System Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠂⠀⢀⠀⠂⠁⢀⠀⠀⣇⠀⠀⠉⢁⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢢⡔⠈⠀⡀⠰⣎⡹⠖⠈⠀⡀⠄⠊⠁⠰⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨⠀⠀⠁⢠⣀⠁⢀⠠⠐⠀⢀⡠⢤⢀⠸⠀⠀⠀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡄⢼⠀⢻⠁⢀⠤⡆⠹⠔⠊⠁⢀⠠⠂⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣀⢼⠀⢸⠀⠰⠒⡟⢤⣤⠔⠈⠀⡠⡄⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨⠀⠀⠀⣸⠀⢸⠀⠰⠂⡇⠈⡏⠀⡔⢩⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⣦⠀⠁⢸⠀⢸⠀⠀⠐⡇⠀⡇⠀⣋⠀⠐⠛⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⡀⢸⠀⠀⠀⡇⠀⡇⠀⠇⢀⠠⠂⠁⢀⠰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠊⠀⠀⡇⠀⡇⠀⠉⠀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⣇⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠁⠂⠤⡀⠀⠀⠘⣇⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠸⢤⣄⣈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠉⠁⢠⡆⠀⠈⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠓⠂⢠⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣠⣤⡶⠒⢚⡿⠋⠉⠉⠙⢄⡠⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⣴⣾⣿⠟⠉⠧⡴⢟⡵⠂⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣾⣿⡿⠁⠀⠀⠀⠀⡟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢋⡀⢀⣠⠄⠠⢤⣀⡀⢰⠁⠈⠱⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⠾⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠓⠢⠀⠉⠙⠦⣀⢤⠚⠉⠉⠛⢿⠋⠉⠛⠿⠻⠛⠛⠛⠋⢉⣁⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⣶⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢿⣦⣀⣀⡀⠀⠀⣀⣀⣤⣦⡴⠶⠿⠉⠉⠁⣀⣬⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠋⠛⢿⣿⣷⣶⣆⠀⠀⠀⠀⠠⣀⠀⠁⠀⠉⠉⠙⠋⠉⠋⠀⠀⣀⣠⠄⠒⠚⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠀⠀⠈⣸⠃⡼⠉⠓⠖⢦⡀⢀⡀⠀⠀⠀⠀⠂⠂⠐⠒⠂⠀⠉⠀⠀⠀⣀⣀⣠⣤⣤⣴⣦⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⡇⢀⡟⠀⠀⠀⠠⠼⠋⠉⠒⠶⢦⣀⣠⣄⣀⣠⣤⣀⣠⠖⠤⠾⢋⡽⣿⡟⠋⢛⡉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢸⡃⠈⡇⢸⡄⠀⠀⠀⠀⣀⠐⠀⢰⠋⠉⡽⠉⠉⡝⠀⠀⠀⠀⠈⠁⠀⢸⠗⠀⣸⠁⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⡇⠘⡅⠀⠈⢐⠖⠀⡨⠁⠀⠀⠁⠀⠈⠀⠀⡀⠣⠀⢀⣀⣤⠴⠖⠚⢛⡆⠀⢸⡆⠀⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⢸⡄⠠⡇⠀⢠⠃⠀⠞⠁⠀⠀⡔⠢⠀⣀⠤⠊⠀⠀⠀⢠⠎⠀⠀⠠⠤⢾⡆⠀⠘⡅⠀⢺⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢸⡇⢀⡗⠀⢸⠀⠀⠀⠀⠀⠊⡠⠂⠀⠀⠀⠀⠀⠀⣀⣼⡀⠀⠀⠀⠀⢰⡄⠀⠀⡅⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠘⡇⠈⢻⠀⠀⢑⣂⣂⣡⠖⠚⠉⠁⠢⣄⠀⠐⠒⠋⢉⡴⠟⠓⠛⠳⣶⣶⠁⠀⠀⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣧⠀⢿⡀⠀⡏⠀⣨⡇⠀⠠⣤⣀⣀⣀⣑⣂⣀⣶⠟⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⡼⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢹⡄⠸⣇⠀⠡⡄⠈⢧⡀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠂⠀⢀⡼⠂⠀⠀⢀⣇⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠀⠀⣧⠀⣟⣀⠴⠁⠀⢸⠏⠒⢤⣤⣀⣀⣠⣠⣤⣤⣶⠀⠀⠐⢤⡾⠀⠀⠀⠀⣘⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣿⣄⡿⠉⠀⠐⠒⠉⠀⠀⡀⠀⠀⢹⠀⠸⡇⠀⣿⠋⠉⠲⡏⠀⠀⠀⢀⣼⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢀⣴⡿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⡆⠀⠇⠀⠀⢠⠀⠀⠁⠀⠀⢠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⠁⠀⠈⠒⠒⠛⠿⢿⣶⣾⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠈⠑⠂⠄⡀⠀⠀⠀⠘⣇⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⢘⣄⣀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠈⠉⠙⠚⠓⠄⣀⡀⠀⠐⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⠀⠀⢀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠤⡴⠊⠉⠉⢁⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣀⣤⣶⡾⡟⠉⢉⡼⠋⠀⡀⣀⣀⠤⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣠⣾⣿⡿⠋⠀⠙⠖⢋⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⣿⠟⠀⠀⠀⠀⠀⠸⠁⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣁⠤⠠⢤⠀⠐⠢⣤⣀⠀⡎⠀⠀⠱⣤⣀⡀⠀⠀⣀⡀⠀⠀⠀⢀⣀⣠⣤⣶⣴⠞⠣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠙⠒⠄⠀⠈⠳⢄⡠⣴⠋⠈⠉⠛⢿⡟⠉⠙⠻⠛⠛⠛⠉⠛⠁⣉⣀⣴⢞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣦⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢷⣄⣀⡀⠀⠀⠀⠀⢀⣀⣴⣤⠶⠶⠿⠉⠉⠁⣀⣨⡵⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠻⣿⣿⣷⣦⣦⠀⠀⠀⠀⠀⢄⠀⠈⠉⠀⠉⠋⠙⠛⠉⠛⠁⠀⠀⢀⣀⠠⠐⠒⠋⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⢹⡏⣹⠏⠳⢤⠦⡀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠠⠤⠶⠀⠐⠊⠉⠀⠀⠀⠀⢀⣄⣀⣠⣦⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢠⡏⠀⣽⠀⠀⠀⠀⢉⡶⠛⠂⠤⣤⣄⠀⢀⣀⠀⠀⣀⡀⠀⣠⡄⣠⣴⠟⣻⣿⡿⠟⠿⠟⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢸⡇⠐⣧⢠⡄⠀⠀⠉⠀⠀⠀⠠⠁⢈⡿⠛⢻⠟⠚⢻⠏⠉⠁⠀⠀⠐⠊⠁⢹⣧⠀⢀⡇⠀⠚⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⢠⡏⠀⠛⠦⢤⡤⠀⢰⠋⠀⠀⠎⠀⠐⠋⠀⠀⣼⠀⠀⠀⠀⠀⣀⣠⡤⠾⡇⠀⢸⡆⠀⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⠁⠀⣿⠀⠀⠠⠋⢀⡴⠁⠀⠀⢠⡀⠀⠀⠀⣠⠎⠀⠁⠂⢒⡾⠋⠁⠀⢀⣰⡇⠀⠀⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⢸⡇⠈⣷⠀⢀⠃⠀⠈⠁⠀⠀⡰⠁⣈⠐⠒⠊⠁⠀⠀⠀⠀⡟⠀⠀⠀⠊⠉⢙⠇⠀⠀⠳⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢸⣇⠰⣿⠀⠈⢧⠀⠀⢀⠀⢈⣠⣎⡀⠀⠀⠀⠀⠀⣀⡤⠶⢳⣄⣀⣀⣀⣀⢘⠇⠀⠀⢸⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⡇⠀⣸⠇⠀⢀⡤⠤⢤⡞⠉⠀⠀⠀⠉⠢⣀⠈⠁⠀⢀⡴⠛⠉⠉⠉⠉⠛⣿⠀⠀⠀⢸⡀⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣿⠀⢹⡆⠀⢨⠀⠠⣼⠀⠀⠐⠶⠤⢤⣤⣤⣭⣤⡼⠟⠁⠀⠀⡀⠀⠀⢸⠃⠀⠀⠀⢸⠁⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠘⣆⠘⣧⠀⠀⢦⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠼⠃⠀⠀⠀⡼⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⡀⣾⣠⠤⠊⠀⠀⡼⠉⠓⠦⢤⣤⣤⡤⣤⣤⣤⢶⡆⣀⠀⠑⣴⠟⠀⠀⠀⠀⣸⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣽⣧⣼⠋⠉⠁⠒⠋⠁⠀⠀⡆⠀⠀⢹⡀⠈⣧⠀⢸⡟⠉⠉⢺⡇⠀⠀⠀⠀⣴⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⢀⣰⡿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠛⠋⠀⠀⠻⣦⡄⠀⠀⠀⠀⠀⠀⢣⠀⠀⢸⠇⠀⠀⠲⠶⠒⠿⣿⣷⣶⣶⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠩⣍⠽⠝⠻⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠑⠃⠀⢀⠸⠈⠀⠀⡁⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⢀⠂⠰⠀⢠⠀⠐⠐⠂⠜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⡀⢄⡀⠐⢰⠀⠐⠀⠸⠠⣳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣴⠛⠿⣿⢻⡿⣿⣦⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠛⢲⠖⢿⠖⠒⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠐⡂⢰⠆⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⣁⣨⡅⢸⢍⣉⡳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢋⣀⠠⠄⠒⡃⢰⠂⠤⠤⡤⠝⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠭⠐⡀⡉⣀⠀⣀⣞⡏⢨⣁⣠⣀⣀⠉⣲⡇⠀⠨⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠀⠁⠀⡉⢋⣀⡤⣞⣛⠠⡇⢨⠤⠄⠬⢤⣤⠄⢂⣈⠀⠒⣄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠈⠑⠒⠊⣁⠴⠚⣁⣐⠢⠄⣈⡇⢰⢚⡩⠖⠢⣉⠩⠂⢄⠈⠒⡦⠁⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠁⡀⠀⣀⡀⡝⢸⢍⡀⠀⣀⠀⠀⠇⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢄⣩⠄⠀⠒⡃⢸⡒⠦⠈⠩⣀⠄⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠖⠀⠤⠀⠖⣃⣣⢸⣍⡀⠀⠐⢤⡂⠀⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠚⠀⠰⠁⠀⠉⡋⢸⡂⠈⠉⣛⠂⠉⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠄⠀⠀⠒⠀⠍⢱⢸⠤⠀⠀⠨⠖⠀⠡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠠⢈⡤⠐⢂⣩⢸⣍⠲⢌⡙⠊⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠄⢀⠤⢀⡩⠗⢰⢄⡉⠂⢍⡐⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠂⡠⠔⢉⡡⢊⢸⡢⣽⡐⠢⡁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢃⡜⠀⣠⣺⢗⢵⢸⠮⡺⠔⣄⠈⠖⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⡡⠀⣰⢭⡗⠀⣹⢸⡁⠈⣇⠨⡌⠚⠦⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⠁⠼⠃⣿⠤⠊⡴⠰⢌⡂⢨⡔⠌⠢⠀⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⠈⠠⢁⡤⠐⡀⠜⢀⡢⠉⠐⠳⡄⢀⢂⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⡎⠀⢁⡎⢠⠀⠙⠐⢿⠇⠠⡘⠄⢰⠈⠢⣄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠁⠆⡠⢨⠀⠈⢆⠼⠀⢘⠀⡆⠃⠘⠀⢳⣄⠠⠙⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⢸⠃⣘⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢸⠀⠐⣄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠃⠈⠸⡉⠆⠀⠀⠀⢠⠃⢣⠀⠀⠀⡠⢟⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⢿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⣾⣿⡟⣾⡿⡁⣈⠺⣿⣿⣿⣿⣦⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣼⠆⣰⣅⡹⣿⣿⢟⡙⢹⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠉⠈⠀⠙⠚⠿⢏⡷⢂⠬⢈⣽⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡟⢿⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⣼⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣞⠀⠀⣳⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏⡳⣿⣦⣢⡆⠀⠀⠀⠀⠀⠀⠀⠈⢻⡾⣶⣳⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢟⢰⡷⢿⠿⢿⣡⢄⣀⠀⠀⠀⠀⠀⠀⠀⢛⠔⠁⠋⡉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠂⠀⠀⠀⠀⠀⠀⠀⣿⡇⡸⠸⠀⠀⠀⠀⠉⢿⣟⣃⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⡇⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⠀⠀⠀⠀⣁⣒⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡇⡀⠀⢰⣷⣿⣏⣴⣇⠀⠀⠀⠀⠀⠀⠀⠀⣮⠍⠉⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⠁⠀⠀⠀⠀⠂⠀⠀⢘⣿⣷⡇⡀⠀⣿⣿⣿⣿⠿⢿⠇⠀⠀⠀⠀⠀⠀⢰⡓⠅⠀⠄⣙⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⣠⣠⠆⣿⣿⣧⡅⣿⡿⠛⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡝⢾⣶⣿⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢺⣷⣦⣻⣿⣿⣷⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠋⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠘⠷⣾⣿⣿⣯⣿⣾⢸⣿⣿⣿⡆⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⠀⢰⣤⣤⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡠⡠⢌⢌⣿⣿⣿⣿⣿⣽⡎⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⡀⠀⠀⠀⣻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣄⣀⢀⡠⣄⢒⠈⠁⠹⣿⣷⣦⣴⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⡸⡌⡈⠎⣿⣶⡄⠀⠀⣻⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡇⢁⣤⢿⣿⣷⣷⣮⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣾⡾⢿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⣿⡿⣿⡿⠋⢛⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⣻⡑⣔⠁⣾⣿⣿⠻⢻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠉⠉⠘⠻⠿⣿⣫⠆⡠⢘⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⣿⣻⠀⠀⠀⠀⠀⠀⠙⢾⣷⣿⣿⠟⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⢠⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⢈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⡝⣿⣶⣴⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⢯⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢻⠀⡇⠻⠛⠻⣶⣤⣄⠀⠀⠀⠀⠀⠀⢑⠠⠈⠠⢙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⢸⡿⢸⢰⠀⠀⠀⠀⠈⢿⠅⠀⠀⠀⠀⠀⠀⠠⠔⢀⣀⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢸⠀⠀⢀⣄⣤⡄⣰⠀⠀⠀⠀⠀⠀⠀⢐⣾⠾⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡄⠀⠀⠀⠀⠀⡀⠀⠀⢼⣿⡽⢨⠀⣸⣿⣿⣷⣿⣆⠀⠀⠀⠀⠀⠀⢸⡗⠀⢀⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣀⢠⢸⣿⣧⡀⢰⣿⠿⠛⠛⠀⠁⠀⠀⠀⠀⠀⠀⢯⣮⣴⣦⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢽⣷⡘⣿⣿⣷⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡁⠉⠻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠳⣵⣿⣿⣿⣿⡇⣿⣿⣿⡆⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢤⣤⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣆⢠⠤⠠⣻⣿⣿⣿⣿⣵⢸⣿⣿⣿⠄⠀⠀⠀⠀⠀⡀⣴⣦⡀⠀⠀⠈⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣷⣼⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣆⢤⢤⢴⡔⠄⠈⠈⣿⣿⣷⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣞⣿⣿⡇⡌⡌⢚⣿⣷⣀⢀⢼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣷⣇⣰⣾⣿⣿⡿⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠋⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⠉⠘⢣⠚⠉⠻⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅⠁⠀⠠⠘⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡯⠀⠀⠁⠘⠀⠰⠀⠁⣈⠑⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣤⣈⣼⣄⣈⣀⠀⢈⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣾⣅⣛⣟⣹⣛⢿⣇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠙⡉⢙⠁⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠠⠄⢨⠅⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠘⠓⠒⣒⡇⢘⡑⠒⠛⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⡁⠰⠒⠈⠁⣠⡦⠸⠄⠉⠙⠋⠬⣩⠝⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⠅⠈⠠⠶⠂⠓⣢⡶⢃⡇⠘⣂⡛⣒⠂⣀⠛⠆⠀⠡⣈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠴⠡⡤⠀⢀⡠⠞⠫⢍⡀⠰⡇⢸⡭⢒⣭⡝⠻⡢⢘⠂⠢⣌⡃⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠔⢠⠎⠁⠀⠒⠉⠀⠁⠉⣧⢸⠒⠁⠀⠀⠁⢡⡄⠑⢦⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣀⠀⠓⡐⠈⠭⠧⢸⢌⣁⠒⢋⢀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡥⠈⢉⡀⠤⠆⢇⢸⡒⠀⠀⢀⡁⢄⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠰⠂⣠⡖⠀⠉⠛⢸⠌⠑⠒⠖⠌⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠶⠀⡂⢵⢸⢅⠀⠀⠡⠀⠐⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠘⡁⠠⠄⣋⢸⣗⠠⣐⠒⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⡤⠀⣈⠵⢰⢌⡑⠠⢁⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⣀⠔⢉⡡⢚⢸⡢⣽⠐⢤⡁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣼⠁⣠⣺⠗⣵⢸⠮⣻⠴⣄⠙⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢀⠂⢠⢫⡟⢂⣽⠸⢁⠈⣇⠠⡘⠒⠄⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠇⡴⠃⠾⠖⢉⠴⠘⢨⡁⠨⡁⠌⠂⠠⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⡄⢀⢂⠖⠈⣀⠚⣠⡖⠀⡈⠳⠈⡌⠄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠜⠀⢁⡞⢠⠀⢈⠐⢿⠂⠀⡘⡂⢃⠈⠰⢄⡈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⠚⣠⢡⡆⠀⠓⠘⠀⢨⠀⠇⠁⠘⠈⡗⠆⡂⠑⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠇⣴⡀⠀⠀⠀⠀⡜⡀⠀⠀⠀⠀⡇⠀⠐⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠐⠐⠁⠘⠀⠀⠀⠰⠀⠛⠀⠀⠠⠊⡇⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠀⠈⢐⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠉⠀⠀⢀⡠⠒⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⣠⠔⠊⠁⠀⠀⠀⢸⠀⠀⣀⠔⠛⠓⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⠁⠀⠀⢀⡤⠒⠁⠀⠀⢀⡤⢶⠀⢀⠸⠖⠉⠀⠀⢀⡤⠚⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⢶⠎⠁⠀⠀⡠⠔⢺⣃⣠⠞⠊⠁⠀⠀⣀⠴⠊⠁⠀⠀⢸⠀⠀⠀⢀⠴⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⠀⠀⢰⠉⢀⡠⠘⠉⠀⠀⢀⡠⠖⠉⠀⠀⢀⣤⠀⠀⢸⡠⠐⠈⠀⠀⢀⡠⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⢸⠀⢘⡏⠲⣤⡄⠊⠁⠀⠀⣀⠴⡎⢁⣸⠤⠊⠁⠀⠀⣀⠤⠚⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⢸⠄⢚⡇⠀⢸⠀⠀⢠⠔⢹⡀⡠⠗⠉⠀⠀⢀⡠⠔⠋⠀⠀⣠⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⢸⠀⣨⡇⠀⢸⠀⠀⢨⠄⢻⠯⣄⣤⣤⠤⠊⠉⠀⠀⣠⡴⠚⢹⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⠀⠀⠘⠉⢸⡇⠀⢸⠀⠀⢀⡀⢸⡇⠘⢿⠟⠀⢀⡠⢶⠉⠈⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⢸⠀⠀⠀⣀⢼⡇⠀⢸⠀⠀⠘⠀⢸⡇⠀⢸⠀⠀⢸⣀⠼⠀⢀⡇⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣾⣦⠒⠉⠀⢸⡇⠀⢸⠀⠀⠰⠖⢻⡇⠀⢸⠀⠀⢸⠁⡀⠔⠛⠙⠲⣼⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⣸⡇⠀⢸⠀⠀⢸⠁⠀⠀⢀⠤⠂⠉⠀⢠⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣄⣸⢀⡠⠔⠊⢸⡇⠀⢸⠀⠀⢸⡀⠔⠊⠁⠀⢀⡠⠔⠊⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⢸⠀⢀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡷⠁







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⢹⠀⡿⢀⣀⡉⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠊⠉⠀⠀⠀⠀⢸⠀⡿⠀⠀⠀⠀⠄⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⠀⠀⠀⠀⠀⢸⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⠀⠀⠀⠀⠀⢸⠀⠁⠀⠈⠉⠉⠉⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠵⠤⣠⠄⠀⠀⠀⠀⠄⠀⠀⡀⢀⠠⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⠸⠄⠀⠀⠐

⣶⣶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⢹⠀⡿⢀⣀⡉⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠊⠉⠀⠀⠀⠀⢸⠀⡿⠀⠀⠀⠀⠄⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⠀⠀⠀⠀⠀⢸⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⠀⠀⠀⠀⠀⢸⠀⠁⠀⠈⠉⠉⠉⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠵⠤⣠⠄⠀⠀⠀⠀⠄⠀⠀⡀⢀⠠⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⠸⠄⠀⠀⠐

⣶⣶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⢿⡿⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠗⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠺⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠘⢿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠃⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⢀⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣤⠀⢀⣤⡄⠀⣤⡄⠀⠀⠀⣿⡇⠀⠀⠹⣿⡆⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⢠⣿⠟⠉⠉⠙⢿⣷⠀⠀⠀⢸⣿⠇⠀⠀⠀⠉⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⣿⡇⠀⠀⢀⣿⡇⠀⠀⠀⣾⡿⠋⠉⠉⠻⣿⡄
⠀⠀⠀⢺⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡗
⠀⠀⠀⢘⣿⣷⣤⣀⣤⣾⣿⠀⠀⠀⢸⣿⡄⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠟⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣷⣤⣀⣤⣾⣿⡃
⠀⠀⣰⣿⠟⠉⠛⠛⠋⠉⢻⣷⡄⠀⢸⣿⣿⡿⠋⠸⣿⣇⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢀⣿⡇⠀⢠⣾⡟⠉⠙⠛⠛⠉⠻⣿⣆
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠘⠿⠋⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣦⣈⣿⡇⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣯⠿⠿⣿⣿⡿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣺⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠤⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⠉⠁⠐⠒⠂⠠⠬⠩⠉⣉⣐⠒⠒⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⡻⢶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣷⣶⣶⣦⣤⣼⣿⣿⡿⢟⣛⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⠿⣿⣿⣿⣿⠿⠿⠷⣾⣩⣿⣶⣿⣿⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠺⠿⠿⠟⢋⡁⠀⢹⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠤⠁⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣉⣒⣒⣚⡣⢼⢠⣶⢸⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠥⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡒⠒⠤⠤⠶⣿⣽⠀⠁⢸⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣤⣍⣀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢭⣉⣉⣭⡏⣺⡰⡿⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡏⠉⠉⣿⣿⣿⣿⣶⡶⠤⣄⣀⡀⠀⠀⠈⣁⣒⠲⠖⢒⣯⣽⣁⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢤⣼⣿⣿⣿⣿⣿⣿⣷⡆⠶⢿⣿⣒⡶⢦⣬⣉⣩⣭⣷⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⢿⣿⡀⠀⣼⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢥⣍⡛⢻⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠁

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢐⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⢲
⢰⠀⠀⢀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⡀⠑⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡞
⠈⢆⠀⠀⠈⢦⣉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠹⠛⠛⠁
⠀⠀⠑⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬



⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⣠⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⡠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡧⠤⡧⢤⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠀⠀⠀⢳
⠀⠀⡀⢴⠃⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣡⠤⠧⢤⣁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠒⠚⠦⢊
⠀⠀⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤
⠀⠀⠀⠀⣠⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣄⠀⠀⢀
⠀⠀⠀⠀⠉⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠁
⠀⢀⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡴⠀⠀⠣⣀⠜
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣁⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠙⠄⠀⠖⠉⠱
⠰⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠋⠉⠉⠉⠁⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣗⣦
⠀⠀⢀⡠⠶⠾⠧⠤⠤⠤⠤⠤⠤⠤⠀⠠⠤⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⠂⠀⠉
⢀⠔⠁⢀⡤⢴⠖⣰⢶⠖⠀⠂⠒⠒⠒⠒⠒⠀⠀⠠⠠⠤⠤⠤⠄⢀⢀⣀⣀⣀⣀⡀⡀⣀⡀⣀⠀⡀⢈⢉⠉⡉⠈⠁⣀⣀⣀⠒⠒⠢⢄⠄⠌
⠈⠀⡰⡝⠀⡏⢠⢵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⢹⠄
⡀⠀⣛⠃⠀⡇⠠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡟⠆⠈⡅
⠀⠀⠴⠀⠀⢋⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡇⢰
⠀⠀⠘⡄⠀⢸⠸⣺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠇⠘
⡇⠀⣉⡅⠀⢐⠆⣖⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡘⠁
⡇⠀⠠⡄⠀⢸⠁⠈⠛⠻⠶⠶⠶⣦⠶⣶⣴⢶⣶⣔⣄⣴⣤⢴⠠⢤⣤⢤⢤⠤⡤⡤⢤⡤⠀⣀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣠⣴⡵⠁⡆⡀⠀⡆
⡇⠀⠰⠁⠀⡎⠀⠃⠀⠤⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠐⠒⠒⠒⠚⠒⠛⠛⠃⠛⠋⠁⠀⠀⡇⠃⠀⡇
⡇⠀⠰⠀⢸⠃⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠂⠤⠄⠀⠀⠆⠃⡇⠀⡇
⠁⠀⠰⠀⡞⠀⠀⢠⠈⠀⠀⠘⠷⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⡀⡇⠀⡇
⠀⠀⠰⢰⠃⠀⠀⡉⠤⠀⢀⠞⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠛⢦⠀⠀⢰⡇⡇
⠀⠀⠔⣯⢠⠀⠀⠙⠶⠤⡤⠤⠚⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠠⠔⠁⡸⠀⠀⠀⢤⡇⢸⠃⠀⠀⠀⣰
⠀⠀⡄⠘⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠒⠒⠀⢰⠀⠂⠒⡶⠒⠒⠒⢲⡆⠀⢠⡤⠀⠀⠠⡄⠀⠀⠀⠀⠀⠉⠒⠤⠄⠊⠁⠀⠀⠀⢸⡇⢠⠀⠀⠀⣴⠃
⠀⢆⠐⠂⡘⠑⠲⠀⠭⠽⠣⠄⡀⢄⠹⢐⠲⣶⣷⣤⣄⣴⣇⣀⣠⣤⡾⠄⡀⣼⣧⣄⣠⠄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠘⡇⢸⠀⣀⡷⡋
⠀⠀⠑⢤⣀⢉⣑⠚⠐⠃⠂⠀⠄⠬⠤⠦⠤⠁⠤⠀⢌⣀⡍⣨⣉⣉⣉⢉⠋⣐⠂⡒⠐⠂⠑⢐⠖⡒⠰⠠⠠⠟⡯⠭⠽⡭⢬⢤⠀⡴⠁⣸⡵⢋⣷
⠀⠀⠀⠀⠘⣹⠉⠈⣽⠉⠀⠐⠒⠒⠒⠂⠰⠀⠤⠤⢀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠐⠒⠒⠀⠀⠠⠬⠤⠥⠤⠤⠥⠃⠐⢊⣠⠶⠋⠙⠛⠁
⠀⠀⠀⠀⠀⠙⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠖⠒⠒⢋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡂⡔⡛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣯⠶⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡜⣧⠀⠀⠀⠀⠀⢀⡴⣊⡵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⢿⡝⣿⠿⠿⢶⢖⡭⢾⣿⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣉⣩⣿⣿⣷
⠀⠀⠀⣴⡾⠿⠿⠿⠿⠿⠶⠶⢶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠶⠿⠟⠛⠛⠋⠉⠁⢀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⠀⢠⡟⠉⣉⣉⣉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠸⣟⠀⢸⡇⢸⠁⡄⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡀⠀⠀⠀⢀⡀⣀⡀⠀⠀⢸⡇
⠀⠀⠀⣯⠀⢸⡇⡏⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⠶⢶⣄⠀⣿⠂⠀⢸⡇⣧⣴⣿⢸⡇⣿⡇⠀⠀⢸⡇
⠀⠀⠀⣿⠀⢸⡇⢧⠈⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⣧⣀⣨⣿⠀⣿⠀⠀⢸⡇⣿⣿⣿⢸⡿⠿⠇⠀⠀⢸⡇
⠀⠀⠀⣿⠀⢸⡇⠘⠓⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠙⠋⠁⢸⣿⠀⠀⠸⡇⠿⠙⠉⠀⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡶⠿⣶⡄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⡀⠀⣸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⡏⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠻⠿⠋⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣶⢶⣦⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⢸⡇⠀⠀⠻⢶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡇⠀⢠⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⠦⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⡼⠁⡇⠀⢸⣿⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇
⠀⠀⠘⠛⠳⠶⢶⣶⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠁⠀⠙⠛⠛⠃⣼⡇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠶⠶⣶⣶⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣠⣤⣶⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣶⣿⠷⠾⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡔⠲⡌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡅⢎⡇
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂⠀⠀⠛⠀⠒⠒⠒⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢊⠴⠁⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠸⠃⠀⠈⡆⡔
⠀⠠⠀⠀⠀⠀⣰⣁⡍⢀⡴⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠓⠒⠒⠒⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱⠀⠀⠀⠀⠀⡠⠂⠉⠉⡇⠢⡀⠀⠀⠀⠀⠀⡇⡤
⠀⠀⠀⠙⠔⠉⠉⠉⠉⠁⡜⢣⠈⠉⠉⠉⠉⠑⠦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢘⠀⠀⢀⡴⢆⠀⢸⠀⠀⠀⠀⢀⠈⡀
⠀⠀⢠⠖⠉⠉⢹⠙⠢⡀⠑⠊⠀⢀⠔⠋⠉⠉⠃⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠙⢦⡀⠀⠈⢹⠈⠁⠀⠀⠸⠛⠀⠀⢀⠔⠁⠀⢈
⠀⢀⠁⠀⠀⢀⠾⡀⠀⢡⠀⠀⢀⠃⠀⠀⢀⠼⣀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠙⠦⣠⡈⢆⡀⠀⣀⠞⢸⡤⠔⠁⠀⢀⠴⠁
⠀⠀⠙⠂⠉⠁⠀⠈⢙⠈⠀⠀⠈⠛⡆⠈⠁⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠀⠉⠈⠉⠀⠁⠀⠀⢀⠔⠁
⠀⠀⠀⢆⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⡠⠒⠛⠂⠘⠓⠢⡀⠀⠀⠀⡤⠖⠛⠃⠘⠋⠢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁





⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠊⠙⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠈⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢣⠀⡴⣔⢉⢩⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡔⠒⢄⠀⠈⠁⠒⢄
⠀⠀⠰⠢⢄⠀⠀⡰⠛⢤⡣⠄⠃⠈⢣⠀⢀⠀⢀⡆⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⠢⡊⠗⢄⠀⠀⠈
⠀⠀⢐⠀⠠⠃⠀⠑⠀⠲⡀⠁⡛⠀⠘⠀⠀⠀⡜⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠙⠦⠴⠃⡀⠜
⢀⠄⣈⠀⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠑⡲⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠲⡀⠑⠄⣀⡠⠎⢠⠒⠄
⠣⠠⠆⠀⠀⠀⠀⠀⡠⠊⠉⠉⠓⢦⠀⠀⠐⠒⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢬⣊⠀⠀⠀⠀⠀⠀⢈⡓⠅⢠⣄⡀
⠀⢠⠄⠤⡀⠀⠀⠠⣁⣀⡠⠚⠤⣀⡇⠀⠀⠀⡠⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢠⠀⠑⢄⠀⠀⠀⡰⠋⠀⠀⢠⠀⠉⢆
⠀⠸⡀⠀⠉⠢⡀⠀⢘⡀⠀⠀⠀⡇⠀⠀⡠⠊⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⠴⠋⠢⣀⣈⠀⠀⠀⣇⣀⣀⠤⠊⠢⢀⣈
⠀⠀⠈⠢⡀⠀⠈⠒⠤⠵⠦⠤⠾⠤⠐⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡁⠀⠀⠀⠀⢨⠈
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣀⣀⣠⠃⠀⠀⠀⠀⡇⠑⣄⣀⣀⣠⠊
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠉⢢⠀⠀⠀⠗⠉⠀⠀⠀⠀⠉⠲⡀
⣀⣀⣀⣀⣀⣀⣀⣘⣤⣄⣀⣤⣤⣤⣜⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣤⣤⣤⣤⣤⣤⣤⣤⣃⣀⣈⣀⣀⣀⣠⣄⣤⣤⣤⣃
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠏⡆⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⢦⡀⠀⠀⠀⠀⠀⠀⡜⡎⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠙⢆⠳⡙⢆⠀⠀⢀⡤⡞⡜⠀⠚⠂⣿⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡜⠀⠀⢀⡱⠬⠖⠛⠉⠈⣻⣰⡥⠤⠖⠊⢫⢳⠉⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠴⠗⠒⠉⠁⣀⡠⠤⠖⠒⠉⠉⠈⢷⡀⠀⠀⠀⢇⢇⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠉⢁⣀⠤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⢀⣀⡬⣧⢀⣀⠤⢾⣿⣠⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡳⣤⡴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⣀⣠⡤⣖⡺⠯⠗⠓⡏⠀⠀⠀⢸⣼⣿⡿⠟⢡⡾⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠜⣧⠼⢄⡀⠳⠗⠀⠀⠀⠀⠀⣀⡤⡴⣖⡹⠷⠛⠋⠁⠀⠀⠀⠀⣇⣀⣠⣴⣾⠟⠋⢀⡴⠋⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⣠⣾⣷⡜⠌⠓⢖⣉⡹⠀⢀⣤⢴⣲⣹⠓⠛⠉⠀⠀⠀⠀⣀⡠⠤⠔⠒⠻⡍⠳⡜⢯⠁⣠⠴⠋⠀⠀⠀⠀⠀⢀⣴⣷⡄
⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠘⠚⠉⠁⠀⢀⣀⡤⠴⠒⢎⠉⠀⠀⠀⠀⠀⠀⠈⢦⡈⢢⡙⣅⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣽⡄
⠀⠀⠀⠀⠻⢿⣿⣿⣿⠀⢀⠀⢀⣀⣀⡠⠤⠶⢶⣯⣭⣭⣭⣭⣭⡭⠭⠶⣒⡤⠄⣀⠀⠀⠀⠑⣄⠙⣌⣳⡀⠀⣀⣴⣿⣿⠟⣡⣾⡿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⠁⠒⠫⠵⠶⠶⠮⠍⠚⢉⣤⣾⣿⡿⠋⣁⣾⠟⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠟⠉⣠⡾⠋⠁⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠛⠉⣀⡴⠚⠁⠀⠀⠀⠀⢀⣴⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⠿⠟⠋⠁⣠⠴⠊⠁⠀⠀⠀⠀⠀⢀⣤⠿⠋⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⠛⠋⠉⢀⡠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠛⠛⠉⠉⢁⣀⣠⡤⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⣠⣶⡿⠋⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⣀⣴⡾⠛⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⠋⠁⠀⢀⣠⣴⠾⠛⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⣤⣶⡶⠶⠞⠛⠋⠁⠀⠀⣀⣠⣴⡾⠟⠋⠁⠀⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣀⣀⣀⣤⣴⣶⡾⠟⠛⠉⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠯⠥⠄⣀⣀⣀⣀⠠⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠴⠾⡿⠿⠛⠛⠛⠛⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠈⠁⠀⣠⣤⠀⢀⣄⣴⣿⣽⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣻⡿⢭⣽⣿⣿⡄
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣠⣯⣶⣿⣿⣿⣿⣿⠿⠿⡿⠛⠂⠉⠉⠙⠉⠈⠀⠀⠀⠀⢸⣿⣿⢻⡇
⠀⠀⠀⠀⢠⡏⠄⠀⠀⠀⠀⠀⣸⣿⣿⡿⠉⠉⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⡇
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣆⣀⣠⣤⣤⣶⣾⢿⣿⡿⣾⡇
⠀⠀⠀⠀⢸⠀⠀⠀⢰⠀⢀⣿⣿⣿⣿⣧⣤⣤⣤⣤⣶⣶⣾⡶⣶⡶⢿⡿⠛⠛⠛⠉⣰⣎⠉⢹⣿⣧⣿⡇
⠀⠀⠀⠀⢸⣾⣭⣀⣧⡀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠈⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠻⢿⣿⣿⣶⣿⣿⣿⣿⣟⠀⣿⣿⣦⠀⠀⠀⠀⠀⠀⣀⠠⡄⠠⠔⠠⣀⣈⣭⣥⣴⠿⢟⡿⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠙⠛⢁⣀⣀⣠⣂⠈⣀⣀⡄⠤⠤⣒⣒⣋⣭⣥⣴⣶⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣷⡶⠿⣛⣋⣉⣩⣭⣽⣿⣶⣾⡿⠿⠿⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⡿⠿⠿⠟⠛⠛⠉⠉⠁

⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⡇⣶⣶⣶⡆⡅⠀⠀⠀⠀⠀⡀⢰⣆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⣧⣿⣧⣤⣇⡇⠀⠀⠀⠀⢸⣿⣡⡮⡃⠀⠀⠀⠀⠀⣿⢀⡀⠀⠸⠿⠀⠀⠀⠀⢸⠟⢸⡀⠘⡆
⠀⠀⠀⠀⠀⠾⠆⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠒⠁

⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢈⣿⣏⠀⠀⠀⠀⠀⢰⠒⡊⠉⠁⠒⠂⠀⠀⠀⠀⢀⣀⣠⣤⣶⡆⠀⠀⠀⠀⠀⠀⠋⠛
⠀⠀⠀⠀⠈⢻⠛⠀⠀⠀⠀⠀⢸⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠻⢿⡏⠀⠀⠀⠀⠀⢠⣇⣸⡄
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠿⡇

⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠶
⠀⠀⠀⣠⠬⢠⢤⢧⡄⠀⠀⠀⢀⠛⠒⣒⠒⡖⠒⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣾⡗⡆
⠀⠀⠀⠈⠈⠋⠉⠛⠁⠀⠀⠀⠸⠛⠛⠰⠺⠟⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆
⠀⠀⠀⠀⠀⠻⢧⡀⠀⠀⠀⠀⢰⣶⣶⠀⠈⠃⠀⠀⠀⠀⠀⠀⣠⣶⠟⠿⠻⡆⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠈⠟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠼⡿⠿⠿⠿⠻⠇⠀⠀⠀⠀⠀⣏⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠚⠉⢉⡉⠉⢉⠉⣉⣉⠁⠂⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠊⠛⠷⠡⠀⠼⢧⠢⠧⠧⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣒⢀⣀⢀⠀⣐⣒⠂⠐⠂⠈⠔⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠸⠀⠀⡀⡀
⠀⠀⠀⠀⠀⢰⠀⠠⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⣀⣀⣤⠄
⠀⠀⠀⠀⠀⠘⠀⠄⠀⢀⡦⢤⡿⢄⠀⠀⠀⠀⠀⠂⠼⠩⢷⡆⠀⠀⠀⠀⠀⠀⢠⠒⡒⣒⢏⠖⠃⠈⠉⠛⠛⠒⠛⢴⡹⡯⣿⡄
⠀⠀⠀⠀⠀⠀⣤⡀⠊⠀⠘⣄⠀⢺⠪⣄⠀⠀⢀⢶⣌⣻⡯⡄⠀⠀⠀⠀⠀⠀⡄⢠⡇⣿⣾⠨⣭⣷⡦⢀⠀⠔⠄⡀⠕⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠱⠂⠀⠰⠋⢉⠂⠉⠃⡀⠀⠀⠀⠀⠀⠈⠁⠛⠉⠁⠑⠀⠠⣤⣤⣤⣤⣤⠆⠠⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠔⠒⢠⢸⠁⠉⢱⢲⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢙⣻⡝⠣⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⡆⢰⡈⢐⡧⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠘⠻⣟⠿⠆⠀⠉⠉⠲⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠋⠋⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⢤⣧⣤⣭⣇⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠈⡄⠈⣿⢻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⢠⡆⠀⠀⠀⠀⠤⠷⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠈⠀⠒⠒⣾⢲⠒⠒⠶⠶⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⣀⢄⣈⡠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠁⠈⠊⠁⠂⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢰⣁⣸⠀⠀⠀⣀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠀⠀⠀⠀⠀⠀⠀⢠⣼⣧⢄⣠⠌⡄⣐⡃⠀⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⡤⠤⠤⠤⠶⠒⠒⠂⠀⠉⠁⠀⠀⠆⢱⣅⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢌⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⢀⠔⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣍⠀⠁⠀⢂⠲⢀⡀⠀⠀⠐⡡⢂⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢊⠈⠁⠀⠀⢠⡀⣋⠀⠈⠉⠚⠠⠅⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠐⠐⠀⠄⢀
⠀⠀⠀⠀⠀⡀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⣼⢤⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⢀⡀⠉⠀⠂⠀⣀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣀⣀⣸⡀⠀⠀⠀⠈⠈⠐⠘⠀⢰⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⡐⠈
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢇⠀⠀⠀⣠⡿⢛⡿⠈⠐⡠⢄⣠⠀⠀⠀⠀⠀⠈⠉⠚⠃⢇⡖⡤⣀⠀⠀⠀⠀⠀⠑⠀⣀⠔⢢⠜⢠⡀
⠀⠀⠀⠀⡠⠸⣽⣿⠿⠗⢛⣴⣵⢄⢘⡶⣶⡇⢀⠄⡀⠀⠀⠉⠀⠀⠒⠠⠀⡀⠀⠀⠀⠀⠁⠙⠽⢃⢂⢤⠀⠀⠄⠠⠃⠀⠂⣴⣿⣦⡀
⠀⠀⠀⢠⠽⡺⢁⡰⣢⣿⡶⢣⡇⠘⡻⢀⣍⡗⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠀⠀⢄⣀⡀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡆
⠀⠀⠀⡾⠄⢡⣳⠉⠹⡿⠂⢴⡟⢲⠁⠀⢖⠅⢀⡐⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠠⠀⠀⠀⠀⠠⠿⠿⠿⠛⠁
⠀⠀⢰⣄⠀⣘⠉⢠⡔⠂⢹⣿⠰⠈⠓⠊⠀⠁⠀⠀⠐⠀⠀⠀⠉
⠀⠀⢸⠀⠩⡇⢐⡺⠋⠸⠃⢐⣤⡟⡄⠀⠀⠈
⠀⠀⠀⠀⠴⠇⢈⡟⢶⣤⠶⢦⣟⣴⠇
⠀⠀⠀⠈⠠⢺⡞⢛⠻⠮⡶⡾⢞⣁⠈
⠀⠀⠀⠀⠘⠿⠗⠀⠀⠒⢃⣍⣄⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠘⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⠄⠀⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣡⡀⢠⠴⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠄⠞⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠂⠒⠂⠒⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠒⠀⠐⠀⠐⠒⠂⠒⠂⠢⡀
⠀⠀⠀⠀⠀⠀⠈⠀⣀⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⡶⢶⡶⣶⣶⣶⠶⠶⠶⠶⠶⣶⣶⣶⠶⠶⠾⢻⡖⣿⠻⠿⠿⠿⡩⠿⡿⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠈⠁⠁⠌⡀⠀⠀⠨⣀⡅⠈⠁⠀⠀⠌⢈⡑⢳⣍⡇⠊⠉⠉⠉⠙⠝⡅⠇⠀⠉⠉⠉⢈⡗⠀⠊⠈⠀⠀⢱⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠄⠀⠀⡀⢈⠇⠄⠀⠀⠀⠀⠀⠀⠈⠢⡑⠃⠀⠀⠀⠀⠀⠈⠆⢸⠀⠀⠀⠀⢘⠃⠄⠀⠀⠀⠀⣹⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠀⣿⠀⠀⠀⠀⡢⢀⠀⢀⠇⠄⠀⠀⠀⡠⠤⠱⠀⠀⠀⠱⠀⠀⠀⠀⠀⡀⠀⠈⣌⠀⠀⠀⠀⠘⠑⠀⠀⠀⠁⠀⠸⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⢑⠀⡂⠑⠠⠋⢰⠀⠀⠐⠰⡋⣐⡃⡃⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠘⠀⠀⠀⠀⠘⠂⠐⠁⠀⡀⠁⠺⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡀⠀⢀⠨⡂⠑⠆⠸⠀⠀⠐⡀⠸⠵⢂⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠃⢰⠐⠠⠀⠀⠠⠀⢼⢘⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠔⠀⠀⠠⠀⢅⠲⡀⢃⠀⠀⠈⠒⠒⠁⠀⠀⢀⠂⡒⠀⠀⠀⠀⢀⡈⠄⠀⠀⠀⠀⡄⠠⠈⠀⠠⢀⠠⠀⣹⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢁⠠⡀⠁⠀⣀⣄⡰⠄⠑⢄⡀⠀⠀⢀⡀⠔⠡⢞⢽⢀⣀⣀⣀⡈⡿⡰⣀⣀⣀⣀⡄⠠⠀⠴⠀⠀⡀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠤⡊⠥⠈⠡⠤⠄⠀⠄⠉⡉⠒⢂⠡⠀⠀⠈⠂⠄⠀⢠⠄⠀⠙⠂⠂⠀⠀⠒⠬⡀⠂⠁⢀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⠐⠁⣠⠄⠄⠀⠀⠀⠀⠈⡀⢑⠗⢸⠄⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⡀⢀⣀⡀⠀⠀⠈⡌⡆⠤⠈⠊⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂⡀⣤⡃⠀⠀⠀⣀⢃⠀⠀⠐⠈⠶⡸⠂⡆⠀⠀⠀⢠⢠⠈⠀⠀⠀⢢⣘⡿⠜⠀⠀⠀⡄⡸⠂⠀⢀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⢄⠖⡆⠀⠂⠀⠀⠐⠀⠀⠀⠀⠂⢙⢂⡂⠁⠀⠀⠀⢀⢰⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢠⠓⠀⠄⠀⡐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠡⠀⠻⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠘⠌⡏⣇⠀⠀⠀⠀⠀⠨⠌⡄⠀⠀⠘⠏⢷⡀⠀⢻⣄⠯⣿⣳⣿⣾⢿⡷⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠡⠔⡘⠁⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠹⡉⠀⠀⠀⠀⠀⠀⢄⡁⠀⠀⢠⣄⠠⢤⠀⠈⢿⢷⡙⠿⢿⣿⣿⡯⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢤⠂⠀⠀⠀⠀⢸⢀⢺⠘⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⢡⠀⢀⣴⡄⣶⣼⢧⢀⣀⠰⣴⡤⢦⢤⣤⠦⠀⢻⠇
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢻⣦⠘⣄⣒⠒⢰⣤⢫⠊⠜⡦⠡⡤⠖⠊⠈⠄⡠⣍⡩⠓⡭⣗⢛⢊⢸⣭⠩⠰⠁⠂⠇⠈⠈⠤⠀⠱⠀⠻⢆⠄
⠀⠀⠀⠀⠀⠀⠀⢱⡄⣀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⢀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠤⠭⣍⣉⣩⣉⣉⣍⣍⣩⣉⣈⣍⣍⣉⡉⣩⣋⢘⡐⡓⣂⢒⢒⡒⡂⣂⢚⢐⠐⡁⡋⡉⢉⢉⢉⡉⡉⣉⡉⣡⡭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠁⠁⠉⠉⠈⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠈⠄⠈⠀⠁⠀⠀⠀⠀⠀⠈⠀⠈⠄⠈⠀⠀⠄⠂⠁⠂⠅⠂⠄⠂⠁⠈⠄⠊⠀⠂⠁⠀⠀⠀⠐⠀⠂⠁⠀⠀⠈⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠓⠶⣤⡀⠀⠀⠀⢀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣸⢃⣠⠖⠶⠶⠶⠖⠛⠉⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠈⠙⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠈⠙⢻⣿⠉⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⣴⠚⠛⠛⠛⠛⠛⠚⠛⠛⠛⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⡀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⣸⠇⠀⠀⣿⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣸⠃⠀⠀⣿⠀⠀⢸⣇⣀⡟
⠀⠄⠀⠀⠀⠀⠈⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣿⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠿⠤⠴⠛

⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣄⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⢀⠀⢀⠀⢻
⠀⠀⠀⠀⠀⠀⣆⠈⠃⠀⠉⠀⠈⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠘⠶⣄⡤⠤⠤⠰⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠃⠀⠀⠀⠀⠓⠒⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣄⠀⠀⠀⠀⠀⠀⢨⡄⠀⠀⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠉⣀⣄⣀⣠⠈⢣
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⣼⠆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢖⢻⢆⡶⠍⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠉⣉⣠⠴⠊
⠀⠀⠀⠀⢠⡤⠀⠀⠂⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⢢⢱⠸⡀⠀⠀⠀⣀⡀⠀⠀⠀⠈⡭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠨⠁⠐⠂⠀⠀⠀⢀⠀⠴⠀⠀⠀⢱⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠰⡀
⠀⠀⠀⠀⠀⠀⠒⠂⠉⠀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⣀⢰⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣃⠀⠀⠠⠉⠋⠀⠀⠀⠀⡇⠀⠀⢠⡤⢤⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⠐⢓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠃⠈⠀⠘⠉⢸⡄⠀⠀⠀⠠⠀⠒⠀⢀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡖⠒⢳⡶⣋⠛⢫⢻⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠼⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠦⡤⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢳⠀⣀⡇⠈⠀⠀⣘⠃⠀⠀⠀⠀⠀⠀⠀⢀⡄⡄⢠⣤⡀⣤⣤⠀⠀⣃⠀⠀⢐⠃⠀⠀⠀⠀⠀⠠⠀⠄⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡷⡇⢸⣸⡇⢸⡇⠀⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠄⢃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⡀⣤⠰⡀⠀⠦
⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠒⠛⠛⠛⠛
⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠁⠀⠀⠀⠀⠀⢠⡤⠶⠤⠤⠤⠤⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠂
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⢸⠀⠀⢀⡶⠲⡄⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡶⡄⠀⠀⠈⢀⡄⠈⠑⠀⠀⢸⡇⠀⠘⠃⠀⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠐⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⢅⠀⠀⠀⣭⠀⠀⢀⡇⠀⠘⠓⠀⠶⣴⡦⠤⠤⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷
⠀⠐⠒⠀⠀⠀⠀⠒⠂⠀⠀⠙⠙⠋⠁⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠊⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⢸⣿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⣿⣄⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣤⡄⠀⠀⣿⡿⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢥⠀⠍⢻⣧⠀⠀⣤⣄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠙⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣄
⠀⠀⠀⣼⣿⣿⣿⡟⠁⠀⠀⠀⣠⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣄⠀⠀⠀⠈⢻⣿⣿⣿⣦
⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⢀⣼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⡀⠀⠀⠀⠻⣿⣿⣿⣧
⠀⣸⣿⣿⣿⠏⠀⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠈⢿⣿⣿⡄⠀⠀⠀⢹⣿⣿⣿⣇
⢠⣿⣿⣿⡟⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⢻⣿⣿⣿
⣸⣿⣿⣿⠃⠀⠀⠀⣾⣿⡿⠀⠀⠀⢠⣿⠏⠀⠀⢀⣴⠄⠀⠀⠀⠀⠠⣦⡀⠀⠀⢻⣿⡄⠀⠀⠀⢿⣿⣷⠀⠀⠀⠸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⣾⡟⠀⠀⢠⡟⠁⠀⠀⢀⡀⠀⠀⠈⢿⡀⠀⠀⢿⣷⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⢸⣿⣿⡆⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⣿⣧⠀⠀⠘⣧⠀⠀⠈⢹⡏⠀⠀⠀⣼⠃⠀⠀⣼⡿⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⣿⣿⣿
⢻⣿⣿⣿⡀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠸⣿⣆⠀⠀⠘⠳⠄⠀⣸⡇⠀⠠⠞⠁⠀⠀⣰⣿⠃⠀⠀⠀⣾⣿⡿⠀⠀⠀⢠⣿⣿⣿
⠸⣿⣿⣿⣧⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠙⢿⣧⡀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⣀⣼⡿⠃⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⣼⣿⣿⣿
⠀⢻⣿⣿⣿⣆⠀⠀⠀⠹⣿⣿⣧⡀⠀⠀⠀⠙⠿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠈⠿⠋⠀⠀⠀⢀⣼⣿⣿⠃⠀⠀⠀⣰⣿⣿⣿⡏
⠀⠀⢿⣿⣿⣿⣆⠀⠀⠀⠘⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠁⠀⠀⠀⣰⣿⣿⣿⡿
⠀⠀⠈⢿⣿⣿⣿⣦⡀⠀⠀⠀⠙⢿⣿⣿⠂⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠐⣿⣿⡿⠋⠀⠀⠀⢀⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣠⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃

⠀⠀⠀⠀⠠⠐⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠤⢀
⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠑⠂⠰⠀⠐⠒⠉⢀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⣀⣀⣄⣀⣠⣀⠀⠐⠂⠂⠂⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠉⠀⠢⡀⠀⠀⢈⠔⠒⠃⠈
⠀⠀⠀⠀⢀⠁⠀⡰⠀⠀⠀⣠⡆⣀⣒⢀⠄⠠⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⠀⠀⠀⡀⠠⡐⠀⠀⠀⠈⠄⠀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠂⠀⢈⢠⠟⠀⠀⠉⠉⠓⠺⡄⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⡀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠔⠅⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠼⠀⢀⡀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠐⠀⡼⠴⠀⠠⢬⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡑⠀⠉⠀⠀⠈⠉⠁⡤⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠴⠦⠆⠀⠔⠋⠁⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⠠⠄⠀⡆⠠⢸⡀⠀⠄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠐⢄⡀⠀⠀⡈⠀⡠⠂⢀⡀⠀⠀⠀⠀⠀⠙⢄⡁⠂⠀⣀⡄⢁⠀⡀⠳⣄⠈⠀⠁⡠⡞⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠢⢄⣀⢥⠚⡇⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠀⠀⠀⠆⠙⠒⠊⣩⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣁⠀⠁⠂⣤⠐⠈⢀⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⠒⠓⢤⡀⠀⡠⠓⣥⠤⣇⠸
⠀⠀⠀⠀⠀⡄⠒⠉⠀⠀⠀⠙⠒⠚⠑⠙⠒⠉⠀⠀⠀⠉⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠁⠂⠤⠤⠅⠀⠈⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⡇⠀⠒⠀⢰⠀⠀⡖⡆⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⡀⡠⠂⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠁⠊⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠦⠔⠒⠒⠢⠴⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⣠⢤⣔⣒⣒⣈⣉⣭⣭⣭⣀⣒⣒⡢⠤⣄⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠛⠉⠙⢻⣿⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣻⣿⣿⡀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣧⠘⢿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣀⣥⣭⣭⣅⢉⣾⣥⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠟⣷⣄⠙⠻⢿⣿⣷⣶⣤⣤⣄⣀⣀⣠⣤⣤⣶⣿⣿⣷⡧⠄⣠⣴⠬⠶⣾⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣄⣀⣿⣿⣷⡄⣀⡉⠙⠿⢛⣻⣿⣿⣿⣿⣛⡛⠿⠛⢉⣀⣴⣾⣿⣿⣀⣠⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠿⡿⣿⣿⣿⣿⣗⡏⠉⠑⠒⠲⠶⠤⠤⠤⠤⠶⠒⠒⠊⠉⢹⣿⣿⣿⣿⣿⢿⠿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⢻⡆⠇⢱⠀⠀⠙⠿⢿⣴⣒⡠⠀⠀⠀⠤⠄⠀⠀⠀⢄⣒⣾⡿⠿⠋⠀⠀⡎⠸⢰⡟⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⢸⡇⢰⠈⡄⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⢠⠀⡇⢸⡇⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⢸⠃⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠸⠀⠀⠀⡰⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⢦⠀⠀⠀⠇⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠤⠶⠒⠒⠒⢚⠉⢩⢡⢄⢳
⠀⠀⠀⠀⠀⠀⢠⣤⣭⣉⣛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⣉⣉⢉⣀⡀⠠⣤⠀⡠⠴⠶⣄⠀⠀⠀⠀⠀⠈⠈⣆
⠀⠀⠀⠀⢀⣴⣞⡄⡟⠀⠀⠀⠀⠀⠀⠀⠲⣞⢢⡀⢋⠀⢹⠹⡉⣙⠙⢞⢆⠱⣽⡄⢸⠀⠻⢶⣤⣦⣄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⣤⢻⣻⣷⡳⣹⡀⠀⠀⠀⠀⠀⠀⠀⢏⢣⡱⣜⡄⠈⣇⢻⠻⠂⠀⠳⣱⠏⢮⣾⠀⠀⠲⡦⠤⠿⠂⠀⠀⣀⣀⡠⠀⠹⡄
⠀⠀⠈⢏⢯⢻⣿⣧⢳⣳⡀⠀⠀⠀⠀⠀⠀⠸⠦⠑⠮⠇⠀⠸⠾⠦⠼⠃⢀⣈⣀⣀⣡⡤⠤⠤⠴⠒⠒⠚⠉⠉⠁⠐⢒⣂⡄⠹⡄
⠀⠀⠀⠈⢾⡎⢿⣿⡷⣳⣳⡀⠘⢓⡒⢒⣛⠓⣒⣒⠚⠉⠉⠉⠉⠉⠉⠉⠩⠔⣒⣂⣒⣒⣛⣀⣠⠤⠤⠔⠒⠒⠚⢉⠁⣀⡄⠀⠙⡄
⠀⠀⠀⠀⠈⣿⡜⡿⣷⡽⣷⡱⡀⠐⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⢒⣋⣉⠩⠭⠭⠥⠤⠴⠶⠖⢢⡀⠐⠶⠚⠘⣚⣉⡠⠥⠴⠂⠀⠘⣄
⠀⠀⠀⠀⠀⠸⣷⢹⣿⣣⡻⡵⡱⡄⠀⠒⠃⠲⠶⠐⠒⠂⠒⠀⠀⠈⢣⠀⠀⡠⡠⡠⢠⠀⠀⠀⢠⢳⡀⠈⠭⠥⢄⣒⣒⣠⠠⠤⠄⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⢻⢣⢳⢿⡳⡹⡝⡜⢄⠈⢉⣉⣈⣉⣉⣁⣀⣀⡀⠀⠀⢧⠀⠐⠱⠱⢱⠁⢠⢃⢇⠏⣷⠀⠈⣉⠁⠤⠤⠐⠒⠒⠋⠁⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠈⡞⡏⣏⣷⠹⣿⡜⢎⢦⠀⠐⣒⣒⣀⣀⣀⣈⢭⡅⠀⠈⣆⠄⣠⢧⢣⢣⢃⣆⣎⣼⣼⠇⠀⠀⢒⣈⣉⣭⠥⠤⠴⠆⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢸⢹⠸⡼⣏⠽⡟⣎⢧⢣⡀⠰⠒⠒⣒⣒⣛⣋⣉⡀⠀⠘⠚⠛⠛⠛⡉⣉⣉⠀⣄⣠⠄⠀⠀⠀⠒⢒⣒⣂⣛⣛⣀⣀⡀⠀⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡎⣇⢣⢻⡖⠽⡘⡌⢧⢱⡀⠉⠭⠤⠴⠖⠒⠒⠒⠀⠀⠙⠚⠛⠀⠛⢉⣁⣀⡄⠀⠀⠀⠀⠀⣈⣩⠤⠤⠤⠤⠤⠤⢄⢀⡂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⡜⣎⣿⡭⢧⣣⠸⡆⢃⠀⢒⣚⡉⠉⠉⠉⠉⠀⠀⠘⢋⣉⣉⣁⣨⠁⠉⠠⢐⠒⡒⢃⣀⡷⣠⢄⡤⠀⣠⣶⢄⡿⣟⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣳⡘⣎⢿⣿⣿⢀⣇⢸⠀⠠⠤⢤⠤⡖⢲⣒⡆⢠⠈⡽⢴⡗⣲⣳⣟⣻⣋⣽⣭⣋⣭⣭⣡⣍⣩⣃⣨⣷⣿⣾⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡳⣌⠳⠬⣿⣿⣫⡟⡄⢰⣖⣟⣩⣯⣏⡼⠤⠟⠒⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢍⣓⣿⣿⣿⠴⠴⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⡠⢀⠄⡠⢖⡤⢊⠴⢂⠔⢊⠴⢂⠴⢊⠤⢊⡴⢂⠔⢃⠔⢂⠔⣃⠔⣀⠔⡠⢂⡤⠂⡠⢂⡴⢂⠔⢀⠔⢁⠔⣠⠞⡠⢂⠔⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠊⠰⠋⠔⠁⠔⠁⠔⢡⠖⢁⠔⠁⠴⠫⠖⢡⠔⢡⠔⢡⠞⢡⠞⠥⠊⡴⠋⡴⢉⡔⢁⠔⢁⠔⠁⠔⠡⠞⠡⠊⠐⠃⠚⠐⠁⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⠓⠶⣄⡀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⣠⣀⣀⣀⣠⠴⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⡤⠖⠛⠛⠲⣼⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠟⠛⠻⢿⣷⡖⠋⠉⠙⠳⣄
⠀⠀⣾⠁⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⢸⡆
⠀⠀⢿⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠈⠳⢤⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠞⠁
⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⢠⠄
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⢠⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢠⠀⠀⢸
⠳⣤⣾⠀⠀⢸⠀⠀⠀⣧⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⡇⠀⠀⢸⠀⠀⢸⡦⠴
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠹⣄⠀⢸⡀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣼⣀⣀⡼⠁
⣶⣶⠶⠾⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⠴⢋⠁⣤⣤⠈⠉⠳⣄⠀⠀⠀⠀⠀⠀⣠⡖⢋⡧⢸⠁⠈⠙⠲⡄⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⢈⡠⢋⠝⢦⡀
⠀⠀⠀⠀⠀⢰⠏⢂⡈⠀⢨⠁⡀⠘⠃⡈⣧⠀⠀⠀⠀⢰⠇⠀⠂⠠⣄⠉⡉⢉⣉⡹⡆⠀⠀⠀⠀⡼⢃⠄⠂⡀⠤⠤⠊⡠⠊⢹⡄
⠀⠀⠀⠀⠀⢾⠀⢅⡨⠀⢀⠤⡄⠻⠃⠀⢸⠀⠀⠀⠀⣿⠤⠤⠄⠒⠃⠘⠋⠘⠒⠂⣻⠀⠀⠀⠀⣟⣁⠄⢊⡀⠄⢂⡡⠔⢊⢵⡇
⠀⠀⠀⠀⠀⠸⡆⡀⡄⠀⠘⠶⠋⠀⠤⢀⡞⠀⠀⠀⠀⠸⡦⠴⠀⠚⠀⠫⠭⠥⠤⢠⡏⠀⠀⠀⠀⢳⡦⢊⠅⠂⠉⠐⠒⡈⠐⣹⠇
⠀⠀⠀⠀⠀⠀⠙⢮⡁⠀⠁⠀⠤⢀⣠⠞⠀⠀⠀⠀⠀⠀⠙⢮⣭⠭⠥⠤⠶⢐⡶⠋⠀⠀⠀⠀⠀⠀⠻⢵⣨⠔⡂⠉⠍⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⣠⢞⡉⠒⠀⠤⠤⠬⠓⢦⠀⠰⠁⠄⡐⢉⣻⣟⣉⠁⠀⠂⠀⠉⢳⣄⠀⠀⠀⠀⠀⠀⡴⠋⣡⠀⠄⣒⡢⠉⠳⣄
⠀⠀⠀⠀⠀⢰⡇⠀⠉⠒⠤⢀⣀⣐⠍⣈⣧⠈⢄⠡⠑⢄⠑⢬⣉⣉⣉⣉⣉⣉⠉⠼⣇⠀⠀⠀⠀⡼⢱⠄⠀⢀⠂⠁⠑⠈⠛⠹⡆
⠀⠀⠀⠀⠀⢸⠉⣑⡒⠀⠤⠤⠤⠤⠤⠀⢾⠀⠀⠑⢄⠙⠟⠢⣄⠀⠲⠄⡀⠰⠐⠒⣿⣷⣄⠀⠀⡇⠀⠠⡠⠌⠀⠀⠂⠘⢃⠀⡇
⠀⠀⠀⠀⠀⠘⣆⠓⡁⠄⠀⠀⠀⢀⡀⢰⡏⠀⠀⠀⠀⠹⡶⢌⡁⠸⠃⡀⠠⠉⡉⠀⠛⢀⠙⠱⡄⢱⡀⡑⠉⠂⠀⠰⣂⢠⠈⣰⠃
⠀⠀⠀⠀⠀⠀⠈⠣⣄⠐⠀⠀⠐⣢⡵⠋⠀⠀⠀⠀⠀⠀⠈⠶⣌⡳⠶⣤⣁⣀⠀⠛⠁⠀⠈⢀⠄⠀⠙⢦⣀⠐⠀⠈⢀⣨⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁⠈⠉⠁⠈⠉⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⢦⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⢤⣤⣀
⠀⠀⠀⠀⠀⠀⣰⠊⠀⠈⠉⡝⠀⢀⡟⢦⡀⠀⠀⠀⠀⠀⡰⠛⠡⠡⠈⢈⡤⠸⢛⢦⠀⠀⠀⠀⠀⢀⡴⠯⠥⠬⠥⠤⠔⢋⠷⣄
⠀⠀⠀⠀⠀⢰⢣⠀⢤⠐⠁⠀⢀⡬⠛⠼⣷⠀⠀⠀⠀⢸⠃⡀⠰⡅⠴⢀⣀⢣⠱⠬⣇⠀⠀⠀⠀⡾⠬⣉⣑⠒⠀⠒⢊⣁⣠⢼⡆
⠀⠀⠀⠀⠀⢸⠀⠁⢰⠉⠉⠲⡌⠒⢦⠀⢸⠀⠀⠀⠀⢟⠢⡍⠀⡊⠉⠁⢎⠘⠲⢒⡿⠀⠀⠀⠀⡯⢑⣒⠂⢀⣩⡽⠴⢒⡪⠭⡟
⠀⠀⠀⠀⠀⠘⣆⠀⠀⢡⠀⡰⠁⠀⠘⣠⠏⠀⠀⠀⠀⠸⣆⠃⠈⢀⠆⢀⠀⠉⠉⣴⠃⠀⠀⠀⠀⠹⣆⢀⠀⠉⠁⣒⡒⠉⠀⣺⠃
⠀⠀⠀⠀⠀⠀⠈⠳⢤⣘⡘⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠈⠣⣤⣀⡠⣈⣳⡽⠞⠁⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⣠⡬⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣴⢂⣴⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⣠⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣠⣴⣾⡟⣼⡇⣾⣿⣿⣿⡇⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠚⢿⠟⠂⠀⠀⠀⠀⠀⣀⡤⣷⣿⣿⣿⣇⣿⣇⢿⣿⣿⣿⣿⣦⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⢖⣵⣿⣷⣾⣿⣿⣿⣿⣿⡸⣿⡌⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⣠⣶⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣦⡙⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⠿⠶⣤⠄
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠓
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⢿⣿⣿⣇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣌⠻⣿⣿⣿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣾⣿⠀⠀⢀⣰⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⠀⣶⣿⣿⣿⣿⣿⣿⡿⠟⢩⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⡀⠙⠟⠉⠁⠈⠉⠁⠀⠀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⣼
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⣼⣿⡟⠀⠈⠙⠉⠸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣀⣰⣦⣀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣧⠹⣿⣆⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠹⣿⡆⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⢻⣦⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣶⣶⣾⣿⣶⡶⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠃⠀⠘⠀⠘⠃⠀⠛⠀⠀⠀⠸⠀⡂⠀⠀⢀⠀⠁⠘⢀⠀⠠⠀⠀⠐⠂⢐⠀⠀⠀⠐⠟⠀⠘⠃⠀⠃⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣬⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣯⣿⣽⣿⣾⣿

⠀⠀⠀⠀⠆⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⡀⠻⣟⡉⣽⣋⠙⠙⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠐⠀⠀⠀⣽⣎⣽⠳⣦⡘⠻⣧⠀⠀⠀⠀⠀⠀⠀⢠⡄⠚⢛⠒⡒⢒⠋⡏⠛⢶⡶⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣾⣆⢻⣷⣸⠀⠀⠀⠀⠀⣠⡼⢶⡗⠤⡠⠈⠀⠡⠘⠠⠘⠨⣑⠈⠉⠙⠛⠲⣄⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⣷⣇⣶⡒⢤⡀⠞⠁⠀⠀⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠐⢛⣏⣀⣀⣀⣴⣲⢪⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⢸⠉⠀⠠⠉⠓⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠛⣆⠀⠀⠀⢸⠺⠭⠭⠽⠀⢹⠏⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⢳⠏⠀⠀⠀⠀⢤⡀⠀⢀⡄⢾⣅⠹⠆⠈⠁⠀⠀⠀⠉⠉⠿⣝⣻⠋⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢦⡀⠀⠀⢀⡔⠁⠀⠀⠀⠀⢀⠲⣝⢆⢦⡙⠒⡌⠋⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣴⣇⣶⣆⡼⠀⠀⠀⠀⢴⡛⢦⡣⣙⠷⠙⠉⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡠⣼⡀⠻⡌⠓⣀⣤⡀⠀⠀⣀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠷⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢠⡄⠹⣎⠳⠄⣠⡾⠋⢾⣥⠶⠛⠉⠀⠀⠀⣾⣩⡿⠿⡍⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⣅⠀⢰⠋⠳⡝⠦⠝⣠⠞⠁⡠⠊⠓⠀⠀⠀⠀⠀⠀⠀⢯⠹⠮⠽⠟⣼⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠻⣌⠷⣤⠏⣠⡞⠁⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠿⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⡀⠀⠀⠀⠈⠉⠉⠉⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⣥⣀⡀⣀⣀⣠⣤⠞⠋

⠀⠀⠀⣀⣀⣀⣀
⠀⠀⢸⡟⠛⠛⢻⡇⠀⠀⠀⢸⠟⠛⠛⠛⠛⢻⡇⠀⠀⠀⣠⠴⣖⣷⠦⣀⠀⠀⠀⠀⠀⢤⠀⢀⣀⣀
⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⣯⢻⢫⠝⡟⢹⠀⠀⠀⠀⠀⢿⠀⢀⣿⣽⡗
⠀⠀⠀⣧⣤⣤⣼⡇⠀⠀⠀⠘⠛⢛⣿⣿⡛⠛⠃⠀⠀⠀⠈⠛⢼⡤⠗⠊⠀⠀⠀⠀⠀⠉⠀⠀⠈⠗
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⣠⣾⣫⣟⣷⣄⠀⠀⠀⠀⠤⢒⣉⣉⡒⠤⠀⠀⠀⠀⠀⢘⣧⡴⢒⣆⠀⠀⠀⢰⠒⠒⡒⠒⣶⣶⡆
⠀⠀⡧⡧⠴⡧⢾⢾⠂⠀⠀⠀⠀⠡⢒⣒⠌⠁⠀⠀⠀⠀⢰⣫⣤⣿⣿⠏⠀⠀⠀⢸⠹⡇⡇⢦⣬⠍⡇
⠀⠀⠙⢿⣾⣷⣻⠟⠀⠀⠀⠀⠀⠀⢱⡦⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣿⡀⠀⠀⠀⠸⠤⠤⠥⠥⠶⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠠⢲⢶⣀⣶⡲⡄⠀⠀⢠⠒⠉⠀⠉⠉⠐⡄
⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⢠⢀⠵⠤⠤⢎⠙⠀⠀⠀⢰⣟⠋⣾⣿⡦⢏⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡏⣿⣿⡿⠁⠀⠀⠀⠈⠛⠒⠒⠒⠒⠛⠀⠀⠀⠈⠹⡶⠾⠿⠷⡾⠀⠀⠀⠀⠉⠒⢦⠔⠀⠉
⠀⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⡀⢀⢄
⠀⡔⠀⣲⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⢸⠄⢠⠀⠀⠀⠀⣦⣼⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠑⠠⠿⢁⠔⡠⠤⣄⠀⠀⠀⡄⠀⠉⠉⠀⢸⠀⠀⠀⠀⢿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠮⠥
⠀⠀⠀⠀⠃⠚⢇⡀⠿⠀⠀⠀⡇⠀⠀⠀⢠⠞⠀⠀⠀⠀⣼⣛⣿⣟⣟⡿⠀⠀⠀⠀⠀⢰⢲⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠘⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠒⠈⠉⠉⠉⠁⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀⣀⣤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡾⣃⠈⢉⣩⠆⠀⠱⡀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⣡⣎⠉⠀⠓⣄⠀⠁⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⡀⢸⠀⠀⠀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠈⠀⢠⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠰⠂⠋⠀⠁⠀⠀⠀⣾⠀⠀⠀⠙⠓⠲⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠉⠉⠉⠉⠉⡹⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠓⢄⡀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣟⢻⠀⠔⠒⠉⠀⠀⠀⠀⢡⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠧⢤⡶⠤⣝⣇⡠⠔⠁⠀⠀⠀⠀⠀⡘⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠈⠓⡪⠋⠁⠀⠀⠀⠀⠀⠀⣠⣜⡀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠘⢠⠴⣶⢢⡤⠀⠠⠔⠊⠀⠁⢨⠗⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⢁⢀⠀⣏⠻⣄⣀⣀⣠⠤⣔⢊⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⢀⠴⢋⡉⠛⠋⠉⠛⠁⠉⠲⡋⠉⠙⣁⣀⢈⠳⢤⠀⢠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⣢⠀⠀⡈⠉⠑⠊⠉⢉⠉⠉⣀⠉⣉⣤⣀⡠⠄⠘⡇⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆⠀⡇⠀⢀⠁⠛⠉⢙⡛⠉⢹⠉⣀⠉⠁⠀⢹⠀⢾⠀⢸⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⡏⠀⢸⠀⢣⣀⠈⡇⠀⣤⠀⣾⠀⠀⡞⢳⠁⡸⠀⢣⠈⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⡷⠀⢸⠀⢎⠀⢀⣱⣄⣀⡰⠛⠤⠴⠇⡾⠀⠁⠀⠌⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⢣⠀⠸⠀⣼⣀⡄⠀⠈⡎⠂⢸⣦⣀⡨⠆⡴⠂⢀⠄⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢨⡀⠀⠁⢈⣅⣀⠠⢺⡤⠰⠊⠀⠉⢠⠔⠁⠀⡹⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠀⠙⡞⠁⠀⠀⢀⠃⠀⠀⠚⢉⣁⠀⣀⠖⠊⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢸⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠸⡀⠀⢉⡎⠁⠀⠉⠀⠀⠀⠀⠀⠀⡇⢸⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠢⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠉⣉⣉⡀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⢃⣴⣶⣿⠆⠀⢀⠀⠙⠿⠿⠷⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟⠁⠘⠋⠉⠀⠀⠀⡉⢆⠀⠀⢰⣷⠀⠀⣸⡉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠩⣸⡀⠀⠀⠀⢸⣿⠀⢀⡇⠸⣆⠀⠀⠉⠀⠀⡏⣁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡈⣃⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⢠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⢀⣄⣀⣤⣴⣶⠀⠀⠀⡎⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠈⢿⡿⠿⠿⡿⠀⢀⡜⠀⣠⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠈⠑⠒⠊⠀⣠⠊⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠓⠠⠤⠤⠤⠊⢁⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠻⡿⣋⠘⡄⠠⣿⠏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠸⡌⢛⡕⢫⠀⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣦⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⠟⠓⠢⡑⠮⣀⡰⠖⠛⠛⠂⣀
⠀⠀⠀⠀⠀⡀⠠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠉⠙⠛⠿⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠈⢦⠌⠀⠀⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⡼⠮⠑⠃⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡤⠀⣠⠤⢤⡀⣽⣿
⠀⠀⠀⠀⣏⠁⠀⠀⠀⢰⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡜⡇⠀⠘⣿⠿⠃⠇
⠀⠀⠀⠀⠓⠤⠤⠤⠤⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⡠⠇⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⣀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠒⠂⠀⠈⠉⠁

⠀⠀⠀⠀⠀⢀⡠⢴⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⢠⢯⢾⣽⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⣚⣻⣿⣿⣛⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠘⠚⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠭⢽⣿⣿⠭⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣰⣄⣿⣿⣿⣿⣿⣇⣦⡀⠀⠀⠀⣷⢾⣿⣿⣿⣿⡷⡷
⠀⠀⠀⠀⠀⡀⣿⣿⣿⣿⣿⠀⡀⠀⠀⠀⢸⡀⣿⣿⣿⣿⣿⠇⡇⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⠂⡇
⠀⠀⠀⠀⠀⢇⠻⣿⣿⣿⡿⢠⠇⠀⠀⠀⠀⢳⡘⠿⠿⠿⢋⡴⠃⠀⠀⠀⠀⠸⣄⠻⢿⡿⠛⣰⠃
⠀⠀⠀⠀⠀⠈⠳⠬⣽⡭⠔⠋⠀⠀⠀⠀⠀⠀⠈⠛⢲⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡟⠋
⠀⠀⠀⠀⠀⠀⠀⣤⣼⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣧⣤

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣗⡂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣲⣖⣿⣿⣿⣿⣗⣶⠂
⠀⠀⠀⠀⢸⢹⣿⣿⣿⣿⡏⢻⠀⠀⠀⠀⢰⢸⣿⣿⣿⣿⣿⢠⡆⠀⠀⠀⠀⢸⡁⣿⣿⣿⣿⡇⡿
⠀⠀⠀⠀⠈⢦⡙⠿⠿⢟⣡⠋⠀⠀⠀⠀⠘⣆⠻⣿⣿⡿⢋⡼⠁⠀⠀⠀⠀⠈⢧⡙⠿⡿⠟⣠⠃
⠀⠀⠀⠀⠀⠀⠉⠙⡟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⡷⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡗⠋⠁
⠀⠀⠀⠀⠀⠠⠤⠴⠧⠤⠤⠀⠀⠀⠀⠀⠀⠀⠠⠤⠷⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠧⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢈⡁⡐⢄
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠁⠘
⠠⠀⡶⠉⢫⠑⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡚⠛⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⢼⣉⠀⠀⢖
⡄⢻⡤⠔⠚⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⠴⠊
⠘⢦⣈⣓⣂⡤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠤⣜⣂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠠



⠀⠀⢠⠖⢦⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠄
⠀⠀⢸⣄⣬⡀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠒⡂⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⡆
⠀⠀⠸⣿⣇⠇⠀⠀⠀⠀⠀⠀⢧⣀⣈⣉⡁⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠳⡅⠀⠜⢀
⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠘⠛⣿⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡶⠄



⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠃⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡍⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡘⢢⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠻⠄⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⢔⠂⢹
⠀⠀⠧⠤⠭⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣒⣒⣒⣲⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⣞⠡⠨⠤⢌⣳⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠈⠉⠉⠉⠐⠒⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⢋⣷⢄⡀
⠀⠀⠀⠀⢀⣀⠠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠛⢁⣴⠟⠁⠀⠉⠢⡀
⠀⠀⠀⠻⣥⠄⠒⠒⣉⣱⢤⡀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠛⠉⢀⡴⠋⠀⠀⠀⠀⠀⣠⣿⣄
⠀⠀⠀⠀⠀⠑⢯⣁⡩⠶⣔⠛⢢⣀⠀⠀⠀⠀⡜⠀⠀⠀⣀⣠⣤⣶⣿⣿⡿⠟⠋⠁⠀⡠⠞⠁⠀⠀⠀⠀⠀⣠⣾⣿⡿⣩⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠢⣖⠊⣛⣥⡨⠷⢄⠀⣼⣶⣶⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⢉⣼⡿⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢤⡤⠝⠶⣊⠙⣻⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⢁⣴⠟⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠉⠙⠿⣖⣒⡭⢿⠤⠤⢐⣒⡺⡍⠑⡔⢢⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠋⣠⠶⠋⠀⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠙⡹⣆⠈⠆⢿⠿⣿⠛⢸⠀⢱⠴⠁⢀⣠⣤⣶⣿⣿⠿⠛⠉⣀⠴⠊⠁⠀⠀⠀⠀⢀⣼⠟⢉
⠀⠀⠀⠀⠀⠀⢈⠀⣀⣀⠤⢄⠀⠀⢡⢻⠀⢸⠘⢃⣉⣶⣼⣤⣾⣶⣿⣿⡿⠟⠛⠉⣀⡤⠔⠋⠀⠀⠀⠀⠀⠀⣠⡶⠋⢁⣴⡿
⠀⠀⡠⢤⣤⡲⢯⠭⠔⠒⠊⠉⠀⠀⢸⡿⠄⢺⣿⣿⠿⣉⡩⠷⢤⡚⢫⣄⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⢀⣴⡿⢛⠇
⠀⠀⠑⠉⠀⠀⠀⠣⡀⠀⠀⠀⠀⢠⠟⠀⠀⠀⢣⣤⣤⣼⠶⣖⠊⢙⣧⣤⠵⢤⡀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⣠⣶⠟⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠖⠁⠀⠀⠀⠀⠀⠫⡀⠀⠀⠀⠙⢤⡤⠜⠳⣎⣉⡦⣤⠔⠚⠉⠀⢀⣠⣴⠟⠋⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣤⣤⣤⡼⠶⣉⣉⠹⠤⠘⠓⢤⣴⠾⠛⠉⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢍⣀⣀⣀⣬⣷⣤⠴⠖⠚⠉⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠫⠥⠀⣀⣀⣀⠠⠤⠔⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣂⡄
⠀⣀⠀⢂⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠉⡙
⠈⠁⠀⠀⠀⡀⠘⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⡁⠀⣉
⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⢛⢒⢻
⠀⠀⠀⠀⠀⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⢨⠁
⠀⠂⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⠸
⠀⠀⠄⠀⠀⠀⠁⠘⡿⢰⠒⢤⠀⣀⢀⣀⣤⠀⡀⠀⠄⠢⡄⠀⡠⠒⢤⡆⡠⢤⣀⠀⠀⣠⠖⠢⠆
⠀⠀⠐⡀⠀⠀⠀⠀⠛⠹⠠⠈⠠⠼⠇⠨⠃⢀⠇⠀⠀⠀⠀⢘⠃⣀⠈⠁⠃⠀⠱⠉⠀⠃⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠑⣠⠐⢳⠀⠈⠀⠁⠈⠀⠀⠀⠉⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣨⡀
⠀⠀⠀⠂⠃⠀⠀⠀⠈⠛⡟⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣓⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢋⠭⢽⠭⢎⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢼⣏⠘⣴⣗⠀⠐
⠀⠀⠾⠄⠘⢂⡐⠀⡯⠀⢹⠁⢼⡾⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣟⣏⠈⠘⣿⠃⠀⡀⡆
⠀⠀⠀⠀⠀⢈⠀⢸⠃⠀⢸⠀⠘⢇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⢉⡤⠈⠠⢀⠀⡆⣤⠃
⠀⠀⠀⠀⠀⡼⡄⢺⠃⠀⢸⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠠⠀⠀⣿⠐
⠀⠀⠀⠀⠸⡇⠃⢸⠃⠀⢸⠒⢄⣏⢀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡁⠀⠁⠈⣀⣾⣤⣅⡠
⠀⠀⠀⠀⠚⠂⠀⠀⠘⡤⢈⠀⠊⠏⠸⡀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⣾⡟⣿⣿⣿⣿
⠀⠀⠀⠀⢠⡄⠀⣨⡀⠁⢨⠰⣄⡀⠀⠘⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢻⣷⣿⣿⣿⡟
⠀⠀⠀⠀⠉⡁⠐⡇⠑⠀⡀⠀⠅⠃⠀⢀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠘⠀⠀⠀⠘⠛⠛⠿⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄⡀⠸⢀⣄⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⣀⣴⣾⡯⠤⠾⠛⠁⠀⠀⠀⢠⣿⣥⣿⣿⡷⠀⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠛⠛⠋⠂⠈⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣇⠄⠈⣾⠇⠀⢀⣀⡀⠀⠀⠀⠀⠀⣸⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠏⠈⡄⠀⡠⠔⢏⠉⢹⣇⣇⣦⠖⠁⠀⣴⣿⣿⣿⡆⠀⠀⠀⠤⣿⡀⠃⣼⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢵⣢⣇⣴⠿⣦⣤⣼⣿⠾⣷⣆⠀⠹⣿⣿⡿⠃⠀⠀⠀⡖⣿⣏⣹⣿⢸
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠄⠄⠀⠀⠈⠁⢻⣄⣽⣿⡿⠧⠤⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⡇⢸⣿⣹
⠀⠀⡤⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⢰⠀⠀⢻⣥⣽⣷⠀⣠⣴⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⡷⡿⡇⢸⢿⢾
⠀⠀⣤⣀⣴⡇⠀⠀⠈⠉⠛⠶⢶⣾⣄⣤⡀⠀⠈⢿⡏⠀⣿⡓⣶⣵⡀⠀⠀⠀⠀⠀⠀⣿⠃⡇⢸⡎⢿⡀
⠀⢾⣿⠋⠙⠳⣄⠀⠙⠓⠶⠟⠋⠀⠈⠉⠉⠀⠀⠈⢿⣤⣴⡝⢿⣟⠁⠀⠀⠀⠀⢀⣾⣹⡿⢹⣾⢹⣿⣷⡄
⠀⢠⣿⣦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡋⠁⠘⣉⠀⠀⠀⠀⠀⠀⠁⣟⣷⢿⣿⢾⣘⠆⠁
⠀⠉⠀⠀⠉⠳⢴⣄⣀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠻⢥⡑⣄⠀⠀⠀⠀⠘⠁⠈⠁⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣈⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢂⠀⠀⠉⠁⠀⠀⠀⠀⣠⡖⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠈⠙⠲⢕⢄⠀⠀⠀⠀⠀⠀⠀⢹⣤⡆⠀⠀⣰⠃⠀⠘⠋⠉⠠⠂⡀⠐⠀⣆
⠀⠀⠀⠠⠐⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⢀⢈⠀⠀⠀⣄⠁⠰⠆⠀⡀⣐⣰⣿
⠀⠀⠀⠀⠘⢠⣤⡀⠀⣠⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⣤⠶⠷⠛⣨⠀⠀⠈⣶⣴⣰⣾⣥⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠃⠀⠛⠏⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡤⠚⠋⠀⣀⣴⣾⣯⣄⣀⠀⠈⠙⠛⠛⠛⠛⠁⠀⡀
⠀⠀⠀⡀⠀⠀⠀⢀⢀⠀⠀⠀⠠⠀⠀⠀⠀⣤⠚⠋⠀⢀⣤⡼⢟⣋⠥⠭⠯⠥⠭⠵⠖⠀⠈
⠀⠀⠈⡄⠀⠀⠀⠈⠁⠀⠀⢀⠐⠁⠀⣠⢾⠀⣀⣴⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠤⣶⣾⠛⠁
⠀⠀⠀⠀⠀⠔⠂⠄⠀⠄⠒⠐⠁⠀⠰⠿⠷⠛⠋⠉⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢰⡇⠀⠀⣴⠶⠤⣤⣤⣤⡤⠶⠶⠛⠉⢹⡇⠀⠀⢸⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠠⠶⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⣀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠉⠙⠲⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟⢷⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢿⣿⣿⣿⢇⣤⣶⣶⣶⣦⣄⠀⠀⠘⣆⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡅⠀⠀⠳⣄⠀⠀⠀⣏⠀⠀⣹⠆⠀⠀⠀⠀⢀⣴⠏⠀⠀⠈⠁⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⣬⣷⣤⣤⡼⠟⠛⠉⠀⠀⢀⣠⡴⠛⠁⠀⠀⢀⠀⠀⣬⣭⣭⣽⣿⣿⣯⣭⣭⡅⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠙⠛⠓⠒⠛⠛⠉⣿⣀⣀⠀⠀⠀⢸⡇⠀⣩⣭⣭⣽⣿⣿⣯⣭⣭⡅⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⣿⣿⣿⣄⡀⠀⠀⠀⢀⣰⣿⣿⣿⠙⠂⠀⢸⡇⠀⣩⣭⣭⣽⣿⣿⣯⣭⣭⡁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⠀⠀⣿⣿⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⢉⣉⣉⣽⣿⣿⣯⣉⣉⡁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⠙⠿⠿⠏⠀⠀⠀⠈⢧⠀⠘⢿⣿⣿⣿⣿⣿⣿⡟⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠉⠛⠻⠛⠛⠉⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣤⣤⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉



⣿⠿⠿⠿⡿⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣻⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠈⠂⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⢀⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡛⠁⠀⠀⠀⠀⠀⠛⣿⣟⠋
⠀⠀⠀⠀⠄⠁⠀⠀⠀⠐⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣷⡄⠀⠀⢀⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⠆



⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢤⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣦⡀
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢣⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿⣿⣧
⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣭⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⠎⣀⣀⣀⠤⠤⠶⣚⡉⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⠿⠛⠛⠉⠽⣶⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠦⠤⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠆

⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠰⠩⠀⠠⠑⡄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣾⣿⣾⣷⡆⠀⠀⠀⠀⠀⢠⣤⠶⡶⢶⡶⣦⣄⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠐⠂⡒⠲
⠀⠀⠀⠀⡐⡄⣦⡄⠀⢃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⡇⠀⠀⠀⠀⠀⠈⢩⣿⠩⡯⣻⣎⠉⢱⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⠀⢸
⠀⠀⠀⠀⠡⠁⠈⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⢀⡇⠀⠀⠀⠀⠀⠀⠿⠿⠷⠾⠿⠿⠖⠃⠀⠀⠀⠀⠀⠀⡸⠅⠤⠤⠼⣀⠌
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⣠⣶⣾⣿⢶⣤⢀⠀⠀⠀⠀⠀⠀⢰⠶⡐⠂⠒⢂⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡍⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⢿⣿⣭⣿⣽⣿⠃⢱⠀⠀⠀⠀⠀⢸⠀⠈⠲⠞⠁⣠⣾⣎⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⡇⣀⣭⣯⣄
⠀⠀⠀⠀⠀⠿⠟⠫⣡⠒⠁⠀⠀⠀⠀⠀⠘⠏⠀⠠⢀⡸⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⡸⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠆⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃



⠀⠀⠀⠀⣿⣿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⡄⣠⣦⡀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠠⠐⠒⣦⡀⠀⠀⠀⠀⠀⠀⡖⠂⠒⣀⠂⠒⢲
⠀⠀⠀⠀⣿⣿⣀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⣘⣟⣁⠀⢃⣙⣟⣁⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⣽⡇⠀⠀⠀⠀⠀⢀⡥⡄⠠⠿⠄⠀⢸
⠀⠀⠀⠀⣍⣉⣉⣉⣉⣉⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇⠺⠿⠿⠿⠂⠀⠀⠀⠀⠀⠿⠷⠶⢤⡤⠤⢼⠅⠀⠀⠀⠀⠀⠸⣿⠟⡿⢿⠿⠟⠛



⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠰⡶⠀⠼⠧⠀⢶⠆⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠿⠿⠿⢿⠀⠀⠀⠀⠀⠀⠀⡛⢍⠉⠉⣉⠝⡇
⠀⠀⠀⠈⠉⠀⣈⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⢠⣤⣤⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⡘⡐⢛⡘⢰⠀⠀⠀⠀⠀⠀⠀⠃⡀⠑⠚⢡⢖⡲⡄
⠀⠀⠀⠀⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣭⣭⣼⠀⠀⠀⠀⠀⠀⠀⠀⣇⠐⠀⠂⡛⠛⣰⠀⠀⠀⠀⠀⠀⠀⠛⠀⠂⠂⢣⣛⡓⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣛⣛⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠛⡇⠀⣀⢀⣀⣤⡄⡀⠀⢀⣀⣄⡀⠈⡗⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⠀⢀⡇⠀⡟⢾⣻⡗⠀⢳⣦⡏⣙⢲⡅⠀⡇⢸⠀⡏⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠈⠉⠀⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢀⡤⠤⠤⠤⢤⡀⠠⠤⠤⠤⠤⠤⠤⠤⠄⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢸⠀⠀⠀⠀⠀⡇⠰⠶⠶⠶⠶⠶⠶⠶⠀⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠘⠒⠒⠒⠒⠚⠁⠐⠒⠒⠒⠒⠒⠒⠒⠂⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠀⠀⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠀⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⠀⠀⣉⣉⣉⣉⣉⣉⡁⢈⣉⣉⣉⣉⣉⣉⡁⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⣇⠸⠤⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠶⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠠⠴⣶⠖⠢⠄⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠂⣠⠶⠛⠒⠆⡀⠐⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠒⠒⠁⡄⢠⡄⢠⠈⠒⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⡦⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⡎⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠌⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠐⠀⢀⣀⣅⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⡀⠀⠆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠁⡌⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⢰⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡄⠠⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⠸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠃⠀⠈⠢⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⠔⠁⠀⠀⡠⠂⠀⠀⠋⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠀⠀⠘⢤⡀⠀⠈⠢⡈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⢀⠔⠁⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠢⡄⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠂⠁⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠈⠐⢄⠈⠳⣄
⠀⠀⠀⠀⢀⡴⠊⠁⡠⠊⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠧⠤⠄⠠⠤⠤⠠⠤⠤⢸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄⠈⠑⢦⡀
⠀⠀⠀⣰⠋⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⠀⠠⠠⡔⢢⠤⠄⠀⠸⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⠢⡀⠙⣆
⠀⠀⠀⢹⠀⠃⠒⠀⠒⠒⠒⠀⠒⠂⠀⠒⠒⠒⠒⠒⠒⠂⠖⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠈⠲⠐⠀⠒⠒⠒⠒⠀⠐⠂⠒⠂⠒⠒⠒⠂⠒⠘⠀⡏
⠀⠀⠀⠺⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠆⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠰⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⡇
⠀⠀⠀⠀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠠⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠚⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠃⢤⣀⣀⡀⣴⢿⡿⢠⣀⣀⣀⠠⡞⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣅⣠⣀⣭⣹⣯⣨⣅⣀⣈⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⡿⠋⣍⣀⣀⣀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿
⠀⠀⢀⣿⡟⣰⡦⠀⠀⠀⠀⠀⠀⠀⢴⣆⢻⣿⡀⠀⠀⣿⣿⡿⠁⣰⠏⢙⣿⣿⣿⣿⣿⣿⣿⡋⠹⣆⠈⢿⣿
⠀⠀⢸⣿⢠⣿⠃⣿⠃⣢⣾⣶⡘⣿⠘⣿⡄⣿⡇⠀⠀⣿⣿⡇⠀⡿⠀⣾⠁⣼⠟⠉⠉⣧⠈⣷⠀⢿⠀⢸⣿
⠀⠀⢸⣿⠘⣿⡄⢿⠄⠻⣿⠟⠡⡿⢠⣿⠃⣿⡇⠀⠀⣿⣿⡇⠀⣿⠀⢿⡀⣻⣄⣀⣠⣟⢀⡿⠀⣿⠀⢸⣿
⠀⠀⠈⢿⣧⡙⠟⠀⠀⢠⣿⡄⠀⠀⠻⠋⣼⡿⠁⠀⠀⣿⣿⣿⡀⠘⣧⣨⣿⣿⣿⠀⢿⣿⣿⣥⣼⠃⢀⣿⣿
⠀⠀⠀⠈⠋⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠙⠁⠀⠀⠀⣿⣿⣿⣷⣴⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣦⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⣠⠤⠔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠀⠀⢠⡤⣼⢾⢚⢚⡹⠀⠀⠀⠀⠀⢀⣀⠠⠆⠀⢠⠛⠠⣤⣄⣀⣀⠀⠀⠀⠀⠙⠃⠀⠚⠁
⠀⠀⠀⠀⣆⣏⣥⡦⠶⠒⠒⠀⠀⠀⠀⠀⠀⠘⠊⠉⠉⠀⢀⣠⠴⣺⠟⠋⠉⠀⠒⠀⠀⠀⠠⠞⢈⣹⠎⠍⣳⡄⠀⠀⠁
⠀⠀⠀⠀⠏⡧⠖⠗⠁⠀⠀⠳⣶⣤⣶⡏⠀⠀⠀⠀⠀⠀⢺⣧⣀⠈⠓⠒⠖⠒⠶⠒⠶⠒⠚⠋⢉⣀⣤⡾⠟⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠔⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠶⠶⠶⠶⠶⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⣀⢤⠶⠋⠁⠀⠀⠀⠀⢀⡀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⢀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⢀⠠⠀⠫⡽⢀⠘⠋⠀⣠⣀⠀⠀⠠⡀⠀⠀⣖⢻⢻⠚⡗⣦⡄⠓⠋⣼⡏⠇⢹⠸⠸⡇
⠀⠀⠀⠴⠶⠚⠃⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⣵⠀⢸⣸⣸⡤⠷⠋⠀⢀⠘⠀⠙⠛⠚⠒⠗⠃
⠀⠀⠀⠀⠀⠀⣀⣬⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠋⠀⢀⠝⠀⠈⠈⠀⠀⠀⠀⠀⣸⣣⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠔⠐⠒⠀⠀⣀⡀⢀⣀⣀⡀⠀⣀⣘⠁⠀⠀⠀⠸⡀⠀⠀⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠉⠓⢤⣸⣷⣦⠀⠀⠀⢀⣴⣶⣦⣰⡆⠀⣾⣷⠿⣿⡟⠃⣾⣯⣭⡍⣿⣏⣽⠇⠀⠀⠀⠀⡗⠒⠒⠀⡖⠀⠂
⠀⠀⠀⠀⣤⣀⠀⠀⠈⠀⠀⠙⠣⠀⠀⢼⡇⣸⣿⠹⣿⣤⣿⠇⠀⣿⡇⠀⣿⣥⣤⡀⣿⡟⠻⠦⠀⠄⣀⠀⢹⠚⠓⢺⡃
⠀⠀⠀⠀⠀⠈⠙⠡⠀⠀⣠⢤⡀⠀⠀⠘⠛⠛⢁⣀⡈⣩⣥⣄⠀⣬⣤⡀⠀⣠⣶⡄⣠⡴⠶⠆⠀⠀⠀⠀⣼⠦⠤⠼⣇⡈⠄⠤
⠀⠀⠀⡤⢤⣤⡄⡀⠀⢸⡁⢀⡽⠀⠀⠀⡀⠀⠿⣤⣅⢹⣧⣾⠗⣿⡾⣇⢸⣏⠀⡀⢿⡷⠚⠃⠀⠀⠀⠀⠀⠁⠒⠀⠉⠁
⠀⠀⠀⠀⠠⢶⣷⣤⡄⠀⠉⠉⠀⠀⠀⠁⡀⠀⠶⠶⠿⠸⠿⠀⠀⠛⠀⠉⠈⠛⠛⠁⠸⠷⠚⠛⠀⠀⠀⠀⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠘⠈⢀⡄⠂⡂⠠
⠀⠀⠀⠀⠰⠄⣀⡉⠀⠀⠀⠀⠀⣀⠤⠆⠚⠀⠈⠒⠂⠀⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢰⡁⠀⡀⠧⠖⠀⠀⠤⠀⣀⢀⡼⠀⠀⠆⠂⠀⢀⠀⡐⢶⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠑⠦⠦⣄⣀⡑⠠⠤⠠⠚⠉⢀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⠀⠰⠀⠑⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠈⣶⠆⠀⢤⠒⠈⠳⣤⡀⠀⠀⠀⠀⠐⠒⠘⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠑⠀⠀⠸⣦⣄⡒⠀⠠⠤⠀⠀⠂⣀⡜
⠀⠀⠀⠸⠇⠀⠀⠘⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠳⠶⠖⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠋⠉⠁⣠⣤⣤⣀⡀⠈⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢟⣽⠿⠿⠀⠀⠀⡀⠈⠉⢉⣉⣉⠀⠀⡿⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠻⡋⠀⠋⢀⢀⣀⠀⠀⠀⡇⠀⡎⣿⣯⠗⠀⢰⠿⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠳⢇⠀⠀⠿⣿⠯⠃⠀⠀⠑⢄⠀⠀⠀⠀⠀⣸⣤⣾⣿⣿⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⡄⠀⠀⠀⣿⣿⡿⣿⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⡼⠛⢋⣵⣾⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡌⣿⣿⣿⣷⣄⠀⠀⠀⠈⠿⣉⣉⠝⠁⢀⡜⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠙⠦⢄⡀⠀⠀⠀⢀⡤⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠛⠋⠉⡆⠀⠀⠀⠀⢻⣿⣿⡟⢟⠛⢻⡿⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡞⠀⠀⠀⠀⠀⢻⡉⠢⡄⠀⠀⡽⢻⣦⢸⣄⣹⣇⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡧⠚⠀⠀⠀⠀⠀⠀⠀⠑⢢⡦⣀⡀⠘⣄⡽⠁⢡⢀⣀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢎⣙⣗⠤⠤⢠⣤⣤⡤⠤⢔⣊⢥⠟⠁⠈⠑⢌⠒⡴⠋⠀⠀⠀⠙⢦⣄
⠀⠀⠀⢠⠤⣸⡉⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠹⡉⠙⠒⠂⠒⠀⠉⠉⠀⡜⠀⠀⠀⠀⠈⡟⡀⠀⠀⠀⠀⠀⠀⡄⢆
⠀⠀⠀⣏⠓⠀⠀⠀⢱⡀⢀⡀⣀⣀⣀⣀⣀⣀⣠⠤⠔⠋⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠳⣁⡃⣠⠒⠉⢳⢤⠇⡼⡀
⠀⠀⢸⠆⠀⠀⠀⠀⡌⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⡆⠀⠈⡖⠊⠀⠇
⠀⠀⠀⠉⠙⠓⠒⠚⠢⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠌⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠘⠢⠔⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⢀⣀⣀⡀⠀⠀⣀⣀⠠⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣿⠤⠤⢤⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⡤⠶⠂⢤⢺⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⢸⠀⠀⠀⠀⠀⠀⡟⠢⢄⣀⣀⣀⡀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣓⣼⣿⣿⣧⣸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠟
⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⡀
⠀⣴⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧
⠀⢿⣿⣿⣿⣿⣿⣧⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⣿⣿
⠀⠈⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠃⠀⠀⠈⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣀⣤⣤⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢸
⢠⣾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⢠⡄⣶⢠⡆⣠⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢹
⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⢻⡌⣧⠹⠈⠃⠛⠘⠃⠛⠸⢃⡞⣠⠇⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣶⡀⠀⠀⢸
⣿⡅⠀⠀⠀⠀⠀⠀⢴⡈⢷⠙⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠾⢡⡞⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸
⣿⡇⠀⠀⠀⠀⣤⡙⢦⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⡴⠂⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸
⣿⡇⠀⠀⠀⠰⢦⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠶⣋⣡⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⠿⠟⠋⠀⠀⠀⢸
⣿⡇⠀⠀⣈⡛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠀⠀⠀⠈⢩⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠠⣬⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠐⠖⣛⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠰⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⡅⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⠀⠀⠀⠀⢸
⣿⡇⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠟⠀⠀⠀⠀⢸
⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⣀⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣋
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣴⠀⠀⣴⠄⠀⣴⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⢧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠆⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣼⣿⣧⣼⣿⣧⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡩⠵⣤⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡴⠭⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣇⡇⠀⠈⢷⡀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⣰⠏⠀⠀⣇⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⢯⡗⠒⠢⡀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡠⠒⠒⣮⠇⠈⠢⣀
⠀⠀⠀⠀⠀⢀⡠⠔⢺⠖⠊⠁⠀⠀⠀⠘⡞⡄⢀⡇⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣇⠀⡼⡜⠀⠀⠀⠀⠉⠒⢾⠒⠤⣀
⠀⠀⠀⢠⢾⠁⠀⠀⠸⣼⡀⠀⠀⠀⠀⠀⠹⡝⡍⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⡝⡽⠁⠀⠀⠀⠀⠀⡸⡜⠀⠀⠀⢹⢦
⠀⠀⠀⠸⡈⡄⠀⠀⠀⠙⣷⣀⠤⠤⠒⠒⠒⠚⠮⣓⢒⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⢒⡪⠞⠒⠒⠒⠢⠤⢄⣰⡗⠁⠀⠀⠀⡞⡸
⠀⠀⠀⠀⠱⣘⣄⣀⡠⢤⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢦⠤⣀⣀⣜⡰⠁
⠀⠀⠀⠀⠀⠀⠈⠈⢆⠈⢣⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠑⣌⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⡔⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠈⠳⣌⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⡴⠋⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠘⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡜⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⡀⠤⠃

⠀⠀⡠⠒⠈⠉⠉⠀⠐⠄⡀⠀⠀⠀⠀⠀⢀⠄⠂⠊⠉⠉⠐⠒⢄⠀⠀⠀⠀⠀⠀⡠⠔⠀⠈⠉⠁⠀⠠⢀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠉⠉⠁⠐⠤⡀
⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⣠⠃⠀⠀⠐⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠂⡀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢐⠜⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠐⠂⠤⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠤⠤⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠤⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠤⠤⠄⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣀⠤⠐⠒⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠐⠦⣔⡬⠔⠒⢳⢀⠀⠀⠀⠠⠀⠤⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⠀⢀
⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠠⠁⠀⠀⡠⣐⠵⠋⠁⠓⡀⣠⠯⠂⢀⠔⠁⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⡐⠊⠀⠀⠀⠀⠀⠀⠈⠢
⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠁⢀⢔⡡⠚⠁⠀⠀⠀⠀⢸⠕⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈
⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣖⡵⠋⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡀
⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⡡⠫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠉⠂⠄⣀⡀⣀⠠⠔⠊⠀⠀⠈⡞⠁⠀⣰⠦⣀⡀⠀⣀⠠⠐⠁⠀⠀⠀⠀⠀⠉⠢⢀⠀⠀⠀⢀⡠⠐⠁⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⣀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠏⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⡀⠃⠐⠒⠀⠀⠀⠀⠀⠒⠘⢀⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠒⠒⢰⣏⣀⣀⣹⡤⠀⠒⠂⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⢀⠀⢘⠠⢨⡿⠭⡓⠆⠃⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡀⠈⠁⠁⠰⠆⠌⠈⠁⢀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⢹⡖⠂⠐⠂⠀⢲⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠰⢀⣾⡇⠀⠀⠀⠀⢸⣷⡀⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⢿⡃⠐⠀⠀⠀⢘⡻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣰⡏⢸⡇⠀⠀⠀⠀⢸⡅⢹⣆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡿⠘⢙⡏⠉⠉⠉⠉⢹⡋⠃⢿⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⢖⠁⠁⡆⢘⡃⠀⠀⠀⠀⢘⡁⢸⠈⠨⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠕⠁⡠⠊⠀⢨⡇⠉⠉⠈⠉⢸⡇⠀⠑⢄⠈⠪⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⠔⢁⠄⠊⠀⠀⠀⢸⣇⢀⡀⢀⡀⢸⡇⠀⠀⠀⠁⠠⡈⠢⡙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⡞⡡⠂⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⢴⡦⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⢳
⠀⠀⠀⠀⠀⠀⠀⣧⣉⣉⣉⣉⣭⣉⣉⣍⣉⣩⡇⠀⠀⢸⡇⠀⠀⢸⣍⣉⣉⣉⣉⣭⣉⣉⣩⣉⣼
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠤⠀⢀⡤⠀⣀⠀⠀⠀⠄⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⣀⠀⢠⡀⠀⠤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠤⣤⠤⠬⡆⢲⠤⢤⣾⡷⣤⠤⠾⢰⠏⠤⣬⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢁⠈⣿⣿⡿⠿⣿⣿⡇⠘⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⣶⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣶⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⣿⣟⣿⣿⡻⣿⣟⣿⣿⣻⣿⣿⣿⡇⣿⣿⣭⣭⣭⣭⣿⣯⣽⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡿⢿⠿⡿⠿⣿⢿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⡇⣿⣷⣶⣶⣾⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣙⣛⣛⣛⣛⣃⣿⣯⣭⣭⣽⣿⣿⣿⣧⣙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢟⣭⣿⣯⣝⢿⣿⣿⣿⣿⣿⣿⡏⣽⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⣾⣿⣿⣿⣿⡎⣿⣿⣿⣿⣿⣿⡇⣿⡟⡛⣫⣭⣵⡆⣿⣭⣽⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢟⡃⢿⣿⣿⣿⣿⡇⣛⢿⣿⣿⣿⣿⡇⣿⣧⢃⠻⠿⢿⡇⣿⠶⢶⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡇⢲⣒⣒⣚⣒⠂⣿⢸⣿⣿⣿⣿⡇⣿⣿⡘⣸⣿⣷⣶⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡜⣷⠹⣿⣿⣿⣿⢱⡟⣼⣿⣿⣿⣿⣧⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠘⠳⣮⣭⣭⡶⠟⠈⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠬⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠭⣽⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⠟⠋⡁⢄⣤⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⡛⢉⡁⢀⣢⣤⢶⣿⣿⣿⣿⣿⠿⠟⠟⣩⣝⢿⣿⣿⣿⠟⠡⠀⠂⣀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣏⠰⠟⣡⣾⣿⣿⣿⣿⣿⣿⠿⠿⠀⠀⠀⡝⢏⢷⣻⣿⠰⠀⢀⠰⡆⠀⣂⣿⡇
⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠛⣿⣿⣷⣨⣿⣿⣿⣿⠛⣉⡀⠀⠀⢀⠀⠀⡄⣇⢸⡄⣏⣿⣷⣄⣈⣐⣆⣬⣿⣿⣷⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⣶⣿⣿⣿⡟⠋⠉⠁⣌⢰⣉⣽⢀⠀⢸⡄⠀⠸⡜⢾⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣾⣿⣿⣴⣿⣿⠉⣟⣡⣴⣧⣤⣤⣶⣿⣼⣇⢹⣿⣿⣿⣿⣶⣶⣾⣢⣭⣿⣿⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⢏⠉⠛⢿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⡿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣽⣿⡿⢻⣿⠟⠿⣿⣿⡻⣿⣿⠛⢻⣿⢍⠚⠀⠈⠉⠑⠈⢻⣿⣿⣿⡄
⠀⠀⠀⢸⡿⣫⡟⠡⠤⠖⢀⠀⠀⠘⢻⣯⣻⣿⡿⣿⣧⣽⣶⡇⢰⣿⣿⣷⣜⠪⠐⢾⣿⣇⠠⠀⠀⠀⠀⠉⢈⣭⡛⢿⠁
⠀⠀⠀⠈⣱⣏⠀⠠⠀⠀⠀⠀⠀⣀⣤⠹⠷⢻⣷⣿⣿⠿⢿⣳⡊⠛⢿⣿⣿⣷⣧⣶⣦⣍⡛⠳⠶⢦⣤⣤⡄⠀⠐⡀⢄
⠀⠀⠀⢠⣿⠒⠖⠀⠀⢀⢀⣀⣠⡟⠈⢘⢸⣿⣿⣷⡝⣅⣀⣭⣭⣁⣟⢡⣿⣟⢻⣿⣿⣄⠙⠛⠳⢶⣦⣬⣤⣀⢀⣠⣾⣫
⠀⠀⠀⠸⣿⣳⣖⡻⢿⣿⡿⠟⠋⠀⠁⠀⣺⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⠘⣿⣿⢿⣿⣿⣿⣿⣶⣿⣿⣿⡟⢿⣿⡿
⠀⠀⠀⠀⢻⣅⡀⠉⠀⠉⠀⠀⡀⢀⣢⣞⣿⣿⣿⡇⡌⣿⣿⣿⣿⣿⣿⣇⢹⣿⣿⣿⣷⠛⠛⠛⢻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⠿⣤⣀⠀⠠⠀⢶⣯⢷⣿⣿⣿⣿⢡⢿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⠿⣛⡀⠀⣸⣿⣿⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢬⣭⣭⣭⣶⣿⣿⣿⣿⡏⣌⣾⣿⣿⡿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣾⣿⣿⣿⢶⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢦⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⡟⣼⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣧⢻⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠙⠛⠛⠛⢇⠛⠛⠛⠛⠉⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⠿⣒⣤⣶⣶⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⢛⣙⢻⣿⠟⣉⡻⣿⡿⣋⡙⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣌⣋⣼⣿⣮⣛⣵⣿⣧⣙⣣⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⡹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⣿⣿⣿⣿⣿⣶⡍⠛⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⡿

⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⣀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⣾⣿⣿⣷⣶⣦⣄⡀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣷
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣿
⣿⣿⣿⢋⣥⣦⣬⣭⣭⣭⣭⣭⣭⣭⣥⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿
⢻⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣮⣭⣍⣛⣛⣛⡛⠿⣿⣿⣿⣿⠈⣿
⠈⢿⣷⣜⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡝⢿⣿⢰⣿
⠀⠀⠹⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⡎⠟⣼⣿
⠀⠀⠀⠀⠉⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢙⠿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⠿⢟⢱⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣾⠿⠸⢿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⣠⣶⣿⣿⣶⣮⣍⣛⠿⠏⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡘⠿⢛⣫⣭⣶⣿⣿⣷⣮⠉⠁
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡿
⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⡧
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠯⠈⢻⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⢸⣿⣿⢀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠸⣿⠑⠀⢀⡠⢤⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠶⠶⠆
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠠⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠄⠈⢋⣀⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⣴⣉⡉⠑⠂⠍⡂⠀⠀⣄⠠⡄⠀⠘
⠀⠀⢀⠀⠀⠀⠀⢠⣤⣼⣿⡄⡰⠤⠐⠄⣀⣀⡈⠙⠒⠦⣀⠁⣣⠀⠀⠀⠈⢦⣐⠀⠀⠀⢨⣵⡄⢰⣀⣷⣆
⠀⠀⠋⠘⠐⠀⠈⠘⠿⣷⡿⠃⠑⠄⠀⠀⠸⠛⠛⣿⣶⣤⡈⠓⡎⠀⠂⡀⡀⢠⡿⠀⠘⠤⣿⠿⠁⠀⠬⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⢤⣄⣈⢹⠿⠊⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠾⡷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠃
⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⢶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣑⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣯⡿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠟⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⣭⣉⣿⣉⣏⣛⣉⣙⣍⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⣿⠿⠿⠿⠿⠿⠿⢿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⢿⣿⡿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡦⣇⣽⣼⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣫⢖⡯⠭⣗⢮⡻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣷⣷⡞⠛⢶⣷⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡅⣹⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡃⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⡋⣻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣭⢫⣵⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣾⡇⢻⣷⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣏⣀⣀⣈⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠙⠁⠑⠀⠂
⠀⠀⢀⡼⠻⡒⠤⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⣶⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠃⠀⢀⢤
⠀⢆⡞⠀⢀⣱⡤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⢆⠀⢀⡱
⢃⠨⠻⣍⡉⠈⠃⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡛⢍⠢⠑
⠀⠓⠤⣌⣉⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠐⣉⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠬⠂



⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢉⣉⣉⡉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣄
⠀⠀⠀⣷⣶⡗⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⣒⣒⠒⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢠⠒⠲⢣⠏
⠀⠀⠀⡻⣿⡷⠁⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠤⣄⠀⢸⣿⣿⣟⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠳⡄⠀⠋⠠
⠀⠀⢀⣠⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠈⠙⣿⠟⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠿




⠀⠀⠀⢠⠀⠀⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⠀⠀⢀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤
⠀⠀⠀⠀⢠⣉⡡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠳⠄⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠦⣀⡤⢖⡉
⠀⠀⠰⠤⠬⠥⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣛⣛⣛⣉⡦⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⡤⠤⠤⣄⣙⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡀⠀⠀⣤⡄⣀⠀⠀⠀⠀⢀⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠀⠀⠀⠀⠀⠀⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣄⣼⣿⡿⠿⠯⠀⠀⠀⠈⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢿⣿⣿⢂⣀⠀⠀⠀⠀⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⠶⢿⡇⢀⣤⣾⣿⣿⣿⣦⠀⠀⠀⠀⢀⣀⠔⠉⠀⠀⢀⡞⢁⣿⣿⣷⡄⠀⠀⠈⠁⠳⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⡀⢀⣴⣿⠋⠀⠀⠀⠀⢸⠁⢸⣿⣿⣿⣧⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⣿⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⢈⠛⠉⢷⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡈⠀⠀⠀⠀⠀⠀⢀⣼⡿⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢃⣆⣤⡦⣮⡄⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣴⣿⡟⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⣾⣭⣀⣠⣿⠻⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠰⠋⠀⠀⣿⣿⣿⣿⣿⡿⠟⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣧⣄⣇⠀⠀⠀⠀⠈⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢻⡛⡛⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⡇⢸⡄⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⡀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⣿⠈⣿⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⠂⢹⡇⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢃⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠻⣿⡆⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⡇⠀⠨⡄⢸⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠚⠛⠛⠃⠀⠿⠇⢀⣠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠛⠀⠀⠀⢘⠃⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⡦⠀⠀⠀⢸⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠪⢄⠀⠀⠀⠀⣇⠀⠀⠀⢀⠀⠀⠀⠈⠈⠓⢄
⠀⠀⠀⢄⠀⠠⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⠃⠀⣶⠀⠹⡀⠀⠀⠀⡁⠀⠀⡑⡀
⠀⠀⠀⠈⠐⢌⠲⣅⢄⡀⠀⠀⠁⠀⠉⠑⠒⠒⠒⠒⠒⡊⠉⠁⠈⠒⠀⡚⠀⠀⣿⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠐⢬⡓⣎⣢⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠴⣤⣤⣿⣤⢤⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠹⡶⡞⠀⠄⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠛⠛⠛⠿⠿
⠀⠀⢀⡤⠴⠀⠀⠃⠀⠿⠐⠀⢞⠻⣕⠢⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠐⠀⠀⠊⠀⠀⠀⠀⠉⠂⡙⢭⡣⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠈⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠀⠀⠀⠀⠂⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠘
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⢨⣆⢳
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠋⠀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢸⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠮⠤⠐⠈
⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⠀⠀⠉⢖⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡅⠀⠀⠀⠑⠢⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⣀⡠⢸⠤⡄⠀⠀⠀⡀⠀⠐⠒⢒⠒⠂⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠃
⠀⠀⠀⢁⠀⠀⠀⠀⠀⠑⡀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢢⠀⠀⢈⡡⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠑⢀⠀⠀⢀⠔⠁⠀⡠⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠈⠁⠀⠀⡐⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡔⢄⣀⡠⢊⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠜⢄⠀⠀⠈⠢⠬⠟⢛⣠⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡶⠤⠤⠵⠦⠤⠠⠤⠤⠾⠟⠒⠒⠒⠚⠿⠛⠛⠛⠛⠟⠻⠟⠛⠛⠛⠛⠿⠿⠟⠿⠛⠻⠛⠦⠄⠤⠄
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠉⡏⠙⡋⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠃⠉⠊⠛⠉⠛⠉⠉⠉⠉⠉⠉⠈⠉⠉⠀⡏⠉⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠰⠄⠀⠐⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠒⡄
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠰⠀⠀⠀⠀⢠⠀⡄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⢀⠁⠀⠀⠀⠀⢸⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣛⣒⣒⣒⣒⣒⣒⣒⣒⣒⠄⠂⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠃⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠍⢸⠀⠀⠅⠉⠀⠀⠀⠁⠉⢠⠏⢨⡏⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠀⡆⠃⠀⠀⠀⠀⢰⡧⢾⠸⠷⡶⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠐⠃⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠘⢇⡐⠀⡆⣦⠀⠀⠀⠃⠤⣀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃⠀⠀⠀⠀⠀⠘⠉⠒⠓⠛⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠛⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⢸⣿⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣿⠀⢸⣿⣿⣏⠻⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣴⣶⣿⣿⡇⢸⣿⠀⢲⣤⠀⢻⣷⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠁⠀⠀⣿⠀⢸⣏⠙⢿⣧⠈⢻⣿⣿⣿
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡆⠈⣿⡄⠈⣿⣿⣿
⠀⠀⠀⠀⠙⢻⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⡇⢸⣿⠀⣠⡿⠃⢰⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣷⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡿⠁⣰⡿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⡾⠃⠀⠉⠛⢿⡇⢸⣿⠀⠉⠀⣠⡿⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠛⣋⣴⣿⣷⣦⣄⢀⣿⠀⣸⣷⣿⡿⣁⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣴⣾⣷⣷⣾⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠲⠄
⠀⠀⠀⠀⡃⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠡⡂⠀⠀⠀⠀⠀⠄⠀⠠⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠠⠄⠀⠀⠠⠀⠄⠠⠉⠁⠀⠀⠀⠀⠀⠀⠒⣭⡤⡷⠁⠀⠀⠀⢈⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠂⠀⠀⠠
⠀⠀⠀⠀⢐⠐⠀⠈⡐⣀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠉⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣈⠁⠛⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗
⠀⠀⡀⠐⠨⠃⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠘⢭⠇⣤⣄⣤⢄⢠⣤⣁⣝⣆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠐⠠⠠⠄⠀⠈⣠⠀⡢
⢸⠐⠆⡄⣀⠀⠀⠈⣀⡈⠁⡈⡭⠀⠀⠀⠀⡈⠠⢌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠠⠢⢄⣀⢀⡀⠷⠠⠉
⢸⢀⡀⠑⠁⠘⠐⢊⡉⠋⠁⠁⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢄⣤⠁⠀⢀⡠⡐⠐⡺
⠈⢟⠓⠊⠀⠀⠂⠨⠯⠷⠄⠉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢶⠀⠀⠖⠂
⠀⠀⠑⢆⡰⠍⠁⠀⡐⣒⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⡠⠀⠃⣀⠂⡀⠀⠈⠀⠢⢅⠄
⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⡡⠂⠀⠀⠀⡐⠀⠀⢀⠱⢆⠒⠦⠀⣤⡀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠆⠈⠃⠀⣠⡝⠢⠰⠄⠀⠐⠀⣌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⣀⡠⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⣀⡀⠀⠂⢡⠀⠀⠀⢠⠀⠦⢎⠙⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠂⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⡶⡤⠀⣤⠀⠀⠀⠀⠀⠱⠀⣀⠀⠲⠸⡆⠐⠚⢿⠗⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠀⠰
⠠⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠁⡌⠥⡀⠶⠁⠀⠀⠀⠀⡀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠃⠀⠈
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣾⣿⡶
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣛⣟⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣦⣄⡀⡀⠀⠀⣠⣶⣿⠿⣷⣦⣄⠀⢠⣴⣶⡿⢿⣶⣦
⠀⠀⠀⣼⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⡟⢸⣿⣿⣇⢹⡇⠀⠀⠀⣀⣤⣤⣸⣿⡿⢻⣿⣿⣿⣿⡿⠃⠀⠀⠀⠙⠿⠿⠿⠿⠿⢿⣤⣿⣛⠿⠿⠿⠿⠛
⠀⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠁⣿⡏⢹⣿⠈⠁⠀⠀⠀⠹⣿⡿⠟⣿⣿⣸⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠏
⠀⠛⠛⠿⣿⡿⠛⠛⠃⠀⠀⠀⠀⠀⠘⡿⠁⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⡅⠀⠀⠀⣤⠀⠀⠀⠀⣥
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠛⠂

⠀⠀⠀⠰⣀⣀⢰⣂⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠙⣤⡄⠀⢀⣀⡀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠫⣿⡯⠋⣠⣤⡝⠆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢘⣶⣿⣿⠛⣿⣿⡛⡄
⠀⠀⠀⠈⢻⣿⣅⣽⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⢠⣾⣿⣿⠇⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⡿⠽⣿⣿⣏⣀⠿⣿⠃
⠀⠀⠘⠀⣿⣿⣿⣿⣿⣿⠀⠃⠀⠀⠀⠀⠀⠰⣜⠿⠟⢁⣾⣿⣦⣄⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢈⣿⣧⣞⣙⠿⣿⣿⠖⠁
⠠⠮⠀⠀⠉⠉⠭⠍⠉⠉⠀⠀⠥⠄⠀⠀⠀⠀⠈⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⠈⠛⠿⠿⡿⠿⠟⠁⠀⠀⠀⠀⠀⠈⠛⠴⠿⠟⠂⢾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗

⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠤⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⡀
⠀⠀⠀⣍⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣯⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠘⠳⣦⠄⢠⣴⣶⣶⣦⡀
⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⡷⠘⣿⣿⠷⣟⢀⡤⣾⡇⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠈⠑⣿⣿⢿⣿⣿⣿
⠀⣤⣾⣿⣻⣿⣦⡄⠀⠀⠀⠀⣰⣿⣿⣯⡻⠿⠤⠼⢟⣶⣿⣿⣆⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠯⣰⣤⣀
⠀⣿⠋⢸⣿⠈⢻⡇⠀⠀⠀⠀⠙⢿⣿⠟⠀⠀⠀⠀⠀⠻⣿⡿⠇⠀⠀⠀⠀⠀⠘⠛⠻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉
⠀⠃⠀⠀⠇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣷⣦⣤⡀
⢀⣾⣿⣿⣿⣹⣿⣶⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣿⣿⢿⣶⣤
⠸⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⢿⣟⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⢀⣀⣴⣹⣛⡟⢻⣛⣏⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢧⠀⠈⠉⠙
⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⢻⣽⣽⢿⣿⣿⠄⠀⠀⠀⠀⠿⣿⣿⣷⣿⣿⣿⣷⣾⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⡸⠠⡄⢆
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣤⣼⣧⣬⡉⠁⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠉⢹⡏⠉⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⡆

⢀⡀
⠸⣷⠶⠟⠻⡿⠻⡗⢶⣆
⠀⠙⠳⢦⡛⠾⣦⡀⠀⢻⣧⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣛⣛⣉⣩⠭⡽⢧⣄⣀⡀
⠀⠀⠀⢶⡷⡶⢆⠝⢳⣄⣍⠙⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⢿⠁⢌⠠⢁⠡⢈⠬⣀⢇⠩⡹⡿⡃⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠉⠻⣷⣯⣆⠹⣿⢧⡇⠀⠀⠀⠀⠀⢀⣤⠾⠿⣯⠍⠆⡔⠀⠂⠀⠂⠐⠀⡂⢃⣂⣹⠀⠀⠈⠉⠙⠳⣦⢀⠄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⢻⣼⢡⣶⡒⢤⡀⠰⠟⠁⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⠻⣿⣀⣀⣀⣀⣰⣒⠶⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡇⣾⠛⠁⠀⠙⠳⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⣆⠀⠀⠀⠀⢸⢩⣟⣛⣛⣻⠉⠉⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠏⠀⠂⠀⠀⠀⢠⣞⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡟⠙⣷⡀⠹⠆⠀⠀⠀⠘⠳⠮⠭⡭⠥⢤⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠘⢦⡀⠀⠰⣅⣈⠻⣦⡌⠃⠀⠀⠀⠀⢀⣤⠀⠀⠀⠉⠒⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡼⠛⢦⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⡀⠰⡙⢷⣝⢮⡳⣄⠉⡳⠈⠁⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣴⣾⣥⣞⡇⣰⠃⠀⠀⠀⠀⠠⣾⠻⣦⡱⣌⣻⠳⠙⠈⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢀⣤⡇⠘⢷⡈⠻⠀⣁⣀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢷⡝⢿⣄⠈⠁⣠⡾⢋⡿⢀⣠⣴⠾⠋⠁⠀⠀⢠⠖⣒⣒⣒⣒⢲⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠇⠀⢀⣤⣤⣙⢦⡀⢹⡄⢉⣴⠾⠋⢀⢬⣛⠋⠉⠀⠀⠀⠀⠀⠀⢸⠀⣟⡭⠴⣿⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠻⢦⣄⢸⣆⠀⠹⡎⠙⢛⣴⠟⠁⠀⡴⠁⠀⠉⡑⠀⠀⠀⠀⠀⠀⠀⠸⣆⣙⣓⣚⣛⣸⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠙⠷⠶⠃⣠⡾⠁⠀⠀⠈⣀⠀⠠⠚⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣹⠃⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢭⣝⡁⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛⠳⠶⠶⠶⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠈⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠒⠀⠴⠤⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡙⣦





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠒⠒⠀⠀⠀⠀⠀⠸⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠮⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤⠀⣀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡀⠀⢈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⣀⡇⠀⠂⢀⡴⠉⠀⠉⠉⠒⢀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠐⠂⠌⢦⡔⠒⠀⣀⣀⣀⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠋⠀⠈⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠛⢄⠀⠃⠈⠀⠀⠐⠒⠒⠒⠒⠈⠁⣀⡤⠐⠃⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢄⠈⠀⠀⠑⠂⠈⠓⠀⠒⠀⠒⠒⠈⢁⡀⠄⠐⠚⢃
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢣⣀⠀⠀⠀⠂⠀⠐⠃⠀⠲⠆⠀⠀⠀⠀⠉⣀⠄⠀⠀⢸⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣆⠀⠀⠆⠀⠀⠀⠐⡄⠀⣤⠐⠀⠛⠛⠁⠀⠁⠀⠠⠀⠸
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠄⠈⠃⢐⡤⠐⠆⠀⡁⠀⠁⢠⡖⠒⠒⠈⠁⠰⡆⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠨⡄⣈⡀⠘⠃⠀⡄⢠⡄⠀⠀⠀⠐⠒⠀⢀⡄⠀⠀⠰⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠈⠁⠘⠀⠠⡀⠈⠓⠀⠴⠄⠄⠀⠋⠀⠀⣀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠑⠂⠀⠀⠄⠀⠀⠁⠀⠈⢀⡶⠚⠁⠀⠐⠀⠙⠁⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠂⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡻⠏⠛⣿⠋⢙⠟⠛⣟⠻⢙⣿⠟⡋⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⣿⣷⣿⣶⣿⣿⣾⣾⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣴⢦⣄⠀⠀⠀⠀⢰⡟⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠖⠒⠛⠛⠛⠛⢷⡀
⠀⣾⠁⠀⠙⠷⠶⠒⠛⣋⠉⠀⠈⠳⡄⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠁⣀⡠⠤⠴⠲⢶⡀⠀⠀⠘⡇
⢸⡇⠀⠀⠀⠀⠐⢾⡋⠉⣻⠀⡿⠚⠁⣌⣼⠀⠀⠀⠀⠀⠀⣠⠤⢾⣁⠀⠀⠀⡞⢁⣤⠀⠀⢀⣤⠙⠛⠋⠉⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠙⢶⡟⢸⡇⢀⣼⣳⡏⠀⠀⠀⠀⠀⣸⢁⠖⠉⠳⢴⡶⠿⠷⠟⠈⠛⠛⠋⠙⠷⠶⠦⣄
⠈⢿⡀⢀⣄⠀⠀⠀⠀⠀⠑⢿⣅⠈⣵⠟⠀⠀⠀⠀⠀⠀⠹⣌⠳⠤⠤⠿⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢷
⠀⠈⣷⠏⡈⢧⡀⠀⠀⠀⠒⣩⠽⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⣾⠛⠀⠀⠀⠙⠦⣄⣀⣤⠴⠃⠀⠀⠀⠘⡆
⠀⣼⠇⠀⠀⠀⠳⣤⣤⣤⣭⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⢠⡷
⠈⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡿⠋⠉⠉⢻⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣸⠁⠀⠀⠀⣠⣇⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣤⣄⠀⠀⢀⣤⣄⣀⡀⣼⡏
⠀⠀⠀⠀⠀⠀⢀⣤⠿⠀⠿⣤⡀⣰⠟⠑⠒⠲⣝⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠟⠛⣋⣉⣤⡀⢻⡄⠀⡿⢀⠤⢭⣉⡛⢳
⠀⠀⠀⠀⠀⢠⡟⠓⠒⠀⠀⠈⠻⡏⠀⠀⠀⠀⠈⡏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢟⠉⠀⠀⠃⠈⣇⢸⢃⣌⡀⠀⠀⠁⣼
⠀⠀⠀⠀⢠⣟⣀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⡇⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢨⣴⠆⠰⠛⠃⢹⡿⠀⢩⣁⠘⠛⢠⡏
⠀⠀⠀⠀⠘⣯⠛⠀⠀⠀⠀⠀⠀⣼⣄⠀⠀⠠⢞⡼⠷⢎⢻⡄⠀⠀⠀⠀⠀⠀⠀⢷⠀⡤⢦⠒⠊⠁⠈⣧⠀⢀⠃⢪⠇⣼⠁
⠀⠀⠀⠀⠀⠘⢦⣤⣤⠄⠀⢀⣴⠃⠈⠙⠒⠛⠉⣴⠖⠋⣼⠁⠀⠀⠀⠀⠀⠀⠀⠸⡆⢹⡭⠄⠀⠀⠀⢸⡄⠀⠀⠸⢠⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠻⣤⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⠂⠀⢀⣀⣠⣬⠟⠛⠲⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠛⠋⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠀⠄⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⣀⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠾⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⠀⠀⠀⡠⠔⠒⠭⢁⠒⢄⠀⠀⠀⠀⢀⠀⠁⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⡀⠀⢀⠀⠐⠛⠲⢦⣀⣀⠨⠐⢵⡀⠈⠖⠉⠳⣍⠠⡀⠀⠀⠀⠢⣄⠀⢀⡀⢠⢄
⠀⢀⣠⠆⠁⡀⠀⠆⢸⣿⡿⣿⣧⢰⠉⠁⠀⠀⣷⣶⣤⣤⣈⠈⠳⢤⡀⡇⠀⠀⠀⠀⠈⢦⡄⠀⡀⠀⣿⣿⡇⠘⠲⠒⠟
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠻⠛⠁⠀⠁⡄⠀⠀⠡⡀⠈⠛⠿⣿⣦⣀⡹⠁⠀⠑⣄⣀⣠⡟⢁⡀⠈⠓⠛⠋⠀⠀⠐⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠦⠠⠈⠿⠋⠀⠀⠀⠀⠀⡀⠀⡙⠯⠧⠀⠀⠘⠂⠀⢠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠤⠀⠀⠀⠀⠀⠲⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠲⢿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠈⠚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢲⣶⣶⣶⣶⣶⣦⣴⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⡀⣶⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣟⡉⠛⠿⣿⡟⢉⣉⣉⣉⣉⣩⣹⣿⢏⣉⣉⣉⣉⣩⣭⣍⠋⠻⣯⣭⣿⠶⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣲⣶⣿⣏⣙⣛⣿⣯⡽⠌⡟⠯⣯⣽⣿⣿⣟⣋⣹⠍⣿⣿⣿⡟⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⢚⣿⠿⠿⣾⣟⢿⣿⣿⣿⣿⣿⣷⣿⡿⢟⣻⣿⡿⠿⠿⣟⣾⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣷⣿⣷⣸⣿⣷⣽⣻⣿⣿⡿⠿⢿⣖⣿⣿⣿⣡⣿⣷⣾⠿⠓⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡇⠀⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣯⣭⣛⣻⣿⣷⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢽⣾⣿⣿⣿⣟⣛⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣛⣛⣛⣛⣿⣿⠷⠖⢲
⠀⠀⠀⠀⠀⠀⠀⢠⢷⣿⣟⣋⣿⣛⣹⣿⣋⣿⣛⣹⣿⣋⣽⣿⣩⣿⣋⣹⣟⣩⣿⣋⣽⣿⣩⣿⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠛⠟⣹⣿⢿⣹⡿⢏⢝⡟⡩⢿⠟⣭⡿⢫⣽⠻⢋⣽⢿⡩⡿⢿⡹⢿⣿⢻⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣯⠵⠇⠈⢟⣭⢈⣟⣥⠐⣋⣥⠘⣫⡄⢛⣡⡄⣚⣥⠚⣋⡼⢹⣫⣌⠀⢈⣶⠟⣿⡀
⠀⠀⠀⠀⠀⠀⢿⣛⠶⢞⡛⠶⠲⠶⣶⠶⠶⠶⠾⠶⠶⠶⠶⠲⠶⠶⠖⠒⠒⠲⣶⡖⠒⠛⢃⣈⣓⣮⡝⣡
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡷⣶⣿⣿⡿⠿⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣯⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡐⠺⢿⡆
⠀⠀⢀⡤⠖⠛⠉⠙⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠄⠀⢀⣀⣀⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠠⡈⠊⠀⠀⢀
⠀⢀⠎⠀⣠⠖⢻⠋⡼⡿⠋⠁⠉⠉⠈⠉⠉⠉⠉⠉⠙⠐⠒⠒⠒⠒⠒⠒⠖⠴⠠⠤⠀⠀⠀⠀⠀⠒⠲⠒⠒⠒⠐⠒⠐⠂⠤⠤⠤⣀⠉⠒⢕⠐⠀⠀⠀⠀⢸
⠀⠈⠀⡘⡎⠀⣿⠀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⡄⠀⠇⠀⠀⠀⠀⢸
⠀⠂⠀⡧⠅⠀⣵⢸⢾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠹⠀⠦⠀⠀⠀⠀⢸
⠀⠀⠀⣛⡀⠀⢸⠀⣽⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⣽⠀⠆⠀⠀⠀⠀⠀⢸
⠀⡀⠀⡅⡇⠀⢸⡇⡭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⢠⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠄⡇⠀⢸⡇⢻⣙⣤⡀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⢻⢸⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠖⡇⠀⢸⠃⠀⠉⠙⠛⠛⠛⠗⠟⠛⠁⠁⠦⠦⠿⠿⠿⠻⠗⠿⠶⠦⠆⡖⡖⣶⣲⣖⡒⡂⡀⡤⡤⠤⠤⣤⢤⣤⢤⣴⣾⠷⠃⢸⢸⠀⢸⠀⠀⠀⠀⢸
⠀⡇⢀⠚⠁⠀⡟⠀⡁⠀⠐⠉⠁⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠃⠈⢸⠀⢸⠀⠀⠀⠀⢸
⠀⡇⢸⠸⠁⣸⠃⠀⠁⢀⠤⠤⢄⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸
⠀⠁⢸⢸⠀⡟⠀⠁⢰⠀⠀⠀⢈⡟⣧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠀⠠⠀⣄⡀⠀⠀⠀⡄⢸⠀⢸⠀⠀⠀⠀⢸
⠀⠀⢸⠘⣼⠃⠀⠀⣤⠓⠀⠠⠊⢀⡟⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡌⠀⠀⠀⢨⢹⡄⠀⠘⣇⢸⠀⠴⠀⠀⠀⠀⡄
⠀⢠⠀⠂⢻⠐⡀⠀⠈⠛⠲⠶⠖⠋⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡆⠀⠀⠀⠀⢧⡂⠠⠒⠁⡰⠁⠀⠀⠸⣾⠀⡆⠀⠀⠀⢠⠏
⠀⠘⠀⢧⢈⢆⡀⠐⣄⣀⣀⣀⣀⢀⣀⣠⠋⠉⠉⢹⠊⠉⠉⡿⠉⠉⠉⢱⡏⠀⠐⡶⠒⠒⠒⢢⡀⠀⠀⠀⠀⠀⠉⠓⠒⠉⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⣰⠎
⠀⠀⠡⡀⠓⢴⣉⡙⢀⠖⠹⢃⠒⡖⠠⠼⠯⠽⢷⠿⡶⢶⡾⣿⣤⣤⣴⣾⢧⣀⡸⠷⠶⠶⢆⣺⠇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡠⠖⠀⡧⠀⡇⢠⣾⠉
⠀⠀⠀⠈⠲⢤⣤⣄⢉⣈⡁⠛⠚⠒⠚⠒⠓⠂⠐⠐⠣⠴⠤⠆⠤⠤⠦⠎⠠⠏⣠⣉⣌⣁⠋⡝⠉⠋⡑⠰⢲⠒⠞⢺⢛⠛⠱⢠⠤⡆⡴⠃⣰⣷⢋⣿⠆
⠀⠀⠀⠀⠀⠀⣟⡀⠀⣼⠁⠈⠉⠉⠉⠉⠁⠐⠒⠒⠂⠠⠤⠤⠤⠄⢀⣀⣀⣀⡀⣀⠀⠀⠀⠈⠉⠉⠁⠒⠒⠒⠒⠒⠚⠒⠐⠚⠂⣉⠠⠞⠋⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⠛⢹⢉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢄⣸⡸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⡄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠱⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠆⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢰⠀⠘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠰⠀⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠠⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⢡⡃⠀⠉⠉⠀⠀⠀⢹⠁⠀⣧⠁⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⣰⠏⠈⢧⢸⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠂⠀⠀⠘⡇⠀⢰⠏⠀⠀⠀⢿⠀⠀⡟⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢀⠃⠀⢀⣀⠀⠀⠆⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⣀⣀⣀⡇⢸⠰⢩⡿⠯⢏⠡⢸⠀⣇⢀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⡀⠁⠱⠦⠦⠄⠁⢹⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡍⠙⡖⠒⠒⠒⠒⠒⣿⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠒⠒⠂⠒⡀⣾⡇⠀⠀⠀⠀⠀⣿⡆⡗⠐⠂⠒⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡀⣠⠀⠖⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⡿⠈⡇⠀⠀⠀⠀⠀⣷⠸⣧⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠈⠃⢤⠀⠒⠀⠈⣀⡈⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠁⠈⢈⡟⢳⠚⡗⠒⠒⠒⠒⠚⣟⢲⠛⣅⠁⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠑⠦⡉⠁⣙⢶⣤⠄⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠴⡫⠀⡀⢈⡇⠀⠀⠀⠀⠀⣿⠈⡈⠩⠳⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠌⠳⣶⡀⠉⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠞⡡⠊⢀⠔⠁⠸⡏⠉⠉⠉⠉⠉⣿⠀⠑⢄⠈⠢⡑⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⠴⠈⠛⢷⣀⠈⡡⢄⠀⠀⠀⠀⠀⢀⡴⢊⡵⠋⡠⠒⠁⠀⠀⠐⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠤⡈⠢⣝⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠲⡂⠈⠉⠳⣄⣀⠱⠆⠀⢀⡴⣋⢔⣁⠀⡎⠀⠀⠀⠀⠀⢸⡇⠀⣀⣄⢀⠀⣿⠀⠀⠀⠀⠀⠈⡆⠈⠑⢌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠈⠉⠻⣧⣉⡡⠬⢷⠦⠄⠐⢚⣁⡙⠦⠀⠐⡗⠲⣖⠒⢒⢒⡎⠀⠀⢸⣿⠀⠀⠈⣖⠒⠒⠒⠒⠒⠓⠒⡒⠒⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠐⡉⢷⡀⠀⢂⢻⣿⡷⡾⠟⠃⢃⠀⢡⠤⠋⠉⠉⠉⡇⠀⠀⢸⣿⠀⠀⠀⡏⠉⠉⠉⠉⠉⡏⢉⣉⠉⠩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣧⠀⠸⠀⠷⠀⠛⣤⡀⡘⢀⡜⠓⠀⠠⠤⡄⣇⣀⣀⣸⣿⣀⣀⣀⡧⡄⠤⠀⠐⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠤⠀⠒⣈⡉⠆⠀⠀⠀⠀⡿⠀⢰⠤⠤⠀⢒⣊⠹⢧⣝⠛⠫⣯⠉⠉⢃⠸⣯⣭⣿⣿⣿⣭⣽⢀⠉⠈⠹⠉⠉⠹
⠀⠀⠀⢀⠀⡶⠖⠙⠒⢂⠈⠉⠀⠀⠀⠀⠀⠀⠀⡾⠋⠉⠀⠀⠀⠀⠙⢤⡄⠒⠙⢻⣄⡈⠵⢄⣘⠀⠉⠉⠉⠉⠉⠉⠉⢘⣀⣀⣀⡀⣀⣀⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣉⠁⢈⠻⣦⡄⠙⠢⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠙⠋⠛⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀⡀⡀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣈⣻⣌⣀⠀⠙⠶⣄⠋⠓⣄⠀⠀⠀⠀⣃⣀⣀⣀⣀⣀⣀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡤⠀⠛⠻⠎⢀⣑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢖⠀⠉⠀⣀⣀⠨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁