AMONGUS Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠉⠉⠉⠛⢶⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠛⠛⠻⢶⣄⠀⠀⠻⣿⠋⠙⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠙⢷⡀⠀⠘⢷⡀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢻⣄⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠹⣦⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠋⠁⠀⠀⠀⠈⠉⠙⠻⣦⡀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠙⣧⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣬⣿⣆⠀⠀⠀⠀⠈⠙⠻⣿⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⣯
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⢀⣾⠋⠉⠉⠉⠀⠀⠈⠉⠛⢷⡄⠀⠀⠀⠀⠀⠹⣧⠀⠀⢀⣀⣠⣴⢿⡄⠀⠀⠀⢹⡇
⠀⠀⠀⠀⣠⣤⣤⣼⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠈⠻⣆⠀⠻⣦⣀⣠⡾⠃
⠀⠀⠀⢸⡏⠁⠀⢸⡇⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣠⡿⠃⠀⠈⠉⠉
⠀⠀⠀⣿⠇⠀⠀⢸⡇⠀⠀⠀⠀⠙⠷⣦⣤⣄⣀⣤⣤⣤⢶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⣿⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢿⡆⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠘⢷⣤⣤⣼⡇⠀⠀⠀⠀⠀⣠⣄⣀⣀⣀⡤⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠉⢹⣿⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢀⣿⠀⠈⣿⡀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⣴⡾⠋⠀⠀⠈⠛⠛⠛⠛⠁

⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣠⠤⠤⡀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⢀⣠⡔⠋⠓⢄⠀⠀⠀⠀⠀⢀⣀⣤⣤⡀
⠀⠀⠀⣰⣁⣠⣬⣷⡄⠀⠀⡴⢚⣡⣬⣚⣧⠀⠀⠀⠀⣠⣼⣦⣼⡀⠀⠘⠷⣗⢈⡉⣉⣱⠀⠀⠀⠀⡇⠈⣀⣄⣹
⠀⠀⣀⡏⡴⠚⠛⠻⡄⠀⢀⣨⠇⡖⠛⠚⠳⡄⠀⢀⣠⠏⡔⠒⠚⠳⡄⠀⣀⡞⢱⠛⠓⠛⣆⠀⠀⣀⡗⣹⠓⠒⠛⡆
⠀⢸⢸⡇⠙⠦⠴⢶⡇⠀⡞⢸⠀⠑⠦⠴⢾⠁⠀⡞⢸⠀⠑⠦⠴⢾⠁⢸⠁⡇⠘⠦⠤⠶⡏⠀⢸⠉⡇⠘⠦⠤⠴⡇
⠀⢸⢸⠃⠀⠀⠀⢸⡇⠀⢧⢸⠀⠀⠀⠀⡼⠀⠀⢧⢸⠀⠀⠀⠀⣸⠀⢸⡀⡇⠀⠀⠀⠀⡇⠀⢸⡀⡇⠀⠀⠀⠀⡇
⠀⠀⠙⣆⢀⡟⣆⡼⠀⠀⠨⣏⢓⡶⠶⢤⡙⢲⡄⠈⠉⣷⡄⢸⣽⠁⠀⢠⡏⠉⣠⠒⠲⣏⢙⡆⠀⢹⡛⣀⡼⠒⡞⠁
⠀⠀⠀⠈⣉⡀⠀⠀⠀⠀⠀⠈⠉⡤⣄⠀⠙⠋⠀⠀⠀⠐⠓⠋⠀⠀⠀⠀⠉⠉⡁⢀⠀⠈⠋⠀⠀⠀⠉⣁⣉⡉⠁
⠀⠀⢠⢋⣡⣬⣳⠀⠀⠀⠀⢠⠊⢉⣉⠳⡄⠀⠀⠀⠀⡴⠿⢛⣢⠀⠀⠀⠰⡔⣉⣩⣆⡀⠀⠀⠀⢸⡉⡇⠀⣇⡀
⠀⢰⣽⢩⠖⠛⠛⢧⠀⠀⢀⣸⢫⡗⠒⠛⢧⠀⠀⢀⣰⠋⡞⠓⠚⢳⡀⠀⢀⡾⢠⠒⠒⠛⢦⠀⠀⢈⣹⢩⠟⠓⠛⢦
⢠⡏⣿⠈⠣⠤⠤⣾⠀⢠⡏⣿⠈⠣⠤⠤⣾⠀⠀⡏⢹⠀⠳⠤⠤⢾⠁⢸⠉⡇⠘⠦⠤⠤⡞⠀⢰⠏⡟⠈⠦⠤⠤⣾
⠸⡇⡏⠀⠀⠀⠀⣿⠀⠸⡇⡟⠀⠀⠀⢀⡼⠀⠀⡇⢸⠀⠀⠀⠀⣸⠀⢸⡀⡇⠀⠀⠀⠀⡇⠀⢸⡀⡇⠀⠀⠀⠀⣿
⠀⠉⢷⠀⢰⢶⣀⡏⠀⠀⢽⡛⢦⡤⠤⣈⠓⢦⠀⠈⠙⣦⡀⢸⣻⠁⠀⢀⡟⠛⢠⠶⠶⣞⠙⡆⠀⢹⡟⠂⣰⠖⢶⠃

⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣠⠴⠋⠓⢄⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡀
⠀⠀⠀⢀⠎⢁⣀⣙⣆⡀⠀⠀⣠⠔⢋⣠⡥⢿⡀⠀⠀⠀⠀⣀⣤⣧⣠⣇⠀⠀⠸⢼⡗⠂⠂⠙⠹⡄⠀⠀⠀⢰⠁⠈⠀⡄⢸
⠀⠀⠀⣼⢋⡵⠿⠿⢯⡁⠀⠀⠘⡞⢩⠽⠿⠿⣇⠀⠀⠀⢀⡟⣩⠵⠮⠿⣄⠀⠀⠀⡿⢩⠽⠿⠿⢧⠀⠀⠀⠘⡦⣺⠯⠭⠿⣄
⠀⢰⠋⡏⠘⣆⣀⣀⣸⠇⠀⣼⢻⡇⠹⣄⣀⣀⣼⠀⠀⡾⢹⡇⠻⣀⣀⣀⡼⠀⢰⠛⡇⠸⣄⣀⣀⣼⠂⠀⡴⢻⡇⢻⣀⣀⣀⣼
⠀⢸⠀⡇⠀⠀⠀⠀⢸⠀⠀⣯⢸⡇⠀⠀⠀⠀⣿⠀⠀⡇⢸⠀⠀⠀⠀⠀⡷⠀⣻⢀⡇⠀⠀⠀⠀⢹⠀⠀⡇⢸⠃⠀⠀⠀⠀⣧
⠀⠘⠦⡇⠀⢠⣤⠀⣸⠀⠀⠙⣼⣇⣀⢀⠀⠸⢥⡀⠀⠳⠼⣄⠀⠀⡤⡴⠃⠀⢸⡶⢧⠀⣀⣀⣠⠾⣄⠀⠻⣼⣤⠀⢠⣀⣠⠏
⠀⠀⠀⠳⠤⠞⠘⠶⠃⠀⠀⠈⠳⠤⠟⠉⠉⠲⣄⡹⠀⠀⠀⠛⣇⣰⠗⠃⠀⠀⠸⢤⣤⠞⠁⠀⠱⣤⠞⠀⠀⠻⠤⠴⢾⣤⠏
⠀⠀⠀⢀⡤⠤⢄⡀⠀⠀⠀⠀⠀⢀⣴⣱⣄⠀⠀⠀⠀⠀⠀⠀⣉⣅⠀⠀⠀⠀⠀⠀⠀⢦⣠⡄⠀⠀⠀⠀⠀⢀⣠⡖⠖⣄
⠀⠀⣠⣏⡤⢦⡦⣇⠀⠀⠀⠀⢰⣉⡠⠤⢌⣱⠀⠀⠀⠀⠀⣼⡛⣫⠭⣆⠀⠀⠀⠙⣏⡤⠴⠧⢤⡀⠀⠀⠀⢹⣠⣇⣀⡽⣄⡀
⠀⣐⣿⠃⡞⠉⠉⠙⣧⠀⠀⣀⣸⠁⡞⠉⠉⠙⣦⠀⠀⣀⣸⠃⡟⠉⠉⠙⢧⠀⢀⣀⡟⢰⠋⠉⠉⠻⡄⠀⢀⣀⡟⢰⠋⠉⠛⢻⡄
⢀⡇⢸⠀⠙⠦⠤⠶⡏⠀⢰⡇⢸⠀⠙⠦⠤⠶⡏⠀⠀⡇⢸⠀⠙⠦⠤⠶⡟⠀⣼⠀⡇⠈⠳⠤⠴⢾⠁⠀⢸⠀⡇⠈⠧⠤⠤⢾⠃
⠀⡇⢸⠀⠀⠀⠀⠀⡇⠀⢸⡇⣿⠀⠀⠀⠀⢀⠇⠀⠀⡇⢸⠀⠀⠀⠀⢀⡗⠀⢻⠀⡇⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸
⠀⠙⢻⠀⠀⡶⡆⢠⠇⠀⠀⣹⠛⠦⣤⣤⣀⠙⠦⣄⠀⠙⠛⣦⡀⢰⣞⡏⠀⠀⢨⠟⠓⢀⡤⠤⣴⠛⢳⡀⠈⢻⠷⠂⢐⡦⢤⠞
⠀⠀⠈⠓⠚⠁⠙⠋⠀⠀⠀⠈⠓⠚⠁⠀⠈⠓⠤⠞⠀⠀⠀⠹⣧⡼⠉⠁⠀⠀⠘⠲⠶⠋⠀⠀⠈⠷⠚⠀⠀⠘⠒⠚⠛⠶⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠉⠉⠉⠉⠉⠛⠻⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣬⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⢀⣴⣿⠿⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣧
⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⢀⣾⣿⠋⠁⠀⣿⣿⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠙⢿⣷⣦⣄⣀⣀⣀⣀⣀⣀⣤⣴⣾⣿⠟⠁
⠀⠀⠀⠀⣿⣿⠁⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⣿⠿⠿⠿⠟⠋⢻⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⣿⣿⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠈⢿⣿⣶⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣶⣦⡤⠄⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢻⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠘⣿⣧⣀⠀⢀⣀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣦⣤⣤⣤⣤⣤⣿⡿⠁⠀⠻⠿⠿⣿⣿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣽⣛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣋⣉⣉⣽⣿⣿⣿⣿⣿⡿⣿⣿⡛⠛⠛⠛⠛⠛⠛⠛⠾⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⡤⣬⣿⣿⣿⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧⡴⠏⠀⠀⠀⠙⠻⠿⠿⠿⠿⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⢟⠉⠁⠀⠀⠀⠀⡟⠉⠛⠻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠁⠀⢀⡷⡖⠶⠒⠒⠋⡟⠲⠤⣤⣀⣈⡽⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣠⠞⠀⠹⣄⣀⣀⡤⡷⠤⠤⠤⢤⣯⣉⠓⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠥⢤⣤⡞⠁⠀⢀⡼⠻⣄⣀⣤⣽⣤⣤⣤⣾⣤⣬⣭⣍⣉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⡇⠉⠳⣞⢉⣴⠾⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⢰⠇⠀⢀⡇⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⣠⣶⠟⠛⠛⢿⡇⠀⠀⣸⠀⠀⡸⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⣼⠏⠘⢦⠀⠀⣾⣧⣄⣠⣏⠤⠾⡗⠋⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⣸⡏⠀⠀⠈⡇⢀⣿⠀⠉⠻⣧⡀⠀⠹⡄⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⣿⢣⡀⠀⢀⠿⢥⣿⠀⠀⠀⠈⢻⣆⠀⣇⡤⠚⠛⢷⣤⣄⣀⣀⠀⠀⠀⠀⢀⣀⣀⣠⣤⢾⣿⣿
⠀⠀⢰⣿⠀⢳⡴⠋⠀⠀⣿⠀⠀⠀⠀⠀⠹⣷⠛⠲⣄⢀⠞⠁⢉⡟⠋⠛⠛⡟⠛⠛⠙⢯⠉⣰⡿⠁⣿
⠀⠀⢸⡇⢀⠞⠓⢦⡀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡆⠀⣸⠿⠤⣄⣾⣤⠤⢤⣄⣷⠤⠖⠒⠚⣿⡟⠀⠀⣿
⠀⠀⢸⣷⡏⠀⠀⠀⠱⡄⣿⠀⠀⠀⠀⠀⠀⠘⣷⣰⠃⠀⠀⣸⠁⠀⡀⢀⠀⣿⠀⠀⠀⢠⡟⠀⠀⠀⣿
⠀⠀⠸⣿⠙⢦⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡧⠤⣄⣀⡏⢸⡀⣇⣿⡗⣿⣰⢀⣀⣾⠃⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⢳⢀⡜⠁⣿⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢹⡟⣩⢿⢿⣿⣿⣿⣏⠉⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⢻⡆⠀⢸⡟⠀⠀⣿⡶⣦⣤⣤⣀⡀⠀⢸⣇⡀⠀⢸⡇⠇⠃⢻⣿⡿⣾⢸⠀⢐⣿⢀⣀⣤⣴⣿
⠀⠀⠀⠘⣷⡀⡜⠉⠙⠲⣿⠀⠀⠀⠈⠉⡟⠳⣾⡏⠉⠙⠺⡗⠚⠚⠚⠛⠣⡿⠖⠚⠙⠿⠛⠉⣁⣸⡇
⠀⠀⠀⠀⠘⣿⡁⠀⠀⠀⣿⠛⠻⠷⠶⠾⢷⣤⣤⣤⣤⣤⣴⣧⣤⣤⣤⣤⣴⣷⠶⠶⠶⠾⠛⠛⠋⣹⡇
⠀⠀⠀⠀⠀⠈⠛⠷⢶⡶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡶⠆⠀⠀⠀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⣇⣀⣠⣤⣤⣴⠶⠾⠛⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠛⣿⠶⠶⠶⠶⠶⠟⠛⣿⠀⠀⠀⢸⡏⠉⠉⣀⣀⣀⣤⣄⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠙⠢⣿⠤⠤⠤⠤⠤⣤⣀⣿⠀⠀⠀⢸⡗⠋⠉⠁⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⡇⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⢿⣄⣀⠀⠀⣀⣀⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣧⣤⣤⣤⣤⡶⠞⠋⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣭⣤⣤⣀⠀⠈⠻⣤⡤⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠉⠀⠀⠀⠙⢷⡄⠀⠙⣦⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠈⢷⡀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠻⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠀⠀⠀⠈⠉⠙⠷⣄⠀⠀⠀⠘⢷⣄⣀⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠹⣦⣴⠏
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣠⣤⣤⡶⠶⠶⣽⣧⡀⠀⠀⠀⠈⠉⢿⡉⠀⠀⠀⠀⠀⣀⠄⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠈⢳⡄⠀⠠⢤⡾⠛⣦⡀⠀⠀⣿
⠀⠀⠀⣴⠞⠛⢻⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⢻⡆⠈⠳⠶⠞⠁
⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠘⢧⣀⡀⠀⠀⢀⣀⣠⣼⠏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁
⠀⠀⢸⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠁⣿
⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⣷⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠘⠳⠶⢾⡇⠀⠀⠀⠀⢰⡶⢶⡴⠖⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣷⣀⣀⣀⣠⡿⠀⠀⠻⠶⠶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠋⠉⠉⠉⠉⠛⢯⡛⠲⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠉⠉⠉⠉⠉⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢸⠀⠀⢾⠁
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⠀⠀⠈⠑⡆
⠀⠀⠀⠀⠀⢸⠤⣄⣀⠀⠀⠀⠀⠀⡏⠀⠀⣀⠤⠶⠶⣤⣄⠀⠀⠀⣀⠤⠶⠶⣤⣄⠀⠀⢸⠇⠀⠀⠀⢸⠀⠀⣀⠀⡇
⢀⡖⠛⠓⠒⣿⢤⣀⡈⠉⠙⠛⠒⠒⢻⠀⢰⡃⠀⠀⠀⢸⠈⡇⠀⢰⡃⠀⠀⠀⢸⠈⡇⠀⣸⠀⠀⠀⠀⢸⣄⡼⠉⠙⡇
⢸⠀⠀⠀⠀⣷⠀⠀⠉⠉⠙⠓⠒⠒⠺⡆⠀⠙⠲⠤⠴⠟⠊⠀⠀⠀⠙⠲⠤⠴⠟⠊⠀⢠⡇⠀⠀⢀⣀⣸⣀⣀⣀⣠⣇⣀⣀
⡿⠦⣄⣀⣀⡏⢣⡀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣠⣶⣄⠀⢀⣴⣦⡀⠀⠀⠀⢀⡞⠀⠀⡴⠉⠉⠉⢣⣀⣀⣀⣀⣀⡿
⡇⠀⠀⠀⠀⡇⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠘⠛⠋⠀⠀⠛⠛⠀⠀⠀⢀⣾⠀⠀⢸⠁⠀⢠⠀⠀⢷⠀⠀⡽
⡇⠀⠀⠀⠀⡇⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⢰⣿⣷⠀⢰⣿⣷⠀⠀⣠⠎⢸⡄⠀⢸⠀⠀⠈⣧⡤⠾⠒⠛⣧
⡇⠀⠀⠀⠀⡇⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠉⠁⠀⠀⠉⢁⡠⠞⠁⣠⠞⡇⠀⠸⡆⠀⢀⡷⠶⠶⠶⠚⢧
⣧⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠚⠋⠁⠀⣠⠞⠁⠀⡇⠀⠀⠳⣄⠀⠓⢦⠤⠴⢶⠋
⢹⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠋⠀⠀⠀⠀⡇⠀⠀⠀⠈⠓⠒⢾⣀⣀⣸
⠈⣇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠈⠙⠋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠓⠛⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢀⣠⠴⠖⠒⠒⠒⠒⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠺⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⠀⢰⡚⠁⠀⠀⠀⢀⣀⡤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠚⠉⠉⠉⠉⠉⠻⣍⢓⣶⣲⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠦⣀
⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠀⠀⠀⠈⠉⠙⠒⢤⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢸⠀⠀⢀⠜⠃
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢸⠀⠀⢣⡀
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⢸⡆
⠀⣀⣀⣀⣀⡏⠓⠲⠤⢤⣄⣀⣀⡇⠀⢠⠞⠉⠉⠹⡗⢆⠀⢀⡔⠋⠉⠙⣟⢢⠀⠀⡇⠀⠀⠀⢸⡀⢠⠦⣼⡇
⢰⠁⠀⠀⢸⡏⠑⠒⠦⠤⠤⣤⣀⣹⠀⠘⢤⣀⣀⣠⣧⠎⠀⠈⠧⣀⣀⣀⣿⠜⠀⢰⠃⠀⠀⠀⢸⠓⠋⠀⢸
⣾⣀⠀⠀⢸⠳⡀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠈⠀⣀⡀⠀⠀⣀⣀⠈⠁⠀⠀⢀⠏⠀⢀⠔⠒⠻⣟⠛⠛⠛⠛⢛
⡇⠈⠉⠉⢹⠀⠱⡄⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠘⠿⠿⠀⠀⠿⠿⠀⠀⠀⢀⡞⠀⠀⡏⠀⢀⠀⠈⣏⠉⠙⡏⠉
⡇⠀⠀⠀⢸⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⢀⣶⣶⠀⢠⣶⣶⠀⠀⣠⠎⢷⠀⠀⡇⠀⠈⣇⣠⠾⠴⠺⡅
⡇⠀⠀⠀⢸⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠑⢦⣀⠙⠉⠀⠀⠙⢉⡠⠞⠁⣠⢿⠀⠀⣇⠀⠀⡷⠤⠤⠤⠶⡅
⣇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠋⠉⢀⡠⠞⠁⢸⠀⠀⠘⣄⠀⠓⢦⠤⠤⣶⠃
⢹⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⣀⣀⠀⠀⣀⣀⣠⠤⠒⠉⠀⠀⠀⢸⠀⠀⠀⠀⠙⠒⠺⣄⣀⡿
⠈⢇⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠉⠉⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣠⠴⢚⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢠⡞⠉⠁⠀⠀⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠤⠤⠤⠤⠴⠊⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠖⠃⠀⠀⠉⢉⡭⠆⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣠⠴⠋⠓⢄⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡀
⠀⠀⠀⢀⠎⢁⣀⣙⣆⡀⠀⠀⣠⠔⢋⣠⡥⢿⡀⠀⠀⠀⠀⣀⣤⣧⣠⣇⠀⠀⠸⢼⡗⠂⠂⠙⠹⡄⠀⠀⠀⢰⠁⠈⠀⡄⢸
⠀⠀⠀⣼⢋⡵⠿⠿⢯⡁⠀⠀⠘⡞⢩⠽⠿⠿⣇⠀⠀⠀⢀⡟⣩⠵⠮⠿⣄⠀⠀⠀⡿⢩⠽⠿⠿⢧⠀⠀⠀⠘⡦⣺⠯⠭⠿⣄
⠀⢰⠋⡏⠘⣆⣀⣀⣸⠇⠀⣼⢻⡇⠹⣄⣀⣀⣼⠀⠀⡾⢹⡇⠻⣀⣀⣀⡼⠀⢰⠛⡇⠸⣄⣀⣀⣼⠂⠀⡴⢻⡇⢻⣀⣀⣀⣼
⠀⢸⠀⡇⠀⠀⠀⠀⢸⠀⠀⣯⢸⡇⠀⠀⠀⠀⣿⠀⠀⡇⢸⠀⠀⠀⠀⠀⡷⠀⣻⢀⡇⠀⠀⠀⠀⢹⠀⠀⡇⢸⠃⠀⠀⠀⠀⣧
⠀⠘⠦⡇⠀⢠⣤⠀⣸⠀⠀⠙⣼⣇⣀⢀⠀⠸⢥⡀⠀⠳⠼⣄⠀⠀⡤⡴⠃⠀⢸⡶⢧⠀⣀⣀⣠⠾⣄⠀⠻⣼⣤⠀⢠⣀⣠⠏
⠀⠀⠀⠳⠤⠞⠘⠶⠃⠀⠀⠈⠳⠤⠟⠉⠉⠲⣄⡹⠀⠀⠀⠛⣇⣰⠗⠃⠀⠀⠸⢤⣤⠞⠁⠀⠱⣤⠞⠀⠀⠻⠤⠴⢾⣤⠏
⠀⠀⠀⢀⡤⠤⢄⡀⠀⠀⠀⠀⠀⢀⣴⣱⣄⠀⠀⠀⠀⠀⠀⠀⣉⣅⠀⠀⠀⠀⠀⠀⠀⢦⣠⡄⠀⠀⠀⠀⠀⢀⣠⡖⠖⣄
⠀⠀⣠⣏⡤⢦⡦⣇⠀⠀⠀⠀⢰⣉⡠⠤⢌⣱⠀⠀⠀⠀⠀⣼⡛⣫⠭⣆⠀⠀⠀⠙⣏⡤⠴⠧⢤⡀⠀⠀⠀⢹⣠⣇⣀⡽⣄⡀
⠀⣐⣿⠃⡞⠉⠉⠙⣧⠀⠀⣀⣸⠁⡞⠉⠉⠙⣦⠀⠀⣀⣸⠃⡟⠉⠉⠙⢧⠀⢀⣀⡟⢰⠋⠉⠉⠻⡄⠀⢀⣀⡟⢰⠋⠉⠛⢻⡄
⢀⡇⢸⠀⠙⠦⠤⠶⡏⠀⢰⡇⢸⠀⠙⠦⠤⠶⡏⠀⠀⡇⢸⠀⠙⠦⠤⠶⡟⠀⣼⠀⡇⠈⠳⠤⠴⢾⠁⠀⢸⠀⡇⠈⠧⠤⠤⢾⠃
⠀⡇⢸⠀⠀⠀⠀⠀⡇⠀⢸⡇⣿⠀⠀⠀⠀⢀⠇⠀⠀⡇⢸⠀⠀⠀⠀⢀⡗⠀⢻⠀⡇⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸
⠀⠙⢻⠀⠀⡶⡆⢠⠇⠀⠀⣹⠛⠦⣤⣤⣀⠙⠦⣄⠀⠙⠛⣦⡀⢰⣞⡏⠀⠀⢨⠟⠓⢀⡤⠤⣴⠛⢳⡀⠈⢻⠷⠂⢐⡦⢤⠞
⠀⠀⠈⠓⠚⠁⠙⠋⠀⠀⠀⠈⠓⠚⠁⠀⠈⠓⠤⠞⠀⠀⠀⠹⣧⡼⠉⠁⠀⠀⠘⠲⠶⠋⠀⠀⠈⠷⠚⠀⠀⠘⠒⠚⠛⠶⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⡏⠉⠁⠀⠀⠀⠉⡟⠉⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢘⣶⠤⠤⠶⠖⢚⡗⠦⢤⣀⣀⣈⣻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⣠⠋⠈⢧⡀⠀⣀⣨⡧⣄⣀⣀⣠⣟⡙⠒⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣄⣀⣠⠞⠁⠀⢀⡼⢯⠉⣉⣀⣧⣤⣤⣤⣼⣥⣭⣙⣒⡚⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢸⠉⠳⢤⡖⢋⣠⡾⠟⠛⠉⠉⠉⠉⠀⠀⠉⠉⠉⠙⠛⠻⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣼⠀⠀⢸⢙⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⢀⣠⣴⠶⢶⣴⣿⠀⠀⠀⡇⠀⠀⡼⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⢠⡾⠛⣆⠀⠀⢸⣯⡀⠀⣸⢃⣀⣼⠥⠾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢠⡿⠁⠀⠘⣆⠀⢸⡏⠛⢷⣿⡉⠁⠙⣆⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⣼⣇⠀⠀⠀⢸⡔⢻⡇⠀⠀⠈⢻⣆⠀⢸⠀⢀⡿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡇
⠀⠀⠀⣿⠈⢧⢀⡴⠋⠉⢻⡇⠀⠀⠀⠀⠹⣧⣼⢿⡁⠀⠀⡽⠛⠻⣶⠶⣶⣶⣶⣶⡶⠶⡶⠿⠛⣹⡟⢻⡇
⠀⠀⢀⣿⠀⢸⢯⡀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢹⣧⠀⠈⢦⣞⠁⠀⡴⠃⠀⠀⠀⣇⠀⣀⣀⣘⣆⣾⠏⠀⢸⡇
⠀⠀⢸⣿⣰⠃⠀⠙⢦⡀⣿⡇⠀⠀⠀⠀⠀⠀⢿⡄⢠⠎⠉⠉⢳⠗⠛⠋⠙⠓⣿⠋⠁⠀⠀⣽⠏⠀⠀⢸⡇
⠀⠀⠘⣿⢧⡀⠀⠀⠀⢧⣿⡇⠀⠀⠀⠀⠀⠀⢸⣷⠏⠀⠀⠀⡏⢠⠀⡆⣤⡄⣿⢠⠀⠀⢸⡟⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠙⣆⠀⢀⡼⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠲⢴⣇⣼⣤⣯⣿⣧⣿⣾⠤⠶⣾⠃⠀⠀⠀⣼⡇
⠀⠀⠀⣿⡄⠀⠸⣆⡞⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠁⡞⡟⣿⣿⣿⢿⢳⠀⠀⣿⠀⠀⠀⠀⣿⠇
⠀⠀⠀⢸⣧⠀⢀⣿⣀⡀⣿⡟⠛⠿⠶⢶⣤⣄⡀⣿⠤⣤⣀⣸⠀⣇⣃⣻⣿⡏⣿⠸⣀⣀⣿⣠⣤⠶⠾⣿
⠀⠀⠀⠀⢻⣆⡾⠀⠀⠉⢻⣇⣀⣀⠀⠀⣸⠉⠛⠿⠀⠀⠈⢹⠋⠉⠉⠉⠉⠙⡟⠉⠁⠀⣛⣉⣀⣤⣴⣿
⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⢸⡏⠉⠙⠛⠛⠛⠿⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠾⠿⠛⠛⠛⠛⠉⠉⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠶⠶⣶⡶⠶⠶⠾⠃⠀⠀⠀⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⣿⣦⣤⣤⣴⡶⠶⠾⠛⠋⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠉⢻⡟⠛⠛⠛⠛⠛⠛⠙⣿⠀⠀⠀⢻⡇⠀⣀⣠⣤⠤⠤⠤⢤⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡍⠙⢾⠧⠶⠶⠶⠶⠦⢤⣠⡿⠀⠀⠀⢸⡟⠉⠁⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢸⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠈⢿⣄⣀⣀⣀⣀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢾⣤⣤⣤⣤⣴⠶⠟⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠠⣤⡄⢀⠀⠀⠀⠀⠔⢰⣿⡂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢒⣲⡄
⠀⠀⢠⣷⡔⠛⠛⣾⠷⠀⠀⠀⢰⠋⣩⠽⢦⡀⠀⠀⠀⢀⡀⢊⡉⢀⡄⠀⠀⠀⢠⡴⡇⢸⡉⠈⢉⡇
⠀⠀⢠⣼⣿⠘⣻⣿⣤⠀⠀⡞⡟⠘⢷⣦⣤⡟⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⢠⡇⠀⠉⠉⢹⡄
⠀⠀⠀⠉⠙⣶⡟⠉⠃⠀⠀⡇⡇⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠠⠄⠀⢸⠐⠀⣤⠼⠾⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠈⢷⠀⡶⣞⣸⠁⠀⠀⠀⠺⠦⠾⠷⠶⠞⠂⠀⠈⠙⠛⠦⣤⠖⠛⠋
⠀⠀⠀⠐⢖⣠⣥⣧⡀⠀⠀⠀⠀⠉⠁⠈⠁⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⣦⡀
⠀⠀⠀⢀⣼⠁⣖⠚⠛⣦⠀⠀⠀⠀⠀⠀⠀⢱⠜⣩⣽⢶⡀⠀⠀⠀⠀⠀⢠⢴⠇⢸⣍⣉⣙⡆
⠀⠀⠀⡏⣿⠀⠙⠻⠛⡏⠀⠀⠀⠀⠀⠀⣴⢻⠀⢿⣦⣤⡾⠀⠀⠀⠀⠀⢸⢸⠀⠈⠛⠛⢻⡅
⠀⠀⠀⢧⣿⠀⢀⣀⣠⡇⠀⠀⠀⠀⠀⠀⢿⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⣼⠀⠀⣀⣀⣸⠁
⠀⠀⠀⠀⠹⢤⠟⠯⠼⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⢰⣶⢲⠇⠀⠀⠀⠀⠀⠀⠘⢧⡼⠛⠧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣩⣤⣤⣿⣿⣿⣿⣿⠛⣿⣯⠉⠉⠉⠉⠉⠙⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⣿⣿⣿⣿⣿⣿⡟⠁⡴⠒⢾⣿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⢸⡇⠀⠈⣿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣆⣸⠇⠀⠀⠻⢿⣿⣿⣿⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠻⡍⠀⠀⠀⠀⣠⡏⠉⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢀⡜⢟⠛⠉⠉⢹⠉⠓⠲⠤⢤⣼⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⡠⠊⠀⢈⣦⠤⠶⠚⡟⠒⠒⢒⡟⠦⢬⣉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠉⢳⠻⢤⣀⡤⠊⣨⣶⠶⠞⠛⠛⠛⠛⠛⠛⠓⠶⢶⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢸⠀⠀⡿⢦⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⣠⣤⠶⢦⣼⡇⠀⠀⡿⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⢀⣾⠛⢦⠀⠀⢸⣇⡀⢠⣃⣠⣾⠴⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣾⠃⠀⠈⢧⢀⣿⠉⠛⢿⣅⠀⠘⡆⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⢠⡿⣄⠀⢀⡼⢯⣿⠀⠀⠀⠙⢷⡀⢹⣠⠞⠛⢶⣤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣠⣤⣾⣿⡇
⠀⠀⢸⡇⠘⣦⠎⠀⠀⣿⠀⠀⠀⠀⠈⢷⡋⠓⢄⢠⠞⠀⢩⠟⠙⠛⢻⠛⠛⠙⢯⠉⣴⠏⢸⡇
⠀⠀⢸⡇⡴⠉⠳⣄⠀⣿⠀⠀⠀⠀⠀⠘⣧⠀⣨⠛⠲⢤⡯⠤⠤⠤⢼⠴⠒⠚⠚⣿⠃⠀⢸⡇
⠀⠀⢸⣿⡁⠀⠀⠈⣆⣿⠀⠀⠀⠀⠀⠀⢻⣴⠃⠀⠀⡜⠀⢠⢀⣄⢸⡂⠀⠀⣸⡏⠀⠀⢸⡇
⠀⠀⢸⡇⠙⢦⠀⢀⡼⣿⠀⠀⠀⠀⠀⠀⢸⡟⠒⠲⢤⣇⣴⣼⣾⣿⣼⣿⡤⠤⣿⠀⠀⠀⢸⡇
⠀⠀⠘⣧⠀⠈⣧⠞⠀⣿⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠁⠎⠏⣿⣿⣿⠳⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⢿⡄⢀⡿⠤⣄⣿⠉⠛⠛⠳⢶⣤⣰⡿⠲⠤⣼⣄⣦⣥⣿⣿⣸⣘⡤⠴⣷⡴⠾⠛⣿⠁
⠀⠀⠀⠈⢷⡜⠀⠀⠀⣿⠶⣤⣤⣄⣼⣀⣙⣃⣀⣀⣸⣇⣀⣀⣀⣀⣽⣁⣠⣤⣥⡴⠶⠾⣿
⠀⠀⠀⠀⠈⠻⢦⣤⣤⣿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣩⡅⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⣿⣉⣉⣁⣠⣤⣤⠶⠟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠳⡶⠶⠶⠶⠶⠶⠟⣿⠀⠀⠀⣿⠉⠉⢉⣉⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠑⢲⡧⠤⠤⠤⠤⠤⣄⣿⠀⠀⠀⢿⡚⠉⠁⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⢸⠀⠀⠀⠀⠀⢀⣸⠏⠀⠀⠀⠘⢧⣄⣀⣀⣀⣤⣴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⢤⣴⡆⠠⣀⠀⠀⠀⠀⠀⠰⣉⠱⡿⡛⣉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠛⣛⣦⡀
⠀⠀⠀⢠⣾⣄⡼⠿⠷⢶⣿⣷⡀⠀⠀⠀⠀⣴⠛⢛⣛⣲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⡑⠀⢀⠀⢀⠀⠀⠀⢀⣀⣾⠀⣼⠋⠉⠉⠛⣆
⠀⠀⠀⠈⠙⣿⡄⣠⣤⣬⣏⠁⠀⠀⠀⣠⣼⡇⢰⣯⠉⠉⠉⣷⠀⠀⠀⠀⠀⠏⠂⠠⠷⠄⠒⠻⠀⠀⠀⠀⠀⣾⠉⡿⠀⠘⠶⣤⡤⣾⠃
⠀⠀⠀⠸⣾⣿⣟⠀⣸⣿⣿⣶⠂⠀⠀⡇⢸⠀⠈⠻⠿⠿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠭⢹⣶⣿⡉⠝⠁⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⡇⠚⠀⣤⡤⠿⠾⠁⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⣤⣤⠶⢸⠃⠀⠀⠀⠀⣤⣧⣀⣸⣶⣄⣠⣴⣦⠀⠀⠈⠛⠶⠶⣤⡀⢀⣤⠶⠶⠋
⠀⠀⠀⠀⠀⣀⣰⠛⠶⢺⠀⠀⠀⠀⠀⠀⠘⠷⠤⠟⠹⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡉⠉⠁
⠀⠀⠀⠀⠀⠙⣄⡴⠶⠾⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢆⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⣉⣙⣷⣄
⠀⠀⠀⠀⠀⣀⡟⠀⣶⠛⠛⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⠏⣉⣉⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠀⢾⣏⠉⠉⠉⣷
⠀⠀⠀⠀⣾⢹⡇⠀⠻⣷⣶⣶⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡇⢸⣿⣀⣀⣈⣿⠀⠀⠀⠀⠀⠀⠀⣾⠀⡏⠀⠘⠿⠿⠿⢿⡏
⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⠀⠙⠛⠿⠻⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠹⢾⡇⠀⢀⣀⣠⢤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠘⠶⣇⠀⠀⣀⣠⡤⢸⠇
⠀⠀⠀⠀⠀⠈⢧⣀⣼⠹⣧⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⣴⢶⠞⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣠⡏⢿⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠋⠘⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠞⠛⠛⠛⠛⠛⠻⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⠛⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡛⢆⠁⣀⠀⠀⡀⠀⠊⠀⠀⠀⠀⠀⠀⠀⣤⠄⡠⠆⣠⣉⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢫⣦⠙⢈⠲⣀⠙⠷⣇⣠⣖⠀⠀⠀⢀⣦⡀⢀⡥⠂⣠⠘⣠⠜⣡⠛⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠐⠰⠷⣮⡑⠎⣱⠤⣈⠁⠉⠀⠀⠐⠊⠁⣈⡠⠄⢢⡀⢚⣠⡞⠄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠈⠉⠐⠈⠘⠂⠮⠬⠋⣏⣈⡉⠧⠸⠂⡉⠥⠒⠉⠁⠀⠀⠀⡀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠘⣂⣀⣠⣤⠄⠀⠀⠀⠚⠓⡒⢂⣠⣤⣴⠶⠾⠿⠿⠟⠚⠛⠻⠾⠶⠷⣦⣤⣈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣤⠤⠚⣋⣩⡥⢤⢶⡔⡚⠋⠉⣭⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⢈⡥⢖⠙⠛⢁⠈⣤⣰⣄⣿⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⢏⠉⢡⡈⣗⣺⠷⠛⢋⣩⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⡏⣌⠀⣋⡫⠝⢋⡤⡶⢮⢾⡈⢿⡃
⠀⠀⠀⠀⣠⡶⠟⠛⣛⢛⣿⢟⢋⡷⢋⣰⢎⡛⢶⣿⡀⢻⠃⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢀⣾⣯⠛⢨⣉⣥⢸⣿⡴⢃⡴⢿⡙⢦⡿⠄⠉⢀⣴⣴⡷⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟
⠀⢀⣾⡅⣉⣉⣉⣉⡉⢙⣟⡰⢏⠻⢮⠷⠀⣡⣤⣾⠟⠋⢁⠄⢠⡞⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⢶⣿⠁
⠀⣼⡟⣧⡼⣆⢀⡽⣿⢻⣿⠷⣦⠧⠈⣠⣴⡿⢋⣀⣀⡁⠀⣶⣿⠁⠘⠷⣬⡝⠛⣿⣶⣶⢶⣶⣶⠶⠶⠶⠞⠛⠛⠋⠉⣠⡶⠃⢹⡀
⢠⣿⣠⣿⢀⠿⣿⡄⢨⢿⣿⡧⠈⣀⣴⡟⢁⠀⣽⠙⢻⣷⣄⡿⠉⠀⠑⠀⠈⠓⠚⠋⠚⠂⠰⠍⠛⣷⣶⡃⠀⢀⣶⡶⠟⠻⡄⠀⢹⡇
⢸⣟⣿⠟⣿⡇⣠⣿⡯⢸⣿⢃⣠⣿⠋⠀⠀⠐⣿⠆⠂⡄⠉⣁⡠⣴⠆⠀⠻⠋⠛⠋⠉⠱⠶⢢⣤⠄⠉⠛⠶⣟⠁⠀⠀⠀⠈⠀⢹⡇
⣼⡇⡷⠀⢸⣿⣿⢐⣷⣼⣯⣸⡿⠁⣞⠛⢿⠶⣿⠀⢀⣴⠈⠙⠒⠀⠀⠀⠀⠈⠀⠀⠠⠀⠘⠀⠀⠀⠹⠓⣤⡀⠀⢤⣀⡀⣀⠄⢸⡇
⣿⡃⣟⢶⣾⠁⢟⣿⠃⣸⣿⡿⠀⠆⠿⣦⣠⠂⠠⠾⠇⠍⠀⠀⠀⠀⠀⠠⠀⣀⢀⣀⠤⠤⠀⡀⠀⠀⠄⠁⠉⠉⢰⣄⠙⢿⡇⠀⢸⡇
⣿⢰⡁⠀⣸⣷⣯⠙⠗⢻⡿⠁⣨⣤⡤⠿⠀⣴⠀⠀⠀⠈⠀⡴⢦⠀⠀⠀⠀⠀⠀⠆⠀⡁⠈⠀⠁⠀⢤⡀⠠⣀⠐⠠⠷⡀⠻⡄⢸⡇
⣿⡎⢣⣴⠁⠈⡷⣴⡟⢻⡇⠘⣌⠉⠴⠀⣾⡠⢄⠀⠀⠀⠀⠀⠀⢀⢀⡀⡠⠄⠒⠀⠀⣵⢤⣀⣀⠊⠀⠃⠀⠀⠀⠀⠀⠴⣆⠀⣼⡇
⢿⡷⠋⠙⣦⣇⠀⠸⠃⢾⣷⡔⠻⣅⠁⣤⠀⠀⠀⢠⡂⠀⠀⢠⠀⠀⢃⠀⠀⠀⠁⠀⠀⢀⠢⡹⠃⠀⢨⠰⠀⡀⡀⠀⠀⠀⠺⠆⢸⡇
⢸⣧⢦⣴⠉⠸⣧⣤⡶⢾⣇⣴⡶⠋⢠⡍⠀⠄⠀⠈⠀⠁⢄⣠⡀⠀⠀⢢⠀⠶⠀⠀⠀⢸⡸⠁⠀⠀⡆⣄⡷⠔⠙⡀⠀⠀⠀⣴⣾⡇
⠈⣿⡉⠘⣦⣁⢾⣻⡄⢾⡿⣄⡀⠃⠈⠁⠀⠀⠈⠄⠠⠀⠀⠙⢧⠀⠀⠈⠖⠂⠀⠀⠀⢾⠃⠀⠀⢸⡝⠋⠀⣠⠇⠀⠀⠀⠈⠰⣿⡇
⠀⢹⣧⣶⢟⣿⣤⣴⣮⣼⡷⠘⢧⡀⠸⠂⠀⠀⠘⠀⠉⠒⠄⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⡏⠀⠀⡠⠃⠀⠀⠜⠊⠛⠙⠀⠐⠀⢠⣾⠇
⠀⠀⢻⣄⣤⣤⣤⣤⣤⣼⣏⠶⢂⣠⡴⠐⣿⠁⠙⠋⢹⠷⠂⣤⡈⠐⠄⠀⠀⠇⠀⠀⠀⠁⠀⠀⠀⢂⣤⠐⢾⠋⢹⠋⠘⠗⢠⣄⣿⠁
⠀⠀⠀⠻⣯⣛⢛⣭⣑⣽⣧⠴⡏⠀⣡⣀⡾⠤⠾⠤⠼⢀⣴⠁⢀⠟⢂⡀⢀⠀⠀⠀⠀⠀⠀⣠⠚⠃⠀⣄⣸⡄⠼⠄⣴⣀⡬⠁⣿
⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⣿⣀⠼⠂⠉⣀⣤⡄⢢⠀⠀⡄⢀⠈⠛⢤⡞⢙⡄⠢⠀⠀⠰⠁⣶⠉⣤⠜⠋⣁⣀⣤⣤⣤⢤⡀⠙⢺⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⠤⠘⠁⢰⠃⢸⡈⡀⢱⢀⡒⠙⡓⠄⠘⢧⠽⡄⠀⠀⢀⠛⣤⠛⣠⡴⢺⡏⢸⣿⠀⢻⡈⣧⢙⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⡀⠀⠂⠤⠀⡼⠣⡁⠦⢤⣀⠀⣓⠀⢢⣀⣞⣃⣠⣄⣬⣼⢁⣼⣿⠇⠈⣧⣸⡿⠀⢈⣧⢸⠎⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠷⢐⠆⠀⢄⡀⠀⠼⠃⣄⢄⡬⠀⠰⢸⣿⠉⠉⠉⠉⠉⣿⠷⡟⢸⣇⢠⠷⣾⠙⡖⡊⢻⠿⠄⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⡦⠌⢷⡀⡠⢐⡅⠀⠀⢠⡑⠀⡠⠖⢿⡟⠀⠀⠀⠀⠀⣿⠀⣷⣾⠉⢥⡶⠘⠒⢭⣥⣬⠲⠳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠉⠙⠆⠀⠀⠀⠀⡝⠲⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠆⣡⣀⡅⢒⡁⠀⠀⠸⢧⠆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡔⠀⠀⠀⠀⠹⠄⠠⠀⣸⠃⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢸⡆⠀⠀⠙⢫⡀⢀⠀⢠⣦⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠃⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠘⠋⠀⠀⠀⠁⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⡀⠀⠀⠀⢀⣠⡼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡁⠀⠀⠀⠀⢀⣴⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠾⠛⠛⠛⠛⠛⠛⠿⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⣏⠘⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⠁⠀⠀⣀⣀⢀⣠⠊⣈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣍⢣⡔⢣⡈⠑⢶⡜⡇⠀⣠⠀⠀⠀⠀⠀⡀⠀⠀⠏⡠⠉⢁⡜⠉⡶⢉⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⢐⠛⣦⡀⠦⣉⠦⣀⠉⠳⠞⠻⠆⠀⠀⣠⠼⠏⠶⠘⠃⣠⠘⢣⡔⢋⣔⠋⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠉⠀⠉⠹⢦⡌⠐⠮⣙⠒⠠⠠⣤⣀⡀⡀⢠⡄⢶⠈⣫⠤⠒⣡⡴⠟⠉⠁⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣙⠒⠃⠸⠤⠖⠀⠓⣈⣁⣤⣐⣈⣀⣀⡀⣀⠀⡀⠄⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⢈⣡⣤⣴⠶⠖⠀⠀⠀⣀⣉⣉⣉⣉⣤⣶⡾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠷⠶⣦⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⡦⠖⣛⣉⣤⢴⠶⠎⠧⢽⠛⠃⢀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⣡⠴⣛⠙⠋⢡⠈⢠⣄⣦⣀⣿⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⡿⡋⠉⣇⠈⣷⣚⡷⠾⠛⢉⣡⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢉⠁⠹⠒⢏⡻⠟⣉⣤⣖⣢⡏⢻⣿
⠀⠀⠀⠀⠀⣀⣤⡶⠶⠷⠶⣾⡷⠾⢳⡤⠋⢡⡴⢾⣅⣽⠀⢳⣽⠂⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢀⣾⢿⠕⠲⠶⢚⢴⣿⣇⡦⠋⣠⡾⢧⡘⣷⠈⠿⠃⢀⡁⢠⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⢠⡿⠻⠿⠤⢾⠶⣿⣼⣿⡟⣠⣾⣅⡻⡄⠹⠏⠁⢠⣤⣾⠿⠟⠋⠉⣻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠋
⠀⢠⣿⡗⠀⣶⠶⠂⣤⣤⣤⣿⡴⣉⢆⠉⠿⠉⠀⣼⣾⠟⠋⢀⠀⢊⣀⣴⠃⢹⣟⠳⢦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡴⠶⠛⣹⠿⣇
⠀⣼⡏⢹⡿⢻⣄⣴⠟⢿⣜⣿⡏⠻⡞⠓⢠⣶⡾⠋⢀⣶⣤⣤⠀⠈⣿⣿⠀⡀⠛⠳⣦⠉⣹⡏⣿⠟⠻⣿⣿⡛⠛⢛⠛⠉⠉⠉⠀⢀⣴⡞⠁⠀⣿
⢠⣿⣤⣿⣇⣬⠻⢿⣄⣰⠛⣿⣿⠀⢀⣤⣾⠏⢠⠀⢸⠄⠙⣿⣷⣴⡇⠀⠀⠘⠀⠀⣈⢉⣉⣈⠉⠀⠘⠂⠉⠻⣿⣿⡀⠀⣨⣿⡿⠛⠛⢇⠀⢀⣿
⢸⣟⣹⡿⠻⣿⢀⣤⠿⡧⠰⣿⡉⣀⣾⡟⠁⠀⡐⠐⣿⠇⠐⠠⠀⠉⣀⠤⣶⠆⠀⠹⠏⠉⠛⠉⠀⠹⠟⠲⣦⠄⢀⠉⠓⠾⡋⠀⠀⠀⠀⠀⠀⠀⣿
⣼⡇⣿⠄⠀⣿⣷⣯⢐⣷⣤⣿⢸⣿⠏⢠⡟⠛⣿⠶⣿⠀⠀⣠⡆⠉⠻⠰⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠘⠁⠀⠀⠘⠿⢲⣄⡀⠀⢤⣀⡀⠀⡄⠀⣿
⣿⡇⢹⣢⣴⡟⢠⢻⣾⠋⣸⣿⣿⠏⠰⠸⣷⣄⣠⠂⢀⣴⣿⢉⠁⠀⠠⠀⠀⠀⢀⠀⠀⣀⣀⣀⠤⢄⡀⠀⠀⠀⢀⠈⠈⠉⠁⣢⡄⠙⢿⡏⠀⠀⣿
⣿⢡⡞⠀⠈⣿⣴⡋⢿⡦⢿⣿⠋⠐⣀⣀⣈⡿⠁⣠⠂⠀⠀⠀⠒⢀⣠⣀⠀⠀⠡⠀⠀⠀⢠⠁⠘⠂⠐⠁⠒⠀⢀⠀⠀⡀⠀⡩⢉⣦⡀⠱⣄⠀⣿
⣿⡍⢇⡀⡼⠋⢳⣅⢄⣶⣼⡿⠠⡏⠛⢛⡁⠀⣴⠋⡁⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⣀⢀⡄⠀⠀⣇⡀⠀⠀⠠⠂⠹⠂⠀⠁⠀⠈⠀⣁⣄⠘⢀⣿
⣿⡇⡤⠛⣇⢸⡜⠉⢻⠇⢸⣿⠀⣸⡦⠨⠁⠘⠋⠈⠁⠂⠀⠀⠀⠀⠀⠀⠐⠔⠂⠈⠁⠤⠀⠀⠀⠋⣙⢮⡶⠀⠀⠄⡀⠀⠀⠀⠄⠀⠩⣹⣆⠘⣿
⢹⡟⣄⠀⡸⠟⣧⠀⠉⣀⣸⣿⠛⣀⣹⡆⢀⠷⠀⠀⠀⢰⠂⠠⠀⠀⠑⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⡷⢈⠞⠀⠀⢀⠃⢀⡐⣀⣤⠀⠀⠀⠈⠉⢀⣿
⢸⣿⢺⠞⡗⠀⡹⣻⣿⠋⢹⣷⠾⠻⣁⠀⢸⠄⠀⠒⢀⠀⠀⠀⠘⠢⣾⡄⠀⠀⠰⠂⠘⠂⠀⠀⠀⣟⡜⠀⠀⠀⣼⣸⡾⠋⠁⡠⠄⠀⡀⢀⡘⢻⣿
⠀⣿⣇⠂⢻⣦⡴⠿⠼⣆⢻⣟⢲⣴⠈⠀⢀⠀⠀⠀⢀⠂⢀⠢⠀⠀⠈⢷⠀⠀⠀⠘⡉⠀⠀⠀⠀⣷⠀⠀⠀⣠⠷⠉⠀⠀⢼⣒⢀⠀⠀⠀⠀⠻⣿
⠀⠸⣿⣼⡟⣹⠷⠶⠾⠧⢼⣿⠘⠛⣤⠀⠘⠁⠀⣀⣈⣀⡉⠉⠓⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⡇⠀⠀⠠⠋⡀⠀⠐⠛⢀⣉⣈⡁⠈⠃⠀⣾⣿
⠀⠀⠹⣧⣤⣤⣤⣤⣤⣤⢴⣿⡰⠛⣀⣴⡆⠘⡿⠁⠘⠋⠈⡿⠛⢰⣤⡈⠑⠄⠀⠀⠛⠀⠀⠀⠀⠀⠀⢀⠀⢊⣠⡄⠘⣿⠁⢹⠋⠈⠿⠃⣤⣸⡿
⠀⠀⠀⠙⢿⣕⠫⢋⡉⣑⣿⣿⡰⢾⠀⠀⣅⣀⣷⣤⣼⡦⣤⡇⢀⣼⠁⢀⠷⢆⡀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠖⢏⠀⢠⡀⣥⡄⣼⡄⣠⣆⣠⠋⢸⡇
⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠉⣿⡀⢀⠶⠘⠉⢁⣀⡄⢤⠠⠄⢀⠈⠉⠚⠿⣀⡼⠛⡄⠰⡀⠀⠀⣰⢀⣤⠉⢀⡰⠷⠚⠉⢀⣀⣠⣀⣀⡉⠙⠦⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢁⡀⠴⠈⢩⠋⢹⠘⠀⠀⡌⠠⡍⠲⢤⠈⠱⣎⢹⡄⠀⠀⠀⠁⢦⣈⡶⠋⣀⣤⣾⡟⣼⣿⠉⢿⡈⣇⠱⣆⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣖⡅⠁⠀⠀⡏⠀⢘⣆⡄⢠⢁⡙⠀⠀⡅⠀⣄⠈⡥⠿⠀⠀⠀⡚⢠⠛⣠⣾⣿⠀⢻⡁⣿⣯⠀⠸⣧⢹⣇⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢑⡀⢀⠀⠁⠀⠉⠀⢳⡜⠉⠓⠂⠐⠳⢀⠸⣶⠿⠾⠷⠿⠿⠻⣷⣔⣿⣟⡏⠀⢸⣃⡿⢧⣀⣠⣿⣶⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠈⠁⢪⢀⠈⠴⢂⡄⠉⠉⠑⠊⠒⢁⠀⠘⣼⣿⠀⠀⠀⠀⠀⠀⣿⠉⣯⢸⡷⡤⠋⠙⣯⣀⠧⠄⠸⢏⣃⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡸⠓⠉⠣⣠⣄⠰⠇⠀⠀⠀⢶⡈⢠⠎⠉⠙⣿⠀⠀⠀⠀⠀⠀⣿⠀⡽⠻⣇⠐⢺⠋⠈⠁⢲⡶⠶⠊⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠁⠈⣃⠀⠀⠀⠀⣰⠉⠙⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠃⡴⠦⣦⢊⡭⠀⠀⠀⠘⠳⠓⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⠂⠀⠀⠀⢀⠸⠦⠀⠄⢀⡏⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⠹⣤⡀⢀⠀⢠⡶⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠛⠁⠀⠀⠀⠁⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣄⡀⠀⠀⠀⠀⠀⢀⣀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡈⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠶⠾⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣤⣤⣴⠾⠋

⠀⠀⠀⠀⠀⠀⢠⣶⡿⠟⠛⠻⠿⠿⢿⣶⣄⣀⣀
⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⢀⣀⣀⣀⣈⣻⣿⣍⠙⢦⣠⠞⡧
⠀⠀⠀⠀⠀⣿⡇⠀⠀⢰⡿⠛⠛⠋⠉⠉⠛⠻⢿⣷⡾⢁⣼⡃
⠀⢀⣴⣶⣾⣿⠁⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡟⠁⠹⣆⣀
⠀⣿⡟⠀⢀⣿⠀⠀⠀⠸⣷⣄⠀⠀⠀⠀⠀⢀⣠⣿⠇⠙⢦⡀⠀⠹⢦⡀
⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠙⠻⠿⠷⠶⠿⠟⠛⣻⡏⠀⠀⠀⠳⣄⠀⢸⠇⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣦⣤⣀
⠀⣿⠇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠳⡾⠀⠀⠀⢠⣶⠟⠋⠉⠀⠀⠀⠀⠀⠈⢻⣷
⠀⣿⡆⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠶⢦⣄
⠀⣿⡇⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⣶⠿⠿⠿⠛⠛⠛⠛⠻⣿⣇⠀⠀⢹⣧
⠀⠘⢿⣦⣴⣿⠀⠀⠀⠀⢠⣤⣤⣤⣤⠄⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⣠⣴⠶⠶⠞⠳⠶⣤⡀⠈⣿⣆⠀⣸⡏
⠀⠀⠀⠈⠈⣿⡄⠀⠀⠀⢸⣿⢹⣿⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢀⣾⣭⣴⡾⠿⠿⣧⠀⠈⣿⠀⢸⣿⣠⣿⠁
⠀⠀⠀⠀⠀⣿⣇⠀⠀⢀⣼⡿⠀⢿⣦⣤⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡟⠉⢉⡉⠀⠠⣤⣿⠀⢀⡾⠃⢸⣿⡿⠃
⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠛⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣤⣾⡁⠀⠀⢸⣿⠶⠾⠃⣠⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣶⣿⣧⡴⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠿⠿⠿⠿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⠛⢷⣄⠀⢠⣴⡄
⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⢀⣤⣴⣶⣶⡶⠶⠶⣾⣿⣿⣷⡄⣹⡿⠋⣰⠇
⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⣼⡿⠉⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣟⢀⣴⠿⣤
⠀⢀⣴⣿⠿⢿⣿⡇⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⡁⠀⠸⠧⣤⡀
⠀⣸⣿⠃⠀⠀⣿⠀⠀⠀⠀⠸⣿⣆⡀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡟⠈⠻⣆⠀⠀⠘⠷⣤
⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⠿⣿⣶⣶⣶⣶⡶⠿⠿⣿⡟⠀⠀⠀⠈⢷⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣄⣀
⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠙⢦⣀⡿⠀⠀⠀⠀⢀⣴⣾⠿⠛⠋⠉⠉⠉⠉⠉⠙⠻⣷⣄
⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⣰⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣷⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣼⣿⡿⠛⠛⢷⣦
⠀⢻⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⡿⠟⠛⠛⠛⠋⠉⠉⠉⠛⠻⣿⣆⠀⠀⠀⣿⡇
⠀⠈⢿⣷⣦⣴⣿⡇⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡄⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⣠⣴⡶⠶⠿⠿⠷⠶⣦⡀⠀⠘⣿⣇⠀⢀⣿⡇
⠀⠀⠀⠈⠉⠙⣿⡇⠀⠀⠀⠀⢸⣿⡟⢻⣿⠁⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⢠⣞⣋⣥⣶⣿⣿⣷⣄⠀⠈⢹⡦⠀⢻⣿⠀⣼⡿
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⢸⣿⡇⢸⣿⣀⠀⠀⣀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣴⡿⠿⠿⠟⠉⠀⣀⣸⣿⠀⠀⢘⣿⠀⢸⣿⣾⡿⠁
⠀⠀⠀⠀⠀⠀⢿⣿⣦⣤⣤⣴⣿⡿⠁⠀⠙⠿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠀⠀⣴⠆⠀⠀⠙⣿⣿⠀⢀⣿⠁⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣾⣿⡀⠀⠀⢀⣾⠟⠛⠛⢁⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠀⢠⣶⣿⣿⣿⡿⠿⠟⠟⠛⠛⠛⠻⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⣿⣿⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠉⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠉⠀⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣀⣀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢱⣿⡟⠉⠀⠀⠉⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⣤⣴⣦⣤⣤⣤⣤⣤⠶⠖⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⣿⡇⠘⣿⣧⡀⠀⠀⣀⣼⣿⠇⠀⠀⠀⠀⣿⣿⡟⠛⢻⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⡇⠀⠘⠿⣿⣿⣿⠿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣿⣿⡇⠀⠘⣿⣿⣤⣤⣤⣤⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⣿⡇⠀⠀⣴⣶⣶⡶⠀⣿⣇⣀⣀⣀⣤⣾⣿⣿⠃⠀⠀⠘⠻⠿⠿⠿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⡇⠀⠀⣿⡇⣿⣇⣀⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣶⣿⠃⠙⠻⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡀⢀⠀⣠⢀⠀⠤⢠⠀⡠⠀⠀⠀⠀⠀⡠⠂
⠀⠑⠂⡇⠉⡌⠀⠀⠞⠀⡇⢀⠀⠀⠀⠀⡑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠌⠣⠐⠀⠃⠀⠒⠀⠘⠁⠄⠀⠀⠐⠄⠁⠔⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠋⠉⠉⠙⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣶⣶⠶⠶⣶⣦⡀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⢸⡿⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠶⠶⠶⠶⠾⠿⠛⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠛⠛⠛⠛⠛⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⣤⣤⣤⣶⣶⠶⠆⠴⠦⠤⠶⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣟⣻⣆⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠟⠒⠂⠐⠒⠲⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣀⣀⣀⣀⠀⠀⠀⠀⢸⣧⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⡏⠉⣿⠀⠀⠀⠀⢸⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢛⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣸⡇⠀⣿⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠻⢶⣶⡶⠾⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⣠⠂⣻⣿⣀⣴⡄⠀⠀⠀⠐⣂⡭⢭⣒⠁⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⣴⠋⣤⠬⠿⣤⡀
⠀⠀⠀⠿⣿⡁⣀⣈⣟⠛⠀⠀⢀⣰⠃⣴⠒⠚⠳⡄⠀⠀⠀⢠⠠⠈⡦⠀⠴⠁⠀⠀⠀⢰⡟⡟⠈⢧⣀⣀⣼⠃
⠀⠀⠐⣶⣿⡇⢉⣿⣿⡶⠀⠀⡟⢽⠀⠙⠿⠶⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣇⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⠽⠿⠏⠉⠁⠀⠀⣧⣸⠀⠀⠀⠀⢸⠃⠀⠀⠀⢀⠀⠀⠤⠀⠀⠗⠊⠀⣶⡾⠟⠁⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⠈⢹⡀⢰⢶⣋⡿⠀⠀⠀⠀⠚⠶⠾⠿⠶⠶⠓⠂⠀⠀⠉⠙⠳⣤⠴⠛⠛⠁
⠀⠀⠀⠀⢲⢊⣨⣥⣆⠀⠀⠀⠀⠀⠉⠉⠀⠉⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠦⣄
⠀⠀⠀⠀⣀⡏⢠⡖⠚⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢣⡜⢛⣛⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡇⢰⣏⠉⠉⢳
⠀⠀⠀⢸⢹⠇⠘⠿⣶⢶⡏⠀⠀⠀⠀⠀⠀⠀⡴⣾⠀⣿⣄⣀⣹⠆⠀⠀⠀⠀⠀⣿⢸⠀⠈⠻⠿⠿⣟
⠀⠀⠀⢸⢸⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠈⠉⠋⣻⠀⠀⠀⠀⠀⠀⢿⣸⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠈⠹⡆⠀⡶⡞⣸⠁⠀⠀⠀⠀⠀⠀⠀⠧⣇⠀⢀⣀⡄⡿⠀⠀⠀⠀⠀⠀⠈⢹⡄⠀⡶⡞⢱⠇
⠀⠀⠀⠀⠀⠉⠋⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠞⠻⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠶⠛⠛⠋⠉⠛⠛⠲⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⠋⠀⠀⠀⠀⣀⠀⠀⠀⠢⠀⠀⠀⠀⠈⠹⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⣉⠄⠐⢤⣀⡆⠀⠁⠀⠀⠀⠀⠀⠀⠖⡠⠚⡠⢢⡝⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠘⢳⣌⠢⣑⠄⡉⠣⠔⠧⠀⠀⣀⠾⠢⠔⢋⡀⢊⡐⢂⡔⠁⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⠉⠳⠌⠐⠩⠇⡂⢠⢤⠄⠄⢤⠐⡊⠥⠘⠠⠞⠋⠀⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⣀⣀⠀⠀⠀⠤⠤⠬⠈⣀⣡⣬⣤⣶⠶⠤⠤⠴⢦⣤⣤⣄⣀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⣠⠴⢛⣋⣭⣥⣤⡶⠶⠚⣣⡷⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣩⣶⠙⠛⢁⢡⣄⣦⣸⣶⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡟⣍⢰⣄⣻⠽⠚⠋⣤⣴⡟
⠀⠀⠀⠀⠀⣀⣀⣤⣀⣿⣐⣄⣩⠭⢊⣡⠖⡟⢛⣧⠹⡇
⠀⠀⠀⣰⣾⠫⣭⠽⢹⣿⢨⠜⣡⢶⡁⡎⠙⡷⠈⠋⣠⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⣾⠛⠣⠴⠶⠾⢼⡿⢡⢾⣍⣧⠙⠋⢀⣤⡶⠷⠛⠉⣹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡞⠃
⠀⣼⣿⣀⣗⠂⢲⣶⢾⣷⣥⣣⡈⢃⢀⣾⠟⠋⠠⠀⢥⣼⠇⢹⣏⣻⠶⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⡴⠶⠚⢋⡽⢻⡇
⢠⣿⣸⡏⢸⣶⡇⢹⢾⣿⡈⠃⣁⣾⠛⠁⣾⠻⣾⣄⢸⠛⠀⠠⠀⠻⠤⠞⠛⠏⠨⡛⢷⣤⡴⠁⠀⣤⣤⡾⣿⠀⢸⡇
⢸⣟⣿⢿⣷⢀⣿⡎⢸⡟⢀⣼⡟⠁⠀⠰⣿⠀⢈⠉⢋⡀⣤⠄⠐⠾⠒⠟⠂⠐⠶⢠⣄⠉⠛⠢⡞⠋⠁⠀⠀⠁⠘⡇
⣾⢿⡇⠀⣿⢾⣅⣷⢼⣇⣿⠏⢰⠛⠻⠶⠧⠀⣠⡆⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠛⢢⣄⠀⠦⣀⣀⡠⠀⡏
⣿⠀⡷⠾⡇⢊⣿⡁⣼⣿⠃⡐⠘⢷⠴⠁⠐⠻⠠⡁⠀⠀⠀⠀⠀⠀⠤⠤⠔⡂⢄⠀⡀⠀⠀⠀⠀⠰⢆⠈⢿⡀⠈⣧
⣿⢸⠀⢠⠟⣷⠈⣫⣿⡇⡠⠶⡶⠚⢀⠾⠀⠀⠀⠀⠐⠱⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡐⢦⠀⠠⠈⠘⠓⢀⠱⠀⡷
⣿⣦⠷⣇⢀⠙⠻⡏⣽⡇⢹⡦⠘⠀⠞⠊⠑⠀⠀⠀⠀⠀⠀⡄⠀⠀⠨⠀⠀⠸⡳⢴⠆⠀⠀⠀⠀⠀⠀⠀⢛⣆⠸⡇
⢻⡧⡀⡸⠛⣄⣈⣄⣿⡟⣅⣹⠆⡸⠀⠀⠀⠖⠀⠀⠈⠀⠀⠀⠀⢀⡀⠀⠀⢸⢨⠊⠀⢀⠃⢥⣀⠤⠀⠀⠀⠉⣠⡇
⠸⣯⠋⢯⡀⣹⢿⡁⣿⣟⠉⠱⠀⠃⠀⠁⠀⠀⠀⠈⠰⣇⠀⠀⠈⡄⠁⠀⠀⢸⠎⠀⠀⢼⡾⠋⠀⡄⠂⠀⠀⠌⣹⡇
⠀⢿⣆⣼⣻⣭⣥⣅⣻⡇⠿⡀⠀⠦⠀⠀⠒⠐⠂⠄⠀⠈⠀⠀⠀⢴⠀⠀⠀⡸⠀⠀⡰⠋⠀⠠⠎⠓⠒⠀⠠⠀⣸⡇
⠀⠘⢯⣡⣤⣤⣄⣤⣼⡇⠖⢈⣤⠐⢶⠃⠛⠃⢲⠖⢠⣌⠀⠄⠀⠸⠀⠀⠀⠀⠀⠀⠐⢀⡄⢲⡖⠹⠏⠰⠖⢠⣸⡇
⠀⠀⠈⠻⣽⣙⣩⣱⣿⡧⠾⠁⢠⣄⠼⠤⠶⠤⠇⢠⡎⣀⠟⢢⡀⠄⠀⠀⠀⠀⠀⣤⠛⡀⢠⡠⠦⠼⠆⠴⣠⡌⣸⡇
⠀⠀⠀⠀⠈⠙⠋⠉⢸⣧⡄⠋⡀⠠⡴⢖⠒⠀⠠⠀⢄⡉⠲⢊⠱⡄⠂⠀⠰⢃⠶⣀⠦⠋⣁⡤⣴⡴⣶⢤⢀⡙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡔⠂⠁⠰⠃⢸⡀⠀⠃⠈⠀⢁⠀⡈⢒⣱⠀⠀⠠⢏⡶⢁⣴⡋⢿⢠⣿⠁⢸⡌⣧⠽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢱⡀⣀⠈⠀⠊⠘⣦⠊⠃⠂⠘⢀⢰⡶⠶⠶⠶⠾⢾⣴⣿⣻⠀⢸⣺⠿⣀⡸⣷⡟⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣁⣃⠀⠑⢡⠈⠁⠈⠉⠋⠀⣈⣼⡇⠀⠀⠀⠀⣿⠉⣇⣿⠲⢃⡉⠤⢶⠂⡘⣭⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡎⠀⠀⠳⠳⡌⠁⠀⠀⢙⠤⡆⠀⢸⡇⠀⠀⠀⠀⢻⡄⢋⢙⠂⠝⠀⠀⠘⣏⡍⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠄⠀⠀⠀⠰⠀⠀⠀⡞⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⡇⠉⠉⠲⣉⠀⠀⢀⣄⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡼⠃⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠛⠁⠀⠀⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣀⣀⣀⣠⡼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣄⡀⠀⠀⠀⣀⡼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⠶⠶⠶⠶⠤⠄⢄⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢰⣾⡍⣠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⡠
⠀⠀⠀⣔⣩⡤⢄⡈⠑⡀⠀⡴⠛⡛⠻⢷⣕⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠇
⠀⠀⠀⡅⠀⠈⢦⠈⠀⠰⣀⣱⣄⢈⡄⠙⠙⣧⠱
⠀⠀⠀⢰⢀⡀⠈⢃⠀⠀⢀⠙⠟⡈⡴⢊⣆⠘⣇⠁⠀⠀⣾⣿⡀⠀⠀⠀⠀⠱⠛⣸⠉⠉⢉⣙⣶⢄
⠀⠀⠀⣼⠉⠁⠀⠀⠓⠤⡤⠄⠚⠻⠷⣿⡛⠂⡿⡀⠀⠀⠻⠟⠁⠀⠀⠀⢀⡇⠐⠂⠒⠋⠉⠁⠀⠀⠳⡄
⠀⠀⢠⠠⠀⠠⠤⠤⢠⠊⠀⠀⢀⠀⢦⡼⠁⢰⢧⠃⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠸⡀⠀⡃⠀⠀⢸⠀⠘⠹⣿⡳⢖⣁⣖⡫⠃⠀⠀⠀⠀⠀⠀⠀⢴⠛⢂⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠑⢄⡑⡀⠀⠈⢆⠠⠀⢯⣉⡉⠉⠁⣀⣀⣀⢤⠤⢲⠊⠉⠁⠀⣣⣾⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡄⠙⢶⠀⠀⠀⠑⠌⣢⡀⠙⡅⠉⠀⠀⠀⢀⡀⢀⣤⣴⠶⠿⠂⠙⠷⣦⣽⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠃⠀⠈⠆⢀⠃⠀⠀⠀⡿⡄⡗⠀⠀⠀⣀⠤⣶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠻⣷⠀⠀⠀⠀⠀⠀⡇
⠀⠦⡀⠀⠈⠀⠀⠀⠁⠀⠀⢀⣼⠑⠟⠤⠤⠤⡈⠀⣼⠁⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⠀⣿⠀⠀⠀⠀⠀⠀⢃
⠀⢀⠈⢦⡀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠰⢲⢀⠇⣰⠇⠀⠀⠀⠀⠀⠀⡴⠛⠁⠀⠀⠀⠃⢧⡀⠀⠀⠀⠀⢸
⠀⠈⠰⠀⠳⣄⢇⠀⠀⠀⢀⡏⠀⠀⠀⠰⠼⠸⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠀⠀⠘
⠀⠀⠀⠳⡀⠫⣠⡁⠀⠐⢻⡀⠀⣀⠴⣠⠀⡇⠀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠑⢄⠈⠙⢦⠰⠸⣇⠎⢁⠴⠿⠀⢹⠀⠈⠻⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⠿⠿⠿⠿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣤⣴⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠻⣿⡄⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⣿⡷⡶⠾⢷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠿⢿⣶⣦⣤⣤⣤⣤⣤⣤⣶⠿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⣤⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⣀⡀⠀⠀⣀⣀⣤⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠛⠛⣩⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⣤⣴⣶⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⣷⣦⡀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣶⣶⠿⠟⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠋⠉⠉⠉⠉⠛⢍⣛⣶⣲⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠢⣄
⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠁⠀⠀⠀⠀⠉⠉⠒⢤⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢳⡀
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⡇⠀⢠⠔⠁
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⡇⠀⠘⢄⡀
⠀⠀⠀⠀⢀⣇⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⡇
⢀⡤⠶⠦⢼⣄⡉⠙⠒⠒⠶⠤⢾⠀⢠⡏⠀⠀⠈⡏⢣⠀⢠⠋⠀⠀⠈⡏⣣⠀⢸⠀⠀⠀⠀⣇⣠⠚⠲⡇
⣸⠀⠀⠀⢸⡀⠈⠉⠓⠒⠒⠒⠲⡇⠀⠑⠦⠤⠴⠗⠃⠀⠀⠑⠦⠤⠴⠗⠁⢀⡏⠀⠀⢀⣀⣇⣁⣀⣸⣃⣀
⡟⠦⢤⣀⣼⠱⡄⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣴⣦⠀⢀⣶⣦⠀⠀⠀⠀⡜⠀⢀⠞⠉⠉⠱⣤⣤⣤⣤⣬
⡇⠀⠀⠀⢸⠀⠙⣄⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢙⡋⠀⠀⣙⡋⠀⠀⢀⡼⡀⠀⢸⠀⠀⡆⠀⢸⠀⣀⠗
⡇⠀⠀⠀⢸⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠘⢦⡀⠀⢿⠿⠀⠘⢿⠟⠀⡠⠊⢀⡇⠀⢸⠀⠀⢸⡒⠋⠉⣈⡇
⡇⠀⠀⠀⢸⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠉⠒⠤⢤⣀⣠⠤⠔⠋⢀⡴⠋⡇⠀⠘⡆⠀⢺⣉⠉⢉⣁⡗
⢧⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠉⠀⠀⣧⠀⠀⠈⠢⢤⣸⡉⠉⢹
⠸⡄⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠙⠒⠒⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣠⠴⡇⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠈⠉⠁⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠊⠉⠉⠉⠀⠀⠉⠉⠒
⠀⠀⠀⠀⠀⠀⠙⠤⠤⠤⠤⠤⠤⠤⠞⠀⠀⠀⠀⠀⠈⠣⠤⠤⠤⠤⠤⠤⠞⠀⠀⠈⠉⣩⠷⠀⠀⠀⠀⠀⢀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠙⠲⡄⠉⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠹⡄
⠀⠀⠀⠈⠉⠙⠓⠬⡉⠙⢢⡀⠀⠀⠀⠈⣳⣄⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⢀⠇⠀⠀⠀⢰⡥⠬⢷⡀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢀⠜⠁⢠⠊⠀⠀⠀⠀⠀⠀⠀⣠⢷⠀⠀⠀⠀⠉⠓⠲⠶⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢰⠃⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠾⠥⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠈⠲⠤⠤⣄⣉⡉⠉⠉⠉⠉⠓⢦⡀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⡓⢄⠀⠀⢀⠀⠻⡤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⡞⢹⡀⠀⡞⢹⡀⡇⡰⠛⡆⠀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠘⡆
⠀⠀⠀⠀⠀⢠⠾⠖⠚⠓⠚⠛⠛⠛⠛⠛⠒⠛⠒⠻⠃⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠶⠋⠀⠀⣷⠴⠚⠉
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡗⠚⠉
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⣠⡴⠋⠀⠀⠀⢳⣀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿

⢀⠄⡄⠄⠠⠐⢠⢀⠀⠀⠀⠀⠀⠂
⠀⠄⠘⠐⠄⢀⠀⠼⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣥⣤⣤⣤⣄⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣿⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣀⣠⣤⠟⠀⠀⣿⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠶⠶⠶⣼⡇⠉⠉⠉⠉⠀⠀⠀⠀⣿⠀⣿⠀⠀⣀⣀⣠⣤⣤⠀⣤⡠⣤⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣻⣦⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠈⠉⠙⠛⠛⠚⠿⠗⠒⠂⠀⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠀⣀⡀⠀⠀⠀⣿⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⣿⠉⡟⠀⠀⠀⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣻⣿⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⡇⠀⣧⣀⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠋⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿

⠀⡀⢀⠀⣠⢀⠀⠤⢠⠀⡠⠀⠀⠀⠀⠀⡠⠂
⠀⠑⠂⡇⠉⡌⠀⠀⠞⠀⡇⢀⠀⠀⠀⠀⡑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠌⠣⠐⠀⠃⠀⠒⠀⠘⠁⠄⠀⠀⠐⠄⠁⠔⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠋⠉⠉⠙⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣶⣶⠶⠶⣶⣦⡀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⢸⡿⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠶⠶⠶⠶⠾⠿⠛⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠛⠛⠛⠛⠛⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⣤⣤⣤⣶⣶⠶⠆⠴⠦⠤⠶⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣟⣻⣆⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠟⠒⠂⠐⠒⠲⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣀⣀⣀⣀⠀⠀⠀⠀⢸⣧⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⡏⠉⣿⠀⠀⠀⠀⢸⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢛⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣸⡇⠀⣿⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠻⢶⣶⡶⠾⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠿⠿⠿⠿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⠛⢷⣄⠀⢠⣴⡄
⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⢀⣤⣴⣶⣶⡶⠶⠶⣾⣿⣿⣷⡄⣹⡿⠋⣰⠇
⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⣼⡿⠉⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣟⢀⣴⠿⣤
⠀⢀⣴⣿⠿⢿⣿⡇⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⡁⠀⠸⠧⣤⡀
⠀⣸⣿⠃⠀⠀⣿⠀⠀⠀⠀⠸⣿⣆⡀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡟⠈⠻⣆⠀⠀⠘⠷⣤
⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⠿⣿⣶⣶⣶⣶⡶⠿⠿⣿⡟⠀⠀⠀⠈⢷⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣄⣀
⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠙⢦⣀⡿⠀⠀⠀⠀⢀⣴⣾⠿⠛⠋⠉⠉⠉⠉⠉⠙⠻⣷⣄
⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⣰⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣷⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣼⣿⡿⠛⠛⢷⣦
⠀⢻⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⡿⠟⠛⠛⠛⠋⠉⠉⠉⠛⠻⣿⣆⠀⠀⠀⣿⡇
⠀⠈⢿⣷⣦⣴⣿⡇⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡄⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⣠⣴⡶⠶⠿⠿⠷⠶⣦⡀⠀⠘⣿⣇⠀⢀⣿⡇
⠀⠀⠀⠈⠉⠙⣿⡇⠀⠀⠀⠀⢸⣿⡟⢻⣿⠁⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⢠⣞⣋⣥⣶⣿⣿⣷⣄⠀⠈⢹⡦⠀⢻⣿⠀⣼⡿
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⢸⣿⡇⢸⣿⣀⠀⠀⣀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣴⡿⠿⠿⠟⠉⠀⣀⣸⣿⠀⠀⢘⣿⠀⢸⣿⣾⡿⠁
⠀⠀⠀⠀⠀⠀⢿⣿⣦⣤⣤⣴⣿⡿⠁⠀⠙⠿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠀⠀⣴⠆⠀⠀⠙⣿⣿⠀⢀⣿⠁⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣾⣿⡀⠀⠀⢀⣾⠟⠛⠛⢁⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⣒⣶⣤⣤⣄⣀⢀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠋⢁⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⢀⠑⠁
⠀⠀⠀⢀⠴⠂⠊⠐⠂⢄⠀⠀⢀⣼⡿⣿⣼⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⡎
⠀⠀⠀⡮⠂⠙⢯⠑⠆⠈⠆⠀⡎⡀⠳⠀⠨⡽⣷⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠁
⠀⠀⠀⢸⠀⠀⠈⣧⠀⠀⠈⠦⠾⣵⡠⠿⡀⣄⠈⢷⠘⡀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣇⣀⢠⣤⣤⣤⣤⣀
⠀⠀⠀⠈⣣⠤⠐⠀⢧⡀⠈⠲⠀⢋⣥⣶⣡⣼⣆⢸⡇⠁⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⡏⠨⠾⠂⣀⣤⠴⠾⠛⢦⡀
⠀⠀⠀⡸⡇⠀⠀⠀⠀⠩⢓⠖⠊⠉⠉⠛⠻⣯⠌⢸⡇⡄⠀⠀⠘⠋⠁⠀⠀⠀⠀⣰⠁⠈⠁⠉⠁⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⡇⢆⢀⠖⠒⠀⠴⠁⠀⢀⣀⡀⠙⢶⠃⠀⡸⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣇⠀⢸⠀⠀⠀⢷⠀⠈⠘⣿⣛⣓⣃⣼⣊⠕⠁⠀⠀⠀⠀⠀⠀⠀⠠⢞⠛⣄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠈⠢⣈⢂⠀⠀⠘⢄⠠⠀⠳⣉⡉⠉⠉⠀⣀⣀⣀⢤⠤⠒⠒⠉⠁⠀⢘⣤⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⢀⠈⠳⣇⠀⠀⠈⠒⢌⡢⡀⠈⢯⠉⠇⠀⠀⠀⢀⣀⡀⣀⣤⣴⡶⠥⠌⠻⢶⣤⣽⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠹⡀⢰⠀⠀⠀⠈⢷⣄⢸⠄⠀⠀⠀⠀⣀⢴⠞⠉⠈⠀⠀⠀⠀⠀⠀⠙⠛⠿⣶⠀⠀⠀⠀⠀⠀⠘⡄
⠀⣤⠀⠀⠀⠂⠀⠀⠑⠊⠀⠀⠀⢠⡇⢼⣎⣀⣀⠀⠒⠉⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠄⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⣠⡍⢩⠆⢠⡏⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⡟⢧⠀⠀⠀⠀⠀⠀⢇
⠀⠸⠠⠈⢳⡀⢠⠀⠀⠀⠀⠀⣾⠁⠀⠀⠸⡀⡿⡼⢠⠛⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠘⠷⡀⠀⠀⠀⠀⢸
⠀⠀⠀⢣⠀⢹⠎⢱⠀⠀⢀⣴⠃⠀⠀⠀⣀⠉⠁⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈
⠀⠀⠀⠈⠣⡀⠑⠾⣄⠀⠀⢸⡇⢀⡴⠞⢣⡇⠀⡇⠀⢦⣘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⢀⠀⠈⢣⡘⠈⣇⡇⢀⠞⠛⠁⠀⢣⠀⠀⠙⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⣒⣶⣤⣤⣄⣀⢀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠋⢁⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⢀⠑⠁
⠀⠀⠀⢀⠴⠂⠊⠐⠂⢄⠀⠀⢀⣼⡿⣿⣼⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⡎
⠀⠀⠀⡮⠂⠙⢯⠑⠆⠈⠆⠀⡎⡀⠳⠀⠨⡽⣷⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠁
⠀⠀⠀⢸⠀⠀⠈⣧⠀⠀⠈⠦⠾⣵⡠⠿⡀⣄⠈⢷⠘⡀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣇⣀⢠⣤⣤⣤⣤⣀
⠀⠀⠀⠈⣣⠤⠐⠀⢧⡀⠈⠲⠀⢋⣥⣶⣡⣼⣆⢸⡇⠁⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⡏⠨⠾⠂⣀⣤⠴⠾⠛⢦⡀
⠀⠀⠀⡸⡇⠀⠀⠀⠀⠩⢓⠖⠊⠉⠉⠛⠻⣯⠌⢸⡇⡄⠀⠀⠘⠋⠁⠀⠀⠀⠀⣰⠁⠈⠁⠉⠁⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⡇⢆⢀⠖⠒⠀⠴⠁⠀⢀⣀⡀⠙⢶⠃⠀⡸⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣇⠀⢸⠀⠀⠀⢷⠀⠈⠘⣿⣛⣓⣃⣼⣊⠕⠁⠀⠀⠀⠀⠀⠀⠀⠠⢞⠛⣄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠈⠢⣈⢂⠀⠀⠘⢄⠠⠀⠳⣉⡉⠉⠉⠀⣀⣀⣀⢤⠤⠒⠒⠉⠁⠀⢘⣤⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⢀⠈⠳⣇⠀⠀⠈⠒⢌⡢⡀⠈⢯⠉⠇⠀⠀⠀⢀⣀⡀⣀⣤⣴⡶⠥⠌⠻⢶⣤⣽⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠹⡀⢰⠀⠀⠀⠈⢷⣄⢸⠄⠀⠀⠀⠀⣀⢴⠞⠉⠈⠀⠀⠀⠀⠀⠀⠙⠛⠿⣶⠀⠀⠀⠀⠀⠀⠘⡄
⠀⣤⠀⠀⠀⠂⠀⠀⠑⠊⠀⠀⠀⢠⡇⢼⣎⣀⣀⠀⠒⠉⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠄⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⣠⡍⢩⠆⢠⡏⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⡟⢧⠀⠀⠀⠀⠀⠀⢇
⠀⠸⠠⠈⢳⡀⢠⠀⠀⠀⠀⠀⣾⠁⠀⠀⠸⡀⡿⡼⢠⠛⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠘⠷⡀⠀⠀⠀⠀⢸
⠀⠀⠀⢣⠀⢹⠎⢱⠀⠀⢀⣴⠃⠀⠀⠀⣀⠉⠁⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈
⠀⠀⠀⠈⠣⡀⠑⠾⣄⠀⠀⢸⡇⢀⡴⠞⢣⡇⠀⡇⠀⢦⣘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⢀⠀⠈⢣⡘⠈⣇⡇⢀⠞⠛⠁⠀⢣⠀⠀⠙⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠙⠲⡄⠉⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠹⡄
⠀⠀⠀⠈⠉⠙⠓⠬⡉⠙⢢⡀⠀⠀⠀⠈⣳⣄⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⢀⠇⠀⠀⠀⢰⡥⠬⢷⡀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢀⠜⠁⢠⠊⠀⠀⠀⠀⠀⠀⠀⣠⢷⠀⠀⠀⠀⠉⠓⠲⠶⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢰⠃⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠾⠥⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠈⠲⠤⠤⣄⣉⡉⠉⠉⠉⠉⠓⢦⡀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⡓⢄⠀⠀⢀⠀⠻⡤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⡞⢹⡀⠀⡞⢹⡀⡇⡰⠛⡆⠀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠘⡆
⠀⠀⠀⠀⠀⢠⠾⠖⠚⠓⠚⠛⠛⠛⠛⠛⠒⠛⠒⠻⠃⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠶⠋⠀⠀⣷⠴⠚⠉
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡗⠚⠉
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⣠⡴⠋⠀⠀⠀⢳⣀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠈⠉⠻⢯⡉⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⣀⣀⣤⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣴⠚⠁⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⠀⠙⢦⠀⠀⠀⠀⠀⢈⡷⣄⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⣸⠀⠀⠀⠀⢠⣯⡤⠬⢧⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣈⣧⠀⠀⠀⠀⠈⠙⠲⠶⠤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠈⠓⠦⠤⠤⠤⠤⠤⠤⣄⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠲⠤⣀⠀⠀⠀⠀⠀⠙⣦⠚⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⣠⣌⠳⡄⠀⢠⣄⠀⠙⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⣇⠀⣰⠃⢸⡀⣹⣰⠃⢹⡀⠀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⣴⠞⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠁⠀⢰⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⢻⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣤⣤⣤⠴⠞⠁⠀⠀⠀⠈⠷⣄⣀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣽⣛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣋⣉⣉⣽⣿⣿⣿⣿⣿⡿⣿⣿⡛⠛⠛⠛⠛⠛⠛⠛⠾⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⡤⣬⣿⣿⣿⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧⡴⠏⠀⠀⠀⠙⠻⠿⠿⠿⠿⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢰⢀⣀⠀⢀⠀⣀⣀⡀⡀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠒⠒⠒⡄⢹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⠘⠒⠐⠒⠒⠒⠒⠂⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⠀⠀⠀⢸⢠⣀⣀⣀⣀⣀⣀⣀⣀⣈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠈⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⢸⣀⣀⣀⣀⣀⣀⣀⢀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡸⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠠⠠⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⣀⣀⣒⣒⣒⡒⣂⣀⣀⢀⠀⠀⠀⠀⠀⠒⠒⢒⡲⢁⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢨⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⠀⠸⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠦⣤⣤⣤⣤⣤⣤⣤⣴⠴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁⠉⠉⠀⠀⠀⠈⠀⠉⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⡉⠉⡉⠉⠉⠉⠁
⠀⠀⠀⡔⢀⠀⠠⢀⠀⢀⡀⠀⣀⠀⢀⡀⠀⡀⠂⢀⠀⣀⡀⠀⣀⣀⢀⡀⡀⡀⠀⢀⠀⢀⡀⠀⢀⠀⠀⡀⠠⠀⠀⣠⠈⢠⡤⣄⠀⠑⣄
⠀⠀⡼⠀⠦⠤⠤⠬⠤⠦⠡⠴⠌⠠⠄⠥⠴⠭⠥⠤⠥⠤⠬⠤⠬⠥⠬⠭⠴⠤⠤⠤⠅⠰⠨⠄⠀⠠⠀⠸⠄⠣⠀⠈⠓⠓⠲⠬⠦⠤⠊
⠀⠀⠳⠖⠲⠒⠒⠒⠶⠲⠖⠒⠒⠖⠒⠒⠒⠒⠒⠒⠒⠒⠖⠒⠒⠒⠒⠖⠲⠒⠒⠒⠲⠒⠒⠶⠒⠒⠒⠖⠶⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠷⣶⡄⠈⣶⡞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠈⠀⢀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣶⣶⣶⣦⣤⣀⣀⠀⠀⠀⠈⠙⠻⣿⣿⣿⡷⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⠟⢿⣦⡀⣀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠙⠻⣿⣿⣿⣿⣿⣿⣧⢹⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⡀⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠘⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡘⣷⣾⣿⣿⣿⣛⠋⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣿⣿⢹⡿⠋⢀⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠿⡎⢰⣷⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡍⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡘⡿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡃⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠒⠚⠙⠉⠙⠙⠒⠲⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⣤⠶⠾⠁⠀⠀⠀⢤⡠⠤⠤⡠⠄⠀⠀⠀⠹⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⡄
⠀⠀⠀⠀⢠⣶⠟⠇⠀⠐⠊⠑⠒⠢⠀⠀⠀⠀⠀⠔⠒⠒⠉⠒⠀⠘⠟⢷⣄⠀⠀⠀⠀⠀⢰⢻⢸⣷
⠀⠀⢀⡤⣾⠷⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠸⣿⣦⡀⠀⠀⠀⢸⣾⢸⣿⡀
⠀⠀⠸⣿⣿⣄⣀⣀⣸⡄⠀⠀⠀⢸⡇⠀⠀⠀⢀⡇⠀⠀⠀⢀⣾⣀⣀⣠⣾⣷⡟⠀⠀⠀⢸⢿⢸⣿⡇
⠀⠀⠀⠈⢿⣿⢿⣿⣿⣿⣷⣶⣶⣿⣿⣶⣶⣶⣿⣿⣷⣶⣶⣿⣿⣿⣿⢿⣿⠋⠀⢀⣠⣶⣿⢸⣿⠀⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠉⠙⠙⠻⣯⣿⠿⠿⣿⣿⣿⡿⣿⣿⣿⡿⠿⢿⣯⡿⠋⠛⠉⠁⠀⠀⠸⠟⠛⢻⠈⣿⠀⡟⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⣤⣤⠖⠛⠛⠛⠛⠛⠉⠉⠉⠹⠛⠛⠛⠛⠳⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠤⠤⠶⣷⣤⣴⣤⣤⣤⣴⡷⠦⠤⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠘⣿⡇⠁⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡇⢿⠀⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⢸⡇⢀⣿⡇⢸⣇⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⡏⡟⠏⠉⠉
⠀⠀⠀⠀⠀⢀⡤⠔⠉⡩⡤⠖⠂⢸⡇⢸⣿⣧⢸⣿⠀⠒⢤⢌⡉⠳⠤⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⣀⣤⣤⣿⣇⡾⣿⠿⣌⣿⣤⣤⣄⡀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⣆⢸⠀⢸⢸
⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⡟⠉⠀⠈⠛⠀⠀⠀⠘⠁⠀⠉⢻⣿⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⢿⠘⣦⠏⣸
⠀⠀⠀⠀⠀⠘⢿⣿⣿⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠘⣧⠈⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢶⡤⢷⣤⣴⢦⡴⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢀⣀⣀⢀⣀⣀
⠀⠠⠀⠋⠋⠛⠚⠛⠛⠓⠠⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠂
⠀⠒⠉⠓⠀⠉⠀⠉⠒⠘⠀⠀⠀⠈⠀⠀⠀⠀⢀⢀⣙⠇⠀⢀⠢⡤⠀⠀⠀⠀⠀⠀⡀⠀⠠⠐⢀⡀⣀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠄⠡⠀⠀⠈⠉⠀⠁⠁⣤⠈
⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠠⠀⠀⠰⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢧⡄⠀⠄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡈⠠⠌⠀⠠⡀⠀⠀⠀⠀⠦
⠀⠀⠀⣤⣀⣤⡀⣀⣀⠒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⢀⡹⠟⠚⠛⠛⡦⠆⠀⠀⠠⠉⠀⠀⠀⠀⣴⣼⣳⣿⡶⣤⡄⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⢰⠂⠠⠤⠂
⠀⠀⠀⠀⠘⠀⠋⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⢀⡞⠀⠈⠀⠀⠀⣀⠁⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢁⠻⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣄⡀⠀⠀⠀⢀⣀⣠⡤⠶⠋⠀⠀⠀⠆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠢⢄⠀⠀⠀⠠⣀⢀⡀⡀⠀⢂⠤⣀⣠⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠻⠻⠟⠶⠓
⠀⠀⠀⠑⠒⠂⠀⠀⠀⠈⠀⠀⠈⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋


⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⢤⣤⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⡟⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡙⢻⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⠿⢿⣶⣶⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣠⠤⠤⣤⣀⣀⡉⠉⠙⠛⠿⠿⣷⣶⣶⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠀⡟⠀⠀⠀⠀⠉⠉⠙⠓⠒⠶⠤⣤⣀⣉⠉⠛⠚⠿⠿⢷⣶⣦⣤⣄⣀
⠀⠀⠀⣀⣠⠤⠖⠛⠉⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⣤⣀⣈⡉⠉⠛⢿⢷
⣴⣾⣋⣉⠀⠀⢀⣠⠴⠒⢻⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡇⠀⡟⣸
⡇⠀⣀⡉⠉⢻⠉⠀⠀⠀⡿⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⢃⡇
⣇⠀⡏⢹⠀⢸⠀⠀⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡾⢸⠁
⣿⠀⣿⢸⠀⢸⠀⠀⠀⢸⠃⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡇⡟
⢿⠀⣿⢸⠀⢸⡆⠀⠀⣾⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣼⢸⠇
⢸⠀⢸⢸⡄⢸⡇⠀⢀⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡇⡾
⢸⡄⠘⠾⠇⠸⡇⠀⢸⠃⠈⠙⠓⠲⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣼⣸⠇
⠘⡇⠰⣿⡇⠀⡇⠀⠈⠙⣒⡶⢤⣀⡀⠀⠈⠉⠙⠒⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰⣏⡿
⠀⠙⠲⠦⢤⣤⣧⠴⠞⠋⠁⠀⠀⠀⢉⣽⠶⠦⣤⣀⡀⠀⠈⠉⠛⠲⠦⣤⣀⣀⠀⠀⢀⡏⠀⣼⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠯⣭⣙⡓⣶⣤⣄⣀⡀⠀⠉⠉⠛⠚⠁⢠⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⢹⡟⠓⠶⠤⣤⣀⣼⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣍⠳⠦⣄⣀⡀⠀⠀⠀⠀⢀⣠⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠬⣭⣭⣛⣛⣛⣭⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠿⠛⠛⠛⠛⣻⣿⣛⡻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⣀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣶⣬⡉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⢰⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⣷⡄⢀⣤⣄⡹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣰⣿⠋⠉⢹⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠀⠀⠀⢰⣿⠁⠀⠀⢸⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠏⠁⠀⠀⠀⠀⣿⡇⠀⠀⢠⠇⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠈⠙⠛⠃⠀⠀⠀⠀⠀⠀⢻⣧⣀⡴⠋⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⣤⣴⣾⣏⠉⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⣾⣿⣿⣿⣿⣿⡿⠇⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠬⠐⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠈⠀⠀⠹⡀⠹⠋⣳⡍⣿⡴⢿⣟⠉⡝⠀⠜⡩⠊⠰⢁⠞⠙⠁⡠⠞⠁
⢤⣄⠀⠀⠀⠀⠀⠁⠈⢢⣴⠦⠤⣤⣄⡀⠀⠀⠀⠀⢠⢖⠛⣧⣄⢸⢱⠟⠋⠀⠀⣻⡜⣁⣤⣀⣁⣀⣀⢄⣀⣤⡈⠀⠀⠀⠀⣠⣔
⣒⠠⠀⠒⠄⠀⠀⠀⠀⣟⡀⠠⢀⠤⣼⠋⠗⠓⢀⢴⢟⡀⠀⠈⠀⠚⠓⡶⠀⠀⠈⠋⠉⠉⠀⠁⠠⠏⠀⢀⠃⠀⢻⡆⡠⠐⠉
⡒⠀⠠⠤⣀⠀⠀⠀⠀⣽⠂⠀⡈⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠈⠀⣀⡾⠃⠀⠀⠀⠀⢠
⠉⠛⠂⠀⠀⠀⠀⠀⠀⠈⢲⡆⠁⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠈⠠⢿⠉⠉⡧⠀⠀⠀⢘⡆⠀⠀⠀⠀⠑⠉
⠤⠄⠀⠀⠀⠀⠀⢀⡠⠤⣴⢳⡤⠀⠀⠀⠀⠀⠁⠰⠐⠀⣀⠀⠀⢢⠈⡄⠊⠁⠀⠁⠀⠀⠀⠀⠘⠃⠀⡴⠚⣫⠄
⠴⠒⠦⠤⠀⢀⡜⠉⠛⠁⠘⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠛⠉⠉⠷⣶⢦⡄⢀
⡒⠖⠂⠀⠀⣸⡏⠁⣀⠔⠚⠀⠈⠒⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣠⠟⣙
⢛⢉⡴⠎⠉⠃⠀⠸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⡷⠀⠀⠀⠀⠀⠀⠀⡀⠶
⠈⣿⣀⡀⠀⢀⠀⢀⡀⠀⢀⠀⠒⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢤⣤
⡒⡚⠻⠷⢶⣟⠀⠈⠳⢴⣎⡠⠀⠀⠀⠀⠀⠈⠀⢠⠀⠄⠆⠀⠀⠀⢀⠀⠠⠄⠈⠈⠀⠀⠀⠀⠀⠁⠀⠄⢰⡄⠀⠀⣀⣀
⢀⣀⡀⠀⠀⠘⠳⣤⣄⣀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠂⢀⠀⠀⠈⠀⠀⠀⠈⠀⠐⠄⠀⠀⠀⠀⠀⢴⢼⠁⢤⠚⠋⠉⠂⢉⣉
⠿⠁⠀⠀⠀⠀⢀⠀⠍⠁⣻⠀⠀⠀⠀⠈⠙⠉⡉⠀⡀⠀⢀⣀⡁⠀⢆⠀⠀⠈⠁⠀⠈⠉⠀⡠⠶⣤⡤⠞⠋⢀⡖⠀⠈⠐⠒⠄⠠
⠈⠀⠀⠀⡠⠂⠁⠀⢀⣴⠿⣄⣄⣡⠴⡄⠀⣠⡷⢚⠂⠀⠀⣹⡍⠂⢢⣄⣀⣤⠟⠦⠴⢧⣀⣀⣶⠏⠓⠲⠞⠋⠁⠀⠈⠀⢄⠐⠠
⣖⠔⠊⠀⠀⡠⠔⠋⠙⠁⠀⠀⠀⠀⠀⠉⠋⣱⠃⠸⣏⣠⣴⠋⢷⡄⠀⠀⢡⠇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠁⠀⢀⡴⢋⡤⠊⢁⣀⣠⢂⡖⢠⢈⡠⠀⢰⢃⣴⣤⡌⢷⡱⢳⣧⣙⡃⢟⢉⠀⠀⠀⠁⠀⢰⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠒⠒⠒⠒⠀⢄
⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⢡
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠖⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠑⠢⠀⢀⣀⡀⠀⠠⠂⠄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠏⠉⠉⠉⠩⠽⠿⠿⠿⠿⣷⠀⠀⠂⠀⡠⠀⠀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠦⠂⠀⠀⢀⢂⠄⢎⣡⠀⠀⠀⠀⠀⢰⡠⠤⠔⠀⠀⠀⣿⠀⠤⠄⠊
⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠉⠑⠢⡀⠀⠄⠀⠀⠰⠾⡀⢸⠣⠃⠀⠀⠀⠀⠨⡁⠀⠀⠀⠀⠀⣿⠀⠀⡠⠀⠂⠈⠀⠀⠈⠐⠢⡀
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠉⠀⠀⠀⠈⠄⠀⠀⠀⠀⠢⣙⡓⠂⠄⠀⠀⠀⠀⠠⡳⣴⠣⠄⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠂⠀⠀⡁⠀⠀⠉⠉⡡⠀⢄⠀⠀⠁⠈⠉⠉⠁⠉⠉⠍⡉⠉⠁⢸⠀⠂
⠀⠀⠀⠀⠀⠑⠢⠄⠀⠀⠀⢀⡀⢎⠀⠀⠀⠀⢐⠒⠐⠄⠀⠀⠀⠀⠀⠀⢈⠊⠀⠓⣤⠀⠀⢸⠀⠈⠂⠀⠀⢀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠆⠈⠀⠀⠀⣠⣄⣤⡀⠀⠀⢰⠀⠀⠈⡳⢄⠴⡓⠀⠀⢸⠀⣠⠊⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⣀⣀⣀⣒⣠⣬⣦⣤⣴⣤⣤⣤⣬⣦⣶⣤⣥⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠤⠠⠄⠀⠄⢠⠈⡗⡺⣾⣽⣧⣗⣛⣗⠒⠚⠛⠛⢭⡷⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⡤⠬⠤⡅⢭⠸⡅⢯⠿⣿⢹⡿⣤⠀⡤⠕⣀⠱⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠤⠤⠤⠤⠤⠍⠉⠁⠨⢭⣁⣉⣉⣩⠭⠽⠬⢤⢬⠭⣬⣥⣭⣿⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠈⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣤⣄⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣁⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣁⣠⣠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣼⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠲⡋⠖⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⢀⡀⠀⢩⠀⢀⡀⠀⠀⠧⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⡀⠀⠉⠁⢀⠶⡀⠉⠀⠀⠠⠂⠀⠈⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠐⠀⠒⠁⠀⠀⠐⠐⠈⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⡒⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠲⢬⠀⠀⡋⠀⠀⡏⠀⠀⢹⠀⠀⡇⠊⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠤⠷⠤⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡭⠩⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡎⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢏⠀⣠⡇⢸⠀⠀⠀⠀⠀⢸⠁⠀⠄⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⡆⠈⠀⠀⠀⠀⠀⢸⠀⣇⠀⢈⠗⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠺⣄⣠⠞⠉⠀⠙⠺⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⡼⠒⠁⠈⠛⢄⠀⡼⢆
⠀⠀⠀⠀⠀⢠⡁⠀⡜⠁⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠹⡄⠀⣠
⠀⠀⠀⠀⢀⠆⠈⡛⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠸⡋⠀⠆
⠀⠀⢠⠊⠉⢉⠒⢧⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡶⠶⠶⠦⠤⠤⠤⠤⠴⠶⠶⠶⠤⠤⠤⠤⠤⠤⠴⠦⢴⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠂⠉⠙⢦
⠀⠠⠚⠈⠁⡈⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠉⠀⠉⠙
⠀⠀⠀⠀⠀⠉⠀⠄⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢹⠀⠀⠀⠀⠀⠀⢸⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠈⠁⠁
⡆⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠒⢺⠀⠀⠀⠀⠀⠀⢸⠐⠐⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠤⠴⠀⠀⠀⠀⠀⠀⠸⠠⠤⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⠴⡇⠀⠀⠀⠀⠀⡶⠤⠄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠈⠀⠀⠉⠓⡄⠀⠀⠀⡴⠋⠁⠈⠁⠈⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠈⠀⠀⠩⠆⠀⠸⠁⠁⠈⠉⠉⠉⠁⠈⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣴⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⡿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣜⢷⣿⣿⢗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡌⣥⢱⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿⠛⠋⠸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠐⠂⠀⠀⣿⣿⣿⢹⡇⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⢳⡽⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⣿⣗⢺⣯⡱⣪⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠾⢷⣥⣾⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⣦⡀⣿⣿⣧⣴⠟⡜⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠈⠿⣿⠟⠟⠋⡅⠀⢿⣿⣇⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠙⠀⠈⠀⠀⠀⠸⣿⣿⢹⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠎⠙⠛⠉⠁⠠⠤⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠒⠒⠒⠺⠇
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣀
⠀⠀⠀⠀⡀⢤⠴⢧⠤⠤⢤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠀⠐⠒⠚⠓⠀⠉⠀⠉⠐⠓⠦⣄
⠀⣠⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣦⡘⠛⠿⠿⠿⠟⠛⣡⣤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞
⠀⠉⠘⠓⣢⣤⣤⡄⠀⠀⠀⣀⠀⠀⠀⣠⣔⣢⠝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣡⡒⢄⡀⠠⣤⠤⠤⠐⠾⡇⠈⢹⡄
⠀⠀⠀⠀⣿⣤⠋⠀⠀⠈⠉⠉⠉⢉⡾⣯⠚⠁⠀⠒⠒⠂⢰⠀⠀⢀⠀⠀⠀⠀⠀⠉⠉⢎⡹⠀⠀⠀⠀⠀⠀⠈⠑⣾⡏⡄
⠀⠀⠀⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⣎⠻⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⢡⠛⣄⠀⠀⠀⠀⠀⠀⠈⢠⢇⡀
⠀⠀⠚⠀⡄⠀⠀⠀⠀⠀⠀⠀⣷⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⡀⠀⠀⠀⠀⠀⠀⢺⠀⢣
⠀⠀⢷⢀⡇⠀⠀⠀⠀⠀⠀⣴⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡇⠀⠀⠀⠀⠀⠀⢨⣠⠆
⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠹⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡈⠃⠀⠀⠀⠀⠀⢀⡙
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢿⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠇⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢋⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⠋⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠛⢿⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠃⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠙⠻⣶⣦⣀⣀⠀⠀⠀⠀⣀⣀⣠⣶⠿⠋⠁⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠈⠉⠉⢻⡟⠛⠛⢿⠉⠉⠀⠀⠀⠀⢀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠂⢤⡤⠄⠀⣸⠀⠀⠀⣿⠃⠠⣤⡤⠐⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⡀⠀⠀⠀⠀⢹⠁⠀⠀⢻⠁⠀⠀⠀⠀⣼⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⣟⠁⠀⠈⠉⠙⠋⢻⡇⠀⠀⢹⣋⡉⠉⠉⠁⠈⢻⡄⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⡿⠛⢦⡀⠀⠀⠀⠘⣃⠀⠀⢻⠉⢃⠀⠀⣠⠔⠿⠁⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⣀⠁⠀⠀⠙⢢⡄⢀⡼⠛⠷⣤⠆⠀⠘⣷⢾⣁⠀⢐⡇⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢀⣽⣀⣀⡀⠴⠧⣤⡼⠳⠦⠀⠀⠀⠀⢰⠆⠀⠀⠈⠛⠛⠦⠄⣀⡀⠀⠀⠷⡀
⠀⠀⠀⠀⠀⢀⡮⠁⠀⠀⠀⠘⠁⠈⠉⠈⠁⠀⠀⠀⢠⣶⣤⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠻⡄
⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣀⣀⣀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠁⠈⠙⠿⢲⡤⠤⣤⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠐⠒⢶⡶⠶⠶⠶⠶⣶⠶⢶⣶⠶⠷⠶⢶⡶⠒⠒⢉⡴⢦⣤⣄⣈⣽⠋⢻⡷⠒⠒⠒⠒⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢧⡸⠦⠤⠤⣾⠀⣄⣀⣀⣘⠛⠁⠀⢛⣋⣉⣥⠀⣇⢀⠀⢼⣆⢨⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⣄⢀⣶⣄⠈⠛⠁⠀⠁⠙⣷⢤⣤⣾⠉⠀⠉⠚⠃⠀⠀⠀⠈⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣼⣿⠖⠺⣿⠉⢹⡿⠳⣄⠀⠀⠀⣾⠇⣸⣇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⢙⣁⣀⠤⠴⡿⠄⢀⣭⠀⠀⢠⡏⢠⠛⠻⣧⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣗⠚⣉⡁⣤⠤⣶⠦⠤⣤⣏⠀⠀⡶⠀⣻⠀⠄⠈⣦⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣉⣉⣀⡀⢰⡏⠀⠀⣸⣧⡄⠸⣄⢰⡋⢹⡷⠀⠀⣦⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡘⣿⠛⠋⠁⠀⠛⠃⠉⠛⠋⠀⠙⣂⠀⠙⠃⠙⡇⠀⠀⢀⠛⠀⠉⠛⠋⠀⠀⠀⠀⠀⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡇⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣦
⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄
⠀⠀⠀⠀⠀⣰⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⢠⡇⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠙⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠘⡇⠀⠀⠀⠙⣤⡀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠈⢁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠧⠤⠤⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠤⢚⣿⣿⣿⡿⠟⣛⣛⣯⣭⠉⣿⡟⢩⣽⣿⣛⡛⠛⢿⡿⠿⣿⡓⠤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⣠⣿⣷⠈⠉⠉⠉⠉⠁⠀⠀⠀⠉⠃⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣡⠤⢀⣀⣀⢀⣀⠠⣦⣉⠷⠋⠱⠞⣡⡤⠄⣀⣀⣀⠀⢠⢤⣜⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠢⣄⡈⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣀⠤⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠙⠳⡂⢌⡛⢶⠤⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠉⠳⡃⠀⡤⠚⠉⠉⠲⣄⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⡀⠈⠲⢌⠉⠓⠊⠙⠳⡴⠚⠓⠶⠉⠈⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⡇⠀⣀⠤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠲⣦⡽⠄⠀⠀⠀⠢⣄⠀⠀⠀⠀⠈⣇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠗⢋⣵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⢿⠄⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡙⠦⡼⠘⢤⠃⡇⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢳⡆⠀⠀⠀⢀⣠⣤⠿⠀⠀⠀⠀⠀⠉⢹⣃⣀⣤⣤⠶⠚⠉⠀⠀⠀⠀⠰⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⢈⣉⡷⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣁⣀⠀⠀⠀⠀⠀⠀⣠⠤⠬⠅⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠆⠀⠀⠐⠒⠛⠻⠶⡄⠀⠀⠀⠀⠀⠀⠀⡤⢝⣷⣌⡉⠁⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠥⢄⠀⠀⠀⠀⠀⠀⠱⡄⠀⠈⠉⠓⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⠔⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⣴⣴⡀⣶⣾⡁⠚⠆⠀⠀⠀⠀⠀⠀⠀⢺⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠏⠘⠇⢷⡏⠃⠙⠀⠀⠀⠀⡔⠢⠤⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⣀⣴⡯⠴⠒⣞⠉⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠙⠒⢄
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠙⠢⢤⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⡤⠖⠁⠀⠀⠰⣕⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣖⠚⠁⠀⠀⠀⠀⠀⠀⠁⠠⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⣶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⡏⠹⡄⠀⠀⠀⢀⡄⠀⠀⠀⡇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡘⠀⠀⠉⠓⠒⠚⠁⠘⠦⠤⠞⠁⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠰⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣴⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣽⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣷⠀⠈⠉⢉⠉⠀⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣁⣀⣈⠉⠉⠉⠉⠙⠛⠛⠛⡉⢉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠆⠛⠂⠄⠠⠄⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⢀⣤⡀⠀⠀⠠⡀⢰⣤⣐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⠀⠀⢀⣶⣿⡟⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢠⣾⣿⡿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠏
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⠀⠀⠀⣾⣿⣿⣿⠃⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠀⠀⡠⠤⠤⠐⠒⠒⠠⢤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⣰⣶⡤⠒⣿⣲⡀⠂⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡲⠁⡘⠀⠀⠀⠁⠀⠀⠀⠁⢢⠋⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⡇⢸⡉⠉⠉⠃⠒⡶⢰⢸⠀⠀⢠⡇
⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣑⡢⠤⠤⠔⠃⢸⠈⠓⢺⠋
⠀⢎⠳⡱⡘⣀⡀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠍⠉⠉⠙⠛⢿⡛⠚⢻⡉
⢎⠢⣳⡵⠋⠉⣗⣲⢦⢲⠒⠒⢲⠃⠀⠀⠅⠀⠀⠀⠀⠀⠱⡀⡤⠷⠤⠤⣄⡀
⠀⠱⡏⠀⠀⠀⡏⠀⢸⢸⠤⢤⡏⠀⠀⠀⠀⠀⠀⠤⢤⠀⠀⡇⠧⠤⣤⢤⡇
⠀⠀⠈⠒⠤⠤⠣⡀⠈⢸⠀⢰⡇⠀⠀⠀⠀⠀⢄⡘⣸⠀⠀⡇⠀⠀⣷⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠑⢕⣤⡈⠀⢸⠀⠀⠀⠀⠀⠠⠀⠠⠠⡀⠀⡇⠀⠀⢸⢀⡇⡇⠀⠀⠀⢠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠇⠀⢸⠀⠀⠀⠐⠀⠉⠁⠈⠒⠁⠀⡇⠀⠀⣿⢸⣇⡇⠀⠀⢀⡨⠖⠛⢻⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡏⠘⠶⠃⠀⠀⠻⣄⡀⠀⢸⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⢄⣀⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⢇⠀⠀⠀⠈⠉⠉⠙⠛⠖⡚⠒⠞⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠉⠙⢦⡀⠀⠀⠀⠀⠀⡰⠐⣄⣀⣀⣰⠡
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢧⠀⠀⠀⠉⠒⠲⠦⠤⠖⠓⠒⠉⠁⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠤⢆⡰⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢶⠃
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⢤⠔⠂⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢘⣀
⠀⠀⠀⠀⠀⣔⡉⠒⠤⣀⢰⣀⠤⠂⢉⡶⠀⠀⠀⠀⠀⢰⠚⠭⣄⣐⠖⢀⡩⠴⠆
⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣙⣅⡤⠖⠁⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠈⠉⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄⣉⠑⠂⢾⠐⠈⣁⡤⠗
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣳⣾⣖⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣁⣴⣿⠟⠋⣠⣴⣿⠟⠉⣠⣴⣿⠟⠉⣠⣾⡿⠟⠉⣠⣾⡿⠟⠁⣠⣾⡿⠟⢁⣠⣾⡿⠛⢁⣠⣾⡿⠛⢁⣠⣾⡿⠛⢁⣤⣾⡿⠋
⠿⠋⠁⠴⠾⠿⠋⠀⠴⠾⠿⠋⠀⠴⠿⠿⠋⠀⠴⠿⠿⠋⠀⠴⠿⠿⠋⠀⠴⠿⠿⠉⠀⠴⠿⠟⠉⠠⠶⠿⠟⠉⠠⠶⠿⠟⠁⠠⠶
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠠⡠⢀⠀⠀⠀⠀⢶⡶⡶⢶⠶⣶⠶⡶⢶⢶⡶
⠀⠀⠀⠀⢰⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠢⠠⠀⠀⠀⢸⡇⡇⢸⠀⣿⠀⡇⢸⢸⡇
⠀⠀⠀⠀⠘⢿⣿⣿⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠘⠠⠑⠅⠃⠀⠀⢸⡧⡷⢼⠤⣿⣿⢻⣿⢼⡧
⠀⠀⠀⠀⠀⠤⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠈⠀⠀⠄⠀⠀⢸⡇⡇⢸⠀⣿⠛⡟⢻⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠀⠀⠼⠷⠷⠾⠶⠿⠶⠷⠾⠾⠧
⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⡀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠡⠍⠁⠀⠄⠠⠀⠄⠀⠀⠀⠀⣿⣧⠀⠀⠀⠉⣉⠙⢻⣶⠄⠀⠀⣿⡿⢿⣿
⠀⠀⠀⠀⠀⠉⠀⠝⢿⣿⣿⣧⠀⠀⠀⠈⠃⠐⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢺⣿⣷⣾⣿⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⠿⠿⠿⠿⣿⣶⣤⣤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⢶⡦⠀⠈⠉⠉⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠠⡒⠤⢂⠀⣀⠀⠀⠀⢀⠀⠠⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠬⠃⠸⢂⠀⠁⣀⠃⡠⠈⠰⠂
⠀⠀⠐⠲⠷⣶⣶⣶⣶⣦⣶⣶⣶⣦⣀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⡀⠀⠜⠁⠐⠀⠘⠁⠈⠁⠡⠄⠨⢂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠
⠀⠀⠀⠀⡀⠆⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⣸⡀⠀⠀⠀⠀⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠃⢰⣦
⠀⢀⡼⠛⠳⣄⠀⠀⢠⠟⠉⠙⢦⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⣿⣿⣿⣿⡷⠀⣠⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢨⣥⠀⢸⣿⡇
⠀⢸⡀⠀⠀⣸⠀⠀⠸⣄⠀⢀⡼⠁⠐⠺⠿⠿⠿⠂⠀⠈⠙⠛⠛⠋⢉⣤⣤⣄⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠛⠀⢠⣭⠁
⠀⠀⠙⠒⠚⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠘⠋
⣁⣴⣿⠿⠋⣀⣴⣿⠟⠋⣀⣴⣿⠟⠋⣠⣴⣿⠟⠁⣠⣴⣿⠟⠁⣠⣼⣿⠟⠁⣠⣾⡿⠟⠁⣠⣾⡿⠟⢁⣠⣾⡿⠛⢁⣠⣾⡿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠰⣄⣀⠀⠀⢀⣠⣶⠾⠼⢶⣤⣤⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢄⣽⣷⣶⣾⣿⣟⣥⣄⠀⠀⠈⠛⠻⢷⣄⡀⢤⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣷⣽⠏⠁⡦⠊⠀⠀⠀⠀⠀⠀⠈⢲⣾⣿⣏⣴⡙⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⡄⠀⠀⠀⡠⠔⠚⠓⠒⠄⠀⠈⠙⠍⠙⢿⣧⣤⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠐⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⣿⠁⠉⠲⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠻⡀⢠⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡦⠀⠀⠀⠀⠉⠒⠒⠀⠀⠀⠀⠀⠀⠀⢀⣦⣼⣧⡑⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣭⣾⣷⣄⢀⡀⢤⣠⢤⡀⠀⠀⠀⡠⠀⠀⠐⢺⣿⠁⠈⠉⠓⠤
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠈⢦⣙⠿⠿⣿⡛⣆⣈⣉⣙⣳⣶⣍⣂⣀⣠⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠁⢈⠛⠻⠿⠛⡋⠉⠺⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⠛⠛⠉⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⡴⠀⠀⠀⣎⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠀⠀⠀⠀⢸⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣒⣻⣀⣀⣀⣀⣸⣛⣛⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠉⠉⠉⠉⠉⠉⠏⠀⠀⢀⣠⠔⠋⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠓⠒⡒⠒⠒⠒⠒⠶⠶⠶⠶⢶⠚⠉⠀⣀⣀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⠉⡇⠶⣿⣷⣲⠀⠀⢰⣶⣾⢷⡆⢸⠀⠀⡼⠀⡼⢙⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣈⡇⠀⠛⠛⠛⠀⠀⠸⠻⠿⠿⠀⢸⠀⠀⢹⡀⣳⣘⠤⠃
⠀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠈⣇⣀⣀⡀⠀⠐⣶⠂⠀⢸⡀⠀⠀⢀⣠⠞
⢠⡉⢙⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢒⡶⠶⠦⢭⣭⣭⣍⣉⣀⣀⣸⣇⡤⠞⠉⠳⣄
⠣⣈⣩⢿⣉⡷⣶⡀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠠⠀⠀⠀⠀⠀⠈⠉⠉⠁⠘⡆⠀⠀⠈⢧
⠀⠀⠀⠀⠈⣦⣇⠙⠲⠤⢤⣤⠤⢴⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣠⠚⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠈⠒⠤⢤⣤⣤⠤⡾⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠹⠶⠶⢶⣤⡄⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⣀⡂⢀⢐⠸⡇⠀⢀⡠⢼⡇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢈⣉⣉⡉⠉⠛⠒⠒⠒⠒⠲⠦⠬⠤⠴⡷⢊⡡⠖⠉⣧⡼⠧⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠒⠒⠒⠒⡏⠁⠀⠀⠀⣇⡷⢦⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⣤⢫⠟⢿⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠶⠶⡶⠤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀⢸⣇⣠⠖⠋⢷⣻⠀⣸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡇⠀⠀⠀⠀⠁⡟⠉⢩⡟⠉⠉⠀⠀⠀⠈⠁⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠶⢾⣷⡀⠀⠀⠀⠀⣯⡀⢬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣚⣉⣀⢀⡴⠋⢀⡇⢀⣠⠔⠋⠀⠉⢻⠟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠬⠭⠽⠶⠚⠉⢰⣛⡒⠦⠤⢤⣤⠏⣠⠏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣷⣾⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡄⠠⣀⡀⢠⠀⠄⡀⠀⠀⠀⠀⠀⠠⢰⡄⠀⠀⠀⠔⠀⠀⠀⠀⠒⠄⠂⠀⠄⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠
⠀⠀⠀⣿⠀⠀⠁⢸⠀⠀⠀⠁⠀⠈⠂⠀⠸⣽⡇⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠈⠑⠠⣀⢄⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠨⢸⡀⠈
⠀⠀⠀⣿⣦⢄⢪⢸⡢⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠄⡀⠢⠆⠸⣸⣄⠀⠀⠀⠀⠀⠁⠊⠐⠤⡪⢔⠁⠢⠀⠄⡀⡀⠀⠀⠀⡇
⠀⠀⠀⣿⠀⠈⠨⠍⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠳⠄⠀⠒⠄⠈⠁⢄⠀⠉⠁⠀⠀⠀⠀⠀⠛⠷⠀⢀⠀⠁⣼⢿⣆⡄⣀⣉⠥⠗⠚⣉⠁⠀⠐
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠑⠋⠶⢯⣶⣟⠬⠗⠚⢉⠠⢀⡜⣧⠹⡆⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠁⢀⠃⠀⠰⡀⠀⠀⠀⠈⠐⠤⠔⠒⠉⠀⣀⡠⠀⠀⠀⢠⠃⠀⠀⠠⠆⠀⠀⠠⠀⠀⠐⠵⠀⣀⡤⠔⠝⣿
⠀⠀⠀⣸⠀⠀⣀⣀⡀⠀⠀⠈⢆⠀⠀⠑⠢⠤⠄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠒⠀⠈⠠⠄⠀⠀⢈⠀⠠⠃⠈⠤⢶⠎⠉⠀⠀⢁⠈⡇
⠀⠀⠀⢸⠀⣠⡿⠽⠷⠄⠀⠀⠈⠓⠠⠄⣀⣠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠁⠀⠙⢤⠀⠀⠀⠀⠀⠀⠠⠀⠈⡇
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢢⡤⠄⠀⠀⠀⣀⢠⠄⠐⡁⠰⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠲⡄⠀⠀⠑⡄⢀⣠⠄
⠀⠀⠀⢾⣗⣂⡀⠀⠀⠀⠀⠄⠀⣄⡐⠯⡀⠀⠭⠂⠒⠀⠀⠁⠀⠘⠓⠤⣄⡀⣀⣀⣠⠚⠀⠀⠀⠘⢦⡀⠀⠘⠁⢀⠀⠀⠀⠀⠨⡇
⠀⠀⠀⠉⠉⠀⠀⠀⢀⢈⠄⠠⠘⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠀⣀⠈⠉⠉⠉⠀⠽⠛⠀⠀⠈⠿⣄⣠⣾⠋⠀⠤⣤⡀⠈⣿⣗⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣈⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣿⠿⠍⠴⠴⣴⣦⠀⠀⠀⠀⠀⠀⢀⣈⡉⠁⠀⠀⠀⠉⢩⣴⣿⡁⣿⠆
⠀⠀⣀⢠⡴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠤⢦⣤⣄⠤⠆⠀⢠⣴⣤⣤⣤⣤⣀⠀⠀⠀⠐⢿⣿⣿⢻⣿
⠀⠉⠁⢸⡇⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣦⣆⡇⣿⢁⡀⠀⠀⠀⢈⠀⠠⠀⠈⣿⠀⠀⠀⠀⠈⠁⠉⠈⠈⠉⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⢸⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠑⠁⠉⠈⠉⠀⠀⠀⢧⣂⡀⠀⣀⡿⢾⡈⠐⠀⢀⠀⠀⠎⠛⡵⠚⠂⠀⠀⢰⢜
⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠔⠵⡡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠭⠮⠂⠈⠑⠢⠔⠀⢨⣶⠅⠀⠀⠐⠀⠀⢸⠃
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠈⠅⡀⣀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠢⠤⠈⠤⠋⠁⠀⢠⠴⠀⠀⢀⠀⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠶⠲⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢞⣫⣵⠶⠿⠛⠛⠛⠛⠛⠛⠷⢦⣍⡻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⣥⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⣩⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢟⣵⠛⣁⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠙⣆⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⣭⣶⡿⠿⠿⠿⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢿⣿⣿⣦⡀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠘⣿⣿⢻⣿⡿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡝⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢿⡇⠀⠀⠀⢹⣷⣼⣿⡇⠿⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢮⣛⡛⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠟⣭⠄⠀⠀⠀⠀⠀⠈⣿⣯⣹⡿⠚⠛⢳⣝⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡷⢸⣧⠀⠀⢀⣀⠀⠀⠀⠀⠀⠉⠉⠀⢺⣿⡆⠀⠀⠀⠀⠀⠀⢹⣿⣿⡣⡎⢠⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⣿⣿⠀⣶⣿⣿⣿⣶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡇⠈⣾⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡅⣿⠀⠉⠋⠈⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡸⠂⠈⢠⡟⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡆⢸⠀⠀⠀⠀⢋⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠁⠀⢀⡾⣱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢷⣼⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠟⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡎⣿⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⡇⠀⠀⠀⠀⢀⡀⣿⠤⠤⠤⣤⠶⠖⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠛⠒⠒⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⡀⢹⢸⣇⠀⠀⠀⠐⢏⠁⠀⠀⠀⠀⠀⠀⠀⠱⡌⠳⣄⠀⠀⠀⢀⠤⠤⠚⠛⠛⠛⠛⢻⣶⢽⡇
⠀⠀⠀⠀⠀⣠⠾⣫⣽⣶⣶⢮⣍⡻⣇⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠙⢦⡔⠚⢻⡄⠀⠤⠤⠤⠤⠄⢸⣿⢸⡇
⠀⠀⠀⠀⣴⢫⡞⢡⠞⠀⠲⢀⡉⢻⣦⢠⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠳⣄⠀⠛⠆⣼⡄⠀⠀⠀⠀⠀⠀⢸⡿⣸
⠀⠀⠀⢰⡏⣾⠀⢈⠀⠀⠀⠀⢳⠀⣿⡌⣷⣌⢻⣆⡀⠀⠀⠀⠀⠀⠀⢶⡋⠁⠀⠀⠹⢦⡀⠀⠀⠳⡄⠀⠀⠀⠀⢀⣾⣧⡝⣦
⠀⠀⠀⠸⣇⢿⠀⠘⣄⠀⠀⢀⡌⠀⣿⢣⡟⠈⣣⠝⣿⣦⣀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢷⠀⠙⠦⠤⣀⣀⣀⣀⣀⡤⠛⢀⣼⠟⣼⠁
⠀⠀⠀⠀⠹⣎⠷⣄⡈⣛⡿⠋⣡⡾⢋⡾⠁⢰⡋⣾⣿⣿⣿⣷⣦⣤⣄⣀⣀⣹⣦⠀⠸⣇⡀⠀⠀⠀⠉⠉⠉⠉⠀⢠⠏⠙⢷⡝⣧
⠀⠀⠀⠀⠀⢈⣷⠎⣙⣿⣰⣿⠻⡆⣿⠀⠀⠈⠛⠶⢮⣍⡛⢻⣿⣿⣿⣿⣿⣿⠙⠀⠸⣇⠙⠂⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢻⡌⢧
⠀⠀⠀⠀⠀⡟⣵⠟⠋⢙⡄⣟⠀⢿⡹⣄⠀⣀⣠⣤⠖⣫⣴⣿⣿⣿⣿⣿⣿⣿⣾⠄⠀⢹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢻⡜⣧
⠀⠀⠀⠀⠀⣷⣸⡧⠚⢛⠄⢛⠀⠈⢻⡌⢛⣭⣷⣶⣾⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠻⣅⠀⠀⠀⠀⠀⠀⠘⠃⠸⠂⠀⠀⠈⢿⡸⣇
⠀⠀⠀⠀⠀⢳⡙⣷⠞⢛⡆⢠⠀⠀⢸⣇⣼⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡹⡆⠲⣀⠀⠈⠳⠶⣄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣧⢹⡀
⠀⠀⠀⠀⠀⢸⡄⢿⡤⠛⣹⡷⠂⠀⠀⢀⠉⠙⠻⠏⠀⠀⠉⠛⠿⠿⠟⠛⠁⢰⡆⠳⣄⣀⣀⡄⠈⡀⠀⠈⠉⠙⠓⠶⠶⡇⠀⠀⠀⣼⣿⡇⢷
⠀⠀⠀⠀⠀⠀⠹⣮⡛⠶⠿⣿⠉⠐⠦⣼⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣼⢿⣿⣿⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡇⣿⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣠⡞⠁⣀⣠⠞⠁⣾⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢻⡆⠀⡴⠃⠀⠀⠀⠀⠀⣠⠟⡌⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⡁⠀⠀⣿⣿⣿⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠻⠿⠷⠶⣶⣶⣶⡖⠚⠁⣰⠁⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢰⣿⣿⣿⣿⡇⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢹⡇⣾⣿⣿⣿⠀⠄⠀⠈⠁⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢸⣿⣿⣿⣿⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⣋⣾⣿⣿⣿⣿⠐⠶⠀⠀⠀⣤⠇⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢳⣾⢿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢩⣶⣿⣿⣿⣿⣿⣿⣅⣼⠂⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠖⠀⠀⠀⠛⠀⠀⠐⠒⢛⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⢿⣿⣿⣿⡿⢟⣭⢸⣇⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣝⣛⣭⠶⠛⠳⣮⡻⢦⣌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⢸⣿⣿⣿⣿⡿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣝⡓⠶⣤⣤⣄⣀⣀⡀⠀⠀⠙⠒⠀⠠⠤⠤⠤⠒⠊⠁⣀⣀⣠⣼⣿⣿⣿⣿⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠭⢭⣍⣛⣛⣛⣛⣛⣛⣒⣒⣚⣛⣛⣛⣛⣫⡭⢭⡜⣿⣿⠟⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠻⣮⣡⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠚⠛⠛⠛⠛⠓⠶⠶⣤⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣞⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣫⡿⠖⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠒⠶⠶⠦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠓⣾⠋⢰⡾⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⠾⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠛⠓⠒⡆⠀⣿⠀⢸⠀⠀⢀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠉⣙⣛⣛⡓⠒⠒⠒⠒⠒⠃⢀⡏⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢻⠀⠀⠀⣿⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⢸⡇⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⡇⠀⠀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⠃⠀⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠇⠀⢰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⢀⡏
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡟⠀⠀⣾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠓⠒⠒⠚⠛⠛⠀⠀⢀⡟⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⢀⡟⠀⣾
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢿⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠁⠀⢸⣇⡴⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡖⢲⠀⠀⠀⠀⣸⠃⠀⢀⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠩⠉⠉⢩⠉⠉⠉⠛⠛⠛⠛⠛⠀⠀⡿⠀⣠⡿⠉
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⣀⡴⠋⠈⠉⠉⠉⠛⡟⢛⡿⠛⠓⣷⡒⠶⠶⠶⠶⢶⣦⢤⣤⣤⣤⣤⣴⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠿⠋⠀⠀⠀⠀⣠⠖⠚⠁⢾⣇⡀⠀⠈⠙⠛⠓⠒⠒⠛⠁⠀⣠⡿⠀⠈⠣⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣧⣀⣀⣀⠈⠻⠯⣿⣶⣶⣶⣶⣶⣒⣒⣾⡽⠟⠁⠀⠀⠀⠀⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡤⠤⡤⢤⡤⠤⣤⢤⣤⣄⣈⣉⣉⣉⡉⠙⠛⠛⠛⠷⠾⠯⢭⣭⣥⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠋⡿⠓⡾⠓⣲⠓⡶⠧⢤⠧⢤⡯⣤⣽⣉⡟⢀⡖⢲⠖⢦⠤⢤⡀⠀⢩⠟⣇⠀⠀⠀⠀⣠⡴⢶⢞⡿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢸⣉⡽⠉⣹⠛⢲⠟⢲⠓⢲⠿⢤⡾⠤⣼⢥⣼⠁⡜⢉⡟⢒⠟⠒⡾⠁⡰⠋⣴⠏⠀⠀⣠⣞⣁⣠⣧⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠤⠤⠬⣉⣉⣉⣙⣛⣒⠛⠒⠛⠲⠾⠤⠴⠧⠴⠃⢸⣉⣽⣉⡟⠓⡾⢀⡜⣡⠞⠁⠀⠀⣰⠋⠀⠀⠀⠈⠉⠲⢄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠓⠒⠶⠶⠶⠯⠭⣭⣭⣉⣓⣒⣒⣒⠒⠒⠦⠤⠤⢭⣭⣉⣠⢋⡼⠃⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠓⠲⠶⠶⠶⠿⠟⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣴⣲⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⣦⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⣼⡗⡜⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢳⢷⠋⣤⡴⣦⣛⣒⣒⣛⣛⣟⣛⢛⡷
⠀⠀⠀⠀⣀⣠⣤⠶⠖⡛⡵⣡⣦⡸⣯⣽⣿⣭⣴⠋⣥⣻⣾⣿⠟⠁
⠀⠀⠰⣟⡉⡵⣢⣉⣹⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢟⣹⣿⣿⠋
⠀⠀⠀⠈⠙⠶⣕⡿⣯⡿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢸⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠾⣵⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡒⠒⠒⠲⠶⣶⠤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢻⣻⣿⣿⣿⠿⣟⠟⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢛⣩⣝⡛⠷⣦⣄⠀⠙⠪⡑⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⣿⡵⠿⠿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣡⣿⣿⣿⣷⣿⣿⣿⣷⣄⠀⠘⢌⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢞⣿⠟⠉⠀⠀⠀⠀⠈⠉⠛⠛⠟⠀⣠⠴⡟⢦⠀⠀⠀⠀⠀⣸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠘⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢏⡁⠀⢸⣦⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠷⡒⣻⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⣸⡽⠈⠙⠢⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡞⠈⣾⣿⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⢫⡀⠀⠀⡐⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣥⣿⠏⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⠀⠀⠈⢿⣾⣭⣟⣛⣛⣛⣛⣩⣭⡶⢿⣘⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⡞⠀⠉⠛⠛⠛⠛⠛⡏⠁⠀⠀⠈⢢⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠓⠒⠒⠲⢦⡀⠀⠀⠳⣄⠀⠀⠐⠀⠀⠉⠓⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⢸⣀⣏⣈⣿⣉⣉⣉⣉⣉⣳⣦⣥⣀⠀⢠⡏⠉⠉⡓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠴⠤⣦⣤⡿⠉⠉⠉⠉⠁⠀⣼⡇⠈⡟⠛⠛⠒⢦⣨⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣇⡼⠁⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠀⠀⠀⣠⡀⠀⠀⠀⠀⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣶⡾⢿⠙⣤⣄⣀⡤⢼⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⣀⡀⣀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠐⢹⠀⠈⢯⡉⣉⡥⠞⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⢀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⢸⠀⠀⠀⢣⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⣠⠎⠀⠀⠀⠘⡄⠀⠀⠀⠀⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠖⠚⠁⠀⠀⠀⠀⠀⠙⠦⣤⡤⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣠⣼⣄⢀⡀⡀⡀⢀⡀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢠⠒⠉⡉⠐⢄⠀⠀⠀⠀⡠⠂⢉⠉⠒⡄⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇⢰⠡⠌⠣⠀⡆⠀⠀⢰⠀⠜⢡⠌⡆⢸⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢹⠀⠀⠀⠀⠣⡈⠑⠒⢁⡴⠁⠀⠀⠈⢦⡈⠒⠊⢁⠌⠀⠀⠀⠀⡏⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠄⢀⡀⣀⡀⢀⣀⡀⠀⣀⡀⣀⣀⣀⡀⠠⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢤⠤⢠⡄⠤⠤⠤⠤⠤⠤⠀⠇⠀⠀⠸⠀⠤⠤⠤⠤⠤⠤⢤⡤⠤⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣧⡀⡀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠆⠀⡆⠀⠀⢸⠦⠤⠤⠤⠤⠤⠤⠤⡤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠴⠇⠀⠄⢰⠀⠠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠛⠐⠃⠘⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠃⠘⠀⠛⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠒⠒⠒⠀⢴⡃⠀⠀⢘⡦⠀⠐⠒⠒⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢱⠀⠀⡎⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠄⠀⠁⠠⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠌⠀⠐⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢻⡟⠉⢙⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡂⠀⠀⠀⡃⠐⠂⢒⠐⠂⡸⠀⡒⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠩⠞⠻⣿⣿⣿⣿⣧
⠠⠀⠀⠊⡠⠀⠉⠀⠐⠀⠀⠀⠀⣀⣤⡄⠀⠁⠀⠀⠐⠁⠀⠀⠀⠀⠐⠀⠑⠄⢈⡉⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣆
⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠠⠂⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠠⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⡡⠤⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⢤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠄⠀⠁⠠⡀⣌⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⢰⣎⣺⡿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⡙⢈⡓⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠄⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⣀⣀⣀⣀⣀⣀⠀⠀⢀⠿⠿⠿⠿⠿⠿⣿⣿⠇
⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣶⡄⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣶⣶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇
⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢯⡀⢀⣯⢦⣄⠀⠀⠀⠀⠀⠀⣠⣦⡄⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣟⠁⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⣠⡾⠛⠛⢷⣼⠛⠋⠙⠻⣦⡀⠀⠀⢠⣿⣿⡿⠀⠀⠸⠿⠏
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢸⢻⡇⠀⠀⠀⢸⡇⠀⠀⠸⣿⣿⠇⠀⠀⢾⡿
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠿⠿⠿⠿⠿⠛⠢⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠸⣿⡇⠀⠀⢀⣼⠃⠀⠀⠀⢛⣫⡀
⠀⠀⠀⠀⠀⠹⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡄⠀⠀⠀⠀⠙⠳⠶⠶⠞⠋⠙⠛⠛⠋⠁⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠉

⣤⠀⣠⣀⣄⣠⠀⣤⣤⣤⣤⠀⣀⣄⣤⣠⠀⣤⣤⣤⣤⠀⣠⣀⣀⣠⠀⣤⣤⣤⣤⠀⣠⣄⣤⣤⠀⣤⣤⣤⣤⠀⣠⣀⣀⣠⠀⣤⣤⣤⣤⠀⣠⣀⣠⣤
⢁⣼⣿⣿⡿⢁⣾⣿⣿⡿⢃⣼⣿⣿⡿⢁⣾⣿⣿⡿⢃⣾⣿⣿⡿⢁⣾⣿⣿⡿⢃⣾⣿⣿⡿⢁⣾⣿⣿⡿⢃⣾⣿⣿⡿⢁⣾⣿⣿⡿⢁⣾⣿⣿⡿⢡⣾

⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢶⣾⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢻⣿⣿⣷⡀⠀⣀⡄⠠⠄⠀⠘⣩⠄⠀⠀⠀⣀⡀⠶⠀⡠⢺⡞⠛⠛⠛⣟⠠
⠀⠀⠀⠀⠀⠀⠻⢻⡿⢣⡀⠁⣠⠄⠀⠞⢑⡴⠒⡖⢸⠞⢹⣿⣎⠨⡀⢴⣿⣬⡙⠢⣹⣖⣀⡀
⠀⠀⠀⠀⠀⠔⠊⠑⠢⡄⠱⡀⠟⣂⣀⣠⡤⡶⠒⠛⢊⡅⢸⣿⣿⣅⣻⠀⠀⠀⠻⣶⣁⠙⠋⠁⠐⠂
⠀⠀⠀⠴⠀⢠⣶⣤⠰⠀⠖⠋⠰⢉⣁⠙⠑⠂⣉⣶⣷⣿⢠⣤⣉⠳⣍⡙⠳⠦⢄⠘⢛⢀⡓⠚
⠀⠀⠀⠀⠠⢨⣯⣿⢷⣦⠄⠎⠰⣃⣩⠟⡟⠈⠁⠉⠻⣧⠀⢿⠸⠄⠈⣔⠀⠀⠺⠧⡉⠛⢡⡦
⠀⠀⠰⠶⢀⡄⢋⣻⠖⠁⠠⠀⠀⠉⠀⠀⠙⣄⡀⣀⢤⣭⠷⡽⠻⣧⡀⠙⠛⣦⠀⠐⢟⢈⣀⠐⠉
⠀⠀⠀⠀⠌⠁⣼⠃⠀⣴⣡⣄⣄⣤⣀⣀⣀⠼⠛⠾⠋⠼⠀⠒⠀⠀⠙⣦⠀⠀⢱⡀⠈⢡⠀⠁
⠀⠠⡆⢀⡇⢀⡏⠀⠆⠋⠻⡏⠠⢀⢈⡡⠀⣀⣈⡀⠠⠄⠰⢆⣤⣤⣄⣿⢚⠀⠰⣷⠀⠀⡟⠂⣀⡀
⠀⠀⠀⡼⠀⡙⠀⠀⠀⡀⠄⢹⡀⠈⢁⣴⣿⣿⣿⣿⣷⣶⢶⣿⣿⣿⣿⡍⣎⠐⠀⢼⠦⣞⣣⠔⠁
⠀⠀⢀⡃⠀⡟⠂⠀⠠⡇⠀⣸⠸⡄⢸⣿⣿⣿⠿⠿⠿⠛⢘⡿⣿⡏⢩⠀⢀⢰⢀⢸⠀⠈⡄⠴
⠀⠀⠨⡍⠀⣄⠤⠂⢰⠀⠀⢈⢰⡇⠀⠉⠉⣀⠀⠀⢸⠆⠀⡿⠻⠿⠆⠰⣶⡸⡄⢘⣄⠀⢃
⠀⠀⠀⣻⠀⢳⡂⠈⢹⠃⠀⣸⣄⢤⠆⠀⠀⠀⢀⠤⠄⠠⠄⠁⡀⣀⣦⣅⢹⡇⣵⡤⠦⠀⡜⠦
⠀⠀⠀⢯⠀⠘⠅⠈⠈⠀⠀⣴⢈⠲⣄⠂⢀⠰⡇⢿⠈⠀⡀⠇⠂⢩⣠⣠⢹⡟⠙⢁⣾⠀⢀⡴
⠀⠀⠈⠘⠃⠂⢰⣀⠀⠀⠀⠘⣯⡗⠸⠉⠟⡴⡠⡞⠴⣸⣄⢲⣦⣀⣿⣿⣸⠛⢄⢶⠉⠙⠺⢅
⠀⠀⠈⡓⠺⣆⠀⠙⢦⠀⠀⡀⠙⣷⣤⡃⠄⣦⠀⣷⣶⢹⣿⣿⣿⠿⡟⠛⠻⣆⣰⣇⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠙⡀⠸⠿⣦⠿⣧⠀⠀⠀⠑⣿⣇⠠⠹⠃⠹⡜⡜⠏⢹⡘⡄⠧⠶⠒⠈⣇⠈⢙⠢⡄⠀⠀⢘⡀
⠀⠀⠀⠘⠄⡠⠖⠁⠀⢻⡆⠀⠀⠀⠼⡙⢧⡄⠰⠀⢇⠁⠀⠟⠁⠀⠀⠀⠀⠀⣽⣆⣸⡄⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⣷⠀⠀⠀⠘⡧⠘⣿⣄⣄⡘⠀⠁⠀⠀⠀⢀⣠⠧⣤⡇⢠⡆⠀⡾⠁⠀⢸
⠀⠀⠀⠰⠀⠀⠀⠀⣠⠞⢻⡆⠃⠀⠀⣥⠀⣿⣿⣈⠁⠒⠶⠒⠲⢏⡅⣶⣸⡿⣷⠛⠛⠋⠀⠀⢀⠂
⠀⠀⠀⡇⠀⠀⠀⣰⠿⣆⠄⣧⡀⠠⠀⢹⠰⡏⢃⡌⠉⠓⠲⣤⣥⠈⢉⠁⠈⣧⣺⣧⣤⣠⣤⠖⠃
⠀⠀⠀⠃⠀⠀⠀⠘⢇⠛⠐⢏⢷⡀⡄⢰⡶⡵⢊⡟⠐⠤⠀⠐⠙⠃⢘⣄⠀⠘⢿⡘⣤⣴⣦⡖⠲
⠀⠀⠀⠈⡄⠀⠀⠀⠈⠛⠧⣬⣞⡇⠠⡼⠚⠁⡾⠙⠖⠂⠠⠼⣦⠠⣠⡜⣄⠀⠈⢧⡼⣿⣿⠛⠃
⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠸⡶⠀⢀⡞⢧⡤⠨⣁⣦⠁⠀⠀⠀⢲⡟⣆⠀⠈⢧⡴⡤⢓⡞
⠀⠀⠀⠀⠀⠈⠱⠤⣀⣀⠀⠀⣀⣀⣤⢺⡏⠙⢄⠐⠂⠀⠙⠃⠐⠺⠗⠞⠁⠏⠂⠀⠀⣤
⠀⠀⠀⠀⠠⠒⠲⠲⠂⠛⠉⠁⡳⠤⠞⠀⠈⠒⠤⠭⠴⠛⠛⠒⠊⠀⠀⠀⠀⠂⠀⠑⠢⢀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⣉⣀⣀⡀⠀⠀⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⠋⠙⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠈⠀⠀⠙⠛⠛⠉⠹⡟⠛⠛⠋⠁⢀⣾⣿⠟⢩⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⣠⣿⡿⠁⠀⣾⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠚⢧⡀⠀⠰⣄⠀⠀⠀⠐⢲⡾⠉⠁⠀⠀⢠⣿⡿⠁⠀⣼⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⡄⠈⠳⡄⠀⢸⣿⣶⣤⣤⣾⣅⡀⠀⠀⠀⣾⣿⠁⢀⣼⣣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣹⣶⣾⠉⠁⣸⣿⣿⣿⣿⣷⣦⣄⣿⣿⢠⢞⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠻⣿⡟⠧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣸⣿⣇⠀⢸⣿⣿⡿⠛⠁⠀⠀⠀⣀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣿⠟⠉⠀⠀⠀⠀⠀⣄⠙⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⡀⠀⠀⠀⠀⠀⠀⠲⣌⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⢷⣄⣴⣿⣿⣋⣿⣶⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣮⢳⡐⡄⠀⠀⠀⠀⠀⠀⡄⣾⠈⢁⡀
⠀⠀⠀⠀⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⣷⣧⢳⡀⢀⣤⣤⣤⣀⣷⡸⢀⣾⡇⠀⠀⢀⣶⣮
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣸⣿⣟⣷⣿⣿⢟⣛⣿⣿⣷⣼⣿⣷⣶⣷⣿⣿⣃⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢻⣏⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢹⣶⣾⣿⣿⣿⡿⢿⣾⣿⣿⣷⣿⣿⣭⣿⡿⣹⣿⡟⠁⠀⢀⠍⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢦⣤⣄⠀⠄⢘⢓⣏⠀⣠⣿⣎⣻⣿⣿⣿⣿⣯⣭⣻⡿⣿⣿⣿⡿⠉⢰⣿⣏⣤⠀⠀⠈⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⢿⣷⣷⡦⣈⠁⠀⣾⡟⣛⢛⣿⡿⡿⣿⣟⣭⣽⣷⣶⣾⣿⣡⠀⠈⢿⡿⣿⣾⣶⣶⣀⣀⠈⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠺⡦⠀⠀⠀⠀⠀⠀⠉⠳⢶⣦⣝⣲⣿⣿⣁⣿⣿⣿⣿⣯⣹⣿⣿⣽⠛⠛⢿⣧⣎⣴⣿⣷⣹⣿⣟⣿⣿⣿⣿⣾⣵⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⣿⣿⣿⣿⣽⣿⣿⣭⣿⠏⠀⢿⡛⠩⠀⠀⠘⠃⣸⣷⣍⢻⣟⣮⢻⣼⣿⠿⠛⢋⣉⣤⣴⠂
⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠁⣀⣿⣿⣿⣿⣿⣿⠎⠋⢐⡂⠈⠐⢶⣶⠄⠘⣧⣽⣭⠛⢻⡟⢛⣉⠭⠔⢒⡈⠉⣉⠡⠤⠴⠶⡂⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣼⡿⣽⣉⣿⣿⡿⡿⠆⣐⣿⠀⠀⠀⠀⠁⠀⢀⡿⢿⣿⠀⣠⢫⣧⣶⣾⣿⣯⣭⣶⣶⠖⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠀⣤⣾⢠⣿⢧⢸⣿⣿⣟⢷⣿⠀⠙⠁⠀⠀⢠⣾⣧⡤⠉⣠⣾⣷⣦⣭⡾⢿⡿⣿⣟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠀⠀⠁⢼⣿⣿⣎⣻⣿⣿⣾⣿⣦⡀⣀⠴⠆⣼⣿⣿⡠⠐⠋⠻⠻⢿⣟⣱⡏⣶⡏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⣽⣿⡿⠻⠛⠛⢉⣨⣿⢇⠈⠉⠁⣩⣿⣿⠿⠛⠀⠀⣠⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡿⣦⣤⣠⣶⡿⣿⣿⣟⡷⣶⣿⣿⣏⡀⠀⠀⠓⠘⢿⣿⡿⠁⣸⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡴⠟⠉⡈⣩⣴⢿⣿⣧⣽⢷⣼⡿⣿⣿⣤⣬⣽⣻⣿⣎⡓⡄⠀⢈⡻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠉⠁⠀⠉⠹⠿⢷⣽⠶⠾⠿⠿⠿⠿⣿⣿⣿⣿⡿⣆⠢⡹⣿⣧⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠟⠀⠀⠀⠀⡈⠀⠀⠀⠙⣧⢻⣎⠻⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⢻⣆⠈⠁⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣅
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⣰⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⣦⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⣦⢄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⣿⠀⠋⢰⡟⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠏⠈⣧⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⡟⠀⠀⢸⠁⠀⠀⠈⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠘⠻⠦⠤⣿⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣠⠖⠛⠛⢶⣀⡇⠀⠀⣼⠀⠀⣀⣀⣀⠈⢳⡄⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢿⡀⠀⠀⣸⢿⡇⠀⠀⡿⢠⡟⠁⠀⠈⢷⠀⢿⠀⠀⠀⠀⠀⠀⠘⢳⠦⣤⣄⣀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⡈⠛⠲⠚⠉⢸⠀⠀⠀⡇⠈⢧⣄⣀⣠⠟⠀⢸⡇⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢩⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠿⠷⠶⠦⢬⣭⣥⣤⣄⣸⡇⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠀⠶⠴⠤⠤⠄⠀⣀⣤⣀⣀⡀⠀⠀⣀⠀⠀⠀⠈⡅⠀⠀⠀⠀⢠⡏⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⡀⢀⣀⣀⣀⣧⣠⣤⣤⣤⣤⣄⣄⣠⣄⣠⣄⣉⣉⣉⣽⣇⣈⣁⣀⠀⠀⠀⣼⠁⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⢸⣧⡅⢸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⢸⠤⢴⠇⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⡏⠀⡆⢸⣿⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠓⠒⠶⣤⡀⠀⠀⠀⠀⠀⣸⡇⢸⠀⡾⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣸⠓⠒⠃⠈⣿⡄⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣿⠃⢸⣺⡃⠀⣀⡴⠋
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢠⡏⠀⠀⠇⠀⢹⣷⡀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣼⡏⠀⢸⠈⠉⠉⠁
⠀⠀⠀⠀⣾⠀⠀⠀⠀⣾⠀⠀⠀⡀⠀⠀⠹⣷⡀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢀⣼⠟⠀⠀⢸
⠀⠀⠀⢰⡇⠀⠀⠀⢰⠇⠀⠀⠀⡇⠀⠀⠀⠙⢿⣦⡀⠈⠳⣄⣀⠀⠀⠀⣀⣤⠞⢁⣠⡿⠋⠀⠀⠀⢸
⠀⢀⣤⣾⣁⡀⠀⠀⡿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⣉⣛⣛⣋⣉⣤⡶⠟⠉⠀⠀⠀⠀⠀⢸
⡴⠋⠀⠀⠈⠉⠙⠛⢳⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⡇⠀⣿⠙⠒⣶⠄⢰⠏⠀⠀⠀⠓⠂⡄⠐⢶⡶⠶⠶⠶⠶⠶⣶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠶⢶⡶⠢⠞
⠀⢻⣀⣻⡄⣰⠋⢠⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⢦⡀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢀⡴⠋
⠀⠀⠉⠉⠉⠛⠲⠟⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⣦⣀⣀⣿⢦⣄⣠⣤⣿⣀⣀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠤⣤⡤⣬⡉⠙⡏⠀⠀⠀⠀⡏⠉⣽⠷⠶⠶⢦⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠾⠃⠀⠀⠀⠈⠷⢶⡇⠀⠀⠀⢀⣷⠞⠃⠀⠀⠀⠈⠓⢶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠖⠛⠉⠉⠙⠛⠶⣄⣿⠀⠀⠀⢸⣠⠶⠛⠋⠉⠙⠛⠶⣤⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠴⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⡅⠱⠀⠀⠀⠀⠀⠀⠙⢟⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢹⡄⠓⠂⣀⠀⠀⠀⠀⠀⢷⠨⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠤⣿⡈⠁⠀⠀⠀⠀⠀⠀⠈⣇⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠋⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⢈⡬⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠐⠂⠀⢠⠀⠀⠰⣿⠀⠖⠀⠀⠀⢠⠀⠀⠀⣼⠀⠀⣁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠷⡾⠄⢀⠇⠀⠐⢰⡇⣀⡀⠀⠀⠀⠀⠀⢸⠀⡟⠀⠀⢩⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⢛⠛⠛⠓⠒⠚⠓⠒⠃⣼⠀⡍⠀⠀⠀⢀⡇⠀⡌⢸⡇⠀⠁⠸⠂⢻
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣯⣭⣵⣶⣅⣀⣤⣨⡥⠴⠖⠚⠋⠻⣿⣿⣿⣿⡿⢾⣷⣶⣷⣿⣯⡉⠝⠀⢡⢸⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⡟⣿⣿⣂⠀⠀⠀⠀⠀⠈⢻⣷⣯⠿⠚⠻⣾⣿⢿⣿⣿⣦⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠻⣇⠈⠁⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣦⣌⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⣼⣿⣿⣿⣿⣿⣿⡟⠁⠉⠛⠶⣽⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠈⠉⢉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⣦⠤⠶⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⣰⣿⠿⠛⢻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠠⠾⠛⢀⡠⠴⠒⠉⠁⠀⠀⠀⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢰⡇⠀⠀⠀⢀⢀⣀⣀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⣠⢤⣿⣿⣿⣿⣿⡷⣤
⠀⠀⢀⣀⣤⣤⣤⠀⢀⣾⠏⠀⠀⠀⠀⠀⠳⢤⣴⣾⣿⡿⣿⣿⡇⡜⣈⡧⠀⠀⣀⣤⡖⠋⢠⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠡⡉⠻⡿⠴⣊⡏⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡝⡓⢸⣿⣿⣭⢤⣴⣺⡽⠛⠉⠀⣴⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⢳⡀⣠⢞⠟⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠷⠃⣏⠉⠉⠉⢩⣾⠉⠁⠀⠀⣠⣾⣿⡿⡿⠿⠿⢿⠟⠁
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢀⡴⢻⣿⣿⠟⠁⠀⠀⠘⡄⠀⠀⣾⡏⠀⠀⠀⠘⠋⠉⠉⠀⠀⠠⢀⠀⢠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡿⠛⠁⠀⠀⠀⠀⠀⣷⠀⠴⢿⣇⠀⠀⠔⢉⢀⣦⣔⠶⠋⠚⠂⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢻⡄⢆⠤⠊⠈⠀⢠⡄⢀⠆⡀⢐⣤⣤⡴⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠜⠀⡏⠐⠘⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣽⣷⣦⣴⣶⣦⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠉⠽⣷⣶⣾⣿⣶⣦⡄⠀⠀⠀⢠⣤⣫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⣤⣀⠛⠃⠞⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠛⠋⠀⠀⠀⠀⠀⠀⣼⡏⢹⣿⣿⣿⣿⣄⠜⢃⡼⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠄⠀⣠⣶⡶⣦⣤⢀⣀⣤⣐⣿⣿⣿⣿⠃⠀⣿⣞⣿⣶⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠻⠿⡿⠟⠁⠈⠻⣿⣿⣿⣿⣿⣿⠀⣀⣻⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠰⠶⣄⣠⡆⠀⠀⣤⣿⣿⣿⡏⠩⠍⠻⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣺⣿⣆⠀⠠⣶⣶⣶⣾⣤⣭⣅⣴⣿⣿⣿⣿⠉⢀⣼⣶⠂⣸⣓⣭⠀⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣔⠦⠘⡿⠿⣿⣿⠿⣿⣿⣿⣿⣸⣿⡀⣾⣿⣿⣿⣿⣿⣏⠀⣰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠾⣿⣿⣇⠹⣿⣧⣰⣿⣿⢿⣯⣾⣿⣿⣿⣿⣿⣿⡏⠀⠋⠀⠹⠏⠩⣏⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢦⢼⣿⡀⠈⠙⣿⣿⣿⣿⡿⢻⣿⣿⣿⣛⣿⣿⣿⣴⣀⡀⠀⠀⣤⣿⠛⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⠀⢸⣿⣿⣦⣰⣿⣿⣿⣿⣶⣿⣿⣿⡏⠻⣿⣿⣿⣧⠿⣭⡄⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠙⠻⣿⣿⣿⣿⣟⢻⣿⣿⡄⠀⠑⠘⢿⣆⢰⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡒⢒⣶⣦⣼⡇⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢉⣀⣤⣤⣤⣤⣀⡉⠻⣆⠀⠀⠀⠀⠀⠀⠀⠙⠂⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣡⠞⠋⣽⣿⣿⣿⣿⣿⣿⣷⣌⣇⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⢡⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠋⡽⠳⠯⠭⣉⣋⠭⠽⠞⢏⠙⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⡸⠁⠀⢀⣀⣀⣀⣀⡀⠀⠘⡆⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡜⠣⠤⢤⠁⠀⠀⢠⠀⢹⠤⠤⠞⢢⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣥⣄⠀⢹⡗⠒⠺⣆⢀⡘⢺⠂⣺⠒⠒⢻⡉⠀⣠⣬⣆
⠀⠀⠀⠀⠀⠀⠀⢀⠴⡪⠁⠀⣹⠗⣿⢳⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⡏⣿⠺⣏⠀⠈⠕⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠻⢾⡶⠋⠉⠁⠀⢻⣼⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣇⡟⠀⠈⠉⠙⢶⡷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣀⣠⡏⢧⣄⣀⣀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈⠉⠈⢹⠁⠘⡍⠙⠋⠁⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠁⠒⠂⡏⠀⠀⢿⠐⠒⠈⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢸⠁⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⣠⠞⠀⠀⠀⠀⠣⣀⣀⣀⠜

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⣁⡢⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣇⠀⠀⠀⠀⠀⠀⠀⡵⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠷⠀⠀⠀⠼⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⡀⠀⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠟⠀⠀⡀⠻⠀⡈⠀⠀⡓⠐⠴⠆⠙⠀⠸⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⡈⡂
⠀⠀⠐⢰⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡯⠋⣾
⠀⠀⠈⠲⡞⡉⠛⠑⠶⣄⢄⡀⠀⠀⠀⠐⠂⠤⠄⠀⣀⣀⣂⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⢀⡤⠊⠀⢔⠏
⠀⠀⠀⠀⠘⢎⠂⠀⠀⠀⠉⠚⠶⣤⡤⠀⠀⠀⠀⠀⠘⠻⠛⠻⠄⠄⠠⠤⠠⣀⣀⡤⠔⠁⠀⢀⢀⡜
⠀⠀⠀⠀⠀⠀⠑⢕⠀⠀⠀⠀⠀⡞⠀⠩⠂⢸⠩⠅⠀⠀⠀⠀⠂⠀⣀⡀⣠⡌⠺⠁⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⢄⠀⠀⠃⠀⠭⠂⢸⠀⠀⠀⢀⠀⠂⠀⠀⠶⠬⠰⠂⠀⣀⡠⣔⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠤⣀⠀⠬⠄⠸⣀⣐⣒⠓⡦⠒⣀⠀⣉⠥⢜⡁⠀⡇⠱⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠒⢽⠀⢆⠄⠀⢠⣤⣤⣤⣄⠀⠀⠀⠴⢝⡡⠄⠸⠆⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⢸⠀⠀⠀⠀⠀⠁⠹⠀⡇⠀⠀⠀⠙⠁⠘⠁⠠⢌⣑⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠭⢱⣐⠀⠀⠀⠀⠘⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⠟⠚⠮⠥⠭⠭⠽⢏⣽⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠉⠀⡉⠲⡀⠀⠀⠀⠠⠒⠈⠉⠓⠢⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠀⠒⠼⣀⣀⣂⣁⣧⡵⠃⠀⠀⠀⠳⢬⣐⡀⠠⠀⠈⡷⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠁⠈⠉⠉⠁⠀⠈⠀⠀⠁⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣴⣿⣶⣾⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢐⣠⣤⣤⣿⣷⣶⣾⢶⣶⣮⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣿⣿⣿⣿⣿⣇⢘⣿⣫⠈⠁⠈⢱⡄
⠀⠀⠀⠀⠀⢀⣼⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠰⣷⣗
⠀⠀⠀⣰⣿⣾⣿⣿⣗⣀⢻⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢿⣿⠁⢸⣿⣯⣿⣿⣷⣛⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣦⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣆
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢻⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢇⢻⣿⣷⣾⠿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣬⣸⣿⣿⢻⣾⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⢷⣿⣿⣷⡿⢱⣿⣿⣿⣿⣿⣿⣏
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣋⣻⣿⣉⡙⠁⠈⢙⣿⣿⣿⣿⡰⠇⣺⣿⡿⡫⣴⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⡅⠐⣶⣾⣾⠿⠟⠚⡛⢿⣿⣻⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠸⠟⣛⣯⡾⢿⣿⣿⣿⣿⡿⢻⡟⣰⣧⣰⢎⣌⣝⢋⣂⣾⡿⠈⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣿⣷⣿⣿⣿⣧⣜⣿⡿⠿⠛⠛⣾⣿⣿⣴⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⠀⣈⣿⣿⣙⣿⡏⠩⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠹⢻⣿⣿⣿⣿⣿⣿⣿⣟⣻⡿⠿⣿⠿⢿⠿⣲⣾⡛⣻⣿⠾⣽⣿⣿⣿⣱⣧⣄⣸⣿⣿⣿⣿⣿⣿⡏⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣽⣏⠁⣼⣤⣤⣶⣿⣿⣁⡨⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢺⣿⡿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⢾⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣹⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⠀⠉
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠟⠹⣿⣿⣼⣿⣿⣿⡿⢿⠿⡿⣿⣿⣿⣿⡻⠟⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡁⢿⠻⣿⣿⣿⣿⣿⣿⡿⡇⠸⢿⣿⣿⣿⣿⣿⣿⢿⡿⢧⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣶⡈⠉⠛⠛⠛⠋⠀⡇⢸⠂⠉⠛⠛⠛⠉⢁⢨⣶⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡙⠇⠀⠀⠀⠀⢶⣀⠃⠈⣀⡶⠀⠀⠀⠀⠘⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⣹⣟⣟⣉⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣿⣿⣿⣿⣿⣿⣄⠀⠀⠁⠙⠛⠛⠻⠟⠛⠛⠃⠘⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⢦⡀⠀⠀⠈⢹⠿⠿⠋⠁⠀⠀⣠⣾⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⢮⣿⣦⣆⡀⠀⠀⠀⠀⢰⣦⢞⠕⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠛⠿⢿⣶⣶⠶⠞⠫⠐⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢀⣼⣷⣻⣆⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡏⢷⡀⠀⠀⢀⣾⣿⣿⣿⣻⣆⠀⠸⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠾⠿⠿⠿⠿⡷⠛⠂⠰⠟⠛⠉⠁

⠛⠛⠛⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⢐⣦⣤⠶⠒⠓⠒⠿⠛⠛⠉⠀⠘⠿⠛⠁⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠓⠟⠒⢶⣴⣿⡿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⠿⣿⣟⡿⡛⠟⠋⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡋⣓⠊⠀⠁⠀⠀⠘⠀⠀⠋⠟⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡶⠀⠀⠠⢀⣀⠀⢀⣀⣀⣀⣠⣤⣤⠤⠄⠐⠶⠚⠛⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣰⡇⢀⠴⠶⣤⣀⠠⠀⢠⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣧⣀⡀⠁⠀⠈⠦⡀⠹⡌⠈⠈⠁⠀⠀⣀⣴⠿⠷⢆⡟⡴⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⢸⣇⡾⠁⠁⠰⠀⠘⠂⠀⠀⠠⢋⣉⢹⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⡝⠢⢦⠀⠀⢘⠀⠈⠀⣰⣤⡴⣿⣿⣏⠀⢩⣽⣿⣷⣧⡴⢸⣿⣿⣿⣿⠿⣋⡹⠀⠟⢀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠈⠀⠀⠈⡁⠀⢜⢻⢿⣼⡿⠀⠈⠀⠁⢈⣉⠄⣾⠋⢾⣿⣿⣿⣿⣇⢅⣳⣦⠱⢇⡆⠑⠀⠠⡆⣤⡀⠀⡀⣠⡄⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡏⠀⠀⠀⠀⠀⠁⠀⢀⢸⡿⣿⣗⠀⠂⠘⣽⠙⠋⠒⠈⠁⠀⢿⣿⣿⣿⣇⢈⡻⣿⡀⣿⠷⡇⠀⠀⣠⣤⣉⣩⣀⠀⠀⠰⠃⠗⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣇⠀⠀⢀⡀⠐⠀⠀⣠⠞⠛⠟⠛⠂⠀⠀⠉⠛⠂⠀⠀⠈⠀⡖⣿⣿⣿⣿⣿⣥⢬⡅⢽⣆⡁⠀⠸⠷⢶⣶⣶⠾⠀⠀⠂⡈⢠⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡎⠀⠈⠀⠀⠀⠐⡟⠀⢀⡃⣄⡀⠀⠀⢀⣴⡛⠛⠋⣷⡀⢰⣿⡿⣿⣿⣿⣿⠈⡇⠀⠫⢻⡀⠀⠀⠀⠀⣠⣀⣀⡄⢼⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⡀⢀⠇⢀⠀⢀⠇⡀⣷⡴⠾⢹⣷⣄⢟⠣⠀⠀⠀⢸⣵⡟⠛⠟⠙⣿⣿⣿⠀⣿⢀⢀⣴⠿⣿⡏⠉⠉⠉⠻⠃⣰⣶⣶⣶⣶⣶⣤⣭⣿⣿⣿⣿
⣿⣿⣿⣿⣷⡈⣄⠀⠀⢡⠀⣷⠎⣀⣷⣿⣟⠋⠈⠁⠀⠀⠀⠈⣾⢈⣀⡱⢾⣿⣿⢿⡁⣸⡿⢼⡅⠀⣹⣧⡖⠈⣰⠲⣮⣹⣿⣿⣿⣿⠿⠟⢛⣛⣻⣿⣿
⣿⣿⠟⠋⢀⣨⣿⠀⠀⠘⣄⣽⣾⣿⡿⠉⠀⠀⠐⠀⡀⠀⠀⣰⠃⣈⡹⣼⠺⣗⣩⣾⠇⠹⣇⠀⠀⢠⡋⡌⠀⡠⠟⠿⣿⠏⠉⠉⣙⣿⣿⣿⣿⣿⣿⣿⣿
⠟⠁⠀⢠⣾⣿⣿⣧⠀⠒⠛⠁⡉⢁⡀⢤⡄⠲⠆⠚⠁⠀⠀⠈⣸⣟⣣⠏⢰⡛⣻⠃⠐⢦⣿⣶⣤⣼⡟⠀⡜⠀⠀⣠⣽⢧⣬⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⢀⠀⠀⠀⠠⣟⡾⣡⠎⠺⢷⣽⣯⣤⡴⣿⡻⢽⡟⣁⡞⠀⠀⠁⢐⡵⣼⢿⣿⣶⣶⣶⣌⠻⣿⣿⢿⣿
⠀⣴⣿⣿⣿⣿⣿⠟⠇⢀⠀⠀⢀⠄⠀⢠⠊⣁⠀⠋⠠⠀⠀⠄⠟⡠⢟⠚⠲⣶⣷⣾⣶⣧⣥⣼⡼⣿⠿⣶⠆⢠⠀⠈⠃⢈⡰⣿⣷⡌⢻⣕⠳⣙⣷⣄⣈
⣴⡏⠀⠀⢹⠟⠃⠀⢀⣿⠀⠀⠀⠁⠀⠀⠈⢀⠆⠀⠂⠀⢀⠴⠂⢀⣤⣶⠾⠟⣻⣿⣿⣿⣿⣿⡆⠁⣾⣿⣦⠚⠁⠀⠀⠀⠉⡙⣿⣿⣿⠿⣶⣿⡿⢻⣿
⣟⠀⠀⠀⠈⠀⠀⢠⡟⣿⣧⠀⠀⠀⠀⠀⠀⠀⠐⣧⣣⢜⠁⣠⣾⣿⠏⠀⠀⠈⣭⣽⣿⣿⣿⣿⣿⣆⠈⣉⣿⡄⠀⠀⠀⠀⠀⠀⢹⡙⣇⠀⠈⠫⣄⠀⠃
⢿⡄⠀⠀⠀⣦⢠⡟⣰⣿⣿⣷⣄⠀⠀⠀⠀⠀⣤⣾⣧⢋⣾⣿⣿⠁⢀⠴⠊⠉⣉⣡⣴⣿⠿⣿⣿⣿⣷⣽⣿⣿⣗⣦⣀⠀⣀⠀⣨⠿⠉⣡⠀⠀⠙⡤⠖
⠀⢹⣇⠀⠀⣱⠏⡰⠛⣿⣿⣿⣿⣿⡗⠖⠐⠛⣧⢠⢶⣿⡿⢿⠿⢶⣄⣀⡀⣸⣿⠿⠯⠀⣐⣿⣿⣿⣿⣿⣿⢮⠟⠋⠍⠀⠉⠀⠗⡐⠛⢙⣆⣀⣤⣿⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣼⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣷⡿⠿⣿⣿⣿⣿⠿⣻⣿⣯⣵⣾⣿⡆
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣯⣝⣻⠿⠏⠀⠀⠀⠉⠀⠈⠐⣦⠀⠹⣯⣶⠿⢟⣛⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⢲⣟⠿⠿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣤⣀⠀⠀⠀⢲⣀⣀⣀⣀⣰⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣤⡼⠿⢷⣶⣶⣿⣿⠛⢻⣟⣋⣥⣤⣤⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣾⠿⢻⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠠⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⢛⣛⣛⣛⣛⣿⣿⡇⠀⠀⢰⢸⣿⣿⣿⣿⣿⣛⡛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠚⠃⠀⡀⢸⣾⣿⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣤⣶⣿⣿⣾⣿⣿⣷⣀⣀⣀⡀
⠤⠤⠤⢤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⡤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢷⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠖⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠲⡄⠀⢸⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠾⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣒⣀⣒⣀⣀⣀⣠⣏⣀⣤⣤⡤⠴⠶⠖⢺
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣴⣒⣖⣒⣶⣒⣒⣖⡒⣲⠖⠒⡶⠖⢤⠤⠭⢭⠭⢭⣍⠉⠉⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡾⢁⠾⢤⠾⢤⡾⠥⣤⠯⢬⡽⠭⢭⠿⠭⣽⢯⣉⣽⣋⡲⣾⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢠⠟⡠⠛⢲⠛⢲⠞⠒⡴⠃⢢⠞⠤⡴⠧⢤⠾⠥⣤⠮⣤⢞⡗⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢠⠟⠐⠿⠿⠿⠿⠿⠿⠿⠷⠾⠟⠖⠾⢷⣶⣿⣶⣺⣗⣲⡽⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⡿⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉
⠀⠀⠀⠀⠙⠛⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠦⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣿⣤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠉⠉⠉⠉⠑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡠⠀⠬⠀⠀⠱⠁⠀⠃⠀⠀⡠⣤⢼⠁⠀⠀⠀⠀⣴⠟⢉⣽⣿⣿⣿⣿⣷⣦⡀⢸
⠀⠀⠀⠀⠀⢂⡀⠀⠁⠀⠈⠀⠈⠂⠀⢘⡜⠉⢻⠀⠀⠀⠀⣼⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠶⠤⢾⡇⠀⢸⡀⠀⠀⢰⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠁⠒⠈⠡⡀⣀⠠⠀⠀⠀⡰⠓⢴⠉⢣⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣜⡗⠉⢣⠀⣠⠞⠁⠀⠀⠳⡀⠑⢄⠘⠿⣿⣿⣿⣿⣿⣿⣿⡿⢟⠕⠁⠀⢀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢋⡽⣦⡀⠀⠀⠀⠈⠢⣄⠙⠲⠬⣙⣛⣿⣿⣯⠭⢞⣁⠀⢀⠤⠏⠀⣸⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢵⡢⠖⠋⠈⠃⠉⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⣀⣠⠟⠚⡆⠀⠉⠻⠀⠀⠀⢱⣀⣀⣠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⢈
⠀⣀⣀⣀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⡴⠞⢀⣀⠤⢆⠀⠀⣀⠀⠉⣹⠁
⢀⣿⠏⠀⠙⠒⠒⠋⠿⣷⣄⠀⢀⠠⣤⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠙⠉⠁⠀⠀⠀⠉⠉⠀⠉⠉⠁
⢸⣿⠀⠀⠀⠀⠀⢰⠀⠀⠀⣊⠄⠈⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⣠⠎
⣼⣿⠀⠀⠀⠠⡠⠈⡀⠀⠈⢇⠀⠀⣀⣔⠁⠀⠀⠀⠀⣀⡤⠚⢇
⣼⡏⠀⠀⡪⣳⠓⠧⣽⠖⠲⠒⠓⠋⠁⠈⣉⡿⠟⠛⠛⢿⡄⠀⢸
⣿⡇⠀⠀⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠙⣄⠀⠀⠀⠈⢳⠆⠊
⣿⡇⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠈⠓⠄⣀⣠⠊
⠙⠛⠒⠉⠀⠀⠀⠀⠀⠀⠀⢾⣷⡀⠀⠀⠀⠀⡴⢅⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠪⡽⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠉⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣧⣄⣠⡼⠇⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣠⡾⠁⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠉⠉⠀⢶⡀⠀⠀⠀⠀⢀⣴⢶⣄⢉⡉⠉⠀⠀⢀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣄⠀⠲⣦⣸⡇⠀⠀⠀⠀⠈⠻⣦⡿⢸⣇⢀⡄⠀⣿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢷⣄⡈⠻⣧⡀⣶⣤⡶⠶⠶⣶⣦⠀⢹⣿⡇⢠⡟⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠻⣦⣈⡿⢻⣧⠶⠶⠟⣫⣥⡀⣿⣿⠃⢸⡇⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡏⠁⠈⠁⠀⠀⢸⣏⣼⠇⢸⣿⣀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⣡⣶⡌⣿⠉⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢿⣼⠟⢹⣆⣀⣤⣤⣴⡾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠉⠉⠉⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛




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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⣴⣷⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⣋⣤⣂⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣼⣿⣧⣿⣟⣿⠈⢿⣛⡧⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣿⣿⣿⠁⠏⢙⡿⠁⠀⠈⠀⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⡄⠀⢻⣧⣾⠾⢷⡦⢰⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠀⢻⣿⣿⣿⣤⠘⣏⠉⠁⠀⣠⣿⣿⢿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⢳⣿⣿⣿⣷⠎⢻⠟⠀⠠⣿⠟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⣤⣾⠀⠀⠀⢻⢤⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⣼⣿⡿⢿⣿⡃⠀⠀⢠⣿⡎⢿⣷⣞⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⣴⣾⣿⣿⣿⡯⠙⢿⣧⠀⢠⣿⣿⣿⡆⢻⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣿⣿⣿⣿⣿⣿⣷⡈⢦⡀⢠⣿⣿⣿⣿⣷⡾⠹⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣷⢻⣿⡄⠉⣿⣿⣿⣿⣿⣿⡇⠀⠙⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣶⣶⣶⣿⣿⣿⣿⣷⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠋⠀⠀⠀⠀⣀⣤⣤⣤⣤⣴⣶⣦⣬⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢠⣿⣿⣿⠿⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠁⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⣽⣿⡇⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⠇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⡿⠿⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣶⣾⣿⣇⠀⠀⠀⠀⠀⠀⣠⣶⣦⣤⣤⣤⣤⣤⡤⠀⠀⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⡿⠻⣿⣿⡟⠉⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⣠⣿⣿⠇⠀⢻⣿⣷⣤⣤⣤⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣶⣿⣿⣿⠟⠀⠀⠀⠉⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⡀⠀⠀⣀⣀⣀⢀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠚⠀⠂⠀⠉⠱⠟⠃⠈⠉⠋⠀⠛⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣞⠉⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡔⠒⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠐⡀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠁⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣋⡄⠀⠠⠆⠀⠀⠀⠀⠀⡂⣺⠦⠋⠄⠀⠄⠀⠉⢍⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⢀⢶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅⣀⣑⠂⠀⠀⠀⠀⠾⠟⠻⣷⡀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣷⠲⠀⠀⠀⠀⠀⠀⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⠂⠐⠒⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠐⠀⠀⠙⠁⡀⡀⠄⣰⠦⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠤⠖⢇⣠⣀⡤⣧⠀⠀⠀⠀⠀⢠⣅⣀⡤⠐⠛⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠑⠊⡀⢿⠃⡀⠀⠀⠀⣿⠉⠑⠋⠉⣉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⣀⣀⡀⠀⠘⠋⠙⠆⢉⠒⠛⠐⠁⢀⡄⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠴⠴⡾⠹⠤⣀⣰⣠⠖⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⣀⠀⠀⠀⣠⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠤⢤⡖⠒⠖⡇⠀⢀⠀⣿⠈⠑⠒⡋⠁⠈⠉⡦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠕⠀⡠⠒⠊⠉⠀⠔⠓⡴⣉⡊⠁⠃⠀⠒⠀⠀⠀⠐⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠰⠀⠀⠔⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣴⡃
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠴⠋⠁⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠂⠒⠐⠒⠒⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠐⠒⠒⠈⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡏⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣹⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⣏⡀
⠀⠀⠀⠀⠀⢀⡼⢋⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡙⢦⡀
⠀⠀⠀⢀⡴⠋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⢦⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⣤⠔⠉⠉⠉⠑⢤⡄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⡣⣴⣶⣶⣶⣶⣤⡻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡗⡍⠁⠀⠀⠂⠀⠉⡛⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣮⡑⠒⠀⠐⠒⣩⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠯⡉⠉⠙⠝⠫⠭⠉⠩⠭⠛⠙⠉⠉⣩⣝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠆⢹⡄⠁⡠⠤⠤⠤⠤⠤⠤⡄⣦⣸⠨⣿⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠈⠠⣼⣧⠀⡂⠀⠀⠀⠀⠀⠀⡁⣾⣟⠀⢙⣿⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠈⣻⣿⠃⠀⠈⠖⠀⠤⠄⠐⠎⠀⢿⣿⠀⠩⣿⣮⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣰⣾⣿⣿⢀⣾⡷⠠⣀⠀⢀⡠⢰⣿⣿⣿⣧⡀⠘⠽⣿⣟⡄
⠀⠀⠀⠀⠀⠀⣀⢌⡀⣀⣤⣿⡿⠙⣿⣿⠋⣄⣠⠉⠋⠋⢡⣀⡌⢻⣿⡟⠱⣀⠈⠸⢻⣯⠄
⠀⠀⠀⢀⡤⠐⠋⠑⣮⣿⣵⠗⠚⢉⣩⣿⣶⣦⣤⣤⣤⣤⣤⡴⢒⡋⠉⠉⠙⠲⠷⣄⢠⠖⠉⠓⠤⣄
⠀⠀⠀⣎⣴⠖⠂⢠⣿⠟⣁⣶⡻⢷⠿⠗⠤⠠⡴⠲⠖⠋⠑⠬⣻⢿⣿⣿⣷⣶⣤⣬⢿⡆⠀⠒⢶⣌⡆
⠀⠀⣔⣇⡱⣡⠆⡼⣁⠤⢐⣂⡹⡎⣆⠀⡄⡀⡀⢠⣀⣄⣄⡀⠈⠑⢯⡢⡀⢀⠀⠈⠳⡽⡄⢦⡱⣁⣗⡄
⠀⠀⠀⠘⠒⠻⣎⣞⣵⣫⡌⣶⣶⣷⣼⣿⠿⢿⢿⣛⠿⠭⠛⢧⠀⢠⣿⣿⣿⣾⣶⣴⣶⣧⣼⣾⡟⠚
⠀⠀⠀⠀⠀⠀⠹⣿⢿⣛⣃⡯⣽⣣⠵⠒⠚⠉⠉⠀⠀⠀⠀⠀⠑⠾⠿⠿⠛⠻⠭⢽⣛⣩⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣤⡤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣴⣶⡷⠶⢶⣶⣦⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⢿⣿⣿⠓⠒⠀⠀⠈⠉⠛⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠻⠃⠀⠨⠽⢶⡆⣤⠠⠿⠿⠿⢓⣴⡮⠞⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠆⠀⠀⠀⠀⠀⠬⠇⠂⠀⠀⠀⠠⡫⠋⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⡞⠉⠁⢿⣶⠀⠀⠀⠠⠠⠔⠡⠀⠀⠀⣿⠃⠀⠀⢀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠀⠀⢰⠁⠀⡆⠈⢿⣦⠀⠀⠀⣹⣦⣄⡀⠀⠀⢩⣘⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⡀⠀⠘⣤⠀⢰⠀⢠⣿⣿⢿⣿⣿⣿⡿⠋⠉⢉⡉⠛⡙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢱⠀⠸⡁⠀⠘⣆⠏⣼⣿⣤⣿⣿⠟⠀⠀⠀⢄⡈⢢⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡇⠀⠙⣄⠀⢻⣼⣿⣿⣿⠟⠁⠀⠀⠀⠐⢤⣹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⠆⢸⡿⠛⠋⢁⠀⠀⢀⣀⣤⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠴⠊⠁⠀⠀⢠⠈⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣧⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣛⣻
⠀⠀⠀⠀⠀⠀⠀⠤⣤⡾⠃⠀⠀⠀⠀⠀⠀⣀⣼⠅⠿⠿⠖⢠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣶⣿⣿⣿⣅⠀⠀⣀⣼⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣧⡄⢠⣾⣷⡄⠀⣴⠿⢷⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠐⠺⣿⣿⣿⣀⣽⣿⣷⠀⣿⣿⣿⡆⠂
⠀⠀⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠙⣿⣿⣿⣿⣿⣧⠙⠋⠉
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠿⠟⠋
⠀⠀⠀⠀⠀⢠⠏⠻⠿⠿⢻⣿⣿⣾⠿⣿⠋⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣴⡀⢸⠀⠀⠀⡀⢉⡩⠏⠀⢸⡿⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣠⣾⣿⣷⠈⠀⠀⠀⠀⠉⠀⠀⠀⢸⠇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⠀⠀⠰⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠙⠛⢿⣿⣿⣿⣤⣤⣤⣤⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣸⣿⣦⢀⠀⠀⠀⠀⢀⣤⣅⣡
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣻⣿⣿⢩⠇⠀⠀⣶⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠻⠉⠉⠉⠉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠣⢿⠀⠀⠀⠸⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠶⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢸⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⢻⣯⣿⣿⣿⣿⣿⣿⣿⣿⡗⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢇⡽⢻⠿⠿⣿⠿⠻⣫⡿⠁⣿⣠⠀⠤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠃⠀⢀⣞⠘⣿⡇⠙⠶⠖⢠⠀⢶⠟⠁⢸⣽⣿⣿⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣠⡞⡯⠀⣄⣀⣀⠀⠀⡘⣶⡘⠀⠀⡀⣼⣿⣿⢠⡄⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⣰⣅⣽⠀⠀⠀⠀⠀⣤⠘⡇⢻⠃⠀⠀⠀⢀⢀⣿⣿⠁⢀⣼⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⢫⣹⣿⡯⣍⣉⡁⢰⠇⠀⢹⣿⠃⠈⠉⠉⠉⢉⣿⣿⡶⠋⠁⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢲⣻⣶⣿⠏⠀⢸⡍⠹⠃⠀⠀⠈⠁⢠⣦⣿⣿⣿⣿⣿⣿⠀⣀⣠⣤⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠉⣿⡟⠀⣧⡘⣧⠀⠀⠀⢀⣶⠀⠈⠡⢿⣿⣿⣿⣿⡇⣺⣭⣤⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⡐⣿⠁⠀⠀⠹⢄⠀⠀⠀⢸⡷⠤⠐⢰⣿⣿⡿⠿⣿⠇⠘⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠉⠁⠀⠀⠃⠀⣮⣿⡆⠀⠀⠒⠚⢻⣿⠿⠋⠀⣰⡟⠀⠈⠁⠐⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣉⢹⡈⠀⠀⢀⠀⣤⣿⣷⡁⢦⡀⠐⣴⠞⠁⣠⣴⣾⣿⣿⠀⣰⡷⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠘⠃⠀⠀⠄⠤⠽⠻⠿⢷⣀⣏⢹⣧⣴⠞⠁⠀⠀⣠⢶⠀⠛⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⣹⡟⠁⣈⡉⣉⡉⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡍⠀⢠⣿⡧⣿⠁⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⢘⣿⠀⠏⠀⠀⠙⠛⠛⠃⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠺⠏⠀⢠⠀⣀⣀⠀⣀⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣳⣿⠀⠀⢈⡛⠻⠿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣄⣐⣿⣿⢿⣆⠀⠀⠉⠉⠛⠛⢋⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡙⠻⣷⣶⣾⣿⠀⡄⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡙⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⠇⠀⠀⠀⠀⣶⣦⣤⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠛⠀⠀⠀⠀⠀⢸⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠟⠃


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⡄⠠⠤⠀⠄⠠⠀⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠒⠀⠉⠀⠁⠀⠀⠐⠒⠚⠛⠁⠉⠉⠙⠒⡠⠤⢄⣤⡭⠖⠦⠀⠀⠀⠀⠀⠀⠀⠠⠈⢁⠞
⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⠀⠀⡀⠁⠀⡀⠀⠀⠀⠀⠀⠛⠠⡀⠀⠐⠀⠀⠀⠈⠂⠀⢀⣈⠀⠀⠒⠤⠤⡀⡄⠂⠀⠀⡠⠃
⠀⠀⠀⠀⠤⢀⣀⣈⣡⣴⣢⠀⠂⠄⠀⠁⠀⠀⠄⠒⠐⠒⣀⠀⠀⠀⠀⠀⠀⣀⠏⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠴⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠈⠃⠒⠀⠨⠄⠤⡄⠠⠀⠀⠐⠖⢈⡀⠀⠠⠐⠤⣄⣂⡱⠖⠃⠀⠘⠻⣋⣀⠀⠀⠀⠀⠉⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⣣⡴⠎⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠈⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀⠀⠀⠀⢀⡀⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢒⣤⡴⣧⣤⣤⣤⣤⣤⣤⣤⣾⡧⠴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⢰⡅⣀⣀⣤⡄⠀⠀⠈⢉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠸⢄⣘⣿⡷⠃⠀⠠⢄⣀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⢸⠇⠀⠲⠖⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⠸⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢀⢀⡇⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⠷⠶⠶⠶⠶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠉⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣼⣀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠊⣨⡟⢻⣏⠉⢹⡇⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠁⢹⡏⠙⢧⣈⡇⡤⠀⠀⠀⠀⢰⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡻⡇⠀⢀⠸⡇⠀⠀⢹⡟⠀⠀⠀⠀⠀⠈⢣⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⡸⠁⡇⠀⠸⠀⡇⠀⠀⣾⠀⠀⠀⠀⢀⠀⠀⢸⠀⠸⡀
⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⠇⠀⡇⠀⡄⠀⡇⠀⢰⠁⠀⠀⠀⠀⠸⠀⠀⠈⣇⠀⣿
⠀⠀⢀⣀⣹⣿⠿⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⣿⠀⠀⡇⠀⠇⠀⡇⠀⡜⠀⠀⠀⠀⠀⠰⠀⠀⠀⣿⠀⢸⡆
⠀⣴⣶⣿⡇⢉⣀⣀⠘⣆⢰⠛⢦⡀⠀⠀⠀⠀⣠⠖⠉⠁⠀⣠⢺⡇⠀⠀⡇⢀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠀⠈⣧
⣷⢿⡟⠁⠀⠻⣄⡈⠓⢾⡿⠀⠠⡈⠳⡶⠖⠋⠁⠀⢀⡴⠊⠁⣼⠁⠀⠀⡇⢸⠀⠀⡇⣸⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⣿⠀⠀⢸
⠙⢦⡀⠀⠀⠀⢠⡙⠆⣼⠃⠀⢀⡇⠀⢇⡀⣠⡴⠖⠉⠀⠀⠀⡿⠀⠀⠀⡇⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠘⡄
⠀⠀⠹⣦⠀⠖⣩⠁⠀⢥⣤⡴⣏⠑⢦⣜⠉⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⡏⠀⠀⠀⣇
⠀⠀⠀⠈⢣⡄⠁⠀⠀⠀⠀⡤⠈⠙⢦⣈⠑⠦⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠻⣄⠠⢄⠀⠈⠀⠀⠀⠀⢻⡟⠃⠸⠷⠀⠀⢠⡏⠀⠀⠀⠀⡟⢧⣄⠀⢸⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡜⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠈⠙⡟⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⡇⠠⢤⣀⣀⡞⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢰⡇⠀⠀⢨⡉⠛⠳⠦⢤⣼⣇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⠃⠀⠀⠀⠈⠓⠶⢀⣀⠀⠀⠉⠛⠛⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⡇⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⡇⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣸⡇⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⠴⠖⠚⢙⡇⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠤⠤⠤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠏⠀⠉⠙⠛⠓⠶⠶⠦⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠶⠶⠶⠖⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠰⠶⠖⠀⠉⠉⠁⠀⠀⠀⣠⣼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠶⠶⠶⠶⠶⠶⠶⠒⠚⠛⠋⠀⢘⣧⡿⠷⠶⠶⠶⢀⣤⡤⠤⠤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⠒⠒⠉⣉⣉⠁⢀⣠⣤⡤⠶⠚⠉⠹⣤⣤⣤⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠄⠀⠀⣀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢰⠆⠀⢰⠀⢸⠀⠀⠐⣦⠀⠀⠀⠀⠀⣿⡎⠀⠀⠀⠘⢧⡄⠀⠀⣾⠀⠀⠀⢀⠀⠀⠀⠀⣠⠃⠀⠀⣠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠃⠀⠀⢸⡇⠀⠃⠀⠀⠈⢳⠀⠀⢀⠀⢹⡇⠀⠀⠀⠀⠘⠷⠀⢰⡟⠀⠀⠀⢸⠀⠀⠀⢀⣿⠆
⠀⠀⠀⠀⠀⡀⠀⠀⠙⠀⣠⠀⠀⠤⢘⣧⡀⠀⣄⣀⡀⠈⢴⢀⣾⠀⢈⣷⠀⠀⣷⠆⠀⠀⣠⣾⠁⠀⠀⠀⢘⣀⡤⢀⣾⠃⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠙⠛⠲⢿⡿⠀⠀⠀⠀⢻⣯⠰⣿⣏⠀⠀⢸⡿⠃⠀⠘⣿⠀⠀⢿⠀⠀⠀⣸⣯⠀⠀⣧⠀⣼⠟⢡⣾⠃⠀⠸⠂⠀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠲⣤⡀⠀⢿⠇⠸⢯⡁⢀⠀⠀⢠⡀⠀⣿⡇⣀⢸⡄⠀⢰⡿⠃⠀⠀⠾⠟⠁⢠⣿⠃⠀⢀⡤⠀⠈⠁⣠⠎
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠀⢀⣀⠀⠻⣦⣌⢠⣦⣀⡁⢸⣿⣶⣿⣷⠿⣿⡿⠿⢶⡿⠠⣿⡇⢸⣶⣾⣴⣤⣾⡿⢹⣆⣀⣀⠄⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠈⠓⠶⣦⡄⢠⣈⣿⠀⠀⠈⢛⣽⣿⠟⠛⡈⠉⠁⡀⢀⡀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠉⣿⡿⢿⣿⠁⠀⣿⡏⠁⠀⢠⣾⠃⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⡀⣀⠞⠻⡟⠆⠁⠀⠀⠀⠁⢀⣀⡀⢀⣀⡋⠀⠀⠀⢀⣐⡀⠻⢁⠀⠙⣇⣠⠎⠛⢀⣤⣿⠟⠁
⠀⠀⠀⠀⢀⣽⣄⡀⠀⠀⢹⠻⣤⣿⠟⢣⡦⠀⠇⠀⠋⠀⢀⠀⢼⣿⣿⢳⣶⣿⣿⡿⠲⢄⣀⡀⠀⢰⡆⠈⠐⢹⣿⣧⠸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠙⢧⠒⢶⣴⡆⢻⠧⠀⠰⢇⡼⠶⠖⣴⠆⠀⠀⠠⠈⠉⠋⠉⠉⠠⠀⠀⠘⢿⠋⠉⢛⣿⣼⡏⠉⠺⡏⢸⣿⠃⢠⣆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠙⠓⣿⡀⢹⡀⢸⣷⠤⠞⠋⠀⠐⣤⠄⣀⣬⡤⣤⣄⡀⠀⠀⠀⠘⠷⣤⣤⡆⠹⣷⠤⠾⢧⠈⣿⠀⡿⠁⠀⢀⡠⠊
⠀⠀⠀⠀⢀⠀⠀⠀⠀⢳⣦⠀⠻⠿⠈⢷⠘⠁⡆⠀⠀⠀⠀⠀⠸⠿⣿⣵⡿⠿⠓⠀⠀⠀⠀⠀⠀⠀⢸⣄⡀⠀⠀⢿⣀⠁⣘⠁⢀⣰⣟⣁⣀
⠀⠀⠀⠀⠘⠿⣟⠀⠀⢦⣿⣷⠇⡤⠀⠀⢿⠀⠀⠀⠀⠀⠈⢀⡀⠀⠠⠀⠀⠖⠀⠀⡤⠀⠠⠀⠀⠀⢸⠟⠀⠀⠀⣸⣃⣴⡿⢿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⣸⠧⢀⣀⣬⣿⣧⡇⠀⠁⠀⠀⢀⣀⣠⣤⣄⠉⠉⠠⣶⣿⣽⣶⠜⠂⢀⣠⣤⣶⣤⣤⣬⠃⠀⠀⠄⠹⣿⣿⡤⢘⣁
⠀⠀⠀⠀⠀⠀⣠⠞⠋⡉⠛⠾⠁⠈⢋⣴⣾⠛⡛⢋⡁⠙⠉⢻⣆⢠⡌⠉⠉⠀⢀⣴⠛⠁⠀⠘⡀⣀⠉⠿⢷⣆⠀⣀⡉⣿⡇⠈⠉⠑⠂⢀⣀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⢦⠀⡆⢰⣿⠁⣀⠴⠿⠛⠛⠛⠛⢧⣸⣿⣄⠀⠀⣴⡏⠡⠞⠛⠛⠛⠛⠛⠛⢦⣀⣹⠆⡈⣸⣿⠄⣀⣿⠿⠿⠀⠁
⠀⠀⠀⠀⠀⠀⠘⠛⢧⡀⠈⢻⣄⠀⣿⢠⣷⠶⢶⣶⠶⠶⠶⣶⠇⢸⣿⡄⠈⢿⡇⢸⣟⠻⠿⠷⠶⠶⠶⣶⠇⣬⠐⢀⣿⣡⡾⢋⡤
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢻⣄⢢⣽⢯⡙⢳⣭⡶⢶⣦⡴⠶⠶⣁⡴⠋⠀⣶⣄⢂⠙⢿⣭⣛⠒⣶⣶⠶⢞⣿⣾⠟⠋⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠘⠛⢿⠏⠀⠈⠙⠻⣿⣤⣤⣳⣦⣼⣋⣠⣤⣼⣧⣿⣬⣤⣀⠙⠽⠛⠛⢿⠟⠛⠋⠁⠀⢠⠄⢹⡏⠁⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠲⣄⡀⠶⠀⠀⠀⠁⠀⠉⠀⠉⡄⠈⢉⣿⣟⣿⣿⣍⠠⠃⠀⠀⠀⠀⠋⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠁⠈⢣⡀⠀⠁⣀⣀⡀⠈⠃⢀⡔⢀⠿⣿⣿⣿⣿⣿⣷⣦⣀⡀⠀⢀⣶⣶⣶⡤⠀⠓⠀⡀⠀⠀⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡀⠙⠷⠛⠋⢻⣿⣄⣤⠜⢃⡿⣿⣿⡿⢻⣿⣿⡿⠙⠿⣿⣦⣼⣿⡏⡸⢷⣄⣀⠀⡀⢀⣴⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⢤⡤⣄⣸⣶⣤⣤⡱⣯⣀⣴⣿⣏⣉⣩⡿⠿⠉⠃⣹⡿⡁⢿⡏⢹⣧⡀⠀⠉⠻⣄⣨⣿⣿⠯⠈⠛⠩⣽⡏⠉⠛⢻⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠑⣻⣀⠀⢹⠟⠁⠀⠀⠈⣹⡟⢹⣿⣇⣀⡀⢰⡉⠀⠁⣸⠀⠀⣾⡁⠀⣀⣴⣿⣿⠻⣅⠐⠀⠀⠘⠁⠀⢤⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡶⠀⢀⠀⠸⣦⣴⠋⢹⣸⣿⡄⠍⠹⣿⡟⣿⠻⣿⡶⡏⢿⣷⣾⣿⣿⣿⡇⠀⢈⣷⡆⠀⢀⠀⠀⣼⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠓⣵⣈⠁⢠⡾⠉⠀⡼⠘⣿⣿⢶⡾⣫⣁⣠⡀⢹⣇⢄⣸⣿⡿⢿⣸⣿⡆⡜⠡⠄⠻⣆⠀⣠⡾⠁⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢻⡲⡾⠁⠀⢀⣄⠂⣿⣿⢻⣿⢿⣿⠀⡍⠀⢿⠈⣿⡀⣴⡀⢣⣼⡃⢸⣆⠈⠀⠙⣷⠋⠠⠄⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⡴⢞⣢⡄⠘⠉⢂⡀⠁⢼⡏⠊⢹⡷⠻⣷⣸⠆⢉⠨⠃⢸⣄⡟⣰⠃⠀⠠⣶⣾⣉⣷⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡼⠃⠚⠛⢳⡄⠀⠀⢿⣲⠘⠁⠀⠀⣀⠀⡝⢻⠀⢀⠀⠀⠈⢡⡈⠋⠐⢀⡼⠋⠀⠀⠙⣷⠶⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢻⡇⠀⠀⠀⢈⠛⣇⡀⠈⣿⡇⢈⠉⡁⠰⢤⣷⣼⠒⠠⡤⠀⠀⣽⠟⠋⣰⠍⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠙⠀⠈⠛⣦⣼⠀⠀⠐⠀⠀⣾⠁⠈⡗⠀⠀⠠⠀⠹⣧⡜⠍⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⢀⡀⠀⠃⠀⠘⢻⡟⠁⠀⠈⠀⠀⣀⢿⣧⠀⠀⠀⠀⠀⠈⠀⣸⡇⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⣠⡼⠃⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠙⢷⣄⠀⠀⠀⠀⢡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣭⣤⡴⠾⠋⣀⣀⠀⣠⣤⣿⣧⡀⠀⣐⣣⡀⠀⣰⢿⣥⠀⣤⠀⣤⡝⠛⠲⠶⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡀⢀⡄⢠⡀⠀⠀⠁⢠⡴⠉⠀⣠⠙⢿⣺⣿⠽⣿⣞⠉⢠⠅⠀⠐⠶⠄⠀⣤⠀⠀⠛⡂⠠⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠉⠀⠛⠀⠠⠴⠶⠴⠾⠳⠀⠙⢩⣶⡏⠀⠐⠛⠛⠖⠛⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣹⣟⣹⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿