Annoying Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣯⠉⠈⢙⣿⣷⡄⠀⠈⠔⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠓⠈⠀⢀⣹⠟⠁⠀⠀⠠⣉⠁
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⡦⢙⡈⠁⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠠⠂⠀⢸⢣⣀⣳⠀⠠⠀⠈⠀⠐⠻⠟⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣧
⠀⠀⠀⠀⢀⠄⠀⠀⠀⠘⠉⠙⠀⠀⡀⢁⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⠉⠻⡛⣿⣆
⠀⠀⠀⠐⠉⡰⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⣤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⣿⣯⡀
⠀⠀⠀⠂⠈⠑⢄⠀⢘⠀⠀⠀⠀⠠⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡀⣿⣿⠦⠠⣾⣿⣿⡄
⠀⠀⠀⠀⠈⠂⠄⡑⠊⢱⣤⠀⠀⢀⠀⠀⠁⠁⠀⠀⠀⠀⠀⢠⠪⡴⠃⠀⠀⣮⣆⣰⠵⠀⠉⠳
⠀⠀⠀⠀⠀⠀⠀⢈⠂⠉⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡀⠄⠂⠀⠀⠈⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒⠂⠁⠐⢊⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⢠⣿⡄⠀⠀⠀⠀⠀⠷⠀⠠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢄⢸⡿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠠⠴⡀⡸⠼⠧⠤⠤⠤⣤⠎⠀⠀⠀⠀⠶⠀⠀⢀⠆⠼⣿⣿⡿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣾⣃⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣴⣶⣶⣶⣶⠶⠿⠿⠛⠛⠉⠀⠀⠀⠀⠈⠛⠻⠷⣶⣶⣦⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⣴⣶⣶⣶⣦⠀⠀⠀⣴⣶⣶⣶⡦⠀⠀⠀⠛⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣾⡆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⢿⣧⠀⠀⠀⠀⠀⠀⣴⠟⠛⠉⠙⠻⠷⠀⠀⠀⠀⠀⣼⡿⠾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣷⣶⣤⣤⣀⣀⣤⣤⣴⣾⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⠛⠉⠉⣿⡏⠉⠉⢹⣿⠉⠉⠛⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠘⢷⣤⣴⡾⠋⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠠⣴⣶⣶⣼⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠀⠀⣠⠤⡞⠲⣄⡠⡭⠵⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣩⠖⠉⠱⡄⡸⠁⡠⠗⢠⡀⠀⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠉⠐⣿⣀⡞⠁⠀⠀⢶⠟⠁⢠⡣⠒⠚⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣀⣀⣀⡩⣿⠀⠀⠀⣠⠏⠀⠀⠛⡃⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠉⠉⠉⠉⠀⠀⠈⢉⠉⠙⢗⠒⠋⠁⠀⠀⠀⠉⠐⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣐⣠⠦⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡂⠠⢤⠤⠀⢤⠄⠠⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠳⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⢸⠀⠀⠀⠀⢸⠉⠒⡤⡀⠀⡠⠔⣫⠴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠧⣄⣸⠀⠀⡀⠀⣸⣀⠤⠗⠊⣡⠔⠊⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣮⣧⣠⢤⠄⠈⠓⠁⠉⠁⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⣠⢊⠿⣑⢢⡀⠀⠀⠀⠀⠀⣀⣀⠈⡰⢸⣆⠀⠈⠀⠄⣀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣇
⠀⠀⠀⢀⡗⡎⣴⣩⢆⠆⢐⠄⡔⣖⢟⠓⡨⠛⠋⡱⢄⠀⠀⢀⠀⡀⠀⠀⢀⡠⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠝
⠀⠀⠀⠈⠛⠧⢻⣏⡰⠀⠤⠤⣈⠀⠀⢿⣀⡔⠉⡉⠣⣠⠔⢣⡐⡈⠈⠉⠁⠀⠀⡠⠚⠃⠀⣀⣠⠔⠋⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢓⣿⠏⠀⠢⠀⢺⠀⡧⢋⡀⢀⢠⠔⣀⠠⠄⠒⠉⠁⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠠⡖⠛⠉⠋⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠖⠐⠤⢟⢹⠐⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣢⣀⠀⠙⣧⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⢪⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠐⠒⠚⢋⠭⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠓⠚⠛⠛⠛⠛⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠄⠀⠠

⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⣀⣀⡀⢀⡠⠄⠒⠄
⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⢸⡉⡉
⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠤⠤⡖⠉⠀⠈⠒⠤⠤⢄
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⣄⣀⠀⠀⠀⠀⠀⣦⣀⡀
⠀⠀⠀⠀⠃⠠⠴⣶⢶⣶⠶⠦⠀⠠⡶⡶⣶⣶⠶⠤⠀⠀⠀⢰⣼⣿⣇⠀⠀⠀⢠⣵⣿⣷
⠀⠀⠀⢸⠀⠀⠀⣧⣾⣿⡄⠀⠀⠀⢰⣧⣿⣿⠀⠀⠀⠀⠠⠾⠯⠉⠛⠀⠀⠀⠘⠋⠩⠿⠦⠀⠀⡀
⠀⠀⠀⠸⡄⠀⠀⣿⣿⣿⠃⠀⠀⠀⠈⣿⣿⣿⠁⠀⠀⢰⠀⣠⣠⠖⢲⢄⡀⢀⣠⠖⢢⡤⢄⠀⢀⢇
⠀⠀⠀⠀⢣⠀⠀⠈⠙⠁⠀⠤⠤⠤⡀⠈⠉⠁⠀⠀⢀⣎⠈⡆⠀⠀⠈⢁⡕⢏⠈⠀⠀⠀⠸⢀⠎⡜
⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠮⣳⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡣⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠀⠀⠀⠀⠉⠑⠲⠠⠤⠤⠄⠤⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠤⣄⠀⢠⢒⢢⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣊⠭⠤⡄⠀⡼⠁⢙⡆⠀⠙⠁⡇⠈⠁⠀⠀⠀⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠤⣔⡦⠀⢠⠎⠁⠀⣠⣿⡜⠁⢰⢅⣀⣹⠶⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠰⠿⣶⠃⠀⠀⠀⢩⠏⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⣤⠤⠍⢓⠢⣤⠴⢲⣯⣀⣀⣠⠴⠃⠀⠀⢨⠽⢒⣊⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⢀⢁⣉⣨⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⡀⠀⠀⠀⢀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⡥⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠤⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⠒⡖⠒⠂⣶⠒⠒⡤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣈⠖⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣿⠀⠀⡇⠀⢸⠉⠉⠲⢄⡀⣀⠤⢊⡡⠶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠾⠤⣀⣿⠀⠀⡇⠀⢸⣀⠠⠶⠒⣉⡤⠚⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣽⣄⡠⣤⠶⠛⠛⠛⠈⠉⠀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⢀⡴⣻⣏⠲⣄⡀⠀⠀⠀⠀⠀⠀⣀⠐⢡⠂⣯⠄⠀⠙⢢⣄⢀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡖
⠀⠀⠀⠀⡼⢤⢡⣜⣸⣋⡬⢢⡄⡤⡶⡾⠣⢘⡵⠛⠂⣍⡀⠀⠀⠀⠀⠉⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵
⠀⠀⠀⠀⠻⢟⡺⣻⢃⡟⢀⣘⣄⠉⠉⠈⢲⠁⢀⠔⠉⠰⢃⢀⡠⣎⢻⢹⠱⡛⠊⠁⠀⠀⡠⠶⠀⠀⠀⣀⡤⠖⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠁⠀⠀⠙⠐⠢⠀⠧⢽⣶⠊⠢⣀⠁⠤⠘⣄⠞⠀⢡⠀⡧⠔⠉⣀⣠⠴⠒⠋⠁⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠂⠀⠀⣀⡸⠶⠷⣞⣲⡼⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⣠⣀⣀⢈⡆⣠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠉⠳⣾⡀⠀⣫⠇⠀⠀⠀⠀⠀⠀⠀⢀⠠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⠒⢷⣆⣠⡤⢿⡋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠰⡒⠛⠲⣤⠀⢀⣀⣀⣀⣀⣀⠤⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠤⣶⣆⡀⠀⠀⠈⠉⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣽⣷⣤⡀⠀⡀⠁⠶⢀⣀⣀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣠⣤⠟⠛⠋⠈⠙⠻⠷⣤⡖⢀⣠⠀⠀⢨⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣶⣤⣀⣀⠀⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣉⡛⠻⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠷⣄⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⣾⠋⣦⠀⠀⠒⠒⠒⠒⠒⠀⠀⣴⣿⣷⣤⡌⠗⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠘⢿⣦⡈⠳⣄⠀⠠⠤⠤⠄⣠⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠙⠿⣦⡈⢣⡀⠀⠀⣠⣿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠈⠻⠿⣷⣤⣴⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⢷⡿⠿⠿⠿⢿⣿⣿⠿⢿⣿⡿⣿⠻⠿⠿⢿⣷⣾⡿⠿⣶⠆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠹⣦⣀⡀⠀⠙⠛⣣⡿⠃⠀⠻⣦⣀⣀⠀⠙⢛⣳⣴⠟⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠙⠋⠉⠋⣻⠟⠁⠀⠀⠀⠀⠉⠛⠛⠚⠛⠉⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣠⠶⠶⣾⡇⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣠⣄
⠀⠀⠀⠀⢰⡇⠀⠐⣿⠁⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠈⠙⡷
⠀⠀⠀⠀⠈⢷⣦⣴⣿⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠛⢷⣤⣤⣤⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠶⠶⠦⠴⣶⡦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠰⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡆⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠛⢿⡶⠶⠶⠦⢶⣶⠶⠲⣶⡞⠛⠟⢻⣿⠛⠛⠛⠋⠉⠁⠀⠀⣽⡟⣧
⠀⠀⠀⠀⠀⠀⡾⠀⣷⠀⠀⠀⣼⠆⠀⠀⠀⠀⡇⠀⠀⣿⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿
⠀⠀⠀⠀⠀⣾⠂⢀⠃⠀⠀⢰⡇⠀⠀⠀⠀⠀⣧⠀⣸⡇⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⢸⡏⠀⠸⣇
⠀⠀⠀⠀⢸⡇⠀⣾⠀⠀⠀⢸⣧⣴⠛⠀⠀⠀⠉⢀⡿⠃⠀⠀⠀⢿⡶⠿⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⢠⡄⠀⣀⡉⠀⠀⠀⠘⠁⠻⣇⠀⠀⠀⢀⣾⡇⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⣿
⠀⣀⠰⢾⡷⠞⠉⣡⣴⣆⠀⠀⠀⠀⠻⣤⠀⠀⢸⡿⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⢉⣁⣨⣿⠇⠀⠀⣿⡀
⠀⢻⣴⠎⠁⢠⡾⠋⢁⣼⡇⠀⠀⠀⠀⠙⣦⠀⣸⣇⢀⣀⣤⣯⣤⣤⣤⣤⣤⣤⡞⠛⠉⠉⣿⡇⠀⠀⢸⣿⠀⠒⢀
⢀⡾⠁⠀⣰⠏⠀⣠⠟⠙⢻⣦⢠⣤⡤⠚⠛⠛⠛⠉⠉⠀⠉⠁⠉⢉⣀⣤⣼⠟⠛⠛⠛⠛⣿⡇⠀⠀⢸⣿⠛⠛⠛⢦⡄
⢿⣀⢠⡾⠃⠀⣴⠉⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣶⠾⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⣿⠈⠳⠦⣾⡀
⠀⢹⡿⠁⢀⡼⠇⢀⣰⣯⣄⣤⣤⣤⣤⣴⠶⠶⢾⡟⠛⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣤⣤⣴⣿⠁⢂⣀⠈⢹
⠀⢸⣧⡀⠻⣧⣤⣾⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⣄⡹⠿⠋
⠀⠀⠘⠛⣿⡏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠈⠁
⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⣀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣼⡿
⠀⠀⠀⠀⠻⢷⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣧⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣶⠿⠿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⣿⠟⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⠋⠉⢹⡏⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⡀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⠼⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠓⠊⢻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠁⠀⠀⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠏⠁⠀⣠⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠠⣶⣄⠀⠀⠀⠉⠙⠿⠷⣤⣤⡀
⠀⢀⣀⠀⠤⠤⠚⠋⠀⣀⣀⣤⣴⠟⠛⣷⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣶⠏⠈⠛⠷⣤⣤⣄⣀⣀⣀⣀⠙⢿⣿
⣀⣘⣿⣷⣶⣶⣶⣶⣎⣉⣋⣁⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣁⣀⣀
⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⠉⠀⠁⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣴⣶⣶⣶⣶⣶⣶⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⣠⠖⠘⠢⡜⢰⢁⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⣠⠀⠐⣀⡧⠞⠹⡀⣇⡸⣋⣴⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠐⡀⣀⠠⢀⣀⣤⠤⢤⠤⡧⢏⠞⣁⣄⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠃⠄⡐⠂⠀⠎⠁⠁⠀⠀⠀⠀⠈⠙⢫⡀⠀⠹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠦⠷⠶⡄⡀⢌⠲⣆⠳⡇⠆⠀⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠑⢄⢀⡉⣲⠠⡕⠘⠆⠃⢸⢎⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠁⠀⠐⠤⡉⠈⠃⠀⠀⠀⢰⣏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⢀⠀⠣⠀⠀⠀⠀⡀⠀⠀⠀⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢊⠀⠀⠀⢀⠦⠀⠒⠶⠆⠀⢐⠄⠀⠈⠩⡣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠑⠠⣠⠃⠀⠙⠢⠄⠒⠈⡀⢶⠀⠀⠀⠡⠸⢂
⠀⠠⢀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⡴⠘⡄⠀⠀⠁⠀⠀⠊⠀⠈⡀⠀⠀⠀⡡⠄⠱⡀
⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠜⢀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⡥⠤⠂⠁⠀⠐⢄⠐⡄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠾⢆⡁⡀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠄⠰
⠀⠀⠀⠀⣄⠀⡀⣀⡀⠤⠊⠀⢀⡀⠤⢌⣀⡀⠠⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠀⠆
⠀⠀⠀⠀⠈⠁⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢤⠄⠀⠐⢄⠀⠀⠀⠀⠀⠀⢽⡀
⠀⠀⠀⠀⠀⠴⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⠀⠁⠐⠠⠀⡀⠔⡀⠈⠑⠒⠂⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣷⣶⣤⣤⣀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⡿⠋⢹⣿⠟⠋⠡⡄⠀⠀⠻⠛⠻⣠⣤⠀⢴⠆⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣷⠀⠈⠁⠀⠐⠦⠀⢀⡘⠀⠀⢰⣿⣿⠗⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢿⡿⣿⣿⣿⡿⠿⠿⣻⠃⠀⠀⠀⠀⠀⢀⠠⣉⣀⠤⣀⠩⣿⡿⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣄⣠⣦⣌⣀⣴⡇⠀⠀⣿⣀⠀⠀⠀⠐⠟⡀⣴⡿⣦⣴⠿⢷⣟⣿⣆⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣛⣿⣌⣉⣽⣯⣄⣴⣶⣤⢿⣿⠷⣀⠘⣦⣀⠔⣿⣧⡾⠿⣶⣼⡿⢿⣦⣦⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣩⣿⣯⣀⣼⣿⣦⣴⠿⣿⢄⣴⢾⡟⣶⡟⣠⣿⡍⣾⡟⣿⣦⣼⠿⢻⡿⣏⣉⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣩⣿⣿⣍⣼⣿⣦⣤⣾⡿⢦⡴⢿⠿⣿⣿⡛⢛⠿⡿⢓⣿⣻⣿⠟⢛⣷⠟⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡷⣦⣴⢿⣧⣤⡾⠿⠶⣶⠟⠋⠸⢄⣛⠛⣻⠻⠟⣷⢾⣇⡿⣫⣷⣯⡝⠉⠉⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⣴⠿⠿⣦⡾⠿⠷⣤⣾⠟⢷⣤⣀⣾⣿⠿⠟⣣⣼⢻⡦⢿⣺⠍⠛⠉⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢧⣴⠿⠿⣶⣶⠿⠿⣦⡾⠿⣿⣷⣿⣿⣿⠿⢛⣷⣞⠛⣻⣿⡇⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⠿⠟⢷⣾⠛⠛⣷⣾⠛⢻⣾⣟⢻⣿⡛⣛⣷⡟⠛⣻⣿⣿⣟⠃⡀⠀⠈⢳⣄⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⠭⢝⠿⣾⡛⠛⣻⣾⣙⢋⣿⠿⢯⡽⠟⢛⡙⣯⢹⡟⢉⡹⠛⠉⣀⡽⣙⠳⢆⠉⠙⠟⢛⣿⣿⣿⡟⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠾⣽⣿⣭⣟⡽⠋⠀⠰⡛⠂⠰⣎⣷⡽⠞⣹⠋⠀⠀⠀⠀⠉⣀⠀⢠⣴⠆⢤⠈⢛⣽⣿⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠁⠊⡙⢁⠘⠁⢠⣶⠄⠀⢸⡛⣷⠋⠀⠨⣷⠷⠰⣶⠺⡏⢩⠉⠀⠈⠁⠀⠈⠀⠈⠙⣯⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠄⠀⢀⡠⠖⠀⠀⣰⣟⣛⠃⠀⢴⠶⣴⠏⠠⠀⠈⠀⠀⣠⡘⠻⣿⣾⣤⠀⠀⠀⡀⣀⣤⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣠⠔⠁⠉⠀⠀⢀⣼⣟⠿⠍⠀⠀⠤⣶⡿⠀⠀⠀⠀⣴⠔⡅⠈⠀⠈⠛⢿⣛⢠⣼⡵⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣀⡴⣁⠚⠀⠀⠀⢠⠾⠇⠉⠠⠄⢠⣷⣾⣿⣩⡄⠀⠀⡔⠀⠀⢄⠀⠘⠀⠧⠀⠹⢆⠀⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠜⣰⣦⣉⡔⠀⠀⠀⢀⡴⠟⠂⠀⣀⢀⣀⠉⣶⡟⣉⣹⠿⣄⠰⠉⠳⣤⣌⣇⠀⢳⡔⠤⠔⠀⠑⣾⡇⠱⣄
⠀⠀⠀⠀⠀⠀⠐⠞⡨⢋⡜⠛⠁⣤⢀⡤⠋⠀⢠⣤⣷⣿⣿⡇⠀⠹⣟⣿⣿⣴⣿⣳⡀⠀⠈⠻⣝⢦⡁⣏⠲⢄⠀⠀⢈⢳⡄⠸⣦
⠀⠀⠀⠀⠀⠀⣠⢌⠰⠋⠓⠲⠠⣼⣏⣄⣿⠀⠰⣿⣿⠟⠁⢻⡀⠀⠙⢿⣤⡿⢛⣿⣿⣆⠀⠀⠹⣆⠙⢿⠀⠈⣷⣄⠈⢦⠻⡜⠛⠳⡄
⠀⠀⠀⠀⣀⠘⡱⠋⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣸⠟⠁⠀⠀⠀⢷⠀⠀⠀⠙⠻⢿⣿⣷⣿⢧⠀⠀⠙⣆⠈⢳⡄⠀⢀⠑⢌⠳⣥⣦⡄⠘⡄
⠀⠀⠀⡔⣡⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠛⠁⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢀⣩⣭⠛⠳⣦⣄⠘⡆⠀⠹⣟⠛⠳⠂⠱⣄⠉⠙⢹⠇
⣴⣄⡻⠿⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠒⠤⢶⣟⠃⢀⣀⡀⠉⠘⢳⠘⡄⠀⠸⡆⠀⠀⠀⣸⣆⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⠀⠀⠀⠉⠻⣿⡆⠀⠈⡀⠘⠄⠀⠙⠀⠀⢰⣿⠛⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣎⢭⣉⡤⢦⡶⣶⢤⢸⣿⡓⣾⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠲⣄⠑⣌⢣⡈⡇⠙⣆⠀⠀⠀⠀⠀⠀⠀⠘⣦⡄⠀⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⡸⢳⡈⠢⣹⣿⣠⣌⣃⠀⠀⠀⠀⠀⠀⠀⠈⠿⣄⠀⣄⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢠⢳⠈⢧⣠⠏⢻⡋⠀⠈⠉⢦⡀⠀⠀⠀⢀⡀⠀⣀⠉⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣸⣠⠄⠀⠉⠀⠀⠛⢷⣦⡀⠠⣿⡀⢀⣴⠏⣀⡼⠧⠖⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣼⡟⣉⠀⢠⡾⠒⠀⠚⠃⠒⠢⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠉⠙⠁⠹⣆⠀⠀⢀⠀⠀⠘⠛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⢀⣠⣤⣶⣶⠿⠿⠿⠿⠿⠿⠿⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢠⣿⡟⢻⣷⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⠛⢿⣿⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⡀⠀⠉⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄
⠀⠀⠀⢀⣠⣴⣾⠿⠇⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡋
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⣧⠀⠀⠀⠀⠀⢰⠿⣇⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⣿⢷⡄⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡛⠛⠙⢧⡀⠀⠀⣠⡟⠀⠹⣆⠀⠀⠀⣠⠇⠈⢳⡄⠀⠀⠀⠀⣿⠀⠙⢦⡀⠀⠀⠀⢀⣼⡏
⠀⢿⣷⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠙⠷⠾⠋⠀⠀⠀⠙⠷⠶⠾⠋⠀⠀⣀⣙⣦⣤⣤⣾⣥⣤⣴⣶⡟⠳⠶⠶⣿⡟
⠀⠈⣿⡷⠶⠶⣼⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢹⡇
⠀⢸⣿⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣿⠀⠀⢹⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠰⠿⠃⠀⠀⠀⢠⠀⠀⠀⠀⣾⡇
⠀⠘⣿⣦⡀⠀⠰⣤⣿⣤⡀⠀⢀⣴⠏⠀⠀⣆⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠣⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠈⠻⣿⣷⣶⣦⣤⣬⡙⠛⠛⠁⠀⠀⠐⠌⠳⠙⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠈⠓⠤⣀⡀⢀⡀⠠⠚⠑⠱⠋⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠉⠙⠛⠛⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠶⢦⡴⠞⠳⠶⠛⠛⠓⠛⠙⠛⠋⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣿⣿⣿⣷⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⣶⣾⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⢻⡏⠉⠀⠈⢿⣄⠀⠀⠙⠛⠻⠿⠿⠿⠿⠛⠛⢛⣿⠉⠉⠉⢹⠟⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⣷⠀⠀⠀⠀⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⡿⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⣹⣦⣴⡄⠀⠀⠀⠉⠓⠶⠦⣤⡤⠶⠖⠛⠁⠀⠀⢠⣦⣼⣇⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⡾⠋⢁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠙⢷⡄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠘⠷⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣴⠾⠃⠀⠀⣀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢷⣶⣶⣶⠿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣶⣶⣶⡾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣁⣀⣀⣀⣀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⣾⡟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣷⡀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣃⣀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⣀⣘⣻⣦
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠙⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠒⠛⠉⠉⠉⠉⠑⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢒⡎⠀⣼⣇⣀⣆⣀⣠⠤⠞⠙⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢹⡤⠬⣅⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⣈⣓⠀⡌⠀⠀⢘⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣎⠠⠤⣽⠇⠀⠀⢈⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⡏
⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣰⠞⠳⢤⣠⠜⠁⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠊⠡⡤⠀⠀⠀⠐⠀⠀⠀⠀⠈⠉⢺⡅⠀⠁⠀⠈⡇⠀⠂⠀⠀⠀⠀⣤⢄⡀
⠀⠀⠈⠀⠐⠂⠀⠁⡠⠞⠋⠁⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢠⠎⠑⠚⠀⠀⢠⣀⣠⠇⠀⠀⢤⣤⡒⠒⠚⠶⢭⣓⠤⢄⠢⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠣⢄⣀⣀⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⣀⠴⠓⠒⠒⠤⠤⠤⠞⠁⠀⠀⠀⠀⠀⠈⢹⠉⠐⠒⠤⠌⡉⠒⠭⢖⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠙⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡉⣉⡀⠙⠛⠒⠒⠾⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⢻⣿⣿⡠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⠏⢰⡧⠞⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡜⢿⠞⠛⡇⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⠔⠀⠊⠀⠀⠘⠋⠉⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⣱⢧⣀⠊⠀⠀⡀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣁⢤⡬⠵⠦⣌⠂⢼⠦⢤⢤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠁⠀⠀⠀⠑⢄⣰⠃⠘⠓⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡤⣄⣀⣀⣀⣀⡠⠤⠒⠊⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⣶⣶⣶⠶⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⠛⡄
⠀⠀⠀⠀⢀⡤⠖⠒⠉⠉⠉⠛⣝⣷⡾⣦⡍⠉⠽⠒⠒⠤⠤⣀⡀
⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠛⠉⠁⠀⠀⠀⠈⠉⠒⠒⠢⠤⠤⢤⠽⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠉⠉⠉⠀⠈⠉⠉⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠄⠈⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⣤⠀⢷⡄⣶⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⡀⠀⣡⠔⠀⠀⠘⠂⠀⠐⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⣀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⠃
⠀⠀⠀⠀⡰⠈⠀⠀⠀⠀⠀⠠⠀⢀⣀⠀⠀⠀⠀⠀⠠⠀⠳⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⣆⠉⠙⠆⠀⠀⠘⠁⢠
⠀⠀⠀⠎⠀⠀⠀⣴⠄⠤⠷⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠘⣠⡄⠈⠉⠀⠰⢟⣁⡀⠄⠀⠀⠸⠋
⠀⠀⢲⣤⠔⠁⠒⠛⠀⠀⠀⠀⡄⠛⡀⠀⠀⠹⠀⠀⠠⢠⡀⠀⠀⠀⠀⠀⢠⡴⠀⠀⠀⣿⠀⠀⠀⠀⢶⣌⡁⠤⠃⠀⠠⢄⡁⠀⠀⠀⠀⠘
⠀⢰⡄⢀⡀⠀⠀⠀⠀⢆⡁⠘⠷⠌⠁⠀⠀⠀⠂⠀⠀⡄⠁⢀⠀⠀⠀⠀⠘⡿⢠⣤⣤⡌⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠊⣤⠐⠀⠀⢸⠇
⠀⠐⣿⠎⠙⣿⢆⡀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠿⠀⠀⠀⠀⠀⢩⡄⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠉⠁⠀⠀⠀⠀⡄
⠀⠀⣿⠀⠁⠀⠈⠋⠀⠀⣀⣀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠙⢆⣸⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠖
⠀⠀⢻⡀⠀⡀⠠⠄⠘⠃⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠁
⠀⠀⠀⢳⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⣦⡀⠠⡖⠒⠀⠁
⠀⠀⠀⠀⠙⠶⣤⡀⠀⠀⠀⠀⠀⢀⣀⠐⠆⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠗⠐⠒⠶⠖⠈⠉⠉⠁⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⠋⠉⡈⠁⠀⡆⠀⠀⠀⠸⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣿⣀⡀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠐⠶⢄
⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⠀⠄⢀⠷⠀⠈⢣⡀⠀⠈⠶⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠠⠀⠀⠉⠙⠛⠛⠋⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠴⠂⠀⢘⡇⠀⡠⠀⠀⠀⠀⡌⠙⠦⣤⡤⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠠⠀⠀⠀⠀⢠⠴⠦⠤⢤⡄⠛⣄
⠀⠀⠀⠌⠁⠀⠈⠳⢬⠓⠀⠁⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢠⣿⠀⠀⢀⣀⡀⠀⠉⠲⣤⡄⠀⠰⢀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡀⠀⠀⠀⠀⠀⣀⠎⠁⠀⠀⠀⠀⠚⠁⠀⠀⠶⠶⠤⠀⣀⣀⣀
⠀⠈⠁⠀⠀⠀⠀⠈⠲⣄⠀⠀⠘⠢⢄⡀⠀⠈⠃⢤⡄⡀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⢠⣆⡰⠇⠟⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡄
⡆⠀⠀⠀⠀⣤⠄⠐⠒⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⣦⠽⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡶⢾⣷⣂⣀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃
⠓⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠐⠿⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⡶⠀⠐⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠊⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⢸⠇⠹
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠲⡤⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⢸⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢠⠀⡄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡘⢀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⡄⢠
⠀⠀⠀⠀⠄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠻⡀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠃
⠀⠀⠀⠈⠳⢄⣀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⠀⠸⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢿⠀⢿⡟⠐⣷⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠘⣷⡀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢻⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⣿⡀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠘⠃⠻⣿⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠰⢿⣛⣤⣤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⢤⡤⠤⠀⠒⡋⠁⠙⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠀⠤⠄⠀⠀⠀⠉⠁⠈⠻⣉⠁⡶
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣴⣶⣶⣶⣶⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⣤⠴⠂⢀⢀⢀⠀⡶⢴⡀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢃⢉⣉⠰⢁⣠⠶⠦⣌⠒⢃⡤⠶⢦⣀⠡⢈⣈⢩⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠬⣍⣮⠏⠀⠀⠀⠀⠨⠀⠀⠀⠀⠙⢬⣨⡭⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠁⠀⠀⣿⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠀⢹⠀⣤⣄⠀⠈⢀⠈⠀⢀⣤⡄⢺⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢸⠄⠈⠛⠀⠀⠀⠀⠀⠘⠋⠀⢌⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠢⡀⠀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡴⠊⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠔⢁⠆⡀⠀⢀⡼⠚⠋⠱⠴⡆⠀⠀⠀⠀⠀⢀⡶⢔⠉⠙⠺⣄⠂⢀⠀⢍⠑⠄
⠀⠀⠀⠀⠀⠀⡀⠡⢀⣀⡶⠋⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⣬⠀⠈⢀⠀⠀⠨⠳⣏⣀⠤⠁
⠀⠀⠀⠀⠀⠀⠈⢀⠀⡓⠁⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠛⡁⢀⠈
⠀⠀⠀⠀⠀⠀⠀⢘⡦⠈⡄⠁⠀⠀⠀⠀⢀⡁⡀⠀⠀⠀⠰⠈⣁⠀⠀⠀⠀⢀⠀⡄⠰⣎
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣄⠘⣜⡀⠀⠀⠀⠀⠑⢢⠀⠀⠀⢰⠐⠀⢀⠀⠀⠀⣈⡜⣀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠒⠠⡔⠀⠀⠘⡄⠀⠀⠀⠢⣴⠏⡵⠐⠊⠍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠁⠐⠂⠠⠆⠼⠢⢤⠤⠤⠄⠊⠀⠐⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡉⠿⠟⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠻⠿⢉⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢰⡿⣿⣿⢿⡆⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣤⣿⣿⣤⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣆⠰⣶⣾⣿⣿⣿⣿⣷⣶⠆⣰⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣷⣤⣙⠛⠛⠛⠛⣋⣤⣾⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⠻⣿⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⣿⣿⣟⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣥⣜⣻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣒⣫⣭⣭⣭⢍⡙⠷⢿⡿⠿⢛⡻⣿⣿⣿⣿⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⣿⠿⠿⢿⡟⢻⡿⠿⠿⣿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⠂⠘⣦⡄⠼⣅⣨⠧⢠⣴⠃⠐⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠅⣠⠴⠾⡷⠦⣄⢰⠀⢨⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣿⣿⣿⡄⠀⣔⠁⠀⠀⠀⠀⠈⣺⠀⢰⣿⣿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣸⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡎⠀⠀⠀⠀⠀⠀⠀⡄⣷⣿⣿⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡄⠀⠀⠀⠀⢠⣿⣿⡿⠉⠈⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠟⠙⠆⢧⡰⠠⠋⠻⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⡤⣔⡉⠉⢉⣒⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡜⡠⠔⠛⠈⣙⠶⠄⠐⠚⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣏⣀⠤⠒⠉⢀⣠⢒⠘⣿⡶⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠤⠤⣼⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⠤⠐⢲⡉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⢀⠠⠐⠈⢈⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡊⣤⠀⠀⢀⣼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⠀⠀⠀⠀⠀⣠⠤⣰⠂⢉⣉⣉⠁⢻⡃⣁⠊⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠆⠈⠁⡐⠋⠉⠩⠳⡀⡉⠈⠀⡧⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⢇⠙⣀⠷⣇⣇⡄⣄⣄⡷⢇⡜⢁⠋⠢⠈⢒⠢⣄⣀
⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠱⡉⠢⢠⡜⡷⡟⣤⢠⠋⡳⠁⠀⢰⠀⠸⠀⠘⠀⠑⠆⠤⡀
⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠑⠲⢌⡖⢅⠕⣯⠴⠊⠀⠀⢀⠆⢠⠃⠀⠸⠀⠀⠀⠀⠀⠑⢄
⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢀⠎⢠⠃⠀⠀⠆⠀⠀⠀⡇⠀⠀⠀⠡⡀
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡐⠁⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⢡
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⠊⠀⠠⠃⢠⠀⠀⢀⠃⠰⠀⠀⠀⠀⠀⠀⡆
⢃⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢀⠔⠁⠀⡰⠁⢀⠃⠀⠀⡌⠀⠇⠀⠀⢰⠀⠀⠀⢰
⠀⠢⡀⠀⠀⠀⡀⠴⠚⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⡠⠒⠁⠀⢀⠊⠀⠠⠃⠀⠀⡐⠀⡌⠀⠀⠀⠨
⣤⣤⣬⣥⣬⣥⣤⣤⣤⣤⣤⣤⣤⣼⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣿⣶⣶⣶⣶⣷⣶⣶⣷⣶⣶⣾⣶⣶⣷⣶⣶⣶⣷⣶⣶⣶⣶
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣿⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠉⠉⠛⢿⣿⣽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⣾⣿⡿⣯⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣼⣿⣿⠿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠿⠿⢛⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣶⣭⣽⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣚⣉⣉⣛⣯⡍⠉⢭⣴⣿⣿⣿⡿⠟⢻⣿⣿⣿⣿⣷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣶⣟⢶⣤⡾⢷⣤⣶⣿⣷⣿⣿⣿⣿⣿⣿⣷⣿⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣟⠙⣿⠉⠉⠉⡿⠇⢸⣿⠉⠉⠉⣿⠋⣻⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣏⠀⠀⣿⠖⠀⠙⢦⡴⠊⠀⠲⢿⠃⠀⣹⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠀⠀⡆⢀⡴⠒⠿⠿⠶⢦⡄⢸⡀⠀⣾⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣿⣿⣿⣇⠀⠀⣇⠋⠀⠀⠀⠀⠀⠀⠙⣸⠁⠀⣸⣿⣿⣿⡷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⢯⣿⣿⣿⣿⡀⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⣿⣿⣿⣿⡷⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⣿⣿⣇⢪⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⣵⣸⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⣿⣞⡀⡆⠀⠀⠀⠁⠀⠀⢸⣀⣧⣿⣿⡟⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⠀⠄⠐⠀⠀⠀⣼⣿⣿⡿⠋⠡⢀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠿⠛⢷⡈⣆⣠⡇⡼⠛⢿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⠟⢿⣯⢸⣿⣿⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⣀⣾⣿⣿⡿⠟⠉⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⣀⣠⣿⣿⣿⣿⣷⣄⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠑⠊⠉⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⣀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠠⠿⠿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⡐⠁⠀⠀⣠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠈⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠃⠈⢿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣿⣟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⣿⣿⡟⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⣤⣴⣿⠟⠋⠙⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣥⡀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣟⠀⢀⣾⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣧⡘⠉
⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⣿⣿⡿⠿⠿⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⡀⠨⠭⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⡉⡉⠉⠁⠩⠅⣀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡏⠀⢨⠀⠔⠈⠀⡀⠚⠛⠁⠀⠀⠀⠀⠘⠛⠓⠀⠀⠀⠢⠀⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠊⠀⣠⠨⠐⠀⢁⣶⣤⠀⠀⣤⣶⡈⠀⠀⠀⣄⠀⠑⠂⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠴⠦⡍⠽⠀⠀⠯⢩⠴⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⢾⣷⡄⢠⣿⡷⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢁⠝⠀⠀⠫⡈⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠠⣀⠀⠀⣀⠄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠄



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠆⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⠀⢐⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠈⡏⠀⠚⠀⢙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠛⠖⠀⠀⠠⠄⣿⣿⣿⣿⣿⡶⣤⠀⠀⠈⢠⣤⡄⢠⡄⠀⢀⡀⠀⠀⠀⣠⣤⣴⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠺⣿⠹⢤⠎⠀⠀⡤⢄⡜⠽⣿⣿⣿⣦⣤⣄⡂⠧⢨⣉⣠⣨⣵⣶⣶⣶⣷⣶⣾⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠺⠹⠂⡠⠒⣠⠔⠥⠎⢌⣁⣻⣽⠻⣿⣿⣿⣿⠟⣻⣿⣟⣹⣿⣿⣿⣿⣿⣿⠿⠟⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⠁⠀⠠⠊⠁⠀⠀⠑⠀⣏⣿⣿⣇⣈⣉⣋⣽⣿⣿⣿⣿⠟⠋⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠔⠈⢈⠁⠙⠁⠁⠉⠀⠀⠀⠃⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⡂⠀⠀⠀⢣⠈⠉⠀⠉⠄⠀⠈⠃⠀⠁⠀⣿⣿⠿⠟⠻⠿⢿⣿⣿⡏⠐⠂⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠉⢀⡆⠀⠀⠈⠲⡄⠄⠃⠀⡀⡀⠀⠐⢁⠈⠁⠀⠀⠀⠀⠀⠠⠈⠛⠁⠀⠀⠀⠀⠀⠇
⠠⠀⠐⠒⠒⠚⠂⢄⠒⠀⠀⠔⠒⡛⠂⡌⠠⠀⡀⠐⠀⢡⡴⠴⠆⢰⠢⡀⠀⠀⠀⠀⠀⠀⠸⠶⡓⠒
⠀⠀⠀⠀⠀⠀⠀⠈⡀⢠⠒⠚⠉⠀⡊⠔⠒⠀⡄⠀⡄⠇⠀⠘⠂⠤⡀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⡱⠂⠀⡆⠀⠀⠈⠂⠄⠠⠀⠀⠀⠐⣅⠀⠀⠀⠀⠣⠐⠀⠑⠂⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠜⠁⠁⢠⠁⠀⠰⠀⠀⠃⠀⠀⢰⠀⢠⠃⠀⠀⠠⠠⠀⠀⠀⠐⠒⠀⠀⠢
⠀⠀⠀⠀⢀⡀⠀⣀⠀⠈⣰⠀⠀⠀⡇⠀⠀⠀⠀⢰⠀⢁⠀⠀⠀⠀⠂⠀⠰⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⠀⢃
⠀⠀⢀⡔⠀⠀⠀⠒⠓⠀⠟⠀⠀⠀⠠⢀⠀⢠⠀⠀⠄⠀⠀⠉⢠⡤⠊⠐⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣤
⠀⠀⠸⠀⠀⠀⢀⠀⣠⡀⠀⠀⠀⠀⠀⠐⠈⣀⠀⠀⣀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠⠀⠀⠂⢋⢀⣠⣢⣿⣿
⠀⠀⠀⠀⠐⡶⠓⠚⠉⢠⠞⠀⢀⡞⠁⠀⠻⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣄⣀⡈⢄⡡⠀⠉⠁⢠⠏⠹⢿
⠀⠀⠀⠰⠊⠀⠀⠀⡰⠉⠀⣰⣿⣏⣁⣀⣀⣀⡀⠒⠀⠀⠀⠀⠀⡠⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠆⡠⠈
⠀⠀⠀⠛⠻⠛⡟⠛⠁⠀⢸⣿⣿⣿⣿⣿⣿⣟⣳⠶⠶⠒⢢⠀⠀⠲⣦⢶⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣬⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢠⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢷⡒⠒⠠⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⠟⠀⠀⠀⠀⠀⠠⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢻⠿⠶⠒⠀⠀⠃⢀⠦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡆⠀⠀⠀⠂⣆⣠⠤⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⢀⠞⠉⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠁⠉⠁⣴⠋⠶⠀⠀⠀⢀⡇⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡆⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠱⠄⣀⠀⣙⣼⣄⠀⠀⣀⡤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠉⠀⠸⢿⠋⠗⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣠⠶⢤⡀⠀⡰⠁⡀⠀⢆⡀⠀⡀⠀⠀⠙⢦⠀⠈⢧⠀⠀⢀⡤⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢻⠘⢥⡮⠏⠑⡾⢍⡢⡀⠉⠛⠓⠒⠒⠚⣫⣤⢤⠈⢣⡜⡡⠾⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠁⠀⠀⠀⠒⠑⠘⣳⡆⠲⠖⢺⠥⢐⠉⠚⠒⣆⠒⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠦⠤⠄⠀⢡⠀⠀⣰⠚⠉⠀⠀⠉⠢⣀⠀⠘⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠀⠘⠓⠳⠤⠄⠀⢤⣄⣧⠀⢸⣀⠀⠀⠀⠀⠀⣈⣠⡇⠀⠀⠔⠉⠀⠀⡠⠛⢧⣤⠤⠖⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⢶⣖⡖⢒⣒⣲⠖⠒⠒⠐⠾⠙⢯⡉⠉⡩⠛⠤⣸⠀⢀⣞⣒⣒⣒⣛⣃⣉⣁⣀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢁⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡇⠀⠀⠙⠉⠉⠉⠉⣦⠼⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⢀⠀⡀⠀⢀⠘⢆⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠓⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠉⠉⠙⠒⢢⠠⡶⠶⠀⠖⠒⠚⠋⠉⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⠤⢄⣀⣀⡇⠠⠇⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠐⢛⣿⠛⠯⢀⡀⠀⣠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢾⣗⣶⠶⠤⢵⣦⣬⣽⣷⣶⣶⣊⣀⣀⣀⣹⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂⠲⠀⠀⠐⠰⠐⠐⠀⠀⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠂⠂⠀⠈⠀⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠒⠒⠒⠒⠲⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡠⢦⣠⠤⡀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⠤⠤⣀⣠⢷⡀⠹⡄⠱⡄⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠃⠀⠀⠀⣹⠀⠱⡀⠱⡀⢹⢡⢾⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢻⡎⠀⠀⠀⢸⠈⢣⠀⢣⠀⢇⠈⣧⠘⣤⠗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢺⡇⠀⠀⠀⣈⣆⠀⢧⠘⠂⠀⠀⠛⠀⡿⢂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡇⠀⠐⠢⢄⣸⠀⠘⠀⠀⠀⠀⠀⢠⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣵⢢⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡀⠉⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠑⣄⠀⠀⠀⠱⣄⠀⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡏⠱⣄⠈⠩⠭⠵⡌⠁⠀⠀⠀⠘⢦⣀
⠀⠀⠀⠀⠀⣀⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⢧⠀⠀⠀⠀⠀⠈⢯⠉⠒⠲⠤⢄⡀
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠀⠀⠀⠀⠀⠘⡄⢀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢣⠀⠣⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠠⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⢟⡀⠀⠀⠀⠠⠜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡛⡞⠉⠉⣇⡸⠞⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠁⠀⠀⢹⠷⠚⠳⡅⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠓⠲⢄⡴⣥⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡞⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⣀⣀⣿⢧⡀⢀⣠⢴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⢐⠋⠁⠀⠻⣁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⡵⣦⣤⣃⡴⣄⠳⢦⣤⣀⣄⣬⡷⠀⠘⡆⢠⢔⣮⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣈⢯⠁⢹⢩⢿⠷⣤⣠⣬⣥⣖⡹⢯⡧⣌⡥⣉⣀⠞⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡤⣤⣄⣀⣇⠀⢸⠜⠊⠉⠙⠣⣸⠀⡽⠀⠀⢋⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠙⠛⠛⠧⠤⠤⠤⠬⠆⠈⣆⣀⣀⣀⣀⣈⣱⠇⠀⡰⠋⠉⣩⠟⢻⡤⠴⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠺⣿⡫⢭⠵⠂⠉⠉⠘⠋⠳⣄⣠⠋⠓⢼⡀⠼⢭⡭⢭⢭⠤⠤⠤⠤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠠⠀⠀⠀⠀⢾⠟⠀⠀⠀⠀⠀⠀⠹⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠉⠓⠒⠒⢤⢤⠤⠤⠤⠴⠒⠒⠋⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠂⠤⣤⣇⣸⣀⡀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣀⣀⣀⣀⡠⣤⣗⣉⣑⠆⠀⠒⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠓⠒⠺⠿⠏⠲⠄

⠠⣤⡤⡄⠀⠂⡄⣤⣤⡄⢠⠤⡄⠀⠀⠀⠀⢠⡀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⠀⠠⡀
⠀⢻⣟⡜⣿⠓⣸⣿⡇⢠⡘⣻⠃⠀⠈⣀⠀⠀⠀⠂⠈⡀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣂⠀⠀⠈⡆⠄⡀⠢
⠀⠀⠓⠒⠀⠓⠀⠒⢛⣂⡐⠒⠀⠀⠀⠁⠀⠀⠰⠁⢼⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠁⢞
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⢰⠀⢤⣀⠪⣕⠍⢻⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⣐⠠⢬⠀⢌⡖⡍⢳
⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠞⠐⠧⢿⣇⡹⢁⡰⠃⠀⠀⠀⡀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠠⠋⢰⣸⣳⣆⡟⢁⡞
⠀⠀⠀⢸⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢣⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠸⠃⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠈⠁
⠀⠀⣞⣶⠀⣶⣶⣶⣶⡶⢶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡦⢀⣤⣤⣤⣤⣤⣶⣤⣤
⠀⠀⠀⢸⠀⣷⡶⣦⣾⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⢳⠐⣫⣭⣍⣿⣧⠉⠉⡁
⠀⠀⢳⠈⠑⠀⠘⢻⣿⢷⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠘⠙⠉⠛⢻⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡔⠀⠀⠀⠇⠠⠀⠀⠀⠀⡀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠸⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⠀⠀⠑⢤⠀⠀⠀⠓⠂⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠁⡄⠀⡀
⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣾⠀⠆⠀⠀⠀⠀⠀⢨⠀⠀⠈⠑⠖⠂⠀⠀⠘⠑⠚⠀⠀⣧⣶⡷⠀⡇
⠀⠀⠀⠀⣒⣀⣀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠢⠤⠀⠒⠂⠀⠀⠀⠀⠔⠢⠞⠰⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠀⠒
⠀⠀⡠⢈⠰⠧⡿⠃⠄⢀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⢈⣂⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢠⣤⣴⢠⣤⣴⣤⣤⣼⣿⡤⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡄⣤⣤⣤⣿⣷⢶⣤⣧⡄⠀⠀⢰
⠀⠈⣉⠀⢾⣭⣭⣽⣿⡏⠉⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡡⠀⣿⣭⣭⣽⣽⡉⠉⠉⠀⠀⠀⠀⢀⠄
⠀⠀⠰⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣰⠞⢩⣿⡛⢿⡷⣦
⠀⠀⢠⣏⣰⣿⣿⣿⣶⣿⡟⣷⣄
⠀⠀⢀⣟⣹⣯⡼⢻⣿⢻⣧⣿⢹
⠀⠀⢸⡟⢿⣟⣿⢿⣿⣾⣿⣷⡟⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠈⢷⡈⠻⢭⣽⣿⡿⠿⠿⠃⠀⠀⠀⣀⡴⠞⠋⠁⢀⣀⣤⣤⠀⣀⠀⣉⡙⠳⣤
⠀⠀⠀⠀⠉⠛⠛⠹⣇⠀⠀⠀⠀⠀⣠⠞⠉⠀⣠⣶⡿⠿⠛⠉⠉⢠⢚⣼⡿⡑⠄⣤⠘⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢠⡾⠁⣀⣰⣿⠟⡁⢅⠂⡥⠌⣠⣴⠿⠋⢂⢃⣼⣿⣦⣄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠠⢤⣾⢟⡠⡔⢁⣐⣬⣶⡿⠛⢁⠌⠒⣡⣾⡟⠀⡙⣿⣤⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣱⡿⠃⣪⣴⡾⢟⡛⢉⠑⠐⠎⣣⣬⡾⠋⠀⠻⣶⣔⠈⢿⣦⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠸⠟⣠⡿⢋⡁⢒⣠⣤⣶⠶⠶⠟⠋⠁⠀⠀⠀⠀⠈⢻⣮⠘⢿⣆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣌⢇⣾⢟⡔⣢⡷⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⢲⡀⢀⡀⠀⢻⣯⠸⣿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡇⣼⡏⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⢻⡇⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣹⣟⣼⠋⠀⣄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣤⠀⠀⠘⣿⣿⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠙⣿⠁⠀⠀⢿⡀⠀⣿⠀⠀⠀⣠⠄⠀⠀⠘⣇⠀⢠⡟⠀⠀⠀⣿⠋⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⣿⠀⠀⠀⠈⠛⠛⠁⠀⠠⣾⣁⠀⠀⠀⠀⡈⠛⠋⠀⠀⠀⠀⢿⣀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡀⠀⠀⠀⠀⣴⠟⠃⠀⠀⠈⠁⠀⠀⠈⠙⢶⡀⠀⠀⠀⠀⣾⣿⣭⣙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⡇⠀⠀⢠⡾⠛⠙⠓⣦⠀⠀⠀⡗⠀⠀⠀⢠⣿⠐⡩⡻⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⡻⢷⠶⣄⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠁⠈⠛⠛⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⠛⠛⠛⡛⣛⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⣍⠙⠛⠁⠀⠈⠋⢋⣽⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠦⣤⡀⠀⠀⠀⡾⠀⠊⠀⠙⠷⣦⣀⣀⣠⣾⠛⠁⠈⣠⠈⣷⠀⠀⠀⣠⡴⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⢀⣬⠿⠛⠀⠀⢹⡆⠀⢀⡇⢨⣄⠀⠀⠀⣿⠉⠉⠈⣿⠀⠀⠀⣠⡀⢹⡄⠀⢰⡃⠀⠀⠛⠻⣯⣀
⠀⠀⠀⠀⢠⡾⠋⢀⣠⣤⢶⡞⢛⢿⡄⢸⡇⢸⡏⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢿⡇⢸⡇⢠⣿⡛⢲⡶⢦⣄⡀⠙⢶⡀
⠀⠀⠀⠀⠻⠴⠞⠋⠁⠀⠀⢿⡀⠊⣿⡄⡇⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⡇⢸⣇⣾⠣⠀⡾⠀⠀⠈⠙⠛⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠜⣿⡇⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢺⡇⢸⣿⠫⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠃⠘⣷⣸⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣾⣧⣾⢃⠇⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠐⠀⠽⠛⠿⠄⠀⠐⢿⠀⠀⠀⡿⠀⠀⠂⠿⠛⠋⠐⠀⠞

⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠘
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠄⠤⠄⠀⠀⠀⠀⠠⠀⠀⠀⠈⠉⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⡤⣦⡄⠀⠀⠀⠀⠀⠀⢀⢀⢶⣶⣆⡀⠀⠈⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⣈⠂⠀⠀⠀⢀⠀⠸⠋⢁⢀⢢⡅⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢓⠀⠀⠀⡀⢀⠃⢀⡄⠀⠀⠛⠻⣿⠀⣀⠈⠑⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⡊⠔⠀⠀⡠⣤⣰⠇⠀⠁⠊⠠⠀⠀⠠⠖⣠⣴⣷⡒⠞⣹⣹⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⡆⠀⠀⠀⠀⠙⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⢠⣤⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⡍⣻⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣟⠈⠐⢄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠉⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⠉⠉⠉⠑⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣀⣀⣀⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠲⢅⠀⠙⠢⡀⠀⠀⢀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡳⣄⠈⠑⢦⡀⠈⠲⣄⢁⡤⢤⠀⠀⠻⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⣀⠀⠀⢸⡄⠈⠓⢄⡀⠙⠂⠀⠈⢿⠀⡏⠀⢠⡶⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡟⠦⣄⠀⠉⠀⠈⠀⠀⠀⠀⡇⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⡼⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⢹⣦⡈⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠀⠀⠀⢠⠇⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡇⠙⠻⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠡⣤⣤⣀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡄⠈⢳⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠳⣄⠀⠀⠀⠀⠀⠀⢿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⢀⡠⠔⢒⣉⣀⢀⣀⣀⢀⣀⠤⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡄⠀⠀⠀⠀⢘⠷⡄⠀⠈⠉⠛⠛⡿⣿⣿⣿⣶⡄
⠀⠀⠉⢢⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣀⠤⠚⠁⠀⢹⠀⠀⠀⡰⠋⠀⠈⢻⣿⣿⡇
⠀⠀⢀⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡯⠞⠊⠁⠀⠀⠀⠀⠈⡇⢀⠎⠀⠀⠀⠀⠀⠙⣿⡇
⠀⠀⠀⠙⣄⢣⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠸⡎⠀⠀⠀⢀⣀⡀⠀⠘⠇
⠀⠀⠀⠀⠈⢆⡇⠀⠀⢀⠴⠒⠉⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⣧⣴⣶⠿⠛⠓⠒
⠀⠀⠀⠀⠀⠘⡇⡠⠞⠁⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠑⢆⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣾⣷⣶⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣿⡏⠀⠀⠀⠿⠿⠿⠿⠿⠈⠿⠿⠿⠿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣶⣤⠄⠀⠀⠀⠙⠉⠉⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⣉⣙⠻⢿⠀⠀⠀⠐⠒⣶⠖⢄⠀⠀⠆⢀⠤⣶⡖⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⠠⠄⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣦⣄⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⣿⣶⣤⣀⣀⠀⠀⠀⣀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⡇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⣀⣀⣀⣀⣀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢦⡤
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠓⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠶⣆⡼⠛⢁⠀⠀⠹⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠞⠓⠒⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠆⠀⠀⠨⠢⠖⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠐⡀⠀⡀⠀⠀⢀⠁⠀⠀⠀⠀⠰⠀⠀⠈⠋⠓⠀⠤⠆⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠉⠊⢄⠄⠀⢀⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠈⠙⠂⠅⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠢⠂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢦⠡⠀⠀⠀⠀⠀⠀⠰⠀⣁⢀⠁⠂⠀⠀⢀⠀⠀⢀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⡠⠊
⠀⠀⠀⠀⠀⢀⠀⠐⡀⡀⠀⢨⠀⠁⠀⠈⠂⠢⠄⠀⠐⢀⠀⠀⠀⠃⠀⠌⠀⠔⠄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⡉⠠
⡀⠀⠠⡆⢠⣛⡧⠆⡥⠚⠀⠒⠀⠈⢨⠀⠈⠄⠀⣠⠀⠀⠉⠀⠈⠀⠀⠀⠀⠠⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁
⠑⡄⠀⠀⡈⠅⠀⠘⠐⠀⠀⠀⠙⠓⠂⠀⠀⠈⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠄⠀⠀⠀⠀⠀⠀⠠⠔⠚⠁
⠀⠈⢀⡀⠈⠈⠁⠈⠀⠉⠀⠀⠐⢶⣨⢃⣀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠄⠔⠄⣀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠨⠉⢀⡀⠀⠀⡀⡀⣀⡀⢀⡈⢉⠂⠀⠐⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠴⡃⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⠀⠀⠀⠲⠒⠪⣁⡚⡽⠋⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠴⠦⠆⠓⠀⠀⠀⢀⡤⢠

⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⣠⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⣀⠀⠀⠀⢀⣠⣴⣶⣶⣦⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣦⣠⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣷⣆⣰⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣠⣶⣿⣷⣦⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡄⠀⠀⠀⣴⣾⣿⣿⣷⣶⣤⡀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣶⡄
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣄⢀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣯⣿⣿⡿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣻⣿⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⢏⣴⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣌⠻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣧⡘⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡾⢰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⢂⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠃⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⣤⣤⣤⣴⣶⣶⣦⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣆⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠁⢀⡀⠄⠀⠈⠉⠙⠛⠛⠛⠿⠹⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⠐⠁⢀⠄⠀⠀⠀⠸⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠃⠀⠐⠁⠀⠄⠀⠀⣠⠎⠀⣀⣀⣀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣿⡿⠁⠀⠀⢀⠄⠀⠀⠔⠈⡹⠿⠿⠛⠋⠉⠁⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡜⠀⠀⠀⢀⠂⠀⡀⠁⢀⠔⠁⠀⡀⠀⠀⠀⠸⣿⡛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠘⠀⠀⠀⠀⠀⠀⠘⠀⢀⠊⠀⠀⠀⠈⠉⠀⠀⠀⡿⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠠⠀⠀⠀⠀⠀⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⣠⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⢀⡴⣇⠀⠀⠀⠀⠀⠀⣀⠜⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⡜⠀⠀⠉⠁⠁⠀⠀⠈⠁⠀⠀⢸⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢰⠣⢀⣀⢀⡀⠀⠀⠀⢀⣀⡠⠴⠊⠁⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠈⠈⡏⠀⠀⠐⠉⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠑⠲⠤⢤⣀⠃⠐⢀⠃⠈⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⢀⠈⡑⠃⠊⠂⣴⣀⡀⠀⢀⣀⣠⠀⠀⠀⠀⠛⠁⠂⠠⣀
⠀⠀⠀⠀⢀⠁⠀⠀⠀⣀⣄⡀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠣⠀⠜⢠⠀⡆⠉
⠀⠀⠀⢀⣠⠄⠀⠀⠀⠉⠘⠁⠛⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠄⠀⠠⠤⠀⠀⠂⠀⠀⠀⠀⢲⠁⠀⠀⠀⠀⠠⢀⠖⡁⠃⠠⠆⠁⠀⠀⣈⡆
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⡄⠀⠁⠀⠹⣆⣤⣀⡆⣀⣀⠀⠐⠂⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⢰⡀⠀⠁⠀⢀⢠⠤⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢼⢦⣠⠤⠼⠁⠀⠄⠀⠄⠀⠀⠀⠉⠁⠈⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠈⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠄⠀⠇⢌⠀⣌⠀⡶⠶⢤⠄⠀⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡖⠀⠀⠀⠀⠄⠀⠈⠸⠂⠀⠈⡃⠒⠀⠀⠀⠀⣠⣠⠇⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⢈⠀⠀⠒⠀⠠⠖⠀⠀⠒⠈⠀⠀⢀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡤⠄⡀⠀⠀⠀⠀⠀⠀⠰⡤⢃⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠈⠀⠃⠚⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠛⠰⢤⠂⠀⡄⡀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⢰⠕⠂⡀⠀⣠⢀⣄⠖⡠⣤⡀⠈⡀⠀⠀⠀⠀⠁⣀⡠⠄⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠐⠌⠋⠀⠐⠠⠀⠀⠀⡠⠁
⠀⠀⠀⠤⠀⠀⢸⢆⡐⠀⠂⠉⠈⠙⣀⣡⡾⠞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⣀⡝⡛⠛⠛⠋⡉⠁⠀⠀⣀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠁⠋⠋⠛⠷⠿⠀⠀⠇⡰⠗⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠾
⠀⠀⠀⢀⠀⠀⡄⢀⠀⠈⣄⡀⠀⠁⠀⠂⠐⠠⠀⢠⣳⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢰⠂⠀⠀⠀⠠⠀⠈⠐⠤⠤⣆⠀⢀
⠀⠀⠀⠀⠀⢈⡀⠀⠄⠀⠇⠀⠀⠀⠐⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣀⠀⠀⠐⠒⢷⠆⠤⠄⠀⡀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀⣀⡀⠀⠠⠿⠃⠀⠀⠀⡀⠀⠀⢀⠀⢀⠀⠀⠄⢀⠑⢀⢂⡀⢀⠔⠁⠀⠀⠀⠀⠀⠁⠂⠀⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠘⢷⠄⠂⠁⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠈⠀⠆⢠⠐⠀⠘⡄⠀⠀⡰⠁⠀⠀⠀⠀⠂⠰⠇⠀⠀⢲⠀⠀⠰
⠀⠀⠀⠐⠀⠀⠰⠀⠀⠠⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⢢⠀⠀⠂⠀⠀⠀⠁⠀⠀⠒⠃⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠘⠒⠂
⠀⠀⠀⡀⠰⡄⣠⠀⢸⡁⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠄⣍⡌⠠
⠀⠀⠀⠁⠀⠠⠀⢰⠟⣤⢠⠍⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠀⠀⠀⠀⡐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠓⣯⠰⠘⠂⠀⠀⠒⣦⢴⣢⠄⢀⣄⣔⠀⠀⠀⠀⠒⠁⠀⠅⡀⠀⠈⠁⠛⣉⢀⡠⠂⠀⠀⠀⢀⠀⠈⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠁⠀⠀⠸⠏⢽⣏⡷⢢⡤⢀⡀⠀⠀⠀⠀⠈⢄⢡⠀⠀⢀⠀⠀⠀⠁⡤⠤⠔⠆
⠀⠀⠀⠀⠀⠀⢀⠠⠦⠀⠠⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠁
⠀⠀⠀⠀⠀⠀⠁⠠⠒⠐⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠤⡄⠀⠀⠀⠀⠑⠀⠀⠀⠀⠐⠤⠤⠖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠿⠿⡿⠻⠿⢿⡟⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡴⢒⣶⠶⣶⢦⡀
⠀⠀⠀⠀⠀⣾⣠⣿⣿⣿⣼⣿⢳⣦
⠀⠀⠀⠀⠀⣿⣹⣯⣞⣿⡟⣿⣸⢻⠆
⠀⠀⠀⠀⠀⣏⢻⣯⣟⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠙⢦⣝⣳⣾⠟⠛⠛⠁⠀⠀⢀⣤⠴⠖⠛⠛⠛⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⢀⡴⠋⢀⣠⣴⠶⠿⠶⢶⣤⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⣴⠟⠉⢀⣀⣤⣄⣀⠈⠻⣦⡈⠑⠛⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣀⡀⣀⡴⠃⠀⠈⠁⣠⡾⠛⠉⠉⠉⠉⠻⣶⡘⠃⢀⣤⣤⣀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⡉⠉⠀⠀⠀⠀⣴⠟⠀⠀⣠⣶⣶⣶⣤⡈⠛⠀⠈⠉⠈⠙⣷⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠸⢶⣤⣤⣤⡾⠋⠀⠀⣰⠿⢦⣍⠳⡌⠻⠄⠶⠶⠶⣄⠀⢹⡇⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⣀⠈⠉⠀⠀⠀⢀⣼⠋⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠘⣧⠈⢿⡀⣧⣀⡤
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⠃⢨⡟⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠞⠀⢸⣆⡈⠳⠦⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠁⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⠀⠀⠘⣿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡼⠛⣿⠀⠀⠈⣇⠀⣸⠁⠀⢀⡴⠀⠀⠀⠸⡀⢀⡾⠀⠀⠀⣿⠛⢷⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⣤⣿⠀⠀⠀⠈⠉⢁⣀⠀⠛⠶⠀⠀⠀⣀⡉⠉⠀⠀⠀⠀⣿⣤⡼⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢿⡀⠀⠀⠀⣰⠋⠁⠀⢀⣀⣀⡀⠀⠈⠙⢷⠀⠀⠀⢀⣿⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⡄⠀⠀⠘⠀⠀⠰⠏⠁⠀⠙⠦⠀⠀⠘⠀⠀⢀⡾⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⣶⡶⢶⣶⡖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⢶⠾⠏⠉⠉⠉⠽⠒⡶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⢀⡞⠁⢀⠹⣧⡀⠀⠀⠀⣠⡾⠁⠆⠙⣧⠀⠀⠀⣠⡤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣽⠷⠂⠀⠙⣆⠀⢸⡇⣠⡥⠠⠌⠛⠛⠾⠛⠋⠐⡄⢠⡁⢹⠀⠀⣾⠁⠀⠐⠿⣯⡀
⠀⠀⠀⠀⢠⠞⠉⣀⣤⠶⣾⠛⢻⣆⢸⡇⣿⠁⢣⡄⠀⠀⠀⠀⠀⠀⢈⢸⡇⢸⡆⣰⡟⠛⡶⠶⣤⣀⠙⢦⡀
⠀⠀⠀⠀⠘⠒⠋⠁⠀⠀⠘⣆⠀⢻⣼⡇⣿⡄⡆⡁⠀⠀⠀⠀⠀⠀⠈⢸⡇⢸⣷⡟⠀⣸⠃⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⢿⡇⢿⡇⢁⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡟⡐⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⠌⢿⢼⡇⢈⠀⣀⠀⣂⡒⠂⠀⢂⣸⣷⡟⠐⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡇⢰⠋⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⣼⠀⠘⠄⠀⠀⡀⠄⠒⠚⠉⠉⠉⠁⠀⠈⠉⠉⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠡⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡴⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⠒⠲⠶⠶⠶⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⡇⠀⠀⢀⡴⠚⠛⠓⢦⣄⢀⣠⡤⠖⠚⠛⠛⠛⠶⣄⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⣠⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠓⠀⣴⠷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣰⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⡀⠀⠀⠀⢀⡴⠖⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠙⠳⠶⠶⠶⠞⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠀⢀⣴⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣇⠀⠀⣠⠶⠒⢶⠖⢶⠖⠶⢶⠖⢚⣛⣫⣭⣤⣤⣤⣤⣤⣤⠴⠶⠛⠁⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦⣀⠻⢤⣤⣼⣤⣼⠧⠶⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠴⠖⢀⣠⣴⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣋⣉⣉⣉⣉⣩⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⢦⠤⠽⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠿⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠢⣀⡛⣄⠀⣀⠸⡓⠀⠀⡠⠔⠒⠃⠀⠀⠀⠀⢠⡤⠒⠋⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠓⠾⠱⣿⡄⠀⠧⢤⣈⠀⠀⠀⠀⠐⠺⣜⣠⠤⠖⠛⠒⠛⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠶⣖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡴⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⣀⣀⡤⠔⠊⠣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣒⠿⠛⠉⣀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠈⠠⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⠈⡙⢲⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠈⡙⠁⠘⡆⠈⢶⢤⣀⠀⠰⡶⢄⠀⡆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢦⠀⢸⡀⠀⠇⠀⠘⣆⠈⠉⠐⠛⠈⢺⠥⠤⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡜⠀⠀⠃⠀⠈⠀⠀⠘⠆⠀⠀⠀⣰⡃⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡧⠤⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢾⣿⡀⢿⠿⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢠⣖⡀⠛⢁⡀⠳⡄⠰⠒⣻⠁⠀⠀⠙⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⣇⣠⣅⠀⢠⠇⠀⠀⠀⠀⠈⠒⡛⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠦⣀⠀⠀⠈⢑⣿⠉⠬⠤⠞⠢⠄⠀⠀⠀⠀⠘⢅⣠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⢿⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⢠⣿⣿⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠿⠖⢺⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣶⣭⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣪⣭⣥⣬⣿⡿⣀⣐⠶⢿⡿⠿⣋⣉⢿⣿⣿⣽⣿⣓⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢿⣿⠿⠽⢿⡟⢻⣿⠯⠿⣿⠿⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⡗⠀⢳⣄⡀⠮⡅⢈⠵⢀⣠⡞⠀⢺⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡦⠀⢨⠁⣀⡤⣼⣧⢤⣀⠀⡄⠀⣸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣶⣿⣿⣇⠀⠸⠰⠃⠀⠀⠀⠀⠘⠂⡇⠀⢸⣿⣿⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⣻⣿⣿⣷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⣿⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣆⠄⠀⠀⠀⠀⠘⠀⠀⠀⢠⣰⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣤⡠⠀⠀⠀⠀⠀⠀⣇⣴⣿⣿⠏⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣆⠀⡀⠀⠀⣠⣿⡟⡟⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠃⢳⡜⠐⠁⠈⠉

⠀⠀⢤⣤⣄⡤⠠⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡇
⠀⠀⢀⠛⢛⡛⠋⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠈⠉⠛⠁⠀⠀⢇⣀⡠⠀⠀⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠒⡆⠒⠂⠀⠀⠀⠤⠤⠤⠤
⣀⣀⣀⣀⣀⣀⣀⠀⠀⡼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⢸⠈⠁⠀⠙⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⢡⣴⣚⣑⣏⣰⣄⢠⣤⣤⣤⣄⣀⣴⣤⣲⣏⣶⡴⣾⣀⠴⠂⠀⠠⠀⣤⣤⣾⣄
⠤⢤⣤⠤⢤⣤⣬⠤⣀⣠⡄⠀⠀⠘⡆⣒⣚⡆⢐⣀⡂⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠹⠉⢻⠙⡏⠹⠙⣅⠨⣏⣙⠒⠲⠶⢛⠁⠋⣉⠃
⠀⣇⢀⠆⢀⣒⣹⢠⣿⢽⠇⠀⠤⠤⠃⠤⠤⠇⠨⠤⠄⠸⠽⠿⠛⠛⠛⠿⠟⠻⠟⠛⠓⠀⠀⠀⠀⠁⠈⠉⠀⡻⠉⠐⣶⠀⢈⡀⠸⣇⡀
⠀⠈⠁⠀⠈⠉⠁⠸⠀⢀⣀⠀⠠⠤⡀⠠⠤⡄⠠⠤⡄⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣴⣶⣶⣶⣶⣶⣾⢿⢃⡀⠉⠀⠀⠂⠸⡐⢲
⠀⢭⠩⠇⠠⠥⢽⠀⣿⢸⣏⠀⠐⠖⡇⠐⠒⡇⠐⠒⢂⢸⣿⣿⣿⣿⣟⣻⢉⠻⣿⣿⣿⣿⣿⡿⢿⠿⢉⣿⣿⣬⡉⣻⣆⠀⠙⠁⠀⠘⢻
⠀⣲⣖⡃⠐⠒⠒⠀⡟⠙⠉⠀⠒⠒⠃⠘⠙⠃⠈⠉⠁⢸⣿⣿⣿⣿⣿⣿⣦⣂⣾⣿⣿⡿⡟⠀⣤⣴⣄⣠⣔⣿⣿⣿⣿⣶⣤⣤⣤⣤⣼
⢐⣛⣿⣻⡀⠀⠀⠀⣦⣤⣤⠀⠀⠀⠄⠀⠐⢦⠀⠒⠂⢸⣿⣿⣿⣽⣿⣿⣷⣽⣿⣿⣷⣿⣦⡄⠀⠛⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣸⠿⠇⠿⣇⣀⡀⠀⣿⣟⣿⠀⠀⠒⠂⠐⠒⠚⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⠟⡡⠀⢀⢹⣿⣷⣄
⣾⠉⠉⠉⠧⡤⠤⠬⣿⣯⡽⡅⠈⠉⠉⠈⠍⠉⠀⠉⠁⢀⡙⢻⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⣿⣿⡿⡟⢟⡟⠉⠉⡉⠁⠒⠀⠀⢠⣿⠿
⠉⠀⠁⠀⠀⠠⠄⠠⠀⠈⠙⣊⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⢿⣿⣿⣻⣿⣿⣿⣯⣿⣷⣯⣘⠀⠀⠀⠀⢈⠛⣿⣯⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡧⠈⠀⢀⣀⣀⣀⣀⣀⣤⣀⣠⣤⣤⣤⣤⡤⣤⠀⢿⣿⣿⣯⡻⣿⣿⣿⣷⣿⣿⣷⣤⠀⠀⠀⠀⣹⣿⣿⣿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣇⠀⠀⢨⣾⣻⣿⣯⣭⣼⣿⡿⢷⣷⠿⣿⣿⠴⠀⢹⣿⣿⣿⣿⣮⡻⣿⣿⣿⣿⣽⢧⣟⡀⣀⣀⣸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠇⢰⣖⣷⣶⣖⡒⠚⢶⣶⣰⣳⢶⣶⣿⠒⡀⢸⣿⣿⣿⣿⣿⣿⣮⣻⣿⣿⣿⣷⣽⣿⣿⣿⣿⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡄⢐⡛⣟⠉⠉⣛⣿⠋⣉⣙⣟⣉⣉⣹⣏⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢨⣅⣨⣍⣯⣤⣤⠤⠯⠤⠤⠴⠧⠤⠴⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠒⠒⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⡉⠀⠀⠀⠀⠀⠀⠠⢝⠢⣄
⠀⠀⠀⠀⠀⠀⢀⠔⠩⠐⠊⠁⠁⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠄⠃⠀⠀⠀⠀⠀⠀⠀⡐⠦⡙⢎⠣⡀
⠀⠀⠀⠀⢀⠔⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢠⠎⠤⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⢄⠱⡙⡄⢣⠱⡄
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄⠀⠀⣠⡀⢠⠏⠌⠁⠀⠀⠀⠀⠀⠀⢀⠐⠁⠐⡄⢢⠡⠘⡀⡆⢧
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠳⣌⠋⠀⠀⠀⠀⠀⠀⠀⢀⠐⠄⠁⠀⠱⠰⠀⡆⠃⢁⢰⢸
⠀⠀⠀⡆⡡⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣳⠃⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢌⢂⠱⠀⢠⠘⡀⠃⡆⢡⢰⠈⢸⢨
⠀⠀⠀⡇⠔⡡⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠈⠱⡢⣄⠀⠄⡐⠘⠈⡌⠀⠈⡴⠀⡇⠀⠁⢸⢸⠀⠼⡀
⠀⠀⠀⠀⠐⡡⢂⠀⠉⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡆⠀⠑⠈⠀⣹⡆⢀⠀⠃⡀⢀⠁⡇⠀⠇⠀⠀⢸⠘⢠⢀⠃
⠀⠀⠀⠰⠐⡱⢁⠔⡀⠈⠀⠊⡔⢁⠔⣠⢂⠔⡠⢀⠆⡄⠀⠀⢞⡟⠁⠀⠐⢣⡾⠋⠀⠀⠀⣄⠣⣽⠀⠟⠀⠆⡄⡆⡈⠀⡈⡌
⠀⠀⠀⠀⠡⠀⢁⠎⢠⠂⡠⠀⡀⠀⠀⠀⢀⠄⢀⠀⡀⢠⢀⢂⠌⣱⡄⠀⠀⠀⢱⣄⠱⡱⠰⠘⢠⠇⡘⡏⠰⢠⢀⠁⠁⡄⠷
⠀⠀⠀⠀⠀⢣⠀⠔⢁⠎⠠⠊⡠⠊⡰⢣⠊⠔⢁⠌⡐⡡⢣⠊⡴⣿⢃⡄⠀⣰⡟⠁⡰⢡⢃⢆⣾⢀⠿⢀⠃⡆⠘⠰⡰⢠⠃
⠀⠀⠀⠀⠀⠀⠡⡀⠈⠰⢁⠜⢀⠌⡐⢁⠊⠠⢃⠌⡐⠠⣡⢞⠞⠁⠈⠁⣴⡟⠀⡴⢡⢊⠎⣼⢃⠎⠇⡜⢠⢠⢃⢃⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠎⢠⠊⡜⠠⠃⡐⢡⠂⠌⠠⢁⠻⣏⠲⠀⠶⠀⢻⡄⠊⣠⢃⠎⡐⡌⡘⡜⢰⢁⠇⡆⡜⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠁⡌⠰⠁⡔⠠⢁⠎⠠⢡⠃⣮⡿⠀⠀⠀⠠⡟⠁⡔⢁⠎⡰⡱⡑⢰⢀⠇⡜⡰⡰⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⡔⠠⢁⠊⢠⢃⠂⡄⡈⠑⣦⡀⠙⠀⠉⣢⡀⠃⠴⠵⠳⢡⠇⠞⡴⣰⢣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢁⠃⡌⠀⠂⡌⠠⢠⢁⠮⡻⠡⠰⠀⣼⠟⢁⠞⡴⠱⢃⠏⠞⠰⠑⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠑⠤⡈⢰⢠⠁⠃⢸⣾⠁⠀⠀⠀⢹⠰⢃⠞⡰⢁⠏⡜⢸⡡⠋⠔⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⣐⠰⢀⢿⡆⣠⡀⠀⠀⢇⠉⡴⡡⠊⢎⠔⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠼⠇⠀⠀⠀⠀⠘⣌⣐⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⢜⠆⠀⣬⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⢀⣀⠀⠰⡦⠈
⠀⠀⠀⠀⢀⠄⡠⠀⡠⢀⠄⡠⢀⠄⢀⠄⡀⣀⠀⡀⠀⡠⢀⠀⡀⠈⡉⠀⠀⡀⢀⠀⢠⠀⡀⢀⠀⡀⢀⢀⠀⢀⠀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠉⠐⠉⠒⠑⠊⠐⠁⠒⠁⠊⠐⠡⠊⠴⠋⠔⠡⠊⠠⠊⠔⠠⠊⠔⠡⠊⠠⠊⠔⠁⠊⠐⠑⠃⠂⠁⠊⠐⠁⠊⠐⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠆⠀⠀⠀⢠⣿⡟⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢸⣇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⡄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣄⠀⠀⠀⠀⠘⣿⣷⣄⠀⠀⠀⠐⠂⠀⠀⠀⣸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⡗⠀⠀⠀⠀⣸⣿⣷⡄⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡰⠀⠀⠀⠀⠀⣹⣿⡿⢿⣿⡿⠟⣠⣶⣿⡏⢻⣤⣾⣥⣼⠻⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣠⣾⡿⣷⠛⢿⡀⠀⣼⣿⣿⣿⡇⣼⠿⠋⠻⢽⡆⢸⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⠀⠀⠀⠀⠀⡴⠡⠶⠒⠹⣦⡙⠻⠿⠿⠇⢉⣠⠬⢤⣬⡣⡨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⣇⠀⠀⠀⣼⡟⠀⠀⠀⢀⣿⣷⠀⢀⣼⠀⠀⠁⠀⠀⠁⠈⠛⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣀⣴⣿⣿⡀⠀⠀⣼⣿⣿⣇⣾⣿⠀⠀⠀⠀⠀⠀⢠⡀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⡆⣼⣿⣿⣿⣿⣟⡛⢓⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⣿⣿⣧⣿⣿⣷⣿⣿⣿⣷⣶⣇⠀⠀⠀⠀⣄⣿⣿⣿⣿⣿⣂⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⠏⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣧⢾⡿⡿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠕⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠋⠉⠀⠉⠘⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠖⠒⠒⠒⠐⠃⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠘⠒⠛⠘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠐⠢⣀⠀⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠉⠉⠁⠶⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⠀⠁⠈⢄⣄⡀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⠈⠁⠈⠉⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢰⡆⠁
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢀⡀⠤⠠⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠘⡑⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠁⠀⠠⠜⠁⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠋⠀⠀⠀⠐⢤⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠈⠁⠀⠀⢀⠀⠉⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠡⠀⠀⠈⠀⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⠰⠀⠁⢀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠘⠀⠀⠀⠀⠀⠹⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢹⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢄⣑⠢⣁⠀⠀⠠⢜⢢⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠈⠀⠀⠀⢨⠀⠀⠀⡎⢠⠀⠀⢈
⠀⠀⠀⠀⢀⠠⠀⡀⢀⠃⠀⠀⠈⢤⣈⡙⢆⠓⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠰⠒⡖⠢⠐⠀⠀⠀⠀⠀⠀⣉
⠀⠀⠀⡠⠆⠀⠀⠀⡌⠀⠀⠀⠀⠀⡄⠉⠂⠁⠀⣁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠉⠉⠀⠡⠀⠀⠒⠀⠰⠀⠀⠀⠀⢰⡎
⠀⠀⠒⠁⠀⠀⠃⠈⠁⠛⠀⠀⠀⠘⠁⠀⠒⠀⠀⠱⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠏⠀⠀⠀⠀⠈⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢀⣀⠄⠀⠀⣀⣀⣀⠠⠆⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢬
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⡄⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣆⣄⣸⣿⣶⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠐⡒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠤⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠠⠴⠤⠤⠤⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠈⠀⡄⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠛⠻⠭⠭⠰⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⢦⣄⠀⠀⠈⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠂⠒⠒⠒⠂⠀⠀⠀⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠂⠘

⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠌⠉⠀⠀⠥⠊⠁⠀⠚⠀⠀⠈⠁⠀⠎⠁⠅⠁⠀⠍⠰⠆⠀⠈⠱⠠⠀⠀⠤⠈⠱⠈⠰⠄⢢⠡⠈⠱⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢒⠩⠿⣁⣩⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣎⠒⠶⢤⠘⣐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⡨⢠⠈⢀⡰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⠁⠀⠈⠑⠢⢤⣠⡏⠀⠀⠁⠃⠀⠨⠓⢄
⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⢴⡾⡋⠢⠤⠄⠀⠀⢀⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⢀⣠⠖⠋⠁⣀⠐⢱⠀⠀⠴⠆⠑⠂⢄⡀⠀⠈⡌⠀⠀⠀⠀⠀⠀⠜⢀⡀⠀⠑⡀
⠀⠈⢻⣄⠀⠘⠄⠸⠃⠈⠢⡀⡀⢀⠤⡪⠀⣈⡴⢻⠁⠀⠀⠀⠀⠀⠘⠀⠀⠑⠀⠀⠈⠠⡀
⠀⠀⢸⠈⠓⢤⡀⠀⡀⠈⢆⣀⠍⠓⣨⠴⠚⠓⠂⠣⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⢄⡠⠣⡀
⠀⠀⢸⠀⠀⠀⠙⠢⡉⠀⢀⣠⠖⠋⠁⠀⢀⠴⠋⠀⠀⠉⠓⠂⢀⡴⡀⠀⠀⠀⠀⠀⠀⢑⡢⡈⡆
⠀⠀⢸⠀⠀⠀⠀⠀⢹⠚⠉⡇⠀⠀⢺⠚⠁⠀⠀⠀⢀⣠⠔⠚⢁⣀⡜⠀⠀⠀⠀⠀⠀⠐⠠⠬⡀
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⠀⠀⠀⢖⠈⣿⠀⠉⠉⣏⢸⡇⠀⠀⠀⠀⠀⠈⠒⠤⠥⠃
⠀⠀⢸⡀⠀⠀⠀⠀⢸⠀⠀⣧⠀⠀⡜⢆⠀⠀⠈⡄⣿⠀⠀⠀⡿⠈⡇⠀⠀⠀⢠⣶⣻⣛⡏⣻⢙⣓⡄
⠀⠀⠈⠳⡄⠀⠀⠀⢸⠀⠀⣿⠀⠀⡇⠈⢆⠀⠀⠸⡇⠀⠀⠀⡇⠀⣷⠀⠀⠀⠈⣿⣟⣏⣹⣉⣯⣿
⠀⠀⠀⠀⠙⢦⠀⠀⢸⠤⠒⠋⠉⢹⠀⠀⠈⢧⠀⠀⢣⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⡀⠈⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠳⣄⢸⠖⠒⠋⠉⠁⠀⠀⠀⠈⢧⣀⠤⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠈⠺⠇⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⢟⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠿⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⣁⣤⣬⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠉⠀⣸⣿⣿⣿⣿⣿⣿⡄⠀⠉⠙⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣷⣀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢀⣀⣠⣴⣶⣾⣿⣿⣿⣿⡿⠿⢟⣛⣛⣿⣿⣛⣛⡛⠿⢿⣷⣦⣄⡀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⡴⣻⣿⣿⣿⣿⣿⣯⣽⣿⣶⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣟⢦⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣉⣃⠀⣉⡉⠉⢹⣿⠟⡻⢿⠿⠿⠿⠿⠿⠛⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣟⣼⢿⣿⣷⠞⢉⣩⢿⣉⡉⠳⢾⡟⢦⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢀⣴⣦⣤⣄⡀⠀⠀⠀⣈⣯⣄⡼⠟⢁⣴⠟⠉⠉⠉⠻⣷⡀⠙⠢⣉⠀⠈⠫⡁⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⣿⡟⠛⢿⣧⣿⡄⠀⡚⣉⣼⠇⣴⠶⠋⠀⠀⢀⠀⠀⠀⠀⠙⠳⣦⡈⣷⣔⢤⡈⠠⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠘⢿⣇⠀⢀⣿⢿⣃⢨⣾⡟⣉⡴⠁⠀⠀⠀⠀⡌⡄⠀⠀⠀⠀⠀⡌⢧⡙⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠈⢻⣿⣿⣿⡟⠁⢸⣿⣼⣿⠅⡀⠀⠀⠀⠀⡄⠃⠀⠀⠈⡄⢀⢸⡌⣧⣿⣿⣿⣿⡕⠄⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⡧⢄⣉⠩⣵⣴⣿⣿⣿⡟⢠⡇⢠⠄⠀⢠⡇⠀⠀⠸⠀⢴⠈⢸⡇⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⣷⠦⠤⠐⣨⣿⣻⣿⣿⣷⢸⡇⣿⠀⠀⣸⢸⠘⠀⢰⠀⡎⠀⣾⣣⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣽⣿⣋⣭⣾⣿⣿⣿⣿⣿⡟⡇⣿⠀⠀⣿⣾⠐⠀⣿⢀⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠹⣧⠀⣿⡇⡇⠀⣿⣸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢻⣦⢸⡇⣡⠀⣿⢹⣿⡘⣿⣿⡿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⣿⣿⣿⣿⣿⡇⣾⣿⠏⣷⢸⣦⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣇⣿⣿⣦⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣤⣿⣼⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣾⣆⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣰⣷⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠾⠇⠙⠃⠙⠉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠋⠘⠋⠸⠷⢶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠿⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⠀⠈⠳⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⠞⠁⠀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠅⠀⢀⣀⣀⠈⠀⢀⣴⠶⠞⠳⠶⣦⡀⠀⠁⣀⣀⡀⠀⠨⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠟⠉⠀⣴⠏⠀⠀⠀⠀⠀⠀⠹⣦⠀⠉⠻⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠆⢸⡇⠀⣀⣀⡀⢀⣀⣀⠀⢸⡇⠰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢻⡾⠋⠉⢻⡟⠉⠙⢷⡟⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣴⠟⠁⠀⢀⣼⣧⡀⠀⠈⠻⣦⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡾⠁⠀⠀⣠⡞⠁⠈⢳⣄⠀⠀⠈⢷⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠼⠋⠀⠀⢀⡾⠋⠀⠀⠀⠀⠙⢷⡀⠀⠀⠙⠧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⣧⣤⣤⣤⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣤⣤⣤⣼⣧⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠞⠛⠛⠛⠛⠳⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠁⣸⠖⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢸⠀⡟⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢸⢸⠀⡇⠀⢀⡟⢳⠶⠶⠶⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣿⠘⠀⠀⠇⠀⢸⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡟⠀⠀⠀⠀⠀⠀⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠲⢄⠀⠀⠀⠀⠀⣰⢿⡟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⣸⠃⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⣤⠞⠋⢀⡼⠃⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⣀⣤⣾⣧⣤⣤⡶⢞⣋⣁⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠇⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠁⠀⠀⣀⣤⡴⠶⠞⠛⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠳⣦⣤⣤⣤⣤⠶⠞⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣤⣤⣀⣠⣤⣾⣿⣿⠟⠋⠉⠉⠉⠉⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠋⠉⠉⠙⠛⠿⠿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⣀⣠⣤⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣦
⠀⠀⠀⣠⣾⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷
⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣦⣀⠀⠀⠀⢀⣤⣾⣿⠿⣿⣿⣷⣤⣀⡀⣀⣀⣠⣤⣶⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠉⠛⠿⣿⣿⣿⡿⠿⠿⠛⠉⠉⠉⠻⣿⣿⣶⣄⣀⣤⣄⠀⢾⣿⣏
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⠛⠀⠈⢿⣿⣷⡄
⠀⠀⣿⣿⡆⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠙⣿⣿
⠀⠀⢹⣿⡇⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⢻⣿
⠀⠀⣸⣿⡿⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⢸⣿
⠀⣼⣿⡿⠁⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⣼⣿
⠰⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇⣀⣀⣠⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⣿⣿⡄⠀⠀⣴⣿⡿
⠀⢿⣿⣄⠀⠀⠀⢠⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣶⡄⠀⢿⣿⣧⣤⣾⣿⠟⠁
⠀⠘⣿⣿⣄⠀⠀⢸⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠈⠛⠿⠿⣿⣿⣆
⠀⠀⠈⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄
⠀⠀⢰⣿⣿⠋⠉⠉⠉⠀⠀⠀⠀⠀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠁
⠀⠀⢾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋
⠀⠀⠘⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⢀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠈⠻⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⡿⠿⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠟⠀⠀⠀⠀⠉⠉⣻⣿⣿⡟⠛⠉⠉⠁⠀⠀⠀⠈⣻⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⡿⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠉⠙⠻⢿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⡿⠿⠛⠋⠁⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⢀⣿⣿⠛⣿⣿⣄⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣷⣄⡀
⠀⠀⠀⣠⣶⣿⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣤⡀⠀⣼⣿⡟⠀⠙⢿⣿⣦⣴⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣄
⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣾⣿⣿⣷⣤⣤⣾⣿⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆
⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⣀⣠⡤⠶⠶⠚⠛⠛⠛⠛⠛⠷⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢠⡾⠃⢻⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⣰⡏⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢆⣀⠴⠷⣄⣴⡆⠀⠀⠀⠀⠄⠀⠀⠀⢻
⠀⠀⠀⠀⣿⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⠀⢀⣤⣤⣀⡤⠚⠳⢤⣴⠛⠙⢶⠟⠁⠀⠩⣤⣤⣾⠉⢿⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠉⠙⠃⢀⣀⡉⠁⠀⠀⠈⠁⠀⠀⢶⡀⠀⠀⠀⠙⣿⡏⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⢶⣿⡗⠒⠊⠉⠉⢻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠊⠁⠀⠀⠀⣹⣦
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢸⡷⢆⣀⣀⡀⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠈
⠀⠀⠀⠀⠘⣷⠀⠀⢀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⢌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣶⣦⣤⠟⠁⠙⠿
⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠈⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠉⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⡶⣶⣀⣠⣄⠈⣷⣤⣀⠀⠀⠀⠀⣖⠉⠉⣁⡈⢱⡆⠀⠀⠀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠘⠛⠁⠙⠟⠋⠀⠙⠷⣦⣀⡀⢸⡇⠈⠛⢁⡬⠁⢠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⣶⣶⡏⠀⠀⢸⡅
⣴⠶⠶⠶⣤⣤⣄⣠⡴⠶⠶⠶⠶⠶⠶⣤⣤⣤⣄⣠⣤⣀⣀⣀⣀⣀⣼⠛⠳⠶⠶⠋⡿⠷⠚⠛⠛⠛⠛⠲⢦⣄
⠙⢷⣄⣀⣠⠄⠉⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣄
⠀⠀⢻⣾⡏⢀⣤⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠸⣧⣼⠏⣹⠟⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠈⠙⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⡄⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⣼⠇⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡶⠾⠁⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠱⣄⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⠀⠉⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠂⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠰⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⢸⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⣀⣸⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⣹⠃⠈⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢼⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡄⢸⠀⠀⢸⣧⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠈⠀⣜⠀⠀⠀⠁⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠶⠶⠶⠶⠿⠶⠶⠶⢄⣠⣄⣼⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠁⠨⠉⠉⠁⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣾⣿⣾⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠉⠉⠛⠓⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠤⢤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⢧⠀⠀⠀⠀⠀⢀⣨⡿⠗⠀⠀⠀⠀⠙⠛⢛⣷⡦⠀⠀⠀⢠⠞⠙⠛⠛⠋
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⣄⠀⠀⠉⠳⣄⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠰⠖⠋⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⡏⠀⠀⠀⠀⠀⣰⣶⢀⡾⣯⠀⣀⠀⠰⣿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠉⠓⠲
⠀⠀⠀⠀⠀⠀⠸⠶⠚⠉⠑⢤⡀⠀⠀⠈⢹⣿⡀⠀⠀⣠⡾⠉⣿⠟⠁⠈⠳⣿⣧⢀⣽⡆⠀⠀⠀⣀⡴⠚⠉⠙⠲⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⡏⣼⠃⣼⠷⢋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⢹⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠸⣾⠇⠀⠹⢦⡀⠀⢀⣠⡴⠋⠀⢸⣿⠇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠹⡇⠀⣄⣀⡀⢠⠀⢀⣀⣠⠀⣸⠏⠀⠈⢳⣄⡀
⠀⠀⢀⣠⣤⠤⠤⠤⠤⠶⠖⠚⠛⠉⠁⠀⠀⠀⣰⠿⡄⠀⠀⠀⣾⡀⠀⠀⠀⢠⠿⣆⠀⠀⠀⠉⠉⠛⠓⠶⠶⠤⠤⠤⠤⣤⣄⡀
⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠖⠚⠛⠋⠁⠀⠹⣦⡀⠀⠴⠦⠀⢀⡴⠋⠀⠈⠙⠛⠓⠲⠶⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⣾⠁⠀⢀⣀⣀⣤⡤⠶⠾⢵⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠚⢿⠗⠲⠖⠺⡿⠓⠒⠒⠲⠶⠶⠶⠶⠶⠶⢶⡮⠿⠶⢤⣤⣀⣀⡀⠀⠈
⠙⢦⣄⠀⠁⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠉⠀⣠⡴
⠀⠀⠈⠙⠲⠦⣤⣀⡀⠀⢸⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⢀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇⠀⠀⠀⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⣿⢿⠿⠿⡿⠿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⣿⣷⣶⠀⠀⢠⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠿⠿⠿⠿⣀⡀⠸⠿⣿⣿⣿⠿⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⡄⠀⣿⡟⠋⠀⣤⣾⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣸⣿⣿⣇⣰⣿⣿⣇⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤
⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⡿⠿⠿⠿⠏⠁⠁⠀⠀⠈⠁⠁⠹⠿⠿⠿⠿⠿⣿⣏⣀⠈⠀⠹⠿⠿⠿⢿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⢠⣴⣿⣿⡟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣤⠀⠀⠀⠀⠀⠈⠛⠛⢻⣿⣿⣦⡄
⠀⠀⠀⣀⣶⣿⣿⢿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⣷⡆⠀⠀⠀⠀⠀⠀⠈⠉⡿⣿⣿⣶⣀
⠀⠀⢸⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⡇
⠀⢰⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡆
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⣶⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡉⢹⣿⣶⡆⠀⠀⠀⠀⢰⣶⣿⡏⢙⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡿⠿⠿⠂⠀⠀⠀⠀⠐⠿⠿⢿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⢸⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⡇
⠀⠀⠈⠛⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠿⢿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡿⠿
⠀⠀⠀⠀⠀⢸⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠿⣿⣿⣶⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣶⣿⣿⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⣿⣿⣷⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣼⣿⣿⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣆⣀⣀⣰⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⢻⣿⣿⡟⠛⠛⠛⠛⠛
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⠿⠛⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⢀⡾⠛⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣸⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠓⠶⣼⢷⡀⠀⠀⠀⠀⠀⢠⠷⠤⣄⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢰⠇⠀⠙⠦⣄⡀⠀⣰⠋⠀⠀⢨⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡟⠀⠀⠀⠀⠀⠉⣹⠃⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡾⠀⠀⠀⣀⣤⠤⣶⠃⠀⠀⠀⣼⡽⠟⠉⠛⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣼⠃⢀⡴⡟⠁⠀⣰⠃⠀⠀⠀⣰⠏⠀⠀⢀⡴⠞⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣰⣣⠶⠋⢸⡇⠀⣰⠇⠀⠀⠀⢠⠏⠀⢀⡴⠋⠀⠀⠀⠀⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢠⠟⠁⠀⠀⢸⡇⢠⡏⠀⠀⠀⢀⡟⢀⡴⠋⠀⠀⠀⠀⠀⣴⠃⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠠⠟⠀⠀⠀⠀⣸⣇⡞⠀⠀⠀⢀⣾⡴⠋⠀⠀⠀⠀⢀⣠⠞⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣿⠃⠀⠀⠀⡼⠉⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⣿⠀⠀⠀⠼⠁⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠶⠋⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠋⣙⣻⣿⣿⣿⣿⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠉⢉⣿⣿⣿⣿⣿⣯⠉⠉⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣛⣯⣭⣷⣶⣿⣶⣶⣮⣭⣛⣻⣷⣦⣤⡀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⢷⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠟⠈⠟⠟⠛⣿⣿⢿⣿⣿⣿⣿⣿⡾⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠈⠉⠉⢙⣿⣿⣿⣿⡤⠞⢓⣶⡛⠳⢤⣿⣧⡁⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣠⣤⣤⣀⠀⠀⠀⢈⣿⣅⡼⠛⣠⠾⠋⠛⠛⢷⣆⠙⠎⢁⠀⠐⢌⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣾⡟⠛⢻⣿⣷⠀⣴⣋⣼⢁⡶⠞⠁⠀⢀⠀⠀⠀⠈⠓⢶⡌⣷⣔⣤⡱⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠹⣧⣀⣨⣿⣿⠤⣾⡏⣥⡎⠀⠀⠀⠀⠼⡄⠀⠀⠀⠀⢦⢱⡝⣿⣿⣷⣄⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⡸⠿⠿⠿⡃⣠⣿⣿⣿⢡⠄⠀⠀⠀⡇⠀⠀⡄⢐⠀⠘⡏⣷⣿⣿⣿⣯⡃⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢯⣁⣒⠉⢿⣿⣿⣿⣧⢸⠀⡏⠀⢠⢧⠀⠀⡇⢸⠀⣰⣇⣿⣿⣿⣿⣿⣿⡀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣹⣦⠄⣒⣽⣿⣿⣿⣿⣿⣸⡇⠀⣾⣾⠠⢠⡇⢸⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢷⡀⢸⡏⠂⢸⣧⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣷⢸⡇⣦⢀⡟⣿⣎⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣻⣿⣿⣿⣿⢹⣿⡇⣿⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣾⣿⣷⣿⢸⣟⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠾⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⢀⡀⠠⠄
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢠⡇⠀⣠⠏⠀⣀⣴⡶⢖⡋⣀⣠⠤⠖⠋⠁⠀⠀⠀⠀⠀⡀⡀⢀
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢀⣴⣮⣴⣿⣵⣾⣿⣯⡷⠾⢛⣋⣉⣤⣤⡶⠷⠶⣓⠚⠋⠉⠉⠉⠁⠈⠉⠉⠁
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣴⣿⣿⣿⣿⣿⣭⡭⠵⢶⣒⣛⣻⣯⣽⠶⠶⠖⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠢⠤⣀⡀
⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⡇⠀⠹⣿⢿⡿⣿⡿⢿⣛⠿⣿⣭⣭⣭⣭⣠⣤⣴⠶⠶⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠀⠀⠀⠸⣮⢻⡄⠻⣆⠙⢷⣄⡉⠛⠿⣥⣀⣀⡀⠤⠤⠀⢀⠈⠉⠒⠒⠤
⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢹⡞⢇⠀⠙⣷⠀⠈⠛⢷⣀
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠓⠀⠈⠣⣀⡀⠀⠈⠙⠒⠒⠒
⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣄⡀⠀⢀⣆⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢲⡆
⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⣠⣾⠟⠀⠀⠀⢀⡀⠀⠀⠁⠢⡄⠀⠀⠀⠀⠀⣀⠤⠐⠚⠻⣇⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠁⣀⡤⠤⢶⠟⠃⠘⣷⠠⠀⢸⡇⠀⠉⠁⠚⠛⣱⣄⠤⠒⠉⠀⠀⠀⢀⡤⠛⠀⠀⣴⡾⠛⠛⠋⠂
⠀⠀⠀⠀⡐⢀⣸⡇⠀⡀⢘⡀⠀⠀⣹⡇⠀⠈⠁⠀⠀⠀⣠⡾⠋⠀⠠⣄⡀⢀⣤⠔⠉⠀⣤⠤⠤⢿⠇⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠘⡻⠋⣿⠡⣾⢀⣠⣴⢾⣿⡿⠁⠀⠀⠀⠀⢠⡾⠋⠀⠀⣐⡀⠀⠉⠉⠀⠀⢀⣄⣸⣧⠀⠀⠀⢠⣆⠀⢻⠶⣴
⠀⠀⠀⢰⠁⠈⠁⠐⢻⡿⠋⠀⣾⡿⠁⠀⠀⢠⣾⣴⡟⠀⠀⠀⢀⡠⠘⠀⠀⢠⢀⡴⠋⠉⣿⠇⠀⣀⣠⣾⣿⣶⠟⢀⠞
⠀⠀⠀⢸⡲⠁⠀⡴⠋⢀⣀⠠⠿⢟⣿⣿⣿⠏⠀⢻⠀⠀⣠⡖⠁⠀⠀⠀⠀⠘⠉⠀⡀⣸⣿⣶⣿⠿⠟⠛⠁⠉⢠⠎
⠀⠀⠀⢸⡀⠀⠚⢶⣴⡏⠀⠀⠀⣿⣿⡏⠀⠀⠀⠈⠇⠊⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⡍⠉⠀⠀⠀⠀⢠⡴⠋
⠀⠀⠀⠀⠳⡄⠀⠀⣿⣏⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣀⣴⡿⠉⠷⣄⡀⠀⠀⢀⣸⣿⣿⡉⠀⠀⠀⠑⠒⣤⠞⠁
⠀⠀⠀⠀⠀⠹⣆⢲⣿⡏⠀⢠⣴⠿⠋⠀⡠⠒⠶⠞⢱⣿⣇⠀⠀⠈⠻⠿⠿⠟⠋⠉⠻⢿⣦⡀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠘⢾⡿⡅⡀⠘⣿⡄⢀⡜⠁⠀⠀⠀⣴⠋⠘⣷⣄⠀⠀⠀⠀⠀⠀⢀⣰⠞⠛⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣧⠀⠈⢻⣾⠀⠀⠀⢠⡤⠽⠆⠀⣼⡿⠀⠀⣦⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⢸⡀⠀⣾⠁⢀⣀⣠⠟⠀⠀⠀⢰⣿⡇⢀⣠⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠷⡾⠛⠀⣿⠈⠉⠀⢿⣇⢀⣸⠿⠿⠛⠋⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⣆⣀⡤⠞⠉⠛⠁⠀⠀⠀⠴⠟⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠾⠋⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠿⠛⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠸⠀⠀⠀⠈⠻⣿⣿⡿⠃⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠲⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⠻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠟⠛⠛⠛⠛⠛⠛⠻⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣀⣀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⣀⡀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢿⣍⣉⣉⣉⣽⡟⠁⠀⠀⠀⠀⠀⠻⣯⣉⣉⣉⣩⣿⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠛⠛⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⣾⣯⣤⡶⠶⠿⠿⠶⢶⣦⣽⣷⡀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⢶⣦⣤⣤⣤⣤⣴⡶⠾⠟⠛⠉



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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢻⡇⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠸⣷⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣯⣿⣷⣄⡀⠀⠀⠀⠀⠀⠉⠉⠙⠻⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠻
⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⠿⠧
⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣶⣿⣦⣀⠀⠀⠀⠀⠀⣾⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣧⣤⣄⠀⠀⠀⢀⣠⣴⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠈⠻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠏⣹⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠈⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⠀⠀⠀⠘⣿⡟⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣿⡇⠘⢿⣿⣿⣿⣿⣿⣿⣦⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣿⣿⣿⢿⣿⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣷⣵⣶⣶⣾⣶⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣵⡄⢀⣬⣶⣾⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢊⣵⣿⠿⣭⣶⣽⣿⣿⠿⠟
⠀⠂⠄⠂⠀⠀⠀⠀⠀⠀⢠⣾⠟⡵⠿⠛⠛⠉⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⢠⣾⣿⣿⣿⣿⡿
⠀⠀⢠⠀⠀⢀⣀⡀⣰⡷⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⢸⣿⣿⡛⢿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⣠⡄⠀⠀⣰⡇⠀⠀⠀⠀⢸⣿⣿⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⡇⠁⠶⠆⠘⠛⠁⠀⠀⠸⡿⠃⠀⠀⠀⠀⣼⢿⣟⠄
⠀⠀⠀⠀⠀⠀⠀⠠⢀⡐⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢠⠖⣇⠀⠀⠀⢰⣿⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠘⠁⡀⠀⠘⠛⠿⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣤⠴⡋⣐⠄⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡟⠋⠁⠀⠊⠁⠀⠀⠀⠀⠈⢧⠀⠘⣆⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⢁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⠄⠀⠠⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡄⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡹⡀⠀⠀⠀⠀⠇⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞
⠀⠀⠀⠀⠀⢀⠤⠔⠚⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡟⠀⣠⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠇⢄⠀⠀⠀⠀⢀⢀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠚⠋⠀⠀⠀⠀⠹⠆⠰⡆
⠀⠀⠀⠀⠀⢁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠒⠒⠤⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠈⠂⡌⢁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⢰⣿⠗⠀⠀⠀⠀⠀⢀⢼⡗⠚
⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⣠⠊⡔⡱⠃
⠀⠀⢀⣀⠀⠀⢸⠁⠀⠀⠀⠆⠀⠠⠊⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⢠⣴⣾⢃⠞⠁
⠀⢺⣿⣿⣇⠀⠈⠑⠦⠄⠀⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠰⠋⠀⠙⠋
⠀⠀⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⣀⠀⠄⠀⠀⢀⠀⢠⠆⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⢁⡆⠀⣖⢰⡏⠀⠇⢠⠠⠀⠀⢀⠀⠐⠐⡃⢸⠁⡂⠀⠆⠀⢀⠉⠆⣆⠆⠀⡉⠀⢠⠂⡀⣄⡖⠀⠀⠀⠰⠀⠀⡐⢄⡠⠀⣶⠀⠂⠒⠂⠁⠆
⠀⠀⠀⠀⠈⠃⠈⠀⠀⠈⠀⠀⠈⠀⠓⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠁⠀⠀⠁⠀⠈⠃⠀⠀⠀⠀⠁⠀⠁⠀⠘⠂⠀⠀⠀⠁⠐⠀⠑⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠘⣧⡀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠇⠙⠉⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠏⢱⠃⠸⠆⠰⠃⢘⠀⡈⠀⠘⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠙⠆⡐⠰⢦⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣤⡀⠀⠖⢾⠁⢇⠀⠢⡘⢆⠀⠀⠀⠆⠀⠘⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⣶⡟⣿⠁⣬⣠⣷⡈⢧⠘⡂⠀⢱⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⡄⣀⢀⢀⠀⡀⣀⢀⣠⢀⠀⣄⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⢹⠀⡁⠠⠛⠉⠀⠁⢜⠃⠁⠠⢄⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠃⠉⠉⠀⠋⠈⠉⠀⠁⠈⠀⠈⠀⠑⠚⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠅⠈⡠⠀⠀⠈⠀⢀⢀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠐⠐⠐⠒⠒⠒⠒⠒⠲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢀⠰⠈⠄⠀⠀⠀⠀⠘⠀⠁⢄⢀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠘⢩⣶⣤⡀⠤⠀⠈⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⡖⢦⡀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡠⠖⠊⠁⠘⣷⠀⠸⢿⣿⣷⣦⣄⣀⣀⡀⢈⡴⠞⠉⠀⠀⠉⢲⡈⠀⠀⠉⢲
⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⣿⠀⠘⠈⣩⠟⢟⠉⠻⡉⠁⢸⠀⡄⠀⠀⠀⠀⠈⣷⡄⠀⠀⠘⠗⢦⠀⠀⠀⠀⠀⠀⡰⠒⠂⠐⠀⣄⣀⡀
⠀⠀⠀⠀⠘⠀⠰⠀⠀⠉⣸⠶⠎⠀⠐⢋⠀⠁⡀⠘⢣⡾⠦⠸⡄⢠⠐⠀⠀⣹⠃⠀⠀⠀⠒⠉⠣⡀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠦⢍⣙⠳⢶⡄⣀⡀⠠⠴
⠀⠀⠀⠀⠀⠇⠀⠀⢀⡜⠁⠀⠀⠠⡄⠀⠀⠀⠀⠀⠛⠁⠀⠀⠘⠷⠤⠤⠚⠛⢄⠁⣤⠀⡀⠀⠀⠑⣄⠀⠀⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠰⠀⢂⠞⠀⠀⠠⠐⡐⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⡆⠀⢘⣿⣧⠀⠀⠀⠀⠘⣆⠉⢇⢸⠊⠀⠀⠀⠀⠀⢀⢀⠀⠠⠄⠀⠈⠃
⠀⠀⠀⠀⠀⠀⢄⠎⠀⠀⠀⣀⠀⢸⠁⠀⠀⠀⠀⠀⠄⠀⢀⡀⠐⠀⠀⢀⡾⠡⠤⣈⣈⣹⣄⠚⠃⠀⣀⡘⠀⠸⣄⠀⠀⠀⠂⠀⠀⡆⢸⠆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢈⠄⠈⢣⣄⡀⢠⣤⠦⠀⠂⠐⠀⠀⠀⠀⠐⢹⣠⠂⠀⡀⠉⢍⠑⠲⠞⠁⠀⠀⠀⠀⠉⠒⢤⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⢠⠋⢾⣷⠘⣷⢻⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠢⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠑⠢⢄⣀⣀⣀⣤⡤⠈
⠀⠀⠀⠀⡎⠀⠀⠀⢲⣾⣿⠀⣈⠋⢱⡏⡀⠈⠠⠀⠀⠀⠀⠀⠀⡀⡀⠀⣠⡤⡏⠉⠙⠒⠂⠑⠦⣀⠀⠈⠁⠒⠒⠒⠀⠀⢠⣤⡤⣀⣀⠀⠁⠁
⠀⠀⠀⠀⠓⢄⠀⠐⠾⠉⠁⠀⠀⠙⠺⢤⣘⠃⠀⠀⠀⠀⠐⠂⠙⢀⡤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠄⠀⠀⠀⠀⢻⣿⣷⢿⡏
⠀⠀⠀⠀⠀⠀⠙⠦⣄⠁⠀⠠⠄⠀⠀⣤⠈⠆⠐⠀⠒⠒⠓⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠐⠀⡉⠂⠌⠀⣈⡃
⠀⠀⠠⡇⠀⠀⠀⠀⠀⡈⢉⠓⠦⢤⡀⠿⢰⠃⠀⠀⠀⠀⠀⠀⠀⠢⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡈⠢⢄
⠀⠀⢀⡄⠀⠀⠀⠀⠀⠱⡀⠢⠐⠀⢙⡋⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡒⠴⢤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠳⣄⠀⠐⠄
⠀⠀⠘⠃⠀⠠⠀⠀⠀⠀⠘⠦⣑⠦⡀⠑⢄⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠐⠁⠀⠉⠉⠘⡟⠃⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⢶⣄⡀⠁⠢
⠀⠀⠀⠀⠀⠀⠄⠁⠄⠀⢠⠒⣩⡟⠒⠒⠨⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠈⠙⢲⣄⣀⠈
⠀⠀⠠⡆⣈⠤⣀⣈⣀⣀⡸⢸⣥⣤⡤⣄⣤⣄⣈⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠤⠤⠂⠀⠀⠀⠀⠀⠀⠳⠠⠤⠠⠤⠴⠢⠶⠤⠬⠭⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠙⣆⣰⠛⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠈⢿⡀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡴⠋⠉⠛⠲⢤⣀⣼⠋⠙⢦⡀⠀⠀⢷⠀⠀⠈⣧⠀⣠⣤⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡾⠁⠀⠀⠀⠀⠀⠈⢹⡄⠀⠈⢷⡀⠀⠈⣧⠀⠀⠸⡏⠁⢀⣬⣧⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣸⠃⠀⠀⠀⠀⠀⠀⣀⣤⣷⠀⠀⠈⢧⠀⠀⠘⣇⠀⠀⢻⢠⡏⠀⣿⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠲⣷⢠⡇⠀⠀⠀⠀⠀⠀⢠⡏⠀⠘⣧⠀⠀⠘⣇⠀⠀⢹⡄⠀⠸⣿⡇⠀⢹⡀⢸⡧⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⣾⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠘⣆⠀⠀⢹⡄⠀⠈⣧⠀⠀⣿⡇⠀⠸⣇⡟⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠸⣆⠀⠈⡇⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿⡷⠓⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⣷⠀⠀⠀⠀⠀⠉⠉⠛⠓⢿⡀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢹⠀⠀⠀⠀⠶⠖⣄⣀⠀⠈⣧⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⡇⣀⠀⠀⠀⠀⠀⠉⠉⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣌⡇⠹⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠷⣄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡆⠀⠀⠈⠃⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⠟⣦⠀⠀⠀⠐⠶⠶⠶⠿⠶⣄⠈⠳⠤⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣾⠀⠈⠳⣄⠀⠀⠀⠶⠤⠴⠶⠞⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡙⠳⠦⣤⣀⣀
⠀⠀⢀⣀⡤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠤⠶⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠈⠉⠙⠛⠲⢦⣄⡀
⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠻⢿⣿⣿⡟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡏⠀⣰⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣀⣀⠀⢠⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⣿⢿⣿⡟⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃⣰⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠃⠘⠛⢻⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣼⣯⣤⡶⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣡⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⣰⣿⡏⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⡀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡧⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣦⡀⠀⠀⠀⠹⣿⣷⠀⠀⠀⠀⠁⠀⠀⠀⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠲⡿⠁⠀⠀⠀⢠⣿⣿⣷⣀⣀⣀⠀⢀⣀⣼⣿⠿⣿⠿⢿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠇⠀⠀⠀⢀⣴⣿⣿⣽⡿⠟⢋⣴⣿⣿⡇⢸⣿⡿⣿⣾⠙⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠐⠟⠛⡹⢃⣙⣧⡀⢾⣿⣿⣿⡇⠾⠋⠠⠈⠺⡇⣸⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣣⡀⠀⠀⠀⣼⡁⠉⠁⠈⢿⣦⠈⠉⢩⠃⠠⠞⠉⠙⠗⠺⣮⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣇⠀⠀⣸⣿⡀⠀⠀⢠⣾⣿⣆⣠⣾⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣿⣿⣷⡀⢀⣼⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠘⢼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣏⣦⣐⡃⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣀⣤⣤⣤⣤⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣆⠀⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⢑⣝⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠈⠉⠙⠋⠉⠉⠉⠉⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣶⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⢹⣯⣟⠿⣯⣛⠿⢿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣠⢈⡑⠶⢟⠺⠿⣴⣶⡉⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⢨⣍⠈⠳⣝⣻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⡿⠛⠛⠉⠈⠉⠛⢻⣿⣿⣿⣿⣿⣿⣦⣽⣶⣿⣾⡿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⣾⣿⣿⣿⣛⣿⣿⣿⣿⣿⡥⢖⡤⢒⣶⠶⠒⢲⡤⢍⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢑⠉⣠⠔⣂⣠⣤⣀⠻⣶⣬⣑⣫⣽⡛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⣠⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⠞⢅⡜⣡⣾⣿⣿⣿⣿⡛⣌⠢⢍⣋⣉⣀⣈⡉⢙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⢰⢁⣶⣶⡌⣿⣿⣿⣿⣿⣿⠃⡰⣾⣾⣿⣿⣿⣿⣿⣿⣿⠈⠳⣤⣤⡠⠤⣤⣤⣤⣬⡐⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢸⢀⡟⣿⣿⣿⣿⣿⡟⠁⠀⡘⠿⠟⢹⣿⠿⣿⣿⣿⣿⣦⣴⣿⡁⠀⣀⣽⣿⣿⣿⣿⣴⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⠈⡄⢿⣿⡿⠋⣩⢀⠀⠀⢀⣀⣠⣿⣷⣞⣿⣿⣿⣿⣾⠫⣿⣗⣾⣿⡿⢿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠈⠸⣇⣼⣿⣤⠀⣿⢸⡗⠈⠉⣽⣿⣿⣿⠟⣋⣙⣿⣿⡏⠀⢻⣿⡿⠉⠐⠾⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⣷⠾⠻⣿⡿⠄⠸⡆⠁⠀⠘⢻⣟⡚⢣⣮⣭⣿⣿⡟⠁⢠⣿⣟⣉⣙⠶⣴⣬⢻⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢠⡁⠀⣠⢿⡷⣀⠀⡇⠀⠀⠀⠶⠞⢷⣶⢇⣻⣿⣿⣤⠀⣿⣿⠯⢯⡿⣿⣿⡿⠿⠦⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠁⠀⠀⠸⣧⣼⡀⢃⠀⠀⢀⠀⠀⠀⠀⢉⣿⣻⡟⠓⠆⣿⣷⡾⣿⣿⡿⠟⠛⠛⣻⠻⣿⣿⣿⣿⣿⣿⣿⣿⣋⠉⠂
⠀⠀⠀⠀⠀⠀⠙⠻⣟⠂⡐⣶⠈⣿⣶⣶⣶⡿⢟⣻⣷⣤⣴⣭⣿⢶⣿⣿⣶⣶⣭⢛⠓⣺⣿⣿⣿⣿⣿⣿⣿⡏⠉⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣠⣿⣀⣾⠏⣼⣿⡿⣻⠉⠾⢽⡶⣽⣯⠀⠀⠀⠀⠙⠋⠛⠥⣾⣽⣿⣿⣿⣿⣿⠏⡇⣰⢋⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⢾⣿⣿⠋⠹⣶⣤⠾⣽⣿⣿⣷⣤⣄⣀⣀⣴⣶⣿⡿⠟⢋⣽⣿⡿⠋⡠⠛⠕⠃
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢤⣤⣿⡸⣿⣿⡛⠦⣴⣟⣀⣤⣿⡿⣩⣹⣋⣿⠿⢫⣽⣿⣰⣾⣷⣿⠿⣿⣀⠜⡱
⠀⠀⠀⠀⠀⠰⡟⣿⠚⠿⠿⣿⣿⣷⡈⠙⢿⣶⣤⣤⣬⣽⣿⡿⠿⠛⠉⣠⣾⣿⣿⣿⣿⠿⣿⣿⣤⢟⡵⠊
⠀⠀⠀⠀⠀⢠⣀⣼⠆⠀⢀⡟⣿⠿⠛⢷⡶⢿⣿⣿⡉⠀⠉⣀⡠⠔⠊⠙⠉⠁⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⢀⠙⢁⣜⡶⠀⡾⠎⠀⠱⡎⠁⠻⡶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣄⣨⣿⡇⢰⣧⠘⢣⡤⠜⠃⢀⠘⣿⣿⣿⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠹⣴⠇⢸⣿⣶⣎⣧⡤⣴⣡⣾⣿⡿⠃⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⡠⠠⠿⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠙⢏⠁
⠀⠀⠀⠀⠀⠄⠀⠀⢀⣤⡤⠀⠈⠉⣉⣉⡉⠉
⠀⠀⢀⣠⣄⠲⣄⠠⠋⠀⠀⠀⠀⠀⠈⣟⣁⣀⠄
⠀⠀⠀⠉⠝⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡅⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠄⠀⠀⠀⣖⡆⠖⢲⢄⡱⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⢆⠄⠀⢀⣡⣴⡞⢵⡎⢣⢋⣧⡤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⢁⣋⣀⣾⣿⣻⣷⣌⣡⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣺⣿⣽⡿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠨⠿⠻⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⠩⣶⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠐⠀⠀⠊
⠀⠀⡀⠐⠀⠀⢈⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⡠⡪⢖⠝⢊⠙⢂⠜⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠈⠀⢀⣀⠠⣀⠀⠀⠁⠀⠀⠀⠀⡠
⠐⠂⢠⠊⠉⠀⢤⠤⠬⠍⠷⠒⠟⠛⣿⣽⣺⠀⠓⡤⠛⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠁⠀⣀
⠀⠀⠈⠢⢄⣁⠈⡅⠙⢆⠲⠀⠀⢰⣩⢭⣥⡕⣠⡔⠈⡐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠊⡱⠋⡲⠂⠀⢰⢀⣀⠤⠤⠒⠒⠤⡀
⠀⠀⠁⠠⠀⡀⠀⡉⠉⠉⠹⣿⡷⢩⣉⠲⣼⠁⣭⣽⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡶⢡⠴⠠⢊⠤⠛⠁⣠⠔⠂⢁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣼⣛⣤⠟⣶⣯⠭⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢰⠁⣜⣠⠎⢀⠜⠁⣀⢖⠉⠀⡤⠄⠘⠀⠀⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠬⠥⢀⡔⠁⡰⢎⠀⡠⠷⠁⣠⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⢠⠎⡠⠪⣀⠔⢫⡠⠐⣣⠐⠂⡠⠊⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠱⣿⣿⣷⣀⡎⠐⢠⡔⠉⠔⠉⠐⠈⣀⡤⠊⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣾⢀⣯⠿⢿⣜⣶⣀⠀⠈⢀⣤⣶⣾⣿⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⣡⢎⡠⣤⡐⠈⣿⣾⣶⣶⣟⣟⣿⣯⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡤⠛⢹⣀⣉⣁⣤⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⡷⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣞⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⠛⡝⢌⠻⠳⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⣢⡀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠀⢡⣈⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠦⣄⠀⠀⠀⠀⠘⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠒⠁⠀⠀⢀⣬⢂⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⡴⢀⡜⢿⡰⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⢿⠘⠄⠰⠇⡹⠉⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⡠⠀⢠⠀⢭⣦⠸⡿⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠂⠀⠀⠀⠀⠐⠠⠀⢀⣼⠢⣙⡖⠀⣄⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠐⠃⠀⢀⡐⢂⣀⠀⠄⢀⠏⠎⢹⢟⢀⠀⠈⠄⠐⢠⢤⡼⣦⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠉⡀⠀⠀⢈⢿⠎⢧⣇⣸⠋⠀⡀⠀⣿⡋⠢⡀⠈⠆⣴⠞⠐⠷⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠴⠀⠀⠀⠀⠈⠀⠒⠦⡜⠟⠀⡴⠃⠀⠂⡒⡇⠙⡄⠈⢉⡶⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠞⠀⠇⢠⠁⣇⢧⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠚⠒⠞⠚⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠰⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠠⠄⣀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢠⠇⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠈⠙⠢⡀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠂⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤⣤⣴⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡛⠿⠿⣓⣛⣿⣻⣿⠿⣟⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣱⣿⡍⡺⣿⣿⣿⣿⣿⣿⣿⣾⡿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣞⢹⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣞⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡻⣿⢻⣿⣿⣿⡿⢿⢿⣿⣿⣟⢿⠿⢯⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣺⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣭⣿⣿⣿⣿⣿⡿⣿⢿⣟⣽⣿⣿⣦⡱⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠿⠟⣾⡿⣿⣟⣭⣿⣏⣿⣟⣿⠿⣟⣻⣿⣿⣿⡞⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣞⠃⣰⠀⢰⣾⣿⣿⣿⢟⣛⢿⣿⣿⣿⡿⣫⣝⣿⡗⣿⣹⣛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣠⣧⡀⣿⣛⣿⣿⣟⣼⣿⣿⣿⣿⣿⣧⣿⣿⣼⣷⡙⣿⡿⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡝⠿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣿⣷⡿⡿⣿⣿⣿⠿⣟⣵⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣶⣝⢿⣿⣿⣿⣿⢿⣿⣿⢿⣮⣿⣿⢏⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣙⣿⣿⣷⣝⠿⣯⣿⠿⢚⣿⣫⡽⣟⣵⡿⣛⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⣫⣽⠸⠿⣦⠶⠦⠾⠾⢧⣿⣷⣾⠟⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢱⣮⣾⢿⣯⣹⣿⣾⣿⣾⣿⣿⢿⢿⣫⣵⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⡏⢹⣋⣫⣺⡿⢽⢸⣿⣿⡞⡿⡏⠄⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⡛⢿⢃⣶⣦⡇⣸⣧⡟⣸⣿⣿⣿⣟⣿⡗⣯⣟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣌⠠⣐⠶⣾⠥⠍⣉⣡⡿⠿⠿⣿⡿⣿⣿⠸⢻⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢿⠶⡭⣒⣠⣭⡅⠹⡇⡿⣿⣿⣿⡸⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠎⠉⠻⠻⢿⣭⣾⣿⡿⠻⣿⣷⣶⣞⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠉⠀⠀⠉⠛⠻⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⠛⠉⠉⠉⠙⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢀⣤⣴⡄⠀⠀⠹⣧
⠀⠀⠀⠀⠀⣾⣿⣛⣤⣤⣤⣤⣞⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢜⣿⣿⣿⣿⣦⠀⠀⣿⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⢿⣻⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⣾⡏⠀⣩⣀⣸⠀⠀⣿⠁
⠀⠀⠀⢀⣼⣿⣿⣿⣇⣀⣇⢀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠙⣿⣟⣿⣿⠟⠀⣰⡟
⠀⠀⠀⠈⠿⣿⣿⣿⣻⣾⣿⣿⣿⣿⣿⠗⠀⠀⢀⣴⠖⠛⠛⠳⣦⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣰⣵⣛⣛⣷⣤⠾⠋
⠀⠀⠀⠀⠀⠈⠉⠙⣛⣿⣿⡛⠛⠉⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣰⣛⣛⡛⠷⢦
⠀⠀⠀⠀⠀⠀⠀⢠⣏⣉⣏⢳⠀⠀⠀⠀⠀⠀⢿⣤⣤⠁⠂⢠⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠛⠿⠿⠽⠟⠂⠀⠀⠀⠀⠀⠈⠿⣿⣶⣶⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⡀⠀⠀⠀⠀⠀⢠⣄⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⡁⠀⠀⠀⠀⠀⢘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣧⡀⠀⠀⠀⢀⣼⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣦⣀⣾⣿⠟⠁
⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠶⠶⠦⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠠⠀⢠⣴⣤⠀⠉⠳⣄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢤⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⠳⢿⣿⣶⣤⣬⢧
⠀⠀⠀⠀⣿⠀⠀⠀⣠⣤⣦⣤⣶⣦⣤⠁⠄⠀⠀⡠⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⡿⢿⣿⣿⡇
⠀⠀⠀⢸⣿⠀⢠⣾⣿⣿⣿⣿⣿⣿⣇⠀⢀⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣴⣿⣿⡇
⠀⠀⠀⢸⣿⡄⣾⣿⣿⣿⣿⣤⣷⣼⣿⡀⠉⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⢀⠠⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠸⣿⡇⠛⠿⣿⣿⣿⣿⣿⣿⡿⠋⣤⡀⣾⣦⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢠⣿⣿⣿⣏⡻⣯⠉⣉⡟⠁
⠀⠀⠀⠀⣿⣇⣀⣀⣀⣈⣉⣉⣉⣁⣠⣾⠋⣿⣿⣿⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⡿⠿⠚⠉
⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠀⠀⢸⣿⣿⣿⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠑⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠀⠀⠠⣤⡟⢻⣥⠄⠀⠀⠀⠀⠀⠀⠐⠤⠀⠁⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠀⠀⠀⠉⢛⠛⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠛⠙⠀⠘⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠰⠄⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠃⠈⡟⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⡀⠈⠐⠠⢐⠂⠈⠀⠀⡜⠁⠆⠀⠀⢠⠀⠀⠀⠂⡀⠀⠀⠀⠠⠀⢸⢀⢀⣠⠞⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠈⢆⠀⠸⢀⡈⠀⠀⠀⠀⡆⠀⠀⠁⠀⠀⠀⠀⠀⢀⠈⠉⠛⠛
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⢢⠞⠀⡰⢿⢄⠀⠀⠀⠀⠇⠀⠱⡀⠀⠀⠀⠀⠀⢈⡁⠒⠀⡛⠋⠁
⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠁⡄⢩⢠⢻⡪⠅⠀⠀⡘⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢆⠚⠑⠤⠒⠒⠃
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⠞⠋⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⢀⠜⠀⠀⠀⠀⢸⡆⠀⠀⠉⠀⠁⢨⡦⠗⠀⠀⠀⠀⠈⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⢸⢀⠀⠋⠆⠀⠀⠀⢱⡀⠀⠀⠀⠀⢀
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠘⡆⠠⠤⠄⠖⠂⠀⠀⠀⠀⠰⠀⠀⠀⢀⠂⠀⠙⠿⠁⠀⠀⠀⠀⢀⢀⠀⠀⢀⡀⠀⠀⡂
⠀⠀⠀⠈⠂⠀⠀⠠⠜⠤⡀⠀⠀⠀⠀⠁⠀⠀⡀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢰⣷⣄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠻⠛⡟⠿⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⢃⠀⠀⠀⠀⠀⠀⢀⠬⠄⠱⠀⠀⠀⠀⠈⠰⠚⠁⠀⠀⠈⠉⡹⠠⡀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠀⢀⡠⠀⣠⠆⠀⠀⢲⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣁⠀⠄⠀⠀⠀⠀⠀⢀⠔⠀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣉⠔⠀⠇⠀⠰⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠋⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠗⠀⠀⢀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢸⠃⠢⡀⢱⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⡀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠤⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⣀⣀⠑⡀⠀⠀⠀⠀⠀⠠⠀⡠⠔⠉
⠀⠀⠀⠀⠁⡀⣀⡴⠉⠘⢶⣤⡀⠈⠀⠀⠀⠀⠘⣛⣛⠃⠠⢄⠛⣉⡽⠁⠐⠀⠁⠀⠠⠄⠀⠌⡡⠈
⠀⠀⠀⢀⠋⠛⣭⡀⠠⣤⠄⢻⠏⠀⠀⠀⡠⠀⠀⠀⠀⢀⠀⠀⠴⠀⠀⠀⠀⣀⠀⢀⡠⠐⠠⠊⠀⣀
⠀⠀⠀⠈⠑⡄⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠣⠀⠠⣤⠄⣖⠒⣖⠒⣶⠒⡆⠀⠐⠂⠈⠀⠘⡀⠄⠊
⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢀⠀⠹⣷⣿⠛⣜⠽⣫⣿⠃⠀⠀⠒⣀⡀
⠀⠀⠀⠀⡀⢆⡙⠰⠴⣺⣷⣇⠀⠀⠀⠀⠀⠀⠢⡀⠈⠚⠾⠩⠞⠛⠁⢀⠄⠐⠦
⠀⠀⠀⠐⠉⠈⠂⠀⠀⠀⠈⠈⠁⠢⡀⠀⠀⠀⠀⣤⡕⠦⣤⣀⢀⢴⠒⣧⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠊⠀⢸⠀⠁⠒⠠⡄⠀⠈⠁⠁⠌⠉⠑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠈⡆⠒⠲⣞⣙⡦⠖⠚⠀⠘⣏⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⠀⢃⠀⠀⡟⠉⣇⠀⠀⠐⠠⠁⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠘⠀⠈⠀⠐⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡀⠀⢀⣀⣠⣤⣤⣴⣾⣁⣀⣀
⠀⠀⠀⠀⠰⣶⣄⡀⠀⠀⠀⠀⠀⠘⠀⠀⣠⢊⣴⣾⢟⣏⣉⠙⠋⠻⠿⠿⠿⣿⣿⣖⠤⣀
⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⠀⠀⠀⠀⣀⣀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣗⣢⣤⡸⠉⠛⣛⣿⣮⣕⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠖⠛⢻⡿⠋⣩⣥⣴⣶⣾⣶⣿⣿⣿⣿⣶⣶⡤⠄⠹⢿⣿⣷⡝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠛⠛⢉⣉⣛⣀⣈⡛⠙⡿⣿⣿⣿⣿⣿⣿⣶⣤⣐⠬⣿⣿⣷⣷⡀
⠀⠀⠀⣤⣤⣤⣶⡶⠶⠖⠀⢠⣾⠀⠴⠚⠛⠙⠛⠉⠉⠋⠙⣷⠷⢾⠀⠙⣿⠻⢿⣿⣿⣦⡔⢬⣿⣿⣿⣿⡄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡠⢣⡏⠠⠴⠶⠒⠛⠛⠉⠟⢛⣿⣗⠒⠋⢰⣶⠟⠀⢐⣿⣿⣿⣟⣀⣬⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠸⠀⣀⣠⣴⠶⢿⣿⡿⣿⣿⣏⡙⠷⠀⠀⠀⠀⢰⣿⣿⣏⠈⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣿⠋⠉⠀⣀⠚⠋⢳⡈⠻⠟⠳⡶⠄⠀⠀⠠⣿⣟⠛⠀⠠⣤⣹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⣰⣿⣷⠙⠶⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠧⠠⣠⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡞⠁⢀⢀⣶⣿⣿⣓⣒⡀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⣠⣾⠀⣴⣿⣷⡾⠿⠿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠙⢆⡈⢻⣿⣿⣯⣤⣤⣭⣑⣄⠀⡿⠀⠀⠀⢀⣤⡾⠟⠓⠀⠙⠛⠉⣾⣿⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣻⠷⠦⠤⠤⠄⠈⠿⠁⣡⠀⠀⢠⠞⠉⠀⠀⠀⣀⣤⢾⣟⠿⠟⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⡀⠀⠀⠀⠐⠒⠛⠁⠀⠀⠁⠀⠀⠀⣀⣴⡟⠻⠦⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣤⣀⣀⣤⠀⡀⣀⣀⣀⣀⣤⣤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⡄
⠀⠀⢀⣴⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣆
⠀⠀⣼⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡀
⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠟⠋⠁⠀⠀⠀⠈⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⣠⣽⣿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣠⣴⣦⠀⠀⠀⠀⢰⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⣾⡿⠛⢻⣿⡇⠀⠀⠀⠀⠻⠿⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡿⠿⠿⠀⠀⠀⠀⣿⣿⣷⣿⣶
⠻⣿⣶⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠀⠀⣀⣀⡀⠀⠀⠀⠹⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⣿⣿⠁⢀⣽
⠀⠀⠉⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⠿⠛
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠁
⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠻⠛⠉⠙⢿⣷⣤⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣶⣦⣤⣤⣤⣤⣶⣶⣿⠿⠟⠋⠁

⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣆⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⡇⢀⠀⠀⠀⠀⠀⠀⢀⡎⣀⣀⡀⠀⢀⡀⢀⣀⠀⠀⣀⢀⣀⣀⢀⠀⣀⢀⣀⢀⠀⡀⢀⡈⠆
⡇⢸⠀⣐⡄⢒⠀⢢⢰⡇⠁⠉⢙⣁⣈⣉⠉⣉⡀⢀⣩⣉⣉⣉⠉⣀⣉⣉⣉⣛⢤⠉⠉⠀⡇
⣧⠜⠋⠉⠙⠉⠓⠛⢦⣧⣀⣀⣈⣉⣀⣈⣀⣉⣁⣈⣁⠀⢉⣁⣀⣉⣉⣉⣉⣉⣈⣀⣀⠤⠃
⣯⠀⢆⠶⠄⠰⠤⠤⠀⣟⣿⡸⣿⠀⠀⠀⠀⠈⡇⠀⠀⠙⠢⣧
⡟⢣⣄⠀⣀⠀⣀⣠⠶⠿⠷⠸⢿⢸⠏⠉⣉⠉⢉⠉⠉⠉⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⡇⢸⣿⣿⣿⣿⣿⡟⠛⠋⠉⢻⣿⢸⡄⠀⠊⠙⠃⠙⠁⠀⡀⠀⠀⠀⠀⣀⣈⡍⠁⠀⠀⠀⠀⣼⡐⠂⣤⠑⡀
⡇⢸⡟⣼⣟⣿⣏⢠⡄⠲⠶⣆⣿⣸⠩⠉⢳⢼⡏⠉⠉⠉⠀⠀⠀⠀⠰⠿⠅⠿⣝⢲⣀⠀⠀⠿⠋⠘⠃⡶
⡇⢸⣋⣙⣛⣙⣛⣿⡀⠀⢀⠏⢢⣍⠂⠀⡈⢠⣟⣩⠉⢻⡁⠀⠀⠀⡀⢀⠦⢀⡼⠛⠐⠃⠀⡀⢂⣨⠔⣄⣀
⡇⢸⡟⢹⣿⠹⣿⢹⠏⣰⢧⢴⡄⠘⡞⠀⠀⠸⣃⠈⢀⣈⠀⠐⠢⠈⣄⣾⣛⣒⣩⣧⣶⢄⡆⣉⣉⠎⡴⢁⠈⣧
⡇⢸⡷⠹⢿⣽⣷⢙⠷⢩⠟⣟⣿⠀⡇⠀⠀⣸⠯⢦⡠⡴⠀⠀⠀⣥⣿⡿⠤⠤⣿⣿⣾⣾⣡⣟⠁⡀⠂⠇⠰⠙⡇
⡇⢸⣶⣶⣾⣧⣮⣭⡙⠶⣃⣉⣭⣴⡇⢰⣿⠇⠀⣀⠘⢠⡆⠀⠀⢸⣾⡏⠀⠀⢻⣽⡘⢿⠋⠠⠐⠉⠒⢴⡀⠴⠃
⡇⢸⣿⡘⣿⡘⣿⣙⣿⠍⡀⠁⠀⠑⢂⢨⠀⠀⣰⠇⠈⢨⠐⠀⠀⠀⠛⣧⠀⠀⢹⣿⣅⡀⠀⢀⠤⠀⠠⠀⢽⢀
⡇⢸⠟⠜⠟⠸⠻⠲⢆⣜⡀⠀⠀⠀⢸⠀⠶⣶⣄⣁⠾⢚⠂⠤⠃⠜⠀⠿⠄⠠⠘⠿⠯⠀⠀⠀⠉⠁⠞⠈⠁
⡇⠈⠉⠉⠉⠉⠉⠉⡙⠀⡘⠓⠒⠐⠞⠃⠀⢷⣤⣬⣶⣿⡀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠐⠀⠀⠀⠀⠀⠀⠂
⡇⠀⡁⠀⠀⠀⠀⠈⡄⠀⡁⡁⢀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢡⠀⠀⠂⠐⠀⠀⠀⠆
⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⢁⠃⠈⠄⠠⠀⠀⣾⣿⣿⠈⣿⣿⣧⠀⠀⠀⠀⡀⠀⢸⣱⠀⠀⢣⠀⡀⠀⢀⢠⠀⢠
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠄⢠⣿⣿⡇⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠘⡀⠀⠀⠀⠈⡀
⠈⠀⠀⠀⠁⠀⠀⠀⢉⡁⠘⡁⠀⠈⠀⢩⢸⣿⣿⠁⠀⠈⣿⣿⣇⠈⠀⠀⣁⠀⠸⠉⢉⠀⠀⠁⣁⣀⢸⠈⢁⣀⡅
⠀⠀⠀⠀⠀⠀⠀⣔⣙⠷⠾⠃⠀⠀⠗⣿⢹⣿⡿⠀⠀⠀⢹⣿⣻⢢⢀⠤⠟⣳⣴⠃⠈⠲⠟⢳⠏⠃⠘⠁⠹⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⠈⠀⠘⠀⠀⠀⠀⠉⠉⠁⠀⠉⠉⠀⠀⠀⠀⠓⠚⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣶⣶⣶⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⢈⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠃⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⠀⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⢈⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠻⣿⣿⣿⡿⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣿⡷⠀⢀⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢀⣴⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⣿⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣿⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠶⠶⠒⠒⠒⠒⠲⠦⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣓⡲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⠯⣍⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢤⣤⣤⣀⣀⠀⠀⠀⢀⡀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⣀⠀⠀⠰⠖⠒⠒⠒⠒⠚⠓⠀⠀⠀⡠⠉⠈⠉⠁⠒⠚⠏⠁⠒⠲
⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠁⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠱⢄⡈⢳⡀⠀⣄⢰⡓⠀⠀⢀⡤⠒⠛⠉⠉⠉⠉⠁⠚⣤⠴⠚⠉⠉⠉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙⠦⢿⠳⡽⡄⠀⠻⣄⣀⣉⡀⢀⣀⣀⠠⠴⣇⣂⣀⣠⠴⠾⠖⠒⠛⠋⠉⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡿⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢋⣩⣭⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠋⠉⠁⣈⡠⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠒⠛⠛⣻⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠖⠒⠋⠁⠀⠀⠉⠒⠤⠀⡀⠀⠀⠀⠀⠀⠘⢉⡵⠚⢁⣀⣠⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢉⠀⠸⠉⢙⣐⠒⠒⠒⠒⠘⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠑⠰⣄⣸⠉⣁⡦⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠢⣄⠀⡉⢳⡇⢈⡑⠒⠀⠄⠐⠒⡖⠂⠀⠀⣀⠀⢠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡀⠈⡙⠁⠘⢦⠀⢳⡦⣄⡀⠀⠐⣷⢦⠘⢻⠁⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠰⣄⠀⢳⠀⠀⢸⠀⠀⢳⡀⠉⠙⠒⠛⠀⠓⣏⣀⠤⠼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠁⠀⡇⠀⠀⣇⠀⠈⠀⠀⠀⢣⡀⠀⠀⠀⠀⡈⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⠀⠀⠀⣈⠄⠀⠀⠀⠀⠀⢳⠀⠀⣠⣾⣿⣿⣿⣿⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢜⣩⣌⡽⠍⠀⣴⠀⠀⣀⣠⣴⠃⠀⠘⢿⠿⠛⠉⠁⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠛⠚⠒⠒⠒⠒⢺⠉⡛⠉⠁⢠⡇⠀⠀⠀⠀⠑⣄⣀⡴⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠤⢶⡴⣷⣦⠀⣾⠀⠀⠀⠀⠀⠀⠈⣡⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⡑⠦⢤⣀⣀⣀⡭⠽⡄⠴⠒⠒⠉⣲⠄⠀⠀⠀⠀⠀⠙⢤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠞⠉⠀⠀⠀⠀⠀⠉⠙⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠢⠐⠄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠐⠐⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠈⠉⠹⠒⠂⠉⠉⠉⠉⢲⠖⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠖⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠖⠊⠉⠑⡌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⢖⠒⠒⠒⠒⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⢣⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⢄⠀⢀⣀⠇⠀⠀⢀⣀⣀⠀⠀⠀⠀⣤⠆⠀⠀⠀⢸⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡂⠑⠚⠀⠀⣠⠤⠬⣍⠉⠀⠀⠀⠀⠀⠖⠉⠉⠂⠀⢯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡉⢀⠁⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣤⣄⣀⠀⠀⠀⢀⣀⣠⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠘⠢⠤⠤⠊⠀⠈⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⣴⠚⠋⠉⠭⠉⠉⠙⡿⠟⡆⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⢼⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡴⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⢩⠉⣽⠀⠀⣾⡁⢹⠉⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣷⠛⠉⠉⠀⠀⠙⠒⠚⢱⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡹⡄⠀⠀⠀⠀⠀⠀⠀⢸⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣤⡤⠤⣤⠤⠤⣤⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⢸⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣵⣶⠾⣤⣞⣳⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⡻⢾⠃⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⡻⣿⠃⠀⠀⠀⠀⡴⠳⣦⢾⠈⣷⠘⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣮⣝⠃⠀⠀⠀⠀⠀⢕⢕⠝⠉⠓⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⡀⢂⠀⠀⢀⣼⡿⢮⡿⠃⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⠁⢢⠸⠀⠀⠀⣾⡿⣿⡶⠁⠀⠀⠀⠈⣀⠀⠠⠄⠈⠠⠀⡒⠁⢠⢤⠚⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠀⠀⠀⠀⠁⠁⠀⣾⣯⣛⠾⠃⠀⡀⠁⠀⠀⢿⠀⠀⠀⠀⠀⡆⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡰⢁⠜⠀⠀⠀⠁⣶⠂⠋⠀⣾⣿⣻⣿⠃⠀⡐⠀⠀⠀⠀⠀⣀⣀⡀⠀⠑⢲⢄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⠙⠿⠁⠀⠀⠄⢀⠠⠐⢂⠼⣿⣿⣿⡆⠀⠈⡇⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠇⡰⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⣿⣿⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢈⢀⠇⡀⠀⠀⠀⠐⢂⠀⠉⡖⡛⠛⡚⠇⡠⠗⠀⠀⠀⠀⢀⡀⡀⠠⡖⠠⠿⠿⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡀⠀⠀⠀⠉⢹⣆⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⡀⢐⣀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠘⠸⡀⠁⠀⠀⠀⠀⠘⠁⠀⠔⠃⠒⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⢀⠠⠠⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠔⠁⠆⠀⠀⡏⢐⡚⠒⠪⠭⢉⣭⡭⠁⠙⠛⠓⠒⠄⠠⣉⡀⢉⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡧⣄⠡⠀⡈⠄⡼⠃⠀⠀⠉⠁⠒⠢⠤⠄⣉⡉⠁⠒⠒⠮⠭⠄⣀⡀⠀⠀⡈⠭⠕⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⣀⣁⡤⠥⢤⡀⠀⢤⣤⠐⠒⠤⠄⠀⠈⠉⠐⠒⠒⠤⠄⣀⡀⠉⠉⠒⠒⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠠⠋⠱⠄⠀⠀⠀⠀⠀⠑⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣠⠖⠒⠒⠒⠒⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣚⠃⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⡄⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⢠⠀⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣸⠀⠀⢠⠃⠀⢠⠃⠀⡌⠀⡘⠀⢸⠀⣧⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⡠⡏⠀⢠⠇⠀⡰⠀⢀⠇⠀⡄⢀⠃⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⣀⠈⡈⠊⠀⠓⠚⠉⠓⠚⠄⠤⠫⣀⠔⠉⠁⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠡⢠⡑⠃⠀⠀⠈⣲⠦⠀⣀⣀⣀⣀⠰⢾⠡⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣴⣀⣴⣀⣉⡒⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⡹⠞⠻⡑⢄⣀⠖⢲⡭⣽⠶⠳⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⡀⠀⠙⠋⠉⠒⠁⡀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠹⡀⠀⠀⠀⠀⢠⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠔⠇⠀⠀⠀⠀⡾⢄⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠓⠂⠀⠀⠀⠀⠓⠊⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡇⠀⠉⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠞⠀⠀⠀⠀⠀⠘⠲⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠾⠿⠛⠛⠻⠿⠶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠞⠛⠛⠛⠿⠶⣶⣤⣄⡀
⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⢀⣴⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣟⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠂⠀⣨⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠸⣯⡀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠙⢿⣄⠀⠙⢿⣄⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⢹⣷⠀⠀⣿⠃⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⣾⠇⠀⣼⣏⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⠀⢰⣿⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠰⠲⢠⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠤⣀⡀⠀⢀⡠⠤⢲
⠀⠀⠀⠀⠀⠀⠈⢒⠥⠒⠈⠉⠉⠉⠉⠑⠂⢄⡀⠀⠀⠀⠀⣀⡅⠀⠀⢹⡉⣉⠀⠀⢸⢄
⠀⠀⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⣠⠊⠀⠧⠤⠒⠁⠀⠈⠒⠤⠸⠄⠙⢄
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢣
⠀⠀⠀⢀⠃⠀⠴⠶⢶⣶⠶⠄⠀⠀⠴⠶⣶⣶⠶⠀⠀⢰⠀⢠⣼⣿⣇⠀⠀⠀⢀⣴⣿⣷⠀⠀⠀⡆
⠀⠀⠀⢰⠀⠀⠀⣠⣾⣿⡄⠀⠀⠀⠀⣤⣿⣿⠀⠀⠀⠀⠤⠼⠯⠉⠙⠀⠀⠀⠈⠉⠩⠽⠦⠀⠀⡇
⠀⠀⠀⠐⡀⠀⠀⢿⣿⣿⠁⠀⠀⠀⠀⢿⣿⡿⠀⠀⠀⢰⠀⡀⣀⠔⢲⣤⣀⢠⣒⠒⢢⠄⢄⠀⢠⢿
⠀⠀⠀⠀⢣⠀⠀⠀⠉⠁⣀⠠⠠⢠⡀⠈⠉⠁⠀⠀⢀⣆⠀⡆⠀⠀⠈⠁⡕⢇⠈⠀⠀⠀⢸⢀⠎⡐
⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡠⠚⠪⢲⣇⠀⠀⠀⠈⡣⢜⠁⠀⠀⠀⣸⠡⠚
⠀⠀⠀⠀⠀⠀⠈⠑⢤⣀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠀⠀⠀⠀⠈⠑⠲⠮⠅⠁⠄⠥⠝⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣯⣿⣟⠻⣟⢿⣿⠿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣻⣿⣿⣮⠿⣿⣿⣿⣷⣝⣿⣾⣿⣿⣿⣟⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⣯⣾⣿⣮⣿⣶⣯⣻⣎⠻⢿⢿⣿⡿⡣⢨⣾⢾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⡀⣽⣿⣿⣿⣿⣿⠿⣿⣷⡙⣧⣄⣻⣟⣾⣹⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣈⡁⣼⣿⣿⣽⣾⣿⠟⣦⣍⣿⣬⡋⠛⠈⢿⣿⡿⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⢻⡄⢻⣿⣿⣿⢿⣿⠈⠋⣿⣧⡻⢷⣴⣦⣾⡆⣠⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⠟⠋⠁⢸⡏⣴⡄⠉⢻⡿⣿⡿⣿⣿⡿⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⢀⠀⠀⡃⠄⠎⢠⠀⣄⡈⠻⣟⡗⣿⣿⡅⠻⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢀⡀⢣⣶⡇⢀⣰⠈⠉⠀⡟⣷⣿⣿⣵⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠎⠀⠀⢀⠀⢠⠀⡄⠀⡟⠇⢸⣯⣷⡄⠀⠗⠻⣿⣿⠏⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⢸⡆⠈⠀⠙⠀⣇⢀⣸⣿⣿⡿⣆⠀⢠⣿⣿⣦⣴⡏⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣤⡘⣿⣦⡉⠀⠙⠶⢻⠀⠘⣿⡍⢻⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢠⣇⠁⠀⠿⣿⣤⠐⠈⣄⠁⠀⠙⣧⢈⣿
⠀⠀⢀⣠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠩⣳⡆⢣⠈⠉⠑⠀⠙⠳⣌⣦⣹⡟⢻⠄
⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠲⢦⣄⣐⣤⣤⣼⣿⣿⡇⣿⡇⡼⠆⠀⠀⠀⢀⣀⡈⢿⣿⡄⠙
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣇⡿⣟⠃⠀⢀⣠⣶⣿⣿⣏⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢶⣿⣿⠃⡿⠢⣶⣿⣿⣿⣿⣿⣿⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⢰⣾⣿⠟⠀⢸⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣾⣿⣿⠀⡀⡆⣦⠀⠀⠀⠀⢹⣿⣘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣿⣿⣿⣿⠇⠀⠀⡇⣿⠀⠀⠀⢰⠈⣿⣻⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⠀⠀⢸⡇⣿⣦⠀⠀⠀⠀⠈⡿⣿⣮⠂
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⠀⠀⠀⠀⠀⠀⠙⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡟⡀⠀⠀⠀⠀⠦⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠀⠀⠀⠀⠀⢀⣾⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣿⣿⣷⣀⢀⣠⣶⣿⣿⡿⠛⠉⠀⠀⠀⠀⣴⣿⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡗⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⢰⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⢀⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⣠⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⡺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢲⣄⠀⠀⠀⠀⠀⠈⣿⠁
⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⢦⡀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠙⠛⠋⠉