Feel good Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⡀⠀⠀⠀⠀⠀⠀⠀⣰⡾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⣼⠁⠀⠀⠀⠀⠀⠀⣼⡏⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⣼⠃⠀⠀⠀⠀⠀⠀⣼⡿⠀⣰⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢀⡾⠁⠀⠀⠀⠀⠀⠀⢰⣿⢃⡴⠃⠀⠁⠀⠀⢀⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡷⠋⠀⣠⣤⡄⣰⣿⢿⠆⣼⡿⠛⠁⣴⡗⢀⣾⢠⣿⡿⣠⣾⢿⣷
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣄⠀⣰⣿⣰⢳⣿⣷⠟⢠⣿⡇⠀⣸⣿⠁⣼⣿⢱⣿⢳⣿⢇⣼⡇
⠀⠀⠀⠀⠀⣠⢻⣿⠟⢿⠋⢹⣿⣀⡼⣿⡅⣠⠏⢿⣇⡴⠙⠿⠞⠀⠁⣿⡧⠊⠛⢋⣿⣇
⠀⠀⠀⠀⠀⠃⢸⣿⠀⢸⠀⠀⠉⠉⠀⠈⠛⠁⠀⠈⠉⠀⠀⠀⠀⠀⢀⣼⡆⣠⠞⠉⣿⠉⠙
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣸⠀⠀⠀⣀⣀⡀⠀⣴⠲⡄⠀⣰⡏⢳⠀⣠⣼⡿⠁⢿⣠⡾⠃
⠀⠀⠀⠀⠀⠀⢸⣿⢀⡏⠀⣠⡾⠋⣿⣇⣾⠟⢷⡧⣼⡟⠻⣾⣾⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⠟⠀⢸⣿⠁⣰⣿⡿⣿⢀⡼⠀⣿⣇⣴⠏⠻⡵⢻⡇⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⢹⣿⡇⠙⠛⠁⠀⠈⠉⠁⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢺⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠕⠊⠀⠀⣀⣤⣄⠀⠀⠀⠀⣀⠀⠀⢠⡟⠀⠀⠀⠁⣤
⠀⠀⠀⠀⠀⠀⠠⠐⢸⡏⠀⠀⢀⣾⠋⠁⣨⠇⢠⡶⠛⠉⡹⢀⣿⠁⠀⠀⠀⣠⡆⠀⠀⣠⠀⠀⠠⠒⡦⠀⢀⣤⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣾⠃⠂⠈⠀⢠⡿⠁⠀⠀⠀⣼⡇⠀⠀⠀⢠⡟⠀⠀⢠⡟⠠⠂⠀⣼⠃⢠⡿⠁⠀⡰⢻⡆
⠀⠀⠀⠀⠀⠀⠮⣉⣿⢤⠐⠁⠹⣆⣀⣠⠔⠈⢷⣀⢀⡠⠀⣿⠁⠀⠀⠀⢾⣇⡴⠁⠸⡷⠁⠀⣸⠇⠀⢼⡇⣀⠔⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢻⡄⠀⡠⠂⠀⠉⠀⠀⠀⢀⡀⠀⣿⣠⠎⠀⠉⠁⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣇⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⣠⠞⠉⢹⠀⠀⠁⢀⡤⢦⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣼⠋⠀⡠⠂⠀⠀⢠⠏⠀⠰⠇⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⣄⠀⣰⠋⠉⠑⢦⢠⡋⠉⠑⢆⠀⢀⣾⠋⠁⢉⡿⢁⠠⠊⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⣼⠏⠀⢀⠜⢹⡇⠈⢲⡤⠤⠘⡇⣿⠒⠒⢸⠃⢹⡇⠀⢀⣾⠋⠁⠀⠀⠀⠀⠀⠀⠈⠳⠤⠠⠔⠁
⠀⠀⠀⠀⠀⠀⢿⣄⡴⠋⢀⣸⡇⠈⢸⣇⣀⡼⠁⢿⣤⣤⠞⠀⠘⢧⡤⠺⣏
⠀⠀⠀⠀⠀⠀⠀⠉⡠⠔⠁⣼⠇⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠸⣀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⣠⡶⠶⣦⠀⢀⡤⠤⡄⢰⡏⠀⠀⠀⢐⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⣰⡏⠀⠐⠁⣴⠏⠀⠀⢁⣿⠁⠀⠀⢠⡟⠀⠀⣾⠃⠀⠀⣸⠇⢠⡾⠋⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠓⣆⠀⢰⢚⣯⡀⠀⠻⣆⠀⡀⠄⢿⡀⠀⡀⢸⡇⠀⠀⢀⣿⠀⠀⢰⣯⠄⠀⣰⠏⠀⣾⠁⢀⠌⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢀⡞⠀⠀⢸⡇⢱⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠸⣇⠀⡀⠀⠙⠋⠀⠀⠁⠀⠀⣿⢀⠄⠙⠛⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⣀⣤⠶⠋⠀⠀⠀⠘⣇⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⡤⣄⠀⠀⣼⠃
⠀⠀⠀⣀⣠⢤⡿⠖⠚⠋⠉⠀⠀⠀⠀⣠⡴⢦⣄⠀⠀⠀⡾⠀⢀⣤⡀⠀⠄⡆⢀⠴⠒⣦⠐⠈⣷⠀⠀⢀⣶⠀⠀⢰⠋⠀⠼⠀⢠⡏
⠀⠀⠈⠀⠀⢸⡇⠀⠀⣠⡄⠀⠀⠀⣼⠃⠀⣠⡿⠀⠀⢸⡇⠠⠋⠀⢿⠀⢠⣷⠃⢀⠔⡿⠀⢸⠇⠀⣠⠂⣿⠀⠀⢸⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⢸⡇⠀⡰⠁⡿⠀⠀⢠⡏⢀⡴⠹⡇⣠⠂⢸⡇⠁⡰⢁⡟⠀⢸⠇⢠⠇⣼⠃⠀⢸⣤⠔⣡⠴⡟⠒⠲⣌⠓⠀⠖⠉
⠀⠀⠀⠀⠀⠘⣇⢠⠃⢸⡇⠀⢀⠂⠙⠋⠀⠀⠉⠁⠀⢸⡇⠀⠳⠋⠀⠀⣿⠀⠘⠛⠁⠀⠀⠀⡠⠊⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠙⠞⠀⢸⡇⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡸⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈⠛⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠛⠛⠋⠉⠉⠙⠛⠛⠒⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠄⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣠⣾⣿⣿⣷⣦⡀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠤⣤⣤⣄⣀⣀⣠⣤⣤⠤⠶⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⣨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣴⡳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⢸⡿⢋⣠⠴⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠶⠖⠒⣿⠋⠉⡭⠿⠦⣄⣀⡄⠀⣠⠋⠀⠀⢀⡁⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣩⣍⠀⢀⡟⠁⢰⠇⠀⠀⠰⣁⣹⣀⣼⣳⡔⣹⠃⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⡾⠋⠀⣾⡕⢯⡠⣻⢀⠄⠸⡄⠀⣹⠃⢯⠍⠸⠔⠉⠋⢸⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⡄⠀⠀⠀⠀⠘⠋⠀⠀⠉⠋
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣥⣤⣤⣤⣶⣶⣶⣶⣶⡶⠶⢶⣿⣿⠿⣿⣿⣯⣿⣿⣿⣿⡿⣻⣿⣟⣻⣿⣿⠃
⠀⠀⠀⠀⠀⣿⣿⣉⡉⣩⣤⣾⣿⣿⣿⣿⡟⢱⣿⡟⣰⣟⣸⣿⢡⡷⣿⠟⣥⣹⣿⢋⣭⢡⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⣛⣿⣿⠁⣿⣿⢋⢖⢋⡄⡿⢡⣿⣿⢠⣿⠿⢿⠃⣜⢃⡉⣄⢃⣿⠃⡾⠃⣼⣿⠏⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⣻⣿⡿⢠⣶⡇⢰⠟⠠⢞⡅⢸⣿⡇⢸⣿⢣⣿⠸⢋⣾⡀⣣⣶⣶⠘⣵⡇⠿⢋⣼⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠽⣿⣇⣸⣿⣿⣶⣿⣶⣿⣿⣿⣿⣿⣤⣴⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠲⠿⠛⠛⣛⣛⣛⣛⣛⣛⣫⣭⡭⠭⠭⠭⣭⣭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠤⠶⠒⠒⠂
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⠸⡄




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⢨⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⢀⡾⠁⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⢠⡾⠁⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⣼⠃⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣴⠏⠀⠀⠀⠀⢀⣤⣄⠀⢸⣿⣡⡞⠁⣀⠀⠀⣀⠀⣴⣶⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠁⠀⣴⡿⣷⢠⣿⠏⣿⠀⣿⡟⠁⠀⣼⡿⠀⣼⣏⢼⣿⡟⣴⣿⢋⣿
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢦⣀⣰⣿⡧⠏⣿⣿⠞⠁⣰⣿⡇⠀⣸⣿⠃⡰⣿⠏⣼⡿⢸⣿⣇⣼⣿
⠀⠀⠀⠀⠀⢀⡞⣹⣿⠛⢻⠋⠸⣿⣇⡴⢻⣿⣀⡴⠁⢿⣧⡴⠉⠻⠟⠁⠀⠀⣿⡗⠉⠙⣉⣼⣟⣀
⠀⠀⠀⠀⠀⠚⠀⣿⡟⠀⢸⠀⠀⠈⠉⠀⠀⠉⠋⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⣶⡆⣰⠟⠉⣿⠇⠈⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣼⠀⠀⠀⠀⣀⣀⠀⢀⣶⢲⡄⠀⣰⣟⢹⡆⢀⣤⣿⡟⠁⣿⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⣿⣇⢀⡏⠀⢠⣶⠟⢻⣿⢠⣿⠻⣦⣇⣴⣿⠻⣶⣧⣿⢋⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣿⡾⠁⢰⣿⠏⠀⣾⣿⣾⡿⠀⣼⠁⣿⡇⢀⡾⠹⣿⡼⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⠞⣿⣿⠘⠿⠾⠁⠀⠙⠿⠟⠁⠀⠉⠀⠘⠧⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠂⠀⠀⣀⣀⣤⣴⣶⣾⣿⡿⠿⠋⠀⠀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⠿⠟⠛⠋⠉⠀⠀⠀⠀⣠⣾⠿⢋⣁⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⣤⣶⣾⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⢛⣭⠶⠾⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⠿⠛⠋⠁⠀⠀⠀⢀⡄⠀⠀⠀⢠⡾⠋⡇⠀⠀⠀⠀⠀⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⡸⠁⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⠀⢠⣿⠃⣰⠃⠸⠋⠀⠀⠀⢀⡀⠈⢀⣴⡶⢶⣄⠀⠁⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢀⣴⢶⡄⢀⣾⢇⡞⢀⣿⠇⡴⠃⣰⠆⠀⣼⠇⡴⣻⠇⢠⣾⠏⣠⣾⡇⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡴⠞⠋⢠⣿⢃⡼⠁⣼⣏⡞⠀⣼⣿⠞⠁⣰⡟⠀⣼⣟⡜⢰⡟⢀⣿⣃⡼⠁⣿⣡⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢿⣿⠁⠀⠀⠀⣾⡧⠞⠁⢠⣿⠋⠀⢠⣿⠃⠀⡴⣿⣃⠞⢹⡟⠀⠸⠷⠋⠙⢋⣤⢾⡟⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢻⡏⠀⠀⢀⣼⡟⠀⣠⠾⣿⣠⠞⠁⠙⠋⢀⡀⠀⠀⠀⣰⠆⢠⡞⢁⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣼⡟⠀⠀⠀⠸⣇⣀⡴⠋⠘⠿⠾⠋⠀⠉⠁⠀⠂⢀⣠⣀⠀⠀⢀⣠⡿⠀⣾⠀⣸⡏⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡀⠈⠁⠀⠀⣀⣠⣀⠀⠀⣴⠛⠳⡄⣟⣀⢹⠀⣰⠏⢹⠇⠀⠘⠾⠋⢀⣈⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣴⡾⠃⠀⠀⣠⣾⠟⠉⢹⣦⠀⣧⡾⢀⣇⣿⣇⣸⣴⡏⣠⣿⠀⠀⠀⣠⠀⣀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⢁⣴⠋⠀⢰⡿⠁⠀⢀⢾⡿⠀⣿⠃⣾⢡⡟⢨⠇⠸⠶⠃⣿⡀⠀⣰⠏⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⣰⡿⠁⠀⢀⣿⠁⠀⡰⠃⣼⣧⢾⡟⣰⠃⠘⠷⠋⠀⡀⠀⠀⠘⠷⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⠀⠀⠸⣿⣠⠞⠀⢀⣿⡏⠈⠛⠁⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠉⠁⠀⣠⢿⣿⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢰⠃⢸⡿⠀⠀⠈⠁⠀⠀⠘⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⡟⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠟



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠁




⠀⠀⠀⠒


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠆⢠⣾⣿⠄⠀⣴⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⠀⠈⣿⣿⠀⠸⡿⠋⠀⠀⣿⡿⠋⠀⢠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢰⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢀⡀⢀⣴⣿⠟⠻⣷⠀⣿⣿⡿⠿⣦⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⢸⣿⣿⣀⣀⣿⠀⣿⣿⣇⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡿⠇⢸⣿⣿⠉⢉⣉⠀⣿⣿⡟⠛⢛⡁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⣿⣿⣶⣿⡿⠀⢻⣿⣿⣶⣿⡿⢸⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⣶⡄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⣿⣿⠃⢀⣠⣄⡀⠀⢀⣀⣀⣀⠀⠀⣠⣤⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠿⣿⣿⢀⣿⣯⣼⣿⣀⣿⣿⣿⣿⡇⣼⣿⣯⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣿⡿⢸⣿⣿⣿⡿⠘⣿⣯⣽⡿⠃⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⣄⣈⣉⣉⣀⣀⣉⣉⣁⣀⣀⣀⣉⣉⣀⣀⣀⣈⣉⣉⣉⣡⣴⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⣿⡏⠁⢠⣾⢿⡆⢠⣾⣿⣿⠀⣴⣿⣿⡀⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣇⠀⣸⣯⣸⡇⢸⣿⢿⣿⠀⣿⣤⣿⠁⣿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠹⠿⠿⠃⠘⠿⠾⠿⠀⠿⠿⠿⠀⠸⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⣤⣾⡆⠀⠀⣶⡀⠀⢹⣷⡀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⢰⣿⣿⡇⠀⠘⣿⡇⠀⠸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡿⢿⠿⢿⠿⢿⣿⡟⠿⢿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠿⠿⠛⠿⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿
⣿⣶⣶⣶⣶⣶⣾⣷⣿⣶⣶⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠁




⠀⠀⠀⠒


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠆⢠⣾⣿⠄⠀⣴⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⠀⠈⣿⣿⠀⠸⡿⠋⠀⠀⣿⡿⠋⠀⢠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢰⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢀⡀⢀⣴⣿⠟⠻⣷⠀⣿⣿⡿⠿⣦⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⢸⣿⣿⣀⣀⣿⠀⣿⣿⣇⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡿⠇⢸⣿⣿⠉⢉⣉⠀⣿⣿⡟⠛⢛⡁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⣿⣿⣶⣿⡿⠀⢻⣿⣿⣶⣿⡿⢸⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⣶⡄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⣿⣿⠃⢀⣠⣄⡀⠀⢀⣀⣀⣀⠀⠀⣠⣤⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠿⣿⣿⢀⣿⣯⣼⣿⣀⣿⣿⣿⣿⡇⣼⣿⣯⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣿⡿⢸⣿⣿⣿⡿⠘⣿⣯⣽⡿⠃⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⣄⣈⣉⣉⣀⣀⣉⣉⣁⣀⣀⣀⣉⣉⣀⣀⣀⣈⣉⣉⣉⣡⣴⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⣿⡏⠁⢠⣾⢿⡆⢠⣾⣿⣿⠀⣴⣿⣿⡀⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣇⠀⣸⣯⣸⡇⢸⣿⢿⣿⠀⣿⣤⣿⠁⣿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠹⠿⠿⠃⠘⠿⠾⠿⠀⠿⠿⠿⠀⠸⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⣤⣾⡆⠀⠀⣶⡀⠀⢹⣷⡀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⢰⣿⣿⡇⠀⠘⣿⡇⠀⠸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡿⢿⠿⢿⠿⢿⣿⡟⠿⢿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠿⠿⠛⠿⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿
⣿⣶⣶⣶⣶⣶⣾⣷⣿⣶⣶⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢰⣊⣷⢣⠀⠀⠀⠀⠀⡐⣡⡮⠋⢉⠙⠣⣎⣢⠀⠀⠀⠀⠀⠠⢳⣏⣲
⠀⠀⠀⠀⠀⠀⠀⠈⡅⠥⠂⠀⠀⠀⠀⢀⣠⣟⠠⡄⠀⠀⡤⢜⣷⣃⠀⠀⠀⠀⠀⠠⠥⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸⣁⠃⠀⠂⠀⠀⠂⠀⢃⣹⠀⠀⠀⠀⠀⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠸⠀⠀⠀⠀⢸⠀⢂⠀⣤⠤⢤⡄⢀⠂⢸⠀⠀⠀⠀⢸⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠑⠐⠀⠤⠬⠀⣀⣢⣌⢒⣊⡤⣀⣀⠬⠠⠄⠀⠒⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠊⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⢴⠀⠀⠀⠀⠀⠀⠀⢸⠄⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⣰⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠉⠛⢀⠀⠀⠀⠈⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⡀⠀⠀⠀⠇⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⡑⠒⠀⠀⡇⠀⠀⠀⣖⡃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⣇⣸⠀⠀⠀⡇⠀⠀⠀⣅⡇⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠀⠀⠀⠁⠀⠀⢰⠉⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⣨⠃⠸⡁⣀⠋⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⠉⠭⠂⠤⠈⠢
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡅⠠⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⣴⢇⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢿⡿⠟⢁⣠⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⢶⠶⠒⠛⣛⣛⣉⡉⠉⢁⡄⠈⢀⡴⠋⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡟⠀⠀⠘⠧⠴⠀⠉⢳⡿⠀⢠⡟⠀⠀⠀⡴⠒⢦⠀⢠⢶⠀⡤⢶⡏⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⡴⢚⣷⠀⣠⡟⠀⠀⣼⠀⠀⣤⣤⠳⡾⣼⢲⠟⡾⣿⡵⣿⠀⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⡟⠀⠀⠸⠷⠋⠳⠞⢸⣃⡔⠀⠸⣄⣀⡴⠋⠘⠖⠁⠘⠋⠁⠀⠀⠻⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⠤⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⡶⠶⢿⣿⣿⡿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡿⢻⣿⣿⣛⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠉⣉⣭⣽⣿⣿⣿⣿⣿⡿⢁⣿⣿⠏⣴⡿⢸⣿⠏⣶⡎⣿⠟⣩⢻⣿⡿⢋⡙⢡⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⢰⣿⣿⠟⣩⢹⢟⣹⣿⢁⣾⣿⠏⣸⣿⣿⣿⡟⠀⣿⠰⠋⡀⠟⣼⡿⢁⣾⠃⣼⣿⡿⢻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣶⣿⣿⡇⢘⣻⡏⢐⣥⠂⢚⡼⠃⣸⣿⣿⠀⣿⣯⡍⣿⢀⣿⢂⡟⢰⠗⣘⣛⠁⡾⣡⠀⣿⡿⢁⣾⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⣿⣿⣇⠘⣫⣀⣋⣴⣆⣿⣿⣿⡀⠿⢋⣼⣿⣈⣡⣿⣷⣤⣾⣿⣿⣤⣴⣿⣄⣉⣤⣾⣟⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠺⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠛⠛⠛⢛⣛⣛⣛⣛⣉⣉⣭⡭⠭⠭⠭⠭⠭⠭⠭⣭⡭⠭⠭⠭⠤⠴⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⠂⢠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁




⡿⠿⠿⠿⠿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿

⡀





⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⣀⣤⣤⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠋⠁⠀⢀⡉⠉⢉⡉⠀⠀⠀⠼⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠛⠁⣠⡄⣰⣿⠋⣻⠀⣿⡟⠛⣦⢠⣄⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠠⣼⣿⡄⣿⣿⠷⠿⠀⣿⡷⠶⠟⢸⣿⠀⠘⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡧⠀⢻⣿⣶⡿⠀⢿⣿⣶⡿⠸⣿⡄⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣿⣿⢠⣾⣻⣷⡄⣴⣿⣿⣷⣀⣾⣿⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣿⡟⣿⣿⣿⣿⠷⣿⣟⣻⡿⠋⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⣉⣉⣉⣁⣈⣉⣉⣁⣀⣈⣉⣉⣀⣀⣈⣉⣉⣉⣉⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣴⣶⣿⣷⣦⣴⣾⣿⣶⣦⣄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢙⡉⠉⠉⠉⠉⠀⠀⠈⠉⠀⠈⠉⠻⠛⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠰⣿⠇⣀⣀⡀⠀⣀⣰⡆⣀⣠⡀⣤⠰⣆⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⣇⢰⣿⣽⡇⣾⡿⣿⡃⣿⣽⣇⠹⣦⡿⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣄⠀⠈⠛⠋⠀⠙⠛⠛⠃⠙⠋⠉⠢⠼⢇⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣄⠀⠀⢀⣠⣴⣶⣦⣄⠀⠀⣀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠙⢿⠟⣻⣿⣿⣿⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠿⠋⠛⠁









⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣭⣭⣭⣭⣬⣭⣽⣤⣤⣽⣭⣥⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢁⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⢀⣴⠶⠶⡄⠀⢀⣤⠤⣄⠀⣾⠃⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠉⢀⡿⠀⠀⢠⣿⣁⠠⠔⠁⣴⠏⠀⠀⠂⣸⡏⠀⠀⠀⢠⡟⠀⠀⣼⠃⠀⠂⢨⡇⠀⣴⠟⠉⢹⣦
⠀⠀⠀⠀⠀⢀⡤⢼⣇⠀⠀⢺⡇⠀⠀⢀⠼⣟⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⠀⡀⢠⣟⡔⠀⢀⡿⠀⢸⡇⠀⣠⠃⣿
⠀⠀⠀⠀⠀⠀⠁⢺⠏⠇⠀⠀⠙⠒⠒⠁⠀⠙⠲⠖⠂⠀⣿⠀⠀⢀⠈⠻⠊⠀⠀⠉⠀⠀⣼⠃⡠⠙⠷⠞⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠔⠁⠀⠀⠀⠀⣠⠴⠲⡄⠙⠚⠁⢀⣀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠘⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⡠⠁⠀⠀⡜⠉⢹⠇⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⢀⠔⠒⠢⣄⢀⠔⠒⠠⡀⠀⢀⡴⠛⠛⢶⡟⢁⡠⠊⠀⠀⠀⢸⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⣼⠋⠀⢠⠚⣇⠘⠦⣀⣀⠨⡎⣷⠤⠤⢵⠂⣾⠃⠀⢠⡿⠒⠁⠀⠀⠀⠀⠀⠈⢦⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠸⣇⣀⡴⠁⢀⣿⠐⠈⣟⠀⣰⠃⢿⣀⣠⡞⠀⢻⣄⡠⢿⠁
⠀⠀⠀⠀⠀⠀⠈⠉⡠⠂⢨⡏⠀⠀⠙⠛⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠘⠧⠤⠊
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢮⡀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⣄⠀⠈⠁⢀⣤⣄⡀⠀⢠⠀⢀⣠⣤⣤⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠲⢦⣼⠃⠀⡀⠈⢳⡄⠐⠋⠀⠀⠙⣦⢀⣴⠋⠁⠀⠀⠙⣆⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⠀⠀⠉⠀⢠⣷⠀⠘⠃⠀⠀⠀⠀⠀⠘⠿⠁⠀⢀⡾⠀⠀⠛⠋⠁⠀⢀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠸⣿⡄⠀⠀⣾⡏⠀⢀⣤⢀⣴⠶⡄⠀⡀⢀⣀⠀⣼⢇⡤⣶⠀⢀⣤⡾⠛⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠲⢶⡆⠁⠀⣰⣿⠃⢠⣿⠁⣿⡁⣰⣧⣾⢳⣏⣿⣴⣿⣯⠖⠋⠀⠩⠵⠾⠛⠁⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡦⠀⠀⠀⠈⡁⠀⢠⣿⡏⠀⠸⣿⣀⣨⠟⠁⣿⠃⣨⡿⢡⣿⢷⣤⣀⠀⠀⠀⠀⠁⡀⠀⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⣴⠞⠛⠛⢻⣿⣿⣄⠀⠀⠈⠉⢀⠀⠀⠻⠟⠋⠀⠾⠃⠀⠙⠻⣿⣶⣶⣾⡿⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⡇⠀⢀⣰⣿⠟⠉⠻⢿⣷⣶⠾⠋⣠⣤⡄⠀⠀⣀⠀⠀⢸⠇⠀⠀⡀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠙⠻⠛⠋⠁⠀⠀⠀⣠⣴⠶⣦⠰⣏⡀⢹⢀⣴⣹⢆⡤⣾⠀⠀⠀⡀⠀⠄⠀⠐⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⡄⠀⣀⠀⠀⠀⢸⣏⣠⠖⣻⣀⣽⠓⡾⢹⣃⡟⢻⣥⠻⣧⣀⡾⠃⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡀⠀⣠⢮⠉⠁⣠⣾⠉⠻⠞⠁⠈⢉⡀⠀⠀⠀⠈⢉⠀⠀⠁⠀⠁⠀⣠⠇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⠃⣸⠁⢸⢁⡟⠀⠀⠀⠀⢠⣏⡇⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡄⢠⡏⠀⠁⣠⣤⠉⠀⢠⡾⣻⠀⣼⡾⠀⠀⢠⡟⠀⠀⣼⣿⠀⠀⣠⡾⠃⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠰⢾⣷⠚⢹⡿⠞⠀⢀⣿⠟⠁⢀⣿⡁⠀⣠⡟⠁⠀⠀⣿⡏⠀⠚⠵⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣼⡇⢸⣇⣀⣴⠟⠻⣦⡴⠋⠈⠛⠛⠉⠀⢠⡶⣶⡿⠀⠀⠀⠀⠀⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠘⠛⠀⠀⠉⠉⣠⠖⢦⡀⣴⠲⡄⢰⡋⢦⢀⡿⠀⣼⣧⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣤⣤⣶⠶⠒⠀⠠⡏⣠⢿⡇⣿⣅⣧⣼⠷⡾⠛⣧⠞⠁⣿⡄⣠⡿⠁⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⢀⣴⢞⠁⠀⠀⠉⢁⣼⡏⣿⣴⠃⠻⡴⠃⠀⠀⠀⠀⠈⠛⠋⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⠀⠀⣰⠏⠀⠀⠀⢠⡏⢸⡇⠀⠀⠀⠀⠀⠀⣠⠿⣆⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⣀⣴⡇⠀⠳⠞⠀⠀⣴⠿⣦⣤⠾⠃⠀⠈⠛⠓⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠈⠻⣤⣄⣤⡴⠞⠁⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⠀⠀⠀⣀⣀⣤⣤⣶⠆⠀⢀⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣠⣤⣶⣾⠿⠟⠛⠉⠁⠀⣀⣴⢟⣋⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠟⠋⠉⠀⠀⠀⢀⡤⡄⠀⠀⠙⢚⣉⠉
⠀⠀⠀⠀⠀⠀⢀⣤⡶⣿⠋⠉⠀⠀⠀⠀⠀⠁⠀⢀⡾⢁⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠉⠁⢀⡿⠀⠀⠀⢀⡀⠀⣰⢿⢀⣾⢁⠎⡘⠁⡀⢀⣤⠀⣴⠟⣷⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡴⠆⣰⢏⡝⢰⣯⠎⣸⡷⠋⣸⠃⣼⡣⢱⢃⣼⣣⢎⣿⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠚⣿⠁⠀⣠⡿⠊⠀⣿⠃⢠⡿⢀⠜⣯⠎⠻⠁⠘⡋⠉⡥⢺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣃⡠⠞⢿⡴⠋⠛⠋⠄⢀⡀⠀⠀⣠⡇⣼⢀⡿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⡀⠉⠀⣀⣤⣄⠀⡴⡙⡆⣯⢸⢠⠞⣽⠀⠘⠚⠠⣍⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢋⠄⠀⣼⠋⠀⣽⠇⣿⢡⢷⡟⡞⢻⡜⣿⠀⢠⠆⠰
⠀⠀⠀⠀⠀⠀⠛⠋⠁⣰⠏⠀⢸⠇⣠⠊⣾⡴⣇⠞⠘⠟⢁⡀⠀⠙⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⠘⠖⠁⣠⣿⠀⠉⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡼⢱⡇⠀⠈⠀⠀⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢖⣎⣆⠀⠀⠀⠀⠀⠀⠀⢀⠔⣩⢟⡶⠶⠖⢶⡨⣕⢄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣱⡲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡼⠾⠟⡸⠀⠀⠀⠀⠀⠀⠀⠇⣾⡾⠋⠀⠐⠒⠀⠙⢷⣷⢣⠀⠀⠀⠀⠀⠀⠀⢅⠻⠷⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠒⠒⡃⠀⠀⠀⠀⠀⠀⢸⣸⣿⠠⠶⠆⠀⠀⠰⠶⠄⣿⣧⡆⠀⠀⠀⠀⠀⠀⢸⠒⠒⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠇⠀⠀⠀⠀⠀⠀⣼⠀⠟⠀⠉⠂⠀⠀⠐⠉⠁⠻⠀⣷⠀⠀⠀⠀⠀⠀⡸⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡀⠀⠀⠀⠀⠀⡇⠉⢇⠀⠀⠀⠀⠆⠀⠀⠀⡸⠉⢸⠀⠀⠀⠀⠀⢀⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢣⣀⠀⠀⠀⠀⠇⠀⠈⢦⠈⠿⣉⣉⠿⠁⡰⠁⠀⡘⠀⠀⠀⠀⣀⡜⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠉⠉⠐⠒⠚⠂⠠⠤⠽⢶⠠⠤⡶⠮⠤⠤⠐⠃⠒⠂⠉⠉⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢄⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡠⠤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⡤⠤⠤⠤⠤⠤⢀⣤⠒⠛⠒⠒⠲⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠓⠢⠤⠤⠄⢼⠀⠀⠀⠀⢠⠦⠇⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢀⠊⠑⡀⠀⢀⠀⢸⠀⠀⠀⠀⢸⠙⡇⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠈⣖⣲⠇⠀⠀⠀⢸⠀⠀⠀⠀⣸⣲⠁⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⢰⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠁⢸⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⢤⠞⠀⢘⠦⠤⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⢴⡃⠀⠈⠲⡤⢞⠀⠀⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠼⠂⠉⠒⠒⠠⠗
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣄⡼⠉⠉⠙⢦⠀⠼⠋⠙⢢⡈⣡⠞⠉⠉⠙⡆⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⡀⠀⠁⠀⣿⠀⠘⠀⠀⠀⠀⠀⠻⠃⠀⢠⠇⠀⠛⠋⠁⢀⡈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣈⠻⠆⠀⣸⡿⠀⣤⢆⡴⠛⡆⣠⣠⣤⢀⣿⡴⣲⠀⣠⠾⣋⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠹⠇⠀⢠⣿⠃⢸⡏⠘⢧⠜⣿⠏⢻⡞⣾⣟⠋⠁⠀⠈⠉⠉⠀⡤⠞
⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⣠⠶⠶⢶⣿⣏⠀⠈⠛⠛⠁⠀⢿⡴⠋⢸⠏⠙⠷⣶⣤⣤⣴⡶⠀⢹⠄
⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⡁⠀⣠⣿⠋⠛⠷⣶⣶⠞⢃⣤⡀⠀⢀⠀⠀⣴⠀⠀⡉⠉⠉⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⠙⠛⠋⠀⠀⠀⣠⠶⢒⡆⢻⣤⣹⣰⢯⣇⡴⣿⠀⠀⡄⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠀⠀⢀⠯⠖⢉⣧⢾⣫⠏⢯⠞⠙⠗⠙⠶⠞⠁⠄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠂⣰⢋⡇⢠⢋⡇⠈⠁⠀⢠⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡁⢠⡇⠘⢁⣌⠚⠀⣴⣶⠀⣿⠏⠀⢠⡆⠀⢰⣿⠀⢀⣴⠆⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠐⢺⢗⠋⡿⠞⠀⣸⡟⠁⣠⣯⣀⣠⠟⠀⠀⣼⡇⠐⠫⠗⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⢸⡼⠀⠷⡤⠞⣉⠳⠚⡁⠈⣉⡁⠀⣴⠛⣿⠀⠀⢠⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣀⣁⣠⠀⢀⡞⢩⡇⢺⠍⡆⣧⣸⢼⣇⡼⢿⡀⢀⡿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢋⣭⠖⠀⠈⠓⢁⣿⢾⣹⠋⣧⠏⠀⠉⠀⠘⠷⠟⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡄⠈⡽⠃⠀⠀⢰⠋⡿⠈⠁⠀⠀⠀⣠⢶⣄⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣠⠴⣇⠈⠛⠁⢀⡜⠳⠶⠞⠁⠀⠉⠛⠛⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠙⠳⠶⠚⠉⠀⠀⠀⠀⠘

⡀




⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠿⠿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠉⠀⢀⣠⣄⠀⣤⣄⡀⠀⠒⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠁⢠⡶⢰⣿⣅⣹⠀⣿⣋⣹⡆⣷⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠘⣿⡟⢸⣿⣍⣭⠀⣿⣏⣩⡄⣿⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠙⠁⠀⠉⠛⠉⠀⠙⠛⠋⠁⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣾⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣿⡇⣴⣞⣶⡄⣴⣶⣶⣄⣠⣾⠓⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣿⠷⣿⣿⣿⠷⣿⣟⣿⠟⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣬⣩⣭⣤⣭⣭⣥⣤⣬⣭⣥⣤⣤⣬⣉⣭⣥⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠇⠀⠀⣀⣀⠀⠀⣀⣀⡀⠀⠐⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠿⠛⠟⠛⠛⠙⠻⠛⠛⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⢠⣶⠀⠀⠀⠀⠀⣠⠀⠀⠀⣀⠀⣄⠈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢨⡃⣸⣟⡿⢰⣿⣿⢰⣟⣿⠙⣧⣿⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄⠁⠙⠛⠃⠘⠷⠛⠀⠛⠛⠣⠬⢏⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣀⡀⢀⣤⣴⣶⣦⣀⠀⣀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠹⣿⣿⣿⣿⣿⣿⣿⡿⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠃⣼⠟⡿⠃⠙⠁








⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣭⣭⣭⣬⣭⣽⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣼⣷⣤⣼⣿⣿⣥⣯⣯⣯⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⠚⠛⠛⠛⠛⠓⠒⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣠⠖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⣴⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⣴⣾⣿⣷⣦⡀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣀⣀⣀⣀⣀⣀⣤⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠤⢤⣤⣤⣤⣤⡤⠤⠶⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠆⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣐⠀⣀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⠀⠆⠀⠀⠌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠡⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠰⠁⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠀⠠⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠉⠒⠢⣄⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣀⠖⠒⠉⠉⠉⠁⠒⠢⢌⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠔⠁⠈⢀⡀⠀⠁⠀⠀⠀⠀⠈⠢⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣽⠢⠀⠑⢄⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡇⠀⠀⡤⠀⠀⠤⠄⢀⣠⣾⡇⠀⠀⠀⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠻⣿⣶⡶⠿⠛⢻⡟⠀⠀⠀⢰⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠉⠻⠄⠀⠔⠋⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠶⠤⠤⠤⡤⣤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⢠⠃
⡄⠀⠀⠀⠀⠀⠀⡀⢀⡀⠠⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢀⡎
⠱⡀⠀⠀⠢⡀⠀⠀⠀⠀⠑⠀⠀⢄⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠘⢎⠀⠀⠈⠂⡤⢆⠀⠀⣀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠑⠦⡀⠄⠀⠀⡀⠀⠀⠁⠂⠤⣀⠈⠢⢄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠁⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠒⠦⠤⡄⠀⠀⠀⠀⠀⠀⠈⠶⠥⣄⠀⠀⠀⠢⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⢄⠀⠀⠈⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡸⠁⣀⠀⠀⣀⠜⠀⠀⠈⠢⣀⠌⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣀⠁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠢⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉⠉⠁⠹⠖⠂⢤⡀⠀⠀⠀⠀⠀⠀⠠⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠑⢦⣀⡤⠞⠉⠱⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠰⡎⢀⡠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠖⠳⠎⠀⢠⠈⠑⠲⠦⢤⣤⡤⢀⠀⢤⡤⢤⡄⠀⠐⣠⡿⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠐⣇⡠⡏⠀⠀⢸⠃⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⡜⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣤⣤⡤⠾⣴⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠉⠁⠀⢀⡜⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠙⢧⡀⠀⠒⠒⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠉⢰
⡀⠀⠀⣠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡐⠰⠆
⠀⠀⢠⡼⠆⠀⠀⡀⠀⠀⠀⢀⡀⠂⠀⠠⠤⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠄⢠⡇⠀⠀⠈⠉⠀⠀⢀
⠈⠁⢠⡜⡀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠲⠲⢤⣌⣉⠒⠠⣄⣀⡙⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⣀⠠⠤⠒⣊⣤⠖⠁⠙⢦⣀⠀⠀⡄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠤⣌⣉⠓⠚⠛⡿⠛⠛⠛⣿⠟⠛⠛⠉⢀⣀⣀⣀⠠⠤⠔⠒⠊⠉⠀⠀⠀⠀⠀⠈⠁⠘⠧⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡙⠀⠈⠙⠉⠁⠴⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠒⠒⢄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⢀⣀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩⠏⠀⢰⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠲⠶⠶⠶⠺⠟⠋⠁⠄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⢀⡁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⢨
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⠤⡀⢠⠙⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⣼⡀⠀⣬⠀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢩⠀⣠⠚⣣
⡇⠘⡁⠀⢷⠄⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣈⠷⠁⡰⠉
⠃⡀⠀⠀⠠⠠⣤⣄⣡⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣴⡇⠀⠛⠀⠼⠁⠀⢀
⠀⢡⠀⠀⠀⠀⠀⠈⠁⢠⠄⠼⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠈⠡⠀⢀⣀⣠⡀⣠⡀⠀⠀⠀⠀⠀⠙⠶⣀⠈⠃⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠳⣀⠀⠀⠀⣀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠈⠛⠀⠀⠋⠀⠀⠠⠟⠀⢹⠀⠀⠀⠀⠀⠀⠀⣹⠀⠘⠳⠄⠀⢀⡔⠦⢀
⠀⠀⠀⠈⠛⡦⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠤⣄⡀⠀⠀⠘⢦⡀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⢰⡄⠸⣧⠀⠀⠀⠀⠀⠀⠀⢴⣆⢀⣀⠠⡤⣤⢤⡄⣀⡀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⢀⠌⠁⠀⠖⠋
⠀⠀⠀⠀⠀⠘⡇⠀⢹⡆⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠲⣄⠀⠀⠀⠀⠻⣄⠀⠀⠀⣰⠋⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣷⠀⠀⢀⡤⠊⠁⠀⢀⣄⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠙⠀⠀⣴⠁⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠈⣦⠀⣀⣧⣠⠛⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⡆⠀⠀⣄⢠⡜⠁⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢸⠛⠉⠉⢨⡏⠀⠀⠀⠀⠀⠀⠐⠀⠈⠃⠀⠀⣀⡤⠞⣋⠀⠀⠀⢰⠀⠀⣿⡏⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⡘⠁⠀⠀⠀⣤⣄⠤⡤⢤⣤⡤⠚⠛⠉⢀⣴⡿⠀⠀⠀⠘⠓⣤⣿⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⡇⠀⠀⠀⠀⣿⣿⡀⠀⠀⢀⣀⣤⡴⠾⢿⣿⠃⠀⠀⢠⣄⢀⡼⠋⠙⢶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢠⡄⠀⠀⠀⢹⣿⣿⣿⣿⣿⢋⣀⣀⣀⣼⠃⠀⠀⠀⢠⡟⠁⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⢣⠀⠀⠀⠀⠙⢿⣟⠉⠀⠀⣀⡤⠏⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠷⣀⠀⠀⠀⠀⠀⠈⠛⠁⠁⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠙⠳⢤⣄⣀⣀⣀⣀⣀⣀⣠⣤⡶⠛⠉⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⡏⠉⠉⠉⠉⠉⠀⠲⠶⠒⠦⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠶⡶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠉⠠⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢣⡀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⢀⣀⣀⣀⣀⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠉⠉⠉⢛⠉⠉⠀⠘⠒⠒⣶⠶⠤⣤⣤⣤⡤⠔⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠸⠃⠀⠀⠀⠀⠀⠘⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⡆⠀⠀⠀⠀⠀⠀⢀⡦⠀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠉⠉⡷⠀⠀⠀⠀⣀⡤⠋⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⣀⣠⠔⠊⠁⠀⠀⠀⠀⢿⡀⢠⡄⠒⠉⠀⠀⠀⢀⢼⡇
⠀⠐⠋⣀⠀⠁⠙⣀⠀⠀⠀⠀⢀⡼⠶⠊⠙⠂⡀⠀⠀⠀⠀⡰⠃⢠⠃
⠀⠈⠲⠄⣀⣉⣉⣉⣉⣉⠉⣰⠛⠤⢀⡀⠀⠀⢡⡄⢀⣀⠐⠷⢀⡙⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣠⣤⣤⣤⣬⣥⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⠻⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣟⣀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠲⣄⠀⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⡀⠀⢨⡛⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠓⢄⠀⣧⡉⣷⣾⣧⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢸⠀⠘⢳⡆⠀⣤⣿⡇⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢆⢸⣸⣧⡰⡶⣏⣳⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠠⣄⣦⢀⡈⢦⣀⣨⣯⣿⣯⣿⣿⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⣻⡄⡤⣀⣻⢿⣷⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣤⡟⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⢈⣷⣠⣿⣴⣤⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣀⠀⣤⣦⣤⣠⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢚⠉⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⡟⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣄⣀⣀⡾⠀⠟⣴⣿⣯⠤⢤⣄
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠛⠛⢻⣄⠀⢉⣡⠀⢀⡶⢸⠏⢀⡶⢀⡟⠻⣦
⠀⠀⠀⠀⠀⢰⡏⠀⠀⣠⣶⣿⠇⣼⠟⠛⠛⡇⠀⠚⠁⡄⠀⢨⣴⠿⢃⣴⣿⠁
⠀⠀⠀⠀⠀⢸⣇⢠⣾⣟⣡⡏⣸⠏⢀⣴⣿⣷⣤⣴⣾⣿⣤⣤⣴⣾⣿⣟⣡⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⣻⠟⠋⠉⠉⣀⠉⣀⠛⢁⣴⠀⣸⠃⢈⡤⢀⡉⠀⣡⠄⢈⠁⢀⡟⠀⣾⣇⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣤⣴⠞⠀⣿⠀⡟⠀⠰⠃⢰⡏⢠⡿⠃⢨⠁⣼⠟⠀⡍⠀⠞⠀⠘⣉⠀⠉⣿⡄
⠀⠀⠀⠀⣿⠙⠶⠟⠋⠀⣸⡇⠀⢿⣤⣴⣦⡔⠁⣦⣤⣶⠇⢰⣤⣴⣾⣧⣴⠎⣰⣿⠟⠻⢷⣿⠇
⠀⠀⠀⠀⠹⣷⣄⣀⣠⣾⣿⣧⣀⣀⣿⡍⡿⠀⢸⣿⢉⡏⠀⣿⠃⠘⠷⠟⠃⣰⣿⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠙⠛⠛⠋⠘⣷⣤⣿⠇⣿⣄⣰⣿⣆⣀⣀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠋⠙⠛⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⡠⠖⢚⣉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡴⠀⢠⡄⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠖⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠰⠁⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⡆⠀⣀⣤⡀⠀⠈⢦⠈⢢⡀
⠀⠁⠀⠀⠀⢿⡉⢢⡀⠀⣠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠷⠄⠀⠠⡶⠿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⢠⣇⡼⠋⣰⠃⠀⠀⠀⢧⠀⢳
⠀⠀⠰⣟⠓⢮⣳⣄⠱⣼⡇⣠⠏⠀⠀⠀⠀⠀⠀⢀⡾⣫⡴⠆⠀⠀⠀⠀⠀⠀⠀⣀⡉⠻⣆⠀⠀⠀⠀⠀⣞⢉⣉⡙⢿⣄⠀⠀⢾⡥⠶⣦⠀⠀⢸
⠀⠀⠀⠈⠓⢄⡙⠁⣠⠟⠁⢳⡀⠀⠀⠀⠀⠀⠀⣸⠇⢁⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡦⠹⣧⠀⠀⠀⠀⠉⠁⠀⡝⠀⠈⢷⠀⠀⡤⠖⠋⠀⠀⢸
⠀⠀⠰⣏⣉⣈⡁⠰⠃⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠙⢦⣄⡀⠀⠀⢀⣠⡴⠛⡇⠀⢹⡆⠀⠀⠀⠀⠀⠀⢷⡀⠀⠈⠁⣤⣉⣱⡄⠀⠀⠏
⠀⠀⠀⠀⢠⠞⣡⣄⣀⣀⣄⠙⢦⡀⠀⠀⠀⠀⣠⣿⡀⢸⣇⠀⠀⠈⠉⠛⠋⠉⠁⠀⠀⣿⠀⢸⡷⣤⡀⠀⠀⠀⣰⠋⢠⡷⠒⠋⠁⠈⠉
⠀⠀⠀⠀⠘⠋⠀⠈⠉⠉⠈⠳⣄⠙⢦⡀⠀⠀⣧⣂⡀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⢀⣀⣸⠇⠀⢠⡞⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠙⢦⠀⠈⣿⠀⠀⢿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠀⠀⣿⠁⢀⡴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠳⢄⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣿⣠⠞⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠈⢧⡈⠉⠙⠛⠛⢛⡿⠃⠀⠀⠀⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⡀⠈⠻⢦⣤⣤⠶⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠙⢦⣀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⠦⣄⣀⠉⠛⠛⠛⠋⠁⠀⣀⣠⡴⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣤⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⢦⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⢿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣀⣼⠃⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⢷⡀⠀⠀⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡿⠀⠀⢀⣠⠶⠛⠙⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠙⢦⡀⠀⣠⠏⣹⠛⠻⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⢻⡙⢦⣀⡴⠋⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠙⠲⠋⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡉⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠐⢦⡈⡷⡄⣾⡰⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⠙⡷⣧⠘⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢤⡴⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⣛⢦⣠⣾⣮⣭⣭⣚⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⡾⠛⣹⡟⣿⠙⡟⢟⠻⢿⣶⣦⣍⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠴⠊⠀⢀⡠⣄⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣴⣿⣏⣠⠞⢁⢣⠃⡧⠇⡈⠇⢀⣿⠻⣷⣍⠺⣾⡣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢯⡀⠀⠋⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⢾⡟⣿⡞⢏⡡⠞⠻⠽⠀⠈⠳⠃⠈⠋⣄⠈⠻⣳⣜⢷⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡏⣾⢡⣹⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠙⣿⠻⣧⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⢠⡏⢸⡇⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠈⠁⠀⡴⣂⠀⢹⡀⢻⡄⡇⠀⠀⠀⠀⠀⠀⡴⠉⠙⣦⠤⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⢨⠱⣾⠇⢠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠁⢸⡇⢸⢷⢸⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⢸⠀⣿⡀⠸⣌⡸⠀⠀⠀⢄⠀⠀⠀⢀⡆⠀⠀⠀⣠⡇⢠⣸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠙⣄⢀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠘⣠⡿⡏⠲⣀⠀⠀⠀⠀⠈⠓⠶⠖⠋⠀⠀⣠⠞⠁⡇⢸⣿⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⣦⣿⠁⡇⠀⠀⠉⠒⠤⣤⣀⣀⣀⢠⡤⠲⣿⡁⠀⠀⠘⠮⣷⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⡏⠀⠀⠀⢀⠴⠊⣿⠀⢀⡠⠴⠳⣄⣷⠙⠦⣀⠀⠀⠈⡶⢻⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡇⣿⡀⠀⠈⠳⠦⣼⣿⡦⠋⠀⠀⠀⠀⢹⡏⠓⠚⠉⠀⠀⡇⢀⠱⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⡇⡇⠀⠀⠀⢀⠇⡇⢥⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⡦⠈⢆⡃⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢸⣵⠃⠇⠀⠀⢀⡌⢸⠀⠸⡄⠀⠀⠀⠀⡟⢨⢣⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡜⢀⠇⠀⠀⠹⣀⣤⣄⣼⠁⢸⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⡜⠀⠀⠀⠀⡟⠟⠻⠿⠆⢸⡀⠂⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠈⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⡄⢰⢠⣏⠟⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠰⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠆⢸⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠶⡄⣄⠇⠈⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠖⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠚⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢧⣠⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⠈⠤⠵⠮⢤⣶⠦⢤⣀⠀⠀⠀⠀⣀⣀⣀⣀⠾⠤⣀⣁⣠⡇⠀⠀⠀⠀⠀⠀⣀⣧⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠈⠉⠉⠉⠉⠀⠈⠙⠿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠼⠁



⣿⣿⡿⠿⣿⡿⢿⢿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣵⣿⣵⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠶⠒⢶⣎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⣸⠟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠋⠉⠁⠀⠈⣿⠹⣆⡀⠀⠀⠀⣀⣀⣤⠞⠋⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⡞⠉⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠈⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⠙⢦⣤⣄⣠⣤⡴⠶⠋⠁⠀⠀⠀⢠⡄⠀⠀⠀⠀⣴⠂⠀⠀⠀⢸⣷⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆⠀⠀⠀⠀⠀⠀⠠⣄⣀⡤⠶⠚⠛⠛⠋⠙⠛⠛⠻⠦⣤⣀⠀⣼⠋⢀⣠⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢠⣀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠉⠻⣧⡀⠀⠈⢉⡴
⠀⠀⠀⠀⠀⠀⠀⣶⡄⠸⣿⡄⠀⠙⣦⠞⠁⣀⣠⣤⣤⣴⠆⠀⠀⠀⠀⠀⠉⠻⣟⠲⠄⠀⠈⠻⣦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠙⠁⠀⣴⣧⠶⠛⣩⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠸⣇
⠀⠀⢀⣠⡤⠶⠦⣤⡀⠉⠛⠶⣼⠃⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡤⠤⠦⣤⣀
⠀⢠⡟⠁⠀⠀⣀⣀⣽⣦⣄⣀⡟⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣀⡀⠀⠀⠙⢷
⠀⣿⠀⣠⡴⠚⠉⠉⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠙⠳⢦⡀⢸
⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⢀⣀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠙⢿
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠷⢦⣄⣀⣀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠶⣤⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⢀⡼⢷⡄⠀⠈⠛⢯⣙⠛⠛⠛⠉⢉⣥⠞⠉⠀⠀⣼⠃⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⣴⠏⠀⠀⢻⡄⠀⠀⠀⠙⠷⣄⣀⡴⠋⠁⠀⠀⢀⡾⠉⢳⡄⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡾⠃⠀⠀⠀⠀⠻⣄⠀⠀⠀⢀⡼⢿⡄⠀⠀⠀⢀⡾⠁⠀⠀⢻⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠹⣆⠀⣠⣿⣁⣀⣽⣦⡀⢠⡞⠁⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠁⠈⠉⠉⠀⠈⠳⠟⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⢀⣀⣀⣠⣤⣤⣤⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⣴⠞⠋⠉⠉⠀⠀⢀⣠⢴⡏⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⣷
⠻⣤⣀⣀⠀⠀⣰⠟⠁⢰⡇⠀⠀⠀⠀⠉⠛⠳⠶⢦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⢾⣏⣠⡴⠋
⠀⠀⠉⠉⠙⠛⠻⣆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠙⠶⣤⣀⣀⣀⣀⣀⣀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠈⠉⠛⠒⠲⠶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠿⠛⠻⠟⠛⠿⠿⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⡥⠄⠀⠀⠀⠀⠀⢤⡀⠈⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠘⣿⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠀⠀⣾⣷⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠉⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡐⠀⠀⢀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣤⡤⢴⠆⠀⠀⠠⠂⠀⠀⡠⠀⠁⣮⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⢌⣀⣠⡔⢈⡀⠠⣤⠐⠈⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⢹⣧⣼⡷⠀⠀⠀⣿⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠠⠐⢻⡀⠀⠀⠀⠙⠈⠁⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣂⠁⠀⢀⠠⠀⠐⠛⠊⡇⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠑⠂⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣶⣶⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠇⠀⢉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠰⠀⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡻⣽⣷⠇⠀⠀⠀⣷⣿⣫⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⡻⠃⠀⠀⠀⠀⠘⠿⠿⠛⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣟⣻⣿⣿⣿⣻⣿⣛⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢀⠀⠀⠀⢱⣠⠔⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠠⠙⡄⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣄⠐⢀⡜⠁⠄⠀⢀⠀⠀⡇⠀⠀⠀⠘⢆⠀⠀⠀⢸⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⢿⠟⣻⠟⡝⡟⢷⣶⣤⣀⣀⠞⠁⠀⢀⠀⠀⠈⠧⠀⠀⢈⣀⡀⠈⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡿⠋⢀⡬⠐⠋⠘⠁⠁⠀⠐⠦⠙⢷⣦⣀⠀⠈⠙⠂⠀⡠⠚⠉⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⣡⡟⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢿⣯⡣⡀⠒⠀⡼⠁⣰⠃⠀⠁⠀⠀⠀⢳⠐⠚
⠀⠀⠀⠀⠀⠀⠐⠟⡴⢡⣇⠛⠀⡀⠀⠀⠀⠐⠂⠀⠈⠀⢀⠀⠛⢸⣻⡷⣵⡀⠀⠱⡀⠹⣄⣀⣀⡤⠀⢠⠏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠹⡄⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢀⠏⠀⠇⠻⠁⠀⠀⣹⢶⠤⢍⡭⠤⠖⠁⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠑⠦⢄⣀⣀⡠⠴⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⠜⠃⠀⣀⠀⢰⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⠦⠤⠤⠤⠤⠔⠚⢿⢄⡀⠀⠀⣀⢀⡠⠊⠁⠀⠀⠀⡇⠀⠀⢇
⠀⢠⠊⢉⣚⠉⢹⠄⠀⢀⣀⣠⠔⠋⡽⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠑⣾⡷⠏⠀⠀⠀⠀⠀⠀⣃⣀⠀⠈
⠀⢿⢠⠋⢀⢱⠈⠓⡄⠀⠞⠿⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢱⣠⠞⠁⠀⠀⠀⠀⠀⣀⣴⡉⠵⠮⣗⠒⣄
⠀⣰⠛⢻⠾⠊⡤⠞⠁⠀⠀⠀⠀⠘⠓⠋⢹⡏⠉⠉⠉⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⣅⢎⣥⡼⢴⠃
⠀⠈⠋⢉⣓⣚⣆⡴⣦⣀⡠⠤⠤⠒⠒⢃⣸⠇⠉⠉⠁⣇⡀⠉⠙⠓⠒⠒⠦⠤⢄⣀⠉⢱⣌⡹⠯⣀⡼
⠀⠀⠀⠈⠓⣮⠿⠛⠉⠁⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⡀
⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢘⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠋⠔⠋⠄⡰⠀⠀⠀⠂⠠⠀⠀⠂⠱⠘⠀⠶⠂⠋⠇⠃⡇⠱⠀⠀⠰⠂⠁⠀⠀⠴⠀⠆⠀⠸⠉⠱⠋⠄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠿⢿⢻⠿⡿⢻⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠠⡄⠀⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠺⠳⢾⡆⠰⠛⡠
⠀⠀⠀⠀⠲⡀⠘⡀⠠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠠⣴⣀⢀⡀⠀⠰⠈
⠀⢰⠀⠀⡀⠑⠄⠀⠄⣀⠀⠀⠀⢀⠄⢀⠔⠀⠀⠈⠑⠒⠀⡀⠉⠉⠉⠆⠀⡀
⠀⠀⠑⢄⠈⠢⣈⡄⠘⠋⠀⢰⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠙⠶⠈⠀⢠⠄⠉⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⢦⡀⠂⡀
⠀⠀⠘⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠙⠦⡄⢀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢙⡷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠘⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⠉⠀⠀⠊⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢂⠀⠀⠀⠀⠀⠀⠀⡰⠋⠑⠶⠀⠉⠉⠉⠛⠻⠶⣿⡄⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⢠⠀⠀⢀⡠⠆⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⢦⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⡄⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠔⠋⠀⣼⣷⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢰⠀⠘⢷⠒⠂⠐⠒⠒⠊⠉⠁⠀⢀⣤⣾⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⢻⠀⠀⠀⢧⣤⣄⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⡷⠀⣀⡀⠀⠀⣀⡀⠐⠈⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⣿⠿⢿⣿⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⡀⠠⠿⢻⡟⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠈⠓⠴⠂⣀⣀⣀⠴⠋⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠐⣂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠂⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠷⣤⣤⣀⣀⣠⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⣤⣤⣤⣠⣤⣤⠴⡾⠋⣀⠀⠷⠈⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢻⡅⠀⠀⡇⠀⠈⠁⠶⠂⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⢄⣀⣀⠀⠀⠀⠀⠀⢀⣀⠴⠛⠁⠀⠘⠀⢆⣤⠁⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠁⠈⠁⠀⠀⠀⠀⠀⣦⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⡀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠈⢛⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠘⠃⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⡀⠀⠀⠀⠀⣠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠛⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠸⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢠⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣄⠀⠀⠀⡸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⣤⡞⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⠳⠶⠞⠁⡇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⠴⠞⠉⠀⠻⣆⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢄⡴⢾⠁⢠⣼⣁⣀⣀⣀⣀⠀⡴⢶⡞⠉⠁⠀⠀⠀⠀⠀⠀⠘⣧⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢇⠀⠁⠀⠀⠀⠀⠀⠀⠄⢐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠡⣀⠀⠀⠀⠀⠀⢠⠂⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠓⢦⣤⣤⡴⠋⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣤⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠋⠁⣰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⣤⣿⣷⣾⣷⣶⣶⣄
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⣼⣿⡏⠀⠀⠀⠉⠉⠈⠙⡇⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⠿⣿⣿⣿⣆
⠀⠀⠀⠀⣿⣿⠟⠋⠛⠛⠿⢿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠏⠋⠉⠈⠁⠀⠈⢻⣿⣿
⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠘⣿⡇
⠀⠀⠀⠈⣜⠀⠀⠀⠁⡀⢀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠗⠎⠀⠄⠀⠁⠄⠠⠀⠀⠀⠀⠐
⠀⠀⠀⢠⣿⣦⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠤⠠⡉⠈⠀⠀⠀⠀⠀⠀⠀⢤⡀⣀
⠀⠀⠀⣸⣿⣿⣿⣿⠗⠐⢿⣿⣷⠀⠀⠀⠂⢀⠄⠹⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡛⠃⠓
⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠈⣿⣿⠇⠀⠀⠀⠀⠀⠈⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢎
⠀⠀⠀⠈⠉⠉⢱⠀⠀⠀⠀⢈⠁⠀⢠⠀⠀⠀⠀⠀⠈⠩⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠑⠁
⠀⠀⠀⠀⠀⠦⠲⠀⠀⠀⠀⠸⠲⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢒⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠥⡄⠈⢴⠀⠀⠉⡙⠀⠀⠀⠀⠈⡤⠏⠘⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠠⢺⠇⠀⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣾⣟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠘⠻⠷⠶⠶⢺⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣠⡤⣄⠀⠀⢀⣴⠄⢀⣴⠆⠀⢠⣾⡟⢀⣤⠀⠀⣠⡀⢀⣾⣏⡼⠁
⠀⢀⡴⠶⠶⣤⡀⠀⠀⢀⣿⡏⣴⣿⡁⠘⡇⢀⣾⡟⠀⣼⡏⠀⣰⣿⠏⢠⣿⠏⠀⣾⡟⠀⣼⡿⠋⠀⡀
⢰⡏⠀⠀⠀⠸⡷⠀⠀⣼⡿⠀⣿⡏⠛⣾⠗⠻⣿⣀⣴⣿⠣⢶⣿⠷⠖⢾⡏⣠⣼⣿⢀⣼⡿⠀⠀⡼⠁
⠸⣇⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⠛⠚⠋⠀⠀⠉⠉⣼⡏⢠⣿⠋⣇⠀⠈⠛⠉⠀⠛⠋⠸⣧⣠⠞⠁
⠀⠈⠛⠶⠶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⣴⠋⠉⢉⣿⠟⠲⢾⡟⠋⣿
⢰⣇⠀⡴⣿⡄⣰⢻⣷⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠸⣧⣠⡏⠀⠀⠀⠀⣼⣄⡀
⢸⣿⣸⠃⣿⣷⠇⢸⣿⠀⢀⣤⡄⣤⡄⠀⠀⢠⢤⠀⠀⠀⠀⣨⠉⠀⣤⠀⠀⢠⣿⣿⡟
⢸⣿⡏⠀⣿⡿⠀⢸⣿⢠⣿⣣⠇⠙⡿⢤⣤⠈⢻⠦⣤⡄⠀⣿⠀⢠⣿⡆⢠⠈⣿⣿⣿⠆
⢸⣿⡁⠀⣿⡇⠀⢸⣿⢸⣿⠁⠀⡰⠁⣼⡿⢀⠏⢠⣿⠇⣸⣿⣄⡞⣿⡷⠋⠘⠻⣿⡏⣤⣀⣤⣦⣄⣀
⠈⠛⠁⠀⠀⠀⠀⠀⠋⠀⠙⠛⠋⠀⠀⣿⣧⠞⠀⢸⣿⡴⠁⢙⣋⢤⣿⡇⠀⠰⣿⠿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⣠⠖⠚⢻⣦⣀⣀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⠀⠀⠀⢰⡋⢀⣼⡿⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⡉
⠀⠸⠥⠂⣠⡿⠋⠉⣸⡷⠀⣰⡿⢡⠇⠀⠀⠀⠀⣀⣀⠀⠀⢙⣛⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⢀⣀⣼⡟⠁⠶⣾⡋⠀⢰⣿⡵⢃⣴⡾⢛⡆⠸⣿⡀⠁⠀⢿⣄⠉⠀⢠⣴⠞⢳⠀⢀⣴⠶⣾⡏
⡼⠉⢸⣿⠙⣦⠀⠈⣿⢀⣿⠋⠀⣼⡟⠓⠋⢡⢞⣘⣿⣆⡠⣋⣻⣷⣠⣿⠟⠚⠋⣰⣿⠃⣸⡟⢀⡖
⢧⡀⠀⠓⠀⠟⠀⢠⣿⢹⡇⢀⡼⠙⠧⡤⠖⠋⠯⡥⠿⠃⠸⢭⠼⠟⠀⠻⢦⠤⠚⠙⠷⠖⠻⣧⠞
⠈⠛⢶⣤⣤⣤⣾⠟⠃⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣧⠀⠀⠀⢀⡀⢰⠇⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠰⡆⢽⢿⢀⣀⡀⢸⠅⡿⠀⡿⠀⠀⠀⢀⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⣆⢠⣀⡄⢻⢸⣿⣸⣿⣯⣾⢰⠃⢰⣇⣤⣦⢄⣾⡟⣠
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡙⣟⣎⣻⣾⣿⣤⡏⣿⣿⣿⢷⡎⣶⣼⣿⣿⣯⢞⢟⡼⠁⠀⠀⠀⢀⡄
⠀⠀⢀⣶⣄⡀⢀⣤⢆⣷⣜⣎⢻⣿⣿⡜⡇⢸⣿⡟⡿⢠⡇⣸⣿⣿⠏⣼⣾⣶⣷⡶⣠⣴⣯⠞
⠀⠀⠀⡙⠻⣝⡮⣝⣿⣿⣿⡻⣇⠹⣿⠇⢣⠸⠉⠀⠃⠸⠁⠜⣿⢃⡞⣽⣿⣿⣯⢞⡵⢋⣡⠆
⠀⠀⠀⠉⠓⢦⣙⠮⣙⠿⣿⣿⡊⢦⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠎⠸⣿⠟⠫⢖⣥⣶⣩⣀
⠀⠀⠀⢀⣼⣶⣿⣿⣟⡦⣌⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣤⣾⣿⣿⡿⢛⣀⣤⣴⣶
⠶⢤⣶⣶⣬⣽⣿⣿⣿⣿⡂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠿⠿⣛⣻⠭⠿⠚⣋⣉
⠀⠰⠦⠤⢭⣭⣉⡛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣏⡍⠁
⠀⠀⠀⢰⣾⣿⣿⣿⣿⣶⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣿⣿⣿⣿⠿⠿
⣀⣀⣠⣬⠭⢽⣿⣿⣿⣿⡻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⣶⠦⢶⡗⠞⠯⠽⢿⣿⣿
⠹⠶⠛⣛⣋⡭⣵⣿⣽⣶⣷⣦⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⢿⣿⣿⣿⣿⣏⡛⠒⠲
⠀⠀⠈⠉⠀⢲⣿⣿⣿⡿⣋⣠⢖⣡⣀⢄⠀⠀⠀⠀⠀⠀⠀⡄⠠⣄⣬⣝⢦⣬⣛⠿⣛⠿⠇
⠀⠀⠀⠀⠀⣈⡵⣾⠵⣪⣿⣵⣿⣿⢫⢎⣰⣴⣀⡄⡇⢲⣠⣜⢦⠘⢿⣿⣷⣿⣦⣝⠻⣽⣲⣄⣀
⠀⠀⠀⠴⣿⠷⢋⣥⠞⣹⣻⣿⣿⢷⣯⣾⣿⣼⣿⣧⢷⠘⣿⣿⣯⣿⢮⢿⡿⣿⡷⠈⠳⠆⠉⠛⠉
⠀⠀⠀⠀⠁⠀⠉⠀⠀⠘⠟⣴⢋⡾⣿⣿⢇⣿⣿⣿⣿⢦⢻⣿⣿⡿⣦⠳⣵⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢡⡟⠀⠻⡿⢸⢼⣿⡟⢻⠸⡿⡜⡿⠛⠈⢧⡙⢮⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠀⠀⠀⠀⣿⠈⠁⠀⢸⡄⢷⣷⠀⠀⠀⠀⠀⠈⠁⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣏⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣆⠂⡀⠄
⠀⡀⠠⠶⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣸⣿⠼⡄
⠛⠱⠟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣷⠕⠿⣆⣦⣤⣄
⠀⠒⡀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⢀⡰⣦⣿⡟⣟⠋
⠐⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡶⢷⠉⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡎⣀⡏⣿⣿⣾⣾⣷⣾⣷⡴⠂⠴⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢂⠀⠶⠮⠁⠀⠀⢻⣿⣿⣿⠛⣻⣿⣿⠃⠀⠀⡉⠉⣴⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠚⠏⠁⣿⣇⣶⣿⢷⣿⠝⠀⠀⠀⠁⠀⠀⠈⠉⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⢈⣻⡆⠀⣀⠉⠀⠀⠉⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠐⠀⠀⠀⠒⠊⠥⣦⣾⠅⠀⠀⠀⠠⠉
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣆
⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⡀⠙⣹⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⡞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠁


⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣀⣀⣤⠶⢾⣿⠿⠿⣿⣶⣤⡀
⠀⢣⡀⣩⡧⣤⠎⠀⠀⠀⠀⡈⠻⣿⣦⡀⠀⠀⠀⠀⠀⢀⣠⠞⢉⡳⢤⣀⣠⣤⣤⣤⠤⢤⣀⡤⠾⡉⠙⢦⣀
⠀⡿⣹⣿⢁⡎⠀⠀⣶⣀⣀⢹⡀⠹⣷⢹⠓⢦⡄⠀⢰⠉⠀⢀⣞⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣸⣀⠀⠀⢱
⠀⡇⢹⠇⡜⠙⠉⠉⠀⠁⠉⠉⠱⡀⢹⡈⡇⡴⠁⢠⠊⠀⡤⠊⢿⠁⢀⠔⠉⠓⠒⠒⠒⠚⠲⡄⠀⡟⠻⣦⠀⠈⢢
⠀⣇⢼⡴⠁⠚⠙⠀⠀⠀⠚⠙⠂⠳⣤⣷⡏⠀⠀⢹⠄⢠⠏⠀⢸⣤⡎⠀⣀⡀⠀⠀⢀⣀⠀⠸⣤⣇⡴⠛⣦⠀⢞
⣠⠊⠀⣇⡀⠘⢗⡒⠾⠖⠒⡾⠀⣤⡏⠸⡄⠀⠀⠸⡄⢸⠀⠀⢧⡀⠀⠀⣁⣁⠠⠄⢈⣈⠀⠀⠀⡸⠀⠀⣳⠀⡞
⢧⠀⡴⣏⠓⠒⠬⡓⠦⠖⡞⠚⠉⣁⢧⠀⡽⠀⠀⠀⠳⠞⠀⠀⠀⣿⡷⣄⣘⢯⣉⣉⣡⣯⠤⢿⣿⠃⠀⠀⠙⠚⠁
⠀⠹⡀⠈⡷⠀⠀⢻⠛⣻⠀⠀⠸⡅⢀⣟⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⠳⡄⠈⢱⡬⢥⠏⠀⢰⠊⢹
⢀⡴⢳⡀⡇⠀⠀⠘⡦⡇⠀⠀⠀⣇⡞⠈⢧⠀⠀⠀⠀⠀⠀⢀⡜⢇⢠⠇⠀⠀⣇⣸⠀⠀⠸⣆⡎⢣
⠛⠠⠤⠽⣇⠀⠀⣸⣁⣱⡀⠀⣠⣏⣀⣀⣨⠇⠀⠀⠀⠀⠀⠛⠲⠼⢿⡀⠀⢀⣇⣹⡀⠀⣠⣏⣠⡤⠗
⣤⣤⣤⣤⣬⣵⣾⣥⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣶⣮⣤⣤⣭⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣤⠴⠖⢦⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡏⠀⠀⠀⠀⠙⡆⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣠⡀⡄⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠳⣄⠀⢀⠚⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣦⢾⡇⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⠶⠶⠶⠾⠛⠛⠛⠛⠶⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠒⠲⠶⣤⡀⠠⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡴⡞⠛⢲⣶⣌⠙⢦⡀⠀⠀⢀⡞⣉⠀⠀⣶⣶⣦⠀⠹⣦⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣾⢼⣿⣿⣿⣿⡿⠀⠀⣿⠀⠀⣿⢰⣿⣷⣾⣿⣿⣿⠀⠀⣿⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣧⠈⠷⠿⠿⠛⠁⠀⠀⣾⠀⠀⣧⠈⠻⢿⣭⡭⠟⠁⠀⠀⣾⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⠘⢧⡀⠀⠀⠀⠀⣀⣼⡏⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠈⢋⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣨⠁⠀⠀⠉⠛⠶⠶⠟⠋⢠⡇⠀⠀⠀⠈⠻⠦⣤⣤⣤⡴⠞⠁⠀⠀⠀⠀⠀⣉⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⣠⠶⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠰⣷⡆⠒⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⠀⢻⠓⠓⣠⠁⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⢉⣧
⠀⠀⢸⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠂⠀⠉⠉⠉⠀⢀⠔⠋⠹⣆
⠀⠀⠈⢻⡈⣷⠀⠀⠀⠀⠀⠀⢠⡶⠛⢦⣈⡙⠛⠒⠲⠦⠶⠖⠊⠉⠙⢲⣦⣤⡴⠚⠁⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠘⡅⢘⠀⠀⠀⠀⠀⢀⡼⠄⠀⠀⠈⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⢸⠇⠘⣦⠀⠀⠀⠀⢀⡤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢷⠀⡇⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣄⣀⣾⡁⠀⢹⣆⣀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠸⡆⣧⠀⣴⠋⠉⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣷⢸⣴⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠸⡀⠛⠀⢀⣤⡶⢿⠀⠀⠀⠀⠒⠛⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡇⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⡇⢠⡴⠋⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠳⠋⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠂⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠅⠀⠀⡸
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣴⣿⣾⣶⣶⣴⣷⣶⣶
⣿⣛⣻⣛⣛⣛⣛⣃⣙⣛⣿⣛⣛⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣯⣭⣭⣭⣭⣽⣿⣯⣭⣭⣽⣭⣭⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠂⠀⠀⠀⢀⠀⡀⡀⡠⢸⠠⡂⢠⠠⡀⡄⡂⠄⠀⢸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢖⠂⣆⠗⠁⠇⠀⠣⠸⠀⠚⠘⠀⠃⠃⠁⠀⠀⡏⢀⠇⠄⠐⢰⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⠤⠄⠤⢄⠈⠁⠀⠈⢁⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠎⠉⠀⠁⠈⠀⢀⣀⠀⠈⠑⠒⠖⠢⠤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠔⠉⠁⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠠⡉⡅
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡀
⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠐⠢⠭⢥⡝⠀⠀⠁⠀⣀⠖⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⠄
⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠷⣮⡍⠁⠈⢣⠀⢀⣀⠆⠁⠀⠈⠙⠔⠆⠤⠤⠦⠀⠀⢀⡀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠘⠀⠀⢸⢸⢀⠆⠋⠁⠀⠀⠀⢹⠉⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠉⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⢇⠀⢁⠏⡼⠀⠀⠀⠀⠀⠠⡘⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠃⢣⠀⠀⠀⠀⠀⢠⠕
⠀⠀⠀⠀⠀⠀⠈⢢⣀⠜⠀⠀⠶⠀⠀⠀⡀⡄⢂⠀⠀⠀⠰⡇⠀⠀⠀⠠⠦⠀⠀⠣⣤⢠⡀⡰⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡙⣦⣀⠀⠀⠀⠀⠀⢀⣠⠞⠓⢤⣀⠀⠐⠐⠀⠀⠀⠀⢀⡠⠔⠁⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠙⣿⠐⠒⢲⣙⠷⡀⠈⠑⠀⠓⠋⠁⠀⠒⠒⠂⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣠⡿⠃⠀⠸⠄⢠⣆⠀⠀⠀⣀⠀⣀⠐⠤⣦⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢄⠀⣈⠀⠛⠂⢸⠀⠉⠉⠂⠴⠗⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⢀⠄⠉⠁⠰⠦⢼⡀⢀⢀⠀⡀⡀⢄⡒⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠐⠃⠴⠻⠦⠐⠛⠒⠀⠀⢘⠁⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠠⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡃⠀⠐⠀⠀⢀⠀⡀⠐⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⡁⠀⢈⢩⡀⠀⠀⡀⣄⡄⠀⢈⡁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢴
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡆⣄⠁
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠡⠀⠀⠀⠀⢹⡀⠀⠀⠐⠂⠀⠀⢀⡀⠀⠸⡋⠁⠋
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠠⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠉⠃⠀⠘⠠⠤⢀⠄
⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⢀⡄⠀⠀⠀⢠⢆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⢠⡞⠃⠀⠀⠤
⠰⠒⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠮⠒⠃⠀⠀⠀⠁⠀⠀⠀⢰⣞⣵⣋⣒⠀⠈
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢀⡔⠊⠙⡐⠀⢰⡗⠞⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡤⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⣶⢖⡶⠶⠶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠃⠀⣠⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⣰⣿⣫⣶⠿⠛⠂⠀⠀⠀⠀⠐⠂⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⠟⠛⠀⠀⠒⠂⠀⠶⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⠟⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡶⠀⠀⣠⡤⡀⠀⠀⠀⠀⠀⠒⠫⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠐⢿⡿⢋⣵⠄⣦⣄⠀⠈⣧⠀⠀⠀⠀⠀⠀⣀⣀⡠⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠈⣇⠀⡀⠀⠀⠳⣈⢀⡴⠛⠉⠱⡄⠘⡑⠒⣒⣛⣋⡩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣽⠋⢀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡆⢹⣿⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠁⠘⠃⠁⢐⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠿⠓⡿⠀⠁⠀⠀⠀⠀⠀⢠⠊⠉⠳⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠄⠀⠀⡉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠚⠁⠀⡄⠀⠀⠀⠀⠀⣸⠄⠲⠖⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠑⢄⡠⠒⠀⠀⠀⡀⠀⡇⠀⢀⠔⠊⣀⠀⠀⠠⠂⢀⠴⠒⢀⡠⠆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣌⠁⠢⣀⣰⠉⡒⠀⣜⣡⣀⠈⠀⢃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⠎⠀⠀⠉⠙⠀⣡⣬⣌⢣⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⣗⢷⡙⠲⠦⠀⠀⠀⠖⢹⣻⠇⠀⠁⣴⠅⢀⡅⠀⠀⠀⣀⠤⠒⣶⠀⠀⠀⠀⠀⡠⠚⠉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣻⣆⣹⣟⠳⣤⠤⢤⡔⣊⢟⡀⠀⠀⠀⠑⠦⣄⣀⡠⠊⠁⠀⠀⣿⠀⠀⠀⡠⠊⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠹⣉⡙⡿⠿⠛⠉⠛⠃⠀⣀⡤⠖⠚⠛⠁⢀⣠⡄⠀⠀⠘⡦⢚⠾⠁⡨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡭⠍⢉⡀⠀⠀⣀⠠⠐⠚⠁⠀⠀⠀⠀⠔⠁⠘⠀⢀⡴⣊⠔⣁⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣠⠞⠁⠀⠀⠁⢀⡥⠚⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡠⠀⠀⠤⠀⠘⠈⠀⠀⠀⠀⠒⢦⡁⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠎⢸⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼⢸⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣠⠇⣼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡛⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⢀⣀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⣀⡀⠀⠀⣀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠉⠓⠶⠿⠟⠾⠿⠛⠲⠮⠛⠛⠻⠛⠛⠳⠂⠈⠛⠒⠲⠦⡄

⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠱⡀⣀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡄⠀⠆
⠀⠀⠀⠀⠙⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠎⠀⠀⠀⠀⠀⠑⠢⣶⣧⠀⠀⠀⠀⠀⠀⣢⣿⡤⡇⠀⠀⠀⠀⠀⢀⣿⠗⠊
⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⢼⣿⡀
⠀⠀⠀⠀⠸⠃⠻⠀⠀⠀⠀⠀⠐⢢⣼⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡇⠀⠀⠀⠀⠀⢸⡟⡇⠀⠀⠀⠀⠀⠀⠸⡜⡇
⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠎⠘⠄⠀⠀⠀⠀⠀⠰⠀⠡⠀⠀⠀⠀⠀⠀⠐⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠀⠀⣠⣀⠔⠁⠀⠀⠘⡀⠀⠀⢀⠄
⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠳⢄⣶⣀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠈⠙⢿⣶⠄⠀⠀⠺⣿⠁⠀⠀⠀⠀⠀⠳⣿⡴⠊
⠀⠀⠀⠓⢲⣾⠆⠒⠈⠀⠀⠀⠀⢿⡏⠑⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢀⣿⡄
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣰⢿⡟⠀⠀⠀⠀⠀⠟⢻⠀⠀⠀⠀⠀⠀⠀⡟⢹⠀⠀⠀⠘⠃⣇⠀⠀⠀⠀⠀⢸⠋⣷
⠀⠀⠀⠀⠚⠟⠃⠀⠀⠀⠀⠀⠸⠈⠛⠐⠆⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿
⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿
⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢊⣿⠖⠳⡀⡴⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⣠⣾⠃⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠘⣡⠿⡃⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠞⢹⡆⠀⠀⢷⣏⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⣠⣄⣤⣤⡾⢻⣦⠌⢀⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣟⣻⣿⡿⣿⣧⣾⡿⠋⣷⣶⡜⠁⠐⠁⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣉⣩⣭⡛⠿⣍⣁⠀⣿⠟⣿⣷⣴⠀⠀⠀⠀⠀⠆⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⢀⠞⣹⣿⣿⣟⣥⣶⣶⣤⣙⠛⠿⠷⠾⢿⣿⢿⣷⠂⠀⢀⣄⣀⡠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠸⣿⣿⣿⣿⣇⣾⣿⣿⣿⣶⣦⣤⡐⠚⢋⣈⣿⣶⣾⣿⡿⠋⠀⠀⡆⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⣾⣦⣽⣿⠃⣿⣿⣿⣿⣿⣿⠾⠿⣿⣷⣆⣉⣹⣿⡿⠟⠋⣡⣤⣶⠔⠀⠀⠀⣀⣤⣾⡟
⠀⠀⠀⠀⠀⣀⣼⣫⣾⣿⣋⣼⣿⣿⣿⣿⣿⡳⣄⠀⠀⢉⣉⣻⣿⣿⣷⣾⣿⣿⣿⣷⣦⣶⡗⠛⠽⠛⠁
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣷⣜⣦⣤⣄⠉⣻⣿⡿⢻⣿⣿⣭⣍⡙⠛⠭⠀⠀⠒
⠀⠀⠀⠀⠛⣿⠀⣸⣿⣿⣿⡿⠋⠀⢉⣀⣿⣿⣿⣿⣿⣯⣿⣿⣛⢃⡞⠉⠉⠁⠈⠁
⠀⠀⠀⢼⣿⣏⣴⣿⡿⠿⢿⡧⣄⣀⣹⣿⣿⣿⣿⣿⣿⠏⠿⢿⠛⢿⣯⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠻⠿⡿⠟⠀⠀⠈⠉⢈⣽⣿⣅⣠⣿⠿⡿⣿⣜⡇⠀⠀⠀⠈⠛⠿⢷⣶⣶⣶⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠚⢻⠟⠓⠧⣽⠿⡟⠤⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠃⠀⠀⠰⣿⠆⢰⠸⡄
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⢸⠁⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿
⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣀⣀⣉⣀⡐⠀⠐⠒⠮⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⢟⣛⣩⣭⣭⣛⣓⡀⠀⠀⠀⢮⣷⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣋⠵⣾⢿⡿⠟⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⢿⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⡼⠁⠈⠀⠀⠀⠀⠐⠒⠲⠄⠀⠀⠀⠀⠀⠀⠸⣷⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠠⠚⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠞⠁⠀⠀⢀⠤⡀⠀⠔⡡⠃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡤⣄⠀⠀⠀⠈⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣧⠃⠀⠈⢀⠡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠁⠀⢰⠚⠆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⡇⠀⠀⠀⠀⣄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⢸⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢙⠂⠤⠞⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣏⠀⠀⠀⠀⠀⠈⣏⠉⠁⠀⣠⠔⣻⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⢠⢂⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠘⢇⡉⣉⡠⣺⠏⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⡠⠚⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠉⠒⠚⠁⠀⠀⠀⠀⣠⠞⠁⠀⣀⠴⣌⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⡀⠀⠀⠀⣀⣠⡴⢾⠁⠒⠒⠉⣁⡠⠟
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⠝⠋⢋⡀⢀⡞⢰⠒⠊⠉⠁
⠀⠀⠀⠀⠀⡠⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠒⠒⠒⠉⣀⣤⡆⠀⠉⠁⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⡴⠂⠤⠤⠤⠤⠖⠚⠦⠔⠒⠋⠁⡼⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡒⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⠃⠈⠙⠲⠤⣀⡉⠑⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠒⠒⠲⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣦⠤⣄⢀⣀⣠⠋⠑⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⡭⢷⡄⠀⢀⣀⣀⣀⣈⠑⠲⠦⢤⣤⣀⣀⡀⠀⣀⡁⠀⠀⠀⠀⠀⢸⠀⡟⠈⠀⠀⠈⠉⠉⠉
⠀⠀⠒⠄⠀⠀⠀⠀⠀⡰⢮⡠⠚⠁⠀⠀⠛⠉⠉⠉⠁⠈⠉⠁⠢⢤⣀⡀⢀⠉⠉⠉⠉⠴⡦⠀⠀⠀⢠⠋⡰⠃
⠘⢌⠢⡀⠈⠑⠢⢄⡘⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡀⠀⠀⠀⣸⠣⠤⠔⣊⡭⠛⠁
⠀⠈⠱⣈⠢⡀⠀⠀⠈⢳⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⠏⠙⠋⠉⠁
⠀⠀⠀⠈⠑⢌⠳⢄⢀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢛⣫⣭⣭⣿⣿⣿⣿⡿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠃⢐⣛⣛⣛⣛⣭⣭⠵⢚⣴⣯⡻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠛⠛⠉⠉⠉⠉⣀⣠⣴⣿⠟⠘⡿⣎⢿⣿⡀
⠀⠀⠀⠀⠀⠀⣴⡿⠋⢀⣀⣤⣤⣴⣶⣶⣿⣿⣿⠿⠛⠁⠀⠀⢻⢻⣆⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢸⡟⢀⣴⣿⣿⠿⣿⣟⣛⠋⠉⠁⠀⢠⡴⠒⠒⠆⠘⡏⣿⣎⠃⠀⠀⠀⠀⠀⢀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⡇⣼⣿⠏⣴⡎⢉⣈⣉⠃⠀⠀⠀⠉⢠⣶⡶⣦⠆⢳⡘⣿⣆⠀⠀⠀⠀⠀⢸⠀⢠⣿
⠀⠀⠀⠀⠀⠀⢻⣿⡏⢸⣿⡟⢏⡹⡿⠃⠀⠀⠀⠀⠁⠉⠝⠃⠀⠈⢷⡈⠻⢷⣤⣀⣀⠴⠃⣠⢞⡏
⠀⠀⠀⠀⠀⠀⠀⢹⣷⢸⣿⣷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠛⢶⣤⣄⣀⣤⡴⠞⣡⠟
⠀⠀⠀⠀⢀⠄⠀⠀⢿⠘⣿⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⢿⣶⣤⣤⣤⣤⡶⠟⠁⠀⣀
⠀⡄⠀⠀⢢⠀⠀⢀⡟⠀⣿⣿⡇⣷⣄⠀⠻⣍⣩⡽⠃⠀⣀⣾⡇⢻⣇⠀⠀⠀⠉⠀⠀⠀⣰⣾⣿⠃
⠀⢱⡀⠀⠀⠙⠛⠉⠀⢠⣟⣾⡇⢸⣿⣷⣦⣄⣀⣠⡤⠚⠹⣿⣧⠈⢿⡄⠀⠀⠀⢳⡄⠀⠉⠉
⠀⠀⠙⢦⣀⡀⠀⢀⣤⣯⠞⢹⠃⢸⡇⣿⣿⣧⠀⠀⠀⠀⠀⠙⢿⣷⡈⠻⣆⠀⠀⠈⡇⠀⠀⡀⠀⢀⡄
⠀⢀⠀⠀⠈⠉⠉⠉⠉⠀⣠⠏⠀⣾⠃⣿⣿⡟⠀⠀⣀⡤⠶⠛⠉⠉⠛⢦⡈⢧⡀⢰⠁⢠⠞⣠⣶⣿⠁
⠀⢸⡀⠀⢀⡀⠀⢀⣠⡾⠃⠀⣼⠏⣸⣿⡿⠁⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠻⡄⠳⣘⢠⡏⣸⣿⡿⠃⡴
⠀⠀⢳⣄⠀⠙⠛⠋⠁⢀⣠⣾⠏⣰⣿⠟⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢻⠘⣇⣛⣁⡤⠞⠁
⢦⣄⡀⠙⠷⣶⣶⣶⣾⣟⣫⣵⡾⠟⠁⠀⣀⣠⠤⠤⠀⠉⠛⢶⣄⣇⠀⠀⠀⠀⠀⢀⡔⣡⡭⠤⠤⡀
⠈⠻⣷⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⢠⡶⠋⠁⠀⣀⣀⡀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⣸⠋⠀⢧⠻⣿⣦⡀
⠀⠀⠀⠁⢄⠰⡠⠚⠙⢲⡀⠀⣰⠏⠀⣤⠾⠋⠉⠉⠉⠛⢦⡄⠈⣿⠀⠀⠀⠀⡰⠃⠀⠀⠀⠑⠪⠝⠛
⠀⠰⠶⠶⠦⠑⢷⠀⠀⠀⠇⠀⣿⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠹⡄⡟⠀⣀⣴⣋⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠻⣄⠻⣦⣀⣀⡄⠀⠀⠀⠀⠀⢇⡤⠞⠋⠉⣀⣀⣀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠉⠛⠻⠛⠋⠀⠀⠀⠀⢀⠔⠉⠀⠀⣠⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣤⣀⣀⣀⣤⠔⠚⠁⠀⠒⠊⠉
⣿⣿⠛⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⠊⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⣀⣀⡀⠀⢰⡿⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢹⡿⠀⠀⠀⣰⡿⠋⠀⣠⠇⢀⣴⠟⠋⠉⡝⢀⣿⠇⠀⠀⠀⠀⣰⠆⠀⠀⣠⡄⠀⠀⠔⢲⡆⠀⢀⣠⡶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢰⣿⠓⠂⠉⠁⢀⣾⠋⠐⠀⠀⠀⣼⡏⠀⠀⠀⠀⣸⡏⠀⠀⢀⣿⠁⠔⠀⢀⣾⠃⢀⣾⠋⠀⢀⠜⣿
⠀⠀⠀⠀⠀⠀⠀⡞⢙⣿⣄⠀⠀⠘⣿⡀⠀⠀⡠⠊⢿⡀⠀⠀⡀⠀⣿⠃⠀⠀⠀⢠⣿⠀⡠⠀⢸⣧⠎⠀⠀⣾⠃⠀⣸⡇⠀⣠⠎⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢸⡿⠈⡆⠀⠀⠈⠙⠋⠁⠀⠀⠀⠙⠓⠋⠀⠀⢿⠀⠀⠀⡀⠀⠙⠋⠁⠀⠀⠁⠀⠀⢸⡏⢀⡔⠈⠛⠚⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠴⠊⠀⠀⠀⠀⠀⠀⣠⠶⠒⡄⠈⠛⠋⠀⢀⣀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⢠⠃⠀⠀⢀⡴⠋⠉⡧⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄⠀⠀⢀⡤⢄⡀⠀⠀⠀⢀⣤⠶⠶⣄⣰⠏⠀⢀⠔⠁⠀⠀⠀⣸⠀⠀⠘⠁⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⡳⣄⠀⣏⠀⠀⠀⠱⡄⢧⡀⠀⠈⢦⣀⢠⣾⠁⠀⠀⣼⣏⠤⠒⠁⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢀⠞⠀⣿⠀⠈⢱⡖⠐⠂⡿⢸⡏⠉⠉⢹⠁⢸⡇⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣤⡴⠋⠀⣠⣿⠀⠀⢸⣇⣀⡼⠃⠘⢷⣤⡴⠏⠀⠘⠿⡤⠞⢻⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠔⠁⢀⡿⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠛⠛⠻⢶⣦⡀⠀⣠⣶⠿⠛⠛⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠉⢿⣿⠋⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⡇⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⢸⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣿⡆⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠙⠿⣦⣄⣀⣴⡾⠛⠁⠀⠀⠀⠀⣀⣀⡀⠀⠀⢰⣿⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠈⠻⣶⣴⣿⠉⠙⢷⣦⡀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⢀⣴⡾⠋⠉⣿⣦⣶⠟⠁⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⡀⠀⠀⠈⠙⢿⣶⣄⠀⠉⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⣠⣶⡿⠋⠁⠀⠀⢀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠉⠻⣷⣄⠀⠈⠙⣷⡄⠀⠀⠀⠀⢠⣾⠋⠁⠀⣠⣾⠟⠉⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠉⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠉⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣿⣇⡀⠀⠀⢀⣸⣿⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⢻⣿⣿⣿⠛⢻⣿⠀⠀⣿⡟⠛⣿⣿⣿⡟⠛⠛⠛⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⠿⢿⣿⠀⢸⣿⠀⠀⣿⡇⠀⣿⡿⠿⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣼⣿⠀⠀⣿⣧⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⢾⣿⣷⠀⠀⠀⣠⣤⣄⡀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⢠⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠙⣿⣷⡀⣼⣿⣿⣿⣷⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠘⣿⣷⣼⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣯⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢸⣿⣿⣿⣿⣿⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡇⠀⢸⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣇⢀⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⣀⡉⣒⡶⢤⣍⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⢦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⣿⣯⣉⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠠⡹⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣾⣿⣿⣿⣿⣿⣿⣧⣿⣭⢫⣞⣿⣿⠐⣷⠻⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠸⢸⡟⠛⠛⠛⠋⠙⠁⠉⠋⠿⢻⣾⣿⣿⣼⠘⢢⡟⣴
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⡆⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣯⣂⠛⢹⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⢿⣴⣯⢟⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣿⣿⣯⣐⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡇⠀⣷⣤⣤⠀⠀⠀⠰⠿⣿⣯⣭⣍⠑⠀⠘⣿⣿⡷⣭⣋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠹⠾⠦⠀⠀⠀⠀⠀⠁⠀⠈⠙⠃⠀⠀⠹⣿⣿⣦⠌⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⡳⣄⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⣻⣿⣿⣮⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⣀⢀⣼⣿⣿⣿⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡖⠛⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⣿⣿⣿⡿⢹⢸⠹⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡇⠰⢸⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⣿⣿⣿⣿⡃⠀⠈⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡲⢤⣀⣀⣀⠠⠒⠉⠀⠀⢠⣿⣿⣿⣿⡇⠀⠁⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡀⠸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⡃⠀⠣⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣧⡰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⢦⣤⡂
⠀⠀⠀⠀⠀⡀⠀⡀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠆⠀⠶⢮⣭⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠊⠢⠙⢿⣷⡖
⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠛⢀⠀⠀⠹⣆⠀⠀⠀⠀⢠⣿⣿⡿⣿⣧⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠈⠘⡉⣀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠹⠀⠀⠀⢀⠃⣿⣿⠃⠘⢿⣷⠀⠀⣀⣀⢿⢿⡀
⠀⠀⠀⡀⠹⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⠀⠀⢀⠂⢠⣿⣡⢖⡒⢚⠻⡈⠋⠉⠀⠀⠂⠃
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢲⠏⠀⠀⠀⠀⢝⠉⠑⠋⠀⠀⠀⠈
⠜⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⢠⡄⠀⣼⠁⠀⠀⠀⢀⣀⡀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣿⣿⣿⡿⣿⡿⠿⣿⣿⣿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣾⣶⣾⣶⣾⣷⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠿⠿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⠿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⣤⣤⣤⣄⡈⠉⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣤⣤⣤⣤⣀⠈⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠀⠀⠀⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣸⣷⡄⠀⠀⠀⠀⢸⣧⠀⠀⣿⠀⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢾⣿⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠈⠛⠟⠃⠀⠀⠀⠀⢸⡏⠀⠀⣿⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⣠⡼⠏⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⢀⣠⡾⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡁⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠈⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠛⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠶⠶⠶⠊




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠿⠶⢶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣥⣴⣶⣶⣶⣶⣦⣤⣉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠏⠙⠛⠛⠛⠛⠛⠟⠻⠿⣿⠂⠀⠀⠀⠀⠀⣿⣿⠉⠁⠀⠀⠀⠈⠉⠁⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⢇
⠀⠀⠀⠀⠀⠀⠀⢀⠿⢿⡿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠀⢠⡄⠀⠀⠀⠈⡞⢙⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠊⣇⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣀⡄⠀⠀⠀⠀⠀⠠⢬⣤⡄⠀⠀⠀⠀⠀⠀⣴⡆⢰⡤⢧⣦
⠀⠀⠀⣀⣢⡷⠠⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⠛⠤⢀⡠⣦⡶⡀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⢀⣴⣿⠧⠘⠀⠐⠛
⠀⠀⠀⠐⠋⠀⠀⠠⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⠌⣁⠠⡉⠁⢴⣤⣤⣤⡀⠀⣠⣤⣶⣾⠟⠋⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠑⠀⡁⠂⢁⠂⠄⢀⠀⠀⠀⠀⢀⠀⠀⠠⡊⠄⢈⠀⠊⠀⠀⠀⠐⢀⡘⠙⠛⠣⠄⠻⠋⠁⢡⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠂⠀⢀⠤⣀⠨⣀⠄⠆⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠂⠀⠀⠀⠐⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⡀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠁⠀⠃⠀⠖⢂⠢
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⢀⠠⠐⠻⣀⠇
⠀⠀⠀⠀⠀⠀⡔⣀⣡⠉⠑⠐⠀⠀⠀⠀⠐⠒⠒⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠠⠮⡄⢨
⠀⠀⠀⠀⠀⠀⠃⠘⠁⠑⢀⠀⠀⠀⢀⠠⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⢣⡄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠂⡡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⣤⣀⠀⠀⠀⢀⡴⢊⣩⣳⣄⡀
⠀⠀⠀⣠⠴⠒⠊⠉⠉⠉⠑⠒⠉⠶⠿⠿⠿⢓⣿⣦⣤⠋⢠⣾⡿⣼⣿⣿⡆
⠀⣠⠊⠀⣀⣠⣤⣶⣶⣶⠀⣴⣶⣶⣶⣶⠟⢁⣿⣿⠃⢠⣿⡿⣵⣿⣿⣿⡗⡍⠉⣷⣄⡀
⢠⢃⣴⠞⢋⣩⡭⠛⠛⠁⢀⡉⠙⢻⣟⠻⡀⠈⢿⠇⠀⠾⣫⡞⠛⣿⣿⣟⣼⠃⠀⣿⣿⢿⡴⣄⡀
⣮⡾⠁⢠⡿⠋⢀⣤⡼⠀⣼⣿⣷⣾⣿⣷⣝⣦⣤⠀⢰⣿⣿⠓⢾⡏⠉⣾⡟⠀⣸⠃⣠⣾⡇⠈⣿⣦⡀
⢹⠁⠀⣿⠁⢠⣿⢯⠇⢀⡿⠿⠿⣏⡹⠋⣠⡎⠙⠀⣼⣿⠇⠀⣼⠁⢰⣿⠃⢀⡇⢀⣿⣿⠀⢠⣿⣿⣿
⡎⠀⢸⣿⣦⣘⣿⡟⠀⠘⠁⣴⡟⠈⠁⣸⡿⢃⣤⠀⣿⡿⠀⠰⠏⠀⣾⡟⠀⣼⡆⡈⣛⡁⠀⣼⣿⣿⡟
⣇⠀⢸⣿⣿⡏⢻⠃⢠⠀⢸⣟⣡⣾⠀⣿⡿⠋⣿⣤⣨⠼⠷⠶⠿⠟⠿⣇⠀⣫⣾⣿⡿⠀⢰⣿⣿⡿⢇⡀
⠸⡄⠀⢿⣿⣧⠎⢀⣾⣄⠹⠟⣁⣼⠷⠶⠖⠚⢛⣒⣒⣒⣒⠒⠒⠶⠦⠼⡟⢻⣿⡿⠁⢠⣿⣿⣿⠀⢠⡟⢶⣄
⠀⠙⠢⣤⣉⣡⣴⠿⣛⣻⣿⠟⠉⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠇⢀⠟⠀⣠⠿⠿⠛⠥⠔⠋⠀⣸⣿⣷
⠀⠀⠀⢈⡽⠋⣠⠞⢉⣴⠃⠀⢰⣿⣿⣿⡿⠟⠋⠉⢀⣠⣄⡀⣠⠔⣚⡿⠀⠘⣀⣴⣿⣶⣤⣤⣤⣤⣴⣾⣿⣿⣿
⠀⠀⢀⠏⠀⣰⠃⠀⣼⣿⣧⡀⠘⢿⣿⠟⢉⣭⣷⡞⠁⣴⠮⡟⠁⣼⣁⠇⢀⣿⡿⢿⣿⡿⠋⠙⠛⠿⠿⠿⠿⠛⠁
⠀⠀⠘⡄⢀⡿⠀⠀⣿⣿⣿⢿⠓⠲⡏⠀⣾⣭⣽⠀⢰⣶⡶⡂⠀⢿⠟⠀⢸⡿⠁⣸⣿⣷⡄
⠀⠀⠀⠈⠻⡇⠀⠀⣿⣿⣿⡜⠀⠀⣇⠀⠿⢟⣾⣆⡈⢛⣽⣿⡶⠶⠿⠷⠖⠒⠛⠻⣿⣿⠏
⠀⠀⠀⠀⠀⠸⡀⠀⠿⣿⠟⣰⠀⢠⣿⣷⠶⢟⣛⣋⣉⣁⣀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣿⣿⣦
⠀⠀⠀⡠⠒⠒⢬⡲⢤⣶⣾⡏⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠛
⠀⢀⠎⠀⢀⣤⣾⣿⣶⡝⡟⠀⣰⣿⣿⣿⠟⠋⠉⠁
⠀⠸⡀⢠⣿⣿⣿⡿⢿⠟⢀⣴⣿⣿⣿⠟
⠀⠀⠳⣌⠻⠿⠿⢚⣡⣴⣿⣿⣿⡿⠋
⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⠿⠋
⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦
⠀⠀⠀⣠⣾⣿⠿⠛⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⢠⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⢰⣿⣯⣤⣄⣼⣿⣿⣁⣀⣴⣷⠄⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⡀⠀⢀⣿⣿⠃⠀⢀⣈⠛⠀⠀⣄⡀⠀⢀⣀⡀⠀⠉⠁
⠀⠀⠀⠉⣈⣽⣿⣿⣿⡿⠿⢿⣿⡏⠀⣠⣿⡿⣿⣷⠀⣠⣾⡿⢿⣿⠀⣼⣿⡏⠀⢠⣿⣿⠇⠀⣼⣿⣿⣴⣿⣿⣿
⠀⠀⠀⢰⡿⢻⣿⣿⠏⠀⠀⠀⠛⢀⣼⣿⠏⢀⣿⠃⣴⣿⡟⢀⣾⠏⢠⣿⡿⠀⢀⣾⣿⠋⠀⣸⣿⣿⡿⢣⣿⣿⠃⢀⣤
⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⢸⣿⡿⠶⠋⠁⣰⣿⡿⠶⠛⠁⣠⣿⣿⠇⢀⣾⣿⡏⢀⣼⣿⣿⡟⠁⣿⣿⠇⢠⡾⠁
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣧⣄⣤⣾⠻⣿⣷⣄⣠⣾⠏⢹⣿⣠⣾⢿⣿⣠⡾⠻⣿⡟⠀⠀⣿⣿⣴⠟⠁
⠀⠀⠀⠀⠀⠻⣿⡧⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠈⠙⣻⠋⢡⡄⠈⠙⠋⠁⠈⠛⠋⠀⠀⠉⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡿⢶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣀⣄⠀⠀⠀⠀⠀⠀⢠⣿⠃⣼⣏⣴⣿⠆⣠⡾⢻⡆⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠉⣿⣿⡿⠃⠀⠀⠀⠀⠀⣼⡇⢰⣿⠏⣼⡏⣰⣿⠵⠋⡴⠁⠀⠀⠀⣰⣿⡿
⠀⠀⠀⢀⣾⣿⡟⠁⠀⠀⢰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠹⠷⠛⠏⠀⠿⠞⠙⠿⠶⠞⠁⠀⠀⠀⢰⣿⣿⠁
⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠈⠛⠀⠀⠀⣠⣾⠿⢷⣦⠀⠀⠀⢠⣶⡿⢿⣶⡀⠀⠀⣠⣴⣶⣤⣿⣿⠃
⠀⢀⣿⣿⡏⠀⠀⣠⣶⣶⣾⣷⣦⣀⣰⣿⠁⠀⢸⣿⡇⢀⣀⣿⡏⠀⠀⣿⣷⠀⣼⣿⡿⠉⣿⣿⠇
⠀⢸⣿⣿⠀⠀⠀⠛⠉⣹⣿⡿⠉⣿⣿⣿⡀⠀⣸⣿⢇⣼⣿⣿⡇⠀⢠⣿⡟⣸⣿⡿⠁⢸⣿⡟⠀⠀⣰⠏
⠀⢿⣿⣿⡀⠀⠀⢀⣼⣿⣿⠇⠀⣿⣿⠙⢷⣶⣿⡿⠟⢿⣿⡍⠻⣦⣾⣿⠿⣿⣿⠁⣀⣾⣿⡇⢀⣼⠏
⠀⠘⣿⣿⣿⣶⣾⡟⣿⣿⡿⠀⠀⠙⠿⣷⡾⠟⠋⠀⠀⠈⠿⣿⣾⠿⠋⠀⠀⠻⣿⡾⠟⠉⢿⣷⠿⠁
⠀⠀⠈⠛⠛⠛⠉⠀⣿⣿⣇⢀⣀⠀⠀⠀⠰⣶⣿⣷⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡟⠛⢿⣿⣄⠀⠀⢸⣿⣿⡇⠀⢀⣤⣄⠀⣴⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⢸⣿⣿⠀⠀⣸⣿⣿⠁⠀⣾⣿⠋⣸⣿⡿
⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡏⢀⣤⣌⠁⢀⣿⣿⣁⣤⣄⠀⠀⠀⣀⣤⣤⣄⠀⢀⣤⣾⣿⠂
⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⡿⠀⣾⣿⡟⠀⣼⣿⣿⠟⢹⣿⡇⢀⣼⣿⠟⣹⡿⢠⣿⣿⠿⠋
⠀⠀⠀⠀⠘⢿⣿⠀⠀⠀⠀⠀⣿⣿⡏⢀⣿⡿⠁⣸⣿⡟⠀⣸⣿⡟⠁⢠⣿⣿⢁⣾⣿⢃⣠⡟⠁⣸⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⣾⡿⠁⢀⣿⡿⠀⣴⣿⣿⠀⢀⣾⣿⠏⣸⣿⡟⠋⠁⢀⣼⠛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⡟⠁⠀⠈⢿⣷⡾⠋⢹⣿⢷⣿⡿⠋⠀⠘⢿⣿⣶⣶⠟⠻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⢀⣀⣤⣤⣤⣥⣤⣄⣀⣉⣀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣶⣶⣶⣶⣶⣿⠏⠀⠀⠉⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⣖⣤⣶⢿⡿⣫⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⢋⣷⢟⣽⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⡿⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦⡽⠛⠁⠀⠀⢀⣴⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢼⣿⡠⡀⠙⠋⣿⣿⣥⣤⣀⠀⠀⠀⠀⣠⣶⣶⣿⡿⢋⡟⢀⡄⠀⠀⠀⢨⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣧⡀⠀⠀⠹⣿⡅⠀⠃⠀⠀⠀⣞⢩⣿⣿⢗⣰⣿⣿⡏⣀⣀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠁⣾⣀⣴⣿⣿⣿⣷⣦⡀⠘⠃⠾⣋⣴⣿⣿⣿⣿⣟⣽⣇⣺⡵⠖⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠘⠛⢿⣿⣿⣿⣿⣿⣞⡁⢀⠀⠀⠀⣀⠀⠀⣰⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠉⠉⠛⢿⣿⣿⣷⡾⣿⣿⣿⣿⣿⣿⣿⣊⣅⣀⠄⠘⠉⢴⡿⠋
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⣿⣿⣿⣿⣿⠁⠀⣠⣴⣿⡿⠟⠁⠀⠈⢈⠽⣿⣿⣿⣿⣿⣯⡷⠀⢀⣈⣍
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣾⡉⢈⣿⣿⣿⣿⠉⠀⠀⠛⠋⠁⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟⠉⠀⠴⠖⠀⠀⠀⠠⠒
⠀⠀⠀⠀⠈⠳⢤⣀⣙⣷⣤⣀⠀⠀⠉⢸⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠐⠿⢻⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠐⢌⣹⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣩⣿⡿⢋⣤⣤⠄⠀⠀⢀⠄⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠽⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣤⣴⣦⠀⠀⠀⠀⠀⠀⢛⣿⣾⠋⠁⣠⣤⡴⠊⠁⠀⠀⢀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠙⠛⠁⠀⠀⠀⠀⠀⣴⣿⡿⣷⠞⢉⡡⠈⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠿⠛⠋⠁⠏⠈⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⣾⣟⣽⠁⠀⠀⠀⠀⠀⢠⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢼⣿⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢠⡆⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⣾⡇⠀⠀⣾⡇⠀⢠⣿⠀⠀⢠⣾⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠸⣿⡄⠀⢻⢿⡄⠀⣟⣧⠀⡼⣿⠀⢀⣿⡏⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⢢⣄⠀⠀⠹⣿⣄⠀⢹⣟⠢⡈⣯⠹⢦⡏⠈⠿⠡⣏⡴⠏⡿⠁⣠⣾⡿⠀⠀⣠⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣀⠙⣿⠙⠢⣯⣠⣼⣷⣶⣿⣷⣷⣶⣶⣿⣤⣹⡷⠊⢡⡞⣀⣤⣾⡿⠁
⠀⠀⠀⠀⠐⣦⣄⡀⠀⠈⠾⣯⠉⢻⣦⣾⣿⣿⡿⢿⡿⠟⠏⠿⠙⢻⠿⡿⣿⣿⣷⣾⡉⢠⡴⠋⠀⣀⣴⣾⠋
⠀⠀⠀⠀⠀⠈⠻⠿⣖⡶⠂⢹⣶⣿⡿⠟⠉⠁⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠈⠋⠻⣿⣿⣾⡓⠙⣋⡽⠛⠁⠀⣀⣀
⠀⠀⠈⠻⣷⣦⣤⣤⠌⢻⣤⣿⣿⠿⠀⠀⣴⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣆⠀⠀⠛⣿⣿⣾⠃⠴⠶⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠉⠛⠷⣆⣰⣿⡿⠟⠀⢠⠎⠁⣀⣀⣀⠀⠀⠠⠈⠀⠀⠀⣀⡀⠈⢱⡄⠀⠻⣿⣿⣦⡞⠋⠁⠀⠀⣀⣀⡀
⠀⠘⠻⢿⣶⣶⣖⠋⠉⣿⣿⠟⠀⠀⢈⣾⣟⣿⣿⣿⣿⣦⡀⢀⣴⣿⣿⣭⣍⠛⢦⠁⠀⠀⠿⣿⣿⡀⣯⣽⡿⠿⠛⠁
⠀⠀⠀⠀⠀⢀⣉⠝⢾⣿⠏⠀⠀⠀⢸⣿⣿⣗⣚⣶⣿⣿⡇⢸⡿⣟⣶⣲⣻⡷⠀⠀⠀⠀⠀⢻⣿⡟⠓⢤⣄⣀⣀⣀⣀
⠐⠒⠿⢿⡿⠿⠯⣤⣸⣿⡇⠀⠀⠀⢨⣿⡋⠛⠛⠋⢹⣿⠇⢸⡇⠈⠁⠉⠉⠀⣀⡀⠂⠀⠀⢹⣿⡧⣾⠓⠛⠟⠛⠋⠉
⠀⠀⠀⠀⠀⣠⡴⢊⠹⣿⣷⡀⠀⢑⠐⢿⣷⡄⠀⢀⣼⠛⣄⠈⠛⡆⠀⠀⢀⣘⡿⠀⠈⠀⣠⣾⣿⣃⣀⡙⣦⣤⣤⣀⡀
⠀⠐⠚⠿⠿⠟⠛⢙⠟⢿⣿⣿⣦⠀⠀⠀⠀⠀⣴⠟⠉⠓⣿⠛⠒⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠉⢦⡈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⣀⣠⣴⣧⡴⢬⣿⣿⣿⡄⠀⠀⠀⠰⣧⣶⡶⠿⠿⠿⠗⣖⣴⡄⠀⠀⠀⠀⢠⣿⣿⡿⠷⠖⠦⣿⣶⣤⣄⡀
⠀⠀⠐⠛⠛⠛⠉⠁⣠⣇⣈⣻⣿⣿⣆⠀⠀⠰⠀⠀⠿⠷⠖⠲⠾⠿⠀⠀⠆⠀⠀⣴⣿⣿⣿⣥⣄⣷⣄⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⢡⡋⢘⣿⣿⣧⡀⠀⠀⠀⣠⡤⡶⠲⠤⡀⠀⠀⠀⣀⣾⣿⣿⣅⡈⣧⠈⠛⠿⢷⣦⡀
⠀⠀⠀⠀⠐⠉⠉⠀⠀⣠⣾⡷⠋⡍⡈⣙⡿⣿⣦⣀⠀⠀⠁⠀⠀⢀⣀⣴⣿⠿⣿⣁⠘⡆⠳⣾⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⠀⢠⣗⡞⠉⣧⣄⡼⡏⢙⣿⠛⡛⡿⣏⠉⡿⣄⠄⡇⠙⣦⣧⡀⠀⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣰⣿⠋⠀⢀⣿⡟⠀⢷⣾⠁⢷⣹⡇⢹⣰⠇⠈⣷⣷⠀⠈⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⢀⣾⠟⠀⠀⣸⣿⠀⢸⣿⡇⠸⣿⡄⠀⠈⠻⣧⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⣿⠃⠀⠸⡿⠀⠀⠙⡇⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠑⠀⠀⠀⠘⠉⠃⠛⠘⠐⠃⠘⠚⠐⠂⠀⠁⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠋⠛⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠙⣿⡇⠀⠀⣼⡟⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣸⡿⠀⠀⣼⡟⠀⠀⡀⠀⠀⠀⢠⣾⠂⠀⠀⢀⣶⠆⠀⠁⢀⣀⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⠷⠶⣾⡿⠀⣠⡾⠛⣿⠂⢠⣿⡗⢩⣿⠀⣾⡧⠊⣿⠂⢻⣿⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⣿⠇⢰⡟⢁⣾⡇⢀⣾⣯⢤⣾⠏⣼⣿⢤⣼⠟⢀⣿⠃⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢿⣄⠈⠛⠁⠘⠛⣽⡏⠙⠋⠁⢰⡿⠙⠛⠉⠀⠀⠛⢋⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⠃⠀⠀⠀⣴⠞⣻⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠙⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⡀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣴⡆⢀⣴⡷⠀⠀⢰⣄⠀⠀⣶⠄⠀⠀⠀⠀⠀⠐⠛⠟⠁
⠀⠀⠀⠀⠀⢠⣶⠿⣿⡟⠉⠙⢦⠀⠁⢀⣤⠉⠉⠉⠛⠛⢻⣾⢿⣶⣿⣯⣤⣀⣠⣼⠟⠀⣼⡟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠹⠅⣰⣿⣀⣀⣤⠏⢀⣠⣌⠉⢰⣛⡆⠀⠀⣼⠏⢠⡿⢁⣭⣭⡍⢉⣡⣤⣴⡟⠀⠀⢠⣴⠶⣤⠀⢀⣴⠆⢠⣶
⠀⠀⠀⠀⠀⠀⣰⡿⠉⠉⠉⢹⡄⢹⡿⠁⢀⣼⢛⡿⠀⣸⡏⠀⣾⡷⢋⣿⠋⣴⠟⠉⢠⡟⠀⠀⣴⡿⢁⣼⠇⢀⣾⠏⣠⣿⠃
⠀⠀⠀⠀⠀⢠⣿⡃⢀⣀⣤⠟⠁⣿⣧⠶⠋⢰⣿⣥⢾⣿⠀⢸⡟⠁⢼⣇⡤⢿⣤⡶⠻⣧⣴⠞⢿⡥⠊⣿⡤⢺⣿⠔⣿⠏
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣴⡟⠀⠐⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠶⠶⠶⢶⣶⣦⣤⣤⣤⣤⣤⣤⣾⣯⣤⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣬⣿⠟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⡾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⣸⠏
⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⣰⠟⠀⢸⠟
⠀⠀⠀⠀⠀⣾⣿⡏⣠⡾⠋⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣸⣿⡏⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⣸⣿⣿⣴⠟⠁⠀⢀⣤⣤⡄⠀⣰⣿⡿⣿⡆⢀⣿⣿⡷⠛⠁⢀⣴⣿⠀⠀⣾⡟⢠⣾⣿⣿⠀⣰⣾⡿⣿⣷
⠀⠀⠀⢀⣿⣿⠟⠁⠀⠀⢠⣿⡟⢹⡿⣸⣿⡿⢠⡿⠁⢸⣿⡿⠀⠀⠀⣾⣿⠏⠀⣸⣿⣣⢃⣿⣿⢇⣾⣿⡟⢀⣿⡏
⠀⠀⢀⣼⣿⣿⠿⣆⠀⣀⣾⣿⣧⠾⠁⣿⣿⡷⠟⠁⢠⣾⣿⡇⠀⠀⣸⣿⡿⠀⣠⢿⣿⠇⣼⣿⠏⢸⣿⣿⢀⣾⣿⡇
⠀⣠⠟⣿⣿⠿⠶⣿⠛⠉⣿⣿⡇⠀⣰⢿⣿⡇⠀⢠⡟⢸⣿⣇⢀⡴⠋⢿⣧⡾⠃⠀⠁⠀⣿⣟⡴⠉⠻⠟⠋⢸⣿⠃
⣰⠋⢠⣿⣿⠀⠀⣿⠀⠀⠘⢿⣷⠾⠃⠘⢿⣷⡴⠋⠀⠀⠻⠿⠟⠁⠀⠀⠉⠀⠀⠀⠀⠀⠛⠋⠀⠀⣠⣴⠶⣿⣿⠳⢦
⠃⠀⢸⣿⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣴⡿⣿⠀⣼⠟⠁⢀⣿⡏
⠀⠀⢸⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣦⠀⠀⠀⣼⡟⠙⣧⠀⠀⣀⣠⣿⡿⠃⢸⣯⡀⣀⣾⠟
⠀⠀⢸⣿⡇⠀⢰⡇⠀⠀⠀⣠⣴⡾⣿⣿⠀⢠⣾⣿⡄⢸⡇⢀⣼⣿⣿⣄⣿⠇⣼⡿⢻⣿⠁⠀⠀⠙⠛⠛⠁
⠀⠀⢸⣿⣷⠀⣾⠁⠀⣠⣾⡿⠋⠀⣿⣿⢠⣿⡿⠈⠛⣿⠷⣾⣿⡇⠈⢻⣿⢿⣿⠃⣼⣿
⠀⠀⠀⠻⣿⣾⠇⠀⢰⣿⣿⠁⠀⣸⣿⣿⢸⣿⡇⠀⣰⡏⠀⣿⣿⡅⢠⣾⠇⢸⣿⣰⠿⣿⠀⠀⢀⡄
⠀⠀⠀⠀⠈⠀⠀⠀⠸⣿⣿⣤⣾⢻⣿⣿⠘⢿⣷⣶⠟⠀⠀⠙⢿⣿⠿⠋⠀⠀⠉⠁⠀⠻⣧⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⢸⣿⣿⣀⣀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢾⣿⡟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⣠⣿⡿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣞⣆⠀⠀⠀⠀⠀⠀⠀⢀⠔⡩⢟⠶⠶⠦⠶⡪⢕⢆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣳⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡼⠾⠟⡸⠀⠀⠀⠀⠀⠀⠀⠇⣮⡾⠋⠈⠒⠒⠀⠙⢷⣵⣡⠀⠀⠀⠀⠀⠀⠀⢇⠻⠷⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠐⠒⡇⠀⠀⠀⠀⠀⠀⢸⣼⣿⠠⠶⠆⠀⠀⠰⠶⠄⣿⣧⡆⠀⠀⠀⠀⠀⠀⢨⠒⠂⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢃⠀⠀⠀⠀⠀⠀⣼⠀⠟⠐⠉⠂⠀⠀⠐⠉⠂⠻⠀⣷⠀⠀⠀⠀⠀⠀⡘⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⡀⠀⠀⠀⠀⠀⡇⠉⢇⠀⠀⠀⠀⠆⠀⠀⠀⡸⠉⢸⠀⠀⠀⠀⠀⢀⠃⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢧⡀⠀⠀⠀⠀⢃⠀⠈⢆⠈⢿⡉⢉⡿⠁⡰⠁⠀⡌⠀⠀⠀⠀⢀⡼⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠉⠉⠑⠒⠚⠠⠤⠤⠵⢦⠤⠤⡴⠮⠤⠤⠔⠓⠒⠊⠉⠉⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠒⠖⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⣄⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡠⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢤⣀⠤⠤⠤⠤⢄⣀⡤⠖⠓⠒⠒⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠷⠤⠄⣀⡀⣼⠀⠀⠀⠀⢸⠦⡇⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢀⡖⠒⡀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠒⡇⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⣧⣴⡏⠉⠉⠉⢹⠀⠀⠀⠀⢸⣴⠃⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡇⢸⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠃⢸⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⢤⡜⠀⠘⠤⠤⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⢤⡃⠀⠈⢦⡤⣜⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⠼⠎⠑⠒⠤⠤⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠋⠉⠙⠉⠉⠛⠲⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠎⠁⠈⣇⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣤⠶⠶⠚⠉⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣀⣰⠇⣴⡶⣿⣶⣄⠀⠀⠀⠐⢾⡵⣶⣬⠇⠀⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣾⠉⣯⠀⠀⢰⡦⠀⠘⣧⠀⠀⠀⠠⣶⠀⠠⣷⠏⠉⡆⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢻⡄⢿⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⢰⡏⠀⣸⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠛⢻⡆⠀⠀⠀⠀⠘⠳⠖⠀⠀⠀⠀⠀⠀⣿⠞⠋⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣠⡴⠛⠋⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⠛⠲⣤⣀⡀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠿⣍⡁⠀⠀⠀⠀⠀⠉⠛⠦⠤⣤⠤⠶⠛⠁⠀⠀⠀⠀⠀⢉⡹⠟⠛⠦⣤⡀
⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠈⠛⠶⢤⣀⣀⠀⢀⣤⣄⣀⣀⣠⣤⡀⠀⢀⣀⣠⠤⠞⠋⠁⠀⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣧⠙⣯⠻⣏⣼⣿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡌⢷⡈⢧⡙⣧⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢾⡟⣆⠳⣄⠻⠜⠷⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⡼⠁⠀⠀⠹⣜⠳⡝⠃⠀⠀⠀⠈⢧⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⣸⠇⠀⠀⠀⠀⢈⣷⣄⡀⠀⠀⠀⠀⢸⣇⡀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⡾⠋⠛⠶⣤⣀⣠⠞⠋⠀⠀⠙⠳⣆⣀⣤⠶⠛⠙⢶⡄⠀⢧⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣼⠟⠋⠀⠀⠀⠀⠀⣨⠟⠀⠀⠀⠀⠀⠀⠻⣅⠀⠀⠀⠀⠀⠙⠷⣼⣇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⢹
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢇⠀⠀⠀⠀⠀⠀⣠⡴⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⣄⠀⠀⠀⠀⠀⢀
⠈⠳⣤⣤⣤⠴⠞⠁⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⠳⢦⣤⣤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠲⠶⢶⡶⠤⣤⣤⣤⣤⣤⣤⣤⡤⠤⣤⠴⠶⠶⠶⡶⠶⠶⢶⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣼⣤⣤⣤⣤⣧⣄⣸⣿⣀⣀⣠⣿⣤⣤⣤⣤⣷⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡹⠁⠀⠀⠈⣟⠀⢹⠁⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⠇⠀⠀⠀⠀⣧⠀⢸⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⣿⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⡆⠀⠀⠀⢻⣿⣿⣇⠀⠀⠘⣿⣿⣇⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⣿⠁⢀⣿⣿⡟⠀⠀⠀⣰⣿⣿⠏⠀⠀⣠⣾⣿⣿⠟⠋⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿
⣿⣿⡀⠀⠀⠈⣿⣿⣿⡀⠀⠀⢻⣿⣿⠀⠀⠘⣿⣿⡇⠀⠀⣾⣿⡏⠀⣼⣿⡟⠀⠀⠀⣰⣿⡿⠃⠀⣠⣾⣿⡿⠋⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⠿⠟⠛⠋
⣿⣿⣧⠀⠀⠀⠘⣿⣿⣧⠀⠀⠘⣿⣿⡆⠀⠀⣿⣿⡇⠀⠀⣿⣿⠇⢀⣿⣿⠁⠀⠀⣰⣿⡿⠁⢀⣾⣿⡿⠋⠀⠀⢀⣤⣾⣿⣿⡿⠟⠋⠁⠀⠀⠀⣀⣠
⢿⣿⣿⣧⠀⠀⠀⠸⣿⣿⣆⠀⠀⢹⣿⣷⠀⠀⢻⣿⡇⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⢰⣿⡟⠀⣠⣿⡿⠋⠀⠀⣠⣾⣿⡿⠟⠋⠁⠀⠀⣀⣤⣴⣾⣿⣿⣿
⠀⢻⣿⣿⣧⠀⠀⠀⠙⣿⣿⡆⠀⠀⢿⣿⡆⠀⠘⣿⣇⠀⠀⣿⣿⠀⣼⣿⠀⠀⢀⣿⡟⠀⣴⣿⠟⠀⠀⣠⣾⡿⠟⠉⠀⠀⣀⣤⣶⣿⣿⣿⠿⠟⠛⠋⠉
⠀⠀⠹⣿⣿⣧⡀⠀⠀⠘⣿⣿⣆⠀⠈⢿⣿⡀⠀⢻⣿⠀⠀⣿⣿⠀⣿⡟⠀⠀⣼⡿⠁⣼⣿⠃⠀⣠⣾⡿⠋⠀⠀⣠⣴⣾⣿⠿⠛⠉⠁
⣄⠀⠀⠈⢿⣿⣷⡄⠀⠀⠈⢿⣿⣆⠀⠈⢿⣷⠀⠘⣿⡇⠀⢹⣿⠀⣿⡇⠀⢠⣿⠃⣸⡿⠁⢀⣾⡿⠋⠀⢀⣴⣾⡿⠟⠉⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿
⣿⣧⡀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠹⣿⣧⡀⠈⢿⣧⠀⠹⣿⡀⠸⣿⠀⣿⡇⠀⢸⡟⢠⡿⠁⣠⡿⠋⠀⣀⣴⡿⠛⠁⠀⠀⣀⣠⣴⣾⣿⣿⡿⠿⠿⠛⠛⠛
⢿⣿⣿⣦⡀⠀⠀⠙⢿⣷⣄⠀⠀⠈⢻⣷⣄⠀⢻⣧⡀⢻⣧⠀⢿⡆⢹⡇⠀⣿⠁⣿⠁⣴⠟⠁⢀⣼⠟⠉⠀⢀⣠⣶⣿⠿⠟⠛⠉⠁⠀⢀⣀⣀⣀⣀⣀
⣀⠉⠻⣿⣿⣦⡀⠀⠀⠙⢿⣷⣄⡀⠀⠙⢿⣦⡀⠙⣷⡄⢻⣆⠸⣇⢸⡇⠀⡟⣸⠇⣼⠏⢀⣴⠟⠁⢀⣤⡾⠟⠋⢉⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣤⡈⠛⢿⣿⣶⣄⠀⠀⠉⠻⢿⣦⣀⠀⠙⠻⣦⡈⠻⣆⠹⣆⢻⡀⣇⠀⡇⡟⣸⠃⣠⠟⢁⣠⠾⢛⣩⣴⡶⠿⠛⠛⠋⠉⠉
⠙⠿⣿⣿⣶⣤⡈⠛⠿⣿⣦⣄⡀⠀⠉⠻⢷⣤⣀⠈⠻⢦⡈⢷⡜⢦⢳⢹⠀⣿⢱⢃⡼⢁⡴⣋⡥⠾⢛⣩⣤⣤⣶⡶⠾⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠉⠛⠿⣿⣷⣦⣄⣉⠛⠿⣷⣦⣄⣀⠈⠙⠳⢦⣄⡙⠲⢌⡪⡣⡷⡀⣻⠁⢊⣴⣿⣪⠵⠖⢛⣉⣉⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠛
⣤⣄⡀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣶⣦⣤⣍⣙⠛⠶⠶⢤⣤⣉⣙⡒⠍⠊⠐⣥⣇⡴⠋⠊⢑⣒⣛⣉⣉⣉⠉⠉⠉⠉⠉⠛⠛⠻⠿⢿⣿⣶⣶⣤⣄⣀
⠿⣿⣿⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠒⠒⠒⠛⢛⣛⠊⢉⠴⢛⠉⠒⠶⠶⣤⣭⣭⣉⣉⠙⠛⠛⠷⢶⣶⣤⣄⣀⠀⠀⠉⠙⠻⠿⣿⣿⣶
⠀⠀⠈⠉⠙⠛⠻⠿⠿⠿⣿⣿⣶⣶⣶⣶⣶⠶⠶⠶⠶⠒⠛⣉⡥⣒⣾⠏⡔⠀⡇⣦⡢⣍⠲⢍⡙⠻⣯⣛⠷⢶⣤⣀⡀⠉⠛⠻⢿⣷⣦⣄⡀⠀⠀⠉⠻
⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠖⢛⡥⢞⡕⣡⢣⠃⡇⣿⠰⡹⣌⠳⣄⠙⠷⣄⠉⠻⢶⣬⡙⠻⣷⣦⣄⡀⠈⠙⠿⣿⣷⣦⣄
⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⠿⠿⠛⠛⢉⣡⣴⠞⣫⡴⢋⡴⢃⡟⢰⠇⣿⡆⢳⠘⣦⠘⢷⡀⠈⢳⣦⡀⠙⠻⣷⣄⠙⠻⣿⣶⣄⡀⠈⠙⠿⣿⣿
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⣀⣀⣤⣤⣶⠿⠛⣉⣴⠞⠋⣠⡟⢁⡾⠁⣾⠀⣿⣿⠘⣧⠘⣧⠀⢻⣦⠀⠙⢿⣆⠀⠈⠻⣿⣦⡈⠙⢿⣿⣦⣄⠀⠈⠻
⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿⠿⠟⠛⢉⣠⣴⡿⠛⠁⣠⡾⠋⢠⡿⠁⢰⡏⠀⣿⣿⡀⢿⡄⠹⣧⠀⠻⣧⡀⠈⢻⣷⡄⠀⠈⢿⣿⣦⠀⠙⢿⣿⣷⣄
⠛⠛⠛⠛⠛⠛⠉⠉⠉⢁⣀⣠⣤⣶⣿⠿⠛⠁⠀⣠⣾⠟⠁⣰⡿⠁⢀⣿⠁⢰⡟⣼⡇⢸⣧⠀⢻⣧⠀⢻⣷⡀⠀⠹⣿⣆⠀⠀⠙⣿⣷⣄⠀⠙⢿⣿⣷
⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⡿⠟⠛⠉⠀⠀⣀⣴⣿⠟⠁⣠⣾⠟⠀⢀⣾⠇⠀⣾⡇⣿⡇⢸⣿⠀⠈⣿⡆⠀⢿⣷⡀⠀⠹⣿⣧⠀⠀⠈⢿⣿⣦⠀⠀⠻⣿
⣿⣿⡿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⣀⣴⣾⡿⠛⠁⢀⣴⡿⠋⠀⢀⣾⡏⠀⣰⣿⠀⣿⡇⠘⣿⡆⠀⢹⣿⡀⠘⣿⣧⠀⠀⠹⣿⣧⠀⠀⠈⢻⣿⣷⡄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⠟⠉⠀⣠⣶⣿⠟⠁⠀⣠⣿⡟⠀⢠⣿⡇⢸⣿⠇⢰⣿⡇⠀⠈⣿⣇⠀⢹⣿⡆⠀⠀⢻⣿⣧⠀⠀⠀⢻⣿⣿⡄
⣀⣀⣀⣤⣤⣴⣶⣿⣿⣿⡿⠛⠉⠀⢀⣴⣾⣿⠟⠁⠀⢀⣴⣿⠏⠀⢠⣿⡟⠀⣾⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⠸⣿⣷⠀⠀⠈⣿⣿⣇⠀⠀⠀⢿⣿⣿⡄
⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⣀⣴⣾⣿⡿⠛⠁⠀⠀⣠⣾⣿⠋⠀⢠⣾⣿⠁⢠⣿⡟⠀⣸⣿⡇⠀⠀⢻⣿⡇⠀⣿⣿⡇⠀⠀⢸⣿⣿⡄⠀⠀⠈⣿⣿⣿
⠛⠛⠉⠁⠀⠀⠀⣀⣤⣶⣿⣿⣿⠟⠋⠀⠀⠀⣠⣾⣿⡟⠁⠀⣠⣿⣿⠃⠀⣾⣿⠃⠀⣿⣿⠃⠀⠀⢸⣿⡇⠀⣿⣿⣧⠀⠀⠀⣿⣿⣷⠀⠀⠀⠸⣿⣿
⠀⢀⣀⣤⣴⣾⣿⣿⣿⡿⠟⠋⠀⠀⠀⢀⣠⣾⣿⡿⠋⠀⢀⣴⣿⡿⠃⠀⣼⣿⡟⠀⢸⣿⣿⠀⠀⠀⣾⣿⡇⠀⢹⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢻⣿
⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⣠⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⡟⠁⠀⣰⣿⣿⠁⠀⣾⣿⡟⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⡀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠘⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢋⣠⣤⣄⡀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠪⠤⢍⡉⣻⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠈⠈⢉⠉⠉⠐⠒⠥⢄⣀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠃⠀⠀⠀⡠⠀⠈⠙⠢⣄⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡆⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⣿⣦⠸⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠃⠀⠈⠁⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠈⠫⠌⡿⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠠⡼⠣⠀⠀⠀⠈⢋⠉⠉⠉⠉⠉⢳⠀⠀⣰⣻⠿⠴⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠕⠭⡁⢸⣦⣄⣀⠀⠀⠑⠤⢀⡠⢔⣁⠔⠺⣅⠀⠀⠀⣀⣀⠀⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⣒⣁⡀⠉⠉⠉⠉⠙⠋⢷⡉⠉⢁⠄⠀⠀⠈⠓⣤⠚⠉⠩⣇⡀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣼⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠉⠹⡁⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣴⣿⣶⣄⡠⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⣿⣿⣦⡤⠤⠔⠾⣿⣿⣿⣿⠃⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠔⠘⡏
⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⢹⠿⠛⠛⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⡩⠉⠉⠁⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠹⡍⠁⠀⠀⠀⢀⡊⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠙⢦⡀⢰⣾⣿⣿⡤⠤⠀⢒⡿⠃⠀⠀⠀⠀⠀⢀⣤⠤⣄⠀⠀⠈⢷⣅⣀⠀⠀⠀⠀⠀⣠⠊⠉⣛⠻⡉⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠤⠶⠚⠉⠀⣘⠲⠦⠀⠀⡴⠉⠑⣷⣄⠱⢤⣀⡜⠁⠀⢉⡶⢦⣀⠔⠛⠓⣾⣉⣹⣇⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⢳⣂⡉⠤⢿⣦⣀⢩⡙⠢⡴⠋⠀⠀⠀⠀⠀⠀⢾⣀⠤⣬⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡒⣉⠵⣽⡁⠈⡞⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⢉⡝⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣊⣩⣷⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⠿⠛⠛⠛⠉⠁⠙⠋⠘⠛⠛⠋⠉⠉⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿
⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿
⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⢀⠐⣦⣶⠶⠿⢿⣿⣶⣄⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⢳⣾⡟⢀⡴⠶⢶⣤⣿⣿⣇⣶⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⡄⠀⠀⠀⠀⠀⠶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢴⡙⣏⠋⠀⠀⠀⡿⡎⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⠻⡀⠠⠴⢠⠀⠀⠀⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢵⡈⢿⣦⠶⢆⠄⠁⢉⣤⢿⣿⠃⠀⣀⣀⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠐⠀⠛⢦⡀⠀⠈⠉⠀⠂⣄⣵⡶⠶⣦⣄⡀⢀⣀⠀⡀⠈⠳⡄⢣⣄⡀⠀⠐⠨⠕⠋⢻⡴⡉⠉⣿⠿⡬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠈⠛⠦⣤⡈⠃⠘⠛⠛⠳⣦⡤⣬⣽⣿⣴⠶⡴⠃⢁⣿⡐⢏⠙⠆⠀⠀⠀⠀⠀⠁⠀⣠⡛⣖⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⢚⣉⣉⡛⠂⠈⠀⣤⡀⠉⣹⣣⠤⠶⣌⡙⣷⠈⢿⠈⣶⣄⣀⢀⣠⠤⣾⡟⠀⠏⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠛⢻⡙⣻⠛⢯⠊⢀⣀⣩⣭⣷⣿⣷⣦⠀⣸⣧⣿⢋⣩⣤⠶⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣤⣾⣿⣷⠀⣿⣿⣾⣷⣴⣿⣁⣴⣀⠸⣿⡿⣿⣿⢻⣿⣿⣿⠋⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣷⣿⣿⣿⠿⣶⠿⣟⣽⣿⡏⢹⣿⠛⠻⣛⠻⣷⢦⡉⣛⠎⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠈⠛⠍⣽⡟⢸⣿⣿⠛⢣⠘⣿⡝⠳⠾⠷⣌⠀⢹⣏⣾⣃⠀⠀⠀⣠⠴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢼⣟⣿⣤⣈⣢⡈⠳⣍⡶⠾⠿⠿⠋⠁⣱⣿⡄⠀⢈⣍⢠⠉⣶⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⣸⡛⢮⣥⣌⣙⠛⠷⠶⠿⠿⣿⣿⣤⡀⠀⠉⢻⣶⣿⡿⢀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⣼⣿⣿⣦⣉⣉⣻⣤⠶⠻⣶⣶⡄⠈⠉⢙⡀⠀⠀⠙⢯⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠠⣴⢿⡟⢻⣄⠀⠀⠀⢈⠀⠀⠀⠀⣀⣹⣿⣿⣿⣧⣍⣭⣤⣶⣿⠿⣿⣿⣾⣆⠀⠹⣦⠠⡀⠸⣿⣻⡄
⠀⠀⠀⠀⠀⡀⠀⣠⣾⡿⡏⢻⡆⠀⠀⠘⡄⠀⠀⣾⢿⣿⣿⣿⣿⣿⣿⣟⡁⠉⠳⠿⠿⣷⣿⣿⡄⠰⣘⡇⠿⣦⡙⠷⡁
⠀⠀⠀⠀⢰⠃⣰⣿⠋⠀⠸⠀⣷⠀⠀⠀⢻⣆⠀⠛⢻⣿⡿⠛⠀⣿⠀⡙⢿⡄⠀⠢⡀⠀⠹⡎⢷⠈⢯⠱⠆⢹⡇⠀⠄⠠⠴⠆
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣰⠃⠀⠀⠀⠀⠉⠒⠚⠋⠉⠀⠀⢀⡿⢀⠰⠀⢻⠀⠀⢻⠀⠀⡀⢸⡀⢸⡇⠳⠈⣧⣤⣀⣿⡶⠆
⠀⠀⠀⠀⠀⢠⡀⣧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⡶⠊⠀⣼⣇⡃⢨⡧⠀⠸⠂⢰⢁⣼⡇⠘⠁⡀⠈⠛⠿⠿⠟
⠀⠀⠀⠀⠀⠀⢳⣌⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣈⣤⡿⠛⠁⣀⣜⡿⣟⣡⡞⠃⠀⠀⢠⣿⣾⣿⠳⠀⣸⠃
⠀⠀⠀⠀⠀⠀⡀⠙⢦⡀⠀⠀⠀⠀⠀⢀⢠⢾⠟⢉⠀⠀⢐⣺⣭⣿⣿⣿⠋⠀⠀⠀⢀⣾⠙⢻⠋⣶⠻⡁⠤⣀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠹⣦⡀⠁⠀⠀⠀⠛⣾⣿⣦⣿⡿⢉⡤⢚⣋⠡⢀⣠⣤⡴⠔⠊⢀⠔⠛⠋⠀⢹⣄⠘⣤⡽⠒⠠⠬⠉⠀⢠⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⡀⠀⠉⢉⣀⠉⣰⣉⠞⣩⣶⣾⡿⠛⠁⣀⢠⡾⣡⢞⠁⠀⠀⠀⢻⣦⡉⠒⠄⠀⠀⠀⠀⠁⠀⣹⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⡏⠀⣿⣃⡼⢋⡟⠉⢠⣄⣼⢁⣿⢰⡏⠠⠀⠀⠀⠀⠀⠙⠧⠙⠶⡤⣴⣦⠈⠠⠂⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢠⣿⠟⢠⡿⡁⠀⠀⠉⣻⣿⣝⢦⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⠤⠴⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠁⠀⠀⠀⢾⣿⢛⡀⠛⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⢠⡀⠀⠀⢸⠋⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡈⢆⢿⣦⣀⣇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠄⢐⣻⣦⣭⡦⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿
⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡆⢹⣿⣿⣿⠠⠏⢀⣼⣿⣡⠏⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⣿⣿⣿⣿⠃⠀⣾⣿⠘⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢸⡇⢸⣽⣿⡏⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⡠⣾⡟⠀⠀⣿⣿⣿⠏⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣳⠀⡇⠈⣿⣿⠀⠀⢺⣿⡏⠀⠀⠀⠀⠀⠀⣠⠞⣱⡟⠀⠀⠀⣟⠻⠏⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⢇⢂⣿⣿⡄⢀⠀⢸⡆⠀⠀⠀⠀⠀⣰⡏⣾⣿⡅⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⡜⢿⣿⣄⢢⠀⠣⠀⠀⠀⠀⠀⢿⡅⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣻⣷⠘⢮⡻⣿⣦⡱⡄⠁⠀⠀⠀⠀⠸⣧⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠿⠟⠃⠀⠳⣌⠻⣿⣾⢆⠀⠀⠀⠀⠀⢿⣾⣿⣿⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⣂⣷⠀⠀⠀⠀⠀⠀⢹⡄⠈⢻⣯⢆⠀⠀⠀⠀⣸⣿⣿⣿⣯⢷⡕⢤⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿⣼⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡙⠷⣝⡢⡀⠀⠀⠀⠀⠙⢿⣷⠦⢄⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⢀⡜⠀⠀⠀⣿⣟⠀⠀⢀⣾⣿⣿⣿⡿⢻⣿⣿⣷⡈⠹⢮⣦⡀⠀⠀⠀⠀⠙⢄⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠞⠀⠀⠀⢠⣿⡆⠀⢠⣿⣿⣵⣿⣯⣤⣌⢿⣿⣿⣇⣀⣈⢿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⢠⣿⡟⠀⠀⣾⣯⣾⣿⣿⣷⣄⠙⣮⢿⣿⣿⣿⣿⣿⣷⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠀⠀⠀⠀⠀⣰⢟⡿⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⠘⡎⢿⣿⣿⣿⣿⣿⡿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⣀⣠⣴⣾⡿⠟⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⠈⢿⣿⣿⡁⠀⠁⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣀⣤⣴⣶⡾⠿⠿⠛⠛⠋⠉⠀⠀⠀⣠⣾⠟⣿⠋⠀⠈⠉⠛⣿⣿⣿⠀⠀⠈⣿⣿⣿⢀⠀⠻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⣴⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⣋⣙⣻⣤⣶⣲⢶⣦⡀⠘⣿⣿⡇⠀⠀⠘⣿⡇⠀⠀⢠⠈⠉⠈⠙⡄
⠀⠀⠀⠀⠀⣰⠟⢁⣼⣿⣿⠟⠉⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢸⣿⡇⠀⠀⢠⣿⣶⣷⡀
⠀⢀⠀⠀⣰⡏⢠⣿⣿⠟⠁⠀⠀⠀⠀⣠⣶⡿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⠇⠀⠀⠀⣿⣿⣿⣿⡆
⠀⡘⠀⢰⡟⢠⣿⣿⠃⠀⠀⠀⢀⣠⣾⢟⣩⣾⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠙⣿⣿⣿⣿⡇⢠⡿⠀⠀⠀⠀⠙⠛⣻⠆⠛
⠀⡇⠀⣿⠃⣼⣿⠇⠀⠀⢀⣴⡿⢛⣵⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠃⡼⠃⠀⠀⠀⠀⠀⠀⠛
⠀⡇⠀⣿⠀⣿⣿⠀⠀⣰⣿⠟⢡⣾⣿⣿⣿⣿⣿⣿⠟⣁⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡇⠀⣿⢸⣿⣿⠀⢸⣿⠏⢀⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⡴⠀⠀⣀⡀⠀⠀⠀⣀⠀⢸
⠀⡇⠀⡏⢸⣿⣿⠀⣼⡇⠀⣸⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⡟⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⢎⣤⠞⠋⠉⢻⣴⣶⣯⢽⣤⡼
⣴⠃⢰⠇⠀⣿⣿⡄⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⠇⠀⣠⡾⠟⠛⢻⡄⠘⣿⣿⠏⠁⠀⠀⠀⠈⠉⠉⠀⠀⣿⠏
⠏⠀⠸⠀⠀⢹⣿⡇⣿⠀⣸⣿⣿⣿⣿⣿⣿⡿⠘⣿⣿⡿⠀⡾⠋⠀⠀⠀⢸⣿⣶⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⠛⠃⣸⣇⠀⠀⠀⠀⠀⠉⠻⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⡀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⠀⣿⣻⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢀⣴⢻⣿⣧⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢀⠈⣜⣾⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣀⡀⢠⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⢉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠉⠁⠀⠀⠁⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠇⢳⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡐⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⡆⢻⠀⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠠⠃⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡜⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣧⠼⢦⡼⠃⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⡐⠁⠀⠀⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠎⠀⠀⠀⢧⣀⣀⣀⣀⣀⣤⣤⣀⢼⠠⠤⠴⠶⠶⠶⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠮⠍⠤⠛⠀⠀⡂⢽⠩⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠳⠀⠀⣠⡆⠀⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⣀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⣧⠇⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣶⡞⠀⠀⣿⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⢣⣀⠀⣀⣀⣠⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⢀⡄⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⡸⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠌⠁⡅⠀⠀⠀⠀⡠⠂⢰⠀⢠⣷⠃⠀⠑⠌⣢⡀⠀⠀⠀⠀⢠⡾⠛⠉⢦⠀⢸⠀⠀⠀⠀⢀⡞⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⢠⠊⡠⠊⠀⠀⡇⠀⠀⣠⠊⠀⠀⠀⡇⠀⣿⣶⣄⡀⠀⠈⠳⢄⠀⠀⠀⣼⠃⢠⠔⠊⠛⢾⣤⡀⠀⠀⡼⣧⠀⠀⠀⣼
⠀⠀⠀⠀⡐⠁⠊⠀⠀⠀⠀⢻⠀⢸⠃⠀⠀⠀⠀⡇⠀⠉⢻⠁⠉⠂⢀⠤⠊⠀⠀⣸⡇⠀⡇⠀⠀⠀⢈⡉⠁⠀⣿⠁⣿⡶⠶⢶⠋
⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠒⠒⠊⠁⠀⠀⢀⢜⡜⡇⢰⠃⠀⠀⠀⣈⡁⠀⠀⣯⠀⢉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⡤⠖⠊⠉⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⠀⠀⠼⣦⣀⣀⣴⠋⢧⠀⠀⠘⡇⢨⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣄⣀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠹⣄⠄⠀⠀⠉⢉⡰⠀⠸⡆⠀⢠⠃⡏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠙⢻⡄⠈⡁⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠉⠱⠖⠊⠉⠁⠀⠀⠀⢻⡀⣼⠀⡇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡇⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢹⡀⣧⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⡇⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢤⣇⡟⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡺⣿⠧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⡴⠚⣇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡛⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⣷⣤⣾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡄⠀⠀⠀⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠈⢀⣀⣤⣴⣶⣿⣿⠿⠟⠛⠋⠁⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⣠⣴⣶⣿⡿⠿⠛⠛⠉⠁⠀⠀⠀⠀⢀⣴⠾⣛⣥⣶⣶⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⣤⣶⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⣠⡶⡄⠀⠀⠀⠐⠋⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣼⡏⢠⠃⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠛⠉⣽⡇⠀⠀⠀⠀⠀⠈⠀⠀⢀⣤⡀⠀⣼⡟⢀⡞⢠⡖⠀⠀⠀⠀⠀⠂⠀⣠⣤⣀⠀⠄⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⢀⣠⣄⠀⢠⣿⢫⠇⣸⡿⢠⠎⢀⡄⠀⣰⡄⣠⣾⠆⣠⡾⠋⣸⣿⠀⠀⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣠⡴⠖⢀⣾⠋⡼⠀⣾⣣⠏⢠⣿⡵⠋⢀⡿⠁⣰⡟⡜⢱⡏⢠⣿⣡⠞⣹⣏⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⡏⠁⠀⠀⣾⣧⠞⠁⢸⡿⠋⠀⣼⡏⠀⣠⣿⢇⡼⢿⡟⠀⢻⡴⠋⠛⢋⣤⣿⠋⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⢺⡏⠀⠀⢀⣿⡇⢀⡴⣿⣇⡴⠃⠘⠋⢀⡀⠀⠀⠀⣴⠀⣰⠋⢸⡟
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿⠀⠀⠀⠸⣇⣠⠴⠁⠹⠷⠞⠁⠈⠉⠀⠒⢀⣠⣄⠀⠀⢀⣰⡇⠀⡇⢀⡿⠁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⡀⠈⢁⠀⢀⣠⣤⡀⠀⢠⠞⠙⡆⣞⣤⢸⠄⣴⠋⣿⠁⠀⠳⠛⠁⣈⡉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣴⠞⠁⠀⠀⣴⡿⠋⠉⣿⡆⢻⣴⠃⣇⣼⣧⣼⢼⠇⣼⣿⠀⠀⢀⡄⢀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠋⣡⡾⠁⠀⣼⡏⠀⠀⡔⣿⠇⣼⠏⣸⢁⡟⣰⠃⠘⠟⠁⣿⡀⢀⡞⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣰⡟⠁⠀⢰⡿⠀⣠⠎⢀⣿⡞⣿⣴⠃⠀⠛⠃⣀⣄⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠘⢷⠞⠁⢀⣼⡿⠀⠈⠁⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡀⠀⢠⠞⣹⡇⠀⠀⠤⠀⠀⢠⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡟⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠟⠛⠛⠿⠟⠁⠈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⣤⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠃⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄⠀⠀⠀⠐⠒⠄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠀⠀⠀⠡⠁⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢹⡄⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⠀⠀⠀⠀⠐⠀⣄⣀⣀⣤⠊⠀⠀⡇⠀⠀⠀⣴⠃⠁⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⡿⠿⡿⢿⠀⠀⢰⡇⠀⠀⠀⢈⡆⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠲⣄⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣞⣀⣀⠀⠀⡼⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠁⢾⠀⠀⠀⠀⠘⠢⠄⣀⠀⠀⢀⣀⡠⠔⠛⢆⠀⢸⣉⣉⣁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⡈⠑⢄⣀⠤⠎⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⣇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠤⠐⠊⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢶⠶⠦⠴⠶⠶⠒⠚⠛⠉⠉⡗⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⠀⠀⠀⠀⠀⠀⣀⣠⣞⠁⢠⡟⠐⠢⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠦⣀⣠⠏⠙⠋⠉⠉⠀⠉⠑⣯⠝⢃⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣰⠋⠉⠀⠀⠀⠀⠀⢀⣴⣯⣵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢤⣉⠓⠶⢤⣀⠀⠀⠀⠈⠉⠛⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂

⡀⠀⡀
⣇⡀⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠓⣤⣾
⣿⣿⠙⣆⣄⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠢⡀⢱⣾⣿⣿
⣿⣿⣿⣿⠿⡉⣀⠀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⢏⢢⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠺⣥⣷⣿⣁⢀⡴⠂⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡁⢡⢈⡢⣄⣾⣆⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡾⣿⣂⣼⣤⣸⡎⡹⠀⢰⡄⣠⠄⠀⣄⢄⠰⡀⣦⠀⡀⢲⣀⠙⢄⣠⣷⣏⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣯⣤⣷⣿⣴⣿⣿⣴⣄⣼⢸⣾⣶⣧⢿⣴⣾⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣍⣀⡀⠀⠐⠒⢭⡓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢟⣫⣭⣭⣭⣭⣇⣀⣀⣀⣐⣾⣾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⡵⡺⠟⠟⠛⠉⠁⠈⠉⠀⠀⠀⠈⠉⠹⣟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠏⠀⠀⠀⠀⠀⠈⠉⠙⠂⠀⠀⠀⠀⠀⠀⢿⡆⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠀⠀⠀⢸⢱⠀⠀⠔⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠘⠙⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⢠⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⣠⠔⣣⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠘⢦⠀⣀⡄⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡄⠀⠀⠀⠀⠀⢤⠐⠚⠉⠀⡀⣿⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢀⢂⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠘⢶⠒⠒⣉⢴⠇⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⠨⠖⠋⠀⠀⠀⠀⣠⠟⠀⢀⡠⡄⡡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣄⣀⠀⠀⠀⠀⣀⣠⠴⡿⠥⠒⠒⢉⣀⠟
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⠝⠋⢍⠀⣠⠃⡆⠒⠉⠉
⠀⠀⠀⠀⢠⡄⠀⠒⡆⠀⠀⠀⠀⠀⠀⣀⣀⡠⡆⠀⠀⢉⣀⠴⠆⠀⠉⠀⠀⢳
⠀⠀⠀⠀⠀⠤⠀⠁⢁⣀⡠⠜⠒⠒⠒⠒⠒⠚⠳⠒⠊⠉⢀⡞⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡬⢌⣉⠒⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠈⠉⠒⢤⣄⣉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠼⣄⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⣯⠙⡆⠴⠖⠂⠤⠤⠥
⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠊⡠⠚⠉⠹⡦⠴⠖⠒⠒⠒⠤⢀⠉⠉⠛⠒⠲⠶⠶⣂⣀⠀⠀⠀⠀⡏⣸⠁
⠠⡀⡀⠀⠠⢄⡀⢀⠎⢩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣇⠀⠀⠀⢀⡏⢀⣀⠤⢚⡴⠃
⠀⠑⢌⠲⡀⠀⠈⠙⠦⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣼⠢⠤⠒⠊⠁
⠀⠀⠀⠑⢌⡒⢄⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣤⣤⣤⣤⣤⣭⣤⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿