Equestrianism Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠄⠐⠐⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠤⠆⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⣄⠄⢀
⠀⠀⠀⠀⠀⠀⢲⣄⠀⢆⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⠀⠀⠘⡆⢀
⠀⠀⠀⠀⠀⠀⠈⡠⠓⠬⠄⡟⢮⠉⠛⠲⠤⣄⣀⠀⠇⠀⢈⣤⠐⣊⡤⠄⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⢸⠁⢀⡤⠖⢿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡡⠤⠈⠁⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⡿⢠⡞⠴⠀⣸⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠒⡀⢿⡀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠶⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠁⠎⠀⠀⠀⡟⠀⣴⡀⣀⠴⠊⠁⠀⠀⠀⠀⠀⢳⡀⢳⡀⠀⠱⢀⠇⢰⣈⡙⠄⠤⣤⣄⣀⡀⡠⠞⡵⠋⣉⣓⠲⠦⠤⠖⣠
⠀⠀⠀⠀⠀⢀⡄⠠⠤⠄⠀⢁⡴⠟⢻⡁⠀⣆⠀⠀⠀⠀⢀⡠⠄⠉⠀⠹⣄⠀⠠⡀⠀⢧⠉⠲⢄⠀⠀⠈⠉⠓⢤⡀⠀⠀⠈⠉⠓⠒⠋⠁
⠀⠀⠀⠀⠀⡼⢀⡄⠠⣆⡀⠁⠀⠀⠀⣷⠀⠈⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠈⠳⠄⠘⢦⠈⣦⠤⠔⠃⠀⢳⡄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠳⠈⠁⣾⠆⠀⠀⠀⠀⠀⢹⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠁⢘⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡇⠀⢰⡀⣹⠆⠀⠀⠀⠃⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠄⠐⢀⡄⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⡞⢸⠃⠀⠘⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠸⡆⢀⡄⠠⠤⣄⠀⠀⠀⠀⣸⠃⡟⠀⠀⠀⠀⠀⣀⡤⠒⠀⢄⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢳⡤⡀⣷⢸⡆⣀⠀⠀⠀⠀⠀⣰⠃⠀⡇⠀⠤⠴⠒⠋⠁⠀⠀⠀⠈⠧⣄⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠱⣄⠸⠀⠃⠈⠙⠓⠂⠴⠚⠉⠉⠁⠀⠀⠀⠐⠦⠤⠤⠤⠴⠒⠚⠓⠀⠉⠉⠀⠀⣦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⢄⠀⡰⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⢸⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⠀⠀⢠⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠳⡀⠀⢸⠄⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣄⠈⠚⢧⣄⡈⠌⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡹⣿⣄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣻⣷⡈⠛⠷⢦⠄⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⠀⠀⢠⣎⢻⣿⣿⣿⠉⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣶⣿⣿⣿⣿⣗⠸⠿⠿⡛⢫⣵⣿⡜⡇⣿⣷⣍⣛⠏⠀⠘⣛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣿⣿⣿⣿⣿⡟⣿⢸⣿⣿⣿⣦⣝⣋⣼⡷⣹⣿⣿⣿⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢻⣿⣿⣿⡿⢱⣿⣦⣭⣭⣭⣭⣭⢩⣭⣴⣿⣿⡏⠃
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣸⣿⡿⠋⠀⠱⣦⡍⠙⠛⠛⠛⠛⠘⠛⠛⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡿⣱⡿⠋⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⠎⠙⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠉⢁⣿⠁⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠈⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⠛⢿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⣷⠁⠀⠀⠀⢀⣠⣶⠚⡿⡚⠆⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠀⢀⡤⠖⢻⣦⠤⠖⠻⣄⡴⠚⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⢿⠟⢻⣿⣧⡼⢉⡤⠟⠉⠀⠀⠀⢰⢿⡄⢤⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣈⠓⢌⣫⣿⠖⠋⠀⠀⠀⠀⠀⢀⣿⠘⣟⣄⠀⢀⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⢿⣿⣿⣿⣿⣿⠋⠛⠿⠿⠦⠤⠤⠤⠤⠤⣤⣾⣙⣆⣨⡿⢯⣉⠜⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⠉⢻⣏⣛⣛⣾⣿⣿⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠙⠻⢯⡐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⣹⣿⣿⠼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⡿⡷⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢠⢃⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢟⡞⠀⢸⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⢀⠀⠀⠀⢀⣠⠋⢸⣧⠀⠀⠀⠀⠀⠹⡄⠀⠀⠐⣿⢿⡿⠇⠀⠀⢻⡆⠀⢀⡀⠀⠀⠀⢀⣀⢰⡏
⣸⡗⠒⠒⠉⠀⠀⣴⣿⡀⠀⠀⠀⠀⢠⣿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠻⣄⣸⣇⠀⠀⠀⠀⠙⠷⣽⣦
⠘⠧⣤⡄⣠⡶⠊⠀⠈⢧⠀⠀⠀⢀⡾⠛⠿⣍⠙⢻⣿⠒⠒⠠⠤⠒⠒⠋⠉⠉⠛⠚⠓⠦⣤⣀⡀⠉⠑⢦
⠀⠀⠉⠙⠁⠀⠀⠀⠀⢸⠂⠀⣠⠞⠀⠀⠀⠘⣇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄⢀⡨⠓⣄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣰⠋⠀⠀⠀⠀⠀⢹⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢾⣄⣀⣀⠞⠙⣇⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡏⠀⠀⠀⠀⠀⠀⠘⠳⣄⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣸⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡚⣳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣄⠀⠀⠀⠀⠀⢀⣠⣴⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠉⠁⠀⠀⠀⢀⣴⣫⣾⣿⣿⣷⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⣼⣿⣿⣿⣿⣿⣿⡩⢟⠿⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣄⠀⣿⣿⣿⢿⣿⣿⣿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡷⠛⢿⡟⠿⠩⣿⣝⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⠟⠀⠑⢿⣿⡻⣿⣷⣦⣤⣍⡀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠛⢉⣻⣿⠦⢤⣀⡀⠙⣷⠘⣿⣿⣿⣿⣿⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⣷⡀⣿⣧⣠⣿⠀⣿⣿⣿⣿⣿⣯⣵⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣦⣤⣶⣿⣿⣿⣿⣿⡿⣷⣿⣧⣹⣿⢿⢻⣾⣿⣿⣿⣷⣿⠾⠋⠁⠀⣿
⠀⠀⠀⠀⠀⢀⣾⣿⡿⡿⠋⣹⣿⣿⣿⣿⣿⡿⢁⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⣶⣦⠀⠀⣀⣿⠇
⠀⠀⠀⠀⠀⣼⣿⣿⡆⠁⠀⢻⣿⣿⣿⣿⣿⠇⢸⣧⣽⣿⣿⣿⡿⠿⢁⣠⡀⠀⣀⣬⣿⠀⠸⠿⠋
⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠈⢿⡿⣿⣿⣿⡄⣼⠏⠀⠀⠈⠀⠀⠀⠙⠛⠻⠟⠋⠁
⠀⠀⢀⣠⣿⣿⣿⣿⢻⠀⠀⢰⣿⣿⣼⣿⡿⠟⠋
⠀⠀⣈⣴⣿⣿⣿⠏⠋⠀⣴⣾⣿⠿⢿⣯⡀
⠀⠰⠾⠿⠟⠹⠋⠀⠀⠀⢻⣿⠁⠀⠀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⢾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢀⣄
⠀⢸⣷⣤⣶⡆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⡟⣿⢀⣤⡀⠀⠀⠀⢀⣠⣴⣿⣯⣶⣿⣷⣿
⠀⠀⢸⣿⣿⡀⠀⢸⡗⢄⣤⣿⣷⠀⣤⡶⠀⣼⣦⣤⡄⠀⡰⣟⡙⠻⢿⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⠄
⠀⠀⠸⠿⠉⠇⠀⠈⠳⠈⠉⠉⠁⣾⣿⣿⣶⣾⣏⠁⠀⠈⡀⠀⠀⠐⡀⠉⠁⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣴⠈⠀⡐⠀⠉⠻⣿⣿⣷⣶⣐⢤⢤⢁⠓⠔⠀⠀⠀⠈⢻⣿⣿⣿⠁⠈⠉
⠀⠀⠀⠀⢀⠠⠦⣄⠀⠀⡀⢀⡜⢁⠀⠀⠀⠈⢿⣿⣿⡿⡃⢲⡈⣿⢀⡀⠀⠀⠀⠀⣿⣿⣿⠄
⠀⠀⠀⡀⠡⣦⠀⠈⢧⡨⠞⢡⠁⡸⠀⠄⠀⠀⠀⠻⣿⣿⠉⢹⠉⢀⣹⣿⡷⢥⣍⠉⠉⠙⠓⢤⡀
⠀⠀⠀⠦⠤⢠⡔⢂⣈⡷⠦⡖⠺⡄⠀⠀⠁⠀⠀⠀⠙⢿⣿⣿⣦⠀⢹⣿⣅⠀⠈⠳⡄⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠻⢶⡠⠉⠑⢄⡀⠀⠁⢳⠤⢄⣄⣀⣀⣀⡠⠴⠋⠉⢿⣇⠀⢿⣿⣷⠀⢀⡇⠈⣢⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⣀⣹⣯⠀⣼⠃⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡀⠀⠃⠀⠀⠀⡠⠃⠀⠈⣷⣌⣿⣿⣿⣿⡁⠀⠀⢠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡇⠉⣿⡏⣳⣿⡿⠀⠀⡜⠐⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣀⣃⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⢸⠀⣿⠇⠛⠛⠃⢀⢾⣁⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠌⠀⠀⠀⠀⠀⠀⠃⢀⠤⠀⠀⠀⡐⢰⡿⠀⣀⣤⡞⠁⠀⠘⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠱⡀⠐⠉⠀⡀⠠⡀⠀⡇⠀⢠⠀⣼⣴⣿⣗⣋⣥⣘⣄⠀⠀⠀⣽⣴⣾⡔⠒⠢⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠙⠦⣄⡴⠋⠉⠉⠛⡇⠐⠁⢀⣿⠿⠿⠿⣿⣿⣿⣿⡧⠀⠀⢻⣾⣿⠶⣄⠀⠇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣴⣿⣿⣷⡀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠁⠀⣘⣀⣐⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⣇⣴⡞⠸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣬⡀⢰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣾⣿⣿⣿⠋⠀⠀⣀⣠⣀⣲⣤⣴⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣁⣬⣿⣶⢟⠋⠁⠀⠀⠁⠉⠻⢦⣦
⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢑⣄⣀⠀⠉⠻⣿⣿⣍⣿⠄⡀⠀⠀⠐⢢⣆⣾⠟⠿⠆
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⠴⠒⠊⠉⠉⠒⢾⡿⣿⣿⣆⣴⡏⠙⠋⠉⣦⠈⢑⣦⣞⢉⡏⢈⠈⠁
⠀⠀⣠⣾⣿⡿⠹⠿⠛⢁⢎⠀⠀⠀⠀⠀⢠⠇⢀⣴⠟⡿⣿⡿⠃⠰⠀⠀⠈⢀⣿⠟⠀⢿⣾⢾⣗⠚⠃
⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⢸⠀⠀⠀⢀⡄⠁⢠⡈⠻⠿⠄⠘⣿⡀⢱⡄⠀⠎⠒⠔⣴⡦⠋⣇⠀⠻⠬⠃
⠀⠀⣼⠿⠋⠀⢀⠤⠤⠣⠂⠀⠀⠐⣩⠴⣶⣦⣝⣂⣀⣀⣀⣈⣿⣾⡿⠦⣄⣤⣾⠹⢑⣾⠋
⠀⠀⠀⠀⠀⠀⡋⢀⡤⣤⡴⠒⠒⠋⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠘⠓⠊⠉
⠀⠀⠀⠀⠀⣠⢷⠃⠀⣿⠇
⠀⠀⢠⣶⠒⡳⢢⣦⣼⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡳⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⢀⢶⡄⢶⡛⠩⠟⠉⠉⠉⠉⠙⠛⠶⢤⡀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠈⣹⣖⢶⡷⢶⠦⢤⣄⠀⠀⠀⠀⣀⠴⠓⠴⠾⡋⠉⠉⠁⠀⡜⠁⠈⠉⠉⡱⠃
⠀⠀⠀⣿⣋⣾⠛⡈⣿⠿⣿⣷⣆⠤⠚⠁⠀⠀⣰⣦⠀⣼⣷⣄⠀⠀⠀⠀⠀⠈
⠀⠀⢸⣇⣸⣿⣀⣗⣡⡶⠟⢻⣿⡆⠙⢷⡤⢞⡩⠀⠀⣸⣿⠃⠀⠀⣠⣤⣤⣤⣶⠆
⠀⠀⢼⢾⣫⠿⠛⠋⠉⠀⠀⠀⢻⣧⡤⢀⡶⠋⠀⠀⢰⣿⣿⣆⠀⣹⣿⠛⠛⠛⠋⢀⡄⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠿⠃⠀⣀⡔⢸⣿⣿⣿⣶⣿⣿⣆⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀⣀⣴⣿⣿⣿⣯⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣸⣿⡿⠋⠀⠀⢹⣿⠿⠿⠿⣿⣿⣆⠀⠀⠀⠀⠘⢦⡲⣦⡀⠀⠀⠀⠠⡙⢿⡿⠿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⡦⣤⡄⠙⢛⠿⡏⠀⠀⠀⣰⣻⡟⠀⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠘⣿⡎⠁⠀⠀⠀⠀⢹⡄⣷⠀⢈⣿⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠙⠳⢶⣤⣭⣥⡌⢀⣤⢞⣵⣿⠁⠀⠀⣀⡀⣾⣿⡇⠀⠀⠀⣠⣴⣿⡧⡄⠀⠀⠀⠀⢸⡇⡿⠀⠘⢿⣿⣿⣏⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⢿⠇⠀⣶⣿⣿⠶⠟⠙⠓⠞⠁⠛⠛⠿⢿⣶⣤⣉⣁⠉⠙⢃⣤⣖⣿⣿⣿⣿⢧⠻⣦⠀⠀⣠⡟⣸⠇⠀⠀⠀⢃⣿⡿⣿⣷⣝⢷
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠙⠻⢷⣶⣤⣀⣠⣴⣤⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣷⣄⢻⡆⠈⠿⠰⠿⠤⣤⣄⠀⣀⣯⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡀⠀⠘⠿⠖⠂⠉⣈⣿⣷⠉⠻⣿⣿⣿⣿⡆
⠀⠄⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠉⢻⣷⡙⣿⣆⠀⠋⠈⢿⢻⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣼⣿⣄⣦⣀⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠙⠿⠿⠿⠿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠻⠆⠀⣴⠟⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠀⣿⣷⣄⠀⠀⠀⠼⠁⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⡄⠀⣦⢠⡆⢀⣴⣶⣿⣿⣿⣿⣥⣾⣿⣿⣿⡀⠀⠀⡆⠀⠀⠀⠀⢸⣧
⠀⠀⢠⣎⠀⠙⠂⠁⠿⠃⠀⠀⠒⠲⢦⣭⣛⢿⣿⣿⣿⣿⡇⢀⡆⠃⠀⠀⢀⠀⣸⣿
⠀⠀⠈⠛⠛⢩⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣬⡻⣿⣿⣷⣿⠃⠀⠀⠀⡖⠰⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠛⢎⠛⠛⠋⠀⣀⢀⣼⠁⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⡄⢷⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣤⣾⡿⠛⠁⠀⠀⠀⣤⠈⢹⣆
⠀⠀⠀⠀⢘⡿⣿⣄⣀⣠⠞⠁⢸⡆⠀⠀⠀⠀⠀⣀⠀⢀⡿⠋⠁⠀⠤⠤⠤⠴⠾⠿⠀⣠⣿
⠀⠀⠀⠀⢸⡇⠈⣹⠟⠁⠀⣀⣸⣧⠤⠶⠒⠋⠉⠀⡠⠋⠀⠀⠀⢀⡤⠖⠋⠀⠀⢺⡟⠙⠟⠁⠀⠀⠀⠤⠤⢤⣄⡀
⠀⠀⠀⠀⢾⣷⣾⠇⠀⠈⠉⠀⠈⣷⠀⠀⠀⠀⢠⡏⣇⣠⣄⠀⠈⠀⠀⣰⠟⠓⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠄
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠸⣧⠹⣿⣿⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠙⢷⣿⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣾⣿⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⣠⣤⣾⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢠⣿⣿⡟⠁⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠉⠉⣠⣶⣶⣤⣤⣤⣤⡤⠄⠀⢻⡄⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠃⠀⢀⡜⠀⠀⢸⠏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⣀⡄⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣠⡶⠟⡋⠁⠀⠀⢀⡾⠁⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣣⡾⠟⣧⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠐⡏⠀⠀⢷⡀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⣿⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⣷⠀⠀⠈⢷⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⢀⣿⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣿⣆⠀⠀⠈⢿⣄⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄⣿⣷⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠀⠀⠀⠀⢿⣦⢈⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⢸⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣀⣀⡀⠀⠀⠀⠀⣼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣿⣿⣆⠀⠀⣼⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢸⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠏⠀⠀⠀⠀⠀⠀⢀⡤⣴⠶⢶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠀⠀⠀⠀⠀⠀⠀⢡⠐⠁⡈⠀⠟⣄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡀⠀⠀⠀⡀⠀⠁⠀⢠⣅⠀⣀⣘⢧⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⠀⠀⠃⣷⠀⠀⣗⣿⠀⠉⠉⠛⠻⣶⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣦⣤⣴⡶⣿⣦⡄⠸⣌⣀⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⣈⣭⡝⠛⠻⠛⠛⢻⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣍⠀⠛⠿⣿⣶⣦⡈⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠾⣿⣿⣷⣶⣤⣈⣿⣿⣶⠖⠁⢀⡀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠙⠛⢯⢹⣿⣿⣿⣿⣿⣿⣿⡀⠀⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⣠⡄⠀⠘⣿⡀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣟⠈⠈⢁⡄⠀⠀⠀⠀⢀⣴⣆
⠀⠀⠀⠀⠀⣠⣴⡾⠟⠁⠀⠈⣡⣄⠀⣠⣄⠁⠀⢈⡀⠀⠀⠀⠀⢦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⠁⠀⣀⣄⣈⡀⢸⡏
⠀⠀⠀⠀⣾⣿⡟⠃⠀⠀⠀⠀⠉⠉⠀⠈⢡⣶⡀⢸⣿⣿⣦⣰⣶⣿⣿⣿⠿⠻⠿⠟⠁⠙⠛⠁⠀⠀⠀⠉⠉⠉⠉⠙⢿⡇
⠀⠀⠀⠀⣿⣿⡃⠀⠀⠀⠀⠀⠀⠉⠀⢤⠀⠀⠀⠈⣿⡍⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⠟⢻⡄
⠀⠀⠀⠀⠘⣿⡷⠀⠀⠀⠀⠀⠀⠈⢃⣄⡀⠃⢀⣈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠈⠙⠐⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⢨⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣷⣿⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡈⠙⠳⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡈⠉⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡌⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠁⢒⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠂⠀⢰⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡋⠰⠀⢴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠻⣿⣧⡀⠀⣼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢸⣿⡇⢻⡎⢛⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡟⢸⣷⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠠⣴⣾⣧⢳⣶⣦⣀⡀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠈⣿⠀⠀⠀⠀⣼⠟⣿⣿⣧⠀⠀⣀⣴⡾⣿⣯⠉⣍⠀⠈⠛⠿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡅⠀⢹⣀⠀⠀⢰⠏⠀⢹⣿⣿⠀⣸⣿⣿⣿⣿⠃⠀⣸⠛⠛⠛⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠷⣤⣀⠉⣻⠶⢯⡀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡏⠀⢰⠏⠀⠀⡀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢼⣿⣿⣆⡀⠀⢱⣾⣿⣿⣿⣿⣿⣿⡏⢇⣼⠟⠄⠘⠇⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣀⡀⠀⠀⢇⠀⠀⠀⢸⣛⣿⣿⣷⣶⣿⣿⣷⣿⣿⣿⣿⣿⠁⣼⣿⡀⢻⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠿⠛⠛⣿⣍⣉⡝⠛⠛⣿⡄⠀⠀⢸⡏⠙⣻⡟⢿⣿⣿⣿⣿⣿⣿⡿⢿⣾⢿⡏⠳⣼⣧⠤⠤⠤⣸⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⣡⣶⣿⣿⣿⣿⠟⢦⣄⠀⣸⢁⠀⢀⡾⠃⣰⢋⣾⣎⢻⣿⣿⣿⣿⣿⣧⢸⣿⢿⡟⠂⢄⣻⡤⠶⠒⢺⡅
⠀⠀⠀⠀⠀⠀⠀⢰⠏⣼⣿⣿⣿⣿⡿⠃⠀⠀⠈⢻⣿⢋⣿⡿⢁⡼⣡⣿⣿⣿⣷⡹⣿⣿⣿⣿⡟⣸⡿⣸⣇⠀⠀⢹⠃⢤⠁⣸⠇
⠀⠀⠀⠀⠀⠀⢀⡟⣼⣿⣿⣿⣿⡟⠁⡠⠊⠠⠀⡿⡘⢸⢏⡥⢋⣼⣿⣿⣿⣿⣿⣿⣎⠻⣿⣿⣧⣿⡗⠡⠛⡆⠀⠘⠳⠶⠴⠋
⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣼⣇⠁⡟⣛⣴⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⣌⠻⠿⠟⣵⡇⠀⡇
⠀⠀⠀⠀⠀⠀⣸⢸⣿⣿⣿⣿⣿⠿⣿⡻⣿⣿⣿⠏⠀⣹⠿⢿⣿⣇⣾⣿⣿⣿⢻⣿⣿⣿⢻⣷⢠⣾⣿⠁⣼⠃
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⡟⠀⣧⡙⢬⡛⣟⣠⡊⠀⣀⢼⣿⣿⡿⣿⡿⢃⣼⣿⣿⣿⢸⡇⢸⣿⣣⡾⢿⡄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡟⢀⣼⣿⣷⣄⠙⠻⢤⣿⣿⣆⠈⣿⣿⡇⢻⣿⣿⣿⣿⡿⢋⡾⢀⣾⣿⣿⡁⠈⢿⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⢠⣾⣿⣛⣿⣿⣿⣶⣤⣀⣈⡁⢻⣧⣸⣇⠀⢻⣿⣿⣄⠠⣌⣴⣿⠿⢷⣬⣍⡓⠄⠉⠳⣄
⠀⠀⠀⠀⢀⢹⣏⠛⠛⣁⡴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣛⡿⠟⠉⠉⠳⣄⠙⢿⣿⣧⡹⡅⠀⠀⠀⠈⠉⠛⠳⢶⡿⣿⣧
⠀⠀⠀⢀⠞⢻⡍⠛⠛⠉⠀⣿⣿⡏⠛⠿⠿⠭⠿⠗⠚⠋⠁⠀⠀⠀⠀⠀⠈⠛⢦⡙⢿⣷⣼⡄⠀⠀⠀⠀⠀⠀⠘⣧⠻⠏
⠀⠀⠀⡈⢀⣼⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢿⣿⣿⢆⠀⠀⠀⠀⠀⡴⠷⣎
⠀⠀⢀⡀⣾⠏⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡸⠀⠀⠀⠀⡸⠁⣰⠃
⣠⡔⠚⡿⠋⠀⢀⣴⣶⣶⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⡶⠀⠀⠀⠀⢰⠁⣰⠏
⠻⣽⠞⠁⠀⠀⠈⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡇⠀⢀⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⣸⠀⠀⢾⡈⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠋⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠻⣿⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢥⡆⠀⠀⠀⠀⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⢟⣃⣀⣴⢆⣀⠈⠓⠀⠀⠀⠀⠀⠀⠈⡷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣀⠀⠘⠹⠿⠭⠭⠭⢭⣥⣟⣿⣤⡀⠀⡠⠊⠀⠀⢀⠊⡱⠛⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⢿⣿⣿⣿
⠀⠀⠰⠦⣸⠷⠀⠀⠀⠀⠀⠀⣰⠃⠀⠉⠙⠷⠏⠁⠀⠀⠀⢀⠞⠀⢠⠎⣰⠂⠀⠀⣀⣠⡤⠤⢤⣴⣾⡟⠋⠁⠀⠀⣿⣿⣿
⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡀⠐⡟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⣿⡟⠘⠦
⠀⠀⢀⣿⡇⠀⣀⣤⠟⢈⡽⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⢰⢳⡘⠾⣿⡄⠁⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠘⡆⠀⠸⢿⡇
⠀⠀⣾⠘⢛⡿⣛⡥⠖⠉⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠘⠦⣍⡒⠈⠻⡀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢀⠇⠀⠀⠀⠣
⠀⠘⠿⡾⠟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⢀⡀⠀⠀⢀⣴⡏⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⣂⣤⠴⠚⠁⠀⠀⠀⠀⠰⠁⠀⠐⠛⢉⣀⡤⠞⠋⠉⠁⠀⠀⠀⠀⢀⡀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⣾⣿⠿⠶⠤⠤⠤⠴⠞⠀⠀⠀⠤⠔⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⣹⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣤⣤⣘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣠⡖⠻⠓⡠⠚⢩⡳
⠀⠀⠀⠀⠀⠀⠀⠐⡿⠿⠿⢿⡇⠀⠀⠀⢀⣀⡴⠋⠀⡀⣸⡇⢄⣤⡀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⢺⣅⠀⠀⠀⢹⣿⠁⠀⠀⠟⠁⠻⡀⠠⠈⠂⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⢺⡟⠋⠉⢀⡔⢾⡏⠀⠀⢰⢴⡰⣄⣱⡶⣠⡞⠉⣄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠑⠀⠀⠀⣷⣤⡳⣄⠀⠈⠐⡀⠘⢋⠻⣧⢤⠓⠖
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠈⠓⢦⡀⠈⠀⣈⣷⢉⡮⠳⠚
⠀⠀⠀⠀⠀⠀⣰⣏⣄⡀⠀⠀⠀⠀⣸⣿⢿⡇⡀⠀⠀⠉⢛⡛⠋⠀⠙⣄⡀
⠀⠀⠀⠀⠀⢠⠀⠘⣦⡀⠀⠀⠀⢺⡞⠉⠘⣇⠉⢤⡈⠛⢋⠈⢳⠄⠘⡠⠉⠱⠄
⠀⠀⠀⠀⠀⢈⠀⠀⠈⡿⠂⢀⠆⢡⠀⠀⠀⣿⠀⠀⠈⠀⠀⠈⠤⠀⠀⠁⠠⡀⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠸⠀⠀⢰⡟⣠⠎⠀⣘⣠⣄⣸⡿⠀⠀⡀⠠⠄⠀⠀⡀⠀⡀⠀⠑⠀⠠⠆⠀⠀⣠⠐⠐
⠀⠀⠀⠀⠖⠸⠂⠀⣋⠀⠈⠀⠀⡯⣼⣿⠿⠱⡀⠀⠈⠀⡀⠐⢫⡅⠤⠈⠁⠀⡄⡀⠀⠀⡐⠋⠀⠆
⠀⠀⠀⠀⠀⠀⠐⢶⢻⣤⣧⣀⣴⣿⣿⣿⡆⢣⡀⢸⠀⠀⠁⢀⣄⠠⠀⠀⠄⠸⠀⠁⣠⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠛⡻⣛⣿⣿⡟⣣⠆⠳⣄⠳⠀⠐⣄⠈⢣⡰⡟⣼⠚⠉⠈⢁⣄⡠⠃⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢛⠋⢸⣿⠷⠋⠠⠦⣀⠑⠶⢆⠠⠀⠀⠙⣩⠃⠀⠀⠀⠊⣀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⡇⠀⠀⢸⡿⣁⠉⠉⠁⠀⢑⣶⣤⣥⣶⣦⡄⠈⠂⢄⠀⠀⠓⠊⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠐⢣⠀⠀⠁⠀⠀⠀⠀⠰⠛⢽⣿⣿⡿⠃⠀⠁⡑
⠀⠀⣠⠞⢛⣿⣾⡏⠀⠀⠀⠘⠳⡴⣄⣤⡴⠄⣀⣀⣰⡤⠙⠁⠀⠀⠀⠀⠈⠳⡄
⠀⠘⣿⣼⣿⣌⣙⢿⠀⠀⠀⠀⢠⣻⣿⣿⣿⣿⣿⣿⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀
⠀⢰⣿⣿⣿⡍⢽⣿⡇⠀⠀⠀⠠⣿⠛⢻⣿⣿⣿⠏⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠆⢘⠄
⠀⠀⠈⠉⠉⠙⣷⣶⣷⠠⠀⠀⢰⡇⠀⠀⢹⡿⣿⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣀⣴⣿⡿⣻⡇⠀⠀⡜⠁⠀⠀⢸⠋⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠤⢴
⠀⠀⠀⠀⣿⣧⣤⣿⡿⠁⠀⡘⠀⠀⠀⠀⢠⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠚
⠀⠀⠀⠰⣿⠋⠉⠀⠅⠀⢸⠀⠀⠀⠀⠀⠢⢄⠘⢄
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠨⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠤⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣹⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣫⣙⣏⣹⣉⣹⣙⣩⣟⣹⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠶⣶⣤⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣴⣆⠀⡑⢄⣠⣤⠀⠀⠀⠀⠀⣠⣼⣿⠟⠛⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣦⡀⠀⣀⡀⠀⠀⣸⡿⠻⣇⣸⠈⠿⠟⠀⣀⠤⠒⠉⠙⠋⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢀⣿⡉⠗⠀⠀⠙⠻⣾⣏⣤⡶⠋⠁⢀⡼⢓⣾⣇⡀⠀⠀⠀⠀⠀⡄⢀⡝⠙
⠀⠀⠀⠀⢨⡀⢠⠀⢼⡀⠀⠄⠀⣿⠟⣠⢆⣀⢀⣀⠫⠉⢘⠳⢤⡀⠀⠀⢀⣷⣿⣧
⠀⠀⠀⠠⣧⣄⠀⠀⠀⠙⣆⢀⠈⠀⠀⢿⠮⠻⡆⢻⡷⠀⠈⣷⣤⣿⡆⠀⣸⣿⣥⣿⣶⠦
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣹⠎⠀⠀⠀⠀⠉⣀⢁⠀⠀⣠⣾⣫⣤⣤⣍⢋⡅⣨⣭⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⡀⠀⢀⣄⣠⣿⣿⡿⠟⠋⠀⠉⠉⠉⠁⠀⠁⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠔⠛⠛⠋⢹⡷⠶⠶⢶⣶⣶⣶⣶⣶⠠⡆⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢱⠇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣀⡀⠀⠀⠀⠀⠀⢀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢣⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣤⣤⠄⠀⣀⣠⠾⠛⣿⠄
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠙⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣡⣿⣿⣿⣴⣿⣿⣿⣷⢀⡛
⠀⠀⠀⠀⠀⠰⠾⠇⢀⣤⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠻⠴⠎⠅⠵⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⠛⡿⠷⠶⠶⠶⠄⠄⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡿⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣱⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠿⢿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⣤⣤⣤⣍⡛⠛⣛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣯⣩⣭⣽⣿⣭⣭⣿⠧⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣂⣙⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⠉⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣡⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠴⠶⠚⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣷⠶⣷⣦
⠀⠀⠀⠀⠀⣠⡶⠿⠿⠿⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⠈⠉⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠀⠈⠙⠀⠲⠄⣀⡀
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⠷⣤⣀⣀⠀⢶⣦⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⣾⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣶⣦⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⠟⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣭⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢀⣠⣜⠛⠓⠶⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⣿⡟⠋⢉⣀⣠⣄⣀⣉⠙⢿⣎⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃
⠀⠀⠀⠀⠈⠀⠹⣷⣴⣶⣿⣿⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢀⣴⣿⠋⠀⣴⠟⠋⠉⠉⠹⢿⣧⡄⠙⣾⠇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⡏⠀⢸⡏⠀⠀⣀⣤⡀⠀⠙⣃⠀⢹⡇⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠈⠛⣇⣼⣿⣿⠟⣸⢿⣤⡀⣿⢋⣤⣤⣷⡿⠀⢸⡇⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣢⡤⣤⣤⣤⣴⡿⠿⠛⢻⣞⣱⣎⠛⣿⡿⠛⢿⣿⠟⣷⢡⣾⣷⣄⡀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢏⡴⠾⣏⠛⢿⣷⣶⣿⣿⣿⣦⡹⣷⣤⣤⣤⣬⠓⢋⣴⣿⣿⣿⡿⠿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⣃⣀⡹⣶⣾⠻⣿⣿⣿⣿⣿⣿⡿⠶⣶⡷⣶⣾⣷⣾⣿⠿⢿⣧⣾⣟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡾⣿⣿⣿⣨⣿⣤⣤⣭⣭⣿⣯⣶⣾⠿⣿⣿⣷⣮⣭⣭⣤⣄⣘⡏⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⣿⣿⡘⣿⣿⣧⡜⠁⠀⢸⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣥⠈⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠟⣱⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⡿⢫⣽⡿⠟⢻⣿⣿⣶⣿⣄⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣿⣿⣧⣾⠟⠛⠿⠿⠿⠿⢿⡇⠀⠀⠀⠸⣿⣿⠀⠈⠹⠗⠚⠉⠉⠈⣉⣉⠛⠋⢷⣒⣀⣒⣒⣂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⣛⡭⣱⣶⣿⠙⣷⣾⡶⢾⣶⣿⠀⠀⠐⣶⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣦⣌⣿⡉⠉⢉⡉⠉⠛⣦⣂
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠏⢳⡋⣼⢛⣿⣿⠀⣴⣿⣷⣼⣿⣿⡆⠀⠀⣿⣿⣿⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢾⣿⣦⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣫⡿⣣⣾⣿⣿⠘⢸⣿⡿⢸⣿⣿⡇⠀⠀⣿⠀⢿⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠻⣿⣷⣶⣿⣍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢹⣗⢹⣿⣿⣿⣌⢻⣿⠓⣾⣋⣿⡇⠀⣀⣿⣤⣾⣿⣿⣿⣿⣯⠉⢻⣿⡏⣠⠄⠹⣿⣿⣿⣟⠀⠀⠻⠿⠟⢹⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣿⣿⣿⣿⣿⣿⡿⢿⣿⢧⣿⣿⣦⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡀⢻⣿⡔⠋⠀⢀⠸⠋⣿⣿⣇⠀⠀⠀⠀⠘⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠟⢻⣻⣟⣻⢷⣾⡟⢸⠿⣱⠾⡅⠀⣿⣿⣿⣇⣽⣿⣿⣿⡇⢸⣿⣿⣦⡜⠁⢰⡎⢻⣿⣿⡄⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⢀⣿⡊⠙⣁⣾⣿⢿⣭⠟⢩⣾⣿⣧⡅⠀⠋⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⠋⠀⠀⢿⣧⢸⣿⣿⣿⡆⠀⠀⠀⡉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⡟⢴⡆⠀⢠⣴⣿⣿⣿⣿⡃⠀⠀⣸⣿⣿⡟⣿⣿⣿⡇⢸⣿⣿⠀⠐⢄⢸⣿⣸⣿⣿⣿⣇⠀⢀⢴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣽⠛⢿⣿⡀⠀⠁⣄⢘⣽⣿⣿⣿⣿⣿⠒⠀⣼⡟⣿⡇⢿⣿⣿⣧⣸⣿⣿⣤⡀⠀⣺⣿⡿⢿⣉⣹⠿⣖⠧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠓⣤⣀⠛⠿⢾⣿⢀⣿⣷⣿⣿⡉⠀⠑⣶⡿⠁⣹⡄⠀⢼⣛⣟⣿⣿⣿⣿⣿⣤⣿⣿⡄⠀⣿⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠋⣻⣿⣿⣿⣿⣿⠛⠲⣤⣄⣉⠩⠛⣻⠿⣷⣶⣦⣤⡴⠗⢿⣤⢄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⣿⣿⣦⣄⣿⣿⣿⣿⡇⠀⣸⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⢿⣿⣿⣿⣿⢃⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠻⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⣰⢟⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠋⣸⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡀⣠⣶⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⡝⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠈⢧⠙⢫⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠠⡆⣼⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠼⠿⠿⠿⠿⠿⠿⠟⠁
⣴⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⢣⡤⣼⡟⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢛⣛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⠟⢿⣿⣷⣦⣤⡴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣀⠀⠈⠓⠀⣉⠻⣍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⣤⢀⣤⣶⣶⣶⣿⣣⣾⣿⡧⠀⠀⢸⠃⠀⢹⣆
⠀⠀⠀⠀⠀⠀⣴⣘⠷⢎⠸⠋⠀⠀⠺⢯⣿⡿⣿⣿⣿⣿⠀⡄⠘⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠉⠉⣼⣿⡇⠀⠀⠀⠀⠀⠈⠙⢮⡝⢿⣿⣾⡇⠀⠀⡐⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⠢⠙⠿⢁⣤⡴⠃⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣀⣀⠜⢸⡇⠀⠀⠀⠀⠀⠀⢰⡷⠛⠉⠀⠀⢀⣴⡇⢘⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠁⣩⠟⠁⣀⣨⣧⠤⠔⠒⠉⢀⡔⠁⠀⠀⣠⠤⠞⠀⠠⣴⠺⠿⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠈⠉⠁⠀⣿⠀⠀⠀⢰⡇⣤⣤⠀⠀⠀⢰⠟⠂⠀⠻⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠘⣧⣹⣿⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠈⠉⢻⠀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠶⠿⠶⠆⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⢀⣾⣿⠏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡠⠚⠛⠛⠒⠒⠒⠂⠀⢻⡀⠀⠀⠀⢀⣠⣴⠞⠋⠀⢠⠂⠀⢸⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣴⠾⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢰⠋⠉⡄⠀⠀⢰⠏⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⢰⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢸⡀⠀⠹⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⢸⣇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⣧⠀⠀⠹⣆⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠿⠟⠀⠀⠀⢹⣇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣷⣦⡀⣀⠀⠀⠀⢀⣴⣾⡿⠋⠁⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣷⡀⢀⣾⣟⠉⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⠟⠋⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣇⠀⠀⢀⣠⣤⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠛⣻⣷⣦⣀⣉⣡⣤⣤⣶⣾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠺⣿⣿⠀⢠⣼⠟⠉⠀⠀⠁⠘⠻⠷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣾⣷⣤⡈⣿⠁⠀⢈⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣦⣤⡷⠒⢩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⣰⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡿⠛⠁⠀⠛⠋
⠀⠀⠀⠀⠀⢠⡴⠶⠛⠉⠀⢻⠹⣿⡟⡹⠋⠻⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⣿⠟⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⡆
⠀⠀⠀⠀⠀⠀⢸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣁⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡅⢠⣾⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⣼⣿⡿⠋⠈⢻⣿⣿⣿⣿⣿⣿⡿⣿⣠⣿⣷⣤⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢸⣤⠆⣀⣼⣿⣻⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣾⣿⣿⡷⣆⠀⠠⢠⡄⠀⠠⠀⠜⢣⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣏⣼⡃⠾⢻⡿⠁⣿⣿⡀⠀⢸⠷⠾⠛⠛⣋⣉⣉⣁⣈⣱⣶⣽⣿⣿⠿⠆⠠⠀⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣯⣤⣴⣿⡇⠀⢛⣀⡅⠀⣴⣶⣿⣿⣿⣿⠟⠛⠛⠿⠿⠿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣿⣿⠿⢿⣿⡿⠟⣋⣭⣤⡄⢰⣿⣿⢿⠀⠿⠿⠟⠛⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⣦⣴⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢹⣷⣿⣿⠶⠾⣛⣿⠿⠟⠧⣬⣇⠈⣁⣠⣤⣶⠆⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣴⣾⣿⣿⣷⣶⡆⠉⣹⠇⠀⠀⠀⠈⠳⣽⣟⣿⣿⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⣻⣟⡿⠏⣹⣿⡿⢸⡏⠀⠀⣄⠀⠀⠀⢹⣿⣏⢙⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠛⠉⠉⠉⠡⠊⠀⢺⡟⠁⠀⠈⣧⡄⠀⠉⠁⠀⢠⣿⣿⣿⣿⣦⠀⣸⣿⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣤⣸⣿⣆⣀⠀⢸⣿⣶⣤⣤⣼⣿⣿⣿⣿⠿⠋⢀⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢻⣿⣿⡇⠀⠿⠟⠛⠛⠛⣛⣫⣭⣴⣶⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⠀⢤⣿⣿⡇⢠⣤⣶⣾⣿⣿⣯⣽⣿⣿⣿⣿⡟⣛⢻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⡶⠿⠿⣿⣇⡶⢿⡿⣿⣿⣿⣿⣿⣅⣹⣿⡟⠉⠉⠻⣿⠟⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⣀⣤⣼⡄⢡⣾⠟⠋⠁⠀⠀⠀⢀⣉⣛⢿⣿⣿⣿⣷⠈⣀⣽⣿⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⣠⣤⣿⣿⣿⣇⣸⠁⠀⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⢨⣿⣷⣷⡀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢰⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣮⣙⣿⣿⣿⠀⣿⣿⣿⣿⣿⣤⣀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣿⣿⣿⣿⣿⢩⠀⠐⡀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣤⣤⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣿⣿⣿⣿⡿⣿⠰⠃⠀⠀⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⢣⣿⡷⠀⠀⠀⠄⠀⠀⢸⣿⣿⣿⣿⣿⠟⣿⣿⡿⢿⡆⠁⣿⠙⣿⡏⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠘⠛⡇⢀⠀⣀⠀⠀⠀⣾⠉⣿⣿⣿⡟⠀⢙⣹⣿⣿⡿⠀⣿⡶⣷⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣭⣭⣥⣀⣾⡇⠀⠀⠀⠁⠀⠀⢻⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⡇⠀⣿⡹⣿⠉⣿⣿⢿⠿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠰⠛⣻⣿⣿⣿⡿⠛⠻⢷⡅⠀⠀⠘⠦⣀⠀⢸⣿⣿⣿⣿⡇⠀⣾⣿⠿⢿⡇⠸⠿⢤⡀⠄⢸⣿⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣟⣻⣿⣿⣿⣇⣠⣄⣈⡿⠀⠀⠀⠀⠀⠁⠚⣿⣿⣿⣿⣿⠀⢿⣟⣶⣘⣿⣾⣦⣾⡴⠇⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⣿⠉⠉⣏⡉⢩⡀⠂⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡆⠀⣹⣿⣿⣿⡏⢿⣿⣧⣀⣛⣭⣽⣽⣷⣶⣦⣤
⠀⠀⢀⢀⠀⢀⣠⣶⣴⣶⣿⣿⡏⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⠁⣽⣿⣿⡀⠹⡿⣿⡻⠃⢸⣿⣿⣩⠿⣿⣿⣿⣿⣷⣄
⠀⠀⡁⠀⠈⠛⢿⡍⠉⢿⣿⡀⠀⠀⠀⠀⠙⠿⢿⡄⠀⠀⣶⣾⣿⣿⣿⠀⣽⣿⣿⣷⡄⠀⠙⠃⠀⣿⣯⣿⠁⠀⠀⠑⢟⣻⣿⡿⠷⡀
⠀⠀⠀⠠⠀⢀⣠⣹⣦⡿⠛⠃⠀⠀⠀⠀⠀⠀⠈⢃⣠⣼⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣷⢠⡀⣀⡔⢿⣿⠋⠀⠀⠀⠀⠀⠻⣿⣿⣆⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠖⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⣿⠇⣿⠿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣦⡏⢠⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⡾⢡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣐⣛⡵⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣉⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢛⡁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⢹⣯⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠃⠀⢼⣿⣿⣿⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⠀⠈⢿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠠⠀⣈⣻⣻⣿⣷⣄⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣄⣹⣿⣿⣿⣿⣿⣿⠠⡶⡠⠀⠀⠀⠀⠀⠔⠛⣫⣩⠖⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⢀⡉⠛⠻⠿⣿⡿⠿⠿⠟⠛⠉⡑⠢⢤⡀⠀⠀⠀⠀⠀⠀⢾⣿⣰⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠀⠻⠿⠆⠀⢰⣾⡀⠀⠀⠀⠀⢺⣄⣸⡟⠁⠀⠀⠀⠀⠀⠀⣾⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣵⡦⣧⢹⣷⣶⡄⣾⣿⣿⡄⠀⠀⠀⢹⣿⣿⡿⢶⣤⣤⣀⣀⣀⡀⠸⠛⢀⡠
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣘⢿⣿⣿⣧⣿⣿⣿⢿⡄⠀⠀⠀⢫⠉⠁⠠⣿⣿⣿⣟⠿⠁⠀⠃⠈⣆⣂⡀
⢀⣀⣠⣄⣀⣄⡀⢸⣿⠀⢣⡀⠙⠿⠽⣿⣿⡅⢸⢻⣄⠀⠀⠈⢆⣰⡄⠀⣈⠉⢁⠁⠀⢠⡴⠶⣗⣶⣶⠤
⠀⠉⠙⠛⠿⢿⣿⣞⣿⣇⠀⠑⢨⠂⠠⠿⣿⠃⠠⣶⣿⣦⣶⠀⠈⡿⣡⡄⠀⠀⠀⢀⣾⣿⣆⠈⢿⢿⣿⣿⣷⣥⡀⠐⡄
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣾⣷⣤⣄⣀⠀⠀⠀⠈⠻⠿⠁⠀⢃⠻⠗⠶⠂⢀⣿⣿⣿⠋⠶⠒⠀⠀⠈⠛⢿⣿⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣮⣍⡓⠶⢄⡔⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⣿⠃⠀⠀⢁⢈⡑⢦⠀⠀⠹⣿⡀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣤⣀⣧⠀⠀⠐⠀⠆⠐⠀⠀⣯⣿⣿⠀⢀⡀⠋⠀⡁⠀⠡⠀⠀⣿⡇⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⢀⢀⣤⣄⡀⠀⠀⠀⢻⣿⣿⡿⠇⢣⡀⠒⠂⢀⠅⠀⠀⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡄⠀⠉⠛⠿⣷⣦⣀⠀⢿⣿⣿⡀⠀⠉⠓⠒⠋⠀⠀⣰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣻⣄⠀⠀⠀⠈⠙⠻⢶⣿⣿⣿⣷⣤⣄⣀⣤⣴⣾⣿⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣧⣶⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣹⣧⣶⣿⣿⣦⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠟⠡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⣓⣷⡄⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠧⢪⣿⣿⣿⣀⣾⡃⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⣤⣰⣿⣿⣿⣿⣿⡿⣛⡐⠁⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣜⣈⣿⣾⠿⢻⣿⣿⣿⣿⣿⠏⣉⣼⣦⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣴⣧⣿⣿⣿⣿⡿⢿⣛⣡⣦⣾⣿⣿⣿⣿⠛⢫⡵⠾⢁⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠯⣿⣭⡼⣡⣲⠊⠉⢉⠧⠍⢸⣿⣿⣿⣷⡶⠶⠆⠈⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠈⢳⣾⠿⠻⠔⣿⣯⣁⠉⠀⣀⠀⠀⢹⣿⣿⣷⠆⠀⣀⣠⠤⠤⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢶⠟⣿⢾⣷⣦⣀⠀⠈⠉⠊⢉⠑⢦⣌⣿⣿⢟⣋⣿⣿⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣰⢶⠖⠿⢟⣿⣿⣿⣿⣶⣤⣀⣀⣁⡦⠉⠛⣉⡯⠉⣁⠀⠀⠠⢄⡈⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⣀⣴⣤⣄⠈⠰⢡⡞⡲⡶⢿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⢀⢬⠟⢿⣶⡈⢀⣱⠶⢆⣛⣶⣶⣿⣧⣤⣤⣀⠀⠀⠀⠈⠐⣦⢀⡀
⠀⠀⢀⣴⢟⣫⣭⢿⣏⣠⣂⣷⣾⠿⣦⣼⣯⣟⣉⣶⠿⠋⡀⠲⠀⠀⢀⣀⣠⣛⠿⠛⠉⢀⣾⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣶⣤⣶⠿⢿⣿⣿⠇
⠀⢠⣟⣽⡿⠁⠡⣼⣿⡿⢻⣷⣼⣦⢹⣸⣯⡽⠋⢁⣈⠧⣽⣀⣤⣴⣿⣿⣿⣿⣦⣴⣾⣿⣿⣿⣿⣿⢟⣻⣿⣿⣿⣾⣿⡿⠃⠀⠠⠀⠙⢏
⠀⣼⣿⣿⠓⠀⣼⣿⣿⠁⣾⣿⣿⣿⣿⣻⣿⡇⠰⣦⣉⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣵⣿⣿⣿⣿⣇⠀⢨⣷⡄⠀⠀⠀⠀⠀⠂
⠀⣿⣿⣿⢠⣼⣿⣿⢿⣯⣿⠏⢸⣿⣿⣿⡷⠀⠀⠀⠉⣺⢃⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣵⡀⠀⠀⠀⠰⣿⣿⢿⡄⠀⠀⠠⠀⠀⡆
⠀⢸⣿⣿⢸⣽⣿⣿⣸⣿⡍⠻⣽⣿⣿⡟⣧⠄⠀⠀⢠⣹⣏⣶⢸⡟⢦⣽⣭⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⡀⠀⠀⠀⢹⣷⣏⣱⣆⠀⠀⠀⠀⠈
⠀⠈⣯⠙⢧⠑⠵⠼⠀⡍⠑⢂⣸⣶⢹⠏⠁⠀⠀⠀⠺⣿⣿⣿⣷⢀⣐⣭⣶⣿⣿⣿⣿⣿⣿⣿⡘⣿⡏⢿⣷⣾⠛⠋⣿⣷⣾⡏⣿⣆⠀⠀⠀⢀⡄
⠀⠀⠘⢧⣂⡳⣄⡿⠀⢸⠋⣙⣿⣿⡏⠀⠀⠀⠀⠠⠀⢿⣿⣿⣿⠏⠉⠙⣦⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣦⡀⠀⢘⡇
⠀⠀⠀⠈⠻⢯⣂⣹⣶⣞⣫⣾⣿⠟⡁⠀⠀⠀⢀⣄⣀⡈⠻⡿⠏⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣻⣯⡻⢿⣿⠿⢁⠼⢻⡿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⢁⡴⣍⠓⠶⠤⡌⠉⠉⠉⠙⠛⠒⠀⠒⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣟⣶⣶⣶⣿⣦⣀⣿⣼⣿⣿⣿⣿⠃⡜
⠀⠂⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣿⣿⣿⣾⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣛⣆⣛⣻⣽⣿⣿⣿⣿⡿⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣉⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⡀⠀⠘⣿⣀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶⣛⣫⣥⣤⣶⡾⣡⣤⠤⠀⠁⠈⠀⢀⠀⠀⠀⠀⢀⡿⢹⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣤⠀⠘⠡⠭⠭⠵⠶⠶⠦⢤⣴⣟⣿⣶⣤⡀⢀⡴⠋⠀⠀⢀⣀⠉⡰⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠶⠄⣸⠷⠂⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠙⠻⠷⢏⠀⠀⠀⠀⠀⢠⠞⠀⠀⡴⠁⣲⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣿⠟⠛⠉⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⢠⡿⠇⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⢶⠻⠿⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠙⠳⡄⠀⠀⠀⠀⣿⣿⠁⠳⡀
⠀⠀⠀⢀⣿⡇⠀⠀⠀⣠⡆⠐⢲⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⠣⣴⣶⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢸⣿⡟
⠀⠀⠀⡞⢿⣦⣤⣴⡾⠛⣠⡴⠋⠀⠙⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⢦⣈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⣻⠀⠀⠈⠙⣧
⠀⠀⣼⣇⣀⣼⢛⡥⠖⠋⠁⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠬⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠘
⠀⠀⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⣠⠄⠀⠀⢀⣤⣾⡇⠀⠀⠀⠀⠀⠀⠘⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣂⣠⠴⠖⠋⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠛⠋⠁⣀⣤⠶⠛⠉⠀⠉⠀⠀⠀⠀⠀⢀⣀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣰⣾⣿⣿⣤⣤⣤⣤⣤⣤⡶⠀⠀⠀⢀⣀⠤⠤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢹⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⣾⡇⠀⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣾⣧⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣿⣿⣿⣿⣛⣋⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠉⠉⢉⣀⣀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠐⣛⣛⡃⠀⣉⡭⢭⢹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⠄⠉⠉⠀⠀⢧⡀⠀⠀⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⠀⠀⠀⢀⠀⣨⢇⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠸⣿⠟⢿⠟⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⠀⠀⡠⠔⠁⠀⠀⠀⡤⠐⠀⢆⣀⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠁⢱⠐⢄⠀⣀⠄⠘⢆⠀⠈⡄⣀⡀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡠⠌⢀⡆⠀⠀⠀⣱⠀⣹⠇⠀⠀⠀⠣⠀⢛⠀⠈⠂⢄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢒⠉⠀⡗⠐⠒⠂⢀⠛⠂⢬⣿⣶⣶⣿⣿⠊⠁⠀⠀⣀⠔⠊⠙⠁⠀⠀⠀⠀⠙⠢⣀⠈⠢⡀
⠀⠀⠀⠀⠀⡠⠂⠁⠉⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠈⠈⠀⣤⠦
⠀⠀⠀⠀⠰⠀⠀⠐⠒⠃⣀⠇⠒⠒⠀⠹⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⢤⣼⣗⠳
⠀⠀⠀⠀⠇⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠈⠛
⠀⠀⠀⢰⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⣿⣿⣿⣇⠀⠀⠀⡄
⠀⠀⠀⠰⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⡇
⠀⠀⠀⡤⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣿⣿⣿⣿
⠀⢀⠌⠀⢀⣵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸
⣴⢟⣷⠢⣼⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠈
⠀⠞⢀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠉⠀⠀⠀⠘⠀⢀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⠀⠀⠀⠀⠂⠁⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡌⠀⠆⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⡙⠒⠤⢀⡤⡄⠀⢰⠀⠀⠀⠀⠀⡇⠀⠘⡀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢠⣸⣿⣿⠉⠢⢀⡘⠀⠋⠐⡆⠀⠀⠀⠀⢰⠉⠗⠈⠣⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠸⡀⣰⠀⠀⡇⠀⠀⠀⠀⣸⠀⢰⠀⠀⢠⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠁⠑⠠⣃⠀⠀⠀⣠⠉⠁⠚⠋⠁⠀⠓⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡶⠊⠁
⣶⣶⣶⠶⡶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡟⠉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠃⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡬⠡⠒⣶⣿⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠂⠀⠀⡿⠀⢀⣤⣶⠆⠀⠀⠀⠘⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡾⠁⢀⠔⠋⢀⣴⢸⢛⣧⠀⠀⠀⠀⠀⣼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣿⡶⣿⠁⠰⣯⣴⣀⠙⠟⡾⠀⢻⣷⡄⠀⢠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠿⠋⢩⣡⣄⢀⢸⡇⠠⣯⣿⣛⡤⢞⢁⣠⣾⣿⣧⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⣰⣹⣿⣿⠇⢸⡇⠈⢉⣏⠉⡀⠘⣷⣿⠟⣿⣿⣿⠟⡝
⠀⠀⠀⠀⠀⠀⣰⡿⠛⠁⢀⣠⣾⣿⣿⣿⡏⢠⠈⢃⣄⣼⣿⣿⣿⠀⠟⠁⠀⠻⣇⠈⢸⣿⣤⡄
⠀⠀⠀⠀⠀⣼⢟⡤⣴⣾⣿⣿⡿⣿⠿⠿⣧⠀⠈⠉⢡⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠱⡄⢈⡟⠁
⠀⠀⠀⠐⠛⣾⣿⠖⠚⣿⡏⠀⠀⠀⠀⠀⢹⠇⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡷⠋
⠀⠀⠀⠀⠀⠘⢯⣷⣾⣿⣿⠀⠀⠀⠀⠀⣼⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⢫⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣟⡷⠀⠈⠀⢠⠋⣰⠏⠀⠀⠀⠠⠀⠀⠀⠀⠀⢿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⣼⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠁⠖⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣤⣴⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⠀⠀⠀⣀⡈⠰⠊⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠒⠀⢀⡀⣀⢀⡀
⠀⠀⢤⣤⣄⠀⠐⠉⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⣄⡶⢦⡀
⠀⠀⣸⡀⠁⠀⠀⠀⠂⠉⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⠈⢁⣤⣼⣶⡀⠀⠀⠀⡀
⠀⠀⠻⠁⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⣰⣿⣿⣿⣿⡗⠈⠀⢠⣷⡄
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠟⢿⣥⠄⠀⠀⢀⠀⢀⠀⠀⠠⠄⠠⠤⠶⠿⠶⠏⠀⢿⣿⣿⠟⠛⠁⢀⣤⡛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⡤⠀⣠⠞⠀⠀⠐⠀⠀⠀⣀⣴⣿⠀⠈⠂⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⡿⣿⠛⠉⠀⠀⠀⠀⠀⠀⢐⠃
⠀⢠⠀⠀⠀⢀⠤⢶⣶⠿⠀⠀⣽⠀⠀⠀⠀⣴⣿⣿⣿⠛⣿⠀⠀⠀⠀⠀⣤⣾⣿⣿⣶⣶⣿⣿⣶⣄⠀⠀⠀⠀⠘⠃⠙⠀⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠱⠂⠀⠀⠀⠀⠀⠀⠀⣈⣴⣿⣿⣶⡟⠚⢿⡁⠝⣿⠀⡇⢀⠀⣿⣦⣄⡙⠉⠀⠘⡟⡉⠙⢻⣿⢿⡄⠠⠀⠂⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠿⡦⢻⡄⠀⠀⠀⠁⠀⠈⢣⠀⠀⠀⢸⣿⣿⣿⣷⣤⡀⠀⠀⠀⣸⡇⠠⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⡼⢷
⠀⠀⠠⣀⡠⠀⠀⠘⣿⣾⣠⠄⠐⣿⡇⠀⠸⠀⠀⠂⠀⢀⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠁⠉⠈⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠙⢼⣻
⠀⠀⡀⠀⠀⠀⠁⠀⠹⣿⡏⠀⠀⠘⠇⠀⠀⠀⣤⡄⠀⠘⠇⠀⠀⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣦⡁⢀⣀⣀⡀⠀⣀⣀⠀⢀⠀⠀⠀⠀⠀⣠⣾⠛⢡
⠀⠀⠀⡀⢠⣤⠀⠀⠀⠹⣿⠄⠀⢀⣤⠀⠀⠀⢻⡇⠀⠀⠠⣤⣴⢿⣿⣿⠋⠛⠁⢸⣿⣿⣿⣿⣿⣦⡙⠉⣀⠀⠀⠈⠁⠀⠀⢀⣤⡰⣾⣿⡋⠀⠸
⠀⠀⠀⠁⣼⣿⡇⠀⠀⠠⢄⣤⣾⣿⣿⠀⠀⠀⠈⣧⣀⣠⣼⣿⠃⢸⠟⠁⠀⠀⠀⣿⣿⣿⣿⠟⢿⣿⣿⣆⠙⣆⠀⠀⠠⣤⡴⣿⣿⣿⣞⠻⠁
⠀⠀⠀⠀⣿⠟⠛⢀⣤⣶⣿⣿⣿⣿⠟⠂⠀⠀⢰⣿⣿⣿⣿⠃⠀⠀⠀⠀⣸⣶⣾⣿⢿⣿⣿⠀⠀⠻⠀⠙⠂⣨⣶⣴⣿⠿⠃⡸⠟⣛⣛⡀⠀⠀⢣
⠀⠀⠠⠀⣠⡤⠾⢟⣛⣿⣭⣭⣭⣥⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⢸⣶⣶⣿⠟⣻⣷⣿⣿⡟⠧⡀⠀⢀⣴⣾⠿⠛⣩⢔⠊⠠⠀⠀⠉⠉⠀⠀⠀⢈
⠀⠀⢐⣫⡴⠶⣛⣻⣽⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⣼⣿⠿⣿⣶⣶⣿⠿⢋⣴⣾⣿⣿⣿⣿⣷⣄⢠⣶⣿⣿⣿⣶⡟⡁⢀⠙⠀⠀⠦⢀⠀⠀⠀⠀⠈
⠀⠀⠈⠁⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡄⢂⠀⣴⣿⣿⣿⣶⣝⣿⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡙⢿⣿⣿⠿⢻⣜⡦⡑⠄⡢⣀⠈
⠀⠀⠀⠀⠸⣿⣿⠟⠉⠀⠀⣿⣿⣿⡇⣿⣾⣾⣿⣿⣿⣿⣿⢧⣿⠏⠀⠀⠉⠙⠻⣿⣿⣿⠿⠟⣛⣃⣤⣤⣴⣶⣾⣿⡜⠈⠀⡀⢈⠱⡤⠈
⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⢈⣿⣿⣷⢹⣿⣿⣿⣿⣿⣿⣟⣼⡟⢀⡀⠀⠀⠀⠀⣼⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⣿⣿⣷⡄⣌⠒⠀⢠⠄
⠀⠀⠀⢀⣿⣿⣧⠴⣦⣀⡤⢈⠟⣿⣿⣸⣿⣿⣿⣿⣿⣏⣆⢉⡀⣼⣇⠂⣼⣆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣾⠿⣿⡿⠁⠈⠁⢠⡆
⠀⠀⢀⣾⣿⠟⣁⠀⠙⣿⣾⣷⣾⣿⣿⣿⣯⣭⣛⣛⣻⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⠁⡀⠀⠀⠀⠉
⠀⠀⡀⠻⢋⣾⣿⣶⣷⠈⠻⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠃⠀⠀⠀⠀⠈⠳⣤⣀⠀⣳
⠀⠀⢹⡇⠀⠀⠀⣈⣿⣆⣠⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⢁⣤⣦⠀⠀⠀⠀⠠⡀⠀⢸⣿⣿⣿
⠀⡄⠚⠁⠀⢀⠀⠙⠛⠛⢿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡙⠿⣿⣿⣿⡿⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠀⠀⠀⠀⠀⣽⣷⣾⣿⣿
⠀⢸⣶⠀⠀⠈⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⡌⣿⣿⣿⣁⣿⣿⡇⠈⣿⣿⡟⠀⢰⣿⣿⣿⣿⣿⣿⣿⣯⣅⠀⠀⣿⣴⣷⣤⣿⣿⣿⣿⣿⣿⠋
⠀⣸⣿⢠⣦⣤⠀⠀⠀⠀⠠⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣾⡇⠀⠹⡿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⢉⠛⢿⣿⣿⣿⣿⠟⠋
⠀⣿⣻⡆⢿⣿⣦⣀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡆⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢠⡞⠓⠛⢻⣿⠏⠁⠀⠀⠰
⠀⠀⠁⠛⠀⠻⠟⠉⠙⢿⡀⠐⠶⢶⣾⣿⣿⣿⣿⣿⣿⡿⠋⠀⢹⣷⢀⠀⠀⢰⡿⠛⠉⢘⣿⣿⣿⣿⣿⣿⣿⣇⢰⣿⡇⠀⣴⣿⡿⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⡆⠀⠀⠈⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣀⡰⠀⣼⣶⡶⠂
⠀⠀⠀⠀⠈⠒⠂⠌⠁⠀⠔⠀⠠⠿⠿⡟⠛⡟⢿⡏⠙⣿⣿⣿⣿⣿⡇⠀⠈⣿⣄⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⠃⣼⣿⠀⣿⡿⣹⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⢋⣾⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡃⢠⣿⣿⣿⡇⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⢿⣿⣿⣿⣷⡀
⠀⠀⣴⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠙⢿⣿⣿⣷
⠀⣼⣿⢿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡆
⢸⣿⠇⣼⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠹⣿⣷
⣿⣿⣸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠻⣿⡆
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⣆⠀⠻⠟⠷⣦
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣨⣿⠀⠀⠈⠀⣹⡇
⢹⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⢹⣟
⠈⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢸⡉⢻⣧⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠻⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠁⠀⠀⠀⣸⠇⠀⠀⠀⢿⣤⡾⠿⣩⣿⠋⠉⣿⠷⣦⡀
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⢸⣿⠀⠀⢀⣿⠀⢀⡟⠀⢸⡇
⠀⠀⠀⠀⢠⡿⢁⣤⣤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣀⣀⣀⣀⣠⣤⡴⠒⢿⡁⠀⠀⠀⠀⠀⠀⣿⢿⡆⠉⠙⠻⠗⠚⠛⢲⣿⠁
⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣉⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⣸⡏⠈⠻⣦⠀⠀⠀⢠⣤⡾⠃
⠀⠀⠀⠀⠀⠙⠷⣤⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⡤⠴⠚⠋⠉⠀⠀⠀⠀⣴⠟⠀⠀⠀⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣠⡾⠛⠛⠷⣦⣼⡇⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠃⠀⢀⠀⠀⣿⠃⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣯⡟⠀⠈⠻⣯⡛⠳⠶⣤⣤⣀⣀⣀⣀⣀⣠⣤⣶⠾⣛⣋⣵⠟⠁⠀⠀⠀⠳⣤⣿⣀⣀⠀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⢰⡟⠁⠙⢷⣤⡾⠀⠘⣷⠀⠀⠀⢈⣿⠋⠉⠉⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⠛⠛⠋⠁
⠀⠀⠀⠀⠀⢰⡟⠛⠶⣄⣠⠿⠃⠀⠈⠻⣦⣄⢀⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠈⠻⣦⣀⣼⠃⠀⣀⡀⠀⠀⠀⠉⠛⠋⠛⢶⣄⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠟⠋⠻⢶⣄⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣀⣤⡾⠋⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠟⠋⠉⠉⠉⠙⠻⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣀⣠⣤⡶⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣴⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡶⣶⡦⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠛⠉⠀⠀⣸⡇⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡾⠛⠋⠉⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠛⠻⢷⠞⢳⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⣼⠧⠤⢀⣀⠀⠛⠿⢴⠿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠙⠷⠶⣤⣅⣈⡛⠒⠾⠄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⢀⣼⠟⠷⣦⣄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⢶⣤⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⡿⠁⠀⢠⠋⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣍⣿⠀⡰⠃⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡦⡼⢁⡼⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢠⠟⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠋⣰⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠋⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡼⠛⢿⢿⡇⢀
⠀⠀⠀⠀⠀⠀⢠⣤⠀⢠⡘⠿⠷⠀⠀⣾⣿⣿
⠀⠀⠀⢀⣤⣌⠲⠟⠒⠚⠧⣶⠷⠾⣿⣿⣿⣅
⠀⠀⠀⢼⡿⢋⡉⠧⠀⠀⠀⠈⠳⣾⣿⡿⢿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣶⣦⡤⣄
⠀⠀⠀⠘⠃⠈⢁⠀⢰⠀⠀⠀⢀⣿⣿⡄⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⣀⠀⣤⠔⠶⣻⠀⠀⢀⠔⠉⣹⣀⠐⠀⠠⠊⠁⠰⠤⠤⠤⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠠⠩⢘⠏⠁⢀⠏⠃⠀⢹⣦⣴⣿⠃⠀⠐⢠⣴⠃⠀⠀⠀⠀⠀⠉⢫⠙⡀⠀⠈⠻⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢀⠀⠀⢸⡀⠀⠀⢰⡿⠿⠃⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⡇⢃⢀⡀⠢⠹⣿⣿⣿⣿⣝⡂
⠀⠀⠀⠀⠂⠄⡀⠋⠀⠀⠀⢸⣧⡀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⢡⡀⡇⠙⠌⠻⢿⣿⣿⡧
⠀⠀⠀⠔⢢⠀⠉⠂⠀⠀⠀⠀⠙⣿⠿⠿⠛⠋⠁⠀⢢⠀⠀⠀⠀⠀⠀⠀⡠⠒⢢⠀⠐⡀⠀⠰⣿⣦
⠀⠀⠀⡀⣰⣧⡀⠀⠀⠀⠀⡆⠀⣿⡆⠀⠀⠀⠀⠀⠀⣁⣀⠀⠀⢀⡠⠞⠁⠀⠀⢆⠀⠬⠛⣿⣿⠛⠁
⠀⠐⣀⣿⠿⠁⠙⢦⣀⣀⢴⣁⡀⠘⣿⣄⢀⣀⡠⠔⠋⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠐⠒⠀⠈⠁
⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣶⣶⣶⣶⠶⠶⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⣀⡹⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣻⣾⣿⣿⣿⣿⣿⠃⢠⡄⢤⣀⣦⡀⠀⠙⣿⣶⢄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣂⣿⣷⣀⣙⣿⣿⣦⣤⣿⣿⡆⢻⣄⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⣸⣿⣿⣿⡿⠿⠿⠿⠚⠛⠉⣉⣩⣿⣿⣿⣿⡟⠋⠉⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣦⠀⠘⣿⣌⠛⠛⠛⠛⠓⢀⣠⣤⣶⣿⣿⣿⣿⡿⠋⣿⣧⠀⢀⣴⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⢸⣿⣿⣷⣶⣿⣿⣿⠿⣻⣿⣿⣿⣿⠟⠁⠀⣿⢸⡀⡛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣾⣿⣿⣿⣿⣿⠟⢡⣾⣿⣿⣿⣿⠋⠀⠀⠀⣿⠀⣯⣻⡿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣉⣛⣛⣽⠟⠁⣴⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢻⠀⢸⣿⡇⢸⡟⢻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠰⣦⣤⣬⣿⠟⠋⢀⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⡆⠀⣿⠀⠀⠁⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣦⣴⣶⣿⣿⣿⣿⡟⠉⠀⠸⣿⡇⠀⠀⠀⠀⢸⣿⣦⣸⣇⣀⣠⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣿⣿⣄⠀⠀⠀⠈⣿⣿⡟⢻⡁⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠻⣦⡀⠀⠸⣿⣿⣦⡀⠀⠀⣿⣿⡇⠀⠃⠀⡀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠈⠻⣦⣀⠻⣿⣿⣿⣷⣦⣸⣿⣷⡇⣀⣸⣷⣽⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⢸⡀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⠋⠉⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢻⣿⡇⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠈⣿⣷⣄⠀⠀⠀⣿⣆⠀⠀⠘⣿⣿⣿⡃⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠸⣿⣿⣷⣤⣤⣿⣿⣷⡀⠀⢿⣿⣿⣿⡀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⣿⣿⠻⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⡦
⠀⠀⠀⢠⡀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣆⠀⡆
⠀⠀⠘⢿⡟⡥⢿⣶⣤⡈⠉⠙⠷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣲⣾⣰⠃
⠀⠀⠀⢿⡇⣐⠈⣿⣿⡿⣷⡄⠀⠀⠉⠻⣦⣄⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿
⠀⠀⠀⠀⠃⠁⠲⣿⣿⣧⡏⢿⡆⠀⠀⠀⠈⠙⢧⣤⡀⠐⣶⣾⣿⣿⣿⡷⣠⣤⡀
⠀⠀⠀⠀⠀⠠⣠⡟⠈⢻⣿⣈⠇⠀⢀⣤⠀⠈⠛⠛⠁⢠⣾⣽⠹⢟⣿⠾⢃⣶⡇⡀⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠛⠾⠃⠀⠸⣿⣿⡀⠠⣿⠟⠀⠀⠀⠀⠀⣾⠟⠁⠠⢉⣴⣿⣿⣿⣇⣩⢆⠀⢈⠙⠋⢷⣶⠶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡇⠀⠀⠀⠀⠀⡼⠋⢀⣤⣵⣿⣿⣿⣿⡿⣿⡏⢶⡄⠀⢭⣒⡾⣿⠇⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠇⠀⠀⠀⡞⣶⡇⠀⣿⣿⣿⣿⣿⣿⢟⣁⣿⣷⣦⠉⡆⠀⠉⢳⣿⠆⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡄⠀⠀⣘⣻⡧⠄⣿⣾⡏⢸⣿⣿⣿⣿⣿⣿⣿⢃⢻⡀⣀⣖⣿⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⣹⣿⠇⢀⣿⣿⣧⠈⢿⣿⠿⠟⠛⠛⢻⣿⣦⠉⣴⣼⣯⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣾⣍⣉⢠⡍⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⠀⠹⣿⡄⠘⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢭⣿⡄⠀⠸⣿⣆⠘⢿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⠆⠀⠉⡼⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡏⠀⡀⠀⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠒⠚⠛⠋⠀⠵⠤⠽⠶⠶⠶⠖⠶⠂⠀⠀⠐⠓⠒⠀⠠⠦⠐⠾⠛⠉⣋⡄⢴⣶⣶⣶⠦⠤

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⡻⣟⡛⣻⡛⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡄⠀⠀⣀⣤⠿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣿⣿⡿⠃⢠⣾⣿⣿⣾⣿⣿⣏⡀⠀⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⢀⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢈⢻⣿
⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠛
⠀⠀⠀⠀⠘⠛⠉⠁⢸⣿⡿⢿⣿⣿⠻⠿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡛⠛⠻⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⠀⠀⣹⡟⢰⣿⣿⡟⠁⠈⢿⡇⠀⠀⢈⡟⠀⠀⠀⠈⣻⠿⣦
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢿⣇⢸⡇⢻⡄⠀⠀⠈⣷⠀⠰⠞⠁⠀⠀⣤⠼⠋⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠙⢾⣧⡄⠻⣄⡀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣭⣽⣀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⢻⡿⠛⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢸⣮⣷⡆⠘⡕⣤⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣼⣯⠀⠀⣹⣦⣽⡏⢩⣀⡷⠈⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡍⣧⣤⣩⠛⡛⡻⠛⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⣧⡇⣷⣻⣶⣿⠃⠀⠘⣧⣼⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢄⡘⠙⣿⣿⣿⣿⠶⠆⠀⢼⣿⣇⠙⠋⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⣄⢻⣿⣿⣅⣀⣀⣀⣼⣿⡿⣆⠀⢀⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡏⠿⣿⣿⣿⣈⣁⣯⣄⣼⢹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠷⢄⣦⠈⢿⣿⣿⣿⣿⣿⠈⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣏⣽⣍⣿⣧⣼⣯⣉⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣅⠀⠀⠈⢷⣼⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣴⣾⣴⣤⣶⣦⣤⣤⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢀⡀⠈⢧⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠠⠊⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣶⣦⣤⣄⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣱⡟⣱⣟⡞⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣾⠁⣈⢋⣅⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡆⣦⣼⣿⣁⣾⠞⠿⠿⣿⣿⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣌⣻⣽⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠈⠁⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣷⣦⣄⠠⣄⣁⠀⣀⢡⡾⠟⠛⠛⠻⠟⠋⠁⢸⡿⣿⣿⣷⡆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣦⡙⢃⣾⣿⡇⢀⡀⠀⠀⠀⠀⠀⣾⢻⣿⣿⣿⣿⠇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢒⣒⣒⣠⣄⣀⠀⠀⠀⠀⠀⡹⢋⣴⡏⢿⣿⡿⠟⠙⠀⠀⠀⠀⣠⣿⣿⣿⣿⣟⣋⠄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⡤⠋⣠⠞⠋⢭⠭⣭⣙⠻⢿⣦⡄⠲⢺⣤⣏⢻⣿⣸⡿⣇⠈⠉⢻⡟⠉⠀⠀⠀⠀⠀⡈⠀⠀⠐⠖
⠀⠀⠀⠀⠀⠀⡴⠀⡾⢡⠊⠀⠀⢠⠀⢈⠟⢦⡙⠋⠀⣾⣿⣿⡌⣿⡏⢳⠟⡄⣲⣄⠀⠀⠀⠀⣅⠀⠀⠈⠀⡴⠂
⠀⠀⠀⠀⠀⢀⡃⢸⠇⠀⠀⠐⢄⡼⠒⠒⢢⠈⠁⣠⢺⣯⣿⣿⣧⣿⢿⠆⢺⣃⣿⣿⡟⣦⡐⢀⡘⢃⣀⠠⠚⠁
⠀⠀⠀⠀⠀⠈⣷⣿⡀⠁⠠⠤⣾⠀⣾⣟⣷⣄⡀⢡⠀⢻⡿⠹⠿⣡⡌⢻⣮⢾⣿⣿⣷⠻⡀⠀⠀⢘⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⢿⣿⣇⢀⠀⢀⠽⣆⡈⠛⠛⢙⠀⠈⠀⢸⣷⠀⢾⣹⣿⠈⣇⣰⠾⣯⣥⣧⡰⠀⠀⠈⡀⠀⠉⠙⠳⢄
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠣⠁⠀⡌⠙⠛⠃⠈⠀⡰⠀⣸⣿⠒⣜⣼⡿⣰⣏⡶⡻⣿⣽⣿⡆⠀⠠⠀⢰⠀⠀⠀⠀⣀⣥⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠂⣀⠀⠀⠁⠀⣠⠇⣰⣿⠃⠈⡷⢀⣀⡀⣶⠘⠷⣼⣿⢿⣇⠄⠀⠀⠀⣤⢤⣦⠈⠙⠛⠻⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⣬⣀⣀⣨⣴⣾⡿⠃⠀⠀⠐⡛⡋⢁⠐⢿⣿⡭⣻⡞⢫⣅⢀⣤⣴⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠻⠛⠉⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠳⢭⣿⣶⣿⣟⠻⢿⠟⢛⠛⠛⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣯⣛⠆⠁⠀⣀⣀⡀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⢹⣷⣦⡀⠙⢩⠉⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠸⣿⣿⣿⠲⢄⢀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠈⠙⣿⡄⠊⡁⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠈⠙⠆⠐⠇⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣑⠀⢀⡀⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣶⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣖⣿⣷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣄
⠀⢀⡀⠀⡠⠖⣋⠭⠭⠉⠉⠉⠙⠳⠦⣄⡀⠀⠀⠀⠀⠀⣠⣝⣿⣿⣿⣿⣿⣿⣿⣶⣶
⠀⠠⣽⣧⣙⡆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⠳⣦⠀⢰⣾⣿⠿⠿⠿⢿⠿⠿⣿⡿⢿⠇
⠀⠀⣾⣶⢻⡟⡝⣿⣿⣶⡶⢄⠀⠀⠀⣀⢼⣺⣅⠾⢝⠂⢀⡀⠀⠠⠃⠀⠀⠀⠔⠁
⠀⢠⣇⢰⡾⠀⢹⠘⠛⢻⣝⣦⡤⠖⠉⣠⣀⣿⡿⠁⠀⠀⢸⣿⠆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⢸⣿⣾⣿⣦⣠⣴⠾⠗⠋⢻⣧⢗⡴⢋⠀⣀⠴⠀⠀⠀⣼⣿⡀⠀⢴⣾⣿⣿⣿⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣓⣶⢘⡇
⠀⢠⣿⣿⠛⠋⠉⠀⠀⠀⠀⠀⢿⣸⣷⢫⣾⡁⢠⠆⣠⢸⣿⣿⣷⣀⣿⣧⠀⠈⢹⣿⣥⣉⠉⠙⠛⣷⣶⣤⣀⠀⠀⠀⣴⢟⣵⠿⡟⢻⡾
⠀⠈⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⡿⢓⠋⣆⣷⣮⢿⣿⣿⣿⣿⣿⣷⠀⠀⠙⡄⢉⠛⢦⡛⠿⠿⣝⡋⠻⣶⣿⣿⣯⣞⣳⣤⢸⣷⣀⢀⣀⣤⣶⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡶⢃⠀⢹⢻⣥⣄⣈⣉⣀⠈⢻⣿⡇⠀⢀⣻⡈⣷⡀⠑⠀⠠⡈⠳⠀⢦⠉⠋⠛⠉⣿⣿⣷⣭⣿⡿⣿⣿⣯⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣾⢝⡓⡒⠂⠀⣉⡛⣷⠄⣸⣷⣿⡞⣿⣿⣿⠿⣥⡀⢀⣿⣿⡀⠀⣿⢣⣿⣷⠈⢶⣄⣿⠀⠀⣸⡷⢰⠀⠀⠻⠳⣽⣿⡝⢶⣾⣷⣘⣧⣾⠇
⠀⠀⠀⠀⠀⠀⢰⣿⠋⠁⠒⠶⣶⣶⣭⣍⣷⡟⣿⣭⣾⣻⣿⣿⣏⣀⣛⣋⠛⠿⠃⠀⣡⣿⠏⠛⣴⡀⢹⣏⠀⣠⡟⢠⡇⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠀⠉⠉
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠰⣾⣿⡷⠿⠛⠛⠛⠋⠉⠉⠉⠙⠛⠿⢿⣿⣁⣉⣉⣥⠴⠾⠛⠛⠻⢮⡙⢷⡀⡇⠸⢛⣻⣿⠁
⠀⠀⠀⠀⠀⠸⣿⡃⠀⠀⠉⠛⠻⢷⣦⣤⣀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣳⡜⠐⡛⣿⣿⣗⣒⡤⠤⢈⣷
⠀⠀⠀⠀⠀⠀⢿⡿⣤⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣿⣽⣮⡭⠭⣵⣖⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠈⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠹⣿⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⣏⣾⣗⣺⣷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡻⣉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⣿⣿⣿⡿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣶⣤⡄⢠⣀⣴⣿⠟⢉⣴⣿⣿⣿⣻⣷⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣁⣉⠉⣉⣠⣾⡿⠿⣄⣠⣿⣿⣿⠋⠙⢻⣿⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⡶⠻⣿⣿⣿⣿⣶⣶⣶⢿⣿⣏⣀⠈⠻⣿⠏⢿⣿⠋⣠⡄⢀⣿⣿⡿⠿⠞⠛⠉⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⣼⣿⡇⢶⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣷⣦⡑⢼⣿⣿⣿⣀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢠⣿⣿⣷⡈⠻⣿⣿⣿⣿⣿⣿⣿⠘⠉⠉⠁⢳⣈⣿⣆⣿⣿⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⢘⣿⣿⣿⣷⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣸⣿⣿⣿⡿⠀⡿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⣸⣿⣿⣿⠿⣿⣾⣿⣿⠏⠉⠛⠿⢿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣅⠻⠁
⠀⠀⠀⠀⠀⣠⣾⣿⠟⢁⣼⣿⣿⠟⠁⠀⠘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⢻⣯⣿⠏⠙⣿⣧⡄
⠀⠀⠀⠀⠀⠉⠉⠀⣸⣿⡟⠋⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇
⠀⣠⣤⣤⣤⣠⣬⣻⣲⣶⣤⣤⣤⣤⣄⡀⣀⢠⣤⣤⣤⣹⣷⣤⣤⣤⣶⣌⣿⡄⡀⣀⣀⡀⣄⡀⣴⣿⣧⣠⣶⣶⣶⣦⣦⣦⣦⣠⣤⡀⡀
⠀⠀⠈⠉⠀⠉⠉⠁⠀⠀⠈⠈⠁⠀⠁⠁⠈⠉⠉⠉⠛⠙⠋⠙⠛⠛⠋⠉⠉⠋⠀⠙⠛⠛⠛⠛⠋⠀⠈⠉⠁⠉⠉⠉⠁⠈⠉⠁⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡉⠆⠀⠀⠀⠀⠀⣤⢾⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡏⣻⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⠻⠿⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣀⣀⣸⣿⣿⣿⣿⣿⣿⡀⠀⠀⡀⠼⠻⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⢴⣾⣿⣿⣧⣽⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣶⡎⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⣀⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠾⠟⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⣀⣰⣿⣿⣿⣾⣏⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⠄⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⢿⢿⣿⣿⠟⣛⣿⣏⣿⣿⡿⢿⣶⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣹⣿⠿⠟⣿⡇
⠀⠀⠀⢀⣴⣿⣿⡟⠁⢹⡟⢿⣿⡿⣸⣿⣿⣽⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠻⠤⠿⠿⣿⣶⣶⣿⡇
⠀⠀⠀⣸⣿⣿⡟⠀⠀⠘⣿⣶⣶⣾⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣿⣧
⠀⠀⠀⢿⣿⣿⣇⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠙⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠀⠹⣧⡀
⠀⠀⠀⠸⣿⣿⣿⣿⣄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣯⣵⡄⠀⠀⠀⠀⠙⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠙⠛⠹⣿⣿⣇⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⢠⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣍⡛⠿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⠙⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣦⡀⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⣠⠁⠈⠀⠘⠀⣨⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣆⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢰⠓⣆⠀⠀⠀⠀⠡⠘⠀⠑⠴⠃⢀⠝⢁⠜⠀⠀⠀⠀⠠⠂⡆⢀⠀⠀⠀⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠆⢤⠈⠝⠁⠀⠀⠀⠀⠀⢡⠀⠀⠀⠂⠡⠊⠀⠀⠀⠀⠀⠔⠘⢄⢣⡀⠰⡀⠀⠪⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠈⢁⠞⡠⠐⡲⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⡤⠐⠒⡚⠲⣄⠙⢄⠀⠈⠠⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠠⠊⠀⠐⠁⠀⠀⠘⡀⠀⡠⠄⠜⠀⠀⢀⡐⠂⠂⠙⠀⠈⠆⠀⠀⣀⠴⠋⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠩⢧⠑⠀⠂⠀⢀⠐⠀⠀⠉⠀⠀⠸⠷⠿⠁
⠀⠀⠀⠂⠀⠂⢁⡀⠀⠀⠀⠈⠹⣿⣒⡀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠁⠀⠀⠈⠒⠉⠀⠀⠀⠠⡔⠁⡀⠀⢀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⠌⠁⠘⠃⠀⠀⢄⡀⠀⣀⣀⣀⣈⣤⣄⣉⣁⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠃⠀⢀⢀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄⠘⠀⡀⠁⢠⣾⣿⠭⠯⠿⠥⠿⠧⠷⠧⠽⠯⠽⠯⠿⠭⢷⣭⣭⡟⠻⢖⡶⠂⠀⠀⠤⠔⠁⠐⠒⡄
⠀⠀⠀⠀⠀⢀⣠⡴⢚⣋⣽⡵⡄⡘⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢤⠀⠀⠀⠀⣤⣠⣶⣄⡀
⠀⠀⠀⢀⡴⢫⡴⠖⠛⠉⡀⠐⠀⠀⠀⠀⢸⠤⠀⠀⠀⠀⠀⡀⠼⠟⠏⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡄⠈⠙⢮⣽⣦
⠀⠀⠀⡼⣶⠋⠀⠀⠀⠀⠀⠀⠀⠤⠤⠰⠀⠄⠰⠀⠀⠀⠞⠀⢀⠀⠤⢠⣵⡰⠂⠢⠒⣉⠆⠀⠀⢀⢀⠀⠀⠀⠠⠂⠁⠤⢀⠈⣯⢘⡇
⠀⠀⢰⡗⡇⠀⠀⠀⠀⡴⢁⡀⢸⣄⣀⢴⠂⠀⠐⡄⠀⠈⠃⠠⡤⠤⠀⢋⡽⠃⠀⠀⠉⢀⢄⡁⠀⠘⠄⡠⠐⠃⠒⠂⠰⠤⢁⡴⢉⡝⡇
⠀⠀⢸⠻⣿⣦⣄⡀⠀⠁⠐⠒⠀⠀⠀⠀⠀⠚⠁⠀⠀⡀⡠⠌⣡⢩⠕⠊⡠⠦⠚⠴⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣫⡴⢋⡔⠁
⠀⠀⠸⣄⡈⠉⠛⠻⠿⢶⣶⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠧⠤⠊⡔⢉⠔⠊⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡶⡶⣾⣻⡯⠟⢛⣡⡶⠋
⠀⠀⠀⣸⠉⠉⣶⠶⠦⠤⠤⢉⣉⡛⠛⠓⠛⠻⠿⠿⠿⢷⣶⣶⣿⣿⣖⣲⣒⣖⣲⣻⡿⠿⠿⠿⠷⠒⢛⣛⣋⣩⡥⠴⢺⡏⠀⢹⡀
⠀⠀⡼⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠓⠒⡶⠒⢶⡖⠒⠒⠒⠒⠒⠒⠒⠚⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⢳⠀⠀⣇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣇⣴⣾⣿⣿⣿⣤⣀⣀⠀⠀⢀⣶⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⡁⠀⠀⠉⠻⣿⣿⣇⢰⡇⣿⣿⣿⣿⣿⢿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠉⣉⢻⣦⠀⠀⠀⠈⣿⣷⡿⢳⢿⣿⣿⣿⣁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⡝⠃⠀⠀⠀⠸⣿⣵⣿⣻⡻⣿⡿⠃⠀⠀⣀⣀⣤⣤⣾⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠗⢾⣇⠀⠤⣀⠀⣈⣿⣶⣭⡛⢿⠃⠒⠅⠀⠛⠿⢿⣿⣿⣿⣿⣯⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠼⠪⠯⣄⠀⢀⣼⣿⣿⡿⣩⣿⣶⠅⢀⡠⠀⠘⠂⠐⢂⠈⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⡀⢀⣼⣿⣿⣿⡿⠁⠘⣿⣿⠶⣭⣤⣶⣶⣶⣿⣦⠓⢆⡀⢿⣿⣧⡀
⠀⠀⠀⠀⠀⣀⣼⣿⡷⠄⠀⠀⠀⢸⣿⣟⣛⡻⠿⠱⣖⣦⣾⣿⣷⣹⣿⣿⣿⣿⣿⣿⠶⠚⠛⠁⠀⠙⢻⡀
⠀⠀⠀⠀⣾⡟⠁⠀⠃⠀⠀⠀⠀⠀⢹⣿⣿⣷⡀⠀⢀⣴⡻⠿⣿⡿⠽⠏⠈⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢹⡷⠄⠀⠀⠄⢀⣀⠀⢺⣸⡈⠙⡟⣼⣺⡅⠻⣿⣿⣿⠂⠈⠻⠆⠐⣠⣶⣦⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠓⠻⠇⢇⠀⢑⣾⣭⣴⣾⡿⠛⠁⠀⠀⠀⠄⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⣾⣱⢿⣿⣿⣿⣻⠎⠀⠈⠁⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣴⣄⠀⠈⠙⠻⠿⣿⡇⠀⠀⠁⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠐⣄⠀⢀⢐⣈⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣔⠨⣝⣻⣿⣿⣿⣿⣿⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⡖⠙⢀⠦⠙⣿⣿⣿⣿⣿⣟⠂⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠁⠀⠀⠀⠈⠈⠁⢼⡟⢹⣿⣿⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠘⠙⠀⠀⢹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣏
⠀⣀⡀⢀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⣀⣉⡍⠀⠀⠀⠀⡀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠂⠀⠄⠀⠠⠀⠀⠀⢀⣠⡤⣾⠄⠬⡧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠽⢧⡄⠀⠀⠀⠀⠀⡠⢞⠁⢰⡇⢙⠡⠳⠤⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⢼⣿⣿⠟⢁⠀⠀⠀⠠⠆⠀⢀⡼⠓⠀⠀⢘⡰⠁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠂⠈⠿⠙⢯⠀⠀⡀⠀⠀⢠⢠⣾⠌⠀⠀⠠⢸⣇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠠⢈⣉⠀⢠⠀⣀⠂⢸⡁⠀⠀⠀⠀⢸⠘⣆⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠀⠀⢡⠀⠀⡈⠁⠂⠲⠮⡀⣏⡆⠀⠀⠀⠀⠈⢧⢻⣆⡴⠛⠉⢙
⠀⠀⠀⠀⠀⠀⡇⠠⠙⠄⠀⠇⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢀⠇⠀⠀⢻⡏⠀⠒⠀⠤⢠⡀⠈⢿⣥⡄
⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⢤⠔⠃⢠⡀⠀⢸⣧⠀⠀⠀⠀⠀⠁⠉⠙⠒⢀⠚⠁
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⢀⠀⠂⠈⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠨⠡⡀
⠀⠀⠀⠀⢀⠀⡠⠂⠀⠀⢀⠀⢀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⢀⠀⠁⠐⠄⡀
⠀⠀⠀⠀⠃⢀⠁⠀⠀⠄⢠⡑⠈⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠐⠢⣄⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠘⡆⢸⠀⠀⠀⠃⠀⠀⠑⢦⡠⠜⠃⠉⣿⠀⠀⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⠀⠀⠀⡤⠊⠀⠀⠀⠀⠍⠀⢀⣷⣶⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣭⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠐⠀⠀⠓⢄⣐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡅⠀⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠠⠒⠀⠀⠂⠀⠀⠀⡀⠠⠀⠠⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠆⠀⠀⠀⠀⠃⠀⠀⠛⠿⠛⠄⡀⠀⢀⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠠⣾⣿⡀⠀⠀⠀⠏⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⢘⠟⠷⠄⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠔⠦⢄⡀⠀⠀⠀⠀⠀⠁⠀⠑⠄⠀⠈⠉⡠⠀⠀⠁⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⡀⠀⠀⡀⠓⠄⠀⠀⠄⠛⠐⠀⠀⠤⠀⠀⠀⠴⠀⠈⠁
⠀⠀⠀⠀⠄⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠒⢀⣀⠉⠉⠉⠉⠉⠉⠉⢁⡀⠒⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠑
⠀⠀⠀⠀⠀⠀⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢂
⠀⠀⠀⠀⢀⠊⡠⠁⠀⠀⠀⡠⠀⠀⠀⠂⠀⢀⡀⠤⠚⠛⠉⠁⠀⡀⠀⠀⠀⣀⠐⠂⠄⣀⠀⠀⠀⠀⠀⠀⢄⠀⠁⡀⠡⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠄⢺⡆⠀⡀⠀⠀⠀⠀⠀⠁⠢⣄⡀⠁⠂⢄⡀⠁⠢⢤⠄⡀⠀⠀⠐⠀⠐⡀⠡
⠀⠀⠀⠀⢠⠁⠀⠀⠠⠀⠀⠀⠀⠀⠔⠁⠀⠀⠹⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠈⠁⠀⠀⠀⠀⣀⠀⠀⠠⠀⢡
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠌⠀⠀⠠⠀⠀⢈⠳⢄⡀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠐⠀⠠⠴⠞⢛⠀⠀⠀⠀⠀⠂⠀⢀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠔⠀⢀⠔⠁⠀⠀⠐⠡⠀⡐⠀⠠⠤⠠⠤⠤⠉⠙⠛⠋⣠⣶⣊⣽⠀⠀⠀⠀⠀⠐⠀⠀⡀
⠀⠇⠀⠀⠀⠀⠀⠀⠄⠀⠦⣠⡀⠊⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⡀⢈⡛⠀⠀⠀⠂⠀⠀⠐⠋⠀⠤⢠⡤⢠⠃⠀⠀⡀⠀⠀⠀⠀⢁
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘
⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡇⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠘⠄
⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠁⠄⠶
⡇⠀⠀⠀⠐⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠚
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢀⠖⠓⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⠀⠈
⠈⡄⠀⠈⠁⡀⠁⠀⠉⠈⠁⠒⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠠⠀⠀⠉⠀⠈⠉⠉⠀⢀⠈⢂⢀
⠘⠵⠀⠁⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠾
⠀⢖⣫⠀⠀⠰⠀⠀⠀⣀⣀⡀⠀⠤⢐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠠⠄⢀⣀⣀⣀⡀⠀⡈⠀⠀⠜⣘
⠀⠈⠒⠒⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡰⠒
⠀⠀⠀⠀⠐⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠾⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠰⢏⣉⣉⣩⣉⠉⣑⠒⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⠘⠃⠀⠀⠀⠀⠈⠉⠁⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⢒⣀⣤⣤⣀⡐⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢀⣀⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡜⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣤⣤⣤⡉⠻⠋
⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⣿⣿⣿⣿⣛⣷⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠙⠋⠉⠉⠉⠀⠀⠤⠄⢘⣷⡆⠀⢀⣠⣴⣟⣉⣿⣶⣤⣭⣹⡇⠀⠀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⣟⢠⣤⣤⣴⠟⠃⠀⠀⡟⢻⣿⣿⣿⣿⡿⠏⠁⠀⠀⢀⠀⠃
⠀⠀⠀⠀⣀⣀⡀⢧⣼⣴⣧⣽⣟⢉⣉⠷⡖⠷⣾⣟⣿⣮⣿⣿⣿⣿⣇⣀⣤⣴⠆⠀⠀⠀⠄⣀⠂
⠀⠀⠀⠈⠈⠉⠉⠁⠹⣿⣿⣯⡛⠉⢉⣁⣾⣿⣿⣿⣠⣿⣾⣿⢟⣿⠿⢩⣿⠿⠖⢠⡄⠉⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⣟⢁⡈⣰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣀⣤⣶⢱⣿⣿⣿⣿⣯⣿⢷⣿⠿⣿⠷⠤⣾⡆⢸⣿⣿⣿⣾⠳⠀⠀⠀⠀⠀⢀⣠⣾⡶⢛⠅⠀⠀⠉⠙⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⡇⣸⣿⣴⠏⠐⣶⣤⣤⣬⣉⣉⣹⠟⠀⠻⠶⠶⣶⣶⣭⣄⠙⠷⣬⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣭⣭⣭⡄⠀⢻⣿⡌⢋⣽⣿⠿⠿⠷⢶⣶⣤⣬⣅⢐⠀⠀⠀⠈⠙⠺⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⡇⠀⠘⣿⣇⢸⣿⣶⠇⠀⠠⠆⢸⣤⣩⡇⠈⡄⠀⠀⠀⠀⠀⠀⢻⠶⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠃⣾⣿⣿⣿⣿⣿⠀⠀⠿⠿⢸⣿⣿⣷⣦⣤⣤⣾⣴⡿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠉⠉⡙⢻⣿⣟⣴⡇⣿⣿⣿⣿⣿⣶⠒⢲⣶⣤⣿⣿⡿⣿⣿⣿⣿⣿⠷⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣵⠄⠀⠀⠸⣦⡿⢿⣏⣹⣮⣿⣿⣿⣿⣿⡀⠀⣿⡇⣿⣿⣿⣿⣯⣿⣿⣷⣶⣶⣤⣤⡀⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠁⠀⠀⢀⣠⣽⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡛⢻⡀⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠙⢿⣷⣾⣿⣿⣿⣿⣏⣾⣿⣿⣏⠀⠀⢾⡿⢿⣿⣿⣿⣿⠛⠋⠙⢿⣧⣼⣅⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⣻⣿⣷⣼⣿⣿⣿⡁⠐⠈⣿⣿⣷⣶⣶⣾⢑⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣠⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢸⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣼⢻⣿⡷⢹⣿⣿⣿⣿⣷⣿⣿⣧⠀⠀⢸⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠟⣯⣾⣿⣿⣿⡆⠀⠘⣿⣛⡧⠀⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣤⣿⡀⠀⢶⡆⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⣿⣼⣿⣉⢿⣿⣆⠀⠀⣿⣯⣻⣠⣿⣿⣿⣿⡟⣿⣿⣷⠤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⢣⣿⣧⣿⣶⣶⡀⠀⢨⣿⣿⣿⣿⣽⣿⣿⣿⣿⣟⡏⠀⠙⣻⡇⠀⢹⣿⣿⣿⣿⣿⣿⡏⠀⢿⣿⣿⠦⠬⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⠿⠇⢀⠤⠙⠋⠛⢿⣿⣿⣿⣿⣿⡿⣾⠃⠙⠶⢼⡇⠀⢸⣿⣿⣿⣿⣿⣷⣝⢳⣾⠘⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⣆⣴⡞⢉⠴⠈⠀⠀⠹⣿⣿⣿⣿⣿⢧⡆⠠⣤⡀⢠⣿⠀⠀⣟⠛⣿⣿⣿⣿⣟⢧⡿⡀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠟⠐⠁⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣸⠇⠀⠀⠙⠋⢹⣄⠀⢿⣿⣿⣿⣿⣿⣿⣿⣵⣧⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⢻⣿⠿⠀⣤⣤⣤⣿⣿⣿⣿⠛⣿⣿⣿⡏⡏⠀⠛⠶⠶⠶⢾⣿⠀⢸⡟⢻⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡌⣡⣿⡼⣿⣿⣿⣿⡿⢉⣿⣾⡿⣿⡟⢻⡵⣤⣀⠀⣀⣀⣤⣷⡄⠀⣿⠉⠛⠛⢿⣿⣿⡿⣿⡁⢈⣿⡧⣤⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡛⠿⣿⣟⣴⣿⣿⣿⣿⢗⣽⣧⣼⣷⣟⣟⠈⠀⠀⠉⠛⠉⠉⠉⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿⣶⣾⣿⣿⣿⣇⢤⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⣬⣟⣿⣿⣿⣿⣿⣷⣾⣿⣯⣉⣁⡀⣀⣈⣙⠛⠛⠛⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢠⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣋⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⠁⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⡿⢋⡿⠿⠿⠛⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠺⠯⣽⣿⣿⣿⣿⣿⣿⣿⣃⣴⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⠻⠿⠟⢁⡾⠁⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⣿⣿⣿⣿⣦⣀⣀⣴⠟⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡉⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣧⡀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠛⠉⠁
⣴⣶⣴⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⢣⡤⣼⡟⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠂⣊⠁⡄⠀⠦⠀⠀⠀⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡡⣠⣤⣾⡇⠐⣾⣶⣀⣀⣀⣀⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠾⠋⠉⢱⡆⠙⢯⣿⠿⣿⢿⡏⣿⡟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢫⡞⠁⠀⣠⠀⢀⠻⣶⡄⠙⠛⠛⠿⠟⣡⠐⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠂⠠⠞⣯⠀⡸⠀⡽⣦⡀⠀⠀⢀⡸⢻⠰⡇⣈⡄⠀⠀⠐⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠒⠒⢒⣱⣿⣾⣿⡽⣄⡔⢿⣦⡴⠿⡃⡾⣴⣿⣿⣶⣀⠊⠐⠂⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢠⠘⣿⡿⢻⣿⣿⠦⢾⣤⠤⠤⣤⠤⡂⣧⠟⠉⠉⠙⠻⢿⣷⣤⣤⣄⡀⠑⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣼⢆⣾⣿⡟⣻⣿⣍⣩⣩⣿⣛⣻⠿⠿⢿⡛⣣⣴⣶⠞⠛⠉⠒⠺⢧⣤⢙⣟⣿⣡⡈⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢿⣿⣿⣿⣿⠟⢯⣉⣩⠍⠀⠈⠹⣏⣳⡛⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣃⣅⣶⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⢿⣿⣿⡇⢠⣤⣤⠆⠀⣀⠄⡘⠉⠙⠓⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠷⠶⠾⠟⢸⡇⢠⠟⠁⣠⣾⠃⣰⣧⠑⢄⠀⠀⠀⠀⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⣿⠧⣀⣿⠟⠁⠀⢮⡀⠀⡄⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⢀⣠⣶⣿⣿⣷⣤⡀⣿⠯⣰⣿⠋⠀⢀⣩⠉⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣍⣽⣿⠹⢿⣿⣿⣿⣿⣿⠏⠀⣼⣁⣀⣰⣿⠡⠞⣿⡯⣿⣥⡀⡀⢘
⠀⠀⠀⠀⠀⢀⣰⠾⢛⣭⠓⠉⠉⠛⠶⣾⣿⣿⣿⣿⡏⠀⣸⣿⣿⣿⣿⣏⡄⢸⢾⣷⠄⠠⠤⠀⡧⠀⢀⣠⡀
⠀⠀⠀⠀⣰⣾⠍⡰⠎⢀⣠⣤⣤⣤⣀⠀⠻⣟⢿⠟⠀⢠⣿⡿⣿⣿⣿⡇⠀⣸⣈⣿⡷⠀⠀⢠⠁⠄⠋⠙⠛
⠀⠀⢀⣼⠟⡆⢦⢁⠔⠁⢠⡿⢈⣿⣿⣷⣣⢉⡾⠀⢸⣿⢻⣟⡿⠿⠷⢇⣼⣿⡈⢿⠐⠂⠀⡖
⠀⠀⣨⡛⠿⠛⡮⠌⠀⠀⢿⠀⢀⣿⣿⣿⣿⡏⠀⠀⣾⠧⡤⢿⣿⣿⡉⣩⣿⣾⣷⢠⢮⣤⡄⣇⡀
⠀⢀⣾⣏⠐⠋⠙⠀⣼⡀⣁⡤⣪⢉⣽⣿⡟⠀⠀⣼⣧⣶⡦⠰⣿⣿⣯⣽⡿⢿⣿⠀⣭⠀⣴⠋⠉⠳⡄
⠀⢈⣧⡆⢘⡁⡇⠠⣌⣷⡿⡄⠩⣤⡵⣿⡄⠀⡐⢹⢛⣏⣹⣆⠠⡌⠀⠀⠀⠀⠾⠀⢿⠆⢺⢻⣿⣆⢇
⠀⢸⣶⣄⣋⠁⡇⢸⣼⣿⡇⣄⢸⣾⣷⣿⡄⢲⡇⢰⠘⣦⠬⣿⣆⠙⢤⣰⠀⣘⣷⣍⡠⠂⠀⠻⣿⣿⡈
⠀⢠⣽⣼⡟⠁⠃⠀⢿⣿⣷⠙⣈⢛⢟⢿⣿⣿⠁⡾⢷⣜⠻⣻⣿⣧⣀⠙⠓⢻⣷⣷⣶⣶⣦⣤⣴⠿⡇
⠀⠀⣿⠾⣿⣋⠌⠀⠈⠙⠉⠱⢭⢮⠕⡿⣿⠗⢘⡃⠀⠉⠙⠚⠛⠛⠋⠿⣿⣿⣿⣿⣿⣷⡈⣷⣄⣰⡇⠃
⠀⠀⢹⣿⢿⣏⡀⠈⡀⠀⠀⠀⠀⠀⡐⢠⠣⢁⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣿⠿⣷⣸⣿⣿⢁
⠀⠀⠀⠻⣿⡿⣷⡀⠑⢤⡀⣀⠀⣪⠔⡡⠁⣨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⡅⠀⠁⠐⠋
⠀⠀⠀⠀⠘⢻⣽⣿⣇⣄⡀⠀⠀⠀⠈⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠙⠳⣿⣿⣷⣟⣰⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠼⠁⠀⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠟⢛⣛⣿⣿⣖⠂⠀⠀⠹⣿⡒⠄⢀⣀⣤⣤⣤⣶⣶⣶⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠠⣀⠀⠶⠟⠀⠞⣛⡛⠋⠉⠉⠛⠛⠛⠿⠿⣿⣿⣿⣿⡿⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠀⠀⠀⠀⠀⢀⠀⠀⠡⠀⠀⠀⣨⣿⣿⠷⣶⠾⠛⠛⠛⠻⠿⠷⢿⣿⣿⣿⣷⣝⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣟⠀⠀⣀⣠⡤⠤⠶⣛⣦⣄⡀⢠⣾⡿⠁⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⡋⢦⣽⣿⡗
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠁⠀⡀⠀⠀⡠⠖⠋⠉⠀⠈⠙⠿⠋⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡯⠀⠀⠀⠀⢳⢰⡝⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⡶⠖⠛⢺⣆⠀⠀⠀⠀⠀⠋⠹⡆
⠀⠀⠀⣠⠄⠀⠀⠴⠶⣿⣿⣧⠀⢰⣿⣷⠞⢸⡇⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠉⠙⠶⣄⡈⠻⣷⣦⣄⣀⡀⠀⠀⠹⡄
⠀⠀⠀⠈⣷⣶⠶⣶⠀⢸⣿⠟⠀⣾⣿⠋⣠⠟⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠙⠳⠦⣌⡙⠻⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠙⠛⠀⡏⠀⠀⠁⠀⠀⣿⠁⠔⠁⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⡇⣷⡉⡿
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⠠⠚⠁⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢒⣲⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠰⣄⡀⠀⠀⣀⣠⡤⠶⠛⣛⣛⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⣛⠟⠉⠉⠀⠐⠋⠉⣡⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠋⠁⠀⠀⢀⣀⣀⣤⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⠶⠶⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠒⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⡦⣤⠄⠊⠁⠀⠀⠀⠀⠀⠀⡖⠊⠉⠁⠀⠈⠉⠉⠘⠣⣀⠀⠀⢀⡆⢑⣦
⠀⠀⠀⠀⠀⠀⢀⢳⣬⡷⠧⣀⡗⠒⠒⠒⠒⠒⠒⡖⠲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠷⠺⠧⠺⣤⣟
⠀⠀⠀⠀⠀⠀⠈⠛⢇⠀⠀⠉⠄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠀⠝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠴⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠘⢄⢀⠀⠰⠆⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠆⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⡄⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⡄⠌⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢖⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢸⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠓⡀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⡀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⣅⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠈⠑⠤⣀⣀⠤⠊⠀⠀⠀⠀⠀⠃⠀⠀⢰
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢨⠀⠀⠉⠉⠘⡆⠐⠒⠒⢅⠀⠀⠀⠀⠈⡢⠤⠤⠆⠒⠂⢙⠂⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠒⠂⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠊⠁⠀⠀⠀⠀⠀⢸⡄⣀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠀⠀⣀⣀⠐⠂⣤⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠠⠆
⠀⢀⠆⠈⠒⢤⣀⠀⠀⣠⢄⠀⠀⠀⠘⡦⣀⠀⠀⠀⢠⢃⠀⠀⠀⠀⠀⠀⠐
⠀⡌⢀⠋⣦⣁⡈⠍⠃⠉⠒⠈⠲⠦⢠⠀⠀⠉⢰⠂⠂⠈⠒⠒⠒⠒⠲⠉
⠰⢀⡴⠋⠀⠈⠉⠓⠤⠀⡀⠀⠀⠀⠀⠀⠀⢀⠊⠘⠂⠐⠃⠀⠀⢀⠃
⠁⠉⠀⠀⠀⠀⠀⢐⡀⠔⠉⠈⠐⠠⠄⣀⣀⠬⢀⡀⠀⠀⠀⠀⠀⠎
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣒⣤⣤⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡙⣋⣄⣤⣤⣴⣶⣶⣷⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠟⡍⠹⡶⠊⠛⠻⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡞⠀⠰⠆⢁⡘⠃⠀⠈⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⣻⠃⢴⡀⠀⠀⡀⠀⠜⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡝⣆⠀⠀⠀⠀⠐⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢳⢉⣼⣶⣤⡄⠀⣶⣿⡶⠀⠀⠀⢀⠀⡀⣀⣆⠔⢰⡷⣐⣄⢀⣴⣾⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠗⡃⣸⣿⣿⡿⠛⢺⡟⠻⣷⡀⣀⣰⣮⣿⣮⣿⡿⡄⣾⣟⣻⣿⣦⣻⣿⡿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠛⣵⣿⣿⣿⣷⣶⣼⣷⣶⣾⣿⣿⣿⣿⣿⣿⢿⡇⡨⢟⠦⣉⣿⣿⣭⣋⠙⠓⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣕⡻⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⣸⡈⡀⢀⣅⢨⣼⣌⡓⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣩⠃⠀⠀⢰⣿⡿⠋⠀⣘⣿⣿⣿⣿⠟⠁⠀⠀⡿⢻⠃⠘⢃⠎⡜⢟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡐⠀⠀⣰⣿⠋⠀⠰⠾⣿⣿⣿⠟⠻⣄⠀⠀⠀⡇⢸⠆⢠⡾⠀⠉⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣟⡟⣃⣠⣶⣿⠇⠀⢀⣴⣿⣿⠟⠁⠀⠀⠈⠳⢤⡀⢳⡀⢟⣧⡶⠤⠤⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠡⠈⠉⠉⡍⢉⣿⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⠙⠲⣿⣿⣯⢴⣶⠆⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⡙⠁⠀⠀⡌⢀⣿⣇⣠⣿⡟⠇⠠⡀⠘⢄⠀⠀⠀⢀⠃⠀⡠⠨⣝⢃⠘⠛⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠸⠁⠀⠀⢰⠀⣸⣿⣿⡿⣿⢰⠀⠀⠈⢂⠈⡄⠀⠀⡄⠀⠠⢇⠀⠉⠟⠛⢄⣂⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠃⠀⠀⠀⡄⠀⣿⣿⣿⣇⣿⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⡠⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⢢⠺⠀⠀⠀⡐⢄⠹⣿⣿⣹⡿⢨⠀⠀⢀⡀⠐⠀⠀⠀⠀⠀⠆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⣁⣵⡀⠀⠐⡀⠈⣿⡿⣻⣶⡆⠀⠠⠄⠈⠑⢲⣶⣤⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⣃⣸⠈⠢⡀⠈⠶⢌⡉⠙⢄⣡⡀⠀⠀⠀⣀⢼⣿⡿⠿⢦⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠙⡅⡸⢠⣴⠙⣄⣼⣎⠀⠀⠈⠁⠈⠀⠉⠓⠂⢠⠀⠉⢢⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢈⠞⠉⠇⠘⣿⣦⡻⢿⣁⣈⡉⠉⢆⠀⠀⠀⠀⠀⡆⠀⢀⠎⠐⢄⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢻⠄⠎⠀⠀⠛⠛⠇⠀⠀⠀⠑⠆⠈⢢⡀⠀⠀⢠⣤⣤⡎⠀⠀⠢⠠⠤⠺⡀
⠀⠀⣠⣄⣇⢂⡄⠀⠀⠀⢀⢛⡅⡔⠁⠀⠀⡄⠀⠇⠀⠀⠀⠀⠳⡀⠁⣹⠀⠀⣾⣿⣿⠀⠀⠀⠰⢀⣀⣀⠔
⠀⣻⣿⣿⣿⣿⣇⠀⠀⣠⣾⣧⠔⠁⠀⠀⠀⢈⠀⣉⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⢙⠃
⠀⠻⠿⢿⠿⠿⠿⠿⠤⠛⠉⠁⠀⠀⠀⠀⠀⠈⠥⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠐⠘⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠖⠒⠒⠒⠲⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⢩⠡⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⢸⡃⠘⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⢠⣶⡵⠮⣡⣆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⡸⣯⢻⡟⡻⢿⡆⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢡⡬⣀⠈⠀⣼⣦⡄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠄⠀⠀⣾⠿⠽⣗⡴⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⢀⠈⡖⠒⢳⣿⣿⢿⣧⣵⣤⣍⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⣿⣷⣴⣿⡿⣿⣷⣿⣇⣿⣭⣿⣿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⠀⣠⡄⠀⠊⣴⡿⡻⣿⠏⠀⢤⣾⡿⡹⣿⣿⣿⣿⢸⣿⠀⠀⠀⠀⣀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢸⣿⣀⣿⡇⠀⠒⣅⠀⠀⠁⠐⠒⠻⣥⣴⣾⢻⣻⣿⡿⢸⣿⢰⣿⡇⠀⣿⣿⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢸⣿⠉⣿⡇⠀⠀⠹⠧⠂⠀⢠⠼⡆⠙⢿⣿⡄⠻⢿⣿⡸⣿⣾⣿⡿⠿⣿⣿⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⣿⠀⣿⡷⠶⠶⠶⣦⣤⣤⣤⣤⣄⣀⣀⣉⣿⠄⠀⠉⢻⣏⢻⣿⡇⠀⣿⣿⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⣟⡛⠛⠛⠛⣿⣿⡇⠀⣿⡏⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⠀⣿⡟⠛⠶⠶⠶⢶⣶⣶⣤⣤⣤⣄⣀⣀⣙⢛⣀⠀⠀⢸⣿⡇⠀⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⢿⣿⣇⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⣀⡀⠀⠀⠀⢀⣀⣀⣤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡠⠞⠑⠦⣄⡤⠖⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣤⠏⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡄⠀⠀⠀⡆⠀⠀⠀⣺⠀⠀⠀⢀⠀⠀⠀⠸⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⡤⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢰⠳⢤⣀⡤⠖⠲⣄⡴⠚⠲⠦⣀⡤⠶⠦⢤⡈⠁⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣡⠖⠓⠲⠦⢥⣀⡔⠛⡩⠶⡞⡷⢾⡙⠳⣤⠔⠒⠛⠲⠦⢤⡈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡾⠁⠀⠀⠀⠀⠀⢸⡀⠀⠠⠄⡿⡇⠤⠀⠀⡸⠀⠀⠀⠀⠀⠀⠑⣾⣷
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠃⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠾⠓⠻⠒⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⣠⢴⡳⣴⠋⡆⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣀⣄⢀⣀
⢸⡙⠆⠁⠀⠀⡽⢦⣀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢠⡇⠘⠋⠼⣻⢤
⠳⣄⡀⠀⣠⠞⠁⠀⠈⠓⢮⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡡⠖⠉⢧⠀⠀⠀⠁⠾⡄
⠀⠀⠙⠫⣅⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡾⠗⠉⠀⠀⠀⠈⠓⠤⣠⠴⠊⠁
⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠈⠻⢽⡒⠲⠤⠤⠤⠤⠤⠤⢒⣒⣯⠔⠋⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀⠀⠀⠀⠀⠀⣼⠉⠒⠲⠶⠖⠒⠚⠋⠉⠀⢳⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⢘⠞⠋⠛⠓⢦⡀
⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣀⠴⠋⠁⠀⠀⠀⠀⢣⡀⠀⠀⠀⣇
⢰⠁⠈⠁⠷⠒⡄⢀⠀⠀⠀⠀⠀⠙⠦⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⢰⣧⡀⠀⠀⠀⠓⢩⠃⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠺⠥⢤⡀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠻⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣀⠀⠀⠀⠀⣠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠒⠒⠢⠤⠤⢤⠤⠴⠒⠒⠒⠒⠒⢹⠗⠒⠒⡏⠉⢻⣦⡀⡰⠁⠷⠋⠿⠖⣦
⠀⠀⠀⠀⠀⠀⣀⣴⡀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢳⠀⠀⣿⡟⡇
⠀⠀⠀⠀⠀⡜⠁⠀⢹⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⢸⠀⠀⡀⡇⢱⣰⠲⣄⡶⢤⠟
⠀⠀⠀⠀⠀⢇⠀⠀⢀⡇⠀⠀⠀⠀⠙⢦⣀⣀⡀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣒⢶⣿⠇⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠈⢦⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠒⠒⠶⣶⣶⡶⠋⠉⠉⠁⠙⠓⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣤⣠⣤⡀⠀⠀⠀⠀⠀⢠⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠙⣦⣞⠺⠃⠀⠀⡽⢠⠒⠢⠴⣤⡜⠀⠀⠀⣿⣶⡶⠒⠲⢤⡀⢀⣤⣠⠤⣄
⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⣿⠀⠀⠸⡙⠋⠀⠀⠀⣤⢼⡗⠀⠀⠀⠛⣿⣷⡖⠢⠗⠘⣷⡄⠀⠀⠘⡶⠇⠀⠀⢸⡀
⠀⠀⠀⠀⣠⣴⣟⣀⠀⣀⣸⡄⠀⠀⡽⢦⡀⠠⣖⡾⠛⠻⢦⡀⠀⢀⡟⠈⠁⠀⠀⠀⠘⡟⠶⣤⠤⠇⠀⠀⠠⣞⠉
⠀⠀⠐⠷⠾⣃⣄⣉⣉⣭⣿⠛⢤⣼⡁⠀⠉⣳⡞⠁⠀⠀⢀⡽⣀⣼⡤⢄⢀⣤⣠⣿⣦⣇⣀⡬⠶⣄⠀⠰⣟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠀⠀⠈⠂
⠀⠀⠀⠀⢀⠀⢀⢄⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠓⠀⡄
⠀⠀⠀⠀⠘⡵⠨⠌⡶⠐⠒⠤⢄⡀⠐⠈⣁⡀⣂⠄⠀⠐⢆
⠀⠀⠀⠀⠸⢀⡔⠊⡇⠀⠀⠀⠀⠈⠀⠀⢒⠂⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠎⠈⠀⠇⢀⠀⣀⠤⠂⠁⠀⠀⢠⠙⡄⠈⠆⢠⢀⠠⠄⢀⣀⠀⠀⣠⢶⠽⡬⢭⣒⡲
⠀⠀⠀⠀⡄⠠⠄⠀⡠⠞⢯⠁⢦⠀⠀⢀⡀⠀⠁⠘⢆⠀⢀⠈⣏⠒⢄⠈⠉⠙⠢⡄⠀⠈⠙⠒⠒⠁
⠀⠀⠀⢸⠰⠀⡖⠀⠀⠀⠈⡇⠀⡠⠖⠉⠀⠀⠀⠀⠀⠑⠀⠳⠘⠆⠊⠀⠘⡆⠀⠈⠂
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠁⠁⢠⠀⠀⠀⠀⠀⠀⠀⢰⢠⠀⣄⢒⡄⠀⠀⠃⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠀⢠⠀⠀⠉⠁⠀⠀⠀⠀⡏⡎⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⣀⠈⡆⣆⠀⠒⠂⠀⠀⡼⢸⠁⢀⣀⡤⠔⠋⠀⠢⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⣅⠃⠈⠑⠒⠀⠠⠞⠓⠀⠀⠁⠠⠤⠤⠤⠤⠖⠈⠓⡂⠀⢆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⢸⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣁⠀⢸⣀⠀⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣿⠻⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⣀⠀⠰⣴⣄⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠰⣦⣄⣐⣶⣿⡿⠿⠛⠛⠛⠛⠛⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⡿⠿⠿⠿⣿⣿⣿⣶⣴⣶⣾⣿⣿⢟⡥⠂⠀⠀⠀⠀⠀⠀⠈⠒⠹⢣⣿⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⡀⠀⠀⠈⠫⢿⣿⣿⣿⣉⡀⣿⣌⡀⠀⠀⠀⠀⠠⠦⠤⠴⣢⣿⠿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⢀⣀⣠⡤⠤⠤⢤⣄⣀⣻⣿⣿⣷⣶⡄⠀⠀⣭⠻⣿⣿⠿⠿⡁⠉⠒⢤⣀⡄⣠⠶⠿⣎⠇⠸⠐⠆⢰
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⠟⠋⠁⠀⠀⠀⠀⠀⡀⠈⠳⠽⢿⢿⣯⣿⣶⣾⣿⠀⢉⠀⠀⠀⢻⡆⠀⣀⣾⣿⠯⣑⠦⣿⠀⠇⡆⠀⢸
⠀⠀⠀⠀⣴⣿⣿⣿⡏⢘⣿⠿⠟⠁⢠⠏⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⣴⣿⠟⣻⣿⣿⣿⠟⠀⠘⡄⠀⠀⠀⠁⠀⣿⣿⠟⡂⠲⣿⣮⣽⣻⡽⠚⠙⠃
⠀⠀⠐⠛⢛⣿⣿⣿⠁⠈⠀⠀⠀⠀⡟⢸⠃⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠿⢿⣷⣼⡁⠈⢿⣷⠀⠀⢦⠀⠀⠀⢠⢂⣬⠕⡡⢀⡈⢠⢿⣿⠀⠙⡿⠀⢶⡇
⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⢸⠇⠸⡆⠀⠀⠀⢀⣴⠖⠀⠀⠘⢧⡀⠀⠙⠉⠁⠀⠈⢿⣧⡀⢈⣷⡀⠀⠋⠀⠀⠊⣴⣿⣿⢉⢸⣿⠀⠀⠉⠓⠋
⠀⠀⠀⣨⣿⠿⠋⠀⠀⢀⣀⣠⡤⠞⡼⠀⠃⠀⠀⠰⢋⣡⣴⣾⣿⣶⣤⣙⠒⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣿⡦⣄⣀⣴⣿⡏⠼⠔⣺⣾⠟
⠀⠀⠈⠉⠁⠀⠀⠀⠀⣾⠐⠀⠀⠀⣀⣀⣀⣠⣤⡴⠛⠉⠀⠈⠁⠉⠉⠛⠛⠛⠻⠿⠿⠛⠛⠛⠋⠉⠀⠀⠀⠀⠉⣿⣿⣀⠴⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠴⠋⢹⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⢶⠏⠀⠀⣸⣿
⠀⠀⠀⠀⣠⣤⡤⠾⣥⡜⣀⣀⣰⣿⡿
⠀⠀⠀⢰⣿⣯⡠⠔⠊⣼⣿⣿⠟⠋⠁

⢠⠄⠀⠀⠀⠀⠀⠐⠂⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄
⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠐
⠀⠀⠆⠀⠀⣀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠬⠃⠀⠐⠁⠐⠀⠄
⠀⠀⠈⠀⠠⠀⠂⢀⠀⢰⡄⠀⠀⠀⠀⣀⢀⠒⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠐⢈⡂⠀⠀⠀⠀⠀⢁⡀
⠀⠀⣀⣄⠀⢢⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⠀⠀⠸⢐⠂⠀⠀⠀⠀⠀⠀⠈⠤⠐⡀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢶⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠑⠂⠓⠀⠀⠌⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠂⠠⠌⡈⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠙⠒⠐⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠘⠛⠻⠿⠿⠿⠟⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣶⣲⡄⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠦⠹⣷⠴⠘⠁⠛⠘⠓⠺⢻⣯⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢻⠗⠀⠀⠀⢀⣀⢀⢀⣀⡀⠀⠀⠘⠧⠌⣡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠁⠀⢀⣴⣿⣿⣿⢸⢸⣿⣿⣷⣄⠀⠀⠘⢯⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⡎⠀⠀⠀⣾⣿⣿⣿⣿⣸⣸⣿⣿⣿⣿⡆⠀⠀⠀⠷⣟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⢽⣿⣿⠻⣟⡛⢛⣛⠻⣿⣿⡷⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⢦⣼⠇⢫⣽⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣛⣿⣄⣰⣦⣶⣄⣰⣟⣿⣦⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⢀
⠀⠀⠀⢠⢎⡈⣠⢌⠄⠀⠀⠀⢀⠤⣶⣿⡙⣿⣟⣿⣿⣷⣿⣿⡿⣻⣿⣿⣿⠶⢄⠀⠀⠀⠐⠛⠓⠂⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣵⣿⣮⣿⣿⣮⣿⣯⣿⣿⣿⢿⣽⣿⣿⣿⣷⣿⣷⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡔⡀
⠀⠀⠀⠀⠀⡀⠤⢤⣄⠼⣿⠿⠟⠛⣛⣛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⡛⣛⠛⠿⢿⡿⢎⣤⠠⠤⡀
⠀⠀⠀⣠⠂⠀⠀⣸⣿⡿⣷⣶⣾⡟⠉⣿⠛⢲⡶⣠⣶⣶⣶⣤⡰⣶⡚⢻⡇⠘⢿⣶⣶⡿⢿⣿⡀⠀⠈⢠⡀
⠀⠀⠸⣿⣶⣾⡿⠟⣋⣴⣿⡿⠏⠀⢀⣿⠛⣿⡇⡟⠁⣀⡀⠙⠇⣿⡟⢻⡇⠀⠀⠻⣿⣿⣌⡛⠿⣿⣶⣾⡿
⠀⠀⠀⢀⡼⠓⠋⠛⠛⠛⠉⠀⠀⠀⣸⣿⠀⣿⣧⡐⠀⠀⠀⠐⣰⣿⡇⢸⣷⡀⠀⠀⠈⠙⠛⠛⠛⠓⠺⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢀⣸⣿⠏⠀⣿⣿⣿⣇⣀⣀⣿⣿⣿⡇⠈⢿⣿⣀⡠⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⢀⣯⣭⣝⣧⣤⣿⣾⣆⣀⣀⣀⣀⣶⣾⣧⣤⣇⣭⣭⣧⠉⢁⡇
⠀⠀⠀⠀⢠⣤⣄⠀⠀⠈⠉⠋⠁⠙⣿⠆⢸⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠀⢸⡿⠁⠉⠋⠉⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠙⠻⠀⠀⠀⠀⠀⠀⠀⣼⠂⢸⣿⣿⠿⠿⢲⠾⠿⢿⣿⣿⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣦⡄⠀⠀⠠⡞⠀⢸⢸⣇⢀⣀⡡⣀⣀⢀⣽⣸⠀⠈⡇⠀⠀⠀⢠⣦⣤⠄⠀⠀⢀
⠀⠀⠀⠀⠀⢄⠀⠀⠈⠙⠃⠤⢀⣀⡇⠀⣾⣾⣿⠿⣿⣿⣿⡿⢿⣿⣿⡇⠀⣇⣀⡀⠀⠘⠋⠁⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣠⢄⠀⠉⠁⠉⠛⠙⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠁⠀⣠⠤⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠛⠒⠀⢸⣧⠄⠀⣶⣦⠀⢰⣶⡆⠀⣶⣷⠀⢸⠇⠀⠙⠚⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣽⣤⠎⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣵⣿⠀⠀⠀⢀⣰⣞⠉⠉⣀⣿⡑⠀⣀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⠴⠞⢩⣤⠽⠛⠋⠁⠈⣳⡞⠁⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠻⡟⠉⣿⣿⡧⠞⣠⠶⠊⠁⠀⠀⠀⠀⢸⠛⣷⡐⠶⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⣉⠢⣜⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⣿⠀⢳⡱⡄⢀⣀⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⠼⣿⣿⣿⣿⣿⣿⡋⠙⠻⠾⠿⠦⠤⠤⠤⠤⠤⢤⣤⣞⣉⣧⣀⣝⡿⢭⣁⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠒⠒⣶⡟⠻⠿⢶⣿⣿⣿⢳⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠛⢼⣏⠠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⣿⣿⣿⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⡽⠉⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣰⠁⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣾⣯⣋⣞⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⢀⡀⠀⠀⢀⣠⡾⠁⠀⣿⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠘⠿⠛⠷⠟⠀⠀⠀⠹⣇⠀⠀⣤⠀⠀⠀⠀⠠⣤⣀⢿⡃
⣘⣏⠛⠋⠈⠁⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⣿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⣿⡀⠀⠀⠀⠀⠀⠙⢷⣽⣦
⠘⠧⣄⣠⠀⣠⣶⠊⠈⠀⢹⡄⠀⠀⠀⠀⣼⠛⠻⣿⡉⠛⢿⣿⠒⠶⠤⠤⠤⠶⠒⠛⠋⠉⠛⠓⠒⠓⢦⣄⣀⡀⠈⠉⠓⢦
⠀⠀⠙⠛⠛⠁⠁⠀⠀⠀⠀⣷⠀⠀⢀⡾⠁⠀⠀⠀⠻⡄⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡈⠉⣹⢶⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⡟⠀⠀⠀⠀⠀⠀⣷⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣩⡏⠀⠙⢁⡼⠻⣄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠶⠎⠀⠀⢹⡀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡙⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣉⣙⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣉⢹⣆
⡶⠒⠲⡶⠶⡶⡶⡶⢶⡶⣶⡶⡶⣿⢿⠿⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣦⣀⡀
⠀⠀⣠⡠⡖⠻⠛⠉⠙⠓⠲⢤⡀⢀⣤⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⢹⣗⣿⢲⣲⣦⡀⠀⡠⠒⠁⡚⢉⣀⠀⠀⠁⠀⠀⠈
⠀⠀⣟⣻⣅⣆⣩⠿⢿⡏⠲⣤⢴⠋⠈⣿⠇⠀⢀⣀⣠⣤
⠀⠀⣿⡿⠛⠉⠀⠀⠘⣷⣆⡴⠁⠀⢰⣿⣦⢠⣿⠉⠛⢃⡄⠀⠀⠤⠤⣀⡀⠀⠀⣀⣴⣶⣶⣄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⡏⣥⣴⠊⠘⣿⣿⡿⣿⣧⠀⠀⠀⠠⣄⣄⠀⠀⠉⠳⣾⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣶⣶⡤⠘⣛⢯⠁⠀⣴⡿⠀⠀⠘⣿⡇⠀⠀⠀⢹⡎⠁⠀⠀⠀⢸⡆⣸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣼⠋⣨⣽⣷⡶⢦⡰⠻⠿⣧⣄⡰⠆⠿⠿⢀⣤⣶⣿⣿⣦⠀⠀⠀⢸⢳⡟⣿⣿⣿
⠀⠀⠀⠀⠰⣿⠀⠈⠛⠷⣤⣄⣠⣤⡀⢠⣬⣍⣽⣶⣾⣿⣩⣭⣹⣧⣼⣧⣤⣤⣿⣤⣷⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⢻⡷⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠻⠧⣅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠂⠉⠉⠉⠉⠙⠛⠛⠛⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣆⠀⠀⠠⠀⣤⠽⠌⠼⣳⡤⢀⠀⠀⠀⢀⡀⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠠⢀⣀⣿⣿⣿⣿⣿⣿⠇⣠⢵⡿⠀⠀⠀⠀⠘⣿⣤⠄⠀⠀⣈⡷⠆⠀⠘⡀
⠀⠀⠀⠀⠀⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣗⠂⢸⡇⠙⠳⠔⠛⠀⣿⡇⠈⡆⠀⣨⣿⣷⣾⣆⢨⡆⢀
⠀⠀⠀⠀⠐⢿⣿⣿⣿⡟⠻⡗⠀⢿⣿⣿⡀⠈⢧⠰⣦⣤⡆⢰⠟⠀⠀⠀⢠⣿⡿⠿⣿⣿⣿⡿⢀⡁⠢⡀
⠀⠀⠀⠀⠀⠍⠛⣻⣿⣿⡶⣾⣷⣿⣿⣿⠓⠄⡀⠀⠙⠋⠐⢁⠠⠀⠀⠀⠺⢿⡁⣀⣈⢻⣿⣷⣿⣧⣤⡈⠯⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⠟⣿⣿⣿⡀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢴⣿⣿⠟⣟⠛⢿⣿⣿⣿⣿⣿⣿⣆⣰⣧
⠀⠀⠀⠀⠀⠄⣀⡀⢀⠿⠉⠀⠀⢹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢩⠀⠈⠂⠀⢀⣀⣿⠉⠉⠛⠛⠓⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⢦⡚⠂⠀⠉⠁⠈⡛⠒⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⠶⠆⠰⠶⠶⠶⠦⠠⠤⢠⣄⡀⢉⣷⠜⠀⠀⠀⠀⠀⠈⠂⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⢁⣴⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠙⠻⠿⠿⠦⠤⢤⣤⣤⣤⠤⠀⠀⠀⡀⠚⠋⠁⠀⠀⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⣼⣷⣠⣤⡶⠶⠂⠀⠀⠀⠀⠀⢄⡀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠶⠇⢀⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⣤⣤⣤⣄⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⠟⢻⢟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⠺⠟⠈⠻⣿⣿⣿⣿⡷⠊
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣃⠀⢻⣿⣿⣿⠉⠀⠀⠀⢢
⠀⠀⠠⠤⠤⠤⠄⠸⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣷⡆⠀⠀⠸⠂
⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⢸⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⠁⢻⣿⣿⣾⠁⠀⠃
⠀⠀⡇⠻⠷⠈⠠⠿⣬⡋⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⡏⢿⣧⣤⠸⣿⣿⡷⠦
⠀⠀⠓⠒⡶⠶⠶⠄⣤⢠⡾⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢁⡟⢻⣶⣩⡅⠀⣿⣿⣷⣀⣠⢷⣦⣤⠒⠄
⠀⠀⠀⠀⠑⢂⡰⣾⣯⣹⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣙⣏⡸⢻⣭⣿⣿⡇⠀⠈⡉⠋⠉⣡⣿⣿⣄⡥⠇
⠀⠀⠀⠀⠀⠛⠉⠛⠉⠻⢺⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡋⢛⣛⡉⠁⣄⣸⣿⣿⣿⠿⠟⣫⠍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡽⢿⣿⣿⠆⠭⠿⠍⠉⠔⠚⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣽⣿⣆⣩⣵⡆⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣏⣿⡿⠿⠟⠋
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠛⠿⢿⣿⣿⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠈⠀⢀⣤⣶⣶⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠆⠀⠀⠈⠋⠀⠀⣤⠀⠀⢀⡀⠀⠋⠑⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠍⠈⠐⠄⠀⠀⢀⠀⠀⠀⠀⠀⠃⣼⣿⣦⣄⠀⠀⠸⠿⠿⠿⢿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠈⠂⡀⠘⢷⠶⠁⡠⠂⠀⣹⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠘⠀⠀⠲⢀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⠀⠀⢀⠀⠈⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠉⣀⣠⣤⣤⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢠⣽⣿⣿⠏⠀⠀⠀⠂⠀⠀⢀⣴⣿⠟⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠂⠐⠒⠀⠀⠀⠀⠀⣀⣌⡿⣿⣧⣀⡀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠂⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⠀⠙⣿⡇⠈⠉⠛⠿⠿⠿⠿⠿⠟⠃⡀⠢⠀⠀⢀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⣤⣠⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠐⠠⠀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢻⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣿⣧⠀⠀⠀⣠⠀⠀⠀⠀⠀⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠓⠀⠘⠛⠃⠀⠁⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⡇
⢄⡀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣆
⠈⢿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡟⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⣈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⢿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢽⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⢘⣦⠸⡏⠑⠀⠀⠀⠈⠉⠛⠲⡄⠀⣶⣾⡿⠿⠿⡿⠿⣿⡿⢿⠁
⠀⠀⢀⣿⢟⣿⢫⣽⣶⣄⠀⢀⠴⠊⠈⢉⢣⢠⣄⠀⠐⠀⠀⢀⠔⠁
⠀⠀⣸⠛⣿⠀⡄⣉⣽⢿⣿⠑⢦⣀⡴⠛⠹⢹⣿⠃⠀⢀⣀⣀⣀⡄
⠀⠀⡿⣿⡿⠟⠛⠉⠀⠀⢿⣦⠈⣵⠋⠀⠀⣾⣿⡄⢈⣿⠛⠛⠟⠀⡀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⢸⣿⡞⣃⣀⡤⠊⣿⣿⣿⣾⣿⣧⠀⠈⠉⠀⢀⠀⠀⠀⠉⠛⠶⣤⣤⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⣿⣧⡿⠟⠀⠀⣸⡟⠋⠉⢻⣿⡇⠀⠀⠀⠈⢷⡽⠆⠀⠀⠀⠈⢻⡅⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢸⡿⠿⢿⣇⣐⠪⡓⠀⣀⣼⣿⠃⠀⠀⠀⣹⣿⠀⠀⠀⢀⣸⣿⡀⠀⠀⠀⠀⢸⡇⣼⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢀⢿⠁⢠⣶⡿⠿⠟⠧⠼⠙⠛⠿⢶⣤⣘⡃⠙⠛⢁⣴⣾⣿⣿⣿⠻⣄⠀⠀⠀⣸⠃⣿⢸⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣗⠀⠈⠉⠛⢷⣦⣄⣠⣶⣄⠀⠤⠤⠬⠭⠭⠭⠭⠭⠥⠤⠭⠷⠤⠿⠤⠤⠤⠿⠤⠼⢾⣿⣿⣿⣷⣂⡀
⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠿⣧⣀
⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣯⣽⣿⣭⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠚⠚⢻⠛⠒⠶⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠘⡄⠀⠁⠀⢸⢀⠀⠀⠀⡞⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠳⡄⠀⠀⠆⡰⢴⠂⠃⠙⠙⠁⠆⣴⣴⠆⠀⢀⡴⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠂⠀⡌⣠⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⢤⣤⠀⠂⠹⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢠⠏⠓⠦⠀⣀⢀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠠⠖⠙⡆
⠀⠀⠀⠀⠀⡔⠁⠀⠈⠉⠢⣀⡟⠀⠀⠀⢈⣉⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠾⠺⠃⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⡦⢤⡄⣤⣠⡌⠉⠛⠲⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⡀⠤⠤⢇
⠀⠀⠀⠘⣇⡀⣀⣀⠀⠔⠉⢀⣄⣀⣰⠦⠈⠑⡀⢀⠀⠀⠀⠉⠙⠻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠃⠚⠙⠁⠀⠀⣸
⠀⠀⠀⠔⢡⣴⡀⡠⠀⠀⣀⣘⣿⣏⣠⣤⣤⣄⠀⠳⣜⢒⡆⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⡠⡄⣀⡀⣸
⠀⠀⣰⡾⢇⠀⠁⠒⠆⢙⠴⠋⠉⠙⠿⠿⠿⠛⠋⠶⠛⠙⢿⠐⡡⢉⡆⠀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⡇
⢀⡼⠁⠁⠀⣨⣷⣿⣿⢻⠀⠐⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⢟⠧⢀⡠⢈⡉⠀⠀⠀⠀⢠⢠⡄⡤⠀⠀⠀⡜
⡟⠁⠀⠀⠀⠹⠿⠋⣻⣏⣠⠀⠸⢃⣀⣠⠣⠤⠠⠀⠔⢁⡰⢏⢡⣿⡬⠁⡀⠀⠀⠀⢀⡨⢴⣒⠻⣯⠭⢭⡿⣓⣲⠤⣄⡀
⠀⠀⠀⠁⠀⠀⠀⢻⣿⣿⣿⣷⣾⣿⣿⣿⣶⣦⣀⠐⠺⠉⠙⠶⣤⠈⠐⠄⠚⢀⡤⣛⠵⠊⠉⠃⠀⠀⠀⡈⡀⠀⠉⠉⡲⢽⡲⣄
⠀⠀⠀⠀⠀⠀⠔⠀⣿⣿⣿⣿⠇⢰⣿⣿⣷⣤⡿⢿⣦⠀⢠⣤⢴⣇⡈⠙⢶⠫⠚⠇⠀⠀⠀⠿⠿⠀⠀⠛⠃⠀⣤⡄⣤⠀⠉⠪⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⢨⣿⡛⢛⣻⣧⢠⣿⣧⠀⢹⣏⠁⠀⠀⢸⠃⠀⠀⠽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠐⠝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⣾⣿⠀⢸⣿⡟⣿⣿⣿⠀⠈⣿⣿⠐⠀⣸⢠⣤⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣃⣰⣽⡇⠀⣿⣿⠀⣿⣿⣿⡆⠀⣹⣏⠀⢀⡿⠈⠉⠁⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⢹⣇⣿⠁⢸⣿⡟⣸⣿⣿⣿⣇⣰⣿⠉⠉⠹⢿⡶⣆⠀⠀⠀⠀⠀⠀⠸⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣸⡏⡋⢀⣿⣿⠇⣿⣿⢁⠟⣽⢹⣭⣶⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⢹⡾⢀⣾⣿⣟⣸⣿⣿⡜⣀⣽⢀⣨⣟⠀⠀⡀⠐⠺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣿⢃⣼⣿⣿⣿⣿⡿⠀⢸⣯⣟⢸⣿⣿⠀⠘⠀⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢃⣼⣿⣿⣿⣿⡿⠁⠀⠀⠻⣿⠻⢸⣇⣀⡘⣷⢐⣂⣀⣛⣓⣒⣀⡤⠀⠀⣀⠀⠀⣠⠤⡀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣾⣿⡿⠋⠀⠈⠁⠀⠀⠀⢸⣿⣷⣯⣟⣩⣿⣿⠿⠿⠟⢛⠛⠟⠿⢿⣿⣦⣟⢄⠀⠈⠉⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣅⣸⣿⠟⠋⠤⠀⠀⠀⠈⠀⠀⠀⠀⠌⠛⢿⣷⣝⢦⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣿⣿⣿⣻⡉⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣟⡋⢿⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠁⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⡟⢛⣶⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⠛⠁⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⣿⣿⣿⣿⣷⢾⠃⠀⠀⠀⠀⠀⠀⠀⠿⠟⠋⠀⠀⠀⠀⠀⠀⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⣿⣿⣿⣿⣥⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣷⡀⢠⠱⣦⡶⠂⠀⠀⠀⠀⣀⣴⣿⠿⠋⢻⣿⢷
⠀⠀⠀⠀⠀⠀⢂⣵⡤⠤⠤⣀⣀⠀⣸⡿⠉⢳⣀⡇⠐⠟⠃⢀⡴⠊⠉⠉⠉⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⣾⡉⠛⠀⠀⠈⠙⠿⣯⣤⡶⠟⠉⠀⠠⢛⣸⣯⣄⠀⠀⠀⠀⠀⢠⠀⣠⡃⠑
⠀⠀⠀⠀⠀⢰⡄⠠⡀⠸⣀⠠⠀⠒⣹⠟⢠⡼⣀⡀⣤⡘⠌⠈⣙⠲⣤⡀⠀⠀⣸⣾⢿⡇
⠀⠀⠀⠀⠀⣧⣄⠀⠀⠀⠈⢣⠀⡀⠀⠀⠹⡓⠝⢷⠸⡿⠆⠀⣹⣦⡼⢿⡀⢠⣿⣧⣾⣷⡿⠆
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⡎⠀⠀⠀⠀⠈⢡⡈⠀⠀⣀⣴⣟⣥⣤⣤⡍⣩⢠⣥⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⡀⠀⢀⣤⣠⣼⣶⣿⠿⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡴⠚⠛⠛⢩⣿⠶⠶⠶⣶⣶⣶⣶⣶⡆⣴⢠⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⡞⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡴⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠘⠁⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠠⠾⠏⠀⣠⡐⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⢉⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠻⣼⡋⠀⠀⠀⠀⢀⡴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣄⢨⡇⠀⠀⢀⡴⠋⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢹⠦⣭⣛⣛⣿⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⠓⠶⢴⣋⠀⢠⡼⠋⡇⠀⢠⡟⠶⣶⣄⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠴⠛⠉⠉⡇⠘⠳⣤⣀⠈⠓⢦⠀⠀⠻⣄⣸⡷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⢠⣤⠀⠀⠀⠀⠀⢿⡀⠀⠀⠈⣽⣆⣸⠃⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡏⣿⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⡇⢹⠇⠀⠀⠀⠀⠾⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⢸⡆⠀⠀⠀⠀⣠⠦⣤⢀⡼⠁⣯⣤⣤⠀⠀⡶⢤⣄⡉⢳⠄
⠀⠀⠀⠀⠀⠀⢀⣟⣠⠟⢠⡞⠁⠀⢀⣤⠞⠁⠀⠀⠈⠓⢦⣽⠀⢸⡇⢰⡇⠀⢸⡇⣼
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢸⡀⢰⠦⣄⡉⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⠃⣼⠀⢀⡞⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠙⣇⣸⡇⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠈⠳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢸⡄⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢿⡀⣧
⠀⠀⠀⠀⠀⠀⠤⠀⠶⠶⠶⠾⠶⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠾⠷⠶⠶⠶⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⣿⣯⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢛⣿⣒⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⡈⠂⣠⣤⣄⠷⠄⠈⡑⣤⡀⠀⠀⠀⢠⣸⣅⡜⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣯⣿⣟⢀⠈⣙⡓⢦⣤⢖⡸⢿⣿⡷⠹⣷⣹⡗⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⠋⠉⠉⠁⢻⣿⡦⢭⣄⡛⠻⣍⣬⢥⡾⢟⣦⡼⢁⠿⢟⢻⣏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡧⠀⠾⠉⣀⡍⠛⠫⠵⢶⡦⠡⣀⣍⣶⡞⠁⢈⣵⠞⠂⠄⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡔⣀⣠⠜⢁⡞⠁⠀⣠⠞⢀⣴⣿⠉⠁⠀⣠⡾⠉⢀⠄⠀⠀⠀⠒⠿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢇⣈⣴⢂⡼⡇⠀⡾⠁⠀⠿⠟⠀⠁⢀⠀⣼⣗⣷⢘⣄⢀⣤⠀⡆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣟⠈⣰⣁⡀⣟⣦⣤⣷⡀⣠⣤⣤⣀⣿⣿⣯⡟⠃⠀⣿⡰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡽⠁⣈⣴⣿⣿⣿⣎⣝⠳⣿⠕⠘⠿⠿⠿⠟⠿⠭⡄⠚⣇⠿⠡⠿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⠰⠞⠋⠉⠁⠈⠉⠙⠋⠁⣜⡀⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⣹⣧⡀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠛⣩⡶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢯⣖⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣷⢶⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠔⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡔⠚⠛⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣴⢋⠠⠎⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⡀⠀⢀⣤⣤⡔⠋⣁⠠⢴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣿⣿⡿⠗⠚⣩⣿⣿⡷⠋⠁⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⡿⠛⠁⠀⡠⠞⠀⣠⢀⠀⣠⡤⠾⠃⠀⢀⣠⠀⠐⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣀⣼⣿⡿⠋⠀⠀⠀⣼⡡⠀⣾⠃⠀⣼⠿⠶⠶⠶⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠁⠉
⠀⠀⠀⠀⠀⣸⣿⣿⠟⠀⠀⢠⣤⣾⠁⣰⣶⡟⠰⣼⠇⠀⠀⠀⢀⠀⢀⣀⣤⡿⠀⠀⠀⣆⡀⣸⣿⡟
⠀⠀⠀⠀⠀⣹⣿⠃⠀⣤⣾⣿⣿⢿⢺⣯⡙⠀⠸⣿⣶⣶⣶⣾⣿⣼⣿⣿⣯⠅⠀⠀⢀⡿⠛⠛⠋
⠀⠀⠀⠀⢸⣿⠇⢀⣼⣿⠏⠁⠀⢸⣿⡇⠀⢙⠂⠈⠉⠛⠛⠛⠛⠛⣿⣷⣿⠀⢀⣀⣾⠃
⠀⠀⠀⠠⣬⣿⡠⠤⢴⣽⡄⠀⢀⣿⡿⡇⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢻⡀⣿⣿⠁
⠀⠀⠀⠈⠉⠻⣤⣤⢾⣿⣯⡤⣿⣿⢸⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣅⠈⠛⢢⡀
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⠀⠘⣧⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠂⣰⠇⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡌⠀⠀⣿⡆⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠊⢻⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⣿⠇⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⠀⠁⢊⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⣀⡄⣀⣀⣀⣠⣤⣾⣾⡤⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠒⠀⠁⠀⠈⠉⠉⠉⠁⠈⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⡀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠐⣶⣄⠀⠀⠀⡜⠁⠀⠘⣆
⠀⠀⠀⠀⡄⠀⡄⣰⠀⣤⣴⣶⣶⣿⣯⣾⣿⣿⡇⠀⠀⡄⠀⠀⠀⢸⡆
⠀⠀⣴⣀⡙⢂⡑⠋⠀⠀⠘⠲⢮⣝⡻⣿⣿⣿⡟⢠⡆⠃⠀⠀⠀⣼⣧
⠀⠀⠈⠉⣱⣿⡿⠀⠀⠀⠀⠀⠀⠈⠙⢮⡙⢿⣿⡿⠁⠀⠀⡜⠘⠿⣿⡀
⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⣆⠀⠀⠀⠀⠀⠈⠢⠉⠀⣠⣦⠼⠁⠀⠀⡘⠷⡄
⠀⠀⠀⠀⣿⣿⠀⠀⣠⠃⢻⠀⠀⠀⠀⠀⠀⠀⢾⠿⠋⠁⣀⣀⣀⣴⡷⠀⣽
⠀⠀⠀⢠⡇⠙⡿⠋⠁⢀⣸⣤⠤⠔⠒⠉⢀⠔⠁⠀⠀⣀⡤⠖⠀⠀⢶⡛⠿⠃⠀⠀⠤⠤⣤⣀⡀
⠀⠀⠀⠸⣷⡿⠃⠀⠉⠀⠸⣇⠀⠀⠀⢰⠫⣠⣤⠀⠀⠀⢠⠟⠛⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠸⣧⣹⣿⡆⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠛⢻⡇⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⣠⣴⣆⣀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⣼⣿⡟⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⡠⣶⣶⡦⠤⠤⠤⠀⠘⣇⠀⠀⠀⠀⠀⣀⣤⡾⠋⠀⢀⡆⠀⠐⡟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢁⣤⡆⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢠⠞⠋⡅⠀⠀⢀⡞⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡷⠛⠉⣧⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢹⠀⠀⢹⡄⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⣿⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣸⣇⠀⠀⠹⣆⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣀⣿⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⢹⣧⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠉⠀⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣤⣦⡀⠀⠀⣰⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢐⡤⠰⠂⠉⠉⠁⠢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠼⠁⡀⠀⠀⠀⠀⠠⢰⠁
⠀⠀⠀⠀⢠⠂⠢⢀⠈⠓⠂⠈⠀⠀⣎⠉⡇⠀⠠⠀⠠⠸⠃⠀⠀⠀⠠⠠⠂
⠀⠀⠀⠀⠀⠑⠶⠛⠄⠀⠀⠀⠀⠀⠘⢝⡀⠀⢀⡀⣁⡈⠀⠀⠀⢀⠝⠲⠂⠁
⠀⠀⠀⠀⠀⠀⠀⢂⠈⢂⠀⠀⠀⠀⠀⢀⠉⠠⣄⠀⠜⠀⠀⠀⢠⠂⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠑⠀⠐⠂⡚⠁⠑⠂⠁⠻⠐⠂⠀⠒⠁⡐⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠉
⠀⠀⠀⠇⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢲⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠒⠒⡆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⠈⢂⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⢂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣆⡨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢼⠀⠀⠀⠀⠀⠘⣠⣽⣿⠵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣧⣼
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⢤⣤⡀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⣄⣠⠤⠄⣀⣀
⠀⠀⠔⡀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠠⠐⠠
⠀⠓⠄⢂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠊⠐⠃
⠀⠀⠀⠀⠋⠉⠀⠋⠉⠒⠒⠒⠂⢤⠤⢤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠄⠒⠒⠒⠊⠉⠑⠀⠉⢃
⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠈
⢇⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⢋⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣿⣿⣷
⠀⠀⠀⠘⣿⡦⣐⣒⣒⡒⠒⠔⣂⣀⡀⡀⡀⠀⠀⠙⠩⢀⡼⢍⢿⣶⡄⠀⠠⣄
⠀⠀⠀⠀⣾⣿⣻⣶⣦⣭⣉⣑⡒⠬⢭⣿⣶⣮⣤⣤⡚⢐⣛⣥⣾⣷⣿⣷⠶⣢⡄
⠀⠀⠀⠀⢴⣿⣿⣷⣯⣿⣿⣿⣿⣿⣷⣶⡿⢟⣻⣿⣿⣿⣿⣾⡽⠿⢋⣵⣿⡿⠁
⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⠉⠻⣿⣿⣿⣿⣿⣿⡿⠿⠁⢀⣽⣇⣴⡇⣏⠉⣿⢤⣤⡬
⠀⠀⠀⢸⣿⣿⣿⡿⠁⣀⣤⠞⠛⠻⢛⡿⠛⠁⠀⠀⢀⣼⣿⣿⣿⣷⣙⣷⡙⣹⣯⣭⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⢪⣿⣿⣯⠍⠉⠈⠀⠀⠀⠀⢀⣷⣦⣴⠟⠃⣾⣿⣿⣿⣿⢿⣏⠋⣅⢻⣿⣿⣯⣿⣦⣰⡄⠀⠐⠀⠀⠠⠊⢐⣀⣲⢤⡐⢄
⠀⠀⠀⠀⠙⠓⠀⠀⠀⠀⠀⠀⣀⡀⠜⠛⠉⣹⣦⣸⣿⣿⣿⣿⣿⣿⣿⡷⡝⣶⢻⣿⣿⣿⣿⣿⡍⠀⠀⢀⡀⠀⣦⣜⠻⣗⢯⠡⡄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢶⣶⣶⣾⠿⠿⣿⣫⣿⣿⢿⣽⣿⣿⣴⣶⣿⣿⣿⣿⣿⣿⣿⣦⣤⡀⠀⠈⢳⣾⠫⣾⡘⠆⢒⢀⠂⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡝⣿⢿⣿⣿⣿⡮⠋⠩⠁⢛⡻⠿⠟⠛⠛⠛⠿⡿⣿⡛⣛⠛⢿⣿⣿⣦⠀⠀⣿⠀⠈⠛⢧⡿⠉⣍⣙⡻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣾⣽⣴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢿⣦⡙⣿⣿⣧⣾⣿⡀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠼⠊⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠿⣿⣾⡿⣿⣿⣽⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠽⣋⠙⣿⣙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣝⢿⣮⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⣷⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⣎⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡿⢿⡽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡿⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⣽⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣐⣤⠀⠀⠀⢀⣰⣿⣦⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢏⣁⢠⠤⣴⡟⣻⣿⡿⠋⠉⠛⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⡿⢻⣧⣤⡀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⡿⢥⣸⣿⣌⠀⠀⣸⣦⣀⣀⣀⣀⠠⢴⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⣼⣿⣿⣿⣿⡿⠁⠈⠛⣿⣿⡷⣶⡉⡈⠡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⣿⣿⣿⣿⣿⠃⠀⠀⠀⢹⣍⢻⣿⣷⢾⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠸⠾⠟⠛⣿⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢻⡈⣾⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢺⣿⠇⠀⠀⠀⠀⣀⠀⠀⠀⠀⣿⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣽⣧⣸⡙⠠⣀⠀⠀⠀⠤⡖⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣿⡿⣿⣆⢿⣀⠘⠀⠐⢁⣄⣰⣤⣀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⢸⣿⣶⣭⠓⣰⣆⣾⣿⡿⢋⡅⣼⣿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣴⠀⠉⠟⠁⠀⠁⣼⣧⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠾⠟⠛⢻⣷⣀⠀⠀⠀⣠⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⣷⠀⠀⠀⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢰⣟⣇⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣻⣿⡟⠀⢀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠋⣿⠏⢀⡼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢰⣿⣱⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢋⣉⡿⣿⣧⣬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣯⣍⣩⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⠃⠈⢿⣿⣿⣿⣿⣿⠛⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢋⡁⡌⠀⢸⣷⣼⣦⣿⣿⣿⣅⣾⡄⣰⢇⠀⠦⠀⣄
⠀⠀⢸⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠉⠀⣨⣾⣷⣃⡀⠹⣿⣿⣯⣿⣠⣾⡟⠷⠃⢸⡖⠀⠀⠀⠀⡀⠀⣀⣰⣦⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣠⠄⠒⠂⡁⠀⠀⠀⠀⠸⢗⣿⡛⠉⢰⠋⠐⠛⣿⠿⠛⣧⣀⣀⣼⣇⣀⠀⣴⠋⠉⠋⢭⣀⠀⠉⠉
⠀⠀⠀⠀⢀⣾⣹⣶⣆⠉⢀⣠⠤⠤⠃⣀⠀⠀⠴⠾⡟⠛⠿⢷⠟⠒⠄⠘⠁⠀⠀⢛⡋⠉⣉⣽⡇⠈⠈⡇⠐⢿⡟⢹⣷⣦⡀
⠀⠀⠀⠸⣿⣿⣿⣿⣫⡻⢿⣯⡀⠀⠛⠉⠀⠀⠀⠐⡇⢰⣷⣾⠀⢐⣧⣤⣀⣠⣤⡿⢿⣻⣾⣿⠶⠶⠏⣤⣤⣾⣿⣇⣿⣿⠁
⠀⠀⠠⠖⠋⠉⠉⠉⠈⠓⠢⠤⠷⣦⣄⠀⠀⣾⣏⡀⢁⢸⠏⠁⠀⠈⢻⡿⠻⣿⣯⣶⣿⡟⠉⢽⡇⠀⠀⠀⠉⠉⠉⠙⠛⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⢾⣿⣿⣿⢾⣿⣿⣿⣿⣷⣌⡹⠿⣿⢡⣤⠤⠄⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡞⢷⣿⠋⠈⠃⠀⠈⢿⣿⣿⢏⡍⠀⡾⢿⣿⣶⣶⡽⠿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢠⠀⢸⣤⠀⠀⡀⢀⣠⡸⣷⣾⡟⠀⠀⡃⢠⢿⡙⡅⢀⣀⣠⠄⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡘⠀⢸⡿⣷⣤⣁⣿⣾⣿⢟⣾⡇⠀⢠⠉⠁⠀⠿⢨⡛⢸⣧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡅⠀⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣆⠀⣀⣤⠘⣇⠊⠀⠀⠀⠀⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⡇⠀⣀⣀⣉⣲⡧⢈⣀⣈⣉⡉⠁⠀⢸⣿⣿⣿⠟⢠⡟⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⠀⠀⣿⣿⣿⣭⣷⣾⣭⣿⣿⡏⠀⠀⢸⣿⣛⣿⠖⠫⣽⠀⠀⠀⠀⠀⢘⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡐⠀⠀⢯⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣽⣹⣿⡇⠀⠀⢶⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⡇⠀⠀⣿⡛⠛⠛⠳⢿⣿⣏⣻⠁⠀⠀⣿⣿⣿⡇⠀⠂⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣾⠿⠛⠀⠀⠀⠀⠈⢻⣏⠀⠀⠀⣿⣿⣽⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠤⢾⣀⣻⣧⣤⣤⣤⣤⣤⣤⣤⡀⡟⠀⠀⢰⣿⣿⣿⣷⣦⡘⢻⡝⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⠤⢌⠡⣸⡏⢉⠉⠀⠀⢀⠸⣿⡷⠸⠏⠙⣾⣿⣿⣿⣿⣿⣥⡇⣷⠂⠐⡶⠦⠔
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣦⡀⢸⠀⡿⠀⠀⠀⠀⠀⣼⣿⣿⣿⠆⠀⠀⣿⣿⣿⢳⣿⠃⣸⣿⣷⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⢸⡇⡀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣾⣿⣷⣿⣿⣿⣧⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠛⠿⡿⠿⢿⡇⣼⠀⠂⠀⠀⠀⢸⣿⡏⢻⣿⠀⠀⢸⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿⠀⠀⠀⠀⠀⣿⣿⠃⣸⣿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⣿⠀⠀⠀⠀⠸⣿⣿⢸⣿⣏⣀⣐⠟⠛⠋⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠀⢠⣶⠀⣿⣿⢸⣿⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⣿⣿⡇⣿⣿⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣿⣿⣦⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠟⠁
⣶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣯⡖⢲⣯⣍⣉⣩⣍⣭⣉⣭⣭⣭⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⠏⢁⠀⠀⣀⣠⣤⣤⣘⣿⣶⣼⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣏⢀⣬⣿⣶⣾⠟⡋⠉⠀⠀⠀⠀⠈⢍⢻⡆⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣍⠉⠁⠉⠙⠟⣿⣿⣿⡿⠛⢻⣇⠋⠀⠀⠀⠀⣀⣀⣀⠀⣼⣼⣿⣇⡀
⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣿⣿⣿⣦⣄⠀⠀⢀⣀⣀⣀⣀⣀⢀⣾⣿⣶⣤⡄⠀⠈⣛⢿⣿⣿⡾⡏⠑⠦⣀⠀⡀⣠⡴⣦⡾⢹⠠⡌⡋
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠿⣿⠗⠚⠉⠀⠀⠀⠀⡈⠙⠯⠽⣿⣟⣿⣦⣾⣿⠀⠉⠁⠀⠸⡶⠀⣀⣿⡾⢝⡢⣽⡁⢸⡌⠀⡅
⠀⠀⠀⢠⣾⣿⣿⡟⢚⡿⠿⠏⢀⡞⡁⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⣾⡿⢛⣿⣿⣿⠟⠀⢸⠀⠀⠀⠈⠀⣿⡿⠟⡐⢿⣷⡽⣾⡽⠚⠙
⠀⠀⠈⠉⣽⣿⣿⠁⠀⠀⠀⠀⣼⠐⡇⠀⠀⠀⠀⠀⠀⠊⠀⡀⠻⢿⣷⣏⠀⠹⣿⡀⠀⢦⠀⠀⢠⠆⠤⢖⠅⣠⣁⡞⣿⠁⠈⣻⡐⡺⠃
⠀⠀⠀⢰⣿⣿⠟⠀⠀⠀⠀⠀⡏⡀⡇⠀⠀⠀⣠⠞⢀⣀⡀⠳⣄⠀⠉⠀⠀⠀⠹⣷⣄⣸⣷⡀⠈⠀⠀⣡⣾⠿⠇⣇⣿⡇⠀⠈⠉⠁
⠀⠀⠠⠾⠛⠁⠀⠀⣠⢖⠒⠋⠐⠃⠁⠀⠀⢊⣤⠾⠻⠿⠿⢷⣦⣥⣄⣀⣀⣤⣤⣼⣿⠿⠟⠛⠳⠶⢾⣷⠒⢛⣩⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⣠⠤⣤⣶⠖⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠛⠛⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣣⡞⠁⠀⣿⡇
⠀⠀⠀⠀⣀⣀⣰⣋⡹⠀⠀⣴⣿⠇
⠀⠀⠀⣼⣿⣁⠠⠚⣵⣿⣿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡻⠟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣬⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠻⣷⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠯⣍⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠛⣯⣿⣿⣯⣱⣎⡛⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⠱⣾⣤⣀⠀⠀⠀⠀⠀⢠⡾⡿⠻⡉⢠⣿⣿⣿⣿⣿⣿⣽⣶⣏⣬⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣌⠉⢉⢩⡁⠈⢰⣤⣿⣿⠁⠀⠁⠾⠁⣿⣿⠉⠦⣈⠉⠛⢿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀⠀⠀⢀⣼⣿⣅⠁⢈⠀⠀⠀⠀⣠⣾⣿⣿⣷⣤⡀⠘⠀⠀⢠⡇⠀⠀⠀⠉⣙⠿⠟⠻⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠁⠀⠀⠀⠈⠙⢿⣾⣿⣟⣿⡗⠐⢾⣷⡦⠀⠘⠑⠽⠿⠿⠏⡈⠉⠛⠛⠒⢾⣗⣒⣒⠒⠌
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡇⠀⣠⡶⠂⠀⠀⠀⠀⢻⣿⣿⣾⣟⠀⠀⢶⣾⣿⠆⠀⠀⠀⠀⠀⢻⣦⠀⠀⢀⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠁⠁⡄⠀⠀⠀⣀⠀⠙⠻⠿⣿⣆⠀⠈⠁⡟⠀⠀⠀⠀⠀⠀⢸⡏⠀⠒⢿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⣧⡀⠀⠀⠀⠉⠉⠐⠒⠺⣿⣷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡋⠁⠀⢀⣬⣶⣦⣾⣧⣴⡄⣤⡀⢰⢿⡏⠀⣰⠀⣰⠀⠀⠀⠀⠀⢀⣀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡀⠀⡸⣿⣿⣿⠟⠻⣿⣿⣿⣷⣸⣾⣷⣏⣹⣾⣿⠀⠀⠀⣠⣶⣿⢏⡴⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⡰⢡⡟⠉⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣏⢸⣿⣿⠀⠀⠠⢿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢀⡜⣱⣿⡏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠛⠛⢻⡄⠀⠇⠸⣿⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⡇⠸⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⠀⠀⠈⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣄⠀⡀⠀⢻⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠘⣻⣦⡀⠈⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢤⣀⣀⣀⣻⣎⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣙⣧⣡⠀⠈⠙⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣯⠇

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⢑⢄
⠀⠀⠀⠀⠀⠀⠀⢠⢃⣤⠴⣚⣳⢆⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠛⠻⢵⣾⠁⠿⣟⠀⣯⠤⠄⠠⠤⠤⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠾⠛⡡⢴⣧⣦⠢⣾⡶⠿⠶⠒⡀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣵⣿⣿⠛⣁⡻⣷⣪⣄⡤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣻⣿⣿⣿⣿⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⡴⣀⡆
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⣸⣟⣿⠸⣿⣿⣿⣿⣟⡛⠛⢿⡄
⠀⠀⠀⠀⠀⠀⢠⣇⣀⣀⣀⣿⣿⣿⣆⣽⣿⠛⡿⣿⣿⡅⠀⢹⣄⣀⣀⣀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⡴⢋⣤⣶⣶⣶⣤⡌⣟⣿⣷⡞⢧⣵⣝⢿⣧⡆⣀⣹⠛⡟⠛⠻⢻⠃⣼⢷
⠀⠀⠀⢠⣮⡾⠟⢿⠿⠿⡿⣿⣿⣿⣜⡟⣿⣿⡿⣻⣍⢻⡿⢾⢟⣧⠟⣄⢐⣼⣼⣧⡟
⠀⠀⠀⠀⠀⠀⠀⣾⡤⠶⣿⣿⣼⣿⡏⢹⣮⣷⡿⣛⠽⣿⣻⣾⣏⣁⠜⡜⢏⣻⣭⡿⣻⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣾⠁⢠⣾⢟⣛⠻⢿⣷⠘⣿⣿⠛⣛⠿⣯⡟⣵⣾⡗⣾⣅⣼⣿⣿⣟⠒⢦⡉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠐⣇⠀⢸⣗⣄⡏⢿⡶⣼⡆⢻⣿⣼⡋⢿⣮⣇⣿⡙⣷⡙⣟⣉⡟⣿⣿⣛⢉⣉⡉⢺⡿⣿⣆
⠀⠀⠀⠀⠀⢠⣿⡀⠈⢧⡸⠛⢾⣷⣿⣷⣾⡿⣿⣿⣶⣿⣿⣯⡻⣾⣬⣻⣭⣼⣿⣿⣿⣿⣿⣿⣿⣷⡘⣿⣧
⣴⣆⠤⠤⣴⣛⣿⣷⣴⣾⣿⡖⢗⣿⣿⣿⣿⢷⡜⣿⣿⣮⣱⣿⣿⣶⣭⡭⠾⠿⠛⢿⣿⣿⣿⣿⣟⠳⣧⠘⢿
⠙⢿⣿⣦⣤⣬⡉⠻⢿⣿⣿⣿⣮⡛⡿⢫⡟⠲⢷⡘⢿⣇⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣷⣾⡾⠟
⠀⠀⠉⠙⠛⠿⣿⡷⠊⠹⢿⣿⣿⣿⣿⣖⠻⡦⡀⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣧⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣟⣿⣿⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣯⣽⣿⣿⣿⣽⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠠⠊⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣶⣦⣤⣄⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣱⡟⣱⣟⡞⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣾⠁⣈⢋⣅⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡆⣦⣼⣿⣁⣾⠞⠿⠿⣿⣿⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣌⣻⣽⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠈⠁⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣷⣦⣄⠠⣄⣁⠀⣀⢡⡾⠟⠛⠛⠻⠟⠋⠁⢸⡿⣿⣿⣷⡆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣦⡙⢃⣾⣿⡇⢀⡀⠀⠀⠀⠀⠀⣾⢻⣿⣿⣿⣿⠇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢒⣒⣒⣠⣄⣀⠀⠀⠀⠀⠀⡹⢋⣴⡏⢿⣿⡿⠟⠙⠀⠀⠀⠀⣠⣿⣿⣿⣿⣟⣋⠄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⡤⠋⣠⠞⠋⢭⠭⣭⣙⠻⢿⣦⡄⠲⢺⣤⣏⢻⣿⣸⡿⣇⠈⠉⢻⡟⠉⠀⠀⠀⠀⠀⡈⠀⠀⠐⠖
⠀⠀⠀⠀⠀⠀⡴⠀⡾⢡⠊⠀⠀⢠⠀⢈⠟⢦⡙⠋⠀⣾⣿⣿⡌⣿⡏⢳⠟⡄⣲⣄⠀⠀⠀⠀⣅⠀⠀⠈⠀⡴⠂
⠀⠀⠀⠀⠀⢀⡃⢸⠇⠀⠀⠐⢄⡼⠒⠒⢢⠈⠁⣠⢺⣯⣿⣿⣧⣿⢿⠆⢺⣃⣿⣿⡟⣦⡐⢀⡘⢃⣀⠠⠚⠁
⠀⠀⠀⠀⠀⠈⣷⣿⡀⠁⠠⠤⣾⠀⣾⣟⣷⣄⡀⢡⠀⢻⡿⠹⠿⣡⡌⢻⣮⢾⣿⣿⣷⠻⡀⠀⠀⢘⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⢿⣿⣇⢀⠀⢀⠽⣆⡈⠛⠛⢙⠀⠈⠀⢸⣷⠀⢾⣹⣿⠈⣇⣰⠾⣯⣥⣧⡰⠀⠀⠈⡀⠀⠉⠙⠳⢄
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠣⠁⠀⡌⠙⠛⠃⠈⠀⡰⠀⣸⣿⠒⣜⣼⡿⣰⣏⡶⡻⣿⣽⣿⡆⠀⠠⠀⢰⠀⠀⠀⠀⣀⣥⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠂⣀⠀⠀⠁⠀⣠⠇⣰⣿⠃⠈⡷⢀⣀⡀⣶⠘⠷⣼⣿⢿⣇⠄⠀⠀⠀⣤⢤⣦⠈⠙⠛⠻⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⣬⣀⣀⣨⣴⣾⡿⠃⠀⠀⠐⡛⡋⢁⠐⢿⣿⡭⣻⡞⢫⣅⢀⣤⣴⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠻⠛⠉⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠳⢭⣿⣶⣿⣟⠻⢿⠟⢛⠛⠛⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣯⣛⠆⠁⠀⣀⣀⡀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⢹⣷⣦⡀⠙⢩⠉⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠸⣿⣿⣿⠲⢄⢀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠈⠙⣿⡄⠊⡁⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠈⠙⠆⠐⠇⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣑⠀⢀⡀⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣶⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣿⡟⠀⠀⠀⠀⠤⠶⠶⠶⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠀⠀⠀⠀⣀⣼⠀⠀⠀⠀⠘⢿⣤⣄⠀⡠⠂⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⣟⣫⣥⣴⣶⠶⠾⠿⠛⠁⣠⣤⣤⠤⠤⠀⠉⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠩⢔⣋⣭⡭⠶⠶⠶⠶⠶⠶⢶⡶⠿⣿⣿⣄⠀⠀⠀⠈⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠁
⠀⠀⠀⢰⣤⠀⠀⠀⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠈⠻⣿⣿⣶⠀⠀⣀⣴⣿⣥⣔⡒⠲⠶⠦⠄⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⣠
⠀⡄⠀⠀⢻⣷⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠙⠿⣿⣶⡾⠋⠀⠀⠀⠉⠉⠓⠦⣄⠀⠀⣠⠊⠀⠀⠀⠀⠀⣠⡾⠋
⠸⠿⠟⢃⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡔⣸⠀⠀⠀⠀⠀⠀⣶⣦⣄⡀
⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⣰⡆⠀⠙⠙⣻⠿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣿⡇⠸⣄⣤⣾⣆⠀⠀⣸⣿⣿⡿⠿⠒⠒⠀⠀⠈⣳
⠀⢀⣾⣿⡇⠀⠀⠀⠀⣠⣾⡿⠁⠀⢀⡴⠃⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠹⣧⠀⠙⢿⣿⣿⡄⠀⢿⣿⣿⡇⠀⢠⣤⣤⣴⣾⣯
⠀⣼⠏⣿⣷⣦⣴⣶⣿⡿⠋⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠙⢷⣄⠈⠻⣿⣿⠀⠈⠻⠿⠃⠀⠈⣿⠀⠹⠿⠛
⢠⣿⠀⠈⠛⢙⡿⠟⢉⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠙⠳⢄⠘⢿⡆⠀⠀⠀⠀⠀⠀⢹
⣾⣿⣄⣤⣶⡿⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⡤⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠸
⠙⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣯⣭⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⣦⠄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣛⡛⠛⠛⠻⠶⠶⢤⣤⣤⣤⣤⣤⡾⠃⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⣠⣾⣧⣴⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣷⣌⡉⠙⠛⠒⠂⠀⠀⠀⠉⠩⢭⣀⡀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣶⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠶⠶⠶⠶⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⡶⠂⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢀⡴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣠⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠟⠛⠛⠛⠛⠿⠶⢤⣀⠀⠀⠀⠀⣀⣀⡤⠶⢿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠞⠛⠛⠉⠉⠉⠙⠛⠒⠲⠬⢕⢠⣶⡾⠿⢥⣤⣤⣴⢿⡷⡄
⠀⠀⠀⠀⠀⠀⣾⣿⠟⠉⠤⣤⣤⣀⡀⠀⠀⣄⠀⠀⢀⡴⡻⠋⠀⠀⡀⠤⣛⢭⣤⢾⣧⣤⣤⣤⡴⠶⠶⠒⠶⣦⣤⣀
⠀⠀⠀⠠⣤⣴⣿⣯⣄⣠⣴⣶⣾⣿⣿⣶⣄⣿⣆⡴⠋⢠⣧⣦⠀⠀⢀⡞⠁⠀⢻⠐⢾⣿⣿⠋⠀⠀⠀⠀⠀⠈⠙⢿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⢹⣿⣿⣿⡙⢮⡙⡏⠙⢿⣿⣿⣿⠋⠀⠀⠈⢧⣿⡆⢀⣊⠀⠀⠀⠀⠀⠀⠻⣿⠀⣄⠀⠀⠀⠀⠀⠀⠀⢹⣿⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠙⠏⠻⣅⠀⠑⣴⣀⠤⠛⢻⠏⢠⠆⣠⠀⠀⡸⠃⠈⠉⠉⠁⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⡟⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣿⡿⠀⠀⠀⡆⠸⡿⠁⠀⣴⣇⡄⣀⡀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⢰⣿⠃⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠿⠀⠀⠀⣰⣿⣏⢹⡏⠀⣀⣠⣤⣾⣿⠿⠟⠛⠉⠉⢹⣧⡀⠀⠀⠀⠘⣿⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠿⣿⣿⣿⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⡀⡄⠀⢻⡀⠈⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢄⡴⠋⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠘⣶⡄⢿⣄⠈⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣏⠞⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⡆⠀⠈⠻⡌⠉⠁⠀⠉⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡆⠀⠀⠀⠀⠀⠀⣠⣃⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⣿⣿⣟⣉⣠⡀⠀⢀⡀⣀⡤⣄⠀⠀⢀⣴⣿⠟⠛⢶
⠀⠀⠀⠀⠀⢦⣤⣾⣿⡿⣛⣻⣿⣿⣿⣿⣏⠀⢛⡼⢋⠳⣾⡿⣿⣿⣿⠁⠀⠁⠾
⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⢿⣿⡛⢙⢿⣿⣿⣷⣌⣱⠇⢰⣿⣇⡋⢙⣹⣧⡤⠟
⠀⠀⠀⠀⠀⠈⢻⡟⢤⣾⠈⣯⢙⠃⠃⢻⣿⣿⢼⣟⣠⡾⢻⠻⢷⣛⣛⣋⣴⣆
⠀⠀⠀⠀⠀⠀⢙⠈⢃⡤⠀⠂⣸⠄⠈⠈⣿⣿⣿⠿⢿⡅⣄⡽⠿⣛⣻⠛⢙⣷⠂
⠀⠀⠀⠀⠀⠀⠀⣷⠿⢾⣯⣟⠀⡄⠀⡄⢸⣿⡧⣾⣼⣿⣿⢋⣀⣼⣇⣪⣾⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⣿⣿⠢⣄⣀⣀⣀⣻⣿⣾⡿⣿⣽⣿⡟⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠠⣫⠴⠟⠚⠛⢃⣤⣀⣄⣠⣀⣌⣉⡉⢙⣿⡟⢪⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣴⠧⠖⠀⢠⡀⠉⠉⡉⠝⢛⡿⣷⢹⢿⡃⣽⣆⣤⣴⡾⠃⠀⣠⣤⣀⣤⣤⣀
⠀⠀⠀⠀⣠⠤⣼⣿⣷⣞⣶⣬⣀⣠⣤⡶⢿⣿⡖⣿⠸⠳⠪⣦⣾⣍⠛⢧⢰⣷⣿⣿⣿⣟⣿⣿⣿⣖⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠠⢾⢟⡟⣽⣿⣿⣶⡿⡶⢟⣿⣉⠷⣸⡿⡆⡗⢇⣠⣽⡿⣧⡀⠙⢟⢿⡿⢿⡟⣿⣿⣿⣿⣧⣤⣴⣿⠗
⠀⠀⠀⠀⠀⣆⡘⣠⣿⣯⣌⢯⣉⠋⠛⣉⠁⣰⢿⠗⣻⡷⣌⣿⡍⢠⡀⣷⡦⡀⠑⢝⡂⡈⢿⣿⣿⣿⠿⠿⣿⠿⣏
⠀⠀⠀⠀⢠⢋⣤⣸⣿⡇⡷⡀⠀⠀⠈⢁⣾⣽⢊⡆⡙⣷⢈⣿⣷⡄⠁⠙⣧⣄⠱⠄⠀⠠⣼⡿⠻⠿⠾⠟⠛⠃
⠀⠀⠀⡠⣨⣷⣿⣿⣿⣿⠟⡁⢢⣤⡚⣻⣿⣡⣾⢿⠟⢹⡟⣇⣘⣷⠀⠀⢹⣇⣤⠶⠐⠛⠚⠁
⠀⣠⠞⢰⡿⠞⠉⠀⣠⠋⣼⣟⡿⢿⣿⣿⣿⣿⣷⣯⠤⣾⣻⣌⠻⣿⣷⡄⢸⣿⣿⡄
⠀⢹⣻⡏⠀⠀⣰⣿⣧⠿⠋⠁⠀⠀⠈⠉⠙⠛⣷⣶⣿⡟⣿⠛⢲⡈⣿⣷⣈⣿⡎⠉
⠀⢸⣿⠀⠀⢀⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⢿⣿⠀⣸⠏⠚⣿⣯⣿⣷⣀⡀
⠀⠨⢹⣷⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢇⡟⣄⢻⣿⣿⣽⣿⣿⣿⠁
⠀⠘⢈⢹⣟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⣞⢠⠏⠀⠀⠉⠛⠉⠉⠉
⠀⠀⠈⠛⣸⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡚⡿⠋⢀⡟⠈
⠀⢀⣶⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠧⢀⣊⡾
⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⣸⠿⢿⠃⢀⣾⡟⢣⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢠⣿⡍⢲⢈⣄⡃⠀⠈⡀