⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣭⣭⣭⣭⣭⣿⣿⣷⣶⣶⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡻⣽⣻⢿⡿⣿⢿⡿⣿⠿⡿⠿⠿⠟⡿⠿⠿⠟⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢡⠊⡕⠢⠙⠊⠓⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠄⢃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⠢⠘⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢰⠈⡆⢉⢀⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢠⠃⡜⢠⠃⡰⢈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢰⢡⠚⡌⢆⡱⠐⢂⠔⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣌⠣⣎⠳⣉⠖⠤⡉⠆⡄⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠰⣌⠳⣌⠳⡥⣊⠵⣈⠒⡠⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠣⡝⣬⠳⣬⢳⡱⢥⠒⡤⢃⠔⠡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠿⠿⢶⣄⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡵⣙⢦⣛⡴⣣⣷⣮⣷⣰⡣⡜⡠⢁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢉⣀⠤⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡘⢶⣩⢶⣻⣾⣿⣿⣿⣿⣿⣿⣿⣷⣥⣊⡄⢁⠀⡄⠐⣠⣴⠞⣡⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⡙⣶⣹⣾⣿⡟⡿⣽⡿⣏⠉⠙⠛⢿⣿⣿⣿⣷⣷⡇⠙⠫⣃⠈⢴⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢢⡝⡲⣟⠫⠑⠈⠙⣿⣽⣾⣷⣶⣶⣤⣝⣿⣾⢯⣿⠇⠀⠀⠀⠠⣿⣿⣿⣿⣿⡗⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⡟⢦⡙⠵⡀⠂⠀⠀⠀⠈⠻⣿⣿⣿⣿⣏⠜⠻⠟⣿⠟⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠧⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⣟⢢⣙⢢⡑⡬⢠⢀⠀⠀⠀⠀⠉⠉⠋⠉⠀⠀⢈⣿⣁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠳⣬⣳⣽⡳⣏⣦⢃⡆⣄⡀⢀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣦⣄⡀⢰⣦⣤⣀⣀⢀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⣜⡳⣧⣟⡾⣽⣻⡼⣯⣿⣟⣿⣿⣿⣿⣿⢿⡻⣿⣿⣿⣾⣿⣿⠿⠊⠉⠛⠻⠿⠿⢾⣷⣯⣟⣶⡡⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⣎⣷⣻⢾⣻⡽⣷⣛⠦⠙⢿⡟⣿⣾⣝⡮⢓⠙⠓⠉⠉⠉⢣⠀⠀⠀⠀⠀⠀⢀⣀⠴⠈⠙⢛⠮⡓⢌⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣸⣟⣮⣟⡿⣽⣻⢷⣏⠷⡀⠀⠹⣶⢯⡿⣿⣿⣾⣶⣶⣶⣤⣤⣴⣶⡶⠿⢟⣛⡿⠁⠀⠀⠀⠀⠂⠡⢂⠂⠄⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢾⣿⡽⣾⣽⣻⣽⣯⡿⣞⣯⢵⢢⢄⠩⣛⢾⣳⢯⣟⡿⣷⣾⣭⣽⣴⣦⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⣿⣹⢷⣏⡿⣷⣏⣿⢿⡾⣏⣷⡎⣷⠈⢶⠉⡾⢉⠿⡉⠿⠹⠏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣯⣟⣾⢿⣯⢿⣾⣻⢿⡽⣾⡽⡾⣝⡦⡑⡄⠃⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⢣⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣟⣿⣿⣾⢯⣿⢾⣟⣷⣯⢿⣻⢷⣻⣽⢯⡷⣝⢦⢣⡐⠌⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⣠⣦⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣿⣟⡾⣿⣿⣟⣯⣿⣟⣾⡽⣿⣽⣻⡽⣾⢯⣟⣯⣟⣧⡝⣦⣑⢂⠆⡠⢀⠀⡀⢀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⣿⣿⠇⠀⠀⠀
⠀⠀⢀⣤⣾⣿⣿⣿⢽⣿⣿⣿⣽⣾⣯⢿⣟⣾⣽⣻⣽⣻⣯⢿⣾⢯⣿⢶⣯⡞⣽⣰⢃⡮⣐⡆⢦⡱⢈⡐⠠⠀⠄⡀⠄⠠⠀⢀⠀⠀⠀⡀⢤⣿⠃⢀⣴⣿⣿⣤⠀
⠀⣴⣿⣿⣿⣿⠿⢻⣧⣛⣿⣿⣿⣷⣿⣟⣿⢾⣳⣟⡾⣷⣻⣯⢿⣻⣽⡿⣞⣿⡷⣯⣟⡾⢧⣛⠣⢆⠡⠄⠡⡈⠄⡐⠠⢀⠡⠀⠠⢈⠐⣸⡿⠋⠀⡾⠛⠿⣿⣠
⣾⣿⣿⣿⠋⠁⠀⠀⢿⣿⣮⠹⢿⣿⣿⣿⣯⣿⡿⣽⣿⡽⣷⣻⣟⣯⡷⣿⣻⢷⡻⢯⠽⣍⡳⣌⠣⢎⡔⢩⠐⡁⠆⡐⣁⠂⠤⢡⢁⣦⡿⠋⠀⣠⡾⠀⠀⠀⠈⠛
⠛⣿⣿⣿⣷⣄⡀⠀⠈⢿⣿⣷⣮⣭⡿⣿⣿⣿⣿⣿⣾⣿⢿⣽⣾⢯⣟⡷⣯⢷⣻⣭⢻⡼⣱⢭⡙⣦⡘⡥⢚⡄⣓⠰⢤⣉⣶⡵⠛⠁⠀⠀⣠⣿⠁
⠀⠀⠙⠿⣿⣿⣿⣶⣤⣸⣿⣿⣿⣿⣷⣭⢟⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⢷⣯⢷⣝⣳⣎⢷⡲⣝⡼⣣⣞⣴⠿⠖⠋⠁⢀⣤⣤⣶⣿⣿⠃⠀⣀⣤⣶⣿⠁
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣷⣮⣽⣝⡻⢛⠛⠿⠿⠿⠿⢿⡿⣿⠿⠾⠷⠿⠟⠛⠛⠛⠉⠁⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⡷⠛
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣠⣀⣀⣀⣀⣠⣶⣦⣄⣄⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢹⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⡟⠉⠻⠏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⢿⣧⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⢹⣿⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿�