Old man Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣦⣤⣤⣀⣀⠀⠀⠀⠀⠈⠉⠙⠛⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠉⠉⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⣶⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢤⣤⠀⣀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡇⢸⠀⠈⠉⠻⠿⢛⣛⡉⠒⠀⠀⠀⠀⠀⠹⣏⠉⠁⠲⠶⢤⣀⡀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢳⡆⠀⠀⠀⠀⠀⠈⠉⠙⠲⠶⢤⣤⣤⣀⠀⠘⣧⡀⠀⠀⠀⠉⠻⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠃⠀⠀⠀⠀⠀⣆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠂⠀⠀⠀⠀⢿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡀⢀⡀⠀⠀⠀⡀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠾⣿⠂⠀⢰⠀⠀⢀⣤⣤⡄⠀⡏⠀⠀⠀⠀⠈⠃⠀⢀⣤⣤⣄⠀⠑⠀⠀⠐⠂⢾⡷⠶⣤
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⣿⠀⠀⠀⠀⠀⠈⢙⣿⡧⠀⠀⠀⢠⠀⠀⠀⠀⠄⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠗
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⠷⠤⠤⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠛⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣏⠻⠑⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣿⠟⠛⠛⠓⠀⠀⣠⡤⠤⠀⠒⠛⠛⠋⢀⣽⣇⡹⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠘⡿⣠⠀⠀⠀⢀⡠⠟⢦⠀⠀⠀⠀⢀⠴⣿⢧⡜⠃⠀⠀⠀⠀⠀⣀⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣦⠀⠀⢡⡿⠃⠀⠀⠈⢀⠀⠀⠁⠀⠀⠀⠀⣤⢻⡎⠁⠀⠀⠀⠀⠀⢠⠏⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⠇⠀⠀⠘⣇⠿⣀⠀⠀⢸⠂⠀⠀⠀⠀⣠⠟⣤⠛⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠀⢸⠀⠀⠀⠀⠘⣤⠙⢦⣀⡀⠀⢀⣀⡰⢞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⡈⠀⠀⠀⠀⠀⠈⠳⣤⡉⠛⠛⠋⣡⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⣀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣾⠋⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠄⠀⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⣤⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡾⠿⣈⣉⣉⠉⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠈⠉⠋⠁⣠⠁
⠀⠀⠀⠀⠀⣾⡿⠒⠉⠁⢩⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠃⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⣼⣋⣠⡆⢰⠀⣿⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠈⠉⠉⠻⣧⣀⣿⠈⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠶⠦⣤⣀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢰⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠥⣤⣤⣄⠀⠉⢻⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠸⡇⠀⠀⠀⠀⠀⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣀⠀⢀⡛⠀⠀⠀⠀⠈⠳⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢱⠀⠀⠀⠀⠀⠀⠈⠉⠒⠖⢀⣄⠀⠀⠀⠒⠒⠚⠛⠃⠘⣧⣀⡀⠀⠀⣠⡞⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡆⢸⡀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠤⠄⠘⠿⢶⣶⡿⠋⢀⡀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⡇⢸⡇⠐⠶⠶⠶⠶⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠴⠶⠶⠶⠶⠀⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢿⡟⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠋⢸⠇⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣼⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠲⠶⠶⠶⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⠀⠀⠀⠀⠚⠓⠂⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣤⣔⣦⣤⣤⣤⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣦⡴⠦⢈⣉⣁⣠⡶⣄⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣉⣉⡉⠉⣉⣉⣻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠖⠉⠉⠉⠁⠐⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣤⣀⣤⣤⣤⣄⣠⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡀⠤⠴⢦⠖⠦⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠈⢻⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣀⠀⠀⠀⠀⠀⠃⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⡆⠀⠀⠀⠀⠀⠈⣧⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⣀⠤⠒⠄⢀⠜⠉⡻⠳⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠎⢀⠀⠀⠀⠀⠈⠉⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣍⠱⠀⠀⠀⠀⠀⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⢀⠀⢀⠀⠀⢸⣈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢈⠀⠀⢂⣰⠋⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠇⠠⠾⠀⠀⣾⡟⠒⠶⡤⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠈⠀⠀⠀⠀⠀⠀⢀⣼⢿⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠀⣈⢀⣈⣈⣤⣨⣇⣀⣤⢶⡿⠛⡉⠉⠉⠁⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⢆⡸⣻⣿⡄⠀⠀⠀⠉⠁⠀⡸⠄⠀⠜⠏⠁⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⠃⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⢀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢀⡠⠊⠀⠀⢀⣶⠶⠛⠛⠛⠋⠁⠙⠷⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠏⠄⠀⠀⠀⣿⡏⠀⠀⠀⠢⢤⠀⠀⠀⠀⠀⣹⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⡇⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⣾⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢳⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠣⣄⡀⠀⣼⣇⣀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⣀⣀⣤⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠿⠟⠻⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡲⠖⢋⠀⣁⣤⣤⠈⢉⢳⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠞⠁⠀⠐⢁⡼⠁⢠⡟⢀⣇⠞⠀⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⢁⡾⠀⠀⠀⠀⠈⠁⠀⢀⠃⠀⠀⠐⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⡌⢀⠀⠀⠀⠀⣀⡀⠀⠈⠀⠀⠈⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡎⡄⠀⣾⡆⢰⠃⠸⠀⠤⠄⠁⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠹⠃⡘⢀⣀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⣿⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠁⠀⠀⠉⢀⣤⡆⠀⠀⠀⠀⠀⠁⠶⣶⠒⠀⠀⠏⢈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢐⠦⠤⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠂⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⢃⠀⠈⠀⠀⠀⠤⠖⠀⠀⠀⠀⢤⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠈⠀⠀⠀⡜⠀⠄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⡼⠡⠂⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆
⠀⠀⠀⠀⠀⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡄
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢳⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡂⠐
⠀⠀⠀⠀⠀⠀⠀⠻⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠁⠁⠀⠀⢠
⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠘⠂⢼⡀
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⣀⣀⠤⠂⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⢁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⢀⣀⣀⣀⣀⣀⣤⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⢀
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠚⢁⠤⠀⠀⠘⠄
⠀⠀⠀⠠⠄⣀⣀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠊⠉⠁⢀⡠⠔⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⠊⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣀⣀⣀⠀⠠⠤⠔⠒⠉⠁⠀⠀⠀⠀⢀⡀⠄⠐⣈⣭⣔⠁⠀⠀⢠⠀⢰
⠀⠠⠀⠀⠀⠀⠃⠀⠀⢠⠇⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠊⠁⠀⠐⠉⠁⠀⣼⠀⠀⠀⠀⡇⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠉⠉⠉⠉⠉⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠧⡀⡠⢤⣄⠀⠀⠀⠀⣤⡠⢄⢀⠼⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢤⠀⡇⠋⢁⡄⠀⠀⠀⠀⢠⠈⠙⢸⢀⡤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣸⡇⠣⣀⣁⣠⠘⠀⠀⠀⣄⣈⣀⠜⢸⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡇⠀⠀⢀⠙⠓⠲⠖⠚⠋⡀⠀⠀⢸⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠙⠛⡛⠊⠁⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⡟⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢰⠀⠀⠈⠙⠲⠤⣄⣀⣀⣠⠤⠖⠋⠁⠀⠀⡀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⡴⠳⠤⣄⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢸⠁⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⢠⢪⣧⣸⢠⣠⢫⢂⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠓⢡⡀⠀⠀⠀⠀⠘⠞⠓⠯⡽⢻⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠢⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠉⠉⣧⣄⡀⠀⠀⠀⠀⠊⢺⡇⠀⠀⠀⠀⠀⢀⣠⡼⠁⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣾⣸⡄⠉⠉⠒⠒⠒⠒⠚⠓⠒⠒⠒⠚⠉⠉⢠⠇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⠉⢳⠀⠀⠀⠀⠀⠐⢲⡖⠂⠀⠀⠀⠀⠀⡟⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠋⠙⠛⠛⢻⡟⠛⠛⠋⠙⠛⢧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠤⠤⠤⠤⠤⠼⠧⠤⠤⠤⠤⠤⠼⠇⠀⠀⠸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠠⠤⠤⠌⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢠⡀⡀⠢⠀⠰⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣩⣄⠀⠀⣠⣶⠆⠀⠀⠀⠀⠻⢇⡀⠁⠆⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⢀⡙⣻⠃⠀⠀⡌⠁⠀⠁⠂⠀⠀⠀⠀⢁⢼⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠈⠀⠈⠀⠈⠲⡖⢚⡥⠀⠀⠄⠠⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠠⠄⠀⠈⠁⠀⠀⠀⠀⡀⢀⠀⢀⢣⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⢆⡠⢐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⣠⠞⠁⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠈⠁⠀⠢⢀⣈⣀⠤⢄⡀⢀⡀⠜⠁⠀⣱⠁⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠒⠒⠶⢄⠀⠀⠐⢀⠀⠀⠀⣬⠀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠂⠃⠀⠀⠀⠀⠀⠀⠉⠄⠀⡌⠀⠀⡀⠯⢣⠃⠀⠀⠀⠀⡱⠗⠂⠀⠀⠀⠀⠹⠀⠀⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠠⠋⠀⠁⠀⠀⠁⡤⠀⠀⠃⠀⣠⠊⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠁⠠⢀⡀⢠⠁⠀⢠⠀⠀⠰⣌⠁⠀⠀⠐⠈⠈⢀⠀⠀⠀⢀⠠⡆⠁⠀⠀⠉⠒⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠔⠉⠑⠠⡀⠀⢸⡀⠀⠀⢑⡀⠀⠀⠀⠀⠀⠈⠑⠀⢉⠠⠊⠀⠀⣀⣀⡀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⠊⠀⠀⠀⠀⠀⠀⠁⣌⣿⠀⠀⠈⠌⠥⠠⠤⠤⠄⠒⣶⠏⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠍⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⡿⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠚⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠈⠀⢈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣀⣀⣀⣀⠠⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠈⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠁⠀⠀⠀⠀⠀⠈⠆⠀⢀⠁⠀⢀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠂⠰⡄⠀⠀⡞⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠉⠀⣠⠂⠀⠛⠀⠄⠃⠰⣆⢲⢰⢢⡤⣀
⠀⠀⠀⠀⢀⡀⠐⠀⠀⠸⢀⡀⠀⠀⠈⠢⢀⡇⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠰⠁⠀⡰⠊⠈⣨⡄⠀⣠⡀⠀⠀⠈⠀⠈⠈⠘⠡⢗⡀
⠀⠀⠀⠔⠁⠀⠁⠀⠀⡃⠀⠈⡖⠀⠀⠀⢘⣄⣀⠀⠀⠀⢠⠏⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠊⠀⠀⣂⠰⢘⡗⡭⠀⠀⠀⠀⠀⡖⠉⠀⠀⠩⠟⠄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⡀⢠⠀⠀⢀⣀⣀⡈⠉⠋⠿⣁⡽⠊⠁⡤⠀⠀⠀⠀⠒⠂⠘⠏⠀⠀⠀⠀⠀⠀⠀⡠⢈⠀⠀⣠⡴⠤⣦⡤⠇⠀⠀⠀⠀⠀⡘
⠀⠀⠈⠢⡀⠀⠄⠀⠀⠀⠀⠁⠀⡈⠀⠀⠈⣁⣀⣀⡤⠀⠑⠤⢊⠜⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⡰⠀⠀⠀⠈⠑⠢⠀⠘⠃⠀⠀⠀⠉⢉⡒⠈⠀⡠⠈
⠀⠀⠀⠀⠈⠐⠀⠈⠀⠂⠀⠀⠀⠈⠐⠂⠀⠀⠀⠘⠁⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠤⠀⢀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡀⠀⠤⠄⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠙⠡⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⣰⠆⠀⠀⠀⣀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠛⠀⠀⠀⠀⠁⠀⠀⠀⡏⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠄⠁⠀⠚⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⠿⡂⠶⠆⠸⠿⣏⣅⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠒⠀⠠⠜⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⢦⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⡐⠃⠀⠀⠀⠀⠄⠀⠤⠔⠊⠁⠐
⠀⠀⠀⢀⠠⠄⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠁
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠉⠁⡈⠁⠀⠠⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠓⠒⠤⠤⠤⠤⡖⠈⣧
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢠⠀⢻⢆
⠀⠀⠀⠀⠢⣄⡀⠙⠠⡄⠀⠠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⡌⢦
⠀⠀⠀⠄⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠑⢀⡀
⠀⠀⠀⣀⠠⠀⡆⠀⠂⠀⢀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠉⢱⡀⠀⠀⠀⠀⣀⢀
⠀⠀⠁⠀⠀⠀⢇⠀⢆⠀⠈⡦⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣰⠀⠀⠀⠀⣾⠁⠀⢤⡀⠀⠈⣆⢡
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠑⠘⣇⠀⠀⠁⠀⠀⠀⠀⠐⠢⠦⠃⠠⣀⣀⣀⣀⡀⠰⠢⠦⠁⢀⠞⠀⠈⠐⠁⠀⠸⠃⠀⣀⣵⠀⢠⠇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠙⢻⠟⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀⠜⠓⠒⠤⠤⣀⣀⠀⠀⠀⠀⠀⢀⣠⠧⢀⠀⠤⠔⠊⠙⠋⠒⠶⠤⠄⣀⡀
⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⣸⣄⣀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢤⣀⠀⠀⠀⡄
⠀⠀⠈⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠐⣒⣒⣒⣀⣤⣤⣤⣿⠁⠀⡀
⣿⣿⣿⡿⣿⡿⠿⠿⣿⢿⠿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣾⣶⣾⣶⣾⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⣿⡄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢈⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⣿⣿⣿⠆⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠈⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢿⣿⣿⣾⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢛⡫⠿⠒⠒⣂⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣡⠖⡫⢔⣩⣴⣶⠿⠯⠭⠭⣝⣛⣛⠾⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠰⠃⠜⠁⠠⠔⠋⠡⠤⠤⠤⠤⠄⣀⠀⠀⠤⢉⡐⠨⠙⢦⣀
⠀⠀⠀⠀⠀⣀⠴⠒⠂⠀⠈⠀⠁⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡑⢌⠳⣄
⠀⠀⠀⠀⣜⠁⡀⠀⠀⠀⠀⠀⠀⢲⠈⣦⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣀⠀⠀⠀⠙⢦⡁⠈⠳⡀
⠀⢀⣴⣾⠿⣑⡄⠀⠀⡴⠀⠀⠀⠘⣧⠘⢧⡀⠹⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢳⡔⠀⡀
⢠⣾⠋⣵⠟⠙⢳⠀⡾⠁⠀⠀⠀⠀⠘⣧⡈⠻⣆⠀⠈⣆⠀⠑⣄⠀⠀⡈⢳⣤⣦⠹⣦⠀⠀⠀⣿⡄
⣾⠃⢠⡏⠀⣀⠈⠸⠁⠀⠀⠀⢠⠀⠀⠈⢷⣄⠈⠀⠀⠘⠀⠀⠈⠳⣄⠙⠦⢱⡙⠂⠹⣧⠀⢰⡼⣿⡀⡇
⡏⠀⢸⡇⢺⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⢀⠀⠀⠑⢄⡀⠈⠳⣶⣦⡉⠀⠀⢹⡆⢸⡇⣿⡇⣿⡄
⠁⠀⠸⡇⠘⠻⠂⠀⠀⠀⠀⠀⠀⠀⢠⡀⢣⠀⠀⠀⠈⠓⢤⣀⣀⠈⢙⡓⠀⠉⠀⠀⠀⢸⡿⢸⡇⢹⣿⣿⠁
⠀⠀⠀⠁⠀⠀⠇⠀⠀⢉⣀⠒⠀⠀⠀⠁⠀⢣⡀⠀⠀⠀⠀⠀⠉⠀⠈⡤⠂⠀⠀⠜⣠⣿⡇⣸⢃⡟⠿⠋
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠑⢄⢀⠒⢤⡉⠂⠀⠀⠀⠐⠀⠀⠰⣹⡟⠠⢿⡟⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⣢⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠂⠀⠀⠀⢀⠀⠀⣀⠀⠀⢡⠀⠰⠋⠀⠀⢄⠄⢀⠝⠃⣴⠀⠀⠐⠶⠾⠿⠿⠿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠉⠈⠳⡀⢰⠚⠓⠒⠂⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡄⡎⠀⠀⠀⠀⠀⠀⠾⠁⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠀⠑⠶⠒⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠰⠃⠀⠀⠀⣻⡆⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⢴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠘⣧⠀⢀⣿⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡤⠄⠀⠠⠤⠤⠀⠀⠐⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⢰⣿⠁⠀⠔⢉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠁⠀⠘⠆⣀⠠⠄⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣿⠃⠀⣠⣴⡿⠃⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⡏⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡧⠖⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠿⣰⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠏⠀⠀⣠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢀⠀⢀⠠⠊⠁⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⠎⣰⠃⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠊⣰⢡⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠞⠁⡀⠿⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠜⠡⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠆⠀⠂⠀⢀⠀⠀⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠁⠀⠈⠋⡀⠀⠀⠀⠁⠀⠀⠀⡸⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⡀⠂⠀⠲⠓⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⣰⣶⣶⡄⢤⡆⠐⣶⣶⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠄⢀⡰⠁⠀⠀⠡⡄⠠⠈⠀⠉⠉⠉⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠈⠃⠀⠀⢀⠖⠉⠀⠀⠀⠀⠀⠀⢀⡠⠚⠠⡀
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠤⠤⠤⠤⠤⠤⠤⠀⠀⠉⠖⠒⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠁⠂⠰⠤⢠⣀⡀⠀⢀⣀⡀⠤⠜⠂
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠠⠤⠤⡤⠀⠠⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡓⠤⣀⠀⠀⠀⠀⠀⠀⣀⢴⠁
⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉⠉⠙⠉⢩⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡄⠀⡀⠘⡟⡄
⠀⠀⠀⠀⠁⠲⠄⠀⠁⠒⠒⠒⠒⠋⠠⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠃⠀⢣⠘⢆
⠀⠀⠂⠀⠤⠀⢀⠂⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡈⠀⠰⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆⠀⠘⡆⠀⠑⠄⡀
⠀⠀⣀⠠⠄⠀⠘⠀⡄⠀⠀⡆⢂⠀⠀⠀⠌⠀⠀⠀⠀⠀⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⡇⠀⠀⢀⠀⣇⡀⠀⠀⠐⠀⢤⠀⡀
⠀⠀⠀⠀⠀⠀⠸⡀⢱⡀⠀⢹⠂⠄⣀⠀⠀⠀⠀⠀⠀⡘⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰⠀⠀⠠⢇⡀⠀⡈⢰⣇⠀⡠⢤⡀⠀⠈⡆⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⢹⠀⠀⠀⠈⠁⠐⠒⠂⠘⠜⠣⠑⠠⢄⣀⣀⣀⡠⠔⠢⠿⠌⠀⡜⠁⠀⠈⠉⠀⠁⠞⠁⢀⣠⣇⠀⣠⠃⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢠⠀⢠⠿⠁⠙⠻⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠘⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢷⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠤⠚⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⢀⡀⠀⠀⠂⠀⠉⠑⠈⠀⠀⠀⠈⠉⠑⠒⠒⠀⠤⠤⠐⠒⢺⡉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠀⠉⠉⠒⠲⠤⣄⡀⠀⠀⠀⠇⠀⡇
⠀⠀⠰⣧⣤⣄⣀⣀⣀⣀⣀⣀⡀⡀⣀⠀⠀⠀⠀⠠⠤⢤⣤⡀⠐⠞⠉⠵⠶⠒⠒⢒⡠⠤⠤⠤⠤⠄⣀⣀⣀⣀⣀⠉⠙⣦⠀⡖⠀⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣾⣶⣷⣤⣤⣦⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡤⠶⠛⠛⠛⠛⠛⠶⠖⠶⢤⢤⣤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡷⠶⠾⠓⠒⠒⠛⠒⠶⣶⣤⣤⣀⡀⠀⠐⠾⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠄⣠⡶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠲⢦⣄⠈⠳⡦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡛⠉⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣼⣇⠀⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⡀⢈⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⣀⡠⠼⠷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢸⠟⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠁⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⠏⠀⠀⠀⠘⢰⣦⣤⣀⠈⠁⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⢠⡴⠊⠉⠙⠲⣄⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⠉⠀⠀⠀⠀⠠⣌⣻⡗⠁⠈⠉⠒⠲⡦⢤⣤⡤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠙⡆⠀⠀⠸⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠙⠒⠎⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⣼⣧⡘⠿⣷⠟⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠃⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⡇⠀⠀⠀⠀⢀⣀⡌⠀⠀⠀⠀⠀⢰⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢀⡿⠑⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⠀⠀⠀⢀⣠⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣰⡟⠁⠀⢀⠄⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠋⠉⠙⠛⠛⠉⠁⢸⡄
⠀⠀⠀⠀⠀⠀⡼⢠⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠻⣤⣀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⡇⠸⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠁⡆⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⡌⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠦⣤⣤⣄⣀⣀⣠⣤⣤⣤⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⠤⠔⠒⠒⠊⠉⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡴⠛⠉⠁⠀⠀⣀⣀⠖⠲⢉⠀⢼⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⢇⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⣰⢄⣀⣀⠰⡶⠁⢹⠀⠀⢈⡇⠀⢿⡈⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡄⠘⣷⡀⠀⠀⠀⠀⣀⠴⠚⠁⠀⣠⠖⡳⠌⢁⡴⠁⣠⠃⠀⡆⢰⣦⠎⠀⢣⣠⣝⠶⣄⡀⠈⠋⠰⠦⠀⣀⡠⠶⠏⠀⢠⡞⣧
⠀⠀⠀⠀⠀⠀⢷⡀⠈⠙⠓⠒⠉⠉⠀⠀⢀⡴⠋⠁⡞⠁⡰⢟⠃⣠⡟⠀⣼⠶⣇⠸⡆⠀⠀⠀⠀⠀⠀⠙⠷⢦⣄⣀⠀⠀⠀⠀⣠⣤⠟⠀⢹
⠀⠀⠀⠀⠀⠸⠆⠙⢧⣀⠀⠀⢀⣀⣤⠶⠋⠀⠀⢸⠃⢰⡇⠈⠉⠉⠑⠚⠁⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠛⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣦⠀⠀⠉⠉⠙⠉⠉⠀⠀⠀⠀⠀⠸⢦⡼⠇⠀⠀⠀⠀⠀⠀⠀⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠴⠿⣿⡟⠢⣀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡇⠀⠀⠀⠀⠀⣠⠔⠁⡴⠋⠉⢉⣷⠀⠈⠆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣸⡇⠀⠀⠓⢄⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣠⠞⠁⠀⠀⡷⠀⠀⠊⠉⡧⠀⠀⠀⠀⠀⠀⠀⣇
⡖⠛⠒⠚⠛⠛⣷⠄⠀⠀⠀⣀⡤⢾⠉⠀⠀⢹⠓⣆⠀⠀⠀⠉⠠⣀⠀⠀⢸⡇⢸⡇⢀⡠⠞⡁⠀⠀⠀⣼⠄⠀⠀⠀⡄⢱⠀⠀⠀⠀⠀⠀⠀⢹⡤⠤⠤⠤⢤
⡏⠀⠀⠀⢠⣥⠠⠴⠶⠶⣟⠁⠀⢘⠀⠀⠀⢸⠀⠈⢷⡄⠀⠀⠀⠀⠈⠀⠴⡇⢸⡟⠠⠶⠀⠀⠀⣠⠾⢻⠀⠀⠀⢀⡇⠸⠷⢄⣀⣀⠀⠀⠀⠀⣃
⣧⣤⣤⣤⣤⣤⣤⣤⣀⣀⣉⣹⣤⣾⠀⠀⠀⣸⡇⠂⠉⣉⢷⣤⣀⡀⠀⠀⢀⡇⢸⡇⠀⣀⣠⡴⠞⠛⢲⡞⠀⠀⠀⢰⣇⠀⡀⣀⣼⠋⠉⠉⠉⠉⠉⠀⠀⠀⢈
⠠⠁⡌⠁⢹⠉⡀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠺⡇⢸⡏⠉⠉⠉⠉⠉⠉⠙⡇⠀⠀⠀⠸⡟⠙⠋⠛⠛⠙⠛⠻⠻⠛⠛⡟⢻⠋⠛
⡆⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠃⠇⠀⠀⢠⡎⡀
⡇⢸⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⡸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡇⢸⣇⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⢸⡇⡇
⡇⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠺⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠘⠧⠾⠋⠁⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠰⣷⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⡇⡇
⡇⢸⠀⠀⠸⠤⠀⠀⠀⠀⠀⢀⡝⠓⠶⠶⠒⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⠙⠛⠛⢫⠀⠀⠀⠀⠀⢣⣼⠀⠀⠀⢸⡇⡇
⡇⡼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠀⣄⠀⠀⠀⠙⠷⣤⣀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣈⠀⠀⠀⣠⠛⠁⠀⠀⠀⣴⠂⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢐⢻⣄⠀⠀⠐⠀⠀⢈⣩⠿⠶⠶⢤⣌⡓⠶⠶⠶⠒⠋⢀⣠⣾⡅⠀⠀⠀⠀⢰⠟⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠉⠓⠦⣄⡀⠀⠘⠀⠀⠀⠀⠀⠀⠙⢶⡄⠀⢀⡴⠋⠁⠀⠈⠃⠄⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣦⣀⠀⠀⠀⠀⠀⠀⠹⣦⠏⠀⠀⠀⠀⠀⠀⢀⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⢦⣤⡤⠶⣿⣤⣤⣤⣤⠴⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣠⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⢠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠠⠆⠀⠀⢸⣿⡷⠀⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠿⢧⣤⡀⠀⠀⣀⣀⠀⠀⠀⣀⣾⣷⠖⠛⠁⣠⣄⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠉⠁⢠⡼⠛⠁⠀⠀⣾⡿⠃⠀⠀⠀⠀⠈⠉⠰⠄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⡆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠘⣿⣦⣤⣀⣀⣀⣤⣤⣤⣤⣄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠃⠀⠀⠀⠀⠸⠆⠀⢸⣿⣿⣿⣿⡏⠉⠻⠀⠀⠀⢰⣿⠉⠁⢹⣿⣿⣿⣿⡏⠉⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⣄⠀⠀⠀⠀⠀⠀⣤⡀⠘⠿⠿⠿⠟⠛⣾⡆⠀⠀⠀⠘⣿⣷⠀⠘⠿⠛⠛⠛⠻⣿⠃⢸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠉⠻⠿⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠾⠉⠀⠀⠀⠀⠀⡟⠉⠀⠀⠀⠀⠀⣤⡄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠚⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠈⠋⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣤⣴⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⣴⣿⡦⢤⣤⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠇⠈⠉⠀⠀⠀⠀⠻⢆⡀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠂⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡇⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⠘⢆⡀⠀⢀⡄⠀⠀⠀⡀⠀⠀⢀⡀⠀⢀⣀⢀⡙⠉⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⠀⠈⠛⠀⠘⠃⠀⣠⠞⠃⠀⠀⠘⠋⠀⠜⠛⠉⢁⣶⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⠀⠀⠀⠀⠀⣠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣟⠀⠰⠶⢀⣀⠀⠀⠀⠀⠀⠀⣀⠖⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⠀⠀⠀⠈⠛⢢⣤⣤⣤⡀⠘⠋⠀⠀⢰⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⡉⠙⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣿⣿⠿⣿⣿⣿⣿⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠛⠋⠉⠉⠉⠁⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠙⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢿⣿⠿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⣠⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄
⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀
⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷
⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠠⠦⠄⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣼⣇⣀⠀⠀⠀⠀⠀⢠⣿⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠉⠀⠀⠀⠈⠉⠀⣿⣷⡆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡤⠤⠤⢤⣴⡾⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⢿⣿⣧⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⢠⣦⠀⢸⡇⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⢿⣿⡀⠀⠀⠀⠀⠀⠈⠛⠿⠤⠾⣿⣦⡿⣇⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠙⢿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⠿⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠙⠻⢿⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⢠⣤⣤⡀⣠⣤⣤⣶⣶⣶⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠿⣿⣿⣿⣿⣿⣿⡷⠂⠐⢿⣿⠁⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠈⠟⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⡷⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣿⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⣀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣿⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⢷⡶⢶⣿⣿⡷⠾⣿⣿⣶⡶⠤⠄⠀⠀⢀⣀⣀⣠⣿⣄⠀⢠⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠈⢿⡿⠛⠛⠛⠛⠋⠉⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⠇⠀⣰⡆⠀⠀⠀⠀⠀⠀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠿⣧⣀⣠⣀⣀⣀⣀⣀⣀⣠⣶⣦⣄⣤⣀⣀⣀⣀⣀⡀⠀⣼⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠉⢻⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀⢀⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣬⣭⣽⣽⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠉⠿⠿⠿⠟⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁⣿⠭⢭⡷⠀⢹⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠂⠀⢶⣲⣿⠇⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡏⠀⢰⠟⠀⠀⠠⡤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⣸⣇⣀⣤⠼⠃⠻⡷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠛⠛⠉⠁⠀⠀⢀⣀⡟⠀⠈⠛⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣾⠁⠁⢷⠀⠀⠀⢰⠃⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠤⠴⠚⡉⢹⡄⠀⢸⡆⠀⠀⠸⡄⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⣿⡜⣷⠀⠈⣧⠀⠀⠀⠻⣦⣠⡶⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠈⢳⣠⠟⣧⠘⣧⠀⢻⠀⠀⠀⣠⠾⣏⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠛⠻⣦⣰⠃⢀⣟⠀⠘⣧⣸⡇⢀⣸⣿⣤⠈⢻⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠠⡏⢴⣿⣿⣯⠀⠀⢸⣯⣛⣀⣹⣿⣧⣴⡿⠾⣷⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣲⡾⠉⠟⠀⠀⠀⠻⢬⣭⣭⣷⣾⠦⣀⣠⣬⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣧⢸⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠸⡇⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣿⠀⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⠉⡟⠃⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⢰⡇⠀⠈⢹⡁⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠸⣇⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠰⣅⠉⠛⠺⢷⠠⠤⠤⣴⡶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠛⠛⠛⠛⠛⠛⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣻⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠉⢹⠀⣴⣩⡤⠿⠂⠀⠀⠀⠀⠀⠐⠻⠤⣍⣳⠀⡟⠉⠉⢹⡀
⠀⠀⠀⠀⠀⢸⡷⢶⡀⢸⠀⠋⠁⢀⣤⠀⠀⠀⠀⠀⠀⠀⣠⡄⠈⠙⠀⣟⠀⡴⢶⡇
⠀⠀⠀⠀⠀⣿⠀⢺⣇⠸⡆⠀⠑⠢⠉⠀⠰⠔⠒⠢⠄⠀⠈⠕⠚⠀⢀⡇⢸⣷⠀⣿
⠀⠀⠀⠀⠀⠘⠷⣬⣏⠀⠙⠶⠶⠒⢶⣃⡀⠀⠀⠀⢀⣘⡶⠖⠶⠶⠋⠀⢸⣧⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⡀⠉⠉⠓⠒⠒⠉⠉⠀⡀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠙⠲⠶⠤⠤⠤⠶⠖⠋⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣀⣨⣷⡀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇
⠀⠀⠀⣠⠶⠋⢡⡟⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⢻⠻⣦⡀
⠀⠀⣴⠃⠀⠀⣼⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠞⠋⠀⠀⠀⡆⠀⠙⠷⣤⡀
⠀⣸⠇⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⣶⠶⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠙⠳⣄
⢀⡿⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⢹⡶⣦⣀⡀⠀⠈⢷
⣸⠃⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠇⣽⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣿⠀⢸⠇⣼⢀⡟⢳⠀⢸
⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡄⢿⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡟⢦⢿⣠⣏⣼⣡⣿⡴⠏
⣿⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠳⠎⠙⠲⠶⣶⠏⣹⢹⠁
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢸
⣿⣀⣀⣀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣿⢸
⠈⣿⠉⠉⠉⢻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠺⠤⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠁⠀⢿⢸
⠀⣿⠀⢠⣶⠸⣧⠉⠙⠲⠶⠶⣤⣤⣤⣤⣄⣀⣿⣤⣄⣤⣤⣤⣤⡶⠶⠞⠛⠉⣹⠏⠀⠀⣿⢸
⠀⠸⣧⠸⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿⢸
⠀⠀⠈⠳⡿⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠶⠶⠶⠦⣤⣤⣤⣿⣦⣤⣤⠴⠶⠶⠶⢶⣾⡁⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇⠀⠀⠀⠀⢿⣼
⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠠⠆⠀⠤⠠⠄⠤⠀⠄⠤⠠⠠⠤⠠⠄⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⣤⣤⣤⣄⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠳⣄⠐⣭⣾⡿⣿⣿⣟⡲⢥⡪⢿⣟⣿⣽⣧⣰⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠿⠿⣷⡘⢦⡀⢭⣿⣿⣿⠿⣿⡷⣯⣿⢷⣿⢾⣿⣿⠏⢠⣄⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠋⠁⠀⠀⠀⠀⠉⠱⠿⢷⣬⡻⣿⣿⣦⡝⢷⣭⣝⠛⠫⠙⠂⠀⠁⠛⠂⢬⣄
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧
⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⢸⣿⡀
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢹⣞⣿⡇
⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠒⠀⠀⠀⡄⠈⣯⣿⣿
⠀⠀⠀⢸⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿
⠀⠀⠠⠸⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢷
⠀⠀⠀⣄⣈⢿⣿⡏⠀⠀⠀⠀⠀⠀⣤⣶⢦⣤⣄⣀⣠⣶⣄⠀⡄⠀⣴⣦⣤⣤⣤⣦⣤⣀⣠⠀⠀⠀⣾⣿⠏⡘
⠀⠀⠀⢸⡿⠄⠻⠁⠀⠀⠀⠀⠠⠈⣁⣤⣤⣶⣿⣿⣿⣯⡈⠀⠀⢠⣼⣼⣿⣿⣽⣤⣤⣉⠓⡀⠀⠀⣿⡿⠐
⠀⠀⠀⠀⠅⢠⠀⠀⠀⠀⢀⠐⣶⡾⠛⠛⠛⠳⣿⣿⣿⡿⠁⠀⠀⣿⣿⣿⣿⣿⠟⠛⠊⠛⠆⠙⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠐⣪⣤⣴⣿⠿⠋⠀⠀⠀⠀⠻⡻⠿⢿⣿⣤⣤⡶⠎⠉⠙⠛⠀⡟⢀
⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣰⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠐⠠⣧⣀⠀⠀⠀⠀⠀⠀⠀⡄⠡⣿⠇
⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠓⠠⡀⠀⠀⢀⣒⣾⡿⠁⠙⣿⣿⣷⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⢀⠙⠶⠿⠿⠟⠁⡀⣸⣆⣌⠻⣿⠄⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⣿⣿⠿⠿⠿⣿⣧⣤⣴⣴⣷⡾⠿⠛⠛⢿⣇⢹⡆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⡀⠈⠀⠀⡿⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣸⠏⣴⠃⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣀⣤⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⣼⠇⠸⠃⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⣟⠿⣇⣰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⠀⡰⠋⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣌⡻⠟⠒⢤⣄⠘⠂⠂⢀⣄⣀⣤⣴⠿⣀⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠐⣤⣴⣶⠾⢛⣿⣟⠉⢹⣿⢋⣴⣿⣵⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⢼⣏⠙⢷⣌⢿⣿⡄⠈⣿⣿⢯⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠻⢾⣿⡀⢹⣿⣿⣿⣧⢸⡿⣿⣿⡿⠃⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡈⣿⣿⣿⣿⣼⣿⣿⠋⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠉⠉⠉⠉⠉⠉⠙⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢛⠉⣹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡤⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢈⡿⠁⠀⠀⣀⣀⡀⠀⠐⠚⠛⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠈⡿⠁⠀⣰⠚⣉⣉⣽⠄⠘⠋⠉⠒⠀⠀⣴⣋⠉⠓⢦⡀⠀⠀⠀⠚⣿⣠⠀⢼⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠈⢹⠇⠀⠠⡷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⢄⣷⠀⠀⠀⠀⢹⡧⠄⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣸⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠛⠀⠀⠀⠀⠈⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣰⠋⡁⢻⠀⠀⡤⠄⢸⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡿⢋⠳⡟
⠀⠀⠀⠀⠀⠀⠀⡏⢨⢿⡟⠀⠀⠐⠃⠀⠉⠉⠀⢀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠲⡉⠓⠀⠀⠀⠀⢰⣿⡉⠀⡇
⠀⠀⠀⠀⠀⠀⣸⠃⠸⡾⠀⠀⠀⠀⠀⣀⣠⠴⠀⠈⠛⠓⠛⠚⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡽⢸⠃
⠀⠀⠀⠀⠀⠀⠹⣄⣠⡇⠀⠀⠀⠈⢹⠋⢠⠏⢠⠀⠀⡄⠀⠀⠀⠀⠈⠛⢶⠖⠂⠀⠀⠀⠀⠀⢠⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢧⠀⠀⠀⠀⢸⠀⠘⠷⠞⠓⠛⠳⠦⢼⣤⡶⠾⠁⢸⠀⠀⠀⠀⠀⠀⠀⣸⢧⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠘⠆⠀⠀⠀⠀⢤⣀⣀⡀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠿⠶⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣀⣀⣤⠴⠛⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢹⠛⠲⢤⣤⢤⣤⡴⠶⠛⣹⠏⠉⠉⠀⣶⣤⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⣴⠀⠀⠀⢸⣄⣀⣤⠟⣻⣅⠀⠀⣰⠇⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠈⢳⣄
⠀⠀⣠⠖⣺⠶⠒⠢⠤⣾⣤⡤⣄⣿⠀⠀⠀⠀⠉⠉⡟⢙⣋⢸⡳⠶⠋⠀⠀⠀⠀⠀⠘⣧⡀⠀⢀⣀⠀⠀⢹⡆
⡴⣾⢁⡞⢡⠖⠉⠀⠀⠈⣆⠙⣦⢹⡄⠀⠀⠀⠀⠀⣿⠻⠟⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⣭⠝⠃⠀⣸⡇
⢇⢹⡘⡇⠞⠀⡾⠛⢳⡄⢸⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣠⠟
⠈⠛⣿⠛⢻⠾⠳⣞⣋⡴⠋⠀⢀⡴⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣰⠋
⠀⠀⢿⠀⢸⠀⠀⠈⠉⠙⠓⠚⠋⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⡟
⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠈⡷⣶
⠀⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⣾⠁⠀⠀⣱⠇
⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠛⠛⠓⠒⠶⠶⠞⠛⠛⠛⠛⠋⠉⠁⠀⠀⡟⠙⠛⠋⠁
⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣶⢶⡄⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢸⠁⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⣶⣒⠒⠲⠀⠀⠀⣧⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠙⣦⣀
⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠙⠳⢤⣀⠀⣿⠀⠀⠀⠀⢹⣦⠤⠤⢤⣀⠀⠀⣠⢿⡄
⠀⠀⣸⣇⣨⣧⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠀⠀⠀⢠⠏⠀⠀⠀⠀⠈⠙⠉⠁⢀⡿
⠀⠀⣟⠀⠀⣸⠄⠀⠀⠸⣄⣀⣀⣀⣀⣀⣀⣤⠦⠤⠴⠃⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢀⡞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⠤⠤⣤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠜⢿⡏⠀⠀⠀⠀⠀⢀⡇⠹⡖⠀⠐
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠒⠂⠠⠿⠤⠀⠀⠀⠀⠀⡼⠳⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣠⣖⢒⠸⠗⢲⡤⢦⠀⡴⢼⣿⢄⣿⣿⠻⠀⠀⡠⠤⠤
⠀⠀⠀⢀⣀⡠⠀⠀⣀⡀⢉⣭⣤⣤⣤⡀⠀⢃⠀⠃⠉⠉⠀⢸⠿⢉⡆⠀⠀⠴⠿
⠀⠀⠀⠸⠾⠛⠛⠀⠀⣤⣬⠫⠒⠚⠉⠋⠙⠉⠉⠀⠀⠀⣘⣟⡃⡈⠀⠀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠂⢤⣙⣻⣴⡯⠀⠀⡸⠈⠡⠄⠐
⠀⠀⠀⠴⠲⠖⠚⠋⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠃⠀⡰⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⠀⠀⠀⠀⠀⠀⠐⠺⠽⢦⣀⡜⠀⠀⠀⠀⠈⠄⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⠿⠋⣁⠀⠀⡀⠀⠀⠀⠀⠇⠀⢠⡄⠀⠀⠐⣁⠀⠀⣸⠐⠛⠒⢀
⠀⠀⠀⠀⠀⢀⠀⠀⡟⠋⠉⠹⢾⣿⣴⣶⣿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠰⠚⠁⠀⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡉⠿⢿⣿⣋⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠙⠻⢶⣟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣴⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣴⣿⣿⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣤⣶⣾⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⣶⡾⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢀⣴⣶⣶⣾⣿⣿⣿⣿⠟⠟⣿⣿
⠀⠀⠹⢿⣿⠟⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣾⣿⡏⣿⣿⣿⣿⣿⠏⠙⠁⠹⣿
⠀⠀⠀⠀⠹⣾⣿⣷⡄⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⢙⣿⡏⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣦⣴⣾⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⠸⣿⡇⢸⡟⠁
⠀⠀⠀⠀⠀⠠⡌⢶⡿⠛⠻⠃⠀⠀⠉⠘⢿⡜⡏⢻⡍⠁⢀⣿⣾⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣿⣿⡍⣦⣿⣧⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠀⣿⣶⣾⡇⠀⢦⣀⠀⠈⢻⣿⣾⣿⡠⠄⢹⡏⢹⣇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠎⣸⣿⣟⣹⡿⣹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣷⠀⠀⠙⠳⣆⠀⢣⣹⣿⡇⠀⠈⣷⢸⣷⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣁⣼⣿⣿⣿⣿⣿⠋⣿
⠀⠀⡀⠀⠀⠀⠀⠀⠙⣿⢿⣿⡗⠀⠀⠀⠙⠻⣦⣿⠻⣧⠀⠀⣿⣞⣿⠙⠛⣿⣿⣿⣿⣷⠰⠿⣩⣿⣿⣿⣿⡿⣫⣤⣄⣿⠂
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠘⣦⣽⣶⠗⠓⠒⠒⠒⠀⠙⣀⣾⣧⣀⣸⣿⣿⣷⠶⣿⣿⠯⣽⣿⣤⢰⣿⣿⣿⣿⣿⣿⣿⣷⣿⡏
⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠘⢺⣿⣴⣾⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⡟⢩⡄⠀⠀⢀⢀⣠⡄
⠀⠀⠀⠀⠀⠉⠋⣤⡀⠀⠀⠀⢻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣼⣿⣿⣿⣇⠈⡀⠀⠀⠀⠉⣽⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⡀⢠⣜⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡏⢸⣿⣿⣿⣿⣿⠀⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣌⡻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⡇⠈⠸⣿⣿⣿⣿⣿⣄⣘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣶⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠹⣿⣿⣾⣿⣿⠿⢿⣿⣿⢿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠐⠓⢿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠁⠀⠄⢠⠟⠁⣴⣶⢤⣀⣉⡈⢻⢹⣿⣿⣿⣿⣿⣿⣿⠿⠀⠸⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⢸⣷⣦⡄⠻⣿⣶⣄⣨⡟⢡⣤⣵⠉⠻⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣧⣴⣿⣿⣿⡇⠀⣼⣿⡿⠏⠀⠀⢻⣿⣆⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⠘⠟⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⡟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠛⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠉⠉⣾⣿⣿⣿⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣯⠄⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⣴⣶⠋⣿⣿⣿⣿⣧⠀⠀⡀⠠⠴⠀⠠⠀⠀⢹⣿⣿⠭⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⢹⣿⣿⣿⣿⠈⠌⠹⠀⠀⠀⠊⢁⠀⢸⣿⣿⡖⠉⡀⠘⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢸⣿⣿⣿⣿⣥⡄⠀⠀⠀⢀⠀⠁⠀⣼⣿⣿⠇⠀⢹⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⣠⠄⠉⢲⣦⡛⢻⣿⣿⣧⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⠀⣼⣿⣷⣶⣾⣿⣿⡟⠀⣿⠻⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠉⠛⢰⣿⣿⣿⣿⣿⣿⣿⠃⠀⡿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⣢⣾⣿⣿⡿⢿⣿⣿⣿⠂⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇⢸⣿⣿⣿⡃⠀⠀⠀⠀⠹⠷⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⠈⣿⣿⣇⠁⠀⠀⠀⠀⠀⠀⠙⠧⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡿⢿⣿⣿⡇⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡿⠃⢸⣿⣿⡁⠘⢿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠙⠁

⠀⢉⡹⠿⠟⠛⠛⠛⠁⠀⠀⠀⣠⣔⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⣿⣿⣿⣿⣿⣾⣿⣿⡇⠀⠀⠛⠒⠛⠛⠋⠻⠻
⣀⣿⠷⠶⠦⠤⠤⠤⠤⢄⢀⣀⡀⠉⠉⠁⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⡿
⣿⠿⠟⠻⠶⠶⠶⠶⣦⡤⠄⠀⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⣶⣦⣀⣽⣿⣿⣿⣿⣿⠿⠁
⣶⣾⣷⣶⣶⣦⣤⣤⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣧⡘⢿⣻⣿⣿⣽⣿⣿⣿⡿⠿⠃
⣿⣿⠛⠛⠑⠛⠻⢬⣉⣉⡁⠀⠀⠈⠙⠃⠀⠒⠀⠀⠀⠸⣿⣿⣿⡆⢿⣿⣿⡿⠿⠿⠟⠷⠄
⣛⡛⢻⣷⣶⣶⣤⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠿⠶⢞⣻⡿⣾⣿⣶⡄
⣶⣭⣄⣈⣙⣛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⢿⣿⣿⣿⣿⡀⢙⣛⣛⣲⣂⣴⣂⣀⣀⣀⣀⣀⣀⡀⡀⢀⠀⣀
⠻⠟⠛⠛⠛⠛⠻⠿⠿⠿⣶⣦⣤⣤⣤⡄⠀⠀⠀⠀⢀⣿⡀⠀⣸⣿⡿⠋⠀⠀⢰⣤⣭⣭⣤⣤⣤⣀⣉⠩⣭⣤⣀⣀⢨⣥⣥
⣶⣶⣮⣤⣤⣴⠼⠶⠶⠦⠄⠤⠥⠀⠤⠚⠋⠙⠛⠋⢀⣹⣿⣿⣿⡿⠁⢀⡆⠀⠈⢿⢟⣛⣿⣯⣿⣷⣶⣾⣿⣿⣛⣿⣿⣿⣿
⣿⣿⣇⣠⡤⠤⠄⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⠿⢷⣾⣿⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣶⣶⡶⢛⠗⠚⢿⣿⣿⣿⡇⠀⠀⠀⠉⠉⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣷⠟⠃⠀⠀⠸⠿⠿⠀⠀⠀⠐⢶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⢿⣽⣿⣷⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣀⣹⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣗⠙⠿⠟⠁⠀⠀⠀⣠⣾⣿⣿⣿⣶⡿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣠⣴⣶⣿⣿⣿⣷⡈⠉⠋⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⡀⠀⠀⠈⠙⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠕⠤⠊⠉⠉⠳⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⣟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣚⡗⠔⢙⠄⠠⠴⣍⡔⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⣀⡠⢤⡀⠀⠀⣈⠀⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣾⣋⣡⠴⠊⠀⡼⠁⣽⠃⠀⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠁⠀⠀⣀⡾⣤⠊⡜⠀⠀⠈⠛⢄⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⢩⡎⠉⠀⠐⠣⡀⠀⠀⠑⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠸⡐⡄⠀⠀⠙⣆⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢀⡧⠘⡄⠀⠀⢹⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⢠⠎⢰⠀⢁⠀⠀⢸⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢀⠔⠁⠀⠀⠀⠐⠀⠀⡜⠁⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢀⠀⣄⠀⠀⠀⠀⠀⠀⣄⣀⠀⠸⣄⠀⠀⠀⣸⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⢀⣀⠀⢳⠎⠀⠀⠀⠀⠀⠀⠀⠙⠒⢲⢋⡓⠒⠚⣷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣀⡇⠀⢱⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠉⡄⢰⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠀⠀⠧⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢧⡀⠀⢀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡃⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡄⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⡼⠀⠀⠀⠀⠀⠀⢠⠖⠉⢈⠉⠽⠷⣇⣀⣀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⡁⠀⠀⠀⠀⠈⣉⣩⠜
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⢒⣊⣉⣭⣭⣍⠙⠛⠒⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣡⣴⣶⣿⣿⣿⣿⣿⣿⠿⠶⠄⠀⠀⢠⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⣤⣾⣛⣛⣻⣿⣿⣿⣿⣿⣟⣃⠀⠀⠀⢿⣿⣿⡿⠁⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢠⣤⣀⡉⠉⢉⠙⠛⠛⠛⠛⠛⠉⠁⠀⠀⣠⣼⣿⠏⢀⣸⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠐⠾⢿⣿⣿⣿⣿⣿⣿⣶⣶⣼⣿⣿⣶⣶⣾⡿⠛⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢺⠋⠉⠉⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠐⠒⠶⠿⢿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢮⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠖⠛⠿⠶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠶⠟⠛⠛⠛⠛⠳⠀⠀⢸⣿⣿⣿⡿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠘⣿⡇⠀⠀⠀⢠⢤⣶⣶⣤⡉⠻⣄⠀⠀⠀⠀⠀⣠⣐⣶⣦⣄⡀⠀⠀⠀⠈⣿⣿⠟⢠⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠱⣸⡇⠀⠀⠀⠛⠋⢿⠿⠟⠙⠀⢻⠀⠀⠀⠀⠸⠧⣿⣿⠟⠛⠋⠀⠀⠀⠀⠀⠋⢠⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢰⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠸⡇⠀⠀⠀⠀⠀⠀⠀⣴⠻⣧⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⣧⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠂⠀⠒⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠙⠀⢶⡏⠉⠛⠛⠛⠛⠋⢉⠟⠀⠀⠟⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠻⣶⣦⠤⠴⠖⠒⠁⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⢆⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⡰⠁⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⣿⠈⢳⡀⠙⠢⢄⣀⣀⣀⣀⡤⠶⠋⠀⠀⢀⠞⠀⠀⡏⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⣿⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⢸⡇⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉⠀⠀⠀⠀⣿⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠳⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠻⢶⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⣼⡃⠀⠀⠀⠀⠀⠀⠙⠻⣿⢶⣤⣀
⠤⠶⠶⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣠⠞⠉⠳⣄⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠙⠻⢶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡞⢻⠀⠀⠀⣹⣦⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠏⠀⢸⠀⠀⢰⡟⠘⢧⠀⠀⠀⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡼⠃⠀⠀⣼⣀⣠⣿⠀⠀⠈⢧⡀⠀⢠⢃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠋⠀⠀⢀⡜⠁⠉⠀⠈⣧⠀⠀⠀⠳⣄⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡞⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⠛⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡎⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢠⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣾⣿⣿⣶⣶⣦⣤⣀⠀⠀⠀⠀⡴⠒⡶⠀⠀⠀⠀⠀⠀⠀⠚⢾
⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⢿⢿⣿⣿⣏⠉
⡀⠀⠀⠀⠄⠀⠀⠛⣿⡄⠀⠀⣿⣿⣿⣿⣿⣻⣷⠀⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣶⣄⡀
⣯⠀⠀⠀⠀⠀⠀⠸⣿⣇⡀⠀⠸⡿⢿⣿⣻⡛⣇⣠⣟⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⣀⣤⣤⢴⣶
⠞⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠀⠀⠀⠀⠙⠛⢿⣿⣿⠟⠁⠀⠀⠀⢠⣤⣼⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣯⣟⣪
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠀⠘⠛⠗⣦⡀⠀⢠⣤⡚⠋⠁⠀⠀⠀⠄⠀⠀⠀⣿⣏⣭⠏⠉⢛
⡎⠉⠈⠉⠉⠉⠒⠠⠔⠒⠒⠒⠒⠀⡈⠀⠀⠀⠀⢀⡇⠘⠋⠈⡿⠧⢀⡀⠀⡠⠦⠔⠃⠤⠤⠬⢽⠙⠀⠀⠉
⣄⡀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣈⡋⢤⠀⠢
⠉⠁⠀⠀⠀⠀⣀⢾⠀⠀⠀⠀⠀⣠⡈⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡴⠖⣋⣀⣾⣿⣫⣷⣷⡀
⠀⠀⠂⠀⠀⠘⠋⠉⠀⡀⠀⠐⠻⣿⣿⣷⣶⠒⠋⠒⠀⠀⠀⠀⠀⠀⠀⠾⣽⡿⠠⣼⠟⢹⠙⠀⡗⢀⡌⢧⡀
⢀⡄⠀⠀⠀⠀⠚⠁⠀⡀⠃⠀⣠⣿⣿⣿⣿⣿⣦⡀⠀⠄⠀⠀⢀⠙⠀⠂⠨⢷⣶⣌⠀⡀⠀⢀⣠⡋⠳⢴⠃
⡀⠉⠙⠀⢀⣰⠿⠖⢩⣤⣦⣴⣿⡿⠋⡤⢨⠻⢷⣿⣌⠤⠀⠀⡤⡦⣴⣾⢧⠴⢿⣿⣾⣧⣤⣮⢿⣀⠤⠄⠠
⠃⠀⠀⠘⠀⠀⠀⠀⠈⠛⠿⠟⠋⠀⢸⣇⡛⠀⠀⢿⣿⣄⠀⠀⠈⠉⠛⠀⠀⢀⣾⣷⣿⣿⣿⣿⣶⣄
⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠈⠻⢟⢣⡀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠛⠿⠿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢲⣶⠒⠀⠉⢹⣾⣰⣿⣿⣿⣿⡿⠁⢠⡾⣦⣀⠀⣾⣿⡀
⠀⠀⠀⢀⣼⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠠⢜⠟⡋⠉⠁⠀⠀⠈⠈⢱⣿⢻⣷⣾⣿⣿⣦
⣀⡀⠀⠀⠉⠀⢠⣤⣄⠀⠀⠴⢦⣤⣤⠀⠀⠀⣤⣷⣬⣿⠏⠐⠾⣷⢷⣠⠴⣾⣄⡀⢀⡀⠀⠈⠉⠛⠙⢟⢻
⣽⠛⠛⠀⠀⣀⣤⣼⠛⣿⠧⡆⠀⠉⠓⢢⡄⠀⠈⠙⠛⠁⠀⠀⠀⠰⣿⠆⠉⠉⠁⢠⡿⠃⠀⠀⠀⠀⠀⠀⠈
⢿⠯⠄⠠⣴⡟⠻⢯⢩⣥⣤⣄⠀⠀⠽⢽⡿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢶⣦⠀⠀⠀⠀⠀⣠
⣤⠄⠀⣿⣻⣿⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠐⠾⢶⣶⣶⣶⠀⠀⠛⠛⠿⠃⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠐⠂⠐⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⣀⣠⠤⢷⣲⠖⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠚⢿⠥⠀⠀⠀⠀⠉⢀⠃⢨⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠈⠃⠀⠀⠀⠀⠀⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢀⢀⠀⠀⠀⡀⠀⠀⡀⡁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠈⠀⠘⣦⠄⠂⠀⢀⠀⠀⠀⠀⢹⣆⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⢀⣀⣀⠀⠀⠈⣹⠟⢻⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣠⠤⠚⠉⠀⠈⢧⢀⠞⠀⠀⠀⠁⠈⠙⠳⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢮⡉⠱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠰⡄⠀⠐⡄⠀⢀⡼⠷⠞⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠄⠀⠀⢹⣠⡟⠀⠀⠀⠀⠀⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠤⠤⡴⠀⠀⠀⣼⣿⠷⠶⠦⢄⡤⣀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠤⠴⠄⣠⠤⠚⠛⣦⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣴⠟⠹⣇
⠀⠀⠀⠀⠀⢠⡖⡏⠀⣀⣀⣀⣠⣤⣤⠶⣤⣿⣀⣄⡴⠾⢿⡟⠉⢁⡀⠀⢀⡀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⢆⠶⣄⠰⠿⣿⣿⡄⠀⠀⠀⠀⠈⠉⠀⠀⠀⣸⡅⠀⢠⡟⡿⠉⠁⢹⣯⣶⠁
⠀⠀⠀⠀⠀⠈⠙⣿⡗⠛⠉⢿⡿⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠈⠁⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢠⣶⣶⠾⠟⠿⠛⠛⠋⠁⠙⠷⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⣠⡎⠁⡀⠀⠀⠀⢶⣿⡟⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⣇⠈⠁⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⠈⠙⠛⠀⠤⢀⠀⠀⢀⣴⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢻⠀⠀⠀⠀⠀⠈⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⣿⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠹⣄⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠈⠓⢦⣤⣤⣾⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣣⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢿⢉⡔⠛⣷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⣿⡧⣶⠛⠛⠓⠶⠿⣿⢿⡷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢹⡯⠟⢀⣀⠀⠀⢀⢺⣿⠁⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣼⡗⢲⡿⢿⡷⢾⢿⣿⣿⡀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠹⣁⢦⠑⡺⣁⣸⠖⢻⣿⠃⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡘⠂⣚⣣⠒⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠛⣦⣀⣠⣼⡯⠶⢿⠏⣿⣦⣄⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣀⣠⢶⡇⠙⢦⡈⢿⣿⣯⡿⠵⠚⠻⣿⣿⣿⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡤⢦⣶⣿⡏⣿⣿⣷⡄⠀⢩⣏⣿⣿⠇⣴⠄⠀⣤⠿⣿⡂⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣠⡞⠙⣎⢿⣿⣿⡇⣿⣿⣿⣿⣄⣾⣹⣿⣟⡀⠀⠠⡴⢀⣴⣿⠃⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⡇⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡦⣈⣡⣾⣿⣿⣧⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢠⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⠟⡏⠀⣣⠙⣿⣋⠹⡟⢿⡄⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⢸⣿⣿⣯⣿⣿⡿⠁⢀⣿⠋⠉⠓⠋⠈⢹⡇⢠⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇⣼⠿⢿⣿⣿⣿⠁⠀⣸⣽⣦⣤⣀⣠⣄⣼⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣴⣷⣿⣾⣿⣿⣥⣼⣿⣭⣤⣤⣬⣿⣷⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣁⣿⣿⣇⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠊⠉⠉⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⢢⢠⣒⠦⠀⠀⠀⠠⢖⣢⢠⠛⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢲⢸⠈⠀⠆⢀⠀⠀⠀⠆⠈⢸⢰⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⠀⠑⠒⠲⠄⣀⡀⠴⠒⠒⠃⣸⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢶⣤⡤⣤⣴⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⡞⢦⡀⠀⠈⠙⠛⠛⠁⠀⠀⣠⠞⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠰⠀⠀⠙⠲⠤⣄⣀⣀⡤⠴⠚⠁⠀⢀⡎⢷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡆⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⢰⠉⠁⠈⡆⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠈⠧⢤⢤⣇⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡤⠼⣄⠀⠀⠀⠀⠀⠄⢸⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣴⣹⠙⠒⠢⠤⠤⠤⠼⠤⠤⠤⠤⠒⠊⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠽⠉⢇⠀⠀⠀⠀⠐⢲⠒⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⣀⣸⣀⣀⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠉⢹⠉⠁⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⡀⠰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣟⣿⣿⠿⢿⣿⣿⣿⣿⢿⣿⣷⣼⣿⣦⣄⠐⣦⣤⠤⢀⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⣷⡾⠛⣿⡃⠀⠙⠋⣉⠀⢀⣠⣴⣿⣿⣿⣿⡟⠛⢿⣧⣾⠋⠁⢴⣶⣤⣀
⠀⠀⠀⠀⠀⡀⣼⡿⠋⠀⢠⡟⠀⢸⣿⡇⠀⣠⠞⠁⠰⣿⣿⢿⣿⡿⣿⠟⠁⠀⠘⣿⡇⠀⢀⡀⠹⣿⢹⣷⡀
⠀⠀⠀⠀⢰⣇⣿⠁⠀⠀⢸⡇⠀⠛⠉⠀⣼⠋⠀⠀⠀⠈⠁⠈⠁⠀⠁⠀⢠⣶⡷⠛⠀⠰⣦⡁⠘⠆⢸⣿
⠀⠀⠀⢀⣼⣷⡜⢷⣄⠀⠈⢡⣤⣤⡀⢰⠇⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⠟⠋⢠⡤⠴⢤⡀⠹⣶⡆⠿⠁
⠀⠀⠀⣿⣿⡏⢈⣦⠙⢦⠠⠘⢿⣿⡇⢸⠀⡇⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⠀⠀⠀⠀⣸⠷⠊⢹⣇
⠀⠀⠀⣿⣿⡷⢿⠇⠀⠀⠀⠀⠀⠛⠃⠈⠀⣧⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⠁⢠⡀⢸⡟
⠀⠀⠀⣿⣿⡇⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣆⠈⠋⣸⡇
⠀⠀⠀⢹⣿⡇⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⣿
⠀⠀⠀⠘⠟⣡⣴⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⢴⣿
⢠⡖⠠⡄⠀⢉⡟⠀⠀⠀⠐⠻⠶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⠏
⢹⣴⣦⠘⣤⣿⠃⠀⠀⠀⠀⣀⣠⡄⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⡶⠿⠀⠀⠀⣀⠀⠀⠀⢀⣤⡀
⢸⠁⢸⣧⠁⠀⠀⠀⠀⢀⠼⣿⢿⣶⣶⣄⠀⢲⡄⠀⠀⠀⣀⠀⠀⢀⣀⣀⣴⠞⣷⠟⣫⡿⢦⡀⠋⣠⣽
⢸⡄⣿⠋⠀⣆⠀⠀⠀⠀⠀⠹⠦⠴⠏⠉⠋⠘⣿⠀⠀⠐⠃⠠⠞⢻⣏⣿⣥⡞⣱⠚⢁⣠⡤⣥⡸⠛⡟
⠈⣧⠘⣦⠀⣿⠀⠀⠀⢀⠄⡄⠀⠐⠀⡴⠀⠀⠙⠀⠀⠀⠀⠀⠀⢈⣽⠟⠙⡇⣯⢰⠋⣡⣴⡈⠷⡄⠇
⠀⢸⣆⠈⢇⢻⡄⠀⠀⠈⠚⠁⠈⠀⠈⠀⠀⣠⠖⠒⠒⢀⡀⠀⠀⣼⠃⠀⠀⢁⣏⣼⣠⣿⣿⡏⠀⣸⡆
⠀⠀⢻⣦⡀⠈⣷⡀⠀⠀⠀⠀⠀⠀⡀⠀⠾⠽⠷⠄⠀⣤⣭⡆⢠⡟⠀⠀⠀⠈⠻⣿⡿⠛⠙⠃⣴⠟
⠀⠀⠀⠉⠻⣧⡘⣧⠀⠀⠀⠀⣞⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⡶⠋⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠈⠁⢸⣧⠀⠀⠀⠀⠉⢀⢀⣠⣤⡶⠶⣶⣦⡀⠀⠈⢷⡀⠀⢸⣏⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠈⠉⠁⣀⠀⠀⠀⢀⠉⠛⠂⠈⣿⣄⠀⠻⣆⠀⠀⠀⢻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⣿⢿⣦⣀⠀⠀⠀⠀⠰⠿⣿⣿⡟⠀⠀⠀⠀⠈⢿⣦⠀⠙⡆⠀⠀⠀⠈⢿⣽⡶⣄
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠹⣧⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⣼⠻⣧⣄⠀⠀⠀⠀⠀⢈⣿⣿⡘⡆
⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⠙⠀⠀⠀⠙⠳⠶⠤⢴⡶⠶⠴⠶⠛⠁⢀⠄⠻⣄⠀⠉⠛⠷⢶⣶⣶⣿⣿⡿⠃⡿
⠀⠾⠿⢻⠟⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠰
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣌⠳⣄⠀⠀⠀⠀⢀⡴⢃⡼⢿⣦⡀⠀⠀⠀⠙⣿⣿⣆⠀⠀⠀⠀⠀⢀
⠀⢤⣄⣀⣀⣀⣀⣤⣴⡾⢿⢿⣬⣙⣿⣷⡌⠂⠀⠀⠀⠊⣠⣞⣡⣾⠿⣿⣦⡀⠀⠀⠘⢿⣿⣷⣄⣀⣀⣴⣊
⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⣈⠉⠙⠛⠛⠿⠿⠿⠿⠿⠟⠛⢉⣠⡤⠉⠛⠛⢶⡤⠤⠈⢻⣿⣿⣿⡛⠛⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣷⣶⣶⣶⣾⠿⠿⠛⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠻⣿⣿⡿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠃⠉⠉⠉⠙⠀⠒⠀⠄⡀⠀⠀⠀⢀⡀⡀⠀⢰⠁⢸⠀⠀⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠘⡀⠘⠄⠀⠀⠘⠀⠎⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠱⡀⠈⠂⠀⠠⠈⠀⢐⣁⠔⠈⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⣀⣀⠀⠀⠀⣠⣀⣀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠸⠄⠐⠀⠀⠀⠀⠈⢀⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⠞⠋⠉⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⡆⠑⠤⠤⠤⢀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⡠⠄⠼⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠉⢀⠀⠀⠀⠀⠉⠈⡦⠄⠀⡈⠑⠤⠤⠤⡤⠔⠁
⠀⠀⠀⠀⠀⢀⣰⠀⠀⠀⠀⠀⢀⠈⣠⡄⠀⠀⠀⠀⠀⣶⡄⠂⠰⠀⠀⠀⠔⠀⠀⡘⠀⠀⠀⠀⠰
⠀⠀⠀⠀⡰⠁⠀⠀⠄⠀⠀⠀⠥⠀⠛⠃⠀⠀⠀⠀⠐⠉⠁⢡⠀⠀⢠⠴⠀⠀⠰⠀⠀⠀⠀⢠⠃
⠀⠀⢀⠎⠀⠀⠀⠀⠀⠤⡄⢀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⠀⠆⠀⠀⠀⠀⠀⠀⠈
⠀⡠⠃⠀⠀⠀⠀⠀⠀⢀⢀⠀⠆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠊⢀⠔⠁⢀⠂⠒⠤⡀⠀⠀⣠⠃
⠀⠁⠀⠀⠀⠀⢀⠐⠒⢲⡀⠑⢄⡀⢀⡀⠀⠀⣀⠀⠠⠀⠖⠈⠀⠀⡠⠊⠆⠀⠀⠈⠐⠈⠁
⠀⡄⠀⠀⠀⠀⡆⠀⠀⠀⠁⢄⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢠
⠀⢣⠀⠀⠀⠀⠘⡄⠀⠸⠀⠀⠁⠂⠄⡤⡀⠀⠀⠀⢀⣦⠂⠀⠀⠀⠀⠀⢸
⠀⠀⢣⠀⠀⠀⠀⢐⠀⠀⢃⠀⠀⠀⠀⠀⠈⠐⢄⣤⠃⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡇⠀⠀⠀⢰⠀⣀⠀⡆⠀⠀⠀⠄⠐⠀⠘⣁⠕⠀⠀⠀⠘⢀⠀⠀⠈⠢⡀
⠀⠀⠐⣧⠠⠀⠀⠘⠋⡠⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠑⠄⣁⠀⠈⠢
⠀⢀⠀⠀⠀⠀⠀⠐⠉⠀⠪⡀⠀⡀⠀⠂⠀⠀⠀⠀⠠⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠈⠦⠀⠀⠀⠀⡰⠂⠲⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⢀⣀⣀⣀⣀⣀⠄⠊
⠀⠀⠣⠔⠁⠀⠀⠀⠀⠣⣠⠀⢠⢀⣀⠀⠀⠤⠄⡰⠂⠐⢄⠀⠀⠀⠀⠠⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠄⠀⠄⠀⠀⠀⠀⡔⠀⠀⠀⠀⠉⢄⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢃⠀⡂⠀⠂⠀⡔⠀⠀⠀⠀⠀⠀⠀⡆⠀⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⠀⢇⣀⠄⠀⣆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠰⡆⢨⠀⠀⠀⠀⠈⠐⡀⠀⠀⠀⠀⣅⠀⠀⠀⠈⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠈⠀⢀⡄⠃⠀⠀⡰⠄⠀⢀⠄⠁⠁⠒⠂⠀⣶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠄⠀⢀⡠⠚⠁⠀⠀⠂⠢⣀⢀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠈⠶⠋⠁⠀⠀⠀⠔⠁⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢈⡀⠀⠀⠀⣀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠉⠉⠉⠉⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠉⠆⣔⡩⠝⠃⠀⠀⠀⠀⠘⠫⢍⣦⢰⠉⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⢲⠀⡇⢁⣀⠴⠀⢀⠀⠀⠀⠀⠦⣀⡈⢸⠀⡖⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣹⠆⠳⣀⣀⣀⡄⠈⠁⠈⠁⢠⣀⣀⣠⠞⠸⣟⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠈⠙⠓⠤⠴⠚⠋⠀⠀⠀⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠒⠒⠒⠴⠒⠂⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⡾⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠸⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠀⠈⡈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡇⠀⠀⠀⠀⠈⠙⠓⠶⢶⡶⠶⠚⠋⠁⠀⠀⠀⠀⢠⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠓⢲⠤⣄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⢋⣿⣀⡇⡸⣰⢁⣇⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠘⠄⢀⠀⠀⠀⠀⠀⠀⠘⠼⠓⠬⢭⡿⠻⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠰⣀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⠤⠴⢧⡀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡈⡟⠲⠤⣄⣀⣀⣀⡀⠉⢸⡇⠀⠀⢀⣀⣀⣠⠤⠖⢻⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢻⣧⣿⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⡏⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁⠀⣧⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠶⠶⠶⠦⠤⠤⣼⣧⠤⠤⠴⠶⠶⢶⣇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠙⠉⠙⠃⠀⠀⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⢙⣿⣯⣽⡽⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠒⠒⠖⠉⠉⠑⠒⠒⠢⣀⡀
⠀⠀⠀⠀⠀⡰⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠀⠀⠘⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠒⠂⠀⠀⡆⠀⠀⠐⠒⠢⠤⣔⠂⠀⠒⢦⡀
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⢀⡴⠃⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⢱
⠀⠀⠀⢀⠇⠀⠀⠈⠢⠤⠐⠒⠁⠈⠉⠉⠢⠤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⡸
⠀⠀⠀⠈⠀⠀⠀⣠⠞⠀⢀⣀⠀⠀⠀⢀⣀⠀⠑⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠔⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⡥⠔⠁
⠀⠀⠀⠸⠤⣀⣀⡁⠀⠀⠋⢁⡀⠀⠀⢀⡈⠁⠀⠈⣂⡠⡴⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⡠⠤⡶⠇⠀⠀⠀⠨⣶⠂⠤⢤⠀⠀⠀⡅
⠀⠀⠀⡇⠀⢃⡸⠉⠐⠒⢾⠁⢘⠖⢶⠁⠈⡶⠒⠉⠹⣀⠃⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⡆⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠚⢣
⠀⠀⠀⠘⢆⡎⠁⠀⠀⢀⡠⠁⠘⠉⠙⠃⠋⣄⠀⠀⠀⢸⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢰⠎⠀⠀⠀⠀⠳⢃⠤⡘⠛⠀⠀⠀⠀⠸⢶⠀⡜
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠈⠑⠒⠀⠀⠒⠉⠀⠁⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⠀⠀⠀⠐⠊⠠⣀⠀⢀⡀⠐⠢⠀⠀⠀⠸⠊
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⢀⠔⡸⠁⠈⡖⠢⠤⠤⣀⡀⣀⡠⠤⠤⠒⢉⠉⠫⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢬⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢦⣀
⠀⠀⢠⠃⡘⠀⠀⠀⢱⡀⠀⡠⠚⢥⡄⠑⠦⡀⠀⡜⠀⠀⢱⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⡰⠁⠀⢀⠋⢫⡉⠒⠒⠒⢊⠟⢹⠀⠀⠀⠡⠉⠢⡀
⠀⠀⠎⢠⠃⠀⠀⠀⠀⠑⠊⠀⠀⢀⡀⠀⠀⠈⠊⠀⠀⠀⠀⡇⠀⠈⠣⡀⠀⠀⠀⠀⢠⠏⠀⠀⢀⠇⠀⠀⠘⡄⠀⠙⢢⣤⡔⠁⠀⡸⠀⠀⠀⠀⢃⠀⠈⠆
⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟⢄⠀⠀⠱⡀⠀⠀⢦⠠⡄⠀⠀⢰⠀⠀⠀⠀⠈⠉⠉⠐⠷⠀⠉⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⡦
⠀⡎⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠱⣄⣀⠱⣀⠔⠉⢲⣈⡒⣲⠚⠀⠀⠀⠀⠀⠀⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⢃
⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢡⡤⡴⠉⢠⠔⠁⠀⠉⠁⠘⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢭⠥⢴⠋
⠐⡇⠀⡘⢹⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠰⣿⣷⣦⡐⠀⠀⠀⠀⠀⠀⡍⠉⠐⠒⠒⠒⠂⠀⠀⠀⠒⠒⠒⠒⠂⠉⠉⡅⠀⡆
⢘⣉⣉⡃⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠈⠿⠟⠀⠀⠀⠀⠀⠀⠀⡸⠒⠂⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠐⠒⠚⣄⡤⠇⢠⡀
⡇⠀⠀⡑⠀⠀⢹⠒⠒⠂⠠⠤⠤⠤⠤⠤⠄⠐⠒⠒⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⡅
⠀⠒⠂⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣾⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠓⠒⠒⡋⡖⠒⠒⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⣄⠀⣀⣄⠀⣀⠀⣀⠀⣀⠀⣀⣀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣗⣒⣐⣴⡾⠃⢓⣶⣐⣀⣒⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣀⣪⣥⡾⠋⢳⣯⣀⣩⣂⡇
⣿⣿⡟⠻⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠀⠀⡤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⢸⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣷⠀⠀⠘⠃⠀⠀⠀⠀⡠⠁⠀⠀⢰⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠔⠃⠀⠀⠀⠱⣄⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣥⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠖⠇⠐⠀⠀⠀⠀⠀⠈⢢⡔⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠐⠲⠂⠀⠈⠁⠈⠁⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⣀⠀⢡⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⢱⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠀⠀⠀⢈⠋⢡⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⠀⠀⠀⡀⠄⠒⠁⠀⠀⢡⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⡣⣝⠰⡄⠂⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⣠⠀⠀⠀⠀⠠⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⢩⠀⠀⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡆⠀⢢⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⢀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠐⠤⢀⡀⠔⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⣀⠴⡀⠀⠀⠀⠀⠀⠀⢈⣀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠉⢀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣿⣿⣿⣿⣛⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣿⣿⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢞⣛⣿⣭⣭⣍⣭⣟⣛⠷⣦⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣭⣝⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣃⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢣⣿⣿⣇⣻⣿⣇⢿⣿⣿⣿⠹⣿⣿⡿⣿⣿⣿⣿⣿⣿⡜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢸⠟⠻⣿⣿⠟⡼⠎⠿⣟⣵⣷⣟⣛⠹⣿⡿⠿⣛⢹⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⠃⠐⠙⣻⣼⠿⢶⣾⣿⣿⣿⣿⣧⣵⣶⣾⣿⣿⣘⢟⡣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡒⠆⠀⠈⣷⣶⣶⣾⣿⣿⣿⣿⣿⣤⣤⣿⣿⡿⣛⣚⣣⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⢤⢜⡻⢿⣿⣿⣭⣭⣍⣿⣿⣿⢛⣫⡜⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠿⢶⠎⠹⠿⠉⢩⠶⢾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢯⣾⢫⣾⠸⣿⢹⡎⢿⡎⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢏⣾⣿⣧⢿⡖⡏⢹⢏⣾⣿⡜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣾⣿⢸⣿⡊⣇⠁⢣⣿⣿⢹⣷⡹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡳⣢⣵⢸⣿⣿⡀⢠⣿⣿⣿⢨⣋⢥⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠻⣷⡰⣶⣶⣨⣭⣭⣩⣸⠟⠶⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡸⠿⢹⣿⣧⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣉⠸⠿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣝⣣⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⣤⠐⠊⠁⡆⠀⠀⠀⢌⠁⠉⠈⠓⢆⠀⡄⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⡰⠁⠀⣀⠤⠊⠀⠀⠀⠀⠀⠙⠶⣖⠠⠬⡆⡀⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠌⠹⣄⠃⣠⠟⣋⣄⠀⠀⠀⠀⠀⠀⠤⡤⡤⣁⠂⢠⠇⢱⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀⠈⢧⣴⠦⠛⠓⠒⢢⣠⠤⣤⡏⠘⠛⠓⢛⠒⡟⠀⣈⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠈⠲⠤⢤⠠⠔⠀⠀⠀⠀⠠⡴⠶⠅⠐⠁⢀⡆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠀⠀⢀⠄⠒⠂⠀⠀⢀⠀⡀⠔⠐⠊⠒⠢⡀⠀⠌⠡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢀⠀⠨⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠐⠠⠀⠀⠠⠔⠛⠓⠀⠐⠊⠁⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠄⡀⠀⠀⠀⠀⠀⠐⠂⠘⠛⠀⠀⠀⠀⣀⠔⠊⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡉⢄⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⢀⠖⠈⠀⠠⠂⢸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠀⠹⡄⠢⡀⠀⠀⠈⠒⠂⠈⠁⠒⠒⠁⠀⡀⠀⠀⢀⠃⡘⠈⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢦⠀⠹⡢⠀⢑⠦⠄⡀⠀⠀⠀⠀⠀⡠⢊⠀⠀⠠⠃⡰⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠈⢦⠀⠰⠐⠇⠀⠀⠘⢄⠀⠀⡠⠊⠀⠀⠈⡔⢁⡐⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⢆⠀⠀⠀⠈⠱⠊⠀⠀⠀⡠⠊⢀⠌⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠑⠀⠀⠠⠁⠀⠀⡠⠎⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠂⣄⠈⡠⠎⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢇⡀⠉⢀⠔⠋
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠇⠹⣿⠀⠀⠀⠀⡴⠓⠀⠀⠤⡄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠿⠀⠀⠀⠉⡝⠀⠀⠀⠀⠠⠀⠀⢀⡐⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡄⠀⣀⣘⠷⠂⠀⠷⣟⣠⠀⢰⠆⢹⡂⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣾⠋⠙⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⡀⠀⣠⡔⠀⠀⠀⠀⣀⡀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠙⢦⡀⠀⠀⠁⣄⣀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣦⡀⠉⠓⠤⣄⠀⢹⡟⠳⢦⣄⣀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠉⠉⢻⡀⠀⠀⢸⣿⣿⣶⣤⣠⠇⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠙⢀⠀⠀⠀⡀⠈⢳⡀⠀⠈⣯⠙⠛⣿⠏⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠂⠀⡀⠀⠌⠀⠀⠀⡛⢦⣄⠨⠙⠋⠁⢀⡜⢣⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠃⠀⠻⡆⠀⠀⠀⠀⠁⠀⠈⢷⡀⠀⠀⠀⠙⡄⠹⡄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢈⠀⢡⠀⠀⠀⠀⠇⠀⠀⡸⠛⠲⣤⣤⠶⠁⠀⠻⠲⡀
⠀⠀⠀⠀⠀⡞⠀⠠⠀⠀⠀⠀⠀⠀⠑⢤⡆⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢇⠙⣆
⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⣜⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⢧
⠀⠀⠀⠀⡌⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡰⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢣
⠀⠀⠀⠀⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⢧
⠀⠀⠀⠀⠇⠀⣺⠀⠀⠀⠀⠀⠂⠀⠀⠈⡏⠉⠉⠁⠘⠓⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⢷⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⡗⢤⣄⣠⣤⢴⠦⠀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠱⠀⠈⣧
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⠛⠀⠁⠀⢸⡇⠀⠠⠀⡄⠃⠀⡆⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠇⠀⠸⡆
⠀⠀⠀⢸⡀⢀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣠⣰⠎⠰⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⠀⠀⢹
⠀⠀⠀⠀⠙⠋⠁⠀⠀⠚⠛⠛⠛⠓⠀⠀⠀⠀⡿⠀⠀⠀⠀⠉⠀⠀⣀⣀⣀⣀⣤⠴⠶⠃⠀⢹⠀⢸⠀⠀⣸⣄⠀⢀⣀
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⣼⢰⠉⣉⠉⠠⠀⠀⠀⠈⣱⡀⣠⣀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢦⢉⠀⠐⡇⢸⡀⡶⠏⣀⡁⢈
⠀⠀⠀⠀⢸⡀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⠈⠙⣁⠘⡖⠊⠉⠁⠀⠈⠁⠉
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⢺⣅⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠤⠶⠶⠶⢶⠦⠶⠀⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠦⣄⣀⣀⣀⡄⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⡄⠀⢷⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠙⠒⠶⠚⠛⠉⠀⠀⠘⡇⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⣀⠀⠀⢸⠀⡇
⠀⠀⢀⣀⠤⠄⠒⠛⠁⠀⠀⠀⠀⠈⠁⠉⠀⠁⠀⠺⣇⢀⣀⣀⣀⣀⣀⣤⣤⠤⠤⠤⠤⠴⠶⠶⠶⠾⣗⢀⡋⠁⢙⡆
⠲⢯⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠴⠖⠚⠧⠔⠚⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠶⠶⠤⠄⣀⣀⣀⣀⣀⣀⣠⣤⡼⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡟⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⠃⠀⠀⢠⣶⣿⣶⡄⠀⠀⠀⢰⣾⣷⣦⡀⠀⠘⣦⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣤⣠⣯⣀⣀⣤⡟⣉⣉⢲⣄⣤⣄⣠⠞⣉⡓⢦⣅⣀⣀⣹⣄⣀⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠚⣿⡽⠋⠉⢿⣿⣜⠉⠈⣻⡇⠀⠀⢿⡚⠉⠛⣼⣿⠏⠉⠉⠻⣿⡿⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠓⣿⡅⠀⠀⠀⠀⠈⠛⠛⣏⢲⡦⠀⠀⠙⠓⠛⠁⠀⠀⠀⠀⠀⣿⡟⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⣧⠀⠀⠀⠀⠀⠼⠁⡈⠉⠁⠀⢀⡙⠦⠀⠀⠀⠀⠀⠀⢀⡟⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⢶⣶⠖⠀⠀⠀⠀⠀⢀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠙⡟⠲⣦⣀⣀⠀⠀⣀⣀⡴⠖⢾⠋⠑⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⠟⠁⠀⠀⠀⣧⠀⠈⢻⣍⠉⠉⣩⡟⠁⠀⣸⠀⠀⠀⠀⠙⢷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢸⠀⠀⠀⠀⠀⠸⡆⠀⠀⣙⣷⣾⣉⠀⠀⢰⠇⠀⠀⠀⠀⠀⢸⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠙⠖⠉⠁⠀⠀⠈⠙⠲⠛⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⣷
⠀⠀⠀⢀⡟⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⡇
⠀⠀⠀⣸⠃⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢿
⠀⠀⠀⡟⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⡀⠀⠀⠀⠘⡆
⠀⠀⠀⢹⡖⠶⠶⡾⠁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠷⣤⠤⠖⢾⡁
⠀⠀⠀⣼⠁⠀⣸⠇⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢻⡀⠀⠘⡇
⠀⠀⠀⡟⠀⢀⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣇⠀⠀⣇
⠀⠀⢰⠃⠀⣸⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⢹⡀
⠀⠀⢸⠀⢠⡇⠀⠀⠀⠻⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡼⠀⠀⠀⠀⣷⠀⢸⡇
⠀⢰⡏⠀⠸⡆⠀⠀⠀⠀⢸⡉⠉⠙⠛⠓⠲⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠒⠛⠛⠉⠉⣸⠀⠀⠀⠀⢀⣼⠂⠈⢷
⢠⡟⠀⡀⢀⡹⡄⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⡋⠀⠀⠀⠈
⣾⢻⡇⡇⣸⢿⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠶⠶⠖⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢰⡏⣿⣄⠀⣀⣠
⠙⣿⣇⣷⡜⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢈⡃⡇⠉⠈⠉⠁
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢸⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠒⠻⠦⢤⣤⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣠⣤⠤⠾⠓⠒⠒⠒⠒⢾⣡⡇
⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⠀⢀⣀⣀⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢄⣀⣀⡀⠀⠀⠀⠀⠀⣸⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⢻⢾⡁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤
⣿⣿⣿⡛⠙⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣴⣞⠿⢻⠷⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡤⢶⠾⢾⣆⠀⠀⠀⢀⣴⣿⣿⣿⠃⠈⠃⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⠴⢾⠿⠾⠋⠀⢀⣤⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡳⠠⠤⢦⣤⣶⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣷⠀⠀⢺⣿⣌⣻⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⡄⢸⣿⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠘⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡋⠀⢿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣶⣶⣶⣦⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⢠⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣴⣾⠿⣯⠿⢿⣿⣿⣿⡟⢛⢿⡿⠻⢤⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢀⣿⣷⠎⠈⠁⠀⣼⣿⣿⠋⢁⣟⣉⣴⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⢿⣿⣿⣿⣿⣿⠀⠠⠙⠃⠀⠀⠀⢠⣾⣿⣿⣿⡁⢚⣵⣾⠞⠁⠙⢻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡇⠀⠀⠁⠀⠀⠀⢀⣿⣿⣿⣿⣿⣾⣿⡗⠁⠀⠀⠀⠀⠉⠪⣽⣿⣤⡤⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣄⠀⢀⡠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣛⣈⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⣿⣿⣿⣟⣥⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠉⠀⠀⠓⢠⣈⡤⢶⣿⣿⣿⢋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠚⢉⣀⠈⣻⡯⠰⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡑⢺⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣯⣴⣾⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠛⠿⠿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⢁⠀⠀⠀⠀⠁⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢀⢈⠉⣀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⣀⡀⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⢀⣤⠀⡒⢴⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡠⠈⠀⠀⠑⠄⠁⠐⠀⠤⡀⣀⣀⠀⠔⠁⠀⠀⠸⡀⠀⠉⠢⡄⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠚⡄⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠁⠢⠤⠘⠂⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⡠⠁⠀⠀⠀⡌⠀⠀⣰⠛⢄⠙⠠⢀⠀⠀⢀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠌⠀⢀⣴⣧⠀⠀⠂⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠘⠂⢐⠕⠁⠀⠑⠄⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠂⠀⠀⠠⠤⠤⠠⠂⠁⠀⠀⢺⡉⢂⠑⡌⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠣⠀⠁⠀⠠⠨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠖⠋⡒⠃⠀⠀⠀⠀⠀⠀⠐⠁⠐⠂⠤⠴⠶⢨⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠃⠀⠀⠐⠖⠒⠒⠈⠒⠒⠀⠎⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⡀⠀⠠⠄⠂⠹⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠜⠂⢲⠆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠒⠃⠀⠀⣀⢀⠀⠀⠀⠐⠒⠃⢠⠆⠀⠀⠠⠀⠿
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠘⢆⠐⠤⠤⠙⢦⠄⡀⠀⠀⠉⠀⠀⠀⠀⠳⡀⠆
⠀⠀⠀⡆⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡠⠂⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⡠⠲
⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢬⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠀⠀⠐⠶⠖⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡠⠤⠤⠤⠤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⣠⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⣠⠞⠡⠠⠤⡜⠢⣀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⡀⠈⠉⠉⠉⠁⠀⡌⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⡇⠀⠀⣀⠒⠒⠀⡇⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠇⠀⠀⠸⠀⠀⠀⣷⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠐⠒⠒⠒⠒⠒⣉⠀⠐⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡖⣴⠀⢸⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣈⠁⡇⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⣰⠀⢸⠁⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⣛⡂⠀⢡⠒⡞⠀⠀⡞⠀⠀⠀⢸
⠄⠀⠀⠀⠀⠤⠄⠀⠄⠄⣤⡌⣤⡀⠀⠘⠓⠃⠀⡄⢠⠇⠀⢀⡇⠀⠀⠀⠸⡄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠁⠄⠀⠀⠀⢠⢇⡸⠀⠀⡸⠀⠀⠀⠀⠀⣇
⠤⠠⠤⣴⣦⣤⣤⣤⣤⠤⠼⢴⣦⣤⣤⣀⣀⡀⠈⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁⠉⠉⠁⠁⠐⠦⠄⠀⠀⠀⠀⢀⠠⠄⠀⠄⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠄⣒⣏⠇⠀⠀⡔⠁⠀⠀⠐⢰⠆⠁⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⢰⠀⠀⡠⠒⠤⠀⠒⢐⠲⠿⠁⠈⡀⠀⠐⠀⠀⠤⠤⠀⠀⠀⠀⠤⢄⠀⢰
⠀⠀⠀⠀⠀⠀⢨⠤⢀⠁⠀⠋⠀⠀⠈⠃⠀⡄⠤⠇⠀⢰⢀⣨⠀⠀⢒⠀⠀⣐⠀⠈⡀⡼⡀
⣰⡿⠝⡀⠀⠀⠐⣀⠃⠀⠀⠈⠠⠠⠁⡀⠀⠐⣀⠆⠀⠠⠰⠆⠀⠙⠤⠼⠩⠤⠋⠁⠱⢁⡆⠀⠀⠀⡰⢽⣦
⠙⠓⠎⡈⢐⠊⠱⠌⡀⠀⠀⠙⠶⠖⠊⠀⠀⢠⠁⠀⠀⠀⠘⡀⠀⠈⠐⠶⠶⠂⠁⠀⢠⠁⠸⠒⠰⡊⢀⠤⠛
⠀⠀⠀⠈⣃⡐⠀⠀⠜⠲⠤⡀⣀⣀⡠⡤⡔⠑⠄⠀⢠⡢⠆⠔⠠⣀⣀⠀⠀⣀⣀⠔⢏⠀⠀⠀⡀⠝
⠀⠀⠀⠀⠀⠁⠂⡌⠀⠂⠄⣀⠸⠽⣀⠅⠁⠄⠈⠖⠋⠀⠌⠀⠀⠇⠑⢄⣀⠈⢀⠀⠀⣂⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⡁⠀⠀⠀⠀⠀⣦⡢⠤⠀⠀⠀⠀⠀⠘⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠀⠀⠀⡅⠀⠀⠀⠀⠀⠅⠀⠀⣃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠆⠀⠀⠀⡀⠰⠀⠀⠀⢣⢀⡉⠀⠀⠀⠀⠁⠀⢉⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡆⠦⠤⠤⠤⠤⠀⠆⠖⠁⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡇⠀⢀⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣂⣁⣸⢍⣀⣁⣢⠀⠀⠀⠀⠀⠀⠣⣤⣀⣄⣄⣠⣠⢄⣠⠬
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠑⠚⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣙⣿⣿⣄⣀⣘⣻⣌⣀⡀⠠⡀⣀⠀⠀⠀⠱⡀⠀⣤⣴⣶⡄⠀⠀⣀⠀⠀⠀⢀⣀⣀⣀⢖⠀⠀⢀⡀⠀⠀⠀⢀⣀
⠀⠉⠩⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢀⡟⣯⣿⠀⢀⣰⠿⠛⣟⢛⣽⣿⣛⣫⣅⣐⣀⣩⣤⣽⣭⣿⣯⣽
⠈⠀⢀⣀⠀⠀⠈⠉⢙⣛⣛⠛⣛⣛⣙⣛⣉⣻⣿⣿⣿⣿⢷⣏⣯⣤⣶⣛⣢⣭⡿⢻⠟⠻⠿⠛⠛⠛⠻⠛⠛⠟⠛
⠀⠀⢀⣀⡀⠀⠀⠀⠉⠀⢀⣈⣉⣉⣉⣽⢿⣿⣟⣿⣶⡼⣿⢿⣟⣿⣿⣿⣷⣿⣗⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀
⠀⠀⠉⠡⠤⠤⠴⠶⠶⢦⢘⣲⠯⣷⣶⣾⣶⣿⣿⣿⣿⣧⣿⣿⣿⣿⡹⠿⢿⣿⣿⣿⣿⣿⡿⠓⠒⠒⠒
⠀⠀⠒⠶⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡋⠙⣿⣟⣷⣆⣒⣟⣿⣿⣿⣿⣟⣛⣒⡀
⠀⠀⠀⠀⠠⠶⠶⠿⠿⠿⠛⢿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣭⣾⣿⣿⣿⣾⣿⡿⠿⠶⠿⠍⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠖⠲⢶⣶⣶⣤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣔⣚⣿⡟⠛⠻⠿⠟⠛⠉⠉⠉⠉⠉⠙⢿⣶⣀⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠶⠦⠤⣤⣀⣀⣀⣉⣻⡯⠭⠭⠤⠀⠀⠀⠀⠤⠤⣄⣀⣀⣄⢠⠽⣿⣟⣉⣛⣓⣒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠠⠀⡀⠀⣠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠠⡀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⡔⠃⠤⡘⠛⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢠⠊⠀⠀⠀⠈⠢⠀⠀⠈⠂⠒⠂⠀⠁⠀⠀⠀⠡⢀⡀⢨⡖⠃⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠆⠀⠀⠀⠀⡠⠓⠀⠀⠠⠁⠀⢠⢤⠠⠄⠀⠀⠀⠩⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠠⠀⠀⣠⡏⠀⠡⠀⠈⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠀⠀⠀⠐⠡⠀⡮⠈⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⣄⡀⠄⠈⠀⠀⣲⠐⠌⢂⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠈⠂⠈⠁⠀⠌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠒⠋⣚⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠒⠶⠊⡅⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠘⠛⠉⠀⠁⠀⠀⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡂⠀⠘⠀⠀⠀⠀⠀⢀⣀⠀⠀⠌⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⡀⠀⠂⠠⣄⠀⠀⠀⠀⠠⠋⠀⠀⠈⡀⠃
⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⠁⢄⠉⠂⡈⠟⠂⠠⠐⠁⠀⠀⠀⠀⠀⠄⠀⠀⢀⡀
⠀⠀⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠐⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠪⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡾⠆⢻⣄⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣦⡈⠀⠀⠀⠀⠀⠀⠀⢻⠀⢀⣰⡏⣦⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢈⡄⢀⠇⠀⠀⢀⠀⠀⠀⢰⢄⡗⢄⣸⠥⣇⠀⠀⠀⠀⠀⠳⡀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⠋⢰⠒⣲⢮⡿⠃⠀⠀⣠⡄⣹⡮⡥⠌⢠⠙⡄⢠⣤⡀⠀⣱⣤⡞⢹⣻⡄
⠀⠀⠀⠀⢠⣤⣀⡀⠀⣯⣤⣼⣎⡉⢆⣀⡼⣀⢀⡀⣠⣧⣤⣩⣷⢬⡓⡤⠥⣼⢧⣅⠸⣏⠛⢿⣿⣾⡉⢿
⠀⠀⠀⣰⣯⠥⡭⢛⡿⠿⣦⣤⣴⣍⣛⡯⢧⣍⣇⣴⣿⣿⣿⣿⡿⣶⢭⢷⠴⣿⡦⠩⠀⢮⠑⣯⡝⣷⡉⣼
⠀⠀⠀⡟⢀⡞⠱⣷⣀⣣⣌⡴⠀⠂⡵⠰⣁⡾⢋⡜⡥⠫⠖⢖⣴⢵⣄⣊⣾⣻⣷⣀⡀⣼⡀⠈⢷⢬⣷⡁
⢀⣀⣀⣻⣎⠼⠗⠝⠻⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣟⣛⣛⣛⣛⣛⠋⣿⡯⠀⣼⠁⠀⠘⡄⠀⠙⡄
⠈⠛⢛⣛⣛⡓⠒⠛⠛⠛⠛⠛⠀⠑⠉⠙⠿⣿⣭⣭⣍⣉⣉⠉⠉⠉⠉⠉⠁⠀⢹⣿⡭⣥⠀⠀⠀⠸⡄⠀⠈⢦⠀⠀⢀⡄
⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠟⠿⢿⣀⣀⣠⡤⠤⠤⠶⠖⠚⠛⠁⠀⢠⡀⠀⠀⠀⢨⠏⠀⣁⠴⠁⠀⠀⢳⠀⠀⠀⠳⡀⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠸⣿⣿⣿⣯⠁⢠⠤⡖⠀⠓⢀⠀⠀⠉⢻⣿⣧⡂⡄⠀⠀⢸⣦⡀⠀⠀⠑⡜⠀⠀⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡠⠀⠈⢆⠀⠀⣼⣿⣟⣈⢀⠀⠀⢸⡇⠱⡄⠀⡠⠉⢞⠀⢢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣷⡀⠁⠀⠀⠈⠆⠈⠣⣿⣿⡧⣠⢀⠀⢸⣷⡁⠈⢦⠁⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠈⠃⠑⠀⠀⠐⠄⠀⠀⢀⠘⢿⣯⠀⠘⠄⢸⣿⠃⠀⠈⠳⡀⠀⠁⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠢⡀⠀⠀⠀⡄⠀⢳⠈⣏⢠⡀⠀⠿⣿⠀⠀⠀⡠⡝⢀⣄⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠒⠂⠤⠀⠀⢮⡁⠀⠀⠀⠀⢄⡘⡄⠀⠀⠀⠀⠀⠀⣿⠂⡀⠀⢀⠂⢀⠔⠁⠀⠿⠡⣆⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡄⠀⠀⠀⠀⠠⢿⡷⢦⣄⠂⠀⠑⠜⠀⠀⢀⣀⣠⡴⠏⠀⠘⠑⠒⠄⠀⠀⠠⢠⠀⢀⠌⠈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣶⣶⣿⣶⣦⣄⣘⠛⠻⠆⠇⠀⠉⠓⠀⠀⠶⠾⢯⣍⡤⠆⠀⠀⠐⠃⠀⡠⠃⠀⠀⢀⡩⢀⣽⣴⠟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡲⠟⠿⠿⠿⢿⡋⡄⠰⠀⠀⠀⠀⠀⢉⡑⣦⠀⠠⠄⠉⠂⠀⠀⣂⠂⠉⢀⣀⣐⣈⣁⣀⢘⡫⠐⠬⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⣚⣛⣙⣩⣥⣶⣶⣶⣤⣀⣀⣀⣘⣒⣈⣥⣤⣄⡒⠒⠈⠡⣶⣾⣿⠿⠿⠟⠛⠋⠉⠀⣀⣀⣤⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣦⡀⠘⠻⠿⠿⣿⣿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⣿⠟⠿⣿⣦⡀⠉⠛⢿⣦⠘⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢣⡿⠘⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠘⠃⠀⠀⠀⢻⡄⢠⡀⠀⠀⠀⠀⠀⣴⡀⢀⠀⢸⣆⣻⣦⣻⡄
⠀⠀⠀⠀⠀⠀⠀⡟⠛⠀⠀⠀⠀⠀⣧⡀⢻⡌⢷⣄⠀⠀⠀⢀⠘⣷⣜⣷⣄⣹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢰⡦⠀⠸⣷⣀⠻⣦⣿⣧⣄⠀⢸⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠰⣆⣴⣾⣿⠀⠀⠈⠻⣿⠞⠻⣿⣿⣿⣮⣿⣿⣮⣙⢿⣿⡟⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠛⠋⠀⠈⠁⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠋⠀⠀⠀⠀⠀⠀⣠⠔⣶⣤⡀⠀⠀⠀⢠⠄⢀⡤⣶⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢾⡰⣄⠀⠀⠀⠀⠀⠀⠀⣀⣉⠉⠀⠀⠀⠀⢾⠀⠀⠀⠘⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣍⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠃⠀⠈⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠠⠶⠦⠤⠤⢤⣶⠆⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠊⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠉⠙⠓⠶⠤⠤⡤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢬⣁⡒⠲⠦⠤⠤⣤⠠⠤⠖⠚⢋⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠲⠶⠶⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⢀⣀⣀⣀⠀⢀⣀⣤⠤⠶⠚⠙⢿⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢛⠏⠁⢠⡽⠛⠉⠁⠀⠀⠠⣤⠀⠀⢳⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣜⣀⡔⢩⣽⡝⢦⡠⣤⠾⣽⣭⠑⣄⣈⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⠏⠈⠧⣀⣈⣠⣾⠂⢸⣆⣈⣀⣠⠋⠈⢩⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⢀⣼⣋⣭⣻⣯⣌⣹⣄⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⢦⣄⡀⠀⠀⢦⣀⢀⡴⠀⠀⢀⣠⠾⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢻⡀⢸⣏⣉⣩⠶⣶⠶⠷⣶⠶⣛⡉⣹⠀⢸⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠸⡇⠀⠉⠙⢯⣄⠈⢳⡼⠃⢀⡽⠋⠋⠀⡿⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⣧⠀⠀⠀⠀⠙⢦⢰⠀⣴⣋⣀⣀⣀⣠⡇⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠈⢿⡾⢹⣅⣀⣀⣀⣹⢻⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⣄⡿⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠉⠈⠉⠉⠀⠸⣇⣠⠴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⢻⡖⠒⠒⠲⠶⠶⠶⠿⠶⠶⠶⠶⠒⠒⠚⣿⠋⠙⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣼⡷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣄⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠙⠲⢶⡶⠒⠋⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣿⢻⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢠⡇⢸⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠇⠈⣇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠳⠤⣤⣤⠴⢿⠀⠀⣿⠦⢤⣤⠤⠞⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣠⣤⣶⡿⠃⠀⠿⣷⣦⣤⣀⣀⣀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤⣬⣉⡉⠉⠉⠉⠉⠉⠉⠉⠙⠲⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠋⠉⠈⠉⠉⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢠⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣷⠀⣿⠉⠛⠛⢿⣿⠿⠟⠒⠀⠀⠀⠀⠀⠈⢻⡞⠛⠂⢤⣤⣀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢹⣄⠁⠀⠀⠀⠀⠀⠙⠛⠶⣤⣄⠀⣀⣀⠀⠀⠹⣆⠀⠀⠀⠀⠙⠻⡆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠶⠿⠷⠀⠀⠀⠀⠀⢀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡿⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⡄⠀⠀⡀⠂⠀⠁⠉⠀⠀⠰⡶⠲⠆⠀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠉⢻⡇⠀⠀⠃⠀⠀⠺⣿⣿⠂⠀⠃⠀⠀⠀⠀⠀⠄⠀⠀⠻⣿⡷⠀⡄⠀⠀⠀⠀⠀⣿⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⢾⠿⠋⠀⠀⠀⢸⠀⠀⠀⠀⠃⠤⠘⢿⠟⠃⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⠛⠛⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⢀⡀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣿⢿⡈⠁⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⢹⡿⠉⠉⠉⠉⠐⠒⣤⠶⠶⠀⠈⠉⠉⠉⢀⣼⢿⡼⠃⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⢸⠇⣤⠀⠀⠀⢀⠴⠋⠲⠀⠀⠀⠀⢀⠔⣭⢻⣼⠃⠀⠀⠀⠀⠀⢀⣼⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⣿⡆⠀⠈⣰⠿⡀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⢀⣰⣉⡿⠁⠀⠀⠀⠀⠀⠀⣾⠁⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⣿⠀⠀⠀⠙⣠⠿⡀⠀⠀⢸⠀⠀⠀⠀⠀⣠⠞⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣿⠀⠀⠀⠀⠛⣄⠛⢦⣀⡈⠀⢀⣀⡰⠾⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠃⠀⠀⠀⠀⠀⠙⢷⣄⠉⠛⠛⠋⣡⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢀⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡞⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠘⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣰⣿⡝⠓⠚⠛⢢⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠌⠛⠿⠓⠀⠘⠃
⠀⠀⠀⠀⠀⢠⣿⡤⠙⠛⠛⢳⣄⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢠⡞⠉⣤⢀⡁⢰⡿⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠘⠷⠾⢷⣤⠁⢺⡇⢿⠋⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣄⣀⡀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠙⠟⢿⣇⣼⣀⣿⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣁⣀⣀⡀⠛⢷⣤⣾⠅
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⠋⠙⠀⠻⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⠀⢸⡟⠉⠛⠛⠻⣷⣆⣤⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠰⣶⣤⣀⡀⠙⠲⠶⣦⣄⣩⣿⣿⣷⣴⣿⣃⡀⠀⠀⠀⣠⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠈⠛⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⠙⠻⢶⠒⣀⣉⣉⠉⠉⠉⠉⢻⣿⣿⣦⣾⣾⠿⠁⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣴⣦⣦⠀⣶⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠐⠚⠿⠟⠛⠛⠛⠛⠛⠛⢛⣛⣉⣛⣻⣿⣿⣤⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣿⠀⣿⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠾⡛⢻⡏⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢨⠇⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠒⠒⠶⠖⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⣠⣄⣶⣤⣤⣤⣄⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣀⠤⠄⠈⢛⣀⢠⣤⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠉⠉⠉⠉⠉⠛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠴⠆⠉⠉⠉⠁⠶⠿⢷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⣀⣀⣀⣀⡀⢀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⣀⢀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠋⠻⢶⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢦⣀⣁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠁⣴⣶⢊⠝⣡⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⠇⢹⡆⢠⡇⠀⠀⠰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣶⡀⠗⠀⠘⡄⢺⠇⠀⠀⢢⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣍⣿⡏⠀⡀⠠⣶⠃⠀⠀⠀⣲⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⠁⠉⠀⠒⠀⠀⠟⠐⠠⡄⠾⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡠⠀⠠⠄⣠⠐⠀⢄⣠⣤⡄⠀⢀⠀⠛⣰⠏⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠙⠁⠀⠀⢳⣴⠟⢹⡟⠻⢿⣿⣧⣠⣼⣀⣼⠇⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠉⣿⡇⠀⢸⡏⢁⠚⢿⣏⠙⣿⣿⣿⠎⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⡄⠀⢸⡇⠹⣷⣤⣙⣧⡈⠽⠥⠆⠔⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⣿⡇⠠⠾⣷⣤⢬⣏⣿⣿⣿⡄⠀⠀⠀⠀⢘⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡀⠀⣄⢈⣿⣾⣃⣀⡤⠿⠛⢦⠰⣄⠀⠀⠁⠉⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠠⢤⣅⣄⡳⠆⠙⠉⠁⠀⠀⠀⠈⣣⠳⡍⠋⠉⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢳⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⢡⣄⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣠⣿⠄⠀⠀⠀⠀⢀⠀⠈⠻⣷⣌⣧⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢾⣦⡤⠬⢿⡿⣁⡀⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣱⠄⣹⡄⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠗⣿⡇⠀⠀⠀⠐⡀⠀⠙⠿⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠰⣿⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣷⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣄⠀⠀⠀⠀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠸⠋⠀⠐⠁⡀⢀⣀⠔⠿⣿⣿⣅⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠀⠓⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠆⠀⠉⠉⠉⠉⠉⠉⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⢉⡤⠶⠂⠉⠉⠉⠉⠁⠲⣄⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⣈⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⠋⠀⠀⣀⣀⠠⠄⠤⠤⢤⠀⡀⠉⠀⠀⠱⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠠⠚⠉⢉⠀⠀⠀⠀⣠⣄⣉⠲⠀⠀⠀⠛⣠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⡀⠂⠉⠀⢈⡗⠀⢰⠊⠀⠈⠙⣗⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣴⡇⠀⠰⡄⠚⠛⣁⣼⠗⠀⠻⣘⠶⠀⢤⡬⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠀⠘⠁⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠈⠀⠀⠀⠠⢤⠤⣄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡟⠀⡜⠃⠀⣠⠞⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⢠⠄⠀⠀⠀⣇⡠⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢰⠀⡘⠃⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⠎⠀⠀⠀⠀⢛⠤⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡠⠞⠛⠓⠐⠒⠶⠀⠀⠈⠉⠀⠳⠎⠁⠀⠀⠀⠀⠀⣿⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠀⠀⠀⠀⢀⡜⠳⢄⠀⠀⠀⠀⠀⠉⠲⢂⡄⠀⠀⣠⣁⣀⡀
⠀⠀⠀⠀⣠⡤⠶⠶⠞⠛⠲⠄⣠⣤⣤⡴⠊⠁⠐⠂⠀⠛⠒⠆⢀⣤⡠⠖⠊⠀⢠⠖⣩⠛⠉⠉⠉⠛⠲⣦
⠀⠀⢠⡾⠙⡆⠀⠀⠀⠀⠀⠀⠀⠈⢣⡘⢷⠂⢶⡀⠀⠀⢀⡀⠀⠀⠀⠀⣤⡞⢁⡼⠁⠀⠀⠀⠀⠀⠀⡏⢹⡄
⠀⢠⡟⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢣⣠⡟⠒⠲⠊⠁⠀⠈⢏⡡⠂⣠⠎⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⢳⡀
⢀⡿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠛⡀⠀⢸⠀⠀⠀⣠⠎⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠸⠁
⢸⠇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠘⡄⢸⠀⣠⠞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡇
⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠻⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠃
⡇⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡷⠴⠾⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠛⢁⣀⣀⡤⢤⡤⢄⡙⢦⡄⣀⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢸⠃
⡇⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡠⠎⠻⡀⣀⠞⠁⢠⡟⠀⣷⠀⢈⡛⢦⡙⠱⢌⡿⠀⠀⠀⠀⠀⡘⠀⠀⠀⢀⡎
⣇⠀⠀⠀⠀⠀⣭⡤⠖⢤⣶⡎⠀⠀⢠⡎⢹⠶⠊⢻⠂⢀⣿⠀⢸⡇⢸⡏⠲⢦⠛⠆⣀⠀⠀⠘⠃⠀⠀⠀⢸⠇
⣿⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⠘⢦⡀⠈⢱⡟⠀⢠⠞⠓⠙⠻⡴⠛⠳⠾⢧⠀⠀⢇⠀⠀⠉⢶⣷⣴⠆⠀⠀⡿
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠞⠉⠈⣅⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⠀⠀⠀⠀⠃⠘⠀⠀⣸⠇
⢻⣄⣀⣀⣀⣤⣤⠶⠖⠛⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠧⣄⠀⠀⠀⠀⠀⣰⠏
⠈⠉⠉⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆⠈⠳⣄⣀⣀⡴⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠶⠶⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠇⠀⠀⠈⠉⠉⠀⠟
⠀⠀⠀⠀⡇⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡷⠚⠛⠛⠓⠒⠴⣦⡀
⠀⠀⠀⠀⣇⠀⢀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠉⠀⠑⠛⠓⢶⡇⠀⢸⣷⠀⠀⠀⠀⠀⠀⢸⡗
⠀⠀⠀⠀⢹⡇⢨⠋⢀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⡾⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⣾⠁⠀⠀⡾⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠸⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠈⠀⡇
⠀⠀⠀⠀⢸⡆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡆⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠘⣷⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⡇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠸⢧⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢀⣠⡇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠦⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠓⠻⡇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡿⠲⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⡇⠐⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⡇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠇⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣾⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣿⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⣿⢠⡿⠀⢀⣧⣀⣼⡇⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⠖⠿⠀⠀⠀⢻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠿⣤⣭⣍⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣩⣭⠿⢧⣤⣤⣤⣼⠟⢩⡵⠲⠖⠒⠒⠒⠒⠒⠒⠲⠶⢬⡽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣄⡀⠀⣀⣠⣤⠴⠶⠖⠒⠛⠛⠛⠛⠛⠓⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⣯⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠻⠷⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣔⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠙⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠒⠚⣳⣆⠀⠀⠀⠀⠀⠀⠈⡿⢲⣶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣖⣿⡏⢻⡀⠀⠀⠀⠀⠀⢠⡏⢸⡷⠖⠚⠋⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠉⢷⡈⠳⣤⣀⣀⣠⡴⠟⢠⡾⠁⠀⠀⠀⠀⢀⣀⣨⡽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣥⣄⣀⣀⣀⣀⣀⣙⣶⣤⣭⣭⣥⣤⣾⣯⣤⡤⠶⠶⠒⠛⠉⠉⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⠴⠖⡚⠋⠉⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠙⠛⣿⠛⠛⠛⠓⠚⠒⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣤⡤⠶⠚⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⠶⣤⣤⣤⣯⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠴⠶⠖⠚⠋⠉⢹⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡏⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣄⣀⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⢾⡗⠛⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠉⠉⠙⠛⢻⡟⠛⠛⠲⠶⠶⠶⠶⠶⠶⠒⠛⠛⠛⠋⠉⠉⠁⠀⠀⠸⡇⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠖⠚⠉⠉⠉⠉⠉⠉⠐⠶⠦⣀⣀⡀⢀⣀⣀⡴⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠲⣄⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣠⣶⠉⠱⠀⠀⠀⠀⠀⠰⠛⢶⡄⠀⠀⠀⡁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡀⠀⠀⠀⠀⠀⠉⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢃⠀⢀⣼⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣴⠀⠀⠀⠀⣶⠄⠉⠉⠀⠀⠀⠀⢠⡆⠁⠀⠀⢴⡖⠀⠈⢠⣼⠁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠈⠀⠀⢸⠇⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠘⠇⠈⡄⠀⠀⠀⣤⠔⠒⠒⠒⠒⠒⠒⠛⠒⠒⠲⢤⡀⠀⢠⠜⠇⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⢄⠀⠏⠀⠀⠀⠀⠩⣉⠉⠛⠿⠛⠒⢦⡀⠹⣦⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠻⠇⠀⠀⠀⠀⠀⢈⣀⣤⠄⠀⠀⣈⡷⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⠀⠀⠤⠤⠄⠁⠤⠤⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣸⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠢⠔⠃⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣴⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⡘⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠈⡄
⠀⠀⢸⠁⠀⠀⠀⠀⡐⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡆⠀⠀⠀⠀⠃
⠀⠀⡌⠀⠀⠀⠀⢠⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢱⡀⠀⠀⠀⠸⡄
⠀⠀⡇⠀⠀⠀⢀⡟⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣷⠀⠀⠀⠀⡅
⠀⠀⠃⠀⠀⠀⡾⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡆
⠀⢸⠀⣀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢱⡀⢀⣀⣼
⠀⢸⡀⠈⠁⣾⠀⠀⠀⠙⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠚⠀⠀⠀⠸⠋⠁⠀⣹
⠀⠈⠙⠳⠶⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠉⠉⠀⠀⣼⠀⠀⠀⠀⠀⣶⠐⠊⠉
⡴⠁⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠸⢁
⠱⣤⣀⡀⠀⣿⡌⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠶⠒⠲⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠠⠛⠀⠁⠐⠂
⠀⠀⠉⠉⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣄⣤⣠⡀
⠀⠀⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠁⣼⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠀⠀⢀⣨⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣀⣈⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠈⠁⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠉⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠈⠇⠀⠤⣀⣀⣤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⡤⠤⠤⠶⠞⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⠀⠀⠀⠀⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⣛⢻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠃⠄⠀⠈⠃⠀⣀⡆⠀⠀⠀⠤⠀⠠⠀⠀⠀⢀⣴⣶⣶⠤⣄⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠄⠁⠛⠿⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⣄⣸⡆⡸⠋⠉⡩⠵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢐⢎⡀⠈⠃⠀⠀⠀⠀⠠⠀⠀⠐⠤⠉⢿⣿⠿⣻⠃⠀⠐⣆⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣌⠛⠖⠊⠀⠀⡀⢉⡷⠈⢠⣾⣶⠀⣴⠀⠀⠀⣠⠞⠛⠻⡷⡌⣏⣴⣌⠙⢿⠁⣙⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠜⠛⡃⠀⠀⠀⠘⣦⣀⠘⣢⣾⣿⣿⡭⠁⠀⠀⣾⡏⡀⠀⢀⣖⠀⣿⣤⡴⠙⣤⡻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣿⠃⠀⠀⣰⣿⣷⠥⢄⢠⣹⣡⠻⣽⣧⣠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⡁⠈⠉⠁⠀⠀⠀⣸⣿⣿⡟⢈⣿⣽⣿⠀⠀⠰⢿⡿⣿⠟⠚⣿⣿⠆⡀⣶⣿⣿⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣧⢀⡀⠀⠀⠀⠀⢹⣿⣿⣧⣾⣿⣿⣿⣇⠀⢰⣿⢟⡟⠀⣼⣿⠃⠀⢡⣤⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣯⡿⠿⠿⠓⠀⠐⠘⣿⣿⣿⡿⣿⣿⣿⣿⡄⣿⣿⡿⣃⣾⣿⣿⣷⣧⣀⣾⡆⣈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡇⡀⠀⠀⠀⠀⠆⢹⠿⣻⣿⡇⠈⢻⣿⣿⣧⢿⣙⡇⢉⣉⣝⣹⣅⡉⠈⣉⣁⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠒⠈⠛⠳⠾⠿⠿⠦⡄⣾⣿⣿⡟⠁⠀⠸⣿⣿⡟⠿⢿⣯⣯⣭⣽⣯⣍⣉⣩⣭⣉⣹⣇⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⣿⣿⠟⠓⢶⠛⠁⠻⢿⡿⠿⠁⠉⠈⠁⠤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⣿⡟⠀⣠⣦⣤⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠈⠉⠉⠉⠉⠁⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠰⢦⡀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡠⠒⠉⠀⠀⠀⠀⠸⠋⠁⠀⠈⠉⠁⣤⣤⠀⠀⠹⡄⢈⡷⢄⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⢀⠇⠀⢴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡅⢈⡇⠀⠀⡖⢶⠃⠈⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⣤⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⢣⡼⠃⠀⠀⣰⠄⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣦⠋⠀⠀⠀⠀⠋⠐⠋⠃⠀⣠⠀⠀⠀⠀⣸⡀⠀⣠⡞⠁⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄⣠⡤⡀⠀⠘⢆⣸⠀⠀⠀⠀⠀⠢⣤⣤⣶⡾⢁⣠⣤⡶⠂⠉⠙⢏⠉⠉⠏⠙⠢⣄
⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠉⠀⣇⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⡡⠜⠓⢉⡀⠀⠀⢱⡄⠀⠨⠀⠀⢸⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠘⠃⠤⢄⣀⣀⠉⠁⠒⢢⠖⠂⠷⣦⡄⠠⠔⠊⠁⢀⡀⠈⠑⢆⡀⠀⠁⠀⢀⠀⠀⡌⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⢤⠏⠀⠀⢠⠋⠀⠀⠀⠀⢀⡉⠑⢢⡀⠈⢡⠀⠰⠀⢸⠀⢀⠃⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣀⢸⠀⠀⠀⠀⠀⢀⠐⣦⣀⢱⡀⢸⠀⠄⢠⠂⢠⣉⣜⣁⡀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⢰⠇⠀⠀⠉⠈⠙⠛⠛⠓⠚⠓⠛⠉⠉⠀⢱⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡏⠀⠀⠀⣀⣤⠄⠀⠀⠀⠀⠤⣤⡀⠀⠀⠸⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠷⡄⠀⡸⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢲⠀⢀⠼⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡇⠀⢸⠋⠁⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠑⣿⠀⠀⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠈⢇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠄⡈⠀⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠎⠁⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠈⠳⣤⠜⠣⢾⠀⠀⠀⠀⠀⣠⠄⠀⠀⠈⠀⠀⠁⠀⠀⠢⡀⠀⠀⠀⠀⣰⠤⠛⢤⡴⠋⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠈⠢⣤⠟⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠛⠠⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠘⣄⡴⠋⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣴⣁⣀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⣀⣀⣱⡄⠀⠀⠘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡌⠀⠀⠀⠀⠉⠉⢉⡿⠦⣀⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣀⠴⠛⠻⡉⠉⠀⠀⠀⠀⠙⡄⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠉⠙⡶⠐⠦⢀⣀⣀⠠⠖⠀⢻⠉⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠶⠆⣀⡀⠀⢀⠏⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⣤⣦⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠈⢧⢀⣀⡀⠶⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢺⡷⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢀⡀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠠⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠒⠀⢬⠭⠁⠀⠐⠒⢒⡒⠒⠒⠀⠀⠩⠭⠄⠐⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠷⠆⠤⠠⡾⢻⠦⠀⠀⠶⢇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣀⣀⣀⣀⣀⠀⡆⢸⡄⣀⣀⣀⣀⣈⣳⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣄⣹⣶⣶⣶⣦⣭⣍⣡⣤⣙⣭⣥⣶⣶⣶⣽⣅⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣷⣾⣃⣼⣯⣹⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⡍⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣰⣁⣠⢟⣟⣀⣀⣜⢻⣂⣈⣆⣰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢨⢏⠈⠛⠥⠼⠀⠘⠧⠼⠛⠁⠙⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠈⠒⢴⠖⢿⡛⣻⠖⠖⠢⠴⠃⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠓⠤⠛⠣⠔⠁⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⡟⠢⡤⠄⣀⣀⠠⢤⠖⢹⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢠⡆⠀⠱⠐⠊⠑⠂⠎⠀⢰⡄⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⣠⠞⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡟⣄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⠆⠀⡆⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⡇⠈⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡸⠀⠀⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠁⠀⢱⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠇⠀⠀⠫⢲⣷⡆⠤⠤⠤⠤⠤⠤⢠⣾⡒⢽⠀⠀⢀⠇⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⢄⣾⠀⠀⢰⠀⠀⠀⠁⢉⠭⠭⡍⠁⠀⠀⠀⢸⠀⢠⣧⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠀⠀⠀⢘⠀⠀⠀⠀⡘⠀⠀⢸⠀⠀⠀⠀⡀⠀⢸⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⡇⠀⠀⠈⡀⠀⠀⠀⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠀⠀⠀⡇⠀⠀⠀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠘⠀⠀⠀⠀⢸⠀⠀⡌⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠉⠉⠈⢉⣁⠇⠀⠀⠀⠀⠘⢆⣉⠉⠈⠁⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠹⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⠓⠚⢻⣶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠠⠶⠶⠶⠶⠶⠶⠶⠶⠦⣀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠉⠰⣦⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⡠⠊⠀⠀⠀⠀⢀⣰⡿⠁⠀⠀⠘⢧⡀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⠞⠁⠀⠀⠀⠀⣤⡾⠋⠀⠀⠀⠀⠀⠈⠹⣆⠀⠙⠢⡄⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄⠋⠀⠀⣀⠤⢎⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠿⠶⠶⠂⣉⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⢀⣴⠞⠋⠀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠛⢦⣤⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣾⠁⠀⠀⢀⠐⠂⢸⣿⡏⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢹⣿⠗⠀⠀⠀⠀⠀⢸⣦⣼⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢻⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢠⣤⡄⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠷⣦⣤⣀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣤⣶⡟⠉⠱⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠛⠉⢹⡿⠷⠖⠚⠛⠛⠟⠛⠛⠛⢠⡍⢋⣹⠇⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡟⠁⠀⠀⠀⠀⢿⣦⣈⠁⠠⠤⠤⠄⠐⠋⣀⡤⢾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁⢸⡇⠀⠀⠀⠀⠀⢸⡇⠉⠓⠶⠶⠶⠶⠶⠚⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⡀⢸⡇⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⡄⠀⢀⡛⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢡⡀⢸⡇⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠘⢣⣾⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡖⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠤⡀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣽⡾⠙⡇⠀⠀⠀⠀⠀⠰⠶⢿⠗⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠈⣦⠀⠀⢀⡀⠠⠴⣯⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⣠⣾⠛⠛⠛⢿⣿⣷⠶⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠛⣴⡶⢋⡻⠶⠶⠿⣄⠘⡇⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣹⣿⡠⠆⠀⠀⠉⣠⡆⠀⠀⠀⣠⠛
⠀⠀⠀⠀⠀⠀⠀⠉⢿⡏⠀⠀⠀⠀⢘⣿⠃⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣧⣰⠂⢀⣼⡿⢀⣤⣤⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣴⣀⣀⣠⡾⠋⢸⡗⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⣤⡷⢸⡏⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡟⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠰⠾⠃⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⢼⣄⣠⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠐⠀⠀⠀⠀⠀⠈⠁⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣠⠖⠀⠀⠀⠀⠀⢀⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⡴⠞⠁⠀⣀⣤⣤⠶⠟⠋⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣷⢶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣴⣾⡾⠿⠷⠾⠛⠛⠉⠀⠀⠀⠀⢀⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠈⠁⠀⠀⣠⡤⠈⠀⠀⠀⣀⣤⣤⠞⠛⠉⠀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⡶⠶⠦⣄⣀⣀⣀⣀⣀⣀⣀⣤⣶⡟⠣⠴⠶⠖⢾⡟⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠉⠉⠉⠙⣿⠉⠉⠁⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⣤⣤⣤⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠃⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠀⠀⠀⠀⣨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣄⣀⣀⡀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣷⣀⡀⠀⠀⠀⠀⣨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠀⠀⠀⠀⠀⠀⣛⠁⠙⠛⢷⣦⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⡀⠀⢈⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⢀⡀⠀⡀⠀⣷⣶⣶⣶⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣧⡀⠀⠀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡷⠯⠤⠤
⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⣿⠟⠉⠉⠉⠈⠀⠀⠀⠐⠻⣿⣿⣿⣿⣿⣿⣿⣷⠶⠶⠒⠒⠒
⠀⠀⠀⠀⢘⣿⡿⠿⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⣗⣐⣒⣀⡀
⠀⠀⠀⢠⣼⣶⣤⣶⡿⠁⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⢿⣿⣿⠿⠿⠖⠦
⠀⠀⠀⠨⢿⣿⣿⣿⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⢿⠿⠭⠭⠄
⠀⠀⠀⠀⢹⣿⣿⣿⡀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣭⣧⣤⣶⣿⣿⡭⠭⠁
⠀⠀⠀⣀⣼⣿⣿⡋⠀⢀⢸⣿⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣖⡒⣒⡂
⠀⠀⢞⣿⣿⣿⣿⣿⣦⣼⣾⣿⠗⢉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣛⣓⠂⠁
⠀⠀⠁⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠉
⠀⠀⢠⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠉⠟⣹⡿⢿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣾⣿⣀⠀⠀⠀⢠
⠀⠀⠀⠺⠋⠀⠈⠹⣿⣿⣿⣶⣾⣿⣶⣦⣄⣾⣦⣸⣿⣿⣿⣷⣀⣰⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉⠉⠉⠁⠉⠉⠁⠀⠀⠉⠉⠁

⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠛⠥⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠉⠈⠹⠃⠀⠀⠀⠐⠃⠀⠀⠀⢸⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠠⠀⠀⢤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣤⣤⣤⢀⣀⡀⠀⣦⣤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⡔⠋⠉⠉⢂⠙⠁⡠⠉⠛⠒⠒⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠉⠊⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠱⡦⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠤⠚⠁⠀⠐⠄
⠀⠀⠀⠀⣠⠤⠄⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⡀
⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠉⠒⠒⠀⠀⠠⠤⠀⠀⠒⠀⠁
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠖⠒⠂⡖⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⢀⣠⢴⡏
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠀⠀⣷
⠀⠀⠀⢀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠆⠀⣿⢳
⠀⠀⠀⠀⠉⠒⠤⠀⠈⠑⠖⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠳⡀
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠈⠢⢀
⠀⠀⠀⢀⣠⠄⠀⠇⠀⠆⠀⢠⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⠀⣇⠀⠀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⣇⠀⡆⠀⠈⡷⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⣤⠀⠀⠀⠄⢰⡏⠀⠀⣀⠀⠀⠈⢧⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠘⠒⠠⢿⠀⠀⠁⠐⠀⠠⠤⠀⠀⠎⠶⡸⢀⡀⠀⠀⠀⠀⢀⡀⡌⢶⡈⠀⢰⠎⠀⠉⠂⠚⠐⠁⡿⠀⠁⠈⣧⠀⠀⡘⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠘⠀⢀⣴⠛⠿⢶⣾⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢧⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⠤⠚⠶⢤⡤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠀⠀⠀⠀⠉⠉⠒⠒⠒⠀⠀⠀⠒⠒⣿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⢧⣠⣀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠠⠤⠟⠋⠵⠶⠴⠶⠖⠀⠤⠤⣤⣤⡤⠀⣀⣀⢀⣀⡈⠉⠓⣦⠀⠐⠀⠐
⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣴⣦⣯⣤⣬⣤⣤⣤⣤
⣿⣟⡥⢼⣟⡛⠛⢛⠛⠛⠛⡛⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠱⣦⡤⠄⠀⠀⢈⣿⣿⡿⠿⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠻⣿⣄⠠⠀⠈⣿⠟⢿⠷⣦⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⣼⣿⣷⠀⠀⠣⠀⢠⠃⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣗⡀⡇⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢻⣿⣧⡤⠀⣼⣿⠆⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⡇⠈⣿⣿⣷⣾⣿⣿⣰⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣶⣧⣄⣿⣿⣿⣿⣿⣿⠏⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⡯⠬⠑⣻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣧⠀⠀⠘⣧⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢹⣿⣿⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡛⢿⣿⣀⠀⠀⢸⣿⣸⣆⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡧⢀⣼⣅⠙⢦⣼⣿⡏⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⣿⡿⠿⢿⣇⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠟⠻⣿⣿⣿⣿⣿⡏⠀⠐⡀⠾⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⢳⣷⡐⢻⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⡀⠀⢿⣧⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏⠀⠀⠤⣀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣏⠁⠀⠸⣿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠟⠀⠀⠄⠀⢁⣀⣀⣀⠀⠀⠀⠘⣿⣿⣿⠓⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠋⣀⡤⠤⠤⠴⣯⣯⣉⣛⣢⣤⢤⠄⠛⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣶⣶⣴⣶⣶⣒⣒⠒⠒⠒⠚⠲⣦⣤⣤⣤⣤⣽⣶⠶⠶⠖⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⠛⠛⠋⠭⠭⠉⠉⠀⠀⠀⠛⠿⠿⠶⠛⠛⠛⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠤⢤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣠⠴⠦⣄⣀⣀⣀⣀⣀⣀⡤⠤⠦⠤⣀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⢠⡶⠿⢶⠀⠀⠀⠀⡴⠾⢶⣄⠀⠀⠸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⢠⠤⢧⠀⢠⠇⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢇⠀⡾⠤⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⣷⠋⠀⢒⡶⠈⠁⠀⢱⠀⠀⠀⠐⠁⠀⠑⢶⡒⠀⠀⢸⢾⠁⠀⢹
⠀⠀⠀⠀⠀⠀⢳⡀⠀⡟⡆⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢧⠹⡀⣠⠖⠉⢉⣉⣩⣭⣭⣭⣉⣉⠉⠑⢦⠀⠀⡼⠁⡎⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠙⠇⢀⠞⠉⠀⠙⠢⠤⠖⠁⠈⠑⣄⠈⣧⠞⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠲⢤⣀⡀⠛⠟⠃⠀⢀⣠⠼⠀⠁⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣀⠉⠉⠉⠉⠉⠁⠀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠛⠉⠉⠁