Sonic the hedgehog Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣤⣤⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠸⣿⣿⣿⣿⣿⡿⠁⠀⠀⣤⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⣯⣽⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠃⠀⠀⠀⠀⢹⣿⠆⢠⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠘⠂⠀⠀⢀⠀⠀⠀⣿⣦⣤⣿⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠘⠀⠀⠀⠀⣀⠀⠈⠀⠀⣠⠊⠀⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠁⠈⠀⢀⠀⢀⠔⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠉⠛⢿⣿⣿⣿⣿
⠀⠀⢄⠰⠄⠀⠀⠀⠀⠀⠀⠌⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣀⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠰⠀⡈⠀⠀⠀⠀⠀⡀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠶⠤⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠲⠀⠀⠀⠀⣰⡟⠀⠀⠀⢰⡿⠁⠀⠀⠀⠈⠙⣿⣿⣿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠁⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⠻⠆⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠃⠀⠙⠁⠀⠐⣧⣀⢀⡄⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣿⣇⠀⠀⠀⠀⠈⠀⠀⠒⠂⠚⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⡆⠀⠀⠀⠀⢀⡤⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠹⣦⣄⣘⢁⣴⣆⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⢹⣿⣿⡄⠀⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠆⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠄⠀⠸⠿⣿⣧⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠉⠛⢦⣄⣠⣄⠀⠀⠀⠀⠀⠀⢀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⣄⠀⠀⠀⠀⢻⣿⣿⣷⣶⣶⣶⡾⠛⠉⠀⠀⠀⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣀⣠⣤⣌⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣠⣿⣿⣿⣿⡇⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠋⠉⠉⠻⣿⣄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⡿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⢀⠀⠆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠁⠈⠛⢛⠂⢀⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢀⣠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠶⣄⠀⠀⠀⠈⢧⣤⠾⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠸⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⢀⠀⠀⠀⠀⠀⢰⠞⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠈⠀⠀⠄⠀⠈⠀⠀⠘⠋⠉⠉⠩⠿⠶⠦⠤⢤⣀⣀⣀⣴⣄⡀⠀⠀⡀⠀⠀⠘⠙⡄⣠⣴⠃⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣻⣿⣦⡀⠀⢄⠀⠀⠀⠀⢠⣿⣬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⠀⡀⠀⠀⠀⡎⠁⠀⠈⠛⢷⣦⣤⠦⣀⣠⠄⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠊⠀⠆⠀⠺⠀⠀⠀⠀⠀⠂⢹⠃⠀⢸⡇⠁⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠀⠀⠀⠀⠀⢠⠎⠀⠀⣸⡇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⢈⠢⠐⢦⣀⡀⢀⡠⠊⠀⠀⣠⡟⠻⠿⠛⠛⠁⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠀⢠⠀⠀⡰⠛⠉⠉⠉⠀⠀⢀⣶⣟⠀⠀⠀⠀⠠⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣿⠀⢀⣧⡀⣀⡀⣀⣠⠤⠚⠉⠉⠀⠀⠀⠀⠰⠷⠀⠘⠣⠄⠀⠀⡴⠋⠀⠀⠀⠊⣦⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⢿⣄⢘⠋⠉⠉⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠐⠊⠀⠀⠈⠠⠖⢶⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢁⣠⠴⠚⠋⠀⠀⠀⠀⠀⠀⠴⠀⣾⠐⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠌⠅⠀⠀⠀⢀⣀⠶⠀⠐⡁⢺⠑⠻⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠛⢋⣅⠀⡛⠈⠁⣘⡦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⣠⢔⡎⢹⣇⡤⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠗⠀⠐⠀⠀⠀⣿⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⢾⣅⣈⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠠⠆⠀⠀⢀⠈⠃⠀⠈⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⠀⠀⠀⠈⠠⠀⠀⣀⣀⣀⣀⣤⠶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⠩⠉⠁⢘⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢠⡔⠁⠀⣤⠀⠀⠀⠀⠀⢄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⡀⢀⡼⠷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠒⠒⠒⠒⠚⠉

⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⡟⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠞⡇
⠀⠀⠀⠀⢤⠈⣄⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⠐⠤⠁⠀⠀⠀⠈⢆⠀⠀⠀⢀⠎⠀⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡌⢠
⠀⠀⠀⠀⠸⡀⠈⢦⡀⠑⠂⠀⠀⢀⡤⢃⠈⡐⠄⠀⠀⠀⠀⠀⠈⢧⠀⢠⠏⠀⠀⠀⠀⠀⠀⠐⡀⢀⠣⣄⠀⠀⠀⠐⠁⣠⠎⠀⡸
⠀⠀⠀⠀⠀⢇⠀⠀⠑⢄⠀⠀⠀⠈⠇⠀⢀⣤⣴⠶⠶⠀⠀⡀⠀⠀⠱⠁⠀⠀⡀⠈⠰⠶⠶⣤⣄⠀⠀⠛⠀⠀⠀⢀⠔⠁⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠑⠀⠀⠀⠀⠀⢠⠛⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠐⡌⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠆
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠈⢨⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢨⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠈⡿⢦⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⢰⣄⡟⠀⢠⠀⠘⣇⣰⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⣠⢾⡍
⠀⠀⠀⠑⡙⢦⡲⢄⡀⠀⣀⣀⡄⠀⠀⠀⠀⢦⡀⠀⠀⠀⠸⢿⠇⠀⣿⡇⠀⢿⠿⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⣄⣀⡀⠀⣀⠴⣢⠚⡑⠁
⠀⠀⠀⠀⠘⣄⠉⠂⢌⣉⠉⠀⠒⠀⠀⠀⠀⠀⠙⢗⠢⠀⣀⢢⣀⡀⢿⠇⣀⣠⡂⠀⠠⢒⠟⠁⠀⠀⠀⠀⠐⠃⠈⢉⣉⠄⠊⢁⡜
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⣓⡤⢄⣀⡀⠀⠀⠀⠀⣀⣀⠤⣔⡁⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠁⡀⠀⠑⢍⠁⢀⡀⢋⠕⠃⠀⡀⠩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠜⡠⠀⠘⢉⡉⠀⢀⠆⢂⠀⠈⣉⠛⠁⠠⡐⠄⠂⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢠⡁⠀⠀⠀⠔⠀⠠⠀⠠⠀⠐⠌⠀⠀⠀⢡⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢀⡟⠀⠀⠳⡀⠂⡴⠣⡄⡤⠳⡄⠂⡰⠃⠀⠘⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠸⠀⠀⠀⠀⠙⡄⠀⠀⢡⠁⠀⠀⠜⠁⢀⠀⠀⠰⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡁⠹⡄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⢀⣤⠏⠀⠀⠀⡼⠀⣥⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⡌⠙⠰⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠠⠶⠐⠙⣄⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠴⡄⠀⠀⠀⠀⠘⠀⠀⡠⡀⠀⠰⠀⠀⠀⠀⠀⡴⠞⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠈⠂⠌⠀⠀⠀⠀⠀⠀⠰⠀⠰⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠀⢠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠠⠄⠀⠀⠀⠀⠀⢀⠃⠀⠀⢃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⡆
⠀⠀⠀⣠⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈⢀⡀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣁⠈
⠀⠰⠾⢯⡼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⠄⣤⠀⡁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡁⢠⡄⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠁⠀⡘⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠻⡛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡻⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠓⡒⠚⢢⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠚⠒⡒⠓⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡀⠀⠄⠀⠸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠄⠀⣚
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠒⠂⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠁⠀⠒⠒⠂⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠂⠈⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠂⠤⣀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⡠⠄⠒⢦⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⠛
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⢁⠞⠁⠀⠀⠀⠀⠀⢀⡠⠔⠒⠢⢄⠀⠰⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠃⠈
⠀⠀⠀⠀⠀⡦⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⠔⠁⡀⠤⡤⠀⡀⠁⠀⠀⠈⠑⢄⢠⠀⡀⠀⠀⠀⠀⠀⡠⠒⠉
⠀⠀⠀⠀⠀⠠⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⡠⢀⡴⠋⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠈⠃⠀⠇⠀⠀⠀⠠⣎
⠀⠀⠀⠀⠀⠀⠾⠁⠀⠂⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢄⡀
⠀⠀⠀⠀⠀⢠⠀⡐⢠⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⡧⠀⠀⢂⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⣾⣿⣇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣧⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢻⣷⡄⠠⠊⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⢀⣴⣦⣄⡀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⣠⡠⠤⠀⢄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣶⡄⠐⠂⠄⢀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠊⠁⠀⠀⠀⠀⡇⠀⠀⢀⠁⠠⡄⠒⠒⠀⠀⠀⠀⠐⠒⠂⠤⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠛⡟⠛⠃⠀⠀⠀⠀⠁⠐⠒⠒⠀⠁⠀⠀⠀⠀⠀⡹⡃⠀⠀⡇⠀⠀⠈⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠊⠀⠁⢀⠔⠀⢀⠆⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠨⢤⡦⠤⢀⡀⠀⠀⣀⠀⠠⠀⠂⠁⡴⠚⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠎⠀⡀⠀⡀⢃⡄⠊⢁⡀⠤⠬⠧⠤⠀⢠⡔⠁⠀⠀⢸⠄⠀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠃⢀⠀⠈⠀⠀⠐⢿⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡔⢁⠀⠀⠀⠈⠑⠤⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠑⠰⠶⠘⢆⡀⠀⠈⠑⠛⠄⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⢰⠀⠣⠀⠀⠀⠀⠀⠈⠢⢀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠕⠢⢄⡀⠀⠠⠙⠆⠄⠀⠀⠨⢦⡤⠀⠀⠁⠀⠘⠀⠁⠀⡘⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡆⠀⣠⢀⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠑⠈⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⢁⠖⠊⠀⠊⢀⠄⠒⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠄⣀⣀⣀⠠⠔⠁⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠉⠢⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⢠⠤⢄⠀⠀⠰⠃⠤⢀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠀⠈⡀⠀⠀⡆⠀⠈⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⢐⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠀⠈⠀⠀⢉⠀⠐⢰⡄⠀⠘⡦⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠈⠉⠉⠉⡩⠒⠉⠀⢰⠀⠀⢡⡔⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠘⡀⠀⠀⠒⠛⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠊⠱⢄⡀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠰⡀
⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠢⡀⠀⠙⠒⠒⠀⠀⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠸⡀⠀⠘⠀⠀⠀⠀⠱⠤⠄⠀⠀⢀⡀⠠⠤⠴⠎⡉⢂
⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡇⠀⠀⠀⡆⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠈⠻⡆
⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⣠⡇⠀⠀⠐⠁⠀⠰⠉⠀⠀⠉⠲⡄⢰
⠣⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠼⠤⠤⠓⠒⠈⠀⢃⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣆⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⣀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠉⠁⠀⠀⠀⡠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠐⠚⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠢⣄⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢴⠖⠻⠛⢭⣍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠆⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⢀⡤⠀⠤⢄⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡼
⠀⠀⠀⢀⠼⠃⠀⠀⠈⠉⢆⢘⡄⢰⡄⠀⠀⠀⠀⠀⠀⠄⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣔⠋
⠀⠀⠜⠉⠀⠀⠀⠀⠀⠄⠬⣍⡒⠦⣀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢀⠀⠀⢀⣀⣙⡍⢲⠆⠭⠈⠉⠢⡀
⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠒⠒⠋⠉⢀⣠⢤⡀⠀⠀⠀⠀⠀⢖⠒⠢⠽⡄⣡⠇⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠉⠉⠢⣀⡀⠀⠀⠀⠄⡤⠤⠔⠒⠊⠁⠀⠀⠙⠢⣀⠀⠀⠀⠘⠖⣒⠞⡫⠃⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⣀⣤⡀⠈⠣⣄⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇
⠀⠀⠀⡴⠉⠉⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⣠⠔⠋⠁⠀⠈⠓⠤⣀⣀⠴⠦⠤⢦⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢀⣀⡴⠋⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢰⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡎⠁⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⣠⡴⠃
⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⢣
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠃
⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠐⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠒⠒⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢰⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠂⠁⠀⠀⠀⠈⠀⠀⠐⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡆
⠀⠀⢸⠀⢰⣤⣠⣄⣀⣴⣦⣤⣦⣤⣴⡇⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠸⠗⠛⠛⠛⠛⠛⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠘⢸⣷⣦⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢁⡤⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠈⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣠⠀⠆⠂⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡟⠀⡀⠀⣰⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠂⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⣴⡶⠀⠈⠃⠁⠑⠀⠐⠃
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠈⠀⠉⠰⠆⠈⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠈⠉⡁⠀⠄⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⣧⡤⠆⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠠⠀⢀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠄⠊⡀⣘⠀⠀⠀⠙⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⡐⠀⠀⠀⠀⠈⢀⠀⠀⠁⠃⢈⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢠⡄⠀⠀⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⢀⠄⠉⠀⠀⠀⠂⠀⠂⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠊⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠢⢄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠓⠀⢄⡀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⡀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠗⠆⠠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠼⠀⠀⠠⠀⠀⠀⣠⠌⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢄⡤⢤⡠⢴⠤⡤⣤⠤⢎⢠⠤⠤⠤⡤⢤⠤⢤⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠒⣬⢰⡆⠁⠘⠀⠀⠁⢰⣬⠈⠃⢸⡆⠇⢸⠐⢫⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠀⠀⠀⠀⣇⠓⢠⡌⢃⡆⣷⣀⠅⣦⠘⢠⣀⡄⢸⡇⣆⣸⠘⢻⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⣿⢸⠀⢶⠠⡄⠋⢰⣈⠁⢶⠀⡇⠈⢰⠈⠃⣦⣸⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣤⢸⠀⠾⠐⢃⣄⠰⢀⡀⠾⡀⡄⢀⠘⢀⣄⠲⣸⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠉⠉⠉⠀⠉⠉⠁⠉⠉⠙⠋⠉⠉⠈⠉⠁⠠⡇
⠀⠀⠸⢦⠤⣤⠤⠤⣤⢤⠤⡤⢤⡤⠤⢤⠤⡤⠤⡤⠤⣤⡤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣄⡀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣾⣿⣿⡿⠋⢹⣿⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣶⣦⣤⣴⣾⣿⣿⣿⡿⠿⠛⠀⠀⢸⣿⠀⠀⠀⠀⠴⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟⠉⠻⠿⠿⠿⢶⣦⣤⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⠿⠿⠟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠛⡁⠀⠀⣠⡶⠚⡁⠂⢄⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠙⣿⣦⣀⣼⣿⢯⠞⠉⠙⢳⣌⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⣾⣿⣿⣿⠁⠁⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣅⠀⠀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣻⢏⡆⠀⠀⠰⠀⠀⢸⡏⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡾⠁⠀⠀⢰⣇⢀⠿⠙⠀⠠⣿⣿⠆⠀⡠⢤⣤⣦⣤⣘⠂⣘⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣧⡄⠀⠀⠀⠀⠛⢀⠀⢀⢀⡴⠟⠁⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣉⡷⣶⣦⣤⡴⠖⠙⠋⠉⠀⠀⣤⣶⣾⣿⣷⣶⣦⣄⠠⠙⢿⣿⣿⡿⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠆⠈⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣛⠛⢦⣤⣀⣤⣀⣀⣀⠀⢀⣴⣿⡅⠂⠄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⣶⣤⣈⣿⠋⣽⣿⠏⠈⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣿⣿⡄⠀⠀⠈⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⢻⣿⣿⣿⡄⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠈⣿⡇⠙⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡀⢸⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣦⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣻⣛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣠⣴⣶⣶⣷⣤⠀⠀⠀⠀⢹⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠿⢿⡿⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠰⣿⣿⣿⣦⣄⡀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠻⠿⣿⣶⣄⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠴⠶⠶⠖⠒⠒⠶⠶⠶⠦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⢤⣀⣤⠴⠖⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠜⠛⠋⢉⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢸⣏⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠐⠐⠉⠀⠙⣅
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣋⣀⡀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⡹⠋⠉⠉⠉⠉⠉⠉⠛⠛⠛⠓⠲⠶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠉⠳⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣁⣀⣄⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⣾⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠼⣀⡀⠀⠸⣿⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡤⠤⠤⠶⠶⢶⡾⠁⠀⠀⠀⣠⣀⠀⢀⣠⣤⣄⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⠑⠚⢣⣤⣤⣶⠄⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠛⢳⠾⠷⠶⠶⢶⠛⠁⠙⣮⠉⠀⣀⣈⠙⠦⣄⠀⢷⡀⠀⠀⠘⣀⣠⠴⠋⠀⣠⡴⠞⠁⢀⡴⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⣸⠇⡀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠸⡷⠚⠉⠈⠙⠢⠼⠀⠀⡩⢻⠻⢭⣉⠀⢀⣠⡟⠁⠲⣄⡠⠎⠀⣠⠆⠈⣧
⠀⠀⠀⠀⠀⢀⣤⠞⢁⠤⠒⠁⢹⠤⡀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⡿⢦⡀⠀⠀⠀⠀⠀⠰⠁⢸⠓⢦⣈⣉⡉⢹⡄⢢⣀⣸⠧⡤⠚⢁⡠⠂⡼
⠀⠀⠀⠀⣰⢏⡵⠊⠁⠀⠀⠀⠘⢷⡠⠄⠀⠀⠀⣀⣠⣶⣛⣋⣀⣠⠞⠀⠀⠀⠀⠀⠀⠁⣠⠋⠀⠀⠈⠉⠉⠉⠻⣄⠈⠀⣲⣬⣶⣋⣠⠞⠁
⠀⠀⠀⡾⡡⠋⠀⠠⠀⠀⠀⢀⡴⠋⠉⠙⠛⠛⢻⡍⠀⠉⠓⠲⠤⢄⣀⠀⠀⠀⢀⣠⠥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠀⠀⢸⢣⡃⠀⠀⠀⠀⠑⢠⠎⠀⢠⠎⠀⠀⢀⣾⣀⠀⠀⠀⠀⢀⡴⠃⠀⣄⠀⠀⠙⠢⣄
⠀⠀⣿⢼⠀⠁⠀⠀⣠⡴⠻⣆⣠⠏⠀⢀⣴⣏⠀⠉⠳⢦⣤⡴⠋⠀⢀⡴⠋⠙⢦⣀⠀⠈⠳⣄
⠀⠀⣿⢸⠀⠀⠀⣴⠋⠀⠀⠀⠙⣦⡴⠟⠁⠈⠛⠶⣤⣀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠉⠳⣄⠀⠘⢷⡀
⠀⠀⢿⢸⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠙⢦⡀⢀⣀⣄
⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⢀⣨⠟⠋⠁⠨⣇⠀⠀⠀⠀⠀⣀⣠⣤⠤⠴⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⣨⣷⠴⠶⠚⠛⠉⠁⠀⠀⠀⡼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢀⣠⡴⠛⠁⠀⠆⠀⣠⠴⠛⠙⡄⢀⡼⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣞⠋⠁⠀⠀⠀⠀⠀⡏⠡⠒⢉⠆⢘⣯⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢘⣆⣉⣥⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡓⠂⠤⠤⠤⠒⢊⣩⡤⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣤⡤⠶⢺⣿⢱⠐⢐⠠⡆⣾⡱⠀⠂⣆⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠒⠒⠓⠒⠒⠒⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠉⠙⠋⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⢀⡤⠖⠛⠛⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⢀⣀⠀⢀⡴⠚⠉⣺⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⢀⡤⠞⠁⠀⠉⠺⠀⢀⡜⠁⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⢀⡿⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣹⡄⢧⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣧
⢸⡄⠀⠀⠀⠀⠈⠀⠀⠸⣄⠘⠿⠿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠲⢤⣄⠀⠀⠀⢹
⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⣀⣤⠊⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠈⠙⢦⡀⢸
⠀⠀⠙⠒⢤⣄⠂⠀⠀⠀⠠⣇⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣦⣄⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠙⣼
⠀⠀⠀⠀⠀⠈⠣⣄⣠⢤⡀⠈⠉⠉⣀⠀⠀⠀⠀⠀⠀⠀⡰⢉⡳⣦⠀⠈⢻⠳⣄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠋⢹⡄⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⢠⡏⠀⠈⢧⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⢧⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⡀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢳⡀⠀⢀⠀⢴⣳⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⣧⠀⠘⡏⠒⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠃⠀⠙⠇⠀⠸⡄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠤⠤⠤⠄⠒⠒⠊⠉⠁⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⢀⠀⠀⡀⡀⠀⡀⢀⠀⢀⡀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠰⢶⡋⠉⠁⠀⠀⠀⡴⠒⠠⣄⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⢀⠃⠀⡀⠴⠑⠀⠀⠀⠀⠈⠓⢤⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠝⠀⠀⠀⠉⠀⠠⠐⠀⠐⢤⡀⠀⠀⠀⢳⡀⡼
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠱⠀⠀⠀⢠⣼
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇⠀⠀⢇⠀⢰⠃⢸⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣰⣃⡠⠤⠤⠤⣄⣀⠀⠀⠀⡠⠤⣄⠘⠟⠀⠀⠀⠈⢲⡟⢀⡇⠀⠀⠀⢠⠞⢁⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⢸⡁⠠⡤⠙⠲⠤⠴⠒⠊⠹⢿⡿⠋⠀⠞⠙⡷⣚⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠱⢄⡈⠒⠀⠀⠀⠀⠀⣀⠜⠁⠀⣰⠌⠢⡽⠒⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⡠⠤⠖⢒⣶⠭⣓⠶⠶⠖⠶⣏⡁⠀⠀⡎⠁⠀⢀⣈⣨⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠊⠁⠀⡠⠒⡩⠔⢊⡃⠀⠀⠀⠀⠘⣌⣓⣋⡽⠢⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠀⡗⠢⠏⢸⡀⠀⠀⠀⠀⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣃⣇⠓⠁⢀⠇⠀⢹⡄⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣹⣶⢏⠀⢀⢾⠰⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢺⠁⡎⠀⢣⠘⣆⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢊⣇⣰⡗⣤⠺⠤⠜⠀⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠰⣏⡆⠀⠀⢀⡴⠋⠉⠉⠛⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢝⠒⠶⠒⢾⠀⡗⠒⠋⠁⡆⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠂⠈⠀⠈⣧⡇⠀⣀⣀⣤⠤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⣠⡤⠖⠋⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡀⡀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠁⠙⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠒⠒⠠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣠⠴⠶⠚⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠤⠤⣄⣀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⢀⣠⠶⢱⡌⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠴⢢⣀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠀⠀⡌⠇⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠇⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢠⣹⣤⠠⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢄⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠸⣄⠐⢠⡯⠀⠱⣆⠒⠒⠢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠳⡀⠘⠀⠈⠁⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠃⢰⠋⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢸⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠸⠀⠀⠀⢈⣦⡀⠀⣀⠀⣠⠆⠀⣵⠀⠀⢀⠁⠀⠀⠀⠀⣠⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠁⠀⠀⠈⠃⠉⣊⣭⣊⣁⠀⠀⠋⠀⠀⠦⢤⠀⠀⠀⠀⠀⢹⡛⠦⣄⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣶⠛⠒⠀⠂⠈⠈⠛⠃⠀⠀⠀⠀⢠⠆⢀⣼⠶⠶⢦⣄⠀⠈⡇⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠋⠁⠀⣿⠈⠓⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⢄⣀⡀⠀⠀⠀⠀⠈⢉⡷⠤⠤⠤⠤⠤⠔⠖⠛⢉⡀⠈⠑⠢⢄⣹⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠔⠒⣄⡀⠁⠈⢢⠀⠀⠀⢀⢮⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⢙⠳⢦⣀⠀⠉⠓⡶⠛⠶⠛⣶⠚⠓⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡠⢀⠁⠀⢠⡟⢦⣠⠔⠁⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⡇⡆⠈⠙⢦⣴⠃⠀⢠⠞⢹⡇⠀⢸⠁⠀⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠀⠸⣤⣴⡇⠀⢸⠇⣠⠔⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢄⠈⠃⠀⢠⠞⠃⡐⠀⠀⠀⠈⢳⣤⣴⠄⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⠀⠀⠐⠘⡆⠀⠂⢠⠀⡟⠁⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⠀⠀⠻⠤⣴⠁⠂⠀⠀⢀⣜⠁⠉⠢⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠠⠐⢁⣠⢀⠶⠴⠃⠀⠀⠀⠀⣻⠤⣀⠀⠀⣀⠀⢿⠉⠁⠒⠇⠀⠀⠀⠈⠛⢦⡤⢴⡉⠉⠢⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⢠⠋⢀⡾⠉⠉⠹⡆⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢣⡀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡼⠁⠀⠀⠀⢹⡄⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣠⣥⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡜⠁⠀⠀⠀⠀⣀⣦⠀⠘⡗⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⡸⠀⠀⠀⠀⠀⡘⢁⡾⡄⠀⢹⣧⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢠⣇⣀⠀⠀⠀⠀⣷⡈⢿⡇⠀⢈⣷⠟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠫⢽⣇⣀⠾⠶⠈⡷⠀⠀⢠⢎⠑⠀⠉⠉⠉⢁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢦⠀⠀⠀⣀⣠⡄⢾⠀⠀⠀⣸⣄⠙⢦⣬⣩⣤⣤⡶⢋⡜⠋⢉⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠼⣦⠤⢤⣤⣤⣤⣦⡞⠀⠀⠀⣧⡎⠳⣀⠈⠉⠉⠀⠠⣀⢀⡾⠋⠁⠉⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠉⠙⠳⢦⣄⣉⠶⣄⠉⠀⠀⣧⠀⠀⠀⠣⢄⠀⠀⢀⠐⠀⢠⠃⢨⡟⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⡴⣆⠀⠀⢹⠀⠀⠀⠀⠀⠉⠓⠞⠣⡀⠸⠓⡟⠀⠀⠀⠀⠀⠀⠀⠠⢇
⠀⠀⠀⠀⠀⠀⢀⡾⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢿⣀⡠⢾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣇⠀⢀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡜⠑⠢⠀⠉⠒⠀⠄⠀⣀⡀⠀⠄⣿⣀⣾⢁⠊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣰⣏⣆⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠦⠤⠤⠤⠯⠿⠓⠉⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠃

⠀⠀⠀⠀⠀⢰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠀⢄
⠀⠀⠀⠀⠀⠀⣃⢣⠠⣀⠄⠠⡀⠀⡠⢔⣲⣦⣤⣭⣭⣭⢭⣴⣷⣌⠀⠀⠀⠨⠃⠰⠀⢄⣀
⠀⠀⠀⠀⠀⠘⠹⢸⢰⠋⢀⡐⡡⣫⣾⣿⣿⣿⣿⣿⣿⣯⣞⣾⣿⣿⣷⣄⣀⡀⠈⠀⠀⠀⠜⠁⢢
⠀⠀⠀⠀⠀⢠⠀⠸⣶⠐⠃⣔⠱⠷⠿⠻⣟⣛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⢖⡒⠒⠚⠲⠤⣄⣀
⠀⠀⠀⠀⠀⣺⣆⠀⢧⠳⡎⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠭⣐⠀⠠⢀⠀⠀⠀⠀⠀⠠⠄⠉⠒⢄⠠⣀⠀⠉⠙⢦⣄
⠀⠀⠀⠀⠀⣸⡌⠙⠆⠀⠙⣦⢄⣀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣀⣙⣦⡀⠊⠐⡄⠀⠀⢀⣀⡘⣄⢀⣱⣌⣳⢰⢻⣿⢽⠆
⠀⠀⠀⠀⠨⡀⠘⢦⡀⠀⠀⢸⣆⡀⠉⢺⣿⣄⠢⡀⠀⠀⢠⡟⣽⣯⠁⠀⣨⠀⠙⣻⣭⣭⣥⣽⣤⣴⢾⡧⠸⠼⢶⡟
⠀⠀⠀⠀⢀⡩⢄⢸⣯⠲⣽⣾⠷⣿⡦⢸⣿⣿⣷⡐⣆⣀⠈⠷⠷⠶⠂⢠⠎⠀⢰⣿⣿⣿⣿⣿⣿⡏⣸⣷⢣⡿⠛⠃
⠀⠀⠀⠀⠁⢠⠚⢸⣸⡀⡟⠿⠸⣿⣿⡦⡿⢁⣼⣻⣆⠠⡙⠀⣒⣢⠜⢉⠈⢠⡘⠟⠛⠿⣻⠟⣋⣼⣯⣿⣯⣀
⠀⠀⠀⠀⡄⠨⠄⠀⡜⣷⣷⣌⣤⠘⠓⢲⣿⡿⢿⣿⡿⢷⣬⣽⣽⠶⠶⠶⠿⠤⢭⢭⣭⢠⣴⣶⣿⣿⣿⡇⢹⣿⣿⡄
⠀⠀⠀⠀⠁⠑⠀⠄⢳⣿⣿⠈⢪⣢⣹⣿⠋⠞⠿⢿⡇⠈⠿⠛⣿⣤⢻⣿⣿⣿⣋⠐⡏⠣⠧⠿⡟⣹⣿⡇⡘⠀⢉⡇
⠀⠀⠀⠀⠀⢸⠀⡄⠘⣿⣿⣠⣾⣽⣿⡏⠀⢀⣇⣼⠇⠀⠀⠀⣯⣶⣤⣭⣤⠖⠀⠐⠈⢲⣾⣿⣶⡿⠋⣠⣣⣤⡾
⠀⠀⠀⠀⠀⠈⠂⠓⠤⠻⠿⠿⠛⠛⢻⣷⣾⣷⡿⠋⢠⢢⣠⡞⠉⠉⠉⠉⠁⠀⠀⠀⠀⠘⠿⣿⣿⣶⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣤⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⣸⣿⣿⣿⣷⣀⣀⡀⠀⠀⣀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠋⠉⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡵⠟⣫⣭⣭⡛⠭⣉⠉⠙⠻⢿⣿⣿⡄⢀⡾⠁⢰⣶⣶⣤⣤⣤⣶⠶⠾⣛⠿⢟⣛⡻⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⣿⣿⣿⣾⣿⣮⡳⡑⡦⠄⣙⣿⣿⣿⡇⠀⣿⣿⡿⠿⠛⠛⠁⡠⠊⠁⣿⣩⣭⣝⣿⣿⣿⢿⣆
⠀⠀⠀⠀⠀⢀⡿⣴⣲⣶⣶⡟⢿⡿⣿⣿⣿⣇⢱⢱⣤⣤⣿⣿⣿⠏⠀⠻⣆⣠⠒⠒⡲⣰⠁⠀⣀⣻⠛⠛⠙⠽⣿⠿⣦⢿⡄
⠀⠀⠀⠀⠀⣸⡇⣿⣿⣿⣿⠗⣀⠸⣿⣿⣿⣾⢈⢶⣭⣽⣿⣿⣷⠀⠀⣸⡏⠛⣿⣿⡃⡇⣾⢩⣿⣍⢸⡉⡱⠸⣿⡿⣸⢸⣷
⠀⠀⠀⠀⢀⣿⣿⡸⣿⣷⣾⣿⣿⣷⣿⣿⢗⣋⣘⡺⢟⢿⣿⣿⣿⠆⠀⠘⣧⠀⠳⠞⠋⣣⠹⣌⣿⣧⣦⢨⣄⣸⣿⣿⡿⣼⣿⣇
⠀⠀⠀⠀⣸⣿⣿⣿⣜⡻⢿⣿⣿⣯⠟⣵⠟⠁⠀⢨⣷⡹⣹⣿⡏⠀⠀⠀⢸⣧⠀⡔⣫⡶⠒⠚⣭⡻⣿⡻⢿⣛⢿⢟⣽⣿⣿⣿⡄
⠀⠀⠀⢀⣿⣿⡇⠈⠛⠿⣷⣶⣶⣶⡄⣇⠀⣀⣴⡿⢋⡇⣿⣿⣿⠀⠀⠀⠻⣏⢸⢼⠋⠀⠀⣼⡿⣣⢨⣭⣭⣥⣶⠿⢿⣿⣿⣿⢿⡀
⠀⠀⠀⣼⢿⣿⠀⠀⠀⠀⣼⣿⣿⣿⣧⡹⢿⣿⣷⣶⢟⣴⣿⣿⡟⠀⠀⠀⠀⢹⣷⡸⣦⣠⣾⣟⣵⡟⣼⣿⣯⠉⠀⠀⠈⢿⣿⣿⡞⣧
⠀⠀⢠⡟⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⢿⣟⠲⠦⣴⣶⣿⡿⠁⠈⠀⠀⠀⠀⠀⠸⢿⣷⣬⣛⣛⣛⣩⣾⣿⣿⣿⣇⠀⠀⠀⠈⢿⣿⣿⠹⡆
⠀⠀⢸⠧⣿⠁⠀⠀⠀⣾⣿⢟⡽⠃⠀⠙⠛⠶⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⣉⣹⣿⠿⠋⠘⣯⡻⣄⠀⠀⠀⠈⢿⣿⡇⣿
⠀⠀⠸⣧⡙⠀⠀⠀⠴⣟⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠚⠋⣠⠟
⠀⠀⠀⠈⠛⠓⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉

⠀⢀⠔⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠢⢄
⠰⠁
⡇
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡟⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣷
⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣤⣀⣼⠿⢿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣀⣿⣿⣇⣀⣨⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⣹⣿⣿⣿⣿⣿⣿⣿⡆
⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠀⠀⠉⠉⢹⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿
⡇⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛
⡇
⡇
⡇
⠰⡀
⠀⠈⠢⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠔⠊

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠈⢇⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⢄⡀⠀⠂⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡐⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡤⢀⡀⠔⠆⠀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⡄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠤⢠⠀⠈⠁⠀⠐⢃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠄⣀⠀⠈⠁
⠀⠀⠤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠲⣀⠀⠀⠀⠠⠀⠀⣀⣀⣀⣀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠃⠔⢠⠂⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠲⠦⢤⣤⣀⣀⣀⣀⣀⣀⡠⠤⠴⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢈⣉⣁⣀⣤⠤⠤⠒⠒⠒⠈⠉⠉
⠀⠈⠓⠲⠶⠤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠲⠶⠤⠤

⣶⡶⠲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣿⣶⣴⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡿⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⢻⣿⣿⣿⣿⣿⣤⣤⣄⣀
⠀⠀⠀⠀⠸⣿⣇⠀⠀⠉⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢻⣿⡄⠀⣴⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠻⣿⣾⣿⠟⠁⠀⠀⠀⠈⠻⣿⣿⣿⠟⠋⠉⠁⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠐⣧⡀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⢸⣿⣶⣄⣸⣿⠀⠀⣀⡀⡀⠀⢀⣀⡄⠀⠤⠀⠀⣤⣤⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠉⠁⠀⠀⠀⣻⡇⠀⠀⠀⠀⢹⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠈⠉⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠛⠂⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠐⠊⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣆⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣶⣶⣶⣶⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⠀⢀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠈⠙⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠂⠘⠃⣠⠹⣿⠟⣹⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠙⡋⠁
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢀⡍⠀⠀⢀⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⡟⠁⠀⠈⠐⢤⡀
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠔⠁⢀⠀⠀⣸⣿⣿⣿⣿⠿⢿⣾⡿⠛⠛⠠⡀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠁⠄⠀⠀⠆⠀⠀⠈⣦⡀⠉⠀⠈⠛⠉⣀⣼⠏⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⠀⠀⠀⠘⣿⣿⣷⣶⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⢀⣠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣿⣿⣿⣿⡟⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⢿⣿⣿⣿⣷⠈⣿⣿⣷⣤⣀⡀⠀⠀⠠⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠄⠀⢻⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⡅⠂⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣿⣿⠿⢇⣘⡿⣿⡿⢿⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⣀⣀⣀⣀⣠⣤⣶⣤⣄⣀⣀⣀⣀⣠⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠊⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠳⡀
⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢀⣈⣂
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣀⣴⣿⣿⣿⣿⣷⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣀⠀⠟⠛⠋⠉⠻⣇⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⠿⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠉⠉⠑⢦⡀
⠀⠀⢀⣀⣠⣤⢤⣤⣾⣷⣤⣀⠀⠀⠀⠀⠑⢤⡀
⢀⡜⠁⠀⠀⠀⢸⠀⡤⢦⠈⠹⣷⣀⠤⠖⠒⠦⢷⣀
⢸⠀⠀⠀⠀⠀⡟⠀⠀⡇⣧⠴⠋⠁⣀⡤⠴⠶⢶⣬⡇
⣸⠀⠀⠀⠀⠀⡇⠀⢴⡇⡧⣤⡶⠋⠁⠀⠀⠀⠀⣳⡀
⡇⠀⠀⠀⢀⡼⠁⠀⢀⠇⢷⣿⡁⠀⢀⣠⠴⠚⠋⠉⠉⢓⣦⡀
⡇⠀⢀⡴⠋⠀⠀⠀⣸⢸⡤⠽⣿⠟⠋⢀⣠⠴⠖⢋⣉⡭⠭⠷⢄⡀
⢹⣶⣏⣠⡤⠴⠒⠋⡇⢸⡀⢸⣷⣶⠊⡿⣷⡤⠖⢉⣁⡤⢤⣤⣤⣷⡀
⢸⠻⣿⣷⡀⠀⠀⠀⡇⣯⠟⠀⣉⡴⠊⢰⣣⣴⣾⣩⠔⠊⣉⡤⠤⣶⣿⣷⣄⡀
⢸⠀⠈⢿⣿⡀⠀⠀⢹⡰⡖⠋⠁⢦⠴⣄⠀⠈⣏⣀⣴⣏⡴⠚⣉⡭⠴⠶⠶⢭⣆
⢸⠀⠀⠈⢿⣷⠴⠚⠉⠱⣜⢦⡀⠀⠀⠘⠖⢄⠈⠁⢸⣃⠴⢿⣷⣤⡴⠒⠋⠉⣉⣉⡳⡄
⢸⠀⠀⠀⠸⣿⡇⠀⠀⠀⠹⡆⢳⡀⠀⠀⠀⠀⠑⠀⠀⣠⣾⡿⢿⣁⣠⡴⠚⠉⢉⣉⣭⣿⣦⡀
⢸⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⣿⠈⡇⠀⠀⠀⠀⢀⡤⣺⠟⠁⠀⢀⣽⣿⣷⣾⣿⣿⡿⠿⠛⠛⠙⠲⢄⡀
⣾⣤⣀⡀⠀⢹⣿⠀⠀⠀⠀⣿⠀⡇⠀⠀⢀⢔⡽⠊⠁⠀⠀⢠⡿⠻⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄
⡟⠲⠬⣍⣛⠺⠿⢧⣤⣄⣀⣿⢠⡇⢀⣴⡷⠋⠀⠀⠀⠀⠀⢻⡣⣀⣀⣀⣠⣤⣴⢐⡀⣄⣠⣀⣆⡰⣴⡳⠽⠊⠱⡄
⠹⣄⡀⠀⠈⠉⠙⠒⠒⠒⠒⢧⠘⣷⢿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠉⠉⠁⠀⠉⠑⠒⠒⠚⠋⠉⢁⣀⣠⠴⢚⣽⠦
⠀⠈⠛⠷⢶⣦⣤⣤⣤⣤⣤⣌⣧⣈⡓⠮⣍⡓⠒⠦⠤⠤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠶⠒⠒⣛⣋⣩⡥⠶⠛⣉⣠⣴
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⣿⣽⣲⠭⣟⣯⣛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢛⣋⣉⣉⣥⣤⣶⣶⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣾⣭⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠂⠂⠀⠀⠀⠀⠀⠀⠈⢄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠄⠀⢀⠀⠀⣄⠀⠈⠈
⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢄⠀⠀⠀⠀⠈⠀⠀⠀⠰⠄⠀⠀⠂
⠀⠀⠘⠀⠀⢀⣤⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠂
⠀⠀⠠⡀⠀⠈⠿⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢄
⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡁
⠀⠀⠀⠀⠀⠁⠒⠂⠐⠀⠀⠀⠠⠤⠤⠄⠀⠀⡀⢀⣀⣀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠓⠦⠀⠀⠀⠰⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠀⠀⢀⠁
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠁⠂⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠁⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠔⠐⠐⠇⠒⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠠⢘⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢅⣀⣴⣀⣀⣀⣐⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡿⠦⣀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⣯⣅⠀⣾⣴⡄⢀⡀⠀⠀⠠⠴⢶⠆⠀⢀⠀⠙⠋⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠙⠛⠾⣧⠃⠀⠜⠀⣠⣀⡦⠀⣤⢡⡈⠻⢀⣤⠤⢹⡶⢦⠀⠁
⠀⠀⠀⠀⠀⢀⣀⣠⣤⢘⠈⠀⢀⠀⠍⢀⡙⢋⠀⡠⠀⣿⠏⠀⢠⠏⣤⣇⡀⣠⣯⣏⣩⡿⡟⠓⠓
⠀⠀⠀⠀⠀⣨⣧⠀⠈⢻⡇⠀⠊⠀⢀⡠⠂⠀⠀⠀⢀⠟⠀⠠⣏⣸⣿⡃⣱⣤⣸⣿⡟⠳⢬⢀⣢⣾⢷⠶
⠀⠀⠀⠀⠀⡿⠋⠀⠀⢠⣷⢀⣠⡴⠛⢥⣴⡆⠀⠀⣮⠴⢀⡚⣿⣿⣾⣧⡛⠹⠉⠻⢿⣤⣌⣻⣿⣿⣧⠳⡀⠀⣀
⠀⠀⠀⠀⠈⠁⠀⣠⣮⡞⢷⢀⡙⢿⣿⣶⣿⣷⣼⡟⠿⠆⠊⡁⢸⣿⣧⣿⣷⣤⡄⠀⠀⠙⢯⣉⢹⣿⢿⣷⠾⠆⠿
⠀⠀⠀⠀⣤⣡⡄⡋⢬⣿⣎⢧⢳⣀⡛⠙⣿⣿⣿⣼⡶⠀⠹⡜⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠹⣟⣯⡟⠁⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠘⠃⠀⠀⠸⣿⣿⡌⠣⠻⡇⠀⠘⢿⣯⣹⡯⠀⣦⣿⣿⣿⣿⡄⣿⣿⠀⠀⠀⠀⠀⢤⠼⣟⡅⠀⢠⡾⠆⠀⣰⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠁⠀⣠⠃⠀⠀⠈⢻⣿⣷⡖⣿⣿⣿⣿⣻⣇⢿⡿⠀⠀⠀⠀⠀⠸⢠⠏⠀⠰⢿⣧⡄⠀⠙⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⡆⠁⠀⠀⠀⠀⠈⣿⣿⣣⠿⣿⣿⣿⠙⢿⡌⣿⣆⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⢸⡇⠀⢰⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠋⡇⠸⣶⣄⡀⠀⠀⣿⠟⠉⠀⣨⠁⠻⣶⣤⣥⣌⣽⣦⣀⠀⢹⠏⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣸⣄⠈⢿⣷⠀⢠⣁⠈⣷⣾⡿⠇⠒⣾⠻⠉⠛⠛⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡏⠛⠻⠁⠀⠀⠉⠑⢿⣿⡇⠀⠀⢹⣦⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⣤⢾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⣇⠀⠀⠈⣿⠀⠀⠀⢠⣄⣼⢳⡶⠀⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡄⠀⠀⠀⠘⣿⡄⠀⠀⠸⣧⡄⠀⢀⣿⠀⠀⢀⣿⣼⣷⠏⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢿⣿⣀⣀⡀⠻⣿⣦⡘⣿⣁⡠⢿⠟⠁⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠉⠁⠉⠛⠁⠈⠉⠁⠹⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢁⡴⢢⠞⡴⠃⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡴⢋⡴⣣⠞⠁⠀⣀⣀⠀⠀⠀⣠⠞⣉⢀⡼⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡞⢡⠞⡴⢋⣠⠴⠛⠁⣹⡄⢀⡾⣡⠞⣡⢞⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⢠⢏⡼⠗⠋⠁⣠⡴⢚⣥⣇⡞⡴⠃⡴⣫⠞⠀⣠⠴⠚⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣷⡟⡼⠁⢀⡴⠞⣡⠞⠋⣼⡟⣰⢁⡾⣱⣧⠖⠋⠁⣀⡴⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣼⢰⣇⡴⢋⣠⣏⠀⠀⢰⡿⢡⣧⡾⣱⠋⠀⣀⡴⢛⣡⠞⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣴⠇⠘⢋⡴⠋⠁⣿⠀⢀⡿⠁⠀⣿⢁⡇⣠⠞⢉⠔⠋⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⣠⠎⡟⠀⢠⠎⠀⠀⠀⣇⣠⠏⠀⠀⢠⠇⢸⠟⣁⡴⠛⡇⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⢀⡴⠃⣼⠁⢠⠏⠀⠀⢀⡤⠛⠁⠀⢀⡔⡿⠀⢀⡴⠋⠀⠀⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⣇⣠⣤⠴⠶⠛⠉⢀⡼⠃⢀⡟⠀⢀⠔⠉⠀⠀⠀⣠⠞⣰⠃⢀⡞⠀⠀⠀⢠⡇⢀⡟
⠀⠀⠀⠀⠀⡀⠄⠂⠁⠀⠀⠻⣄⣀⣀⣀⣠⠴⠋⠀⠀⣞⣠⡎⠀⢀⣀⣠⡤⠞⠁⣠⠏⠀⡾⠀⠀⠀⠀⢸⡇⣸⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣀⡠⠞⠉⠘⣷⠛⠉⠉⠀⠀⣀⡴⠋⠀⡼⠁⠀⠀⠀⠀⢸⢀⡏
⠀⠀⠀⠢⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉⠁⠀⠀⠀⠀⠈⠙⠒⠒⠚⠋⠁⠀⠀⡸⠁⠀⠀⠀⠀⠀⢸⣼
⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠒⠒⠚⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠲⠤⠤⠤⣤⣤⣤⡤⠤⠴⠒⠋⠁



⣤⣤⣤⣴⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣬⣴⣬⣽⣴⣾⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⢐⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⡐⢀⠁⠀⠀⠀⠀⣨⠑⠒⣂⠐⢀⡚⠒⢉⡀⠄⠀⠀⠀⠀⠛⠛⠀⠀⠀⠄⠀⠀⠀⠀⢠⡠⠀⠀⠀⠀⠀⢄⠠⡀
⠀⠀⢀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠐⢁⣂⣉⠀⢈⣁⡢⠓⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢠⡔⠇⠐⠂⠂⠀⢄⠠⡄
⠀⠀⠘⠐⠁⠦⠤⠤⠤⠤⠤⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⢈⠋⠁⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⠀⢤⡤⠀⢀⢠⠁⠀⠀⠀⠀⠀⠀⠤⠁⠀⠈⠠
⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠁⠁⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢀⠤⠄⠀⢠⠀⠄⠅⠀⠀⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠴⠰⠤⠤⠤⠤⠤⡴⠴⠄⡀⠀⠀⠀⠈⠀⠈⠁⠁⠈⠀⠃⠀⠀⢠⠈⠁⡄⠀⢀⠀⠤⠐⢈⡁
⠀⠀⢘⠁⠀⠅⠀⠀⠀⠀⠀⠀⡇⡀⠘⠄⠀⠀⠈⠀⠔⠢⠈⠉⠁⠉⠉⣀⠒⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⠀⠒⠈
⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇⢀⠈⢘⠀⠀⠄⠑⠄⠐⠀⢁⠀⢀⠠⠀⢒⠀⠀⠀⠀⠀⡀⠀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠘⠠⠂⠂⢀⢀⠀⢀⠀⠀⠁⠺⠗⢈⠀⠀⠈⠀⠀⠒⠀⠉⠁⠉⠁⠐⠂⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⢤
⠀⠀⠄⠀⠐⠀⠉⠈⠁⠈⠈⠁⠐⠀⠀⠄⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠘⠄⠀⠀⠀⠠⢀⠀⠀⠀⠠⢊⠠⠀⣤⠀⣤⠀⢴⢵⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠂⢄⠐⠀⡀⡀⠘⡁⠂⠂⡁⠀⠀⠀⢀⠤⡀⠀⠀⠄⠀⠆⠀⢀⠀⠒⠀⢀⣤⣤⣤⡀⠈⠊⠀⡀
⠀⠐⠃⠃⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠳⠀⠀⣂⠐⠀⠐⢡⠉⠁⠁⢄⠁⢄⠄⢁⠀⠀⠀⠐⠤⠈⠀⠉⠄⠁
⠀⠀⠖⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡷⠢⠀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⢙⠀⠀⠁⠀⠀⠀⣠⣄⠘⠀⠃⣀⣄⠀⠀⠐⠀⢀⠀⠀⠁⠀⠈⠀⠀⢀⠀⠁
⠀⠀⢀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠄⣀⡀⡀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡁⢈⠂⠀⠀⠀⠀⠖⡁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠢⡀⠀⠀⠀⠀⡤⠒⠒⢄⠀⠀⠀⠐⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠈⠁⠀⠈⠁⠉⠀⠀⠉⠠⠂
⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠌⠄⠉⠀⠀⣄⡀⣠⢠⠀⠜⠃⠈⠀⠀⠸⠿⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢠⠒⢢⠄⠀⠀⠀⠀⠀⠀⢀⡔⢢⡄⠀⠀⠀⠀⠈⠀⡼⢀⢀⠀⢨⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢄⠀⠀⠉⠁⠀⠤⠀⠄
⠀⠀⠀⢑⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⡁⡡⠀⣉⢘⠁⠀⠀⠀⠀⠂⠔⠀⠒⠀⠒⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠛⠃⠘⠃⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣝⢿⣿⢮⠻⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢮⡑⠌⠳⡀⠙⡄⢻⣿⠸⠟⣿⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠯⣍⡙⠻⢮⡂⠀⠀⠀⠀⠀⢻⠀⠀⢿⡟⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣟⣛⡛⠛⠒⠢⠄⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢻⢫⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣛⣫⣭⣭⡥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢣⠞⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣛⡫⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡄⣤⢄⠀⠀⠀⠀⠁⣠⡾⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡠⣀⣤⠔⠂⠀⠀⠀⠀⠀⢸⡏⡟⠻⠜⠧⠘⠋⠀⠀⠀⠀⠐⣉⣴⠿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣾⠟⣉⠀⠀⠀⠀⠀⠀⡄⢠⣶⣶⡇⢸⡇⠀⠀⠀⠀⠀⠠⠔⠋⠉⠚
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡾⠋⢀⠀⠀⠀⠀⠀⠑⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣪⣭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⡴⢣⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣛⣛⣯⣭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣣⡧⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠐⠢⠤⣤⣤⣬⣽⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣼⣷⠀⠀⣧⠀⠀⠀⠀⠀⠨⡳⣦⣬⣙⣲⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣿⣿⣴⡄⣿⣧⠘⣄⠈⢦⡐⢌⡳⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⣼⣦⡳⣿⣷⣝⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠴⠄⠀⠀⠀⠀⣀⡤⠄⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠒⠉⠉⠉⡩⢟⣿⣷⣶⣶⣤⣀⡠⣴⣖⡒⠋⠉⣉⣉⣉⡀⣀⠀⠀⠚⠋⠙⠢⠤⢤⣄⣀⡀
⠀⠀⠀⠀⢠⡴⠚⠉⢉⣠⣤⣤⣭⣌⠁⢾⢶⠻⠛⠛⠻⣿⣿⡭⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢤⡀⠛⣰⢤⠀⠀⠀⠀⠈⣃⣶⡀
⠀⠀⠀⠀⣿⠁⠀⢶⣾⣿⠟⠛⡻⣿⣷⡄⢼⠀⠀⠀⠈⣁⣸⡄⠀⢀⣤⣤⠀⣀⣴⣾⣿⠿⢛⡿⣿⣄⠈⣷⠢⠼⠃⠔⣠⣴⣶⣾⣿⠁
⠀⠀⠀⠀⠘⢧⣀⣸⣿⡇⠸⢿⠇⢹⣿⣷⠈⡑⠤⢀⣀⡠⠠⠀⠀⠀⠀⠀⠀⢈⣿⣿⡇⢰⣿⠆⢸⣿⣧⠀⠀⠃⢀⠈⢿⣿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣤⣤⣶⣿⣏⡼⠣⣬⣤⣀⣀⣀⣀⣀⣤⠤⠤⠤⠤⠤⠽⠿⣿⣤⣍⣤⣾⣟⡿⠦⠤⣶⣶⣶⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠀⠀⠀⠈⠙⠛⠋⠁

⢿⣿⣷⣦⣀⠉⠪⡳⣤⡈⠳⣕⢄⠙⢿⣧⡀⠳⡱⡱⠱⢱⡱⢻⡀⠀⠇⡇⠘⡇⠀⢸⠀⠸⠈⢀⡟⢌⣎⠎⢎⢎⠞⢀⣼⡽⡿⡫⣪⠞⢁⣴⢞⠕⠋⣀⣴⣾⡿
⣄⡀⠉⠙⠿⣗⣤⡀⠙⠝⢦⡈⠢⡑⢀⠙⣟⣄⠑⠄⠀⠁⠳⠈⢇⠀⢠⢻⠀⡇⠀⠘⠀⡆⠀⣸⠁⠜⠈⠀⠠⠊⡠⣻⠫⠊⠈⠜⢁⡴⠫⠃⢁⣤⣺⠿⠋⣁⣴
⢈⠙⠳⢤⣀⠈⠙⠪⣖⠀⠀⠙⠲⣄⠀⠀⠈⠻⡢⡈⠀⠀⠀⠁⠘⡀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠃⠘⠀⠀⠀⢁⢔⠝⠁⠀⠀⣠⠴⠋⠀⠠⣲⠕⠋⠁⣀⣬⣾⡿
⢄⡀⠀⠀⠈⠛⠦⣄⡀⠉⠂⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠐⠈⠀⡠⠔⠪⠛⠋⠁⢀
⣦⣌⣉⠂⠀⠀⠀⠀⠉⠒⠤⡀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⣈⣥
⠀⠀⠉⠙⠓⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠚⠉⣁⣠
⠷⠶⢶⣤⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⣚⣋⡭⠥
⣉⣀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠈⠁⠀⠀⢀⣀
⠓⠶⠶⠭⠥⠄⠂⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⠁
⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤
⠶⠶⢶⣒⡒⠬⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⣀
⠉⠉⠀⠀⠀⢀⡀⠀⣀⣀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉
⣶⣖⣾⣿⡯⠷⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶
⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢄⡀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠋⠁⠀⠀⢀⡤⠖⠋⠀⡀⠄⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⡀⠀⠀⠢⢄⠀⠀⠂⢄⡀⢀
⢁⣠⣴⡚⠉⢀⣠⢖⠏⠂⠀⣠⠖⠋⠀⠀⢀⣴⠕⠀⠀⠀⠀⡄⢠⠁⠀⢸⠀⠀⠀⠀⡃⠀⠈⡄⢁⠀⠀⠀⠀⠨⣦⡀⠀⠈⠻⣶⣄⠀⠀⠹⡲⣄⡀⠈⢓⠯⣷
⡿⠟⢉⣠⣶⡯⠛⢁⢠⣪⠞⢁⠔⡡⠀⣰⡻⠫⡢⠂⠀⠀⡼⢀⡏⠀⠐⣼⡄⡇⠀⢠⣇⠂⠀⢹⡀⢣⠀⠀⠀⢔⠝⣽⣕⢄⠈⢮⡫⣷⣕⡄⡈⠛⢽⣶⣤⡉⠻
⣤⣾⡿⠟⠉⣠⢔⡵⠛⢁⡴⡻⠊⣠⣾⡟⡕⡵⡱⠀⠀⡼⡱⣼⠁⡀⡆⡇⣿⡇⠀⢸⢸⢰⢀⠈⣧⢎⢧⣆⢆⠈⢮⢪⢻⣷⣷⣄⠙⢮⡻⣟⢮⡢⣄⠉⠻⢿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⠁⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢐⢦⡐⠉⠉⠉⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⡆⠹⠆⠁⠘⠒⣂⡤⠔⠀⠈⠓⠀⡀⠀⠀⠀⢀⣴⠟⣉⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠆⠰⣊⡥⠀⣀⠤⠖⠉⠁⠀⠀⢤⣀⠀⠀⠈⠑⠲⠶⠿⠛⠉⢁⡔⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠘⠁⠃⠀⣈⠥⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⢀⠴⠋⠀⠈
⠀⠀⠀⠀⠀⠠⠤⠤⠒⠂⠀⡠⠂⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢀⡠⠚⠁⢀⠴⠚⠉⡇
⠀⠀⠀⠀⣈⣉⣓⣒⣒⠂⠀⠐⢦⡀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠤⠚⠁⠀⠀⠐⣁⣀⠤⢒⠀⡄
⠀⠀⡇⠀⢀⣉⠉⠉⠛⠉⠂⠀⠀⠙⢂⣙⠲⠄⠀⠀⢀⣀⠤⠤⠤⠀⠒⢒⡲⠒⠒⠒⠒⣒⠤⠦⠒⠊⠅⠐⠂⠉⠁⠀⠸⡄
⠀⠀⢧⡀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠋⠉⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠐⠒⠒⠈⠉⠈⠃
⠀⠀⠀⠈⠑⠀⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢐⣿⣷⣦⣠⣄⡀⣤⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣠⣴⣦⡀
⠀⠀⠀⠀⠀⠀⡔⢠⣿⣟⣽⣿⣿⡵⣶⣾⡗⢻⣆⠀⠀⠀⠀⠀⠀⣰⡿⣷⡆⣼⣿⡿⣿⣿⣷
⠀⠀⠀⠀⠰⣿⣤⣿⣿⣿⠟⠁⠀⡀⠈⠋⠁⢸⠿⠳⡀⠀⠀⣠⢾⣿⣿⠦⠰⢿⣿⡛⠋⠻⣿⡆
⠀⠀⠀⠀⣾⢩⣿⣿⣿⣿⡄⡉⢻⣿⠂⠀⢀⡄⠀⠀⢸⣠⡞⣵⣿⠿⠧⣤⣦⡑⡘⣿⣷⡄⢸⣿⡀
⠀⠀⢠⠀⣭⡟⠿⠇⠹⡟⢳⣀⠀⠀⠁⢠⡾⠁⠀⢀⣾⣿⡧⣿⡇⢇⠀⠘⠛⠀⢠⣿⣿⡛⢾⣿⣧
⠀⠀⠈⠷⣟⣻⡶⣰⠿⠀⠘⢿⣿⣿⣿⡿⠁⠀⢠⡾⠻⣿⣧⠈⠻⣄⠁⠀⠤⠞⠓⣹⡿⣻⡆⢻⣿
⠀⠀⠀⠀⠈⣽⣿⣤⠀⣰⠓⢴⠞⢻⠟⡸⠃⣠⠟⠀⠀⢻⣿⡆⠀⣇⠀⠀⢀⣴⡊⣹⢿⣻⣧⠾⠋
⠀⠀⠀⠀⣰⣿⣿⡿⣧⠈⢷⣄⢄⢬⣼⠇⣰⡟⠀⠀⠀⠘⣿⡇⠀⠃⢳⡄⢧⡾⣣⣭⣼⣯⠁
⠀⠀⠀⢀⡿⢹⣿⣷⣌⣛⡟⣿⣿⣦⣌⡑⠻⣀⠀⠀⠀⠀⣿⡇⠀⣦⣼⠻⠾⠟⣿⣿⣼⢿⡄
⠀⠀⠀⣼⠇⢿⣿⡐⠛⠛⢛⣿⡍⢿⡏⢹⠇⠈⣤⠀⠀⠀⢿⡇⠀⢹⣿⣋⣁⣿⡶⢻⣿⡛⣧
⠀⠀⢠⡿⠶⢿⠿⣿⠻⣿⣿⡿⣿⣿⣦⣎⡀⠀⠛⠀⠀⠀⣿⡄⠀⣾⡿⡿⢛⣁⣴⣿⣿⡄⣿
⠀⢀⣼⡇⠀⠅⣶⠛⢻⣿⠃⢴⣯⣿⠙⣏⠙⠷⣄⠀⠀⣰⣿⣇⡴⠟⠋⣠⣿⠟⢩⣹⣿⡟⣾
⠀⣸⣿⠀⣴⣦⠬⠭⣿⣿⠟⠚⡋⡟⢀⣿⡀⠀⠈⠱⣴⣿⣿⠥⣶⣾⡿⠶⠿⠗⢻⣻⡟⢡⡟
⠀⣿⣿⣄⠈⠁⣠⣾⣿⣶⣦⠴⡿⡇⣸⠛⠙⠶⠤⢀⡼⢋⣠⠶⠿⢛⡓⠚⠛⠒⠛⠳⠇⢼⠃
⠀⣿⢹⣯⠷⠾⠿⠿⢿⣿⣿⣷⣄⢶⡟⠀⠀⠀⢰⡾⣷⡟⣛⣉⣩⣤⣿⣿⣿⣿⣟⣶⣶⡤⣤⢄⣀
⠀⣿⡎⣿⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠀⢀⡀⢰⢿⣿⣯⠄⣤⣝⠛⠋⠉⠉⠑⠚⠛⢿⡿⠀⠃⠑⢿⡆
⠀⢿⣷⡘⣧⡀⠀⠀⠀⠀⠀⢠⠞⠁⢠⣾⣿⣷⣾⣿⡏⣴⣾⣿⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠁
⠀⠘⣿⣷⡌⠳⢦⣄⣀⣠⡶⠋⡴⠀⠻⣯⠥⠋⠠⠙⣿⣿⣿⡏⠠⣤⣀⣀⣀⣴⣟⡄
⠀⠀⠸⢿⣿⣶⣦⣤⣥⣤⣴⠞⠁⠀⠀⠀⠀⣠⣤⠀⠙⢿⣿⣷⣤⣾⣿⣿⣿⡿⠛

⠀⠀⠀⠈⠉⠙⠛⠉⠛⠉⠁⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠈⢻⣟⠛⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠛⢶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⠑⠒⠒⠠⠀⣀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠦⡀⠀⠀⠀⠀⠩⣕⣶⣤⠏
⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⢿⣧⡍⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠹⣍⠀⠀⠘⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠙⣦⣴⣾⠿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠾⡿⣶⡖⠂⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⡇⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⡇⠀⠀⠀⠀⠨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠛⣧
⠀⠀⠀⠀⠀⢶⣤⡀⠀⠀⣠⡄⠁⠙⠃⠀⠀⣀⣤⡾⠿⠿⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣿⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⢧⡁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠟⡇⠀⠉⠒⢄⡀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣁⣀⣀⣀⣤⣾⠿⠋⢀⣐⡒⠛⠿⠯⡁⠂⠀⢀⣴⡃⠈⠀⣽⠀⠀⠀⠀⠘⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣩⠟⠙⣫⣴⣾⠿⠟⠉⠉⠑⠂⠀⠘⠂⢀⠈⢿⣞⠀⠀⣿⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢁⢰⣿⠟⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⣤⡉⠀⠈⣿⡀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡞⠁⠀⠀⢠⣤⠆⠀⠀⠀⠀⠀⠀⠈⠛⣦⢀⡼⠙⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠠⢄⠐⢾⣧⣶⠿⠉⠀⠀⠀⢀⣶⣦⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠈⠳⣆⠉⠀⠀⠀⣀⠀⠀⠈⠙⠟⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⠀⠀⠀⠀⠀⠀⠈⢷⣖⣈⡉⠁⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⡟⠃⠀⠀⢀⣀⣀⠠⣴⣿⣅⡙⢶⣀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠻⣷⡝⠆⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢿⣸⠇⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣸⣿⣿⣿⠟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡾⣿⣏⡈⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢙⡀⢁⣭⣿⡿⠃⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣹⣆⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡋⠁⠈⠙⢢⣄⣀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⡿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣀⣉⠀⠀⠀⢠⣄⣬⣀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠘⠛⠁⠀⠀⠀⢈⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢰⣮⡥⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢖⡎⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⣳⡚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡮⠭⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠭⠭⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣭⣭⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠩⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠔⣛⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠬⠽⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⣎⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠭⣟⡉

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

⠀⠀⡀⠀⠀⠀⢀⠀⡀⢀⡀⠀⠀⢀⢀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⢠⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠸⡆⠀⠀⢀⠀⠀⠀⡾⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠰⠀⢠⠚⠻⣷⡄⠀⣇⠀⠀⢸⠀⠀⠀⠃⠀⠀⠸⠁⠀⠀⣰⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢸⠀⠀⢸⡇⠀⢸⠀⠀⢸⠀⠀⢠⠀⠀⢀⠀⠀⠀⠰⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⣄⠀⠀⢸⡄⠀⠰⣿⠀⠀⣀⣀⡘⠀⠀⡿⠀⢀⡟⢠⠖⠳⣦⠀⢀⡴⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿⡀⣼⠉⠈⠉⣷⡴⠶⢦⣄⠀⡏⠀⢀⣿⠀⠋⠀⠀⠀⠀⠉⠀⠀⠀⢀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⣴⠾⠛⠛⠛⠋⠉⢉⣭⣭⣭⣭⣭⣭⡇⠀⠀⠀⣿⠀⠈⠁⣿⣭⣭⣭⣭⣍⠉⠉⠙⠛⠛⠻⢶⡄⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣠⠖⢂⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⢀⡀⢸⣿⠀⡀⢰⣿⣿⣿⣿⣿⣿⠀⠀⣠⣶⡒⢦⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠀⡾⠰⢡⣄⡸⡟⣀⣀⢸⣿⣿⣿⣿⣿⣿⣿⣇⣠⣿⣿⣏⣀⣿⣿⣿⣿⣿⣿⣿⢠⣤⡝⠛⣡⣤⡱⠸⡆⠴⠒
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⡇⠀⠘⠛⣩⣥⠛⠛⣸⣿⣟⣩⠴⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠛⣣⣦⡙⠛⠁⠀⡇⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠒⠀⡇⠀⠀⠀⠙⠛⣰⠟⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣙⠛⠁⠀⠀⠀⡇⠐⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣷⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢹⡀⠀⠀⠀⠉⣿⠀⠀⠀⠛⣿⣿⣿⣿⢿⣿⣿⣿⠿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠀⠈⠉⠁⠀⠀⠀⣸⠁⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⢧⣀⡀⢀⠀⢹⡀⠀⠀⠀⢸⣧⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⣀⣀⡀
⠀⠀⠀⠐⠒⠛⠋⠉⠉⠀⠀⠀⠀⢉⡛⠛⠛⠛⣧⠀⠀⠀⠈⠉⠉⠙⠛⠿⠛⢿⠿⡿⠿⠿⠿⠿⣿⣿⠛⠛⠛⠛⠛⠛⢛⡋⠁⠀⠀⠀⠈⠉⠉⠛⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠖⠛⠉⠁⠀⣀⡄⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢀⠈⠙⠦⣄⡀⠀⠉⠀⠀⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠀⣀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢀⠀⠑⠂⠀⠀⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠀⠀⠀⠠⠚⠁⠀⠀⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠈⠁⠀⠀⠀⠢⣄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⣠⠆⠀⠀⢸⡿⠋⠁⠀⠀⠀⠀⠀⠀⠶⣿⡟⠁⠘⠆⠀⠀⢦⡀⠀⠈⠳⠀⠀⠀⠀⠀⠙⠲⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣘⣁⣀⣀⢀⣞⣁⣀⣠⣆⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣧⣀⣀⣀⣦⣀⣀⣙⣆⣀⣀⣀⣙⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢰⠾⠉⣉⣉⣩⣉⣉⠉⠿⠿⠉⣉⣉⣩⣉⣍⠉⠿⠉⣉⣍⠙⠟⢉⣉⣍⠉⢉⣉⣉⣉⣍⣉⣩⡉⠉⢉⣉⣉⣉⣉⣍⡉⠹⣶⣀
⠀⢰⡶⠶⠶⠶⠾⠂⣶⣿⠉⠉⠉⠻⠿⠀⠀⣾⣿⠉⠈⠉⣿⣶⠀⠠⣿⣿⣧⣴⣾⣿⡿⠀⢸⣿⡏⠉⠉⠀⠉⠁⠀⢸⣿⡏⠉⠁⢹⣿⡆⠈⠻⠶⠶⠶⣶⣀
⠀⢸⡇⠸⠿⠿⠂⠀⣿⣿⠀⠀⠺⣿⣿⠀⠀⣿⣿⣤⣤⣤⣿⣿⠀⠈⣿⣿⠘⠟⢸⣿⡇⠀⢸⣿⡿⠾⠿⠿⠂⠀⠀⢸⣿⡿⠿⠷⢿⣯⡀⠀⠺⠿⠿⠀⠈⣿
⠀⠘⢻⣦⣤⣤⣤⡀⠛⣿⣤⣤⣤⠟⣿⠀⠀⣿⣿⠁⠀⠈⣿⣿⠀⢀⣿⣿⠀⠀⢸⣿⣇⠀⢸⣿⣧⣤⣤⣤⣤⡄⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⣠⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠘⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣻⣮⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠿⣠⣤⣤⣶⣶⠈⠉⠉⠁⠀⠀⢠⡤⣤⣤⣤⣀⡀⢀⣾⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠋⠁⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣿⣹⣿⣿⣿⡿⠋⠀⠉⠙⠻⢿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣷⡄⢀⣀⣀⣘⠿⠟⠛⠛⣤⣤⣀⣀⠀⠿⠿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢛⡟⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡙⠉⠉⠀⢻⣿⣿⣿⡀⠀⠀⠠⣿⣿⠯⠇⠀⠀⠀⠀⢨⣷⣦⣤⣀⡀⠀⠀⢀⣠⣶⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠺⢿⡿⢆⣀⣀⣤⣽⣿⡿⠶⠒⠚⠛⠋⠁⠀⠀⢀⣀⣀⡐⠛⣿⢿⣿⣿⡿⠛⠘⠻⣿⣿⣿⣿⣿⢿⣄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⣉⣩⡗⠲⣄⠀⣼⡿⠁⠀⠀⠀⠐⢠⣬⡉⠒⢬⡙⢯⣅⠀⠀⠀⠀⠀⠙⠻⠟⠉⠁⠀⠀⢻⣧⣿
⠀⠀⡠⠤⠀⠀⠀⠀⠀⠠⠄⢐⠒⠒⠦⠤⢻⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⢾⣷⡄⢙⡦⠈⠛⢓⠒⠄⠒⠿⣿⣦⡀⠀⠀⠀⠠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣘⡄⠀⠀⢀⡀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠠⠴⠀⠈⠉⠉⠀⠀⠀⠀⠀⠑⡄⠀⠀⡈⠛⠛⠀⣤⣴⣶⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢿⠁⠀⠀⠀⠀⠐⢤⣄⡀⠉⠓⠲⢤⣉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠘⢆⠀⣂⣤⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⣦⣌⡻⢷⡀⣶⣯⣿⡄⢬⠻⣿⣉⣿⣶⢦⣌⡙⠲⣭⡳⢤⣀⣀⣀⣿⠀⠸⡆⢹⠘⡷⣿⡇⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠿⣶⣵⣿⡿⣿⣤⠸⣆⠈⠉⠛⠛⠺⠿⠿⣶⣄⡙⢦⡉⠙⠛⠉⠀⠀⠑⠈⢀⠟⢿⡀⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠉⠛⠿⣿⠿⢯⠄⠈⠛⠦⠄⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⠀⠀⠀⢀⣴⣖⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⢾⣷⣶⡀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠒⠂⣸⡷⢂⡾⢛⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⠀⠠⠀⠀⠀⠀⠀⠐⠒⣛⠋⠁⠀⠐⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡜⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡈⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⣠⡄⢠⣿⣿⣿⣿⣇⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⡄⣴⣿⣷⣿⣿⡟⢿⣿⣿⣿⣿⣿⣇⢠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⣿⣿⡇⠈⢿⣿⣿⣿⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣿⣿⣿⡟⠈⢻⠇⠀⠘⠋⢻⣿⣿⣿⠞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⣀⣠⡏⣙⠇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣤⣾⡸⣆⣀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢡⠱⢻⡀⣼⣿⣿⡇⣿⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢿⣿⡇⣿⣿⣧⣀⡟⠎⡌⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢫⡳⣄⠰⢦⠘⣀⣄⣿⣿⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⣿⣿⣿⣿⢰⣀⠃⡴⠂⣠⢞⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠃⠀⠱⣭⣾⣿⣿⣇⣝⠷⢹⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠟⣻⣿⣿⣷⣭⠎⠀⠘⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⣳⣌⢻⣿⣿⢿⣹⣷⣌⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⡿⣿⣿⣟⣡⣞⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⠗⠛⣛⣛⠛⠛⢾⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⣿⣿⡛⠺⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⡿⠞⠁⣀⣀⣀⠀⢀⡀⠀⠹⠶⠶⠶⠶⠶⠶⠶⠶⠏⠀⢀⣀⡀⠀⣀⠀⠈⠳⣿⡻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⢀⡯⠟⢹⡀⠈⠉⠀⠀⠀⠀⢀⣀⣈⡀⠀⠀⠀⠠⠋⠉⠀⣞⠉⣹⠀⠀⠈⠻⣟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡏⡀⢸⣏⡉⠳⣤⠛⢉⣽⡇⠀⢀⣀⣀⠻⠭⠭⠟⣀⣀⡀⠀⢠⡞⠓⡎⠛⢡⠞⠳⡄⠀⢸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢰⠇⠘⠓⠳⡞⠈⢳⠞⠚⠃⣴⢫⠀⠈⢳⡄⢀⡞⡁⠀⠙⣦⠈⠓⠞⣡⠶⢬⠳⠞⠁⠀⠈⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡏⠸⠀⠀⠀⠀⠷⠷⠾⠀⠀⠀⠻⣜⠳⢂⣼⣿⣿⣧⡛⠦⣠⠟⠀⠀⠀⠻⢤⠞⠀⠀⠀⠀⣆⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠁⢱⠀⠀⠀⠀⠀⠀⠠⠄⣠⣴⠆⣈⣭⣭⣤⣭⣭⣤⣭⣭⣁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢾⡿⣼⠟⢡⢞⣁⣀⢀⣀⡀⣀⠀⣀⢀⣈⠳⡌⠻⣦⣄⠀⠀⠀⠀⠀⠀⠐⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣷⠀⠀⠀⠀⠀⠀⣠⡾⢃⡴⠋⢸⡇⣤⣸⣭⡗⣿⣾⣿⢸⡧⠀⠙⢦⡘⢿⣷⠀⠀⠀⠀⠀⠀⣠⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸⣿⠀⠀⠀⠀⠀⠰⠏⣠⠞⠀⠀⠘⠳⠟⠘⠀⠛⠛⠋⠛⠘⠳⠂⠀⠀⠳⣄⠹⠀⠀⠀⡀⠀⠀⣿⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⣿⣷⣦⣀⣀⠀⢀⣾⠁⠀⠀⠀⠀⢉⡿⢸⡟⣿⣿⣶⡇⣿⣉⠀⠀⠀⠀⠈⣳⡀⠀⣘⣿⣩⣾⣿⡏⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡙⠛⠟⠛⣉⣴⣻⣦⢳⡀⠀⠀⠀⢼⣧⢸⣧⡿⢿⠹⡇⣿⡤⠀⠀⠀⢀⡞⣵⣟⣦⣈⠛⠻⠛⢋⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⢿⣿⣿⣧⣙⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣋⣼⣿⣿⡿⠉⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠊⠀⠐⢌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣠⠖⠁⠀⠈⢳⡄⠉⠀⠀⠀⠀⠈⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠳⡈⠀⡀⠤⣀⣀⠀⠛⠀⠀⠀⠀⢀⡴⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⣄⣉⡙⠛⠒⠀⠀⠀⠉⠑⠂⠤⠤⠒⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠨⠦⠬⠍⠉⠀⣀⠀⠤⠤⠒⣁⠀⠀⠀⢀⣀⡤⠊⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⣋⠙⠂⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠉⠉⣩⠔⢋⡤⠊⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠹⣮⠈⠉⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⡫⠔⠊⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⡠⠔⢊⠁⠀⠈⠆⢠⠈⠀⠀⡜⠁⡀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠀⣀⠤⠒⠉
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠑⢤⡀⠀⠊⠀⠀⡸⠀⠀⠈⠑⠦⡀⢠⡺⠛⠁⢀⡠⠔⠊
⠀⠀⠀⡀⠐⠤⣀⡀⠀⠀⠀⠈⢀⣀⠤⠚⠁⠀⠀⠀⠀⣠⠔⠉⠀⡠⠒⠉
⠀⠀⠰⡙⠦⣀⡀⠀⠉⠉⠉⠉⠀⠀⠀⠀⢀⣀⡠⠔⠊⢁⡠⠖⠉
⠀⠀⠀⠙⠦⣀⡈⠉⠉⠒⠒⠒⠀⠐⠊⠉⣁⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣠⣴⣶⣾⣿⣿⣦⠡⠀⠀⠀⠀⠈⠀⠀⣞⠀⠀⠀⠈⠻⣿⣧⠀⠀⠀⠀⠀⠑⢄
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠁⠀⠀⢀⡤⠋⡇⠀⠀⠀⠀⠙⢿⣧⠀⠀⠀⠀⠀⠈⠣⡀⠀⣠⣾⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⡒⠒⠒⠉⠀⣼⡃⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⢀⠀⣀⣾⣿⡿
⣶⣟⡻⠿⣿⣿⣿⣿⣿⣧⣙⢂⣀⣠⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣾⣥⠯⠙⠛⠛⠛⠛⠛⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⣏
⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⢀⠈⡄
⠀⠀⢀⣀⣠⣤⣤⣤⣤⣶⣦⣦⣤⣤⣿⡁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡴⠻⡧⠇
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠐⡆⣀⣒⣊⣢⡀⠀⠀⠀⠀⠀⠀⠀⡇⢠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢹⣿⣷⣯⣿⣾⣿⣿⡟⠀⡀⠀⠀⡇⢸
⢄⠀⠀⢹⡟⠙⠛⠻⢿⣿⣿⣿⡿⠿⠟⠻⣿⡄⠀⠀⠀⢿⣿⣿⣿⠋⢁⣀⠙⠘⠀⠀⣰⡇⠄
⣄⠙⠒⠞⢧⡀⠀⠀⡠⠈⠉⠀⠀⢀⣠⣴⣿⡇⠀⠀⠀⠘⣿⣿⡏⢠⣿⣏⠀⠀⠠⠈⣸⠇
⠈⠳⢾⡇⠀⠙⡶⢊⣀⣤⣤⣶⣾⣿⣿⣿⣿⣷⣤⣤⣤⣴⣿⣿⡇⠘⣿⣿⡄⠀⠀⠀⠀⢠⣤⣤⡄
⢦⣄⡀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠿⣀⠈⠙⠁⠀⠀⠠
⠀⣹⡷⣴⡟⠛⠋⠉⠉⠻⢙⣻⣿⣿⣿⣿⣿⣿⠟⣥⠌⠀⠀⠀⠀⣧
⠀⣽⠁⠀⠈⢻⣦⡀⠀⠀⠛⠿⣿⣇⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⡍⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⣿⣽⣿⡻⣿⡀⠀⠀⠀⠀⠀⢀⢴⡇⡀⠄⠀⠁⠀⠀⠀⠉⠉⠀⠐⢰⡾⠋
⣾⠃⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⣿⣿⣿⣿⣿⡄⠀⠀⢀⡠⠂⠡⠈⠁⢀⣠⣴⣾⣶⣶⣦⣤⣤⣀⣰⣿
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣿⣿⣿⣿⣿⣿⣤⣴⠿⠐⠁⣀⣴⣾⣿⡿⠛⠉⠈⠉⠛⠻⢿⣿⣿⣿⠀⢀
⡇⠀⠀⠀⠀⠐⢦⣄⣀⣀⡀⢀⣿⡯⠟⠛⠻⣿⡿⢃⣠⣴⣿⠿⠟⠉⠀⠸⡄⠀⠀⠀⠀⠀⠀⢈⢹⣷⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿⡿⠋⣁⠀⠀⣀⠀⠀⢹⡄⠀⠀⠀⢴⣷⣤⣿⣿⣿⣿⡄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⡄⠀⠀⠀⢹⡁⠙⠀⣿⣴⣿⣿⡀⠀⠀⢳⡀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿
⣦⣤⣑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⢳⡀⠀⠈⢻⣿⣿⣷⣦⣤⣈⣧⣤⣤⣶⣿⠟⠈⣿⣿⣿⣿
⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠱⡄⠀⠀⠙⢿⣿⡟⣹⠿⠿⠿⠿⠛⠁⣠⣾⡿⠛⠋⢉
⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠘⢦⡀⠀⠀⠙⠃⠱⢦⣤⣀⣀⠄⣮⡿⣿⣡⣤⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣴⡀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠙⢦⡀⠀⠀⠘⢾⣿⣿⠿⠟⠓⠊⢻⣿⣫⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠙⠲⣤⣄⣀⡁⢀⣤⣶⣤⡀⠀⠈⠙⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠘⠻⣿⣶⣤⣤⣼⣿⣿⣿⣿⣿⣿⡄⠙⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣶⣄⡀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠙⣿⠟⠛⣍

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠐⠒⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠒⠒⠀⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁⠀⠀⠀⠀⠀⢀⠠⠄⠐⠒⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠤⣠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠠⠒⠈⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠂
⠀⠀⠀⠀⢀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠠⠐⠀⡀⠀⠈⠂⠀⠀⠠⡀⢀⠀⡀⠀⠀⠀⠀⢀⠄⠊⠁
⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠔⡡⠂⠀⠀⠀⠀⠢⡀⠀⠀⠈⠚⠀⡇⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣀
⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⢀⠈⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⣠⣄⡀⠈⢿⠇⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⣀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠘⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠆⠀⠀⡈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠙⠀⠀⠆⠀⠠⠁⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠁⠀⢀⠌⠀⠠⠁⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣁⠤⡈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠉⠈⠩⢉⠔⠂⠀⣐⣢⣈⠁⢀⡘⠁⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠁⠀⢴⡃⠀⠂⠀⠀⠀⠀⠀⠈⠁⠢⡠⢎⠀⠀⠀⠀⠄⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⠂⠀⠀⠉⠘⠄⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠣⠀⠀⠀⠀⠈⠂⣀
⠀⠀⠀⠀⠀⠀⡠⠃⠀⡄⠀⠀⠘⠂⠀⠀⠘⠳⠆⠀⠀⠀⠀⠀⢀⠘⠀⠀⢁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠘⠄⠠⠄⠵⡀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠌⠀⠒⠐⢁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠄⠀⠀⠀⠁⠢⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡈⠠⢈⣁⡀⠀⠀⠆⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠈⡀⠀⠀⠀⠉⠐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢨⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠤⠄⠀⠠⣀⠀⠾⠀⠀⠓⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠄⠀⠠⠤⠦⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠠⠢⡀
⠀⠀⠀⠀⠀⡠⠐⠉⠠⡀⠈⠱⠒⠀⠀⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅
⠀⠀⠀⠐⠊⠀⠀⠀⠀⠈⡀⠀⠁⠀⠀⠀⠱⠤⠄⠀⢀⡀⠠⠤⠶⢉⢂
⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⡁⠀⠀⠠⠈⠀⣀⠀⢀⡀⠘⠆
⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠀⡀⠔⡄⠀⠠⠁⠀⠈⠀⠀⠀⠈⠢⠘
⠀⠀⠚⠀⠀⠀⠀⠐⠀⠀⠉⠀⠀⠀⠀⠀⠐⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢤⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⢠⠄⢠⡬⢁⠀⡀⣤⠄⢤


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠉⢁⣠⠤⣄⠀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⡶⠋⠀⠀⠀⠐⠫⡔⠀⡎⡇⠀⠀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣞⠀⠀⠀⢠⠄⠠⢄⠀⠑⠃⣸⠀⠮⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠄⠀⠀⢠⠁⢠⡤⡀⢃⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⡆⠀⡆⠀⠸⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡠⠐⢾⣬⣛⣊⠀⠀⠀⢀⡠⠐⠢⡀⠀⠀⣤⠤⠤⠤⢤⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⡼⠅⢰⠈⢻⠟⠀⠁⠒⠂⠁⢀⠝⠀⠇⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠤⠝⠒⠜⣦⠈⠳⣄⣀⡀⠀⠈⣁⣤⣊⡀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⣨⠀⡰⠣⠤⢰⡟⠉⠉⠉⢯⣉⠛⠯⡉⠓⢦⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⠐⡟⠀⠀⠀⢠⠸⣄⠝⢆⢰⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⡞⠀⠏⠀⢧⢞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢯⡬⣄⠀⠈⠢⢦⡊⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡞⠀⢸⡓⡆⠊⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠾⠴⠷⢀⠴⣇⣻⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢅⡀⠀⣐⠀⡆⠀⠀⠐⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠐⡄⠈⡅⢳⠀⡞⢱⠋⢡⠎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢰⠀⣂⡼⠀⣇⡆⠀⠃⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⠶⠶⠖⠚⠋⠁⠀⠀⠈⠙⠲⠦⣄⣀⣀⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠆⠤⠠⠤⠤⠠⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠛⠛⢻⠟⠭⣫⡙⣿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠌⠋⠛⠛⠛⠛⡛⠁⠀⠀⠈⢻⢸⡟⠛⣷⠶⠶⠾⠿⠿⠷⠶⠶⠶⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠈⠉⠉⠁⠀⠀⢠⡆⠀⠀⠀⠀⠀⢷⣶⣶⣶⣿⣿⠀⠰⣤⣿⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠂⡤⠤⠤⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⣼⡷⠤⢐⣶⣾⣾⣏⠛⠛⣉⣁⣼⣿⣆⣿⢛⣦⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⢆⣿⠀⠀⠀⠀⠀⢸⣧
⠀⢀⡀⠀⠀⠼⠄⡀⠀⠀⠀⠀⡌⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠭⠾⠼⠧⢽⣿⣿⣯⢻⣿⠙⣿⣿⣿⣿⣿⣷⣇⠀⢀⢸⣾⣿⡆⠀⠀⠀⠀⢸⣿
⢀⡏⠀⠀⠀⠀⢰⣿⣄⣀⡀⠌⣀⣀⣀⣈⣿⣿⣿⠟⣿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⢸⣿⠀⢰⣿⣿⣿⣿⣿⡏⢀⠔⠛⠉⠙⠃⢢⡀⠀⠀⢸⣿
⢸⠀⠀⠀⠀⠀⢸⣿⣿⠿⠁⣰⣿⣿⣿⣿⣿⣿⡇⠸⠿⠤⠶⠶⠶⢶⣶⣤⣤⣾⡿⠈⠀⣠⣿⣿⣿⣿⣿⣿⡇⡎⠀⠀⠀⠀⠀⣾⡗⠀⠀⢸⣿
⢸⠀⠀⠀⠀⠀⢸⣿⣿⡿⠷⠾⠯⢽⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣻⣿⣿⣿⣿⢿⠶⡇⠀⠀⠀⠀⢠⣿⣿⢀⡀⢸⢿
⣼⠀⠀⠀⠀⠀⣿⣿⣿⡷⢸⣶⣶⣶⣾⣦⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣤⣶⣏⣉⣉⣉⣉⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄
⢿⣀⣀⣀⣀⣤⣿⣿⠿⢧⣾⣿⡿⠿⠿⠿⢿⣿⡆⢂⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⡿⠿⠿⠟⠁
⢸⣿⣿⣿⣿⣿⣿⡿⠀⠈⠁⠀⠈⠉⠉⠉⠉⠀⠂⠀⠄⣀⣀⣀⣀⣀⢴⣿⣿⣿⠿⠒⠒⠒⠒⠒⠒⠒⠒⢪⢙⣇⡀⣀⣀⣀⣼⣿⡇
⠀⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠤⠤⠤⠤⠤⠈⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢠⠎⠀⠀⠀⠀⠀⣠⣴⣾⣿⣯⣽⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣆⢀⣴⣿⠿⣷⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣽⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⡆⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⡿⣿⣻⣿⣿⣿⣿⣿⡿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢹⢿⣿⣿⣿⣿⡇⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠉⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⠀⠻⣿⣿⣿⣷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣩⠀⠀⢸⠋⢹⣽⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠠⣤⠀⠁⠀⠘⠙⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡼⠶⣄⣀⣀⡤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠛⠛⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⢣⡀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡁⠀⠠⢤⠐⠻⢿⣆⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠑⢄⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⠁⡔⠀⠀⠀⠀⠀⠠⣼⣿⡆⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠑⢄⠀⠈⠉⠛⢿⣿⣿⣿⣿⡇⠸⠀⡇⠀⠀⠀⠀⠀⠐⣿⣿⣿⣄⣀⠀⡠⠊
⠀⠀⠀⠀⠀⠑⠢⢄⠀⠀⠀⣿⣿⣿⣧⡀⠀⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢌⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⠿⠿⠟⢿⣿⣿⣿⣧⡀⠀⠀⢀⡴⠋
⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠻⢿⣿⣿⣿⣿⣷⣄⠀⠈⢻⣿⣿⣿⣷⣀⠴⠋
⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣦⠀⠀⠹⣿⣿⣿⣧⠀⢄
⠀⠀⠀⠀⠀⠁⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡤⠐⠉⠹⣿⣿⣿⡴⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠀⠤⠄⠀⠠⠀⠄⠠⠆⢸⣿⣿⣿⣿⠀⢸⠀⠀⠛⠛⠉⠀⠜⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠄⡞⢄⠀⠀⠀⠀⠀⢀⡤⠚⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠈⠉⠉⠉⠀⠠⠇⠀⠈⠀⠀⣀⠴⣧⣀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⠁⠉⠀⠀⠹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣶⠀⠈⠁⡄⠰⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣾⣿⡤⠔⠒⠒⠸⠀⠀⠈⠲⢌⣐⠒⠀⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢢⠀⣼⠟⠁⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡙⣄⠀⠀⢀⣠⣴⣶⣀⣐⣞⣧⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠦⠀⢀⡤⠊⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠘⢷⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠠⢲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠛⠀⣀⠤⠒⠉⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⡠⠒⠁⠁⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁⠊⠀⠀⠀⠀⠠⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠐⠀⠀⣠⠄⠀⢉⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠁⠀⠀⠀⠠⠎⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⡁⡀⢤⠀⠀⠀⠀⠀⢠⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⣿⠆⠀⠀⠀⠐⠂⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⣠⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠑⢀⡰⠁⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠤⠀⠴⠀⢒⠥⠀⠤⠀⠀⡤⠀⠄⠀⠚⡲⢇⠀⠀⡀⠀⠈⠓⡄⢠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠃⠈⠀⠀⠠⢤⡤⠤⠤⠤⣤⣤⠁⠘⢦⡀⠘⠲⠶⢾⠀⠃⠀⠀⢸
⠀⠀⠀⠀⢀⡔⢩⡅⠀⠉⠲⡀⠀⢰⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠆⠀⠀⠈⠑⢦⡀⠈⢳⠐⠂⢀⡎
⠀⠀⠀⠀⡜⠀⠀⠰⣠⡔⣄⠰⢄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⠶⣤⡴⣤⣀⣹⠶⠶⠛⠛⠋
⠀⠀⠀⠀⠳⡄⢠⡗⠀⠀⠉⠑⠂⠸⠇⠀⠀⣀⠂⠀⠀⡶⠠⢤⡄⢀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠻⢶⣇⠀⠀⠀⠳⠄⠀⠀⠘⠀⠁⠀⠀⡘⠋⠛⠰⠦⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠛⢷⠺⣦⠀⠀⠀⠰⠂⠤⠤⠤⢄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⡈⠀⠈⢷⡀⠀⠸⡷⠲⡖⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⠇⠀⠀⠀⠀⢵⡀⠀⠁⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⢤⡄⠄⠀⠰⠋⢳⡀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠊⠀⠀⠀⢳⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⢷⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⢀⠀⠀⠈⢧⠀⠀⠱⡎⠉⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⠀⠓⠒⠛⣇⠀⠀⠘⣶⡄⠈⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢄⠋⣆⠀⠀⠘⠟⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⣾⠀⢸⣇⠀⠀⠀⠀⠀⠀⡝⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢹⠀⠘⠟⠀⠀⠀⠀⠀⡀⠈⠔⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡤⠀⠀⠀⣠⠊⠀⢀⣀⣁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠋⠁⠀⠀⠠⠀⢷⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⣀⠘⠟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡠⠆⠉⠀⠀⠙⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢤⠶⣄
⠀⢠⣶⡖⠒⠒⠚⠛⠉⢉⣁⣠⠼⣀⠈⠛⠶⢤⡀
⠀⢸⠉⠻⢶⣶⣉⡉⢩⢍⠑⠀⡀⠈⠑⠦⣄⣸⡇
⠀⠸⣄⡀⠀⠈⠉⣿⢄⡹⠝⢀⣨⣵⣶⡶⠟⣿⠁
⠀⠀⠘⡏⠓⠲⣤⠿⠶⠾⡟⠛⣿⠁⠈⢧⣀⣽⡦⡤⢤⣤⣄
⠀⠀⠀⠸⡄⠀⢻⡄⠀⠀⠳⣄⣹⠶⠞⠋⠉⠀⡐⢀⣾⣿⣿
⠀⠀⠀⠀⠙⢦⡀⠻⠦⠴⠞⠉⠀⠨⢀⠀⠀⢀⠀⢸⠛⢿⠛⢶⣶⣄
⠀⠀⢀⣤⣶⣾⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠈⠀⢸⣴⣧⣸⣿⢿⣿
⠀⣰⣿⣿⣿⣿⣻⣿⣿⣿⣿⡽⠁⠀⠀⠀⠀⠀⠀⠈⠻⣁⠙⢟⢿⠟⠉⡉⡳⣦⣀⣤⣶⣶⣦⣄
⠀⣿⣿⡟⣿⣿⣧⣿⣿⣿⣿⣷⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢸⠀⠓⢠⠘⠻⢿⣿⣿⣿⡆
⠀⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠟⠦⣄⡉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⣀⣉⡡⠊⠻⣿⣿⣿⣀⡀
⠀⠈⠿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠉⠛⠲⢤⣤⣿⡿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠉⠉⢉⣳
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⢄⣀⡠⠆⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣿⣿⣿⣿⣿⡿⢿⡧⠤⣀⣀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠉⠙⠒⠛⠛⠿⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠒⠂⠤⢀⡀
⠀⠀⠀⠀⠀⠴⣂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⢀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⡀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠢⣅⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣂⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⢀⠈⠂
⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡜⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⣧⣿⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠘⣿⠇⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⡆
⠀⠐⠐⠂⠀⠉⠉⠀⠀⠉⠉⢑⠆⠀⠀⠀⠀⠰⠉⠉⠙⠂⢀⡀⠀⠀⠀⠀⠀⢀⠀⢸⣋⡤⢾⣦⠀⠀⠀⠀⠀⡀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢃⠀⠠⢶⠀⠀⠈⠁⠒⠒⠒⠉⠀⠀⠸⠿⠶⠟⠁⠀⠀⠀⠀⠈⡗⢁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠑⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⣠⠀⢆⣀⡸⠀⠐⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣢⣄⡀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⢠⠴⠀⠀⠊⠉⠅⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠉⠀⣠⠍⠀⢻⣉⠝⠛⠀⠀⠉⠛⠆⢄⠀⠀⠀⠀⣄⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠈⠀⣀⠤⢪⠏⠀⠀⠀⠀⠀⠀⠀⢢⠈⠐⠒⠁⢈⠷⠄⡠⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣾⠉⠀⢰⠋⠣⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣆⠀⢱⠀⠏⠋⠀⠠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⠸⠗⠁⠀⠀⡘⠀⠀⠈⡆⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠂⠤⣀⣀⠔⠁⠀⠀⠀⡖⠒⣒⣲⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠟⠠⠄⠀⣀⢼⠁⠈⠃⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠈⡀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠃⠀⠄⠀⠀⠀⣧⠀⣻⠆⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠠⠜⠓⠀⣃⠀⠉⠁⠀⠀⠀⠈⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⠀⠀⣠⠊⠘⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⢀⣀⠀⠀⣠⠄⠀⠠⠄⠀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠘⠀⠀⠈⠀⡀⠀⠀⠀⢀⣀⣸⣄⣘⣂⣈⣁⣀⣀⣀⣀⣐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠤⠊⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⣀⠠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠛⠋⠉⠉⠁⠀⡠⣄⡀⠀⠈⠉⠙⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣀⠀⠀⠀⠀⠀⣸⠐⠀⠨⣑⢦⡀⠀⠀⠀⠈⠙⠢⣄⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡇⠂⣀⠤⠚⠃⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⠉⠉⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠗⠀⠀⠃⠈⠁⠀⢀⠔⠊⠉⠑⠢⣄⠀⠀⠀⠀⠈⢳⡀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢀⡀⠀⠈⢆⠀⠀⠀⠀⢈⡙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡇⠀⠀⠸⡆⠀⠀⢠⠏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⢳⡀⢀⡏⠀⣸⠁⠀⠀⠀⠀⠀⣀⡤⣄
⠀⠀⠀⠀⠀⠀⢠⣞⣤⡤⠤⠤⠤⠤⢤⣤⡀⠀⠀⠀⢀⡤⠦⢦⣀⠘⠋⠀⠀⠀⠀⠀⠉⣿⠃⣠⡇⠀⠀⠀⠀⢀⡜⠁⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣟⠀⢀⣤⠈⠳⠦⣤⣤⠤⠖⠚⠉⣷⣯⣿⠟⠀⢠⠞⠛⣮⠔⣚⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠹⣆⠀⠘⠢⢄⣀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⣊⠰⣀⢹⣎⠭⢽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢈⣓⣦⣄⡀⠀⠀⠀⠀⣀⣠⠴⠋⠀⠀⢀⠼⠉⠀⠈⠻⣁⠠⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣠⠴⠚⠋⢉⣽⠯⠖⣦⡩⠛⠙⠉⠉⠻⣦⣄⡀⠀⢀⣿⠀⠀⠀⠤⣤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠚⠉⠀⠀⢀⠔⠋⣠⠴⠚⣙⠀⠀⠀⠀⠀⠀⠈⣆⡉⠛⢉⣠⠗⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⢸⠲⣄⡸⠉⢳⠀⠀⠀⠀⠀⠀⣹⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡜⢣⡸⠀⠈⡇⠀⠛⢤⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠔⠷⣀⡀⠀⣰⠇⠀⠀⢸⠇⠀⢠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⡿⢥⠀⢀⣰⢻⠒⣆⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢲⡏⠁⡏⠀⢸⡄⠸⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣼⠀⢸⣅⠀⣠⣇⠀⣻⠍⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠳⠤⠞⠀⣹⡅⠉⠉⠁⠀⠈⢹⣦⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠁⠀⠀⠀⠀⢘⡇⡏⠀⠀⠀⣀⣴⠋⠀⠀⠀⠀⠈⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠫⡙⠓⢶⠶⠒⣯⠀⣹⠓⠚⠉⠁⢸⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠘⢆⠀⠂⠀⠘⡇⣏⠀⠀⠀⠀⠈⣀⣀⣤⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠸⠀⢐⣀⡴⠏⠻⠤⠶⠒⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⣀⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡤⡆⠐⠒⣤⢠⡒⠄⠐⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⡀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠐⠀⠀⠀⠀⢄
⠀⠈⠒⠤⢤⠤⢄⣀⣁⡀⠀⠀⠀⠠⠤⠄⣀⢣⠀⠀⠀⠀⣀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⣀⡀⣑⠀⠀⠀⠘⠆⢧⡄⠀⠀⡱⠊⢍⡉⠉⠐⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠄⠀⠀⠀⠠⡀⡀⡀⠀⣄⢀⠀⠠⣀⠀⡀⠀⠃⠔⢱⢸⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠴⡄⠐⠒⠁⠇⠀⠠⡀⢠⠀⣀⠀⠠⠂⠉⠻⣄⠀⠀⡸⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠘⠚⠀⣬⡀⠀⢀⣱⢀⢀⣀⠞⣧⣤⣤⣴⠏⡷⠊⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠤⢾⠈⠙⠿⠍⠉⣼⠄⢠⡀⠀⠈⠉⢀⣼⡅⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣼⣦⣤⣠⣤⣶⠗⠛⠻⣌⣐⣓⣫⠽⠷⣄⠠⡀⠀⠀⠀⠀⠀⠈⣇⠄
⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢰⡟⠀⠈⠉⠉⠀⠄⠀⠀⠀⠘⠛⠒⠒⠒⠾⢷⣾⣦⠀⠀⠀⠀⠀⣿⣶
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡍⢥⠀⠀⠀⠀⢻⠿
⠀⠀⠀⠀⠀⠀⢇⢀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠠⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠘⢎⠓⠤⡄⠤⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣈⢉⣹⠧⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠴⠇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠔⠒⠈⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠵⢶⣒⣠⣤⣄⣀⡀⠀⢀⡠⠀⠀⡥⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⣉⠠⠄⠞⢼⣿⣯⢿⣷⡶⢰⢡⣶⣿⡢⠷⠋⠡⠀⣀⠉⢐⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣽⣥⠈⠁⡁⠂⠲⢤⣥⣟⠀⠰⠜⠛⠋⢀⡄⠀⢀⣾⢼⠃⠈
⠀⠀⠀⠀⠀⠀⢰⠻⠀⠀⠀⣿⣽⢗⣾⢿⠅⡀⠀⠀⠉⠉⠁⠶⣴⣾⢆⣾⡇⠀⡟⠁⣬⠆
⠀⠀⠀⠀⠠⠤⠈⠁⠀⠀⠀⠀⠀⠈⢡⠔⠈⠀⠀⠀⠀⣀⢐⠄⣽⣻⡎⢿⡀⠾⠤⠚⠁
⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠌⠂⠈⠀⠀⠀⣰⣻⡭⢂⠈⢋⠉⠚⠁
⠀⠀⠀⢻⣷⠄⠀⠉⠙⠛⠒⠠⠌⠀⠀⠀⠀⠒⢰⢱⡾⢀⣴⠋⠁
⠀⠀⠀⠀⠈⠑⠠⢀⡀⠀⢀⣤⣀⡀⠨⠀⣀⢰⣃⠸⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣪⠗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⣾⣿⣿⡌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣿⣿⡽⣿⣆⠘⡦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⠟⠀⢰⢀⣿⣿⣶⢄⡀⠀⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢃⣿⣿⡏⠀⠀⠀⠸⢸⣿⣿⣿⠀⠙⢾⣾⣷⣤⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣉⣡⣠⣶⣿⣿⠟⠀⠀⠀⢠⢃⣿⣿⣿⣿⠀⠀⠘⣿⣿⣿⡿⠼⠦⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠃⠚⠋⣩⡭⠿⠛⠛⠋⠁⠀⠀⠀⢠⠃⣼⣿⣿⣿⠃⠀⠀⢸⣿⠋⠁⠀⠀⠀⠀⠀⡱⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⢤⢠⡞⢁⣾⠁⠀⠀⠀⠀⠀⠀⢀⢃⣾⣿⣿⡿⠃⠀⠀⢠⣿⢻⣇⢀⣄⣀⣠⣤⡴⢁⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣱⡟⠺⠷⣯⠟⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⠏⠀⠀⢀⣴⣿⡫⣜⠛⣨⢜⡭⡼⢶⣖⠀⡀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡏⢨⢿⠀⣗⣦⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠁⠀⢀⣴⡾⢻⠊⠀⢹⠓⢡⡏⢀⡠⠼⢿⣇⠀⠀⡠⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣸⠇⢐⠦⢇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⢀⣴⠟⡉⠀⡇⡆⠀⢀⣴⢯⠁⡴⠄⠚⢿⣟⣆⠚⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣘⠖⢁⣴⡧⣾⢒⠆⠀⠀⠀⠀⠀⠀⣸⠇⠀⣰⡿⢻⣆⠹⣦⡙⠿⠓⠁⣹⠟⣰⠷⠖⠈⣛⣻⣿⣧⡒⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⣴⢫⣌⣣⡼⠾⠀⠀⠀⠀⠀⠀⢰⡟⠀⣼⡟⠀⠀⠙⢦⣅⢈⡢⠔⠊⡡⢾⣿⡶⠞⠓⠈⣛⣻⣏⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⢮⣀⠈⢄⣹⣿⠂⠀⠀⠀⠀⠀⢀⡿⠀⣼⡏⠀⠀⠀⠀⠀⡙⣋⡤⡶⠉⠀⠀⠿⣿⡟⢋⡁⠈⣭⣿⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢀⠸⢋⣛⢳⠋⠀⠀⠀⠀⢀⠀⣾⠃⣰⡟⠀⠀⠀⠀⠀⢠⠏⢘⣷⠸⡆⠀⠀⠈⢪⡫⢸⠗⠶⣿⠍⠀⢳⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣷⡾⣋⣿⠽⠉⠀⣀⠀⠀⢀⠎⣼⠃⣴⠋⠀⠀⠀⠀⠀⠀⠈⢰⣼⣿⠀⣧⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠈⢼⣿⣧
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠈⣀⠔⠊⣹⣿⣧⣬⣷⡾⣡⡞⠁⠀⠀⡀⠀⠀⠀⠈⠁⣿⣿⣿⡀⣿⣿⣷⢮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⣿⡆
⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⣠⢞⣡⣾⣿⣿⣿⣿⡿⣋⣾⢟⣡⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⢸⣿⣿⣿⣿⣦⡢⡀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⣾⣏⠣⣀⣀⠤⢊⣵⣿⣿⣿⣿⣿⢟⣡⡾⠛⢋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⢻⣿⣿⣿⣷⣿⣮⣵⣢⡤⠄⣂⣰⣿⣿⣿
⠀⠀⠀⠀⠹⣿⣷⣶⣶⣿⣿⣿⣿⡿⣟⣽⣾⣉⣭⣶⠊⠉⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡏
⠀⠀⠀⠀⠀⠉⠛⢿⣿⣯⣿⣿⣿⣯⣿⣷⣾⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣍⡛⠿⠿⠿⠿⢿⣿⣿⣟⣿⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠾⠿⠿⠿⠛⠛⠋⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢁⡈⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠁⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⢀⡦⠀⠀⠀⠀⠀⠀⠀⣀⠔⢁⠄⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠠⠀⠀⠀⠀⠀⠀⢠⠞⠡⠖⠁⠀⠀⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠘⠀⢠⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠘⠃⠀⠀⠀⠄⢐⠈⡃
⠀⠀⠀⠀⠀⠀⢀⠈⠒⠈⠁⠠⡍⠀⠀⠀⠀⡠⠋⣠⠆⠀⠀⠀⠈⡀⠀⠂⠀⠂⢉⢑
⠀⠀⠀⠀⠀⠐⠀⠀⠉⡁⠀⡈⠀⠀⠀⢀⠜⢁⠔⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠂⢩⣀
⠀⠀⠀⠠⠤⠀⠤⢀⡀⠀⠉⠁⠀⣀⠔⠁⢠⠏⠀⡁⠢⣀⠀⠀⠀⠀⢀⠀⠈⢉⠁⠂⠀⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡤⠔⠊⠀⠀⢰⠃⠀⠀⠑⠀⠠⠑⢄⠀⠀⠰⠄⢀⣉⠀⠤⠤⠈⠀⢥⣄⡀
⠀⠘⠤⠄⠠⠤⠤⠒⠊⠁⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠸⠑⠤⢀⡔⢉⠀⠀⢀⣀⠠⠄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠄⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠐⠁



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

⠀⠀⠀⠀⠀⡞⠛⠓⠲⠶⢤⣄⡀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠴⠶⣄
⠀⠀⠀⠀⠀⣇⠀⡔⠒⠦⠤⣄⣙⠳⣤⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠲⣤⡤⠒⠋⠉⢀⠀⠤⡄⠀⢸
⠀⠀⠀⠀⠀⣿⠀⢡⠀⠀⠀⢀⣬⠿⠥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠽⠶⠶⣤⡀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⢸⡄⠈⡆⢠⠞⠉⠀⢄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣠⠀⠀⠙⢦⢠⠇⢀⡟
⠀⠀⠀⠀⠀⠀⢷⠀⣸⠋⠀⣀⡤⠊⠱⡄⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⡞⠉⠓⢦⡀⠈⢿⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⢳⠃⢀⡞⠉⠀⠀⠀⠙⣆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠹⣄⠘⡟⢥⡀
⠀⠀⠀⠀⣠⠖⠋⠉⢠⡏⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠸⡆⠃⠀⠙⢦⡀
⠀⠀⠀⣰⠃⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡔⢻⣄⠀⠀⠀⠀⠀⠀⠀⣸⡋⢦⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢳⡄
⠀⠀⢠⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢰⢱⣿⡟⣦⡀⠀⣠⣄⠀⣰⣿⣿⣼⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⡄
⠀⠀⢸⠁⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠘⡾⣿⡇⡇⠙⢺⣿⣿⡞⢧⢻⣿⣿⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢷
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠱⣌⡰⣃⣠⣼⣿⣿⡧⣬⣦⡵⠃⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⢸⡄⠀⠀⠀⠀⠀⢈⣷⣤⣤⣤⣤⣤⠤⠶⠛⠉⠀⠘⢿⣿⠃⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⢦⡀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠈⣇⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣷⠰⡖⠒⠦⠤⣄⡿
⠀⠀⠀⠹⣄⡤⠖⠒⢾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢨⠇⣰⠇⠀⢹⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠉⠀⠀⠀⡎⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠓⣩⠞⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠉⡻⠗⠒⠶⠶⠶⠶⠖⠒⠛⠛⠯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠂⠯⠇⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢧
⠀⠀⠀⠀⠀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⢾⢾
⠀⠀⠀⠀⠀⠘⡆⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡞
⠀⠀⠀⠀⠀⠀⣟⣠⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣤⣵⡀
⠀⠀⠀⠀⠀⠀⠘⣄⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⢦⡀⠸⡅⠈⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣆⣬⢽⡀⠀⠀⣠⣤⡴⠦⡿⣅⠈⢸⡇⢠⡇⠀⡇⢀⣴⣶⣾⡿⠿⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢰⠃⠀⠰⣩⣯⡉⠰⣇⣼⣇⠀⢳⣸⣿⣦⡷⠋⢩⣴⣶⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣧⣤⣷⣤⣀⣿⣿⣷⠀⠻⢿⣿⣦⣤⠽⣯⣷⣶⣶⡿⣋⣵⢖⠛⣻⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢰⣭⠾⠟⣛⣭⣴⣶⣮⣭⣿⣦⡶⠞⣿⣷⣤⠀⢿⣿⣿⣿⣵⣿⢿⣼⣾⣿⢟⣛⣻⣿⣦
⠀⠀⠀⠀⠀⠀⡞⣀⣰⣿⣻⣿⣟⣛⠿⣿⣿⣿⣷⣦⠈⠻⣿⣷⣦⣿⣿⣿⣿⣴⣼⣾⡿⠿⣿⣶⣿⡟⣿⣧
⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⣯⣭⢻⣿⣿⣮⢿⣿⣿⡿⠿⠾⠿⠋⢻⡉⠻⠿⢿⣾⣿⣿⣿⣿⣼⣿⡿⣱⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⠿⢛⣙⠿⣵⣦⡌⣿⣿⣿⣿⣿⣶⣤⣶⣤⡿⠤⢤⣤⣄⣀⡏⠀⠟⢻⣿⣿⣵⣿⣿⣿⡿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⣗⢿⣿⣷⣿⣿⣷⣿⣿⣿⡿⢛⣛⢻⣿⣟⢴⣿⣷⣿⣿⠟⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡙⣆
⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣇⣿⣻⣷⣿⣿⣾⣿⣿⠟⠁⢀⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣭⣭⣭⣭⣾⣿⣿⣿⣿⣿⣿⣯⣿⢿⣿⣿⡯⠀⢀⡎⠀⠀⠀⠙⠮⣽⣿⡿⠿⠿⣿⠟⡼
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡿⠻⣿⣿⠛⠋⠀⣽⣿⢤⣤⣂⡼⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠴⠊
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⡿⢣⢻⢦⣀⣹⣧⣀⠀⢜⣻⠁⣀⣽⠇
⠀⠀⠀⠀⠀⠈⡿⣿⣿⣿⣶⡇⠀⢠⠃⠈⡷⠤⢤⣾⣟⣛⠛⠛⣋⡩⠏
⠀⠀⠀⠀⠀⠀⠙⢬⣛⠿⠿⣁⡴⠋⠀⠀⠉⠋⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⠄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠁⠈⠁⠉⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠶⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠆⠀⠀⠀⡿⣷⢄⣄⣠⠸⡇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⡳⢾⡄⠀⠀⢤⡔⠀⠀⠀⠊⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⠛⠃⠀⠀⠀⢀⡒⡆⠀⢸⣁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡀⠀⠀⠀⠀⠀⣀⡤⠞⡏⣰⠃⠀⡀⠀⠄⠉⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠙⠂⣙⠢⡀⠈⡧⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠘⣆⢠⠜⠃⠀⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠉⠘⠀⠆⠀⠀⠠⠤⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠃⡀⠀⠘⠦⠀⠘⢆⠀⠀⠀⠀⠀⢠⡂⠥⡀⠀⠰⢠⡤⠂
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢄⠱⠀⠀⠡⠖⠀⠀⠀⠰⠄⠀⠈⢁⣀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋⠙⠀⡀
⠀⠀⠀⠀⢀⠃⠠⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠀⡔
⠀⠀⠀⠀⢨⠤⠀⡀⢀⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠐⠔⠀⠀⠈⣃⠰⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⢼⡀⠀⣸⠒⢄⠀⠆⢉⣀⠀⣸⠆⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢠⡀⠁⠈⠁⢀⡌⠀⠗⡄⠉⠁⠀⠀⡔⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠤⠘⠀⠀⠀⠀⠀⢁⡀⠀⠰⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⢢⢁⡀⠀⠀⠀⠀⡀⠀⠀⠈⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠐⠦
⠀⠀⠀⠀⢀⠜⠁⠀⠈⠐⢄⠀⠀⠈⠀⠀⠀⠇⠀⠋⠀⠀⠀⠀⡠⠄⠂⠀⠀⠐⢆⡆
⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⡀⢀⠖⠀⠀⠀⠀⠀⢀⠢⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠪⡀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠠⠔⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢠⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠋⠈⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠙⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⣤⡘⠁⠀⠀⠀⠀⠀⠀⠻⣃⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢤⡄⢠⣤⣶⣶⣶⣶⣤⠄⠀⠉⠁⠀⠄⠀⠀⠀⠀⠀⠀⠐⠘⠂⠀⠀⠀⠀⢀⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡀⠠⠘⠊⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠛⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠔⠞⠐⠈⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠉⠻⠀⣀⠀⠀⠀⠈⢑⡂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⠁⠀⢠⡇⠀⡇⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⢸⡆⢸⣷⡀⠀⠹⣷⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠁⠀⠀⡿⠋⠀⣷⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡼⠀⠈⠰⠇⠀⠀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⠀⠀⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⢣⣶⠷⠾⢷⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠞⠃⢘⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠀⠀⠀⠀⢀⣀⣀⢀⢀⡴⠃⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠋⠁⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⠔⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣤⣄⡀⢀⠀⠐⣒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠿⠟⠋⠀⠀⠙⠓⠤⠄⣀⣀⣀⣀⣠⣿⡀⠀⠒⠲⠖⠂⠐⠒
⠀⠀⠀⠙⢷⣦⣄⡀⠀⠀⠀⠀⢀⡈⠙⠀⠂⠷⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⡀⠀⠈⠉⠀⠀⠀⠀⠀⠲⢶⣞
⠀⠀⠀⠀⣄⠉⠙⠛⠋⠀⠀⠀⠀⠀⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⢤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠘⠛⢲⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢀⣤⣴⣤⣤⣤⣤⠤⠊
⠀⠀⢶⣠⣤⣀⡀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠀⠀⡂⣄⣀⡀⣀⣠⣤⣤⣤⣀⣄⣀⣴⣿⡦⠀⠀⠀⠀⠀⠑⡘⢢⠀⠀⠀⠉⠀⠀⢶⡦⠄
⠀⠀⠀⠈⠉⡙⠛⠋⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⢮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠄⠷⣶⡦⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢎⣴⠁⠀⠀⠀⠠⣿⣿⠛⠛⠓⠚⠒⠛⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠄⠀⠠⠤⠤⢤⣤⣤⣠⠤
⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠺⡣⠊⠁⠀⠀⠀⠀⠀⠋⠙⠀⠀⠀⠀⠀⠀⠸⠛⠁
⠀⠀⠀⠀⠈⠙⠛⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣢⣮⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠙⠛⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣚⠿⠟⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣦⣰⣦⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣤⣤⣽⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⡿⠉⠁
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣼⣷⠟⠑⠂⠀⠀⠀⠀⠀⣤⣀⡀⢀⣀⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣤⣎⣁⣠⣤⣤⠠⢬⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⣿⣩⢹⡿⣿⡟⢡⣾⣿⣫⣿⣥⣼⢿⣿⣿⣧⣀⢹⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠈⣿⣿⣿⣿⡟⢻⣇⣿⣦⡿⣼⣿⣿⣾⣿⣿⣿⡿⣿⣛⣿⣽⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢻⣿⠉⣿⠀⢸⣿⣎⢿⡇⣿⣿⣿⣷⣤⣼⣿⣿⠉⢵⣶⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠘⣿⣧⣿⣄⢠⣿⣿⣷⡇⣿⣿⣏⡻⠉⣿⣿⣿⡇⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⢿⣿⣿⣿⣾⣿⣿⣿⣿⣾⣯⣿⣿⠛⢻⣿⣿⣃⣴⣶⣶⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⢿⣿⣿⡉⢸⣿⣿⡹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣾⢿⣿⣤⣬⣏⣖⡿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⣿⠛⠛⠉⣿⣤⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣿⣿⣷⣿⣶⣶⣿⣿⣽⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣾⣿⡿⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⢀⡀
⠀⠀⠀⢰⣦⡀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡆⣼⣷
⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠈⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠹⠿⠿⠛⠛⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡗⣿⡦⠚⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢹⡇⣿⣿⣿⣿⣿⣿⣿⣷⠶⠿⠃⣸⡀⢠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣇⣸⣇⠿⠿⢿⡿⢿⠿⣿⣃⣴⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣐⠉⢠⡿⢈⣩⣤⣶⣾⣿⣿⣿⣿⣷⣾⣿⡟⣋⣼⣿⡿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢿⣿⣦⣴⣿⠿⣛⣋⠭⠿⠭⣍⡩⢿⡿⠟⢷⣥⣾⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣛⣿⡿⢳⡂⣬⡿⠺⢿⠶⣮⠟⠿⢧⡾⠛⣥⡎⠛⡉⠉⠛⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣿⠿⠛⠷⢤⡉⠈⣱⠞⢊⠛⣷⣶⣿⣿⣷⡞⣽⠛⠋⠀⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⣴⣿⣾⣷⣄⠸⣡⠖⣸⣿⡿⣋⣉⡡⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣾⣿⠇⠀⢹⣿⡄⠇⣸⣿⢣⠘⡤⠙⢻⠂⣿⡇⠀⠀⠀⠀⠀⠈⣴⣦⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣤⣿⡇⠘⠓⠸⣿⣏⣸⣿⠓⠈⠞⢳⡄⠉⢰⣿⡇⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⣿⣿⣾⣿⣿⢿⣿⣿⣿⣿⣶⣶⣿⣦⣤⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠙⣿⡄⢉⣥⣬⣭⣍⡀⠀⠻⣿⠟⠉⢉⣿⡏⠀⠀⠀⠀⠀⢠⣶⡆⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢀⣶⣿⣿⣿⣿⠉⢫⠿⢿⣶⣴⣶⣶⣿⣿⠿⣿⣿⣆⠐⠀⠀⢸⣿⡇⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⢻⣷⡛⢻⣿⣦⣤⣤⣤⣬⣭⣥⡀⣠⠀⢠⡄⢈⣿⡇⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣀⣿⣿⣦⣙⢿⣿⣿⣿⣿⠿⠛⠁⢀⣀⣤⣴⣾⣟⡁⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣿⢿⣿⢋⣿⠻⣿⣿⣿⣿⣷⣾⣿⣿⡿⠿⠋⠉⠈⣿⣦⣿⠿⣶⣄⡀
⠀⠀⠀⣤⣤⣤⣄⡀⠀⣠⣴⡾⠟⠛⠛⠛⠹⣿⠀⢀⣸⡿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣧⠀⠐⠂⠀⣿⣿⡁⠀⠈⠙⢿⣦⡀
⠀⠀⠀⣿⣿⣯⠛⠻⠿⣿⡇⠀⠀⢾⡆⠀⢀⣿⣿⡿⠛⠁⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠸⣿⡟⢿⣦⡄⢀⣠⣽⣿⣷⣄
⠀⠀⠘⣿⣿⠏⠀⠀⠀⠸⣿⡄⠀⣸⣧⣤⡾⢻⣿⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⣿⡇⠀⠙⣿⣿⠛⣉⣭⣽⣿⣧⡀
⠀⠀⠀⠿⣷⡾⢀⠀⠀⢀⣿⣧⣴⠟⠉⠉⠀⣼⡿⠀⠀⠀⠈⢿⣦⣄⣀⣠⣴⡿⠏⠀⠀⠀⠀⠀⣿⣷⠀⠀⣿⣷⣾⠟⠉⢉⣩⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠸⠿⣿⣷⡾⠟⠛⠉⠁⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣀⣉⡛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⣀⠈⣿⣇⣠⣶⡿⠛⠉⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⣴⣿⠿⠟⠿⠿⠿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠇⠘⠻⣿⡏⠀⠀⠀⠀⠀⠀⢀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠉⠙⢿⣧⡀⠀⠀⠀⠀⢹⣿⡦⠀⢠⣾⣿⣿⣿⡇⣤⡄⢰⣶⣼⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⢻⣿⡄⠘⠛⠋⠁⣿⣇⣿⣿⣄⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣤⣤⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠻⢿⠿⠉⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⢿⣿⡻⣿⣍⣉⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣶⣶⣶⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⢻⣦⡀⠘⣿⡎⢙⣛⣻⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣤⣤⣤⣤⣼⣿⠿⢿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣽⣷⣶⡿⠿⠿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣛⠛⣿⡟⠋⢠⣿⡏⠉⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣷⣶⣾⣿⣦⣤⣤⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠙⠋⠛⠛⠛⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣾⣷⣤⣤⣤⣤⣤⣄⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣴⣦⣴⣶⣶⣦⣤⣤⣶⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣿⣿⣿⠟⠛⣛⢿⡿⠛⠛⠛⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣡⣤⣤⣤⣭⣭⣍⡙⠛⠻⠿⢿⣿⣿⣇⣀⣤⣾⣿⣿⣦⣀⠀⢁⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠻⠿⠿⠿⢿⣿⣷⣶⣤⣄⡘⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⢶⣶⣤⣤⣤⣄⣀⠀⠉⠙⠛⠿⣿⣿⣿⠻⢿⣧⣀⠀⠀⣀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⣷⣤⡀⠀⠀⠀⠉⠳⣾⣿⣿⣿⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣿⣿⣷⣄⣤⣶⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⠀⠀⠀⠀⣠⣴⡾⠿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⠞⠛⠉⠀⠀⠀⠈⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣭⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠟⠋⠉⠻⢿⣿⣿⣿⣿⣿⣿⠿⠙⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣦⡀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣾⣶⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⢀⣠⡤⢄⣀⣠⣤⣴⠶⠛⠉⠀⢠⣄⣀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣟⣩⣶⣿⠿⢟⣫⣥⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠤⠄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢌⠛⠉⠀⠺⡿⠿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠁⠀⠀⠛⠻⣦⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣀⣀⣀⣼⠁⠀⠐⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⡉⠉⠛⠲⣤⣀⠀⠀⠀⠀⠂⠘⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⢀⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⡀⠀⠠⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢶⠃⠀⠀⢀⠀⠈⠀⣀⣈⣉⣙⢶⣄⡀⠀⠀⠀⠈⠉⠉⠙⠛⠦⢄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠒⠶⠖⢉⣴⠀⠀⠀⢻⣿⣿⣿⠿⣿⠄⠀⠀⠀⠂⠀⢄⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⣴⣃⣤⠔⠂⠀⠹⣧⠀⠀⠀⠋⠙⣿⣷⡄⠀⠀⠀⠀⠀⠀⠘⠷⠆
⠀⠀⠀⠤⠀⠀⠄⠀⠀⠀⠀⠀⠀⣄⣀⣻⡿⠁⠀⠤⠤⠤⠴⠀⠀⢀⠀⠨⡋⠩⠿⠶⡄⠀⠀⠀⠀⠀⠀⠙⢠
⠀⠀⠀⠀⠀⠀⠘⢋⠠⠂⠀⢀⣠⠟⠉⠉⠀⠐⠔⣶⣦⣉⣍⠓⠀⠘⠛⠒⠀⠀⡸⢀⣰⣖⠚⠃
⠀⠀⠀⠀⠀⠀⡠⢋⠀⢠⠞⣽⠟⠀⠀⠀⠀⠀⠀⠙⢿⠟⣹⣿⡆⠀⠀⢀⡠⠖⠐⠿⡉⠑⠢⠆⠀⠂⠈⠒
⠀⠀⠀⢀⠴⠚⣓⠿⠖⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⣭⣾⣷⠀⢰⡉⠀⠀⠰⡄⠘⠗
⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⡀⠀⠸⡿⣟⠋⠂⠀⣷⡄⠀⠐⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠰
⠀⠈⠀⠒⠄⠀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠉⠾⣿⣯⣁⣀⠉⠄⢠⣀⠀⠶⠿⢿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢤⡄⠉⠛⠿⢿⣶⣶⣧⡀⠀⠀⣻⣿⣿⣄⠀⢠⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⣶⣬⡛⢿⡄⠦⠀⡸⣿⣿⣿⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠠⣾⣿⣿⣷⣦⣀⡀⢁⣸⣿⣿⣷⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡄⣠⡄⠀⣠⣈⠻⢿⣿⣿⣿⣿⣷⣮⣉⠙⣻⠀⣴⠀⠀⠀⠀⡄
⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠇⠤⠉⡉⠀⠀⠙⠻⠿⣿⣿⣿⣿⠿⠃⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡄⠈⠢⠀⠀⠀⠀⠀⡀⡀⡠⠂⠀⠘⠁⠀⢀⡴⠤⠤⠦
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⠀⠀⠱⡀⠀⠀⠀⠀⢀⠈⠀⠘⡀⠀⠀⢸⠋⠹⠇⠀⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠈⢢⣼⣄⡀⠀⠰⠆⠀⢸⡀⠀⡆⠀⠀⠈⠁⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠗⢾⠁⠀⠈⢧⣠⣴⡦⠤⠾⠀⠀⣹⡄⢀⠔⢀⠀⠦⠸⣦⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⣀⣤⣴⣾⡟⠟⢿⠐⠀⠙⠢⣀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢰⠀⠀⢘⣦⣤⣴⣾⣿⣿⣿⣿⣿⣄⠴⣬⣧⡀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣄⡼⠀⠂⠀⣸⡟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣄⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡄⠀⠒⠊⠀⣿⣇⣸⣦⣿⣿⣿⣿⣽⠿⠛⠛⠉⠀⠀⠀⢀⣾⣿⣷⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡅⠀⠀⠀⢸⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠙⢿⣿⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠱⡀⢀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡘⠌⢂⠀⢸⣿⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣙⣏⣍⣹⣏⣽⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣄⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⡿⠛⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣹⣦⡀
⢸⣿⠛⣷⣶⠖⢛⣷⣶⡞⠒⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠉⠒⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠉⠉⠉
⢸⣡⣾⡿⠃⣰⣿⡿⠋⢠⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⢿⣿⡟⠁
⢸⣿⠟⢠⣾⣿⠟⠁⣴⣿⡿⢻⡿⠻⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣻⣾⣄⡀
⢸⠃⣴⣿⡿⠋⣠⣾⣿⠟⢁⣼⡇⠀⣿⣿⣿⣿⣿⠋⣠⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣟⢧⣄
⢸⣾⣿⠟⢁⣼⣿⡿⠃⣠⣾⣿⣇⠀⣿⣿⣿⡿⠁⠀⣿⡇⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣄
⢸⡿⠁⣠⣿⠋⢹⣤⣾⣿⠿⣿⣿⣷⡞⠻⢭⣄⣀⣀⠻⢕⠒⠚⠿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣮⣳⣄
⢸⣇⣾⣿⡿⡆⠀⢻⡿⠋⣠⣾⣿⢯⣅⠀⠀⠀⣉⣀⡴⢪⣹⠆⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⣿⣿⢿⣿⣿⣿⣿⣿⣿⡻⣦
⢸⣿⡿⢋⣴⡿⠶⠶⢽⠾⣿⡟⠁⣴⣿⣷⣦⣤⣤⣭⣭⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣧⠀⠀⠀⠈⠉⠙⠻⢿⣽⣷
⢸⡟⣠⡿⢋⣤⠾⠛⠻⡇⣻⣠⣾⣿⣿⡿⠋⠞⠉⠉⠻⣿⡏⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣧⠀⠀⠀⠀⠀⠀⠀⠈⠻
⢸⣿⣿⡗⢀⣤⡴⢞⠛⢷⠹⣿⣿⠟⣻⡇⠀⠀⠀⠀⠀⣿⣿⣆⠀⠈⠻⣿⣿⣽⣿⣿⣿⣿⣿⣿⣷⢿⡄
⢸⡿⠛⣷⡈⢩⣤⡴⢿⢿⡀⢻⣧⣀⣿⡇⠀⠀⠀⠀⢰⣿⣿⣿⣷⣄⠀⠈⢻⣿⣿⣷⠙⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣤⣶
⢸⣧⣾⣿⣇⢠⣤⣴⣾⣼⣧⣟⣿⣿⣿⣷⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠹⣿⣿⣇⠀⠙⢿⣿⣇⣿⠀⠀⠀⠀⠀⠀⣰⣿
⢾⣿⣿⣿⣽⣿⣿⣿⡿⠿⠿⢿⣾⠿⠟⠻⠧⠀⠀⠾⠿⠿⠿⠿⠻⠛⠻⠀⠀⠀⠛⠃⠁⠀⠀⠀⠙⢿⣿⠀⠲⠶⠆⠐⠶⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠀⣀⡀⠀⡀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⡀⠈⢁⠀⢛⡝⠾⠁⠀⠀⣀⣤⣤⣤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠿⠀⢸⡴⠀⠀⢀⣀⣤⣿⠿⠛⠉⠙⢆⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠈⢁⠀⠿⣄⣠⣶⣿⡿⠋⠁⠀⠀⠀⠀⠈⣯⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣞⣉⠉⠉⠑⡆⠀⠐⠒⢤⣤⡄⢁⣴⡏⠁⠠⢤⡉⢙⣥⣀⣀⣴⠋⢠⣾⢦⠀⣸⣟⣦⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⡿⠛⠋⠉⠉⠁⠀⣀⡇⠀⠀⠀⠈⠟⣤⣼⡿⠃⠘⠛⠚⠛⠋⢀⣤⠎⠀⢠⡿⠉⠉⠉⠑⠿⣿⡿⠰
⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⠟⠉⢹⣿⠃⠀⠀⠀⠀⠀⠀⡸⣍⣁⣀⠀⠀⠢⢤⢙⡄⠘⠂⠀⠀⠀⠀⠙⠁⠀⣰⡿⠁⠀⠀⠀⠀⠀⠈⠋
⠀⠲⣿⣿⣿⣿⣿⣿⣿⢿⡿⠃⠀⠀⢸⣿⣦⣀⡂⠀⠀⠀⠀⢸⠀⠀⠈⠉⠓⠂⠙⠨⡅⠀⠀⠀⠀⠈⢻⣦⣤⣿⠋
⠀⠈⠛⠿⠿⠟⠉⠽⠗⠊⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣏⠀⠀⢨⢸⣶⣄⡀⠀⠀⢸⡇⢠⡖⠈⠀⠀⢀⡈⢿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⡈⠁⠀⠘⢰⡮⠟⣿⣶⣦⣼⡇⠀⣱⢀⠔⠀⣉⢀⣼⢗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣿⠀⠀⠉⠛⠿⠿⣷⣿⣻⠄⠈⠻⣿⣋⠈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠙⠛⠛⠋⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠶⢀⣈⣁⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣤⡷⠀⠈⠛⢿⣦⣤⣤⣤⡚⠆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢠⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⠶⠿⠿⠿⠛⠛⠛⠿⠿⠷⠶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣙⣷⣦
⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠛⠛⠉⠉⠉⠉⠁
⠀⢸⡟⠳⣦⣄⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⣿⠀⠀⠀⠲⠶⠶⢴⣾⣟⣉
⠀⠈⣧⠀⠀⠙⢳⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢉⡴⢶⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣦⣄
⠀⠀⢿⡀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⠔⠋⠀⣸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄
⠀⠀⠸⣇⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠣⠤⠄⣀⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀
⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⢀⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠲⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⣴⠟⠉⠛⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠻⣦⡀
⣿⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⣰⠚⠉⠀⠈⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠈⠉⠉⠉⠛⠻⠶⣦⣌⢿⡄
⠘⣧⣨⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⠤⡀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿
⠀⠈⠻⡆⠙⣆⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢸⣴⡜⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⢻⣄⠘⣷⣀⡄⠀⠀⢀⡴⠋⠀⠀⠀⢸⢸⣷⢀⣠⠾⠛⠛⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⣀⣠⡽⠦⣈⣿⣟⣲⣶⣋⠀⠀⠀⠀⠀⠀⣣⠴⠋⠁⠀⢀⡀⠀⢈⡿⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⢿⣿⣶⣶⣾⣿⣿⡷⠀⠀⠉⠓⠒⠒⠒⠋⠁⠀⣀⡤⠖⠉⠀⢀⡾⠷⠛⠛⠉⠉⠉⠉⠉⠙⠛⠶⣦⣄⠀⠀⢸⡇
⠀⠀⠀⠈⠉⠛⠋⠉⠻⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠶⠦⠴⠶⠶⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⣼⢇⠀⠀⠀⠀⠀⢀⡾⠁⠘⣧⠀⠀⠀⠀⠀⢀⡞⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠇⢉⣈⡷⣦⣄⡀⠀⣼⡃⠈⠀⢻⡇⠀⢀⣠⡶⢎⣀⡁⢸⠀⠇
⠀⠀⠀⠀⢻⣏⠍⣙⠛⠒⠲⠦⠤⣇⣀⣿⣆⠁⢉⡌⢻⣿⣿⣿⡇⠀⠐⠸⣿⣾⣿⠟⡁⠀⠈⣰⣷⣀⡤⠤⠤⠖⠒⠛⣋⢭⣿⡇
⠀⠀⠀⠀⠘⡟⣥⢤⣬⣑⠠⠀⠀⠀⠈⠻⢿⡆⠀⠁⠀⠹⢿⣿⠃⠀⠀⢠⣿⡟⠇⠀⠀⠀⣪⡿⠋⠁⠀⠂⠀⠐⣚⣭⡤⡌⡿
⠀⠀⠀⠀⠀⢻⠸⡌⠆⠉⠳⣦⡳⣶⡀⠀⡬⠒⠶⡄⠀⠀⠀⠉⠁⠀⠀⠀⠘⠀⠀⠀⣠⠴⠋⡥⠀⣤⡶⣺⡶⠛⠉⢶⣰⢳⠇
⠀⠀⠀⠀⠀⠈⣇⢳⡱⡀⠀⠘⢿⣾⣿⣯⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣩⣾⡟⠀⠀⢠⢣⢇⡟
⠀⠀⠀⠀⠀⠀⠘⢆⢣⠱⡄⠀⠸⣿⣯⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠩⢹⣿⡿⠁⠀⣠⢃⢏⡞
⠀⠀⠀⠀⠀⠀⢀⣼⣧⣱⡙⡄⣬⢻⣿⡉⠀⠀⠀⠀⠀⠤⣶⣆⠀⠘⠂⠀⠲⢤⠄⠀⠀⠀⠀⠀⢩⣿⠿⣄⡴⣡⢋⣾⣤⡀
⠀⠀⠀⠀⢀⣴⣛⣾⣿⣷⣷⣼⣿⣛⠀⠀⠀⠀⠀⠀⢀⠀⢀⡚⡄⠀⠀⠀⠴⠂⢄⢄⠄⠁⠀⠀⡀⠀⣛⣾⣥⣷⣿⣿⣿⣻⡦⡀
⠀⠀⠀⢰⡛⠊⢉⡽⠟⠋⡧⣄⣠⣤⣬⣽⣗⣦⣄⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠀⣀⣴⣾⣿⣭⣭⣭⡤⢉⠝⠻⢯⡉⠉⣛⠦
⠀⠀⠀⠀⠈⢳⠋⡠⠬⣀⣭⣿⠛⠛⠋⠉⠛⠻⢿⣮⡵⢖⠀⠀⡄⠀⠀⡀⠀⠈⢤⣶⣷⡿⠟⠛⠋⠋⠻⠿⣿⣤⣦⠜⢄⠱⡎⠁
⠀⠀⠀⠀⡰⠃⠊⢠⣽⣏⢉⢿⡄⠀⠀⠀⠀⠀⠀⠈⠙⢷⣁⣰⡇⢀⡀⢹⣄⣴⡿⠫⠁⠀⠀⠀⠀⠠⠀⣠⡟⠉⣹⣧⡠⠀⠘⢆
⠀⠀⠀⡼⣁⣤⣤⣀⡀⠻⣆⠁⠹⣆⠀⠀⠀⠀⢀⢀⡀⠀⠉⣿⣿⣈⣩⣿⡟⠉⠀⢀⠀⡄⠀⠀⠀⠀⣰⠏⠀⣸⠏⠀⣀⣤⣀⡈⢣
⠀⠀⠀⠛⣿⣻⣿⣽⢷⣦⣿⣧⣀⣈⠳⢦⡀⠀⢸⣄⡀⠀⠀⢹⠟⠿⠿⢿⡇⠀⠀⣄⣼⠁⠀⢀⡼⠞⣁⣠⣾⣿⣠⠾⣯⣿⣿⣿⠛
⠀⠀⠀⠀⢸⠡⢯⣾⣿⣿⣃⣿⠋⠛⠛⠳⠭⣓⡲⠛⢃⣠⠴⠋⠀⣠⣄⠀⠻⢆⡀⣛⠯⠔⣚⡭⠾⠛⠛⠛⣿⣀⣻⣷⣶⣯⠈⡇
⠀⠀⠀⠀⢸⣐⣿⣿⣿⡩⠟⠿⣆⣠⡀⠀⠀⠈⠉⠓⠦⠀⠀⠀⢸⣷⣾⡇⠀⠀⠀⠰⠶⠋⠁⠀⠀⢀⣠⣴⠿⠛⠆⠿⣿⣿⣆⡅
⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣧⡴⣮⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣥⣶⣿⣴⣿⣿⡟⢿⡇
⠀⠀⠀⠀⠸⠇⢸⣿⣿⣿⣾⣿⢛⣾⡿⠿⣿⣄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣒⣿⠿⣿⣿⣿⣿⠿⣿⣿⣿⠇⠈⠇
⠀⠀⠀⠀⠀⠀⢸⢹⣿⣿⣿⣷⣿⠋⠀⠀⠀⠙⠷⣦⣍⣓⠲⠶⢿⣾⣿⣿⠷⠖⢒⣫⣵⢾⠏⠁⠀⠀⠙⣷⣸⣿⠮⣿⡇⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡤⠀⠀⠀⠀⠒⠈⠩⠛⣿⣷⠶⣶⣶⣶⡾⠿⠟⠓⠂⠁⠀⠀⠀⠀⣰⣿⣿⣿⣻⣾⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⡏⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣾⣿⢟⢻⣿⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢃⠀⠼⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠓⠉⡼⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⣿⣿⣿⣶⣦⣤⣀⡀⠀⠸⡇⢀⣀⣀⣤⣴⣿⣿⣿⡛⢁⠀⡂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣿⠀⣿⣿⣿⡿⠋⠀⠀⠉⠛⠷⣼⣧⠾⠛⠉⠀⠀⠙⢿⣿⣿⣿⠀⣇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧⡴⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠛⠛⠉⠉⠉⠉⠉⠛⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧⠤⣄⣀⢀⣀⡤⠴⠒⠒⠂⠀⠀⠐⠀⠀⠀⠈⣉⠳⠴⠒⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣽⡂⡲
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠉⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠰⠯⠤⠤⣤⠤⠤⠤⠤⠤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⡀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⡀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠝⠁⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡸
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠢⢄⣀⣿⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⢉⡭⣍⣿⣀⣀⣠⠴⠲⡤⠔⠒⠲⢤⡀⠀⣼⠀⠀⠐⠀⠀⣨⠟⠛⣛⡴⠚⢓⠦⣄
⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⣏⠀⠈⠁⠀⠀⠑⡄⠀⢹⡤⠖⠛⠢⢌⠆⠈⠳⡶⢖⡚⠉⠀⡴⠋⢅⢀⡴⠁⣠⠈⢧
⠀⠀⠀⠀⢀⠴⢋⠤⠂⠘⡧⠀⠀⠀⠀⠀⠀⢰⢀⣬⠷⡄⠀⠀⠀⠀⠀⠀⡟⠢⣍⣉⣉⣇⠠⣀⠿⢤⠊⣁⠔⣸
⠀⠀⢀⡴⡡⠊⠀⠀⠀⠀⢹⢦⣀⣀⣀⡤⠶⣯⠭⠤⠔⠁⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠉⠉⢦⣀⡬⠞⠛⠥⠞⠁
⠀⠀⡞⡌⠀⠀⠐⢄⢀⠞⠁⡀⠀⠀⣨⠃⠀⠀⠉⠁⡲⠂⠀⠀⠲⢏⠁
⠀⢰⣧⠁⠀⠀⢀⡤⢿⡀⡜⠀⢀⣴⠋⠓⢦⣀⣠⠞⠁⣠⠞⠲⣄⠀⠙⢦⡀
⠀⢸⣿⠀⠀⢠⠏⠀⠀⠉⠧⠴⠋⠈⠙⠲⢤⣀⣀⣠⠞⠁⠀⠀⠈⠳⣄⠀⠹⣄
⠀⠘⣧⣂⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠳⣄⡤⠶⡄
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡦⠚⠁⠀⠀⢳⣄⣀⣤⡴⠖⠚⠉⠉⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⢀⣠⠴⠋⠉⠀⢀⡤⠒⢆⠀⣼⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾⠉⠀⠀⠀⠀⢸⠁⣂⡱⢈⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⠀⣀⡠⠵⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠈⣉⣩⣤⣶⠋⠡⢀⢀⢤⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠈⠀⠈⠈⠈⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠤⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠁⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⢀⠔⢋⠤⡈⠳⡀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠘⠁⠤⠁⠀⡘⡄⠹⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣠⠤⠶⠦⢄⡀⠀⠀⠉⠀⠁⠀⠀⠉⠉⠉⠉⠑⠒⠦⢄⡀
⠀⢀⡀⠤⠀⢐⡎⠀⠀⠀⠀⠜⣡⠤⠤⠤⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⢇⠀⠀⠀⡞⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠈⢆⣠⠼⢧⡀⠀⠀⠀⢸⠁⠀⠀⢰⡷⡄⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⠇
⠀⠀⠈⡇⢀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣧⣿⡆⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁
⠀⠀⠀⢧⢺⠈⠳⡀⠀⠀⢸⠀⠀⠀⠀⠘⢿⣿⠀⢸⣠⠔⠒⢦⡀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠈⢮⣆⠀⠹⣦⣀⡜⠀⠀⠀⠀⠀⠀⠁⣠⠟⠁⢀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠙⢆⠀⠹⣿⠀⣀⣀⣀⣀⣀⡤⠚⠁⠀⢀⠜⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⢰⣾⣭⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠫⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠒⠈⠉⠉⠉⠉⠉⠉⠉⠉⠓⠲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⠤⠤⠴⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠰⠲⠄⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⣀⠀⠀⣤⠤⣴⡟⠉⢙⣶⠾⠛⠲⣄⠀⣾⠟⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⣁⣀⣀⡀⠈⠓⢾⣇⣀⣿⠁⠀⢸⡇⠀⠀⠀⠘⡄⣿⣀⣤⠴⠗⠒⠀⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠚⠉⣥⣶⡍⠙⢷⣄⠀⠉⠉⠛⠛⠛⠛⠛⠓⠚⠛⠛⠛⢻⠉⢀⠴⠞⢛⣛⡛⠶⣌⣆
⠀⠀⠀⠀⠀⠀⢀⡟⢀⣤⡀⠙⠿⠃⣀⣄⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢿⣿⠇⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⣸⠁⠻⣿⠏⠀⠀⠐⣿⣿⠀⡇⠠⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣾⣿⡇⠀⠀⢠⣿⣿⠘⣷
⠀⠀⠀⠀⠀⠀⣿⠀⢦⡀⠀⣾⣿⡆⠀⠀⠘⠀⠀⠀⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⡈⠉⠁⣴⣶⡀⠙⠋⠰⢻
⠀⠀⠀⠀⠀⢀⣿⠀⠀⠙⠲⠾⠭⠤⣠⠖⣂⣀⡀⠀⣷⠀⠀⠠⠈⠠⡀⠀⢀⠤⢒⡂⠙⠦⣄⣻⠟⢁⡀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠁⡾⠉⠉⢻⡄⣿⡆⠀⠀⠀⠀⠘⣆⠀⡼⠋⠙⢷⠀⡄⠀⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣠⡾⢡⣏⣿⡄⠀⠀⠀⠔⠉⠳⣷⣀⣀⡾⢡⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣸⠲⠬⠤⠶⠋⠀⠈⠻⣤⡀⠀⠀⠀⠀⠙⢏⣩⣴⣿⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣴⣿⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠈⣧⡈⣿⣿⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢀⡜⠀⢿⣇⠀⠀⠀⠀⠀⠀⣰⣿⢿⡇⠀⠀⠀⠀⠚⠁⣿⠏⢿⣷⡀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠈⠻⢦⡤⠴⠋⠀⠀⠸⣿⡀⠀⠀⠀⠀⢠⣿⠃⠀⢻⡀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣿⣄⡀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⣾⣿⣦⠈⠢⣀⣠⡴⣶⡟⠲⣄
⠀⠀⠀⠀⣸⣿⣿⣿⣦⡀⠈⠻⢿⣻⣿⣅⠈⠢⡀
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣳⡀⠀⠀⠙⢿⣿⣆⠀⠘⢦⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠙⢿⡆⠀⠀⠙⣄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠙⢔⠠⡀⠈⠳⣄
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢻⡆⡀⠀⠀⠀⠈⠳⡙⢦⡀⠈⢢⡀
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣾⣿⡌⢢⡀⠀⠀⠀⠈⠢⡑⠄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⣿⣿⣿⣿⣿⣄⠹⣄⠀⠀⠀⠀⠈⠢⣄⣀⣀⣉⣲⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⠻⣿⣿⣿⣿⣿⣦⡘⢦⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠘⢿⣿⣿⣿⣿⣿⣦⣵⣀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣻⠿⠿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠿⠷⠶⠿⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠚⠋⠉⠉⠉⠉⢉⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠡⢄⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠲⢄⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣤⠤⠤⠬⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢶⣫⡏⠁⠉⠀⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣔⠉⠉⠓⢄⣀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜
⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠈⢣⣨⠦⠘⠆⠀⠀⠀⠀⠀⠀⠀⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⣁⡀
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠸⠧⣛⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠌⣫⡉⡹⠂⠂⠀⠈⠑⢄
⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠓⠒⠊⠉⠀⣀⡤⢤⡀⠀⠀⠀⠀⠀⢳⠐⠒⠠⢧⠀⡼⠀⠀⠀⠀⠀⠀⢈⢣
⠀⠀⠀⠈⠉⠳⢄⣀⠀⠠⠐⠀⠄⣠⠤⠔⠒⠚⠉⠁⠀⠀⠙⢤⡀⠀⠀⠀⠈⠖⢒⡸⢛⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠢⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⣀⡀⠈⠣⣄⣸⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⢠⠒⠊⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⢀⠤⠒⠉⠀⠉⠢⢄⡀⠁⢀⡤⢀⢠⡄⠀⠀⠀⠀⠀⢸
⠀⠀⢀⡋⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠱⠀⠀⠀⠀⢀⡟
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠦⢄⡀⡔⠒⠈⠉⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⡴⠁
⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⠂⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠉
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡇
⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠋
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠈⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠒⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠁
⢀⣀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⢀⡀
⠀⠈⠙⠲⢤⣀⠀⠀⠀⠠⣄⡉⠲⢄⠙⠢⣄
⢄⡀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠉⠲⢤⡉⠂⠈⠑⢄
⠤⣄⣁⢦⣤⣀⠀⠀⠈⠙⠳⠦⣄⠀⠈⠓⠄⢀⡀
⠀⠀⠀⠉⠓⠛⠙⠒⠦⢄⡀⢀⣈⠙⠻⠿⣶⣤⣉⠲⢄⡀⢀⡀
⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢄⣀⡈⠉⠒⠢⢄⡉⠛⠷⢦⣈⠐⠉⠢⢄
⠤⠬⢍⣉⡙⠛⠓⠶⠦⠤⣤⣀⡉⠉⠒⠢⠤⠀⠀⠐⢤⣌⡑⠢⠑⠢⢁
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣀
⠉⠉⠉⠉⠉⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⣒
⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠒⠒⠂⠐⠒⠒⠒⠩⠭⠭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⠀⠄
⠤⠤⠤⠶⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠐⢊⣉⡉⠤⠀⠈⠉⠀⢀⠤⠔⠂
⠔⠒⣒⣀⡀⠄⠀⠀⠀⠀⠀⠀⣀⣀⣤⠽⠟⠒⠉
⠉⠉⠁⠀⠀⠀⠀⢀⡴⠖⠛⠉⠉⠁
⠀⣀⡤⠴⠒⣊⡩⠥⠒⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⠔⡀
⠀⣤⡶⠞⠋⠁⠀⠀⠀⠀⠀⣀⡤⠒⠉⢀⡠⠖⣩⠔⠋
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⠐⠊⠁
⠀⠀⢀⡠
⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠇⡟⢳⢼⠋⢸⡏⠁⡟⢻⠀⢸⠂⢸⡇⣿⣸⢸⡋⠱⣏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣙⡆⡟⠋⢸⡉⢸⡏⠁⣇⣸⠀⢸⡀⢸⡇⡏⣿⢸⡋⢠⣝⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠈⠉⠈⠉⠁⠉⠉⠀⠈⠉⠉⠁⠁⠈⠈⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⣴⣶⣶⣶⣤⣄⣀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠃
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⢸⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⣿⡿⠀⠈⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⢸⣿⡇⠀⠀⢹⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠸⣿⡇⠀⠀⠘⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣴⣤⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⣿⡁⠀⠀⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣼⣭⣿⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⢸⡇⠀⠀⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠙⣿⣿⣿⡄
⠀⠈⣷⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣠⣿⣿⣿⣿
⣰⣾⣿⣷⣶⣤⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⣠⡾⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿
⠙⠿⣿⣿⣿⣿⣿⣿⣟⠛⠦⣄⣀⡀⠀⠀⢀⣀⣤⠾⠋⠀⠀⣀⣀⣀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢀⡴⠋⠉⠉⠉⠉⠉⠛⠛
⠀⠀⠀⠙⢯⡛⠛⠛⠋⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣨⡟⠻⢀⣿⣿⠟⠉⠉⠙⣿⡿⠴⠶⠶⢴⠋
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠀⢀⣾⣿⡏⠀⠀⢀⠞⠁⠀⢀⣀⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡇⠀⠀⣼⢀⡴⠚⠉⠉⠉⠉⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣄⣀⣀⠀⠀⢀⣀⣀⣿⣿⣿⣿⣿⣿⣿⠟⢶⣤⢿⡏⣀⡤⠴⠶⢤⣄⠀⣿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣴⣶⡞⠂⠀⣀⣤⠴⠤⡄⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⣤⣶⠁⢀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠘⢬⡉⠉⢷⡶⠞⠁⣀⠠⣀⣄⣲⣯⣿⠿⠷⢦⡄⠀⢀⣠⣶⣶⠀⢀⣀⠀⣀⠄⠈⠩⠷⠶⢫⣿⠿⠤⣆
⠀⠀⠀⠀⠐⠶⠿⠟⠋⠀⠁⠀⠀⠀⢸⡾⣥⣶⡄⠀⠀⢈⣶⣰⣿⣿⡝⠋⠁⠀⠀⠀⠀⠛⠻⠛⠁⣽⠀⢘⣧⣼⣿⣊⣷⣆⠀⢠⡨⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣚⢯⣾⠿⠃⠀⢰⢿⣿⣿⣧⣿⡚⠀⠀⠀⠀⠀⠀⠀⢀⢶⡖⠥⠞⠋⠀⠈⠙⠛⢯⡹⠓⠞⠁
⠀⠀⠀⠀⠀⠀⢠⠖⣋⠄⠀⣀⠤⣚⡿⠝⠉⠉⠙⠷⡆⠀⠀⠀⡀⢻⡌⠛⠢⢀⡀⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠘⢀
⠀⠀⠀⢀⡠⠔⡁⠈⠀⠀⠈⢡⠌⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠲⠳⡀⠀⠀⠉⠀⠘⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠤⣯⡅⢀⡠⢀⣀⣀⣀⣀⣀⢀⣀⣀⣢⣀⠀⣀⣈⠂⠀⠀⠠⠤⠀⠀⢠⣤⣄⢠⣤⢤⣤⡤⣅⣤⣤⠤⠤⠤⠤⣤⠀⠄
⠄⠄⠀⠉⠓⠓⠛⠛⠛⠒⠒⠐⠙⠛⠓⠛⠛⠋⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠉⠋⠉⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡏⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢒⠖⠉⡩⠝⠂⠠⠴⣊⠁⠀⠀⣀⡰⠃⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠂⠀⣘⡁⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣠⠊⠉⢁⣀⡀⠁⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠈⠒⡌⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠞⠁⠉⠉⠀⢀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢀⠴⠂⠉⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠴⠈⠋⠀⢀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⡴⠛⠂⡀⠀⠀⠀⠀⠈⠱⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢌⡁⡀⠤⠤⠤⠤⠣⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⡠⢊⠌⢠⡤⠐⠚⡉⠁⠉⠁⠒⠀⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢂⠄⠘⠀⠁⠀⠰⠠⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⡠⠊⠀⠊⠀⠈⠛⠀⠀⠀⠀⠤⠾⠂⣈⡉⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠑⢤⠒⠈⠉⠉⢉⠉⠊⠀⠀⠀⠀⠀⠀⢀⠞⢁⠔⡠⠂⠀⠀⠀⠀⠘⠖⠀⠀⠠⠀⠀⠤⡌⠒⠚⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠗⢀⣀⢤⠤⠄⠀⣸⠇⠀⠀⠀⠀⠀⠀⣰⠋⠠⣢⠞⠀⠀⠀⠀⠀⠀⠀⠰⡧⠤⠀⠠⠤⠄⠀⠑⣯⡵⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠤⣈⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⠞⠁⣀⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⣈⣁⠀⠘⠁⠠⢴⡆⡄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢨⠈⠙⠁⠁⠀⠀⠀⣀⡔⠁⠀⣨⠏⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠁⠒⢲⣈⣉⣉⣠⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠑⠢⢌⣆⠀⠀⠀⢀⡤⠚⠁⠀⣀⢼⠏⠀⠀⢳⣄⠀⠉⠢⣀⠀⠀⠀⠀⠀⠋⠃⠨⢟⣭⣿⠯⠥⠭⠭⠤⢬⡽⣦
⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣦⠴⠚⠉⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠙⠍⡓⠢⢌⠣⣀⠀⠀⠀⠀⢹⠈⢉⣉⣀⣤⣤⣤⣤⣤⣤⣽⣷⣷⣤⣀⡀
⠀⠀⢧⣀⠀⠀⠀⠀⢀⣀⣀⡤⠴⠚⠉⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠁⣽⠣⣄⡀⠀⠀⣠⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠋⠈⠢⡀
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠉⠑⠚⠒⠤⠤⠄⠐⠒⠒⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠡⢄⣀⣀⣀⣀⣀⣠⡤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠂⠀⠀⠀⠐⠒⠒⠂⠉⠉

⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙
⣿⣿⣿⣿⡇
⠀⠀⠀⡀⠁⠀⠀⢹⣶⣴⠺⡏⠉⣽⣶⣾⡇
⡀⢳⣦⣤⣀⠀⢀⣼⣿⣯⡀⠹⣼⠃⣨⡿⢷⣄⣀⣀⡴⡾⠁
⢠⣅⠙⣿⡻⠿⠟⣡⢾⡟⢿⣦⣴⡾⠛⡿⢀⡛⠟⠉⠔⣡⢾⡏
⠈⢿⣿⣾⣿⣶⣿⡿⠘⣿⡶⠟⢛⡳⣾⡇⠈⠻⢶⢞⠿⠋⡼⠁
⠀⠈⢳⣌⢽⣿⡿⣁⣠⣿⣿⠛⡟⣹⣿⣅⡀⠀⠀⠁⠀⡜⠁
⠀⠀⠀⢹⣿⣿⡟⠋⠙⠁⠙⠲⠜⠋⠀⠀⠉⠓⠦⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠸
⠀⠀⣶⣶⡇

⠀⠈⠒⠄⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠈⠂⢄⠀⠀⠈⠠⡀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⠈⠂⢀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⠄⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠒⠒⠈⢈⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡀⠀⠠⠤⠀⠒⠂⠈⠉⠉⠉⠀⠀⠀⠀⠀⡀⠤⠐⠊⡡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠔⠀⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⣠⠞⠀⠀⠀⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⡠⠖⠉⠀⡠⠂⠀⠀⢀⠞⠁⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢀⡤⠚⠁⠀⢀⠔⠋⠀⠀⢀⠔⠁⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⣿⣷⣶⣾⣾⡿⣾⣶⣶⣾⣿⣾⣾⣶⣶⣶⣶⣿⣶⣾⣷⣶⣾⣶⣶⣿⣶⣷⣾⣷⣿⣶⣶⣶⣾⣿⣶⣾⣷⣶⣶⣶⣾⣷⡷⣷⡾⢾⣾⢷⣶⣷⠾⠶⡿⠶⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠁⠀⢸⣿⣿⣿⣿⣷⣀⣀⣀⣀⣀⣀⣠⣶⡾⠛⠛⢻⣿⣶⣦⣄⣀⣀⣀⣀⣀⣰⣿⣿⣁⣀⣀⣻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣠⡾⠿⣛⣛⣛⠻⢭⣍⡉⠉⠙⠛⠿⣿⣿⣿⣆⠀⣼⣿⣿⣷⣿⣿⡿⠛⠉⢉⣩⠭⢟⣛⣛⡻⠿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁⠀⣾⣭⣽⣭⣻⣿⣮⡙⢮⡲⣄⣀⣈⣛⣿⣿⣿⠋⠉⢁⣀⣀⣀⠀⢀⠔⠉⠀⢰⡏⣩⣭⡙⣾⣷⣽⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⢳⣽⣀⣀⣀⣿⣟⣿⠀⣾⣇⣀⣀⡼⢠⠏⠀⠀⠀⢸⡟⢿⣿⢟⣿⣿⣿⣷⢻⣧
⠀⠀⠀⠀⠀⠀⠀⣸⡇⡞⣻⣿⣿⣿⡙⠿⠛⣹⣿⣿⣿⢳⡌⣇⡿⠿⣿⣿⣿⣿⠀⠈⡹⠿⢿⡁⡏⣼⠛⣻⣿⡛⢃⡤⢤⡘⢛⣿⣟⠛⡆⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡄⡇⣿⣿⣿⣿⠇⣀⡘⣿⣿⣿⣿⢸⡇⡟⣳⣶⣾⣻⣿⣿⡀⠀⠳⠶⠾⠃⡇⣿⠸⣿⣿⠇⠘⢄⡠⠇⠺⣿⡿⠀⡇⣼⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣧⢹⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⠿⣋⣀⣙⣛⠿⣿⣿⣿⣿⣷⡄⠀⡠⢖⣚⣁⣈⠻⢿⣿⣽⡇⣤⣤⣰⢿⣿⣿⣿⢡⣿⣷
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣦⡻⣿⣿⣿⢿⣿⣿⢿⡿⣱⡾⠛⠁⠀⢨⣷⣌⢧⣹⣿⡿⢣⢞⣴⡿⠋⠀⠀⢩⣶⡹⣿⡟⢿⡿⢿⣿⣿⡿⣫⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠻⣿⣦⣍⣙⠓⠒⠒⣋⠀⡏⠀⠀⠀⣰⣿⡟⣻⢸⡿⣿⠀⣾⢸⡏⠀⠀⢀⣴⣿⠟⡇⢈⡛⠒⠒⢚⣉⣭⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠈⠙⠻⠿⣿⣿⣿⣿⡆⢿⣦⣠⣾⡟⣋⣴⡿⣸⣿⣿⣿⣿⡸⣷⣄⣴⣾⣟⣫⣼⠇⣾⣿⣿⣿⠿⠿⠛⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⢻⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣮⡙⠿⠿⠿⠿⢋⣴⣿⡟⠛⠛⣿⣷⣌⠻⠿⠿⠿⠟⣩⣾⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⣿⣿⢻⡄
⠀⠀⠀⠀⠀⡿⣾⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⣿⡟⠛⠶⣶⣿⣿⣿⠏⠀⠀⠀⠈⢿⣿⠛⠳⢶⣾⣿⣿⡿⢿⣿⣿⡆⠀⠀⠀⠀⢹⣿⣿⣿⠸⡇
⠀⠀⠀⠀⢸⡇⣿⡟⠀⠀⠀⠀⢠⣿⣿⣿⣻⠏⠀⠈⠛⠳⠶⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠲⠾⠿⠛⠉⠀⠈⢷⡹⣧⠀⠀⠀⠀⠀⢿⣿⣿⡇⣿
⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⣸⣿⡟⣵⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡌⠃⠀⠀⠀⠀⠘⣿⣿⡇⣿
⠀⠀⠀⠀⠘⢧⡈⠓⠀⠀⠀⠚⣛⡵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠚⠛⠋⣠⠏
⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠖⠛⠉