Burgle Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠈⠉⠉⠁⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠲⣶⣶⣶⣶⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠈⠋⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡽⠙⢧⣀⣀⣀⣼⣧⣄⣠⣼⠿⠙⢏⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢈⠝⠉⠀⠀⠉⠫⡀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠘⠀⠀⠀⠈⠂⠤⣀⡀⠤⠚⠁⠀⢀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⡠⠒⣾⣶⠒⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠒⣶⣷⠒⢄
⣀⠄⠒⠚⢿⠖⠛⣷⣦⣤⣽⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣏⣤⣤⡞⠛⢲⡿⠓⠒⠠
⠣⣖⠟⡀⢰⠃⢀⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠀⠘⡄⢰⠷⣢
⠀⠀⠀⠑⠊⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣃⠀⠑⠈
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡏⠉⠙⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠋⢹
⠀⠀⠀⠀⠀⠀⢧⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡼
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⠈⠉⠉⠉⠉⠉⠉⠁⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⣄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣽⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⢉⠉⠙⠛⠒⠒⠦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⡙⠉⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⢀⣴⣿⢦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣯⣿⣿⡿⢿⢻⣿⣷⣶⣤⣀⡐⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡩⠍⠙⢻⣿⣺⣿⣿⣧⠀⢽⠿⣿⣿⣿⣿⣿⣷⣦⣄⣐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠈⠙⢿⣿⣽⢦⡀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠙⢿⣸⣿⣦⣀⠀⠀⠈⣙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⠿⣷⡆
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⢀⣽⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠛⠛⠿⠟⠛⠛⠛⢿⣿⣧⡀⠀⠀⣠⣿⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⠤⣄⠀⠞⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡀⠀⠠⠻⢟⣽⣶⣿⣿⠟
⠀⠀⠑⠤⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠘⠲⣁⠀⢀⠂⠈⠓⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠈⠓⠦⣀⠀⠄⠀⠀⠈⠉⢙⠒⠒⠶⠶⣴⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣛⣷⡀⠀⠀⢈⣿⣿⣦⣄⠀⠀⠈⠉⠒⠦⢬⣀⣀⣂⣀⠀⠠⠞⠁⡰⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻⣿⣷⠀⣴⣛⣻⠿⢿⣿⢟⣢⣄⡀⢀⠀⠀⠀⠀⠸⡁⠀⠀⠐⢯⠁⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢿⣿⡿⠃⠀⠀⠁⠛⠛⠻⢿⣿⣿⢆⣘⠁⠀⠙⠀⠠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠉⠉⠉⠉⠁⠒⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠮⠤⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠃⢤⣀⠀⠀⠀⠀⠰⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣼⣿⣿⣿⣿⡿⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠘⠀⠈⠻⣿⣿⣿⣇⠘⠛⠿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⠀⠀⠀⠈⠻⣿⣿⣦⣀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣷⣦⣀
⢀⡤⠒⠀⠈⠐⢤⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣶⣿⣿⣿⣿⣿⢿⡿⣿⣿⣷⡀
⡎⠀⠀⠪⡤⡄⠀⢡⠀⠀⠀⠀⠀⢀⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠻⣿⣦⣀⣠⣾⣿⠗
⠣⡀⠀⠀⠘⠁⠀⠈⡀⣀⠔⠒⠤⣎⠀⠙⣄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠂⠀⠀⠙⠿⢿⣿⣿⡃
⠀⠈⠑⠂⠤⠄⢀⡔⠫⣱⠖⠑⢦⡈⢆⡀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⡸⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠂⠡⠀⠀⠀⡇⠀⠑⢄⠀⠀⠉⠓⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠑⣤⡀⠀⠙⠢⢄⡀⠀⠀⠈⠉⠓⠒⠖⢲⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡄⠀⣠⣿⣿⣶⣤⣀⠀⠈⠑⠒⠢⠤⡤⢴⠒⢁⠴⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣾⣿⠟⠉⠛⠿⣿⣿⣶⣦⣤⠤⠊⠑⣄⢉⡨⠖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⣿⣿⢠⣿⣿⡇⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣰⣿⣿⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣿⣿⣿⣾⣿⣿⣷⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣇⡀⣴⡿⢿⣄⠀⠀⠀⣠⣶⡄⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠸⣿⡄⢀⣴⠟⠈⣿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠙⠿⠛⠁⠀⠀⣿⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⣿⣇⣀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣷⠀⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠹⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⡟⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠈⠻⣷⣶⣦⣤⣤⣴⠶⢶⣾⣿⣿⡟⠉⠉⢻⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣿⡿⣿⣿⣿⡇⠀⠀⣿⣿⠿⢷⣤⣤⣿⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣿⠀⠈⠙⠻⣿⡶⠟⠋⠁⠀⠀⠈⣿⡟⢨⡇⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠛⠛⠋⠀⢸⡇⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠸⣧⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣆⠀⠀⣾⠁⠀⠙⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠹⣆⠀⠀⢀⣾⣿⣿⣿⣿⣿⠏⠀⢰⠏⠀⠀⠐⠚⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠘⢧⣀⠘⠛⠿⠿⠛⠉⢀⣠⡴⠋⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠉⠛⠒⠒⠚⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣠⣄⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣷⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠙⠛⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⣿⡅⠀⠀⠀⣀⣀⣴⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣀⠀⠀⠀⠀⠀⢀⣤⡄⢸⣿⠀⠀⠀⠀⣠⣴⣿⡿⠛⠁⠀⠙⣿⣤⡴⠟⠋⠹⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⡀⠀⠈⠛⠃⢸⣿⠀⣀⣴⣿⡿⠟⠉⠀⠀⠀⠀⣼⢿⣿⠁⠀⠀⠀⢹⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠻⠶⣦⣤⣼⣿⡾⠿⣿⣅⠀⠀⠀⠀⠀⠀⠘⠃⠀⠛⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢿⣦⡀⠀⠀⠀⠀⢀⣤⣾⣏⣀⣀⣴⡾⠟⠋⠻⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣧⣄⣀⠙⠻⢷⣦⣤⡴⠟⠉⠉⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠛⠉⠉⢻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣤⡀⠀⠀⠀⢀⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠁⢀⣠⣴⣿⣿⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠛⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠋⠉⠉⠉⠙⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣏⠁⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣀⣀⣀⣀⣀⣠⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⢤⣀
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠦⣄
⠀⠀⢀⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣄
⠀⣰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢧⡀
⢰⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣷
⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠑⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠁
⠀⠀⠀⠀⠈⠉⠓⠒⠶⠤⠤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⡤⠤⠴⠖⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠶⠶⠶⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⣛⣯⣵⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣯⣝⡻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣽⣿⣭⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡩⠀⠁⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⠷⠮⠩⡛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠄⠀⠀⠈⣿⡇⠈⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡁⠀⠿⣿⠃⢄⠀⢀⠟⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡻⡀⠀⢀⡿⡇⠸⠿⠿⠿⠿⣿⣍⡙⠛⠛⠛⠛⠛⠛⠛⢛⣉⣭⣿⣿⠛⠋⠉⠁⠀⠀⠘⢀⡼⢠⢣⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡲⠈⠀⠀⠀⠀⠀⢀⠂⠰⣿⡿⠟⠃⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠀⡤⠃⠀⠀⠀⠀⠈⢀⢡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣣⠠⡆⠀⠀⠀⠈⠓⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣭⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣷⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡼⡄⠀⠀⠀⠀⠀⣤⠀⠀⠈⠂⠀⠤⠤⠀⠈⠀⠀⢀⣴⠀⠀⠀⠀⠀⢠⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡹⣄⠀⠀⠀⠀⢸⣿⠀⠐⠒⠒⠢⠀⠐⠒⠂⠉⠽⠏⠀⠀⠀⠀⡰⢣⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⢧⡀⠀⠀⠀⠑⠄⢀⠀⠒⠒⠒⠒⠀⢁⠠⠊⠀⠀⢀⣠⣞⠱⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⣋⣴⣿⣦⣀⡀⠀⠀⠠⢀⠉⠁⠉⠁⠀⠀⠀⠀⣰⣶⣿⣿⣿⣿⣷⣎⢿⡄
⠀⠀⠀⠀⠀⣤⡶⢶⣤⣀⠀⠀⠀⣠⡾⣡⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠁⠀⠉⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡞⣷⡀
⠀⠀⠀⠀⢸⣧⠂⠀⠈⢙⢷⣄⣰⢟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠄⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡻⢦⣄
⠀⠀⠀⢠⡿⠡⠷⣄⠀⢠⣆⡿⣣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣝⠷⣶⠶⢟⠷⣆
⠀⠀⠀⣼⢇⣄⣀⠀⠱⠀⠱⣾⣿⣟⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠂⠉⠀⠷⣽⠂
⠀⠀⢸⡧⢆⠀⠙⠋⠀⠀⠀⠈⠙⠉⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣹⣿⣿⡿⠋⡀⠔⠛⠂⢿⡄
⠀⠀⣼⣳⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣆⠀⠙⠛⠁⠀⠘⠡⢤⡤⢾⣎⢷⣄
⠀⠀⠸⣧⡳⣬⠆⠀⠀⠀⢀⠤⢴⣿⣿⣿⡿⣉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡙⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⢻⡆
⠀⠀⠀⠈⢿⡸⣶⣖⠒⠂⢀⣠⣾⣿⣿⡿⣱⢿⡜⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣜⢦⡀⠀⠀⠀⠀⠉⠁⣸⣿⣿⣾⡇
⠀⠀⠀⠀⠘⣧⡻⣿⣿⣿⣿⣿⣿⣿⢟⣼⠏⠀⢿⡜⣿⣿⣿⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣜⢿⣦⣀⠀⢐⣧⣶⣿⣿⡟⣼⠇
⠀⠀⠀⠀⠀⠈⠻⣮⣝⣛⡿⠿⢛⣵⠟⠁⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣾⣿⣿⣿⣿⣿⣿⣿⢸⡇⠙⢦⣝⡿⢿⣿⣿⠿⣛⣵⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠁⠀⠀⠀⠀⢸⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣸⠇⠀⠀⠉⠛⠻⠶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣝⣛⡿⠿⠿⠿⣿⣿⡿⠿⠿⠿⢿⣛⣫⣭⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠤⡤⣤⢤⡤⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠋⠛⠻⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⢸⠋⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⣄⠀⠀⠀⢀⣼⣄⠀⠀⢀⣾⡿⠻⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠉⠛⢶⠶⠋⠁⠉⠛⠛⠛⠉⠀⠀⠈⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠀⠀⠈⠁⣄⠀⠀⠀⠁⠀⠀⠀⠀⠠⠚⠁⠀⠀⣀⠔⠋⠀⠀⠘⡆⠀⠀⡤⠞⠄⠄⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⣶⣤⣄⣀⣀⣀⣀⣀⣠⣤⣶⣾⣿⠀⠀⠀⠀⠀⡑⠀⠀⠳⣀⠉⠀⠐⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣤⣄⡼⣇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⡿⣤⣤⣶⠛⠷⠖⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣁⢠⣿⣿⠀⠀⠹⣦⡀⠀⠀⠈⠻⣿⣿⣿⠟⠛⠻⠛⠻⣿⣿⣿⡿⠃⠀⠀⢀⣴⡟⠁⠀⣿⣿⡆⠊
⠀⠀⠀⢠⠔⠀⢀⠀⠉⢛⠉⠉⣹⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⠉⠛⠀⠀⠀⠀⠀⠘⠉⣁⣀⣤⣴⣶⣿⣿⣶⣾⣿⣿⠁
⠀⠀⠀⠈⠳⣆⠈⠀⠀⠊⠀⠀⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣟⣛⣛⣛⣿⣿⣿⣿⣛⣛⣻⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀⣀⣀⡀⠀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠙⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠠⠀⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠒⠠⠄⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⠈⣴⣦⣄⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠗⣶⣦⡀⠀⠀⢿⡟⠋⠻⢿⣿⠿⠟⠛⠛⢁⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢸⣿⣿⣿⣄⠀⢸⡇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⣀⣀⡀⠀⠘⣠⣴⣿⣿⣿⣿⣿⣷⠌⠑⢶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠈⠉⠉⠉⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡞⡋⠈⢿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠛⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢓⣀⣈⣻⣿⣿⠟⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⢀⣴⣶⣤⡀⠙⠻⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠿⣀⠀⠀⠀⠀⠈⠛⠿⠟⢿⣤⡀⣤⡀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠀⠀⠀⠙⠤⢸⢣⣿⣿⡟⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠼⠷⢖⡂⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠈⢿⡀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠉⠁⠒⠂⢤⣄⣀⠀⠀⠑⡄⠀⠀⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⠀⡤⠂⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠊⠙⠋⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⢀⢁⠠⢊⡠⠊
⠀⠀⠀⠀⠀⠀⣰⣷⣦⡆⠀⠉⠉⢀⣀⠀⠤⠄⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠔⠁
⠀⠀⠀⠀⠀⣠⣿⡿⠛⣃⠤⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⠖⠘
⠀⠀⠀⠀⢸⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣆
⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠊⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⣀⣤⣂⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠿⠛⠛⠉⠉⠉⠉⠙⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣠⠤⠒⠋⠉⠉⠁⠈⠉⠉⠛⠓⠲⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠙⠃⢁⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⣿⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣈⠙⠶⣴⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⢷⣾⣿⣿⣿⣿⣿⣿⣿⣏⠻⠇⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡄⠀⣀⣤⣿⣿⣿⣿⣿⣿⣤⣤⡀
⠀⣠⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⢀⣤⡤⠼⣿⠛⠿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣶⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⡿⠟⠉⠙⣷⠆⠈⣿
⣾⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣶⣤⡀⢸⡿⠶⠶⢹⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣼⠃⣀⡼⠁
⣿⣿⣿⣇⠀⠈⣿⣿⣿⣿⡿⠿⣿⣿⡄⠻⣄⣀⣸⠀⠀⠀⣠⠖⠋⠉⠉⠛⠿⣿⣿⡿⢛⣵⡶⠞⠋⠀⠀⠀⠀⠀⠀⢨⣿⡉⠉⠁⠀⠀⠀⠀⡴⠃⠈⠁
⢿⣿⠹⣿⣷⣄⠸⣿⣿⣿⣿⠀⠹⣿⠃⠀⠈⠙⡻⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⣀⣀⡠⠾⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠉⢷⡄⠀⠀⠀⣠⡞⠁
⠸⣿⣆⠈⢿⣧⡀⢻⣿⠻⡿⠀⠀⠀⠀⠀⠀⢀⡁⠀⢸⡇⠀⢠⠞⠛⢻⠛⢛⣿⢿⡄⠀⠛⠿⢶⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠙⢶⡶⠞⠋⡀
⠀⠹⣿⣦⠈⠻⣿⣶⣥⣀⡀⠀⠀⢀⣀⡀⠀⠘⣧⠀⠀⠀⢠⡏⠀⠀⢼⣤⠟⣡⠎⣷⠀⠀⠀⠨⣇⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠙⠿⣷⣤⣀⠙⠛⠿⠿⠿⠿⠟⠛⠛⠻⣷⡈⢧⡀⠀⠈⠀⠀⣿⠎⢷⣞⢁⡴⠛⣧⣀⠀⠀⠈⠛⠶⠶⢤⣤⠜⠁⠀⢀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠉⠙⠻⢷⣶⣦⣤⣤⡄⠀⠀⠀⠘⢷⡀⠙⢦⡀⠀⠀⠙⢦⣄⡉⠻⢦⣴⣋⣽⠛⢶⣶⣤⣄⣀⠀⠀⠀⢀⡴⠋⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⠟⠁⠀⠀⠀⠀⠀⠙⣦⡀⠙⠳⣄⠀⠀⠙⠿⣧⡀⠀⠉⠛⠛⠛⣋⣀⠈⠉⠀⢀⡴⠋⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡈⠙⢶⣄⡀⠈⠙⠦⣤⡤⠒⠉⠉⠛⠋⠀⠀⠈⠀⠀⠰⠖⠉
⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠉⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡈⠙⠳⢶⣤⣤⣄⣀⣀⣀⣄⣤⣤⡤⠶⠚⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⣤⠊⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣤⣉⣉⡉⠉⠉⠉⠀⠀⠀⢀⣀⠀⠀⠀⣿⣿⣿⠛⠻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⢿⣅⠀⠀⠀⣿⢿⣿⣆⠀⠘⣿⣿⣿⣿⠙⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣿⠀⢸⡟⠿⠆⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠠⢶⣄⠀⠙⢷⣿⠀⠈⡧⠀⠀⠈⠿⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠶⢶⣶⣶⣤⣤⣴⣶⡾⠟⠁⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠈⢣⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠢⣀⡴⠊⠀⠀⢀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⡠⠔⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢸⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠐⠦⡀⠈⡁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠉⡄⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⡀⣰⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⣛⣛⣛⣟⣡⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⣏⣩⣿⣭⣿⣿⣯⣭⣭⣭⣿⣍⣩

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠨⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡴⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡙⢷⡄⠀⠀⠀⣠⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⣴⢋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣍⣻⣆⢹⡄⢀⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡟⠉⠀⢀⣠⣆⣄⠀⠀⠀⠀⠀⠀⣶⣾⣿⣯⣽⣿⣷⣥⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠂⠀⢠⠶⠃⠀⢠⣿⡟⡝⠈⠳⠀⢀⡠⢖⣩⣿⣿⣿⣿⣿⡿⠛⠷⣚⣿⣿⣿⣿⣷⣿⣿⠁⠀⠀⠉
⠀⠀⠀⠀⠀⣉⠀⠁⠉⠀⠀⠀⠀⢹⠀⠈⠀⠀⣿⣿⣤⡀⠀⠀⠋⢠⡟⣿⠛⣉⣽⣿⣿⠁⢀⣼⡟⠛⢻⣿⣿⣿⡟⠁⠀⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣶⣸⡟⠁⢸⡗⠀⠀⢀⣿⣿⣿⣿⣽⣿⣿⣿⣥⣶⣿⣷⣤⣞⠻⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢹⠿⠷⠶⠟⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢧⡀⢀⣀⣀⣀⣈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠭⠿⣿⣿⣯⣷⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠀⣄⣀⡀⣰⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠜⡁⠀⠀⢻⣿⣿⣄⣇⣀⣂⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠙⢷⣄⡀⠀⠀⠀⠀⠀⣠⣴⣯⡍⠀⠙⢿⣿⣿⣿⣿⣿⣿⣏⠉⠙⠛⠛⠛⠿⠛⠛⠟⠋⠉⠉⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣽⣿⣽⣿⣯⠿⢻⠀⠀⠀⢀⣬⣉⠉⠉⠉⠙⢦⡈⠁⠂⠁⠀⢂⣀⡒⠒⠒⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠈⠁⠂⢌⠁⠀⠒⠒⠂⠀⠈⢠⡘⣆⠀⠀⠀⠉⠿⠀⡀⠀⠀⠈⠳⢦⣙⠂⢀⣀⣠⣆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠓⠾⠿⢷⡾⣷⢶⣦⣤⣤⣀⢬⣅⠹⣆⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠙⠶⣄⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⢧⣄⣀⣀⡀⠀⠀⢹⡄⠀⠀⠠⠤⠄⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠸⠅⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⢯⢻⣿⡇⠀⠀⢠⠇⠀⠀⠀⠀⠄⠐⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⠀⠀⢤⣤⣀⣀⣀⢤⣤⡀⠀⠤⠤⠄⠘⣦⣀⡀⠀⠀⠸⡤⡏⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠛⠛⠿⣿⣿⡄⢰⠆⠀⢀⠜⠉⠋⠀⠀⢀⡾⠃⢿⠀⠀⠀⢮⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢀⡴⠋⠀⠁⣿⠀⠀⠀⣻⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡴⠋⠀⠀⠀⢀⡴⣫⣄⣀⣀⡀⣿⣤⣶⣾⣿⣿⣷⣶⣄⣾⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠄⣠⡴⠋⠀⠀⠀⢀⣤⣟⣀⠁⢀⣭⣭⣽⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢺⣿⣿⣦⣄⠀⣴⣿⣤⣾⠅⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡴⣿⠤⠤⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣄⣿⣿⣿⣿⣿⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠉⠈⠉⠉⠐⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠇⠈⢀⡍⠉⡉⣀⠀⡠⠤⢤⡄⠶⠚⠀⠀⠉⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠑⠄⢈⠡⠔⠀⠀⠀⠀⠖⠁⠀⠀⢠⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⢊⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⠒⢢⠤⣄⡀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⡠⠂⠁⠀⠀⡙⠓⢤⡀⠀⠀⠀⡆⠈⠉⠓
⠀⠀⠀⠀⢀⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠤⠤⠤⠤⠀⠒⠈⣀⠠⠒⠁⠀⠀⠀⠀⡰⠁⠀⠀⠈⢣⣀⣀⡇⢀⡌
⠀⠀⠀⡰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠠⠤⠤⠤⠤⠤⠀⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⢊⠀⠈⢇⡚⠢⡀
⠀⠠⣞⠀⠀⠀⠀⠀⠀⡀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠂⠉⠛⠓⠲⢤⣀⣰⢌⠠⠄⠀⣁⢀⡗
⢀⠔⠉⠀⠀⠠⢀⢀⡠⠚⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢼⠀⠀⠀⠀⠀⠀⠈⠁⠀⡿⢤⠂⠁⡜
⢇⡀⠀⠀⠴⣄⣸⠉⠀⠀⢠⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠘⢇
⠀⠱⠞⠤⠜⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠒⠒⢲⡋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡤⠚⠉⠉⠉⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⣀⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⢤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣅⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡋⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢻⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠘⢄
⣦⣶⣤⣦⣤⣤⣦⣤⣤⣤⣤⣤⣿⣿⣿⣿⣧⣴⣤⣤⣤⣤⣤⣤⣤⣤⣿⣶⣷⣾⣧⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⠀⠀⠠⠤⠠⠀⠐⡒⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠤⢄⡀⠀⠀⢀⠄⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⢀⠀⡰⠁
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠈⠤⠃⡀
⠀⠀⠀⢀⠜⠀⠀⢀⡀⠒⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠉⠚⢱⡥⠐⡖
⠀⠀⠀⠐⢄⡀⡒⠊⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠊⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠎⠀⠀⠀⠀⠀⠀⡷⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡁⠠⠀⠀⠀⠀⠀⠀⣀⣡
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠘⢛⢃⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠒⢒⠓⠓⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⢉⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠐⠚⠘⠘⣤⡀⠀⠀⠄⢀⣀⠀⠀⠀⠀⢰⠶⢀⣠⡴⠄⠁⠀⠀⣍⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⡤⠆⠲⡄⠀⣌⡓⠀⠤⠔⠒⠒⠉⠉⠉⠀⠈⠉⠉⢉⠙⢒⠤⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⡀⠀⢀⠀⣀⣀⣷⠀⠁⠀⠀⠀⠀⠀⢀⡀⠄⠤⠶⢶⡟⠛⠁⢀⣠⣧⡬⢄⣀⠈⠁⠒⠆⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠨⠁⢀⡾⠀⠀⠀⠀⢀⠀⣈⣒⡤⢓⣂⠄⢐⡀⠄⢈⡀⠠⠥⠀⠠⢉⡉⠥⠄⠌⠐⣀⡠⠽⠧
⠀⠀⠀⠀⠀⡍⠀⠀⠐⠊⣳⢸⠀⠈⡀⠈⠀⠀⢈⢉⠉⢰⠒⣶⣌⣘⠳⡖⡭⠐⠚⠭⠤⣍⡉⠚⢯⠥⢞⡩⠤⣀⠈⠠⣄⣲⣤⠶⠍⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠀⠀⢀⣈⣁⣸⣄⣌⠂⠤⠤⠤⠤⠤⠆⠹⠒⠀⠀⢆⠻⡇⢈⠁⠀⠈⠁⠀⠈⠑⠒⠠⠔⠠⠀⠐⠀⠈⠁⠤⣐⠤⣐⡢⢀⣀⣵⠠⠔
⠀⠀⠀⠀⠈⠁⠒⠒⢺⠙⠉⠐⠒⢜⠂⠀⠀⠀⠀⢀⠁⢾⣣⠄⠀⠰⣒⡟⢛⠾⠛⠴⠖⠢⠾⢷⣶⣦⣴⣄⡀⠇⠴⠖⢒⣉⡁⠥⠔⠒⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠥⠬⠼⠦⠤⣀⣀⣀⣠⡴⠋⠉⠉⠋⠉⣕⠖⠅⠿⠁⠀⠀⢀⠀⠀⠈⠈⠉⠓⠑⠠⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠉⠀⠀⠈⡍⠁⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠐⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠁⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠁⠀⠀⠀⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⣀⣀⣠⣄⡀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠛⠻⣿⣯⣽⣿⣶⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣦⣤⢀⣤⣄⣀⡉⠉⠉⣍⢀⠠⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠀⠀⡐⠈⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⡠⠊⣀⣴⣿⡿⠉⠑⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠛⠉⠺⣿⠿⢿⣋⠀⣀⣀⣴⣿⣷⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢄⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⡭⠀⠀⠀⠀⣃⣀⠿⣿⣿⣿⣿⣿⡿⠋⠀⠈⢠⡀⠀⠀⠀⠀⠀⠄⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠒⠚⡳⢎⢰⣄⣤⣿⣿⣄⣀⣀⣠⡋⠁⠀⢀⣠⣾⣿⣿⣦⡀⠀⠀⠀⠂⡄⠀⠛⢿⣻⡆⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⣹⣿⣿⠿⠿⠟⡸⣿⣿⣿⣿⣦⣶⣾⣿⣿⣿⡿⠛⠁⠀⠄⠀⠀⠈⢄⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂⣀⠃⠀⠈⣀⣀⣤⣧⡀⠈⠁⣸⡿⠿⠿⠛⠋⠁⠀⠀⣀⣤⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣦⣤⣴⣿⣿⡿⢿⣿⣷⣾⢃⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠿⠟⣇⠀⣰⣇⠉⠛⢣⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣠⣾⡿⢿⣿⣿⣦⠈⡈⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣀⣤⣶⣿⣷⠄⡀⠀⠀⠀⠀⢀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠋⠀⠈⠛⠛⠁⠀⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠿⠻⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠟⠛⠛⠋⠉⠁⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠒⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢤⠐⠣⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠁⢀⠀⠀⠂⠀⠀⠀⠀⠈⡄⠈⡀⡀⣠⣀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠓⠀⠀⠀⠀⠀⠄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠐⢀⡀⠈⠀⠀⠀⠄
⠀⡀⠀⠀⣀⣀⡤⠆⠀⠈⠃
⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⡅
⠀⠁⠒⠀⠄⠀⠀⠀⠠⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⣀⣀⣠⣤⣤⡤⠤⠀⠀⠀⠀⠀⠀⠠⠤⠤⠼⢀⡀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡀⠤⠐⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⡀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣿⠟⠛⢿⣿⣵⣾⣿⣿⡿⠛⢿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠹⣦⣶⣿⣿⣿⣿⣿⣹⣿⣿⣟⠀⠀⢠⣿⣿⣿⣿⣿⡧⠀⢸⣿⣿⡇⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠊⠀⠀⠀⠘⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣆⣠⣾⣿⡿⠛⠛⢿⣷⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠛⠛⢿⠛⠉⠉⠉⠉⠉⠛⠋⠉⠀⠈⡱⠀⠈⠉⠉⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇⠀⠀⠀⢀⡟⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠻⢦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⡇⠀⠀⠀⣼⢡⡞⠨⠁⡉⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⣀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠤⠅⠀⠀⣰⠃⠈⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠥⠦⡀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣶⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠓⣤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣴⠚⣧⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠶⣶⣶⡤⣤⣶⣶⣶⣶⠶⠻⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣦⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⡸⠃⠀⠀⠈⠉⠛⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣿⣧⡴⠛⢶⣀⣴⠎⠀⠀⠀⣀⡴⠷⣤⣤⣤⡴⠶⠟⠷⠤⣤⣤⣄⣀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⢿⡀⠀⠀⠙⠁⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣙⡷⠾⠋
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⠷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠈⢳⣄⠈⣿⡄⢠⣿⠂
⠀⠀⠀⠀⣀⣤⣴⠶⠶⠿⠷⠶⣦⣄⡀⠀⠀⠀⢀⣘⣿⣧⢀⣠⣴⠿⠞⠛⠳⠾⣏⡀
⠀⠀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⡈⠻⠿⢿⣿⣿⣯⡀⠀⠀⡀⠀⠀⠈⢻⣆
⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣼⡟⠙⠿⣿⣷⠦⣼⣀⣀⠴⠌⣿⡄
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠶⠶⠿⣿⡇⠀⠀⠈⠙⡳⠾⣿⣿⠙⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⢿⣿⣷⣄⠀⠀⠘⢿⡄⠀⠀⠀⠙⠦⠀⠉⢉⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡋⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠙⠷⣤⣀⣀⣀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣤⡀⠀⠀⠙⢿⣿⣷⡶⠞⠛⢹⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠈⠙⠻⢿⣿⣿⣶⣄⣠⣶⠟⠉⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⣸⣟⡀⠀⠀⠀⠀⠉⠛⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⢀⣀⣤⡶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣦⣤⣾⠏⣠⣴⠾⠛⠋⠉⠙⠛⠶⣄⠉⠛⠓⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠸⣿⠁⠉⠉⠉⠛⠛⠛⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠶⣶⠶⠶⠶⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣠⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣽⣽⣽⣯⣭⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⣭⣽⣿⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⡲⠈⠐⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣂⣥⣾⡷⠛⠁⠀⠀⠀⠀⣠⣭⡒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠙⠲⢿⡛⠻⢥⡀⠀⠀⠀⢀⣤⣮⡥⢋⡴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢐⣠⣭⣿⣶⣤⣀⠀⠀⠀⠈⠙⠲⢽⣦⣴⣾⠿⣛⣥⣺⢵⣚⣭⣶⡆
⠀⠀⠀⠀⠀⠀⣀⣠⢤⢲⣻⠿⠟⠋⠉⠘⢦⡉⠛⢦⣄⡀⠀⠀⠀⠈⣉⢲⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠐⠠⢉⡛⠋⠁⠀⠀⠀⠀⢀⣠⣤⣿⣷⡶⣞⣻⣷⡆⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢄⡀⠀⠀⣊⠕⡢⢄⡠⢶⣯⡥⢞⣛⣭⣶⣿⣿⣿⣽⡇⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⢷⣶⣍⡂⠀⠉⠓⢶⣾⣿⣿⣿⣿⣿⡿⢿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠑⢤⡀⠨⡛⢿⣷⣤⣀⢸⣿⡿⢿⣻⣭⣷⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠈⢹⣾⣭⣶⣿⣿⠿⣛⣭⣷⣿⡿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠢⣀⠈⡳⢦⣸⡿⢟⣛⣭⢶⣻⣿⣿⣿⡿⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡂⠙⠿⣟⣭⣾⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢸⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡚⠛⠉⠉⠉⠉⠒⠤⣔⣲⠶⣤⡀⠀⠀⠀⠀⠀⢀⣤⠞⣉⣉⣽⡶⠆
⠀⠀⠀⠀⠙⠛⠿⣯⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣽⣗⡢⢄⣀⠀⠠⢀⡀⠈⠉⠓⡓⠤⢤⣀⣴⣾⣯⣷⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⣿⠒⠦⣌⡙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣈⣉⣲⣤⣤⠝⠒⠋⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⢸⡷⠦⣄⡉⠙⠲⢤⣉⡛⢒⣛⣛⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⡀⣻⡿⠿⠿⠿⠿⠯⠅⠀⠐⠒
⠀⠀⠀⠀⠀⠈⣗⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⣤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣟⡒⠤⣀⠀⠀⠀⠀⣹⣟⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⢍⡓⠦⢍⣀⠒⠠⠄⠙⣩⠵⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠈⠁⠀⠉⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣍⣙⠒⠒⠖⠋⣱⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠐⠒⠛⢣⣄⡀⠀⠈⠐⠛⠷⠆⠀⠀⠀⣀⣠⣤⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⢟⠂⠤⣀⠀⠀⠀⠀⠀⠈⠙⢋⣉⣐⢤⡶⠒⠋⠀⣻⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢤⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠉⠒⠲⠶⣾⣿⣿⣿⣿⣿⣽⡇⢀⣀⣤⣿⣿⣯⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⣉⡒⠾⣿⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠤⣀⠀⠈⢿⣿⣿⣿⠿⠟⢛⣳⣯⠛⠉⢁⣼⣿⣯⡽
⠀⠀⠀⠀⠀⢈⡀⠠⢤⣴⣿⠢⠬⣭⣿⣿⣿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢤⣀⡈⠀⠒⠛⠉⢀⣴⣾⣿⣿⡿⠒⠚⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠬⢭⣛⣿⣊⣁⡒⠺⣿⡿⣿⡀⠤⢤⣴⣿⡆⠀⠀⠀⠀⢸⡏⠑⠒⠤⢍⣀⡐⠀⠀⢼⣻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢀⣑⠒⠶⣿⡟⠢⢬⣽⣿⣿⣿⠌⣙⣻⣿⣛⣧⠄⠀⠀⠀⠈⠣⠄⣀⠀⠀⠀⠉⠉⣒⣭⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠬⢉⣛⣿⣛⡁⠒⠲⢿⣯⣿⡓⠲⢾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠐⠓⠶⢾⣿⣧⠬⢐⣒⣶⣿⣿⡅⠙⠛⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠲⢾⣿⣿⠿






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄
⠀⠀⠀⠀⣶⣶⣶⣶⣶⠄⠀⠀⠀⠀⠀⠀⣀⣤⣶⠗⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⠶⠶⠶⠂⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡤
⠀⠀⠀⠀⠿⠀⠀⠿⠿⠂⠀⠀⠀⠀⠀⠀⠉⠉⠱⣄⣀⣠⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠀⢰⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠆⠾⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣀⢾⡁
⠀⠀⠀⠀⠠⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⣿
⠀⠀⠀⠐⢶⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣙⣿⣿⣋⣛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠁⠀⠀⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⣒⣂⣒⣒⣉⣄
⠀⠀⠀⠀⣰⠆⠀⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⣿⣿⣿⣿⣿⣿⣥⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⣿⣤⢀⣿⠦⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡟
⠀⠀⠀⢠⡿⡇⢀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣶⠶⣶⠃⠀⢇
⠀⠀⠀⠈⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⢸⡤⠀⢠⣴⠥⠀⠀⠀⠀⠀⠀⠀⡯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠖
⠀⠀⠀⣸⠀⠀⢃⠼⡇⠀⠀⠀⠀⠀⠀⢀⢷⠀⡤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠒⠈⠉⠖⠒⠒⠢⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠠⠐⠈⠀⠀⠀⠀⠀⠈⠀⠀⠈⠓⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⡒⠛⠒⠒⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⣴⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣿⣿⣿⣿⡿⣿⣖⣶⣤⣤⣀⠀⠀⠀⠌⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣿⣿⣿⣼⣿⣿⡏⠙⢿⣿⣿⣿⣿⣷⣾⣤⣀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠘⠁⠀⠈⠻⢧⣿⣿⣿⣷⠀⠙⠋⠻⢿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠉⠻⣿⣿⢱⣄⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣷⣦⣄⣀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⠻⠟⠛⣿⣷
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⡼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⠙⠛⠛⠋⠉⠉⠛⢿⣿⣧⡀⠀⢀⣴⣿⡇
⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠤⢤⠀⠞⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⢀⠀⠀⠘⠛⢿⣿⣿⣿⣿⡟
⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠲⣄⠀⠀⠠⠈⠙⠲⠤⣄⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠈⠑⠦⣁⠀⠀⠀⠀⠀⠉⠙⠒⠶⠦⢤⡤⠤⣤⠤⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢷⣦⠀⠀⠀⠀⣿⣷⣦⣀⠀⠀⠈⠙⠲⠤⣀⣐⠀⠀⠐⠀⠀⠀⣠⠞⠁⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢾⣿⣧⠀⢀⣴⢿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠈⠉⠙⢛⡞⠁⠀⠀⣤⡚⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣿⣿⡾⠁⠈⠛⠻⣾⣿⣿⣲⣶⣦⡀⠠⠖⠀⠑⠄⠀⢀⡠⠇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠑⢿⣿⢜⣶⣤⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣮⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡤⠀⠶⢦⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠁⠀⠀⠀⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠠⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡼⠁⣀⠉⠰⢿⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠁⣴⡿⠛⠒⠈⠉⠀⠀⠘⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢘⠋⣠⠖⢀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡏⣰⠋⠘⠁⠀⠐⢶⣄⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠩⠀⠀⠀⠀⠀⣸⣿⣧⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡖⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠘⢢⡀⠀⡴⣿⣿⠁⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠆⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣍⡁⠀⡀⢹⣤⣾⡿⠃⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⢈⠀⠁⢈⣉⣩⣿⣉⣽⡋⠘⠂⠀⠀⠀⠀⠀⠀⠈⠉⠒⠈⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠛⠻⢿⡓⠦⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⣿⣿⣿⠧⡀⢀⣶⠀⣰⡆⠘⠋⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠖⠛⠖⢶⣤⣀⣠⣽⣿⡿⢿⠻⠷⠞⠛⠛⠂⠈⣀⣀⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠉⠉⣩⣿⠟⢀⣼⣿⠟⠋⠉⠉⠉⠉⠉⠀⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠁⣴⣿⡿⢃⣴⡿⠟⠛⠋⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠈⠀⠰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢣⡿⣿⡿⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣠⣶⢄⣤⣵⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡌⠀⠀⠀⠀⣶⣶⣶⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠴⣿⣿⠟⢛⡿⠛⠿⡿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠰⣿⡇⢠⠟⠁⠀⠀⠀⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⣽⣿⣿⣇⠀⠀⠀⠀⠺⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣟⠻⣇⠀⠀⠀⠀⠈⠙⢻⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠢⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⡿⠱⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠛⠋⣿⣟⣀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⣸⣧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀⠀⠀⠀⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣰⠶⢆⣀⠾⢦⣀⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣧⣠⡜⠿⣀⣤⡿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠂⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣿⣿⣶⣶⣿⣿⠇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠁⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠤⢤⠫⠀⠘⠿⠟⠛⠿⠛⠀⠸⢣⠤⠄⢀⠀⢀⡀⠀⠀⠀⠂
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⡁⠂⠠⠤⠤⠄⠀⠐⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠤⠀⠐
⠀⢀⠀⠀⠀⠀⠀⢀⠘⢏⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢉⠟
⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠈⠁
⠀⡐⠠⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡯⡷⣿⣿
⠀⠀⠀⣀⠀⠀⠁
⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠃⠐⠃⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⣀⢘⣀⢐⡘⢀⣀⣠⣤⣤⣤⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣐⣛⣛⣛⣛⣛⣛⣛⣛⣻⣸⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⣋⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢴⡶⢫⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣭⡍⢩⡍⡉⠉⡿⠉⡍⢩⣷⣿⣿⣿⠿⡟⢸⣿⠟⣥⣯⣭⣭⣿⣿⣿⣿⣿⣿⣶⠸⣧⠻⠴⠛⠚⢻⢀⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣣⠾⠛⠓⠈⠁⢀⠈⠉⠁⠈⠉⣉⠝⣡⡖⢁⡇⡟⣵⠋⠀⢋⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠛⠓⠀⢸⢸⢿⡵⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠜⠁⠀⠀⢀⣤⠠⢄⠀⠀⠀⠀⡜⣠⣾⣿⢃⡼⢱⡿⣿⠀⠀⢸⠀⠀⠀⠀⣄⣠⣤⣠⢤⠀⠈⠀⠀⠀⠀⡜⡜⣼⣏⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⢎⡴⠁⠠⠄⠀⠀⠸⣌⣉⠴⠀⠀⠀⢸⣵⣿⣟⣡⢃⡆⣼⣼⣿⡄⠀⡼⠀⠀⠀⣸⢻⣿⠿⢯⠏⠀⠀⠀⠀⠀⡰⢱⠡⣽⣾⣏⡝
⠀⠀⠀⠀⠀⠀⡠⢣⠎⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡻⣿⡿⢁⢾⡇⣿⣾⡿⠁⡴⠁⠀⠀⣆⠀⢀⠄⣀⡀⣤⣤⣤⣴⡖⣰⢣⢣⡜⣽⣿⡿⠁
⠀⠀⠀⠀⠀⡔⣱⣿⣶⣶⣶⡖⠰⡆⣴⣠⣞⣪⠉⡥⣤⣤⣤⣿⡟⢠⢋⣾⡇⣿⣧⣶⡧⢤⣤⢤⣤⢀⣤⣤⠄⠤⢤⡥⣭⠵⠒⠺⢃⣯⢻⣰⣿⡿⠁
⠀⠀⠀⠀⢾⢤⠴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦⣤⣤⣭⣉⣉⣉⣹⣡⣏⣤⣼⣄⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣭⣭⣭⡴⠶⠒⣒⣛⡿⣱⣷⣿⡿⠁
⠀⠀⠀⠀⢘⡿⣿⡿⢗⣒⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡷⠤⢤⣬⣿⣿⣋⠈⠀⡐⠒⠲⠶⠶⠶⢴⣶⣿⣽⣿⠃
⠀⠀⠀⠀⠸⣯⣭⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣽⢿⣯⠿⠿⠿⠿⠿⠿⠦⠶⠶⠿⢿⣾⣿⣿⣿⣿⣿⢿⣻⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣲⠶⠤⢤⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠛⠉⠉⠀⠀⠰⡅⢃⠴⣣⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠲⠭⠙⠛⠫⠷⢦⣄⠀⠀⣠⣾⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⡿⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡐⠌⠊⠁⠀⠣⡠⠀⠐⢉⠝⣓⡶⢤⢠⠐⣸⣿⣿⠿⠏⣆⣴⣾⡆
⠀⠀⠀⠀⠀⣠⠐⢊⣥⠞⠁⠛⠀⠀⠀⠀⠀⠈⠉⠉⢉⢥⡪⣾⣿⡀⣳⣯⣿⢍⣡⣾⣿⣿⢏⡇
⠀⠀⠀⠀⠀⣾⢛⠒⠤⣼⣒⠤⣁⠀⡀⠤⣤⠄⢤⣲⣵⣿⣿⣿⡿⢐⣿⡛⠈⠀⡉⣿⡾⣻⢏⡀
⠀⠀⠀⠀⠀⡷⠈⠹⠷⣒⠭⠹⣶⣯⣠⣚⣭⣶⣽⣾⣿⣿⣿⡿⢟⣋⣀⡀⢠⢴⡃⣿⣿⠿⣤⠉⠹⡖⠤⢄⡀
⠀⠀⠀⠀⠀⠑⠢⠤⣐⡢⠭⣝⡂⠝⡓⠮⣿⣯⣷⣿⠿⠋⢉⣁⡠⢴⢻⣻⢮⠷⢻⡿⠖⠚⠹⣧⢤⡐⢋⣵⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣗⡲⢬⣙⡒⢧⣿⣿⡋⠭⠃⠀⠀⠀⡔⣵⣿⣿⠲⣱⡋⠀⢀⠔⠀⠀⠁⢚⠻⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⣤⠒⠩⠍⠁⣂⡐⠀⢍⡛⠑⠁⠀⣀⢀⣀⡠⣪⣾⣿⣿⣿⣼⠁⠀⠒⠁⠀⠀⠀⠤⠀⠀⠀⠃⠰⣨⠉⣲⡄
⠀⠀⠀⠀⠀⠠⣗⠂⠈⠩⠥⠐⣂⡈⠓⠲⠤⢔⣉⣐⣊⢔⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣀⢕⡤⢖⡫⢔⣷
⠀⠀⠀⠀⠀⠀⢿⣊⡠⠬⠍⠁⠐⠒⢒⣀⣤⣬⢐⣶⣂⣭⣾⣿⣿⣧⣤⡠⠤⣤⠄⣀⠀⢀⡀⠠⣐⡮⢝⣪⣵⣾⣿⣾⢿⡃
⠀⠀⠀⠀⠀⠀⠉⣶⢖⠶⡮⢭⠥⡤⡥⣤⢶⢖⡶⡶⣲⢻⣿⡫⡃⠀⠙⠻⣖⠬⣙⣠⣭⣶⣦⣟⣾⣿⣿⡿⣟⣊⣥⣒⣿
⠀⠀⠀⠀⠀⠀⠈⡯⠁⠑⠊⠂⢄⣈⡘⠐⠑⠠⠊⠄⠔⠔⡠⣻⣮⡀⠀⠀⠉⠯⢃⣬⣭⣍⣬⣿⣻⢿⣷⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠈⢻⢦⡉⠭⠽⢓⣾⡿⣿⣿⣿⢿⠶⠄⡀⠀⠰⣷⡶⣶⡷⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⢯⢍⠑⢾⣻⣷⣿⣿⠟⠋⠁⠀⠀⠀⠑⠒⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⠃⠐⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠢⠰⢀⠄⡠⡀⢀⠀⠀⢐⠅⡨⢀⠀⡠⠀⠔⠁⠀⢀⠔⠡⠊⠀⢀⢀⠑⡡⠂⢀⠔⡁⡠⢀⢔⢕⡔⡰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠡⠪⠀⠅⠨⠀⠕⡨⢂⠔⠀⡨⠀⠀⡠⠪⢀⢀⠔⡡⠂⠂⠐⠁⠊⠌⠐⠁⠃⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠀⠀⠉⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠧⠬⠥⠤⠤⠤⠤⠤⠤⠤⠬⠭⠭⠥⠬⠼⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠰⣟⣛⣻⡆⠀⠀⠀⠀⢰⣟⣛⣻⠆⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢦⣬⣩⣥⠴⠖⠛⠛⠳⢦⣬⣍⣥⡴⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠙⠲⠦⣤⣤⠴⠶⠋⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠛⠋⠉⠛⠓⠶⠦⢤⣤⣤⣤⠴⠶⠞⠋⠉⠙⠛⠲⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣥⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣬⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠉⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⠁⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠶⠦⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠴⠖⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣃⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⢀⣘⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠙⠛⠲⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠛⠛⠉⠉⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠷⠦⣤⣤⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣤⡤⠶⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠀⠀⠈⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣠⣴⠶⠿⠿⠿⢿⣤⡀⠀⠀⡀⠀⠀⠀⠀⣉⣉⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⢿⡿⢋⠀⠰⠄⠀⠀⠒⣈⣉⣤⣤⣴⣶⠶⠿⠟⠛⠋⢿⣇
⠀⠀⠀⠀⠀⠀⣈⣤⣴⣶⠿⠛⠋⠁⠀⢀⣉⣉⣥⣤⣶⡶⠾⠛⠛⠿⢿⣭⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠉⣡⣤⣤⣶⡶⠿⠿⠛⠛⠉⣹⡟⡁⠀⢐⣦⣀⠀⠀⠈⢻⣆⠀⠀⡀⢀⣀⡀⡀⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⣩⣥⣤⣤⣤⣤⣤⣬⣭⣬⣤⣤⣭⣭⣬⣥⣤⣤⣥⣭⣤⣤⣥⣭⣭⣥⣥⣭⣭⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⣩⡿⠛⠉⠉⡉⠉⠙⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡿⠀⠀⢀⡀⠀⠀⠀⣼⠋⠀⠀⣠⠾⡗⣦⠀⠀⠈⢿⡄⠀⠀⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⣿⡟⠀⢠⣶⣦⠀⠀⢸⡏⠀⠀⠀⠻⣶⣇⡉⠀⠀⠀⠘⣷⠀⠀⢠⣶⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣿⣷⠀⠈⠛⠋⠀⠀⢸⣧⠀⠀⠀⣤⡌⡟⢻⡆⠀⠀⢠⡿⠀⠀⠈⠛⠋⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠙⠲⡷⠛⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣠⣀⣀⣀⣀⣀⣀⣀⣹⣷⣦⣄⣀⣁⣀⣤⣾⣟⣡⣄⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣫⠟⠋⠉⠳⢬⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡵⠞⠉⠙⠿⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠻⠟⠻⠿⠷⣷⡶⣶⣄⠀⠀⠀⠀⠀⢀⣴⣶⠶⠶⠾⠻⠿⠶⣾⣷⣝⢶⣄
⠀⠀⠀⠀⠀⣠⢾⣿⣏⣀⣠⣤⣤⣀⣀⣀⣀⠀⠈⠹⣧⠀⠀⠀⠀⣾⠋⠀⠀⣀⣀⣀⣀⣤⣤⣈⣙⣿⣿⣷⡄
⠀⠀⠀⢠⠾⠛⠛⠉⢁⣀⣠⣤⣤⣤⣤⣭⣉⣙⠛⠓⠻⠶⠶⠶⠶⠟⠚⠛⣋⣉⣥⣤⣤⣤⣤⣄⣀⡉⠉⠛⠛⠶⡄
⠀⠀⠀⢹⠀⠀⢀⡞⠉⠁⠀⠀⠀⠀⠀⠀⣀⠉⠙⢦⡀⠀⠀⠀⠀⢠⡶⠋⠉⠀⠀⠀⠀⠀⢀⡀⠈⠙⢷⡀⠀⢀⡇
⠀⠀⠀⠈⢳⡄⢸⡇⠀⠀⠀⠀⣰⠟⢀⡾⠁⠀⠀⢨⡇⢀⡴⣤⠀⢸⡀⠀⠀⠀⠀⣠⠞⢠⡞⠁⠀⠀⢸⠇⢰⠟⠁
⠀⠀⠀⠀⠀⣷⠀⣧⠀⠀⠠⠞⠁⣴⠋⠀⠀⠀⠀⣼⠁⡾⠁⠈⣧⠘⣧⠀⠀⠀⣴⠋⣰⠏⠀⠀⠀⠀⣾⠀⡾
⠀⠀⠀⠀⠀⠸⣆⠸⡆⠀⠀⣠⠞⠁⠀⠀⠀⢀⡼⠃⣼⠃⠀⠀⠘⣆⠘⣦⠀⠀⠀⡼⠃⠀⠀⠀⠀⣰⠇⣸⠃
⠀⠀⠀⠀⠀⠀⠹⣆⠙⢦⣄⣀⣀⣀⣀⣠⡴⠟⣁⣾⡇⠀⠀⠀⠀⢸⣧⡈⠳⢦⣄⣀⣀⣀⣀⣠⡴⠋⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣬⣍⣉⣉⣭⣤⠴⠞⠉⣼⠇⠀⠀⠀⠀⠸⣇⠙⠳⠦⣤⣭⣉⣉⣩⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⣁⠀⠀⠀⠀⠀⠀⣘⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣫⠁⡀⠀⢉⠛⢦⣀⣀⡴⢋⠉⠐⣦⠉⢙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⠋⣸⠃⠀⡿⠀⢰⠌⠁⠀⠹⡆⠀⠘⣇⠈⣧⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⣋⣀⣉⣀⣀⣃⣀⣈⣀⣀⣀⣀⣁⣀⣀⣉⣀⣙⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠙⢳⣶⡤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠶⠶⠖⠒⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣧⡴⠛⠹⣄⣤⠞⣿⡙⢲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠀⣴⡒⢢⣴⠊⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⢸⣟⣹⡧⠤⠴⠿⠧⠬⠭⣿⣟⣿⣿⣝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⢠⡦⢘⣿⣿⠛⠀⠀⠀⠀⠀⠀⠙⢦⡀⢀⡠⢼⣥⣤⣤⣶⣶⣶⣶⣦⣤⣄⡀⠈⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢈⡷⢯⣤⠿⠁⠀⠀⠀⠀⠀⢈⡳⠌⢳⣿⡲⠀⠁⠉⠛⢿⣿⣿⣿⣟⣻⣿⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠉⠓⢦⣹⣯⣁⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢉⣀⣉⣙⣶⣤⣀⠈⠹⣏⠳⣤⠐⠀⠀⢠⣦⣖⣚⢻⣃⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣭⡀⣈⡷⠚⠉⢿⡈⠻⣧⠀⠀⠀⠈⠉⣻⠟⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣶⣶⣛⣿⣿⣥⣤⣥⣤⣬⠿⠿⡟⠃⠀⠀⠀⣀⣯⣀⣨⣥⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣽⣿⠏⠁⣿⣿⣧⣼⣿⣿⣶⡞⠓⠒⠚⢿⣩⢭⣷⣸⠟⠿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣥⠴⠿⣻⢿⣿⣿⣥⣀⠀⠈⠙⠻⢿⣷⣦⣄⣠⠛⠋⠉⢻⣶⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣤⣤⣤⣤⣄⠀⠀⠀⠀⢠⣾⡁⠀⠈⠙⠻⢿⣿⣶⣤⣀⠀⠉⠻⢿⣧⣤⣷⣞⣛⢻⡟⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠟⠛⠛⣿⣿⡿⠀⠀⢀⡴⠻⣿⣿⣷⣄⡀⠀⠀⠉⠛⠿⣿⣷⣦⣄⠀⠉⢻⡿⢻⣾⣿⠿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣼⣷⣦⣄⡙⠻⢿⣿⣷⣤⣀⠀⠀⠈⠙⠻⣿⣷⡶⠋⠀⠀⢻⣀⣠⣽⣿⡇
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣠⣾⣿⣿⣿⣿⣿⣶⣄⡈⠙⠻⠿⣿⣶⣤⣀⢀⣴⠟⠀⠀⠀⠀⠀⢻⡿⠟⢹⡇
⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣾⣥⣭⣿⣿⣥⣤⣤⣤⣤⣤⣤⣄⠑⠶⠊
⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠁⠈⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠉⠓⠲⠤⣀⣀⡀⠀⠀⠀⠀⠛⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠤⢤⣤⣤⣤⣤⡤⠴⠶⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣼⣏⠀⠀⣰⠶⣶⠶⣶⠶⠶⠶⠶⣶⠶⡶⠶⡄⠀⢀⣹⣦⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣭⣤⣤⣀⣀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠶⠿⢶⣷⣶⣶⣶⣶⣿⠶⠷⠶⠿⠟⠛⠛⠉⠁⠀⠀⢀⣀⣀⣤⣬⣭⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡿⠛⠛⣻⣿⣿⣿⣛⠛⠻⣛⣿⣿⣿⣟⠛⠛⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣀⣤⣿⡇⠀⠀⠶⠻⠿⢿⣷⠀⠀⣿⡿⠿⠛⠶⠀⠀⣿⣿⣤⢰⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣇⢿⣿⣿⠀⢀⣀⣀⣠⡭⠟⠛⠛⣯⣍⣀⠀⠀⠀⠀⣿⣿⡟⣸⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⡏⢸⣏⣙⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠛⠛⣛⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣡⡽⣾⣿⣟⡛⠛⠛⠒⠒⠒⠀⠀⠀⠀⣾⣿⣿⠋⠀⢀⣴⠾⢻⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⣴⠟⢡⡞⣿⣿⡛⠛⠛⠶⠶⠦⠤⠀⠀⠀⢠⣿⡿⠁⠀⠰⠋⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠞⣡⠖⢹⠀⣿⣿⡛⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢠⣾⡿⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠴⠚⠁⠀⣸⠀⡟⣿⣿⣄⠙⠳⠦⣤⣀⠀⠀⠀⠘⣿⡄⠀⠀⢀⡿⠋⣴⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠇⣸⡿⢿⣧⠀⠀⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠞⠀⡼⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣀⠴⠃⢻⡀⠹⣧⠀⠰⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠳⡄⠘⠇⠀⢻⡄⠀⠀⠀⠈⡇⠀⠀⠐⠃⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠈⣿⡄⠀⠀⠀⠃⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢻⣷⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⣿⡇⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠖⠋



⠠⠀⠄⠠

⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⡀
⠀⠀⢿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⠃⠀⠀⠀⣴⣿⣿⠿⢿⣿⣿⡿⠿⣿⣿⣧⠀⠀⠀⢿⣶⣶⣶⣤⣄⠀⢀⣤⣴⣶⣶⣾⠇
⠀⠀⠀⢹⡿⠿⢿⣿⣿⡿⠿⢿⣿⠀⠀⠀⠀⠘⣿⣿⣷⣶⣶⣿⣿⣶⣶⣾⣿⣿⠃⠀⠀⠈⢿⣯⣀⣉⣻⣿⣿⣋⣁⣨⣿⠏
⠀⠀⠀⠸⣿⣶⣴⣿⣿⣶⣴⣾⡿⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⣿⣿⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠙⠛⠿⠿⢿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣶⣄⠀⠀⣠⣶⣿⣿⣿⠋⠀⠀⠀⣄⣤⣶⣶⣶⣶⣶⣶⣶⣦⣄⡄
⠀⠀⣀⣤⣾⣿⣿⣶⣶⣾⣿⣿⣦⣄⡀⠀⠀⠀⢿⣿⣿⠛⠛⣷⣾⠛⠛⣿⣿⡿⠀⠀⠀⠀⣿⡿⠿⠿⢿⣿⣿⠿⠿⠿⣿⡗
⠀⠘⢿⣿⣿⣅⣀⣹⣿⣁⣀⣿⣿⣿⠟⠀⠀⠀⠀⣼⣿⣷⣾⡟⢻⣷⣾⣿⣧⠀⠀⠀⠀⠀⢻⣿⣦⣤⣤⣿⣯⣤⣤⣾⣿⠇
⠀⠀⠘⠙⠿⠻⡿⣿⣿⡿⡿⠻⠟⠙⠀⠀⠀⠀⠀⣸⠟⠛⠋⠀⠀⠙⠛⠻⣇⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⢴⣶⣶⣿⣿⣿⣿⣷⣶⣶⠀⠀⠀⠀⢶⣿⡿⠿⠿⣿⣶⣶⣿⠿⠿⢿⣿⡶⠀⠀⠀⢰⣶⣤⡀⠀⣀⡀⠀⣠⣴⣾
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠻⣷⣶⣶⡿⠛⠛⢿⣶⣶⣾⠟⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣄⠉⣿⣿⡋⣁⣼⡿⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠸⣿⡋⠉⢻⣿⣿⠟⠉⣻⡿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⢩⣭⣤⣬⣿⣿⣿⣿⡿⠃
⠀⠀⣀⠀⠉⠙⠛⠓⠘⠛⠛⠉⠁⢀⡀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⠀⠙⠛⠟⠛⠛⠟⠋
⠀⢀⣻⣶⣶⣶⣦⣄⢀⣤⣶⣶⣶⣾⣃⠀⠀⠀⠀⢻⣿⣷⣶⣶⣶⣶⣾⣿⠟⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣽⣿⣿⡛⠛⢿⣿⠟⠛⣻⣿⣿⡅⠀⠀⠀⠀⢸⡿⠛⠻⣿⣿⠟⠛⢿⡇⠀⠀⠀⠀⢸⣿⡿⠿⣷⣦⣄⣤⣶⡿⠿⣿⣿
⠀⠀⠈⠙⠛⠻⢿⣿⣿⣿⠿⠛⠛⠉⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣾⣿⡇⠀⠀⠀⠀⠀⢿⣷⣦⣤⣼⣿⣿⣤⣤⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠙⠋⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠙⠛⠟⠛⠉⠙⠛⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣤⣀⠀⠀⢀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠳⠶⠶⠶⠶⠦⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠀⠀⠉⢀⣀⣤⣤⣤⣶⣶⣶⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣿⡿⠟⠛⠶⣿⣿⣿⣿⣿⠿⠿⢿⡿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡁⢀⡀⠀⠠⠤⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⢈⠀⠈⠻⢿⣧⣄⡀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣐⢡⠀⢀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠠⠎⠁⠀⠀⠀⠀⠳⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡠
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠉⠀⠁⢄⢀⠊
⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠐⠀⢂⠀⠀⡈
⠀⠀⠀⠀⠉⠃⠀⠀⠀⣀⠀⠠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠹⡄⠀⠀⠀⣀⠜⠛⠛⠤⠄⠚
⠀⠀⠀⠀⠀⠀⠀⢀⠚⠛⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠀⠀⠀⠑⠐⠀⠁
⠀⠀⠀⠀⠘⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⣀⢀⡀⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠲⠋⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⡀⠀⠀⡘⠁⠀⠡⣤⠐⠉⠙⠓⠲⠂⠀⠤⠤⠄⠄⠀⠒⠀⠀⠈⠁⢀⠐⠁⠀⠃
⠐⠀⠐⠁⠀⠀⠀⠀⠑⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢸⠀⠀⠀⠂⠀⠀⠀⠀⠘⠅⠒⠀⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⢀⠀⠀⠀⠀⠘⠀⢂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠒
⠀⠀⠀⢄⣀⣀⡀⠀⠠⠤⠀⠐⠒⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⢉⡁⢰⣶⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⠋⣉⡠⠤⠖⠒⠋⠉⠈⢇⠀⠿⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠒⠚⣉⣁⠤⠤⠶⢊⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠘⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠂⣉⣉⡤⠤⠔⢂⠟⠋⠁⢀⣤⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧
⠀⠀⠀⠀⠀⡔⠒⢈⠉⢠⡤⠤⠒⠊⠉⠁⠀⠀⠀⡔⠁⠀⠀⣴⣾⣿⡿⣷⡆⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀
⠀⠀⠀⠀⠀⢱⠀⠘⣢⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⣿⡁⢹⣷⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠘⡆⢀⠥⠔⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠈⠻⣿⣾⣿⣦⣤⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹
⠀⠀⠀⠀⠀⠀⢡⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⠙⢻⣷⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠀⠏⡇
⠀⠀⠀⠀⠀⠀⠈⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢠⣿⡇⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠰⡇⠀⢧⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠿⣷⣶⣿⣿⡿⠋⠀⠀⣠⠋⠀⠀⠀⠀⣀⣠⠤⠄⠓⠀⣉⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⠛⠁⢀⣠⣞⠥⠤⠒⠒⢉⣁⠤⠤⠖⠒⠈⠉⠀⣳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⡲⠶⠶⠒⠊⣉⣁⠤⠔⠒⢺⠋⠁⠀⠀⠀⠀⠀⢀⠞⢁⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢴⠀⠙⡆⠀⣀⣀⠤⠄⠒⠚⢉⣁⠠⠤⠶⣶⣿⣿⠟⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⢸⡄⠈⡳⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⠇⠀⢉⣉⡁⠤⠔⠒⠋⠉⠓⢤⣀⠀⠀⠀⠈⠉⢀⣀⣴⣋⡠⠤⠤⠒⠒⠂⢋⣉⣉⡡⠤⠤⠰⡖⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠧⢲⠊⠙⡍⠀⠀⠀⠀⠀⣀⣀⣀⠠⠬⠽⠛⠒⠊⢉⣉⣁⡠⠤⠤⢲⠖⠂⠉⠉⠉⠀⠀⠀⠀⣀⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⡷⠀⠻⠒⠒⠊⣉⣁⣤⠤⠤⢤⠒⠒⠊⠉⠻⠟⠉⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠦⠤⠖⠲⠶⠬⠭⠥⠤⠤⠤⠤⠤⠤⠭⠶⠦⠤⠤⠤⠤⠶⠮⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠨⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⣉⣉⡭⠥⢤⣙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠿⠙⠛⠋⣁⡤⠄⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢳⣶⣾⣟⠛⣿⣶⡶⢚⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠹⡏⢀⡹⠻⠶⠟⠋⢀⠉⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡀⣈⡥⢀⠲⡀⠀⠀⢸⠷⢛⣱⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⢄⣀⣀⡈⢀⣀⣀⣄⣤⣾⡾⠟⢛⣡⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢿⠿⡿⠿⠿⠿⠿⢛⣛⣩⣭⣴⣶⡿⢿⣟⣭⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣶⣶⣶⡿⠿⠿⠛⢛⣋⣭⣵⣶⠾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣤⣤⣶⣶⣶⠶⠿⢟⣛⣫⣭⣷⡾⠿⣿⣫⣥⣶⠿⢃⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣭⣭⣿⢷⣶⡶⠿⠿⠛⣛⣫⣭⣴⣶⡿⠛⠻⢭⡶⠞⡋⣀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣴⠃⠀⠙⠲⣿⣿⣛⣻⣯⣭⣾⣏⠀⠀⠀⠀⠙⣿⡜⠋⢷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡀⢸⠶⠤⣎⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢱⣞⣟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢿⣿⣀⣗⡼⠦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠸⣅⣂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠘⠁⢿⡖⣒⣀⣿⣿⣿⣿⣿⠟⠋⠉⠉⠙⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣘⠀⢀⣀⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⣤⣤⣌⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⢀⣤⡐⠆⠀⠀⠘⠛⠛⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢶⡙⣒⢸⠛⠛⢿
⠀⠀⠠⣾⣄⠻⢷⣶⣄⣀⣀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣧⠙⢿⣶⣧⣀⣡⡴⠋
⠀⣰⣦⡜⠻⠷⣶⣬⣭⠉⠩⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣜⠻⠿⣶⣧⣭⡭⣩⣧⣴
⢠⣤⣭⣿⠿⠖⣻⢩⡭⠂⠈⢰⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣤⣬⣽⡿⠿⣺⣭⣭⡟⠿⢿⣇⣀⣀⣀⣠⡀
⠈⣩⣤⡶⠷⢛⣣⣾⡧⠠⠀⠈⠐⠿⡛⣛⡿⢿⣂⠋⠀⠀⣏⣥⣶⠿⠞⣯⣼⣧⢧⡤⠼⣷⠿⣟⣿⣿⠿⣖⡞
⠀⠋⣤⣶⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠈⠀⠀⠀⠙⣡⣴⣾⣿⣿⡿⣡⣿⣾⣷⣿⣿⡈⢧⠀⠀⠈⠁
⠀⠀⠈⠹⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⡿⣸⠂
⢀⣤⣤⣾⣿⣿⡿⠂⠀⠀⠀⠀⢀⣤⡚⠁⠀⠀⠀⠀⠀⢀⣤⣤⣶⣿⣿⣿⠟⠺⠭⠭⠭⢭⣤⣞⠁
⢠⡟⠻⠟⠛⠉⠀⠀⠀⠀⠀⠀⠘⠋⠻⠶⠄⠀⠀⠀⠀⢀⣾⠛⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠛⠙⠶⠆

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⢻⡏⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣤⣾⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠁⢹⣿⣿⣿⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡗⠐⠀⠒⠀⠘⣿⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣇⣤⣀⣠⣀⣠⣿⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⡿⠛⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣟⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⡇⠀⠀⣠⣤⣿⣿⣷⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢈⣹⣿⣿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⡤⢶⣖⣚⣛⣛⣓⣲⡶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⡞⣽⣯⣽⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣝⣶⡄
⠀⠀⠀⠀⠀⠀⢹⣿⠻⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣹⡄
⠀⠀⠀⠀⠀⠀⢸⣱⣿⡀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⢻⠇
⠀⠀⠀⠀⢀⡴⣻⠽⣛⣷⣤⣄⣀⣀⣠⣶⡿⣷⣄⢰⣿⣿⣟⣯⠷⠋⣀⣤⠴⣖⡿⠿⠭⠭⠽⠿⣳⡦⣄⡀
⠀⠀⠀⢰⢯⠊⠔⡗⠊⠉⠻⣿⣿⣿⡀⢼⣿⣶⡝⢸⣿⠿⠙⢳⢶⡻⠝⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢝⢦⡀
⠀⠀⠀⣿⡇⠠⡄⣁⠀⠠⠄⠸⠻⣿⣿⣶⣾⣿⡇⠈⠇⠈⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡽⣄
⠀⠀⠀⣹⢀⢠⡀⠀⠈⠐⠤⣀⠀⠀⠉⠉⢉⢝⠇⠀⣼⠐⠋⠀⣀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⡆
⠀⠀⡼⡿⠁⢼⣀⠐⠠⠄⢀⡀⠉⠒⠒⢉⡡⠊⠀⠀⢸⢛⣤⣬⣕⠒⠚⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣷
⠀⠘⣟⢆⠀⠀⠈⠉⠒⠂⠀⠀⠒⠒⠊⠁⠀⠀⠀⢀⡾⠁⠙⣿⣿⣧⠀⣿⣿⠟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠈⠳⣟⡄⣀⠀⠀⣀⣀⣠⠤⡤⣤⠤⠤⣴⠞⠁⢣⡀⠀⢹⣿⣿⣴⡿⠏⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⡏
⠀⠀⠀⠀⠈⠙⠳⠾⣯⢿⢁⣞⡑⠀⡹⢧⡍⠀⣀⣤⣶⣿⡶⠖⠋⠁⣠⣷⣶⣾⣿⡿⡄⠀⠀⠀⠀⠀⠀⠀⢠⢮⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⠖⠲⣤⢿⢷⣮⣿⠿⠿⠛⠉⢀⣠⣴⣾⢿⡿⣛⣫⠭⢾⣅⠀⠀⠀⠀⠀⢀⣴⣵⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡟⡵⡁⠀⢸⢗⣿⠏⢹⣱⣤⣶⣶⣿⣿⣿⠿⡇⠂⠀⠀⠀⠀⢸⣿⣀⣀⠠⣤⣲⡵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣟⠤⢈⣀⡎⡏⠀⠀⢸⣿⣭⣭⣥⣤⣤⣤⣤⣧⠀⠀⠀⠀⣠⣫⡞⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠚⠁⠀⠀⠈⢧⣟⡿⢿⣿⣿⣿⣿⣿⡿⢶⣤⣪⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠒⠒⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠉⣛⣛⣟⡛⢛⣭⣭⣤⣀⢂
⠀⠀⠀⠀⠀⢀⡜⠛⠿⠛⠉⡛⠷⠶⠿⠛⠁⡅
⠀⠀⠀⠀⠀⣼⠀⠈⠀⠠⠤⠐⠂⠀⠀⠀⠀⢱
⠀⣀⠀⠀⠀⠘⢤⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣤⣧⣀⠀⠀⠀⠀⣀⣴⡾⡆
⢸⠇⢠⣄⣀⣀⣀⣤⣴⣿⣯⠽⠿⠯⠭⣟⣻⣿⣿⣿⣿⣿⣿⡿⢃⣏⡇⢱⠀⣀⠀⠀⣀⠠⠂
⠋⡀⠀⠙⠻⢿⣿⠿⢫⣿⣿⣭⠭⢭⣭⣴⣿⣿⣿⣿⡉⠛⢋⣷⣾⣼⣿⣾⣿⣿⣦⡖
⠈⢳⡄⠀⠀⠀⠀⠀⣼⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣧⠀⣼⣿⣿⣿⣿⡇⣿⣿⣿⣿⠟⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡆⠻⢿⣿⣿⣿⡇⣿⣿⡯⣿⡄⠀⢠⠂⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣿⣿⣿⣿⠟⣿⣿⣿⡟⠉⠀⠀⠀⠈⠉⠛⠰⣿⠿⠇⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠁⢸⣿⣿⣿⣧⣀⠀⣀⣠⢴⠂⠘⠀⠀⠀⠀⠀⠘⡼⡆⠀⠔⠑⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠿⠋⠁⠀⠀⠀⠻⣿⣿⣿⣿⠿⠛⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠳⡣
⠀⠀⠀⠠⣄⣀⡴⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⢿⡿⣿⣿⣶⣶⣦⣤⣤⡄⢠⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣘⣿⣯⣲⣦⠀⠀⠀⠀⢀⠄⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠈⣧⠀⠀⢠⣿⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⢿⣿⣿⣆⣿⡇⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣽⣿⣿⣿⣿⣿⠷⠚⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⣩⣿⣿⣿⣾⣶⡶⢲⣾⡑⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣧⣤⣾⣿⣿⣿⣿⠟⣸⠃⠈⠻⡿⣄⠁⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⡀⠋⢠⡄⣤⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⠀⢀⣴⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣽⣯⠀⠀⣿⣿⡟⡇⣿⠿⠛⠃⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡗⡀⠀⢿⣿⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡇⠀⠀⠛⢹⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠀⢠⣤⣿⢸⡟⣿⣿⡄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢈⣹⣻⣿⣿⣷⡶⠶⢦⣤⣄⣀⣀⡀⠀⠀⠀⢀⠠⠀⠀⠀⢀⣿⣿⣼⣧⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣷⣤⣮⣿⣿⣿⣿⣿⣿⣷⣂⣀⡤⠖⣾⣿⣏⢻⣿⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⢷
⠀⠀⠀⠀⢶⣾⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣽⣿⠿⢛⣁⣀⣀⣿⣿⣷⡬⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠛⠛⣿⣯⣍⣉⣉⣉⡙⠛⠛⠻⠿⡯⠔⠚⠋⠉⠉⣙⡻⣿⣿⣧⢀⡀⢰⣁⠀⠀⠀⡆⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣿⣿⣻⣻⣶⣶⣦⣤⣄⣀⡀⠐⠉⠁⣀⣿⣿⣿⣧⡬⠙⠿⠿⡿⠀⣂⣠⡤⠞⠋⠁
⠀⠀⠀⠀⢸⣶⣦⣤⣀⣀⡀⠊⠉⠉⠛⠛⠻⠿⣿⣯⡤⠴⣚⣉⣽⡿⠿⣿⣿⣿⣿⣿⣻⣟⠁⡿⠟
⠀⠀⠀⠀⠙⠿⠷⠾⣯⣿⣿⣿⣶⣶⣦⣤⣀⣤⠼⠗⠊⢉⣉⡠⠼⠁⠀⠀⣌⠁⠀⠀⢠⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣧⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⡼⢽⡧⠀⠼⣏⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣻⡗⢀⡚⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠷⢦⣯⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣦⡤⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣒⣒⣢⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠄⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⣀⣀⣠⣤⣤⡤⠤⠀⠀⠀⠀⠀⠀⠤⠤⠤⠼⢀⡀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⡀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣀⣠⣤⣶⡶⣶⣾⣿⡿⠛⢿⣿⣽⣾⣿⣿⡿⠛⢿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢦⣶⣿⣿⣿⣿⣿⣵⣿⣿⣿⠀⠀⢠⣿⣿⣿⣿⣿⡧⠀⢨⣿⣿⡇⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠊⠀⠀⠀⠘⣿⣿⣿⣿⣿⡯⣯⣿⣿⣿⡆⣠⣾⣿⡿⠛⠛⢿⣷⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⢿⠘⠉⠉⠉⠉⠉⠛⠋⠉⠀⠈⡱⠀⠈⠉⠉⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇⠀⠀⠀⢀⡟⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠹⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⡇⠀⠀⠀⣼⢡⡞⠨⠁⡉⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⣀⠀⠈⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡀⠀⠤⠅⠀⠀⣰⠃⠀⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⠦⡄⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣶⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠓⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢆⣴⠒⣧⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠶⣶⣶⣤⣤⣶⣶⣶⣶⠶⠻⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣦⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⡰⠃⠀⠀⠉⠉⠛⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣿⣦⡴⠋⢦⣀⣴⠎⠀⠀⠀⢀⡴⠷⢤⣤⣤⠴⠶⠛⠷⠤⢤⣤⣄⣀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢿⡀⠀⠀⠙⠁⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣙⡷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⢵⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠡⣾⣷⡀⠀⠈⠹⣿⣿⣿⣿⠏⠀⠀⢀⣾⣷⠈⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣻⣿⣿⣶⣶⣶⣼⣿⣿⣧⣶⣶⣶⣿⣿⣏⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⢰⠁⠈⣽⣻⠛⠛⢹⣿⣿⡏⠛⠛⣟⣯⠁⢈⡆⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣤⣑⠤⠝⠋⠀⢀⣼⣿⣿⣧⡀⠀⠙⠫⠴⣊⣤⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠠⣀⣡⣤⣶⢶⣤⣼⣡⠆⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣷⣴⠛⠃⠈⢁⠈⠁⠘⠛⣤⣾⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⣛⣀⣤⣾⣿⣿⣷⣤⣀⣛⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡵⠋⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡺⠛⠛⠓⠶⣤⠀⠀⢸⡟⠲⠦⠤⠤⠴⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣋⠀⠀⠀⠀⣀⣼⡇⠀⣿⣝⠷⠶⠤⠤⠶⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣧⣉⡙⠛⠛⢋⣹⡿⠃⠀⢻⣍⡙⠛⠛⠛⠋⣿⡟⠀⠀⠀⣀⣤⠴⠚⣋⣥⣙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣥⣍⣉⣛⣫⣽⣿⠀⠀⠀⠀⠈⠙⠛⢛⣛⣫⣥⣤⠶⠚⠋⠉⠀⠀⠉⠁⠀⠉⠓⠌⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣀⣀⣤⠾⠛⢀⣀⣀⣤⡤⠶⠒⠛⠻⣿⣿⣿⣷⣄⠐⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣟⠋⠉⠉⠀⠀⠀⠀⠀⠀⣷⠈⠻⣿⣿⣿⣷⣄⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠈⠻⣿⣿⣿⣿⣦⡛⠀⠀⠀⢀⣠⡴⢚⣩⡴⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡛⢦⡀⠈⠳⣄⡀⢄⡀⠀⠀⠀⠉⠻⠷⣦⣬⠙⢿⣿⣿⣿⣦⡴⢞⣫⡵⣞⣫⠵⢶⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡛⢦⡙⢦⣄⠈⠛⢦⡙⠶⣤⣤⠤⠤⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⠿⢛⣋⡭⠶⣚⣻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢦⡙⢦⡙⠷⣄⠀⠙⠶⠶⠶⣒⣛⣯⠽⣷⣿⡿⢿⣿⣿⣿⠶⣛⣭⠵⠖⠋⢩⡴⣞⠛⠛⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⢦⣙⢦⣈⠳⠶⠖⢛⣋⣭⠵⠖⣛⣭⠴⢖⣻⣿⣿⠟⠋⠁⠀⠀⠀⠀⣿⣜⣃⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢦⣝⠷⠶⢚⣋⣥⠴⢞⣫⡭⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⢻⣤⣉⠙⠛⢋⣉⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⠷⣚⣫⡥⠖⠛⠉⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⣿⠦⣤⣍⣉⣩⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠒⠁⠀⠀⠈⠙⣆⠀⠙⠶⣤⣤⣤⣤⡤⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⢤⣤⠤⠶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣤⡤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣼⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢩⣧⠤⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣤⣴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠻⣿⣦⡙⠻⢿⣿⣿⣿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢿⣧⢌⣽⠾⠓⣨⢋⠈⢤⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⠿⠶⣶⣿⣿⣎⣡⡿⠀⢇⠘⣧⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣾⠿⠛⣿⣿⣳⣾⡿⠈⡎⢧⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣻⣧⣴⡿⠎⣿⢛⣿⣧⣾⣧⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡽⣿⣤⣾⣿⣤⣶⣿⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠻⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣧⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡆⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡟⠁⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠏⠀⠀⠀⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣼⣿⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣻⣿⡿⠏⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⣿⣿⢛⢿⠛⠟⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢿⣿⣿⠿⢿⣿⠟⣣⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⣵⣶⣶⣶⠞⣢⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣡⣾⣿⠿⣋⣵⣾⣿⠟⣋⡛⠿⠿⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣋⣼⡿⢋⣵⣾⣿⠟⣋⣴⣿⣿⣿⣿⣿⣷⡌⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢟⣡⣾⣿⣿⣿⠻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢟⣩⣶⣿⡿⢛⣥⣿⣿⠃⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢠⣾⠟⣫⣴⣿⡿⢟⣡⣶⣿⣿⣿⣿⣄⠙⠛⢉⣉⠙⢿⣿⣿⣿⣿⣿⠿⣋⣴⣿⡿⠟⠉
⠀⠀⠀⢀⣴⡿⢱⣿⣿⣿⣿⡐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⠿⣋⣵⣾⣿⠟⠉
⠀⠀⠀⣾⣿⣿⡘⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣉⡀⢺⣿⣿⣾⣿⠟⠋
⠀⠀⠀⠻⣿⣿⣿⣌⢿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣾⣿⠿⠋
⠀⠀⠀⠀⠈⠙⢿⣿⣦⡹⣿⣷⡜⢿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡜⢿⣿⣮⣥⣤⣭⡛⣩⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠻⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠿⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣹⠋⠉⠉⢯⠉⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⣰⣇⣀⣀⣀⣈⣧⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⢀⠟⣛⣛⣉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣙⣛⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢒⡏⠁⠀⠀⣾⠀⠀⠀⠈⠉⢳⠦⣼
⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢸⠃⢀⣀⣀⣷⣀⣀⣀⣀⣀⡞⠀⠈⣷⡀
⠀⠀⠀⠀⠸⣿⠿⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⣉⡉⠛⢻⠁⠀⠀⠀⠀⠀⠀⠀⡾⠀⣠⠞⠋⣍⠁⠀⠀⠀⠀⠀⠈⣉⡝⠢⣴⠁⢱
⠀⠀⣀⣤⣤⣇⣈⣙⣛⣒⣲⠶⠄⠀⠀⠰⠒⢛⣋⣉⣭⣥⣤⣼⠖⠲⢦⡄⠀⠀⠀⣀⣇⣼⣥⣶⣶⣾⣿⣶⣄⠀⠠⢖⣋⣡⣤⣀⣈⣇⡼
⠀⣿⡁⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⡿⠿⠿⢿⣿⣿⣀⣁⣠⡿⠀⠀⡞⠉⢹⣿⣿⠟⠉⣉⠛⣿⣿⣷⣾⣿⠟⣋⠉⠛⣿⣿⠗⢦⡀
⠀⠈⠙⠛⣻⣿⣿⣅⢀⣴⣶⠀⣹⡿⠋⠉⣿⣇⣶⣿⣦⣾⡿⣿⠉⠉⠀⠀⠀⠀⠙⠒⢻⠙⢿⣧⣤⣥⣤⣿⣿⠛⣿⣧⣀⣩⣤⣶⣿⡿⠒⠊
⠀⠀⠀⠀⣿⠘⢿⣿⣿⣿⣿⡿⢿⠇⠀⠀⢹⣿⣿⣿⠿⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⠛⠿⠿⠿⢻⠃⠀⢸⡟⠛⠿⠿⠛⢩⠇
⠀⠀⠀⠀⣿⠀⠀⠈⠉⠉⠉⠀⣿⣤⠤⠤⠾⠃⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⡀⠀⠀⠈⠉⠉⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠉⠉⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠚⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣠⠶⠚⠻⠿⢿⣿⣋⣉⣉⠉⠉⠉⠉⠉⣩⣿⢟⣛⢻⡿⠛⠛⢦⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢨⡇
⠀⠀⣠⣤⣼⣉⣉⣭⣭⣭⣭⣭⣉⠉⢉⣩⣭⣿⣿⣿⣷⣦⣼⡷⠶⢶⡄⠀⠀⠈⠷⣼⣿⣿⡟⠉⢡⣤⡄⠀⢹⣿⣿⣿⡏⠀⢰⣦⠀⢈⣿⣿⣿⠋
⠀⢾⡁⠀⣽⣿⡿⠛⣻⠛⠛⣿⣿⣿⣿⣿⣏⠉⢭⠉⣹⣿⣿⣿⢀⣸⠇⠀⠀⠀⠀⢸⣿⣿⣧⣄⣈⣋⣤⣴⣿⣿⢿⡿⣿⣷⣶⣶⣶⣿⣿⣿⠃
⠀⠈⠙⠒⣿⣿⣿⣶⣾⣶⣿⡿⢿⡏⠻⣿⣿⣿⣶⣿⡿⠟⠋⢸⡏⠁⠀⠀⠀⠀⠀⠸⡏⠛⠿⣿⣿⡿⠿⠟⠋⡁⣸⠇⠈⠙⠛⠛⠛⠛⠉⡟
⠀⠀⠀⠀⣿⠈⠉⠛⠛⠉⠁⠀⠿⠖⠂⠀⠈⠉⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠆⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠠⠦⡴⠶⠆⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠈⠳⠦⣄⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⣠⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠖⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣀⣀⣀⣀⣠⡤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠐⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠋⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⡭⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡻⢭⣟⣿⣿
⣿⣿⣿⣿⢁⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣇⡀⠘⣩⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠒⠚⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣺⣄⡀⠋⣻⣿
⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠛⠿⣿⣿⣿⣿⡟⠿⠿⢿⣿⡿⠿⢿⣻⣿⠿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣷⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣄⣀⣠⣾⢽⡿⠟⠀⠀⣄⠀⠀⣤⠀⢰⣿⣿⣆⣀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠴⠆⢉⢱⡦⢸⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣤⠀⠀⠀⠉⠀⠀⠀⠀⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠟⣛⣓⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣥⡀⠀⠈⠉⠃⣀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⢀⠞⢯⢿⡛⣿⡿⣿⣿⣶⣭⣥⣶⣿⣟⣻⣿⣿⣿⣿⣿⣿⠟⢻⣿
⣿⣿⡟⡟⠉⠹⠧⣿⣿⣿⣿⣿⣯⣤⣤⣤⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣆⢸⣶⣿⣷⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⡿⣥⣴⣿⣿
⣿⣿⣷⣝⠗⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⡃⣀⣀⣀⣀⠀⠀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣟⣻⠿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣦⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠋⠉⠉⠁⠁⠉⠉⠉⠁⠀⠀⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣾⣿⣿⣿⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⢻⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠉⠉⠉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣽⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠈⠁⢹⣿⣿⣯⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⠀⠀⣨⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠛⠀⠘⠻⠿⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣘⣆⠀⠀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⠤⠖⠒⠛⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠀⣞⠉⠀⠀⠀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣦⣤⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⣤⣶⣾⣿⣿⣿⣿⣟⠉⢻⣿⣿⣿⡅⠈⣿⡿⠀⠀⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣿⠟⠋⠻⠷⠾⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠒⠉⠀⠀⠀⠀⢀⣶⣿⣿⡆⠀⠀⡸⢀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠒⠢⡀⠀⠀⠀⠈⠙⢧⠀⠀⠀⠀⠀⠸⠛⠛⠙⠃⠀⢠⠃⢏⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠪⢤⠀⠹⡀⠀⠀⠀⠠⣈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡨⠉⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⠀⠈⠀⠀⣇⠤⠎⠀⣀⠀⠉⠛⢶⠲⠶⠶⠖⠒⠒⢚⠁⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⡎⠉⠑⠜⠉⠀⠈⠂⣄⠀⠁⠈⠉⠉⠀⠀⠈⢷⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠯⠇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣀⣀⠀⠀⠀⣀⣀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⣸⠍⠉⠛⠛⠛⠛⠛⠋⣹⠛⠒⠲⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣯⠞⠳⣤⠞⠁⠀⣠⠶⠶⠶⠶⠒⠛⠷⠤⣤⡄⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣰⢯⣤⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠈⠉⠀⠰⠀⢀⠂⠀⢶⡔⠡⠐⡄⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠁⣀⠐⣄⠈⠲⣦⠀⠀⠆⠀⠀⠀⢀⠈⠀⡐⠀⠈⠂⠀⠀⠄⢮⣱
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠘⠂⠀⠈⠁⢀⣀⠀⠀⠄⠡⠀⠀⠀⠁⠀⠐⠀⠀⢀⠈⠱⡃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⣀⠑⢐⣀⣀⠠⠀⢠⠁⠁⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠱⡄⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⣀⣁⣤⣼⡶⠂⠀⠀⠀⠈⢐⡄⠀⠀⠐⠀⢀⠄
⠀⠀⠀⠀⠀⡀⡀⠀⠈⠀⠀⠀⠀⠄⠀⠈⠉⠘⠿⠋⠊⣀⠀⠘⠀⣄⠀⠁⠀⠀⡀⠀⠀⠀⢠⠐⠀⠤⠀⠀⠀⠀⢐⠀⠀⢀
⠀⠀⠀⠀⢐⠀⠀⠀⡄⠀⠀⠀⠀⠒⠃⠉⠀⠀⠀⠁⠐⠡⠀⠀⠀⠀⠂⠀⠈⠘⡤⢔⡊⠁⠘⠠⠀⠀⠀⠂⠀⠀⠈⠠⡀⠘
⠠⠀⠌⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠁⢈⣦⠞⠡⠀⠉⢦⡀⠀⠈⠁⠀⠀⠠⠑⠂
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⢈⣠⠄⠀⠀⠀⢠⠈⣂⠀⠀⠀⠀⠀⠀⠀⣴⡟⠣⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠒⠂⣤⣀⣀⣀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠂⠀⠀⠁⠀⢀⣴⣾⡏⠁⠀⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⢀⠀⡄⠀⠤⢀⠠
⠀⠀⠀⠀⠈⠙⠛⣿⣿⡆⠀⠀⠀⠀⠐⠠⠄⠂⠀⠐⠒⠠⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣰⣶⠶⣶⣶⣦⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠁⠉⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠉⠙⠛⠿⣿⣿⣿⣿⠿⠛⠋⠉⠛⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⣀⣤⣿⣿⣤⣀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠋⠀⠀⠹⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣏⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣄⡀⢀⣴⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡳⢴⡶⠧⡶⠴⡃⠧⣶⠖⡖⢼⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠠⠤⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠈⠒⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠖⠖⢑⠀⠀⠀⡸⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⢐⡿⠀⠀⠈⠀⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠵⠋⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡀⠀⠀⠀⠀⠀⠀⢀⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠⠃⠀⠀⠀⠀⣠⣚⠡⠃⠃⢱
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡎⠀⣠⠋⠇⠀⢣⣠⠞⠉⠀⠀⠀⠀⡘
⠀⠀⠀⠠⠒⡱⢂⠀⡄⠀⠀⠀⠀⠣⠐⠁⠀⠈⢄⠀⠁⠀⠀⠀⠀⢀⡴⠓⠒⠀⠤⣄
⠀⠀⠀⠠⡑⡆⢅⣀⢱⠄⣀⣀⠀⣀⠀⠀⠀⢀⠀⣑⡦⠤⠄⠒⠊⠉⢲⠒⠒⠒⠠⢄⡉⠲⢄
⠀⠀⠀⠀⠣⠉⠹⠇⠚⠢⠤⠀⠀⠩⠤⠤⠤⡤⠔⠉⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠈⠒⢄⡑⢄⡀⠀⠀⠀⠀⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠙⡎⢀⣀⡀⠠⡐⢤⡜⠀⠀⢀⣀⣠⠤⠐⠒⠉⠁⠀⠀⠉⠐⢄
⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⡀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡇⠙⡎⠣⣀⡸⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠦⠀⠀⠀⠀⠀⠸⠀⢐⡄⠈⠒⢄⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⣠⡇⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⡄⠀⡀⠀⠀⡇⠀⢸⠱⣄⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⣀⣠⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠈⠢⡈⠢⣴⠀⠀⠇⠀⠈⠑⢄⡀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠸⡀⢸⠂⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊
⠀⠀⠀⠈⠑⡌⢆⣸⠀⠀⠀⠀⠀⠈⠁⠒⠒⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠀⠀⠀⠀⢀⢀⠳⡈⠑⣄⠀⠀⠀⠀⠀⠉⠑⠒⠀⠤⠤⠤⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠣⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠒⠚⢅⠀⠙⠢⡀⠀⡄⡘⠀⠹⡄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⠳⢀⠃⠀⠀⠹⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠁⠀⠀⠀⠀⠘⣤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠋⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⡁⠀⠀⠀⠀⠀⠀⢀⣴⣴⣶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠘⢦⣤⠀⢰⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⣿⣷⣜⣧⣾⣵⠀⢠⡀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢀⣤⣶⣾⣿⣿⢿⠟⠛⠷⣤⣿⣶⣦⣦⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢉⣾⢻⢏⡇⠞⠈⡠⡀⠀⠆⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⠀⠸⣿⠛⠘⠀⣴⠀⠀⡀⣤⣾⣤⣤⣤⣤⡔
⠀⠀⠀⠀⠀⢀⣤⣲⣖⣂⡖⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⠈⠀⠀⢿⡷⠖⠋⠉⠅
⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⢠⣶⡀⢀⡀⢰⣿⣀⣤⣤⣤⡾⠁
⠀⠀⢸⣿⡿⢿⣿⣿⣿⡇⡇⠈⣄⣀⡀⠀⠀⠀⠀⣠⣼⠏⠀⠀⢀⣴⣿⣿⢿⠿⠟⡩⣥⠀⠀⠀⠀⠘⡄
⠀⠀⢘⠛⠀⠈⠈⢠⢸⡇⡇⠀⢻⣀⣀⡀⢠⣴⣿⣿⠋⠀⠀⠀⠸⣿⣅⣿⢰⣄⡀⣠⣿⣤⣤⣤⣴⣶⡇
⠀⠀⣼⣀⣧⣧⣼⣾⣿⣇⢳⠀⠸⣿⣛⣿⣭⣽⣿⠇⠀⠀⠀⠀⠀⠙⢿⣿⣼⣟⠙⠉⠉⠉⠉⠉⠉⠉⢱
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠸⡀⠀⢻⣿⠿⠿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡇⢧⠀⠘⡇⠀⠀⢸⡃⠀⠀⠀⠀⠀⢠⣶⡄⣿⠀⣿⢘⣿⡄⠀⠀⠀⠀⢤
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡸⣄⠀⠙⠀⠀⢺⣵⣄⠀⠀⠀⠀⠈⠻⣿⣿⣦⣿⣾⠿⠁⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣷⣻⠀⠀⠀⠀⣄⣿⣿⣿⣶⡄⠀⠀⠀⠀⠻⠉⠿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠆
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣦⣤⣤⣤⣄⡀⣀⣀⠀⢀⣀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁


⣾⣷⠿⠿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢉⠛⠉⠛⠃⣐⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢀⠋⠘⠁⢀⠌⣀⠈⠀⠌⢙⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠚⡔⠠⢀⡆⠠⠁⠀⠀⢀⡙⠢⡈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢠⡆⠀⢀⠀⠇⠀⠀⠀⡉⠀⠀⠈⠳⣏⠹⣆
⠀⠀⠀⠀⢀⠠⠀⠒⠐⠒⠂⠠⣀⠑⢼⠅⠀⠀⠀⡄⠀⠀⠤⠀⠤⠀⠴⠒⠘⠘⢣⡀
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⠐⠘⢿⣿⣿⣿⣶⣶⣶⠤⢤⣄⣀⣀⣀⣀⡽
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠈⠻⣿⣿⣿⣿⣠⣾⣿⣷⣼⠇
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠁⢀⡜
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸
⠀⠀⠈⠐⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀
⠀⠀⠀⠀⠸⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠆⡀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⢪⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠚⢣
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⡰⣄⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢍⡶⢌⡓⠤⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣌⡑⠪⠆⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⡉⣿⠃⠀⠀⠀⢀⡤⢤⡀⠀⠀⢀⠀⠀⠈⢮⡑⢲⠀⠀⠀⠀⢃⡀
⠀⠀⠀⠀⠀⣶⠶⠋⠉⣿⡟⠁⠀⠀⠀⢸⠇⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⠙⢾⡀⠀⠀⠀⣀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡟⠇⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠟⠒⠒⠊⠁⠉⠒⠤⣀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡅⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⣀⡠⠄⠊⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢀⠇
⠀⢠⠋⠀⠁⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢰
⠀⠸⠀⠀⠀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⢾⠁
⠀⠠⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡀⠀⢺
⠀⠀⡀⠀⠀⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣆
⠀⠀⡃⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣱⡀⠀⠹⡄
⣠⣤⣶⣤⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⢀⣤⠴⠶⠒⠒⠛⠛⠛⠛⠛⣛⠳⢦⡀⠀⠀⠀⢀⣤⣄⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠈⠁⢀⡜⡉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⡳⣎⢳⡀⢀⣶⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⠈⠀⢀⣠⣤⡀⠀⠀⠀⠀⣴⣶⣾⣭⣿⣧⣥⣾⣿⣿⣿⣿⣿⣿⠀⠀⡂
⠀⠀⠀⠀⢀⠀⠉⠉⠉⠁⠀⡶⠋⠀⢰⣿⣳⠁⠙⠀⣠⢖⣩⣿⣿⣿⣿⡿⠋⢓⣻⣿⣿⣿⣷⣿⠇
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⡇⠀⢀⢀⣿⠳⣦⠀⠀⠁⣾⣽⣭⣴⣿⣿⡆⢀⣾⣍⢉⣿⣿⠿⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢻⣾⣅⣠⡟⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣇⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠘⠀⠉⠁⠀⢀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⡆⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠖⣀⢰⡧⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⣿⣿⣿⣯⢔⡂⠀⠘⣿⣿⣽⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠘⠳⢤⡀⠀⣀⣀⣀⣶⣾⣷⠀⠙⠿⠿⣿⣿⣿⣿⡉⢉⡉⡉⠛⠛⠛⠛⠁⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⢶⢿⡛⢛⠛⠹⡀⠀⠀⠶⣶⠀⠀⠀⠙⢆⡀⠤⠀⠀⠶⠭⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠬⠤⠤⣤⣤⣥⣤⣤⣀⣀⡀⣐⡃⢳⡀⠀⠀⠉⠀⢆⠀⠀⠀⠙⢦⡑⠃⠛⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢰⠉⠀⠉⠉⠉⠁⠉⠀⢳⡀⠀⠀⠀⠸⣄⣀⣀⠀⠀⠈⡇⠠⠀⠤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠸⡽⣿⡇⠀⠀⡗⠀⠀⠀⠀⠐⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠐⠢⣢⣦⣴⣶⠦⠀⠐⠒⠀⣷⣤⠄⠀⠀⣷⢯⠀⠀⢀⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⠘⠃⡠⠂⠁⠀⠀⢠⠞⠁⣸⠀⠀⢸⠇⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⠞⠁⠀⠀⣠⢖⡁⠀⠀⣿⣀⣤⣴⣷⣤⣄⡀⣿⣿⣿⣿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⠀⣠⠞⠁⠀⠀⣠⣴⣃⠉⣉⣙⣻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⣿⣷⣤⣀⣴⡧⠶⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣾⡗⠒⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣶⣾⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠋⠙⠛⠋⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠁⠀⠀⠀⠀⠀⡴⠂⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⡴⣊⡤⠤⠀⢧⣀⣠⠇
⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⣀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠉⠉⠉⠙⠲⢄⢰⠁
⠀⠀⣴⠶⣶⣾⣤⣶⣶⣶⣶⣿⣿⡿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣡⡍⠢⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢀⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠋⣀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠾⠏⠘⢦⡀⠀⠀⠀⠀⠒⣋⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣋⠁⠀⠀⠀⠀⢀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⢀⡰⡀⠳⠀⠀⠀⠀⠀⠀⠈⢻⢲⠒⠒⠚⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⡀⠀⠀⠀⠀⠀⠀⠀⢐⡧⠰⢤⡀⠄⠠⠄⠠⠸⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢫⡳⣄⠀⠀⠀⢀⠔⣹⠦⢤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡥⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠱⡄⢙⣦⡞⠁⠀⠉⠀⠉⠀⠀⠀⠀⠀⡠⠴⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠋⠀⠈⠳⢄⣀⠤⠤⠤⠔⠒⠒⠛⠢⠤⠴⠊⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡆
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢧
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠸⡄
⠀⠀⠀⣠⠔⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢧
⠀⠀⠀⠓⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⡩⢽⡡⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠉
⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣽⣷⣦⣶⣶⣦⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠠⢂⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⢂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢁⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠁⠁⢀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⢀⡊
⠀⠀⠀⢄⠀⠘⢶⣄⠀⠻⡅⠀⠀⠀⠀⠈⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠈⢀⠀⠀⠀⠀⡽⠁⢀⣴⠞⠀⢀⠄
⠀⠀⠀⠈⠢⠀⠀⠁⠠⠄⡠⡀⠀⡀⠈⠓⠀⣂⣀⡀⠈⠀⠀⠀⠀⠀⠈⠀⣀⣀⡀⠐⠉⠀⡀⠀⡠⠀⠤⠀⠀⠀⠠⠊
⠀⠀⠀⠀⢸⠷⣤⣈⣒⣀⣠⣨⡴⠖⠛⠋⠉⠉⠉⠉⠉⠛⠲⢤⠴⠚⠋⠉⠉⠉⠉⠉⠛⠛⠶⣬⣠⣀⣀⣊⣠⡴⢿
⠀⠀⠀⠀⢸⡆⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⣾
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠉⠉⠛⠲⣄⡀⠨⠿⠯⠀⣀⡴⠚⠋⠉⠉⠉⠛⠲⢤⣀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⠀⠀⣾⡁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠈⠙⠲⠤⣤⣤⡤⠶⠚⠛⠁⠀⠀⠙⠙⠳⠦⣤⣤⡤⠴⠚⠉⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠶⠶⠶⠒⠛⠉⠉⠁⠀⠀⠉⠉⠙⠓⠲⠶⠶⠶⠶⠶⠚⠋⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠷⣦⣤⣀⣠⣤⣶⢿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠈⠟⢀⠀⠉⣉⠉⠁⢀⣼⡟⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡄⠀⠀⠘⠁⠊⠛⠈⠁⠉⠀⠀⢀⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⣦⣤⣤⣤⣤⡶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⠉⠁⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠋⠙⠻⠷⣶⣶⠿⠛⠉⠛⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣡⣠⣄⣀⣀⣀⣀⣤⣠⣄⣄⣀⣠⣀⣄⣹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡗⠲⢤⣀
⡇⠀⢠⣈⠉⠓⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⡇⠀⢸⠉⠙⠓⠦⣄⣈⠙⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⡇⠀⢸⠀⠀⠀⠀⠀⠉⠙⠲⠤⣀⡉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣀⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢿⡟⠛⠛⠛⠉⠉⣿
⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⢲⠦⣄⣀⠀⠀⠀⢀⣀⣤⣤⣴⣾⣿⣿⣿⣾⣿⣷⣶⣶⣾⣷⣿
⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⢸⠀⣄⣉⠛⠲⢴⣟⠛⢻⣿⠿⢿⣿⣿⠻⣿⣿⣿⡿⠋⢿⣿⣿
⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⢸⠀⡇⠈⠙⠲⠤⣌⡁⠿⠃⡴⠺⢿⡏⠀⠉⠉⠉⠀⠀⠀⠁⠘⣇
⡇⠀⣿⠀⠀⠀⢀⡴⠖⠒⠲⠤⣄⡀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠉⠓⠀⢳⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⡇⠀⣿⠀⠀⠀⠀⠳⣤⣀⠀⠀⠀⢹⡆⠀⠀⢸⠀⣿⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⣇
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⠲⠾⠄
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⣤⣤⢤⡿
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠛⠁⠀⢀⡇⢀⣀
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣷⠶⣤⣄⡀⠀⠀⠀⢸⠇⣿⣿⣿⣿⣶⣦⣄
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣄⣼⣿⣿⣶⣤⣼⠀⠈⠉⠉⠛⠻⣿⣿⣿⣦
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⢸⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣨⠀⠀⠸⠀⢠⠞⣫⣳⡄⠸⠀⠃⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠓⠦⣄⠀⠀⠀⠀⠀⢹⣿⣷
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠶⠤⠤⠴⠞⢸⠹⣍⣀⣹⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠤⠄⢠⢟⣻⡋⢳⣄⣠⠟⢻⣀
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⡛⠛⢛⠋⠻⣟⠦⢞⡿⢁⠀⣠⡤⠶⠶⠒⢲⣶⠒⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡛⣿⠇⣸⠧⢿⣄⡼⠥
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⠀⣶⠈⠓⠋⠁⢸⢸⡁⣄⡀⠸⠀⣠⠉⢁⣼⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣇⠀⢀⡹⠛⠋⠉
⡇⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⡇⠀⢸⠀⣿⠀⠀⠀⠀⢸⠀⣧⣈⣛⣀⢚⣋⣀⣉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡴⠋
⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢸⠀⣿⠀⠀⠀⠀⢸⠀⠙⢯⡉⠉⠉⠉⠉⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣇
⡇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⠀⠀⠀⠀⠈⠋⠉⠉⠉⠉⠉⠁⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⢻⡄
⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢈⣧
⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣿⣿⡄
⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷
⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀
⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠤⠤⠤⠄⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣿⣿⠓⠒⠒⠒⠒⠒⠒⠒⠲⠶⠦⠤⠄⠀⠀⠀⠀⠛⢿⣷⣄
⠃⠴⠷⠔⠒⠒⠒⠊⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⡦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⠶⣿⣿⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣛⣁⣀⣀⣀⣀
⣿⣿⡿⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⢿⣿⣿⠿⠿⢿⣧⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢰⡶⢀⡀⠸⠿⣸⣿⠀⠀⠀⢀⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣶⠶⠟⠿⠷⣾⣿⠇⢀⣤⣶⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣅⣘⠛⢛⣠⡾⠋⠀⠙⢿⣿⡏⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣏⣽⣿⠇⠈⠙⣛⣛⡋⣁⣀⣤⡀⠀⠈⠉⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠉⣿⣦⣤⠘⠛⠛⠛⠛⠛⠻⢣⣦⠰⣷⣮⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⣧⣄⣉⣿⣶⣶⣶⣶⣶⣶⡖⢿⠏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠉⠙⣋⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣦⡾⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠿⠋⠠⣭⣭⣭⣭⣭⣭⣭⣭⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣤⣤⣀⣀⣀⣿⣿⣿⣿⣿⠛⠻⠿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠉⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⣀⣠⠤⠶⠖⠒⠚⠛⠛⠛⠛⠛⠛⠛⠚⠳⣆⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⢠⠴⠛⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⣀⣤⣴⣶⣿⣿⡿⠛⢿⣿⣿⣿⡿⠛⢿⣿⡏⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣗⠀⣸⣿⣿⣿⣯⣀⣼⣿⠇⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⣿⠿⠿⠿⠿⠟⠛⠋⠀⡀⠙⠛⠛⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⢘⡆⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣈⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⠀⠀⢀⠇⡴⠖⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠉⠓⣄⠀⠀⠀⠀⠈⠙⣇⠀⠀⠀⠀⠀⠘⠋⠉⠉⠉⠀⠀⡞⠘⢄⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠡⢢⠀⠘⡄⠀⠀⢀⠠⠤⡘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⡦⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⣇⡠⢆⠁⣀⣄⠀⠉⠻⢖⠒⠶⠶⠒⠒⠒⢛⡓⠀⠒⠚⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⡞⠉⠑⡦⠋⠁⠀⠑⠢⣀⠀⠉⠉⠉⠉⠉⠉⠀⢹⡄⠀⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢖⡣⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣿⠛⠻⠶⠶⠶⠶⠶⠿⠛⡿⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢻⣦⡀⢀⡴⠋⠀⢀⣠⣀⣀⣀⣠⣤⣞⣁⠀⠀⠀⠉⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠹⡏⠀⠙⠋⠀⢀⡴⠋⠀⠈⠉⠁⠀⠀⠀⠉⣩⠟⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣽⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⣭⣭⠿⠀⠀⠀⢠⠈⠡⠤⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠠⢜⠀⠀⠶⠖⣿⠍⠡⠤⠤⢵⠀⢚⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣉⣿⣉⠘⣛⣛⣽⡇⠀⢒⣚⡿⠀⠥⠵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢤⣷⠀⢈⣁⣀⢽⡇⠈⣉⣭⣿⡀⢒⣲⡧⠀⠀⡀
⠀⠀⠀⠀⢀⠀⡀⡤⢄⡐⢒⡲⠒⠚⠓⠒⠛⠂⠀⠂⠤⢾⡃⠀⠶⣺⣯⠄⣉⡥⣿⢉⣽⡇
⠀⠀⠀⠀⠘⠙⠓⠻⢶⠀⠈⠀⠀⠀⠀⣀⣠⠴⠖⠒⠈⢉⠉⠈⢻⣽⡻⡤⠤⣶⣷⢴⣿⣧⡆⢀⣤⣤
⠀⠀⠀⠀⢀⡀⠀⢈⢹⠀⠀⠀⠀⠀⠀⡿⠿⠷⢤⣬⣦⡶⣰⣭⣿⣶⣒⣖⣒⣲⣾⣽⠿⠿⣯⣽⣿⣿⠷⠂
⠀⠀⠀⠀⠈⠙⠊⠛⠻⢦⣀⠀⠀⠀⠀⣏⠂⠀⠀⠀⠀⡏⠁⠉⠈⠉⠩⣏⠉⠉⠾⣁⣐⣐⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠤⠹⠷⢶⣦⣄⣀⣇⣠⣤⣶⣾⠽⠏⠂⠐⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣣⠚⠉⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠄⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢰⣄⣴⣾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄⠐⡈⢉⡕⣶⣠⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠘⣿⣿⣿⣿⠿⢿⣇⣀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣄⣠⠀⢨⣼⣮⡿⢡⣿⣿⣆⣸⣿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣷⡀⠀⠼⠿⢿⡏⠀⠀⠈⠛⠷⣿⣷⣄⠀⠀⠀⠀⢀⠀⣀⣸⡟⠁⠀⣀⣽⣿⡿⣿⡹⣿⣿⣿⣿⣿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⢈⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣧⣿⣷⣶⣶⣶⣿⣿⣿⣿⣦⡀⣱⣟⡀⠹⣿⣧⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢈⣠⣾⣿⣿⣿⣿⡿⠀⠛⣄⡀⣱⡾⠿⠿⠿⠿⠛⠛⠋⣴⣿⣿⣿⣿⣏⣀⢀⣽⢈⢡⠀⣿⣟⡫⣿⢝⢏⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠔⠈⢿⣿⣟⣿⣿⣿⣿⣿⣿⡀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⣠⣠⣶⣿⣿⡟⣿⣿⣿⣿⣿⣾⣿⣆⢸⣿⣷⣷⣿⠄⡰⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⣀⣤⣶⣠⣶⣿⡿⣷⢿⣿⣿⡿⠿⠋⠛⠛⡿⠛⣿⣿⠀⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⢻⣤⣴⣿⣿⣿⣿⣿⡿⢃⣼⠛⣿⣿⣿⠇⠤⡄⠀⢰⢦⣆⣽⣽⡦⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇⠿⣿⣿⣿⣿⠟⠁⠀⣾⠃⠀⣿⠿⠋⠀⠀⣹⢻⣾⠿⣿⣿⣿⡏⢘⣿⠿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡿⠛⣿⡆⢸⡭⠛⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢰⡟⣸⡇⠀⢻⣿⣿⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⠏⠀⠀⡾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠛⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⢀⡤⠴⠶⠦⠤⢤⣖⠢⣤⣀⡀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠉⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠁⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣾⣶⣾⣾⣷⡀⠀⠀⠀⠀⠀⣠⠴⠶⡿⠷⠦⣄⡀
⠀⠀⠐⠦⠀⠀⠀⠈⢳⣀⣈⣉⢀⠞⠀⠀⠀⣀⠴⠚⠁⠀⠈⠉⠉⠉⠉⠀⠙⠢⣄⠀⠀⠀⠈⢦⣀⣀⣀⣀⡜⠁⠀⠀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⣀⠼⢿⠿⢿⠿⣄⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢀⠿⢿⡿⠿⠿⢷⡀
⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⢀⡤⠶⠧⢤⠀⠀⠀⠀⠀⠀⠈⢷⠁⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⡴⠁⠀⠀⠀⣠⠤⣄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣠⠦⡄⠀⠀⠀⠈⢢⡀
⠀⠀⢸⠁⠀⠀⠀⠀⢧⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⢆⠀⠀⠀⠀⠀⠀⠀⢸⡆⠸⣅⣀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢙⠆⠀⣇⢀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⣹⠀⠀⠀⠀⠀⡏
⠀⠀⠘⡄⠀⠀⠀⠀⠈⠛⠉⠀⠀⠘⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠉⠛⠁⠀⠀⠀⢀⡜⠁
⠀⠀⠩⢞⡦⢄⣀⣀⣀⣀⣀⣀⣀⣠⣼⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣴⡯⣀⡀⠀⠀⠀⠀⢀⣠⡔⣋⣀⣀
⠀⠀⠀⠀⠀⠞⠁⠼⠋⡽⢋⠜⢋⠔⢁⠔⢁⠞⣩⢟⡿⢛⡿⠋⣹⠟⣹⠋⡹⢋⡽⠁⠎⠰⠋⠞⠙⢫⠟⡽⠁⠞⠙⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠋⠀⠈⠀⠀⠁⠀⠁⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣖⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⣩⢶⣏⠙⢿⣆⠙⢷⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⣧⡀⠻⣷⡀⠹⣷⢀⣥⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡹⣧⡘⢿⣄⠘⣿⣠⣶⣿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⢳⡌⣷⠈⠋⠙⠛⠛⠻⠿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠓⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣬⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢾⣧⡸⠟⢉⣹⣶⡤⠄⠀⠀⠀⠠⢤⣤⣴⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡟⢻⣧⡀⢻⣃⣶⠟⠉⣡⡤⢶⠿⠇⢀⠰⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠿⣶⣿⠹⣷⡀⠻⠗⠠⠿⠋⢠⣾⠋⣠⡾⢛⠅⠀⠀⠘⠋⠛⠻⢛⣽⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢈⢹⡆⣹⣷⠶⣶⣶⣶⣶⣶⣥⣾⠋⣴⠏⣼⠇⡀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⢿⡘⣋⣤⣤⣤⣤⣤⣄⣀⣀⣀⡡⠞⣡⣾⠃⡼⠇⠀⠀⠀⠀⠠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⢿⣿⢎⡛⣋⣉⣉⣉⣉⣉⣉⣥⣤⣤⣤⡾⠛⠁⠈⣠⣶⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⡀⢀⣼⣿⢦⡿⠿⠿⠿⠿⠿⠷⠶⣶⣤⣤⣬⣤⣴⣶⠿⠛⣡⡶⠀⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠟⠿⠛⠁⢴⣶⣶⣶⣶⣶⣶⣶⣤⣤⡀⠉⣉⣉⣀⣤⣶⡾⠟⣵⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣄⣀⣉⠙⠛⠛⢛⣛⣉⣉⣠⣴⠾⣡⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣉⣉⣉⣉⣉⡉⠙⢻⣟⣋⡛⢛⣛⣋⣉⣥⡴⠞⣋⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠛⢛⣛⡿⢋⣿⠟⢚⣛⠛⣛⣝⣩⣽⣥⣤⠴⠞⠛⣉⣭⣶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠾⣻⣿⠋⢠⣿⠏⢀⣾⠏⢀⣿⠇⢸⣿⠶⣶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⣰⣿⠃⢀⣾⠏⢀⣾⠏⢀⣿⡏⠆⠉⠐⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣀⡾⠟⢁⣴⡿⠋⣠⡾⠋⣠⣾⢏⣀⣀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠷⣾⣿⣤⠾⠟⠁⣴⡿⠛⠛⠛⠛⠛⠛⠉⣉⣀⣤⣿⠉⠉⠙⠻⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣤⣄⣹⣶⣶⣶⣿⣷⣶⣶⣶⣶⣾⣿⠿⢿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡉⠙⠛⠻⠿⠿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠛⠻⠿⠿⣶⣶⣤⣤⣤⣤⣈⣉⣉⣉⠉⠉⠛⠛⠛⠃
⡶⢶⢶⠶⡶⠶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⠶⡶⣶⠶⣶⢶

⠀⠀⢠⣤⣤⣤⠤⠤⣤⣀⣀⣀⣀⣤⠤⠤⢤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠲⣤⡀⠀⠀⢀⡴⠖⠛⠒⢦⡀
⠀⠀⠈⢻⠁⠀⠀⠀⠀⠈⡟⢹⡇⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⢴⡏⠀⠀⠀⠀⠈⢷⠖⠲⡾⠀⠀⠀⠀⠀⢹⡦
⠀⠀⠀⠘⣇⠀⠀⠀⠀⡰⠃⠀⢣⡀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢀⡾⠀⠀⢷⡀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⢈⣙⡒⠒⠋⣁⣀⣀⣀⡙⠒⠒⣚⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠴⣛⣀⡀⢀⣈⣓⠦⠤⠴⠞⠁
⠀⠀⠀⠀⢺⣤⣥⣴⣿⣿⣿⢿⣿⣿⣶⣤⣥⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠉⠛⠛⠛⠛⠉


⠀⠀⣴⣶⠖⠒⠛⠓⠒⠲⣶⣶⡶⠒⠒⠛⠓⠒⢶⣶⡄⠀⠀⠀⠀⠀⣶⡟⠋⠉⠉⠉⠛⢷⣤⣤⣶⠛⠋⠉⠉⠉⠛⣷⡆
⠀⠀⠉⢹⠀⠀⠀⠀⠀⢀⡏⠙⡄⠀⠀⠀⠀⠀⡼⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠨⠀⠀⠙⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠣⣀⡀⣀⣠⠞⠀⠀⠙⢦⣀⠀⣀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣀⠀⠀⠀⣠⣴⣄⣠⣴⣤⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣤⣶⣦⡀
⠀⠀⠀⠀⢻⣶⣶⣿⣿⣿⡿⠻⣿⣿⣿⣶⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣴⣿⣿⣿⡿⠻⣿⣿⣿⣶⣴⠆
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠉⠉

⠀⢀⣀⡤⠔⠒⠒⠂⢤⣀⣀⣀⣀⡠⠄⠒⠒⠲⢤⣄⣀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⠄⣀⣒⣀⣀⡠⠤⠤⠠⠤⣄⣀
⠀⠘⡿⠀⠀⠀⠀⠀⠀⢹⡏⠉⡟⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⡟⠛⡏⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠹⡀⠀⠀⠀⠀⠀⡸⠀⠀⢣⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⡰⠃⠀⢳⡀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠙⠢⠤⠤⠤⠚⢁⡀⠀⣀⠑⠦⠤⠤⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠚⠁⠀⠀⠀⠙⠢⠤⠤⠔⠃
⠀⠀⠀⠀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣷⣦⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⣴⣾⣿⣿⣷⣾⣿⣿⣷⣶⣤⣤⠂
⠀⠀⠀⠀⠈⠻⠿⣿⠿⠿⠋⠙⠻⠿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠛⠛⠛⠛⠉

⡟⠒⠤⣀
⡇⢰⠒⠤⣍⡑⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣦⣤⣀
⡇⢸⠀⠀⠀⠈⠙⠒⠤⣉⡒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠆⢹⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣽⣭⣥⣤⣤⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⣈⣑⠢⢴⡛⢻⡿⢿⣿⠟⣿⣿⡿⠻⣿⣇
⡇⢸⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⡇⠀⠉⠒⠤⣟⢸⠉⡿⠀⠈⠁⠀⠀⠀⠸⡄
⡇⢸⠀⠀⠘⢧⣀⠀⠉⢲⠀⠀⠄⢸⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠈⢳⠇⠀⠀⠀⠀⠀⠀⠀⢳
⡇⢸⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢠⢤⣈⣇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⣀⣀⡾⠀⠈⠉
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠾⣀⡀⠀⠀⠘⠁⠀⡇⣀⣀⣀
⡇⢸⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣯⡙⣲⣤⣄⣀⠇⠛⠻⢿⣿⣶⣄
⡇⢸⠀⠀⠀⠀⠀⠀⠀⡇⡿⠀⠀⠰⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣯⢀⠀⠀⠀⠈⠙⢿⣷⡀
⡇⢸⠀⠀⠀⠀⠀⠀⠀⡷⢆⠀⣀⣰⣻⡿⠆⣧⣤⣤⣴⣶⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣆⠉⠒⣤⡤⣄⠈⣿⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⡇⠒⠒⠒⠲⡿⢽⡿⠉⣉⠭⠭⠭⣥⠤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠈⢸⣿⡗⣸⠽⣇⡼
⡇⢸⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠇⢸⠈⠉⠀⠄⡇⢤⡘⣠⠌⣤⡟⠛⠛⠛⠛⠛⠛⠛⠛⡆⣠⠏⠉⠁
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠓⠚⠀⡀⢸⠀⠀⠀⠃⠺⡖⠒⠒⠒⢺⠃⠀⠀⠀⠀⠀⠀⢀⡴⢻
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠉⠉⠉⠉⠉⠀⢸⠀⠀⠀⠀⠀⠀⠐⠉⠀⠈⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠖⠒⠒⠒⠒⠒⠒⠒⢶⣿⡀
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⣀⣀⣀⣀⣀⣀⣠⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠙⣿⣧
⠧⠼⠤⠴⠐⠒⠒⠒⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡦
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣯⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠘⠘⠛⠀⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠘⠁⠛
⠀⠀⠀⣶⣶⣶⠂⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢴⣶⣶⠒
⠀⠀⠀⢈⣀⣁⡀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⣤⣤⣶⣶⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⢀⣀⣁⣀
⠀⠀⠀⠻⠛⠟⠀⠛⡛⠉⠉⠉⠉⠉⠉⠉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⢛⠋⠹⠟⠿⠁
⠀⠀⠀⣰⣦⣶⠆⡤⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢤⢰⣶⣶⠆
⠀⠀⠀⠀⠁⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⡉⣀
⠀⠀⠀⠿⠯⠏⠁⠒⠒⠒⠒⠒⠒⠒⠒⣿⣿⣿⣿⡏⢁⣬⡉⠙⠋⠁⡴⣤⢸⣿⡗⠒⠒⠒⠒⠒⠒⠒⠒⠸⠏⠿⠁
⠀⠀⠀⣤⣤⠄⠄⢀⣀⣀⣀⣀⣀⣀⣈⣿⣿⣿⣿⣧⣈⠛⢋⣠⣤⣀⣙⣉⣼⣿⣁⣀⣀⣀⣀⣀⣀⣀⡀⢠⠠⣤⠦
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⢻⠿⠇⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠻⣿⣿⣧⣀⠀⠀⠀⢀⣠⡿⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠘⠀⡿⠉
⠀⠀⠀⣠⣤⡄⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠈⠙⠛⠿⠿⠿⠟⠛⠉⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢠⠀⣤⠤
⠀⠀⠀⠙⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣼⣿⣿⣷⣾⣿⣿⣷⣾⣿⣿⣧⣾⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛
⠀⠀⠀⢺⡿⣿⠀⠀⠀⠀⠀⠀⠀⣴⣄⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⣠⣦⠀⠀⠀⠀⠀⠀⢰⡾⣾⠁
⠀⠀⠀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⢠⣤⣄⡄
⠀⠀⠀⠙⠓⠋⠀⠉⡀⠀⢀⠁⠀⣰⣿⡏⠉⠿⡿⢿⣿⣿⡿⣿⢿⣿⢿⣿⠿⠈⢹⣿⣆⠀⠀⠀⠀⢁⠉⠙⠃⠛
⠀⠀⠀⣶⣷⣶⠃⠤⠤⠤⠤⠤⠠⣿⡟⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠀⠀⠀⢻⣿⠄⠤⠤⠤⠤⠤⢴⣷⣷⠓
⠀⠀⠀⢀⣀⣀⡀⠀⠒⠒⠒⠂⠀⠀⠀⠙⠀⣴⣿⣯⣷⣯⣿⣽⣼⣮⣼⣿⣦⠀⠉⠀⠀⠀⠐⠒⠒⠒⠂⢀⣀⣀⣀
⠀⠀⠀⠙⠓⠛⠀⠉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢈⣉⣉⣉⣉⣉⣉⣉⣉⠉⠙⠃⠛
⠀⠀⠀⣠⣆⣶⠂⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡤⠤⣤⣤⣤⣤⣤⣤⣤⣤⢠⡶⣦⠒
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠦⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠾⠧⣦⣤⣤⣀⣀⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣤⣄⣀⣀⡀⠀⠉⠉⠉⠛⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡟⠛⣛⠛⠻⣿⣶⣦⣄⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣶⣤⣶⣾⡿⠟⠛⠿⡏⠀⢀⣀⣠⡤⠤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢉⣁⠀⠀⢈⣤⡶⠗⠚⠉⠁⠠⠀⠀⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⡠⠊⠐⠀⠀⠀⢀⣄⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠛⡏⠀⠀⣀⣠⣤⡾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠸⡇
⠀⠀⢰⠒⢄⠀⠀⠀⠀⠀⢀⡙⠘⠀⠙⣻⣿⣿⣿⡟⠀⠹⠟⠉⣧⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠇
⠀⠀⢼⠀⠈⢆⠀⠀⠀⠁⣃⡕⠂⣠⣾⣾⡿⠿⢿⣿⣷⡄⣠⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠏
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠈⠁⠸⣿⣿⠟⢻⡶⠉⠻⣿⣿⣿⠟⠁⢘⣗⠦⠤⣤⣄⣠⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⣸⣤⣤⣄⢀⣀⣤⣼⡇⠀⢸⣷⣆⡼⠛⠉⠀⢀⣴⣿⡿⠓⢤⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⣿⣿⠏⠱⣤⣴⣶⣿⣿⠟⠁⢠⣾⣿
⠀⠀⠀⠀⠀⠀⠹⣿⣏⠿⠛⠛⠉⠁⠀⠀⠈⠁⠀⠀⠹⡿⠛⠋⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⣶⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡡⡀⢭⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠛⠋⠁⠀⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠥⠜⠀⣀⠤⠒⠋⠉⠀⢀⡠⠄⠒⠂⠉⠀⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣞⣶⡄
⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⣠⢴⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠈⢷⣻⣦⣄⠀⢰⡏⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⠛⠒⠒⠛⠉⠀⠀⠹⣿⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣄⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢀⣀⣀⡀⠤⠔⠒⠉⠁⠀⠈⠙⣷⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣾⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢻⣷⡄
⠀⠀⠀⠀⠀⠀⡀⠀⣴⣾⣿⣷⣌⡻⣆⠀⣀⠠⠄⣀
⠀⠀⢄⠀⠀⡠⠄⢸⣿⣷⣏⢷⣽⣿⣮⡇⠀⠀⠀⠈⣷⠞⠉⠑⠢⣄⠀⢀⡀
⠀⠀⠀⠑⠄⣈⠀⣸⣿⡟⣟⣸⢿⠇⢸⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣼⠋⠁⠉⠑⣄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⣶⡿⠄⠛⠠⠤⠄⣀⠼⠄⣄⡀⠀⠀⡏⠀⠀⠀⠀⣸⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠛⠙⠛⡟⠀⠀⠀⠀⠀⠸⠁⠀⠀⠠⠁⠀⢸⠁⠀⠀⠀⢠⡏⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠈⢀⠀⠂⠀⠲⡀⠀⠀⠀⠀⠰⠧⣀⣴⢇⡜⠀⡟⠀⠀⠀⢀⡞⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠀⠲⣤⣀⣀⣀⣤⣴⠋⠀⢸⡇⠀⠀⢠⡞⠀⠀⠀⣴⣷⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⣿⡿⡟⠹⡿⢶⢶⣿⣷⣴⢶⣯⣀⣀⣠⣾⣽⣏⢻⣾⣿⡿⠋⣡⠄⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠞⠁⠃⠀⠀⠈⡇⠛⢿⡀⠀⠉⢍⣍⠙⢿⣯⣻⣫⣾⡿⠀⣮⡈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢻⢢⠈⠻⣿⣿⣿⡇⢸⣿⢃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠓⡄⠀⠀⠀⢡⡀⠙⠹⣿⠀⠘⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⣀⣀⡀⠀⢀⣀⣼⣣⠀⠈⠀⠀⣀⠀⣟⠦⣀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠙⠛⠛⠛⠋⠥⢤⣤⣤⣤⡼⠋⠦⠖⠚⠙⠓⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠓⠲⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇