Horseshoes Emojis & Symbols

⠀⠀⠀⢀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣆
⠀⠀⠐⣼⣿⣿⣿⣿⣿⣶⢄⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⢲⣿⣿⣿⣿⣿⣿⣶
⠀⠀⣰⣿⣿⣿⣿⣿⣿⠏⣾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡄⢳⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠻⠿⣿⣿⣟⡻⠏⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠛⣛⣹⣽⡿⠞⠁
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠛⠛⠉⠁
⠀⠀⠀⠀⣀⣀⣀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⢀⣀⣀⡀
⢀⠀⠀⣾⠉⡈⠉⢻⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⡋⢉⡉⢹⡆
⠸⠀⢰⡇⣿⣉⢠⡟⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⣇⢈⣹⡧⣿
⡇⠀⣼⠁⠀⠁⢸⠇⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢿⠀⠀⠀⢹⡆
⡇⠀⣿⠀⣀⡀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⣀⡀⢸⡇
⡇⠀⢻⠸⣯⡤⢸⡄⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣸⠡⣬⡿⢸⡇
⣧⠀⢸⡆⠀⠀⠘⡇⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⡿⠀⠀⠀⣾⠀⢰
⢸⡀⠀⣧⠀⢀⣀⢿⡀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⣸⢃⣂⢀⢰⡏⠀⣸
⠈⣇⠀⠘⣆⢿⣉⠉⢧⠀⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⢠⠇⢈⣹⢇⡿⠀⢀⡏
⠀⠹⡆⠀⠹⣆⠀⠀⠈⢳⡀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⡠⠋⠀⠀⢀⡾⠁⠀⡼⠁
⠀⠀⢻⡄⠀⠘⢇⠈⢡⠖⠐⢄⠀⠙⠻⣿⣲⣤⣄⣀⣀⣀⣀⠤⣴⣺⡿⠛⠁⠀⠘⠱⢶⡀⢠⠞⠀⠀⡼⠃
⠀⠀⠀⠻⣄⠀⠀⠳⣜⠓⠁⡠⠀⠀⠀⢀⣈⣉⣛⣛⠓⢚⣛⣋⣉⣀⠀⠀⠀⠀⠐⠑⣛⡵⠃⠀⢀⡼⠁
⠀⠀⠀⠀⠙⢦⡀⠀⠈⠁⠀⠀⠀⠀⣼⡿⠛⠛⠿⠿⠿⠿⠿⠛⠛⠻⣿⡄⠀⠀⠀⠉⠉⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠘⠿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡼⠟⠀⠀⢀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⣤⣄⣀⠀⠀⠀⠀⠀⠈⠈⠉⠉⢀⣀⣤⣴⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢿⡿⣷⣶⣶⣶⣶⣿⣿⣿⣿⢛⡿

⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⢉⣭⣭⣍⠉⠉⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⣩⣭⣍⡉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠈⠃⠀⠙⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⠃⠀⠘⠁⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⠇⠛⠉⠉⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠉⠉⠛⠘⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡤⣄⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢠⢤⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡜⢀⠎⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠘⡄⠱⡀⠱⡀
⠀⠀⠀⠀⠀⠀⢀⡞⢠⠟⠳⡞⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⢹⠶⠛⣆⠱⡀
⠀⠀⠀⠀⠀⢀⡞⢀⣟⠀⢰⠃⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⢇⠀⣹⡄⠱⡀
⠀⠀⠀⠀⠀⡼⠀⢠⡟⣻⠃⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠘⢿⠻⡇⠀⢳⡀
⠀⠀⠀⠀⣸⠃⠀⡼⢠⡏⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠸⡆⢻⠀⠈⣧
⠀⠀⠀⢀⡟⠀⣰⣧⣼⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣼⣧⠀⢸⡄
⠀⠀⠀⢸⠇⠀⡟⠀⢸⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠃⠀⢸⡀⠈⣇
⠀⠀⠀⣾⠀⠀⣷⢤⡎⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠸⡤⢾⠇⠀⣿
⠀⠀⠀⣿⡀⠀⡇⠰⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢸⠀⠀⣿
⠀⠀⠀⣿⡆⠀⣧⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⡇⢸⠀⠀⣿
⠀⠀⠀⣿⡇⠀⣿⠛⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠛⣿⠂⢸⣿
⠀⠀⠀⢸⣷⠀⠸⣇⣀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣠⡏⠀⣾⡏
⠀⠀⠀⠈⣿⣇⠀⠙⣏⠙⣆⠀⠀⠐⢄⡙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⣠⠆⠀⠀⣠⠏⣹⠟⠀⢰⣿⠁
⠀⠀⠀⠀⠸⣿⣆⠀⠈⢧⡈⢳⣄⠀⠀⠙⢦⣌⡙⠻⠶⠶⣤⣤⣤⣶⡶⠶⠞⢛⣩⡴⠚⠁⠀⣠⡴⢋⡴⠃⠀⢠⣿⠇
⠀⠀⠀⠀⠀⢻⣿⣦⠀⠀⠹⣟⠁⠈⢢⡀⠀⠈⠛⠻⠷⢶⣤⣤⣤⣶⡶⠶⠿⠛⠉⠀⢀⡴⠊⠀⣹⡿⠁⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠈⠛⢶⣿⡉⠲⠤⣄⣀⠀⢀⣀⣀⣀⣀⡀⠀⢀⣠⡤⠖⢋⣹⡶⠞⠁⠀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣀⠀⠀⠈⠛⠲⢤⣽⠇⢸⡇⠀⠀⢸⡇⠀⢯⡤⠶⠚⠉⠀⠀⢀⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣦⣤⣀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⡜⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⡼⠀⠀⢄⡐⢤⣰⢃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢱⡤⠒⠈⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠈⠓⣶⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢠⠃⠀⣰⠛⠃⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⢳
⠀⠀⠀⠀⡞⠀⢠⠃⠤⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⢹⡀⠈⡄
⠀⠀⠀⢠⡇⠀⡸⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢧
⠀⠀⠀⢸⠁⠀⣷⠛⠂⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡇⠀⢸
⠀⠀⠀⢸⠀⠀⡏⠂⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸
⠀⠀⠀⢸⠀⠀⡇⢀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠈⠀⠀⡷⠏⠃⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠴⡄⡇⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢠⠀⠀⠹⠀⠀⠘
⠀⠀⠀⠀⢇⠀⠀⠂⡤⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠆⡀⢀⠇⠀⢀⠇
⠀⠀⠀⠀⠸⡄⠀⠀⠑⠀⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⢀⠹⡞⠀⠀⡼
⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⡄⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡀⠀⢀⠜⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠈⠑⠦⣤⣀⡀⠀⠀⣀⣀⠤⠀⠉⠀⠀⠀⠌⠀⡿⠋⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⣚⡰⠋⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡜⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣄⠀⠀⠀⠀⠀⠀⠀⠉⣭⡽⣟⢿⠋⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠟⠂⠛⠿⠀⠀⠀⠀⢀⡀⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠂⠀⠀⠄⠀⠀⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣤⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿⠿⣿⣦⣄⡀⠀⢀⣀⣀⡀⠀⢀⣠⣴⣿⠿⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣯⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢠⠤⣴⣶⣶⣶⣶⣤⠤⣤⣄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠨⣤⣘⠻⠋⢾⡋⠉⢻⣿⣿⣿⣿⣷⣮⣑⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⣀⣴⣶⣾⣦⠉⠳⡀⠘⡿⣆⡘⣿⣿⢿⣻⣿⣥⣝⠱⣮⡑⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⣿⣿⡀⣻⣿⠟⠀⡀⠇⠀⢱⣿⢩⣿⡿⠘⣿⣿⠋⣻⣿⣦⡹⡮⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⣿⣿⣧⡾⠛⠁⠀⣀⣤⠦⢤⣤⣬⣿⣬⣿⣦⣀⠈⠛⢾⣿⣿⡹⣿⣦⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠂⠀⣼⣯⡈⡹⠋⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀⠙⢦⣞⢉⠹⣧⠀⢻
⠀⠀⠀⠀⠀⡌⠀⣸⣿⣿⡟⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⢿⣶⠘⢻⡆⠘⡇
⠀⠀⠀⠀⠀⠀⢀⡿⠿⣿⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢻⣤⠼⢷⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣷⣶⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣿⣖⢻⡆⢸
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣹⠿⣸⡇⣼
⠀⠀⠀⠀⠀⠀⢸⣿⠿⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣹⡀⣸⠃⡿
⠀⠀⠀⠀⠀⠀⠀⢷⣶⣧⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⣉⡿⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡐⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⡿⣿⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣏⣶⠏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⠿⡥⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠃⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⣝⣂⡈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡡⢻⠟⠻⣟⠁⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣴⣧⣰⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠳⡞⢏⣠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠻⣄⡸⣃⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡿⠿⠟⢀⣴⣿⠿⢿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠃⣠⣾⣿⣿⣿⣶⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣇⣸⣿⡿⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆
⠀⠀⠈⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⣿⣿⣿⠟⠀⢾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠈⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣴⣿⡿⠃⣰⣦⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⢸⣿⡋⣻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⣿⣿⡿⠁⠈⢿⣿⣿⠿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⢙⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣴⣿⣿⠃⠀⠀⠘⣿⣿⣦⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣿⣿⣉⣹⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣏⣉⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⣄⣼⣿⣿⡀⠀⠀⢀⣿⣿⣧⣠⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣿⣟⣉⣿⣿⣧⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⢀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣼⣿⣿⡉⣻⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢻⣿⡟⠉⠛⢁⣴⣿⣿⣟⠉⣻⣿⡟⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣅⣸⣿⣿⣿⣶⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⠟⢀⣤⣤⣶⣿⣿⣿⣇⣨⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⡟⠙⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⡴⣿⣿⣿⣿⠋⢻⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣾⣿⣿⣿⣧⣠⣿⣿⣿⣿⡿⠟⢋⣠⣴⣿⣀⣼⣿⣿⣿⣷⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⠉⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠆⠀⠀⠀⠀⠀⠀⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⣑⣷⣦⣄⣀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣄⣠⠤⠖⠊⡡⠌⢇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠈⠛⢟⡾⠩⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠋⢰⠺⠉⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣰⣶⡦⡄⠀⢠⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠌⠀⣠⣶⢶⣆⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⢰⣟⠈⢠⠇⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢸⠏⡨⡘⡄⠈⢧
⠀⠀⠀⠀⠀⠀⢠⠇⠀⣸⡛⣀⡘⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡆⠀⠟⠇⠂⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⣸⠀⠀⡟⠀⠈⠃⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⠀⢰⠐⢸⢱⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡿⠀⢸⡇⠀⠀⡄⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠈⢸⡀⠀⣷
⠀⠀⠀⠀⠀⠀⣿⠀⢸⣷⡃⠀⠇⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⣖⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣇⠀⢸⡗⠘⠃⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠣⡇⠀⢸⠀⠘⠁⠀⠀⣹
⠀⠀⠀⠀⠀⠀⢸⠀⠈⣧⠀⠀⠀⠀⠀⠁⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡺⠀⠀⡾⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⢹⠂⡀⠀⠀⠀⠀⢈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢡⠃⠀⢰⡁⠀⠀⠎⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⣿⢃⣠⠖⡀⠀⠀⠩⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠑⠁⠴⢠⡏⠣⠀⠡⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠄⠁⠈⠹⣄⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠁⠀⠀⠀⣠⠟⠀⠀⢠⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠘⢦⠀⡠⠈⢢⠀⠀⠀⠀⠉⠙⠓⠒⠦⠤⠤⠴⠖⠒⠛⠉⠀⠀⠀⠀⢀⣴⠁⢄⠀⡴⠃⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠓⢉⠀⠀⠙⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡋⠀⠀⡊⠊⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠙⠒⠊⡰⠈⠀⠈⠁⣤⣄⣀⣐⣶⣖⣀⠀⣤⣭⠵⠆⡅⢄⠑⠐⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⣢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣔⡯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠮⢭⣔⣒⣒⣒⣒⣒⣒⣒⣒⣬⠭⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⣟⢿⡲⣄⠀⠀⠀⠀⠀⠀⣠⣶⠟⠛⢟⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠈⣧⣷⠈⡇⠀⠀⠀⠀⢸⢹⠃⠀⠀⠈⠳⣌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⠀⠀⣠⣯⠏⣰⠃⠀⠀⠀⠀⠘⣞⢦⠀⠀⠀⢀⣼⣓⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡙⢦⡀⢾⡽⣡⠞⠁⠀⠀⠀⠀⠀⠀⠈⠻⡷⢀⡴⠋⠈⢳⡈⢦
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⡴⢻⠯⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢳⡀⠀⠀⢳⠈⢧
⠀⠀⠀⠀⠀⠀⢰⠃⢀⠀⣸⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢀⠈⣇⠸⡆
⠀⠀⠀⠀⠀⠀⣾⠀⠛⠀⡟⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠀⢹⠀⣷
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡆⢸
⠀⠀⠀⠀⠀⠀⡇⠐⠿⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠺⠇⢸⡇⢸
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⡄⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⠀⣼
⠀⠀⠀⠀⠀⠀⢸⡀⠺⠇⠀⢷⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠿⠀⡟⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⣄⠈⢳⡀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⣠⡀⠀⣸⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⠛⠁⠀⠙⠦⣌⡳⣄⡀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠛⠃⣰⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢠⣶⡄⠀⠉⠛⠛⠛⠒⠒⠚⠉⠁⠀⣶⡆⠀⠀⡰⢃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠉⠀⠀⢀⣤⣄⠀⠀⢠⣤⡄⠀⠀⠉⠁⣠⠞⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠘⠿⠟⠀⠀⠘⠿⠃⠀⢀⣠⢞⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⣀⣀⣤⣴⡶⠿⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠣⠉⠑⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⡉⡁⠜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⣄⡀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⡴⡤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠐⠰⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢫⠜⠀⠁⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣰⢣⡖⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢲⠙⠆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣿⣤⡇⢐⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⡧⢣⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣏⠇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡅⢸⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢘⢀⣰⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠆⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡟⠙⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢠⠁⠐⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢇⠣⠃⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠀⣨⢀⠸⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠘⡕⠛⣀⡊⠪⢷⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠝⠙⡼⠋⢋⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⢦⣄⢡⡄⠀⠁⠉⠳⠶⠶⠶⡖⠲⠋⠀⠀⢀⡞⢳⡰⠃⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠙⠤⠝⡠⠀⠠⠄⠀⠀⠀⠂⢀⡀⠤⠀⠉⠠⠊⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣶⣄⡂⣰⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⢤⣄⣀⣀⡤⠀⢢⣔⣂⠥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⣴⣶⣶⣆⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⣤⣰⣶⣶⣆
⠀⠀⠀⠀⠀⢠⣿⡷⢶⡌⠁⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⢡⠶⢾⣿⡆
⠀⠀⠀⠀⠀⢸⣟⠷⢄⣀⡀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⢀⣀⡠⠾⣻⡇
⠀⠀⠀⠀⠀⢸⢿⡿⢷⣶⣾⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣷⣶⡾⢿⡿⡇
⠀⠀⠀⠀⠀⣺⣯⣾⣷⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣵⣾⣷⣿⣷
⠀⠀⠀⠀⢠⣿⣿⣶⣽⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⣶⣿⣿⡄
⠀⠀⠀⢀⢿⣿⣿⢻⣿⡿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⣿⡟⣿⣿⡿⡀
⠀⠀⠀⠈⣶⢸⠏⢙⣛⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣛⡋⠹⣧⣶⠃
⠀⠀⠀⣹⣿⣿⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡇⠀⣿⣿⣟
⠀⠀⢀⣽⣯⡇⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⢸⣹⣯⡀
⠀⠀⢰⣿⣿⡇⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣿⡄
⠀⠀⢸⣿⣿⣿⣄⢸⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣠⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣏⣩⣾⣿⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣷⣍⣹⣿⣿⡇
⠀⠀⢸⠛⠛⡏⠈⠻⠿⠇⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠸⠿⠛⠁⢻⠛⠛⡇
⠀⠀⢸⣹⣿⣿⠀⠀⢻⣿⣞⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⣿⡟⠀⠀⣿⣿⣏⡇
⠀⠀⠈⡏⣿⣿⡆⠀⠈⣿⣿⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢗⣾⣿⠁⠀⢰⣿⣿⢹⠁
⠀⠀⠀⠸⣹⣷⣻⡄⠀⠘⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣿⣿⠃⠀⢠⣟⣿⣏⠏
⠀⠀⠀⠀⢣⢻⣧⢻⣶⣤⣘⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣵⣿⣿⣃⣤⣶⡟⣾⡟⡜
⠀⠀⠀⠀⠀⢳⣻⣷⡻⣿⣀⣸⣿⣿⣿⣿⡷⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⢿⣿⣿⣿⣿⣇⣀⣿⢟⣾⣟⡞
⠀⠀⠀⠀⠀⠀⠱⣝⣿⣌⢿⣿⣿⡁⠙⠯⢶⣿⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣻⣿⣿⣿⡶⠽⠋⢈⣿⣿⡿⣡⣿⣯⠎
⠀⠀⠀⠀⠀⠀⠀⠈⢺⡛⣃⣝⡿⣿⣦⣄⣀⡉⢽⣿⣿⡽⣛⣯⡶⢶⣿⣛⢯⣿⣿⡿⢉⣀⣠⣴⣿⢟⣫⣘⢛⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣾⣝⡿⣿⣦⣴⣿⣿⣶⡾⠁⠀⠀⠈⢷⣶⣹⣿⣦⣴⣿⢿⣻⣵⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷⣿⣯⣿⣮⣻⡇⠀⠀⠀⠀⢸⣟⣵⣿⣽⣿⣾⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢯⣽⣿⣿⡆⠀⠀⢰⣿⡿⣧⡽⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠒⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠎⠀⡀⠀⠀⡀⣸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⣾⢞⠜⢁⠁⠉⢢⠏⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡏⣦⠀⠀⠀⠘⡀⠸⡆
⠀⠀⠀⠀⠀⠀⡟⢮⡀⡡⡠⣱⠏⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣜⡘⢇⠀⠀⠂⢀⣾⡷⣄
⠀⠀⠀⠀⠀⡰⠻⡺⣯⣟⣛⣥⡿⣿⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⡙⣺⣛⣭⠿⡇⠜⠑⢄
⠀⠀⠀⠀⡰⠁⠀⣀⣭⣯⠛⠵⣼⣯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢍⢫⠋⠁⠀⡔⣦⣄⡀⠣⡀
⠀⠀⠀⢠⠁⢠⣾⡝⠋⡝⠀⣰⣣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡕⣄⠀⢿⡊⣩⣝⢆⣱⡀
⠀⠀⠀⡎⢠⣿⡫⢳⡜⠀⣰⢫⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢆⠘⣿⡟⢻⡾⣧⢛⡀
⠀⠀⠐⢠⣿⢿⡧⢊⠃⢀⢏⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⡶⣸⡿⠴⡗⢼⡞⣇
⠀⠀⠇⣼⡯⠀⠀⡌⠀⣸⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⡠⣷⡤⢍⡘⣿⣹⡀
⠀⠐⢀⣷⣿⢦⠀⢀⢠⣯⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢨⣄⣿⣱⣶⣯⣿⠺⡇
⢀⠃⣸⣿⣿⣠⣷⡀⣼⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠉⢹⣷⠈⣿⠉⡀⢰
⢸⣿⣿⢭⡿⢿⣿⠨⡾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢸⡟⠒⠋⠀⡇⢸
⣸⡮⣿⠾⣟⡡⣿⠗⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢸⢌⠀⠀⠀⠃⢸
⡇⠈⣿⣿⠟⡆⡇⠀⡯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣿⣢⣄⠀⠀⠀⢸
⡇⠀⣷⢿⣦⠇⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢰⡷⠉⣯⡇⡘⠀⡆
⠇⠀⣿⣂⠀⠀⢸⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢀⡿⣧⡼⠛⢰⣧⡜
⢀⠀⢸⣧⣀⣀⠈⡄⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⣠⣯⡃⠀⠀⢀⢿⣵
⢸⡄⠈⢿⣿⡋⠢⡰⣀⡀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⢠⣞⡵⠛⣷⡄⡠⣏⣰⠃
⠈⡳⡀⠈⢿⣷⡤⠃⠹⡐⢾⠊⠙⠲⡤⣠⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠒⠂⠉⠀⢀⢤⣾⡫⣏⡠⣊⣿⡜⢁⣸⠇
⠀⠰⡑⡄⠀⠙⢶⣆⣀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⡀⠀⢂⡼⡷⡟⢊⣃⣤⠼⠻⣝⢁⣿⠏
⠀⠀⠱⡈⢢⡀⠀⠈⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⢢⢆⠔⠔⠕⠄⢔⠕⢄⠍⠝⢍⠨⠔⡡⣍⡴⣿⠋
⠀⠀⠀⠈⠳⣍⡦⣀⠀⠀⠀⢀⠀⠀⠀⠐⠀⡠⢀⢄⠄⠔⠁⡡⡢⠢⠊⡨⢂⠊⠂⠡⠠⠂⡈⢐⢴⣩⠞⠁
⠀⠀⠀⠀⠀⠈⠑⢾⣑⠦⣤⣑⡄⡠⠀⢂⠈⠀⠁⠡⠂⠈⢀⠔⠔⡄⡑⡱⡡⠡⠑⡀⢐⠠⢚⡡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣬⡹⡷⠷⠧⢦⣎⣔⣐⣡⡠⣠⢂⢊⣌⣔⣴⡴⡴⠔⣿⠥⠚⠉
⠀⠀⠀⠀⠀⢀⠀⣤⠤⠴⠦⠶⢨⢽⣻⣶⣶⣬⣤⣄⣭⣍⣽⣩⣭⣭⣥⣷⣖⣻⣭⣠⣤⣤⣤⣤⣤⠤⠤⠀⣤⠤⣀⣀⣀⣀⣤⠤⠠

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⠉⠉⣩⣭⣍⡈⠉⠉⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⠉⠉⣩⣭⣉⠉⠉⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠃⠀⠘⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠠⠀⠘⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡇⠾⠛⠛⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠙⠛⠳⢸⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣀⣀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⢀⣀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⣰⠋⡹⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢃⠘⣦⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⡴⣧⣴⠁⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⢧⣼⣷⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣾⠀⢈⠏⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠈⡀⠀⢻⠀⢣
⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⡷⣦⡜⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⢣⣤⢾⡇⠈⢧
⠀⠀⠀⠀⠀⠀⣸⠇⠀⣸⠁⡾⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⢻⡈⣧⠀⠘⣧
⠀⠀⠀⠀⠀⢰⡟⠀⢠⡏⢸⠇⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠘⣇⢸⡆⠀⢹⡄
⠀⠀⠀⠀⠀⢸⠁⠀⡟⠳⢿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡃⠀⠟⠚⢿⡄⠘⣧
⠀⠀⠀⠀⢀⣿⠀⢸⡇⠀⠸⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢁⠀⠸⡇⠀⢻
⠀⠀⠀⠀⢸⣿⠀⠘⡿⢻⡟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡟⢻⠃⠀⢸
⠀⠀⠀⠀⢈⡟⠀⠀⡇⢸⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡇⢸⡆⠀⢸⠃
⠀⠀⠀⠀⢸⣿⠀⠀⣇⢸⣇⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢰⡇⢸⠃⠀⣿⡇
⠀⠀⠀⠀⠸⣿⡄⠐⣿⠛⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠏⠛⣿⠇⢠⣿⠇
⠀⠀⠀⠀⠀⣿⣇⠀⢹⣆⣀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣠⡟⠀⣸⣿
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠻⣏⠙⣆⠀⠀⠐⢦⡈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢃⡠⠂⠀⠀⢠⠋⢹⠟⠁⢠⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠙⢆⠘⢧⣀⠀⠀⠙⠶⣈⠙⠻⢶⣶⣦⣤⣄⣤⣤⣤⡶⠶⠞⢛⣁⡤⠛⠀⠀⢀⡴⠁⣴⠃⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⠀⠀⢳⣾⠁⠙⠦⡀⠀⠈⠛⠳⢶⣦⣤⣬⣉⣀⣬⣤⣤⣶⠾⠛⠉⠀⢀⡤⠊⠈⢷⡞⠁⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⡀⠀⠙⠲⣤⣴⠛⠶⣤⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⣀⣠⠤⠛⣧⣤⠖⠉⠀⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣦⣄⠀⠈⠉⠑⠶⢤⣌⡙⣳⠀⢰⡆⠉⠁⢸⡗⠀⣾⢋⣉⡤⠶⠊⠉⠁⠀⣀⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣦⣄⡀⠀⠀⠈⠉⠉⠀⢸⣷⣶⣶⣾⡷⠀⠈⠉⠁⠀⠀⢀⣠⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣷⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⣿⣿⣶⣤⣤⣤⣤⣤⢤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⠔⢺⣿⣿⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣇⢿⣟⣾⣿⡟⣿⣿⣿⣦⠀⢻⣷⣄⠀⢀⣴⣿⠏⢠⡾⠋⠀⢀⡇⠀⠻⢋⣇⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣮⣿⣿⣏⣿⣧⡙⢻⣿⣿⣇⣸⢣⢏⣷⣯⢳⢹⡄⡟⠀⠀⣠⡿⠀⠰⣠⡾⣫⠞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣯⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣼⢻⡇⣿⢻⣼⣿⠃⠀⢠⠏⠀⠠⠊⠁⣴⣶⡏⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣾⡇⣿⣿⣿⣿⠀⠀⡏⠀⠀⠀⠀⢀⠁⠉⠁⢫⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣳⣿⣟⡽⣳⣿⣿⣿⣿⣿⣸⢻⣿⣿⣿⣿⣇⣿⢿⣿⣿⡆⣔⢧⡀⠀⠀⠀⠈⢆⠢⠀⠘⢄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⡿⣻⣿⣯⡍⠻⣿⣿⣿⣵⣷⣿⣿⣸⣿⣀⣿⣏⣸⣿⣸⣧⠈⢈⢳⣤⣀⣀⣴⡖⠀⠀⢦⡀⢣⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣻⣯⡾⣿⣿⣵⣾⠟⠁⠀⣸⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠣⡈⣿⡇⠈⠙⢧⣝⠂⠀⢀⠹⠀⣧⣙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⡿⠛⠉⠀⠀⠀⣰⡿⠋⠺⠗⢀⣤⣶⣶⣖⣶⣶⣢⣄⠰⡶⢌⢻⣷⡄⠀⠀⠈⠑⠲⣬⣧⡀⠿⢯⡏
⠀⠀⠀⠀⣀⣀⣀⣀⣈⡛⠻⠵⠼⠁⠀⠀⣀⣤⣾⡟⡑⠛⢀⠞⢁⣽⣿⣥⠿⢧⣽⣿⣅⠑⢄⠻⠢⠹⣿⣦⣄⡀⠀⠀⠸⠤⠽⠶⢚⣀⣀⣀⣀⡀
⠀⠴⠚⠉⠁⠀⠀⠈⠉⠙⠛⠿⣿⣿⡿⠿⠿⠛⡟⡴⠖⢠⣯⡴⣛⡥⢞⣻⣷⣿⣛⠷⣝⡳⣬⣆⢰⣧⢹⡛⠿⠿⠿⣿⣿⡿⠟⠛⠉⠉⠁⠀⠈⠉⠙⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢾⠑⣀⠀⡾⠿⠿⠿⠛⠛⠉⠀⠈⠙⠛⠻⠿⠿⠿⡄⣀⡆⣷⠶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠛⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡤⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠻⠂⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠑⠿⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⡠⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⡴⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣒⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⡻⠿⠛⠻⠿⡿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣡⡾⢃⢄⡀⠀⠀⢀⣤⠞⢿⡏⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣄⠫⣱⠌⡢⡴⢋⢔⡩⠃⠿⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣮⠢⡖⢌⢧⡪⠂
⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⠀⠀⠀⠀⠀⢀⡾⢙⢷⡕⣌⣆⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢰⣡⡶⢇⠀⠀⠀⠀⠀⠀⣾⢁⠂⠍⣿⡼⡀⡆⢇⠀⠀⠀⠀⠀⠀⢠⣿⡓⠢⡀
⠀⠀⠀⠀⠀⠸⣏⢔⢊⢆⠀⠀⠀⠀⢰⡇⢈⡹⠸⢸⡇⡗⢲⢸⠀⠀⠀⠀⠀⣠⡿⡩⡛⠷⠃
⠀⠀⠀⠀⠀⠀⠹⣎⢋⠣⡣⡀⠀⠀⠸⡇⠈⠁⢠⢸⣇⡉⠉⢸⠀⠀⠀⠀⣴⢟⠜⠦⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⢍⠪⡂⢄⡀⢻⣦⣄⠈⢿⣿⣯⣄⡎⠀⢀⣠⡾⡣⠺⡔⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡑⠫⣑⢢⠬⡽⡔⠳⡀⠙⠿⢿⣤⠶⢛⠥⢎⠤⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠄⣈⣁⣈⠰⢵⣄⡀⠰⠂⠩⠄⣂⠥⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠛⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠉⠙⢿⣿⣿⡟⠛⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣾⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠙⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠞⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢉⠋⠉⠐⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠚⡆⠀⠀⠀⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⢎⣤⢀⡴⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⠃⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⡩⠓⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠃⣤⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢄⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⠇⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠃⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡸⢠⡤⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡇⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠈⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠉⠛⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⡀⠀⠀⠀⠀⢠⠃⡘⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠰⠿⠀⡇⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⣱⠀⠀⠀⢀⠎⡰⢁⣄⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢤⠀⠀⠀⢸⢸⠀⠀⠀⠇⠀⠀⠀⡼⠱⡄⢀⠀⠀⠚⠁⠀⠀⢀⡞⢠⠁⠈⠉⡞
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⢣⠀⠀⢸⡘⡀⠀⠀⢸⡀⠀⠀⠹⠦⠦⡌⢳⠠⡆⢣⠀⢠⠎⢠⠃⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠰⡇⠠⣀⣀⠀⢀⠔⠁⠀⠀⠀⢇⢣⠺⠷⠀⢳⡀⠀⠀⠀⠀⢸⠀⢧⠀⣈⠖⠁⡰⠁⢾⠆⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⢸⣠⣼⠀⠀⠀⠀⠀⠈⢎⠆⠀⠀⠀⠙⢶⠤⢤⣀⡤⠧⠜⠋⢁⡠⠊⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡇⠘⡆⠀⠀⠀⠀⠀⠈⢎⢢⠀⢠⣶⠀⠁⠒⠂⠀⠀⠒⠈⠁⢰⡶⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢱⠀⢱⠀⠀⠀⠀⠀⠀⠀⠑⠷⣄⡁⠀⠀⠀⠀⢴⡇⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⠆⣀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠤⠤⠤⠖⣲⠉⠡⢔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡉⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢧⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠇⣠⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⡄⢤⡄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⠎⠀⠉⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡘⡄⠀⠀⠈⠢⣄⠀⠀⣀⣀⣀⠤⠒⡁⠔⠁⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⠢⡀⠀⣠⡀⠉⠁⠀⠉⠀⠒⠈⠀⠀⠛⢂⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡢⢌⠀⠀⠀⠀⠀⠶⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠒⠒⠊⠉⠁

⠀⢀⠀⠤⠂⠄⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡀⠀⠐
⠀⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢀⠄⠀⠀⠁⠀⠽⠀⠠⢤
⠀⠀⠀⠀⠀⡌⠘⢶⠑⡆⠐⠂⠈⠃⠀⠀⠀⠀⠀⢰⠀⢀⡆⠀⠀⢀⢀⡀⠄⠀⣲⡟⣔⣿⣤⣾⡦⡀⠄⠀⠦⠼⠟
⠀⠀⠀⠀⠀⡁⠠⠦⡀⢠⡶⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡀⠀⠐⠠⢝⠦⠄⠠⠏⠀⠁⠀⠀⠀⢘⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⢀⣈⣉⠁⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⢸⡗⣠⡴⢰⠦⢶⠀⡆⠰⡫⠀⠀⠀⠀⢽
⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⢀⠀⢀⡄⣃⢀⡲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠤⠠⠄⠀⠂⣀⣢⣘⣜⣃⡠⡠⣷⣤⢴⣷⢶⣤⣴⠤⠠⠀⠄⢒⡀⢀⡀⡤
⠀⠰⢵⣄⡺⢿⣬⠛⠞⠳⠫⠿⠑⠒⠉⠓⠛⠑⠉⡁⡉⠀⠈⠀⠀⠈⠁
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢄⠰⠾⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⠀⡈⢠⡀⠀⠀⠐⠐⠊⠁⠉⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠆⠀⠠⠀⠀⠤⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄⠓⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠙⠂⠊⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⣀⠠⣤⠀⠀⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠃⠘⠃⠀⠀⠈
⣿⡿⠿⠿⠻⠿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠒⠂⠀⢀⡼⠤⠀⠀⠀⢀⡨⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣠⠴⠟⠒⠒⡂⠋⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣀⣴⠞⠋⠀⠀⠀⠀⠀⠑⠒⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣠⣄⣰⠏⠀⠀⠀⠀⠘⣧⣰⠻⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡤⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⢀⣴⣿⠋⠳⡀⠀⠀⠀⢠⡞⢉⣯⠻⡄⠀⠀⣠⠶⠦⠞⠃⠸⣟⠎⠳⢄⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠙⣦⣤⡴⠋⢰⡿⣌⣢⠈⠳⢮⣁⣀⣀⠀⠀⠀⠹⡗⢀⣀⣵⢶⣷⡾⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣠⠴⣿⠁⠀⠀⠀⠘⢧⣀⡈⠑⠀⠀⠀⠀⠈⠑⠄⠀⠀⠉⠉⠁⠀⠸⣅⠙⠲⠦⢤⡴⢺⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⢀⣿⡄⠀⠀⠀⠀⣤⣻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⠇⢸⡄⠀⠀⡞⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠸⣟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⢠⠘⣆⣿⣇⠀⡼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠀⢹⠀⠈⠙⠢⠀⣤⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⡆⢸⣿⣿⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢄⠀⠀⣾⡇⠸⡀⠀⠀⠀⢠⣿⡆⠀⠀⢸⣿⣿⡂⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣿⣦⣿⣷⠀⠙⢂⠀⠀⢸⣿⣇⠀⠀⣼⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡴⢄⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣾⣧⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠘⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠸⠈⠃⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠀⠀⠀⠔⡀⠀⡠⠀⠁⠀⠀⠲⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠐⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠠⠈⠀⠈⠢⠔⠊⠁⠀⠀⡀⠄⠂⠀⠀⢎⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠀⢀⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠤⠀⠀⣀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠉⠐⠢⢀⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣎⠠⠂⠁⠀⡀⠀⠀⠀⠀⠈⠁⠀⠡⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⣄⣠⣬⣭⣷
⠀⠀⠀⠀⠀⡏⠡⠀⣀⣀⣀⣤⣄⣀⠀⠀⣠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠐⠋⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡿⠿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⠛⠳⢦⣠⣄⡀⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣤⣾⣦⡐⠛⠉⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⠁⠀⣀⣤⣾⣷⣦⣤⣄⣀⣀⣤⣦⣴⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠄⠺⠦⠴⠶⠴⠶⣤⣤⡤⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣈⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⠖⠊⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣬⣭⣭⢽⠶⠶⠶⠦⢤⠄⢀⣀⠀⠀⠀⠈⠉⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⣿⠶⠶⣦⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠅⠀⠀⠀⠀⠀⠀⠉⠙⢩⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⡒⠶⣿⡙⠻⣿⣍⠉⠻⣿⡙⢿⡖⢶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠺⢿⣭⣉⡛⠷⢮⣝⡲⢬⣙⠳⢤⡉⠳⢤⡈⠳⣄⠹⡌⢧⢹⢸⢇⡟⣹⠖⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣛⡿⠷⠶⢤⠤⠭⡙⠳⠦⣬⣙⠳⢮⣝⡲⠬⡓⠤⣙⠢⠈⠃⠈⠀⠈⠈⠈⠐⠡⢞⡡⠞⣋⣷⢤⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠈⢲⡀⠀⣰⠃⠀⣨⠉⠓⣤⣌⣙⣒⠦⢍⡓⠤⢁⠀⠁⠀⠀⢠⠞⠉⠉⠉⠳⡄⠀⠐⠈⠥⠴⠶⢛⣳⣄
⠀⠀⢀⣴⣫⠔⠢⣅⣠⡞⠉⠳⢄⡀⠉⠀⠠⠉⠁⢀⣼⠷⢻⣿⣷⢶⡌⠉⠂⠀⢀⡴⢲⡇⠀⠀⠀⠀⠀⢸⣀⣠⠞⠛⣯⡩⠭⠭⠭⣷⣄
⠀⢠⣾⣿⣶⣾⣿⣿⣿⣷⣄⠰⠞⣉⡦⠴⠞⣳⠖⠋⠁⠀⠘⣿⡏⠀⠈⠓⣲⡯⣍⡀⣨⢷⠴⠚⠓⢦⣤⣾⣾⡃⠀⣼⡿⠻⢏⣹⢷⠚⠙⢦
⢠⣿⣿⣿⣿⣿⣇⣤⡤⣉⡛⠳⢿⡀⣀⣴⣞⠁⢀⠀⠀⠀⣩⣿⡿⠀⢀⡟⠙⠲⡀⠉⠳⢤⣴⡶⠾⡿⠟⡻⠉⠀⠀⣘⣷⠀⠀⢙⣷⣯⣴⣾⣇
⣼⣿⣿⣿⣿⣷⠟⣽⣻⣯⣿⣷⣶⡉⠛⢿⣿⣿⣶⡄⠀⣴⣿⠓⠀⣠⣿⣤⣤⣠⠞⢀⣀⢤⣬⣻⣮⣄⠈⠀⠀⢀⣠⣿⣿⣤⣾⣿⡿⣿⣟⣿⣿
⣿⣿⡟⣷⡿⣿⢻⡼⣧⣷⣿⣶⣿⣿⣶⣄⣌⡙⠻⢿⣼⣧⣤⣤⣾⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⢬⣿⣿⢿⡟⣿⣾⣿⣿⣻⣯⣿⣿
⣿⣿⢿⣿⣷⣞⣚⣿⣿⣭⣿⣿⣿⢻⣿⣶⡬⠝⠳⣤⣈⣙⠻⢿⣿⣿⣿⣟⡷⠾⠿⠛⢛⣋⣭⣭⣷⣶⣾⣷⣿⣿⠿⣾⣝⣗⣯⣷⣿⣿⢾⣿⣿
⢹⣿⣾⣿⣷⣮⣿⣧⣿⣭⣻⣏⡽⣻⣿⣿⣿⣷⣄⡙⠻⣿⡷⢯⣤⡿⠉⣿⣀⡀⠝⠻⠿⠷⠶⠯⢿⣧⢾⣛⣿⣶⡿⣿⣿⣿⣷⣿⣿⣹⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⣿⣆⣤⣍⣳⣯⡳⣄⣈⠙⠛⠻⠶⣬⣵⣾⣿⠿⠷⠿⠯⠿⢶⣿⣟⣿⣞⢻⡿⣽⣿⣿⣿⠃
⠀⠀⠹⣿⡽⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣗⣺⠭⣿⡷⣷⣿⣾⣳⣾⣿⣦⣽⡷⠀⠀⠀⠈⠹⣿⣿⣿⣯⠭⠽⢿⣟⣒⣶⣭⣝⣛⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣷⣾⣷⣺⣭⣿⣿⣿⣿⣭⢟⡻⢿⡲⢭⣓⢪⣽⣿⡿⠟⠋⠁⠀⠀⠀⡀⠀⠀⢸⣿⣛⣛⣯⣭⡿⢾⣛⣛⣿⣭⣭⣭⡽⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣞⣛⡻⣍⣯⣭⣝⣿⣮⡓⣭⣷⡾⠟⠋⠁⠀⠀⠤⠀⠀⠀⡀⠀⠀⢀⣼⡿⣟⣻⠽⢗⣛⣭⠭⠴⣶⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣽⣹⣯⣷⡿⣿⣭⠁⠄⠀⠀⠀⠒⡤⠀⠁⠀⠴⠀⣠⡶⣛⠽⣛⣭⠶⣚⡭⢵⣲⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣯⣶⣶⣖⢐⣠⣶⠖⠔⠐⠀⢢⣾⣷⣻⣵⣟⣯⣶⠿⠗⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠛⠛⠛⠛⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠙⠛⠋⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠷⠶⣶⡆⠀⣴⡶⠶⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⡀⠀⠀⠀⣶⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣄⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣷⠀⣀⡀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠋⢹⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⠷⠛⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⠶⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠹⠶⠋⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⡤⣦⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣯⣀⣼⢱⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣾⠉⢹⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠈⠉⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡌⠛⠋⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢰⡶⣿⣷⣤⣀⡀⠀⣀⣀⣤⣴⡾⠋⣴⠛⢿⡄⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠻⣤⣸⡇⠉⠛⠛⠛⢛⣛⣉⠁⠀⠀⢿⣤⠞⢡⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣁⠀⠀⠀⠀⠀⢿⡁⣹⡆⠀⠀⠀⣠⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢷⣶⣤⣤⣀⣈⣉⣉⣤⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣶⣾⠟⠉⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⡶⢶⣶⣤⣄⡀
⠀⠁⠀⠀⠠⠀⠀⠀⡀⠀⢛⣈⠀⡀⣈⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠫⢦⡀⠀⠉⠻⢿⣿⣶⣤⡀
⠠⠀⠀⠂⠀⡀⠀⠠⠀⠐⠀⠈⠁⠀⠀⠀⠀⡨⢭⡉⠉⠉⠛⠻⠿⢿⣿⢿⠋⣁⢀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠈⠻⠿⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠂⠙⡆⠉⠃⠀⠀⠀⠀⠀⠈⠓⠶⢧⠀⠀⠠⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⣀⣀⣠⠤⠀⠖⠚⠉⠉⠂⠀⠀⣐⢤⢀⣤⣤⡄⠀⠀⠀⢀⣠⣀⣀⣠⣿⣿⣿⡶⠶⣾⣦⣅⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠉⠉⢉⣤⣤⣦⡄⣑⣱⣶⣶⠈⠒⠃⠠⣾⣷⣀⣘⠛⣿⠀⠰⠶⠄⠀⠛⠛⠛⠛⠛⠛⠛⠁⠀⠛⠋⠉⠁⠁⠀⠁⠁⠁⠀⠈⠈⠀⠀⠈⠘
⠀⠀⠀⠳⡌⢸⣼⣷⣿⣿⠿⠿⠿⠓⠀⠋⠈⠉
⠀⠀⠐⠰⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⡀⠀⡀⠀⠀⠀⢀⡀⢀⠀⢀⡀⠀⣀
⠀⠰⣦⣄⣶⣤⡦⠄⠀⠀⠠⠷⠿⠚⠛⠺⠳⠓⠖⠁⠷⠿⠄⠀⠀⠐⠄⠁⣸⠬⠥⠤⠛⠉⠉⠉⡉⠈⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁

⣷⣶⣰⣶⣧⣤⣀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠈⡱⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⢄
⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⠘⢳⠊⠀⢠⣖⡀⠀⠋⠧⠞⠃⠀⠀⠀⠀⠀⠎⠀⠀⠄⠀⠀⠀⢁⠀⠐⡦⠀⠀⠰⠷⣠⠄⠀⡐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⢈⡄⠀⠀⠠⡚⠳⡄⠉⠺⣾⣿⣷⢾⣷⢆⡤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⢀⡀⠀⠄⢷⣿⣷⣤⣔⠉⠀⣀⢠⡖⠓⠑⠈⠄⠀⠀⠉⠛⠻⠿⠽⢷
⣿⣿⢿⣿⣿⣶⣾⣿⣿⣿⣿⣯⠉⠩⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣽⣿⣿⣟⣶⣿⣧⣴⡧
⢿⢿⣷⣬⣭⣭⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡈⠀⣤⣤⡄⢤
⣶⣶⣵⠯⣿⣭⣿⣿⣿⣿⣿⣇⡀⢀⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣾⣿⣿⣥⣄⣈⣷⣤⣤⣤⣀
⣿⠏⢻⣿⣿⣿⡿⡉⠉⡻⣟⠛⠛⠃⠤⠀⠀⠀⠀⠒⠤⣄⣠⣀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⠉⠉⠛⠿⠿⢤⢈⠀⠻⠟⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣷⣶⣾⡿⢿⣿⣿⣷⣤⣤⣦⣤⣄⡀⠀⢀⡀⡛⠃⠀⠀⠀⣀⣈⡉⠉⢁⡂⠀⣤⣄⠀⠀⠀⠀⢀⣀⠀⠀⠀⣄
⣿⣿⣿⣿⣿⣿⣿⠿⠟⣻⣿⠿⠿⣿⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⢿⠿⢯⣭⣿⣿⣿⣿⣿⣿⣾⡿⢿⢷⡿⣷⣴⣼⢿⣦⣶⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣷⣴⣶⣾⣿⣿⣿⣻⣿⣿⣷⣆⣉⣭⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣱⣶⣿⣿⣿⣿⣟⠿⣿⡿⣿
⣥⣽⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣻⣿⣿⣿⣿⣴⣿⣅⣿
⠹⠟⠛⠟⠿⢿⣭⣿⣽⣤⣤⣤⣠⣤⣬⡽⠋⠉⠁⢸⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠟⠻⠿⣿⣿⣛⠛⢛⣳⣖⣖⠶⠶⠶⠶⠶⠶⠴⠯⠭⠙⠻⠿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣇⡁⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⢷⡾⢶⣶⠶⠤⣀⡢⠤⣤⣤⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠃⢀⠀⠈⠛⢻⣈⣁⣀⡈⠙⠛⠛⠛⠛⠿⠿⠷⣶⣶⣿
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠀⠀⠒⠦⢄⣠⣦⠈⠀⠀⠀⠀⣶⡦⢄⣀⠀⣀⣤⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠄⡀⢈⡉⠓⠀⠀⠀⠀⠀⠐⠻⢷⣇⡱⣀⡀⠐⠀⠁⣾⣾⡛⠛⠻⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣠⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣦⣆⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢶⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣟⠛⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣓⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠠⠄⠀⠀⠀⢤⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⣠⠾⠛⠃⠄⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡴⠟⠉⠀⢀⣾⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣴⣷⣧⢀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠃
⣿⣿⣿⣿⣿⣿⣷⣿⣠⡀⣦⣷⢠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣧⣧⣼⣰⣇⢠⡀⣿⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⣠⢸⣷⣿⣦
⣿⣿⣿⣿⢿⢿⡿⠿⠛⠛⠟⠿⠛⠋⠛⠉⠉⠋⠉⠉⠋⠈⠉⠛⠁⠀⠋⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⣸⣧⡆⢠⢀⣿⣾⣷⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⣾⡧⢀⡀⣸⡇⣰⣠⣧⣴⣰⣿⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣾⣧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣸⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠻⣿⣿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠈⠁⠈⠀⠁⠈⠈⠉⠈⠉⠙⠛⠋⠉⠉⠛⠻⠟⠿⢻⡿⣿⠿
⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⣿⠃⢹⣿⣿⣿⣿⣿⡃⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡟⠀⢐⣻⣿⣿⣿⣿⣷⣼⣿⣧⡀⢀⣦⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢸⣿⣯
⠉⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⡄⠀⢰⣾⣿⡇⠀⢰⡆⠀⠀⣿⣿⣿⡁⠀⠀⣾⠄⠀⠀⠀⢰⡧
⠂⠀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢺⣿⣿⣶⣤⣾⣿⠀⣘⣿⣿⣿⣶⠆⢸⣿⡇⠀⠀⠀⣿⣟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣘⣿⣿⣿⡟⢻⣿⣿⣦⣽⣿⣿⣿⣿⡥⣿⡿⠿⠀⠀⣽⣿⣿⠇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣶⣤⣿⣿⣿⣷⡤⠀⠀⢰⡇
⠀⠀⠀⣤⣽⣿⣿⣿⡧⠀⠀⢽⣿⣿⣿⡿⢻⣿⣿⠦⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠠⢽⣿⣿⣿⣿⣿⣧⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠼⣿⡟⠓⠀⠀⠙⢿⣿⣿⡃⠀⣿⡟⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠙⢿⣿⣿⣿⣿⣷⡖⣸⣿⣿⡅⠀⢀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠒⠚⣉⠁⠀⠀⢤⣿⣿⡟⠀⠀⠹⡏⠁⠀⠀⣤⣿⣿⡿⠁⢿⣿⣿⣏⠁⠀⠀⢤⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢁⣾⣇⠀⣸⣿⣟
⠀⠀⠀⠀⠠⠴⠒⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⠀⠘⣿⣟⠀⠀⠀⠀⠐⢺⠻⣿⣿⠁⢻⣿⣿⡟⢻⣿⣽⣿⣧⣠⣿⣿⣿⣃⣾⠀⠀⣴
⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢒⣿⠁⠀⢈⣿⣿⠀⠈⣿⠙⣿⣿⣿⠇⢸⣿⣿⣿⡆⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⣉⠅⠀⠀⠀⠙⠀⠀⠀⠋⠀⣿⣿⡿⠀⠈⣿⣿⣿⣷⣾⣿⣿

⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⡿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀
⣼⠟⢤⠀⠀⠀⠀⠀⣀⣀⣠⣴⣽⣿⣿⣿⠟⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡟⢠⣯⡄⠀⠀⠘⠛⠻⠿⠿⣧⣀⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⢿⣄
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣟⡉⢛⣉⡏⡀⣀⣸⣿⣿⡶⠿⠒
⠀⣾⡻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣄⡄⠀⠀⠀⠀⠀⢸⡏⣿⣿⠟⢿⣿⠿⢿⣿⣿⣿⡀⢀⣠⣶⡀⣠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣀⣄
⡟⠀⠙⠃⠀⠀⠀⠀⠀⢸⡿⢯⣿⣤⣀⣻⣆⣠⠿⣿⠟⣿⣯⣨⣿⣹⡟⣹⣿⣿⢿⣋⡀⣄⣽⢳⣻⡿⠁⠀⠀⠀⠀⣼⣇
⠅⠀⢰⡀⠀⠀⠀⠀⠀⠈⢄⣈⣭⣿⣿⣿⣣⡿⠷⣯⣟⣯⣅⠉⣻⣿⣾⣿⡿⠿⣫⡻⣿⠿⢻⣿⣼⣧⠀⠀⠀⠀⠀⠈⢿⣧⡀
⡟⠀⢺⡇⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⡼⡟⢶⣅⣄⣀⣿⣾⣿⣷⣿⣿⣿⣍⣴⣶⡿⠏⣥⣤⢿⣷⠖⢯⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠁⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢗⡋⢱⣐⣯⣶⣽⣿⠿⠿⣻⣿⣿⣭⣔⡻⠿⢿⣾⣷⣚⣺⣛⣸⣗⡋⣻⡖⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣿⣿⡇⢠⡄⠀⠀⠀⠾⣷⡿⠷⠛⢫⣽⠋⠡⡶⡖⠀⠛⠿⣿⣿⣿⣥⣤⡀⠈⠉⢻⣿⣿⣿⣿⣵⣾⣷⠄⠀⠀⠀⠀⠀⠀⠈
⢸⣿⡿⣰⣿⡇⠀⠀⠀⠈⠛⠿⣿⠿⣿⣯⣤⣤⡍⢵⣶⣿⣿⣿⣿⣿⣿⡿⠟⠙⣛⠋⠛⠋⢻⣾⡿⠟⠛
⣿⣿⣧⣿⣿⠃⠀⠀⠀⠀⢀⠀⠈⠙⢿⣿⣿⣻⣶⣿⣿⣿⣿⡿⠋⠁⠀⣠⣤⣤⣤⡼⣿⣿⡫⢋⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⢿⠇⡀⠀⢸⣿⢿⣽⣿⣿⣿⣿⢿⣿⣶⡶⣾⠟⡓⣿⣫⣿⣿⠋⢀⣈⠛⠃⠀⠀⠀⠀⠀⠀⢰⣾⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡀⠉⠀⠈⣿⠐⠙⣿⣥⣿⣇⣦⢻⠻⠐⡿⣾⣿⣧⣼⢿⣿⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⠀⢿⣰⣿⣿⣿⣿⣿⣷⣼⠀⣰⣿⣿⣿⣿⠀⢘⣿⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⣦⡉⠙⣿⣿⣿⢿⣿⣿⠀⣿⣿⡿⠛⠃⠀⢘⣿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡏⢀⣶⣦⠀⠀⠀⠀⠈⠀⠀⠻⠟⠁⠀⣿⣿⣿⠀⢻⣿⣶⣿⠏⠀⠀⡀⠀⢸⣿⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝
⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣿⠀⢸⡟⠟⠡⣿⣿⡟⠀⠀⠘⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⠿⠿⠀⠘⠇⠀⠀⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠻⣶⡄⠀⠀⠀⠀⠀⣄⠘⣦⣄⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠉⠳⣈⡟⠀⠀⠀⢀⣠⣿⣷⣿⣿⡓⢶⠿⠿⣿⣷⣦⣼⡇⡴⠋⠉⠀⠙⣦
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⡾⠁⠀⢠⣶⣿⣟⣿⠟⢋⡻⣇⠈⠂⠀⠀⠈⠉⠛⢷⡀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢠⡞⠁⣴⣦⡀⠀⣰⠏⠀⠀⢠⣿⢻⣼⠏⡀⢀⠀⢉⠈⠀⢠⡀⠀⠀⠀⠀⠈⢻⣆⠀⠀⣤⣶⡄⢻⡄
⠀⠀⢠⡟⠀⠀⠻⠿⢃⡼⠁⠀⠀⠀⠘⣿⣾⠁⢦⣭⣯⣄⠛⠁⠀⠲⡝⣆⠀⠀⠀⠀⣼⣿⣷⡀⠛⠿⠃⠀⢻⡄
⠀⢀⡾⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠠⢼⡿⠁⡀⠼⠿⠻⢄⠀⠀⠀⠀⢹⡎⠀⠀⢀⣼⣿⠿⣿⣷⡀⠀⠀⠀⠀⢿⡀
⠀⣸⠇⢀⣤⡄⠀⡾⠀⠀⠀⠀⠀⢀⡞⠁⡴⠃⠀⠀⡰⠃⠀⠀⠀⠀⠀⣷⣤⠀⣿⠋⡿⣫⣽⣿⣷⠀⠀⣤⣄⠘⣧
⠀⡿⠀⣸⠈⡇⢰⡇⠀⠀⠀⠀⢠⠎⠠⠎⠀⠀⠖⣴⠁⠀⠀⠀⠀⠀⢀⣟⡟⠠⠃⠚⣽⣿⢿⣿⣿⡇⠀⠏⢸⡀⢻
⢰⡇⠀⣿⠀⠁⢸⡃⠀⠀⠀⢠⠃⢠⠗⠀⠀⠴⠻⣧⡀⠀⠀⠀⠀⢀⣾⣏⠀⠀⣠⠾⢋⣠⣾⣿⣿⡇⠀⠀⢸⡇⢸
⢸⡇⠀⣷⣀⣀⢸⡇⠀⠀⢀⣿⠀⡀⠀⠀⢀⢀⡴⠟⢻⣤⣤⣤⢾⣿⣿⠃⠀⠛⣡⣾⣿⣿⣿⣿⣿⡇⢀⣀⣀⡇⢸
⠘⡇⠀⢻⣿⡿⠀⡇⠀⠀⣾⣷⣜⡇⠠⠖⣉⣊⣤⠶⠛⠉⠉⠀⣸⡿⠃⠀⠀⣰⣿⣿⢿⣿⣿⣿⣿⠁⠸⣿⢿⡇⢸
⠀⣿⠀⢸⡀⠀⠀⢸⡀⠀⠸⣷⠻⢃⡼⢫⡾⠉⠀⠀⠀⠀⠀⣴⡟⢀⣠⣴⠿⣻⠟⠁⣾⣿⣿⣿⡏⠀⠀⠀⣸⠀⣾
⠀⠸⡇⠈⣧⣠⣄⠀⢣⠀⠀⠙⠿⠾⣧⠿⠋⠀⠀⠀⠀⢠⣾⠏⢠⣚⣩⣤⣾⣥⣤⢾⣿⣿⣿⡿⠀⢠⣤⣠⡏⢰⡏
⠀⠀⢻⡄⠘⣿⡿⠆⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣃⣤⣾⠟⣩⣤⣴⣶⣾⣿⡿⣿⡿⠁⠀⠻⢿⡟⢀⡿
⠀⠀⠈⢿⡄⠙⣄⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⣠⣞⣫⣭⣽⣟⣡⢞⣫⣥⣤⣤⣤⣴⣾⡟⠁⠀⠀⢀⡞⢀⣾⠁
⠀⠀⠀⠈⢿⡄⠘⢶⣿⣷⠀⠙⢦⡀⠀⣀⣤⣾⡿⣿⣿⣿⣿⠿⠛⣋⣭⣤⣶⣿⣿⡿⠋⠀⢰⣿⣷⠞⢠⡾⠁
⠀⠀⠀⠀⠀⠻⣆⠈⠻⡉⣠⣤⡀⠙⠿⣿⣿⣶⣶⣶⣤⣤⣤⣶⣿⣿⣿⣿⣿⡿⠋⠀⣠⣤⡀⡹⠋⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⢿⣿⡿⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠋⠁⠀⠀⠸⣿⣿⠏⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣶⡄⠘⠛⠀⣠⣶⣶⣿⣷⣾⣷⢦⣶⠆⠈⠙⠛⠷⠶⠄
⠀⠀⠀⠀⢸⡿⠊⠉⠉⠙⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣦⠾⠿⠛⠃⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⣶⡿⠀⠀⠀⢀⣾⠛⠋⠻⣿⣿⣤⣄⣀⣀⡰⠖⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣶⣤⣴⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⠻⣿⣶⣷⡆
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣠⣤
⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⢀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠲⣦⠀⠀⠀⠀⠸⠆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠛⠛⠀⠂⠙⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠾⠐⠀⠆⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢠⣤⣾⡟⠁⠀⢀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⢤⣄⣠⣀⣀⡀⠀⢀⣠⣤⣶⣦⢀⣤⣶⡿⠟⢱⠟⣉⣅⠀⠀⢀⣼⡿⠛⠋⠀⣐⠕⠖⠚⠗⠾⢿⣿⣿⣿⡿⠿⠿⠿⠻⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠂⠰⠞⠛⠛⠛⠛⠛⠋⠈⠙⠚⠋⠐⠓⠲⠶⡾⠋⠈⠃⠴⠿⠛⠁⠀⠀⠐⠹⠑⠾⠀⣠⣆⡄⠀⠀⠀⠀⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⢂⡄⠀⢀⠀⢀⡀⠒⠀⠛⠒⠀⠀⠀⢒⣀⣀⡀⠀⠀⠀⠀⠖⠂⠉⠁⣠⣩⠰⡆⣶⠶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠁⠀⠉⠀⡒⢀⠐⠀⠀⠸⢛⠛⠋⠋⠁⠁⠀⠀⠴⠒⠋⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⡃⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠦⠍⠹⠧⠥⠬⠄⣀⡀⢀⣀⢀⣶⣀⣤⢤⡶⣿⢸⣗⠀⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⠋⠁⢨⡇⣿⡨⡛⣼⡦⢿⡆⠀⠀⠀⠀⠀⠀⢀⣄⣂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⢹⣿⣿⣿⠽⠉⢿⣷⣿⣇⠖⣿⣇⠸⠷⠀⢀⡀⣀⣤⣤⣴⣿⡧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠀⠤⠄⢀⣀⡀⢀⠀⣀⣴⣴⣻⣯⣭⣶⣶⣿⣿⣿⣿⣟⣹⢟⢰⣴⣾⣿⣿⣿⣿⠿⢿⢺⠓⠚⠿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⡿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠙⣿⠉⣹⠉⠀⣽⠀⣀⡠⠔⠊⣁⡀⠀⠀⡀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠀⣿⡄⠀⣿⣙⠀⣠⣾⣷⣄⢉⠳⠦⣄⣁⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣿⡇⠀⣿⠟⠈⠈⠙⠋⠛⠻⢷⡶⠀⠉⠛⣶⡀⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣿⣟⠀⣿⡟⠀⠀⠀⠀⠀⠀⡦⢹⡆⠖⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠤⣤⠤⢤⠤⠿⠤⠌⠁⢀⣿⣿⠄⠠⠤⠄⠀⠴⠁⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠂⠀⠐⠀⣈⠁⠀⣿⣿⡗⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣤⣤⣤⠬⢻⠁

⠀⠀⡀
⠀⡀⡇⠀⠀⡀⣼⡀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐
⢰⡇⣿⣿⠀⣧⣿⡇⢰⢾⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠛⠿⣿⣀⣿⣿⡇⣼⣾⡆⣷⢀⡇⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⡧⠇⢄⡀
⠀⠀⠀⠀⠓⠉⠛⠓⠿⠿⢯⣿⣤⣧⣾⣠⡆⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⠀⠀⢳⠀⢸⠄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠑⠄⠀⡬⡆⢀⠂⢠⠈
⠀⠀⠆⠠⠒⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⠀⠀⠒⠣⣀⡤⡀⣸⣭⠆⠠
⠐⡆⢰⣛⣒⡀⣤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⠌⢨⠅⣀⠹⣷⣥⣕⠂⢓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡅
⠀⠀⠀⠀⠀⠀⢠⡀⡀⠀⠀⠀⠀⠠⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠐⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠋⠈⠁⠀⠀⠐⣞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢰⣤⡶⠾⠞⠛⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⣼⡭⠇⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣿⡝⠂
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣼⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣿⡀⠹
⠀⠀⠀⠀⠀⠀⡬⠀⢀⠀⠀⡏⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣇⠀⠇
⠀⠀⠀⠀⠀⠀⣵⠂⠀⠀⢸⡇⠀⠀⠀⠀⠈⠙⠒⠠⠤⠀⢀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠟⠀⠈⣷
⠀⠀⠀⠀⠀⢸⣋⠁⡀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠐⠀⠄⠀⠠⠀⠀⠀⠀⠀⣰⠀⠀⣟⡓
⠀⠀⠀⠀⠀⠈⠁⠀⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡈⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡟
⠀⠀⠀⠀⠀⠨⠄⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠⠅⠀⠀⠀⠑⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⡃
⠀⠀⠀⠀⠀⠐⣻⠤⡄⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠀⠀⢰⢇
⠀⠀⠀⠀⠀⠀⠯⠄⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣤⠥⠁
⠀⠀⠀⠀⠀⠀⠰⣠⠄⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⣸⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢱⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡞⣡⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠢⢄⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⡛⣟⣙⠛⣿⢛⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⣿⣿⣛⡛⣿⣫⣽⣿⣿⣿⣿⣿⣿⣿⡿⠿⣟⣻⣧⣶⣍⣽⣿⡟⢿⣿⠛⣿⣿⣿⣿
⠀⠀⠀⠀⣷⣾⣿⣿⣿⣿⢯⣿⣵⣺⣿⣿⣿⣿⣿⣧⡼⢷⣟⡻⡾⣷⡿⣿⣿⡹⢏⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣾⢾⣿⣿⢿⣷⣿⣛⣿⣿⣿⣿⣿⣿⣧⣾⣝⣷⡷⣾⣿⣿⣿⣿⣿⠋⣿⣟⢻⣿
⠀⠀⠀⠀⣿⣿⣳⣿⡿⣷⡿⠯⣽⣿⣿⠿⡫⠲⣾⣯⣭⣽⣗⡾⡻⠿⣿⣿⣷⣿⣿⣿⣷⣾⣿⣿
⠀⠀⠀⠀⠻⠻⠿⢳⣷⣿⣿⣿⣿⡵⢶⠊⢀⠀⠙⢿⢿⣿⣿⠟⠁⠀⠉⡟⠿⠿⠿⠟⢿⣯⡿⣿
⠀⠀⠀⠀⠅⠀⠀⠄⢹⠇⡀⠀⠈⠁⢺⢸⠭⡀⠀⠈⣎⣙⠄⡜⣭⣥⠀⠁⣿⣿⠀⠀⢸⠤⠽⠿
⠀⠀⠀⠀⠄⠀⡀⣀⣸⣀⣀⣀⣀⣀⣉⣘⣚⠃⠤⠬⠤⠤⠤⠧⠧⠴⠤⠴⠠⠀⠄⠀⠚⠒⠂
⠀⠀⠀⠀⣦⣤⠤⠤⠤⠤⠤⠤⠬⠥⠤⠤⣤⣁⣀⣀⣒⣂⣀⡀⡄⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⢀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠑⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠈
⠀⠀⠀⠀⠀⠀⡤⠚⡩⠛⠉⠉⠙⢢⠀⠀⠀⠀⢸⠀⠀⠀⢼⡃⠀⠈⠑⢄⠀⠀⠀⠀⠀⠈⢧⠒
⠀⠀⠀⠀⠀⣸⠀⡘⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⢸⡄⣋⠂⠛⠀⠄⢢⠠⠀⢧⠀⠀⠀⠀⠀⠈⢯
⠀⠀⠀⠀⠀⡗⠀⡷⠶⠆⠀⠀⠀⠀⡇⠀⠀⠀⠸⡇⣄⡸⠀⢘⢀⡓⠀⠁⢸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣤⣗⣾⡥⠜⢴⡂⣤⠄⠃⠱⡒⠚⠛⠋⠡⣍⡝⡏⢨⠉⠢⠈⠀⢸
⠀⠀⠀⠀⠃⠬⠉⠭⠂⠀⠁⠀⠀⠀⠀⠑⠠⠀⠀⠀⠈⠓⢙⡲⣦⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠁⠀⠀⠀⠀⠈⠀⠈⠉⠉⠀⠀⠀⠈⠉⠁⠁⠀⠈

⠀⠀⠀⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⣩⡄⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠠⠀⡀⠀⠀⣤⣀⠈⢸⠘⢻⠛⠟⠛
⠀⠠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠐⠀⠀⠱⡄⢸⣿⡿⠟⢻⡆⠖
⢲⣾⣻⠯⢀⣀⡀⢀⣄⠀⠀⢀⠄⠂⠀⢠⠀⠀⠀⠐⡄⢠⡈⠑⠄⣀⠀⢹⢨⣤⣤⡈⢻⠐⣿⣷⣦
⠘⡇⢨⠀⠫⠀⠀⢹⣿⠀⠀⣠⠀⠀⠴⠋⠀⣾⠀⠀⡅⠀⠁⠀⠀⠻⣤⡀⢢⠀⠙⠑⢾⣠⠐⠈⠁
⠠⠶⠾⠶⠴⣤⡄⠉⡉⠀⠘⣿⠗⠀⠈⠀⣤⣤⣤⣦⡥⣤⣤⣤⣥⣴⣴⣶⣶⣷⣶⣶⣼⡷⠶⠛⢻⣯
⢰⣦⢞⣀⣠⡀⡀⣀⣀⣒⣸⣿⣧⣤⣤⡤⠬⣿⠭⢭⠽⠟⢻⣿⣿⣿⣻⣿⣿⣿⡟⡙⣹⡄⢠⡀⣾⣿
⣀⡶⢚⣻⢲⣿⠌⣱⣿⠟⣿⡿⠿⠿⠃⠀⠀⠠⠴⠫⣷⣖⠶⢿⢿⣿⣽⠟⠉⠈⢰⡟⣿⡥⣿⣄⣿⣿⠄
⠡⠺⠯⠖⠁⠀⠀⠿⠷⠏⠁⠀⠐⠂⢚⡁⢀⣉⡉⠉⡉⠙⠛⠿⠓⡬⣁⠉⠃⠀⣷⡀⣿⣗⠥⣸⣿⣿⣦
⠀⠀⢐⣒⣀⠀⠀⠀⠀⠀⠒⠀⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⠦⠉⠀⠀⢠⣿⠀⣼⠁⠀⣿⣿⣿⣧
⢴⣾⠏⣽⡿⢿⡶⣴⣏⣿⠦⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢟⡃⢹⠅⠈⣻⡿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠀⠀⠀⠃⠂⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣄⣧⣀⣼⡿⠈⡀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⣏⡿⣧⠀⠇⢰⢸⡇⠀⠁⠀⡄⠀⠀⠊⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⣿⣿⣿⡿⣿⡳⡄⢀⠸⣾⡿⠢⠆⠰⡗⠀⠀⠀⠤⠄⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠐⠚⠋⢹⣿⣿⣿⠝⣧⣼⣾⣿⣧⡀⠃⠀⡇⣤⠙⠖⣿⣾⢀⣯⢡
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢘⢻⣬⣿⣿⢿⣦⣭⣿⣿⡿⢿⣿⢀⢀⡷⢻⣎⣖⣉⣀⣀⣇⣠⣀
⠀⠀⠀⠀⠀⠀⢘⠄⠀⠀⠀⣀⠀⠸⠛⠛⢑⣿⣿⣿⣿⣿⣷⠿⢿⣿⡟⠟⢻⡏⠈⣼⡇⠉⣹⣻⠏⠁⢹⡿⠧⠤⠴⠖
⠀⠀⠰⡄⢀⠄⡌⠂⠀⠀⢈⡷⠀⠀⢰⡶⣁⣾⡛⠟⠃⠻⠼⠁⠘⠋⠃⠱⣆⠁⠀⠻⠀⠈⢹⣟⢤⣾⠐⡧⠤⠄⠶⣶
⠀⠀⢀⣷⡎⣴⣇⠀⣄⢠⡟⣠⣤⢔⡿⠃⠛⣛⣧⡴⠃⠄⢦⠀⢦⡄⠀⠀⠉⠀⠐⠘⠀⠂⠐⠀⠀⠃⠀⡷⠴⠃⠐⠂
⣇⠀⢈⣿⠳⣾⣯⣾⡏⣸⡶⠟⠋⣹⡵⠖⠛⣩⠏⠀⣀⠀⢀⠀⠀⠀⠤⠀⠠⠀⢀⣠⠀⡤⢶⠖⢸⠇⠘⡗⠂⠀⠐⢶⣶⠦⢤⡄
⡟⠀⣼⠸⣷⣿⣿⣵⡿⣿⠗⢋⣸⣇⣤⡾⠿⢓⣦⡂⠁⠀⡀⠀⢀⣀⡀⠀⠀⠀⠀⣀⠀⠇⠀⡄⠀⠀⠀⡇⠀⠀⠀⠀⠁
⠺⢀⡷⣠⣿⣾⢣⠟⣤⣾⢒⣿⠿⢋⠝⠀⣀⣾⣏⠉⢀⣈⣠⣶⡿⠤⠍⠒⠒⢀⠀⣿⡿⠀⠼⠀⠀⢀⣰⡇⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢺⣷⣿⣿⣵⣾⠟⣱⣯⣿⣷⠢⣈⣠⣾⣿⣻⣿⡟⢻⣭⣯⣉⣙⣛⣏⡛⠉⣵⣷⡿⠁⠀⠀⠀⠀⠘⠀⠀⠀⢀⠾⡟⢷⣄⢀⣤⣾⣿⣦⣄⡈
⠀⢸⡿⡳⣿⣿⠕⣫⣾⣿⣿⣿⣿⣿⣻⣟⣧⣽⣦⣔⣦⣭⣽⣿⣿⣽⠿⣶⣾⣿⣿⣧⣀⣀⣀⣤⣠⣤⣤⣄⣠⢋⡀⡃⠈⠻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⢠⣿⢀⣷⡿⣫⣾⢿⣻⡟⢨⠿⠛⢾⣅⣹⣿⣯⣿⠿⢶⣟⡛⠖⠙⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣒⡀⡠⠆⣿⢿⡿⡿⠟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢉⣿⣾⠑⣿⣏⠻⠞⣿⣿⠛⠒⠓⢸⡍⡉⣉⣻⣟⣛⢷⣁⢈⣳⣦⣬⣭⣿⣿⣻⣿⣿⣿⣿⣿⣿⣛⣿⣻⣳⣿⣾⡓⣗⣿⣾⣾⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⣿⣿⠿⠿⠾⠶⠾⠶⣶⣶⣾⣷⣾⡦⠤⡿⢷⡤⠼⢶⣶⣦⣤⣯⣿⢷⡿⠧⠧⠿⠿⠿⠿⢿⣿⣿⠽⠿⢿⣭⣿⣿⣻⣙⣛⢹⣟⣓⠻⠺⠺⠗⠿⠫
⠀⠀⠈⠙⠉⠉⠁⠀⠀⠀⠀⠁⠂⠀⠜⠃⠒⠖⢙⢤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠉⠛⠛⠛⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠁⠀⠀⢠⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠁⠀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣀
⠀⠀⠀⡀⠀⠴⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣧⢄⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣚⣳⣛⣝⣱⣫⣬⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⢀⡀⡀⣀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣤⣼⣢⠞⠀⠀⠘⢦⣸⠃⠀⠀⠀⠹⡄⠀⡄⠀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⣀⣻⡀⠈⢿⡀⠀⢀⣠⡖⢷⡄⢻⠞⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⠈⡏⠉⣩⠍⠻⣄⠀⣷⠞⠋⠳⢤⠀⢳⣈⡆⠀⠀⢸⣀⣠⠀⠀⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣷⠚⠁⠀⠀⠙⢾⡇⠀⡀⠀⣄⠀⠈⣿⣀⣤⠶⠛⠙⠀⠁⠀⠀⢀⡤⠚⠻⣝⢩⠻⡏⣏⡙⠟⢦⣀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠄⠀⠀⠀⠀⣿⡤⠞⠁⠀⠘⣇⣸⠋⠁⠖⡄⠀⡃⠀⠀⠀⡴⢋⣀⣀⠀⠈⢻⣴⣬⣤⣧⣷⣧⣽⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⣇⣤⣦⣶⣠⣏⣀⣀⣠⣶⠀⣸⠏⠀⡄⠡⣧⣤⢶⡤⣄⣼⡇⣯⢉⢹⡇⠀⠀⡇⣶⢶⠲⣆⠀⠀⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣬⣤⣤⣤⣤⣤⣥⣄⣉⣉⠉⠉⠛⠛⠛⠛⠛⢳⠲⠶⠷⠦⠼⣯⣍⣇⣹⡈⠀⡇⠀⠀⡇⠿⠾⠴⠟⠀⠀⣿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠙⢓⠲⠷⠦⣤⣤⣀⡀⠀⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡶⠶⠶⠶⠴⠦⠶⠤⠤⣤⣤⣄⣤⣀⣀⡀⠠⢀⠀⠀⠈⠉⠙⠻⢦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠳⣌⠠⠀⠀⠀⢀⠀⠈⠉⠉⠉⠙⠛⡓⠲⠦⠤⢤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⡞⠳⣬⡛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣶⣶⠓⠚⡛⠓⠓⠺⠶⡶⢴⣤⣄⡀⠀⢂⠌⡃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠃⠀⢸⡳⣄⡞⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡌⠁⣀⡀⠁⡀⢀⠈⡀⣵⠟⢳⡄⢍⡓⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣞⣯⣹⢦⡀⠈⢇⣸⢉⡽⠋⠀⠀⢀⣀⣠⣤⣤⣤⡄⠀⠠⠀⠙⠲⢤⣄⠠⠀⣀⠀⣀⠙⢳⡟⠂⠠⡸⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡈⢹⡃⡴⠛⣿⣾⡟⣭⡄⢀⡤⢾⡏⠸⣇⡼⠿⡿⠁⠀⠀⠁⠀⢰⡟⠻⡟⢦⣠⣄⢀⡀⣹⡏⣈⡴⢦⣴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⣟⠁⡴⢻⠍⣿⡿⣠⢾⡇⢸⣧⣞⡉⣩⠟⠁⠀⠀⠀⠀⠀⠈⡿⠛⠁⠈⣧⡶⣄⣤⣼⣏⣉⠷⣴⠛⠖⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡙⠲⢿⠞⠁⡼⠀⢸⣠⢿⣠⠿⢯⣤⡤⠛⠁⠀⠄⠀⠀⡀⠀⠀⢰⣇⣤⡀⠀⢸⣿⠉⡄⢼⡇⢉⣨⣏⢉⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠲⢬⣇⠼⡁⠘⢿⡿⠿⠛⠛⠉⠁⠀⠀⢐⠀⠀⣻⡟⠛⢦⠀⣸⢿⣬⡿⠀⢸⣿⠤⠴⢾⡇⣤⠼⣯⢠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢴⣤⣿⣾⠷⣦⣄⣻⣠⣤⢀⣄⡀⠂⠈⡆⠈⠳⢹⣷⣊⡼⠀⢻⠀⣿⠀⠀⢸⢿⠄⠰⢸⡧⠄⠄⡧⠤⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠈⠹⡇⠀⠀⠉⠻⣏⣿⠟⠋⢠⠀⢀⡗⠀⢠⠏⠠⢻⠀⠀⢸⠀⢸⠀⠀⢸⢸⠖⠒⢺⡗⠶⠒⡷⠲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣁⣀⡀⡠⣠⡰⣠⣮⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⠀⠀⠀⡀⡄⡆⣯⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢣⢌⢄⢌⢏⢜⣼⢫⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡉⡜⣶⡄⢃⢣⢡⢠⠸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡛⣷⣱⡳⢢⢢⣿⢇⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣱⠘⢿⡄⠄⡄⡎⣧⣿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠻⣿⣽⣻⢻⢋⣿⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣜⣻⢳⡟⣟⣯⣿⡗⡝⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⣻⣟⡻⣻⣝⣿⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡙⢿⠟⠛⠙⢀⣤⣤⣀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⡸⠀⢀⣴⡻⡿⢻⠃⠈⣽⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡕⣄⠀⠸⣷⡽⢟⡓⣄⠐⡄
⠀⠀⠀⠀⠀⠀⢠⠃⢠⣿⣿⣧⢀⠃⠀⡜⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣎⢣⠀⢻⣶⡿⣿⣎⣷⣿⡄
⠀⠀⠀⠀⠀⠀⡎⢰⣿⣿⠋⡹⡏⠀⣸⣯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢃⠀⢿⣧⠈⣿⢽⣏⣿⡀
⠀⠀⠀⠀⠀⢸⠀⣿⣿⠛⠊⢸⠀⢀⣏⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⣯⡾⣟⠛⣯⡛⣿⡱⣇
⠀⠀⠀⠀⠀⠇⢸⣿⡇⠀⢀⠇⠀⣸⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⣟⣿⣿⠐⢬⣽⣿⣿
⠀⠀⠀⠀⡸⠀⣿⣽⡿⢧⠸⠠⢢⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣾⣿⣽⢿⣿⣿⡏⠹⡇
⠀⠀⠀⠀⠇⣰⣿⣿⣇⣼⣿⠃⣽⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠉⢹⣿⠀⢹⡇⢡⠀⢃
⠀⠀⠀⢸⣿⣿⣯⣿⠿⣷⣿⢪⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢸⣿⠶⠟⠁⢸⠀⢸
⠀⠀⠀⣾⣿⣿⣿⣿⢛⣻⣿⡷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣟⡁⠀⠀⢸⠀⢸
⠀⠀⠀⡟⠻⣿⣿⣾⣯⢹⠋⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⣾⡅⡀⠀⠀⢸⠀⢸
⠀⠀⠀⡇⠀⣯⣿⣏⢸⠸⡆⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⢠⣿⠾⣿⣦⠀⡎⠀⡇
⠀⠀⠀⡇⠀⢻⡿⠻⠋⠀⡇⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⣼⡟⠀⣿⠟⢰⠁⡰
⠀⠀⠀⡇⠀⢸⣿⡅⠀⠀⢱⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⣼⢟⠷⠞⠋⢠⣿⢿⠃
⠀⠀⠀⢠⠀⠈⣿⣦⣤⣄⠀⡆⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⡾⣯⡁⠀⠀⢀⣾⢿⠇
⠀⠀⠀⢸⡄⠀⢹⣿⣿⠈⠳⠘⣄⡤⠈⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⢀⣴⡿⠞⢹⣿⠆⣠⣟⣥⡟
⠀⠀⠀⠈⡷⡀⠀⠻⣿⣦⡰⠀⠘⡔⢷⠅⠈⡹⣶⣤⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠴⠒⠋⠁⠀⠀⣄⣴⡟⢿⣁⡴⣫⣿⣼⢣⢨⡿
⠀⠀⠀⠀⠰⡙⣄⠀⠉⠿⣶⣀⠀⠙⡄⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡠⠠⢂⣾⣿⣟⠭⠛⣀⣬⡿⡻⣽⠃⣺⡿⠁
⠀⠀⠀⠀⠀⠱⡈⢢⡀⠀⠀⠙⠛⠒⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡴⡠⠢⢊⢌⠔⡕⠝⡑⡻⡻⡟⣫⡻⠝⡰⡍⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠘⢶⣙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠜⠕⡡⠊⢚⠔⡑⡡⠪⢪⢎⠌⠜⢔⡔⡴⢕⠎⡜⡡⢟⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⢲⢤⣀⡀⡠⡦⡰⢁⠔⡀⢀⠎⠔⡡⢂⠆⠔⡡⢊⢊⠔⠰⡑⡡⢢⢊⢎⢎⢞⠎⢈⡴⢊⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣸⡟⡿⣼⣠⣁⠊⡔⡡⢊⠜⡀⠁⢂⠔⠔⡡⢁⢎⠔⡐⡡⢣⢢⢃⣧⡼⣞⣧⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠧⣧⣏⣛⢻⢳⢷⢾⢴⣥⢧⣮⣼⣴⣡⣮⢾⠼⠷⢛⣛⣭⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⡶⠶⣖⠲⠖⣒⠶⢶⡿⢿⡿⣾⣼⣶⡼⡼⣼⣼⣼⣼⣾⡾⢾⣿⡭⠥⣶⠤⠶⣶⠤⢶⡶⠶⠖⢒⣀⠶⠶⣦⣤⠤⣤⡤⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠈⠀⠀⠀⣹⡀⠀⠀⠀⣷⠀⠀⣴⣀⣿⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⣠⠶⠛⠉⠁⠀⡀⠀⣿⣷⢠⡿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠸⣇⠀⣿⣿⡾⢡⡏⡿⠈⠙⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⡄⠀⢀⠔⠀⠐⠂⠐
⡄⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢁⣿⣿⠃⠀⠁⠈⢈⡀⢠⣤⣤⣤⣧⡔⠁⢸⠋⡴⠁⡀⢸⡇
⠂⠀⠀⠀⠀⠄⠀⠀⠈⣿⢾⣼⣻⡏⣠⣴⡆⣪⣶⣗⠉⣉⠛⣯⣝⠳⢄⣾⢉⣁⣠⢀⠏⠈⠐⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⣹⣿⣿⣿⣿⣿⡟⠟⠛⠛⠒⠛⠛⢯⡛⠓⣾⢿⣍⣀⣠⠟⠀⡀⠀⠀⠀⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟⠁⢸⡀⠀⠀⠀⠀⠀⠹⢾⣃⠘⠓⡾⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠋⠀⠁⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢈⡄⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣄⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⠀⠀⠀⢸⠁
⠛⠒⠀⠘⠀⠼⢹⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠄
⡀⠀⢠⠀⢀⡀⣾⣿⣿⣿⠀⡀⢠⠈⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠙⣶⠞⢣⠸⡇⡸⠋⠉⠙⣨⣡⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣆⠀⠀⢠⠶⠆⡒⣂⠀⢀⠀⣔⢉⠰
⡷⠛⣿⡾⢻⣶⣧⡄⣠⠃⡟⢁⡶⠂⠀⢠⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠚⠀⠌⠘⠛⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⡐⠋⠀⠀⠉⠘⠻⡟⠛⠂
⣁⣤⣯⣧⣾⣿⣾⡖⣹⣰⢴⠋⠀⠀⠀⠀⢙⣆⠙⣄⣀⣀⣀⣀⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣈⣡⣄⣈⢀⣀⣀⣄⣀⡀⢀⣄⣀⠀⣀
⠉⡁⣿⢹⠗⣿⣷⣧⣽⣿⡏⣉⡔⠁⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠤⢭⣝⣯⡙⠛⣛⠛⠛⠛⠧⠷
⠀⢰⢻⢸⡄⣿⠂⣿⢟⣸⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠘⡇⠸⣿⠃⠈⣯⣌⠃⠖⡾
⠲⣶⣼⣆⢻⢠⣾⡟⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⣼⣾⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡠⠆
⣽⠿⠿⢿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠁⠈
⠀⠀⠀⠘⣋⢩⣿⡛⣯⠈⠙⢢⣴⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠⠮⠂⣴⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⡀⠀⠐⢒⠉⢤⣼⣿⣾⣷⣽⣿⡯⠭⢭⠹⣿⣿⡂⢀⣴⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⠋⠀⠀⠿⠤⠄⠠⠀⠘⠀⠀⠀⢸⣻⡅⢀⣀⠺⠄
⡄⠔⠀⠀⠀⡀⢭⣥⣥⣨⣿⣷⣤⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣠⢆⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡
⣄⣸⢀⣶⣭⣀⡨⠙⣻⣿⣯⣿⣇⣠⣼⣶⣾⠋⢻⠿⠿⢟⣿⣻⣿⡿⢿⣿⣿⣿⠋⣩⡿⠉⠀⠀⡀⢀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠁
⣯⢿⣿⣿⣿⣿⣿⣿⣾⠿⠻⠛⠙⣿⣾⣿⡿⣿⣿⠤⣴⢿⣭⣭⣿⣿⣿⣿⣿⣿⠀⠟⢿⣧⣤⠛⠉⠙⠺⠇⠀⠀⢈⣤⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣟⣿⣿⣿⢿⣿⠋⡭⣄⠀⣀⣀⣼⣿⣿⣍⣟⠉⣃⡾⢣⣴⣶⠶⣶⡿⣿⣿⡏⠉⠉⠀⠀⠻⡇⠀⠠⣤⣤⢆⠀⠀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⣀⣀
⣿⣿⣿⣷⠿⠁⢀⣀⣙⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⣾⡿⠛⠛⠛⠻⣿⣿⣿⣠⣤⣤⣤⣀⣀⣼⣶⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠙⠛⠿⠿
⣻⣿⣟⣱⣟⠻⣧⡄⠀⠀⢈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡶⠒⠒⢛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⣠⣤⣤⣤⣤⣴⣶⣟⠋⠉⠀⢄⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡷⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⢉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⢸⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣧⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⣼⣿⡟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢿⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣽⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⠠⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣾⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠳⣼⣿⣦⣀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⡴⣛⡵⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠿⠷⠀⢀⣤⣶⣶⣶⣶⣶⣶⡿⠷⡀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣬⣳⣦⣤⣤⣤⣤⣤⣶⣶⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣀⠤⠂⠊
⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⡤⠋⠀⢀⠴⠃
⠀⠀⠹⠘⠀⠀⠀⠀⠀⠀⠀⢀⣾⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠰⡀⠀⠀⠀⠀⡀⣠⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠒⠒⢦⡄⠷⡀⠀⠀⠀⡸⣿⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁
⠀⠀⠀⠙⢷⣄⠀⠀⢸⣿⡏⠀⠙⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡁⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠙⣷⣾⣿⣹⠃⠀⠘⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡅⠀⢀⣴⡆⠀⠛⠛⢧⡀⠱⣄⠀⠀⠀⢀⣴⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣯⡃⠀⢊⡥⠐⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢰⣿⠟⠉⠀⠀⠈⢹⣿⠄⠀⠀⠀⠉⠀⣠⠀⠀⠀⠀⠋⠘⠁
⠀⠀⠀⠀⠀⠀⣼⣿⡟⢡⠖⠉⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠻⡏⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠺⡷
⠀⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⣸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⣰⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈⠉⠉⡛⠛⠲⠶⢤⣄⣀⡀
⠀⠀⣰⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠠⠄⣀⠉⠛⠛⠳⠾⠷⠶⠒⠒
⠀⣰⣿⣿⣿⣍⢀⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣴⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣤⣤⣀⡀
⠿⠿⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣦⣤⣄
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿
⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠙⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⡄⠀⠀⡤⠤⠄⢀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠐⠀⠀⠀⠀⠀⠷⠄⠄⠉⢀⠠⠀⢈⣄⠈
⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⡂⠀⢀⢀⠀⡀⠀⠀⠀⠀⢀⡈⠆⠜⠆⡨⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⣰⡛⢻⣳⢛⣡⠀⠀⠀⠘⠒⢲⣀⠁⠠⠞⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢠⣾⣄⣀⣈⣣⠀⠀⠴⠥⠿⠺⠚⠒⠓⠆⠀⢰⡈⠉⠈⠙⢄⡀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⠀⣆⡀⠀⢄⠄
⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡃⠈⠃⠁⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡍⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢠⠤⠀⠈⠀⠠⠉⠀⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁
⠀⠂⠄⠃⠁⠀⠀⠈⠁⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠈⠀⠁
⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡄⢡⠀⡆⠀⠤⠄⠀⠰⠆⠁⠀⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠔⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠈⢊⠺
⠀⠀⢠⠀⠐⠃⠀⡀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠈⠀⣤⠦⠀⠐⠀⠀⠀⠀⠀⠀⠠⠤⠖⠂⠀⠀⠈⠀⠐⠐
⠀⢀⠰⠤⠘⠀⠀⠄⠀⠐⠁⠀⠀⠀⢰⠀⢀⣲⡋⠉⠀⠀⠀⣠⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣈⣀⠉⢀⡀⢀⡔
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣈⣀⣀⣉⣈⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠙⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⣤⠛⠙⣿⣄⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣶⣤⡤⠀⡀⣠⣿⣿⣿⠋⠀⡀⠀⣠⣴⣞⠁⡀⠀⠀⠙⣳⣶⣿⣿⣿⠷⠦⠀⠸⠇⣷⣤⡄
⠀⠀⠀⠤⠈⣋⣾⣿⣯⣿⣿⣿⣿⡟⣿⣿⣶⠿⢛⡿⠟⠁⢤⣾⣿⣿⣿⣿⣯⠱⠶⢶⣄⠀⢀⠈⠁⠀⠄
⠀⠀⠀⠰⣾⣯⣿⣿⣾⣾⣿⣿⣿⣷⣿⣿⣿⣷⣾⣷⣾⣾⡆⠉⢻⣽⣿⣿⠿⠁⠀⢀⣨⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣟⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠻⠿⡿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⡛⠀⠂⢀⣀⣠⣬⣯⣭⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣶⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⠟⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣦⣍⡛⠿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿⢛⣩⣴⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣦⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣴⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠁⠈⢻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⠁⠈⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣤⣤⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⣤⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠛⠻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠛⠛⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣆⡀⣀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣀⣀⣰⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠁⠉⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⣴⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣦⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⡀⠀⣠⣠⣶⣻⣽⣿⣷⣀⢺⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢺⣿⣾⣾⣉⣟⣹⣿⣧⠙⢿⠫⠈⣵⡿⣶⡂⠴⠶⠶⠶⠶⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠰⢤⡀⠀⠀⠀⠀⡀⢤⣮⡧⣍⢛⡉⠀⠉⠉⠁⢙⡿⠛⡛⣱⣢⣸⠋⠱⠒⢀⡀⠀⢤⡴⠬⠿⠄
⠀⠀⠀⠆⢀⠰⠀⠛⢻⢷⡀⢨⠗⠚⢻⣿⣿⣷⣄⡒⣢⡠⣽⢃⡛⢿⡇⠐⠀⠐⠂⢿⣴⠀⠀⠉⣻⣿⠀⠷⠀⠀⠀⠆⡀⠐⢂⣠⣄⡀
⠀⠀⠀⠒⢦⡌⠉⠁⠀⠘⠛⠀⠘⠄⠁⠀⠀⠑⠛⠛⠓⠛⠋⠈⠃⠈⠙⠀⠀⠁⠀⠉⢉⠑⠂⠁⠈⠙⣰⣴⡞⠽⠶⡾⣿⣶⣿⣿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠠⠀⠨⠀⠀⠀⠀⢀⡤⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠁⠀⠠⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠠⠀⠀⢴⠦⠀⠀⠺⠖⠒⠀⠀⠴⠀⢠⡀⠀⠀⣠⡀
⠀⠀⠐⠆⠷⣾⠇⢶⢾⠞⠠⠇⣾⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⢀⣄⡀⡐⡀⢠⡄⢦⠴⠂⢮⣄⡄⠶⣀⡄⢤⢰⣸⠈⢀⡅
⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⠓⠈⠁⠃⠀⠘⠀⠀⠁⠈⠀⠀⠉⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡾⠿⠻⠛⠻⠉⠉⠉⠁⠁⠉⠙⠋⠋⢻⡿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠐⠛⣻⣠⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠁⡿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣀⢀⣀⣒⣶⣶⡦⢀⢀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣠⣴⣦⣦⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⡦⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⢈⣿⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⣀⠀⠐⠟⠻⠀⠀⠀⣼⣶⣦⡄⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⢿⢿⣿⣷⣾⢿⣿⣻⡻⠈⠁⠘⠛⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣟⢿⣽⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣧⣸⠁⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣳⣿⡡⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠏⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⢤⡄⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡘⠁⠀⢀⣠⣴⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⠷⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀⠀⣀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⢾⠞⠁⣄⠙⣾⠁⠀⠀⣹⡀⣆⣠⠂⣤
⠀⠀⠀⠀⠀⢠⣀⠘⡟⢛⡛⣦⠈⣧⠴⠯⡁⢣⣸⠄⠀⣿⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠓⡷⠋⠀⠈⢳⠇⣀⠀⣄⢀⣟⡤⠖⠋⠉⠀⠀⢀⠔⠛⢯⡉⠛⢋⢛⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣀⡀⣼⠚⠁⣀⠘⣾⠃⠀⣁⣀⡀⠀⣰⢡⠤⢦⠀⢹⢛⣛⣛⠛⠛⡇⡀
⠀⠀⠀⠀⠀⢘⣩⣉⣩⣭⣉⣉⣛⣙⠛⠛⠒⠳⠞⠧⠭⠿⠭⣿⣻⣸⡀⣸⠀⢸⠻⠼⠼⠀⠀⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠲⠦⠤⣀⡟⡇
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡉⠉⠉⠉⠉⠉⠙⠓⠓⠒⠲⠤⠤⢤⣄⣀⠉⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⠤⣤⣀⣀⣀⡀⠀⠀⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢄⣴⢻⣯⣇⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀⣠⣄⣄⣈⣉⢉⠉⠒⠒⠦⠆
⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⡾⢌⡟⣚⡇⠀⠀⠀⠀⠀⠀⠙⢿⡛⣁⠐⠀⠀⠀⢸⡿⣯⠓⠦⣄⡂
⠀⠀⠀⠀⠀⠀⢀⡾⣯⠗⣄⡘⣼⡩⠏⠀⣀⣤⡤⣤⣴⡄⠀⠀⠙⢲⣤⡀⠀⠀⠌⣳⡋⠠⠝⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠦⢼⠜⢠⠿⢯⡾⣠⡞⢹⣠⠷⢯⠟⠀⠀⠀⠀⢿⡽⠋⣶⣂⠂⢺⡷⢾⣻⡇⡇
⠀⠀⠀⠀⠀⠀⠸⡓⠤⡗⢁⡆⢸⣱⣇⠷⣯⡭⠟⠁⠀⠀⣀⠀⠀⣮⣄⡀⢸⣿⠉⢽⡇⣈⣇⡲⡇
⠀⠀⠀⠀⠀⢀⠀⠈⠓⢷⣪⣄⠙⡟⠋⠉⠁⠀⢀⠀⡀⣿⡋⢳⢠⡟⡾⠃⢸⣷⠐⢺⡒⠚⡇⠤⡇
⠀⠀⠀⠀⠀⢸⠠⠀⠙⠛⡏⠉⠻⢾⣿⡶⠗⡀⢈⡅⢈⡝⢻⠋⠀⡇⢻⠀⢸⢿⠤⢼⠥⠤⡇⡁⡇
⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠁⠀⠀⠈⠉⠀⠈⠉⠈⠁⠉⠉⠉⠁⠀⠁⠉⠀⠈⠉⠀⠈⠀⠈⠁⠀⠁

⠀⠀⠀⠀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣦
⠀⠀⠠⣱⣿⣿⣿⣿⣿⣶⣦⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢶⣾⣿⣿⣿⣿⣿⣯⠄
⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⢣⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡈⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠺⣿⣿⣿⣿⠿⣿⢃⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡈⠻⠿⣛⣻⣽⡿⠖
⠀⠀⠀⠀⠈⠙⠛⠻⠶⢾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠷⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⢠⠞⠛⠛⢶⡄⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢠⡶⠛⠛⠳⣄
⢸⠀⠀⡿⢠⠖⠂⢰⡇⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠸⡆⠰⠲⣆⢿⡀
⡌⠀⢸⡇⠘⠓⠃⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠘⠚⠋⢹⡇
⡇⠀⢸⡃⠀⠂⠠⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡀⠀⠀⠐⣇
⡇⠀⢸⢦⡴⠶⠸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⠇⠴⢦⠀⡿
⣇⠀⢸⡇⠛⠒⠀⣧⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣸⠀⠒⠛⢰⡇
⢿⠀⠘⣇⠀⠀⡀⢹⡀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⢀⡟⠀⠀⠀⣸⠃
⠸⡆⠀⢹⡀⢠⣤⡞⣧⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⡼⢡⣤⡐⢠⡟⠀⢰
⠀⢷⠀⠀⢷⡸⠦⠄⠘⣇⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠀⡰⠃⠠⡴⢃⡿⠁⠀⡾
⠀⠘⣧⠀⠈⢷⡀⠀⠀⠈⠧⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⢀⡜⠁⠀⠀⢀⡾⠁⠀⣸⠃
⠀⠀⠘⣆⠀⠀⠳⡄⢡⡞⠓⠑⢄⠀⠉⠻⢯⣗⣶⠤⣤⣤⣤⡤⠤⣶⣾⡽⠟⠉⠀⠀⠈⠚⢳⡀⢠⠞⠀⠀⣰⠇
⠀⠀⠀⠘⣦⠀⠀⠈⠦⡙⠋⢀⠐⠀⠀⠀⣀⣠⣍⣉⣛⣛⣛⣛⣉⣩⣄⣀⠀⠀⠀⠡⠀⢉⣋⡴⠁⠀⢀⣴⠃
⠀⠀⠀⠀⠈⢷⣄⠀⠀⠈⠈⠀⠀⠀⠀⣾⡿⠛⠛⠻⠿⠿⠿⠿⠛⠛⠛⢻⣷⠀⠀⠀⠀⠉⠁⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⣿⡇⠀⠀⠀⠀⠀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠘⢯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡽⠃⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣄⣀⠀⠉⠉⠀⠀⠀⠀⠀⠂⠈⠉⠁⣀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠾⠧⣽⣽⣿⣿⣿⣿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⡾⢛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⠋⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⢲⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⢹⣾⣿⣿⣷⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⣼⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⡀
⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡀
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣧⣀⠉⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣦⣀⣠⡄⠀⠀⡀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣇⣠⡄⠊⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣉⣉⢛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠿⣿⡿⠿⠿⠟⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⡿⠛⠋⢛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠄⠀⠀⠀⠐⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢻⡇⠀⠀⠀⡀⣀⣤⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣥⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⡇⣸⣤⣤⣶⣾⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣶⣤⣴⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⡀⠀⠐⡀⠀⠀⠀⠀⠈⠁⢰⠀⠀⡀⠀⠐⠼⠁⠀⠀⠀⠀⢀⠂
⠀⠈⠄⠀⠰⡄⠀⠀⠁⢀⡤⠀⠀⠀⠀⠀⠀⢸⡄⡞⠀⠀⠀⠀⢆⢤⠀⠐⠀⠆⠐
⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢀⣦⠘⢄⠒⠲⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⡇⡄⠀⠀⠀⠊⠁⠔⠂⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⠀⠐⠀⡀⠀⠀⠈
⠀⠀⠀⡀⠸⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠥⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⡀⠠⠀⠀⡀⠀⢀⠖⠀⠊⢀⠁
⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠄⠀⠀⠀⠀⡦⠃⠀⢠⠀⡇⢀⠀⢰⡁⠀⢸⠀⡰⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠱⠠⠀⠀⠘⠁
⠀⠀⠤⠀⡤⠤⠒⡠⠔⠀⠀⠀⠀⠂⠀⡀⢀⡒⢦⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠀⠤⠠⠠⣄
⠐⢀⣠⡀⠄⠀⠀⡀⠀⠀⣀⠐⠁⢀⠔⠁⠠⠉⠋⠁⠀⠀⠀⠀⠀⢀⣻⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠐⠰⠲
⠀⠘⠛⠉⠀⠈⠀⠀⠉⠁⠙⠶⠛⠛⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠈⠢⣄
⠀⠀⠀⠀⠀⠤⢠⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⠀⠀⠀⠁
⠀⡄⢠⠀⠀⠀⢐⡀⣠⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡾⠋
⠀⠻⠶⠘⡆⠀⠀⡼⠘⠛⠾⠧⠢⠀⠠⠄⠀⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠄⠀⠘⠂⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢀⠀⢀⢡⢀⠀⠀⠀⠀⣠⠀⠀⠠⠶⠒⠒⠒⢦⣄⣀⠀⠀⠀⣦⠀⠀⢠⡄⠀⢷⠀⠀⠀⠀⠀⡀
⢸⡇⢸⢿⣼⣄⣾⠰⡇⢻⢸⣶⣄⣀⡀⡀⠲⠶⠉⠉⠁⠀⠀⢿⢿⠄⢸⡇⡆⢸⠀⠀⠀⠠⠀⠀⠀⠃
⠸⠇⠈⠸⠃⠸⠀⠀⠁⠈⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣠⣀⣀⣀⣄⣀⣀⣀⣤⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠈⣤⣤⠤⣤⡀
⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠞⠉⠠⠤⢈⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠆⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠁⠐⡟⠐⠠⠀⢼⡳⠽
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⡀⠱⠀⠁⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⣀⠈⢀⠀⠁⠻⠗⠂
⠸⠇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡙⠀⠀⠆⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⡀⣁⠀⠀⠀⠀⢀
⠘⢋⣍⠁⠀⠀⠄⠀⡀⠀⠀⡆⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠄⠈⠉
⠐⣋⡉⣇⠀⠀⠀⠸⠉⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠯⣀⠠⡄⠠⠀⠊⠀⡀⢦⠤⠤⠄⠀⠀⡀⢀⠀⢀⠀⡀⠀⠄⠀⠀⡀⠀⢀⠀⢆⡀⠀⣄
⠸⠟⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢄⠈⠀⠐⠂⠐⠐⠃⠈⠉⠈⠁⣄⠛⠉⣿⠁⡡⢀⣦⣙⠁⡬⣜⠒⠀⣬⠀⠀⡇⠀⣪⠀⢀⠀⠀⡇
⢀⣀⡠⠀⠀⠀⠀⠀⠄⠌⠀⠀⣀⠀⠀⠉⣄⡤⠘⠋⠀⢠⣶⠀⢀⠀⠀⠀⠀⠀⠰⠗⠚⠀⠀⠀⣤⠖⡗⠻⠂⠐⠀⡧⠀⢸⠀⠐⠂⠀⢈⠀⠰⠂⠀⡀
⠋⠀⠀⠀⠀⠀⣀⣀⠄⠀⠀⠀⠕⠠⠀⠀⠙⠀⠂⠆⠠⠈⠉⠀⠘⠀⠀⠀⠀⠀⠀⣶⠘⠆⢀⡀⠀⠀⣃⢸⡆⢀⠀⢖⠀⠀⡂⠠⠦⠀⠈⠀⠀⠆⠀⠁
⠀⠀⠐⠁⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⢴⠨⠄⠈⠀⠄⠀⠋⢸⡇⠐⠀⠐⠉⢠⡇⠀⠀⠀⠀⠀⠀⠇⠀⠂
⠀⠀⠀⠀⠀⠄⢠⠀⢱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⢸⣅⡄⠀⠀⠀⠀⢀⢸⡆⠐⠀⡠⡀⠐⠁⢠⠄⠀⠈⠀⠀⠀⠤⠤⠃
⢀⠀⠀⠀⢂⡈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠈⠦⠄⢠⠏⢠⡤⠐⠃⠀⠀⣦⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠄⠀⠀⠀⠐⠉⠁
⠠⠳⠄⠀⠠⠂⣀⠀⠀⠀⠀⢀⠀⠀⡀⡐⣀⢠⠤⠖⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⠤⠐⠂⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⡟⠀⠀⡰⢤⠠⠞⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢠⣀⡀⠠⠔⠒⠚⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣠⣤⣀⣀⣀⣀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠁⠈⠉⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠀⠈⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢶⡀⠀⠀⠀⢸⠟⠀⠀⢳⣿⣿⣿⣦⡀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⣾⣿⣦⠊⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⡀⠀⢶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠠⣤⣹⣿⣿⡆⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⢰⡀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣶⠖⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠁⢀⠀⠠⢈⣷⣄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣿⣇⠀⠐⣿
⣸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠞⠃⣾⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡧⠀⠀⠀⢀⣼⣿⣿⣷⣄⡈⠙⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣿⣿⠄⢻⣿
⣿⡆⠀⢠⣧⠀⠀⡀⠀⠀⠀⢨⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⢠⣿⠿⠛⣻⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠈⠛⠉⠙⣿⣿⠉⠉⠉⣽⣿⣿⡤⣾⣿
⣿⡗⠀⣸⣿⡀⢰⡇⠀⠀⢀⣸⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⢾⣿⣿⠶⣿⣿
⣿⣿⡀⣻⣿⡃⣼⣧⠀⠀⠠⣽⣿⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⠀⠘⢿⣿⣿⡿⣿⣿
⣿⣿⣽⣿⣿⣿⣿⣿⠄⠀⠴⣾⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⣤⣶⣾⣿⣿⣿⣿⣿
⣿⣿⣭⣿⣿⣿⣿⣿⡃⢀⣤⣾⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡶⠼⢿⣿⣷⡄⢀⣤⠤⠶⣶⣤⣴⣾⣿⣿⣿⣣⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣴⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠛⡋⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⠿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⣿⣿⣟⣉⣹⣿⣿⠟⠁⠀⣿⣿⣿⣿⠀⠈⠻⣿⣿⣿⣿⡿⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⢸⣿⣿⡿⠋⠉⠉⠁⠀⢀⣠⣿⣿⣿⣿⣇⡀⠀⠀⠙⠳⢿⣿⣟⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⢀⡤⠖⠘⠙⡏⣿⣿⡿⣿⡿⢿⣿⡟⣶⣦⣄⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣿⣷⡀⣀⣠⣴⣿⣿⣟⣿⣦⣼⣟⣋⡻⢁⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⣯⣻⡹⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⣴⠋⢣⣾⣿⣿⣿⣾⣴⡇⣝⡦⢿⣿⣾⣿⣿⣿⣿⣯⣽⣿⣿⣵⣿⣷⣿⣿⣿⣿⣿⣧⡀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣄⣩⣿⣷⣿⣿⡯⣻⣷⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣟⣿⣿⣟⣿⣿⣿⣯⡿⣤⣏⣻⡆
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠁⠈⠙⠓⠿⢷⣿⣿⣿⠟⠋⠁⠀⠈⠉⠉⠁⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣾⠏
⠀⠀⠀⠀⣤⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠰⣿⣶⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠛⣿⣿⣿⣿⣿⣧
⠀⠀⣴⣷⣶⣿⠛⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⡀⠀⠀⠀⠀⠈⣿⣿⡶⣿⣿⣿⣿⡿⡿⡄⠒
⠀⠀⠉⣹⣿⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⣀⠀⢸⣿⣇⣿⣿⣿⣿⣿⣿⣍⠛
⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⣿⣿⢿⡿⣿⣿⣿⣿⡟
⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⣾⣎⣿⣿⣿⣿⠁
⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⡯⣿⣿⣿⡇
⣾⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣷⣿⣿⣿⡇
⠙⠻⢿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡟⠿⣿
⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠷⢹⡿⣿⣿⣿⡄⢹
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠐⣼⣧⡿⠙⢸⣿⣼
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⣿⣿⣷⡈⠙⠛⢿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢃⣸⣿⣿⡝⠀⢸⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⣶⣿⣿⣿⣧⠀⠀⠀⠀⠉⠛⠿⢿⣷⣶⣤⣀⠀⠀⠀⢰⣿⣿⢸⣿⣿⠟⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣶⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣶⣶⣿⣿⡇⠀⠁⣉⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢹⣿⣿⣾⣿⡇⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣫⢯⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⢸⣿⣿⡙⢻⣿⡐⠚
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣾⣿⣣⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠟⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢲⡄⠀⠀⠀⠀⠀⠀⣀⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣄⡀⠀⠀⠀⣰⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢉⣿⣿⠿⣿⡷⢤⣼⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣠⣤⣤⠀⠒⠒⠒⠛⢛⡤⠚⠛⠈⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡀
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⢀⣼⣿⣿⡷⣿⣿⣿⣿⣿⣿⡦
⠀⠀⢸⡄⠘⣇⠐⣆⣀⡀⠀⠀⠚⠻⠋⠙⠋⠉⠛⠉⠀⡉⠙⣻⣿⣿⡟⠀⠀⠀⠀⠀⠀⢰⡁⡀⢀⡀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠈⠁⠀⠁⠀⠁⠀⠀⠉⠈⠒⠂⠤⢀⢀⡀⠀⠈⣀⠢⣿⣿⣿⣟⡀⢀⢀⢀⣀⣠⣤⣴⠏⠉⣿⡎⠈⢹⣿⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⢃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡷⢶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠿⠯⠀⢸⣫⣿⣯⣿⣿⣟⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣒⣤⣤⠿⣿⣿⣿⣀⡀⢸⡇⠰⠇⠀⠀⠀⠀⠀⠀⣰⢠⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⠿⠁
⠀⠀⢀⣀⣀⣤⣤⣤⠤⠐⠒⣀⢀⣉⣠⣠⣤⠶⠶⠚⣁⣈⣰⣟⣙⣻⣛⡿⠿⣿⠷⣶⣶⣶⣶⣴⣤⣼⢷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠶⠿⠛⠉⠃⠀⣄⣰⡴⠶⠗⠋⢉⡉⢉⣡⣴⣶⠶⠾⠟⠛⠉⣉⣉⣬⣷⣦⡶⠺⠚⢒⠉⡀⠉⢹⡿⠻⠿⢿⠿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠂
⢤⣄⣶⣤⠿⠟⠟⠋⢈⣀⢀⣐⣾⡿⠿⠟⠉⢁⣄⣠⣶⣼⣿⣿⠿⠟⠋⢉⣤⣶⣧⣿⣶⣿⣦⣿⣿⣻⡥⠼⣯⢄⡥⢀⡀
⠛⠛⠋⠀⣀⣀⣦⣠⣦⡿⠋⠛⢩⢤⣴⣴⣽⣷⣿⡟⠋⡉⠋⢀⣰⣶⢶⣼⣿⣿⡿⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⣳⡍⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠒⠈⠉⠉⠉⠀⠒⠂⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣠⠞⢋⣼⠏⣠⣴⠶⠺⠟⠛⠛⠛⠛⠛⠷⠶⣦⣄⡘⢿⣍⠛⠢⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⣠⠞⣤⡶⢋⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣝⢾⠀⢈⣦⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡞⠀⣴⠟⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡕⢧⡈⢳⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢠⡟⢠⡾⢃⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠻⡄⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⡟⢻⡿⠁⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡆⠱⠜⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣾⠤⡟⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢻⡈⢧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠃⢸⡇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡇⠈⣇⣸⡆⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⢤⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠹⠀⢨⡠⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⢨⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⢰⠖⠦⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢿⠀⡇⢨⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣀⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡜⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠸⡄⢱⠉⠷⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢰⠋⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⣶⠚⡆⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡾⢀⡶⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⣆⢹⡄⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠘⠧⣾⠃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠈⠋⠁⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⢀⡜⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡒⠊⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠉⠀⠀⠉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠉⡓⡢⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢤⡐⡏⠏⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⣄⡄⠐⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠅⢠⠲⡄⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠃⣾⣺⠇⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡏⢦⠸⡆
⠀⠀⠀⠀⠀⠀⠀⡘⢠⣿⣿⡆⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡅⢸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠘⢝⣨⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢰⠈⠃⢸
⠀⠀⠀⠀⠀⠀⠀⢁⠀⡟⢙⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⡀⢻⡅⡁⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⣨⣠⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢎⠉⠴⠀⠛⢳⢤⣀⠀⠀⠀⠀⢀⣠⠴⠉⠑⡼⠁⡡⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠳⡦⠑⡄⠀⠀⠉⠙⠛⠛⡋⠉⠀⠀⢠⠞⢹⠝⢁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠉⠀⠀⠈⠁⠀⠀⠀⠢⠴⣂⡉⢀⢉⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⢀⢀⡀⣀⡺⣿⡯⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠒⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⢠⡾⠓⢤⠀⠀⠀⠀⠀⠀⠀⢽⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢨⠤⠖⣂⠂⠉⠓⠻⠤⠶⠄
⠀⠀⠀⠀⡀⠀⠀⠀⡀⠈⢀⣴⠟⠓⢹⠚⠑⢖⠀⠀⠆⠀⠐⢾⣧⡀⣀⡀⠈⢛⠹⠏⠜⢤⠀⠀⢈⠀⠀⠈⣆⠀⠀⢤⡀⡀
⠀⠠⡤⠀⠨⠤⠤⠤⠰⢠⡞⠁⠀⠀⠉⠀⠀⠈⢣⣤⡄⠀⠀⢿⣧⣀⣀⣀⡀⢁⡎⠁⠀⠈⢧⠀⠄⢰⣄⣴⣿⣀⣀⣼⣇⡜
⠀⠀⣀⠀⢄⣀⣀⢀⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢹⣿⣿⣿⣿⡇⠸⠀⠀⢠⣀⣧⠆⠀⠘⠉⠁⠠⠚⠛⠉⠀⠤⠄
⠀⠀⠀⠀⠀⠁⣖⡋⣉⢀⡀⢴⠀⠂⡖⢸⠀⠶⢸⣿⡇⠀⠀⢹⣿⡿⠟⠋⡡⣆⣁⣹⡞⠛⠁
⠀⠀⠀⣀⡥⠄⡝⠃⣿⠘⣁⣀⣀⠀⠦⢸⣀⠀⢸⣿⣇⠀⣠⠟⠋⠁⢀⡐⠁⠀⠀⠙⢹⠢⢄
⠀⠀⠀⠀⠀⠐⠒⠒⠀⡈⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠉⠉⠀⠒⠐⠒⠛⠋⠉⠙⠓⠒⠲⠦⠤⠤⢤⣀⣀⣀⡀
⠀⠀⠀⠒⠀⠐⠂⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠉
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡠
⠀⠀⠀⠀⠀⠈⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠨⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡀⠀⠀⠀⠠⢄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠈⠉⠁⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠙⠲⢬⡐⢂⡤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠕⠆⠉⠒⠀⣈⢀⢂⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⢀⡠⡖⠍⢢⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⢀⣴⢫⡁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣀⠀⠀⢀⡴⡋⡤⠂⢀⡠⠄⠦⢄⠀⠀⠀⠀⣠⠞⠉⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠙⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⡞⠣⠀⢠⢎⢡⠌⠀⠀⢘⠀⠀⠀⠀⠱⡄⠀⢰⠃⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍⣉⣛⣂⠐⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢠⡟⣰⠁⢠⠫⡨⠃⠀⠀⠔⠋⠀⢀⠀⠀⠀⠘⡀⡻⠀⠀⢀⡀⠀⠉⠙⢢⠀⠀⠀⠀⠀⠀⠻⠀⠨⠍⣁⢄
⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⡟⢡⠃⢠⡏⢣⠁⠀⠀⡏⠀⠀⠀⠀⠑⢄⡄⠀⢳⡱⠁⡰⠏⡀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠐⠈⡄
⠀⠀⠀⠀⠀⠀⣸⠁⢀⡾⢵⠏⢀⠳⡘⠆⠀⠀⠀⢣⠠⠄⠀⠀⠀⠈⠳⢤⣩⣿⣼⢃⠄⠐⣀⡠⠖⠁⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠁⢰
⠀⠀⠀⠀⠀⢀⠃⠀⣼⢡⡜⠀⣜⢣⡘⠀⠀⠀⠀⠀⠙⠒⠲⠖⠧⣄⣁⣮⣹⣟⢷⣶⠿⠛⠷⠶⠤⣄⠀⠀⠀⠀⠀⠀⣛⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡌⠀⢠⡷⢻⠁⠀⡟⢤⡇⠀⠀⠀⠀⠀⠀⢀⡤⠒⠹⡂⢈⢵⡝⣿⠛⠳⢌⣠⠂⠀⠀⠈⠳⠀⠀⠀⠀⣰⠥⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢠⠁⠀⢸⣇⣸⠀⢰⡏⢶⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⡸⠇⠈⠀⣿⡙⠔⠊⠉⠒⠀⠀⣀⡴⠀⠀⠀⣰⣃⠈⢡⡖⠀⠀⠀⡿
⠀⠀⠀⠀⠘⠀⠀⢸⡉⣁⠀⠸⣏⢺⠀⠀⠀⠀⠀⠀⠘⢄⣀⣀⡀⠀⠀⠀⢀⠣⠀⠀⠀⠀⠀⠀⢛⠁⠀⠀⠀⣰⠡⠤⢠⣏⢀⠂⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣽⡟⡀⠀⢷⣙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⢀⠎⠀⢧⡀⠂⠄⠀⢀⠆⠀⠀⠀⡼⠃⠂⣠⡶⢌⠇⠀⣰⣷
⠀⠀⠀⠀⣷⠀⠀⠸⣇⠘⢇⠀⠘⣎⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠹⣛⣖⠛⠁⠀⠀⢠⠾⠁⠀⣰⠟⢀⠎⠀⣰⢧⠂
⠀⠀⠀⠀⢻⡄⠀⠀⢿⣤⠚⣄⠀⠘⢟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⠳⣄⢀⡴⣉⠉⢠⣞⠛⣹⠋⠀⣴⢁⡜
⠀⠀⠀⠀⠸⠣⠀⠀⠈⠲⠀⠈⠂⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⡬⠋⠀⢀⣶⣥⢉⠞⠁⢀⣾⢸⡜⠁
⠀⠀⠀⠀⠀⢧⣆⢀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⣀⣴⣿⣞⡳⠋⠀⣠⠞⢸⠞
⠀⠀⠀⠀⠀⠘⡌⠾⢇⠀⠈⢢⣄⡀⠀⠀⠀⠀⠀⠈⠳⠦⣀⠀⠀⢀⣀⣀⣤⠴⠒⠈⠀⣀⣴⣼⠅⠙⡾⠋⠀⢀⡔⠋⢰⡾⡓⡒⠒⠒⠒⡂
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠳⣄⠀⠈⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣀⣰⠿⠟⠛⣢⠗⠉⠀⢀⡰⡏⣄⡾⠉⠉⠛⠓⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠑⢤⡀⠀⠙⠶⣦⡄⢀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⢹⣄⣠⠔⠋⠁⠀⢀⡴⢊⢑⡥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠉⠢⢄⡐⠤⢈⠛⠺⠅⢤⡀⢀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣠⢀⠏⣠⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣁⢂⠄⡈⠑⠢⢤⡄⠀⣀⣽⡇⣂⠀⠀⠀⠀⣀⣠⡤⢚⠏⣂⣸⠚⠁⠀⢀⠀⠀⡀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠪⣔⠕⡠⢀⠀⠀⣩⣾⣉⣿⡤⡴⢒⠟⡝⣑⣡⠼⠊⠉⠉⠉⠀⠈⠒⠉⠀⠒⠨⠔⠊⠁⠂⠙⠊
⠀⠀⠀⠀⡀⠔⣀⠠⠐⠀⠠⠔⠀⡀⠄⠀⣀⠉⠑⠓⠾⠴⢥⣍⣌⣈⡰⠥⠧⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠄⢀⡀⠀⡀⠀⣀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠐⠀⠀⠔⠂⠩⠔⠂⡡⠔⣀⠔⢀⡠⢐⠠⠒⡠⠔⣂⠤⢒⠡⠒⡋⠴⢂⠡⠒⣉⠤⢒⡩⠔⣊⠥⢒⠥⠒⠩⠔⠊⠔⠊⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠉⠀⠈⠀⠈⠁⠀⠁⠀⠈⠁