Pointed Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⢉⣻⣿⣶⣤⣤⣀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⣶⣶⡶⠈⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁⠀⠀⠉⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠚⠁⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⢃⡶⠴⠾⠿⠧⠄⠒⢬⡣⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠒⠈⠉⠁⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⡼⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡄⠀⠀⠀⠀⠀⣼⣿⣿⣷⠃⠀⠈⠉⠉⠉⠉⠚⠆⠀⣭
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠰⣿⣿⣿⡇⠀⠀⠀⠀⢺⣿⣿⣿⣿⠀⠠⠴⠇⠀⠀⠀⠀⠀⢶⡏
⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣀⣼⣿⣿⣿⡇⠠⠀⠀⠀⠀⠀⠀⡀⠀⢸
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣹⠇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠺⢿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣶⣤⣤⣤⠴⣺⣿⠇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢐⣿⠀⠀⠀⠀⠀⢈⡿⣿⠀⠀⠐⠒⠚⠛⠮⠄⠀⢸⣿⣿⠏⠀⠉⡅⠀⣿⣿
⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸⣿⡀⠀⠀⠀⠀⡾⠓⣿⠀⠀⠀⢀⣀⠀⠛⠒⠲⢾⣿⡟⠀⣠⡾⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠱⡀⣤⣾⣿⣇⠀⠀⠐⣶⣶⣶⣿⡀⣤⣴⣿⣿⡷⠀⣀⣤⣿⣿⣷⣾⢛⣷⠆⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣄⣠⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣷⣾⣿⠿⠛⠛⠛⠣⠄⢀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣠⠚⠉⠛⠒⢦⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣼⠃⠀⠀⠀⠀⣾⠀⠀⡰⠀⠀⠀⢰⣿⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⢧⣀⣀⣀⣀⡼⠛⠂⢰⠁⠀⠀⢠⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢯⣀⣀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⢤⡀⠠⠤⢤⣤⠤⣤⠔⠂⠆⠄⠀⠛⠐⠏⠀⠛⠿⢷⣤⣀
⠀⠀⠀⠀⡔⠉⠀⠀⠰⢲⠖⠀⠒⢰⡄⠋⠉⠹⣼⠃⠀⠀⠀⠠⠠⡄⡠⠀⣀⠠⠊⠀⢉⣿⢷⣦⣀⠀⠀⠀⢀⡤⠀⠀⠠⣖⣶⣶⣤⣤⣤⡀
⠀⠀⠀⠀⠣⣄⣀⣀⣀⣀⣀⣀⣀⣀⡃⠀⠀⣰⢇⡄⠀⠀⠐⣾⣷⣿⣿⠯⠄⠀⠀⠀⠀⠉⡤⡿⣿⣿⣶⣶⣼⣷⠖⡟⠑⠙⡞⣿⣿⠻⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡟⠐⠊⠀⢀⠄⠊⢀⠙⠍⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠼⠈⡇⠃⢷⠸⡎⠀⠀⠀⢹⢹⠋⢣⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠉⠉⠉⠂⠀⠀⣠⠴⡋⠩⠻⣯⣀⡀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠈⠀⡇⠀⠀⠀⠐⡇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⣀⡤⠖⠉⠀⠀⠀⠀⠀⣸⠿⣏⠈⢁⠀⠀⢸⣿⠀⢀⠀⠄⠀⡇⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠀⠈⠀⠀⠀⢀⠴⠋⠛⠣⣻⠆⠈⠁⠃⢸⣃⠊⠈⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⡠⢖⠁⠄⠀⠀⣠⠏⠀⠀⠀⠀⠈⠍⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣾⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠠⠀⠀⢀⣐⠀⠄⢲⡿⠀⠀⠀⠀⠀⣿⠘⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⣈⣀⣀⠤⠤⠄⠐⠒⠈⠁⠀⠀⠀⠈⠃⠀⠒⠂⠐⣶⡏⢸⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠛⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠂⠐⠒⠲⠶⣶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣣⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣏⠉⠉⠛⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣱⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣧⡀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⢛⡝⢀⣿⣁⡴⠀⠀⠀⠀⢀⡀⠀⠀⠈⠿⢿⣿⣷⡈⡄⠀⠀⢀⠹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠋⠀⢸⣿⡿⠁⠀⢀⠴⢊⣁⣤⣤⡄⠀⠀⣀⠈⢿⣇⠀⠈⡆⠈⡆⠘⠻⡓⣄
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡇⠀⠀⢁⡴⠋⢁⡤⡶⠷⠀⠀⠈⠻⣦⣿⠇⠀⡇⠀⢸⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⢠⡟⡠⠞⠉⠀⠀⠀⠀⠀⠰⣶⣿⣿⡇⠀⢸⢀⡄⠀⠀⠀⠀⠀⢬⣳⡀
⠀⠀⠀⠀⠀⢸⡅⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢝⣿⣷⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⢸⠹⡄⠀⣀⠀⠀⠀⠀⠙⠳⣄⡀
⠀⠀⠀⠀⠀⢸⡗⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠅⠀⠘⠀⢇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⣠⠚⠟⠀⢀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣾⣿⡀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣊⠀⠀⠀⢸⣤⣿⠓⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⡼⠁⠀⠀⠀⠘⢷⡌⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣮⣤⣀⠄⠀⣀⣤⣾⣿⣿⠟⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⢆⠀⣷
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠉⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⡿⣿⣿⣿⣿⣿⠿⢿⣿⣦⣤⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢆⢹⡄
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠢⢤⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣿⣿⣿⣄⡀⠈⠻⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⢿⠻⡿⣶⡄⠀⠀⠀⠀⠀⣻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦⣀⠀⢿⣿⣷⢇⣀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⣇
⠀⠀⠀⢸⡇⣀⠀⠈⠀⠀⠀⠀⠀⠀⢻⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡿⠏⠉⠉⠀⣿⣿⣏⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡆
⠀⠀⠀⣼⠋⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⠀⠀⠀⢠⣿⡿⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⡄
⠀⠀⠀⣿⠀⠀⠀⠠⢦⣶⣄⠀⠀⠀⠀⠀⠚⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿
⠀⠀⠀⠹⣄⡀⠀⠀⠀⠨⢿⣧⣄⠀⠀⠀⠀⠈⠿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⣴⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣈⢢⠐⢄⠀⢭⡿
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠐⢉⠿⣿⣷⣄⠀⠀⢀⠀⠀⠀⢀⡠⠊⡁⣠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠐⠊⠈⠈⠀⠃⠠⣅⣼⠇
⠀⠀⠀⠀⠀⠘⢿⣦⡄⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠊⠽⢿⣿⣿⣯⣀⡠⢂⣔⡥⣠⣼⣿⣿⠟⠃⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠙⢻⣷⣄⠀⠀⠀⠐⢋⢟⣿⣿⣿⣾⣿⣿⣿⡿⠛⣽⡟⠁⠀⠀⠀⢀⣴⠟⠛⠳⣄⢀⣦⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⠀⠀⠙⢿⣦⣀⠀⠀⠀⠩⠞⠛⠡⢋⠟⣹⣿⠁⢰⣿⡅⡀⠀⠀⣶⠟⠁⠀⠐⣤⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⠀⢙⢿⣷⣦⣀⢀⠀⣀⢔⣡⣴⣿⠏⠀⠸⣿⣿⣇⠀⠸⠁⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠘⠛⠿⢿⣿⣵⣿⣿⠟⠁⠀⠀⠀⠹⣿⣿⣆⢳⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣦⣤⣊⣤⣼⠟⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⣀⡤⠤⠔⠒⠚⠉⠉⠙⠫⢭⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⢳
⠀⠀⠀⠀⣼⠉⠀⠀⢠⡿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣤⡤⠤⠤⠤⠤⠤⠶⠶⠶⠖⠒⠋
⠀⠀⠀⢀⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣼⠦⢤⣤⠀⠀⢾⣉⣉⣶⡄
⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⢿⢦⡜⠀⠉⠓⠒⠶⠖⢋⡏
⠀⠀⠀⢸⡇⠀⠀⠀⡏⠀⠀⠀⠀⠒⠒⠚⠻⢿⠛⢷⡞⠀⠙⠒⠒⠒⠒⢲⠒⠋
⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠉⠉⠒⠶⢤⣄⣀⡠⠞
⠀⠀⠀⠸⡇⠀⠙⠀⣿⣧⣄⡀⠀⠀⠀⠀⠀⢀⣀⣹⣶⣤⣀⣀⡼⠃
⠀⠀⠀⠀⠻⠶⠶⠶⠾⠁⠀⠉⠉⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠚⠉⠁⠀⢀⣠⣤⣰⡄⠀⢦⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠲⢾⣀⣠⣤⠤⠤⠤⠤⠤⠴⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⢿⡿⠉⠀⢀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⢀⣀⣤⠀⠀⢀⠹⠥⡀
⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢉⣿⣿⠉⠉⣍⠉⠣⣄⣈⣐⣰⣾⠄⢀
⠀⠀⠀⠀⠀⣾⣶⣿⣿⣾⣿⡇⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡓⠲⠾⢤⣤⣤⡭⠿⠿⠶⠞⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢹⣷⢤⣤⣀⣤⡤⠤⢐⣨⣿⣿⣿⡷⣤⣼⠀⠀⠀⠀⠀⠀⠀⠢⡧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⡈⣿⣄⡀⠀⠀⠀⠚⠋⠙⢻⣿⣿⣦⡸⠉⠓⠶⠶⠶⠒⠒⢺⠛⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠷⠟⢸⡟⠙⢶⣤⣀⣀⣀⣠⣿⣿⠿⠛⠛⠲⢤⣤⣤⣤⣤⡴⠋
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠻⠟⠟⠛⠛⠻⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣶⣾⣭⣭⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏⠛⠛⠉⠁⠈⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠘⡿⡿⢿⠋⠀⠀⣰⡶⢶⠀⡴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢀⠀⠀⠀⠀⠀⠘⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢦⡀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠁⠀⠙⠳⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠢⡀⠀⠀⢤⣈⠉⣃⡀⣄⡀
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠘⢆⠀⠀⠈⢣⡛⢿⡸⡈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⣼⢄⣤⠄⠀⠀⠀⠑⠀⠠⠀⠱⣿⣿⣧⠐⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠸⡄⠧⣀⠀⠘⢿⣿⡆⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⣠⡈⠃
⠀⠀⠀⠀⣾⣿⡿⠇⠀⠀⠀⠀⠀⠈⢇⠈⠑⢤⡈⢿⣿⡀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢀⡠⠚⠉
⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠑⣄⠻⡇⠀⢀⡴⠀⠀⠀⠈⡇⠘⠁
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠈⠂⣀⠔⠉⠀⠀⠀⠀⠀⢑⡆
⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣠⢘⣷⠖⠚⠁⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣤⡀⠀⠀⢠⠎⣿⣿⢹⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣎⢿⣿⣿⣦⡀⠈⠀⠟⠛⠀⠀⠀⠀⢀⣠⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⠙⢿⣿⣿⣶⣄⡀⠀⠀⠀⢀⣴⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⠙⠻⣟⠉⠛⠓⢦⣴⣿⡟⠁⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠈⠙⢀⡀⠀⠈⠉
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⠿⠿⠿⠋⣡⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠈⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠁⢚⡓⠀⠀⠶⠦⠀⠀⠿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠁⠀⠀⠒⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢦⣈⡀⢀⡄⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠘⢦⡤⠛⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⣶⣿⠳⣤⣤⣤⣴⠾⠡⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⢾⡿⠿⠿⣷⣾⣿⠀⠀⠀⠈⠉⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠉⠀⠀⠀⠀⢰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⣴⣿⡟⠛⠋⠉⠀⠀⠀⠀⠀⠀⢀⠴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠙⠶⣤⣀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣀⣠⠴⠚⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠙⠻⣶⣤⠼⣣⠞⠉⠉⠉⠉⣿⡀
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⣠⠾⢯⣷⡶⠶⠞⠛⠉⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⣀⣀⠀⠀⢐⢺⡇
⠀⠀⠐⠇⠰⢄⡤⠤⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠏⠉⠛⠶⢬⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠲⠶⠶⠶⠶⠖⠚⠋⠁⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠈⠉⠙⣦⡀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⣿⡟⠶⣄⠀⠀⠀⠀⠀⠹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⢰⠟⣇⠀⠈⠻⣦⡀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠈⠀⠘⣆⠀⠀⠈⠻⠷⢶⣾⣿⣿⡿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣠⣤⣴⠟⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠟⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⣾⡿⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢳⢟⣯⣭⣝⢻⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⣿⣿⣿⣿⣸⣬⣶⣶⣮⣵⢸⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⢹⣿⣿⣿⣧⡻⢿⣿⠿⣫⣶⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣧⢿⣿⣿⣿⣷⡰⣶⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⢻⣿⣿⣿⡎⢿⠿⣿⣿⣿⣾⣝⠻⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⢻⡿⢿⣿⣮⣛⠿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣬⣛⠿⠿⠛⢀⣬⣽⣶⣶⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁⢀⣾⡿⠿⢿⣿⣿⣿⡟⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢱⣾⣿⠸⠿⠿⠿⠘⠉

⡔⠀⠀⠉⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠣⢀⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠉⠀⠈⠁⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⡤⢄⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⢋⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⡠⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠰⠆⠃⠀⡀⠀⠈⠀⠀⢠⠀⠀⠢⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠁⠀⠀⠀⠀⢀⠴⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⠊⢉⡀⢀⠠⠤⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣲⠒⡋⠁⠀⠀⠀⠀⠀⠀⠀⣀⠈⠙⠚⠋⠉⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⣼⡅⠀⠀⠀⠀⠀⠀⠂⠀⠙⢾⡇⠀⠀⠰⠄⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢠⡟⢃⠀⢀⣠⡤⠤⠤⣀⠐⠈⣯⣷⠀⠀⠀⠅⠆⠀⠀⢐⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⠀⠀⠀⠸⣼⢾⠀⣾⣓⣻⣤⣀⣀⡙⢆⢻⣿⠄⠀⠀⠀⠻⠄⠀⠀⣼⠧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡏⠀⠀⠆⠀⢿⣾⠀⠉⠉⠀⠀⠀⠀⠉⠙⣷⢽⡀⠀⠀⠀⠀⠀⠀⠀⢍⡤⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⠀⠀⡀⠀⠀⠘⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⠁⠀⠀⠀⠀⠀⠀⢸⠎⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠐⠀⠀⠀⡱⠀⡄⠀⠈⣧⡀⠀⠀⠀⠀⠀⠀⠄⣧⡟⠀⠀⠀⠀⠀⣰⠀⠀⠀⡐⠀⠀⠠⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⢸⠀⠀⢀⡀⣿⣟⢦⣄⣀⣠⣠⠷⠞⠋⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠁⠀⣸⡴⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣀⠀⣀⠀⢈⣙⣻⣧⣼⣷⠛⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠐⡻⠁⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠂⠀⠀⠀⠀⠈⠑⢽⣤⣤⡿⠻⠫⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⠀⠀⠀⠀⡜⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠈⢙⡒⠿⢾⡇⣾⠓⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⢰⠇⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⠀⠈⠉⠙⢸⡇⠁⠺⡟⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⡟⠀⠀⠀⣠⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁⠀⠀⠀⢀⠴⠒⠉⠉⠹⣷⣶⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣼⠃⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠙⠻⢿⣅⢀⣳⢤⡀⢀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⢰⡇⠀⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠇⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⠙⠁⠠⢕⣢⣤⠴⠞⠁⠀⠀⠀⢰⠏⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠂⣴⠆⣰⡟⢄⡆⢄⣠⠀⠀⠀⠀⠀⠘⠻⡗⠶⠿⣏⠁⠀⠀⠀⠀⠀⠀⣰⠋⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣁⣐⠛⢿⡾⢣⣾⠋⠙⠳⢦⣄⡀⠀⠀⣿⠀⠀⠈⠳⣦⣬⣐⣀⣠⠞⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠻⠦⣄⠀⠰⠈⠉⢣⠀⢸⠄⠀⠀⠀⠈⠓⠮⣽⣣⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣁⡀⣼⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡊⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡃⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⢉⠄⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣼⣗⠀⠀⠀⠀⠀⠱⣄
⣠⠴⠒⠒⠋⠉⢉⠉⠉⠉⠀⣨⢀⠀⠀⠉⢉⡇⠀⠀⢠⡄⠼⢛⡇⢸⠀⠀⠀⠀⠀⠀⠈⢧⡀
⢿⣿⣾⣾⣴⣴⣿⣿⣷⣾⣶⣿⣿⣦⣴⣦⣾⣄⣀⡰⣬⣿⣀⢸⡏⠝⢀⣦⠂⠀⠀⠀⠀⠀⠻⣄
⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⠀⠉⠛⠛⠛⠛⠛⠋⠉⠽⢿⣖⢻⣾⠇⢂⠆⠀⠀⠀⠀⠈⠷⠶⢴⣶⣶⣤⣤⣄⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⣀⠀⣤⣶⠀⠀⠀⣶⣿⣿⣞⣽⣾⣿⢡⡄⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠿⠿⣿⣿⠿⠟⠋⠉⠀⣈⣹⣿⣆⠙⣿⣿⡕⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣀⠀⠀⣠⣦⣤⣬⣿⣿⣮⠛⠶⢬⡉⠛⠃⠠⠤⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠿⠟⠛⠋⠉⣠⣼⣿⡟⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⢶⣷⣤⣀⣿⣿⣯⡦⡪⣀⣀⣀⡀⠀⠀⠀⠀⡀⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣿⣦⡥⢾⣼⣿⣸⣿⣿⣿⣿⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⣟⣿⣿⣿⣿⣿⣷⡀
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⠚⠛⠒⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣁⣴⠾⣿⠇⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠓⠚⠛⠗⠋⠀⣤⡶⣿⣿⣿⣿⣤⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⣤⣶⡾⠃⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢹⣿⣯⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣇⡄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣮⣹⠇⠐⠋⢙⣻⣿⣿⠷⠀⠀⣀⣾⣟⣻⣋⡙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣸⠿⠀⠀⠀⠉⠛⢻⡿⠀⠀⠀⣿⣿⠝⠋⠉⠁⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⣸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⣰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢿⡄⠀⠀⠀⠀⠈⠃⠐⠿⠛⠁⣤⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠓⠀⠀⠶⠦⠴⣶⡶⠶⠶⠆⠟⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠰⠶⠶⠀⠀⣸⣿⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⣷⡀⠰⣤⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⢀⣿⠹⣄⠈⠙⠷⣦⣤⣤⣶⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⣤⠴⠖⣻⠏⠀⣀⣀⣀⣸⣧⣤⠬⢷⠶⣤⡤⠭⠿⢿⣧⢶⣷⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣞⣛⣉⣉⣉⣁⣀⣀⣀⣀⣴⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡟⣏⠉⠛⠛⠿⠿⠟
⠀⠀⠀⠀⠀⠀⣠⠏⠈⠉⠉⠉⠉⠉⠉⢿⡟⠁⡄⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⡄
⠀⠀⠀⠀⠀⢰⣇⡀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢰⠇⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠉⢻⣿⠶⠂⠀⠀⠀⠀⢳⣦⡟⠀⢸⡀⠀⠀⢠⣆⣀⣀⣀⣀⣠⣶⣶⣦⣤⡟⢸⡇
⠀⠀⠀⠀⠀⢠⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠧⣤⣸⡇⠀⢠⡿⠶⣍⠉⡿⢹⣿⣿⣿⣿⠀⡧⢸⡇
⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠈⢧⣀⣘⣁⣴⠋⢰⡇⣿⣿⣿⣿⣿⣠⡇⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠉⠁⠀⠈⠀⠙⠛⠛⠛⠛⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣈⣛⡛⡻⣿⢷⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠾⠛⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠉⠉⠙⠛⠛⠋⠁⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⡠⠶⠦⣤⡀⠀⣰⢞⣛⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠘⠛⠉⠓⠀⠀⠊⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡇⠀⠀⠀⠈⠤⠄⠠⠂⠐⡄⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠂⢠⣤⡤⣴⠖⣲⠞⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠉⠃⣀⣀⠀⠀⠀⣠⠿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣷⣄⡀⠀⠀⠀⠀⠀⣼⠋⠀⠇⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡇⢿⣿⣿⣶⣤⣤⣴⠾⠃⠀⡜⠀⠀⡟⠙⠓⠦⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠫⢸⣿⣷⣌⠻⢯⡿⠛⠋⠀⠀⡠⠎⠀⠀⣠⠃⠀⠀⠀⠀⠀⠉⠉⠛⠶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠉⠻⣷⡙⠆⠀⠈⠉⢠⣤⣶⣦⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⡼⣿⣿⡿⢧⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡾⠀⣼⣿⣧⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡰⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⡥⠴⠋⠀⠀⠢⠀⠀⣼⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢰⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠐⣟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣹⡄⠀⡄⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠛⢷⡄⠀⠀⠀⠀⠀⠀⡈⠀⠀⣿⢀⣾⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⣠⡿⠛⠁⠀⠀⠀⢻⣆⠤⠀⣐⣶⠿⠁⠀⠀⣼⣿⠇⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠘⣿⠞⠋⠉⠀⠀⠀⠀⢀⣿⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢀⡞⢡⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⠀⠀⠀⠀⣰⠟⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣦⠀⠀⢸⡏⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⡈⠁⠿⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣷⡄⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⣿⣿⣿⣿⠟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⢻⡿⡿⠃⣠⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠞⠉⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠁

⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣠⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣛⡛⠟⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⠿⠿⠿⣿⣿⣿⣿⡿⠿⠿⠛⠟⠛⠛⠛⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⣿⣿⣿⣿⣿⣿⠟⠋
⣿⣿⣿⠿⠋⠁
⣿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇⠀⠀⠀⠄⠀⠀⠀⠐
⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡇
⡇⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐
⡇⠀⠀⠀⠀⠠⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⡇⠀⠀⠀⣐⠢⣤⢤⡄⠀⣴⣠⢶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣧⣄⣀⣀⣼⣴⡟⣶⣿⣯⣹⣳⣦⣷⡷⣄⣀⣤⣄⣀⣄⣠⣄⣀⣀⣀⣀⣀⣠⣄⣄⣀⣠⡄⢀⢀⠳⢂⠀⠀⣤⣠⣄⣤⣄⣤⣠⣠⣤⣤⣄⣤⣤⣠⣄⣤⣄
⣿⣿⣧⣾⣮⣿⣽⣿⣶⣿⣽⣽⣿⣯⣮⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣉⠛⢉⣉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠖⠒⠊⠉⠉⠀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠉⠉⠉⠉⠉⠉⠉⠫⣅⠀⠀⠀⠀⣀⡤⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⣠⠖⠲⣄⣈⣳⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠀⢀⡠⡾⠁⠀⠀⠀⠀⠀⠀⣠⢊⣡⠦⠒⠀⠉⠉⡆
⠀⠀⠀⠀⢠⠔⠒⠚⠉⠉⢠⣾⠿⠛⠉⢰⠁⠀⠀⠀⠀⠀⠀⠁⡼⣏⡀⣀⣀⠀⠀⠸⡅
⠀⠀⠀⠀⣸⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢀⠎⢢⡴⠋⠁⠀⠀⠀⢠⡇
⠀⠀⠀⢀⡇⠀⠀⠀⠀⢿⡀⠠⠤⢐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠳⠴⠒⠊⠁⠀⢀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⣧⠀⠀⠁⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⣤⡤⠔⠋
⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⠸⣆⠀⢀⣤⣴⠾⠦⠤⢤⣄⣀⣀⣀⣀⠤⠗⠋⠁
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠙⠻⠿⠟⡟
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⢀⣠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠠⠆⡀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠴⠀⠯⡄⠀⠀⠀⠢⡀⠀⠀⠀⠀⠑⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠠⠀⠘⠆⠠⠂⠀⠘⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠐⠳⠂⠀⠀⠀⠁
⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠃⠋⠀⠀⢀⡀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠐⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣤⡀⣷⠀⣼⢀⡴⠎⠀⠀⠀⠀⢐⢺⡏⠄⠀⢀⠉⢲⠀⠀⠀⠀⠀⠙⠛⢶⣄⠀⢰⠟⠀⠘⣆⢣⣈⣴⢀⡔⠀⠀⠐⣿⣄⠀⠀⠀⢀⣹⣧⣤⣤⣤⣾⠟
⠀⠀⠀⠈⢷⣼⣼⣷⢟⣀⣀⠀⠠⢤⠔⠀⣹⣇⠄⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠻⣦⠓⠀⠐⡖⣲⠀⢟⢃⣿⡵⠂⠀⠀⢸⣿⣤⣶⣿⠿⠟⢛⠉⠉⠉⠁
⠀⠀⠀⣀⠀⠙⠿⠟⠛⠉⠁⠀⠈⢻⣏⡀⠨⣧⠄⠀⢆⢀⡀⠀⠊⠵⠀⠀⠀⠲⠀⠀⣟⠀⠀⠀⠐⠈⡖⣆⢨⠄⠁⠀⠀⢀⣼⣿⡿⠋⠀⣀⣤⣬⣷⣷⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⣿⡠⠐⣷⠄⠘⣾⣫⠄⠀⠉⠐⣀⠀⠀⠀⡆⠀⠀⠈⠡⠀⠀⠘⠿⠿⢃⠀⠀⠀⢴⣿⣿⠏⠀⠀⣀⣿⣿⣿⣿⡿⠓
⠀⠀⠀⠀⢁⠄⠀⢀⠀⠀⠁⠀⠀⠀⠠⠿⣬⣿⠊⢱⡿⠄⢀⠀⠀⠀⠉⠀⠤⠀⠈⣀⠠⠲⠀⠀⠀⠰⠖⠚⠟⠀⠀⠀⠀⠉⠁⠀⠀⠀⣸⠿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⣿⣹⠛⠂⢠⡉⠦⡒⠤⠄⠀⠐⡸⠐⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⣀⢀⡀⡀⠀⠀⢱⡄⠀⠇⠀⠀⠈⠿⠉⠁⠀⠰⢐⠣⠼⠀⠆⠠⠈⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⠛⣿⡇⠀⣴
⠀⢀⣀⡀⠀⢀⡀⠁⠀⠀⢨⣵⢀⣟⠀⣰⠀⠀⣀⣤⣄⡀⠈⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⣀⠀⠀⠀⠐⠂⠀⢰⠀⠀⠀⠀⠀⣿⡇⠀⡿⠀⠀⠀⠀⠈
⠀⣿⣿⣿⡀⠸⣿⣾⣷⠀⢨⣶⢾⣭⢴⡍⠀⢀⣛⣓⣉⣵⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠠⡄⠀⠐⠁⠀⠀⠀⠀⠀⣿⣧⡾⠃⠀⠀⠀⠠⠀⠄
⠀⢹⣿⣿⠇⢠⣿⡿⠃⠀⠀⢰⣿⡿⡦⠀⠀⠘⣿⠿⠷⠾⣟⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣿⠀⣿⡆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠻⠿⢀⡤⠁⠀⠀⠀⠀⠀⣭⣻⡰⡞⠁⠀⢰⠶⢖⣻⣯⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣿⡇⠀⠀⠀⠀⠀⠎
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡾⠏⠋⠀⠀⠀⠘⢿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠃⠀⠀⠀⠀⠐⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⠀⠀⡀⠀⠀⠀⠀⣶⣾⣷⣀⠀⢀⡀⠴⢃⡄⢀⣠⡄⠀⠦⠀⠀⠀⠐⣄⠀⠄⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠑
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣴⡇⠀⡆⠀⠀⣿⣿⣿⣿⡀⢲⡙⡘⣆⣃⠀⡁⡦⢀⡃⠈⠀⠀⠠⡀⢂⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⢶⠘⡁⠈⠁⠀⠘⡛⡴⠁⣠⣶⣤⠀⠈⠆⣡⠀⠠⢀
⠀⢠⣭⣏⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢿⣿⣿⡏⠀⣀⠈⠁⠀⠈⠀⠀⠁⠀⢸⣿⣿⣿⡇⠀⠀⠈⠛⠟⠋
⠀⠈⠖⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠂⠀⠀⠀⠀⢹⣿⣿⠃⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠄
⠀⠀⠍⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⢨⣿⣿⠀⠀⠀⠀⠀⠙⠀⠀⠠⠄⠀⢸⣿⣿⣿⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣠⣼⣦⠀⠀⠀⠐⠀⠛⡛⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⢀⣤⢀⡀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⢠⠰⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⢀⡄⣸⣿⡼⠇⠀⠀⢸⣇⢀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠔⡆⠉⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⣀⡏⣦⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦
⠀⠀⠸⣿⣿⡯⠀⠀⢀⠀⢨⣷⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠖⠀⠀⠀⠸⣄⣧⢰⡄⣿⠃⡏⣤⠀⠀⠀⡄⢀⡠⠂⠀⠀⢀⣤⢄⡀
⠀⠀⠀⠙⣿⡇⠀⠀⠼⢷⣘⣏⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢻⡄⣇⣿⢠⡇⣿⠀⠀⠀⢃⠀⢀⠀⠀⢀⠿⠇⡋⢿
⠀⠀⠀⠀⠈⠁⠀⠀⠀⢻⣶⢿⠃⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢚⣇⠰⠄⠀⠀⠀⢣⢹⣿⣸⣇⠇⠀⠀⠀⡎⡀⣻⠀⠀⠈⡆⠃⠠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠅⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣌⡁⠀⢫⡔⡮⠁⠀⠀⠀⢻⣿⣿⠞⠀⠀⠀⠀⠰⣃⣿⠀⠀⠀⠑⡀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠟⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠠⠀⢀⡿⡿⢗⢐⡱⠿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠸⣻⣿⠀⠀⠀⠀⠱⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠏⠁⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⢰⣁⠀⠀⠀⠙⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠃⠀⠰⠀⠣⢸⡀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠔⢊⠊⠀⠀⡀⠀⠉⠀⣢⠜⡄⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⠩⠢⡀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠃⠈⡆⠀⠀⠀⠀⠘⢯⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠕⠁⠀⠀⠀⠀⠀⢢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢣⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡐⠛⢳⠜⠀⠢⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠢⡀⠀⠀⠀⠈⢆⡀⠀⢀⣀⣀
⠀⠀⠀⠀⢀⡠⠤⠒⠀⡕⠉⠀⠀⠀⠻⡗⢄⠀⠀⠀⠀⠀⠀⠈⠆⠈⠓⠄⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢖⣅⠌⠪⠗⢤⡀
⠀⠀⣀⠖⠁⠐⠃⢠⢀⠇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠘⡄⠀⠐⠢⢼⡦⢄⡀⢀⡤⠤⡀⠀⠀⢀⠔⠚⠋⠀⠂⠈⠑⠄⠀⠀⠈⠓⠦⣀⡀
⠔⠋⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠂⠀⠀⢀⡤⠦⡟⠁⠙⠆⠀⢑⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠛⠈⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣇⡶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠻⡄⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠒⣦⣵⣸⡟⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢆⡁⢂⠘⠈⡌⠘⢀⠠⠆⠀⠀⠀⠀⠔⢁⠄⠊⡁⠄⠂⢘⠻⣜⣻⠶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢤⡀⠙⢦⡸⣿⡤⠋⠀⠴⡳⣖⠠⣄⡎⠠⢂⡤⠊⠀⡠⠊⠁⢠⢹⣮⣳⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⣢⣌⡙⣿⣿⣿⣾⠧⠒⢖⠾⠎⢣⠀⠱⣧⠊⠀⢀⠜⠀⠀⢀⠃⠈⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢦⡤⠀⣡⢚⡟⢇⡉⣗⠢⡎⡄⠀⢡⠀⠘⣤⠐⠁⠀⢀⠀⠁⡰⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⠃⡼⡵⠛⠁⡎⢁⠈⠁⣾⠀⠀⠰⠀⠀⢣⠀⠰⠄⠄⠀⠁⡠⠞⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠭⠭⢉⡚⠀⠀⢡⠀⠲⠀⠈⠟⠀⠈⠀⠀⠀⠀⠀⠀⠆⠀⣶⣤⣢⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠤⠤⠀⠀⠀⡅⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⠂⢤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢫⡁⠒⠠⠤⠄⢚⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠣⣁⡂⠐⢒⡩⡆⠇⠀⠃⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠠⠐⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⠈⠀⠀⢸⠀⠀⠸⠀⠀⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣧⠰⡀⠠⠀⠀⠀⠇⠀⢸⠀⠀⡄⠀⢠⠁⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣄⣥⣀⣆⠀⡀⢰⡀⢨⠀⢀⣇⣀⣼⣤⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠷⣧⣭⣩⣍⣿⣍⣏⣉⣋⣍⣩⣥⣤⠴⠟⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⠀⣀⣀⣀⣀⣀⣠⡤⠤⠤⠔⠒⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢸⣿⢀⣾⠟⠋⠉⠙⢻⣦⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⠿⠿⠋⠀⠀⠀⠀⠀⢿⣿⠿⠛⠛⠻⢷⣄⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢸⣿⠀⢠⣤⣤⣀⣻⣏⠁⢈⣉⡙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠈⠹⣿⠀⠉⠉⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠳⡄⠀⢿⡇⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢱⣦⣙⣦⣸⣇⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡾⠋⠉⠻⡏⠛⠛⠛⠛⠛⠛⠻⣿⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡿⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⢠⡿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠀⠀⠀⠀⢤⣀⠀⠀⠀⢀⣠⡾⣿⣀⠀⣀⣴⡿⠻⣷⣀⣀⣠⡟⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠙⣿⡟⠙⠛⠣⣄⡈⠻⠟⠛⠿⣄⠀⠀⠉⠉⠁⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⣦⠀⠀⠀⠙⢦⡀⠀⠀⠈⠛⠢⠄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣠⠿⡿⠀⠈⠱⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠊⠁⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡿⢋⣿⣿⠷⣶⣶⣶⣶⠶⠶⠚⠋⠡⣤⡤⠶⠶⠿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠙⠻⠶⣤⣄⣀⡀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠐⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠻⠿⣶⣦⣄⣀⠀⠈⠉⠛⠻⠶⢶⣿⣤⣀⣀⣀⣀⣀⣀⣿⣛⠛⠿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠙⠛⠿⢶⣦⣤⣄⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⣠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠶⢶⣶⣶⣶⣶⣶⡾⠿⠟⠋⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⣩⣿⣹⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠠⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠶⢤⠤⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠋⠁⠀⠀⠀⢶⣦⣤⣤⣤⣴⡄⠀⠀⡀⢿⡄⢀⣰⣆⣄⣀⣠⣄⣤⣸⣦⣤⣶⣾⡦⠴⠖⠋
⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣄⠀⢧⡘⣿⣾⡿⠿⠿⠚⢿⡛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⢛⡛⠉⢉⡄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣵⣿⣿⣿⣏⡀⠀⠷⣀⣼⣧⣤⣤⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⠁⠈⣻⣻⣿⠉⠉⠉⠉⠙⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⣶⡶⠶⣻⣿⣿⣿⡿⠛⠛⠛⠻⠿⠿⠶⠶⣴⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⢠⠇⠟⠇⢠⣄⠂⠉⠀⠈⠀⠛⣷⣦⣶⣗⣰⠇⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠐⢄⣘⣚⣠⣥⣥⣤⣴⡶⠶⣾⣿⠿⣦⣷⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⡷⠄⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠾⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣿⡟⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⠀⢰⣿⡇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢿⣿⣧⢸⣿⣇⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⣶⡄⢀⢰⡇⢠
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡄⢰⢀⡇⡇⠀⠀⠀⢿⣿⣙⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⡇⣼⠈⣷⣼⠀⣶⣦
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢸⣧⢸⣸⣿⠃⡇⠀⠀⠈⠛⣿⣿⡿⠿⣿⡿⠋⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣾⣿⡏
⠀⠀⢸⠀⠀⢰⣿⠁⠀⠀⢸⣿⣿⣿⠏⠸⠇⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣶⣾⣾⣦⣼⣿⡇⠀⠰⣸⣿⣿⣿⠀⢤⣼⠀⠀⠀⣸⣿⡿⡇⢸⣿⣇⢀⣠⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡓⣿⣿⡖⣿⣿⣿⣿⣿⣿⣾⣿⣷⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⢿⣿⢟⣷⣿⣿⣿⣿⡿⢯⣬⢄⡹⣿⣿⣿⣿⣿⣿⡿⢹⣿⡿⣭⠞⡁⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⡬⣯⣡⣽⠂⡽⠏⠟⣟⣷⣾⣇⡀⠀⢹⣿⠛⠻⣿⡿⢋⣹⣿⡁⠁⠐⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢰⣿⢻⠅⣈⣥⠹⡇⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠈⠃⠀⢾⣻⣿⣿⣿⣷⣾⡶⢶⡿⠿⠿⢿⠿⡿⠿⠿⠿⣿⣿
⢸⣿⠷⠀⠈⠉⠀⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠈⣿
⠰⠆⠒⠤⠤⠠⠌⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠶⠶⠶⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⢄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠈⠤⠈⠀⠀⠀⠀⣀⣀⣀⣀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢁⡾⠤⠒⠒⠚⠛⠛⠓⠒⠒⠤⢄⡀⠉
⠀⠀⠠⢤⡀⠀⠀⠀⠐⠢⠄⣸⢀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⡐
⠀⠀⠀⠀⠀⡇⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠉⠒⠤⠤⠒
⠀⠀⠀⠀⠀⢱⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠠⠐⠈⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣠⠒⠁
⠀⠐⠓⠲⠄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠙⢦⡀
⠀⠀⠀⠀⠈⠢⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⢤⣀⣀⡄
⠀⠀⠀⠀⡠⠔⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠜
⠀⠀⢠⠊⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣀⠀⠄⠂⠁
⠀⢠⠇⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⡄
⠀⠸⠈⠁⠉⠐⠠⢀⣀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠔⠂⢙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡈⠀⠈⠙⠲⠦⢤⣀⣀⣀⣀⣀⣀⣤⡤⠖⠋⠁⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⣀⠤⠚⠩⡀⠀⠀⢀⠞⠉⠢⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡴⠋⠁⠀⠀⠀⠀⠑⢄⡀⠏⠀⠀⠀⠈⠑⠒⠠⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠁⢻⣇⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠺⣿⡁⢰⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡖⠁⢀⣤⠼⠟⢐⣿⠁⢳⣄⠀⠀⠀⢀⠀⠘⣿⡅⠀⠀⢀⣾⣿⠇⢨⣿⡓⠳⢄⣀
⠀⠀⠀⠀⠀⡠⢀⡼⠁⠘⠉⠀⠀⢀⣽⠏⠀⢸⠟⠛⠦⠂⠀⠀⠴⠛⠁⠐⠈⠉⡠⢖⡚⠛⠛⠁⠀⠀⣹⣦⡀
⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⡀⠀⣤⠈⠀⠀⠠⠕⡋⡀⢰⢠⡄⢀⠀⠀⠀⠀⠀⠨⠒⠈⢻⣶⣤⣄⡀⠈⠉⠈⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣇⢰⣿⠆⢰⠀⠀⠘⠓⠛⠟⠿⠷⠾⠄⣴⡀⣄⢀⠀⠀⠀⠀⠈⠁⠉⠉⠓
⠀⠀⠀⠀⠀⢠⡇⢰⣿⣾⣿⣿⣿⣧⣿⡆⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠃⠂⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠚⠛⠚⠛⠛⠛⠛⠛⠛⠛⠓⠚⠳⠤⠄⠀⠤⠆⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⣺⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⣀⡀⠀⢀⠀⠈⠊⠀⠀⠀⠀⢀⣀⣇⠀⢠⣿⣿⢀⣿⡄⢸⡄
⠀⠀⠀⠀⠀⠀⠠⠠⠄⠤⠀⠀⠀⠀⠀⠀⢴⡏⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠚⠻⠷⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡷⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⢀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⣀⣠⣤⣬⣥⣤⣄⣀⠀⢴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡦⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢴⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠾⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠈⠉⠛⠛⠻⠟⠛⠛⠉⠁⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠒⠒⢲⡖⣲⠛⡟⣻⢓⣲⣀⣠⡤⠤⠶⠒⠒⠚⠛⠛⠉⠉⠉⠉⠉⠉⠛⠒⠲⢤⣀⡀
⠀⠀⢸⠀⡇⣸⢰⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⣀
⠀⠀⠀⠀⠁⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⣀⠀⠰⡄⠀⠀⠀⠀⠀⠈⠙⠲⡦⢄⡀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡉⠉⠒⠦⠤⠤⢤⣀⠀⠀⠙⠒⣹
⠀⠀⠀⠀⠀⣸⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠋⠁
⠀⠀⠀⡀⡀⣿⣀⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠱⣄⡠⠞
⠀⢰⠸⡇⡇⢹⠉⠉⠙⡿⠲⠦⠤⠤⠤⠤⠤⣤⣀⡀⠈⠙⢦⡀⢀⣠⠇
⣀⣸⣀⣇⣻⣸⡄⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠈⢦⡙⠲⠀⣸⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⣿⣿⣿⠆⠀⢂⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣼⣿⡇⢿⣿⠆⠀⠀⠀⠢⠀⣴⣿⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇⢡⣼⣿⠋⢠⣼⣿⣧⡀⠀⠀⠀⢸⣧⣇⡀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣿⣿⠀⠈⠛⣿⣷⡈⣿⢿⣿⣏⠀⠀⠀⠀⢦⣻⡇⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⣙⣿⣄⠀⠀⠸⣿⡁⣿⣧⣿⠛⠁⠀⠀⠀⠈⠻⣿⣄⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⣾⡋⣿⣿⠷⠀⠀⠹⣷⣿⢹⣿⣆⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠻⣿⣷⣿⣧⡀⠀⠀⠀⠈⠻⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⠈⠉⠁⠙⠿⣷⠀⠀⠀⠀⠈⠻⢧⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣤⣦⣄⣴⣀⣀⣴⣀⡀⠈⠓⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠻⠷⠷⠶⠤⠀⢀⣀⣀⣀⡀⠀⢠⡴⠶⠾⠿⠾⠿⠿⠿⠿⠷⠿⠷⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡶⠶⠶⠶⠶⠒⠈
⠀⠀⠀⠀⠐⠒⠉⠉⠉⠉⠛⠛⠛⠿⠿⢿⠶⠶⠶⠖⠂⠀⢉⣙⣻⣿⣿⣿⠶⠄⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣟⣉⣉⡁⠀⠀⠀⠀⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⣈⣉⠛⠛⠻⢿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⣿⣿⡿⠃⢀⣠⣤⣤⣶⣶⣶⣶⣶⣶⡶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠋⠉⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠐⠚⠛⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢿⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠿⢷⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⠀⠀⢸⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⣿⠀⠀⠸⡄⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⠋⢿⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⡇⣿⠀⣷⠀⣿⠀⣴⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠈⠃⢸⡷⠘⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⡂⡼⠀⠛⠀⣻⡄⠀⠀⣠⣾⣮⣴⣆⠀⠀⢰⠀⣿⡀⢰⡟⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⡇⢹⣾⠆⣼⠛⡇⠀⢀⣛⣹⣿⣿⣿⣤⠀⠈⢠⠏⠃⠈⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⡇⢹⣇⣼⣡⣤⣀⣶⡄⠀⢹⣿⣿⣿⣿⣠⣿⢠⡀⢸⠀⠆⢸⡇⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⣿⢻⣿⡿⢿⠞⣷⠀⣆⣿⢹⣿⢻⡉⣿⠿⠁⠸⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢻⣼⡏⣧⣸⠁⠈⠀⠆⢾⣿⣿⢸⣿⢹⣷⣿⡀⠀⡾⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢸⣿⠃⠟⢹⠀⠀⠀⢰⠸⣧⡏⣸⣿⣸⣧⠿⠧⢘⡇⢠⢠⡿⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⣿⠀⡀⢙⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣶⣿⡿⣦⣸⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣿⠀⠁⢸⠀⡄⢰⡆⠀⢸⣇⣿⣿⡧⡻⢇⣿⡛⡧⠙⣿⡁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⠀⠀⢸⡀⠀⠈⠇⠇⢸⣿⣿⡟⠁⡇⢈⢸⠁⡇⠸⢻⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠃⠸⡇⢠⠀⡆⠀⢸⢻⣿⡇⢀⠃⠘⢌⠀⠁⠘⢸⡇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠁⠀⡇⠀⠀⢿⠀⢸⣿⣿⡇⠸⠀⠀⠈⠀⠀⡆⢸⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⠐⡄⢿⠀⠀⣼⠀⠸⢿⣿⡟⠀⠂⠀⡜⠀⠀⢸⣿⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢱⠸⡇⠀⢸⠀⢸⣿⣿⡇⠈⠀⢸⡇⠀⣀⢰⣿⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠘⡆⢰⠀⢸⠀⠸⣿⣿⣧⠈⡀⠐⡆⠀⢉⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠁⠘⠀⢮⠀⣸⣿⣿⣿⠀⡇⢰⣇⠀⡄⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⣿⡄⢀⠟⣸⠀⢿⣿⣿⣿⡆⡗⣸⡏⣰⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣻⣿⣿⣿⣿⠈⣐⢻⣥⣾⣿⣿⣿⣿⣴⣿⣷⣿⣿⣿⣿⣿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⡾⠟⠟⠯⠬⠭⠍⠿⢭⠭⠍⠉⠉⠙⠛⠛⠛⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⡙⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡗⠦
⣤⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣽⣿⡿⠿⠧⣽⣿⣿⣿⣿⣿⢿⣿⣷⣶⣶⣦⣤⣤⣄⣽⣿⣷
⠛⢻⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠶⠶⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢳⡀⠀⠀⡇⠀⢸⠃⠀⢀⠞⠁⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠐⢄⠈⢢⠀⡀⢣⠰⠀⡇⠆⢏⠄⣠⠋⠀⣠⠖⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣀⠀⡑⢄⠑⠌⠀⠂⠀⡇⠘⠌⡔⡡⡢⠞⢁⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⡀⠁⡂⠀⠀⠀⠀⣀⠀⠁⠀⠪⡈⢀⠠⢚⡡⠀⣀⡤⠴⠂
⠀⠀⠀⠀⠀⠰⠶⠦⠤⢤⣀⣀⡉⠑⠀⠀⡐⢡⠊⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢉⣀⣠⣄⣠⣤⡴⠒⠦⠤⠴⠒⠶⠄
⠀⣀⣀⣀⣀⡤⠴⠤⠤⠤⢄⡀⠀⠄⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣁⠀⣀⠴⠉⠉
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⢄⣀⣀⠨⠤⠤⠤⠤⠔⠊⠉⠀⠀⣀⠜⢁⡤⠂⠀⠐⠀⢀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠁⠐⠄⠄⠀⣀⠐⠄⠀⠀⠵⢦⣒⠂⠀⠀⠐⠒⡴⠊⢉⠁⠂⠉⠀⠶⠀⣒⠍⣁⣀
⠀⠀⠒⠐⠒⠐⠀⠐⠂⠐⠒⠀⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⣰⠚⣡⣀⣄⣖⡰⠀⠀⠂⠁⠚⠀⠘⠲⠗⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⢤⣭⠴⠞⠟⠋⣁⣉⠁⣀⣈⠛⠛⠛⠋⠉⠉⠓⠒⠛⠛⠚⠓
⠀⠀⠀⠠⠶⠒⠒⠛⠒⠛⠓⢾⣍⠉⠁⠀⠩⠤⠬⢒⣐⠈⠁⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠰⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⠤⠒⠋⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⡰⠲⠖⠂⢰⣦⣤⡤⠶⠋⠁⠀⠀⠀⠀⠀⠀⢀⡤⠤⠴⠚⠛⠃⠤⠤⠤⣤⣀⣀⣀⣀⣀
⠀⠀⠀⢰⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡂
⠀⠀⠀⡜⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡅⠀⠙⠓⠶⠶⠶⢷⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠁
⠀⠀⠀⢢⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⠛⠳⢦⣀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠘⡆⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⢆⡀⠂⠈⠉⠛⠲⣤⠴⠋
⠀⠀⠀⠀⢁⡀⢀⣀⣘⡶⠶⠤⣬⣀⣀⠀⠀⠀⠻⣄⠀⠙⠒⠦⣄⣀⣤⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠚⠳⢦⣤⠴⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⠉⠉⠉⠉⠛⠓⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⠀⠀⢀⡤⢀⣀⣀⡀
⠀⠀⠀⠀⠠⠀⠒⠒⠒⠒⠚⠛⠋⠉⠉⠁⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⢣⠀⠈⠀⢱⡀
⠀⠀⠀⠠⠀⡀⡀⠀⣀⣀⣀⣄⣀⣀⣀⣠⣀⣀⣀⣠⡴⠊⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠉⠉⠀⠀⠀⢀⣴⣶⡤⢤⣤⣤⠴⠀⠀⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⣀⣤⠴⠞⢋⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠀⠀⠀⢀⣠⡴⠋⠉⡥⠀⠀⠀⠀⠀⠀⢀⣀⣀⠇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢺⡁⠀⢀⣠⣞⣁⣠⣤⠴⠶⠒⠋⠉⠀⠙⠒⠒⠀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣠⠴⠖⠖⠉⠁⠀⠀⠀⠀⠈⠐⠒⠂⠀⣀
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣯⣿⣇⢻⡅⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠹⢴⣀⣠⣄⢠⣄⣀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠀⠀⠀⠀⠀⡀⢀⣀⣴⣿⣿⢉⠈⠛⠦⠄⠀⠀⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠄⠀⢬⣠⡿⠾⠿⠟⠩⡟⠉⠀⣅⡀⠀⠀⠀⢀⢀⠄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡠⠛⠃⣄⠀⠉⠉⠛⠒⠒⢲⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡄⠀⢄⣀⡀⠀⠀⠘⣌⠀⠀⠙⠲⠶⣤⡤⠤⠂
⠀⠀⠻⠿⠿⠿⠿⠿⠟⠛⠛⠉⠀⠀⠛⠿⠿⠦⠴⠴⠛⠓⠒⠢⠤⠀⠁


⣴⣶⣶⣶⣶⣴⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶

⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⡀
⠀⠀⠀⠈⠚⠉⠳⠀⠀⠀⠀⢀⢪⠉⠋⠄⠀⠀⠀⢠⣵⠌⠦⢡⠸⠀⠀⠀⠀⠊⢠⠀⠀⠀⢠⡾⣿⢆
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠗⠁⠀⠀⠀⣸⡑⡆⢰⠀⠀⠀⠀⠀⡄⠀⠘⠀⠀⠀⢠⠁⡍⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡴⠀⠂⡄⢚⣧⣥⣬⡴⡏⠀⠠⠀⠁⠀⠀⠀⠀⠀⠂⡅⡇⣷
⠀⠀⠀⠀⠀⠀⡆⠀⠠⠀⢰⠀⡠⡀⠀⡃⠀⢠⠃⠀⠀⠉⠉⠀⠀⠀⠠⠀⡀⠀⠀⡀⠀⡀⠐⣡⣇⡇
⠀⠀⠀⠀⢱⠀⠇⠀⢠⠃⠀⠀⠑⠤⠒⠒⡌⠀⠀⠀⢀⣠⣶⣀⠀⠀⠀⠐⠄⠀⠀⠏⠀⠀⠀⠹⠿⠁
⠀⠀⠀⡀⠙⠀⡤⠠⠃⠀⠀⠀⠀⢂⠀⠀⠄⠀⠀⡜⢡⠊⠉⠑⠙⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⢀⠞⠁⠉⠢⠀⠀⠑⠌⣄⣀⡠⠖⠁⠀⠀⢠⠀⠀⣠
⠀⠀⠀⠀⠈⣦⠃⠀⢀⡠⢅⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠙⠀⠀⠐⡤⠊⠉⠈⠑⡄⠤⠚⠀⠀⠀⢠⠂⠢⠀⠀⠀⠀⠉⢀⠀⢀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠒⠆⠀⠬⠇⠀⠀⠀⠀⢆⠀⠀⣰
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⢄⡀⠀⡰⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠃
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⣩⣴⡀⠀⡀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠟⡀⠀⠀⠀⣀⠴⣈⡷⠖⡢⡄⠀⠀⠲⠃⠀⠈⡋⠀⠀⠀⠀⠐⠂⠀⢠⡀⠜⠄⡀
⠀⠀⠀⢠⡌⣷⡄⠃⠀⠀⡜⠁⠂⠀⠁⠀⠈⠸⠀⠀⠀⡠⠄⠀⠁⠀⠀⠀⠀⠀⠀⠠⡀⠺⡇⡼⣂⠂
⠀⠀⠀⠘⠹⠇⡅⠀⠀⠀⡇⠀⠀⠀⠈⠀⢀⢀⠃⢀⡠⣷⢣⠀⠄⠀⠀⠀⠀⠀⠀⠀⠋⠁⠛⠋⠉⠁
⠀⠀⠀⠀⢰⠀⠇⡀⠀⠀⠰⡀⠂⠀⠀⠀⡌⡜⠀⠃⠀⠈⢣⢠⠀⠀⠀⣀⠀⣀⠀⣀⠴⡶⡄⢀⢠⠒
⠀⠀⠀⠰⣎⠀⢸⡷⠀⠀⠀⠑⢄⢆⢐⣰⠜⠀⠀⢃⠀⠀⢸⠸⠀⠀⠸⣇⢀⠾⡖⣇⡄⡇⢇⣸⢨⣱⠄
⠀⠀⠀⠀⠚⣧⣸⠃⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠠⡀⢙⡆⠀⠀⠀⢀⠀⣄⣦⣀⠀⠄⢀⠀⡠⠄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠀⣈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⠀⠋⠀⠀⢹⡗⠂⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢠⡿⠃⣽⣿⠀⠀⠀⠀⠈⠁⠀⠀⠀⠛⡘⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡞⣉⢰⣾⢰⣸⣿⣟⣤⡀⠀⣄⠀⠀⠃⡀⠀⠀⢳⣾⣘⣧⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠘⠓⣪⡇⢸⣿⡿⣷⣿⣿⠸⠇⠀⠈⠀⠐⠌⠛⠖⢄⢼⠻⠳⠁⠈⢳⠄
⠀⠀⠀⠀⠀⠀⣼⡷⢶⡾⠛⠀⠀⢯⣟⢻⣿⣻⡷⠂⠀⠀⢢⠀⣆⢠⠐⠀⣽⡶⣶⣶⣶⡇⠂⢠⠀⠚⣆⡀
⠀⣠⡆⠀⢠⣼⠟⠃⢸⣧⡎⢀⣠⣼⡿⣯⣄⠀⡀⠀⠀⠀⠄⢷⡌⢸⡍⢀⠁⠸⡷⣿⣿⣏⠂⠋⠀⠀⠀⠛⠦
⠰⢾⣇⠀⠀⠀⠀⠀⣠⠉⢀⣨⣧⣿⣾⣿⣿⣇⠐⠲⠄⣠⡌⢆⠀⣯⣧⠀⣤⢀⣼⣿⣿⣿⣷⡄⢦
⠀⠛⣿⡄⠀⢀⣴⣾⠃⠀⠠⢙⠛⠉⠙⠻⢿⣏⣄⠀⠳⠇⠑⢿⡇⠀⠹⠖⢋⣼⡿⣻⣿⣾⣥⡏⢐⡀⠵⢀⡀
⠀⠀⠘⢿⠶⣿⣿⣿⡀⠠⠀⢌⠳⡀⠀⠀⠈⠛⠿⠰⡀⢤⠂⠀⠉⠀⠀⠀⠘⢁⣾⡿⠟⣹⡿⠓⠀⠁⠀⠀⠃
⠀⣀⣴⣿⣏⣴⣷⣿⡟⠀⠀⠀⢁⠐⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡀⣴⠇⠀⠂
⠘⠿⠋⠋⠿⣿⢒⡼⠋⠈⠁⠐⠈⠛⣄⡀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠲⢹⣟⣷⡇⡀
⠀⠀⠀⠀⠀⠻⠾⠿⠶⠤⢀⡀⠀⠀⠈⠻⠦⠀⠒⠤⠉⠢⠀⠀⢄⡈⠙⠛⠀⠂⠲⠄⠋⠹⠁⠈⠂⠂⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⣄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡿⠋⡝⠁⢹⠉⢻⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⡘⢠⠀⡎⠀⡈⡻⣧⡀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠻⠛⢷⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠇⠀⣇⠈⢰⡇⠀⠁⠱⢈⣷⠀⠀⠀⠀⠀⣤⡿⡼⠀⠀⡆⢸⠛⣶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⡇⠃⢸⡇⠀⢠⠰⡏⠸⡷⠁⠀⠀⠀⣾⠁⡇⢰⠀⢁⠀⡇⢹⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠀⣄⡇⠀⢸⡇⠀⠎⢠⡆⠀⣿⠀⠀⠀⠙⣿⠐⣧⠀⠀⢸⠰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⢸⡀⠉⡇⠀⢸⡇⠀⡆⢐⠇⠀⣿⠂⠀⠀⠀⣿⠀⢸⠀⠈⠸⠀⠘⠀⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⢶⢷⣄⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠃⠀⡇⠃⠀⠃⠀⠀⢀⡇⠀⢹⠀⠀⠀⠀⣿⠀⠻⡀⠀⣀⠇⠘⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⠃⢸⡆⠹⢳⡔⠀⠀⠀⠀⠀⢸⡇⢸⠇⠀⡇⠀⠀⣴⠇⠀⢸⠇⡔⣸⡇⠀⠀⠀⢹⡄⠀⡅⠠⢫⠀⣼⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢾⡁⢰⡄⣿⣄⡤⠀⠀⠠⣼⡇⠈⠀⢰⡇⠰⠀⡇⠀⠀⠀⠀⠀⢹⠂⠀⠀⠀⣸⡗⠀⡆⠀⢸⠀⣯⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⠀⡘⠁⢸⠇⡏⣿⠀⠀⠀⠀⢸⡇⡀⠀⠀⡇⠀⠀⡇⢀⠆⠀⠀⠀⣾⠀⠀⠀⠀⢸⣇⠀⡝⠀⢸⢀⡗⢠⡗
⠀⠀⠀⠀⠀⠀⢠⣯⠀⢹⠀⠇⡠⢟⠀⡇⣿⡀⠀⠀⠀⢸⡇⡇⠲⠀⡇⢀⠀⡀⠀⠀⠐⠀⢀⣿⠃⠀⠀⠦⣸⠏⢶⠀⠀⡜⢻⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⠀⡃⠀⠸⠀⡷⣿⠀⠀⠀⠀⢸⡇⡇⠀⠀⡇⠈⠀⡅⠀⠀⢰⡀⠀⣿⠀⠀⠀⣨⡟⠀⡜⢀⣴⠃⡬⢠⡿
⠀⠀⠀⠀⠀⠀⠰⣧⠀⢸⢢⠙⡆⢀⣄⡆⣿⠀⠀⠀⠀⢸⡇⡇⠀⠀⡇⠓⠀⠃⠀⠀⣸⠇⠐⣿⣀⣤⠾⠋⣠⡎⠀⣠⠃⡰⢁⣾⠁
⠀⠀⠀⠀⠀⠀⠰⣿⠀⢰⠀⢰⡀⠘⡇⠆⣿⠀⠀⠀⠈⢺⡇⡿⢆⠀⡇⠀⢠⡄⠇⠀⣧⠀⢠⡿⠭⠼⠒⢉⡳⠤⠚⣃⠟⣡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠸⣆⠘⣱⠀⠇⢰⢿⡄⠀⠀⠀⢸⡇⡇⠀⠀⡇⠂⢸⡇⠀⠀⣿⠀⢸⡟⠒⢻⠋⣉⡹⠶⢺⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢣⠀⢻⠀⢸⠘⡜⢷⡇⢠⣠⣸⡇⡇⠀⢠⣇⠀⢸⡇⡄⠀⣻⡄⢸⣏⣹⣯⣥⣴⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⣇⠘⢦⠈⢧⡈⠫⠿⠿⠿⢹⡇⡇⠀⠀⠏⠀⢘⠃⠈⠀⣿⠁⢸⡭⠍⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠘⡗⠈⠳⢄⣉⠚⠳⣆⡉⠨⣷⡇⠐⣄⠀⠀⠀⠀⠀⠀⣿⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠘⠷⣤⡤⢉⣉⣉⡩⠴⣲⣿⡇⠀⢹⠀⠀⠈⣦⠂⠀⡹⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣤⣄⣀⣀⣀⣴⠶⠛⠁⢸⣿⠀⢸⠀⡄⠸⠏⠀⠀⢷⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢿⢸⠀⢀⠀⠁⠰⠀⢠⠀⠼⢣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⠀⠸⢦⠀⡀⠙⠀⢸⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠐⠈⠀⣷⠀⢰⠸⡆⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠀⢸⠀⠀⠿⠀⠀⢸⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢘⠀⠀⢰⡄⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣀⠀⢸⠁⠀⢸⠀⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠀⠰⠈⠆⠘⠀⠘⢸⠀⣿
⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢸⡀⠀⠿⠀⠀⣈⠀⣿⠆
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⢾⣿⠀⢸⠃⠀⣟⠀⡔⡏⠀⡿⢶⣤⢤⣤⣀⡀
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠚⠋⣩⡀⠀⠁⢸⣿⠀⠈⠀⡄⣿⠀⠀⡇⢸⡇⠀⠰⠄⠠⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠠⠄⠀⠒⠂⠻⢿⡷⠂⡆⠀⡿⢀⣀⣷⠟⠀⠠⠒⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠉⠀⠙⢻⡷⠶⠿⠟⠋⠀⡀⠀⠙⠀⠰
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠀⢀⡀⠀⠈⠥⠤⠀⠀⠀⠤⠄
⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣶⣶⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⢀⣠⣶⣿⠿⠛⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠙⣿⣦⡀
⠀⠀⠀⣰⣿⣿⠟⠁⠀⠰⠛⠁⢺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠈⠻⣿⣦⣀
⠀⠀⢰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⣠⣴⠶⠶⢶⣄⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣾⠟⠁⠀⣀⣤⣿⠧⠀⠀⢹⡇⠀⠀⠀⢰⣆⠀⠀⠀⠉⠻⣷⣄
⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⢀⣴⡿⠛⠉⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⢿⣷
⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠁⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠘⣿⡇
⣼⡿⠀⠀⠀⠀⠀⣄⣀⣼⠂⠀⠀⠀⠀⠘⢷⣤⣀⣀⣀⣠⣴⣾⣿⠟⠛⢷⣤⣀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⣿⡇⠀⠀⠀⠀⠀⠈⠉⠻⣷⣄⠀⠀⠀⠀⠀⠉⠙⠋⠉⠉⠉⢿⣧⠀⠀⠀⠈⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⣼⣿⠀⠀⣤⡆⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡿⠛⠛⠛⠃⠀⠀⠀⠀⠀⢻⣿
⣿⡇⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⣿⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠘⣿⣧⡀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠈⠻⣿⣶⣄⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠈⠛⠿⠷⠶⢶⣾⣿⣿⡟⠀⠀⠀⠀⢀⣀⣀⡀⣀⣴⣾⡿⠟⠉
⠀⠀⠀⠈⠛⢿⣷⣤⡀⠀⠀⠙⠻⣶⣄⡀⠀⠀⠀⠀⠀⢀⣾⣿⢻⣿⡇⠀⢀⣤⡾⠛⠉⠙⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣤⣀⠀⠀⠙⠻⢷⣶⣤⡶⢾⣿⡿⠃⠘⣿⣇⠀⢿⣏⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣶⣤⣀⡀⠀⢀⣴⣿⠟⠁⠀⠀⠘⢿⣷⣬⣿⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢸⠀⠀⠀⠀⠱⡄⠀⠀⠀⣰⠋⠸⡇⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢿⡇⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⡎⠀⠀⠀⠀⠀⠘⢦⠀⣰⠃⠀⠀⡇⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⣀⣴⠞⠉⠀⠀⠈⢧⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢀⣴⣷⣤⣄⡀⢃⠀⠀⢀⣴⣿⡄⠀⠀⠙⠢⣤⣶⣿⣿⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠑⠤⣀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢀⣴⣿⣿⠋⢹⡇⠙⢮⣀⢠⣾⣿⡏⢳⠀⠀⠀⠀⠀⠙⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⠓
⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠴⠿⠿⠟⠁⠀⢸⡇⠀⠀⠈⠈⣿⣿⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⣙⣶⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣰⣿⡟⠀⠀⠸⡀⠀⢀⢄⣠⣶⡿⠋⠀⡇⠈⠳⢄⠀⠀⠀⣀⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⣠⣶⣿⠿⠋⠀⠀⠀⢧⠀⠀⠀⠙⢦⡀⠀⠈⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠃⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠚⠛⠉⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠙⠦⣀⡟
⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠃⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣦⡄⣀⡀⠀⠀⠀⠈
⣿⣿⣆⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠠⡈⠋⠑⠆⠀⢀⣠⣤⣤⣤⣀⣀⡀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡆⠀⠉⠉⠒⠤⢄⣀
⣿⣿⣿⣷⣄⠀⢝⡛⠿⠿⠿⠛⡹⠃⠀⠀⠀⠀⠀⠀⠀⠙⠀⣀⣴⣾⣿⣿⠃⠈⢇⠉⠻⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠠⢀
⣿⣿⣿⣿⣿⣷⡄⠉⠛⠲⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⠇⣀⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⣿⣿⣷⣤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠿⣦⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⡟⣧⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠀⠀⣀⡴⠞⡻⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡇⣿⠀⠀⠀⠀⠀⠀⣿⢸⣿⣇⠀⠀⠀⢠⡾⢋⡴⢋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⢻⣄⠀⠀⠀⠀⠀⣿⢸⡇⣿⡀⠀⠀⣾⠀⡾⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⡀⠀⠀⠀⠀⢹⣄⠻⣄⠉⠛⢶⣤⡀⢸⡇⢸⡇⠸⣇⠀⠀⢻⡄⣷⠈⣷⠀⠀⠀⠀⠀⢀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠲⣤⡀⠀⠈⠻⢦⣈⠙⠲⣄⠈⢿⣾⠁⢸⡇⠀⢿⡄⢀⣼⠃⣿⠀⢹⡆⠀⠀⣀⡴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡑⢤⡙⠳⢤⣀⠀⠙⣦⠀⢻⣆⣸⣿⣤⣼⣧⣤⣼⣿⡟⠉⣠⠟⢀⣾⣃⣤⠞⢩⡖⣿⠋
⢀⣤⣴⣦⣄⠀⠀⠀⠀⠀⠀⠈⠳⡄⠙⢦⡀⠉⠓⢶⣿⡶⠟⠋⠉⠁⠀⠀⠀⠀⠈⠉⠙⠻⢿⣤⣾⠟⠋⣠⡴⢋⡼⠃⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⣶
⠾⠛⠳⢦⡈⣷⣄⠀⠀⠀⠀⢀⣀⣹⣆⠀⠻⣦⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡾⠋⢠⡞⠁⠀⢀⣴⠞⢛⣉⣉⣉⠙⠛⣩⡿
⠀⠀⠀⠈⢷⡹⣍⡛⠿⠿⠛⠋⠉⠙⠻⢷⣴⠟⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣴⣿⣀⣀⣴⠟⣡⠞⢩⡴⠶⠶⠶⠟⠋
⠀⠀⠀⠀⠈⠻⣦⣉⡛⠚⠛⠛⠛⠲⣦⣼⠏⠀⣠⠶⠚⠋⠻⠿⢷⣴⣿⠀⠀⣾⣆⣤⣶⣟⣿⠓⠶⣦⣄⠹⣏⠉⠉⣁⡴⠃⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⣦⣼⠇⠀⣀⠁⢠⣶⣿⣿⣿⣿⣿⣿⠂⠀⢻⣿⣿⣿⣿⢿⣿⣦⠈⠙⣆⢻⡟⠛⠉⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠖⠛⣿⠀⠀⠻⠷⠿⢿⣷⣯⣻⡼⠟⢿⠀⠀⢸⡏⠿⣿⣧⣽⣽⠿⠳⠞⠃⠈⣿⠻⠿⢧⣤⣀⣀
⢀⣀⣠⣤⣶⣒⣚⣛⣉⣀⣀⣀⣀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⠀⢸⣷⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⣀⣈⣉⣙⣛⣿⣶⣦⣤⣤
⠈⠙⠛⠛⠻⠿⠧⣤⣤⣀⣀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⣏⠀⠀⠀⣿⢙⣦⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⣀⣀⣤⡤⠴⠶⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡽⠿⠛⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠶⠟⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⡶⠚⠛⠛⣷⡀⠀⠀⠀⠀⠀⣠⣀⣀⣠⣴⣤⣤⢶⣤⣄⣤⣄⠀⠀⠀⠀⠀⢠⣿⣄⠀⠻⢤⣤⣤⣤⣄⣀
⠀⠀⠀⣀⣀⣀⣠⣼⠇⣰⠏⣠⡤⠴⣾⠟⠳⣄⠀⠀⠀⠀⠟⠛⠛⢻⣿⠿⠷⠛⢻⡟⠛⠛⠀⠀⠀⠀⣠⣿⡀⠉⠳⠶⠤⠶⠶⠶⢬⣙⠳⣄
⠀⣴⢟⡭⠥⠤⡤⠴⠞⢁⡾⠃⠀⣰⠏⠀⢀⣿⢦⣄⠀⠀⠀⠀⠀⠀⠙⠷⣶⠶⠟⠁⠀⠀⠀⠀⣠⣾⡃⠈⢿⣦⣤⣠⣤⠴⠶⠾⠶⣌⢣⣹⡆
⢸⣷⡶⠟⠛⠓⠶⠶⠟⠋⠀⠀⣰⠏⠀⣴⠟⠁⣀⡽⠷⣤⣀⡀⣂⣤⡀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣅⠈⠙⢦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠙⢦⣽⣟⣳⣶
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⠞⢁⣴⢾⡟⠀⢠⡟⠉⣙⣿⡿⠿⣶⡶⠶⢾⠟⢻⡇⢀⡟⠉⠻⣦⣀⠙⢦⡘⢧⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣼⣥⠞⠋⠀⠘⣷⠀⣾⠀⣸⠏⠈⣿⠀⢹⠀⠀⣿⡄⠈⢧⠈⢷⣄⠀⠀⠙⠳⢦⣉⢮⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⢸⡇⢸⡆⣿⠀⠀⢹⡆⢸⠀⢰⡟⢳⣄⠈⠳⢤⡉⢷⡄⠀⠀⠀⠈⠛⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⢀⣼⠃⡼⢀⡿⠀⠀⠘⣧⢸⠀⣾⠁⠀⠉⠛⠷⣦⠹⡄⢿⠀⠀⠀⠀⠀⠀⠈⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⡜⢁⡾⠃⠀⠀⠀⢻⣼⢰⡇⠀⠀⠀⠀⠀⢸⡆⠇⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢰⣯⡾⠋⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⢸⣰⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠈⢿⣗⢸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠓⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠛⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠈⣧⠀⣀⣤⠤⣤⡀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣾⠟⠁⠀⠀⠀⣿⣆⠀⢀⡿⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⣿⣿⡀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠈⠻⢧⣸⡇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⡿⣧⠀⠀⠀⠀⠀⠀⠀⠈⢿⡃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠘⢧⡀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⡤⠞⣧⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣠⡤⠒⠉⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀⠀⠀⠀⠀⠀⠀⠈⣿⠆⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠙⠛⠶⠶⠶⢤⣴⡿⠶⠶⠞⠛⠛⠛⠻⣦⠀⣿
⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠈⣷⡍
⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠉⠉⠛⠛⠷⠶⠶⠶⠶⠶⠛⠛⠉⠉⠉⠙⢷⣸⡇
⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢻⣿⠇⠀⠀⠀⠀⢀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⢸⡟
⢹⣿⡆⠀⠀⠀⠀⠀⠀⢸⣷⣶⣾⣿⠿⠿⠷⠶⢦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢀⣿⠁
⠈⣿⣧⠀⠀⠀⠀⠀⢠⡾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠘⢿⣇⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣶⣶⣶⣶⣶⣶⣿⡿⠟⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠉⠉⠀⠀⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⢠⣠⣀⡄⠀⠀⢆
⠀⠀⠀⡍⡉⠉⠉⣙⣛⣛⣛⣋⢙⡟⠓⠤⡤⠊⠁⠀⠀⠀⠀⠀⠀⣠⣼⣿⠟⠻⣷⣤⡀⠸⣍⠀⠀⢠⣀⣀⠈⢀⠀⠐⠀⠠
⠀⠀⣀⣴⣦⣠⣴⣤⡈⠉⠏⠀⣾⠀⠰⠀⡇⠀⠀⡀⠀⠀⠀⠀⢠⣽⣿⠿⠷⠶⠾⠿⠿⠶⢦⡟⠿⠾⠿⠭⠧⠽⠿⢶⣶⠾
⠀⢠⣿⣿⣿⣿⣿⣿⣷⣶⣦⣾⣿⠀⢸⠀⡁⠀⠀⠀⠀⠀⢀⣰⣾⣧⣬⡶⣀⣤⠀⡀⢀⠀⠀⠈⡃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠠⠄⠀⠀⡀⠈⠁⣴⠯⣾⣭⣯⣉⡉⠛⠻⠶⠾⢿⣿⠷⠖⠁
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣧⠀⠀⠚⠃⠉⠉⠁⠀⣻⡟⠓⠛⠿⣷⣦⣤⣀⣀⢨
⠀⣿⣿⣿⣿⡿⢿⣿⠿⢿⣿⣿⡍⢅⡿⣀⠀⣠⢀⠀⠀⠀⠠⣿⣿⣶⣦⣄⣀⡉⠙⡋⠋⠁
⠀⢿⣿⣿⣇⠀⣰⣏⠐⢀⣽⣿⣶⠋⠀⠉⠑⠂⠤⠔⣂⡀⢠⣀⣙⠉⠹⠿⠟⠛⠛
⠀⠘⠻⠿⠽⠿⠭⠽⢿⣯⣭⣯⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠐⠒⠒⠤⢤⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠒⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠉⠀⠀⠀⡀⠀⠀⠀⠀⠐⠁⠀⠈⡂⢤⣶⣾⣿⣆⣤⣶⣶⣶⣶⣶⡾⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠟⠀⠀⠀⠀⠀⠲⣷⣶⣦⠤⡤⠄⣀⡀⠪⠀⢀⡿⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣿⣿⠷⣀⠀⠀⠀⠉⠑⢺⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠺⣶⣿⣿⣯⠀⠀⠀⠉⠉⠙⠻⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⡛⠉⠉⠹⠒⠀⢄⡀⠀⢼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠈⠁⠀⢰⣆⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢨⣍⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣷⣶⣴⣴⣾⣿⣿⣶⣶⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⠁⠦⠠⠤⠤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠰⠷⠿⠛⠋⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⣴⢖⠒⠂⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⣳⣼⡧⠀⠀⠀⠤⠤⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠱⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠉⠟⠛⠛⠉⠙⡃⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁⠐⡆⢰⢃⠀⣰⣾⡆⠀⠀⢀⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣹⣿⡟⠂⠀⠀⠤⠶⠴⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠀⠀⠠⠆⠸⠿⠿⣿⣿⣦⡀⠀⠁⠢⡀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠛⢇⡀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢠⢹⠁⠀⠴⢾⡇⠀⠀⠀⣿⣿⣿⣿⡀⠀⡄⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠉⠓⣺⣿⡏
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠑⠠⠄⡀⣠⣿⠸⡀⠀⠀⠎⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⢸⣦⣤⣤⣿⣿⣤⡀⠀⠀⠀⠀⠀⣸⣧⣤⣤⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⠀⠀⠀⠀⠀⢀⡌⠉⠙⢳⠓⠀⠊⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣗⡫⠟⠁⠀⠈⠉⠙⠛⠻⠾⠷⠶⠿⠿⠟⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠉⠓⠶⠲⠤⢼⣿⣶⠦⠂⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣠⣧⣤⣄⣤⣜⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⣿⣿⣶⣶⣶⣶⣤⣀⠀⠀⠀⢀⣠⣤⠶⠛⠉⠀⠀⠀⠀⠉⠉⠛⠳⢦⣤⣀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⡀⣀⡀
⠀⣿⣿⣿⣿⣿⠟⢻⣿⣧⣴⡾⠏⠉⠀⠀⠀⠀⡀⣀⣀⣀⣀⣠⣄⡀⠀⠉⡻⠿⠷⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉
⠀⣯⠽⣛⡿⠁⣰⣿⠏⠀⠀⠀⠀⡀⠀⡀⣀⣤⣜⣿⣿⣿⣿⡿⢿⣿⣦⣤⣓⣶⡶⣄
⠀⢰⣺⣽⠃⢠⣿⣿⣄⠠⡀⠰⣤⢸⣺⣿⣽⣽⡻⣯⣿⣿⣿⣤⣤⣷⠉⠉⠉⠁⠀⠀⠀⢀⡰
⠀⢾⣿⣿⠀⢸⣿⣿⣾⣴⣴⣷⣧⣯⣿⣿⣿⣯⣿⣿⣿⣿⣯⣰⣿⣿⣿⣶⣦⣤⣤⣄⡶⠞⠁
⠀⣺⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣠⣿⠻⢿⣿⣷⣀⠀⠀⠀⠀⢰
⠀⣿⣿⡇⠂⠀⣿⣿⣿⣿⣯⣋⢻⣮⣿⣿⣾⣿⣿⣿⣿⣯⡻⣤⠀⠙⠛⠿⢷⣶⠶⠖⠁
⠨⣿⣿⣇⠀⠀⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⣰⠂
⠀⣿⣿⣿⠀⣠⡈⡿⢧⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠉⠙⠛⠿⠿⠟⠃
⠀⣿⣿⣿⣆⡺⠿⢇⡴⠃⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣶⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⢠⣾⣿⣿⡆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⠀⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠀⠹⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⣀⣀⡀⢀⡀⣀⣀⣀⢀⣀⠀⣀⣀⣀⣀⣀⣀⠀⣀⡀⠀⢀⢀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣼⣿⣼⣟⠋⣹⣾⣿⠹⢏⣿⣿⣿⡄⣿⡏⠉⣿⠉⠉⢸⣿⣧⠀⣿⢸⣿⣆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠈⠁⢿⡟⢿⣶⣿⠏⠻⣿⣿⠟⢸⣿⢻⣿⡇⠀⢿⠄⠀⣿⠋⢻⡇⣿⠸⣿⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⠀⣤⣤⣤⣀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣿⠉⠉⠉⠀⣿⣟⠙⣻⣷⠀⠀⢴⣿⣿⣿⡅⠀⠀⢰⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⣼⣿⣧⣿⣷⠀⠀⢸⣿⡄⠀⢘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⣿⡏⠹⣿⣧⠀⢠⣿⡿⠁⠹⣿⣧⠀⢸⣿⡇⠀⢸⣿⣦⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠈⠉⠁⠈⠉⠁⠀⠀⠉⠉⠀⠈⠉⠁⠀⠀⠉⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠆
⠀⠀⠀⠲⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣆⡼⢠⠆
⠀⢀⠀⠢⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠠⠇⠀⠉⠁
⠀⠀⠀⢆⠆⠀⠀⣸⠆⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⡅⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⢠⠀⠰⡄⠀⢀⢀⡎
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠹⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⡆⠀⡇⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢘⢣⡄⢠⡀⠀⠘⠆
⠀⠀⠀⢤⣠⡄⠧⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⣰⠀⠀⠀⠀⡿⠛⡷⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠈⠻⡄⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠀⠀⢀⠀⠀⠀⠀⢦⠀⠀⠈⠁⠀⠀⢀⣠⠁⠀⢸⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⡇⢠⡆⠀⣰⠧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⣆⠀⠀⠀⡈⣤⣀⣰⠇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠱⣌⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠈⠙⠓⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠰⠀⠀⠀⠀⠀⠀⠁⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠿⠻⠛⣋⣉⣻⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⠿⣿⢯⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠋⠉⣽⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠛⠿⠷⣦⠀⠀⠰⣶⣿⠛⠛⠓⠆⠀⠀⠹⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠤⠖⢻⠿⣿⠂⠀⠀⢞⠡⢤⣤⣄⣀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠙⢹⠟⣡⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡴⠀⠀⠈⠁⠉⠒⠁⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠰⢦⣤⠤⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢙⠓⠒⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠒⠀⢀⡘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠴⠛⠛⠁⠀⠀⠀⢳⠄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠴⠚⠋⡀⠙⢶⠀⢠⠀⠀⠀⠀⠀⠀⠀⠐⠶⢤⣤⣀⣀⣠⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢧
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠠⠞⠋⠉⠙⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠐⢿⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⢸⣴⠏⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⡟⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠋⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡏⠁⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⣿⣿⡇⠘⠏⢿⣷⡀⠀⠀⣀⣤⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⣿⡇⠀⠀⣠⣿⠃⠰⣿⠟⠋⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡴⠃⠙⣦⢠⡿⣦⡀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡞⠁⠀⠀⢸⣿⠁⠸⣷⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠠⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠙⣧⣠⠞⠁
⠀⠀⠀⠀⠀⣠⡄⠀⠀⠈⠳⣤⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⣰⠋⠹⣆⠀⠀⠀⠀⠙⢿⣿⡆
⠀⠀⠀⠀⡏⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠖⠉⣭⠿⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣎⣀⣼⡠⠃⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣛⠯⣭⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⢆⠀⠀⠄⠐⢄⠹⡂⢠⠈⢖⣠⠂⢠⠀⡀⣀⡀⠤⢀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠑⠦⣸⡄⢄⠻⣇⡜⢣⣸⢁⢂⣾⠟⠋⠁⣀⠔⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⡀⠰⠈⢻⣬⣶⣿⠞⠳⣿⣿⡏⠀⠰⢀⣼⡉⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢚⠛⠻⣦⣴⣯⠝⠉⠀⠀⠀⠀⠉⢋⣷⡶⣋⠥⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢈⣑⠶⠔⣿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣥⠶⠿⠒⠉⠉⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠠⠐⠉⠀⠔⠚⢅⣻⣿⠀⠙⠒⠂⠀⠀⠑⠚⠋⢈⣿⢌⡵⠒⢀⣤⠄⠂⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⣀⣠⠔⢲⡿⢋⡹⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣒⠻⢍⠑⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠖⠁⢂⣡⡞⠟⢭⣷⣤⣀⣀⣀⣠⣴⣯⠟⠉⠻⣦⡅⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡴⢋⠘⠂⣡⣿⣿⡟⠷⢻⡟⢯⠻⣅⡘⠀⠘⣷⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⠄⠉⣀⣠⣴⢾⠏⠌⣸⡏⢶⠋⣿⣄⠣⠹⡏⢦⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠉⠁⠀⠀⠐⠀⡔⠁⢧⠀⠀⠸⣄⠑⠄⢩⠀⠌⠢⠑⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⡶⢯⠀⠀⠀⠁⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢐⠙⠃⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⢾⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠙⠶⠄


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⡟⡛⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣶⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠋⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡺⠋⠋⠛⢂⠀⠀⠀⠀⠀⢠⢡⢋⠉⠇⢡⠣
⠀⠀⠀⠀⠀⠀⠀⢸⢠⠐⢸⠀⡇⠁⡄⠀⠀⠀⠀⠀⠀⠐⢀⠁⡆⠀⢰⠘⠀⠀⠀⠀⠀⠆⡆⠘⢰⠸⠈⡀
⠀⠀⠀⠀⠀⠀⠀⡆⠈⢠⢸⠀⡇⢀⢠⠀⠀⠀⠀⠀⠀⡇⠸⠀⠁⠀⠈⠀⡆⠀⠀⠀⠘⢠⠀⠀⠘⠀⡀⠇⠆
⠀⠀⠀⠀⠀⠀⢠⠀⡇⠈⢸⠀⡇⠸⢸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⠀⡆⠀⠀⠀⠀⢸⠀⠀⢀⠀⡇⢠⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠄⡄⢀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⠀⡇⠘⠀⡇⢸⢸⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠘⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⡀⡇⢸⠀⠃⠸⠀⠇⠀⠀⠀⠀⢸⠀⠇⡁⢠⠀⢸⢸⠀⠀⠀⠀⠰⠹⠀⢁⢩⠣
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠃⡇⢸⢠⠀⠀⠀⠀⠀⠀⠀⠂⠘⢠⠀⠃⠸⠀⡄⠘⠀⠀⠀⠀⡇⠀⢨⢸⠈⠀⡇
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡇⢰⢸⠀⡇⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢀⠀⡄⠀⠀⠀⡄⡆⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡇⢸⠀⠀⠇⠀⠀⠀⠀⠀⡀⢁⠈⡀⠆⠸⠀⢡⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠃⡄⠈⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⡇⠈⠀⠀⢠⠀⠀⠀⠀⠀⠆⠸⠀⢁⢸⠀⠇⠠⠀⠀⢸⢰⠀⡇⠀⠀⢸⠀⢠⠀⡄⠀⠀⢠⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠃⠀⠀⠃⢸⠀⠀⠀⠀⠀⠰⡀⢣⠸⡀⠆⠸⡀⠣⡀⠀⠀⠀⠃⠀⡇⠀⡄⠈⠀⡇⠀⠀⠸⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⡆⢠⠀⢰⠀⡄⢸⠀⠀⠀⠀⠀⠀⠱⡀⢆⠱⡈⢆⠑⢄⡈⢣⠀⠀⢠⠀⠁⠀⡇⢠⠀⡆⠀⠀⡄⢀⠀⠸⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠂⢀⠀⡇⢸⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠱⡈⢆⠱⣄⠑⠠⠌⣹⠀⠀⢸⠀⠀⠀⡇⢸⠀⡇⠀⠀⠇⢸⠀⡆⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠑⢌⡑⠢⠄⣸⠀⡆⠀⠀⡇⠀⡇⢈⠀⠀⠀⢠⠀⡆⢠⠀⡆⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠘⠀⢸⠀⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢈⡐⠂⠰⠀⡇⠀⠀⡇⠀⠀⢸⢀⠀⢠⠃⡘⠀⡆⠀⠁⠘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⢸⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡆⡇⢠⠀⠃⠀⠂⢸⢀⡦⠁⡰⠃⡘⠀⡐⠀⠇⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸⠀⠀⠀⠀⡸⠊⢀⠎⢀⠜⠀⡰⠁⡜⢠⠃
⠀⠀⠀⠀⠀⠀⡀⢠⠀⢠⠀⡇⠘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⢸⠀⠀⢀⠖⢁⠔⠁⡠⠋⠀⡔⠁⡸⠀⠎
⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢁⠊⡔⠁⡠⠊⠀⢠⠊⠀⡰⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⠾⡘⢀⠎⠀⢀⠔⠁⡠⠊⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡇⠀⡀⣧⢂⡠⠒⢁⠠⠊⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡇⠀⠀⡝⠻⡄⣉⠠⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⡇⢰⠀⡇⠀⠇
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⢸⠀⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⡄⢸⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠃⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣗⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠞⢉⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠇⠀⢀⡾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣽⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣿⣿⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢙⡁⣴⢁⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠏⣾⠁⠆⠀⠠⠞⢦⡀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠔⢡⣿⡶⠀⠀⠀⠛⣴⣆⠀⠀⠀⣠⢀⡞⠀⠀⢹⣷⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣷⡿⢃⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢉⣴⣿⠆⠀⠀⠀⠉⠀⠀⣼⢧⣎⣰⣆⠀⠀⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢫⣵⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡿⢠⡶⠀⠀⠀⠀⣴⠏⣸⣧⣽⣿⣆⣀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣤⡎⠉⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⢾⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣯⣿⣿⠗⣼⠇
⠀⠀⠀⠀⠀⣰⡗⠀⠀⠀⠠⠼⣿⣧⡀⠀⠀⠀⠀⠀⠘⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⠘⣿⡿⢟⣵⠟⠁
⠀⠀⠀⠀⠀⡟⢠⣤⣤⠀⠀⠀⠀⠹⢯⠄⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⢠⣽⣽⣇⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠁⠋⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⣠⡀⠀⠀⠐
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⣟⡄⠀⠀⠀⠀⠀⠀⠠⠞⠀⢷⠛⢠⣾⡿⠃⠀⠀⠀⠀⠀⠈⠹⢿⠟⣵⡀⠘⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⣴⠞⣹⠿⠏⠀⠀⠄⠀⠀⠐⠁⠀⠀⠠⢳⢿⣥⢟⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⢴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⠁⠀⠀⠀⠀⠁⠀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⠉⠟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇
⠀⠀⠀⠀⠀⠀⠀⣠⠶⣹⡾⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⢿⣷⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠱⣄
⠀⠀⠀⠀⠀⠀⠠⠟⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣎⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠣⠤⠄⠀⠀⠀⠀⣠⠀⠀⠘⣩⡟⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠁⠀⠀⣸⡿⠂⠄⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢘⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⠄⢀⠀⠀⠀⠀⠀⣠⣴⠾⡁⠀⠀⠀⠀⠠⠟⠃⠀⠀⠈⠀⠙⢃⡠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠛⠃⠀⠀⠀⠀⠟⣥⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢋⡵⠎⣠⠖⠀⠀⠳⠄⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖⠀⠀⠀⠀⠀⣡⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠂⠀⠀⠀⠉⠳⠠⣄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⠵⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢩⡀⠀⠀⠀⠀⠀⠀⠀⠛⠇⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡀⠀⠀⠀⠛⠐⠀⠐⠈⠁⠀⠀⠀⠀⠘⠃⠞⠁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠋⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠿⠛⠛⠋⠙⠻⡿⡿⠷⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠁⠀⠀⣀⣤⣤⣤⣤⡄⢻⣶⣧⣄⠀⠀⠉⠉⠒
⡛⠂⠈⠁⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠉⠙⠛⠿⣷⣦⣄⠀⠀⠀⠀⢦⣤⣤⣤⣀⡀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠗⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠳⣤⣀⡀⠀⠀⠁⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣄⠀⠀⠀⣴⣿⡿⠀⢀⣴⣿⣆⠀⠀⠀⠀⠈⡀⠀⠀⠈⠻⣿⣷⡄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠛⠛⣿⣿⣿⣷⣶⣾⣿⣷⣤⣴⣿⣿⡏⠁⠁⡀⠀⠀⡹⠛⠿⠒⠂⠀⠙⢿⣿⣷⠂⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠀⠠⣧⠠⠤⠐⢦⠲⠤⠰⠿⠿⣿⠙⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠙⠳⡖⠒⢸⣿⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣼⣦⣀⠀⢸⠇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠻⠿⠿⠟⠛⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⣠⣾⣿⣧⠀⠈⢳⣤⣾⡟⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣾⡿⡀⠀⠉⠑⣿⣿⣿⣿⠀⠀⠀⢻⣷⣆⡀⠲⡄⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠉⠀⠹⣿⣯⣿⣷⠀⠀⠘⡛⣿⣿⣿⠀⠀⠀⣸⣿⠋⠄⢨⣽⣶⣬⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠈⣻⣿⣿⣽⡇⢠⢮⢠⣾⣿⣿⣤⣶⣿⣿⣿⣴⣲⡇⠈⣉⣩⣠⣤⡑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠆⠀⠀⢿⣿⣿⣿⣃⡌⠀⢸⣿⣿⢿⣷⡿⢿⣿⣿⣿⡧⢁⣬⣿⠿⡛⠻⢷⠟⠷⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⢸⣿⣿⡵⢋⡀⠀⢾⣿⣩⣿⠟⠃⠹⠓⣹⡿⠛⢿⣿⢟⣿⠿⠂⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡧⠀⠀⢸⣿⣯⠑⠇⡁⢀⣻⡿⢋⣿⠀⠀⠀⠂⠈⣿⡋⠈⢫⡸⣿⠀⠀⠀⠀⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⢸⣿⠯⠀⠂⢀⠴⣿⡅⠿⠉⠁⠀⠀⠀⠀⣿⠗⠀⠀⠉⠛⠺⠺⠖⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡗⠀⠆⠹⢿⣦⠀⠀⠡⠀⣿⣷⣶⣿⣿⣗⣒⠦⢄⣿⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠙⢆⠀⠀⠀⢹⣿⣿⠋⠀⠀⠈⠙⣦⠈⡑⠲⠮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣿⣿⣷⡴⠶⠶⠞⢷⣄⣀⣸⡿⣏⠀⠀⠀⠀⢠⡟⡤⠁⠀⣠⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠿⠛⠓⠶⢤⣤⣌⢹⣿⣷⣌⠓⠒⠶⠶⠛⠁⠀⠀⡰⠋⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⣿⣿⣿⣏⠀⠀⠀⠀⠠⡙⠂⠀⠙⢿⡛⠿⣶⣦⣤⣤⣤⣄⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠟⠉⣰⣿⡿⠋⡝⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣁⠈⠁⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣇⣀⣼⣟⠋⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⣀⢄⠀⣠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠧⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠻⢟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣤⣤⣄⣤⣀⣤⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣾⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠟⣿⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⡻⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠶⠦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⢇⡄⠀⣰⠆⠀⠀⣺⠁⠀⡀⠉⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠹⠿⠦⢤⣴⡞⠀⠙⠉⠁⠀⠀⢸⠃⡆⠘⣷⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⢸⣇⣧⠐⡏⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⢢⠀⠀⠀⠠⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠹⣿⣿⣶⣧⡟
⠐⠀⠀⠀⠀⠀⠀⣿⣁⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠛⠛⠋⠀⠀⠀⠘⠗
⣸⠇⠠⠄⠀⠁⢀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠰⠟⠻⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠚⠇⠀⢰⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡦
⠁⠀⣧⠀⠋⠀⡀⢠⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⣀⠀⠦⠄⠀⠀⠀⠁⠀⠀⠀⢀⠀⡀⠀⠀⠈⠉
⠀⠀⢸⡖⠂⠈⠀⡼⠀⠀⠀⠀⠀⠀⠰⢼⣦⠀⢸⡇⠀⠀⢀⠇⠀⠀⠀⢨⠁⢢⢠⠀⠀⠀⢸⣀⠀⠀⠸⠶⠃
⠀⠀⠀⠀⠰⢪⡿⠃⠀⠀⠀⠀⠀⠀⠀⠐⣷⣶⢾⣧⣼⠷⠃⠀⠀⠀⠀⠀⠀⠘⠊⠀⠀⡤⠚⠋⠀⠀⡆⢠
⢠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠀⠘⣧⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻
⢛⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡅⠀⠀⢸⠀⠀⠀⠈⠋⢀⠠⠀⠀⠰⣄⡴⠀⠓⠂⢀⣀⣀⠰⠁
⠸⡤⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⡀⠀⣆⣸⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢰⠀⡀⠛⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⢠⡄
⡀⠀⣀⡀⠀⠀⠀⢠⠂⠀⠀⠀⣠⣀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⣠⡼⠃⠀⠀⠀⠀⠀⠀⠈⢢
⠧⢀⠈⣇⠀⠘⠠⡇⠀⠀⠀⠀⠛⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠠⡀⠀⠀⠀⠀⠀⡔⠀⠄⠀⠀⠀⠀⢦
⢷⣸⠀⠹⠃⠀⠀⠁⠀⠀⢠⢏⡆⢰⡞⡇⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠈⢀⠠⡁⠀⠀⠀⠀⠀⡀⠀⢸⠇
⢀⠹⠀⢠⡄⠀⠀⠀⠀⠀⣹⣼⡆⣾⢰⠇⠀⠀⠀⠀⠀⠀⠀⠘⣇⣧⢡⠀⣇⡿⠀⠀⠀⠀⠀⠀⣘⠟⠀⢀⠠⠀⠀⠙⣿
⠾⠃⠀⡾⠛⠀⠀⠀⠀⠀⠻⣿⣷⣷⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⠳⠷⠋⠁⠀⠀⠀⠀⠀⢀⡀⠈⠀⣀⠀⠂⠀⠀⢻
⡅⠀⢸⠃⠠⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⠄⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠰⠀⠀⢠⡜⠀⠀⠈
⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⣀⣥⠞
⠐⠀⠚⠛⠁⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣼⠀⠀⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⠿⠿⠿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⢷⣄⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⡇⠀⠙⢿⡆⠙⣦⣴⣄⡀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣴⣿⣿⠙⣧⡀⣠⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠛⠀⠈⠁⠈⠻⣦⣄⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⣠⣶⣿⣿⣿⡏⠀⠈⣿⣿⣿⡿⢫⣾⣿⣿⡇⠀⠀⠀⣤⡀⠀⠀⠱⣦⡀⠙⢷⣦⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣇⣠⣾⣿⣿⣿⣿⠟⠀⠀⠀⣿⣿⠟⣱⣿⢏⣿⣿⣷⠀⣄⠀⢹⣿⣦⡀⠀⠘⣿⣦⣄⠈⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⢋⣴⡇⠀⢸⡿⢋⣼⣿⠋⣼⣿⣿⣿⠀⢻⣦⡘⣿⣿⣿⣦⡀⠈⢿⣿⣷⣄⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⣡⣾⣿⠃⠀⠛⢠⣾⣿⠃⣼⣿⣿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣿⣿⣦⣌⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⣰⣿⡿⠁⣸⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡏⢀⣼⣿⡟⠀⣰⣿⣿⣿⣿⠏⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⣾⣿⠟⠀⢰⣿⣿⣿⣿⡏⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣯⣤⣴⣿⣿⣿⣿⡟⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶⣶⢶⠶⢤⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣯⣭⣭⣤⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠁⠉⠉⠉⠉⠀⠀⣼⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠿⠶⣦⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠖⣤⣼⡆⠀⠸⣭⣄⣀⠀⠈⠀⠀⠀⢹⣿⠟⠻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⡰⠃⠀⠀⠉⠁⠿⠛⠖⠀⠀⠈⠿⢡⠌⢇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠰⠀⠤⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠐⡖⠦⡄⢀⣀⣀⡘⠆⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢫⡁⠓⣤⠒⣆⠀⠀⠀⠀⠀⠀⠀⠘⣷⡛⠷⠶⣦⣴⠟⠁⠀⠀⠀⠀⠀⣰⠟⣿⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠰⠷⢾⣌⠁⢸⡄⠘⢦⡀⠀⠀⠀⠀⠀⠀⢹⠙⠒⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠀⡿⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠶⠶⢤⡀⠈⢃⣸⣿⠀⢀⡅⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⡼⠁⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⣤⣀⣤⠀⢿⡄⠸⠟⠃⠄⢸⠃⠀⠀⠀⢀⣠⣴⡞⢻⣶⣦⣤⣤⣶⣾⠃⠀⠀⠀⢀⡾⠁⠀⢀⡀⠙⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠘⣿⣿⣧⡈⠻⠟⠀⠀⢀⣿⠖⠛⠛⠛⠉⠁⢸⡇⠸⣿⣿⣿⡯⠉⠀⠀⠀⢀⡴⠋⠀⠀⢀⡞⠀⠀⠀⠀⠉⠓⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠛⣻⡿⠃⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠈⣧⠀⠙⢷⣾⣧⡀⢀⣠⠞⠉⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣶⠗⠈⣀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠹⣿⡴⢿⡀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠡⢿⣿⠀⠀⠍⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠈⣇⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠈⠑⠐⠤⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢻⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠈⠑⢀⠔⠀⠀⢀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠘⠇⠀⠈⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠉⠑⠀⠈⠀⠀⠀⠀⠀⠙⠀⠀⡠⠞
⠀⠀⠀⠀⠀⠀⠠⠀⠘⠆⠀⠀⠈⠇⠀⠀⠿⢸⡇⠀⠀⣄⠀⣷⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡀⠀⠀⠀⠀⠞⠀⡇⠀⠀⡁⢠⡏⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠤⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢐⡄⠑⣤⠀⢀⠀⢰⠀⠀⠀⠀⡇⢸⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇⠀⠛⠀⠈⠀⢠⠀⠀⠀⠀⠁⢸⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⣀⠀⠀⠀⠤
⠀⠀⠀⠀⠂⠀⣿⠀⠀⢸⠇⢠⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⢤⡀⠀⠀⠀⠀⡄⠀⡇⠀⠠⠀⡇⠈⢺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢠⠀⠃⠀⣈⠀⠇⠀⠀⠀⠀⠀⠀⢀⠀⣠⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠸⣇⠀⠀⢿⠀⠄⠀⠀⠀⠀⣆⠀⠀⣠⡟⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠳⣄⡈⠃⠤⣈⡲⢀⡀⠀⠀⠀⠁⠀⠾⢋⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⢀⣌⣀⣀⣀⡀
⠀⠀⠀⠐⠃⠘⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⢿⣾⠿⠿⠿⠿⠟⠿⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⡿⠾⠿⠿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣄⣲⠄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠈⠋⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⢀⡀⠊⠀⠈⠙⢦⡀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠰⠀⠀⠀⠀⣾⣿⡏⣴⡆⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠈⡧⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⡶
⠀⠀⠀⠀⠀⠀⠀⡄⠀⣾⡿⢀⠀⠀⠀⠀⣿⡟⢡⣿⡇⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⢸⠀⠀⠀⠀⣿⠀⢹⣿⠀⠀⢿⣿⠟⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⢀⠀⣿⡄⠀⢹⣿⡇⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⣿⣿⠀⡄⣠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢁⠀⠀⢀⠀⠀⠀⠀⠀⠠
⠆⠀⠀⣿⣧⣿⣇⠀⠈⣿⡇⠀⠘⠀⠀⠀⣿⡀⠸⣿⠀⠈⣿⠇⢰⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠈⠀⠀⠀⠀⠀⡃
⡄⠀⠀⢻⣿⢿⣿⠀⢁⢻⣧⠀⢸⠀⠀⢰⡟⠃⢸⡟⠀⢰⡟⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⡀⠰⠀⠀⠀⠀⢰⡇
⠰⡆⠁⣸⣿⣆⠙⣧⠘⠀⣿⡀⢸⠄⠀⢸⡇⠀⣿⠃⠀⢾⠁⣴⣿⠟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁
⠀⠸⣄⠛⢿⣿⣧⡈⢣⠀⠸⢧⠈⠀⠆⢸⡇⢰⡟⠀⠀⠀⣴⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠈⠂⡀⠙⢿⣦⠠⠖⠀⠈⠀⠠⠀⠀⠁⡞⠃⢀⣠⡾⠋⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡀⠀⠈⢦⣀⠙⢷⣄⡀⠈⢤⠀⠀⠰⠂⠀⠀⠛⠉⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠁⠐⠒⠈⠉⠱⠦⣬⠻⠷⡄⠡⠀⠰⠀⢀⣤⠶⠚⠁⠈⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣶⣿⣶⣤⣤⣤⣼⣿⣿⣧⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣶⣾⣿⣀⢀⣀⣨⣿⣿⣷⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⠛⠻⠿⠿⠿⠿⠿⠿⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠘⠓⠘⠛⠈⠉⠃⠐⠀⠈⠉⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠠⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢾⣄⣱⠖⢀⠄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⣄⡠⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡖⢠⠚⠑⠶⠚⣯⣹⡖⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠐⢢⠜⠉⠀⠈⠀⠀⠠⠀⠀⠈⠁⠐⠠
⠀⠀⠀⠀⠀⢀⠘⠁⠀⠀⡐⠈⠀⠁⠠⠎⣿⠙⣏⡉⠳⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠈⠡⡀
⠀⠀⠀⠀⠰⠃⠀⠈⠀⠰⠁⠈⠁⠀⣉⠰⡏⠘⠃⠈⠱⠞⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠆
⠀⠀⠀⠰⠂⢰⡁⠀⠸⠀⠄⠀⠀⠈⠁⠀⠀⠀⠀⠸⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⢠⡆⠀⡉⠛⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠄⠀⣷⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠃⠀⢧⡤⠐⠇⠐⠁⠀⠀⠀⡄⠀⢀⠀⠀⠀⠠⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀
⠀⠀⢠⡄⠀⠀⠀⠀⢠⠀⠆⠀⢢⠀⠁⠀⠈⡀⠐⠂⠀⠁⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⢁⠀⠰⠏⠀⠈⢃⠀⠀⠀⢀⠀⠀⠀⠱⠀⠀⠀⠀⠉⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠦
⠀⠀⠀⠘⠃⠀⠰⡶⠀⠈⠡⣄⠀⠀⠃⠀⠀⠰⠀⠶⠘⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⢀⠚⠃⠀⠀⡇⠀⠀⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⡄⠈⠂⠀⠀⠘⢄⡀⠀⡄⠀⠰⢀⡀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠰⡷⠀⠀⠀⠀⠀⠀⡇⠀⡀⠈⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠻⠀⠈⠻⠀⠀⢀⣁⣈⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠤⣤⣴⣾⣿⣷⣶⣷⣶⣿⣿⠿⡿⠟⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣦⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠀⠀⠉⠉⠁

⠀⠀⠀⢠⣄⣰⣆⣠⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣆⡠⠀⠀⠀⠀⠀⡄⣐⣰⣂⡀
⠀⠀⠀⣤⣿⣿⣿⣿⣤⠀⠀⠀⠠⣿⣿⣿⡧⠀⠀⠀⠀⠬⣿⣿⣿⡎⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⣽⣿⣿⣿⡧⠀⠀⠀⠐⣾⣿⣿⣿⡒
⠀⠀⠀⢠⠿⢿⡿⢯⠀⠀⠀⠀⠀⠹⠿⠟⠃⠀⠀⠀⠀⠈⠽⠿⠿⠓⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠐⢚⠿⡿⢟⠓⠀⠀⠀⠈⠝⢿⠿⠛⠌
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⣄⣂⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⠀⠀⠀⠀⡄⣂⡐⢀⠄
⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⢾⣿⣿⣿⣿⡇⠀⠀⢀⢹⣿⣿⣿⡧⠄⠀⠀⠀⠠⣿⣿⡯⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡆⠀⠀⢒⢾⣿⣿⢀⡒
⠀⠀⠀⠻⣿⣿⣿⣿⠏⠀⠀⠀⠹⣿⣿⣿⡿⠁⠀⠀⠀⠚⡻⢿⠿⡓⠂⠀⠀⠀⠀⠋⠛⠑⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠁⠀⠀⠁⠆⡉⢁⠲⡌⠁
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⢤⣰⣦⡆⡀⠀⠀⠀⡀⢦⣴⣤⠆⡀⠀⠀⠀⡀⢆⣶⣠⢄⠀⠀⠀⢀⡄⣴⣇⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣂⣅⡂⠄
⠀⠀⠀⣽⣿⣿⣿⣿⡥⠀⠀⠠⣽⣿⣿⣿⣿⠥⠀⠀⢠⢽⣿⣿⣿⡯⡄⠀⠀⣀⣻⠁⠀⠙⡟⠆⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠅⣾⣿⣿⣿⡎⠄
⠀⠀⠀⠚⣻⢿⣿⡟⠂⠀⠀⠀⠊⣻⢿⢿⡛⠂⠀⠀⠀⠊⡻⠿⢿⡓⠂⠀⠀⠈⣙⣷⢤⡞⠻⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠂⠝⠿⠿⠟⡔
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁
⠀⠀⠀⠠⡰⠘⠰⢢⠀⠀⠀⠀⢀⣠⣤⣆⡀⠀⠀⠀⠀⣠⣢⣖⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣷⣶⣄⠀⠀⠀⢀⠰⣀⣆⡐⢀
⠀⠀⠠⠅⢰⣿⣷⡄⠩⠀⠀⠀⢺⡏⠀⠈⣿⡂⠀⠀⢸⣹⠁⠀⠈⣳⡅⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⠀⠀⠌⣿⣿⣿⣿⡅
⠀⠀⠀⢫⢌⠙⢋⠠⠑⠀⠀⠀⠙⠷⡦⣶⠟⠀⠀⠀⠘⢭⡷⢤⡴⡿⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⠿⠀⠀⠀⠐⢙⠿⠿⢟⠐
⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣵⣭⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣷⣿⣿⣾⣽⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠐⠛⢛⢻⠷⢶⣦⡤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠔⣁⠀⠀⠉⠁⠀⠀⢩⣺⣷⡰⡄⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⣀⡠⠴⢚⡟⣏⡞⢁⠄⢠⠔⣊⡩⠉⠉⠛⢿⣿⣿⡔⡀⢂⠘⣦⣀
⠀⠀⠀⢀⡴⠋⠀⠀⠈⠋⠐⣿⡰⡃⢀⡔⢉⡤⠤⢄⠐⠦⣭⣻⣿⣧⣀⠀⠂⠈⠻⡷⣄
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢥⣾⣷⠁⢋⠞⢁⡴⠞⠉⠁⠀⠤⣜⣿⣇⢹⡰⣶⠀⠀⠉⢸⣷⡄
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣨⣺⣯⢳⢏⣴⣫⠆⠀⠀⠀⠀⢐⣿⢸⣗⠘⡿⡏⢠⡔⠀⠀⠈⠻⣄
⠀⠀⢹⣦⠑⢄⠀⠀⠀⠀⠚⢴⢿⡼⣿⠋⠀⠀⠀⠀⠀⠀⣶⣿⣸⣷⢸⡇⢳⠈⣇⠄⠀⠀⠀⡘⠑⢄
⢀⡴⠻⢧⢷⡈⠆⠀⠀⠀⠀⠐⢽⣿⣷⡀⠀⠀⢀⠀⡀⣿⣿⣣⣿⣟⡈⠁⢺⡄⣿⡩⠀⠀⠀⠑⠄⢘⡇
⡾⠀⠀⠀⠘⠻⡌⠂⡀⠀⠀⠀⠀⠋⠿⣝⣳⣶⣷⣞⣚⣭⣿⣿⣿⣽⡃⠀⢸⣿⡷⡖⠀⠀⠀⠀⠘⣆⣧
⣷⡔⠀⠀⠀⠀⠁⢀⡇⠀⠀⠀⠀⠀⠀⠘⡜⡽⡽⡹⠹⠻⡻⣿⣛⠋⠙⣷⣾⣿⣷⡀⠀⠀⠀⠀⠀⠐⢿⡄
⢹⣿⣿⣦⠀⠀⠈⠭⢙⣾⢀⠀⠀⠀⠀⠀⠈⠝⡱⢣⣷⢰⣷⣼⡯⠤⢤⡘⣿⣿⢥⣒⡀⠀⠀⠀⠀⢙⣷⣷⡀
⢨⣇⠈⠻⣇⠀⠀⠀⠙⠿⢿⡆⠀⠀⠀⠀⠀⠈⢡⣿⢃⣿⠻⠋⡿⡎⠁⠈⠊⣿⣿⣤⠠⠤⡀⠀⠀⠀⠛⠿⣧
⢸⠾⠟⢄⣺⣇⡀⠀⠀⠈⢉⢳⣴⢂⠀⠀⠀⠀⠐⣣⠋⠃⠁⠐⠃⠀⠀⢀⣳⣿⣗⣂⠀⠀⠀⠀⠈⠉⠉⢛⣚
⠸⡄⠀⠀⠉⠻⣿⣄⠀⠀⠀⠠⢹⣾⡔⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⣠⣿⠯⠷⠦⠀⠀⠀⠀⢀⠀⡄⠀⡀⢀
⠀⠹⣗⡄⠀⠀⠙⣿⣧⠀⠀⠀⠑⠽⣷⣶⠇⠀⢀⠀⠀⢀⢀⣶⣿⣧⣿⣫⠥⠀⠀⠀⠀⠀⠰⠪⠓⠊⠆⢨⡠
⠀⠀⠈⠻⣵⣂⠄⠘⢝⣿⣄⠀⠀⠈⣪⢿⣿⣦⡮⠶⠴⣧⣿⣿⣾⡿⣓⡚⠃⠀⡄⣠⣤⠴⠀⠠⣵⡾⠖⠊⠁
⠀⠀⠀⠀⠈⠑⢿⣢⣀⠪⢿⣧⣄⠀⠐⢵⣿⣿⣿⣿⣿⢿⡿⠉⢰⣏⡖⠀⢠⡴⠋⠁⠈⣲⡽⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⣀⠈⢿⣾⣀⢀⡉⢀⡽⠟⣡⢞⠃⠀⠘⣗⠏⠀⠋⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣬⡋⢿⣿⣿⣿⣶⠟⡡⠀⠀⠀⠀⠹⣭⡆⠀⠀⠀⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣲⣾⣿⣽⣟⣋⣀⣀⣀⣀⣀⣀⣈⣙⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⢸⣿⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⣯⣼⠟⠉⠀⠈⠹⣷⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠛⢋⡀⠀⠀⠀⠀⢻⣟⠉⠉⠉⠛⣷⡶⠶⠶⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⣤⣿⠀⠀⠀⠀⠀⣸⡇⠀⠘⠓⠲⢾⣷⠀⠤⠦⢼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⢆⠀⢹⣿⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢤⡈⢳⣀⣿⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢀⣴⠞⠛⢿⠛⠿⠷⠶⠶⠶⢾⣧⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣿⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣼⡇⠀⠀⠀⣸⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⣤⣀⡀⢀⣀⣠⡾⣧⣀⣀⣠⡾⠛⢷⣄⣀⣴⠋⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢹⣿⠉⠉⠓⢤⡈⠛⠛⠛⢧⣀⠀⠉⠉⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⢀⣾⡟⠷⣄⠀⠀⠙⢦⡀⠀⠀⠈⠙⠂⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣠⠞⣹⠃⠀⠈⠆⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢛⣿⡷⣦⣤⣤⣤⡤⠶⠖⠂⠀⣤⡤⠴⠶⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠈⠙⠳⢦⣤⣀⡀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠘⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠿⢷⣦⣤⣀⠀⠉⠙⠛⠶⠾⢷⣤⣤⣀⣀⣀⣀⣿⣉⠛⢿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠉⠛⠻⠷⢶⣦⣤⣀⣀⣀⠀⠈⠉⠉⠉⢁⣀⣀⣼⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠿⠿⠿⠛⠛⠉⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⣷⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡠⠀⡡⠋⣿⣧⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢠⣿⣤⡀⢀⣿⣷⣶⣷⡚⠋⠏⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⣿⠖⢼⡿⠋⠀⢾⢿⡿⢿⣿⣿⣶⣄⡿⣿⣿⣿⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢃⠰⠊⣄⠀⣀⣴⡾⢇⣿⣿⣿⠿⠉⠐⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣀⣀⣈⠀⣨⠔⣺⣿⣿⠟⡩⡢⢺⣿⣿⣷⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠳⠤⠻⠿⠾⠿⠟⠟⠛⠋⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡿⢀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠠⠋⢠⡄⠡⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢃⠀⠰⣦⠙⣾⡿⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠈⡆⠀⠘⠛⣿⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠛⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠟⠿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⡙⠀⠀⠐⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠛⠋⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⢀⡙⠲⠦⠀⠀⣀⣀⠄⠈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠙⠙⠛⠂⠀⠀⠻⢥⣤⡄⠀⣿⠟⠁
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠙⠛⠓⢒⠒⠀⠀⣠⠋
⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⡀⠈⠙⠀⢀⣠⣶⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢀⡴⠖⠒⠲⠦⣀⠀⠀⠀⠀⠀⠁⠀⠙⠲⠶⣶⠿⢻⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣴⠋⠀⠀⠀⠀⠀⠈⠛⠹⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠀⠀⠀⠤⣀
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⡄⢰⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣷⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⢀⡀⠀⠀⠀⣠⣿⣿⡿⠁⠀⠀⠀⠀⠑⠠⠤⣀⣀⡀⠀⠀⠀⠀⢀⣠⠋⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠛⠿⣻⡋⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⢸⡏⠀⠐⠀⢀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡴⠀⡀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⠧⢻⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⠀⠻⠂⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠛⠁⢁⣀⣀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⢤⣀⣠⣤⢶⡾⣿⣿⠦⠶⠛⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣋⣏⣹⣹⣿⣭⣽⣻⣋⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣭⣭⣭⣿⣿⣭⣭⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⢸⣿⡀⢸⣦⡀⣿⡏⠀⣾⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠹⣶⣦⣿⣷⣤⣿⣿⣿⣷⣼⣿⣧⣾⡟⠀⣤⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣼⣿⡿⠟⠛⠋⠉⠉⠙⠛⠿⣿⣿⣧⣾⣷⣀⣴⣾⠟⠁
⠀⠀⠀⠒⠶⢶⣤⣤⣀⠀⠙⠾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⡿⠀⢀⣀⣀⣀⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣉⣩⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣤⠤
⠀⠀⠒⠛⠛⠛⠛⠛⠛⢛⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣭⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠿⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⡛⠛⠛⠻⠿⠷⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠟⠓⠂
⠀⠀⠀⠀⠀⣠⣶⠿⠟⠋⠉⢸⣿⠟⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡟⠻⢿⣦⣄
⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⠋⢁⣴⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⢦⠀⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⢻⡟⠀⣿⡿⣿⣿⠻⣿⡿⣿⣧⠈⢻⣷⡀⠀⠀⠀⠀⠀⠈⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠈⠀⣸⣿⠁⠉⡏⠀⣿⣇⠀⠘⠂⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠹⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⡉⠉⢻⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣥⣤⣀⣤⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⡿⠋⠁⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣉⠉⠙⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⠟⠁⠀⢀⠴⢶⣶⣶⣄⠀⠀⣠⣴⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣴⣶⠾⣤⡾⢿⠟⢿⠛⣧⠐⢿⠇⠾⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢛⣙⠛⢿⡟⠀⠀⠀⠘⣇⠀⠀⠀⢸⡏⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣞⡼⠁⠀⠁⠀⠀⠀⠀⠈⠙⠒⠚⠉⢠⣀⣤⣀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡜⠺⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡦⣶⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠛⠷⣦⣤⣀⡀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠃⣀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣸⣷⠀⢸⠓⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠙⠹⠏⠀⡼⢀⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣷⣶⣮⠊⣠⠊⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠀⠈⠛⠲⢤⣀⡀⠀⠀⠀⠀⣯⠀⠚⠉⠻⢟⢿⣿⡾⠟⢳
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⢀⣢⠀⣮⣄⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀⠀⠸⡆⠀⠀⠀⠈⠀⠀⠀⢀⡿
⠀⠀⠀⠀⣠⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠻⣄⠀⠀⠀⠀⠀⠀⢸⠃
⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⢸⡿⢻⣿⣿⣇⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠈⢷⡀⠀⠀⠀⠀⢹⡀
⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣼⣧⣤⡴⠶⠞⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⣿⠋⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠈⠻⣦⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⡇⠈⠄⠀⣿⠀⠀⣿⠆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣰⣿⠀⠃⠀⠀⡠⠃⠀⠀⢿⡀⢀⢀⡤⣾⡃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠇⠀⠀⣠⠾⠉⠀⣠⠀⢀⠀⠀⡀⠀⠀⠀⠗⠀⠻⡇⠙⢧⠀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠀⠀⠀⠜⠁⢀⣀⡾⠇⠀⠀⠀⠀⣹⠆⠀⡔⢃⠀⠀⠈⠂⠈⠁⠀⢸⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⠁⠀⠀⠀⠁⠀⠔⠉⠀⠀⠀⠀⠀⠀⠄⠀⢀⣼⠁⠈⡀⠀⠀⠐⠀⠀⠀⠀⠡⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠉⠀⡰⠃⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡀⠀⢰⡀⠀⠈⠉⠉⠑⠒⠀⠀⠤⢄⣀⡀⢀⡠⠄⠀⠒⠀⠉⠉⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢐⣿⣿⣂⠀⣼⡅⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣲⣿⣿⡷⢠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⡗⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠠⣽⣿⣿⣷⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣸⣿⣿⣿⣅⣺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠚⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣼⣿⣿⣿⡶⢿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠚⠛⠛⠛⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢠⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀⣠⣤⢶⢒⠞⠉⠁⠀⠀⠀⠀⠀⠈⠉⠘⠒⠐⠢⢀⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣯⠹⣷⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣂⢆⢄⡤⠀⣠⢀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣷⡞⠃⠒⢦⣄⡀⠀⠀⠩⣏⠉⠉⠉⠉⠉⠉⠉
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠦⠄⢴⣶⣦⣶⣿⢿⣿⠿⠆⠢⠘⠀⠊⠉⠀⠀⠈⠁⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⠀⠠⠗⠶⠛⠋⠉⠉⠈⣐⡇⠀⠁⢘⣶⣤⣆⣀⣀⣀⣲⡴⠃
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⣔⠁⠈⠲⢄⡀⠀⠀⠀⠉⠉⠩⠛
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⢻⢧⠀⠐⣴⣄⡄⠀⠀⢄⢸⣎⠀⠀⠈⠙⠻⠶⣶⠦⠥⠔⠁
⠀⠀⠹⠿⢿⣿⡿⠿⠿⠿⠟⠛⠛⠊⠁⠁⠀⠿⠿⢿⢶⣤⠥⠬⠿⠓⠒⠦⠤⠤⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡀⠀⠐⠀⠀⠀⠀⠀⣀⡀⠘⢃⣀⣉⠀⠰⠇⠀⠀⠀⠀⠀⠀⣄⠹⣇⠹⠀⠀⡜⢀⡀
⠀⠀⠀⠀⠀⡀⣄⣴⠟⢯⡓⣄⠀⠀⠀⠀⠀⠀⠙⠁⡄⠀⡀⠀⡀⠀⠀⠶⠂⠀⠀⠀⠲⠀⠁⢈⣤⣤⣄⠐⣡⡴⡆
⠀⠀⠀⠀⠀⢀⠸⡧⣸⣅⣼⣇⠀⠀⠀⠀⠀⠛⠃⠰⠀⠢⣀⣀⡡⠄⠁⠠⡤⠀⠀⠨⠭⠄⠀⣿⣿⣿⣿⡇⠀⠰⠶
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠛⠏⠀⠀⠀⠀⠀⠀⠀⡴⠄⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠶⠒⣡⠌⠛⠛⠋⠀⢌⡙⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⣤⠀⠲⠄⠀⠀⠀⠀⠀⠀⠺⠃⠀⡆⢰⠀⠘⠦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠲⠄⠸⡄⠀⢠⡟⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢃⣀⠃⢠⠆
⠀⠀⠀⠐⠶⣄⠢⠄⠐⠈⡔⢀⣤⡄⠀⠀⠀⠀⠀⢠⠿⣟⣛⡛⠿⣷⡀⠀⠀⠀⠀⠀⠐⠄⢠⣾⣿⣿⣿⣦⡀⠤
⠀⠀⠀⠒⠀⠀⠂⠀⠀⠀⠠⠄⠀⢤⠄⠀⠀⠀⠀⢿⣜⢾⣜⣟⡶⣿⠁⠀⠀⠀⠀⠀⠐⠂⣿⣿⣿⣿⣿⣿⡇⠤⠄
⠀⠀⠀⠙⠋⠁⠔⡀⠄⣀⠢⠘⠲⠄⠀⠀⠀⠀⠀⠈⠮⣯⣿⣿⡵⠃⠀⠀⠀⠁⠀⠀⠐⠂⡘⠿⣿⣿⣯⠟⠡⢄
⠀⠀⠀⠀⠚⠁⡸⠁⠀⠘⡄⠐⠦⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠀⢀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢀⡀⠀⢠⡄⠀⣠⠆
⠀⠀⠀⠀⢠⣀⢻⠢⡸⠣⠞⣏⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣤⣀⣹⠟⠀⠀⠀⠀⠑⠧⢤⣤⣤⠀⠀⠀⠀⠀⢀⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢛⠍⠫⣝⢦
⠀⠀⠀⠀⠈⣲⠄⠀⠀⠀⠀⠀⠺⣍⣀⠀⠀⠀⠀⠀⢰⠡⠄⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⢆⢞⣻⣾⣿
⠀⠀⠀⠒⠛⣡⣟⣀⠀⠀⠀⠀⣏⠈⠁⠀⠀⠀⠀⠀⠈⠢⣈⣋⠠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠧⣂⠥⣕⣵⠏
⠀⠀⠀⠀⠘⠉⠀⡶⠉⣦⠲⡏⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈