Vertigo Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠋⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁
⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⠃⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤
⣿⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢀⣿⣿⣿⣿⠇⠀⠀⢠⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⡀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⢠⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⠁⠀⢠⣿⣿⣿⣿⠟⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⡀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡆⠀⠀⣿⣿⣿⡟⠀⢠⣿⣿⣿⣿⠋⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠉⠉⠉
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧⠀⠀⢹⣿⣿⡇⠀⣼⣿⣿⡿⠁⠀⣰⣾⣿⣿⣿⣿⡿⠟⠋⠁
⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠘⣿⣿⣿⣿⡄⠀⢸⣿⣿⠁⢰⣿⣿⡟⠀⣠⣾⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⢿⣿⣿⣷⠀⠈⣿⣿⠀⣼⣿⡿⢀⣴⣿⣿⡿⠋⠁⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠈⠻⣿⣿⣧⠀⢻⣿⠀⣿⡿⢡⣾⣿⠟⠁⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠙⢿⣿⣧⡘⣿⡄⣿⣧⣿⠟⢁⣤⣶⣿⣿⠿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡙⢿⣷⣼⣧⣿⣿⣯⣶⠿⠛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣶⣤⣄⣀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⢿⣿⣿⡿⠟⠋⠉⠉⠙⢿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠟⠉⠀⠀⠀⠀⢀⡀⠀⠛⠙⠿⣶⣦⣄⡉⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢩⠉⠁⠀⠀⣀⣴⠄⣰⠧⠌⠛⠛⠃⢄⠈⠛⢿⣿⣶⣄⡀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⡿⠿⠿⠿⠛⠛⠉⠁⠀⠀⢀⠃⠀⠀⡠⠊⣰⣩⡾⠋⠀⠀⠀⠀⠀⠀⠑⠀⠀⠙⣿⣿⣿⣦⣄⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⡇⠀⢀⠞⠁⢐⡿⠋⠠⠀⢀⠤⠒⣈⣁⠒⠀⢎⡑⠀⠈⠻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠃⣾⣇⣶⠁⠀⡰⠋⢥⡘⠃⢀⡼⢂⠟⠁⣀⣴⠆⠀⠀⢀⡟⢧⠈⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿
⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⡿⠚⢻⣿⡏⠀⡴⠁⠀⠘⣧⠠⠊⣁⠩⠴⠋⢉⡁⠀⠀⡐⢩⡶⢾⠀⠀⠻⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠙⢿⣿⣿
⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⠀⠰⠀⠀⠀⠐⢫⡄⠈⠀⢀⡠⠊⠀⢣⠀⠀⠀⠀⣠⡶⠁⠀⠀⠹⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠈⠻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⡀⠀⣀⡋⢀⠂⠀⠀⠀⢠⠊⢀⡠⠔⠉⠀⠀⠀⠀⠘⢆⠀⠶⠟⠖⢠⡧⡀⠀⠸⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠘
⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠇⡰⠁⠀⠉⢀⣠⡤⣤⣄⢤⡞⠃⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠃⠀⠀⠄⠀⢹⣿⣿⣿⣿⣿⣆
⠛⠋⠉⠀⠀⠀⠀⠀⠀⢀⣧⡀⠀⠀⠀⢉⣿⡄⠘⠤⠄⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⢀⡄⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡶⣄⠀⠀⠹⡼⠀⢀⠤⠸⢷⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⡠⠽⠛⡛⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀
⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⠿⠋⠀⠈⠑⣦⡀⠈⠁⠖⠐⢶⡜⠓⡄⠉⠑⠲⠤⠖⠊⠉⠀⠀⡤⠞⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⣠⣾⣿⣿⣦⢀⣸⡆⠀⢳⣄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⠏⢻⡏⣦⠀⠀⢣⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇
⠋⠉⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⡿⠁⠀⠈⠀⡿⠀⠀⠀⠱⡘⣳⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢠⠅⠀⠀⠀⠀⠸⣷⡈⡆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⣿⠘⣾⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢀⡤⠞⠁⠀⠠⠄⠀⠀⠤⢄⣙⠀⠈⠃⠀⠀⠀⣰⠎⠁⠢⣄⣀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢀⡤⠞⠁⣠⠖⠂⠓⠳⠶⣶⠠⢄⡀⠉⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠈⠑⣶⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⡔⢀⡤⠚⠉⠁⠀⠀⠀⠀⠀⠈⠑⠢⢌⢦⡀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢺⣤⣿⣿⣿⣿⣿⣿⣿⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣶⣤⡀⠀⠀⠈⠻⣿⣿⣿⣧⡀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿
⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠉⠛⠻⢿⣷⣦⣀⠀⠈⠻⣿⣿⣷⡄⠀⠀⠀⢻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠈⠛⠿⣷⣤⡀⠈⠻⣿⣿⣆⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠛⠻⠿⢿⣿⣶⣤⣈⠙⢿⣦⡀⠘⢿⣿⣆⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⡙⢿⣦⠀⢻⣿⡄⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠏
⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀⠙⢿⣦⠹⣧⠀⢿⣷⠀⠀⣿⣿⣿⡏⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠿⣿⣶⡄⠹⣇⢹⡇⢸⡿⠀⢠⣿⣿⡿⠁⠀⠀⢀⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠙⢿⡄⡿⣸⢃⡿⠃⣠⣾⣿⡿⠁⠀⠀⣠⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢠⣾
⠀⣠⣴⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⣧⣷⡵⢋⣠⣾⣿⠿⠋⠀⠀⣠⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⢀⣴⣿⣿
⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⠟⠋⠁⠀⣀⣀⣤⣤⢭⣷⢿⣓⠛⠛⠉⠉⠀⢀⣠⣴⣾⣿⣿⣿⠟⠋⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⣿⣿⣿⠟⠁⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠀⠀⣠⣶⣿⡿⠋⣡⢞⢽⢻⡄⠛⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠟⠋
⣿⡿⠃⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠋⠀⠀⢀⣾⣿⡿⠋⢠⣾⢡⡏⣾⠘⣷⣄⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⡿⠛⠁
⠟⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠁⠀⠀⢀⣾⣿⣿⠃⠀⣾⡇⢸⣇⢻⣆⠘⠿⣿⣶⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠁⠀⠀⠀⣸⣿⣿⣿⠀⠀⢿⣷⠀⢻⣆⠻⣷⣄⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠘⣿⣧⠀⠻⣷⣌⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿
⠀⠀⣴⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠹⣿⣷⡀⠈⠻⣷⣄⡉⠛⢿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣾⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠹⣿⣿⣦⡀⠈⠛⢿⣶⣤⡈⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧⠀⠀⠀⠘⢿⣿⣿⣦⡀⠀⠉⠻⢿⣷⣦⣄⣀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⢻⣿⣿⣿⣦⡀⠀⠀⠈⠛⠿⣿⣿⣷⣶⣤⣄⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⠴⠖⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠶⠟⠋⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣶⣶⣦⣤⣬⣍⣐⡲⢦⡄
⠀⠀⠀⠀⠀⠀⣀⠰⠟⠉⠀⠀⣠⣴⣶⡼⠟⠛⠉⢉⣉⣁⠀⣉⣉⡁⢈⡛⠛⠉⢿⣿⣿⣶⣼⣷⣦
⠀⠀⠀⠀⠠⠞⠉⠀⠀⣠⣾⣧⠾⢋⣡⣄⠰⠿⠟⠘⠛⣉⡀⣉⣉⡁⣛⣛⠇⠴⣶⡬⢀⡛⢿⡿⢷⣾⣷⣦⡀
⠀⠀⢠⠖⠀⠀⠀⡰⢂⡴⢋⣵⠶⠘⠉⠀⡀⢤⣶⣦⣿⣿⡷⣿⣿⢷⣿⣿⣵⣿⣶⢁⣙⠿⠟⣢⣿⢿⣿⣯⣤⣦⡀
⠀⠀⠁⠀⠀⠀⠈⣠⣯⡶⠈⠁⡤⠂⠀⣨⣶⠟⢛⣋⡌⡭⠇⢲⣶⢒⡯⠅⠌⡛⠛⢿⣿⣇⣤⣝⠿⢻⣤⢿⣿⣿⠟⢀
⠀⠀⠀⠀⠀⠀⣼⡧⠉⢀⡴⠃⢀⣴⠟⣩⠰⢞⡌⢵⣂⠩⠍⠸⠋⠈⠉⠈⠙⠀⠤⠀⠀⠙⠿⣏⢰⣜⢻⢿⡍⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣼⠛⠀⣴⠇⢀⣰⠟⣥⠜⠩⢖⡌⢺⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡍⢤⡘⣿⣟⠿⣿⣋⣤
⠀⠀⠀⠀⠸⠋⠀⣴⠍⡐⣿⢫⡶⣁⠚⡵⣨⠺⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣆⢛⡈⣽⡖⣿⣿⣿⣇
⠀⠀⠀⢰⠇⠀⣌⠃⠰⣿⢷⡎⣠⢋⠞⢶⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠻⠘⢿⡘⣿⣧⣤
⠀⠀⠀⡌⠀⠸⠏⠀⣿⣿⡮⢠⡍⡚⡚⠶⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣧⢸⡇⢿⣿⣿
⠀⠀⠰⠇⠀⣾⠀⢰⣾⣯⡅⣩⢈⣃⣋⣛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡀⣿⢨⣿⣤
⠀⠀⠐⠀⠀⣧⠀⢈⠹⣟⡁⣉⢨⡅⡭⡭⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⣿⠸⣿⠹
⠀⠀⠀⠀⠀⢿⠀⠀⠀⣟⡃⢙⡈⣥⢷⠲⣛⣩⠄⡀⠀⠀⠀⠀⠀⠀⡠⡻⢝⠪⢀⠆⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⢱⡿⢰⣿⣴
⠀⠀⠀⠀⠀⢸⡄⠀⠀⢹⠗⠈⣃⠨⡄⢧⡣⡵⢛⡴⣡⢢⢦⢀⣲⡼⡝⡌⢆⠑⣀⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣼⡇⣸⡏⠿
⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢻⠇⠈⢥⡐⠆⠉⣌⠃⣔⠣⠝⣚⢘⡇⣇⢒⠨⠄⢊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣰⡿⢀⣿⢱⣶
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠙⠇⡈⠱⣄⠛⢤⡙⠢⢭⢑⡒⠸⠅⠧⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢁⣾⠃⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠙⢀⡀⠓⠄⣀⠓⠠⠤⢁⡉⢀⡉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢁⣾⢃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠈⠀⠀⠈⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣠⡾⢁⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⢁⡼⠋⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠟⢋⣠⡾⠋⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⣶⣶⣶⣶⣶⣶⣶⠾⠿⠛⠋⣉⣤⠶⠋⢁⣤⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠖⠚⠋⠁⣀⡀⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣯⠽⢷⣒⣒⣒⣒⣒⣒⠭⠍⠒⠩⠑⠬⠂⠕⠨⠢⠑⢝⢮⠢⡳⡱⡙⡝⡼⡼⡼⡜⡞⡟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⣟⣻⡽⣗⣚⡭⠭⢗⣒⣒⣒⣒⡒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠈⠘⠑⠱⢱⢳⢹⢹⢹⢹⢸⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣫⡽⢖⣚⡭⠽⣒⣒⣒⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠢⠣⡣⡫⡫⣿⣿⢟⡿⣻⣿
⣿⣿⣿⣿⣿⣿⣿⣶⣟⣯⠶⣛⡭⠖⣚⡭⠵⣒⡪⠭⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡨⡪⣪⣾⢗⣫⡾⣫⣿
⣿⣿⣿⣿⣿⣿⡯⢭⡭⠶⣛⡥⢞⣫⠵⢚⣩⠵⣒⡭⢅⡂⠤⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⢪⢞⢜⡵⣫⡵⠾⣋⣵
⣿⠿⠶⡾⢿⣭⣭⣿⣿⣿⣭⡶⢋⡵⢞⣩⠖⣋⠵⢊⡥⢒⠭⣂⠅⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡔⡱⢫⢞⡼⣫⡶⠟⣛⣭
⣿⣟⣛⣛⣛⣛⣛⣛⣶⣶⣶⣚⣋⡴⢋⡴⢚⡥⢞⡥⢚⡥⣊⢔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠞⡵⣣⠞⡴⢋⣴⠿⢛⣫
⠿⣿⣛⣛⣛⣛⣛⣻⣿⣭⣭⣭⣭⡶⢋⡴⢋⡴⢋⡴⣫⢞⡵⡡⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡡⢎⡴⢣⡞⣱⢟⣵⣾⠿⣛
⣿⠿⢿⣛⣛⣫⣭⣭⣭⣥⣶⡶⠶⠞⢋⣵⠏⡴⢋⠜⡡⢫⢎⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢋⡴⢋⡼⣡⠟⣩⣶⡿⢿
⣶⣿⣿⠿⢛⣛⣩⣭⣥⣴⣶⠶⠾⠛⣋⣥⠞⡵⢫⢞⡔⡡⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠰⡐⡄⠀⠀⠀⠀⠀⠀⠀⠨⣙⣉⡴⢋⡴⢫⡾⢋⣵⣾⠿
⢟⣩⣶⣿⡿⠟⢛⣩⣥⣴⣶⠶⠟⢛⣩⡴⢞⡴⡣⢪⢊⠈⠀⠀⠀⠀⠀⠀⠀⡀⣌⠸⡌⢣⡙⢌⢊⢂⠀⠀⠀⠀⠀⠀⠮⣥⡴⢋⣵⢏⣴⢟⣩⣶⣿
⣿⣿⠟⣩⣴⣿⡿⢟⣩⣥⣶⠾⠟⣋⣥⠶⣫⢞⢔⠕⠁⠀⠀⠀⠀⠀⠀⣰⠀⣇⠘⢦⡙⠦⡙⢮⡣⡱⡡⠀⠀⠀⠀⠀⣑⠶⠶⢛⣡⠞⣱⠿⣫⣵⣾
⡿⣡⣾⣿⢟⣡⣶⡿⢟⣩⣴⠾⢛⣩⡴⣪⢕⢕⠅⠀⠀⠀⠀⠀⣠⢰⡇⠹⣦⡈⠳⢦⣍⠳⣌⠳⣕⢔⢕⡄⠀⠀⠀⠀⠪⡓⠶⢛⣡⡾⣫⡾⣟⣽⣾
⣱⣿⡿⣡⣿⡿⣫⣶⡿⢋⣵⡾⣋⡵⣺⢕⠕⠁⠀⠀⠀⠠⢠⠃⣿⡈⠻⢦⣌⡙⠳⣦⡈⢷⡌⢷⡘⣎⢯⢪⢖⢀⠢⠀⢐⢍⣛⣛⣡⡾⣫⣾⢟⣽⣾
⣿⣿⣽⣿⢏⣾⡿⣫⣾⢟⣥⢞⡭⣺⣟⡗⠀⠀⠀⣠⢰⡇⢻⣦⣈⠛⢷⣦⡈⢻⣦⠘⣿⡌⢿⡌⣷⢹⡜⣏⢯⢟⢜⢦⡪⡳⣍⣉⣭⡾⢛⣵⣿⣿⣿
⣿⣿⣿⢯⣿⢟⣼⢟⣵⢟⣵⡯⣮⡿⢛⡡⡠⣰⠁⣧⠘⢿⣦⣌⠛⢿⣦⠙⣿⣆⢻⣷⠘⣿⡘⣷⠸⣇⢻⡸⡞⣾⣷⣵⢟⢝⠦⣭⣥⡾⢟⣵⣿⣿⣿
⣿⣿⣿⣿⣯⣾⣫⣾⣵⣿⣿⣿⡿⡻⣱⢱⡇⢿⣄⠙⢿⣦⣌⠻⣷⣌⢻⣷⡘⣿⣆⢿⣧⢹⣧⢹⣧⢻⡜⣷⢹⡼⣿⣿⣿⡿⣗⠶⠶⠾⣻⣵⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⡽⣹⢱⣷⠸⣷⣌⡛⢿⣦⡙⢿⣧⡹⣿⣆⢻⣷⠸⣿⡌⣿⡎⣿⡆⣿⡎⣿⡼⣧⢻⡽⣿⣿⣿⣮⣟⣻⣟⣯⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣳⣧⢻⣦⠻⣷⣮⡻⣿⣦⡹⣿⣮⣿⣷⣽⣿⣮⠿⣧⣿⣷⣽⣿⡸⣿⡘⣿⡜⣿⡽⣷⣻⣿⣿⣿⣟⢮⣭⣯⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠙⠛⠛⠛⠲⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠈⢹⠉⠉⣻⠛⠛⠛⡟⠛⠛⢿⠛⠛⢻⠛⠛⢻⠛⠛⢹⡏⠉⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢾⣦⣼⣄⣀⣽⣀⣀⣠⣇⣀⣀⣼⣀⣀⣸⣀⣀⣸⣄⣀⣼⣄⣤⣧⠾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠧⠤⢤⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣁⡠⠿⣆
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣼⠷⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠘⠛⠛⣶⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣷⠀⠀⠀⠀⣴⠟⠛⠃⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡏⠀⣼⠋⠉⠉⠉⡏⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⠟⠀⠀⢿⣤⣤⣤⣤⣿⣤⣤⣤⠾⠁
⠀⠀⠀⠀⢀⣤⠶⠶⢶⣤⣤⣭⣍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡴⠶⢶⣤⡀
⠀⠀⠀⠀⣟⠀⠀⠀⡼⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠛⠁⠈⢹⣧⠀⢀⡼⠛⢶⣄
⠀⠀⠀⠀⠙⢦⣄⣀⣳⡀⠀⠀⢀⣰⠟⢀⣀⠀⠀⣿⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠙⢦⣠⡾⠋⠳⣮⡀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢛⣻⣷⠞⠋⠉⢻⡄⠹⣦⡀⢠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢿⡀⠀⠀⣠⡾⠋⠀⠀⠀⠈⠻⢦⣄⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠈⢧⠀⠀⣠⠟⠀⠈⠻⣧⡀⠀⠀⠀⠀⢹⡇⠀⠀⠈⠛⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣀⣼⠷⠛⠁⠀⠀⠀⠀⠈⠻⢦⣀⣀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⡀
⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣶⣶⣶⣶⡶⠶⣶⣶⣶⣶⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⡿⠿⢰⣇⠸⠿⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣇⣛⢻⡿⢛⣃⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣘⣃⣸⣿⣿⣿⡇⠀⠀⣿⡴⠖⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣡⣴⣾⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣧⡴⠶⠶⠶⠶⣿⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠙⠳⠶⠶⠦⠴⢿⣿⣿⣿⡿⠟



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠪⢒⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠊⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠀⠀⠐⠒⠶⣶⣄⡀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠉⠻⢷⡂⠤⠒⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⢼⡇
⢀⡄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢸⣦⡀⠀⠀⠀⠀⢀⡠⠛⠲⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⡈
⡟⠁⠀⠀⠘⡄⠀⠠⠄⠀⠀⠀⠀⣻⠀⠈⠳⡀⠀⢠⠔⠁⠀⠀⠀⠀⠙⢄⢀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠁
⢹⡄⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢳⣤⣶⣷⣦⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁
⠀⢹⡄⢰⣄⢤⣼⡄⠀⠀⠀⠀⠀⣸⣽⡄⠀⣀⢀⠆⡶⣄⡀⢐⡤⠀⣀⣀⣨⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠃⠀⠀⠀⠀⠀⣰⠄
⠀⠀⢻⣿⣷⡈⢿⣿⡆⠀⠀⠀⣠⡿⠻⠃⠚⠉⠉⠈⠐⠋⠻⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠟⢋⣡⣤⣶⠄⠀⢀⣠⡴⠞⠋
⠀⠀⠀⢿⣿⣷⡈⢻⣿⡄⠈⠈⠍⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠉⠉⠣⠛⡄⠀⠀⠀⠀⠀⠘⣧⣴⣾⣿⣿⣿⡿⠷⠛⠉
⠀⠀⠀⠈⢿⣿⣷⣼⠛⡇⢀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⢸⡉⠀⠀⠀⠠⡙⣶⣤⡀⠀⠀⠀⣀⣀⡀⣠⡴⠔⠋⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⢢⣼⠿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣿⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣠⡀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡱⠂⠈⠛⢿⣿⣤⣌⠻⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠚⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢆⠀⠙⠻⣿⣧⣄⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⢀⢂⢀
⠀⠀⠀⠀⢺⢆⡀⠀⠀⠀⠀⣀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠈
⠀⠀⠀⠀⠀⠢⣍⣿⣷⣏⣍⣇⡴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠀⠈⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⣀⡤⠖⢒⣶⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠘⠁⢀⠤⠚⣛⡛⢟⣟⣻⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⡄⠀⠀⠀⠁⡰⠟⢛⣿⣮⣭⣧⣤⣤⣝⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢰⢱⣤⣆⡴⢊⠔⢺⣿⣏⡙⣿⣿⣿⣿⣿⣿⢦⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠛⢿⡴⠁⠀⠀⢻⠀⠀⢹⣿⣿⣿⣿⣿⡇⡟⢶
⠀⠀⠀⠀⢀⣠⡤⠶⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠺⠤⠈⠷⡦⠾⠆⠀⣠⣿⣿⣿⣿⣟⡏⢱⣾
⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣤⣤⠄⠀⠄⠀⠘⠻⣿⣿⣿⣿⡟⠀⣾⡟
⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⣉⢿⣿⣀⠀⢀⠀⠀⡀⠀⠈⠙⣿⣿⣷⡦⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠆⠀⠀⠀⠀⠀⠀⠻⢆⣿⡿⠋⠉⠁⠉⠁⢀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢈⠳⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠈⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠈⠳⣦⣌⡉⠙⠛⠛⠛⠛⠛⢉⡉⠤⠀⡾⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⡀⠀⠀⠀⠐⡆⠀⠀⠀⢹⡤⡤⠐⠁⠀⠀⠀⣆⡀⠀⠀⠀⢸⠤
⠀⠀⠈⠙⠒⠷⠠⠀⠠⠈⠄⠄⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⠀⢀⣧⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣦⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠐⢶⠶⠆⠀⠠⠀⠀⠀⠀⠙⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣛⣛⣲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠞⠻⢦⣤⣀⣉⣉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣁⣬⠟⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⠶⣶⣤⣬⣭⣭⣉⣉⣉⣉⣉⣉⣉⣉⣩⣥⣤⣶⡾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⢸⡇⢸⡏⢹⡏⢹⡏⢹⡏⢹⡏⠉⡏⠉⡇⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣄⣸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⠀⡇⢀⣇⣠⣧⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠙⠛⠛⠻⠿⠷⠾⠷⠾⠷⠾⠿⠿⠛⠛⠛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⡾⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣴⣦⣤⣤⣀⣀⣀⣀⣤⣤⣤⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣤⣤⣤⣀⣀⣤⣤⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣉⣉⣉⣛⣛⡋⠉⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇⠀⠀⠀⠀⠀⠀⢀⣴⠶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠛⠿⢿⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⠿⠟⠛⢻⡇⠀⠀⠀⠀⢀⣴⣟⡱⠋⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠛⠻⡿⠷⠶⣦⡀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⢰⡿⠃⠈⠹⣦⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⣠⡇⠉⠛⢸⡇⠀⠀⢻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡇⠀⠸⣧⣀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⢿⣿⣷⣾⠟⠁⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢟⣛⡛⣟⠛⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠉⣀⣿⣀⣀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠥⠤⠠⠤⠴⡆
⠀⠀⠀⠀⠀⠀⣠⠤⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⡏
⠀⠀⠀⢶⣒⠊⠀⢀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠤⠤⠤⢶⠗
⠀⠀⠀⠀⠈⠉⠉⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠎⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢳⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢳⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢠⣎⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣷⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⡇⠘⠳⠞⠀⠻⠶⠃⠀⠳⠞⠀⡇⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⢠⣏⣀⣀⣀⣀⣀⣀⣀⣀⣷⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣀⣀⣀⣀⣀⣷
⠀⠀⠀⠀⢸⡇⠙⠒⠃⠘⠒⠃⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠛⠚⠃⠙⠒⠃⢸
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠠⠀⠀⢸⣤⣤⡀⠀⣠⣤⡀⡇⣠⣄⡀⠀⣠⣤⡄⠀⣠⣠⣄⡇⢠⣤⣄⠀⢠⣠⣼⠀⠀⢀⠀⠠⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠒⠒⠀⠀⢸⠀⠀⢧⣤⡇⠀⣧⣧⡇⠀⢧⣤⡇⠀⢧⣤⡗⠀⢯⣧⡼⠀⢿⣤⡼⠀⢸⠀⠀⠈⠒⠚⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠈⢢⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠔⠢⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠐⠊⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢻⠉⠀⠀⠀⠉⡟⠦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠂⠐⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⡴⠃⠀⢸⠀⠀⠀⠀⠀⡇⠀⢹⣆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⡄⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⢸⠸⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠃⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⢀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣇⣀⣀⣀⣸⣀⣀⣀⣀⣀⣇⣀⣸⣀⣷⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠉⠀⠈⠈⠁⠈⠀⠁⠈⠁⠀⠁⠈⠀⠀⠀⠀⠀⠁⠈

⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣽⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠓⠀⠀⠠
⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣉⣉⣉⠉⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿
⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣤⠀⠒⠀⠁⠀⠀⠀⢀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⣴⣮⡃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣄⣶⣤⣤⣤⣤⡤⠴⠶⠖⠒⠋⠁⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣽⣃⣢⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⡤⢤⣤⣥⣤⡤⠶⠶⠶⠒⠛⠛⠛⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿
⢿⡋⠙⠏⠀⠀⠀⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠌⠉⠀⢿⣿⣿⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁





⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠳⡇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢠⠞⣧⠀⠀⢀⣀⡤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⣠⠋⠀⠘⠚⠉⠉⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠈⠙⠲⢤⠞⠉⠉⠙⠣⡀⠀⠉⠳⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⡈⣧⠀⠀⠀⠀⡰⣋⣞⠉⣱⣄⡀⢸⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⡼⠁
⠙⣄⠀⠈⠉⠉⠉⠀⠈⠉⠉⠀⠀⢠⡾⣽⠣⣼⠃⢌⣇⡜⡸⠀⢀⡤⠲⡈⠳⣄⠀⠀⠀⠀⠘⠧⠤⠤⢤⣤⣀⣀⣀
⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠒⠲⢧⡦⢴⠛⣬⣶⠁⡰⠛⢢⠰⢇⣠⠟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠈⢾⣠⣿⠏⠻⣅⣀⠼⣀⡠⠃⢱⠛⡆⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⣨⠆⠀⠀⢠⡇⠀⠀⠀⠀⠠⠁⠀⣼⠟⠁⠀⠀⠘⠷⣤⢼⠒⠒⡇⢀⠇⠀⠀⠀⠀⠀⠐⠛⠲⠤⣄⣀
⠀⢀⡠⠖⠋⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠛⠳⠿⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠽⠗
⠶⣉⡀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉
⠀⠀⠉⠙⠒⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡛⢏⣙⡦⢵⣈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠉⢫⠈⣇⣸⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡏⣧⣞⡁⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣀⣀⣀⣈⣶⡄
⠀⠀⠀⣴⣁⣤⠤⠤⠶⠒⣶⠂⠀⠀⠀⡠⢿⡀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠉⠙⠲⢤⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣠⠔⠋⠀⠀⠙⠓⠒⠚⠋⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠑⠢⢌⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡠⠞⠁⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡽⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⢀⡴⠋⠸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣥⡄⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣈⡋⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⢀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⢶⣾⡟⢹⡟⠛⣷⣶⢶⣦⣄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣤⣶⢿⣿⣿⠇⠀⠻⣿⣅⣸⣇⣀⣤⡿⠀⠘⣿⣿⣶⣤⣀
⢀⠀⢠⣾⠟⠀⠀⢀⣴⠾⠿⠉⠁⠘⣧⣀⣀⡀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⣸⣿⠃⠈⢻⣷⣄⠰⣶⡄⠀⢀⣦
⢿⡀⠸⣿⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠤⢤⣤⣶⡶⠾⠿⠷⠟⠋⠁⠀⠀⠈⠉⠙⢷⡌⠉⠀⠈⣿
⠈⢷⣄⠙⠂⠀⣺⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⢠⣴⠟
⠀⠀⠉⠀⠀⠀⠹⣿⠦⣄⣀⣀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⢀⣀⣤⣾⠟⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⣈⠉⠙⠛⠻⠦⠤⢤⣤⣤⣴⣾⣿⣿⣶⣤⣭⣭⡭⠵⠶⠾⢟⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣶⣤⣤⣴⣶⡾⠿⠛⠁⠀⠀⢀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣼⡇⠀⠀⠀⠀⢀⡾⠋⠙⠋⠉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠘⣷⣀⣘⣿⠆⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⢉⣉⣁⣠⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⣤⣤⣤⣤⣤⣶⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠦⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⢤⡀
⠀⠀⠀⠠⢠⡆⢠⡆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠈⠀⠀⢠⣄⠁
⠀⠀⠀⡄⢸⡇⢸⡇⢠⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⡿⣷⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣶⣄⢹⣯
⠀⠀⠀⣷⣸⣟⣸⣗⣼⡄⠀⠀⠀⠀⠀⣸⣿⠀⣶⣿⣿⣦⢠⣿⡇⠀⠀⠀⢀⣀⠀⣿⠀⣿⣿⠀⣿⠀⣀⡀⠀⠀⣰⣆⢸⣷⢿⡿⡿⡮⡿⢠⣦
⠀⠀⠀⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣠⣿⣿⢴⣶⣿⣷⣶⣾⣿⣷⡀⠀⠀⢹⣯⣼⣿⣿⣿⣿⣿⣿⣧⣿⡇⠀⠀⢻⣿⣶⣾⣾⣷⣷⡗⣶⣿⣿
⠀⠀⠀⢸⣿⡏⢹⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⠟⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢦⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠇⢀⡀
⠀⠀⢰⣶⣶⡆⠰⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⢀⡈⣾⣦⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣾⡆⠀⠓
⠀⠀⠠⣯⣼⣤⣤⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣇⣏⣧⣼⠁⠀⠀⠀⠀⠀⠀⢻⣟⣀⠀⣿⣿⢀⣀⣻⡗⠀⠀⠀⠀⠀⠀⣦⡼⡿⢿⣴⠀⡀
⠀⣠⣠⣿⣿⠛⡟⣿⣿⣅⣄⠀⠀⠀⠀⣀⣺⣀⡟⠛⠛⠿⣰⣇⡀⠀⠀⠀⢀⣾⣿⣉⣉⣿⣿⢉⣉⡿⣇⡀⠀⠀⠀⢸⣧⣬⣻⣿⣟⣡⣤⣷
⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⡏⠀⠀⠀⠀⠹⣿⣿⣷⡶⣷⣾⣿⣿⠃⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠸⠿⠿⠿⠇⠈⠿⠿⠿⠿⠀⠀⠀⠀⠐⠿⠿⠿⠃⠸⠿⠿⠿⠀⠀⠀⠀⠾⠿⠿⠿⠿⠇⠸⠿⠿⠿⠿⠷⠀⠀⠀⠀⠿⠿⠿⠁⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠃⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⢴⠄⠀⠐⠆⠀⣶
⠀⠀⠀⠀⢰⡇⢹⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⢀⠠⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢶⣶⠀⣿⡇⠀⠀⠀⠀⠀⠠⠾⢧⠀⣼⠄⠰⠿⠆
⠀⠀⠀⣶⢸⣷⣾⣇⢰⠀⠀⠀⠀⠀⠀⠀⠀⣧⢼⡿⣿⢧⡇⠀⠀⠀⠀⠀⠀⠀⣠⣤⠁⣾⣟⢠⣿⡇⠀⠀⠀⠀⠀⠘⣟⡟⣤⣿⣥⢹⣻⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣀⣸⣿⣾⣷⣿⣞⣿⣀⣀⡀⠀⠀⢠⡄⢸⣿⣶⣿⣿⣼⣿⡇⢠⠄⠀⠀⠀⣘⣛⣟⣛⣟⣛⣻⣟⣓⡀
⠀⠠⣤⣿⣿⣿⣿⣿⣿⣤⡄⠀⠀⠀⢘⣟⣿⣛⣟⣛⣛⣟⣻⣛⣻⡁⠀⠀⢰⣷⣾⣿⣿⣿⣿⣿⣿⣷⣾⡄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⢀⣿⡟⣿⡿⢻⣿⢻⣿⡇⠀⠀⠀⣸⣿⣿⣿⣿⣟⡿⠛⠟⠛⢙⠁⠀⠀⠈⣿⡟⠻⣟⠿⢿⢿⡟⢿⣿⠀⠀⠀⠀⢰⣿⣿⣿⡏⣿⣿⣿⣇
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠉⠉⠁
⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⣤⣤⠀⠀⠈⣿⣿⡇⠀⠀⣠⣤⠀⠀⠀⢴⣶⡦⢀⣿⣿⠀⣴⣶⡆⠀⠀⠀⠀⣤⠄⣈⣿⣿⣿⣿⡀⠠⣤⠀⠀⠀⠀⠀⠀⢰⣦⢀⣿⡦⢼⣿⠁
⢸⣿⣀⣀⣀⣿⣿⣇⣀⣀⣸⣿⠀⠀⠀⢸⣿⠷⠶⠿⠿⠶⣿⡾⡇⠀⠀⠀⠀⣿⡀⣟⣛⣛⣛⣛⣛⢀⣿⠀⠀⠀⠀⣿⡇⢸⣯⢸⣿⡇⢸⣿⠂
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠠⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣯⣤⠀⠀⠀⣿⣷⣶⣷⣶⣷⣶⣾⣿⠇
⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⡟⣻⣿⣿⣿⡇⠀⠀⠀⣸⣿⣿⢻⣿⣿⣿⣿⠟⣿⣿⣇⠀⠀⠀⣾⣿⣿⣿⣿⣿⡏⢹⣿
⠉⠉⠀⠀⠈⠁⠀⠁⠀⠀⠈⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠁⠀⠉⠉⠉⠉⠀⠈⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠟⠉⠁⠀⠀⠀⠙⠃⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡆⠀⠘⠁⠀⠈⠁⠀⢠⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⡏⠈⠀⠒
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠇⠀⠀⢀⠀⠄⠄⠀⠀⠀⠀⠀⠘⣇⠐⠘⠂⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⡀⠁⣀⡉⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⢀⠠⠈
⠀⠀⠀⠀⠀⠀⠢⠀⠈⡁⠀⠀⠀⠀⠀⠢⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠠⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠰⠊⠀⠒⢢⠄⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⠀⡀⠀⠀⠘⠰⠄⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠸⣄⠀⠂⠀⢀⡇⠀⡏⠀⠑⠀⠀⠁⠈⡀⠑⠢⢀⠀⠀⠂⡀
⠀⠀⠀⠈⠳⣄⠀⠘⡁⢰⠃⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡈⠗⠣⠑⠂⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠈
⡠⠀⡀⠒⠐⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠇⠐⠛⠋⠁⢨⠒⠰⠦⠤⠀⠀⠀⠀⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⣠⣁⣀⣀⠀⠀⠀⠀⢘⡀⠀⠀⢀⣘⡀
⠀⠀⠀⠀⠈⠉⢙⠁⠀⠀⠘⠛⠋⠃⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡀⠸⣠⡤⠄⡀⢀⡔⠀⠑⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠁⠀⣀⣀⣠⠀⠀⡀⢉⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠡⠀⠤⠐⠈⠁⠀⠸⠛⡋⠁⠀⡎⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⠀⢧⠀⠀⡇⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⡍⠀⡐⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠰⠇⠰⠅⠀⡃⠀⢠⡑
⠀⠀⠀⠀⠀⠄⠐⠈⠉⠡⠀⠀⠀⠀⢀⡀⡀⠋⠐⠊
⠀⠀⠀⠀⠰⣄⣀⣀⠀⠤⠧⠐⠒⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣤⣤⣴⣶⣶⡄⠀⠀⠀⢹⠿⠁⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠿⡟⠂⢠⣶⣤⣤⣄
⠀⠀⠀⠀⠺⡿⠇⠀⣾⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠛⠛⠙⠛⠛⠻⠿⣶⣤⣀⠀⠀⠀⠀⠈⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⣴⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆⠀⣼⡟⠻⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⢻⣇⠀⠠⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⢛⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠀⠀⢸⣿⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡷⠀⣿⡅⠀⠀⣴⡿⠛⠉⠙⠻⢿⣶⣤⣀⡀⣀⣤⣴⡾⠿⠛⣁⣀⡀⠀⠀⠀⢸⣿⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣿⡇⠀⢰⣿⠁⠀⠀⢀⣀⣠⣤⣿⡿⠿⢿⣿⣧⣶⣶⣿⠟⠛⣿⣄⠀⠀⢸⡿⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣶⣾⣿⡶⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⠿⣿⣶⣶⣿⣷⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⡿⠀⠀⠀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⣤⣤⣀⠀⠀⠀⠀⣿⣿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠂⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⣠⣾⠟⠛⠛⠛⠻⢿⣦⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣄⣀⣀⣀⣀⣠⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠉

⠀⠀⢀⠀⢤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⡀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⣤⣸⣾⣿⣷⣏⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⢾⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⣶⣶⣤⣀
⠀⣠⣿⣿⣿⣿⣿⣠⡀⠀⠀⠀⢠⠐⠒⠁⠀⠀⠱⠿⡻⡍⠀⠀⠀⠀⠃⠀⣠⣿⡿⠤⡁⠀⠀⠀⠀⡔⠒⠋⠀⠀⠈⠮⠟⣿⣿
⠀⠀⠸⠛⣻⠛⢧⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠀⠀⠤⠃⠀⠀⠀⠀⠐⠲⠿⠟⠠⠤⠤⠬⠆⠀⠀⠀⠣⠤⠤⠤⠤⠤⠤⠜
⠀⠀⠀⠀⠈

⠀⠀⢀⣄⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣦⣤⡄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣠⣀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣶⣤⣄
⠀⢠⢿⣿⣧⠀⠿⠟⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠼⢦⠿⡼⠗⠐⠻⠓⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠁
⠀⢺⡸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠶⢀⠀⠶⠀⠀⠀⠀⠀⠺⢋⣯⡹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠂⠀⠂
⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠁

⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⡀
⢀⣴⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⡀⠀⠂⣤⠀⡀⠀⡀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠁⣶⢈⢀⠀⣶⢀
⠀⠙⠛⠛⣛⡻⠛⠋⠀⠀⠀⠀⠀⣀⠀⣼⣿⣷⡀⣀⠀⠀⠀⠀⠀⠉⠛⢛⣿⠛⠛⠋⠁⠀⠀⠀⠀⠀⢇⣿⢸⢸⢐⣿⢸
⠀⠀⢀⢸⣓⠯⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠘⣻⠋⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣻⠸⣻⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣬⣭⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⣀
⣴⣾⣿⣿⣷⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠜⢀⣝⡁⠘⠋
⠈⢩⠋⠉⠀⠀⠀⠓⣢⣀⡸⠂⠀⠀⠀⠀⠘⠿⣿⣿⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢉⢆⡼⠁⠉⠉⠁
⠀⠀⠐⠀⠀⠀⠒⠒⠋⡃⠀⠀⠀⠀⠀⠀⠀⠼⠟⠁⠀⠀⠀⠀⢠⣴⣷⣶⣿⣷⣾⣿⣤⠀⠀⠀⠀⠀⢠⣤⣾⣧⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠘⢁⠄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠉⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⢠⡝⠐⠞⠋⠄⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣋⠠⠄
⠀⠀⠀⠊⠀⠂⠐⠢⠀⠀⣿⡅⠀⠀⠠⡀⠀⠀⠩⠄⠁⠉⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡳⠤⠄
⠀⠈⣉⠁⢀⠶⠤⢠⡀⠀⢉⠁⡄⠀⠀⠍⠀⢠⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⣙⠀⠀⡶⠈⠀⠄⠀⢠⡜⠁
⠀⠀⡄⡀⢀⠀⡶⠀⢠⡄⠀⢰⠀⠀⡀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡐⠀⢠⠁⠀⠀⠠⣽⣇⠲⢀⠁
⠀⠀⡿⠄⠈⠳⠀⠀⣿⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⡇⠀⠁⢸⠀⠀⠀⡰⠑⠌
⠀⠀⢰⠀⠈⠀⠀⢰⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⠀⠀⢰⠃
⠀⠀⠸⡄⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢸⠄⢀⣏⣀⣠⣤⣤⡀
⣀⣀⡀⣧⢀⣀⡀⣾⡯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠃⠈⠛⠛⠉⢉⣉⠉⠉
⠀⣞⣆⣿⣗⡙⠉⣿⡅⠈⠁⠀⢀⠀⠉⠛⠋⠉⠀⠀⠈⠉⣤⠤⢍⣀⠀⠀⠀⠁⠀⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐
⣾⣼⣿⣿⣿⣷⣿⣿⠃⣶⣦⢀⠈⠀⠠⠄⠐⡖⣦⣴⣤⣀⣀⣴⣿⣿⣶⣆⡠⣴⣦⣴⣿⡷⠒
⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣷⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠉
⣿⠿⢿⣿⠟⠛⣻⡟⠀⠉⠉⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠠⠿⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣥⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠉⠉⠙⠛⠛⠻⠿⢿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠉⠻⣿⣦⣄⣠⣤⣤⣴⣶⣶⣶⣾⡿⠿⠿⠿⠟⠛⠻⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠙⠻⣿⣯⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣽⣿⣷⣄⡀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣟⡛⠛⠋⠉⠉⠉⠙⠻⣿⣶⣤⣀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣻⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠉⠀⠈⠙⠻⣿⣷⣤⣀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⣀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣷⣶⣶⣶⣶⣿⣿⠁
⠀⠀⠀⠀⠀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣿⠃
⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣶⣶⣤⣤⣤⣤⣤⣤⡾⠋
⠀⠀⠀⠀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠉⠉⠉⠉⠁
⠀⠀⠀⠀⣿⣿⡿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣦⣄⣀⣠⣾⠟⣿⣿
⠀⠀⠀⠀⠘⣿⣧⡈⠙⠻⣿⣿⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣿⣿⣿⠟⠋⢠⣿⣿
⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠈⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⣿⡀⠀⠀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢶⠀⠀⠖⢒⣻⣷⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⢀⣸⡆⡀⠋⠉⢻⡏⢀⣀⣸⡆⠀⠀⠀⠀⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠘⢻⡇⣧⠀⠀⣸⣷⢸⠘⢻⡇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⢠⣼⣧⢰⡄⣸⢿⣿⣼⣄⣼⣧⠀⠀⠀⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣤⣿⣷⡄⠀⢲⣿⣿⡾⣰⡟⣏⣿⣿⣿⢿⣿⠀⢠⣮⣤⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣿⢠⡀⠃⣹⣿⢉⢹⣿⣿⣿⣿⡏⢸⣿⢀⡄⣤⣤⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢀⣿⡏⣹⣿⠠⣞⡟⠾⠿⡟⡿⢙⣿⢃⣿⣆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠙⡇⣻⡿⠀⣀⣤⣤⣀⠀⠀⢻⣿⡟⣹⣿⣧⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢹⡃⠀⢘⡇⢸⡇⢸⣿⣿⣿⣿⠂⠘⠀⣿⡧⣻⣿⡏⣽⣇⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⡇⣶⣿⣿⣄⣼⡇⣹⡇⠠⣿⣿⢻⣿⠀⠒⢈⣿⣧⣾⣿⣿⣿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠶⢶⣶⣿⣧⣶⣻⣿⣛⣻⡷⠞⠋⠉⠉⠉⠳⢾⣛⣛⣿⣓⣦⣼⣿⣶⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣽⣿⠁⠀⠀⠀⠀⠀⠀⠀⠙⢯⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠋⠉⠙⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⠛⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠥⠤⠤⣀⡀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠏⠀⠀⠀⠀⠀⠉⠓⣄⠀⢀⡼⠁
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡞
⢀⡴⢏⠉⠋⠉⠙⠓⠦⣄⠀⣷⣀⣀⣀⣀⣀⣤⡤⠤⠤⠶⠒⠚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⣾⡇⠀⠑⢦⡀⠢⡀⠀⠈⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⣯⠱⡀⠀⠀⠙⢦⡈⠢⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠹⡄⠱⣄⠀⠄⣀⠙⠢⡘⢦⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠐⠀⢀⡾⠁
⠀⠙⣆⠈⠣⣄⠀⠀⠀⠈⠢⡙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠈⠛⢦⣀⠉⠒⠦⠤⠤⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠶⠴⠶⠚⠉⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⣠⠴⠚⠋⠉⠁⢀⡠⠞⠋⣻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⢀⣴⠋⠁⠀⠈⠁⡠⠖⠉⣠⠔⠋⠁⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠈⠛⠲⠶⠶⠒⠋⠁⠀⠀⠀⠀⣼⠃⠀⠀⣠⠔⠁⢀⡴⠊⠡⠀⠀⢀⢼
⠀⠀⠀⠀⣠⡞⠻⢭⣀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠦⣴⠋⠀⣀⠔⠁⠀⢀⣀⠤⠞⢁⡾
⠀⠀⢀⡾⠉⠙⠲⢤⣈⠉⠒⠤⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡉⠉⠉⠋⠉⣉⣉⣀⡤⠶⠋
⠀⠀⢸⣧⣀⠀⠀⠀⠈⠙⠲⢤⣀⠉⠒⠤⣄⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢿⡈⠓⠦⢄⡀⠀⠀⠀⠈⠙⠲⢤⣀⣉⣳⠖⢻⡆
⠀⠀⠀⠈⠳⣄⠀⠀⠈⠉⠑⠒⠒⠒⠒⠚⠋⠉⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣶⣶⣶⣶⣾⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡈⠻⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢷⣌⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⣿⠈⠻⣦⣈⣻⣶⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⠀⣷⣄⠙⢷⠌⠉⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠻⣯⡙⠃⣿⡏⠻⠿⠛⠛⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣄⠰⣶⣾⡄⢻⣤⣤⣤⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋⠻⣦⡈⠻⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢶⣄⠙⢷⣌⠻⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣦⡈⠻⣦⣨⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣦⡙⢷⣄⠙⠷⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣤⣤⣶⣤⣸⣿⢠⣌⣻⣦⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠛⠛⠛⣧⠘⡿⠿⠆⠙⠃⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠿⣯⡉⠻⣦⡀⣿⡈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣀⡐⢷⣄⠙⢿⠀⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣦⡈⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⡙⢷⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⡿⠟⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣅⣿⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠷⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣾⣿⣿⣿⣿⣷⣾⣷⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⣤⠌⣋⣉⣻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⡿⣿⣿⣿⣟⣿⣿⣯⡇⣼⠀⠀⠀⠀⠀⠀⠘⠋⡖⠀⠀⠉⠈⠻⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡽⡇⣿⢹⣿⣟⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣽⣿⡇⣿⣿⢿⣿⣇⣿⠆⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡤⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⣿⣿⣿⣿⣿⣿⣿⠯⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⠈⠉⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣯⡤⠤⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢠⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠿⠏⠀⠀⠀⡀⡄⣴⣦⣹⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⡈⠐⠚⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢠⠀⠉⠻⢿⣿⣿
⣿⣿⣿⣿⣿⠂⢀⢠⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⠿⠂⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣧⣠⣿⣾⣧⣰⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣤⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⣠⣤⣿⣸⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠖⠂⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⢀⢀⣿⣿⠧⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠉⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣉⡝⣸⣿⢯⠄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⡁⢠⡴⠞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⢻⢱⣿⣿⡇⠀⣃⣈⠁⠲⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡎⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢾⡿⡟⠀⡰⠭⣛⠷⣶⣄⡈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠈⠀⠀⠀⠀⢤⣤⣤⣶⣶⣶⣶⣖⣒⣿⣄⣾⠗⡅⢠⠁⠀⠀⠉⠘⠻⢿⣷⣤⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡕⠀⠀⠀⠈⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠏⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⡙⢿⣮⡓⢭⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⢠⡶⢋⡭⠒⠀⠀⠐⠀⣝⠻⣷⣬⣑⠢⢅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢀
⠀⠀⠀⠀⠀⠀⢿⠀⠸⡀⠀⠀⠀⠀⢀⠙⢷⣮⣙⠻⢿⣶⣬⣍⣒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠤⠤⠖⠋⣁⠔⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⠢⣀⠀⠀⠀⠀⢿⣮⣝⡻⠶⣮⣍⣛⠿⠿⣷⣶⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⢠⠀⢌⠙⠦⣌⠁⠢⢄⠀⡈⠻⢯⣟⠷⢦⣍⣉⠙⠃⠀⠈⠉⠉⠛⠛⠛⠛⠿⠟⠛⠛⠛⠋⠉⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣁⠀⠈⠙⠦⣄⡀⠑⠘⢶⣭⣛⠲⠤⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠄⡈⠁⢂⠈⡻⠿⣿⣷⣶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⣠⠴⢫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡈⠄⠨⣷⣦⢭⣉⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋⠀⠄⣒⣈⣩⠤⢖⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⣮⣍⣛⠶⠤⠄⠀⠀⠀⠀⠀⢀⠔⠉⣉⠉⠉⠉⠉⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠙⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣓⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠒⠒⣦⣤⡄⠀⠀⠀⠀⠀⣿⡏⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠿⠿⠉⢭⣤⣤⣶⣦⣤⣄⣯⠏
⠠⠤⠴⠶⠶⠻⠋⢛⡂⠨⠤⠴⠖⠒⠀⠀⠐⣒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣐⡒⠘⠒⣛⡭⠄
⠀⠀⠀⠀⠀⠀⠔⠂⠁⠀⠈⠈⠁⠒⠉⠥⠔⣒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠂⡆⢰⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣘⣀⣁⣃⣘⣘⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⣀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢸⠉⢉⣉⣉⣉⣈⡀⠀⠀⠀⡴⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢨⡇⠀⢀⠎⠈⠢⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠘⣀⠤⡄⠒⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠤⠤⠤⠤⠤⠤⡇⢠⠊⠀⠀⢱⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠣⢀⣀⣀⣀⠠⠔⠊





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣿⣟⠀⠀⢳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠰⡀⠀⠀⠀⠀⠀⣡⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠡⠃⡂⠀⢀⣼⢻⠟⣀⣀⠀⠀⣼⣿⣿⣿⠿⠛⠻⠿⣷⣄⡀⠻⠀⠀⠀⠀⠀⠄⠀⠀⠀⣴⠀⠀⣼⠏⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡆⣴⠀⢀⣾⠃⢸⠇⠘⠇⠀⢺⣿⣿⣿⣯⣤⣤⣄⣄⡀⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⣸⡿⠀⢹⣷⡀
⠀⠀⠀⠀⠀⡀⢀⡟⣀⠀⠸⣵⡏⠀⠀⠩⡀⠚⠀⠈⡷⠰⣾⣿⠿⠟⠛⠉⠉⠉⠙⠙⠟⠀⣠⣦⡀⠀⠀⣤⠀⠀⠻⡆⠉⣶⣿⡇⠀⠀⠙⣧
⠀⠀⠀⠀⣼⠁⣾⠡⠁⠀⢸⣿⠀⠀⠀⠀⠙⣿⣿⡿⠉⠭⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠇⠀⠀⠀⢹⣷⣿⡇⠃⠀⠀⠀⠈⢀
⠀⠀⠀⠀⢻⣻⣿⣿⣿⣿⣿⡿⡟⢓⠒⠖⠶⢿⡿⠁⠀⢀⣹⣿⠀⠀⠓⠀⠀⠀⢰⠀⣷⣶⣦⣀⡀⣀⣀⣀⣀⣀⠈⢀⢙⣿⠀⢀⣀⣀⣀⣀⣀⣆
⠀⠀⠀⠀⢸⣿⣧⠬⢴⣯⣾⣿⠀⠀⣠⡄⠀⠚⠀⠀⠀⠀⢉⣷⠀⠀⠀⠀⠀⠀⠀⠀⣫⠏⠛⢻⣟⠛⠛⠛⣟⠉⠉⢹⡋⠉⠂⠸⣿⡍⠉⠉⠉⠛
⠀⠀⠀⠀⠈⣛⣽⣆⠈⠛⣿⣿⠀⠀⠹⠃⣴⡶⠖⢿⣿⣿⣿⣿⠀⣀⠀⠀⠀⠀⣄⠀⡟⠀⠀⢘⠛⢀⣀⣈⣛⣀⣀⣈⡉⢀⡆⢸⡿⠷⠆⠙⠛⠁
⠀⠀⠀⠀⠀⣬⣿⡟⢸⣿⢽⣷⠀⠀⠀⢀⡙⠆⠐⠖⠂⠙⢛⣿⡀⠙⠀⠀⠀⠀⠉⢀⠁⠀⠈⠻⠿⠿⠃⠙⠈⠉⠉⠙⠙⡻⠇⢸⡿
⠀⠀⠀⠀⠀⡹⢿⣿⠸⠿⣿⣿⠀⣿⠀⠸⣄⣀⣀⣠⣤⣤⣬⣿⡇⠀⠀⠀⠀⠀⠀⠹⠿⠁⢰⠬⠃⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣈⢟
⠀⠀⠀⠀⠀⠿⣽⣿⢐⡻⢯⣿⠀⠀⠀⠀⢽⡟⠉⠉⣿⣿⣿⣽⡇⢸⡆⠀⠀⠀⡗⠈⠀⠀⢸⠶⣬⣤⠀⠀⠀⣤⣤⠀⠀⠯⣻⣛⠿⢀
⠀⠀⠀⠀⠀⣰⣾⣿⠀⣋⣹⣿⠀⠀⠀⠀⢘⡇⢠⠀⢸⣿⣿⣞⣇⠸⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⣥⣭⠀⠀⠀⣷⣽⠀⠀⣶⡿⣿⠅
⠀⠀⠀⠀⠀⣈⣚⣿⠀⣿⣿⣿⠀⣿⠀⠀⢨⡇⠀⠀⠀⠀⠫⣼⣿⠀⡀⠀⠀⠀⠀⠀⠹⣋⢟⡭⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣳⣖⣂
⠀⠀⠀⠀⠀⠈⠛⣿⡏⠼⣿⣿⡄⠀⠀⣿⣿⣛⠀⠀⠐⡿⢿⣷⣿⢠⡄⠀⠀⠀⣶⠀⠀⡟⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⢠⡿⡿⣿⡃⠀⠀⠈⠉⣿⠀⠀⠰⣿⣾⣟⣿⠐⡇⠀⢀⠀⠛⠀⠀⣷⠀⠀⢰⡄⠀⠀⠀⢴⡆⠀⠀⠸⢿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⠾⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠒⠘⠃⠀⠀⠀⠘⠁⠀⠀⢸⣇⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣥⣈⡅⠀⠀⠀⢀⣿⠀⠀⠀⢐⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣝⣧⢄⣀⣤⣤⣼⣤⣤⣡⣬⡷⡿⠿⠧⡤⢤⠤⠀⠀⡀⠾⠛⠛⠿⠿⠿⠷⠶⠶⠶⠶⠦⠴⠾⠛⠛⠛⠂
⠀⠀⠀⠀⣤⡶⠶⠻⠟⠛⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢦⠀⠀⠀⠑⢦⣁⠚⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢀⠀⠘⡐⠓⡀⠀⠀⠀⢻⠻⣄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠂⠀⠂⠀⠋⠀⠀⠀⠀⠀⢿⠏⠳⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡔⠈⠂⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠀⠠⠀⠀⠀⠁⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢰⣄⠀⠀⠁⠂⠀⡀
⠀⠀⠀⠀⠀⣱⡀⠜⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠓⢀⠀⠀⠈⠉⠰⠆
⠀⠀⠀⠀⠀⠀⢉⠀⢄⠀⠀⠀⠀⠀⠀⠉⢀⡀⠄⠀⠘
⠀⠀⠀⠀⠀⠀⢸⡀⢈⡆⠀⠀⠀⠀⠠⡄⠀⢛⣆⠀⠸⣅⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠇⠈⠁⠀⠀⠀⠀⠈⢠⠄⠘⢀⠱⡀⠀⢰⠄⠀⠀⠀⠀⢐⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠙⣷⠀⠀⠐⠁⠀⠁⠈⠀⠀⠄⠀⠀⠲⣄⠀⠀⢐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠳⣮⣾⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠙⣆⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣴⠏⠳⣄⠀⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣂⣀⣀⣀⣈⣁⣀⣀⣀⣀⣠⣚⣁⣀⣀⣌⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⢀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡤⠸⡈⢉⠧⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⡈⠤⠐⠂⠉⠈⠡⠌⠁⠐⠂⠼⢏⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠐⠒⠀⠠⠶⠄⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠐⡧⠂⠁⠀⠘⡀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠀⣌⠂⢄
⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣔⠚⠋⠀⢀⠞⣡⠃⠀⠀⠀⠀⠀⠰⡀⢉⡀⠀⢰⠀⠀⡐⠀⠀⡠⠊⠀⠑⢬⠦⡀
⠀⠀⠀⠀⡠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠁⠀⠀⢠⠂⠌⠀⠑⠄⡀⠘⢀⠀⠀⢡⠀⡀⠀⠰⠀⠀⠀⢀⠔⠀⡀⠀⠀⠀⠡⠘⡄
⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⡹⠤⠳⡀⠀⠀⠀⢀⠷⠎⠀⠀⠀⠀⠈⠢⢀⠀⢤⡀⠃⠸⠄⡄⠠⠄⡠⠂⠀⠂⠈⠀⠀⠀⣀⢥⠘
⠀⢀⠽⠀⠀⠀⠀⠀⠖⠊⠉⢡⠀⠀⠔⠁⠀⠙⡄⠈⢆⠀⠀⡌⢘⠀⠀⠈⠰⠀⠄⠀⢀⠑⠄⡩⠚⢁⠀⠉⠒⢎⠠⠞⠀⠀⠠⠐⠀⠁⠀⠀⠆⢃
⠀⠈⠉⢻⠀⠀⠀⠀⠠⡗⠀⠡⠒⠚⡄⡀⠀⠀⡀⠀⠈⢧⠀⡀⡌⠀⠒⠂⠀⠄⠤⠀⠉⠉⡐⠀⠀⢀⠤⡈⠀⠀⠰⢂⣠⠀⠀⠆⠘⠂⠀⠀⢸⢹
⠀⠀⠀⠈⠀⠒⠂⠄⣀⠘⡀⠠⠀⠘⣡⢀⡀⠔⠀⠐⠲⡘⠀⠇⡇⠀⠀⠀⠀⢀⠀⠀⠤⠆⡆⠀⠀⠰⠀⠌⠀⠀⠰⠨⠤⠀⣀⠀⠀⠀⠀⠀⢸⢰
⠀⠀⠀⠀⠀⠀⡔⣩⢍⠉⠁⠀⠈⠙⠁⠀⠀⠀⠀⠀⠈⣰⠀⡗⠃⠀⠀⠀⠀⢀⡀⠀⠒⢈⡍⢄⠀⠀⠀⠰⠀⠠⠁⠒⠖⠀⢀⠀⡀⠁⠀⠁⡙⠘
⠀⠀⠀⠀⠀⠀⠦⠤⠤⠔⠚⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠸⠀⡤⠀⠐⠀⠀⠀⠀⠀⠉⡠⠋⡤⢐⢂⡈⢇⠲⣤⠁⠠⡀⠀⠀⠁⠀⠀⢠⣁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⡁⢀⡀⠠⠤⠄⠒⠂⠉⠀⠀⠀⠀⠀⢡⠐⡀⠀⠀⠠⠤⠈⢀⠎⠀⠈⠁⡈⠀⠒⠈⡄⠀⠀⠄⠈⠒⢄⠀⠀⢀⠊⡝
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠈⠊⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠱⣘⣄⠀⠀⠀⡐⠁⠀⠀⠂⠀⠇⠀⠠⠄⠐⡀⠀⠀⠥⠀⠀⠑⡤⢃⠌
⠀⠀⠀⠀⠀⠀⠠⡏⠀⡀⠀⢦⢠⣷⣀⡠⢀⢾⡃⡊⠀⠀⠀⠀⠀⠀⠈⠢⡑⢤⠊⠀⠀⠀⠚⠁⠰⠀⠀⠀⠖⠀⠡⠀⠀⠀⠀⣠⢊⡴⠁
⠀⠀⠀⠀⠀⡀⠀⡌⡉⠀⢀⠀⠉⠈⠁⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢌⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⠀⡀⠜⠁
⠀⠀⠀⠀⠘⡙⢛⠰⠳⠲⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠃⠬⢀⠁⣀⣀⣂⢰⠀⠥⠐⠂⠉
⠀⠀⠀⠀⢠⠏⠁⠀⠚⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣹⠃⠀⣀⣀⣠⠀⣇⣀⣸⠀⠀⢀⣀⣀⡠⢰⠀⡆
⠀⠀⠀⠀⣬⣀⣀⣲⣞⣈⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⡊⠇⠀⠐⠒⠒⠒⠀⠀⠀⠂⠒⠀⠀⠀⠀⠀⠰⠄⠇
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣿⣧⣀⣠⣼⣇⣀⣴⣄⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⡀⠀⣿⣷⣶⣿⣿⣿⣿⣿⣯⡿⡿⠼⢿⠿⣿⣿⣿⣿⣇⠀⢰⣶⣶⡄
⠀⠀⣀⣠⣤⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣷⠿⠿⠿⠿⠿⣿⡟⢻⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠶⢖⢲⣶⣞⣾⣶⣲⣾⣿⣿⣷⣿⣿⣲⢒⣶⣶⣶⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠿⢿⣬⢭⣭⣍⣭⣽⣿⣛⣿⣿⣿⣿⣾⣿⣭⣿⣿⣿⣿⡿⡝⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⣿⡟⠛⠻⣿⢿⣿⡿⢟⣿⣻⣿⣿⣿⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣾⣆⣩⣴⡺⣿⣾⡶⠶⠟⠋⠉⠉⠅⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣟⣻⡿⣿⠿⠿⠿⣿⡿⢿⣟⣻⢿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠶⢮⣭⣭⣭⣭⣭⣭⣭⣭⡵⠶⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡶⠛⠛⢶⡞⠉⠀⠈⠉⢉⣭⣭⣿⣿⡛⠉⠉⠉⣴⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣼⣷⡀⠀⠀⠹⢦⣤⣤⣤⣴⣿⣿⣿⣿⣇⣀⣀⣤⣿⡿⣿⠟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠻⢶⣄⣠⣶⣽⣶⡉⠁⢻⣿⣿⣿⡟⠉⠉⠀⣨⣽⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣀⠀⠀⠀⠀⣩⣛⣿⣿⣇⣠⣴⠿⠿⢿⣻⣿⣶⣤⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣙⠻⠶⣦⣤⣿⣿⣿⣿⡟⠉⠀⠀⠀⢻⣿⣿⣿⠇⠀⠉⣉⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣌⣉⡙⠛⠋⣄⣀⣠⣤⠶⠟⢛⣉⣀⣤⣶⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠈⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⢀⠁⣀⣤⠀⠀⠣⠀⠀⢀⠔⠀⠀⠀⠀⠄⣉⡀⠀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠳⡥⣴⠠⢀⡰⠎⠀⠀⠀⠀⠀⠀⠠⠤⡀⢠⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠁⠁⠘⢆⠀⠀⢀⣀⠀⠀⠀⠀⠘⠂⠀⠈⠀⢀⠀⢀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠓⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⢀⠀⠄⢂⠈⠆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⣀⠷⡀⠀⠠⠀⠀⠀⠀⢀⡠⠲⢠⠀⠀⢀⠄⠀⠀⢠⡀⠈⠉⢡⢌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠀⡆⠀⠀⠀⠀⢾⠀⠠⠞⢁⠤⠊⠄⠀⠀⠀⠑⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡋⠒⠂⠤⠤⠤⡄⠒⣮⣤⢤⣄⢠⠀⠀⠈⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⠀⠀⠀⠀⣤⡴⠁⠁⠀⢀⡘⠀⢻⡛⢄⠀⠈⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠄⢻⠁⠀⢦⠀⣀⡘⠁⠀⠈⣅⠀⠱⠄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢀⡠⠜⢄⠀⠀⠈⠀⠀⠀⠀⠀⢿⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠉⠁⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠈⠈⡢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠂⠀⠀⠀⠀⠀⠈⠓⡔⠂⠐⠀⠒⢀⡉⠄⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⠀⠀⠸⡀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡠⠁⠀⠀⠀⠈⢄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡰⠁⠀⠀⠀⠀⠀⠀⠑⢄⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⢀⢴⠁⠀⠀⠀⠀⡸⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣉⣉⣉⣉⡉⠤⠔⠀⠀⠀⠀⠀⢇⠀⠉⠀⠒⠀⠉⢀⡠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣠⡉⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢀⠰⠀⠒⠄⠀⠀⠀⠀⠀⠠⠰⠀⠀⠄⠆⢀
⠀⠀⠀⠀⠀⠀⠸⠤⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⢃⢀⠞⡍⠌⣂⠄⡀⠀⠀⠀⠀⠐⢯⢨⠭⠅⠭⠝⠹⡆
⠀⠀⠀⠀⠀⠓⠖⠀⢀⡤⠃⠀⠀⠀⠀⠀⠀⠀⡤⢁⡀⣤⠀⠀⡖⠒⠀⠀⠀⠀⠀⢀⠉⠲⡠⠔⣇⡄⠴⠀⠀⠀⠀⠀⠇⡀⠀⢆⠂⠀⠠⠇
⠀⠀⠀⠀⠀⠰⠉⠈⠉⠡⠀⠀⠀⠀⠀⠀⠀⠨⠀⠨⠤⠄⠅⠀⠑⠀⠀⠀⠀⠀⠀⠈⠚⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠱

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠀⠀⠀⠈⠀⠠⢀⠄⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡉⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠛⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠠⢥⠄⠀⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠊⠀⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠰⣤⠚⡖⣰⠒⠀⠀⠀⠀⠀⠀⢠⣾⠒⢠⡘⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠈⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⠀⠀⠀⡐⠁⠠⡢⠀⣠⠀⡔⠄⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠐⠀⠀⠈⢬⡱⡀⠰⠹⠀⠀⠀⠀⡔⠁⠀⠈⣤⡀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠰⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⣀⠄⣈⠉⠀⡀⠡⠂⠀⠀⠀⠀⡄⠀⣠⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⢀⠁⠐⢄⠀⡠⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⢦⠀⢤⠂⠀⠀⠀⠀⠀⠀⢀⡔⠓⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⣠⣤⣾⠿⠿⢿⣿⣭⣤⣤⣄⣀⡀⠀⠰⠆
⢿⣠⣴⠶⣶⡟⠁⡈⠃⠀⠀⠀⣠⣤⣤⣤⣀⣉⡉⠛⠛⠶⠶⢦⣤⣄⣀⠐⠲⢤⣤⡀
⢸⡿⣤⡀⠈⠻⣤⡀⠀⠀⠀⠋⠀⠀⠀⠀⣤⣉⣉⣙⡛⠲⠶⣤⣤⣈⡉⠛⠲⠶⣤⣌⣉⠒⠂
⠸⣇⠈⠛⢦⣄⡈⠙⠳⢶⣤⣄⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠒⠶⢤⣍⠙⠳⠶⣦⣤⣈⡉⠙⠶⢦⣤⣀⡀⠀⠀⢤⡀
⠀⠙⠳⣄⡐⠈⠙⠲⣦⣄⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠉⠙⠳⢶⣤⣄⣉⠙⠶⣤⣄⡈⠁
⢀⡀⢠⠜⠉⠒⠦⣄⣒⡩⠝⠛⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣈⡙⠲⣤⣀⠀⠀⠀⠀⢀⡀
⠈⠙⢷⣄⣀⠀⠀⠀⠉⠑⠚⠡⠄⢀⠉⠉⠛⠒⠤⠄⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⣀⠀⠀⠀⠤⢀⣀⡤⠀⠀⣀⣉⠙⠷⣤⡉⠓⠤⡀⢢⠈⠁
⠀⠀⠘⢷⡉⠛⠂⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠦⣤⣀⣉⡐⠛⠻⠥⢤⣤⣀⣈⣿⠆⠀⠀⠈⣷
⠀⠀⠀⣀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠲⠶⠤⠤⠤⠤⠤⠤⠤⠤⠛⣦
⠀⠀⠠⠀⢀⡤⠈⠙⠛⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢀⣰⠏
⠀⠀⠸⣦⡞⠁⠠⢀⡀⠀⠀⠉⠉⠛⠛⠳⠶⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠉⠛⠛⠓⠲⠶⠶⣶⠖⠛⠋⠁
⠀⠀⠀⠈⢿⡶⣤⣄⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠛⠛⠓⠶⠶⠶⠶⠶⣾⠶⠞⠃
⠀⠀⠀⠀⠀⠻⣦⡈⠙⠛⠲⠶⠦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠶⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠞⠁
⠀⠀⠀⢉⠀⠀⠀⠙⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣤
⠀⠀⠠⣿⠀⠠⣇⡠⠆⠀⠉⠙⠳⠶⢦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⢛⣻⡿⠋
⠀⠀⠀⠻⣦⡀⠈⢿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠶⠶⠶⠶⠶⠾⠛⢋⣹⡖
⠀⠀⠀⠀⠈⠛⠶⣼⣆⠈⠉⠛⠛⠶⠶⢤⡤⢀⠀⠈⠉⠀⠀⠐⠒⠂⠒⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠞⢋⣽⢧⣄⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣖⠲⠦⣤⣤⣀⣀⣀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠶⠶⠾⠛⠋⢉⣀⣤⠶⠋⠁⠀⣨⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⣀⡈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠒⠊⠉⠀⠀⢀⣤⠞⠋⣸⣇⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡙⠙⠛⠻⠶⠶⠶⠦⠤⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠋⢁⣠⠾⠋⠉⠛⣦⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢳⣤⣤⣄⣀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠀⠐⠂⠀⠀⠀⡀⠠⠄⠀⠠⠔⠋⠁⠀⠀⣠⡴⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣍⣉⣉⠉⠉⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠴⠛⢉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡉⠙⠛⢛⠟⠛⠛⠛⣛⡉⠀⠀⠀⠀⣀⠢⠤⠚⠀⢀⠐⠂⢀⡠⠾⠿⣷⠶⢶⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⢤⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⣠⠶⠛⠁⢀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠷⣦⣤⣤⣤⠴⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠉⠛⠉⠁⠀⣠⣾⣷⡶⣦⣄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢷⣤⣄⣀⡀⠐⠲⠆⠀⠈⠀⠀⠀⠀⠀⣠⡴⠞⠁⠀⣀⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⡿⠛⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⢀⣤⠶⠞⢋⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⢾⣤⣤⠤⠔⠒⠂⠀⠀⠀⢀⠀⠈⠁⢀⣤⣴⡟⠁⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣶⡤⠄⠀⠀⠀⡶⠀⠀⠀⠈⠉⠀⠈⠻⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⠐⠀⠒⠂⢀⣀⣠⣤⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣨⡿⠷⠖⠂⠀⠀⠀⠀⠀⢉⣉⣡⠤⠖⠋⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣿⠤⣄⣀⠛⠀⠀⠀⠀⠀⠉⠉⠙⣶⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⣤⠶⠿⠷⠦⠄⠁⠀⣀⠀⠀⣰⠒⠛⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣟⠳⢦⣀⡀⠀⠀⠀⠁⣤⡤⠍⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠁⠁⠀⠀⠉⠙⠛⠳⠶⠶⢼⣷⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣷⡄⠀⠀⠀⠀⠀⠀⢲⣦⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠴⠄⠀⠀⣀⣈⡓⣶⡿⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⠑⠀⠀⠀⣀⣀⢸⣇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠈⢲⣶⠖⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢘⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣾⡿⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣹⡿⠷⠛⢻⡇⡟⠿⠼⣿⣟⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢺⣿⣾⣿⣿⣿⣿⠀⠀⠻⣧⠏⠀⢰⣿⣿⣿⣿⣾⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⡿⢿⣿⣿⣿⢿⡿⢿⡆⠀⣶⢠⢦⠀⣾⡿⣿⠿⢿⣿⣿⣾⣿⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡞⣿⠋⠀⠀⠙⢿⣷⢸⡹⢺⣷⠀⠛⠘⠚⢸⣿⣗⣹⣘⣸⡿⠋⠀⠙⣾⣷⡄
⠀⠀⠀⠀⠀⢠⡿⡿⠁⠀⠀⢀⣤⠀⠙⢿⣿⣿⣿⡆⠀⠀⢀⣿⣿⣿⣿⡿⢋⣀⢀⡀⠀⠈⢷⣿⡄
⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠸⠼⠁⠀⠈⠻⣿⣿⣷⣀⣀⣸⣿⣿⠟⠉⠀⢸⣺⢼⡹⠀⠀⠘⣯⣷
⠀⠀⠀⠀⢰⣧⣿⣿⣿⣷⣶⣦⣤⣀⡀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣿⣻⡇
⠀⠀⠀⠀⢸⣻⣿⣿⣿⡐⣹⠴⣹⣿⣿⣿⣷⣿⣟⢹⠋⡻⡟⢻⣿⣶⣾⣿⣿⢹⣟⣽⢯⢻⣿⣿⣿⡇
⠀⠀⠀⠀⢸⡿⣿⣿⣿⣷⣿⣶⡿⠿⠟⠛⠋⣿⣿⣶⣾⣷⣿⣾⣿⠛⠻⠿⣿⣼⣧⣽⣮⣾⣿⣿⣷⡇
⠀⠀⠀⠀⠘⣿⣿⠛⠋⠉⢁⡄⢠⣄⠀⠀⢀⣬⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣽⠃
⠀⠀⠀⠀⠀⢹⣽⣆⠀⠀⠸⡽⠻⠜⢀⣴⣿⣿⣿⡇⠀⠀⢻⣿⣿⣷⣄⠀⠀⣎⡇⠀⠀⠀⣰⣛⡟
⠀⠀⠀⠀⠀⠀⢻⣹⣄⠀⠀⠀⢀⣴⠿⣿⠿⣿⡿⠀⠀⠀⠘⣿⣿⢿⣿⢷⣄⠈⠀⠀⠀⣰⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣟⣧⣀⣴⣿⣿⣃⣧⣋⣿⠃⢀⠀⣀⡀⢻⡿⣸⡳⢎⣿⣷⣄⢀⡼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⣿⣿⣿⣿⣿⣿⡟⠀⢿⡖⢧⠇⠘⣿⣿⣿⣿⣿⣿⡿⣿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣿⣿⢿⣿⡁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣛⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣾⣿⣯⣽⣿⣟⣿⣿⣹⣿⣷⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⣿⡒⠚⠋⠉⠉⠉⠉⠉⠉⠉⠛⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⣀⣤⣤⣴⣶⣶⣦⣤⣤⣀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⠟⠛⠿⣿⣿⡟⣠⠴⠚⠉⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⠲⠤⣬⣍⣉⣉⣉⣩⣭⡭⠤⣿⠗⣸⣿⣷⣶⣶⡄⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢰⣶⣶⣶⡶⠒⠶⣶⣶⣶⣤⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠿⠟⠋⣁⠤⠀⠀⠈⠉⠉⠉⠻⢬⡉⠛⠻⠇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⢠⡴⠦⡀⠀⠀⠀⡤⢶⣤⠀⠀⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⣤⣅⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣍⠛⠓⠒⠛⣋⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣬⣙⣛⣛⣛⣩⣴⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠤⠤⠤⠤⢤⢤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⣀⣸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠉⠉⠉⠉⠈⠹⢿⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠤⠤⠤⠤⠶⠾⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠏⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⡌⠐⢲⡶⠲⠂⠒⠒⢒⣾⢻⠀⠀⠀⠀⠀⠀⠀⣧⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⡔⠺⣾⡟⢀⣦⣀⡸⠟⡀⣸⠀⠀⠀⠀⠀⠀⠀⣥⣿⡆⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⢈⣿⡋⢁⣼⡁⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⣽⣿⡇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠘⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠓⠲⣶⠶⣾⠟⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠃⠀⠀⢤⣧⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢙⡆⠀⠀⣞⠉⣷⠀⠀⣚⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⣀⣀⣠⣤⣬⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠈⠋⠁⣀⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⢸⡇⣼⣿⣿⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⢀⠀⡤⠠⡄⠀⠀⣠⠒⢠⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⠉⠉⠉⠉⠉⠉⢸⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠼⠃⠀⠀⠢⠴⠃⠀⠀⠙⠶⠞⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣋⣸⡇⠀⠀⢨⣀⣵⠀⠀⠺⣤⡿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠐⠀⠀⢠⣤⣤⣤⣤⣤⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⡄⠀⠀⣔⠀⣆⠀⠂⢠⡁⣰⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⢹⠙⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠙⠛⠃⠀⠀⠈⠛⠋⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣀⣈⣀⣀⣼⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁

⣷⠛⠛⣶⢶⡶⣶⠶⠶⣾⠶⢶⠶⠶⣾⠷⠶⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⢶⡶⡶⣾⢿⣿⣿⠾⠿⠷⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠘⠒⠒⠲⠤⠠⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢤⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢄⠀⠀⠀⠀⠀⠐⢉⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠹⢛⠒⠀⠀⠀⢭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⠅⠀⠸⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣐⣛⣚⡀⢸⠂⠀⠀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⡀⢀⠀⠈⡀⠁⠂⣼⡿⠆⢠⣜⡄⠈⠀⠀⠀⠀⠤⠄⠄⡀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠊⠀⠀⠀⢲⡀⠀⢒⣿⢇⠀⠐⠀⠱⠀⠀⠀⡀⠊⠀⠰⠃⠂⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠂⠠⡈⢄⠀⢀⣈⣿⡋⢘⣽⡆⢇⠂⠐⠁⠀⠀⠀⠢⠀⠠⣈⣈⡤⠃
⠀⠀⠀⠀⠀⠀⢀⠐⠀⠄⠁⠀⠀⠈⠀⠙⠲⣤⡝⢿⣞⣍⢀⠺⠄⠀⠀⠦⠀⠀⠈⡆⢘⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠶⠀⠀⠀⠀⠀⠀⠈⠑⠀⠙⣿⡟⠒⡴⠀⠀⠀⠔⠔⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣦⣤
⠀⠀⠀⡄⢀⣰⠂⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡁⣰⡟⠀⠀⠠⠆⠀⠉⠁⠀⠀⠀⠀⠀⣰⡞⠀⡀⠀⢀
⠀⠀⠀⠱⣾⠈⠀⠀⠉⠀⠘⠃⠀⠀⣦⡀⠀⠀⠻⠇⠀⢴⠠⠂⢓⠀⡄⠀⢠⡀⠀⠀⢷⡏⠁⢴⢄⡌⢠⠄⣅⠀⢰⣄
⠀⠀⠀⠀⠉⠀⠀⠘⠃⠠⠖⠀⢀⡆⠃⠃⠀⠀⠀⠀⠀⠀⠀⣉⡈⠀⠀⣀⡞⢻⠀⠀⠀⠁⠀⠑⠀⠀⠃⡀⠈⢰⡞⠙⠄
⠀⠀⠀⠀⠀⠀⠰⢤⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠲⣦⣄⠀⠀⠀⠐⠛⠀⠘⠀⠀⠀⠀⠀⣤⡤⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠽⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⡠⠊⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠚⢁⠀⠀⠠⣄⠲⣄⠀⠀⠀⠀⠀⠀⠐⠋⡀⠀⠀⢠⡐⢦⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡀⠀⠲⣄
⠀⠀⠀⡄⠀⡀⠀⠐⠁⠀⠈⠂⠈⠃⠀⠂⠀⢸⠀⢀⠀⠀⠊⠐⢢⠑⠀⠑⠀⠑⡀⠀⠀⠀⠟⠀⠀⠀⠾⢄⠉⡀⢀⠁⢡
⠀⠀⠀⠡⣀⠱⣄⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⢦⡈⢦⡀⠂⠐⠀⠂⠀⠀⠀⠀⠀⠀⠐⣤⡀⠀⠰⡄⠙⠀⠀⠀⣸
⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠠⠊⡠⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⢀⠔⢁⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣀⡀⠀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠝⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣽⣿⣿⣭⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣯⣽⣽⣭⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⣉⡉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⠙⢛⣛⣛⣿⣿⣿⣿⣟⣛⠛⠛⠻⠿⠿⣷⣶⡌⠉⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⣫⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡭⠽⣒⣀⠠⣤⣄⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣵⣿⣿⣽⣿⣿⣿⣛⣭⣽⣿⣿⣿⣿⣟⣓⡛⠿⠿⠛⡿⢿⣟⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⣾⣿⣿⣿⣿⣿⣟⡋⠙⢛⡉⢉⠁⠀⠻⣿⣛⣫⠉⢒⡀⠉⠛⠾⣍⡂⠉⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⣿⣿⣿⠿⠻⣿⣷⣶⣶⣶⣧⣄⠓⢈⡙⢂⡀⠈⡙⠊⠙⢶⣌⠳⣌⡙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⣿⣿⣿⣤⡾⢸⣿⡷⠿⣿⣿⣝⢿⣷⡄⡅⢠⠙⠆⢈⠳⠀⠀⠈⠻⣮⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢿⢿⣿⣿⡟⠀⢹⣽⣟⣛⢸⣿⣿⡏⣿⣿⢹⠈⠇⢢⡈⢣⠀⢠⠀⠀⠈⠳⡌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⡈⣿⣧⣳⡓⠈⠻⠤⠻⢛⣿⣿⢣⣿⡿⠈⠀⢠⡀⠧⠠⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡹⣎⠻⣿⣧⣝⣈⡃⠤⣿⣛⣵⣿⡿⠁⡄⢠⡀⢳⡀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡳⠌⠛⠿⠿⣿⣿⡿⠿⠟⠉⣠⠀⡠⠸⠃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢉⣉⠀⠴⠀⠐⣚⠁⠈⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠀⢐⣒⣋⣁⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣘⣛⣛⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠐⢤⣄⣀⣂⠰⢬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠁
⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢶⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⢇⠈⢧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣧⠈⢇⠈⢆⠈⢇⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠠⣦⠘⢇⠘⣇⠘⣇⠙⢧⠈⢆⠈⢆⠈⢆⠈⣦⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢿⣈⣿⣿⣿⣿⣿⣿⡙⡇⠘⣄⠈⣆⠘⣆⠈⣆⠸⡧⠸⠇⠸⠇⠸⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠛⠋⠈⠛⠛⠿⣿⣿⣿⣿⣧⠘⡄⠈⡄⠘⡄⠈⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⢋⣬⠐⡄⠘⣄⠸⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣿⡷⠞⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡠⠞⠋⠁⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⣀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⠀⢶⣦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠈⣿⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠟⠁




⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⣋⣭⠴⠖⠛⠉⠁⠀⢀⣀⣀⣤⣌⡙⢦⠘
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣉⡥⠖⠚⠉⠁⢀⣀⡤⠴⠖⢚⣋⣩⣥⠄⢀⡴⠃⠘
⠀⠀⠀⠀⠀⣠⠞⣉⠴⠚⠋⢁⣀⡤⠴⠒⣋⣩⠤⣶⣺⠿⠟⣋⣡⠴⠚⣉⣀⣠⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠸⡇⠺⠥⠴⠒⣛⣉⠥⢖⣚⡭⠷⠒⣋⣡⣴⣶⠿⠷⠒⣛⣉⡭⠤⠶⠒⠒⠲⢤⣙⠲⣄
⠀⠀⠀⠀⠀⠉⠓⠒⣛⡭⠽⣒⣋⣭⠤⠶⠚⢛⣉⣩⠤⠴⠒⠛⠉⠉⠀⢀⡀⡀⠀⠀⠀⠀⠈⠳⣌⠳⡀⠈⠳⣄⠹⠄
⠀⠀⠀⡀⢰⡇⢰⡋⢁⣀⣈⣩⠥⠴⠒⠚⠋⠉⠁⠀⢀⡴⠒⠚⠛⣦⣀⣈⠙⠦⠀⠀⠀⠀⠀⠀⠘⢦⠹⡄⠀⢹⡀
⠀⠀⠀⠁⠈⠣⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠛⠛⠉⠉⠉⠉⠉⠳⡄⠀⡀⢀⠀⠀⠀⠀⠈⢧⠹⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⣀⠤⠐⠒⠀⠀⠀⠀⢠⡳⣄⠉⠦⠀⠀⠀⠀⠀⢸⡆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠎⠳⡄⠀⢤⠀⠀⠀⠀⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⡆⠘⡇⠀⣆⣸⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢰⡧⣤⣤⣤⣤⣤⣤⡤⠤⣤⣴⠒⣺⠛⢯⣸⣧⣿⠀⣧⠀⣸⢃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⢸⣤⣾⣁⣳⣤⣏⣬⣷⣴⡧⠬⠷⠛⠚⠋⠉⡄⢸⢀⣇⡼⢡⠞⠀⠀⠀⢀⣀⣤⡤⣔⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⡆⠘⣇⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡟⢠⠟⣠⣯⠤⠖⢚⣋⣩⡤⠀⣠⠞⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠙⣎⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡼⠁⠸⠤⠤⠴⢒⣫⠽⢛⣡⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⣀⡤⠖⢊⣩⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⠒⠚⠃⠈⠁⠀⠀⢠⢸⠀⡴⠋⠁⠴⠾⠗⠒⠒⣒⣛⣋⣉⡉⠙⣶⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠠⢏⡏⠀⠙⠒⠒⠚⠛⠉⠉⣉⡭⠾⢛⣩⡤⠞⠁⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠓⢤⣀⠀⢉⡠⠏⠀⠀⠀⠀⠀⣀⡤⠖⣋⡥⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣄⡀⠈⠙⣿⠀⠀⠀⠀⣠⠖⢋⣥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠁⠀⠀⣄⠳⠤⣬⠭⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤


⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢒⠒⢩⣝⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠘⢉⣔⡂⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠩⠴⠦⣭⡝⣣⠔⢺⢋⡠⠤⣀⠀⠀⠀⢀⡠⢿
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣠⠒⢒⡖⢪⠞⠁⠀⡷⠁⠀⠀⠀⠈⠁⠉⠁⣠⠎
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡀⢠⠊⠵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⡸⠀⢠⠔⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⢠⠃⠀⡆⠀⣰⠁⡸⠢⠤⠀⠀⠠⠤⠤⠔⠒⠂⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡎⠀⢀⡇⠀⢲⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠸⠁⠀⢸⠁⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡴⠒⢾⠁
⠀⠀⠀⢀⠀⠀⠀⣾⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠌⠀⠀⠈⡂
⠀⠀⢀⠎⠀⠀⠀⡆⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⠀⠀⡰
⠀⠀⡜⠀⠀⠀⠀⠇⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠤⠚⠁
⠀⣰⠁⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⢀⠇⠀⠀⠀⠀⡸⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡇
⡘⠀⠀⠀⠀⠀⠓⠤⣀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡇
⢇⠀⠀⠀⠀⠀⠀⠀⠀⡩⠒⠒⠢⠤⠤⣈⣦⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⡇⠀⣀⡠⠤⠤⠤⢄⡀
⠈⠲⢤⡀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⡠⠛⠒⠒⠚⠙⡋⠀⠀⠀⠀⢠⠋⠉⠀⠀⠀⠀⠀⣸⠉⠳⡄
⠀⠀⠀⠈⠑⠦⣀⡏⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠈⠧⣀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⡆⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠃⠘⣆⣀⡀⠀⠀⠀⢀
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠘⠋⣡⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢈⣤⡀⠈⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡼⠟⠿⣿⡀⠈⠑
⠀⠀⠀⠀⠀⠀⡛⠁⠀⠀⢠⣤⣤⡌⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠁⠈⠀⠢⡀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣈⣥⡦
⠀⠀⠀⠀⢀⡀⡁⠀⠀⠠⠿⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣄⠋⠁⠀⠀⠻
⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⡂⠀⠀⠀⠀⠀⠀⠃⠻⠀⠀⠀⠀⠀⢰⣶



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣙⣋⣍⣹⣟⣽⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠶⣚⣻⣭⣽⠿⠿⠿⢛⣛⣛⣉⣩⡭⠭⠭⠭⠤⠶⠶⠶⠶⠶⠶⠦⢭⣭⣭⣽⣛⣿⣿⣷
⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⣩⠥⠖⠚⠋⠩⠅⠐⠒⠂⠉⠉⠉⠀⠀⠀⠀⠒⠒⠒⠂⠀⠉⠉⠉⣉⣉⣛⣛⣓⣒⣿⣿⣿⣿⣿⣿
⠀⢀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠶⠒⠒⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣒⣐⣒⣒⣀⣀⣭⣿⡿⠛⠁
⣴⡿⠋⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠛⠋⠉⠀⢀⣀⡠⠠⠀⢀⣀⣀⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⢿⣧⣀⠀⠀⣠⣴⣶⡿⠟⠋⢁⣀⣤⡴⠶⣚⣫⣭⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠁
⠀⠙⡛⠿⢿⣿⣿⣧⣤⣶⣾⣿⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠉⠉⠁
⠀⠀⠙⢿⣶⣶⣶⣦⣭⣭⠭⠭⠭⠭⠭⠭⠍⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠙⠻⢿⣿⣟⠯⠭⢭⣛⣛⣛⣛⣛⡛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣶⣶⡶⠦⠤⠤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡯⠭⣍⣛⣛⣛⡒⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡶⢶⣮⣭⣭⣭⣭⣭⣉⣉⣉⣩⣥⣤⣤⣤⡤⠤⠶⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣶⡶⠶⠦⠤⣤⣤⣤⣤⡤⠤⠤⠴⠶⠒⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣛⣓⣒⣒⣒⣛⣛⣋⣩⡥⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣓⣚⠫⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣶⣶⣖⣒⠒⠒⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣶⣒⣒⣒⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⠌⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢟⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣟⠮⠽⠋⠁

⠀⠀⠀⠀⠀⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠢⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢰⡇⠀⠀⠀⠀⠉⢳⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⡐⡇⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠻⣤⡀⠈⠳⣶⣤⣈⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡥⠀⠀⢀
⠀⠀⠀⠈⠙⠷⣾⣿⣿⣿⣷⣦⣉⣻⣟⠂⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣼⣷⣔⡁⠀⠀⠀⢀⡸
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣟⣋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⣿⠡⣓⣤⡻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢜⠁⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⣀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⠉⠁⢀⣤⠏⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢷⠀⢀⠀⠀⢀⠀⣀⣴⣿
⣿⣶⠦⠄⠀⠂⠀⠈⠀⠀⣀⡠⠴⠚⠋⠁⠀⠀⠀⠀⢠⣄⣀⣀⣀⣤⣆⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢉⣻⣿⣿⡿⠃
⠙⠻⠶⣤⣤⠤⠤⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⠄
⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠒⠐⠚⡆⠀⠀⠀⠀⠀⠀⢠⣾⣿⠞⠓⠂⠐
⠀⠀⠀⠉⢁⢤⠁⠀⠀⠀⠀⠀⠀⠘⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣟⡽⠁
⠀⠀⣾⡤⠴⠐⢦⠀⠀⠀⠀⠀⠀⠀⢻⡧⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⡀⢸⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⢳⡴⠀⠚⠇⠰⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠸⡇⢹⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⢹⣿⢀⣄⣄⣰⡶⠀⠀⠀⠀⠀⠀⢹⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⢸⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠻⣏⡭⠵⢿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⣭⣿⡇⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣀⡠⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠶⠦⠴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠬⣁⠀⠈⠁⠀⣧⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⡀⣆
⠀⢠⣤⠀⠀⠀⡀⠀⣾⡆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠮⢿⣿⣿⡛⠛⡄⣼⡇⠜⢳⣾⣯⣿⡁⠤⠀⠑⠆⠀⢀⠄
⠀⠀⠈⠀⠀⠈⢁⠀⠛⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⡿⢿⣿⡇⠆⣿⣿⣸⢈⣿⠫⢿⣏⠉⠀⠲⠀⠀⡀⢰⡀
⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣇⣹⣿⡿⠅⣿⡇⠍⢹⣿⣧⢸⡿⣶⠀⡴⠄⠀⠀⠀⠁
⠀⢠⣤⣴⡤⠰⢵⡄⠀⠀⠀⠀⢰⠀⠰⠀⠀⠀⠀⣤⣿⡇⣿⣿⢟⣻⣿⡿⣿⣦⣘⡿⣿⣾⡇⣿⠀⢱⣿
⠀⡌⠛⢋⣶⡀⠘⠛⠀⠀⠂⠄⠌⠀⠀⠀⠀⠀⠀⢹⣿⠟⢿⣿⣸⣼⡿⠢⠿⣿⣣⣤⠹⡿⡁⣿⣤⡬⣷⡆
⢠⣾⣿⣿⣿⣿⣾⠋⡅⠀⠀⢠⡀⠀⠀⠀⠀⠂⠀⢸⣿⠃⣿⣯⣻⣤⣦⣭⣤⣉⣿⣿⣦⣼⣟⣿⠥⡭⡤⠤⠄⠀⠉⠁
⠀⠉⢻⣿⣿⣿⡏⠀⡇⠄⠀⠸⡇⠚⠛⠃⠀⠀⠙⣶⣟⣀⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢟⡿⡿⣿
⠀⠀⠈⠉⠙⠻⠂⠀⠇⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠙⠉⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠚⠛⠛⠛⠒⠒⠒⠶⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⣠⠤⠤⠶⠶⠶⠶⠶⠤⠤⢤⣀⣀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⠦⣄⣠⡤⠤⠤⠤⠤⠤⠤⣤⣀⡈⠉⠲⢤⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠈⠉⠓⠒⠲⠦⠤⠤⠤⠤⠥⠶⠒⢚⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⣀⡉⠙⠒⠲⠤⠤⠤⠤⠤⠤⠤⠤⠖⠚⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡉⠙⠒⠲⠦⠤⠤⠤⠤⠤⠴⠖⠒⢋⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣍⡙⠒⠒⠶⠤⠤⠤⠤⠴⠶⠒⠛⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣉⠙⠒⠒⠒⠶⠶⠖⠒⠒⠛⣉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣉⠓⠒⠒⠒⠒⠒⠒⠚⣉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢍⣙⡓⠒⠒⢒⣛⣩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢭⣭⣭⡭⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠖⢚⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂
⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⡆⠀⠀⢀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⢰⠄⢈
⠀⠀⠀⠀⠁⠀⠀⠠⡀⠀⠈⠀⠁⠀⠀⠀⡀⠀⠀⠘⠐⡀⠀⠀⠀⠆⠀⠂⠀⠀⠁⠀⠀⠁
⠀⠤⣀⠀⠀⠀⠠⠀⠈⠈⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠠⢠⠀⡨⠎⡀⠰⠄⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⠀⠰⠀⠐⠈⠅⠄⠀⠠⠨⠀⠑⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠿⠌⠀⠡⠤⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢰⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⢻⣧⡀⠀⢀
⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⠀⠐⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢰⠀⡆⠀⠀⠂⠠⠀⠀⠀⡀⠀⠀⠠⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢷
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠮⠨⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⣉⣛⣋⣑⣨⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣼⣿⣾⣿⠷⠼⠭⠭⠴⠷⣞⢛⣳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠶⠶⠶⠶⠾⠿⣿⣿⣿⠿⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠉⠉⠉⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣖⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣮⣥⣭⣁⣒⣒⣒⣒⣒⣀⣩⣭⣥⣶⠿⢟⡛⠭⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢻⣿⣿⣷⠠⠍⠉⠉⣭⣭⣭⡍⠩⠭⠄⠐⠂⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣗⣴⣦⡄⠀⠀⠀⠐⠲⣖⢂⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣶⣖⠀⠀⣰⣦⣭⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣆⣲⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠒⠞⣿⣏⡗⣶⣤⣴⣬⣥⣭⣵⣤⣤⣤⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠋⠙⠉⠉⠉⠉⠉⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠙⠿⡿⠟⢛⠻⠟⠻⠟⠛⠟⠛⠛⠛⠿⠛⠛⠻⢿⠟⣻⠿⠿⠟⣿⣿⣿⣿⣯⠙⠉⠑⠾⠇⠀⠨⠉⠉⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣽⣇⠀⠀⢠⣴⣽⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠸⠿⢿⡟⠀⠀⠀⢀⣀⣈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⡀⣠⣄⢀⡀⣸⣇⠀⠀⠀⠈⠁⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⡿⢁⠉⠈⠶⢹⡿⠁⠀⢀⣠⣠⣄⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣈⠙⣿⣥⣌⠄⠀⠀⣼⡇⠀⠀⠀⣿⣿⣿⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⣇⠈⢻⡀⠉⢠⡟⠐⢀⠀⢸⣿⡄⠀⠀⠸⣻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢛⣦⣄⡈⠁⠀⢸⠀⢀⠘⢳⡦⠀⠀⢈⣿⡇⠀⠀⠀⠀⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢠⣼⣿⣿⡟⠗⣴⣶⣿⠖⢀⣀⢀⡁⣀⡀⣈⡋⠃⠀⠀⠀⠀⢀⠀⣼⡲⢤⣤⣶⡶⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡀⢀⣹⣿⡏⠁⠀⢀⡅⠀⠀⠈⣋⠈⡉⢩⣿⣿⣿⣿⣞⠀⠀⠀⠀⠀⣭⡁⠀⣨⣯⠥⠤⠤⣴⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣨⣿⠟⠃⠀⠀⠀⡇⠀⠀⠀⢉⣈⣏⣵⡿⠟⠉⠛⢿⣄⣀⣀⠀⠀⠁⠀⠈⣿⠁⠚⢓⡀⢀⣀⠰⣦⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠰⣿⣄⣄⣿⣄⣉⢉⣿⣷⠀⠀⣶⠰⣿⡏⠁⣀⣀⣸⠇⠀⢹⠛⠛⢶⣩⣄⢉⣀⣨⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⠀⠀⠠⣤⠀⠀⣿⡗⠒⠚⠋⠛⢰⣿⠛⠋⠉⢻⠏⢹⣷⣠⣿⢿⡿⠂⠀⣾⠋⠉⠉⠡⠌⣙⠛⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⢸⣶⠶⢆⣿⠀⠈⣿⡯⠿⣭⠅⠀⢉⠀⠄⢈⡍⢹⠀⣸⡏⢸⣿⣿⣟⠀⠀⣤⣠⣄⣷⣤⠄⣤⣰⡶⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⢻⢿⣶⣿⡿⠀⠀⣿⣷⢦⣻⡄⣽⠌⢧⠀⢈⣉⠒⠴⠿⣧⢸⣿⣿⣿⣿⠀⣽⣿⣿⡿⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠲⢿⡿⠿⠉⠀⠀⢿⣏⢈⡙⣸⣿⠀⠘⠿⠿⣿⣷⠀⠀⢸⣯⣿⠸⢛⣿⠀⣻⣿⣿⣿⣷⣶⣤⣤⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢸⠇⠀⡄⠀⠀⣶⡾⠍⠉⠉⠻⣏⣃⣐⣈⠍⠛⠛⠀⠀⠸⣯⠿⠀⠩⠉⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⡆⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⣀⡙⠋⠀⠀⠀⠻⠛⠓⠢⡈⠀⠒⠒⠒⠺⣿⣿⣿⣿⣿⡟⠉⠀⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⢇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⣈⠁⠀⣰⡆⠀⠀⠤⠹⣶⠀⢿⣎⠀⣶⣶⣤⠀⠉⠻⠗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢈⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠀⠀⢸⠇⠀⠀⠀⣁⠙⢂⠘⠛⠀⠙⠿⠟⠀⠀⠚⠂⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⣴⠀⢰⣆⣀⡆⠀⠀⠘⠓⠲⠄⠀⠀⠀⠠⠀⠁⠀⠃⠀⠀⠈⠀⠀⣠⡄⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢸⡀⠈⠽⠛⠓⠠⣶⣤⠀⠀⠀⠠⡄⠀⠀⠀⡀⠀⢠⠀⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠶⠞⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠁⠐⠂⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠐⠲⠤⠤⠤⠤⢤⠤⠤⠴⣶⢶⢾⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠸⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡽⠦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣾⣟⣲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠁⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠛⣛⣿⢿⣿⠑⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣙⡎⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⠿⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠹⠀⡆⡆⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣱⢳⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡆⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠦⠴⠶⠞⠳⠻⠛⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡖⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠤⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇⢰⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡼⠇⠸⣤⣄⣄⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣤⣤⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣮⠉⢧⠀⠀⡎⠘⢿⣿⠉⠉⢏⠉⠉⠉⠉⢩⠉⣴⣦⡶⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢀⣡⡀⣷⠀⣸⣿⡖⠀⢈⢄⠀⠀⠀⣸⢴⡿⠈⢅⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣿⡀⠉⠻⣿⡻⣿⡇⣠⢞⣺⣧⣤⡰⠣⠋⠁⡐⠕⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄⠀⠈⠻⣾⣿⡿⠋⠁⠀⣨⡧⠁⠀⡰⠋⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣟⣷⠤⣠⣿⣿⣀⡠⣤⣾⢟⣷⣀⣠⣦⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⠿⠶⠾⠿⠛⠛⠛⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠾⠿⠿⠿⠿⠿⢷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠁⠀⣀⣠⣤⣤⣤⣀⣀⣀⣀⣉⠉⠛⠯⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⡹⠃⣀⡤⠔⣚⣉⣭⣥⣴⣾⣟⣋⣉⠉⣛⠻⢿⡓⠢⢦⣿⣷⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠏⠀⡴⣂⣭⡶⠞⠋⠉⡁⠠⠀⠂⣀⣈⠉⠉⠁⠂⠀⠄⣚⠲⣄⡈⠙⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠚⣀⡵⣿⢛⠁⢀⣤⣶⣟⡾⠿⠛⠋⠉⠙⠛⠛⠛⠷⠦⣄⡈⠱⣎⣿⢦⣹⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⢠⠊⣿⢿⣇⣤⡾⢋⡴⣞⠵⠖⠒⣒⡒⠒⠲⣶⡒⠦⠤⡀⠀⠙⢦⣨⣹⡿⣻⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡙⢦⣝⡿⣿⣧⣆⣿⣫⠴⠚⢋⡐⠒⢲⣦⣤⣀⠈⠣⡀⣀⣱⣴⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠻⣮⣽⣿⣿⣿⣿⣶⣾⣭⣭⣭⣭⣭⣼⣿⣯⣿⣿⣾⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣿⡟⣿⣿⣿⣿⣿⠯⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢼⣟⡿⢚⣻⣿⠿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣿⠿⢛⣛⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠼⣥⣽⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⢶⣶⠒⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡋⣻⣿⣿⣿⣯⡥⣽⣿⣿⣿⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡟⠛⠛⠛⠛⢻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣠⣿⣿⣿⣧⡀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡽⢿⣿⣿⡿⣻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢺⣿⣿⣿⣿⣿⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿⢤⢤⣀⣀⣀⣀⣀⣀⣀⣠⡽⡇
⠀⠀⠀⠀⠀⠀⣀⠴⠚⠙⢧⡀⠻⣤⣀⣀⠀⠀⠀⠁⢈⣀⣀⡴⠃
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠙⣆⡧⢀⣈⡉⠉⠉⠉⠉⠁⠀⣶⣛⣉⢉⣙⣦
⠀⠀⠀⢠⣶⠋⠙⢫⡳⡀⠀⠀⠘⡇⠀⠀⢩⢭⠉⢹⠀⠀⢸⡷⠤⠤⠤⠤⢾⣀
⠀⠀⠀⠇⡇⠀⣠⠤⣷⢹⠀⠀⠀⣷⠀⠀⠘⡫⠆⢸⠀⠀⢸⠀⠀⠀⠀⠀⠈⡇⠉⠓⠢⢄⡀
⠀⠀⠀⡇⢧⡠⠗⣶⣹⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⣾⠑⠒⠲⠶⠒⠊⡇⠀⠀⢀⠔⠉⡆
⠀⠀⠀⠘⢾⣤⣞⣡⡧⠏⠀⠀⠀⢸⠉⠒⠦⠤⠴⠚⠀⠀⢻⠀⠀⠀⠀⠀⠀⡇⡌⡀⡜⢰⠁⢩⣦
⠀⠀⠀⠀⠀⠈⢉⣈⣱⡦⠤⠤⠴⠚⠓⠤⣄⣀⣀⣀⡤⠔⢻⡄⣀⣀⢀⣀⣠⠏⠙⠓⠷⠞⠛⠛⠉⠑⠢⢄⡀
⠀⠀⡠⠤⢤⠴⢅⣀⡤⠟⠀⠀⠀⠀⠀⡴⠶⠶⣆⠀⠀⠀⠀⠉⠛⠛⠚⠛⠁⠀⢠⠤⠖⠢⠤⢤⡀⠀⠀⠀⠀⠁
⠀⠀⠑⠲⠦⠤⠤⠃⠀⠀⠀⡆⢉⠉⣹⡯⠭⢭⡿⠶⡉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠄⣤⡤⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠭⠭⠟⠁⠀⣎⠀⣀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣬⣭⣬⣭⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⣀⣀⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⢶⣶⡶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⠛⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⠿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣤⣤⣬⣥⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣉⣉⣩⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣉⣩⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣉⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣶⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⠋⠛⠛⠋⣛⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣤⣼⣿⣿⡀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣻⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⣿⣿⡅
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣖⣚⣉⣉⣉⣭⣥⣀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣖⣺⣯⣵⡶⡟⠟⠛⢓⣒⣶⣶⣶⣶⣶⣮⣟⡚⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣪⠷⡻⠿⢛⡩⠤⠒⣂⣭⣿⣭⡥⣤⣤⣤⣤⣤⣉⡙⠫⣝⣯⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡰⠟⠉⣠⡪⠔⢈⣡⣤⠶⠿⠛⣛⠿⠟⠛⠛⠛⠛⠲⠦⢬⣉⠙⣾⢿⣿⢣
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⡴⠋⢁⡤⢾⠟⢉⡀⠔⠂⠉⣀⣀⣠⣤⣄⡀⣀⣠⣤⣤⣌⣽⣿⣯⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣿⡟⢀⠶⠏⣠⡾⡩⣮⠥⣂⣥⠤⠶⠾⠛⠛⠉⠛⠛⢢⣔⣫⠽⣒⣯⣤⠴⠛⣿⣿⣿⢽
⠀⠀⠀⠀⠀⠀⡿⣷⠗⢷⣾⣿⡾⢞⣭⣶⢿⣥⡤⢤⣤⡶⣶⣞⡿⢿⣿⣶⣿⣿⣶⣭⣬⡷⣿⣥⠏⣿⡟
⠀⠀⠀⠀⠀⠀⠉⠘⠛⡿⣿⣿⣿⣿⣿⣿⣶⣶⣿⠿⣶⣿⣿⠿⢿⣿⣿⣿⣿⣿⣷⣶⡶⣛⣿⠶⡿⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣙⡿⠿⣿⡧⣭⣿⡿⣿⣿⣟⣿⣷⣺⣿⣿⣿⣿⣿⡿⣿⣟⣻⣿⣮⡽⠾⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⡿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠷⣶⣶⠶⠛⢻⣹⣹⡿⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣛⣛⣶⣶⣶⣶⣾⣧⣶⣶⣶⣶⣞⡫⠥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢽⣉⣛⡻⠿⠿⣿⡿⣿⣯⣭⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣟⣻⣭⣿⣛⣻⣷⡾⠿⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⣿⣷⢿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠁⠠⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠀⠒⠂⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠐⠐⠐⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠤⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⠀⠁⠈⠀⠁⠈⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⠀⡄⣠
⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠈



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠈⠻⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⢀⣴⡿⠋⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠻⣷⡄⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣾⣿⣄⠠⣾⠉⠀⣤⣤⣶⣶⣤⣤⠀⠀⣿⣆⣴⣿⣿⣿⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡻⠿⠷⠶⣬⣭⣭⣭⣭⣥⡶⠿⠿⠟⣛⣫⣭⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⢶⣴⣟⣒⣒⠒⠒⠒⠒⠚⠛⠛⢋⣉⣡⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣭⣍⣛⡛⠛⠿⠿⠿⠿⠛⠛⢛⣩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⣍⡛⠛⠛⠻⠷⠶⠶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣍⣛⠿⠿⠶⠾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣍⣛⣿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣉⣛⠛⠋⠁
⠀⠋⠃⠀⠂⠒⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠣⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢀⣐⡂⡀⠠⡀⠀⠀⣀⣤⠀⠈⠂⠀⠠⠤⠄⠊
⠀⠀⠀⠀⠀⢉⠀⠪⢀⣘⠠⠃⠀⠀⠽⣿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠄
⠀⠀⠀⠀⠐⢶⡖⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣀⡀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠙⠋⠉⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠛⠛⠻⠀⠀⠀⠀⠰⣶⠀⠀⠀⠉⡁⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠻⠀⠀⠀⢀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠄⠀⠀⠀⠈⠀⠈⠀⠀⢶⣀⣀⣀⢀⣀⡀⢈⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠈⠉⠀⠘⠋⠩⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢄⣀⣄⣀⣀⣀⣀⣤⠶⠚⠶⡙⠋⠁⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣨⢶⠁⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠑⡄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠺⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢀⣀⡤⢴⠊⠉⠓⠤⢤⠽⠶⡀⠀⠀⠀⠠⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠋⡟⠁⠀⠀⢻⡘⣇⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠘⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡤⠚⠀⠀⠀⠀⠈⢷⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠘⢇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⢀⠼⠃⠲⠶⠄⠴⠚⠃⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠓⠦⠤⠤⠤⠒⠳⢄⠀⠀⠀⠀⠈⡄⠀⠀⠀⣀⣀⣀⣀⣀⡴⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢧⣀⡀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠉⠾⢛⡋⠉⠉⠉⠀⠀⠀⠀⠀⠆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠙⠓⠶⠶⠞⢛⡻⠁⠀⠀⠀⠀⠀⠀⠙⠒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠰⠀⠀⠀⠀⠁⠐⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢸⠄⠀⠀⢤⠀⠀⠀⠀⠀⠀⠐⠥⢿⠄⣹⡄⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡀⠀⠂⠜⣇⠀⠀⠀⠀⠀⠀⣦⢽⠀⠿⡇⠀⠀⠀⡀⠀⠀⡀⠀⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠙⠇⠀⢱⡟⠏⠀⠀⠀⢀⡴⠛⡇⣦⡷⢅⢇⠀⠀⡒⣻⠀⠀⠲⠶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⣼⣿⠀⣠⡀⠣⡀⠀⠀⣽⠀⠀⠐⣟⡀⠀⣀⠀⠀⠀⢑⠂⠀⢸⣿⣤⣄⢀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢱⣿⡡⠁⠀⣹⣦⠀⢼⠀⣀⠀⠘⠿⠾⠳⠿⠿⠿⢯⠀⢠⡆⢨⣧⢤⠱⡤⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣟⠑⣶⡖⢻⣿⡇⢘⣿⣭⠙⠆⢠⣤⡀⠀⠀⠀⣂⣀⣰⣿⢤⡇⠀⠀⣇⠤⣺⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡍⠀⠈⠈⠡⠀⠀⠬⢸⡧⠸⠇⠄⠰⠔⠯⠄⠠⠠⢶⠶⠏⠙⠋⢳⡏⣍⡄⠈⠉⠀⢻⡧⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢤⠀⠁⠀⠀⠀⢸⠀⠀⠀⡂⣔⡀⠀⠂⠀⠐⢺⠀⠃⠀⠀⢸⡃⠀⠀⢀⠀⠀⠈⠃⠀⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠒⠋⠀⠐⣴⠀⠀⠀⠀⢸⠀⠀⠀⣿⠁⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⢸⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠄⠀⠀⠀⠀⠈⠒⠶⢄⣨⡅⠀⢸⣧⠀⠀⢠⣟⠀⡘⠉⡇⠀⠀⢀⠎⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⣄⢀⣠⠐⢄⡀⢄⢀⣠⢍⠳⡄⣸⠙⡛⠋⠉⢽⡍⠉⣉⠉⣁⠔⠁⠀⡀⢄⠀⠀⡀⠀⠀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡀⠉⡀⡀⠀⠀⢀⣀⠄⠀⠀⣹⡛⢒⠂⠀⠀⠘⣟⡉⠀⠙⠃⠐⢚⠊⠁⠀⠉⠋⠈⠓⠀⠈⠉⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣼⠟⠓⠁⢙⠀⢀⠂⠀⠋⡅⡨⢃⠁⠀⠀⠀⠀⠀⠘⣟⠂⡰⢄⢰⢌⡁⠀⠠⠀⠐⠢⠀⠀⡠⣦⠤⣄⠺⡌
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠈⠁⠀⠀⠀⠤⠴⠃⠀⠀⠀⠆⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠛⠒⠂⠬⠖⡋⢡⣀⡀⠀⠜⡞⠁⠀⠣⡀⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠘⠐⠦⢄⠀⢀⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠄⢁⠄⠀⠀⠁⠁⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠬⠀⠀⠀⠀⠈⠉⠈⠀⠳⢄⡔⠐⠤⠦⢦⡀⠀⠀⠀⠀⠈⠄⢸⡃⠀⠀⠀⠀⠀⢸⣦
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠋⢀⠀⠀⠀⠀⠀⠀⢘⢷⡀⠀⠀⠀⠀⠈⣇⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⠀⠈⠀⢀⢙⡂⠈⡆
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡁⠀⠁⠀⠂⠀⠀⠀⠀⢀⠀⣼⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣇⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⢙⡆⠀⠀⠀⠈⠈⢆⡀⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠑⠊⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠆⠉⠉⠉⠐⠲⣤⡀⠀⠠⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠠⠀⠀⠀⠀⠀⠙⢦⡀⠡⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⣧⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⡴⠃⠀⠀⠙⠶⣘⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠀⢹⠗⠚⠀⠀⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁⠀⠀⢀⠀⠀⠀⡌⠀⠘⠀⠔⠀⠀⢀⣧⠀⠀⠀⣀⠀⠀⠀⠰⠀⠙⠀⠀⠀⣼⠀⠀⢀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⣀⠀⢰⠂⠦⠘⠂⠀⠀⠀⠈⠀⠀⠹⣇⠀⠀⠻⠁⠀⠀⠀⠂⠀⠛⠀⣰⠟⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠉⠉⠉⠙⢿⠁⠀⠀⠀⠠⠈⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠡⡀⠙⠷⣀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠘⠛⠷⠆⠀⠀⢘⠛⠃⠀⠠⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠈⠛⠲⠶⠤⠂⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠈⠃⠖⠂⠉⢩⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⡀⠈⠀⡀⠁⠀⡂⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠁⠀⠈⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠐⠒⠢⠤⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⢀⣠⠁⠀⠃⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠃⠉⠶⠂⢴⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⡆⣶⠚⠀⠀⠖⣤⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠄⢀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠂⠈⠁⠀⢸⡇⠀⡄⠀⠀⠀⠐⠆⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⢠⠁
⠀⠀⠀⠀⡠⠐⠂⠁⣀⠀⠤⢤⡆⠀⢸⡆⢠⡇⠀⠀⠀⣈⠀⠘⠧⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠆
⠀⠀⠀⠻⣄⠀⠘⠳⠀⠀⢀⣼⠃⠀⣾⠀⠀⠱⢄⡀⠰⠋⠀⢨⡀⠈⠱⢄⠀⠀⠀⠠⠞⠀⠚
⠀⠀⠀⠀⠈⠳⣄⠀⠃⠀⣸⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⢀⠀⠉⠐⠶⠁⠀⡸
⠀⠀⠀⠀⠀⠀⠈⠳⡄⢀⡙⢧⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠈⠑⠂⣀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡠⠜⠋⢃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠈⠃⠘
⢀⡤⠤⠤⠤⠀⡤⠁⠀⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠫⢀⣉⣀⣤⣤⣤⡴⠃⢽⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠘
⠀⠻⠂⠈⠉⠀⠀⠀⠀⢸⠉⠁⠛⠓⠒⠀⠀⠀⠀⠀⠀⠒⠀⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⣤⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⣀⢀⢈⡄
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠁⠀⠀⠀⠈⣿⠷⠿⠟⠀⠀⠀⠀⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⢠⡄⠀⠀⢀⠁⠀⠀⠀⠀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠰⣁⠠⠤⠼⠧⠴⠗⠻⠤⢀⡀⣠⠊⠁⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠖⠋⠀⠀⠀⠀⠀⣀⠘⠄⠀⠈⠀⢹⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⡀⣠⡀⠶⠂⠉⠙⣿⣯⣴⠲⠒⠋⠹⠟⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠿⠀⠀⠀⣗⢀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠺⠇⠀⢿⡄⠀⠀⣯⣈⠀⢀⣸⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⠀⣼⠁⠀⢰⠃⠀⠀⠨⣅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢀⠁⠀⠀⠀⠀⠀⠀⣆⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠎⠉⠻⠤⠼⠋⠁⠦⣄⠀⠀⠰⡜⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠐⠂⠉⠉⠙⠅⠀⠀⠉⠀⠀⠀⠀⠀⠀⣜⠻⠂⠤⠗
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⢈⣁⠀⠠⢤⣄⡐⠶⠌⠛⠈
⣀⣀⣀⣀⣀⣀⣙⣿⣿⣶⣶⣶⣆⣄⣉⣉⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⣿⣿⡿⢿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣆⣥⣤⣿⣄⣴⣇⣠⣷⣈⣴⣾⣁⣾⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠚⠋⢉⢉⣿⡉⠉⠉⠙⠓⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⢹⣼⣿⣧⡀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠸⣿⣶⠶⠄⠀⠀⠀⢤⡏⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⣸⣆⠀⣿⡀⢠⣶⣀⡀⢀⣾⣧⡀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⡄⠀⠈⡟⣺⡾⢿⣗⣿⣿⣿⣧⠀⡿⠈⠀⠀⠀⠀⣸⣿⣧
⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠉⠁⢰⡇⠀⠀⡇⣻⡷⣾⣿⣿⣿⣿⢉⣤⣿⢰⠀⠒⠈⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢠⣿⠛⠦⠀⡇⣿⣯⣿⣿⡷⠦⣬⡄⠘⣿⠘⢀⣇⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢸⡟⠛⡛⠾⠿⠿⣛⣻⣿⠀⢢⣾⣧⣷⣿⠀⣾⣿⣦⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢸⡇⠂⣿⣿⢻⡏⣻⣿⣿⠁⠈⢹⠙⢍⠙⠀⢻⡇⠄⠀⡀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⣀⢸⣧⣴⣿⣿⣿⣿⣿⣿⣿⡄⢀⠂⣤⢀⡀⠀⢸⡇⣁⣴⣧⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣞⣦⡀⢹⡐⠀⠋⠧⢸⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⢀⠈⠙⠻⣾⡟⠙⢳⡄⢴⢀⣬⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡟⣀⠙⢤⣈⢻⣷⣈⣷⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣶⣦⣿⡿⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⡇⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣸⣿⣀⣀⣸⣇⣀⣀⣿⣇⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣴⠟⠋⠉⠉⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣀⣀⣿⠀⠀⣿⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠉⠉⠉⠀⠀⠉⠉⠉⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠿⠶⠶⣶⠀⠀⣶⠶⠶⠿⠀⠀⠀⠀⠀⣿⣷⠀⠀⢀⣀⣀⣤⣤⣴⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡿⠛⠛⠋⠉⠉⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⣿⡀⢶⡄⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢸⣧⠈⠻⣦⣤⣤⡀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠟⠁⠀⠀⠙⢷⣤⣀⣀⣀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣠⡀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢧⢰⢻⡇⣾⣩⠆⣶⠙⡇⢰⢠⡎⠁⢐⣩⡆⢹⠋⢷⢠⠎⢱⡄⡖⠙⡆⢾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘⠃⠙⠒⠂⠙⠒⠃⠘⠀⠓⠒⠙⠒⠃⠘⠂⠙⠈⠓⠚⠁⠃⠀⠃⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠂⠀⢀⠀⠀⠀⠀⠀⠘⢣⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠉⠉⠀⠂⠀⠀⠠⢄⣀⠤⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⢤⣀⠡⠄⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡠⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠒⠛⠂⠐⢢⣀⢈⢁⡒⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠭⠀⠀⠀⠀⠄⠐⠀⠒⠐⠶⢶⢎⠀⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡉⠉⠙⣫⣍⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡟⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⡦⠄⠠⠠⢠⣄⡀⠀⠀⠀⠀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣔⣤⣀⣩⠥⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠦⠉⠤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠚⠋⠁⠐⠶⣀⠂⢀⢙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠎⠥⠌⠀⠀⠀⠸⠀⠀⠈⠠⡥⡍⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡠⠟⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢹⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⣃⣥⡠⠤⠤⠀⠀⢀⡤⠔⠋⢠⠤⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⣔⣓⡶⠒⠒⠀⠀⡄⠄⠦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁⠀⠀⢀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠰⣯⡽⠇⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠓⠒⠐⢊⠔⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢉⢣⡀⠀⠀⠀⠀⢀⠈⠒⠄⠠⠀⢑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠒⢆⠀⠀⠀⠀⠀⠀⠁⠀⠲⢜⡀⡃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣚⡂⣀⣀⢀⣀⡀⠀⠀⠀⢙⡓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠕⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣲⣶⣠⣞⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠘⣿⠉⠉⠉⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⡀⠀⣀⢻⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠷⢹⡇⠀⠘⢸⣏⢁⣶⣆⡀⠈⠉⠉⠛⠂⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡤⠖⣛⣀⣤⣤⡶⠾⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⢶⣦⠀⠉
⠀⠀⠀⠀⠀⠀⠋⠀⣠⣴⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠶⠖⠚⠓⠀⠀⠀⠀⠀⠀⣿⢰⡆
⠀⠀⠀⠀⠀⠀⢠⡀⣿⡁⢀⣠⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣿⡌⡇⠀⢀⣀
⠀⠀⠀⠀⣤⠀⠘⡇⢹⡇⠈⢀⣀⡄⠀⠀⠀⠀⠀⠀⠤⠶⠒⠂⠀⠀⠉⠉⠁⠀⠀⢻⡇⣷⠀⠈⠉⠉
⠀⠀⠀⠀⢻⠀⠀⣧⠸⣷⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠈⠁⢠⠸⡆⢿⡄⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠋
⠀⠀⠀⠀⠀⠀⠀⢇⣇⢸⣧⠀⠀⢀⣀⡀⠀⠀⢠⠤⠤⠶⠶⠶⠀⠀⠶⠶⠶⠤⠄⣼⠇⣤⠀⢰
⠀⠀⠀⠀⢀⣠⠀⠀⠙⠀⢻⣖⠛⠉⠉⠀⢠⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠃
⠀⠀⣠⠶⠋⠀⠀⠸⠀⠐⢦⡻⣦⡀⠀⠀⠀⢀⣀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣼⠏⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠳⡜⢿⣄⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⠀⠀⣠⡾⢫⠞⠁⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠻⣦⡀⢔⡀⠀⠀⠀⠀⠐⠀⢀⣴⠟⣱⠏⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠰⣌⠻⣦⡀⠀⠀⠀⠀⠀⣰⡿⢁⠘⣣⠖
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠻⣦⣄⠀⣠⡾⢫⡞⢋⡞⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠶⠋⠴⠃⠀⠺⠿⠛⠋⢠⠄⠐⠋⠵⠛⠚⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣀⣀⣀⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣽⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⢀
⠀⠀⠀⠮⠏⠀⠀⢀⠀⠀⡋⠁⢀⠐⣔
⠀⠀⠀⠀⠀⡀⠐⠢⠳⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⢀⠁⠀⠀⠀⡰⠂⠉
⠀⠀⠀⠀⠒⠜⠀⠀⠀⠀⢀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⢄⡀⠆⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠉⠃⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠴⠛⠓⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⡀⠀⠀⠀⡜⠔⠁⠀⠀⠀⠀⠱⣕⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠙⡄⠘⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢎⡞⠉⢐⢾⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠥⠔⠦⣤⣤⣄⣀⡤⡇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠦⠤⠤⠼⠘⠇⣀⣁⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⡄⠀⡄⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⡅⠀⢠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠐⣒⠐⢶⣄⡀⠀⠀⠀⢀⡄⠀⠀⠀⠂⠀⠰⠶⠀⠀⠀⠀⣆⡐⢲⣶⡆⠀⢲⡶⠀⡆⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠰⠈⠁⠀⠀⠀⠀⠀⠀⠈⢻⠄⠀⠀⠁⠀⠀⠀⢲⠀⣠⣄⣍⢙⡟⠿⣿⣦⡏⢻⣥⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⠀⢀⣰⠶⠂⡀⢆⣴⠧⠎⠀⠀⠀⢀⠲⠀⠀⠀⡀⢨⣉⢿⣿⣤⣷⣼⣟⣷⣾⢿⡇⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣶⣤⣵⡿⠛⣳⡿⠻⡀⠀⠀⢀⣀⠻⢿⣧⡶⣅⣂⣀⣏⠀⠰⣤⣼⡙⣿⣿⢃⣼⠖⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⡇⣿⠁⣨⡟⠁⢰⠟⡿⢀⣴⣿⢥⣴⣿⡿⣶⣌⣱⣿⣾⠀⠁⣿⣙⣿⣿⣼⢣⠄⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢦⡀⣿⣻⣷⡿⠀⢠⠟⠈⢻⡯⡌⣷⡜⣿⣻⢇⠹⠉⡿⣾⠈⠀⠀⠸⠿⣿⣿⡿⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣷⣻⡟⣿⡀⠀⡜⠀⢠⠘⢀⢱⣸⠀⣿⢅⠈⠀⠁⠄⢸⣇⢓⣷⣶⠂⠸⣿⡇⠀⠆⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢈⠿⣋⢴⣿⣶⢰⠁⢞⠻⡀⠈⣹⣽⣴⣡⣾⠄⢠⠠⠖⢘⠙⡼⡏⠀⢀⣴⡿⣏⠘⠂⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣯⣁⠉⣾⣿⡄⠠⠾⡇⣡⣴⣿⣿⣿⣿⣿⣿⣿⣦⣀⣬⢀⣧⣱⡀⣿⣿⣿⡞⣷⠂⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⢿⣿⣿⢃⡀⠀⣿⠿⢿⣿⣿⣷⣞⣻⣼⣿⢻⣻⣿⣎⠟⢻⣿⣿⣿⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⡻⣿⣾⣿⣿⣺⣿⣿⣳⡶⣿⡻⢿⣿⣿⣿⣿⠹⣯⢿⣷⣻⣿⣿⣿⣿⣿⣿⣟⠹⡇⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⡿⢿⣿⣿⣿⣿⣿⡿⣿⣷⣿⡃⢿⡏⢛⠁⠉⣠⣿⣟⡛⢫⣿⣿⣿⣿⣿⡟⡃⠀⠇⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⠉⣤⣿⣿⣿⣇⠈⣇⣼⣿⣿⣟⣢⣊⣥⣁⡞⣹⣿⠿⣷⣼⣦⣾⣿⣿⣟⡁⣻⢠⣤⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠒⣷⣿⣿⡿⣿⣿⣿⠋⠟⠉⡻⢿⣿⣿⣿⢾⢿⠁⠀⠉⡿⡿⢿⠀⠉⠀⠀⡞⣼⡇⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⠛⣹⡿⠃⠤⡿⠟⠈⠀⠠⣗⢬⣑⠛⠟⡣⢾⡀⠀⠀⠃⠃⠀⠀⠀⣠⣾⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠗⡹⣆⢀⡇⠈⠍⠂⠀⠀⢀⢔⣿⢟⠟⢱⡟⣏⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⡀⢀⣀⢀⣤⠀⠀⠀⢠⣶⠞⣠⡿⠛⠁⠈⠹⢷⣤⠹⣦⠰⠇⠟⠀⠸⠀⠘⠁⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⡷⠀⠀⣼⣿⣾⢋⡐⠈⠁⠀⠀⣠⢽⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣇⠀⠀⢺⡿⣯⠈⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⠛⠿⣿⣿⣿⣿⣄⠀⢸⣷⣽⡧⠀⠀⠀⠀⠀⠀⠟⠿⠿⠀⠀⠀⢀⠀⠀⢀⠸⣷⡇⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⣴⡄⠀⠈⠽⡿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⢧⣤⣷⣿⡗⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢺⣿⣾⢁⢀⡐⠛⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣦⠀⢀⠀⠀⣤⣧⣼⣿⣾⣿⣿⣿⡗⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠻⠿⠿⠿⠾⠿⠦⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠠⣄⣀⡀⠀⢀⠀⢀⣴⢶⡚⠳⠀⢀⣀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠾⢶⣦⣽⠀⠈⠉⠙⠁⠂⠉⠁⢱⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠠⠐⢢⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠱⠀⠀⣸⢳⡖⠀⠀⠈⠠⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⢀⠃⠀⠀⡿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡤⢺⠁⠉⠙⠛⠁⠀⠀⢀⡎⠀⢠⠴⠃⠀⢉⠓⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢠⠀⠛⠛⠟⠀⠀⠠⣀⠄⠀⠀⣸⠃⠀⠀⠀⠰⠋⠘
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠠⢟⡧⣀⣀⣀⠰⡄⠀⠀⠀⠀⠐⣾⠶⠴⠛⠈⣳⢄⡀⡄⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⡿⠇⠀⠀⢱⣌⣹⠋⠈⠙⠄⢰⣶⣄⣀⣅⠀⢀⣠⣴⢧⠀⠀⠈⠁
⠀⠀⠀⠠⢄⠀⠀⠀⠀⢀⠀⠀⠐⠤⡄⢶⠾⠛⠛⠀⠀⠀⠈⣟⠉⠉⢽⣛⣛⣯⡅⠀⢣⠁⣠
⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢻⣋⠙⠻⠿⢿⡄⠀⠈⠒⠁⠀⠀⡀
⠀⠀⠀⠤⠴⠂⠀⠀⠀⡠⠐⢢⣄⠀⣀⠀⠀⠀⠹⣀⠀⢠⠾⣿⢽⣯⡷⠄⠀⠀⠉⡄⠀⢠⣄⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠹⠀⠈⠀⠒⠒⢂⠀⠉⡁⠈⠀⠀⠀⡳⠳⠾⠛⠋⠀⠉⠀⠈⠀⠀⠈⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠁⠀⠀⠀⠀⠀⢀⡴⠀⡠⡗⠀⠀⢁⡀⠀⡄⠀⠀⠀⠀⠀⢀⠄⠀⠀⢠⡔⠊⠀⠀⡀
⠀⠀⠀⠠⠂⠀⠀⠀⠹⠿⠄⠀⠀⠀⠀⠀⠀⠜⠀⠀⡁⠁⠀⠀⠀⣠⡴⠀⠀⠀⠐⠂⠀⠀⠀⠀⠐⠁⠀⠠⡀
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠋⠔⠉⠀⢠⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⡷⠾⠛⢹⡄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⢇⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠒⠚⠛⠛⠛⠛⠛⠛⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠲⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⢿
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠫⣉⠀⠸⣄⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠀⠀⣀⠟
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠳⡄⢸⠙⠛⠒⠒⠶⠦⠴⠤⠶⠶⠖⠚⠛⡃⣠⡤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⣜⣀⣤⣀⡀⠀⠀⠀⠀⠹⣼⡇⠂⠤⢤⣄⣀⣀⣀⡀⠀⠀⠀⢰⣿⠧⠤⠤⠤⠤⠴⢿
⠀⠀⡴⡿⠋⠉⠙⢦⡹⣄⠀⠀⠀⠀⢹⡇⠀⠀⠀⡤⠤⡄⠀⣿⠀⠀⠀⢸⡿⠶⠤⠤⠤⠴⠶⣏⣀
⠀⡼⢰⡇⠀⠀⢀⣈⣧⠘⡆⠀⠀⠀⠀⡇⠀⠀⠀⠷⢮⡤⠀⣿⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⡏⠉⠓⠦⢤⣀
⠀⣇⢸⡇⠀⢾⣁⣈⢿⡄⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠀⢸⠀⠀⠀⣿⠤⣀⣀⠀⠀⢀⣀⡤⡇⠀⠀⠀⠀⠈⣹⠦⡄
⠀⢹⡀⢧⠔⠋⢩⠿⣼⢡⡏⠀⠀⠀⠀⣧⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠉⠉⠉⠉⠁⠀⡧⠀⡀⠀⠀⡞⢁⡀⣿⡀
⠀⠀⠳⣜⡦⢴⣋⣴⣣⠞⠀⠀⠀⠀⠀⢿⠈⠙⠓⠶⠶⠶⠖⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⠁⡄⣸⠀⡏⠀⢀⣽⡆
⠀⠀⠀⠈⠉⠉⠉⠉⠳⢤⣀⣀⣀⣀⡤⠾⠦⣄⣀⠀⠀⠀⠀⠀⠀⣀⡤⣿⡀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠲⢧⣿⣤⠽⠳⠾⠋⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠤⠒⠻⠉⢹⡄⠉⠉⠀⠀⠀⠀⠀⣉⣉⣉⡛⠛⠋⠉⠀⠀⠈⠳⣭⣓⣒⣒⣒⡪⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠄
⡞⠋⠉⡛⠛⠓⢦⡴⠒⠋⠀⠀⠀⢀⣀⣀⡀⡾⣒⣂⣉⣹⡆⠀⣀⣀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡖⠉⠉⠉⢹⠉⠓⢦⡀
⠉⠒⠲⠧⠤⠤⠼⠁⠀⠀⠀⢠⡎⠁⡸⠉⢉⡷⠦⠭⠭⣾⠖⠫⡁⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣤⣼⡤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣙⣛⣛⡩⠇⠀⠀⡼⠁⠀⠀⣰⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⢤⣤⣀⡀⢠⣶⣿⣿⣿⣿⣿⣷⣦⡀⢀⣠⣤⠤⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⢀⣰⣶⣦⣭⣝⠻⠟⣫⣭⣴⣶⣷⣦⣤⣤⡄⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⢀⣾⣿⡿⠟⣛⣭⣶⣿⣿⣶⣬⣝⡻⢿⣿⣿⡟⠀⠖⠁⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⠀⠀⠾⢛⣭⣶⣿⣍⠻⣿⣿⣿⣿⣿⣿⣿⣷⣮⣙⠶⠀⠀⠘⣿⣿⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢋⡠⠆⢺⣿⣿⣿⣿⡿⣱⣶⣭⣛⣛⣛⢛⣭⣵⣾⣿⡷⠐⢦⡀⠛⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠈⣛⣛⢩⣵⢾⡿⣿⣿⣿⡿⠿⠷⣮⡍⣛⣛⡁⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠀⢸⣿⣿⢸⣇⢯⣚⠎⣿⣿⠸⣩⢱⣸⡇⣿⣿⡇⠀⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢸⣿⣷⣶⣾⣿⣿⣷⣶⣿⣿⡷⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢸⣿⣿⠟⠿⣿⠟⠿⣿⣿⣿⡇⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣿⣷⣶⠀⠀⠻⣿⣿⣷⣶⣿⣷⣶⣿⠟⠀⠀⠀⢾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⡀⠀⠀⠀⠈⠙⡻⠿⠿⢟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣾⣿⣷⣝⠿⣿⣿⣿⣿⡿⣫⣾⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣝⣫⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⢀⣼⠇⠘⡆⠀⠤⠄⠀⠀⠀⠀⠀⠀⢰⠃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⢂⠂⠀⢻⣰⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⠚⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⣸⢰⠁⠀⢸⣷⡆⠀⠀⠀⠀⠀⠔⢺⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⣏⠘⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⣀⣿⡀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⢀⣻⠀⠀⣸⡿⢠⣀⣤⣼⣟⣷⠀⢀⡀⠀⠀⣿⣿⠃⠀⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠇⠀⠀⠀⠀⣼⣿⢛⣷⣿⣁⡛⠛⠛⠛⠉⢻⣿⣿⠧⠀⠀⢻⣿⣿⣴⡴⣷⢠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣬⡄⠀⠀⠀⣸⣿⣯⣿⣿⣟⣑⣀⣑⡒⠮⠗⢿⣿⡛⠟⠛⠟⠻⣩⣒⠆⡮⢰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⡿⡿⢦⣤⣾⣛⣛⣫⣿⡿⠿⡟⣛⠛⠖⠒⠒⠼⣿⣷⣦⣤⣀⣐⣿⣿⣿⡉⣉⣁⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡂⠘⠀⠐⢸⣉⢩⡏⠉⢻⢻⠱⠇⣠⠍⠐⠂⠀⠀⠈⢉⣓⠀⠸⠂⠀⡍⠀⢐⠗⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣯⣤⣶⡴⣅⡒⡀⡀⣀⢠⣰⣤⡆⣧⢾⡰⣦⣆⢶⣦⣬⣴⢠⣤⣄⣄⣻⣤⡄⣤⣴⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠸⠐⡻⣿⣿⠻⠹⠩⣿⣿⣿⡟⢉⠄⢃⠁⠹⢘⡹⣿⣿⣯⠟⡿⠻⣯⠶⡗⠙⠓⠋⠟⠟
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⣿⢁⠀⠀⠀⡹⣶⣷⠛⣋⠉⡉⠀⠀⢼⠇⢻⡿⣿⣶⣡⡀⡇⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣿⠘⠹⡀⠀⢡⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⢸⣿⣻⠛⢿⣏⢠⣍⠀⠀⠀⠀⠀⢸⣇⣀⣀
⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⣿⣄⠈⢀⡀⢀⣷⣇⣽⡏⠛⠀⢀⠀⠀⠀⣸⢿⡃⡄⠀⡾⢰⡯⠅⡀⠂⠀⠀⢈⠈⠉⠉
⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠆⠀⡀⡀⢀⡇⠀⠀⠀⠀⠛⡇⢀⠘⢄⢠⣗⡿⠷⠶⠊⠀⠀⣶⠀⠂⢺⡽⣅⠋⠨⡇⠴⠓⠀⠀⠀⣠⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠒⠚⠢⠥⠦⠆⠰⠀⠄⠀⣀⠃⠁⠀⠘⠹⣿⡉⢛⣶⠀⠀⠀⠉⠀⠀⣼⣷⠉⠃⠀⡧⢸⢂⠁⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⣂⢀⠀⠀⡭⠄⠀⠠⠀⢀⣯⡿⠽⠁⠀⠀⠀⠀⠀⠀⢸⣼⣇⣀⣀⣿⣿⡾⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠇⣴⠀⠄⠀⢷⡗⠀⢰⠂⢘⠻⣥⠀⣸⠉⠀⠠⠀⠀⠀⠈⠃⣽⣿⣯⣿⣿⡿⠃⠘⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠄⠁⠘⢼⠅⠉⠀⠄⠸⣿⣿⠀⠉⠀⠁⠀⠀⠀⠀⢸⡏⣿⡿⣿⡇⠛⠃⠐⡞⣂⠙⠀⠂⠃
⠀⠀⠀⠀⠄⠄⠠⠀⠤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠒⠂⠈⠛⠛⠁⠀⠀⠀⠀⠀⠤⠀⠌⠇⠉⠀⠈⠁⠀⠃⠈⠀⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⢺⠓⡀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠊⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡉⢀⢬⠝⠀⠀⠀⠀⠀⠀⠤⠤⡀⣃⠀⢠⠔⢓⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠾⡄⠀⠀⠀⠀⠀⠀⠈⠢⢻⣟⡔⠉⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⡚⣇⠴⠃⠀⠀⠀⠀⠀⠀⢀⣧⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡍⢏⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⣖⣲⠖⢒⣒⡒⠲⣲⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢒⠲⣆⠀⠘⢆⠹⣤⠤⠤⠤⢤⣤⠋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠿⠃⡸⢀⣠⡈⠓⢌⡻⣦⠼⢛⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡒⠛⢧⡏⡹⠁⠀⡴⢛⡴⢎⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣣⠤⣿⠃⢠⠞⣴⡯⠠⢤⣵⣌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠠⢼⣤⠋⣼⢉⠉⠉⢉⣉⣙⣆⢹⡀⠀⠀⢀⣨⣙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢨⠓⠛⠛⠒⠒⠒⠚⢻⡄⡗⢦⣀⠜⣡⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠘⡇⠤⠒⠨⠀⠀⠲⣾⠃⡇⠸⢡⣾⣩⣾⢄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠱⣤⠤⠤⠤⢤⣴⠇⡼⡡⠖⠣⡀⠳⠵⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⡙⢮⠽⢋⡤⠞⠊⢱⡂⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡥⢆⠛⢄⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣠⠿⠅⠨⠷⣌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣼⣓⣲⠖⠒⢲⡾⣆⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣸⠁⠀⠀⠀⠀⠀⠀⠘⡆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⢲⠾⡀⠀⠀⠀⠀⠀⠀⠀⠰⠟⡴⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣌⣦⢧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠊⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⠦⠦⡍⠃⠀⠀⠀⠀⠀⠀⠀⠘⢢⠤⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣂⠓⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠆⠀⠀⠀⠒⠲⢄⡀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠙⠏⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⠔⠂⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⠇⠀⠰⠆⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠘⠄⠁⠀⠀⠀⣧⠀⠀⡀⣠⠀⠀⠀⠰⠀⠈⠀⠀⠀⣸⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⣠⠀⠰⠂⠶⠀⠂⠀⠀⠀⠀⠀⠀⠹⣏⠁⠀⠙⠀⠀⠀⠀⠀⠀⠰⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠉⠙⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⣀⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠛⠷⠆⠀⠀⢘⠛⠁⠀⠀⡀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡲⠶⠤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⢠⡄⠀⠀⠀⠀⠄⠀⠀⠈⠑⠒⠂⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠘⠀⡀⠀⠀⣂⠠⠀⠐⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠄⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣄⠀⣠⠃⠀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠋⠉⠒⠣⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡎⢶⠘⠀⠀⠆⢤⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⡇⠀⡄⠀⠀⠀⢸⡆⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢠⠂
⠀⠀⠀⢀⡠⠀⠀⠀⡀⠠⠄⢰⡖⠀⢸⠇⠀⡇⠀⠀⠀⡈⠀⢸⠣⣄⠀⠀⠀⠀⠀⠀⢀⠃⠀⠆
⠀⠀⠀⠻⣄⠀⠀⠓⠀⠀⠀⣾⠁⠀⣾⠀⠀⠑⠀⢀⠠⠁⠀⢸⠀⠈⠑⢄⡀⠀⠀⢀⠎⠀⡘
⠀⠀⠀⠀⠈⠳⣄⠀⠠⠀⢸⣿⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠐⠠⠀⠈⠀⠶⠁⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⢰⣼⣿⣠⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠢⣄⠀⢠⡅
⠀⠀⠀⠀⠀⠀⠀⣠⠜⠛⢣⡄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠈⠓⠌
⢠⡔⠒⠒⠀⠠⡼⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠓⢨⡥⠒⣶⣶⠢⠔⠈⢢⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠐
⠀⠚⠘⠀⠈⠀⠀⠀⠀⡈⠉⠙⠛⠛⠃⠠⠤⠤⠀⠤⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣤⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡀⣀⣘⡆
⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⠃⠀⠀⠀⠀⣿⠿⠟⠛⠀⠀⠀⠀⡜⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠇⠀⠀⢰⡄⠀⠀⠀⠃⠀⠀⠀⠀⣀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣐⢀⠄⠀⠸⠃⠸⡟⠛⠂⣠⠀⣰⠃⠀⠀⠡⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠞⠋⠀⠀⠀⣀⣀⣀⠐⠀⠀⠀⠀⢹⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⣀⡀⠤⠀⠛⠁⠈⣿⠷⡶⠋⠚⠉⢻⠛⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⡇⠀⠀⠀⡟⢈⠀⠀⢠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠧⠄⢿⡆⠀⠠⣏⠠⠤⠀⡄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⡘⠁⠀⣰⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢀⠅⠀⠀⠆⠀⠀⠀⡘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠴⠲⢏⠉⠛⠒⠚⠃⠈⠢⣦⠀⠀⢀⠓
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠃⠀⠁⠈⠡⠀⠀⠀⠀⠀⠀⠀⢀⡀⢌⠟⠁⠴⠋
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⣧⣀⡀⠒⠦⡌⠓⠂⠉
⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠉⠙⠛⠒⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠖⠒⠒⠒⠲⠶⢦⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⢛⣛⣻⣭⣭⣭⣭⣭⣉⡛⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⠶⣚⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⢉⣙⡵⠂⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠫⠭⢭⣭⣭⣭⣭⣥⣴⣶⣿⣿⠏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣖⣒⣒⠒⠒⠒⠒⠛⠋⠉⠁⠀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⣙⣒⣒⠲⠶⠶⣾⣛⣛⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡾⠭⣭⣭⣭⣩⣭⠥⠤⠒⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⣭⣉⣉⣩⣭⣭⣵⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣒⠶⠶⠶⠶⠖⠒⠋⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣒⣺⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠭⠭⠍⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣚⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠄⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠳⠄⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢏⣉⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⣤⣄⣀⡀⠀⣀⣤⣤⣠⣾⣿⣿⣷⣶⣿⣿⣷⡄⢴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⠀⠈⠉⢰⣿⣿⣿⣟⠋⣤⡙⣻⣿⣿⣿⣿⣟⠸⣿⣿⣿⣿⣦⠠⠴⠶⠶⠖⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡄⢠⣶⣶⣦⣿⣿⣿⣿⣇⣭⣨⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣶⣦⡀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣭⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠟⣴⣿⣿⣿⣿⠿⢟⡁⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⣶⣬⣙⡻⠿⠿⢟⣭⣙⣛⣫⣼⣿⣷⡝⠟⣛⣭⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⢛⣛⣫⣴⣷⣶⣬⣝⠛⠿⢛⣫⣭⣶⣾⠸⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⢀⣽⣿⣿⣿⡿⢋⣼⠿⠿⣛⣛⣭⣥⣶⣾⣿⣶⣦⣭⣙⣛⠧⠹⢿⣿⣿⣿⣿⣷⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠷⠤⠀⣛⣛⣋⣭⣭⣥⡄⣶⣿⣿⣭⣭⣙⣿⣿⣿⣿⣿⣛⣭⣭⣽⣿⡇⣶⣶⣬⣭⣍⡁⠤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣟⡰⣿⣿⣿⣛⠛⣛⣿⣿⣿⣿⣿⣟⡛⢛⣻⣿⣿⢌⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⡿⢃⣾⣿⣿⣯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠁⠀⠈⡛⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠈⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣬⣝⣛⣛⣩⣵⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣦⢤⢤⣴⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⢠⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣼⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠀⠁⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⣿⣿⠿⠿⠿⣼⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡷⢶⣦⢉⡉⡋⠉⠁⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⡧⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⠉⠓⢥⡀⠄⣪⣽⡄
⠀⠀⠀⠀⠀⠀⢸⣿⠶⠴⡶⢶⠒⣻⠂⠀⠀⠀⢀⣹⡥⡀⠀⠀⠹⡿⠿⠿⠿⢿⣶⣶⡶⢶⣤⣤⣇⡀⢸⠉⠙⠓⠒⢲⠀⠀⢱⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣀⠸⠀⠿⡂⠀⢠⣾⡿⠛⠿⣖⢄⠀⣠⣷⣄⠀⠀⠀⠀⠈⠉⠲⡄⠈⠙⣿⣿⡄⠀⠀⠀⢸⠀⠀⢸⢸⣿⣿⣇
⠀⠀⠀⠀⢀⣤⠟⢳⣯⡿⠯⢿⣄⣶⣛⣦⣼⣿⣦⣤⣤⣾⣿⣟⢋⣡⣾⡁⠀⢀⣹⣿⣿⣷⣷⣦⣶⣿⣿⠁⠀⠀⠀⢸⠀⠀⢸⣼⣿⣿⡟
⠀⠀⠀⠀⡿⢥⣀⣀⠈⠉⠉⠉⢡⣿⣤⣤⣠⡤⣼⣟⣭⢫⡉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⣽⢫⣿⡏⢸⠀⠀⠀⠀⢸⠀⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⡇⠀⠀⠈⠉⠉⠉⠉⢹⣿⣿⡆⡇⠀⢸⡟⣸⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⡇⠸⠀⡀⠀⠀⢸⠀⠀⢸⡟⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡏⠓⠒⠒⠒⠒⠚⣿⢛⣿⠻⠛⣿⡇⣿⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠻⣿⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⣠⣾⣛⠋⣘⣃⢨⡳⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠂⠉⠉⠉⠀⠒⣦⡀⠀⠀⠀⠀⡇⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⣞⣿⣿⣿⣾⣿⣿⣗⠙⡘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠛⠛⠒⠚⠋⢹⡇⠀⠀⠀⠀⡇⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠳⠤⠤⢤⣤⠤⠴⠚⡇⠀⠀⠀⠀⡇⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡇⠀⢉⠉⠉⠉⢉⣉⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡉⠛⠒⠤⠶⠶⠶⠖⠙⢙⣆⡠⠤⢤⣧⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡇⠀⠘⢒⡶⠻⣟⠛⠀⠀⣿⣝⡢⢤⣄⣀⣀⣀⣀⣀⡤⣿⣝⣲⠲⠤⠤⠤⠤⠒⣎⣭⡏⠀⠀⠀⠀⠈⠉⠙⠓⠲⠦⢤⣄⣀⡀
⠀⠀⠀⠀⣧⠀⠓⠤⠤⠤⢤⡇⠀⢰⡋⠀⠀⠈⣱⠆⠀⣿⣿⡿⠋⠓⠓⠒⠒⠒⠒⣋⣿⠀⠉⠉⠛⠒⠛⠋⠉⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠽
⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⢀⡇⠀⢀⠙⢦⣤⠞⢁⠀⠀⢿⣿⡇⠒⠢⢤⠖⠂⠛⢿⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡇⠀⠈⠙⠋⠛⠋⠋⠀⠀⣸⠿⣇⣀⡀⢸⣶⣶⣲⠷⡳⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣦⣤⣀⣀⠴⠊⠁⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣷⣶⣦⣤⣤⣴⣶⡆⢹⣿⠀⠀⠈⢉⡹⣻⡉⢈⠇⢱⣿⣄⣀⠀⠀⠀⠀⠀⠀⣀⡤⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡤⠞⠉
⠀⠀⠀⢠⠶⠶⠖⠒⠒⠒⠒⠛⠉⠉⠉⠙⣿⣟⣛⣡⠴⠃⣴⣆⣢⠿⠿⣿⣏⣁⣠⣿⣽⠀⠉⠙⠛⠒⠒⠛⠋⠁⠀⣿⠉⠙⠻⠿⢿⣿⣿⣇⡤⠞⠉
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠄⠀⡄⠀⡈⠉⠛⠂⠀⠼⠈⠉⠉⠀⠘⠧⣄⣀⣀⠀⠀⢀⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣗⠺⢿⡶⠦⠛⠲⣎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠸⡧⠤⠤⠶⠒⠒⠒⠒⠒⠚⠛⠋⠉⠉⢻⠀⠈⠳⠖⠚⠉⠁
⠀⠀⠀⠀⣧⡤⠤⠤⠤⠴⠶⠶⠒⠒⠒⠒⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠿⣿⣶⠶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣬⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣴⠶⠶⣄⠀⠉⠉⣹⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠋⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣧⣤⣤⣼⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣴⠖⢶⡄⠀⠀⢸⣷⢶⣶⣶⡄⠀⢰⣶⠿⠿⢷⣦⠀⠀⠀⣴⡿⠿⠷⣶⡀⣿⣷⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣾⣁⣀⣀⣻⡄⠀⢸⡇⠀⠀⠸⠷⣶⣿⠇⠀⠀⠘⢿⣶⣶⣴⣿⠀⠀⠀⠻⢿⣿⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠉⠉⠉⠉⠁⠀⢸⡇⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡏⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⢻⡆⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⠓⠶⠶⠾⠁⠀⠀⠀⠀⠀⠀⠀⠘⠷⢤⣤⣼⠃⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣿⡇⢠⣶⠶⠶⣶⡆⠀⠀⢸⣿⠶⠶⠶⢶⣦⠀⠀⠀⣴⣶⣶⣶⣶⣶⣦⠀⠀⢠⣦⣤⣤⣤⣼⣿⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⢿⣿⣧⣼⡿⠀⠀⠸⣿⣶⣶⣾⠏⠀⠀⠀⠈⣿⣦⣤⣤⣿⠃⠀⠀⠀⠈⣿⣤⣤⣿⡇⠀⠀⠈⣿⣿⣸⣿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠛⠋⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⣴⠶⠶⠲⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠒⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⣿⡀⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⢀⣿⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠉⠉⢉⣁⣀⣀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⢀⣀⣀⣈⣉⣉⣉⠁⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⢀⣀⣽⣧⡀⠀⠀⠠⣿⠿⠛⠛⠛⢻⣿⠀⠀⠀⢠⣿⠉⠉⠉⠀⠀⠈⣿⡀⠀⠀⢀⡀⣼⡟⠛⠛⠛⠛⢻⣧⠀⠀⢠⣴⣿⣀
⠀⠀⠀⠀⢰⣿⠛⠋⠉⢹⣷⣠⣤⣤⣿⠀⠀⠀⠀⠘⢿⡶⠾⠿⠿⠋⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠘⣿⣦⣤⣿⠏⠙⠛⠿⣶⣦
⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⣿⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⣿⠀⠀⠀⠀⠀⢸⡟
⠀⠐⠀⠀⠀⠀⣿⡄⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣦⣤⣤⣤⣤⠀⠀⠀⠀⠻⠶⠶⠶⠟⠀⠀⠀⠀⠀⣾⡇
⠀⠈⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢯⣭⣭⣭⣭⣭⣭⣭⣥⠶⣤⠶⠶⡶⠶⣶⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠰⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡸⡇⠀⢸⠀⢸⠀⠀⣧⠀⢸⠀⢠⠁⠀⣿⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⢺⠀⢸⠀⠀⡟⠀⢸⠀⢸⠀⠀⡟⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⢸⠀⢸⠀⠀⡇⠀⢸⠀⢸⠀⠀⡏⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠘⠀⠀⠀⠀⠀⠀⣿⣏⣉⣭⣭⣭⣭⣭⣭⣭⣭⣿⣧⣿⣄⣼⣤⣼⣤⣠⣧⣠⣸⣀⣸⣀⣠⣧⣼⣟⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣈⣀⣿⡇
⠀⢰⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣛⣛⣿⡿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣚⣛⣉⣉⣩⣭⣭⣭⣍⣉⣉⣛⣛⣛⠛⠿⠿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠟⠛⢛⣛⣛⣛⣉⣉⣉⣉⣉⣛⣛⠛⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣬⣉⣛⠛⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣬⣭⣉⣛⡛⠿⠿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣛⣉⣉⣩⣭⣭⣭⣭⣭⣭⣉⣉⣙⣛⠛⠻⠿⠿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣥⣭⣤⣭⣭⣉⣉⣉⣛⡛⠻⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣠⣤⣤⣤⣤⣭⣍⣉⣉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣛⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⢶⣶⡶⢶⡶⢶

⣤⣶⣒⢹⣏⣉⣕⣠⠄⠀⠀⠀⠾⣯⡀⠒⠠⣤⡀⠀⠀⠀⢠⢢⢄⠄⣀⡀⣐⣔⠄⠀⠀⢤⣖⣂⠉⠉⠉⠒⠢
⠀⠹⣟⣛⣛⡛⠟⠁⠀⠀⠀⠀⠘⣯⠟⠀⠒⣶⡇⠀⠀⠀⠀⠙⢻⣿⣿⣾⠿⠃⠀⠀⠀⠀⢴⣿⠧⠁
⠀⠀⠈⠑⠦⣬⠌⠁⠀⠀⠀⠀⠀⠲⣄⠀⠀⡏⠀⠀⠀⠀⠀⠀⢠⣿⣿⠖⠋⠀⠀⠀⠀⠀⠀⢹⡷⠂
⠀⠀⠀⠀⢀⣈⣷⣤⠄⠀⠀⠀⠀⠀⠈⢢⠀⡇⠀⠀⠀⠀⠀⠀⠈⢟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡆
⠀⠀⠀⠠⡟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠠⠼⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡠⣄⠀⠀⠀⢰⠄⠀⠀⠀⠀⣴⢊⡁⢀⣼⠇⠀⠀⠀⢰⣾⣋⣩⣍⣉⠀⠀⠀⠀⠀⣤⠶⠛⠓⠂⠒⢒
⠀⠀⣞⣸⣀⣠⡤⢞⡽⠀⠀⠀⠠⣬⣭⣵⡾⢟⣡⠆⠀⠀⠀⠸⣿⣡⠤⠤⠀⠀⠀⠀⠀⢞⡡⠄⠀⠀⠀⣩⡝
⠀⠀⠈⠉⠑⢒⣾⠽⡪⠀⠀⠀⠀⠀⠰⠶⠾⠟⠁⠀⠀⠀⠀⠀⠘⣷⡖⠂⠀⠀⠀⠀⠀⣄⠳⡥⠀⠀⠜⠉
⠀⠀⠀⠀⠀⠀⢀⡛⠂⠀⠀⠀⠀⠀⠈⣛⡛⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⠀⠀⠀⠀⠀⠀⠀⢏⠀⠘⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


⠀⠀⠀⢀⣠⡤⠶⠒⠒⡲⡄⠀⠀⣠⠒⡀⢀⣀⡒⠲⡄⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢾⣿⣿⣷⣶⡄
⠀⠀⠰⠫⣥⢴⠾⣟⣾⣿⠃⠀⠀⠸⡒⠈⠭⠭⠁⢈⠀⠀⠀⠙⢓⡒⢂⣀⡠⠔⠀⠀⠀⠀⠻⡿⠛⠛⠛⢫⡵
⠀⠀⠀⠈⠛⣿⣿⣿⣿⠇⠀⠀⠀⠀⢟⠀⠀⠀⣺⠁⠀⠀⠀⠀⠀⠩⠭⠐⠂⠁⠀⠀⠀⠀⠀⠘⠳⡄⠀⠈⡗
⠀⠀⠀⠀⢀⣿⣿⠟⠋⠀⠀⠀⠀⠀⠈⠣⣀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠉⠍⠁⠀⠀⠀⠀⠀⠀⠀⡰⢁⡤⠚⠁
⠀⠀⠀⠀⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⡶⠆⠸⠿⠿⠀⠿⢿⣷⣤⣶⣦⡄⢀⡀⠲⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠚⠛⠉⠀⢀⣀⣄⣠⣴⣶⣤⣶⣦⣤⣤⣬⣉⠙⠛⠿⢷⣀⣍⡛⢦⡀
⠀⠀⠀⠀⠀⣠⠄⠋⠁⢀⡀⢴⠾⠃⠙⢉⣙⣋⣭⣥⡤⣤⣤⡤⣌⣉⡉⠛⠿⡶⣀⣌⠙⢿⣂⣌⠢⡀
⠀⠀⠀⡀⠚⠁⠀⣀⠰⠛⠁⡀⢤⣶⡦⠟⣛⡉⢩⣤⡄⣤⣤⠄⣭⣭⢁⡻⠷⢠⣌⡛⠟⣠⣌⠻⡷⠈⠢⡀
⠀⠀⠞⠁⠀⠠⠞⠉⢀⡰⠞⠋⣚⠿⠚⠀⢁⣥⠄⠶⡆⢲⣒⢐⣶⠆⣬⣟⢃⢶⣭⠋⢴⣌⠻⢃⣀⠻⣿⡄
⠀⠀⠀⢠⠞⠁⢀⠰⠟⢁⣄⠛⣩⣤⠚⣫⡄⢶⣃⠹⠭⢸⡏⠸⠍⠨⡗⠂⢬⡕⣠⣝⠷⢉⢰⣮⠻⠷⠉⣤⣮⡀
⠀⠀⣰⠂⠀⠰⠟⢁⣤⠛⢁⡴⣍⠰⣗⡌⢾⡌⠻⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠂⠙⠫⡰⣼⡷⡉⢥⡜⣿⡌⠿⠓⡀
⠀⠀⠁⠀⣼⠃⢀⡘⠁⣴⢃⣜⠋⡶⣭⠺⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡝⠎⢛⡈⣤⡄⢿⣷
⠀⠃⠀⣰⠀⢀⠛⠁⣶⠀⡜⣑⠏⢶⣝⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⠇⠿⠘⠟⠈⣉
⠀⠀⢀⡉⠀⠼⠁⣴⠂⣬⢁⡋⡞⠶⠬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣧⢹⡇⢻
⠀⠀⠸⠀⢀⡇⠀⡆⢠⡅⣘⢘⣛⣛⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢩⠀⣥⢠
⠀⠀⠰⠀⢠⡄⠀⡅⢨⡅⣩⢨⣬⡭⠥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡆⠿⠸
⠀⠀⠀⠀⠈⡁⠀⣃⠈⡇⢨⡀⡆⢶⢛⣫⢔⡀⠀⠀⠀⠀⠀⠀⡬⡓⢕⠄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡷⢰
⠀⠀⠀⠀⠀⠃⠀⠛⠀⢙⡈⢥⠰⠎⢫⡱⣋⡕⣱⢣⢦⢥⡽⡺⡜⣜⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢈⡁⠚
⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⢣⡀⢲⠘⢢⡐⢎⡐⢩⡞⡸⠨⠅⠅⠅⢐⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⡿⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⣄⠙⢂⡀⠲⢀⡑⠰⠍⠬⠁⠍⠨⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠃⣼⢇⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈⠐⠆⢀⡉⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡼⠣⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⢀⠞⣡⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠞⣡⣌⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⡀⣺⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⡀⢰⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠉