠢⠤⢀⣠⣤⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠀⠀⠂⣰⣿⣍⣠⣽⣿⣏⠀⠀⠀⠀⡼⢛⣛⡽⠋⠑⣮⣽⣁⣤⣵⣾⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⢦⡀⢿⣭⣥⡈⠙⠷⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜⠿⡯⠤⠄⠀⠀⢍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡈⠒⠍⡻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣗⠶⡖⠶⣶⣦⣤⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠻⣶⣤⡀⠘⣧⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⡇⠃⠀⠈⢻⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⢸⢿⡿⣠⡿⣧⡀⠀⢠⣶⠶⣶⣶⠂⠀⠀⢀⠞⠙⠚⠉⠀⠀⠀⠀⠞⡡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⡼⠞⠡⠋⣰⣿⣿⣦⡀⠈⠛⠋⠁⠀⣠⣴⣟⣵⣦⣤⣤⣤⠄⢀⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠀⠀⠀⠀⠀⠈⠉⢸⡏⣿⣿⣶⣤⣤⣴⡾⣿⢛⣿⠋⠃⠀⠩⣤⣶⠃⠈⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢶⡆⢰⠄⣠⠀⠠⣶⣿⠀⢏⣿⣿⣿⣿⣿⡟⠀⠘⠻⠀⠀⣀⣦⣟⣯⣶⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣸⡿⢝⡛⠘⠉⡽⢶⣿⠇⣠⣾⣿⢻⣿⣿⣿⣇⣦⢠⠌⣦⣾⠿⠛⠉⠉⠉⠙⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠕⡡⠊⠀⠀⠀⠀⢀⣰⠟⠛⠳⣖⡟⠀⠛⠁⠉⡉⢀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⢡⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⣠⡴⠚⢋⣱⣬⡉⠨⠀⠻⢃⡟⢁⠞⠀⢀⢈⣦⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆⠀⣀⢠⠈⣿⣦⣀⡴⠋⠈⠈⠀⠀⣀⢻⣶⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣿⡿⠈⠋⣃⠃⠁⠊⠀⠀⠀⠀⣧⠐⢨⡆⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⡀⠴⠿⡀⠀⠀⠀⢀⣴⣾⣿⢀⡟⢠⡿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢠⣆⡀⠀⠁⡰⢀⣴⠿⠛⠫⢋⠜⠒⠛⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠀⠠⠚⠁⠀⠀⠀⠁⠀⠀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣄⣶⡐⠨⠁⠈⠑⠄⠁⡤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⡒⠃⠀⠀⠀⠀⠀⢂⠀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⢈⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢀⡀⠀⠈⠂⠀⠐⠀⠀⡠⠄⠀⠈⡆⠀⠀⢀⠈⠈⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣀⠠⠪⠀⠀⠀⣠⡁⠀⠀⠈⢑⠀⡀⠠⡇⢀⠔⠁⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣐⡄⢰⣦⠀⠀⠘⣿⠀⠀⠀⢀⠢⠀⠆⠘⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠈⠿⠃⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠃⠐⠀⢠⠊⢀⠰⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢠⠀⢉⠠⣥⠒⢁⠌⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠃⢠⠀⠀⢀⣀⠀⠤⠄⠒⠂⠉⠄⣦⣤⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡕⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⠈⠉⠉⠉⡙⠀⠀⠘⠋⠉⢹⠤⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠄⠀⢸⠷⠿⠿⢿⠀⡀⢀⠄⠂⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⢠⣷⢶⣤⣤⣼⡆⡀⠌⠳⠤⢔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠙⠿⢿⠠⢍⣯⣽⣌⢁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠈⠋⠉⠉⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠁⠚⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠒⠀⠁⢢⣀⡀⠀⠀⢡⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡔⠁⠑⣼⠄⠀⠈⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⡶⠃⢀⡇⢀⠥⠀⠊⣡⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢀⣠⠶⠋⠀⠰⠀⡀⠊⠁⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠀⠀⠀⠀⠀⠁⠈⠈⠀⠉⠄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣒⣭⣷⣶⣿⣛⣟⡻⣿⣷⣾⣭⣖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⣾⣿⣿⣿⣿⢿⣿⣿⣧⡘⢿⣿⡏⠹⣷⣾⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠻⠿⠿⣿⣯⣴⣿⣿⣿⣿⣿⣶⣮⣍⣀⢛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⡿⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⠿⠿⠧⠀⠀⠀⠀⠤⠬⠭⡙⢿⣿⣇⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⡿⢁⣤⣤⡀⢶⡀⠀⠀⢀⣤⣀⡀⠀⢿⣿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢁⣹⣿⣿⡇⠘⠋⠀⠀⠀⠁⠀⠀⠉⠀⠈⠉⠀⢸⣿⣿⣄⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⢿⠀⠀⠀⠀⠀⢾⣇⠰⠄⢀⠀⠀⠀⠀⣾⣿⣿⣿⢾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣾⡇⠀⠀⢀⣤⠤⠀⠶⠦⣌⢳⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠈⠻⢿⠿⣿⣿⣷⠀⠀⠘⣿⣿⣿⣿⣿⡾⠋⠀⠀⣸⣿⣿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⣴⣞⠀⠀⣠⣿⣿⣿⣆⠀⠀⠈⣉⣉⡋⠍⠀⠀⠀⢠⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠉⠁⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⠾⠟⠈⣿⣿⣿⣿⣉⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣷⣤⣴⣤⣤⣤⣿⣿⣿⡟⠻⣿⠋⠉⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣿⡿⣿⣿⣿⣿⣿⣿⣿⠃⠃⠀⠛⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⢈⢷⣹⣿⣿⡏⠘⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠙⢿⣿⣿⣿⠟⠛⠁⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢀⣦⡸⡞⣇⢻⣿⣦⡀⠑⠺⢻⣆⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⢸⣿⢿⠟⠀⠀⣀⡄⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣾⣿⣿⣿⣹⡜⣿⣿⣿⣷⣦⣄⡙⠳⡄⠀⠀⢀⣼⠟⠁⢀⣠⣶⣿⡿⠀⠀⠀⣼⣿⣵⠏⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣻⣿⣧⢻⡏⢻⣿⣿⢿⣿⣿⣷⣌⠢⣠⣞⣡⣶⣾⣿⠛⡉⡁⠀⠀⣇⣼⣿⣿⡏⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠃⠈⠋⠛⠁⠂⢻⠛⣿⣷⣽⣏⠋⠉⠁⠀⠈⠀⡐⠀⣆⣿⣿⣿⡾⠀⣤⣤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠈⠀⠀⠐⠀⠈⠃⠈⠃⠘⠛⠀⠀⠀⠀⠀⠘⠁⠀⠛⠛⠛⠛⠁⠘⠙⠋⠐⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣘⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⣄⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡷⢣⡸⡶⢿⣲⣄⠀⠈⡳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣿⣿⣾⣿⡇⣿⣷⣏⣳⢶⡿⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠒⠻⢿⡿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠤⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⣿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⣒⡆⠀⠀⣴⣖⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢉⠃⢣⡀⢀⡼⢹⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⢸⠀⣠⣿⣿⣄⢸⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠈⠆⠀⠀⠀⠋⠀⣿⣿⠈⠋⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢀⠀⠀⠀⠀⢠⣿⣿⡀⠀⠐⠀⠀⠆⡆⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢸⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⢠⠀⠀⠀⢃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣦⡤⣀⣄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠋⠀⢀⠀⠀⢤⡤⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠉⠳⣤⣀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡃⢽⡶⣄⠈⠑⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡀⠀⠀⠀⠀⠰⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⢯⡛⠶⣿⣯⣌⣲⣤⣄⠲⣴⡆⣀⡀⢤⣤⠤⠀⣀⡀⠀⡠⠀⠀⠈⣊⣤⠎⠳⠠⠄⠀⠙⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠓⠊⢤⣴⡒⠦⣛⣴⣤⣤⣄⡀⠀⠈⠓⣤⣤⣀⠞⠀⠀⡀⠚⢩⡼⠂⠀⠁⠀⣀⠴⢚⠤⣴⠶⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠁⠀⠀⡐⠦⣤⣀⡀⠀⠈⠛⠷⠾⠇⣻⢿⣤⣿⣿⣟⡷⣮⣞⠉⠉⠒⠊⠀⠀⣀⣀⠠⠀⠂⠩⠐⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠆⠠⠄⣸⠍⠙⠻⠻⢏⣹⠷⢋⣙⠛⠛⠁⠀⠀⠠⠖⠋⠉⠀⠀⠀⠒⠀⠁⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⠤⡀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠈⠀⠈⠉⠉⠴⠶⠾⠀⠀⠉⠀⠀⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠙⠛⠓⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠊⠳⡈⠳⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠤⠤⠀⠀⠀⠒⠂⠈⠉⠉⠉⠉⠐⣀⠤⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⡀⠸⡄⠀⡀⠐⠠⣄⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⣖⠴⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠁⠀⣧⣄⣈⣓⠒⠀⠉⠉⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠀⠤⠄⠉⠉⠉⠈⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠼⠴⢦
⠀⠀⠀⠀⠀⠀⠀⡴⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⣴⢉⡁⠁⣦⣄⠶⡄
⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡂⠀⠀⠠⣀⠋
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠁⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣝⡛⣋⡙⢻⡟⣫⣿⢻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣤⣤⣶⣤⣄⣀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣻⣶⠄⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠟⠛⠁⠀⠈⠉⠙⢻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠉⣽⣍⣡⣦⣤⣶⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢈⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠤⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣄⠀⠀⢠⣶⣿⡿⠿⠷⠆⢻⣿⡇
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⡛⠀⠀⠙⣿⢿⠿⢿⣿⠃⣸⣿⡃⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⠉⠁⠈⠁⠀⠀⠀⠀⠐⠚⠋⠀⠀⢸⡟⢹⣿⣦
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⣸⣿⣿⣿⣆
⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡀⠀⠀⠀⠛⠿⠾⠛⠀⠀⠀⠀⠀⠠⠃⠀⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣇⠀⣠⣶⣶⢶⣦⣶⣤⡀⠀⠀⠀⢀⣤⣀⣽⣅⣠⣿⣿⡄
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡉⠻⣥⣤⣤⡬⠉⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⡉⢁⠀⠀⣀⣀⣴⣿⣿⣿⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⠟⠁
⠀⠀⠀⢸⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠘⢶⣿⣿⣿⣿⣿⠿⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⢸⠿⠋⠁⠉⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⠿⠟⠃
⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⢀⡀⢸⣿⡇⢰⡆⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣤⣤⣀⠀⣀⠀⡎⢿⣾⣿⡇⠼⠿⡇⢸⣇⡀⢠⡄⣤⣸⣷⠀⠠⠀⠀⣴⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⣴⣿⣠⣿⣿⣿⣧⣧⣸⣷⣿⣿⢋⣿⡇⠀⣷⣾⣹⣷⣾⣧⣿⣿⢁⣾⠀⣀⣠⣿⡇⣶⢀⠘⡄⣀⣀⣿⡀
⠀⠀⠀⢀⡀⣼⣿⣿⣿⣿⠿⣿⡏⣿⣿⣿⣿⣿⣿⡿⣿⣿⣆⣹⣿⣿⣿⣿⣯⠈⠙⣻⣧⡄⣿⠿⠟⣛⣿⣿⣶⣿⢻⣿⣿⡿⣿⣿⣤⣤⣤
⣤⣷⣿⣿⡿⠿⣿⣿⡿⢿⣿⣿⣷⡿⢿⣿⣷⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠿⠟⠻
⠀⠉⢻⠃⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠁⠘⠋⠀⠀⠈⠉⠛⠿⢿⡿⠉⠉⠀⠀⠀⠀⠉⠙⠟⠋⣻⣿⣿⠟⠛⠿⠉⠉⠀⠛⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⠀⠀⠀⠀⠀⢸⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡏⠀⠀⠀⠀⠀⡿⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢀⣀⠀⢰⠛⢦⠀⠀⣀⡤⠖⠚⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠓⠶⣤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠇⢀⡴⠛⠁
⠀⠀⠀⠀⠀⡏⠉⠳⣾⠀⢸⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⡴⠶⠦⣄⡀⠀⠀⠸⠶⠋
⠀⠀⠀⠀⠀⢻⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⣀⣤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠞⠈⣇
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⡴⣄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⡴⠋⠀⠀⢹
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⢰⣄⣀⡸⠁⠈⠒⠀⠚⠁⠈⠑⠦⠴⠈⠳⢄⣀⣀⣀⣠⡶⠋⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢘
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⢀⡴⠒⠒⠦⡀⠀⠀⠀⠀⠘⡆⢀⡤⠚⠛⢤⡘⡇⠀⠀⠀⠀⠸⠧⠶⠶⠒⠛⠋⠉
⡏⠀⠀⠦⠼⠂⠀⠀⠀⠢⡄⠀⠐⣦⣤⠜⠀⠀⡎⠀⣠⣤⡀⠹⠀⠀⠀⠀⠀⢹⡏⠀⣶⣶⠀⢷⡇
⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢁⠀⠀⠀⠀⢧⠀⠙⠿⠃⢠⠇⠀⠀⠀⠀⠈⣧⡀⠙⠋⢀⡿⡇
⢻⡀⠀⢀⠀⠀⢠⠞⠉⠀⠈⠦⠤⠎⠀⠀⠀⠀⠈⠳⠤⠤⠴⠋⠀⠀⠀⠀⠀⠀⣧⠉⠒⠒⠉⠀⢹⠀⠀⠀⠀⠠⣶⠒⠒⠲⠶⠤⢤
⠈⣧⠀⠀⠁⠀⡇⠀⢀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣸
⠀⠈⢧⡀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠳⡄⠀⢰⠇
⠀⠀⠀⠙⢦⣀⠀⠙⠓⢲⡆⠀⠀⠀⠀⠀⢸⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠹⣦⡏
⠀⠀⠀⠀⠀⠉⠛⢦⡄⢸⠇⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠶⣯⡞⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⠙⠒⠤⢤⣄⣀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⠀⠀⢀⡾⠃
⠀⠀⠀⠀⢀⣤⠤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢑⡞⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⣴⠋⠀⢰⠋⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡾⢄⠀⠀⠀⢠⠞⠁
⠀⠀⢰⡇⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⠜⠀⠀⢠⠏
⠀⠀⠀⢷⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠘⡇⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⢠⡞⠳⣄⡀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠑⢄⡀⠀⢀⡴⣏⠀⠀⠈⠙⠳⢤⣄⣀⣠⡴⠟
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⣢⠞⠀⠘⡆
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣠⣾⣁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣧⢀⣠⠞⠁⠀⠈⠙⠶⣄⣸⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢛⣫⣭⣭⣿⣿⣿⣿⡿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠃⢐⣛⣛⣛⣛⣭⣭⠵⢚⣴⣯⡻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠛⠛⠉⠉⠉⠉⣀⣠⣴⣿⠟⠘⡿⣎⢿⣿⡀
⠀⠀⠀⠀⠀⠀⣴⡿⠋⢀⣀⣤⣤⣴⣶⣶⣿⣿⣿⠿⠛⠁⠀⠀⢻⢻⣆⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢸⡟⢀⣴⣿⣿⠿⣿⣟⣛⠋⠉⠁⠀⢠⡴⠒⠒⠆⠘⡏⣿⣎⠃⠀⠀⠀⠀⠀⢀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⡇⣼⣿⠏⣴⡎⢉⣈⣉⠃⠀⠀⠀⠉⢠⣶⡶⣦⠆⢳⡘⣿⣆⠀⠀⠀⠀⠀⢸⠀⢠⣿
⠀⠀⠀⠀⠀⠀⢻⣿⡏⢸⣿⡟⢏⡹⡿⠃⠀⠀⠀⠀⠁⠉⠝⠃⠀⠈⢷⡈⠻⢷⣤⣀⣀⠴⠃⣠⢞⡏
⠀⠀⠀⠀⠀⠀⠀⢹⣷⢸⣿⣷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠛⢶⣤⣄⣀⣤⡴⠞⣡⠟
⠀⠀⠀⠀⢀⠄⠀⠀⢿⠘⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⢿⣶⣤⣤⣤⣤⡶⠟⠁⠀⣀
⠀⡄⠀⠀⢢⠀⠀⢀⡟⠀⣿⣿⡇⣷⣄⠀⠻⣍⣩⡽⠃⠀⣀⣾⡇⢻⣇⠀⠀⠀⠉⠀⠀⠀⣰⣾⣿⠃
⠀⢱⡀⠀⠀⠙⠛⠉⠀⢠⣟⣾⡇⢸⣿⣷⣦⣄⣀⣠⡤⠚⠹⣿⣧⠈⢿⡄⠀⠀⠀⢳⡄⠀⠉⠉
⠀⠀⠙⢦⣀⡀⠀⢀⣤⣯⠞⢹⠃⢸⡇⣿⣿⣧⠀⠀⠀⠀⠀⠙⢿⣷⡈⠻⣆⠀⠀⠈⡇⠀⠀⡀⠀⢀⡄
⠀⢀⠀⠀⠈⠉⠉⠉⠉⠀⣠⠏⠀⣾⠃⣿⣿⡟⠀⠀⣀⡤⠶⠛⠉⠉⠛⢦⡈⢧⡀⢰⠁⢠⠞⣠⣶⣿⠁
⠀⢸⡀⠀⢀⡀⠀⢀⣠⡾⠃⠀⣼⠏⣸⣿⡿⠁⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠻⡄⠳⣘⢠⡏⣸⣿⡿⠃⡴
⠀⠀⢳⣄⠀⠙⠛⠋⠁⢀⣠⣾⠏⣰⣿⠟⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢻⠘⣇⣛⣁⡤⠞⠁
⢦⣄⡀⠙⠷⣶⣶⣶⣾⣟⣫⣵⡾⠟⠁⠀⣀⣠⠤⠤⠀⠉⠛⢶⣄⣇⠀⠀⠀⠀⠀⢀⡔⣡⡭⠤⠤⡀
⠈⠻⣷⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⢠⡶⠋⠁⠀⣀⣀⡀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⣸⠋⠀⢧⠻⣿⣦⡀
⠀⠀⠀⠁⢄⠰⡠⠚⠙⢲⡀⠀⣰⠏⠀⣤⠾⠋⠉⠉⠉⠛⢦⡄⠈⣿⠀⠀⠀⠀⡰⠃⠀⠀⠀⠑⠪⠝⠛
⠀⠰⠶⠶⠦⠑⢷⠀⠀⠀⠇⠀⣿⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠹⡄⡟⠀⣀⣴⣋⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠻⣄⠻⣦⣀⣀⡄⠀⠀⠀⠀⠀⢇⡤⠞⠋⠉⣀⣀⣀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠉⠛⠻⠛⠋⠀⠀⠀⠀⢀⠔⠉⠀⠀⣠⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣤⣀⣀⣀⣤⠔⠚⠁⠀⠒⠊⠉
⣿⣿⠛⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⢶⣄
⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠿⣷
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⠀⢠⡿
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠘⠷⠀⢠⡴⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⠶⣶⡾⠋⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠨⣽⣿⣿⣿⠍⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣛⣛⣉⡛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠓⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⣼⡟⠛⠶⠶⣦⣤⣄⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡏⠀⠀⠀⠀⠀⢀⣀⣠⣼⡇
⠀⠀⢀⣿⠛⠛⠛⠛⠳⠶⠁⠀⠀⠀⠻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠏⠀⠀⠀⠀⠀⢛⣋⣭⣥⣾⣷
⠀⠀⢸⡟⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠽⠿⠛⢻⡏
⠀⠀⣼⡷⠶⠶⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠾⣿
⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀
⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣷
⢠⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⣀⣸⣇⣀⣀⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡀
⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠟
⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠙⠻⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢹⠶⠂⠦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠒⠷⣦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⣿⠖⣤⡜⠛⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠈⣟⢦⣽⣎⣒⢠⣈⡉⢄⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠫⠆⡠⢄⣀⠀⠈⠶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠓⠪⠋⣙⡯⣭⡹⣠⣉⣀⡉⠉⠉⠣⣀⣁⠀⠔⠁⠄⠠⠞⣡⠅⠀⠀⠀⠀⡮⢄⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠑⠀⠀⠐⢤⣄⡀⠀⠈⠛⠷⠼⡏⣻⣯⣽⣿⣷⣶⣼⡭⠙⠂⠒⠀⠀⠀⢁⡀⠀⠐⠃⠅⠂⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠳⣄⠀⠀⠀⠀⠀⡀⠈⣙⠳⢂⣦⣜⣎⡙⠛⠻⣛⣿⢾⣍⢛⡛⠁⠀⠀⠠⠚⠉⠁⠀⠀⠂⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⢤⣀⠀⠀⠀⠀⠀⠙⠒⠀⠈⠀⠀⢈⠐⡓⠺⠛⠃⠙⠁⠠⠤⠀⣀⣤⣅⡀⠀⠈⠀⡘⠋⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢧⠈⢷⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠓⠒⠒⠂⣀⠈⠉⠉⠀⠀⠈⠀⠀⣤⡐⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⢧⠀⡇⠀⢠⣈⠒⢦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠸⡗⠃⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠟⠓⠲⠤⠭⠀⠀⠀⠈⠶⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⢄⡀⣀⡈⠁⠉
⠀⠀⠀⠀⠀⠀⡉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠍⠉⠀⡈⠝⠀⠁
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠈⠃⠀⠀⠈

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⢾⣿⣷⠀⠀⠀⣠⣤⣄⡀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⢠⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠙⣿⣷⡀⣼⣿⣿⣿⣷⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠘⣿⣷⣼⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣯⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢸⣿⣿⣿⣿⣿⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡇⠀⢸⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣇⢀⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣶⡆⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⣿⣿⣧⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢠⣿⣿⣿⣶⠀⠀⠀⠈⢻⣿⠿⣿⣿⡟⣱⣷⣶⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⡿⣿⣿⣿⡄⠀⠀⠀⠀⢁⣤⣻⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢠⣶⣿⡇⠋⠀⢸⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⣨⣍⡛⢧⡹⣿⣿⣿⣿⣇⡇⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠐⠻⢿⣿⢿⡏⢸⣿⣿⣿⣷⣄⣸⣿⣿⣿⢹⡇⠀⣸⣦⣽⣿⣿⣿⣿⡟⠿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣯⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡿⢻⣿⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣷⣿⣿⡿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣯⣥⣷⣾⣧⣀⠀⣠⢽⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⡽⠖⠀⠀⠘⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣰⣿⣍⠳⣜⠿⣿⠿⣟⣛⣛⡿⣿⣿⡿⣫⡵⢋⣥⣶⣿⡧⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣍⡛⢛⣋⣭⣭⣭⣙⠲⠶⠚⣫⣴⣿⡿⠋⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣠⣤⣴⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡄
⠀⢺⣿⣿⠟⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⠏⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⢠⣾⡿⠋⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⡿⢰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿
⠘⣿⠇⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⡇⠈⠟⠛⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣠⣤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⣋⣉⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣾⡿⠿⠛⢛⣉⣩⣤⡴⠶⣚⣛⣭⡭⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣀⣀⣤⣤⣶⣾⣿⠿⠟⠛⢛⣉⣥⡤⠶⠖⣛⣫⣭⠵⠶⠛⠛⠉⣉⣀⢠⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡿⠿⠟⠛⣉⣩⣤⠴⠶⢞⣛⣭⡭⠶⠖⠛⠋⠉⣁⣤⣤⡀⠐⣾⣿⣿⠿⠆⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⠿⠿⠛⢋⣉⣡⣤⠶⢖⣛⣫⣭⠵⠶⠚⠛⠉⠉⠀⣠⣴⣶⣄⠀⢸⣿⡿⢿⣿⡆⢻⣿⣧⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⢰⣾⡿⠿⠟⠛⣋⣉⣤⡴⠶⢚⣛⣭⡭⠶⠖⠛⠋⠉⠀⠀⠀⣴⣶⡆⠀⠀⢼⣿⡟⢻⣿⣧⠸⣿⣷⣼⡟⠋⠈⣿⣿⣶⣿⡆⠸⣿⡄
⠀⠀⠀⠀⠀⠀⢤⠶⠶⣛⣫⣭⠵⠶⠚⠛⠉⠉⠀⢠⣤⣦⠰⣿⣿⡀⠀⠀⠸⣿⣿⠀⠀⢸⣿⣷⠈⣿⣿⡄⠙⠿⣿⣿⣶⡄⢹⣿⣯⠁⠀⠀⢻⠇
⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠁⣠⣤⠀⣴⣿⣿⣿⣦⠘⣿⣿⡄⢿⣿⣇⠀⠀⠀⢿⣿⣇⠀⠀⣿⣿⡆⢹⣿⣷⢠⣴⣶⠙⣿⣿⡌⣿⣿⣄⣤⡄⢰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢲⣿⣧⠘⣿⣿⠸⣿⣿⠈⣿⣿⡆⢹⣿⣷⠘⣿⣿⡄⠀⠀⠘⣿⣿⡀⠀⢸⣿⣿⠀⣿⣿⡆⢻⣿⣧⣿⣿⠇⢹⡿⠿⠟⠛⠈⠉⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣿⣿⠀⢿⣿⣇⢸⣿⣿⠈⣿⣿⡄⢻⣿⣧⠀⠀⠀⢻⣿⣇⣀⣀⢿⣿⣷⣿⣿⠇⠀⠙⠛⠛⠉⠀⣀⣀⣤⡤⠶⢖⣛⣫⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⠸⣿⣿⡀⢿⣿⡇⢹⣿⣷⠘⣿⣿⠀⠀⠀⠘⣿⣿⡿⠿⠆⠉⠉⠉⢁⣀⣠⣤⠴⠶⣚⣛⣭⡭⠶⠖⠛⣋⣉⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⢻⣿⣇⢸⣿⣿⠀⢻⣿⣿⣿⠟⠀⠀⠀⠀⠉⢀⣀⣤⡤⠶⢖⣛⣫⣭⠷⠶⠛⢛⣉⣩⣤⣶⣶⣿⣿⡿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠈⠻⣿⣿⡿⠋⠀⠀⠈⠁⣀⣠⣤⠴⠶⣚⣛⣭⡭⠶⠖⠛⣋⣩⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠀⠀⠀⣀⣀⣤⣤⠶⢖⣛⣫⣭⠵⠶⠚⢛⣉⣡⣤⣴⣶⣿⡿⠿⠛⠛⠉⢉⣴⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⢚⣛⣭⡭⠶⠖⠛⣋⣉⣥⣴⣶⣾⡿⠿⠟⠛⠋⠉⠁⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠵⠶⠚⢛⣉⣡⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⠿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⣖⣤⣶⢿⡿⣫⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⢋⣷⢟⣽⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⡿⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦⡽⠛⠁⠀⠀⢀⣴⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢼⣿⡠⡀⠙⠋⣿⣿⣥⣤⣀⠀⠀⠀⠀⣠⣶⣶⣿⡿⢋⡟⢀⡄⠀⠀⠀⢨⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣧⡀⠀⠀⠹⣿⡅⠀⠃⠀⠀⠀⣞⢩⣿⣿⢗⣰⣿⣿⡏⣀⣀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠁⣾⣀⣴⣿⣿⣿⣷⣦⡀⠘⠃⠾⣋⣴⣿⣿⣿⣿⣟⣽⣇⣺⡵⠖⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠘⠛⢿⣿⣿⣿⣿⣿⣞⡁⢀⠀⠀⠀⣀⠀⠀⣰⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠉⠉⠛⢿⣿⣿⣷⡾⣿⣿⣿⣿⣿⣿⣿⣊⣅⣀⠄⠘⠉⢴⡿⠋
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⣿⣿⣿⣿⣿⠁⠀⣠⣴⣿⡿⠟⠁⠀⠈⢈⠽⣿⣿⣿⣿⣿⣯⡷⠀⢀⣈⣍
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣾⡉⢈⣿⣿⣿⣿⠉⠀⠀⠛⠋⠁⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟⠉⠀⠴⠖⠀⠀⠀⠠⠒
⠀⠀⠀⠀⠈⠳⢤⣀⣙⣷⣤⣀⠀⠀⠉⢸⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠐⠿⢻⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠐⢌⣹⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣩⣿⡿⢋⣤⣤⠄⠀⠀⢀⠄⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠽⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣤⣴⣦⠀⠀⠀⠀⠀⠀⢛⣿⣾⠋⠁⣠⣤⡴⠊⠁⠀⠀⢀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠙⠛⠁⠀⠀⠀⠀⠀⣴⣿⡿⣷⠞⢉⡡⠈⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠿⠛⠋⠁⠏⠈⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⣾⣟⣽⠁⠀⠀⠀⠀⠀⢠⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢼⣿⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠼⠟⠙⠃⠙⠃⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⠠⠤⠄⠠⠄⢀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠅⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⣀⡴⠏⠀⠀⠀⠠⠄⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠴⠒⠒⠚⠛⠛⠉⠀⠀⠀⠀⠀⢢⣀⡀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡏⠀⠀⠀⠀⠀⣠⠖⠋⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢤⡀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠃⠀⠀⠀⠀⢰⠃⠀⠀⠀⣀⠀⠈⡆⠀⠀⠀⢠⠖⠉⠁⠉⠓⢄⠙⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠀⠀⠀⠀⠀⠸⣀⠀⠀⣾⣿⣤⢀⠇⠀⠀⠀⡇⠀⠀⣠⣤⠀⠈⡆⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠉⠒⠛⠿⠿⠊⠀⠀⠀⠀⠇⠀⠘⣿⣿⡷⠠⠃⠀⠀⠀⠀⡟⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢰⣧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠘⢧⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢦⣀⡀⠀⠘⠛⠲⣤⠀⠀⠀⠀⠀⠀⠀⠉⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠀⠀⠀⠀⠀⠀⣠⠖⠂⠉⠁⢁⣀⣼⠇
⠀⠀⠀⣀⡬⠛⠁⡤⠶⢦⡸⣇⡀⠀⠀⠀⠀⠀⢀⣠⠞⠉⢻⡓⠶⠤⢤⣤⠤⠤⠴⣶⠚⠻⣄⡀⠀⠀⠀⠀⠀⢀⣸⢇⡴⠶⠤⠈⠛⢧⣀
⢠⠶⣫⠵⠀⠀⠀⠀⠀⢀⡗⠀⠙⠓⠶⠶⠶⠚⠋⠁⠀⠀⣸⠱⡀⠶⢯⣿⠷⠤⡔⢹⡄⠀⠀⠙⠓⠶⠶⠶⠒⠋⠁⢸⡀⠀⠀⠀⠀⠀⠪⣝⡲
⣿⣫⠴⠚⠁⠀⣀⡤⠖⠿⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠚⢹⠀⠀⠀⢸⡿⠀⠀⠀⠀⣿⠦⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⠽⠲⢤⣀⠀⠈⠛⠦⣝
⠹⣇⢀⡠⠆⠉⠁⠀⠀⠀⠀⠉⠉⠉⠀⠐⠛⠋⠉⠀⠀⠀⢨⡀⠀⠀⣿⣇⠀⠀⠀⠀⢨⡀⠀⠉⠉⠀⠓⠀⠉⠉⠉⠀⠀⠀⠀⠀⠉⠀⢤⡀⣨
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣻⣷⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠿⣿⠟⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢀⡀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⡏⠉⡇⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⡇⢸⠇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⡼⠀⠸⡆⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⡀⢸⡇⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⠏⢿⠀⠉⠙⢿⡆
⠀⠀⠀⠸⣿⡇⣿⣿⡟⠘⠋⠋⠅⠀⠀⠀⠉⠀⠈⡇⠀⠀⢀
⠀⡀⠀⢀⣿⣧⣸⣿⡇⠀⠀⠀⡆⠀⠀⠀⠂⠁⠀⡇⠀⠀⡄⠂⡄
⢆⠀⣠⣾⣿⣿⠏⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⡠⠀⢧
⠈⠛⠛⠛⠛⠁⠀⠀⢹⣄⢀⠀⠀⢸⣶⣶⡇⠀⠀⡸⠀⢀⠁⠇
⠀⠀⠀⠨⠐⠄⠀⠀⠘⠮⣅⡀⠀⠈⠁⠁⠀⣀⡜⠓⠆⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠄
⠀⠀⠀⠐⠂⠨⡢⠀⠀⡀⡔⠉⠑⠒⠒⠒⠋⠁⠘⡆⠁⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠛⠀⠠⠖
⠀⠀⠀⠀⠀⠀⠈⠂⠄⠂⠉⢲⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠶⠎⠀⠀⠁⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣤⠤⠤⠶⢲⠾⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⠒⠲⣶⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⢀⣀⣴⣏⣀⣾⠄⡼⠀⠀⠀⠀⢸⣿⣭⣭⣍⣉⣹⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢑⢦⠞⠉⠉⠁⠀⠀⣠⢣⠎⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠱⡀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣵
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⣰⡦⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠁⠀⠏⢀⣴⠞⠋⠉⠁⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠲⠶⠦⣄⡉⠀⠀⠀⢛⠃⠠⠀⠀⠈⠀⠀⠀⠀⠀⣤⣶⣾⢿⣦⡀
⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢿⡄⠀⢵⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢿⣇
⠀⠀⣾⡏⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡆⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⢀⣄⣠⣀⣤⣶⣶⣿⣿⣿⣿⣿⡿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣼⡟
⠀⠸⣿⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⣾⠟⠁
⠀⠀⢻⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠙⠿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⠏

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⢠⡈⠳⣄⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠱⠂⣠⡞⠁
⠀⠀⠀⠀⠱⣄⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡞⠉
⠀⠀⠀⠀⠀⠈⢡⠊⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⠷⢦⡄⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⢿⣦⠀⠀⠀⠀⠀⢸⠃⡀⠀⢀⡀⢿⡄⠀⠀⠀⠀⣠⡾⠋⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠹⣷⡀⠀⠀⠠⢾⠀⠁⠂⠈⠁⣸⢧⠀⠀⣴⡾⠋⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠈⠻⢦⡀⠀⢻⡄⠺⣯⡗⢀⡿⠃⢀⣾⠟⠀⠀⣠⠞⣠⣴⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠙⢦⡀⢻⠦⣤⡴⣿⣠⣴⡿⠋⠀⣠⣾⢫⡞⡀⠀⣀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⠀⠀⠈⢷⡈⢰⣦⣴⠛⠛⠁⠀⢀⣾⠛⠁⢸⠀⣉⣁⡀⠀⢙⠆⠀⢤⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠁⠈⠀⠉⠀⠀⠀⢀⡾⠁⠀⠀⠸⣆⠻⠾⢁⢠⡋⠀⠀⡂⢿⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠈⢓⡖⠉⠸⣀⠀⠀⢹⡄⠀⡼⠁
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣆⣤⣶⣋⢳⣤⡾⠉⠳⣶⠟⠀⡸⠁
⠀⠘⡕⢦⣄⣏⢳⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣻⣿⢋⠀⠀⠀⠙⢿⣿⠀⠤⠤⠄⠒⠒⣲⡾⠁⠀⣼⠃
⠀⠀⠙⣦⡈⠁⠘⣇⡀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣼⣿⡇⢠⡄⡄⢠⡦⠸⣿⣿⡆⢒⣛⣛⣻⠏⠀⢀⡾⠃
⠀⠀⠀⠈⠙⠶⣤⡈⠛⠷⣤⣄⡀⠀⠀⢿⣇⠀⠀⠀⣿⣿⡇⠈⠁⠀⠀⠀⠀⢻⣿⡏⣉⣁⣼⠏⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠉⠛⠷⣦⣸⣿⠲⢤⣀⣙⢿⣿⡀⢿⣥⣽⠃⣰⣿⠟⢀⣠⡿⠃⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠙⠿⣧⣤⣀⣀⠀⠉⠳⣄⣉⣡⢴⡏⠀⣴⣿⡋⠙⢦⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⡿⠙⢻⣿⣷⣦⣼⠉⠁⢘⣧⣾⣿⣿⣿⣦⡾⠋⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢧⣠⣿⣿⣿⣿⡏⠛⣶⠛⠀⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠉⠻⣿⣿⣿⣷⣾⣿⣷⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⣶⡄⠀⣾⣿⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⢀⣽⡇⣿⢳⠀⣿⣿⠀⠂⢹⣷⡆⠀⠀⡦⠀⠀⣇⠀⠀⣾⣿⠀⡄⠀⣾⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡄⡄⠸⡶⣿⣿⣿⣃⣿⠚⠂⢿⣿⢹⡇⢀⣿⣶⣿⡄⢏⠀⠀⣿⣷⣤⣿⣿⣶⡇⣰⣿⣰⣾⣤⣠⡦⣤⡀⠠
⠀⣠⣀⡀⠀⣸⡇⣀⣠⣿⣿⣿⣾⣿⣯⣶⣼⣇⣼⣾⣇⣠⣿⣿⣿⣷⣶⣾⣸⣿⣿⣿⣿⣿⣿⣿⡟⢛⣿⣻⣿⣿⡇⣨⣇⠀⠄⠀⠀⠀⣼⠇
⠷⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠻⠿⠿⠿⠿⠿⠯⠭⠿⠿⠿⠿⠿⠿⠿⠷⣿⠶⠖
⠀⠀⠀⠀⠸⠉⠉⠉⠉⠀⠀⠈⠛⠛⠿⠿⣿⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣭⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢆⡴⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣫⠀⣸⠁⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣥⣀⢤⡑⢄⠀⠘⡆⠀⠀⠀⠀⠀⢐⣁⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠑⣌⠣⡀⠹⡄⠠⠤⢀⡔⠉⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢡⠀⢳⢀⠤⣼⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠒⠢⢤⡀⠀⢸⢽⠇⢈⡇⠀⣸⢀⡀⠀⢀⠏⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢹⠲⡌⢦⣾⠟⠀⠘⢧⡾⠙⠛⠗⠒⠓⠒⠒⠒⠒⠚⠋⠉⠉⠁⠉⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢌⡉⠈⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠛⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠲⢤⡀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⢤⡤⠤⠤⠥⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣇⢩⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⢸⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠙⣯⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⣆⠂⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢰⢋⣍⡽⠛⠒⣲⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠘⢆⠀⠘⢤⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠈⢮⡑⢤⣴⣩⠤⠬⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⢧⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠈⠓⣒⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠏⡀⠀⠀⠀⠀⠈⢣⡀⠀⠈⠳⣄⠀⠀⠀⣠⣾⣅⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⠎⣠⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡎⢀⡇⠀⠀⠀⢸⡀⠀⢳⡀⠀⠀⠈⢧⡠⢾⣹⠀⠀⠀⣼⣿⠄⣀⣀⡠⠤⠚⣡⠞⠁
⠀⠀⠀⠀⠀⠀⡼⠀⢸⠁⠀⠀⠀⠈⡇⠀⠀⠱⠔⠉⢀⣼⢦⣇⠉⠀⠀⣸⢻⠇⠤⠤⠤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠐⠧⣄⣈⠉⣿⠿⡖⠚⠃⠀⠀⠀⠀⠀⠀⠈⢸⠀⠉⠓⠢⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠈⢙⡀⡗⠒⠒⠒⠒⠂⠀⢤⠠⡴⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠉⠉⠀⠀⠀⠀⠀⠀⠀⡇⡇⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠴⠞⢾⠃⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠃⠀⠀⠀⠀⣀⣀⡠⡾⠤⢤⡀⢾⡤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠰⠦⠾⠶⠤⠶⠖⠛⠙⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⠂⢠⣤⣤⣄⣀⠀⠀⣰⣧⣬⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣽⣿⣶⣿⣿⣿⣿⣿⣆⡀⣿⣯⣄⢠⣀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠹⠇⠹⢿⣿⣿⣿⣷⠸⡅⣩⠀⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡉⠀⣘⠃⠸⣿⣿⣿⣿⠤⣧⡙⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣦⣁⡴⠀⠻⣿⣿⣿⣶⠟⣿⠟⢀⣾⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠻⢻⠯⣤⡟⠀⠀⣀⣈⣋⠀⠀⠀⠘⣇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠄⣌⢠⢠⠀⢈⠄⠀⠐⠙⠿⠈⠳⡆⠀⠀⢻⣿⣿⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⡾⠃⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢇⠀⠀⢸⣿⣿⡆⠀⢌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣰⣤⣞⠰⣧⠚⣷⣄⣄⠀⠀⠃⠀⢀⣲⡏⢳⣿⠀⢸⢸⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⢿⡟⠁⢠⣿⣿⡿⠀⢰⠄⠀⣨⢷⡅⢈⣻⠀⢸⣾⣿⣿⡇⢠⣟⠃⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⣠⠟⠈⢻⡀⣰⠃⠀⣴⠋⣾⣯⣨⣿⠀⢘⣿⣿⣿⡇⠀⠈⢁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠻⠶⢀⠈⠃⠋⠐⠈⣧⠚⠉⠛⢿⡿⢀⣾⠛⠉⢹⡇⠐⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠓⠧⣀⣠⡀⡇⠀⢀⠄⠈⡠⠊⠃⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡛⣡⣤⣴⡾⠁⠀⠀⢀⣴⡟⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⢠⣤⣴⠋⠀⠀⠀⠀⠠⠽⠋⠙⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣰⡏⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠀⠀⠀⢠⠀⠀⠀⠀⠱⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣀⢀⡀⠉⠀⣀⠴⠦⠀⠀⠀⠀⠀⠀⠕⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠠⣻⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠇⠀⡟⠃⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣭⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⣗⡆⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣿⣧⡀⢫⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⡛⢃⣺⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡟⠉⠀⢠⡤⢠⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠛⠻⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢋⣠⣤⣄⡀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠪⠤⢍⡉⣻⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠈⠈⢉⠉⠉⠐⠒⠥⢄⣀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠃⠀⠀⠀⡠⠀⠈⠙⠢⣄⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡆⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⣿⣦⠸⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠃⠀⠈⠁⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠈⠫⠌⡿⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠠⡼⠣⠀⠀⠀⠈⢋⠉⠉⠉⠉⠉⢳⠀⠀⣰⣻⠿⠴⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠕⠭⡁⢸⣦⣄⣀⠀⠀⠑⠤⢀⡠⢔⣁⠔⠺⣅⠀⠀⠀⣀⣀⠀⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⣒⣁⡀⠉⠉⠉⠉⠙⠋⢷⡉⠉⢁⠄⠀⠀⠈⠓⣤⠚⠉⠩⣇⡀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣼⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠉⠹⡁⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣴⣿⣶⣄⡠⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⣿⣿⣦⡤⠤⠔⠾⣿⣿⣿⣿⠃⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠔⠘⡏
⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⢹⠿⠛⠛⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⡩⠉⠉⠁⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠹⡍⠁⠀⠀⠀⢀⡊⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠙⢦⡀⢰⣾⣿⣿⡤⠤⠀⢒⡿⠃⠀⠀⠀⠀⠀⢀⣤⠤⣄⠀⠀⠈⢷⣅⣀⠀⠀⠀⠀⠀⣠⠊⠉⣛⠻⡉⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠤⠶⠚⠉⠀⣘⠲⠦⠀⠀⡴⠉⠑⣷⣄⠱⢤⣀⡜⠁⠀⢉⡶⢦⣀⠔⠛⠓⣾⣉⣹⣇⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⢳⣂⡉⠤⢿⣦⣀⢩⡙⠢⡴⠋⠀⠀⠀⠀⠀⠀⢾⣀⠤⣬⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡒⣉⠵⣽⡁⠈⡞⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⢉⡝⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣊⣩⣷⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠋⠉⠉⣷⡞⠉⠉⢳⣤⠖⠒⢦⣤⡤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠴⠒⠒⣾⠓⠛⠠⢤⡯⠽⠦⣤⠏⠽⠤⣴⠟⣯⣀⢉⣷⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠹⣎⠁⠁⠻⣏⠒⠓⠺⡗⠒⠀⢿⠤⠤⢼⡧⠤⠤⢤⣄⣀⠤⠒⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛⠁⠀⠈⠻⠃⠀⠈⠲⡦⠘⠦⣤⡀⢳⣀⡀⠀⠀⢸⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠋⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡅⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠆⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠛⠒⠂⠠⠤⠤⠤⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡂⢀⢀⣠⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠒⠒⠒⠒⠠⠤⠤⠴⠓⠠⠤⠤⠶⠊⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣯⣿⣿⣽⣯⣽⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣿⣿⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡴⠶⠒⠒⠲⠶⠤⠤⣤⣀⡉⠛⢯⣑⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣉⣵⣶⣤⡀⠀⠀⠀⠀⠉⠉⠓⠪⠷⣦⡀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⣠⡴⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠳⢤⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠒⠲⠦⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠒⠒⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠒⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⠤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣦⣄⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢱⡄⠀⠀⠀⠀⣾⠉⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡈⢢⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠒⠒⠒⠋⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⣄⣩⠗⠀⠀⠀⠀⠀⠸⡄⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⡴⠊⠉⠉⠑⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣧⡤⠖⠊⣉⡉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠫⠄⠛⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠶⣖⡒⠒⠒⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢦⣀⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡃⡸⠁⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠦⢤⣤⣀⣀⣀⣀⣀⣋⣁⣠⡤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⣻⡟⢻⣟⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢋⡙⠁⠈⠉⠉⠀⠘⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣄⡰⠋⣀⣀⡀⠀⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡝⢻⣟⣑⣮⡅⠀⠛⠒⣇⣀⣀⣀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⡗⣿⠀⠁⠚⣫⣶⣀⠬⠍⣿⣿⣿⠪⣑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⣼⠋⠀⠀⠀⠀⣯⠶⣄⢀⡇⠀⡿⠙⠺⠇⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣊⡥⢄⠀⠀⠀⠀⠈⠲⡌⠉⠀⠀⣧⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⣀⡴⠋⡦⠀⠀⡴⠊⠁⡀⠀⠀⢣⠀⠀⠀⢀⣇⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⢳⡎⠹⢲⡄⠙⠒⠮⠤⠾⣁⡸⢀⡦⣶⢞⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠁⠀⠀⠀⠙⢳⣿⣄⣠⠟⠒⠀⣀⣈⣰⡏⠀⡟⡎⡆⠀⠀⠀⠀⠀⠀⠠⠉⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠛⠮⣙⣟⣦⣀⠀⠀⠀⠈⠛⡗⠫⢷⣒⣾⠁⠈⠁⠀⣸⡄⢻⢹⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⢰⠉⠛⠁⠀⠙⠓⠢⣤⣀⣧⠀⠀⠙⠳⠦⢤⣤⣼⡭⠗⣸⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢿⣿⣷⣖⣒⣒⣒⣋⣩⠥⠖⠁⠀⠀⠀⠀⠀⠀⢯⠀⠉⢩⢟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣀⡀⢠⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⢉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠉⠁⠀⠀⠁⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠇⢳⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡐⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⡆⢻⠀⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠠⠃⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡜⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣧⠼⢦⡼⠃⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⡐⠁⠀⠀⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠎⠀⠀⠀⢧⣀⣀⣀⣀⣀⣤⣤⣀⢼⠠⠤⠴⠶⠶⠶⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠮⠍⠤⠛⠀⠀⡂⢽⠩⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠳⠀⠀⣠⡆⠀⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⣀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⣧⠇⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣶⡞⠀⠀⣿⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⢣⣀⠀⣀⣀⣠⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⢀⡄⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⡸⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠌⠁⡅⠀⠀⠀⠀⡠⠂⢰⠀⢠⣷⠃⠀⠑⠌⣢⡀⠀⠀⠀⠀⢠⡾⠛⠉⢦⠀⢸⠀⠀⠀⠀⢀⡞⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⢠⠊⡠⠊⠀⠀⡇⠀⠀⣠⠊⠀⠀⠀⡇⠀⣿⣶⣄⡀⠀⠈⠳⢄⠀⠀⠀⣼⠃⢠⠔⠊⠛⢾⣤⡀⠀⠀⡼⣧⠀⠀⠀⣼
⠀⠀⠀⠀⡐⠁⠊⠀⠀⠀⠀⢻⠀⢸⠃⠀⠀⠀⠀⡇⠀⠉⢻⠁⠉⠂⢀⠤⠊⠀⠀⣸⡇⠀⡇⠀⠀⠀⢈⡉⠁⠀⣿⠁⣿⡶⠶⢶⠋
⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠒⠒⠊⠁⠀⠀⢀⢜⡜⡇⢰⠃⠀⠀⠀⣈⡁⠀⠀⣯⠀⢉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⡤⠖⠊⠉⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⠀⠀⠼⣦⣀⣀⣴⠋⢧⠀⠀⠘⡇⢨⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣄⣀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠹⣄⠄⠀⠀⠉⢉⡰⠀⠸⡆⠀⢠⠃⡏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠙⢻⡄⠈⡁⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠉⠱⠖⠊⠉⠁⠀⠀⠀⢻⡀⣼⠀⡇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡇⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢹⡀⣧⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⡇⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢤⣇⡟⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡺⣿⠧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⡴⠚⣇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡛⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⣷⣤⣾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁

⠀⠀⠠⠤⢤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⣲⣶⡶⠶⠤⠤⢤⣀⣀⣀
⠀⠀⠀⢰⡷⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠗⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⣶⠶⠦⢤⣤⣀⣀⡀
⠀⠀⠀⡒⣿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⣥
⠀⠀⠀⠄⠀⠀⠀⠀⠀⢴⠶⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀
⠀⠀⠀⣦⣀⠀⠀⡀⣀⣤⣤⡀⠀⢸⡿⠿⢻⠇⠀⠀⠀⠀⠀⠀⠀⠐⣭⠁⠀⠀⣤⣤⣴⣶⣾⠿⣿
⠀⠀⠀⣿⣿⣇⣾⣶⣾⣻⣻⣿⣦⡾⣠⡔⠫⢶⣤⣤⣾⣿⣷⣤⢤⣿⣧⣄⢴⣷⡿⢻⣿⣿⣁⣠⣯⣤⣤⡀
⠀⠀⠀⣿⣿⣿⣽⣿⣿⣿⣿⡿⠋⠉⢷⡖⣠⠄⠀⠠⢿⣿⣿⣧⠶⠋⠋⠙⠀⢈⣡⠄⠹⣿⠉⢉⣉⣀⣀⡀
⠀⠀⠀⠈⠛⠉⣿⣿⣿⠃⠈⠳⡶⢶⣞⣁⣽⣷⡄⢀⣠⠛⠻⣧⣦⣀⠀⠀⣠⠈⣧⠀⠸⣿⡟⠛⠋⠉⠉
⠀⠀⠀⠀⠔⢀⣋⣿⣿⣄⣠⣴⣷⢸⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⡟⢿⣿⣿⡿⢰⣿⣷⣶⣿⠃
⠀⠀⠀⠚⠛⠛⠛⢻⣿⣿⣿⣿⣿⢸⣿⣿⡏⢸⡇⠈⢻⣿⣿⣿⣧⣈⣿⣿⡇⢸⣿⣿⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠛⠻⠎⠿⣿⣷⠞⠁⠑⠀⠻⠿⠛⠻⠿⠟⠛⠟⠈⠛⠟⠻

⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⠀⣀⣴⣞⣡⡄
⠀⠀⠀⠀⠀⠀⡔⣿⣿⣮⢞⡵⣫⠝⠁
⠀⠀⠀⠀⢠⢖⡣⣞⡿⠁⢀⣜⣡
⠀⠀⠀⠀⣎⣿⣿⣿⣷⣠⠜⠻⡀⠀⠀⢀⡀⠀⠀⠀⠈⠁⢀⣀⡀
⠀⠀⠀⠀⠙⠿⣿⣿⠿⣇⠀⠀⠱⣄⠀⢉⡤⢤⣀⠀⢀⠞⠁⠀⢸⠂
⠀⠀⠀⠀⠙⠓⣼⣅⠀⠀⠑⢢⡄⠀⠙⠺⣆⠀⠈⠳⣼⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢠⡎⢻⣷⣄⠀⢀⣭⡓⢤⡀⠈⢣⡀⠀⠈⢳⡭⠥⠤⠤⠤⠤⠔⠒⠚⠉⠁⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠳⢿⡈⢮⣑⢮⡀⠈⠙⢿⣦⡀⠙⠲⢤⠤⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⢀⡤⢔⡛⠿⠆⠀⠀⠀⠃⠀⠀⢹⡉⠁⠐⠒⠒⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠘⠋⣉⠴⠚⢇⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠉⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠈⢆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠻⠻⡖⠚⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⢣⠀⠀⠙⠛⠓⠿⣶⡄⢀⣤⣶⠿⠛⣁⣸⠾⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠹⡭⠞⠋⠀⠀⠀⢣⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠧⣄⠀⠀⣾⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠷⢖⠤⣈⣋⢿⠢⣄⠀⠀⠀⠀⠀⠀⠀⠸⡁⢠⡼⠂⠀⠙⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⠔⣊⡥⠄⠒⠂⠀⠀⢲⡀⢱⠀⠁⠀⠱⢬⡑⢤⡀⠀⠀⠀⠀⠀⣈⡦⡇⠀⠀⠈⠳⡞⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣈⠳⢄⡠⢊⡵⠊⠁⠀⠀⠀⠀⠀⢰⡣⠷⠞⠀⠀⠀⠀⠈⢯⠓⢬⡒⠒⣒⡩⠥⠒⠊⠹⣤⣀⡀⠠⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠑⣒⣻⡀⠀⣀⠉⣉⣉⡹⠂⠀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡩⠕⠊⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣤⣀⣀⣀⣀⣀⣀⢤⠤⣴⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣻⠧⡔⠈⢋⠫⠏⠉⢖⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣍⠋⢀⢿⣾⣿⡛⠺⣧⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣷⣶⠷⠞⠿⢳⣾⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣀⡴⢩⠃⠀⠀⣠⠎⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⡀⠈⣛⣦⣀⡀⣀⣨⠇⣴⡉⠀⠀⢉⣑⣾⣖⠒⣒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣏⡫⢟⣽⡩⠊⠀⠀⠉⠒⠭⣓⠿⣏⡩⣟⣿⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠽⠷⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠟⠛⠛⠿⠟⠁⠈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⣤⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠃⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄⠀⠀⠀⠐⠒⠄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠀⠀⠀⠡⠁⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢹⡄⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⠀⠀⠀⠀⠐⠀⣄⣀⣀⣤⠊⠀⠀⡇⠀⠀⠀⣴⠃⠁⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⡿⠿⡿⢿⠀⠀⢰⡇⠀⠀⠀⢈⡆⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠲⣄⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣞⣀⣀⠀⠀⡼⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠁⢾⠀⠀⠀⠀⠘⠢⠄⣀⠀⠀⢀⣀⡠⠔⠛⢆⠀⢸⣉⣉⣁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⡈⠑⢄⣀⠤⠎⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⣇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠤⠐⠊⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢶⠶⠦⠴⠶⠶⠒⠚⠛⠉⠉⡗⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⠀⠀⠀⠀⠀⠀⣀⣠⣞⠁⢠⡟⠐⠢⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠦⣀⣠⠏⠙⠋⠉⠉⠀⠉⠑⣯⠝⢃⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣰⠋⠉⠀⠀⠀⠀⠀⢀⣴⣯⣵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢤⣉⠓⠶⢤⣀⠀⠀⠀⠈⠉⠛⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠛⢿⣦
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣿⡇⠀⠀⠀⠀⣿⣧⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀
⠀⠀⠀⠀⠀⢰⣿⠋⠀⠈⠉⢻⡇⠀⠀⢀⣴⡿⠋⠉⠀⠈⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⠀⢈⣿
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠘⣿⡀⢠⡿⠋⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠛⢷⣾⠃
⠀⠀⠀⠀⠀⠀⠛⣷⣦⣄⣀⡀⠙⣷⣿⣷⡾⠷⢶⣶⣤⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⣰⡿⠁
⠀⠀⠀⠀⠀⢀⣾⠟⠋⠉⠛⠛⠛⠛⣿⡟⣿⡄⠀⠀⠉⠛⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣼⠟
⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⣠⣾⠏⠀⠸⣿⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⣀⣼⠟⠷⣦⡾⠋
⠀⠀⠀⠀⠀⠈⢿⣦⣀⣠⣤⣾⡟⠁⠀⠀⠀⣿⣧⣤⣀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⢈⣿⠏⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢻⣇⠀⠀⠀⢀⣿⠏⠉⠉⠉⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⣾⠏⣠⣾⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣶⡿⠋⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⣼⣏⣴⠟⠁⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠿⠟⠁⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣾⡇⠀⠀⠀⠀⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣿⡀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠙⠿⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡀⠀⡀⠄
⠀⠀⣀⠀⠀⠂⢠⡔⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⠀⡀⢤
⠀⠈⢹⠀⠸⡇⠘⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢲⡘⠀⠞
⠀⠐⠀⠀⠀⠤⠄⣠⡁⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢴⡄⠈⠋⠠⠖⠐⢀⠖
⠀⠀⢣⠀⠀⠀⠀⠀⠁⡰⠶⠾⠀⠀⠀⠀⠀⠰⣶⠀⣀⠀⠐⣀⠀⠶⠄⠠⢆⠀⠀⠀⠀⠈⠓⢦⠀⢥⠀⠀⠀⢀⠻⢦⡀
⠀⠀⠀⠓⣤⡀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⠀⠀⠁⠀⠀⠞⠁⠈⣃⠀⠀⠀⠀⠀⢸⡀⠀⠙⠀⠀⡜⠁⠒⠛
⠀⠀⠀⠀⠀⠹⠂⠸⡄⠀⠀⠀⠀⠀⠀⢀⠰⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣐⠆⠀⠀⠀⢱⠄⢀⡠⠴⠁
⠀⠀⠀⠀⠀⠀⣧⠀⢿⡀⠀⠀⠀⠀⠀⣘⡷⠄⠚⠓⠂⠘⠛⠒⠤⣀⠀⠀⠀⠀⢞⠁⠀⠀⠀⡰⠃⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠙⡄⠘⣧⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠳⢄⠀⠀⠈⢓⠀⢀⡞⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢃⡀⢘⣄⣠⠏⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⠈⠆⠀⠀⣸⣄⠎⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⢰⡞⠉⠉⢹⠇⠀⠀⠀⠀⠀⠊⠁⠙⠀⠀⣀⡤⠚⡁⠀⠀⠸⠀⠀⣿⠏⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡎⠀⠀⠀⣤⣤⠤⠤⠤⠔⠒⠋⠁⣠⣼⡇⠀⠀⠐⠓⢶⣟⡀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡇⠀⠀⠀⢻⣿⣄⣀⣤⣤⠤⠚⠛⢻⡟⠀⠀⠀⢦⡤⠎⠀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢰⠀⠀⠀⠈⢿⣿⡿⠿⠇⠀⠀⣸⠋⠀⠀⠀⢠⠟⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢳⡀⠀⠀⠀⠙⠓⠀⠴⠖⠊⠀⠀⠀⢀⣴⠋⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⠢⣄⣀⡀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠙⠛⠒⠒⠢⠤⠤⣄⣀⣀⣀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠰⠒⠒⠂⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠉⠉⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢣⡀⠀⢀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠁⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣤⠤⠒⣇⣀⣀⠀⠀⠀⠀⠀⠀⢀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⡄⠀⠀⠉⠉⢹⠉⠉⠚⠟⠓⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠇⠀⠀⠀⠀⠸⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠀⠀⠀⠀⠀⠀⣧⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⣳⠀⠀⠀⠀⣠⠜⠁⠀⢠⡄
⠀⠀⠀⣀⠀⠀⢀⣤⠔⠊⠁⠀⠀⠀⢻⣀⠠⠔⠊⠁⠀⠀⠀⢸
⠀⠀⢘⠁⠄⢀⣈⣃⣀⣀⠀⢀⡠⠛⠊⠉⠣⠀⠀⠀⠀⡀⠁⢰
⠀⠀⠈⠁⠀⠤⠤⠤⠤⠤⠀⠿⣈⠀⠄⠀⠀⠰⠀⠤⠄⣛⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠄⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠬⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠰⢂⠍⠀⠩⠙⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢄⣤⠀⡜⡀⠀⠘⣀⠭⠤⡁⠀⠀⠀⡆⠀⠠⢦⠔⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠃⠊⠐⣠⠁⢀⠎⠀⠀⠀⠈⢢⠀⢸⡜⠀⢐⠀⠂⢘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢎⠉⠲⣿⠀⢸⠀⢀⣀⣀⠀⠀⠀⣸⣋⠔⠉⣱⠆⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠈⠢⣈⠉⠁⠀⠀⠀⣀⠔⠉⠀⡠⠊⠁
⠀⠀⠀⣀⡠⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢰⠀⠈⠡⣀⡠⠈⠀⠀⡔⠊
⠀⡰⠊⠁⠀⠀⠀⠀⢀⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠒⠦⢄
⣸⠁⠀⠀⠀⠠⠤⠖⣋⡤⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⣀⣀⣀⡀⢀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠱
⡇⠀⠀⠀⠀⢰⠀⠀⠀⠰⠂⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⠉⠲⡄⡠⠖⠒⠒⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣀⠳⣄
⢣⠀⠀⢰⠉⠻⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠃⠃⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⠈⢳
⠀⢣⡀⠀⠑⠂⠀⢠⡇⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣉⣉⣉⣀⣀⣀⣈⣉⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡀⠀⢠⠀⡇
⠀⠀⠑⠄⢠⡇⠀⠀⠀⠀⠘⢁⠇⠀⠀⠀⠄⠀⠎⣹⠷⢎⠀⠶⠀⠒⠐⠂⠸⠭⠂⠶⠠⡄⢠⡖⡄⠀⠀⠀⠀⠀⠀⠀⠀⣶⣽⣝⠀⠈⠁⡇⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠙⣁⣀⣀⡄⠠⠔⠊⠀⠀⠀⠈⠀⠬⠀⠈⠋⢁⡀⠠⣶⠀⠰⠆⠰⠦⠀⠠⠄⠀⠶⠈⠀⠀⠄⠀⠀⠀⠀⠀⠰⠉⢅⡎⠑⠂⣲⠑⠤⠶⠶⢯⡀
⠀⠀⠀⡰⠁⠢⠤⠤⠬⢣⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠐⠢⢀⠤⡀⢀⣀⠀⣀⠀⢀⡀⠀⠀⢘⡴⠿⡀⠐⡀⠀⠀⠀⠀⢰⠀⢊⠁⣠⠞⠁⠀⠀⠀⠀⠀⢣
⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⡀⠀⠀⠑⠒⠒⠁⠈⠉⠀⠈⠉⠀⠉⠁⠈⠋⠀⠛⠈⠦⣤⠔⠀⠰⡀⠀⠀⠀⠘⡆⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⣼⢃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣶⡄⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⢱⣾⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⣤⣶⠟⠋⠁⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠈⢿⠀⠀⣠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣇⣼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣿⣿⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠁⣸⣿
⠀⠀⠀⠀⠀⢿⣏⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠈⠙⢶⣤⠀⠀⠀⠀⠀⠀⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢴⡟⠿⠿⣷⡄⢀⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠲⡄⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣷⡤⠜⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢠⡇⠀⠀⠀⠃⠀⠀⠄
⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣀⠤⠶⠒⠒⢤⣄⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠠⣤⣀⠀⠀⠀⣰⠏⠀⠀⢀⠊⠀⠀⠀⠀⠀⡴⠀⠈⠢⡀
⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠛⠢⠚⠋⠀⠀⠀⠀⠂⠁⠀⠀⠀⣰⣃⣄⣠⠖⠛⢇
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠂⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠿⠿⣆⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠤⠄⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣥⣤⣤⣽⡧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⡏⠉⠉⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣏⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣟⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠒⠶⠶⠾⠿⠛⠻⠿⠿⠿⠿⠷⠶⠶⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⡿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡼⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠒⠛⠛⠛⠛⠛⠓⠲⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⣴⠶⣄⠀⣿⠀⢹⡆⢀⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠈⠙⣿⠀⠸⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡴⠛⠉⠛⠲⣤⡀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢀⣠⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⢠⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠚⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢹⡄
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⣷
⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢦⣀⠀⠀⣠⠏⠳⣄⠀⠀⢀⠿⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢸
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡧⣤⣤⠎⠀⠀⠈⠈⠉⠁⠀⠀⠈⠙⠓⢋⠀⠈⠳⠤⣤⣤⣤⠴⣞⠁⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⣀⣀⣸
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢠⠴⠛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠘⡆⠀⡤⠒⠋⠉⢶⡀⣿⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠛⠛⠉⠉⠁
⣿⠀⠀⠰⣄⡸⠂⠀⠀⠀⠐⢆⣠⠀⠀⢰⣄⣀⡴⠁⠀⢀⡇⠀⢀⣤⣀⠀⠱⠀⠀⠀⠀⠀⠀⢹⡼⠀⣠⣶⣦⠀⢻⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠸⡇⠀⠺⣿⡿⠀⢀⡇⠀⠀⠀⠀⠀⠈⣧⠀⠙⠿⠛⠀⣸⣿
⢿⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠲⣆⠀⢀⡇⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⣀⡞⠀⠀⠀⠀⠀⠀⠀⢹⠳⢤⣀⡀⠴⠋⢹⡄⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⣀⡀⣀
⠸⣇⠀⠀⠢⠀⠀⢰⠋⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠹⣧⠀⠀⠈⠉⠉⠉⢹
⠀⢹⡄⠀⠀⠀⠀⢸⠀⠀⠠⠽⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⡾
⠀⠀⠹⣦⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⣼⠁
⠀⠀⠀⠈⠳⣤⡀⠀⠈⠛⠲⠤⡤⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠚⠁⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣰⠇
⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡇⣼⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⣲⠶⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⣀⣤⠶⢦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⢉⡟⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⢀⡾⠋⠀⢀⡾⠁⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⢀⡾⢄⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⢸⠃⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⣸⠇⠀⠀⢠⡞⠁
⠀⠀⠀⢸⡆⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⢠⡟
⠀⠀⠀⠀⢷⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣠⡟⠉⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡼⣯⠀⠀⠀⠀⠉⠛⠶⣤⣤⣤⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠉⠳⣴⠏⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⠟⠉⠛⠶⣤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣆⣴⠟⠁⠀⠀⠀⠀⠀⠉⠛⠶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⢷⣶⠏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⣤⣤⣼⣿⣽⣦⣾⣿⣤⣿⣾⣧⣤⣤⡀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣷⣿⡿⠛⢻⣿⣄⣻⢿⣣⣼⡿⠟⢿⣿⣶⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣟⡿⠋⣠⣶⣿⣿⣿⣿⣫⣿⣿⣿⣷⣄⠙⢿⣛⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡟⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠸⢿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠓⢻⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣏⣵⣿⡟⠻⠃⠉⡀⢹⣿⣷⣝⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠛⠃⣼⣿⣿⣏⡉⠁⠉⠩⠈⢿⣿⣧⡀⠛⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠁⢀⣶⣸⣿⣿⠀⠀⠀⠐⠛⠀⠸⣿⣟⣰⣄⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣶⣦⠀⠀⠈⠉⠉⢹⡏⠉⠁⠀⢠⣤⣆⠀⣨⠉⠉⠁⠀⠀⣤⣶⣦⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠘⠿⠶⠤⢶⣿⣗⠚⠃⠀⠀⣀⡀⢀⣿⣷⢦⠶⠾⠇⠸⡟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⣘⢿⣿⣿⣿⣤⣴⣆⣚⣁⣀⣾⣿⣿⡿⣛⣠⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠿⣅⣸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣠⡿⢿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣬⣴⣿⣟⣳⣶⣿⣿⢛⣻⡾⡛⢿⣿⣶⢾⣻⣯⣶⣬⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⢾⣿⣿⣿⣻⣿⣿⠿⠛⢾⣿⢻⣷⠟⠿⢿⣿⣿⣻⣿⣿⣿⠖
⠀⠀⠀⢀⣤⣶⣶⣶⣤⡀⠀⠐⠛⠻⠿⠿⠟⠋⠀⠀⢀⢴⢯⡯⢶⠄⠀⠀⠈⠛⠿⠿⠿⠛⠂
⠀⠀⢰⣿⣿⣿⠿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈
⠀⠀⢸⣿⣿⣁⠀⠉⣿⣿⡇
⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠈⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠉⠉⠉⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠈⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⢀⣀⣛⣛⡛⠻⠿⢯⣛⠿⢦⠀⢦⣼
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠘⢁⣀⠀⠀⠚⠋⣉⣩⣭⣟⠛⠶⠀⠈⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢠⣿⣡⠴⠀⠀⠸⠟⣦⣀⣀⣠⣴⣶⡄⠀⠀⠀⠀⣠⡶⠶⣤⠀⣿
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢠⡟⣡⠾⠋⠀⠀⠀⠀⢉⣩⢴⣿⡽⠂⠀⠀⠠⡾⠋⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢸⠃⣠⠆⠀⠀⣀⣾⣯⣶⣯⡵⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢻⣇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣴⠶⣠⣾⠃⠀⠀⠀⣉⡵⢞⣽⠏⠀⠀⠀⠀⠀⠀⠀⠿⠶⡶⠿⠦⠀⠙⢷⣄⠀⠀⣠⣄
⠀⠀⠀⣀⡴⠿⠀⠀⣠⢴⡟⠀⢸⠇⡿⠀⠀⠀⠀⠉⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⡟⠉⠻⢶⣤⡶⢶⣄⣀
⠀⠀⣾⠋⣰⠟⢡⣾⡟⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⠀⠀⠀⠀⠙⠷⢾⣧⣽⡿⣦⠀⠀⣠⠶⡆
⠀⠸⣏⠀⠿⠀⡿⠘⠇⠀⠀⠰⣄⡀⠀⠀⠀⠲⣦⣀⣀⣀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢿⣶⣞⣡⠞
⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣦⣤⣴⢿⣯⣩⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⣙⠶⢤⣤⣤⠶⠋⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢧⣤⣤⣤⡤⠤⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡖⢀⡟
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠙⠻⠶⠖⠂⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠿⣤⡌⠻⠓⢶⣦⣤⣀⡀⠀⠀⠀⢠⡏⢀⡾
⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⣴⠏⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡅⠀⠀⠈⢻⡄⠀⠀⠓⠀⠈⣹⠟⠲⠦⣼⣁⡾⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⣶⣶⠶⠛⠁⠀⣀⣼⣄⣠⣄⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠈⣿⠆⠀⠀⢀⡾⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠉⠛⠉⠈⠙⢷⣶⣶⣤⣴⣟⠉⠙⠛⠓⠶⠶⠶⠶⠿⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠄⠶⢦⣤⣄⣀⠀⠀⠙⠿⠟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠉⠙⠻⠶⣤⡀⠀⠸⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⣴⠶⠛⠂⠀⠀⣀⠀⠀⠈⠙⢦⠀⣤⣿⣧⡀⠀⣀⡠⣄⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠴⠋⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⢶⣶⡉⣿⠟⢋⣡⡏⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⠗⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⠀⠀⠈⢻⣽⡏⠀⡈⢻⡿⠳⠶⢦⣤⣤⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⣿⠛⠁⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠛⣅⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⣼⠂⠀⠀⠀⠀⢿⣿⣋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣗⠀⠈⠙⠷⣦⡀⠀⠀⢀⣴⣿⠋⠀⠚⠁⠀⠀⠀⠀⠀⠈⢿⡉⠉⠛⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠛⠛⣷⠀⠈⠙⠆⣴⡟⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠙⠳⢶⡶⣦⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⣰⠿⡅⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⣼⡃⠀⠀⠀⠀⠀⣼⠃⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢿⣀⡞⠁⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠉⠉⣉⢉⣭⡍⢩⣭⡉⣉⠉⠉⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣴⣆⣿⣬⣿⣇⣸⣿⣴⣿⣰⣦⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢸⡇⠈⢻⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⡟⠁⢸⡇⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⡇⢸⡇⠀⠀⢻⣿⣿⣿⡏⣡⣿⣿⣿⡟⠀⠀⢸⡇⢸⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠘⣷⡈⠿⣦⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⣴⠿⢁⣾⠃⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠈⠻⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠟⠁⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡄⣾⣿⡆⠀⠀⢰⣿⣷⢠⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢠⣄⡀⠀⠀⠀⠀⢠⣜⡿⠹⠏⠻⠿⠁⠀⠀⠈⠿⠟⠹⠟⢿⣣⡄⠀⠀⠀⠀⢀⣠⡄⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠙⢿⣦⠀⠀⠀⠶⣡⣶⣾⣿⣿⣷⠀⠀⠀⠀⣾⣿⣿⣷⣶⣌⠶⠀⠀⠀⣴⡿⠋⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢻⣧⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡆⠀⠀⣸⡟⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⣿⡇⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢸⣿⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⣿⡇⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢸⣿⠀⠀⠀⠘⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠃⠀⠀⠀⣿⡇⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡘⣿⠀⠀⠀⠀⢹⣿⣿⣿⣷⡀⠀⠀⢀⣾⣿⣿⣿⡏⠀⠀⠀⠀⣿⢃⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡀⠀⠙⠿⠿⠟⠁⠀⠀⠈⠻⠿⠿⠋⠀⢀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⠿⠿⢟⣛⣛⣭⣭⣭⣭⣝⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢗⡲⣶⣿⣿⣿⣿⠫⣵⡤⣶⠝⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣮⣵⣿⣿⡿⠿⢿⣿⣮⡃⠫⠾⠇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣛⣫⢍⢠⣚⠿⢸⣿⣿⣿⣮⡻⢏⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣝⡋⠾⠇⢟⣣⣿⣿⣿⠿⠿⣿⣿⣿⣀⠀⣤⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⢟⣼⣿⣿⣿⡤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⣿⣿⢰⣭⡝⠻⠿⣿⣿⣿⣿⠛⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣮⣣⣿⣿⣌⢵⡞⣴⡹⣶⢎⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣤⣭⣥⣶⣿⣿⡿

⠀⢀⣤⣀⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣿⣿⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣦⣤⣶⣦⣶⣶⣶⣶⣶⣶⣤⣤⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠉⠉⠛⢛⡉⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣦⣾⣿⡌⢀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣷⡏⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠉⠉⢛⣿⣧⡀⠀⠈⣽⣿⣿⣿⣿⣿⣿⣿⡆
⢸⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⢻⣿⣧⣤⡄⠹⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⣤⣸⣿⣿⣿⣿⣿⣿⡇
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢺⣿⣿⣯⣄⡀⢻⣿⣿⣿⣿⣷
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⣴⣿⠿⠛⠛⠿⠶⠀⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⡿
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠉⣤⣶⡶⣤⣄⠀⠈⣿⣿⣿⣿⣿⣷⢺⣿⣿⣿⣧
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠃⠠⠄⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢈⣿⣿⡿⣿⡿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣾⣿⣿⡷⠛⠁⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⣼⠉⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⡇⢠⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣽⣏⣴⣿⣿⡿⠿⣿⣿⣿⣿⣿⡿⠤⢬⠛⣿⠇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠻⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣤⣾⡿⠁⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣤⣤⢻⣿⣿⣿⣿⣧⣤⣤⣤⣶⣾⣿⠟⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡄⢁⣽⡿⠟⠋⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢇⣻⡀⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣷⣀⢻⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢻⣿⣿⣿⣿⡟⣿⣿⣿⣿⣦⠀⢾⣆⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣄⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠘⠻⣿⣿⢿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⢿⣿⣿⣿
⠀⠀⢰⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⠀⠀⠀⠙⠛⠀⠘⠿⢿⣿⣿⣿⣿⣿⣼⡗⢻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠶⡦⠤⠄
⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⢉⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣤⣿⣷⣿⣿⣿⣿⣿⣿⡟⣿⠁
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⡟
⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠀⠉⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⣫⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⣡⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⢉⣡⡴⢛⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣠⣾⣿⣿⣿⠿⢛⣋⣩⣥⣤⣶⣶⣶⠶⠿⠛⠛⢉⣴⡟⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⣾⣿⣿⣿⣽⣶⣿⠿⠟⠛⠋⠉⠉⢀⣀⣠⣤⣶⡾⠟⠉⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⣿⣿⣿⡿⠟⠋⠁⢀⣠⣤⣴⣶⡿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⠟⠉⢀⣠⡄⠈⣩⡥⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠶⢤⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⢀⣴⣿⣿⠀⠀⣀⣀⢀⠀⠉⠙⠃⠀⠀⠀⠀⠘⠋⠉⣀⣀⣀⠀⣿⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⣴⣿⡟⠈⢿⡄⠰⣿⠛⣿⣿⡟⠦⠀⠀⠀⠀⠀⠠⢾⣿⣿⠛⡿⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⣾⣿⣿⣷⡀⠀⠁⠀⠈⠓⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠊⠁⣸⣧⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣼⢻⣿⣿⣿⡟⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⣿⣿⣿⣿⣿⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠤⠴⠀⠀⠀⠀⠀⣸⣿⠟⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢷⢸⣿⣿⡹⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⣀⣠⣄⣀⣤⣀⡀⠀⠀⣰⡟⠁⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠈⢧⡙⢿⣿⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠹⢯⣍⣉⡴⠋⠀⢀⣾⠋⢀⣴⡿⠟⠁⠀⠀⠀⢠⡞⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆⠈⢻⣷⣶⣾⣷⣿⣿⣿⣇⠙⢷⣤⣀⠀⠀⠈⠁⠀⣠⣶⣿⠃⢠⣿⠋⠀⡠⢤⣄⠀⢀⣿⢡⣿⡟⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⢻⣷⡹⣿⣿⣿⣿⣿⠀⠀⠉⠻⢷⣶⣶⣶⣾⣿⣿⣿⠀⢸⣯⠀⠀⠀⠀⢹⡇⠸⣿⣼⡟⠀⢀⠀⠀⢸⣿
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⢻⣷⢹⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠉⠉⠸⣿⣿⣿⣧⠀⠻⣷⣤⣀⣤⣿⠃⠀⢿⣿⠁⣠⣏⠀⠀⣿⣿
⠀⠰⣌⠙⠻⣦⡀⠀⠀⠀⠀⠀⠘⢿⣿⣧⠀⢿⡇⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⣌⡉⠛⠉⠀⠀⠀⠀⠻⢠⣟⡿⠀⠀⣿⡿
⠀⠀⠘⣷⡀⢙⢿⡆⠀⢰⡄⢀⠀⠘⣿⣿⣧⢸⣧⠸⠟⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⣯⣙⠛⠿⣿⣶⣌⡓⠶⣄⠀⣼⣻⡇⠀⢰⣿⠁
⠀⠀⠀⠙⢿⣌⠪⡻⣆⠸⡷⠋⠀⠀⠘⣿⣿⡌⡏⠀⠀⠀⢀⠶⠋⠉⠉⠛⢶⣄⠀⠀⠀⠙⢦⣙⢿⣿⣿⣦⣄⠉⠻⣿⣆⠀⠀⢿⡿⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠙⠳⠮⠳⢄⠁⠀⠀⠀⠀⢻⣿⢻⡇⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠈⠓⢌⡻⣿⣿⣷⡄⠈⢿⣇⠀⠘⢁⠴⠋⣠⣴⡶⠿⠿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⢸⣿⢸⡇⠀⠀⠀⠁⠀⣠⠂⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠙⠮⡻⣿⣿⡄⠀⣿⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⢠⠖⠈⠀⠀⠀⠀⠀⠀⢀⣿⣿⡸⣿⡀⠀⠀⠀⣼⠃⠀⣠⣴⡟⠀⠀⠀⠻⣆⠀⠀⠀⠠⢄⣀⠙⡌⢿⣿⡀⢹⠆⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠀⠹⣷⡀⠀⢰⡏⠀⣴⣿⡟⠁⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠉⠳⣬⡘⣿⡇⣿⠀⠀⢀⣀⣀⣀⣀⠐⢤⣤⣤⠾⠋
⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⠀⠀⠈⠻⣄⠈⠇⢰⠿⠋⣀⣤⠤⠤⠄⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⢷⡜⠃⠃⢀⣦⠭⠭⠭⠉⠻⢷⣦
⠀⠀⠀⠀⠀⠀⠙⠻⠶⠾⠟⠛⠉⠀⠀⠀⠘⠻⠿⠿⠶⠦⠁⣶⣦⢀⣈⣀⣀⣀⠀⠀⠀⠀⠈⣣⡴⠖⠒⠄⠀⠀⠀⢿⡀⠀⠀⠙⠛⠿⠿⠶⠶⠆⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠓⢀⣤⠉⡅⢲⣬⣙⠛⠛⠛⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⣠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⢸⣿⡄⠻⣿⣷⣄⠀⠀⠀⠀⢹⣿⣄⡀⠀⣀⣠⣾⠃⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠀⢸⣿⣧⠀⠈⠻⢿⣷⡀⠀⠀⠀⠙⠿⠿⠿⠟⠋⠁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠁⠀⠀⠸⣿⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣻⣿⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠌⠄⠀⠀⠀⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢤⡀⢀⣤⡀⢀⡤
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣄⣴⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⡇
⠀⠀⠀⠀⠀⠀⠈⠋⣿⠀⠀⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠐⠒⠒⠒⠉⠉⠉⠉⠉⠉⠉⠉⠐⠒⠒⠲⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⠒⠲⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠻⢟⣉⡉⠉⠉⠉⠉⣑⣢⣬⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀
⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠒⡽⠋⠙⠲⠦⣤⣭⣿⣽⣧⠥⠷⠞⠛⢟⠳⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⢁⣤⠞⠀⠀⠀⠀⠀⢹⢠⠀⠃⡇⠀⠀⠀⠀⠈⠳⣄⠈⠙⢿⡷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿⣻⠋⠀⢰⡏⠁⠀⠀⠀⠀⠀⠀⢸⠘⠀⡃⡅⠀⠀⠀⠀⠀⠀⢈⡷⠀⠀⠙⢮⣿⣷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠘⡇⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠁⠁⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣾⠁⠘⣇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠸⠀⢸⡄⠀⠀⠀⢓⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡜
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡻⡀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠸⡁⠀⠄
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠦⠀⠀⠁⢣⠀⠀⠀⠘⣆⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⡇⠃⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠸⢸⡆⠀⠀⠀⠸⡄⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢠⠹⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢳⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠆⠀⠀⢰⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⡀⠀⢀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⢣
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢃⠀⠸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢸⠀⢐⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⡀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⣄⡬⠿⠤⡆⢠⣤⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠛⢢⣄⣠⠃⠀⠀⠀⠀⠀⠀⠀⠉⢡⡅⡈⢙⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢀⡠⠔⠚⡃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⣀⣷⠋⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠈⢵⠀⠀⠗
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡖⠊⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡛⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡾⢇⡼⠁
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⢭⣍⣙⣛⡓⠞⠷⠶⣶⠶⢖⠖⠶⠞⠛⠛⡛⣉⣥⠴⠞⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉
⡛⣒⣒⡒⣒⣒⡒⣒⢒⡒⣲⢒⣶⣶⠶⠶⢶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣀⣠
⣿⣿⡟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⢀⡤⠔⠒⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢠⡄⢀⡀⠀⠀⢀⣿⠃
⠀⠁⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⢈⠇⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⡄⠀⠀⣀⡀⠀⠀⢄⠈⠢⡀
⠀⠎⠀⠀⠀⢰⠓⢦⡀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣧⡀⠀⢀⣠⣴⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣠⠞⢉⡿⠀⠀⠀⢳⠀⠹
⠀⠀⢠⡴⠤⣈⢳⡀⠳⡄⣾⠁⣠⠟⠀⠀⠀⠀⠀⠀⠀⣴⠟⣉⡀⠀⠀⠀⠀⠈⠉⠀⠉⠙⢷⣄⠀⠀⠀⠀⠀⢀⡴⠒⠒⢦⣟⠈⠙⠃⣴⢛⣀⣀⠀⠀⠘⡇
⠀⠀⠀⠱⢦⡈⠳⠟⠁⣹⠿⢰⡃⠀⠀⠀⠀⠀⠀⠀⣸⠇⠚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣦⡙⣧⠀⠀⠀⠀⠘⠷⠒⠛⡦⠈⠻⣦⠀⠘⢋⣠⡼⠃⠀⠀⡇
⠀⠀⢀⡤⠴⠨⠗⠀⡾⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡏⢧⣀⠀⠀⠀⠀⠀⠀⢀⣴⢿⠁⠸⣇⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠘⠆⠀⠛⢤⡀⠀⠀⢸⠇
⠀⠀⠈⠑⠒⢲⠖⠀⠁⠀⠀⠰⢏⠀⠀⠀⠀⠀⠀⢸⡟⠀⢸⡇⠀⠙⠓⠶⠤⠤⠴⠖⠛⠁⢸⡀⠀⣿⡀⠀⠀⠀⠀⠀⢀⡽⠂⣤⣠⡴⠟⠓⠲⠃⠀⠀⠉
⠀⠀⠀⠀⠰⡥⠞⠛⠶⠶⠞⢧⡀⠳⣄⠀⠀⠀⢠⡞⡛⠂⠸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠍⢳⠀⠀⠀⣰⠏⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⣄⠀⠈⢳⡶⠀⠀⣿⣿⣷⣤⣄⣀⣀⣀⣀⣤⣴⣿⡇⠀⢲⣴⠟⠀⢠⡞⠁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠈⠳⣄⠈⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣗⢀⡴⠋⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠳⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣷⠞⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣿⠀⠀⠀⢹⡟⠿⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣇⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⣡⡿⠃⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣄⠀⠙⠳⠶⠶⠞⠋⠀⡀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⡀⠀⠈⠳⣦⣄⣀⣀⣠⣴⠞⠁⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠳⠦⣄⣀⣉⠉⠉⠉⢀⣀⣀⣤⠴⠚⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⣠⣤⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⢠⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣠⢦⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠢⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡇⠘⣇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣧⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢿⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⡟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠙⣆⠀⠀⠀⠸⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⢀⣠⠶⠛⠙⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠳⣄⠀⠀⢠⠟⣻⠻⠶⠶⢶⣶⣶⣶⣶⡶⠶⠾⠟⠻⣏⠻⣄⢀⡴⠛⠁⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠈⠳⠶⠋⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣈⠋⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢀⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠛⠋⠁

⠖⠋⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣷⠀⢰⣿⢼
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣟⣯⣽⣿⡿⠏⠀⠀⢿⣿⠇⠀⠀⠁
⣠⣶⣿⣦⠀⠀⣴⣿⣿⡿⣿⣿⡿⠀⣀⣤⣶⣶⣶⣤⣀⡀⠀⠀⠻⣿⠃⠀⣿⣿⣿⣿⣿⡟⢠⣾⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉⢀⣠⣤⣤⣀⣀
⣿⣿⡿⣿⡇⣼⣿⣿⠋⣼⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠉⠀⢰⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⣿⣷⣄
⣿⣿⢡⣿⣿⣿⡿⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣤⣴⣶⣦⣄⠈⠙⠿⣿⣿⣿⠃⢠
⣿⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠉⠁⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢿⣿⣶⣾⣿⣿⣷⣶⣄⡀⠀⠉⠛⠿⠿⠿⠿⠟⠁⠀⢀⣀⣤⣄⡀⠀⠸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⡁⠀⠀⠾⣿⣶⣦⣄⡙⢿⣿⣄⠀⠀⢼⠿⠟⠋⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⠆
⠛⠻⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣦⣄⠀⠈⠙⢿⣿⣦⣽⣿⣦⣄⠀⢠⣤⣤⣀⠀⠀⠀⠀⠙⠿⣿⣽⣿⣿⣿⡟⠁⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠛⠋⠉
⣄⠀⠘⢿⣿⣿⣷⣦⣄⣈⠙⣿⣿⡿⠀⠀⣠⣄⠈⠙⠛⠿⠿⠟⠀⠀⢻⣿⣿⣿⣶⣦⣤⣀⣀⡀⠉⠉⠛⠉⠀⠀⠀⢀⣴⣾⣿⣿⡿⠋⢀⣠⣴⣾⣿⣿⣿
⠏⠀⠀⠀⠈⠛⠿⡻⠟⠟⠙⣿⣿⡇⠀⠀⢹⣿⣆⠀⠀⠀⠀⣠⣄⡀⠀⠙⠿⣿⣿⡿⠟⡻⢾⣯⣿⣦⡀⠀⠀⣴⣾⣿⣿⣿⣟⣫⣤⣾⣿⣿⣿⣿⣷⣶⣿
⠀⠀⢀⣀⣀⣀⡀⠀⠉⠓⠛⠟⠛⠀⠀⠀⣼⣿⣿⡆⠀⠀⢈⣿⣿⣿⣷⣄⠀⠀⠉⠛⠂⠐⠀⠾⠟⠋⠁⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠉⠉
⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠇⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢿⣿⣿⠇⠀⣼⣿⣿⣿⡇⣿⣿⣿⣷⡀⠀⠀⢀⣴⣿⣿⣿⣿⠋⠀⣾⣿⣿⣿⠿⢿⣿⣶⣦⣤⣴⡤⠀⢀⣾⣿
⣿⣿⣿⠟⢋⣽⣿⣿⡿⠋⠉⠀⠈⠉⠀⢀⡀⠉⠉⠀⢰⣿⣿⣿⣿⡇⣿⣿⣿⣿⣧⠀⠀⣾⣿⣿⠿⠋⠁⠀⠀⠀⠉⠙⢿⣿⣦⣌⣉⣩⡵⠋⠀⢠⣿⣿⣿
⣿⡿⠁⢠⣿⣿⣿⣿⠀⠀⠀⢀⠀⠀⠀⣾⣷⣄⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡧⠀⠉⠀⠀⠀⣴⣶⣶⣶⣤⣄⠀⠀⠉⠛⠛⠋⠁⠀⠀⢀⣾⣿⣿⣿
⡿⠁⠀⢸⣿⣿⣿⡇⠀⠀⢠⣿⣧⠀⠀⣿⣿⣿⣆⠀⢸⣿⣿⣽⣿⣧⠈⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿
⠁⠀⠀⣿⣿⣿⣿⠁⠀⠀⢸⣿⣿⡄⠀⢸⣿⣿⣿⡀⠈⢿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣦⣄⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠀⠀⣠⣾⣶⡄⢼⣿⣿⣿⣿⣿
⠀⠀⣰⣿⣿⣿⠏⠀⠀⠀⣿⣿⣿⡇⠀⠀⢻⣿⣿⠇⠀⠀⠈⠉⠉⠉⠀⠀⠘⢿⣷⣽⣿⣿⣿⣿⣷⣦⠀⠻⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⡀⠉⠉⠉⠁
⠀⠸⢿⠿⠟⠋⠀⠀⠀⣸⣿⣿⣿⠃⠀⠀⠀⠙⠛⠀⠀⠀⣠⣴⣶⣶⣶⣤⡀⠀⠙⢿⣿⣿⣿⣿⡿⠿⠃⠀⠀⢹⣿⣿⣿⣿⣿⣿⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡏⠀⠀⣠⣤⣤⣤⣄⣀⠀⠈⠉⠉⠻⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⢸⣿⣿⠃
⣀⣀⣀⣠⣤⣴⣿⣿⣿⣿⡿⠋⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠈⠃⠹⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⣿⣿⡟⣠⣶⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠘⣽⣿⡂⠀⠀⠀⣴⣶⣶⣦⣄⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿
⠈⠛⠛⠛⠛⠋⠉⠀⠀⠀⢀⣠⣤⣤⣤⣄⣀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣧⡀⠀⠀⠈⠹⠇⠀⢠⣴⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⡿
⣠⣴⣿⣶⣶⣤⡀⠀⠀⢰⣿⣿⣿⠿⢿⣿⣿⣿⣶⣤⣀⠀⠙⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠺⣿⡋⢻⣶⣤⣈⡙⠿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠟
⣿⣿⡿⠟⠛⠿⠟⠀⠀⠘⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣷⣦⣈⡛⢿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠉⠓⠤⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⢀⣠⣾
⠟⠉⠀⠀⠀⢀⣀⣀⡀⠀⠀⠙⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠙⢿⡿⠋⠁⠀⣤⣾⣿⣿⣿
⣠⣴⣤⣾⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠛⠛⠉⠁⠀⢀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣶⣄⠈⠉⠛⠋⠉
⣿⣿⣿⠿⢛⣯⠿⠿⠛⠛⠻⢿⡷⠀⠀⠀⢠⣶⣾⣿⣿⣶⣦⣤⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇⢀⣤⣶⣿⣿⣿⣿⣷⡄⠀⠀⣿⣿⡆⠀⠀⢀⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠋⠈⠉⠚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣁⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠓⠀⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⠁⠢⠀⠀⡀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠴⢊⠉⡑⠂⠌⠳⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢇⠈⡀⢀⠁⠠⢤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠉⠈⠓⠲⠂⠀⠃⢛⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠤⠀⠈⠂⢀⡞⡠⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡤⠆⠀⠀⢠⠁⠐⢄⣎⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠺⡤⠀⠀⠀⠀⡈⡺⠐⣄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢥⣤⣀⠔⠁⢀⠐⢁⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠀⠉⠈⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⢆⠉⠂⠦⢀⠀⠀⠀⠀⠢⠄⣁⠠⠐⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣀⡁⠈⠀⣖⠀⠀⠘⡎⠳⠀⠀⠀⠀⠀⢀⠠⠄⣀⢣⡀⠀⠀⠱⡀⠀⠀⠀⠀⡆⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⢀⢄⣐⠁⠀⢀⣀⠂⢼⠄⠀⠀⢃⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠃⣸⣧⠑⠘⠢⠄⡬⠁⢨⠈⠁⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⡠⠏⡀⠀⢢⡢⠀⠀⠀⠁⠘⣁⡤⡠⠋⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠑⠢⢄⡈⣙⠁⠭⠀⠡⠲⣀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡉⠂⠀⣠⠊⠐⢈⢲⠀⠈⠀⠈⠀⡔⠂⡆⡀⠀⠀⠜⠃⠀⠪⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡨⠗⢒⠀⠈⠉⠀⠀⠀⠀⣔⠀⣀⠌⣂⢬⢀⡚⠠⡐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠤⠖⠋⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠠⠂⡀⠀⠀⠈⢆⠀⢰⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠲⠉⠠⡴⠁⢀⡀⠀⡈⠀⠠⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠐⠂⠀⢀⠜⠈⡴⢏⢁⠸⠍⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⡆⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄