Outbreak Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⡀⠀⠄⢠⠔⠀⠀⠀⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠄⠀⠁⠀⠀⠒⠬⠒⠂⠀⠀⠧⠀⠀⠄⠀⠈⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢤⠀⠀⢀⣀⠣⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢈⡖⠠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠢⡀⠈⢡⠀⠀⠓⠐⠦⠃⠄⠈⠐
⠀⠀⠁⠀⠀⡴⠃⠀⠀⠠⠉⢳⡐⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠘⠄⢃⢀⡁⠠⠀⠀⠋⡘⠢⠀⢀
⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠄⠀⠈⠀⠀⠄⠀⠀⠀⢈⠂⠀⠉⠤⠢⠀⠀⢑⠰⡄⡀⠀⠈
⠀⠀⠠⠀⠄⠙⣠⠄⢀⣀⡰⠊⠀⠀⠀⠀⠀⡀⠀⡀⡠⠀⡄⡠⠂⠈⠀⠃⠀⠀⠠⠐⠀⠀⠀⠐⠴
⠀⠀⢅⠀⠀⠀⠁⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠉⠠⠀⠀⢄⠎⠠⠐⢀⠌⠀⠀⠰⠁⠀⠤⠀⠀⠀⡀
⠀⠀⠀⠁⠐⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠎⠺⡿⠀⠉⠀⠀⢈⡙⠀⠀⠀⠘⠂⡀⠅
⠀⠀⢀⠀⠀⠄⠀⢑⠠⠀⠀⠀⠐⣤⡀⠀⠫⠄⠀⠀⠀⣌⠐⠀⠀⠀⠀⢀⠈⠁⠀⠀⠠⠄⠈⢀⣀⠒⠢⠂
⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠁⠀⠀⠀⠉⠈⠈⢛⠔⣀⠞⠊⡀⠀⠀⠀⠠⠀⠘⠀⠀⠀⠀⠀⠂⡠⠊
⠀⠁⡀⠀⡁⠀⠀⠠⠄⠐⠀⠀⢀⠀⡆⠐⠈⠁⢦⠙⢁⠔⡈⡁⠀⣠⠃⠀⠀⠀⠀⠀⠀⡡⠊⠙⡤⠲
⠀⠀⠀⡀⢀⠀⠀⢠⡈⢂⠐⢃⡀⠄⠽⠄⠀⣀⡠⠆⠀⠀⠁⢀⠁⠡⠈⠀⠀⠀⠀⠀⠋⠀⠔⡠⠀⠀⠄
⠀⠀⠀⠠⠠⠁⠀⠆⡀⠂⠔⠁⠀⠀⠅⢀⣀⡐⠀⠠⠅⠀⠀⠠⠨⣄⡐⡀⠀⠀⠀⠀⠈⠁
⠀⠀⢀⠐⠢⠐⠀⠀⠁⠁⠠⠀⡀⠐⠀⠀⠘⠢⢄⠁⠐⠀⡠⠁⡀⠊⠀⠁⠢⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠤⠀⠀⠀⠁⠀⠐⠀⠠⢀⠀⠐⠀⠑⠄⠀⠀⠈⠀⠀⠂⠀⠀⠔⠆⢠⡀⠀⠀⠈⡀⠈⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠠⡀⠂⠁⡐⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡀⡛⠁⠂⠀⠉⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⣓⠂⠁⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⡃⠀⠀⠀⠐⠈⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠠⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⡐⠔⠄⠀⠀⠖⠀⠀⠎⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠑⠀⠀⠀⠀⠐⠃⠀⠀⠀⠈
⠀⠀⠀⡄⠀⠁⢀⠔⠊⠀⠀⢤⠀⠀⠀⠀⠉⢀⠀⢄⠀⠒⠉⠀⠀⠀⠀⠀⠁⠒⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠆⠀⠡⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐⠉⠀⠁⠀⠀⠀⠀⠐⢂⡠⠀⠀⠀⠜⠂⠀⠀⡀⢀⠠⠀⠀⠀⠂⠄⠀⠂⠐⠀⢀⠀⠀⠀⠉
⠀⠀⠀⠀⠠⠁⢠⠀⠀⠀⠎⠑⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠁⠐⠂⠁⠀⠂⠀⠀⠁⠀⠐⡀⠀⠀⠀⠀⠀⠁⠂⠀⠀⢉⠀⠠
⠀⠊⠁⠑⠄⠐⡁⠀⠤⠔⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠈
⠀⠐⠄⠔⠀⠉⠀⠑⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢠⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠊⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠈⠁⡀⠚⠀⡀⠒
⠀⠀⠈⠀⠀⠀⠔⠂⠀⠀⠉⡄⠀⠀⠉⡦⡂⠈⠀⠀⠠⣀⠀⡠⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⠐⠠⠠⠀⠁⣀
⠀⠀⠀⠂⠂⡀⠄⢀⠀⠀⠀⠃⠀⠀⢸⣛⡻⠄⠀⢠⡶⢶⡶⡶⣶⢶⢶⡶⣶⡄⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠊⢀⡠⠀⠒⠒⠐⠀⠀⠀⠻
⠀⠈⠃⠑⠆⠀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣡⣟⣧⣧⣡⣿⣧⣿⡇⠀⠀⠀⠀⠑⣄⠀⠀⠀⠈⠀⠀⠀⠈⠀⠤⠀⡀
⠀⠀⠀⠀⢒⠃⠀⠀⠠⠋⠘⠄⠀⠀⠀⣶⠄⠀⠀⠀⣍⠉⣉⣩⡁⣥⣭⡈⣤⠀⡄⣤⣤⣤⠁⠀⠉⠀⠀⠀⠃⠀⠀⠀⢀⣀⠀⠐⠃⠉⠀⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢿⣸⡇⣽⡇⣿⣼⡧⣿⠀⣿⠹⢦⣍⠀⠀⠀⠠⠀⠠⠦⠀⢀⠀⠠⠄⠀⠀⣀
⠀⣦⡤⢤⣤⣸⠀⠀⠈⠑⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢻⡇⣿⠸⣆⢿⣤⡟⢸⣦⣿⠀⠀⠀⠀⠂⢀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠈⠈⠁⠀⠀⠀⠀⠂⠀⠀⣀⠉⢀⡀⡀⢀⢠⢀⣀⡀⡀⣀⠀⠀⠐⠠⠄⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠈⠀⠠⠄⠇⠀⠸⠸⠬⠁⠷⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡴⠀⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠤⡀⠸⡀⠀⠀⠀⠀⠀⢸⡀⢀⣀⠑⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⡀⠈⠁⠀⠀⠀⢱⠀⡀⠀⠀⠀⠀⠀⠸⠗⠀⠊⠀⠀⠀⣄⡄⢠⠀⠀⡀⠀⠀⠀⠀⢀⢜⠁⠀⠀⠐⠀⠀⠀⠀⢸⠀⠀⠂⡀⠀⣀⠀⠀⠀⠂
⢰⠀⠂⠀⠀⠀⠀⠁⠀⠀⢀⢀⣄⠠⠀⠀⠀⠀⠀⡀⠠⠏⠀⠀⠀⠡⢔⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⡀⠀⠀⠀⡄⢸⠀⠀⠈⠄⠁⠐⠁⠀⠁
⠀⠆⠀⠀⡠⠀⠀⠀⠁⠦⢙⠆⠀⠈⠂⠀⠀⠀⠀⠀⢨⠂⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⡈⠁⠚⠀⠀⠀⠈⠀⢀⣀⣀⣀⣀⣈⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠈⠳⡀⠑⠨⡠⠟⠀⠀⠀⠀⠀⠂⠐⠷⠐⡒⠜⠀⠀⢀⠀⠀⠀⠀⠀⠠⠂⠂⠀⠡⠠⠀⠀⠁⠂⠀⠀⠘⠁⠉⣩⡈⠉⠃
⠀⢀⠀⠀⢠⠀⠀⠀⠀⠠⠋⠑⠝⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈
⠀⠈⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠄⠀⠔⠀⠀⠀⢠⠁⣠⣀⠉⠀⠀⠀⠠⠀⢀⢀⠀⢤⠀⠀⠀⢰⣌⣀⣈⣀⣡⡄
⠀⣿⢹⡆⣿⣋⠀⢰⠻⠄⡞⢻⡇⢸⣟⡃⣾⣛⠀⠀⠀⢀⠀⠃⠀⠁⢀⠀⠀⠀⢄⠙⢋⡐⠀⠀⠀⠀⠀⠂⠁⠀⠐⠁⢰⠀⠀⠐⠀⠒⠀⠂
⠀⣿⢿⡄⣿⠉⠀⢘⢳⡄⡟⢺⡇⢸⡏⠁⣿⠉⠀⠀⠀⠐⢂⠂⠀⠀⠂⠀⠀⠀⠈⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠨⠀⠈⠀⠅
⠀⠙⠋⠀⠉⠉⠀⠈⠋⠁⠁⠈⠁⠈⠁⠀⠙⠋⠀⠀⠀⠀⠁⠀⠀⠀⠐⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠴⠀⠠⠄⠄⠠⠆
⡿⠉⢹⡟⠻⠿⠿⠟⠿⡛⠿⠻⠿⠿⠟⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠐⠒⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠒⠒⢢⢤⣄
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢀⣀⠠⠤⠐⠒⠒⠈⠉⠉⠉⠉⠉⠉⠉⠒⡊⠑⠁⣿⠀⠀⡐⠉⠙⠙⠲⡄
⠀⠀⠀⠀⠀⠀⡆⠀⠀⡿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⢽⡇⠀⠇⢀⠤⠤⢤⠇⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣗⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠉⠳⡄⢀⠔⣻⡇⠀⠀⢁⣀⣀⣀⠀⠀⠀⠀⠀⠰⠉⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⠀⡇⠀⠀⠀⢀⣿⢁⣠⣿⠷⠚⠛⠉⠉⠉⠉⠙⠛⠲⠦⣀⠾⡀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⢸⡀⠀⢾⠀⠈⠢⢤⠐⠋⠈⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⢈⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢰⠒⠒⠚⠁⠀⠈⠋⠙⢹⡄⢣⡀⠀⠀⡥⠀⠀⠀⠀⢠⠞⠉⠉⣦⠀⠀⠢⢄⣠⡾⠿⢦
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠟⠀⠀⠀⠘⠀⠂⠰⠀⢀⢰⠖⠓⠛⢡⡿⠓⠐⠋⠀⠀⠀⠀⠀⣇⠀⠀⠠⡾⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢘⠀⠀⠀⠀⠀⠀⠐⣃⣀⣼⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠐⢠⠀⢱⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡠⣄⢸⡇⠀⠀⠀⢀⡤⠶⢤⡀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⡀⠀⠀⠀⠐⣇⣀⣠⣼⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⢀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠊⢻⡇⠀⠀⠀⡏⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢸⡄⠀⠀⠀⡏⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠙⢦⡀⠀⠀⠀⠀⠀⠑⠤⠗⢸⡇⠀⠀⠀⠃⠠⠤⠊⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀⣠⠞⠀⣐⣠⣲⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⢈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢺⡂⣀⠀⠀⠀⠀⠀⠀⣒⣒⣾⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠈⠢⡀⠀⠀⠀⢀⣼⡿⢷⠀⠀⠀⠠⡀⠀⠀⠀⠨⡀⠀⠀⣩⠀⠀⠀⠀⢀⣀⣀⣠⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠀⠀⣠⠞⠋⠀⢈⡧⠊⠀⠀⣸⠀⠀⠀⠀⠈⠒⠛⠁⠀⠀⠀⢀⡏⠀⠈⠙⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡞⠁⠀⠠⠊⠉⠉⢢⣤⣾⡁⠀⠀⠀⠀⠀⢠⠀⢐⠀⠀⠀⣀⡽⠓⠶⣴⠋⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠉⡀⠄⠂⠀⠀⠐⢦⣀⣤⣼⣿⣿⣿⣶⣦⣤⣄⣀⠜⠀⠀⠱⡖⠋⠁⠀⠀⠀⠸⣄⣀⣠⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠉⠉⠙⠛⢦⣀⣤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⠙⠂⠀⠙⠿⠿⢶⣆
⠀⠀⠀⠀⠀⠻⠫⢿⣷⣄⠀⡓⠂⠀⠀⠀⡠⠈⠁⢀⣀⣤⣑⣄⡀⠀⠀⢼⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠠⠠⣮⢁⠽⣿⣿⣿⣿⣿⣦⣼⣿⣿⡛⠇⠀⠀⠀⠀⠀⠀⠀⠠
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠁⣠⣒⣶⣶⣶⣵⣾⣿⣿⣿⣿⠈⡻⣿⣿⠯⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⡤⠆
⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⠊⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠾⠏⠁
⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢰⡗⢦⣆⣀⣀⣀⣠⣿⣿⣆⣤⡽⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠒⠙⣩⡴⠛⣹⣿⣿⠟⣿⠛⠿⣿⠿⢆⣀⣘⠛⠿⣿⣛⣫⡵⠿⠋⠁⠀⠀⠀⠀⠀⢀⣠⡤⠶⠄
⣤⣤⣤⣤⣤⣤⣄⣠⣤⣈⣧⣤⣤⣴⠟⣉⡴⣾⣿⣿⣟⠐⠿⣦⢠⣶⣶⣞⠟⠉⠉⢻⢿⡏⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠉⠁⠀⠀⢀⣀⣴⣧⣞⣉⠃⠈⢱⢬⣶⣔⡚⢡⣿⣿⣗⡁⡈⠉⠚⢿⣿⢏⣴⡞⣠⠫⠿⡿⠍⠾⠿⠟⠱⠝⠖⠑⠒⠱⣤⠂⠀⠈⠉⠀⠛
⠀⠀⠀⠀⣤⣶⢿⡏⠓⠦⠀⠀⠀⠂⠨⠿⢿⣾⣿⣿⣿⠄⠀⠈⠹⢿⡿⢟⠑⣼⠿⢀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢦⣾
⣤⣶⡀⠰⠛⠁⠀⠀⠀⠀⠐⠀⠀⢀⣀⣤⣶⣿⣿⣷⡾⢷⣄⠀⢀⠀⠀⣡⣿⣿⡤⠀⠀⠈⠁⠀⠐⢄⠀⠒⠠⠄⢀⣤⣤⢀⣀⣳⣄⡀⢸
⡉⢩⣿⠆⠀⠀⠀⠀⣠⣄⣨⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⢾⣿⣿⡶⠖⠹⣿⠋⠯⡥⣄⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠁⠀⠸⠯⡽
⣿⣾⣿⣶⡀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⡽⠉⣻⢟⣾⠙⢷⡦⠀⠀⠹⠆⠀⠀⠈⠈⠙⠳⠶⢲⢠⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⠩⠀⠀⠘⠀⠈⠁⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡔⠁⠀⠀⠑⠀⠀⠀⠀⠰⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢰⠄⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠠⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠂⠀⠀⠑⡂
⢠⠊⠈⠂⠐⣁⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠂⠈
⠀⠑⠒⠁⠀⠀⠁⠀⠀⢀⡀⠀⠔⠀⠀⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠠⣀⡀⠄⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⣶⢶⡀⠀⣠⣭⣤⣭⣤⣤⣤⣤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠴⠄⠀⠀⠀⠷
⠀⠺⠈⠂⠁⠀⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⣪⣿⣷⣷⣽⣧⣿⠀⠀⠠⡈⢠⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀
⠀⠀⠀⢈⠁⠀⠀⢪⠨⠀⠀⠀⠰⠆⠀⠀⢨⡉⡍⣭⢩⡤⣍⣤⢠⣤⡶⣦⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠣
⠀⡀⠀⠀⢠⠀⠀⡸⣀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡇⣿⢸⣿⡏⣿⢸⣏⡟⢷⠀⠀⠈⠀⠈⠃⠀⠀⠀⠀⠀⠠⠄
⠀⠓⠓⠛⠚⠀⢀⠄⠒⠒⠀⠀⠀⠀⠀⠀⠀⠛⠁⠛⠘⠃⠛⠙⠛⠁⠛⠛⠠⣠⠄⠀⠐⠄⠀⠀⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠨⠿⠈⠸⠿⠹⠸⠼⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠠⠆⠀⠠
⠀⠀⣀⠀⠀⠲⡄⠃⠀⠀⠀⠀⢠⣇⠠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⣡⠑⠂⠐⢀⠀⠀⠀⠀⠀⢱⡦⠀⠀⠀⠀⠀⠀⠀⠄⠀⡂⢐
⠀⡀⠀⢠⠀⠀⠀⢠⡺⠍⠉⠢⠀⠀⠀⠀⠈⡄⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠼
⠀⠀⠀⠘⠀⠀⠀⠰⡄⠀⢄⠰⠀⠀⠀⠀⠀⠀⠂⠠⠆⠁⠀⣀⠀⠀⠀⠀⠠⠀⠀⠈⠈⠀⠀⠐⠀⠀⠛⠉⣩⡉⠙
⠀⢀⡀⣠⠀⠀⠀⠀⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠀⡐
⠀⡤⡅⢠⡤⠀⣠⡀⣠⣤⡀⣠⡄⢠⣤⠀⠀⠀⢠⠆⢰⠀⠀⠀⠀⠰⡦⢀⠀⠀⠀⠀⡠⠗⠘⡆⠀⠀⠷⠲⢒⠒⠲
⠀⣧⡇⢸⠒⠀⠿⡍⣷⢼⡇⣿⠂⣽⠒⠀⠀⠀⠄⠀⠀⠌⠀⠀⠀⠂⠄⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⡂
⠀⠓⠃⠘⠓⠀⠛⠋⠋⠘⠀⠛⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡾⠿⠿⠿⠿⠿⠿⠷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄
⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣷⣄
⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠘⢿⣦⡀
⠀⠀⢀⣾⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠛⠛⠿⣷⣤⣀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠻⣷⡀
⠀⢀⣿⠏⠈⢹⣿⠿⠿⣶⣶⣶⣶⣶⡾⠿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣴⣿⣋⣤⣤⣶⣶⣶⠿⠿⠿⣿⡀
⠀⣾⡟⠀⠀⣼⡟⠀⠀⢸⣿⠀⠀⠀⠀⣰⡿⠁⠀⢀⣴⣶⣶⣦⣀⠀⠀⢀⣼⠿⠛⠻⠿⣿⣯⡉⠻⣷⣆⠀⠀⢹⣷
⢸⣿⠁⠀⢰⣿⠃⠀⠀⢸⣿⠀⠀⢀⣼⡟⠁⠀⣰⣿⠋⠀⠀⠙⠻⣷⣶⡿⠋⠀⠀⠀⠀⠀⠹⣿⡄⠈⢿⣇⠀⠀⣿
⣾⡏⠀⠀⣾⡏⠀⠀⠀⣼⡟⠀⠀⠈⢿⣶⣶⡾⠟⠁⠀⠀⠀⢀⣴⡿⠻⢷⣦⡀⠀⠀⠀⠀⠀⢹⣷⠀⠘⣿⡀⠀⢹
⣿⡇⠀⢰⣿⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠸⣿⡀⠀⣿⡇⠀⢸
⣿⡇⠀⣾⡏⠀⠀⠀⣼⡿⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⢶⣦⣀⠀⠀⠉⠻⣷⣄⠀⠀⣿⡇⠀⢸⣿⠀⢸
⢿⡇⢸⣿⠀⠀⠀⣰⣿⣧⣄⣠⣾⠟⠛⢿⣦⣠⣶⣿⣯⣄⢀⣀⠀⠀⠙⠻⣷⣄⠀⠀⠈⣿⡇⠀⣿⡇⠀⢸⣿⠀⣸
⢸⣿⣿⣇⣀⠀⣴⣿⣧⣼⣿⡟⠁⠀⠀⣠⣿⡿⠋⠀⠉⣿⣎⡻⢿⣦⣀⠀⠈⠻⢿⣶⣾⣿⣧⣾⠟⠀⢀⣾⡏⠀⣿
⠀⢿⣏⠙⠛⠿⠟⠀⠉⠉⠻⣷⣤⣤⣾⡿⠋⠀⠀⣠⣾⠟⠛⣿⡆⠉⠻⢷⣦⣤⣾⡟⠀⠛⢿⣶⣤⣴⡿⠟⠀⣸⡿
⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⢈⣽⡿⠻⣷⣄⣠⣾⠟⠁⢀⣼⡿⣷⣤⣤⣾⠟⠉⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⣰⣿⠃
⠀⠀⠘⢿⣆⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠈⠙⠛⠿⣷⣾⠿⢿⣶⠿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠈⢻⣷⡀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠙⢿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣶⣶⣶⣶⣶⣶⣶⣶⠿⠿⠛⠋⠉
⣿⣿⠿⢿⢻⠿⡿⢻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⣀⠞⡃⠀⠀⠴⠀⠀⠆⠁⠂⠀⠀⠀⠀⠒⠀⠀⠀⠈⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠠
⠀⠀⠀⡄⠀⠁⢀⡔⠊⠀⢈⢠⠀⠀⠀⠀⠀⢀⡐⢤⠀⠚⠈⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠘⢀⡀
⠀⠀⠀⠀⢀⠀⠡⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠉⠠⠁⢄⠀⠀⠀⠀⠂⡠⠀⠀⠀⡸⠃⠀⠀⢡⠂⡀⠄⠀⠀⠛⠀⠀⠒⠀⠀⠀⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠠⠀⢠⠁⠐⠀⢎⠑⡄⠀⠀⠀⠌⠂⠀⠀⠐⡀⠀⠀⡀⠀⠀⠀⢐⠒⠀⠀⠀⠀⠀⠀⠀⠈⢒⠂⠀⠄⠀⠀⠈⡐⠀⠀⠨⡁⠀⠔
⠀⠎⠁⠁⡄⠑⣃⠀⠤⠔⠊⡠⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠋⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⢂⠀⠁
⠈⠳⠠⠔⠀⠈⠀⠑⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⢀⠦⠂⠀⠀⢄⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠈⠀⠀⡀⠉⢠⠀⠒⢀⠐⠂
⠀⠀⠄⠀⠀⢀⠌⠢⠀⠀⠉⡂⠀⠀⠈⠦⢆⠈⠀⠈⠀⣄⢀⡠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡆⠀⠀⢠⡀⠊⠄⠀⠌⢀⡀
⠀⠀⡀⡈⠒⡁⠀⢀⠀⠀⠀⠁⠀⠀⢸⣛⡩⠇⠀⠀⣴⢶⡶⣶⢶⢶⣶⡶⣶⡆⠀⠀⠁⢠⡄⠀⠂⠁⠀⠀⢀⠄⠁⣀⠄⠘⠓⠃⠂⠂⠂⠐⠇
⠀⠈⠇⠉⢆⠀⠀⣀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠄⠀⣿⣡⣟⣧⣻⣼⣜⣧⣿⡇⠀⠀⠸⡀⠐⢤⡀⠀⡀⠄⠁⠀⠀⢀⠀⠠⠄
⠀⠀⢀⠀⢒⠃⠀⠀⠠⣓⣘⡂⠀⠀⠀⢴⡄⠀⠀⠀⣉⠉⣉⢉⡉⣉⣭⣍⢩⡀⣤⢠⣤⣤⠀⠀⠉⠀⠀⠀⠉⠀⠀⠠⠀⢀⡀⠤⠧⠀⠐
⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡟⢸⡇⣿⣤⡿⢸⡇⣿⠘⢷⣬⡁⠀⠀⠐⠄⠀⠴⠀⠀⠀⠀⠤⠀⠀⢀⡀
⠀⣦⣤⢤⢤⢺⠀⠀⠐⠉⠑⠀⠀⠘⠠⠤⠤⠐⠀⠀⠘⣿⡇⢸⡇⣿⠘⣧⢸⣧⣿⢸⣧⣼⡇⠀⠀⠀⠰⠀⡀⠀⠈⠀⠒⠀⠐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠉⠀⠉⠀⠀⠀⡀⠒⠀⠂⢀⠉⠀⣈⣀⢈⢠⡀⡀⡀⢀⢀⠀⠀⠀⠂⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁
⠀⠀⠐⠀⠄⠀⠀⠀⠡⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠁⠐⠼⠧⠠⠸⠷⠇⠇⠿⠼⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⠆⠀⠀⣀
⠀⠀⠀⠀⠀⡀⠤⡄⠸⡀⠀⠀⡀⠄⠀⠰⡇⢀⣀⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠄⠤⠐⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠈
⠀⠄⠈⠁⠀⠀⠀⢱⠀⡀⠀⠀⠀⠀⠀⠸⠷⠀⠖⠀⠀⠀⢄⣂⢈⢄⠀⠀⠀⠀⢀⠀⠄⡴⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢀⠀⢀⠀⠂⠀⠐
⢠⠀⠀⠀⠀⠀⠀⠁⠈⠄⢀⣔⣀⡀⠀⠀⠀⠀⠀⠠⢀⠜⠀⠀⠀⠉⢴⠀⠀⠀⠈⠀⠀⠈⠚⠂⠀⠀⠀⠀⠀⢰⠀⠀⡅⠀⠠⠂⠐⠊⠀⠇
⠀⠆⠀⠀⢠⠀⠀⠀⠁⠢⢙⠆⠀⠈⢆⠀⠀⠀⠀⠀⢐⡃⠀⠀⠀⠀⡬⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠻⠀⠀⠀⠀⠀⣀⢀⣀⢀⣀⡀
⠀⠀⠀⠀⠚⠀⠀⠀⠈⠱⡀⠑⠨⡁⠖⠀⠀⠀⠀⠀⠂⠀⠱⠠⢖⢄⠉⠀⢄⠂⠀⠀⠀⠀⠀⠆⠐⠀⠈⠔⠕⠀⠉⠑⠀⠀⠘⠋⠉⢩⣍⠉⢹
⠀⢀⠀⠀⢠⠀⠀⠀⠈⠠⠈⠉⠎⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⢀⠐⠂⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠁⠀⢸
⠀⠈⠓⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡤⠀⡄⠀⠀⠀⠀⠈⢁⣌⠉⠀⠀⠀⠀⠌⠀⣐⡀⠤⡀⠀⠀⠠⣤⣁⣈⣁⣈⣴
⠀⣿⢹⡆⣿⣛⠀⢰⠿⡆⡾⠻⡇⢰⣟⡃⢸⣛⠂⠀⠀⢀⠂⠃⠀⠋⢀⠀⠀⠀⢀⡙⠋⡀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠃⠀⠀⠀⠁⠂⠀⠂⢐⠘
⠀⣿⢿⡄⣿⠉⠀⢘⢳⡄⡿⠚⡇⢸⡏⠁⢸⠉⠁⠀⠀⠠⠂⠀⠀⡀⠢⠀⠀⠀⠀⠑⠂⠁⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠀⠀⠀⠅⠀⠁⢨⠐
⠀⠛⠛⠀⠉⠉⠀⠈⠛⠁⠁⠀⠃⠈⠁⠀⠘⠛⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠰⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⢤⠀⢠⠀⠁⠀⢀⡄⠠⡄⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠫⢻⡰⢿⠔⢰⠆⢎⣷⢂⠏⠟⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⢀⢘⡿⠀⠀⢙⠈⠀⠈⠃⠐⠸⠋⠀⠀⣿⢁⠠⠊⢂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠕⠨⢎⡠⠛⠁⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢀⡧⡅⡨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠅⠢⠀⠀⠈⠝⠀⠀⠀⠀⠨⠄⠀⠀⠀⠐⡡⠀⠀⢀⠀⡌⢄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠐⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡑⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠀⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢀⠠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣨⢁⢁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡨⡅⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠠⠆⠀⠀⠀⠘⠸⠀⠀⠀⠀⠀⠀⢐⡂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⡀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠨⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠠⠤⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠳⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠘⣀⣠⠀⢀⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢃⠀⢀⣀⡈⠪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡘⠀⠁⠊⠀⠀⠀⠀⠀⠀⢖⡂⠀⠀⠀⠀⠀⠀⠃⠃⠀⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⡋⠈⠉⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⡏⠂⣵⠢⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢂⠚⠇⠠⢒⠒⡆⡼⢲⠂⡀⡜⠃⡰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⠘⠀⠱⠆⠘⠃⠈⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠘⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣠⠤⣄⢀⡤⢤⡄⢤⠀⢠⢤⡄⣤⠤⣄⠀⠀⢀⣤⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣔⠋⠉⣃⣾⣀⠀⠀⠀⠀⣏⠀⣀⢸⡇⠀⣿⠘⣧⡟⢸⡇⣿⠀⣹⠠⡤⠈⣿⠀⠳⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⡎⠙⢏⡸⠋⢱⠈⠙⠄⠀⠀⠈⠛⠁⠀⠙⠋⠁⠀⠉⠁⠈⠁⠉⠉⠁⠀⠀⠀⠉⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⣼⣧⠠⠋⠀⠀⠀⠀⠀⠂⠆⠀⠠⠸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠠⠀⠇⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠁⠈⠓⠢⠀⠀⠀⠀⠀⠄⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠐⠀⠀⢠⠩⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠈⠀⠀⡊⠀⠀⠈⠁⠀⠀⠀⠐⠂⠑⠀⠀⠀⠀⠀⡀⠀⠀⠰⠀⠀⢀⢀⠀⠀⠀⠆⠀⠠⠀⠀⠀⠀⠀⠒
⠀⢀⠀⠀⠀⡄⠀⠀⡖⠤⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⢒⠀⠀⠀⠀⠈⠀⠐⡂⠀⠀⠀⠈⡐⠀⠀⠅
⠰⠁⠀⠂⡈⢄⡐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠈⠉⠀⠀⠀⠁⠀⠀⠠⠀⠐⠀⠀⠀⠈⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠢⠀⠀⢂⠁⠀⠢⢄⠐⠀⠁⢀⡀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡀⠁⠀⠁⣀
⠀⣀⢀⠂⠄⠠⠀⠀⠀⠀⠀⠨⠭⠇⠀⢰⡶⣶⢶⡶⢶⡶⣶⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠠⠀⠘⠀⠀⠀⠀⠛
⠀⠈⠀⢡⠄⠀⠀⡄⣀⠀⠀⠀⡀⠀⠀⠸⠷⠿⠷⠷⠿⠿⠿⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠐⠀⢒
⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠈⠉⠀⠀⠘⣇⡿⣾⢸⣟⡧⣿⢸⠸⣏⡇⠀⠀⢀⠀⢀⠀⠀⠀⡀⠀⠉
⠀⣆⣤⣤⡖⠀⠐⢓⣂⡀⠀⠀⠀⠀⠀⠀⢿⡇⣿⢸⡟⣧⣿⣼⠸⣧⡿⠀⠀⠀⠀⡀⠀⠂⠐⠀⠀⠈⠉⢀⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢤⠄⢠⣤⢠⢠⢠⠀⠀⠁⠃⠁⠀⠈⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠈⠈⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⣰⠀⣀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢈⡄⠁⠀⠀⠐⠞⠂⠀⡖⠉⢱⠈⠁⢰⠊⠑
⠀⠀⠉⠀⠀⢈⠀⠀⠀⠀⠀⠀⠛⠀⠂⠀⢀⠠⠄⢠⠀⠀⠀⢀⠈⠧⣩⣉⠀⣈⣉⢙⡆⠹⣤⢚⣤⣠⣌⣇⣴
⠀⠀⠀⠀⠀⠀⠀⡀⢴⠒⠰⡀⠀⠀⠀⠀⣈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠀⠀⣄⠀⢀⡇⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠸⠀⠀⠀⢮⠀⠀⠀⡅⠀⠀⠀⠀⠁⠤⢠⡄⠚⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠲⠒⢒⡒⠒
⠀⡀⠀⣠⠀⠀⠀⠀⠑⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠁⡀
⠀⣬⡉⣁⡀⠀⣀⠀⣀⣀⢀⣀⢀⣀⡀⠀⠀⢀⡆⢰⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⢀⠴⠐⢢⠀⠀⠴⠤⠤⠤⠴
⠀⣧⡇⣿⠂⠐⢯⡁⡧⣿⢸⡷⢸⡷⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠢⠴⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂
⠈⠓⠃⠛⠂⠀⠛⠋⠃⠛⠘⠃⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⡇⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣰⣧⡀⠀⠀⠀⢡⣤⣼⣿⣶⣦⣤⣴⡇⠀⢀⡀⠀⠀⣄⠀⠀⠀⢠⣶⡆⠀⠀⢠⠄⠀⢀⠀⠀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⠶⠿⣿⣿⠉⠀⠉⣿⣿⣍⣲⣄⣀⡤⠞⠛⠂⠀⠀⠀⠈⠙⣆⣀⣀⡦⣤⣴⣤⣾⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠀⣸⣿⣤⣴⣿⣿⣦⣤⣴⣿⡿⠟⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⣿⢋⣿⣿⣿⣧⣬⣿⣶⠉⠀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⣀⠀⠀⣸⡿⠿⣿⣿⣿⣿⣿⡟⠉⢻⣿⠉⠁⠉
⠀⠀⠀⠀⠀⠶⠀⠲⣧⣿⣿⣿⣿⣿⣧⣤⣾⣿⣧⣄⣤⣿⣿⡁⠈⠀⠀⠀⠙⠉⠉⢻⣧⣀⣼⣿⣧⣼⣿⣷⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣈⣿⣿⡿⠿⣿⣿⣿⣿⡏⠈⣿⡏⠉⠑⠾⠀⠀⠀⠰⠆⠙⣯⣿⣿⣿⣿⠋⠉⠙⣿⣫⠋⠉⠛
⠀⠀⠀⣀⣀⣠⡴⠞⠛⠻⢿⣿⣿⠀⠀⣸⣿⣿⣿⣿⣿⠋⠀⠲⠀⠀⠀⠀⠀⠀⠀⣠⠟⠻⢿⣤⣽⣶⣤⡾⠿⢧⠈
⠀⠀⠀⠘⢿⠋⠀⠀⠀⢠⠊⠙⠻⣿⠿⠿⣷⠾⠿⣿⡁⠐⠀⠀⠀⠀⠀⠀⠐⣿⡟⠁⠀⠁⢠⠉⢹⠉⣡⠀⠀⠀⢹⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣇⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠆⠀⠀⠉⠀⠀⠀⢈⣿⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣼⣷⣭⣿⣦⣵⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠺⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢀⡈⢻⡀⢺⠂⢰⡇⠀⠀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣈⣣⣼⣷⣾⣿⣿⣷⣶⣼⡃⢰⠶⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢦⡀⠀⣼⣿⠿⣿⣿⣿⣅⣈⣿⣿⣿⣿⣷⣤⠞⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢙⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣏⠀⢹⣿⣶⠛⠃⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣈⣉⣿⣿⣿⣿⣿⣿⡿⠻⢿⣿⣿⣿⣿⣷⣿⣿⣿⡖⠛⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣉⣿⣿⠉⣿⣿⣿⣀⠀⣸⣿⣿⠟⠿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⣁⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣠⣿⡟⢻⣿⡋⣉⠙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢩⡬⢿⣿⣿⡋⠉⢻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣯⡈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⢿⣷⣶⣾⣿⣿⣏⣉⣿⣿⣿⡿⢯⡀⠙⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⣰⠋⢻⠿⣿⡿⡿⠿⣿⠋⠉⠳⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠿⠂⣿⠐⠷⠀⢹⡀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⣀⡀⠘⢿⣿⠟⠀⣀⡀⠀⢰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣧⡀⠘⢿⡟⠀⢸⣿⠀⠘⣿⠟⠀⣸⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⣤⣄⠀⢻⣷⣀⣨⣷⣶⣾⣿⣶⣾⣧⣀⣴⡿⠃⢀⣤⣄⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣷⡀⠈⠛⠻⣤⣴⣿⣿⣿⣿⡏⠀⠀⠈⠙⣿⣿⣿⣷⣤⡼⠛⠋⠀⣀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⣶⣤⣴⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣴⣿⣿⣿⣿⣿⣷⣤⣴⡿⠟⠋⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠈⣿⡿⠋⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⡇⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⠛⠛⢻⣿⠀⠀⢀⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⠀⠀⠘⣿⣿⠛⠛⠛⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣶⣶⣶⣾⣿⣦⣤⣾⣿⣿⡟⠁⠀⠀⠈⢻⣿⣿⣿⣿⣄⠀⢠⣿⣿⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠋⠁⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠙⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⠉⠛⢿⣿⣿⣿⣿⣿⠿⠛⠉⠉⢻⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣶⡿⠛⠉⠀⣨⣿⣿⣷⣤⣾⣿⣿⣿⣿⣏⠀⠀⢀⣠⣾⣿⣏⡀⠈⠙⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⠿⠁⠀⠰⣿⡿⠀⢨⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣏⠀⠸⣿⡷⠀⠀⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⠏⠀⣰⡏⠈⢹⣿⠉⠙⣧⡀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠘⠿⠟⠀⢸⣿⠀⠘⠿⠟⠀⠀⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⠀⠀⠿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡿⠟




⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⢒⠄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡄⠀⠘⠀⠁⠀⢀⠀⡀⢀⣀⡀⡂⢰⠄⢁⠘⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠑⢠⣀⠀⠔⠒⠈⠓⠘⠁⠉⠉⢍⣡⠃⠒⠮⢅⡔⠠⠠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢀⡀⠀⠈⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⠢⣏⠑⠀⠰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠌⠸⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠨⠀⠄⠀⠀⠉⠀⠀⠀⠑⢆⠀⠉
⠀⠀⠀⠀⠀⠀⣎⠀⠀⣠⠤⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⡘⠁⠀⠀⠠⠥⠀⠀⠀⠀⠀⠀⠰⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠐⠉⠑⡄⠔⠠⠄
⠀⠀⠀⠀⠀⠀⠈⢙⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠄⠀⠀⠈⠚⠉⠀⠀⢉⠄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⢸⠁⡀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⠀⠀⡀
⠀⠀⠀⢀⡠⠄⡆⠀⡠⠀⠀⠀⠀⠒⠂⠀⠀⠀⠠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢎⡠⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠈⡇⠀⠑⠁
⠀⠀⠀⠀⠀⣀⡇⠀⠀⢀⡀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠨⣥⠄
⠀⠀⠀⠀⠀⠧⠇⠀⠀⢆⡩⠴⠀⠀⠀⠀⠀⠀⡎⣱⠐⠂⠀⠀⠀⠀⠐⠊⠀⠀⠀⠓⠀⠀⠀⠀⣟⠆⠀⣀⠀⠀⠦⠤⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡔⠲⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠇⢐⡆⠒
⠀⠀⠀⢠⡡⠀⠘⡀⠀⠀⠀⠀⠀⠲⣀⠤⠀⠀⠀⠀⢀⠀⠀⠈⠉⠀⠀⠀⠣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜
⠀⠀⠀⠀⠀⠀⠀⣳⡀⢈⢀⠀⠀⠑⠃⠀⠀⢠⡐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⣼⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠉⠀⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⣼⣼⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠙⢶⣀⢀⠀⢴⠂⡀⠀⠀⠀⠀⢠⠀⢇⠀⠀⠉⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣠⠾⠁⠀⣌⠇
⠀⠀⠀⠀⠀⠀⠙⠎⠀⠀⢠⠛⣽⣦⡈⠉⠁⠀⠀⢀⡰⠀⠉⠁⠀⡁⢀⠀⠀⠀⠋⠁⠀⠐⠂⣠⡾⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣹⠿⣦⣤⡴⡅⠈⠓⠚⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⣀⣼⠟⠁⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⡉⠀⢀⡧⠤⢻⡷⢦⣤⣄⣀⣠⣤⡄⠡⣤⡶⠖⠛⠙⠦⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠧⠔⠀⠘⠀⠀⣄⣁⠀⢸⠀⠉⠈⠀⡛⢀⠀⠀⠦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠃⠀⠀⠀⢀⣀⡀⠀⠀⠀⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠙⠛⠛⠛⠲⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠈⢹⠉⠉⣻⠛⠛⠛⡟⠛⠛⢿⠛⠛⢻⠛⠛⢻⠛⠛⢹⡏⠉⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢾⣦⣼⣄⣀⣽⣀⣀⣠⣇⣀⣀⣼⣀⣀⣸⣀⣀⣸⣄⣀⣼⣄⣤⣧⠾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠧⠤⢤⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣁⡠⠿⣆
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣼⠷⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠘⠛⠛⣶⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣷⠀⠀⠀⠀⣴⠟⠛⠃⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡏⠀⣼⠋⠉⠉⠉⡏⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⠟⠀⠀⢿⣤⣤⣤⣤⣿⣤⣤⣤⠾⠁
⠀⠀⠀⠀⢀⣤⠶⠶⢶⣤⣤⣭⣍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡴⠶⢶⣤⡀
⠀⠀⠀⠀⣟⠀⠀⠀⡼⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠛⠁⠈⢹⣧⠀⢀⡼⠛⢶⣄
⠀⠀⠀⠀⠙⢦⣄⣀⣳⡀⠀⠀⢀⣰⠟⢀⣀⠀⠀⣿⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠙⢦⣠⡾⠋⠳⣮⡀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢛⣻⣷⠞⠋⠉⢻⡄⠹⣦⡀⢠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢿⡀⠀⠀⣠⡾⠋⠀⠀⠀⠈⠻⢦⣄⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠈⢧⠀⠀⣠⠟⠀⠈⠻⣧⡀⠀⠀⠀⠀⢹⡇⠀⠀⠈⠛⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣀⣼⠷⠛⠁⠀⠀⠀⠀⠈⠻⢦⣀⣀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠚⠑⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢀⣄⠠⠀⠄⠀⠦⠬⢠⠁⡠⠀⠀⠐⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠈⠉⠐⠄⡀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠡⡄⠀⠂
⠀⠀⠀⠀⠀⡄⠀⢀⠦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⣀⠃⠀⠀⠐⠄⠁⠀⠀⠀⠀⠊⠤⠀⠀⠀⠀⣀⠀⠀⠀⠀⢂⠀⠀⣈⠉⠱⠀⠐⠃
⠀⠀⠀⠀⠀⢠⠌⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⢐⡀
⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠠
⠀⠀⠀⠈⠁⡅⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⣓
⠀⠀⠀⠀⠨⠇⠀⠠⣈⠀⠀⠀⠀⠀⠀⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠖⠀⠀⠀⠀⠬⠀⣀
⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⡗⠂
⠀⠀⠀⠖⠀⢈⡀⠀⠀⠀⠀⣁⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠳⡄⠢⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡰⠇⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⡸⣦⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢑⡿⠃
⠀⠀⠀⠀⠀⠲⠎⠀⠈⠛⣲⣄⠙⠂⠀⠀⠀⠄⠐⠊⠀⠀⠀⠀⠀⠒⠀⠀⠠⢀⣴⠛⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡽⠷⣤⡤⣁⡈⠂⠁⠀⠀⠑⠈⠀⠀⠀⢀⣠⠿⠋⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⢀⠜⠒⢹⠗⠶⣦⠤⢤⠤⣀⡰⢶⠛⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢀⣀⠀⠐⠆⠀⠸⠀⠀⠀⠈⡠⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⡀
⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣶⣶⣶⣶⡶⠶⣶⣶⣶⣶⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⡿⠿⢰⣇⠸⠿⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣇⣛⢻⡿⢛⣃⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣘⣃⣸⣿⣿⣿⡇⠀⠀⣿⡴⠖⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣡⣴⣾⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣧⡴⠶⠶⠶⠶⣿⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠙⠳⠶⠶⠦⠴⢿⣿⣿⣿⡿⠟



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠪⢒⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠊⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠀⠀⠐⠒⠶⣶⣄⡀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠉⠻⢷⡂⠤⠒⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⢼⡇
⢀⡄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢸⣦⡀⠀⠀⠀⠀⢀⡠⠛⠲⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⡈
⡟⠁⠀⠀⠘⡄⠀⠠⠄⠀⠀⠀⠀⣻⠀⠈⠳⡀⠀⢠⠔⠁⠀⠀⠀⠀⠙⢄⢀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠁
⢹⡄⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢳⣤⣶⣷⣦⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁
⠀⢹⡄⢰⣄⢤⣼⡄⠀⠀⠀⠀⠀⣸⣽⡄⠀⣀⢀⠆⡶⣄⡀⢐⡤⠀⣀⣀⣨⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠃⠀⠀⠀⠀⠀⣰⠄
⠀⠀⢻⣿⣷⡈⢿⣿⡆⠀⠀⠀⣠⡿⠻⠃⠚⠉⠉⠈⠐⠋⠻⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠟⢋⣡⣤⣶⠄⠀⢀⣠⡴⠞⠋
⠀⠀⠀⢿⣿⣷⡈⢻⣿⡄⠈⠈⠍⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠉⠉⠣⠛⡄⠀⠀⠀⠀⠀⠘⣧⣴⣾⣿⣿⣿⡿⠷⠛⠉
⠀⠀⠀⠈⢿⣿⣷⣼⠛⡇⢀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⢸⡉⠀⠀⠀⠠⡙⣶⣤⡀⠀⠀⠀⣀⣀⡀⣠⡴⠔⠋⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⢢⣼⠿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣿⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣠⡀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡱⠂⠈⠛⢿⣿⣤⣌⠻⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠚⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢆⠀⠙⠻⣿⣧⣄⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⢀⢂⢀
⠀⠀⠀⠀⢺⢆⡀⠀⠀⠀⠀⣀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠈
⠀⠀⠀⠀⠀⠢⣍⣿⣷⣏⣍⣇⡴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣅⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠇⣸⠏⠀⠀⢀⣤⣴⡶⠒⢦⠀⠀⠉⣻⡿⠋⠁⠀⠀⠀⠀⣿⣍⢳⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣏⣀⡾⢿⡀⠀⠀⠻⣟⣩⣤⣶⣯⡤⠶⢦⣏⡇⣀⡀⠀⠀⠀⠀⣹⣿⣯⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠲⣝⢦⡴⢶⣏⠉⢁⡼⠋⠀⠀⠈⠸⡄⠉⠉⢻⡏⠙⢿⡟⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣴⣛⠂⠀⠀⠙⢻⣭⠀⠀⠀⣀⣀⣀⣁⠀⠀⠘⣷⣀⣀⣻⣦⡀⢰⣿⠈⢷⡀
⠀⠀⠀⠀⣰⡏⣸⡀⠀⢠⡾⠛⢙⡇⠀⠀⠀⠀⠋⢀⣴⡛⠁⠀⢨⡿⠃⠀⡜⡽⠉⠉⠁⢀⡽⢛⢿⣇⣸⡇
⠀⠀⠀⠀⣿⣠⠿⠍⢻⡟⠀⢀⣾⠀⠀⠀⠀⠀⠀⠈⠑⠛⠻⣿⡏⠀⠀⣀⣄⡀⠀⠀⠀⢫⠞⠁⢦⡉⠉⠀⠀⣴⣶⣆
⠀⠀⠀⠀⠈⠁⠀⠀⢘⡷⠖⠛⠮⡑⠀⠀⢀⣴⢶⡆⠀⠀⠀⠁⠀⠀⠸⣇⠀⠙⣦⡀⠀⠀⢰⠿⣦⠙⣀⠤⠶⣿⣽⣿
⠀⠀⠀⠀⠀⣠⢦⡀⡾⠀⠀⠀⠀⠀⠀⢰⡟⠁⣰⠃⠀⠀⠀⠀⠀⠀⠀⢿⠶⠤⠿⠃⠀⢀⣼⡀⢸⡆⢰⡏⠁⠙⠿⠋
⠀⠀⠀⠀⢠⡇⠀⣷⣒⡒⠒⠂⠀⠀⠀⠺⣷⠾⠛⠃⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠙⠿⠁⠈⡇⠀⢀⡀
⠀⠀⠀⠀⠸⣇⣰⢫⡏⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠈⢻⡆⠀⠀⠀⣠⠖⢲⡀⠀⠀⢀⣀⣠⣇⣴⠟⢹⡄
⠀⠀⠀⠀⠀⠈⠁⢸⡇⢰⡟⠻⣦⠖⡊⠁⠀⠀⠀⠀⠀⠘⣟⢶⣴⠟⠀⠀⢠⣿⡇⠀⢸⠃⠀⠀⠤⠖⢲⡾⣿⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡃⢀⡟⠋⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⢧⣴⣏⣀⣠⢤⡀⠀⢸⠇⠙⠟⠁
⠀⠀⠀⠀⠀⣴⣄⠀⢿⡈⠷⠞⠁⠀⠀⡞⢹⠛⢿⣧⡀⠀⠀⠀⠀⣤⣄⣀⣀⠀⠀⠀⠙⡿⠁⣰⠃⢀⡿
⠀⠀⠀⠀⢸⡷⠈⢿⣞⣡⠄⠀⠀⠀⠀⠹⣬⣳⣤⣿⠃⠀⠀⢀⡴⠋⠉⢉⣽⠇⢤⠀⠘⠧⠞⠁⢠⡾⠃
⠀⠀⠀⠀⠘⣷⡀⣼⠋⠙⢦⡀⠀⢀⡄⠀⠀⠉⠉⠁⠀⢠⡄⠘⠷⠶⠚⠋⠁⠸⣟⣧⣤⡀⣌⡓⢤⣤⡞⠉⡷
⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⣴⠿⢦⣎⡔⠀⠀⠀⢀⣠⠴⠾⠧⣄⡀⠀⠀⠀⢀⣾⠋⣠⢟⡴⠋⠉⢶⡏⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠙⣷⠦⣄⡀⠘⢧⣀⣀⣀⣼⡿⠀⡶⣄⣀⣙⠿⠛⠉⠀⠀⠀⢻⣧⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠦⣾⠀⠀⠉⣳⣶⠾⠿⠿⠷⠒⠒⣧⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠉⠉⢳⡄⠀⠀⢰⡋⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠉⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠀⠈⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⣀⡤⠖⢒⣶⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠘⠁⢀⠤⠚⣛⡛⢟⣟⣻⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⡄⠀⠀⠀⠁⡰⠟⢛⣿⣮⣭⣧⣤⣤⣝⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢰⢱⣤⣆⡴⢊⠔⢺⣿⣏⡙⣿⣿⣿⣿⣿⣿⢦⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠛⢿⡴⠁⠀⠀⢻⠀⠀⢹⣿⣿⣿⣿⣿⡇⡟⢶
⠀⠀⠀⠀⢀⣠⡤⠶⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠺⠤⠈⠷⡦⠾⠆⠀⣠⣿⣿⣿⣿⣟⡏⢱⣾
⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣤⣤⠄⠀⠄⠀⠘⠻⣿⣿⣿⣿⡟⠀⣾⡟
⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⣉⢿⣿⣀⠀⢀⠀⠀⡀⠀⠈⠙⣿⣿⣷⡦⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠆⠀⠀⠀⠀⠀⠀⠻⢆⣿⡿⠋⠉⠁⠉⠁⢀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢈⠳⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠈⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠈⠳⣦⣌⡉⠙⠛⠛⠛⠛⠛⢉⡉⠤⠀⡾⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⡀⠀⠀⠀⠐⡆⠀⠀⠀⢹⡤⡤⠐⠁⠀⠀⠀⣆⡀⠀⠀⠀⢸⠤
⠀⠀⠈⠙⠒⠷⠠⠀⠠⠈⠄⠄⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⠀⢀⣧⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣦⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠐⢶⠶⠆⠀⠠⠀⠀⠀⠀⠙⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣛⣛⣲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠞⠻⢦⣤⣀⣉⣉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣁⣬⠟⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⠶⣶⣤⣬⣭⣭⣉⣉⣉⣉⣉⣉⣉⣉⣩⣥⣤⣶⡾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⢸⡇⢸⡏⢹⡏⢹⡏⢹⡏⢹⡏⠉⡏⠉⡇⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣄⣸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⠀⡇⢀⣇⣠⣧⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠙⠛⠛⠻⠿⠷⠾⠷⠾⠷⠾⠿⠿⠛⠛⠛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⡾⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣴⣦⣤⣤⣀⣀⣀⣀⣤⣤⣤⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣤⣤⣤⣀⣀⣤⣤⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣉⣉⣉⣛⣛⡋⠉⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇⠀⠀⠀⠀⠀⠀⢀⣴⠶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠛⠿⢿⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⠿⠟⠛⢻⡇⠀⠀⠀⠀⢀⣴⣟⡱⠋⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠛⠻⡿⠷⠶⣦⡀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⢰⡿⠃⠈⠹⣦⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⣠⡇⠉⠛⢸⡇⠀⠀⢻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡇⠀⠸⣧⣀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⢿⣿⣷⣾⠟⠁⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢟⣛⡛⣟⠛⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠉⣀⣿⣀⣀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠚⠓⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠙⡏⢹⠋⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⢀⣨⠀⠀⠀⢀⡇⢸⡀⠀⠀⠀⣇⡀⠉⠳⡄
⠀⠀⠀⠀⠀⣠⠞⠉⠹⡄⠀⠀⠘⠦⠾⣍⣧⠴⠒⠚⠉⠓⠚⠉⠓⠒⠦⣼⣩⠷⠴⠃⠀⠀⢠⠏⠉⠳⣄
⠀⠀⠀⠀⣰⠁⠀⠀⢰⣇⠀⠀⠀⣠⠔⠉⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠉⠢⣄⠀⠀⠀⣸⡆⠀⠀⠈⣆
⠀⠀⠀⠀⢿⠀⢀⡴⠧⢌⡓⠶⢶⠁⠀⠀⣞⠀⠀⡽⠀⠀⠀⠀⢮⠀⠀⣳⠀⠀⠈⡶⠖⢚⡡⠼⢦⡀⠀⡿
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢙⡶⠃⠀⠀⠀⠈⠙⢻⣳⠄⠀⠀⠀⣜⡟⠊⠁⠀⠀⠀⠘⢶⡋⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⡞⠀⡹⢤⡇⠀⢰⠋⠙⣆⣠⡄⠀⠀⡰⠋⠙⢆⠀⠀⢀⣄⣰⠊⠑⡆⠀⢸⡤⢏⠀⢳
⠀⠀⠀⠀⠀⠀⢧⠀⡱⠚⡇⠀⠸⢄⡠⠋⠙⠁⠀⠀⠱⢄⡠⠆⠀⠀⠈⠋⠹⢄⡠⠇⠀⢸⠓⢎⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠉
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣠⠷⡄⠀⠀⠀⢀⡤⣼⡝⠀⠀⠀⠀⢫⣧⢤⡀⠀⠀⠀⢠⠾⣅⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣾⠀⠈⠳⡖⢊⡥⠖⠺⡀⠀⠀⢯⡀⠀⡳⠀⠀⠀⠀⣞⠀⢀⡽⠀⠀⢀⠗⠲⢬⡑⢲⠞⠁⠀⣷
⠀⠀⠀⠀⠹⡀⠀⠀⠸⡏⠀⠀⠀⠈⠢⣄⠀⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠀⣠⠔⠁⠀⠀⠀⢹⠇⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠙⢦⣀⡰⠃⠀⠀⢠⠖⢶⣋⡟⠒⠤⢤⣀⡤⢤⣀⡤⠤⠒⢻⣙⠶⠲⡄⠀⠀⠘⢆⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠈⣹⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⣏⠁⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣠⣇⣸⣄⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠞⠁

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠳⡇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢠⠞⣧⠀⠀⢀⣀⡤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⣠⠋⠀⠘⠚⠉⠉⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠈⠙⠲⢤⠞⠉⠉⠙⠣⡀⠀⠉⠳⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⡈⣧⠀⠀⠀⠀⡰⣋⣞⠉⣱⣄⡀⢸⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⡼⠁
⠙⣄⠀⠈⠉⠉⠉⠀⠈⠉⠉⠀⠀⢠⡾⣽⠣⣼⠃⢌⣇⡜⡸⠀⢀⡤⠲⡈⠳⣄⠀⠀⠀⠀⠘⠧⠤⠤⢤⣤⣀⣀⣀
⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠒⠲⢧⡦⢴⠛⣬⣶⠁⡰⠛⢢⠰⢇⣠⠟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠈⢾⣠⣿⠏⠻⣅⣀⠼⣀⡠⠃⢱⠛⡆⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⣨⠆⠀⠀⢠⡇⠀⠀⠀⠀⠠⠁⠀⣼⠟⠁⠀⠀⠘⠷⣤⢼⠒⠒⡇⢀⠇⠀⠀⠀⠀⠀⠐⠛⠲⠤⣄⣀
⠀⢀⡠⠖⠋⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠛⠳⠿⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠽⠗
⠶⣉⡀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉
⠀⠀⠉⠙⠒⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡛⢏⣙⡦⢵⣈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠉⢫⠈⣇⣸⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡏⣧⣞⡁⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣀⣀⣀⣈⣶⡄
⠀⠀⠀⣴⣁⣤⠤⠤⠶⠒⣶⠂⠀⠀⠀⡠⢿⡀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠉⠙⠲⢤⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣠⠔⠋⠀⠀⠙⠓⠒⠚⠋⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠑⠢⢌⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡠⠞⠁⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡽⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⢀⡴⠋⠸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡊⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢲⡀⠀⠀⠀⢀⢤⡀⠀⢀⣱⣸⡁⠀⠀⢀⣀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣌⣇⢀⣠⠴⠓⠛⠉⠉⠉⠉⠉⠉⠉⠙⠓⠦⣽⡀⠀⢠⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡩⠞⠉⢠⠖⢦⠀⠀⣀⣀⠀⠀⠀⠔⠙⠆⠀⠀⠉⠳⣗⠋⠁
⠀⠀⠀⠀⠀⠀⢀⢤⡔⠋⠀⠀⠀⠈⠀⠀⠀⢺⡀⢈⠇⠀⠀⠀⠀⠀⡠⠊⢳⠀⡨⠋⢙⡆⢀⣀
⠀⡠⢢⠀⠀⠀⡿⠁⢠⠚⠙⢦⠀⠀⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠉⢻⡁⡸⠃
⠀⢧⣄⣳⣤⡜⠀⠀⠈⠑⠲⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡎⠀⠀⠀⠀⠀⠀⣀⣀⢻⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠈⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠇⠀⠀⠀⠀⠰⢊⡁⢸⠆⢧⢤⣊⣤⣄⡇
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢒⣿⡆⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣸⣇⣀
⠀⠀⠀⢿⠇⠀⢳⠲⠤⣄⣀⣀⠀⠀⠀⠀⠴⣎⣁⠀⠈⠻⢃⡰⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⣤⠝
⡔⠙⢒⣲⡄⠀⠀⢢⡀⠀⠻⢟⠭⠋⠀⠀⠀⠀⠀⠉⠩⠭⠤⠒⠁⠀⠀⠀⠀⠀⠀⠠⠤⠃⠀⠀⣇⠘⡆
⠉⠊⢀⡄⡇⠀⠀⠦⣍⣉⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠖⠒⠂⠀⠀⠀⠀⠀⠀⠈⠳⡅
⠀⠀⠀⠓⢿⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⡄⠀⠀⣀⠤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠸⣆⠯⡤⠀⠀⢀⢤⡀⠹⣄⡽⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⢳⢠⠟⠓⡄⠀⠘⡆
⠀⠀⠀⠀⢧⣸⣆⠀⠀⠀⠹⣄⡷⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⣱⠀⠀⠀⠀⠀⠀⠀⠛⢡⠟⣄⠀⠳⣄⡀⢹
⠀⠀⠀⠀⠀⡄⠾⢦⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡠⠻⡠⠞⠀⠀⠀⡟⠈⠑⢦⡀
⠀⠀⠀⠀⠀⣃⡠⢚⡽⢦⡀⠀⠀⢸⡁⣻⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠓⠒⠁⢀⡠⢎⢲⣀⢣⠀⠀⠀⢠⢇⠴⡄⢆⣧
⠀⠀⠀⠀⢺⣀⣀⠎⠀⠀⣟⠳⢤⣌⣉⠁⠀⠀⠀⠘⠦⠏⠀⠀⣀⣀⣠⣔⢾⡉⠀⠘⣆⠸⠘⠀⠀⠀⠈⠟⠀⢹⡎⠛
⠀⠀⠀⠀⠀⠎⡰⠀⠀⠀⢻⠀⡠⢊⠏⠉⢹⢛⠟⠻⢛⠋⠯⠍⠉⠹⠘⢤⡀⠙⢦⡀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⢸⠀⢱⡲⡄⠀⠸⡄⠧⠼⠃⣠⠃⠀⠀⢄⡸⠀⠀⠀⠀⠈⠓⠒⠁⠀⠀⠙⢢⡄⠀⠀⡇
⠀⠀⠀⠀⣾⠰⣤⠧⣄⠀⠀⢷⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢹
⠀⠀⠀⠓⠈⠑⠚⠛⠁⠀⠀⠸⢦⠀⠈⠉⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢒⣨⠗⠲⠔⠦⠟⣠⠳⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠏⠀⠁⠤⠀⠤⠠⢤⡐⢄
⠀⠀⠀⠀⠀⠀⠀⢰⡿⡿⡱⠚⠦⠴⠋⠓⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠒⠚⠳⠤⠔⠛⠰⣄⣀⢤⡿⢆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣷
⣿⣿⣉⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⠉⠉⠙⠓⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⠉⠉⠉⠉⠙⠓⠶⢤⣀⡞⠁⠀⢀⣀⠀⠀⠹⡄⠀⠈⠙⠲⣄⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⢀⣠⠴⠒⠒⠒⠒⠦⢤⡀⠀⡾⠀⠀⢀⡞⠙⣆⠀⠀⠙⣆⠀⠀⠀⠈⢧⠀⠀⠈
⠀⠀⠀⠀⡠⠞⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡇⠀⠀⣾⠓⢦⡘⢦⠀⠀⠘⢦⠀⠀⠀⠈⡇
⠀⠀⡠⠊⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⣀⡤⢤⣄⠀⠀⠀⠸⡇⠀⠀⢿⠀⠀⠙⢮⣧⡀⠀⠈⢧⠀⠀⢀⡇
⣠⠞⠁⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡾⠁⠀⠀⠈⢳⠀⠀⠀⢷⠀⠀⠘⢆⠀⠀⠀⠙⣷⡀⠀⠈⢳⣀⡞⠀⠀⢀
⠙⣄⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢧⠀⠀⠀⠀⣸⠁⠀⣠⢬⣧⡀⠀⠈⠳⢤⣀⣠⠞⢳⡄⠀⠀⠉⠀⠀⢀⠞
⠀⠈⠳⣄⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠙⠲⠖⠚⠁⢀⣄⣷⠀⢻⣷⣤⡀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠈⠳⣄⡀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠻⣧⠀⢿⣧⠙⢶⣦⣤⣀⣀⣀⣠⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠈⠓⠦⢤⣤⣤⡤⠴⠚⢧⠀⢿⣇⠈⢿⣆⠈⣿⡄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⣀⣀⢀⣀⣀⣠⠼⢧⠈⢿⣆⠘⣿⡆⠘⠿⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣀⣀⡀⠘⣆⠘⠿⠀⠈⠁⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠙⢦⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢀⣠⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⡴⠞⠉⠁
⣿⣿⠛⢛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠆⠀⠀⠓⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠒⠒⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⣿⣆⠀⠀⠀⠀⣼⣿⣿⣿⣶⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⣾⣿⣦⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⠀⠰⠿⠿⠿⠿⠿⠿⠿⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⣷⠀⢠⣤⣤⣤⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢸⣿⣿⣿⡿⠃⢿⣿⣿⣿⡟⠀⣴⣿⣿⣦⢰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣻⡇⢠⢭⣭⣭⣥⣬⣭⣭⣭⣤⠔⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣿⡇⢸⣛⣛⣛⣛⣛⣛⣛⣻⣧⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⣿⣿⣿⣿⣧⣠⣶⣶⣾⣿⣿⣿⡇⢸⡶⡶⠶⣶⣶⣶⣶⣶⣶⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠛⢡⣿⣿⣿⣿⣿⣿⣿⡇⠸⢭⣭⣭⣭⣭⣭⣭⣭⢭⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⣠⢄⠀⠀⢈⡶⣏⠀⠀⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣀⣨⣇⣀⠀⠀⣠⡄
⠀⠀⠀⠓⢺⠒⠋⠉⠀⠈⠉⠒⢺⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡷⠫⣦⠐⠷⢠⣭⠲⡉⠁
⠀⢀⡀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⡀⠀⠀⠀⡀⠀⠀⣀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢶⠆⣤⡘⠗⢠⣤⠮⠿⡀⢀
⠀⢮⡽⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⣹⠀⠀⠘⣿⠖⠉⡷⡀⠀⠈⣙⢶⡟⠀⠀⠀⠛⠛⣧⣶⢤⣭⠘⣗⣴⡅⠾⢆⠏⠛⠃
⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⣰⠃⣤⣄⡁⠀⠀⠘⠝⠁⠱⡀⠀⠀⠀⠀⠘⣄⢴⠅⣈⡛⣀⡸⢇⠞
⠀⠀⠀⣖⢺⠒⠤⢄⡀⣀⠤⠖⢺⢒⡄⠀⠰⡶⡇⠀⡈⠑⠀⢰⢢⠀⢀⠀⠀⣷⡶⠀⠀⠀⣶⠋⠑⠦⢭⡥⠽⠓⠉⢶⠆
⠀⠀⠀⠈⠉⠀⠀⢠⡟⣧⠀⠀⠈⠉⠀⠀⠀⠀⠹⡿⠒⠀⢀⠈⠁⠠⠢⢿⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⢠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⠄⠀⠀⠀⠀⠀⠀⠀⢠⣷⢄⡀⢸⣶⠁⠀⣀⣴⣧⠀⠀⠀⠀⠀⠀⢰⣋⣆⢀⣀⣨⠷⡁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠺⡀⠀⣀⣸⣀⡀⠀⡸⠂⠀⠀⠀⠈⠁⠀⠈⠉⣿⡞⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⡸⠊⠁⣎⣽⠀⠈⠑⠦⣻⠼⠂
⠀⠀⢀⠀⠀⢀⠟⣩⡄⢴⡆⣨⡝⢄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡖⣄⠼⠁⠀⠀⠘⠈⠃⠀⠀⠀⢧
⠀⠀⠘⠒⢲⢧⣦⢨⣄⠾⠂⣨⡱⠾⢷⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡆⠀⠀⠀⠀⠀⠛⠉⣇⠀⠀⢀⠀⠀⠀⠠⣳⠴⡜⣤⠶⡄
⠀⠀⠀⠀⢸⣠⡄⣨⡁⠾⢢⣬⠱⡶⢸⠀⠀⠀⠀⠀⢠⡲⣄⠴⠒⠉⠁⠉⠙⠲⢤⡠⣲⠀⠀⢸⠄⢰⡚⣞⠁⠀⠀⠘⢚⠏⠉⠚⠁
⠀⠀⠰⠒⠉⢧⢡⣭⠁⡘⠃⣀⠾⢢⠏⠑⠲⠀⠀⠀⢀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⠀⢯⡹⠉⠲⢍⡁⣀⣀⡠⢄⣇⣀
⠀⠀⠀⠀⠀⢀⡽⠧⢼⣋⣨⠿⠶⡅⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢠⠗⣅⠀⠀⠘⠧⠋
⠀⠀⠀⠀⠀⠛⠀⠀⠀⢸⠀⠀⠀⠘⠃⠀⠀⠀⠺⠭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠿⠆⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡉⠁⠀⠸⣉⡗⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠣⠴⣲⢞⠘⠢⣄⠀⡔⢦⠀⠀⡳⣀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣇⠀⠀⠀⢴⡀⣹⠤⠽⠤⣽⡁⣦
⠀⠀⠀⠀⠀⣀⠀⢠⠎⠀⠈⢳⠺⡀⠀⠀⠉⡖⠉⠀⠘⠚⠉⠑⠢⠤⡄⡤⠤⠒⠉⠙⠚⠀⠰⢣⡔⠋⣶⠄⠪⠇⣴⡎⠣⡴⠆
⠀⠀⠀⠀⠰⣅⠵⡉⠀⠀⠀⠀⠀⠀⠀⣄⠀⢱⠀⢀⠀⠀⠀⠀⠀⠀⠳⠇⠀⠀⠀⠀⠀⠀⠓⡞⣔⡆⡠⡄⠿⠂⣨⡄⠿⢿⠜⠆
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣰⣀⠄⠀⠐⢺⣩⡯⠯⣩⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠚⣇⣤⡀⣉⠁⠿⢂⣬⠁⣶⢄⠗⠇
⠀⠀⠀⠀⠀⠀⡤⢜⢤⡘⠬⠗⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⢳⡉⣨⡟⠁⠪⠇⠉⢖⠆⡼⠳⠆
⠀⠀⠀⠀⠀⠀⠑⠊⠀⠈⠑⣄⡴⠒⠚⢳⠑⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⣑⠮⣁⠺⠗⠐⣿⡤⡎⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠬⠇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠾⠉⣽⠉⠵⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢠⡼⢀⢤⠄⡄⡠⠀⠠⣶⠀⡤⡄⣠⠯⢄⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠁⠈⠉⠃⢸⠁⠀⠀⠚⠈⠉⠉⠀⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢸⠀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⡾⢠⠏⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⣸⢃⡏⣸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢤⠂⠀⢠⠏⡞⢰⠇⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⣠⠀⠈⠉⠙⢡⡏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢰⠇⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠳⠀⢠⣀⣈⡓⠠⡤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠘⣷⠀⢀⠀⠢⢀⠀⡼⠀⢀⠀⠀⢠⠀⣾⣿⣿⣿⡿⣿⣦⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢷⡀⠂⠀⠀⣼⠁⠀⠈⠉⠀⢠⣿⢿⣿⠿⣿⣷⣿⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⠀⠀⠸⣇⠀⠀⣰⠃⠀⠀⠀⠀⠀⠘⣿⣾⣿⣿⣿⣿⣿⣿⡏⠀⡀
⠀⠀⠀⢀⠠⠐⠈⠀⠀⢧⣀⠀⠀⠰⠋⠀⣰⠏⠀⠀⠀⠀⠈⠒⠁⠹⢽⣿⣿⣙⣿⣿⢯⡁⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣷⣶⠦⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠉⠉⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣩⣭⣭⣤⣍⡉⢉⣩⣭⡉⢋⣩⣤⣭⣭⣍⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⡘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⡇⠿⠿⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⡿⢋⣵⣶⡿⢿⣶⣮⡙⢿⣿⡇⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⢱⣿⣿⣿⢰⡆⣿⣿⣿⡎⣿⡇⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢿⣿⡇⢿⣿⠰⠶⢾⡷⠶⠆⣿⡿⣸⡇⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⡘⣿⣿⣿⠸⠇⣿⣿⣿⢃⣿⡇⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣮⣛⠿⠿⠿⠿⣛⣵⣿⣿⡇⠿⠿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⠸⠇⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⠶⣶⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣌⠃⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣄⠀⠀⠀⠀⢀⡄⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣥⠴⡞⠀⠀⠀⠀⠘⡖⢲⠏⠀⠀⠀⣴⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢇⢠⣖⣲⠄⢸⠁⡜⠀⠀⠀⠀⡸⠲⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⠞⠙⡄⠀⣠⢘⣿⠀⠀⠀⣠⠇⠘⠦⠏⠘⠒⠞⠀⢧⣀⠀⢀⡜⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠙⣄⠉⠻⡌⠷⠖⠋⠁⣤⠀⠀⠀⣠⠶⠶⠦⣄⠀⠉⠋⠀⢯⡀⠀⢠⡖⢤⡀⠀⠀⢀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⠓⠒⠃⠀⠀⠀⠀⠉⠀⢀⡞⢡⡆⠀⠀⠘⡆⠀⠀⠀⠀⠙⠒⠋⣹⠶⠃⠀⢀⡼⣆⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡶⠃⠀⠀⣠⠖⠛⠲⣦⠀⠀⢸⡅⠀⠀⠀⠀⢠⠇⠀⠀⠀⣀⡀⠀⠀⢧⣀⣠⠔⣋⡤⠼⠓⠃
⠀⠀⠀⠀⠀⠀⡴⢦⡀⠀⠀⠘⢷⠃⠀⠀⢰⡏⠀⠀⠀⢸⡇⠀⠀⠙⠦⣤⣤⠴⠋⠀⠀⠀⡞⠉⢹⠀⠀⠀⢉⡴⠊⠁
⠀⠀⠀⠀⠀⠀⡀⣸⠉⠓⠒⠒⠋⠀⠀⠀⠈⢧⣀⣀⣠⠞⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠛⣶⠀⠀⢾⠇⠀⠀⠉⠉⠁⠀⠀⣿⣀⡿⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠘⣇⣀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⣠⠶⠒⠒⠶⣄⠀⠀⠉⠀⠀⠀⢀⡴⠞⠋⠉⠉⠛⠳⢦⡀⠉⠀⠀⠀⢉⡭⠽⠾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠃⠀⠀⢠⡟⠁⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠰⡷⠀⠀⢳⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⠗⢦⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠘⠒⠒⢶⡖⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⢦⠀⠛⠀⠀⠀⣠⠟⠀⠛⠃⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣴⠛⠓⠦⡞⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠉⣀⠀⠀⠈⠙⠶⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⢳⣤⡄⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠸⠍⠹⣤⠴⠊⠉⠱⡄⠀⠀⠀⢀⣀⠀⠀⠀⣠⠴⣤⡀⠀⠀⠈⠙⠛⠒⠚⠛⠉⠀⠿⠂⡴⠋⠻⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠦⠏⠀⠀⢀⣀⡼⠀⠀⠀⠈⠋⠀⠀⠰⣇⡀⣸⠇⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⣀⣀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠙⣦⠞⢹⡶⣄⣀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠈⠁⠀⠀⢀⣀⠀⣾⠁⠈⢶⠋⠹⠂
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠙⠚⢉⡇⢠⣤⣀⣀⠀⠀⠀⣀⣀⣤⡀⡞⠉⠈⢳⡘⣆⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⣰⠃⠀⠀⠉⢳⠀⡾⠉⠀⠀⣧⣽⣄⠀⠀⢷⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡍⠙⢿⠀⠀⠀⠀⢸⠀⡇⠀⠀⠘⠳⠒⠋⠀⠀⣸⠛⠉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⡾⣤⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠘⠁

⢀⠔⠒⣒⡒⠒⠒⠒⠒⠒⠒⠒⠒⢤⠀⡠⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⣒⠢⡀⢠⠔⠒⠒⠒⡲⠶⢶⠒⠒⠒⠒⠒⣄
⢸⠀⢈⠿⡉⠉⠉⠳⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢀⢤⣤⡲⣶⠭⠛⠒⡦⠀⡇⢸⠀⣀⠔⡫⠖⠉⠉⠀⠀⠀⠀⠀⢸
⢸⡠⠊⠉⠑⡄⠀⠀⠀⠀⠂⠀⠀⢸⠀⡇⠀⠀⢸⢇⣠⡄⣿⠀⠀⠀⡇⠀⡇⢸⢊⡠⠊⠀⠠⠀⠀⠁⠀⠀⠀⠀⢸
⢸⡟⠀⠀⠀⠃⠀⠠⡲⠤⠄⢒⡒⢽⠀⡇⠀⠤⢚⡇⠀⠀⣿⠆⠀⠀⡇⠀⡇⢸⠉⠀⠀⠀⠀⣀⠤⡦⠄⢐⣒⡒⢿
⢸⡇⠀⠀⠀⠇⣠⣒⠭⠓⠀⠀⠀⢸⠀⡇⠀⠀⢘⡕⠒⠀⣿⠀⠀⠀⡇⠀⡇⢸⠀⠘⠃⠀⡎⡠⣒⠭⠂⠀⠀⠀⢸
⢸⡟⠒⠒⠒⣣⣟⡲⢭⡂⢀⡠⠒⢹⠀⡇⠛⠉⢱⣣⡤⠤⠿⠄⢀⣀⡗⠀⡇⢸⠀⢐⡰⠀⢀⣏⡲⢭⡂⢀⡠⠒⢹
⠘⢌⣉⣉⣉⣁⣈⣙⣗⣊⣁⣀⣀⡼⠀⠣⣀⣀⣀⣀⣀⣈⣉⣉⣐⣒⣒⡠⠃⠸⣄⣀⣀⣀⣀⣉⣺⣕⣊⣁⣀⣀⠜
⠀⡠⠤⠤⠤⠤⢤⣤⣤⡤⠤⠤⠤⣀⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⠀⢀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀
⢸⠀⠀⡴⠊⠉⠁⠀⠀⠉⠲⡀⠀⢸⠀⡇⠀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢀⠔⠦⡀⠀⠀⠀⠀⠀⡤⠲⢄⢸
⢸⠀⢸⠀⡠⢄⡴⢄⡶⣄⠀⠸⠀⢸⠀⡇⡼⠠⠤⠧⣀⠀⠀⠀⠀⣀⣀⠀⡇⢸⠘⢤⡤⠃⠠⡦⣠⣤⠀⠧⣠⠜⢸
⢸⠀⢸⣰⠁⣬⠁⡀⢀⡌⠑⣦⠀⢸⠀⣷⠇⠀⠀⠀⠀⠉⠉⠐⢾⠁⠀⢳⡇⢸⢠⠊⠓⡄⣈⣁⣁⣉⡀⡖⠙⢦⢸
⢸⠀⠣⡉⡏⠉⠀⠀⠉⠉⡏⣡⠇⢸⠀⡏⡇⠀⠀⠀⠀⠀⠀⠀⣽⠀⡠⠃⡇⢸⢸⠀⠀⡇⠉⠉⠉⠉⠁⡇⠀⢸⢸
⢸⠀⠀⠘⠧⣀⠀⠀⢀⡠⠟⠀⠀⢸⠀⡇⠘⢄⡀⠀⠀⠀⣀⡠⠟⠊⠀⠀⡇⢸⢸⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⡾⢸
⠸⡀⠀⣠⠤⢼⡉⠉⢙⠤⢤⡀⠀⣸⠀⢇⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢀⠇⢸⡈⠛⠛⠁⠀⠀⠀⠀⠀⠙⠛⠃⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠉⠉⠙⠛⠛⠻⠿⢿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠉⠻⣿⣦⣄⣠⣤⣤⣴⣶⣶⣶⣾⡿⠿⠿⠿⠟⠛⠻⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠙⠻⣿⣯⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣽⣿⣷⣄⡀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣟⡛⠛⠋⠉⠉⠉⠙⠻⣿⣶⣤⣀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣻⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠉⠀⠈⠙⠻⣿⣷⣤⣀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⣀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣷⣶⣶⣶⣶⣿⣿⠁
⠀⠀⠀⠀⠀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣿⠃
⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣶⣶⣤⣤⣤⣤⣤⣤⡾⠋
⠀⠀⠀⠀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠉⠉⠉⠉⠁
⠀⠀⠀⠀⣿⣿⡿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣦⣄⣀⣠⣾⠟⣿⣿
⠀⠀⠀⠀⠘⣿⣧⡈⠙⠻⣿⣿⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣿⣿⣿⠟⠋⢠⣿⣿
⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠈⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉

⣿⡟⣽⢿⡿⣿⣯⣿⣿⣿⣿⣿⣻⣾⣿⣯⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⡀
⠛⠋⠙⠛⠙⠋⠛⠛⠛⠛⠃⠋⠁⠛⠘⠃⠛⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⣲⠀⠀⣀⣀⣀⡙⣷⣶⡶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡲⣿⣱⣿⣿⣿⣿⣿⡟⣿⡟⣴⣤⣀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⣠⣶⣷⠀⠀⢠⣴⣿⣿⣾⣧⣿⣿⣿⣿⣿⣧⣟⣵⣿⣯⣙⣷⣦⣀⣿⢿⣷
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣄⣴⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⡟⠛⠛⢿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠘⠛⠉⢉⣿⣿⣿⣿⡀⠀⣰⣿⣿⣿⢿⣿⣿⣧⡀⠀⢀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⡆⠀⢠⠶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣥⣶⣿⠀⢸
⠀⠀⠀⠀⡞⠙⣆⡸⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠈⠙⣄⡜
⠀⠀⠀⠀⡇⠀⣿⠿⣛⣫⣿⡿⠉⠙⣿⣿⣿⣿⣿⣶⣤⣴⣾⣿⣿⡟⠁⠀⢹⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠘⠾⠃⠀⢹⣿⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢰⣿⢟⣻⢿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣄⣙⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣽⡟⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⠟⢙⣿⡿⠁⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⠀⠙⢿⣿⣿⣿⢿⢿⣿⣤⣤⣠⣤⣾⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠙⠛⣿⡧⢿⣿⣿⠿⠿⠟⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠾⠿⢧⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠴⠛⠀⠀⠀⠀⠘⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠀⠀⠖⠆⣄⠀⠼⠄⠓⠊⠡⠒⠢⠋⢴⠀⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣆⠁⠦⠒⠀⠀⠀⠀⠀⢌⠀⠀⠀⠀⠀⠀⠀⠉⠠⣃⠀⠐
⠀⠀⠀⠀⠀⡀⢄⠀⡀⡀⠆⠈⠀⠀⠀⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⢠⠬⠃⣀⡀⠀⡄⠠
⠀⠀⠀⠀⢀⠈⠪⣦⡉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠒⠀⠙⡤⠦⠀⠀⠁
⠀⠀⠀⠈⠀⢅⠒⠀⠈⠐⠀⠀⠠⠈⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠀⠄
⠀⠀⠀⠀⠂⠀⠘⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠣⣤⠠
⠀⠁⡀⣀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⠈⠀⠐
⠀⠀⠠⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠒⠜
⠀⠀⠈⠀⢐⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣁⡀⡓
⠀⠀⠀⠈⠼⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠐⠰⠀⠀⠃⠀⠀⠀⠀⠄⠀⠀⠐⠂⠀⠀⡏⠔
⠀⠀⠒⠀⡀⠳⡀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠧⡄⠀⡀
⠀⠀⠀⠠⠀⠀⠕⠀⠀⠀⠀⠉⢠⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠄⠀⠀⢌⠠⠐⠠
⠀⠀⠀⠀⠀⠀⠚⠊⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠜⠀⠑
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⠓⣉⣠⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠄⠀⠄⣀⠐⢋⠪⠐
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⢊⣁⠶⠂⣠⡄⢀⣀⠀⠀⠀⠀⣀⡀⢄⠠⠖⢃⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠁⠀⠈⠉⠀⠛⠁⠁⠠⠰⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⣉⣍⣉⠉⠉⠉⠉⠉⣿⡏⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠸⣷⣤⣤⣤⣤⣾⠋⠉⢹⣧⣤⣤⣤⣤⡿⠃⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⢸⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢻⣿⠟⠃⠀⠀⠀⠀⠀⣿⣿⡿⣣⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⢿⣫⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠋⠉⠙⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⠛⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠥⠤⠤⣀⡀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠏⠀⠀⠀⠀⠀⠉⠓⣄⠀⢀⡼⠁
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡞
⢀⡴⢏⠉⠋⠉⠙⠓⠦⣄⠀⣷⣀⣀⣀⣀⣀⣤⡤⠤⠤⠶⠒⠚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⣾⡇⠀⠑⢦⡀⠢⡀⠀⠈⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⣯⠱⡀⠀⠀⠙⢦⡈⠢⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠹⡄⠱⣄⠀⠄⣀⠙⠢⡘⢦⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠐⠀⢀⡾⠁
⠀⠙⣆⠈⠣⣄⠀⠀⠀⠈⠢⡙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠈⠛⢦⣀⠉⠒⠦⠤⠤⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠶⠴⠶⠚⠉⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⣠⠴⠚⠋⠉⠁⢀⡠⠞⠋⣻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⢀⣴⠋⠁⠀⠈⠁⡠⠖⠉⣠⠔⠋⠁⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠈⠛⠲⠶⠶⠒⠋⠁⠀⠀⠀⠀⣼⠃⠀⠀⣠⠔⠁⢀⡴⠊⠡⠀⠀⢀⢼
⠀⠀⠀⠀⣠⡞⠻⢭⣀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠦⣴⠋⠀⣀⠔⠁⠀⢀⣀⠤⠞⢁⡾
⠀⠀⢀⡾⠉⠙⠲⢤⣈⠉⠒⠤⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡉⠉⠉⠋⠉⣉⣉⣀⡤⠶⠋
⠀⠀⢸⣧⣀⠀⠀⠀⠈⠙⠲⢤⣀⠉⠒⠤⣄⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢿⡈⠓⠦⢄⡀⠀⠀⠀⠈⠙⠲⢤⣀⣉⣳⠖⢻⡆
⠀⠀⠀⠈⠳⣄⠀⠀⠈⠉⠑⠒⠒⠒⠒⠚⠋⠉⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣶⣶⣶⣶⣾⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢘⣀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠉⠀⠀⠀⠀⠈⠈⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⢀⠘⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠆
⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⣀⣀⣀⡀⠁⠀⣀⣀⠀⠀⠀⢀⣀⡀⢀⣀⡀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣤⠀⠀⠀⢀⣠⣄⡀
⠀⠀⢠⣾⣿⠿⠿⣿⣧⠀⣰⣿⡿⠿⢿⣿⣦⠀⢻⣿⣇⠀⢠⣿⣿⠁⣿⣿⡇⠀⣿⣿⡿⠿⣿⣷⡀⠀⢈⡯⠀⣠⣴⣿⣿⠀⠀⣰⣿⠟⠻⣿⣆
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠇⡀⠀⣿⣿⡇⠈⣿⣿⡀⣼⣿⠇⠀⣻⣿⡇⠀⣿⣿⡇⠀⢹⣿⡇⢀⣀⣑⠀⠛⠋⣿⣿⠀⠀⢻⣿⣄⣠⣿⣿
⠀⠀⠸⣿⣷⣀⣀⣾⣶⠀⢿⣿⣧⣀⣠⣿⣿⠃⠀⠸⣿⣷⣿⡏⠀⠀⣿⣿⡇⠀⣿⣿⣇⣀⣿⣿⠇⠻⠿⠿⠀⠀⠀⣿⣿⠀⠀⢀⣉⡛⢉⣿⡿
⠀⠀⠀⠙⠻⠿⠿⠟⠁⠀⠀⠙⠿⠿⠿⠛⠁⠀⠀⠀⠻⠿⠿⠀⠀⠀⠻⠿⠇⠀⠿⠿⠿⠿⠛⠋⠀⠀⠀⠁⠀⠀⠀⠻⠿⠀⠀⠈⠻⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠢⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢣⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠶⠀⠤⠤⢠⡄⠤⠀⠂⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⠆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡶⣿⣿⣿⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣻⣿⣿⣿⠋⣀⣬⣥⣀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢟⣽⣿⣟⣿⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣽⡏⢰⣿⣿⣿⣿⡆⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⣿⣿⣿⣿⡿⢿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠈⠻⣿⣿⠟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠃⣸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠈⠻⣿⣿⣿⣷⣄⡐⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣾⣿⣦⣠⣉⡈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⡁⣉⠉⢁⡈⠻⣽⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠿⢷⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣾⣿⡿⣿⣿⣿⣿⠁⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠿⣿⢟⢾⣻⡁⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⡈⣀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢻⣿⣿⣿⣿⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣯⣀⣤⣷⣤⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠇⠈⢛⣿⡟⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢷⣿⣿⣻⣿⣟⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡽⣿⣿⣷⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠈⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠃⠀⠙⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠁⠀⠀⠀⠀⠹⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⢀⣾⣿⣿⣷⡄⠀⣴⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁⠀⠀⠀⢀⣶⣿⣷⡄⢸⣿⣿⣿⣿⠀⠸⣿⣿⣿⣿⢇⣾⣿⣿⣿⢫⣶⢠⣾⣿⣷⡄⠀⠀⠀⠈⣿⣿⣿⣧
⠀⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⠀⠘⢿⣿⣿⠏⠈⠻⣿⡿⣋⣤⣤⣬⣿⡿⣡⣿⣿⣿⡿⢱⡟⠃⠘⣿⣿⣿⠇⠀⠀⠀⠀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⣾⣿⣿⡿⠀⠀⠀⣠⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⠟⣰⣿⣿⣿⣿⣿⣽⣷⣶⣶⣦⠀⠀⠀⢿⣿⣿⣷
⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢏⣼⣿⣿⣿⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢫⣾⣿⣿⡿⣣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⡟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠙⢿⣿⣿⣿⣿⢿⣿⣿⣿⠏⣴⣿⣿⣿⢏⣼⣿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡟⠋⠀⠀⠀⣾⣿⣿⡿
⠀⠀⠀⠀⠸⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢇⣾⣿⣿⣿⢣⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣣⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡄⠀⠀⠙⠿⠿⠟⠉⣱⣿⣿⣿⠟⠰⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⡿⠛⠻⠿⠟⠋⠀⠀⢠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣆⠀⠀⠀⠀⢀⣼⣿⣿⣿⠏⠀⠀⠙⠿⠿⠿⠿⠋⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣄⠀⢠⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠂⡆⢰⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣘⣀⣁⣃⣘⣘⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⣀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢸⠉⢉⣉⣉⣉⣈⡀⠀⠀⠀⡴⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢨⡇⠀⢀⠎⠈⠢⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠘⣀⠤⡄⠒⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠤⠤⠤⠤⠤⠤⡇⢠⠊⠀⠀⢱⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠣⢀⣀⣀⣀⠠⠔⠊





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⡠⠤⣀⡀⠀⠀⠀⠀⠀⢀⡠⠤⢄⣀⠀⠀⠀⠀⠀⢀⣀⠤⠄⣀⠀⠀⠀⠀⠀⠀⣀⡠⠤⣀⡀
⠀⠀⡠⠊⠀⠀⠀⠀⠈⠢⡀⠀⢀⠞⢡⣤⢀⠀⠈⠑⡄⠀⠀⡔⠉⠀⠀⠀⠀⠙⢆⠀⠀⡠⠊⠀⢀⣠⣠⠈⠢⡀
⠀⠀⠃⢸⠁⠈⡅⠀⢸⠄⢃⠀⡎⠀⠯⠽⠄⢀⣄⠀⢸⠀⢸⠐⢒⠀⠀⠀⠐⠒⡎⡆⢀⠃⠀⠀⠀⠻⠟⠒⠀⠃
⠀⠀⢇⠸⠤⠠⠇⠈⠸⢀⡆⠀⢣⠀⣒⣒⠂⠸⠉⠉⡸⠀⠸⡈⠲⠤⠤⠤⠤⠒⢡⠃⠀⣆⠀⠀⠀⠀⢱⡂⢧⡆
⠀⠀⠈⠢⢄⣀⣀⣀⠤⠊⠀⠀⠀⠑⢤⣀⣀⣀⡠⠞⠁⠀⠀⠙⠦⣀⣀⣀⣀⠔⠁⠀⠀⠈⠢⢄⣀⣀⣀⠤⠊

⠀⠀⢀⠔⠊⠉⢉⠉⠒⢄⠀⠀⠀⡠⠊⠉⠉⠉⠑⢦⡀⠀⠀⣠⠖⠉⠉⠉⠉⠢⣀⠀⠀⢀⡔⡊⢉⠉⠉⠲⣄
⠀⠀⠏⢀⡠⠒⠒⠐⠄⠈⠇⠀⡜⠀⢶⠆⠀⠀⣶⠀⢱⠀⢰⠁⠀⠀⢀⣠⣄⣡⠸⡆⢠⡏⢶⣿⣿⡶⠀⠀⠈⣧
⠀⠀⡄⠀⢘⢠⣦⡞⠐⠄⡌⠀⢇⢰⣿⡇⠀⢰⣿⡇⢸⠀⢸⠀⢀⡰⠥⠔⠀⠀⢀⠆⠰⡆⠈⠟⠿⠰⣶⣶⠀⡿
⠀⠀⠑⢌⠀⠀⠙⠒⢒⠔⠁⠀⠈⢮⡉⠁⠀⠈⢉⡡⠃⠀⠀⠣⣉⠀⠀⠀⠀⣠⠎⠀⠀⠑⢄⠀⠀⠀⠉⢁⠔⠁
⠀⠀⠀⠀⠉⠑⠒⠉⠁⠀⠀⠀⠀⠀⠈⠁⠒⠊⠉⠀⠀⠀⠀⠀⠈⠉⠒⠂⠉⠀⠀⠀⠀⠀⠀⠉⠑⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠉⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢶⣄⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢸⡇
⠀⠀⣀⣤⣤⣤⣤⣤⡄⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣿⡇⠀⠀⠀⠀⢸⡇
⢠⡾⠋⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠸⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⡶⡶⠿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⣾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡿⠀⠀⠀⣿⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢨⡇⠀⢹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠘
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣄⠀⠀⠀⣀⣠⣄⣤⣼⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⢠
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⣀⣀⣀⣀⣸⣇⣠⡾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⣿⡏⠉⠉⢻⡟⠉⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠚⠛⠛⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⣀⣼⠇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠉⠁
⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⢀⣴⠟⠋⠁⠀⠀⠈⠙⠳⣦⠀⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣾⠃⠀⠀⠀⠀⠀⢀⡶⠀⠘⣧⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⡇⠀⠠⣄⠀⠀⣠⡟⠁⠀⠀⢹⡆⠀⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⣧⠀⠀⠙⢷⣾⠋⠀⠀⠀⢀⣿⠁⠀⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠶⠟⠉⠀⠀⠀⢀⡇
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠈⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠟
⣴⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣤⣦⣶⣴⣶⣴⣶
⣿⣿⣇⣀⣁⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⣀⡆⢀⢤⠀⡀⣠⠀⢀⡤⠀⣠⣄⢀⣼⠥⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠃⠘⠀⠛⠛⠂⢱⠃⠀⠀⠞⠃⠓⠓⠂⠃⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢲⠀⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⣼⢀⡖⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢰⠇⡜⢡⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠌⠀⢠⡟⣸⢁⡟⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⡠⠾⢴⠇⡼⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢠⠇⠀⠀⠀⠀⠘⢃⡟⠀⠀⠀⠀⢠⡀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣠⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢁⣠⣷⣶⣶⣤⡜⠊⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢸⣧⡀⠠⠀⠉⠢⢠⡏⠀⠠⣄⠄⠐⣧⣿⣿⣯⣿⣿⣛⣿⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣿⡛⣿⣏⣿⣿⣿⣟⣿⡏
⠀⠀⠀⠀⠀⠀⢀⡠⠔⢃⠀⠀⠀⠀⢸⣧⠀⢀⡾⠁⠀⠀⠀⢀⠀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢠⣀⡀
⠀⠀⠀⡀⠄⠂⠁⠀⠀⠘⢦⣄⡀⠀⠊⠀⢀⣼⠃⠀⠀⠀⠀⠀⠉⠀⠈⠻⣿⣿⣍⣽⣿⠟⠧⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣷⠾⠖⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠘⠊
⠀⠀⠀⠀⠀⠀⠀⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⢻⣷⣶⣶⣾⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⠿⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢰⣶⣶⣶⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⣶⣶⣶⡆⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⣶⣶⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⡟⣩⣭⣤⣿⣿⣤⣭⣍⢻⣿⣿⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣇⠛⠛⠻⣿⣿⠟⠛⠛⣸⣿⣿⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡙⢿⣿⠟⣡⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣆⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡝⢿⣿⠟⣡⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣦⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣟⣻⡿⣿⠿⠿⠿⣿⡿⢿⣟⣻⢿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠶⢮⣭⣭⣭⣭⣭⣭⣭⣭⡵⠶⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡶⠛⠛⢶⡞⠉⠀⠈⠉⢉⣭⣭⣿⣿⡛⠉⠉⠉⣴⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣼⣷⡀⠀⠀⠹⢦⣤⣤⣤⣴⣿⣿⣿⣿⣇⣀⣀⣤⣿⡿⣿⠟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠻⢶⣄⣠⣶⣽⣶⡉⠁⢻⣿⣿⣿⡟⠉⠉⠀⣨⣽⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣀⠀⠀⠀⠀⣩⣛⣿⣿⣇⣠⣴⠿⠿⢿⣻⣿⣶⣤⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣙⠻⠶⣦⣤⣿⣿⣿⣿⡟⠉⠀⠀⠀⢻⣿⣿⣿⠇⠀⠉⣉⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣌⣉⡙⠛⠋⣄⣀⣠⣤⠶⠟⢛⣉⣀⣤⣶⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⣴⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠿⠇⠉⠄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣊⣉⣀⠀⠀⠀⠘⣿⡗⠀⠀⠚⠃⠄⠀⠀⠀⣶⣶⣶⣎⡀⠀⠄
⠀⠀⠀⠀⡀⠀⢀⣥⣬⣷⣶⣄⠀⠀⠀⣾⣿⣷⣶⣦⠀⣿⣿⣿⠀⠀⠀⢠⣥⣤⡄⢸⣶⣶⡆⠀⠀⠀⢹⠿⢿⣿⠇⠀⠀⣀
⠀⠀⠀⠀⠢⠀⠸⣿⣿⣿⣿⡿⠀⠀⠀⠛⠛⠻⠿⠋⠀⠄⠀⠔⠀⠀⠀⠘⠉⠉⠃⠈⠉⠉⠁⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠔⠂⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⢸⠉⠒⠲⠠⢤⠀⠀⠀⠀⠀⠀⢆⠀⣾⣿⣷
⠀⠀⠀⠀⠁⠴⠃⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠔⢾⠋⠈⠀⠀⢠⣶⣾⠀⠀⢀⢀⣸⣤⠿⠆⠀⠀⢸⡟⢈⣻⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡱⠀⠀⣠⠂⠀⠊⠁⠀⠀⠀⠀⠀⠀⢹⠘⠒⢀⢀⣰⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛

⠀⠀⠀⠀⠀⠀⣐⣚⣛⣓⡀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤⡀⠀⠀⠀⠀⣴⠶⣶⠶⢶⠶⣶⠶⢶⡀
⠀⠀⠀⠀⠀⠀⡿⠉⠉⠹⡇⠀⠀⠀⠀⠀⠀⠿⠉⠈⠹⠇⠀⠀⠀⠀⠐⢛⣛⡛⠛⡛⠛⠓⠂⠀⠀⣿⢠⣿⡄⣨⢀⣽⡅⢸⠃
⠀⠀⠀⠀⠀⢀⣠⣿⣿⣷⣜⠀⠀⠀⠀⠀⢀⣴⣯⣯⣷⣄⠀⠀⠀⠀⠀⠘⠟⠇⢘⣟⠀⠀⠀⠀⠀⣿⢀⣿⡀⢸⠀⣻⡂⢸⡁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡄⡀⠐⠒⠿⠃⢠⠀⠀⠀⠉⠉⠩⠽⠿⠯⠭⠉⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⡒⠦⣄⠀⠀⠀⠀⠀⢀⡤⠖⠒⠦⣄
⠀⠀⠀⠀⣴⣿⣿⣦⠀⣠⣤⡀⠀⠀⠀⣀⡀⠀⡦⣀⡀⢀⣀⠀⠀⢀⡞⣷⣆⣻⣇⣴⡎⢧⠀⠀⠀⡴⢻⣶⡄⠀⣴⣦⠱⡄
⠀⠀⠀⠀⠷⣾⣷⠏⠸⣿⣿⠇⠀⠀⢀⣹⡃⣬⣥⣤⣤⢘⣝⡀⠀⢸⠀⣾⣷⡿⢿⣾⣷⢸⠀⠀⠀⡇⢨⣿⡝⢦⣽⣯⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢾⡷⠀⠀⠀⠀⠀⠘⠛⠓⠀⠀⠀⠀⠘⠛⠃⠀⠈⢧⡀⣾⣷⣾⣿⣦⠞⠀⠀⠀⠱⣾⠿⠇⠀⠿⢿⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠚⠁⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡀⠀⠀⠀⢀⣀⣤⣤⣿⣷⣶⣦⣤⣄⣰⡏⠀⠀⠀⠀⠀⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣴⣾⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣶⣄⠀⣠⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⣿⣿⡿⢟⣣⡤⢄⣠⣤⣤⡲⣶⣄⣀
⠀⠀⠀⠈⠻⠿⠻⠷⣶⣤⣄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡿⣫⣶⣿⢋⣴⡿⠿⢿⣿⠇⠾⠟⣛⣛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⡟⡡⠋⣡⡴⠂⠜⠁⠀⠙⠛⠻⢿⣿⣿⡲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣸⣿⣿⣿⣿⣿⠏⢎⣾⡟⢀⣴⣾⠟⣰⡄⠀⠀⠀⠀⠀⠀⠀⠙⣟⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣼⣿⣴⣿⣿⢏⣼⣿⠇⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⣴⣿⣿⣿⣄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠟⣝⣵⣿⢏⣿⠀⠀⠒⢉⣭⡍⠛⠀⠀⠰⠾⣿
⠀⠀⠀⠀⠀⠀⠘⠿⠏⠁⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣙⣿⠁⠈⠉⢽⣶⠿⠁⠀⠀⠀⠛⠿⠓⠀⠀⣀⣳⢦⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠋⠙⣿⣿⣿⣿⣿⠏⡿⡄⠼⡆⠀⠐⠢⣠⣴⣶⣶⣶⣶⣶⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠻⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⡇⣿⣄⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⠿⣛⣫⣽⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⠋⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠻⣷⣶⢄⠀⠀⢹⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠛⠛⠛⠛⠉⣿⡀⠀⠀⠀⠙⣿⡈⢣⣄⣼⣿⣿⣿⣯⣭⣿⣛⣛⣭⣭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⣼⣷⡀⠀⠀⠀⠘⡇⠀⠙⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠻⣿⣿⣿⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿

⠀⠿⠿⢿⣦⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⡸⠧⡀⠀⠀⠀⣀⣀⣀⠀⠄⣴⡄⠀⠀⠀⠀⣀⣀⣀⡠⠠⡐
⠀⢒⠀⠨⠙⣁⣈⠃⠀⠀⠀⢀⣠⣤⣆⣀⡀⢸⣤⣤⡄⠀⠀⠀⠸⠿⠏⠀⢀⣉⣁⡀⠀⠀⠀⣿⣿⣿⣧⡄⠀⠠
⠠⠀⠀⣴⣶⣾⣿⣿⡆⠀⠀⢸⣿⣿⣿⣿⡇⢸⠛⠛⠃⠀⠀⠀⣼⣿⣿⡆⢿⣿⢿⠇⠀⠀⠀⢸⠙⠻⠟⢃⠔⠁
⠀⠑⢀⠹⠿⠿⠿⠿⠇⠀⠀⠀⠉⠉⠉⠉⠠⠈⠀⠀⠀⠀⠀⠀⠅⠀⠀⠄⠠⠀⠠⠀⠀⠀⠀⠀⠉⠂⠄⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⢐⠁⢨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣬⡣⠀⠀⠀⠀⠀⠀⠈⡏⠉⢲⢀⣀⠀⠀⠀⣀⣥⢀⣿⣿⣿
⠀⠱⠋⠀⠀⠀⠀⠀⡴⠄⠀⠀⠀⠀⠀⠐⣙⠏⠁⠀⠀⠀⠸⠿⣥⣀⠀⡕⠲⠶⠞⠛⠀⠀⠀⢹⣇⣠⣽⣿⣏
⠀⠀⠀⠈⠀⠀⠀⣸⢅⡠⠃⠀⠠⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡧⠤⠞⠀⠀⠀⠀⠀⠈⠛⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⢠⣭⣭⣭⣥⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣶⣄⡀⠀⠀⠀⡟⠛⣿⠛⢻⠛⣿⠛
⠀⠀⠸⢇⣀⣀⡸⡄⠀⠀⠀⠀⠀⠘⢇⣂⣀⠸⠁⠀⠀⠀⠀⢹⢩⣭⡍⢉⣍⠉⢹⠁⠀⠀⡧⠰⣿⠆⢼⠠⣿⠦
⠀⠀⣢⣾⣾⣿⣷⡜⠀⠀⠀⠀⠀⣰⣾⣿⣿⣷⡄⠀⠀⠀⠀⢸⠘⠙⠁⢸⣯⠀⢸⠀⠀⠀⣇⣠⣿⣄⣸⣀⣿⣄
⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⠤⠬⠝⠟⠛⠤⠼⠀⠀⠀⠀⠀⠒⠚⠛⠓⠒

⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣭⡓⠲⢄⠀⠀⠀⠀⢀⣤⠖⠋⠙⠓⢦⡀
⣰⠿⣿⢿⡆⣠⣶⣦⡀⠀⠀⢠⣤⠀⢸⣦⢤⠀⣠⣄⠀⠀⢰⠋⣿⢦⣿⡅⢿⠏⢧⠀⠀⠀⡞⢽⣷⣄⠀⣾⣷⠹
⠘⠿⠿⠿⣁⠻⢿⠿⠃⠀⠀⣨⣮⡰⠶⠶⠶⠶⣸⣮⠀⠀⢸⠸⠿⣿⠟⢿⡿⠷⣸⠀⠀⠘⡇⣼⣿⡌⠳⣾⣷⢀
⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠛⠛⠀⠀⠈⠣⣰⠿⠷⠿⢿⡶⠃⠀⠀⠀⠙⣟⠛⠃⠀⠛⣻⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠤⠤⠤⠤⢤⢤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⣀⣸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠉⠉⠉⠉⠈⠹⢿⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠤⠤⠤⠤⠶⠾⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠏⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⡌⠐⢲⡶⠲⠂⠒⠒⢒⣾⢻⠀⠀⠀⠀⠀⠀⠀⣧⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⡔⠺⣾⡟⢀⣦⣀⡸⠟⡀⣸⠀⠀⠀⠀⠀⠀⠀⣥⣿⡆⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⢈⣿⡋⢁⣼⡁⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⣽⣿⡇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠘⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠓⠲⣶⠶⣾⠟⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠃⠀⠀⢤⣧⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢙⡆⠀⠀⣞⠉⣷⠀⠀⣚⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⣀⣀⣠⣤⣬⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠈⠋⠁⣀⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⢸⡇⣼⣿⣿⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⢀⠀⡤⠠⡄⠀⠀⣠⠒⢠⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⠉⠉⠉⠉⠉⠉⢸⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠼⠃⠀⠀⠢⠴⠃⠀⠀⠙⠶⠞⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣋⣸⡇⠀⠀⢨⣀⣵⠀⠀⠺⣤⡿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠐⠀⠀⢠⣤⣤⣤⣤⣤⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⡄⠀⠀⣔⠀⣆⠀⠂⢠⡁⣰⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⢹⠙⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠙⠛⠃⠀⠀⠈⠛⠋⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣀⣈⣀⣀⣼⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁

⣷⠛⠛⣶⢶⡶⣶⠶⠶⣾⠶⢶⠶⠶⣾⠷⠶⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⢶⡶⡶⣾⢿⣿⣿⠾⠿⠷⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⢌⠁⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⢂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢠⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠒⠂⠉⠉⠉⠑⠀⠶⠤⣀⠀⠀⠀⠀⠠⡄⠈⢣
⠀⠀⠀⠀⠈⠒⠂⠁⠂⢌⠂⢄⡀⠀⠀⠀⠀⠀⣐⠀⠈⡑⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠙⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠉⢀⣀⠀⠀⠈⠙⠏⠆⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⢠⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⣆⠀⠉⢳⠀⠀⠀⠀⠀⢳⡀⢀⠀⠀⠀⠀⠨⢦⡀⡘
⠀⠀⠀⠀⠀⢀⠐⠢⡀⠀⠀⠀⠀⠘⠂⠈⠐⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠀⠉⣉
⠀⠀⠀⠀⠀⢠⠀⠀⠍⠀⢘⢀⣀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⢤⠀⠉⠀⠀⠀⢀⠀⠀⠀⠠⠅⢀⠂⠉⠇⠀⠀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁⠉⠇⠀⠀⠘⠦⠀⠀⠒⠀⠤⠄⠀⡆⠀⠈
⠀⠀⠀⠠⣄⣀⠀⠠⠐⡀⠀⢤⡀⠀⠀⠀⠲⢤⡀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠳⠤⠂
⠀⠀⢧⠀⠘⡄⠀⠀⠀⠀⠀⠀⠳⡀⠀⠣⣄⣀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠈⠳⠄⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣆⡀⠀⠀⠀⠈⠂⢌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠲⣄⠈⠑⠢⢤⣀⠀⡀⠀⠀⣄⠀⠀⠸⠐⡀⠀⠀⠀⠀⠁⠢⡐⠠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠸⠀⠀⠀⠀⢠⠀⠉⠉⠉⠈⠀⠀⠀⢀⢡⣀⡀⠀⠀⠀⠀⠈⠀⠌⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠙⠀⠀⠀⠀⠀⠰⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧
⠀⠀⠀⢀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠛⠛⠛⢻⣿
⠀⠀⠠⡀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⣤⣤⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⡀
⠀⠀⠀⠱⠈⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣼⣿⣿⣧⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠛⠛⠢⡈⢀⢂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠂⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣠⣴⣾⣶⣦⣤⠋
⠀⡤⠊⠉⠉⠉⠉⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠤⡀
⠀⢃⠀⠀⢀⠛⡄⠀⠀⡸⠀⠀⠀⠀⠀⠀⠉⠀⢻⣿⣿⣿⣿⣿⣿⡏⠈⠉⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠐⡄
⠀⠀⠉⠉⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⠿⣿⣿⠿⠛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⡀⠀⠀⠈

⠀⠀⠀⠀⠀⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠢⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢰⡇⠀⠀⠀⠀⠉⢳⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⡐⡇⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠻⣤⡀⠈⠳⣶⣤⣈⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡥⠀⠀⢀
⠀⠀⠀⠈⠙⠷⣾⣿⣿⣿⣷⣦⣉⣻⣟⠂⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣼⣷⣔⡁⠀⠀⠀⢀⡸
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣟⣋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⣿⠡⣓⣤⡻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢜⠁⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⣀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⠉⠁⢀⣤⠏⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢷⠀⢀⠀⠀⢀⠀⣀⣴⣿
⣿⣶⠦⠄⠀⠂⠀⠈⠀⠀⣀⡠⠴⠚⠋⠁⠀⠀⠀⠀⢠⣄⣀⣀⣀⣤⣆⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢉⣻⣿⣿⡿⠃
⠙⠻⠶⣤⣤⠤⠤⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⠄
⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠒⠐⠚⡆⠀⠀⠀⠀⠀⠀⢠⣾⣿⠞⠓⠂⠐
⠀⠀⠀⠉⢁⢤⠁⠀⠀⠀⠀⠀⠀⠘⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣟⡽⠁
⠀⠀⣾⡤⠴⠐⢦⠀⠀⠀⠀⠀⠀⠀⢻⡧⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⡀⢸⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⢳⡴⠀⠚⠇⠰⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠸⡇⢹⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⢹⣿⢀⣄⣄⣰⡶⠀⠀⠀⠀⠀⠀⢹⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⢸⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠻⣏⡭⠵⢿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⣭⣿⡇⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣀⡠⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠶⠦⠴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣛⣿⣟⣿⣿⣟⠲⠲⠒⢒⠶⠶⠒⠶⠆⠀⢀⠐⠶⠶⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣤⣿⣿⠿⢿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣶⣦⣤⣴⣶⣶⣧⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣖⣫⣷⣶⣦⣆⠨⣾⣭⣭⣭⣉⣉⣉⣉⣉⣉⠡⡟⡛⠛⢛⣫⣭⣟⣄
⠀⠀⠀⢠⣴⣤⣤⠤⠀⠐⠀⠸⢿⡟⠋⠻⠇⠠⠿⠿⠿⢿⣛⡿⢿⣿⣿⣿⢺⣿⣿⣿⣿⢏⣟⣯⡾⢉⣴⢦⣤⣶
⠀⠀⠀⠀⠀⣽⣿⣿⡿⣾⣀⡗⣿⣟⣛⣛⣢⢦⡀⠐⡄⠀⠀⣤⠄⠁⠀⢈⠱⠿⠽⠟⣣⣛⣉⣿⠚⠿⠺⠿⢸⣿
⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⣽⣧⣿⠾⠿⣵⣿⣯⣿⢷⣿⡆⠒⢢⣴⣢⣤⣤⣤⡄⠀⡘⢀⡀⠀⢹⠇⣀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⡀⠀⠀⠈⣍⣉⣋⣉⣭⣬⡌⠉⠈⠉⠀⠂⠀⡁⠀⠸⣢⣴⠿⠋⠐⠼⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠋⣙⣧⢸⣿⡏⣏⣛⠲⠶⢬⣭⣭⣭⣭⣭⠽⠶⠒⠀⠀⢠⡄⣧⡀⣸⠟⠁⣶⣿⣾
⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⡟⡷⠼⣿⢻⣽⣶⣿⣿⣲⠶⠴⠰⠄⡤⣀⢰⢀⣠⣼⠀⣿⡇⠗⠀⠘⢱⡿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡏⢽⡇⠀⢠⣿⣼⣿⣿⣿⣿⣿⣶⣴⣦⣤⣯⣏⣭⣹⡐⡾⢰⢹⡇⠀⠀⠀⣸⣖⣿⣰⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣔⣿⣿⣯⣿⣿⣿⣿⣟⣿⣿⣿⣿⣞⣗⣴⣿⣿⣿⠘⢺⣧⣌⣅⠀⢈⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠇⣾⣧⣝⡛⣿⠿⣿⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣻⣽⣿⣿⠀⢸⣿⠟⢁⣴⣿⣷⣿⣿⠇
⠀⠀⠀⠀⠀⠀⡠⣠⣮⣽⣯⣿⣉⣿⣿⡿⣿⣷⣿⣯⣶⣵⣿⣿⣽⣿⣟⣿⣿⣿⣄⣸⣇⣴⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢠⢊⡼⢿⡷⣾⠮⣯⣭⣿⣹⣟⣻⡿⢿⡷⣿⠶⣿⣼⣯⣽⣟⣻⣿⡟⢁⣼⣭⣻⣿⡿⠛⠉⠛⠁
⠀⠀⠀⣰⠵⣿⣿⣛⡛⠛⢲⠷⠾⠿⣼⣯⣽⣋⣻⡿⣿⠶⣿⢾⣯⣽⣯⣿⠏⣰⣿⣿⣿⡿⠋
⠀⠀⠀⢺⡷⣶⣾⣿⡻⠻⠿⠿⢷⣶⣶⣶⣤⣭⣍⣉⣉⣛⡛⠚⠓⠶⠿⢃⣼⣿⣿⣿⡟
⠀⠀⠂⠉⠛⠓⠒⠫⠹⠭⠍⠓⠒⠒⠢⠬⢭⣍⣙⣛⠛⢻⣿⣿⣛⣀⣠⣿⣿⣿⣿⣾⣵⣶⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠓⠒⠒⠠⠍⢯⣍⣩⣹⣿⣿⣿⣿⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠠⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠐⠲⠤⠤⠤⠤⢤⠤⠤⠴⣶⢶⢾⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠸⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡽⠦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣾⣟⣲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠁⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠛⣛⣿⢿⣿⠑⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣙⡎⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⠿⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠹⠀⡆⡆⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣱⢳⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡆⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠦⠴⠶⠞⠳⠻⠛⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡖⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠤⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇⢰⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡼⠇⠸⣤⣄⣄⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣤⣤⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣮⠉⢧⠀⠀⡎⠘⢿⣿⠉⠉⢏⠉⠉⠉⠉⢩⠉⣴⣦⡶⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢀⣡⡀⣷⠀⣸⣿⡖⠀⢈⢄⠀⠀⠀⣸⢴⡿⠈⢅⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣿⡀⠉⠻⣿⡻⣿⡇⣠⢞⣺⣧⣤⡰⠣⠋⠁⡐⠕⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄⠀⠈⠻⣾⣿⡿⠋⠁⠀⣨⡧⠁⠀⡰⠋⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣟⣷⠤⣠⣿⣿⣀⡠⣤⣾⢟⣷⣀⣠⣦⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⠿⠶⠾⠿⠛⠛⠛⠉


⡿⠿⠿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣛⣿⣿⣿⣿⣿⣿

⠀⣠⠴⠤⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀⠀⢀⡤⠦⠤⠤⠤⠤⠤⠤⠤⠤⠶⠤⠤⠤⠦⣄⠀⠀⡤⠤⠤⠶⠤⠤⣴⣤⣤⠦⠦⠦⠤⠤⠶⢤⡀
⠀⡇⠀⠸⣭⠗⠒⠒⠒⣦⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⠀⠀⠀⠀⠀⢀⣀⣀⠤⢄⣒⣒⡪⡍⣿⢸⡆⢸⡇⠀⠀⣀⠴⢋⠤⠤⠼⠃⠀⠀⠀⠀⠀⠀⡇
⠀⡇⠀⣸⣉⣇⠀⠀⠀⠈⠔⠠⠀⠀⠀⠀⠀⣿⠀⢻⠀⠀⠀⢰⣫⠗⠒⠊⣿⠁⠀⠀⠀⠃⠗⢸⡁⢸⣇⡤⠚⣡⠞⠁⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⡇
⠀⣧⠋⠀⠀⠀⢹⡄⠀⢐⠈⠀⠀⠁⠀⠀⠀⣿⠀⣿⠀⠀⠀⠸⢛⡤⠴⠆⣿⠂⠀⠀⠀⠀⣷⢸⠂⢸⣇⡤⠊⠀⠀⠀⠂⠀⠂⠐⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠋⠉⠉⠉⢹⡆⠀⠨⣸⠭⠭⢭⣉⣉⡭⣿⠀⣿⠀⠠⠤⢚⣱⠀⠀⠀⣿⡿⠀⠀⠀⠃⣯⢸⠂⢸⡏⠀⠀⠀⠀⠠⠀⢠⠤⣺⠭⣍⣉⣉⣒⡩⡇
⠀⣿⠀⠀⠀⠀⢸⡇⣀⠤⢊⡵⠆⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠸⣚⠤⠄⠀⣿⠁⠀⠀⠀⠃⣿⢸⡆⢸⡇⠀⠰⠆⠀⢰⠉⣁⠤⢛⡶⠆⠀⠀⠀⠀⡇
⠀⣿⣤⣤⣤⣤⣼⢃⣗⣫⠥⣂⠀⠀⠀⢀⡤⣿⠀⣿⢠⡦⠤⢚⣱⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⢸⠇⢸⡇⠀⡠⢄⠀⡎⢨⢗⣫⠥⣂⠀⠀⠀⢀⡤⡇
⠀⡿⣄⣀⣀⣀⣸⠿⢥⣒⡫⠵⣂⡤⠖⠁⠀⣿⠀⣿⠀⠀⠀⠘⠮⠭⠭⢍⣛⣓⣒⡒⠒⠖⡿⢸⠇⢸⡇⠀⠓⠚⠀⠈⠺⣥⢒⡫⠵⣂⡤⠞⠁⠀⡇
⠀⠳⢤⣤⣤⣤⣤⣤⣤⣬⣴⣭⣤⣤⣤⣤⡤⠏⠀⠘⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⡥⠞⠀⠈⠧⣤⣤⣤⣤⣤⣤⣤⣭⣶⣭⣤⣤⣤⣤⠴⠃
⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⡏⠀⠀⢀⠔⠒⠒⠋⠉⠉⠙⠲⢄⠀⠀⠀⣷⠀⣼⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢰⡇⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⡇
⠀⡇⠀⢰⠁⠀⠀⠀⣀⠀⢠⠀⠀⠀⢳⠀⠀⣿⠀⣿⠀⡰⠋⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⡏⠀⠈⣷⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⡆⡇
⠀⡇⠀⡇⠀⡸⠳⣤⡏⠢⠼⠑⢄⡀⠀⡇⠀⣿⠀⣿⢸⠉⠉⠉⠛⠢⢤⣀⣀⠀⠀⢀⡤⠤⣄⢸⡃⢸⡇⠙⠲⠖⠁⠀⣩⡃⡖⡶⡆⠈⠓⠶⠚⠁⡇
⠀⡇⠀⣇⣰⠁⢠⡄⢀⣀⠀⣴⠀⠉⣆⡇⠀⣿⠀⢻⡎⠀⠐⠠⠄⠀⠀⠀⠈⠉⠉⡇⠀⠀⠈⣾⡁⢸⠇⡞⠉⠙⣦⢀⣉⣁⣀⣀⣄⢠⠋⠉⠙⡄⡇
⠀⡇⠐⣅⠉⡟⠉⠉⠉⠉⠉⠉⠙⡏⠉⡽⠀⣿⠀⣻⢰⠐⠠⢀⠀⠀⠀⠀⢀⠀⠀⡇⠀⢀⡼⢸⠀⢸⡇⡇⠀⠀⡿⠘⠋⠉⠉⠉⠋⢸⡀⠀⠀⡇⡇
⠀⡇⠀⠈⠳⣇⠀⠀⠀⠀⠀⠀⠀⣷⠊⠀⠀⣿⠀⣿⠈⣆⠄⣀⠀⠀⠀⠀⡀⠀⠀⣇⠴⠊⠀⢸⡇⢸⡇⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⡇
⠀⡇⠀⠀⠀⠈⠓⢦⠤⠤⠤⡖⠋⠀⠀⠀⠀⣿⠀⣻⠀⠈⠓⠦⠤⠤⠤⠤⠔⠒⠉⠀⠀⠀⠀⢸⡇⢸⡇⣧⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⣇⣀⣸⠃⡇
⠀⢧⡀⣀⣤⣒⣒⣇⣀⣀⣀⣝⣒⣒⣄⣀⣀⠟⠀⠹⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼⠁⠘⢇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢁⠀⠐⠐⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⡀⢀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⢀⢈⠀⠈⡀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠈⢹⠴⠬⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢈⠁⠂⠀⠀⠀⡀⠀⠀⠰⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣅⢀⡀⠀⢀⡂⠀⣀⢀⡄⠀⣀⠀⣀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⣠⠞⡁⠀⠀⠀⢨⠃⠀⠛⠃⠴⠆⠙⠤⢀⣀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠳⢤⡐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠃⠀⠀⠐⠂⣶⠆⠀⠒⠠⠆⠀⠠⠄⠛⠘⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠟⠶⠔⣄⡁⠀⠀⠁⠀⠀⣀⣠⠞⠉⠀⠄⠀⠀⠀⠀⠀⠀⠐⠶⣆⠀⢠⠄⠀⠠⣀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠬⠁⠋⠉⡟⠙⠋⠂⠉⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠹⠑⠒⠆⢚⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⠀⠐⠀⠢⠁⠄⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠉⠲⠗⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠
⠀⠀⠀⠀⠀⠈⠦⣆⡀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠋⠈⠹⡇⠀⢀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢀⡀⠀⠀⠀⢀⡀⠂⢀⡐⠐⠂⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠂⠀⠀⠀⢀⡠⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣆⠀⡀⠀⠐⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡏⠀⡀⠀⠀⢠⠁⠀⣠⠀⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣰⠎⠀⠀⠀⠀⠀⣰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠙⢲⣄⠓⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠀⠀⣀⠜⠃⠀⠈⠀⠀⠀⠀⠀⢗⠀⠀⠈⠀⠂⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠛⣖⣧⣌⣀⠀⠀⠀⢀⣀⣠⠴⡞⠁⠀⠄⠀⠀⠀⠀⠀⠀⠁⠘⠓⠦⢀⡠⠶⠀⠰⣆⢴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠅⠘⠀⠭⠉⠩⠉⠈⠘⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠨⠉⠛⠓⠋⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣿⣶⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⣿⡟⠻⠟⠻⠟⣿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⢁⡤⠅⠀⠀⠀⠀⠘⠛⠀⣸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢠⣶⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡏
⠀⠀⠀⠀⠀⢀⡤⠶⠒⠚⠛⠛⠛⠓⣦⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢿⠶⣄
⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⣤⣤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⣄⣀⡰⠀⣀⣤⣤⡀⣠⡾⡋⠁⠀⠈⢳⣄
⠀⠀⠀⢀⣞⡿⣧⣭⣥⣤⣼⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⣸⣟⠯⠊⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⢸⢸⡅⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣰⠆⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠈⢧⡽⠏⠉⠉⠙⢳⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠶⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⠀⠀⢀⡴⠛⢳⠀⠀⠀⠀⠀⣠⣴⡋⢠⡾⠁⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠘⣷
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⣾⠤⠖⠋⠀⠀⠀⣴⠞⠁⣈⣿⠏⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢻⡄
⣏⡉⠳⣄⠀⢠⠖⢦⡀⠀⢀⡇⠀⢸⣤⣤⣤⣄⣠⣤⣤⣤⠴⠿⣤⡾⠋⠉⠻⣦⡀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠉⠓⠺⠇⠈⠲⢤⣳⡀⣸⠇⢀⣼⣶⣄⣀⠈⠁⠀⠀⠀⢀⣀⣿⣄⠀⠀⠀⠈⠛⢦⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠈⢳⣦⣄⠉⠙⠛⠛⠛⠛⠉⠉⠁⠙⢦⡀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢀⣀⣀⠀⠀⢻⣿⣞⣛⣋⡉⠉⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣬⡿⢦⣄⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢯⣡⠭⠟⠃⠈⣿⣟⣛⣩⡅⠀⠀⠀⢀⣴⠟⠉⠉⠉⠉⠉⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⢀⣠⢤⡄⠀⠀⠀⠀⣀⢹⣿⣭⣵⠞⣃⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣞⠉⣠⠎⠀⠀⠀⣠⠚⡿⠀⠀⠀⠙⠛⠉⠀⠀⣶⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⣤⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⠾
⠈⠋⠁⠀⠀⠀⠸⣅⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣈⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣰⣠⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠆⣠⠶⣆⢠⠀⡴⢀⠀⣠⡤⣰⠃⠀⠀⠀⠀⠀⠀⢰⠏⠳⠶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣤⠀⢯⠴⠃⠘⡾⠁⠼⠰⣯⠞⠻⡤⠀⠀⠀⠀⢀⣠⠏⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⢰⡋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡴⠁⠀⠀⢀⠤⢄⠀⠈⡿⠛⠁⠀⡎⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢀⣀⣽⣶⣾⣶⣶⣷⣤⣀⣀⠟⠛⠃⠀⠀⢾⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⢀⡤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠲⡄⠀⠀⠸⣇⣀⣽⣿⣿⣿⣿⣦⣼⣿⣿⣿⠛⠻⣿⣿⣿⢶⢋⡔⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠉⠒⠤⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣵⣿⣿⣦⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢿⣿⣿⣿⠉⢻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣴⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣦⣾⣫⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢄⣀⣀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⡗⠊⠙⣯⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠈⢷⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠷⣤⣿⣿⣿⣿⣇⠀⢨⣿⣿⣿⣿⣿⣧⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠿⠿⣿⣿⣿⣿⣿⢿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠚⡄⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿⢤⢤⣀⣀⣀⣀⣀⣀⣀⣠⡽⡇
⠀⠀⠀⠀⠀⠀⣀⠴⠚⠙⢧⡀⠻⣤⣀⣀⠀⠀⠀⠁⢈⣀⣀⡴⠃
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠙⣆⡧⢀⣈⡉⠉⠉⠉⠉⠁⠀⣶⣛⣉⢉⣙⣦
⠀⠀⠀⢠⣶⠋⠙⢫⡳⡀⠀⠀⠘⡇⠀⠀⢩⢭⠉⢹⠀⠀⢸⡷⠤⠤⠤⠤⢾⣀
⠀⠀⠀⠇⡇⠀⣠⠤⣷⢹⠀⠀⠀⣷⠀⠀⠘⡫⠆⢸⠀⠀⢸⠀⠀⠀⠀⠀⠈⡇⠉⠓⠢⢄⡀
⠀⠀⠀⡇⢧⡠⠗⣶⣹⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⣾⠑⠒⠲⠶⠒⠊⡇⠀⠀⢀⠔⠉⡆
⠀⠀⠀⠘⢾⣤⣞⣡⡧⠏⠀⠀⠀⢸⠉⠒⠦⠤⠴⠚⠀⠀⢻⠀⠀⠀⠀⠀⠀⡇⡌⡀⡜⢰⠁⢩⣦
⠀⠀⠀⠀⠀⠈⢉⣈⣱⡦⠤⠤⠴⠚⠓⠤⣄⣀⣀⣀⡤⠔⢻⡄⣀⣀⢀⣀⣠⠏⠙⠓⠷⠞⠛⠛⠉⠑⠢⢄⡀
⠀⠀⡠⠤⢤⠴⢅⣀⡤⠟⠀⠀⠀⠀⠀⡴⠶⠶⣆⠀⠀⠀⠀⠉⠛⠛⠚⠛⠁⠀⢠⠤⠖⠢⠤⢤⡀⠀⠀⠀⠀⠁
⠀⠀⠑⠲⠦⠤⠤⠃⠀⠀⠀⡆⢉⠉⣹⡯⠭⢭⡿⠶⡉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠄⣤⡤⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠭⠭⠟⠁⠀⣎⠀⣀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣬⣭⣬⣭⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⡄⠀⠀⠀⣸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣤⣄⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣼⣿⣿⣿⢸⣿⣿⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⢸⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⢰⣶⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣽⣷⣾⣿⣶⣿⣽⣿⣿⣿⣿⣿⣿⢸⣿⣿⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣮⣭⢹⣿⢩⣭⣾⣿⣿⣿⣿⣿⣿⠸⠿⠿⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠿⠿⠿⠿⠿⠬⠭⠼⠿⠿⠿⠿⢿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⡿⣛⣛⡻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⢏⣾⣿⡿⣻⣎⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣜⢿⣌⣵⣿⢏⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣭⣵⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⢀⡀⢀⡀⣀⢀⢀⣀⢀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠼⠻⠼⠳⠣⠶⠟⠿⠫⠼⠩⠖⠏⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠄⠀⠀⠀⠀⠀⠀⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣽⡉⠀⠀⠀⠀⠀⠈⢣⠀⠀⠘⣍⠂⢠⠇⠀⠀⠀⠀⢀⡬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⢸⡡⠤⠄⡗⠒⠉⠢⡮⢆⣀⠠⠎⠀⠠⠤⠀⠀⠀⠀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠐⡻⣖⠉⠖⡀⠀⠂⠀⠀⠀⠃⠀⠈⠓⢄⡴⠋⠀⠀⢀⡤⠟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢅⣀⠀⠘⠀⠀⠘⠆⠀⢱⡂⠀⠠⠀⠀⠀⠀⠀⡀⠈⠑⠄⢠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠇⠀⠀⠀⠀⡴⡈⠀⠀⠐⠂⠀⠀⠰⠃⠀⠀⠀⢠⣷⠶⠂⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠘⠗⠲⠤⣄⢠⠃⡐⢄⣀⡀⠀⠉⠢⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠒⠁⠈⣦⠴⠒⠉⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠄⠀⡀⠀⠠⠤⠤⣤⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠓⠈⠉⢹⠙⠂⠀⠀⠀⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⢨⠤⢤⣤⣀⣀⣀⢠
⠀⠀⠀⠀⠀⠀⠀⡀⠄⣐⠡⡠⠊⠁⠀⠠⢀⠐⠁⠀⠀⢀⠀⠀⠀⠀⠈⠳⠂⠀⠠⢦⣅⡀⠀⠀⠈⠉⠓⠞⠃
⠀⠀⠀⠠⣏⡙⠂⠉⠀⠀⠀⡸⡖⠂⠀⠈⠉⠀⠀⢰⠀⠀⠀⠘⡄⠀⠀⠤⣅⡀⢀⣿⡀⠉⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠌⠀⣸⠷⢄⡀⠄⠀⢠⠋⠀⠀⠀⠀⢹⡄⠀⠀⠈⠳⣎⠀⠉⠲⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡞⠁⠀⠀⠜⠂⠀⢀⡀⢰⠀⠀⠀⠀⡀⣀⠤⠐⢤⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠲⠥⠀⠀⢁⠀⠈⡟⠁⠐⣖⠊⠙⡆⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⢔⡀⠀⠀⢹⠀⠀⡹⡄⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⠀⠀⠈⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⡦⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣠⡀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠀⠀⠛⠃⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣦⡀
⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⢠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⣠⣤⠀⠀⣤⣤⣤⣤⡀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⣀⣀⣸⣿⡏⠉⣿⣿⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡿⠿⠿⠃⠀⠻⠿⠿⣿⣷⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣧⣤⣤⡀⠀⢀⣤⣤⣿⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠛⠛⢻⣿⡇⠀⣿⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⠀⠀⢾⡷⠀⠰⣿⣿⠆⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠸⢿⣿⣿⣿⡿⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⠀⠉⠻⣿⣆⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⢀⣴⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣈⣿⣷⣄⣀⣀⣀⣀⣀⣰⣿⣟⣀⣀⣀⣀⠀⠀⠀⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣦⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⣸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣇⠀⠀⢹⣿⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⠀⠀⣸⣿⡿⠛⠛⠿⣿⣦
⠀⠀⠀⠀⠀⣸⣿⠀⠀⠈⢿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡿⠀⠀⣿⣿⣤⣤⣤⣤⣿⣿
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⣿⡆⠀⠀⠀⠀⠀⠀⢼⡟⠛⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⡀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡿⣠⡶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡹⣷⡾⠟⠛⠛⠿⣢⡿⢣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⣷⡀⠀⠀⠀⠀⣠⣴⠆⣼⠇⠈⡛⠃⠀⠀⠛⠋⠁⢸⣟⠻⣦⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠿⢷⣤⣀⢀⣴⠟⢁⡀⠉⢰⡿⠛⢿⡆⠀⠀⠀⣀⣀⣉⠀⠈⠻⡦⠀⢀⣼⠟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⣾⣦⣤⣈⠙⠛⠛⠛⠛⠁⠀⠘⢷⣤⡾⠃⠀⠀⢰⡟⠉⢹⡇⠙⠷⠶⢿⣛⣥⣤⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠉⠙⠻⢶⡄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠈⠻⠶⠟⠁⠀⠀⣴⢿⣏⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢿⡄⠀⠀⠀⠀⠀⣼⠟⠁⠀⢹⣧⠀⠀⠀⠀⠀⠀⠋⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣄⠀⣠⣴⢶⣄⠀⢿⣄⠀⢀⣼⠏⠀⠀⠀⠀⠀⢸⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣿⡿⠋⠀⣿⣅⣀⣿⠀⠈⠛⠛⠛⠁⣀⡀⠀⠀⠀⠀⠀⠙⢷⣤⣀⡀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠙⠛⣫⣭⡶⠶⢶⡌⠙⠋⠁⠀⠀⠀⠀⢠⡿⠛⢻⣆⠀⣀⣤⣶⣦⣤⣈⡉⠛⠿⠟⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣼⠏⠁⠀⢺⣦⡀⢀⣀⠀⠀⠀⠀⠀⠈⢿⣦⡾⠃⣀⠉⢁⣴⠟⠉⠙⠻⣶⡆⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⠿⣦⣿⡆⢀⣤⡄⠀⠀⣤⣄⡀⢸⡏⣴⠟⠁⠀⠀⠀⠀⠸⣷⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⣱⡟⢡⣤⣤⣤⣴⡿⣿⡼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠁⣿⠀⠀⠀⠀⠀⠀⠀⢸⣷⠙⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣿⠇⠀⠀⠀⠀⠀⠀⢺⣧⣤⡾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⡇⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣸⣿⣀⣀⣸⣇⣀⣀⣿⣇⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣴⠟⠋⠉⠉⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣀⣀⣿⠀⠀⣿⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠉⠉⠉⠀⠀⠉⠉⠉⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠿⠶⠶⣶⠀⠀⣶⠶⠶⠿⠀⠀⠀⠀⠀⣿⣷⠀⠀⢀⣀⣀⣤⣤⣴⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡿⠛⠛⠋⠉⠉⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⣿⡀⢶⡄⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢸⣧⠈⠻⣦⣤⣤⡀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠟⠁⠀⠀⠙⢷⣤⣀⣀⣀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣠⡀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢧⢰⢻⡇⣾⣩⠆⣶⠙⡇⢰⢠⡎⠁⢐⣩⡆⢹⠋⢷⢠⠎⢱⡄⡖⠙⡆⢾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘⠃⠙⠒⠂⠙⠒⠃⠘⠀⠓⠒⠙⠒⠃⠘⠂⠙⠈⠓⠚⠁⠃⠀⠃⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⡆⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣧⠀⠀⠀⠀⠈⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣜⠦⠤⠤⠤⠔⠹⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣛⠦⢄⡴⠋⠁⠀⠀⠀⠰⣏⡶⠈⠳⣄⡹⢈⡇⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠑⠤⣀
⠀⠀⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⠉⡹⠛⠓⠀⠀⠐⢄⠀⠀⡠⠄⠀⠖⣩⣴⠋⠀⠀⠀⠀⠀⠀⢸⣁⡜⣿⣤⢄⣀⠀⠉⠑⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠹⢤⣙⣒⣒⠲⡴⠃⢀⣤⡀⠸⣽⣶⡄⠠⣶⣯⠇⠀⠀⠈⣆⠀⠀⠀⠀⠀⠶⠄⡀⠉⠈⢡⢿⣠⠃⡀⢀⡄⣠⡄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⢠⡒⣲⠀⠀⢻⣀⠀⠀⠀⡸⠀⠀⠈⠑⠢⢄⠀⠀⠀⠀⡼⢀⡟⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢻⣀⠤⠀⠀⠀⠀⠑⠒⠚⠀⠀⠉⠁⢠⡙⢯⡈⠣⡀⢀⠇⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠤⠭⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠤⢴⡏⠁⡔⢋⡴⡋⠀⠀⠀⡴⠤⣄⠀⠀⠀⠀⠀⠀⡽⢦⡙⣦⢸⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠑⠚⠀⣹⠀⡀⠀⠙⠒⠚⠀⠐⡄⢷⡠⢦⢣⠀⠉⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⢩⠖⠋⢉⡗⡠⢦⠤⢤⠖⢳⣸⠅⠈⣧⠓⣤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠀⡠⢊⡞⠁⢸⠀⢸⡄⠀⠀⠀⠀⠙⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠀⠀⣼⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠟⠀⠀⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠛⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣿⣿⣷⣶⣶⣿⣿⣿⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣰⣶⣶⣾⣿⠀⠀⣿⣷⣶⣶⣆⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⣀⣀⡀⠀⠀⢀⣀⣀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠙⠿⠿⢿⣿⠀⠀⣿⡿⠿⠿⠋⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⣶⣦⡀⠀⣠⣾⡿⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢸⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⠘⢿⣷⣾⡿⠋⠀⠀⠛⠛⠟⠛⠛⠛⠛⠛⠛⠻⠛⠃⠀⢸⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⣀⡀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⢻⣿⣄⣠⣾⡿⠋⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⢸⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⠀⠙⣿⡿⠋⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠀⠀⠀⠀⠠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣱⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠿⣿⣿⠿⣦⡀⢐⠀⠀⠀⠀⠀⠀⠙⢛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⣬⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣂⠀⢀⠀⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠰⠌⠀⢉⣄⠀⠀⠠⠀⠈⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠠⠤⢤⢾⣿⣿⣿⢃⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠺⢤⣾⣿⣿⣾⣿⣦⠰⠀⠀⠁⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⣾⡿⣿⣏⣿⡟⢹⡶⢇⡀⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⢻⡍⠻⡐⢳⡇⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠠⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠱⠀⠈⠁⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠀⢲⠀⠀⠒⠀⠰⠃⠀⠀⠀⠀⠤⠐⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡸⠁⠸⠃⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢆⣠⣾⣦⣀⣆⠀⠘⠁⠀⠀⠀⢠⠀⠀⠰⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⢲⡠⢿⣿⣿⣿⣿⠏⢠⡀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣷⠠⣴⣿⣿⢿⣶⣾⣿⣤⠀⢤⠀⠀⠈⠀⢀⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣿⡿⢻⣿⣿⣿⠿⣿⣟⢿⢤⣆⠠⠀⠀⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠋⠉⣷⣿⡟⣿⣿⣀⡿⣏⠠⠾⣥⠺⣤⡀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠫⣍⡙⢳⡃⠖⢱⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠰⠏⠀⠀⣶⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄⢠⠤⣄⢠⠀⣠⢠⡀⡤⢤⡀⠀⠀⣤⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣀⡤⢿⣀⡼⠈⣷⠏⢸⠀⣇⣠⠇⠀⠈⢽⠈⣓⡿

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣿⡛⣛⣛⣛⣙⣛⣿⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣏⣻⣯⣽⣙⣿⣿⣛⣛⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠀⣴⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠤⣤⡄⣠⣀⣴⠟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡄⣄⠘⡿⣆⡡⡾⢹⢃⣀⣠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⠈⠻⠿⠿⣿⢃⣤⠿⠃⠀⠀⠀⠀⠀⠀⠠⢤⡄⢠⣾⣧⣶⢻⡏⣿⢋⡿⢿⣝⣵⢄⡤⠖
⠀⠀⠀⠀⠀⣠⡈⠀⠀⠀⠀⠉⠀⢀⢲⣴⣿⣿⣿⣴⠂⠈⠀⠀⠀⠀⢠⣀⡀⠾⣷⡟⢷⡹⣼⣷⣧⣎⠀⢋⠽⢿⡽⠞⢁⣀
⠀⠀⠀⠀⠀⢀⣡⡀⣠⡀⠀⠀⢀⢄⣼⡿⠿⣿⣷⡄⡖⠑⠀⠀⠀⠀⠀⠘⠺⣧⣯⣝⠣⣵⣿⣿⣿⡿⠧⠀⣀⡘⣷⢯⣍⡩
⠀⠀⠀⠀⠀⠸⣿⣷⣤⣶⡶⣶⣿⣾⣿⠃⣰⣿⣿⣿⣃⠈⠀⠀⠀⠀⠲⣶⣶⣾⣯⣬⣭⣿⣿⣿⣿⡆⠀⠀⠭⢄⣻⣬⣙⠳⠃
⠀⠀⠀⠀⠂⡀⢴⣻⣏⠙⣻⣿⣿⡟⠿⣿⣿⢏⡉⣹⡇⣀⠀⠀⠀⠀⢀⣀⠠⣲⣿⡶⢛⣯⢟⢿⣿⠃⡀⣌⠂⣨⢋⠛⢭⣀
⠀⠀⠀⠀⠀⠂⣈⠿⣻⣾⣿⣿⣿⣷⣾⣿⣿⣿⣷⡏⠂⠀⠀⠀⠀⠀⠀⠁⠒⡿⠒⠙⢿⣵⢋⣾⣿⠀⣕⣰⡶⣿⡺⣃⣀⠉
⠀⠀⠀⠀⠀⠀⠁⢀⠟⡻⠻⡉⠹⣾⣿⡧⢀⢼⡃⠘⠂⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⣴⡷⠏⣿⣿⣿⣷⣫⣿⢮⣿⣀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⢸⡋⠉⠘⠓⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⣸⡿⠇⢱⠈⢡⣷⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠋⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠠⣄⣀⡀⠀⢀⠀⢀⣴⢶⡚⠳⠀⢀⣀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠾⢶⣦⣽⠀⠈⠉⠙⠁⠂⠉⠁⢱⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠠⠐⢢⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠱⠀⠀⣸⢳⡖⠀⠀⠈⠠⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⢀⠃⠀⠀⡿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡤⢺⠁⠉⠙⠛⠁⠀⠀⢀⡎⠀⢠⠴⠃⠀⢉⠓⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢠⠀⠛⠛⠟⠀⠀⠠⣀⠄⠀⠀⣸⠃⠀⠀⠀⠰⠋⠘
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠠⢟⡧⣀⣀⣀⠰⡄⠀⠀⠀⠀⠐⣾⠶⠴⠛⠈⣳⢄⡀⡄⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⡿⠇⠀⠀⢱⣌⣹⠋⠈⠙⠄⢰⣶⣄⣀⣅⠀⢀⣠⣴⢧⠀⠀⠈⠁
⠀⠀⠀⠠⢄⠀⠀⠀⠀⢀⠀⠀⠐⠤⡄⢶⠾⠛⠛⠀⠀⠀⠈⣟⠉⠉⢽⣛⣛⣯⡅⠀⢣⠁⣠
⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢻⣋⠙⠻⠿⢿⡄⠀⠈⠒⠁⠀⠀⡀
⠀⠀⠀⠤⠴⠂⠀⠀⠀⡠⠐⢢⣄⠀⣀⠀⠀⠀⠹⣀⠀⢠⠾⣿⢽⣯⡷⠄⠀⠀⠉⡄⠀⢠⣄⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠹⠀⠈⠀⠒⠒⢂⠀⠉⡁⠈⠀⠀⠀⡳⠳⠾⠛⠋⠀⠉⠀⠈⠀⠀⠈⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠁⠀⠀⠀⠀⠀⢀⡴⠀⡠⡗⠀⠀⢁⡀⠀⡄⠀⠀⠀⠀⠀⢀⠄⠀⠀⢠⡔⠊⠀⠀⡀
⠀⠀⠀⠠⠂⠀⠀⠀⠹⠿⠄⠀⠀⠀⠀⠀⠀⠜⠀⠀⡁⠁⠀⠀⠀⣠⡴⠀⠀⠀⠐⠂⠀⠀⠀⠀⠐⠁⠀⠠⡀
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠋⠔⠉⠀⢠⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⡷⠾⠛⢹⡄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⢇⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠒⠚⠛⠛⠛⠛⠛⠛⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠲⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⢿
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠫⣉⠀⠸⣄⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠀⠀⣀⠟
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠳⡄⢸⠙⠛⠒⠒⠶⠦⠴⠤⠶⠶⠖⠚⠛⡃⣠⡤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⣜⣀⣤⣀⡀⠀⠀⠀⠀⠹⣼⡇⠂⠤⢤⣄⣀⣀⣀⡀⠀⠀⠀⢰⣿⠧⠤⠤⠤⠤⠴⢿
⠀⠀⡴⡿⠋⠉⠙⢦⡹⣄⠀⠀⠀⠀⢹⡇⠀⠀⠀⡤⠤⡄⠀⣿⠀⠀⠀⢸⡿⠶⠤⠤⠤⠴⠶⣏⣀
⠀⡼⢰⡇⠀⠀⢀⣈⣧⠘⡆⠀⠀⠀⠀⡇⠀⠀⠀⠷⢮⡤⠀⣿⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⡏⠉⠓⠦⢤⣀
⠀⣇⢸⡇⠀⢾⣁⣈⢿⡄⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠀⢸⠀⠀⠀⣿⠤⣀⣀⠀⠀⢀⣀⡤⡇⠀⠀⠀⠀⠈⣹⠦⡄
⠀⢹⡀⢧⠔⠋⢩⠿⣼⢡⡏⠀⠀⠀⠀⣧⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠉⠉⠉⠉⠁⠀⡧⠀⡀⠀⠀⡞⢁⡀⣿⡀
⠀⠀⠳⣜⡦⢴⣋⣴⣣⠞⠀⠀⠀⠀⠀⢿⠈⠙⠓⠶⠶⠶⠖⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⠁⡄⣸⠀⡏⠀⢀⣽⡆
⠀⠀⠀⠈⠉⠉⠉⠉⠳⢤⣀⣀⣀⣀⡤⠾⠦⣄⣀⠀⠀⠀⠀⠀⠀⣀⡤⣿⡀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠲⢧⣿⣤⠽⠳⠾⠋⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠤⠒⠻⠉⢹⡄⠉⠉⠀⠀⠀⠀⠀⣉⣉⣉⡛⠛⠋⠉⠀⠀⠈⠳⣭⣓⣒⣒⣒⡪⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠄
⡞⠋⠉⡛⠛⠓⢦⡴⠒⠋⠀⠀⠀⢀⣀⣀⡀⡾⣒⣂⣉⣹⡆⠀⣀⣀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡖⠉⠉⠉⢹⠉⠓⢦⡀
⠉⠒⠲⠧⠤⠤⠼⠁⠀⠀⠀⢠⡎⠁⡸⠉⢉⡷⠦⠭⠭⣾⠖⠫⡁⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣤⣼⡤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣙⣛⣛⡩⠇⠀⠀⡼⠁⠀⠀⣰⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣴⠶⠾⠛⠉⠉⠀⠀⠀⠀⠈⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⣀⣤⣤⣤⡶⣶⠀⠀⠀⠀⣶⢦⣤⣤⣤⣤⣤⣤⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣿⠀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⣠⣴⠶⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢻⠛⠟⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⣣⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡁⠀⡏⢹⣇⣴⠟⠛⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣈⡁⠀⠀⠀⠁⠀⠸⣇⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⣠⡶⠶⣤⣀⡀⣾⡁⠀⠀⠉⠛⠷⢦⣤⣀⠀⠀⠙⣧⡀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⢰⣯⠀⠀⠀⠉⠙⠿⢷⣦⣄⣀⠀⠀⠀⠈⠉⠛⠶⣦⣌⣻⣦⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠙⠷⢶⣄⣀⠀⠀⠀⠈⢉⣛⣿⣶⣦⣄⡀⠀⠀⠀⠈⠙⠛⢷⣄⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⢠⡾⠋⠙⠻⢶⣦⣾⠋⠉⠉⠉⠙⠛⠛⠛⠷⠶⠶⠶⣶⣦⣤⣤⣤⣄⣀⣀⡈⢻⣄
⠀⠀⠀⠀⠀⠸⣧⣀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠛⢷⣦⣄⡈⠛⠶⠶⠶⢶⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⠙⠛⣿⣶⣤⠶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣧⡴⠶⠛⠋⠉⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⢀⣠⣤⣶⠾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⢻⣄⠀⣀⣠⣤⡶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⢀⣠⣤⡿⠿⠛⠉⠁⠀⠀⠀⢀⣀⣤⣶⡾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⢠⡾⠛⠉⠀⠀⠀⠀⣀⣠⣤⡶⠾⠛⠋⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠟⠉
⠀⠘⣷⣀⣀⣤⣴⣾⠿⠛⠋⠀⠀⠀⠀⢀⣀⣤⣶⡾⠟⠛⠃⠀⠀⠀⠀⢀⣠⣤⡶⠾⠛⠉⠁
⠀⠀⠀⠉⠉⠁⢸⣇⠀⠀⣀⣠⣴⣶⠿⠛⠋⠉⠀⠀⠀⠀⣀⣠⣴⠶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⢸⡇⠀⠀⢀⣀⣤⡴⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠟⠛⠉


⠀⠀⠀⠄⡗⢠⡀⣰⠒⠂⡤⢤⡀⠄⠐⠂⠀⣠⠀⣧⠀⠀⠀⡄⢠⠀⡄⠀⠠⠀⠰⠀⠴⡄⠈⠀⡘⠀⡀⢪⠦⢐⠀⠰⠂⣀⡴⡄
⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠒⢠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠈⢀⢀⣀⣀⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠎⣁⠑⠊⠉⠑⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠀⠤⢿⣧⠤⢀⠔⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣫⡿⣿⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠈⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠝⠿⠋

⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⡀⡐⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⡀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠠⠜⠈⠀⠈⠢⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⡀⠀⠀⠆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⡀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠄⠀⠀⠐⠄⠰⠄⠰
⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠠⡀⢰⢀⠅⢠⠀⠀⠀⠀⡀⢄⠈⠈⠀⠛⠠⠀⠀⠀⠀⠢⠀⠂⠀⠂⠀⠒⠆⠀⠀⠀⠀⠘⢉⡠
⠀⠀⡀⠀⠄⠄⢀⠀⠀⠂⠀⠀⠀⢢⠄⠁⠀⠈⠑⡀⠁⠀⠀⠠⡤⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠈⠁⣨⠀⠀⠀⠀⠀⠘⠄⡀⠀⠀⠀⠢⠀⢄⠄⠁⠀⡀⠀⡼⣘⠤⢤⡁⡀⠀⠀⠉⢀⠀⢠⠀⢤⠀⠁
⠀⠠⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⢈⡈⢠⣀⠀⢀⡠⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠌⡇⠀⠀⡡⠐⡀⠀⠀⠈
⠐⠀⠀⠠⠀⠀⡀⠤⠠⠄⡄⠀⠀⠀⠈⠀⠎⠱⠀⠒⠀⠀⡀⡀⠀⠀⠂⠆⠀⠀⠀⠀⠀⠅⣌⠒⡊⠁⡐⠀⠀⡀⠰⠀⠀⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠂⠀⠀⠁⠐⠁⢂⠀⠀⠀⠀⠀⠀⠂⠣⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⡀⠈⠀⡈⢁⠤⠒⠂⠤⡎⠀⠤⠀⠁
⠀⠁⠀⠀⣅⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⡀⠀⠐⡀⠠⡐⠂⠁⠃⠀⠂⠔⡐⠀⠀⠠⠊⠄⠀⢀⡀⠈⠈⠀⠀⠀⠀⠀⠢⠤
⠀⠂⠀⠀⠐⢄⡀⠀⠀⠀⡀⠆⠀⠀⠀⠀⠘⠈⠀⠤⠃⠀⠀⠀⠀⠀⠄⡀⠀⠀⠃⡀⠈⠀⠀⠀⠀⠠⠰⡀⠀⠀⠀⠀⠀⠬⠀⠤⠁
⠀⠀⠀⠀⠀⠀⠁⠑⠀⠀⠁⠀⠀⠀⠤⠀⡀⣀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠄⢀⠀⠄⠀⠀⠔⠐⠤⠀⣀⡠⠐⠁⠂
⠀⡀⠁⠐⠁⣀⠀⠀⠀⡀⡀⠀⠀⠈⠀⠀⠐⠀⢄⠀⠀⡀⠀⠀⠀⠐⠐⠀⠊⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠃⠀⠃⠀⠐
⢠⠄⠀⠀⠀⡀⠀⢀⡌⠁⠈⠉⠄⠀⠀⠀⠄⢀⣂⠀⠀⠁⠀⢠⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⠀⠡⠅⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠂⠀⠀⠀⠀⠀⢆⡀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈
⠀⠀⠀⢀⢀⠀⡁⠀⠀⠈⠊⠀⠀⠄⠀⠀⡤⠀⠀⠀⠉⠆⡀⡀⠀⣀⠀⢀⠤⠀⠃⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠔⠀⠀⠀⠁⠁⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢂⠌⠀⠀⠀⠁⠈⠐⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠒⠀⠀⠀⠉⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠁⡄⠀⡀⠄⠐⠀⠀⠀⢈⢐⠠⠓⠊⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⠀⠀⠀⠀⠀⡧⠂⠀⠀⠐⠀⠀⠀⢀⠠⠄⠤⢀⡀⠀⠄⠀⠀⠀⠀⠀⠄⠀⠆⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠐⢂⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⠀⠠
⠀⠀⠀⠂⠁⠀⠈⠀⠀⠀⠀⠀⠀⠗⢂⠠⡖⠠⠈⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡠⠀⢀⠂⠀⠀⠂⠁⢀⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠘⠀⠐⠠⠀⠀⠀⠈⠐⠂⠐⠈⠁⠀⠀⡀⠀⡀⠀⠀⠀⠀⡀⠀⠒⠀⠀⠰⠄
⠀⠀⠀⠀⠂⠀⠁⠀⠀⠠⠌⠊⠀⠒⠁⠄⠀⠐⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠐⠆⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⢀⢄⢄⠀⠀⠀⠀⠀⠤⡨⠒⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⢠⣾⣎⣾⣦⡈⠀⠀⠀⠀⢨⣈⣡⠃⠀⠀⠀⠀⠈⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⣿⠇⢹⣿⣇⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢰⠉⠀⠑⠄⠀⠀⢰⣦⣴⣤⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠸⡩⠊⠢⡤⠎⠀⠀⠀⠘⣿⣀⣸⡟⠀⠀⠀⢀⣼⣿⣶
⠀⠀⠀⠀⠀⠢⡀⣀⠇⠀⠀⠀⠀⠀⠈⠛⠏⠀⠀⠀⠀⠘⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣠⠿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⣶⣾⣿⣿⣷⡀⠀⠀⠀⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⢰⠖⠒⠒⠒⠲⡆⠀⠀⠀⠻⡿⠿⡿⠃⠀⠀⢠⡞⠡⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⣸⠸⡆⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⢀⡤⢦⣄⠀⠀⠀⠸⣇⠀⠁⣄⣸⠃
⠀⠀⠀⠀⠀⠀⢧⣀⡼⠀⠀⠀⠀⠈⠴⠿⠷⠉⠀⠀⠀⠀⠛⠛⠛⠛⠃⠀⠀⠀⠈⠓⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣦⡀⠀⠀⠀⢸⣿⣧⣽⣿⠂⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡴⣏⣩⠙⢢⡀
⠀⠀⠀⠀⠀⠀⣿⢔⢼⡇⠀⠀⠀⠘⣿⠃⢹⣿⠀⠀⠀⠀⠀⠈⠣⡀⠀⠂⠀⠸⡇⠚⠻⣗⢠⡇
⠀⠀⠀⠀⠀⠀⠻⠶⠾⠃⠀⠀⠀⠀⠿⠇⠼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠬⠼⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠄⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢄⠀⢀⣀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⡲⠀⠀⠈⣲⣯⣀⢸⣤⡜⠀⡀⡋⣨⠇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⡀⠀⠈⢙⣿⢔⡊⠁⠀⠃⠀⠐⢀⡷⡆⠈⠹⡧⠢⢄⢇⠭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡷⠤⠊⠁⠈⠆⠀⠀⠀⠀⠀⠀⢎⠉⣕⠄⢺⢉⡢⠀⠚⣢⢊⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⡠⠚⠏⡴⠋⡆⠀⠀⢠⡊⠩⡱⠀⠀⠰⠷⣧⠀⣀⣾⡀⠀⠉⣸⠏⢣⣀⠋⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢄⡱⢎⠀⠀⠀⠉⠙⠟⠒⠀⠀⠀⠑⠜⢂⡀⢀⠉⠀⢸⡀⠀⡱⠠⡺⡠⠣⠂⠰⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠆⠁⠀⢀⣀⠀⠙⠃⠀⠀⠀⠀⠀⠧⡀⢄⡕⠀⢈⣰⡯⢀⢔⣕⡕⡁⠬⢝⣉⡳
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠁⠀⠀⣎⡘⠤⠀⠀⠀⠀⡀⠀⠀⠀⠩⠼⠅⢀⠌⠈⣣⠁⠽⢁⠑⠨⡘⡡⢈⣇⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠉⣵⠄⠀⠀⠙⠉⢽⠂⠀⠀⡎⠈⣁⠀⡠⠀⠐⣤⠄⡫⢛⠕⠴⡋⡓⠢⢊⡨⣊⠠⠝⢤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡇⠹⢰⠒⠄⠀⠀⠈⠀⠀⠀⠈⠲⠿⠂⣳⡮⠓⠅⠊⣂⡑⠈⢣⣡⠀⠬⡙⢰⣚⠔⠧⢤
⠀⠀⠀⠀⠀⠀⠀⢰⠒⣴⣤⡀⠈⠂⠛⡿⠇⠀⡔⠤⢂⣤⠀⠀⠐⢉⠤⢰⠈⣁⡦⡘⢆⠥⣪⠚⢕⡙⡩⢕⡱⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⡇⢀⠀⠀⠀⠀⠀⠱⠾⡮⢗⠨⡢⠃⠣⠘⡼⢉⡉⣍⣊⠕⡪⢔⢍⡆⣈⡏⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⣵⢲⠦⠀⢠⡄⣀⠑⢮⡳⢟⡩⣚⠽⣫⠴⢗⣽⣠⠜⠁⡺⠔⢥⢝⠀⠖⢬⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠉⠳⠙⣏⣉⠎⢑⣢⡭⡠⠀⣨⡴⢢⢸⣂⡥⠒⡰⡿⠺⠽⢲⠾⢇⢜⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⢢⠖⠺⣡⠆⣐⢧⡤⠕⠹⢥⠺⣇⣠⠝⢀⣧⠱⡧⠴⠴⠆⡴⠋⠀⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⣖⡉⡾⢏⣃⠤⢒⠈⠐⠩⠂⠔⡪⢀⠷⣲⢿⣶⠶⡜⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⡆⡕⠮⡇⢎⣠⢵⡎⠥⠞⠲⡘⢄⡀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠬⠘⠧⠤⠃⠀⠀⠀⠀⠐⠷⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠤⣂⠤⣀⠠⢀⡠⢔⡠⠔⣂⠤⢒⡀⢄⠢⠄⡲⠄⡒⠭⢒⠨⠔⡨⠓⠈⠅⠒⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠁⠀⠁

⠀⠿⠿⢿⣦⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡸⠧⡀⠀⠀⠀⣀⣀⣀⠀⠀⣤⡄⠀⠀⠀⠀⣀⣀⣀⡠⠠⡐
⠀⢒⠠⠬⠛⣁⡈⠁⠀⠀⠀⠀⣀⣤⣆⣀⠀⢠⣤⣤⡄⠀⠀⠀⠘⠿⢏⠀⢀⣉⣁⡄⠀⠀⠀⣿⣿⣿⣧⡄⠀⠠
⠀⠀⠀⣴⣶⣾⣿⣿⡆⠀⠀⢸⣿⣿⣿⣿⡗⠸⠛⠛⠃⠀⠀⠀⣸⣷⣾⡆⢾⣿⣿⡇⠀⠀⠀⢸⡙⠻⠟⠁⠔⠁⠶
⠀⠀⢀⠹⠿⠿⠿⠿⠃⠀⠀⠈⠉⠉⠉⠉⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠁⠄⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⢐⠉⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣢⠀⠀⠀⠀⢘⠀⠈⡟⠑⠢⢀⣀⠀⠀⠀⣀⣇⢀⣿⣿⣿
⠀⠱⠊⠀⠀⠀⠀⠄⡴⢄⠀⠀⠀⠀⠀⠐⢙⠋⠃⠀⠀⠀⠸⢿⣝⡀⠀⠅⠠⠴⠾⠻⠀⠀⠀⢹⣇⣠⣽⣻⣏
⠀⠀⠀⠀⠀⠀⠈⣰⣁⡠⠃⠀⢀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢉⣧⠤⠜⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⢠⣨⣭⣭⣥⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⣀⣤⣶⣷⣦⣄⡀⠀⠀⠀⡖⠒⣶⠒⢒⠒⣶⡒⢲
⠀⠀⠸⢇⣀⣀⠸⡄⠀⠀⠀⠀⠀⠘⢇⣀⣀⠸⠁⠀⠀⠀⠀⠘⢩⣩⡉⢉⣉⠉⢹⠁⠀⠀⡧⠴⣿⠦⢼⠤⣿⠦⢼
⠀⠀⣠⣶⣿⣿⣶⡄⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⡄⠀⠀⠀⠀⠸⠘⢛⢃⢸⣿⠀⢸⠀⠀⠀⣇⣠⣿⣄⣸⣀⣿⣅⣸
⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⠤⠬⠼⠾⠿⠤⠼⠀⠀⠀⠀⠀⠤⠼⠿⠧⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣒⡒⠢⣄⠀⠀⠀⠀⠀⣠⠔⠒⠒⠲⢤⡀
⣰⢿⣿⢷⡄⢠⣴⣤⠀⠀⠀⢀⣀⠀⢐⢆⣤⠀⢀⣀⠀⠀⢠⠋⣷⣆⣿⡅⣶⠎⢧⠀⠀⠀⡜⢹⣶⡄⠀⣶⣶⠹⡄
⠘⢿⣿⡾⠃⠻⣿⡿⠃⠀⠀⢨⣏⠠⣤⣤⣤⡦⣩⣏⠀⠀⢸⠰⡿⣾⠟⢷⡾⡷⢸⠀⠀⠀⡇⣼⣿⡝⠢⣽⣿⠀⡇
⠀⠀⠀⠸⣿⠆⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠛⠛⠀⠀⠈⢦⣰⣿⡷⣾⣷⡦⠋⠀⠀⠀⠱⣟⠛⠃⠀⠛⢻⡞
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣷⣶⣶⣯⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣾⣥⣤⣤
⣿⣯⣼⣭⣬⣭⣯⣿⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡇⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢝⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⡀
⡇⠀⠀⠀⠻⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠁⠀⠀⠀⠀⠀⠀⢠⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠻⢿⣿⣿⣿⣿⣿⣿⣿⠻⡕⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣏⣯⣵⣿⣿⣿⣿⣟⠅⣿⣿⣛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣾⣿⣿⣧⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⢯⣿⠇⡹⣿⣿⣿⣦⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣠⣴⣶⡌⢶⣇⣽⣿⣿⣷⣌⣀⣨⠻⣿⣿⣿⣿⠿⢟⣋⣭⣽⣿⣽⣿⣿⣿⡇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢋⣠⣿⣿⣧⠀⠻⡿⣿⣿⣿⣿⢿⣿⣦⠼⠟⣋⣴⠞⣩⠿⢷⢸⣿⣿⣿⢿⣿⣷⣄
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠙⠉⠿⠿⠆⠚⠀⠈⠙⠙⣿⣿⣭⣤⣶⣿⣿⣿⣿⣁⣀⣴⡿⠻⣻⣿⣿⣿⣿⣿⣿⣄
⠀⣠⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣟⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣋⣭⠴⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⣼⣿⣿⣿⣿⣿⣿⣿⡄⠐⠺⡏⠛⠿⣼⡗⠀⠀⠀⠄⠀⠀⠀⠈⢻⡶⣦⡟⢿⣿⣿⣿⣿⣿⣿⣭⣥⠄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠙⠻⣿⣿⣿⣿⣿⣿⣷⣄⣀⣠⣶⠀⠃⠁⠀⢀⣄⣢⠀⠀⠀⠀⠀⢳⣶⣶⣶⣿⡿⠿⠿⣿⣿⡟⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⠷⢄⡾⠾⠿⠛⣉⣠⣤⣤⣶⣾⣿⠟⣿⠻⠂⠀⠀⣠⣾⣏⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠐⠻⣿⣿⣿⡏⠍⠉⡉⢠⣠⠿⠇⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣸⣿⣿⣗⠀⠀⠈⡏⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣼⣿⣿⡿⠀⡀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠸⠏⠉⠉⠢⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣿⣿⣿⣷⣶⣶⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣏⣿⣹⣏⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⡟⢛⣿⠛⠛⠛⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠟⣛⣩⣭⣙⡛⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠛⠛⠛⠛⠒⣠⣾⣧⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⣱⡞⠋⠁⠀⠉⠛⢶⡙⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣰⣶⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣶⣶⣦⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢰⡏⠀⠀⠀⠀⠀⠀⠈⣷⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⣀⣤⣾⠃⠀⠀⠀⠀⢸⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠸⣧⣤⣤⣤⣤⣤⣤⣤⡿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡟⢷⣄⣭⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⢠⣤⡿⣿⣿⣿⣧⣙⠏⠀⢠⣤⠀⠀⢿⣡⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣿⣧⠚⠉⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⣿⡟⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⣴⣿⣿⣷⡄⢸⣿⣿⣿⡿⢣
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⢠⣼⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⡇⠸⢧⣿⣿⣇⠿⠀⣿⡿⢋⣴⣿
⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠗⣺⠁⠀⢸⣿⣿⣿⠀⠀⢹⡒⠛⠛⠛
⠀⠀⠀⠀⣀⣶⣛⣶⣖⣲⣀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢸⣿⢻⣿⠀⠀⠘⡇
⠀⠀⠀⠀⠙⠋⠉⠉⠉⠉⠉⠛⠛⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢛⢻⣿⡇⠀⢠⣶⣦⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠃⠀⠀⠀⠀⠀⢸⣿⣦⣀⣴⣿⡇⠀⣸⣿⣿⣅⡀⠀⠀⠀⠺⠛⠿⠛⠛⠛⠛⠛⠛⠛⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠀⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⣴⣶⣾⣿⣿⣿⣿⣶⣿⢾⣿⢿⣿⣿⣿⠋⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠛⠛⢻⣿⣿⣿⣿⣿⣾⣾⣿⢾⣿⣻⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⡇⡇
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⠉⠁⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡖⢻⠑⡆⢯⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠘⠛⠀⠰⣏⡗
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⡿⣩⣭⡹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢈⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⡹⢿⣿⡇
⠀⠀⠀⠀⠀⠸⢇⣿⣿⣇⠿⠀⠀⠀⠀⠀⠀⢀⣤⣦⠸⣿⣿⠇⠀⠀⠀⠀⢰⡿⠛⠉⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡻⠟⠻⠷⠻⠾⠿⠃
⠀⠀⠀⠀⢸⢳⣿⣿⣿⣿⣷⢻⠀⠀⠀⠀⠀⣈⢿⣿⣧⣰⣷⣦⣀⡀⠀⠀⠙⠷⣶⠄⢻⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⠈⢿⣿⣿⡻⣿⡷⠶⠿⠁⠐⠒⠂⠒⠒⠂⠤⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠿⠛⢆⣤⣶⡝⢋⣾⡿
⠀⠀⠀⠀⠀⢙⢻⣿⣿⣿⢛⠉⠀⠀⠀⠀⠀⣿⣀⣈⣻⣛⣁⣸⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣿⣧⣤⣦⣴⣿⣿⢟⣵⣿⠟⠁
⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣰⣾⣿⡇⡇⠀⠀⠀⠀⢠⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⢸⢸⣿⣻⣿⢸⠀⠀⠀⠀⢀⡀⣿⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣋⣛⣁⣀⡀⣇⡀⣀⡀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠒⠚⠸⠿⠻⠿⠘⠒⠂⠀⠀⠀⠀⠻⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠟⠈⠀⠀⠀⠛⠿⠿⠛⠛⠻⠿⠛⠛⠛⠛⠛⠛⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣦⢤⢤⣴⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⢠⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣼⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠀⠁⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⣿⣿⠿⠿⠿⣼⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡷⢶⣦⢉⡉⡋⠉⠁⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⡧⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⠉⠓⢥⡀⠄⣪⣽⡄
⠀⠀⠀⠀⠀⠀⢸⣿⠶⠴⡶⢶⠒⣻⠂⠀⠀⠀⢀⣹⡥⡀⠀⠀⠹⡿⠿⠿⠿⢿⣶⣶⡶⢶⣤⣤⣇⡀⢸⠉⠙⠓⠒⢲⠀⠀⢱⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣀⠸⠀⠿⡂⠀⢠⣾⡿⠛⠿⣖⢄⠀⣠⣷⣄⠀⠀⠀⠀⠈⠉⠲⡄⠈⠙⣿⣿⡄⠀⠀⠀⢸⠀⠀⢸⢸⣿⣿⣇
⠀⠀⠀⠀⢀⣤⠟⢳⣯⡿⠯⢿⣄⣶⣛⣦⣼⣿⣦⣤⣤⣾⣿⣟⢋⣡⣾⡁⠀⢀⣹⣿⣿⣷⣷⣦⣶⣿⣿⠁⠀⠀⠀⢸⠀⠀⢸⣼⣿⣿⡟
⠀⠀⠀⠀⡿⢥⣀⣀⠈⠉⠉⠉⢡⣿⣤⣤⣠⡤⣼⣟⣭⢫⡉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⣽⢫⣿⡏⢸⠀⠀⠀⠀⢸⠀⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⡇⠀⠀⠈⠉⠉⠉⠉⢹⣿⣿⡆⡇⠀⢸⡟⣸⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⡇⠸⠀⡀⠀⠀⢸⠀⠀⢸⡟⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡏⠓⠒⠒⠒⠒⠚⣿⢛⣿⠻⠛⣿⡇⣿⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠻⣿⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⣠⣾⣛⠋⣘⣃⢨⡳⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠂⠉⠉⠉⠀⠒⣦⡀⠀⠀⠀⠀⡇⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⣞⣿⣿⣿⣾⣿⣿⣗⠙⡘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠛⠛⠒⠚⠋⢹⡇⠀⠀⠀⠀⡇⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠳⠤⠤⢤⣤⠤⠴⠚⡇⠀⠀⠀⠀⡇⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡇⠀⢉⠉⠉⠉⢉⣉⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡉⠛⠒⠤⠶⠶⠶⠖⠙⢙⣆⡠⠤⢤⣧⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡇⠀⠘⢒⡶⠻⣟⠛⠀⠀⣿⣝⡢⢤⣄⣀⣀⣀⣀⣀⡤⣿⣝⣲⠲⠤⠤⠤⠤⠒⣎⣭⡏⠀⠀⠀⠀⠈⠉⠙⠓⠲⠦⢤⣄⣀⡀
⠀⠀⠀⠀⣧⠀⠓⠤⠤⠤⢤⡇⠀⢰⡋⠀⠀⠈⣱⠆⠀⣿⣿⡿⠋⠓⠓⠒⠒⠒⠒⣋⣿⠀⠉⠉⠛⠒⠛⠋⠉⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠽
⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⢀⡇⠀⢀⠙⢦⣤⠞⢁⠀⠀⢿⣿⡇⠒⠢⢤⠖⠂⠛⢿⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡇⠀⠈⠙⠋⠛⠋⠋⠀⠀⣸⠿⣇⣀⡀⢸⣶⣶⣲⠷⡳⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣦⣤⣀⣀⠴⠊⠁⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣷⣶⣦⣤⣤⣴⣶⡆⢹⣿⠀⠀⠈⢉⡹⣻⡉⢈⠇⢱⣿⣄⣀⠀⠀⠀⠀⠀⠀⣀⡤⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡤⠞⠉
⠀⠀⠀⢠⠶⠶⠖⠒⠒⠒⠒⠛⠉⠉⠉⠙⣿⣟⣛⣡⠴⠃⣴⣆⣢⠿⠿⣿⣏⣁⣠⣿⣽⠀⠉⠙⠛⠒⠒⠛⠋⠁⠀⣿⠉⠙⠻⠿⢿⣿⣿⣇⡤⠞⠉
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠄⠀⡄⠀⡈⠉⠛⠂⠀⠼⠈⠉⠉⠀⠘⠧⣄⣀⣀⠀⠀⢀⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣗⠺⢿⡶⠦⠛⠲⣎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠸⡧⠤⠤⠶⠒⠒⠒⠒⠒⠚⠛⠋⠉⠉⢻⠀⠈⠳⠖⠚⠉⠁
⠀⠀⠀⠀⣧⡤⠤⠤⠤⠴⠶⠶⠒⠒⠒⠒⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⢒⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⡀⠖⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡈⠄⠀⠀⠈⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠐⠀⠀⠀⠀⠀⠴⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠠⠀⡀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠂⢀⡀⠐⠁⠀⠀⠀⠄⠂⠠⠄⠀⠀⠀⠀⠠⢀⠀⠀⠤
⢀⠔⠐⠀⠀⠔⠁⢀⢀⡕⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠄⠐⠂⠀⠀⠈
⠐⢀⢀⠌⠀⠀⠢⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠊⠀⠠⠁⠀⠆
⠀⠀⠀⠀⠀⠀⢤⠀⠀⠒⠎⠁⠀⠠⠤⠀⠂⠈⠀⣀⡀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⢀⠈
⠀⠀⠀⠰⢀⡀⢀⠀⠀⠀⠂⠀⠀⣚⣛⣱⠀⠀⢠⡤⣥⣤⢤⣤⣤⣤⣤⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠰⠧⠀⠀⠠⠀⠧
⠀⠸⠈⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣷⣿⣺⣼⣼⣷⣿⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠐⡚⠀⠀⠠⠈⠀⠀⠀⠀⠀⣶⠀⠀⠀⢨⡉⢩⢉⡍⣉⣭⣍⣩⠀⣄⣤⣤⡌⠁⠈⠁⠀⠀⠉⠀⠀⠀⠀⣀⠀⠠⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣿⢸⣇⢿⢶⡿⢸⠀⣿⢙⠶⣍⠀⠀⠀⠂⠀⠐⠀⠀⠄⠀⠆⠀⠀⣀⡀
⠰⡇⡔⠲⠠⠇⠀⠈⣁⣈⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠸⡿⢸⠈⣧⠸⣶⡷⠻⠶⠿⠀⠀⠀⠈⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠠⠄⠀⢤⠠⠠⢠⣄⡄⡄⡄⡤⠀⠀⠈⠐⠈⢀⠀⠈⠀⠀⠀⠀⡀⠀⠐⠂
⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠘⠀⠘⠋⠃⠃⠉⠛⠀⣀⠂⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⢀⠐⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠂⠄⠀⠘⠀⠀⠐
⠀⠀⢀⠀⠀⠀⢒⠀⠃⠀⠀⠂⠀⠀⢸⡄⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⠀⠀⠀⠈⠁⠈⠀⠀⡀⡐⠒⠀⠡⢀⠀⠀⠀⠀⠀⠘⢕⠄⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⢐⠀⢒⠀⢠
⢀⠀⠀⢀⠀⠀⠀⠀⢁⠰⡊⠐⠪⡀⠀⠀⠀⠀⢀⢈⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠀⡴⠀⠠⠃⠄⠈⠀⠀⠀⠔
⠀⠁⠀⠀⠇⠀⠀⠀⢚⠐⠠⡀⢀⡅⠀⠀⠀⠀⠀⠋⢤⡄⡠⡀⠋⠀⠀⡀⠀⠀⠀⠀⢀⠠⠄⠰⠠⠐⠀⠂⠠⠀⠀⠴⠴⠶⠲⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⠠⠓⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠓⠴⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢀⠒⢀⠐⠃⠀⠀⠀⡐⠀⠠⡀⣀⠀⠀⠀⢀⣀⣂⣀⣀⣂
⢰⡟⣦⢰⡖⠂⢠⡶⡆⡴⠲⡆⢰⡖⠂⣶⠒⠀⠀⠀⡀⠐⠀⠒⢀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠠⠙⠁⠀⠇⠀⠀⠈⠁⠎⠰⠁⠌
⢸⡷⣏⢸⡏⠁⠈⠳⡄⣿⠲⡇⢸⡏⠁⣟⠉⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠈⠐⠊⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⡁⡉⠈⡁
⣈⣛⣃⣈⣉⣀⣈⣛⣁⣁⣈⣁⣘⣁⣀⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣂⣀
⣿⣿⣋⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠰⣦⣀⠀⠀⠰⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠀⠀⠀⣀⣴
⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠉⠳⠆⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢴⠞⠉⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄⠉⠀⠀⠀⠀⠘⠉⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿