⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡒⣤⣤⣀⣀⣠⢀⡀⠀⢨⣿⣧⠹⣛⣦⣤⣄⣀⣀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣀⡐⠆⠀⠀⠀⠀⠐⢦⡜⠀⠀⠀⢿⣱⢯⣝⡻⢿⣿⣿⣶⣼⡻⣿⣷⣝⢮⢷⣿⣿⣿⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢯⣞⡽⣭⣛⢿⣻⢿⡿⣿⣽⠿⣆⠻⣛⣿⢿⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡾⣵⢻⡼⡮⢟⣷⣿⣷⣿⣿⢿⣿⣭⠿⣞⣷⣻⣮⣽⣻⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢣⠀⠀⠀⠀⠀⠈⠹⣷⢯⣳⣽⣾⣿⣿⣻⠟⣿⠿⣿⣻⣷⢻⣯⣯⣻⣿⣿⣿⣻⣿⣿⡿⣾⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠀⠀⠀⢀⣘⣯⡽⣛⣯⢿⣹⢿⣦⡀⢀⢌⡽⠟⢣⣿⣿⣿⣟⣳⣿⣭⣻⣿⡻⢭⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡀⠀⣙⢿⣽⣿⣿⣿⣿⣯⣿⣿⣿⣾⣿⡹⣧⢽⣿⣻⣾⠿⢻⣿⣷⣿⢽⣫⢻⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣫⡽⣽⣳⣽⣻⣿⣷⣯⣿⣿⢿⣷⣜⠻⣿⣟⢧⣾⣯⣛⣷⠃⠖⠀⢠⣽⣿⢎⣵⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣷⢾⡿⣽⣿⣷⣿⣿⣶⣞⣿⠟⣛⠿⣿⢿⣷⡝⣮⣷⣾⣧⣻⣿⣿⣶⣼⢿⡿⢳⣞⣿⣿⣿⣻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣯⣿⣿⣿⣯⣿⣿⣟⣿⣾⣭⠻⣽⡒⢾⢦⣞⠆⣙⠿⡿⢷⡷⢽⣮⣿⣿⣷⣗⢯⡿⣿⣽⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⢿⣷⣯⣿⣿⣷⣿⣿⣯⣿⣿⣿⢿⣻⣦⡟⣿⡾⣞⣻⣿⣿⣿⣿⢋⢦⡛⣬⢱⣾⣿⣾⣻⣽⣿⣿⣽⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣻⣿⣿⣿⣞⣿⣿⣽⣷⣿⣿⣟⣿⣿⣿⣿⢿⣯⣟⡿⢿⡼⣿⣿⣏⣎⢗⣺⣶⣿⣿⣻⡆⢻⣾⣿⢿⣿⣯⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣟⣟⣾⣟⣿⣿⣯⣛⢿⣿⣿⣽⣿⣿⣟⣿⣿⣿⣿⢿⣿⣿⣷⡷⣝⣪⣾⣿⣿⣿⢿⣳⡿⣤⢻⣟⣿⣿⣿⡟⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣯⣿⣿⡽⣿⣿⣟⣿⣟⡾⡼⢿⣟⣿⣾⣿⣿⢿⣾⣿⣿⣿⣯⣿⣿⢮⣽⣿⡿⣿⣾⢿⣽⣳⣮⢧⠍⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⣿⣿⣯⣿⣿⣯⣽⣿⣿⢿⣿⡿⣷⣯⣟⠮⣹⢬⣛⠿⣿⣿⣿⣻⣾⣿⡿⣿⣿⣿⣿⣿⣿⢯⣿⡟⡿⣿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣷⣿⡿⣿⣻⣿⣿⣿⣿⣿⣭⠓⠀⠸⢼⡻⢶⣭⢻⠟⡿⢿⣿⣿⢿⠷⣯⣛⢮⢟⡶⣻⣷⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣼⢻⣿⣿⣾⣿⣟⣿⣿⣿⢿⣷⣿⣻⠞⠀⠀⣠⢞⡬⣟⢳⡞⣶⣿⣿⣷⡼⣬⣭⢻⣵⣫⣟⣾⣿⣿⣳⣿⣟⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡹⢎⣿⣿⣟⣿⣽⣿⣿⣯⣿⣿⣿⣟⢯⣴⣦⠒⣿⢸⣇⣟⣺⡝⣿⠿⣟⠿⢹⣃⣚⣫⡟⣿⣿⢿⡟⣾⣽⣿⣷⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢚⣽⡗⣼⣿⡿⣿⣿⢿⣷⣿⣿⣿⣽⣮⠟⣿⣿⣷⣾⣦⣧⣼⣤⣷⣼⣷⣾⡿⣿⠻⣿⣛⣵⣿⣾⣿⣾⣿⣿⣯⣿⣟⣷⡄⢀⣠⣤⣤⣶⣶⣶⣤⣤⡀⠀⠀
⠀⠀⠀⢬⢷⡝⣦⢿⣿⣿⣿⣿⣿⢿⣷⣿⣿⣿⣿⣶⣭⣩⣍⣛⣙⣛⡛⣋⣯⣝⣮⣷⣿⣿⣷⣿⣿⢿⣻⣽⣿⣿⡷⢿⣻⣽⡼⡿⠿⠟⠻⠻⠟⡿⣿⣿⣿⣿⣦⠀
⠀⠀⠀⠈⠷⢯⡼⣼⣻⣿⣾⣿⣻⣟⣻⣽⣳⣟⣻⡿⣿⡶⣝⣮⢷⣽⣾⣿⣿⣿⣿⣿⢿⣿⣻⣿⣿⣿⢿⣛⣽⣶⣿⠿⣛⣭⣗⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⣿⠀
⠀⠀⠀⠀⠀⣀⣹⡔⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣿⣿⣷⣿⣿⣿⣿⢿⣛⣧⣽⣾⣿⠻⣭⢶⣿⢟⡵⢯⡴⣄⣤⣄⣤⣀⣄⣠⣼⣷⣿⠟⠀
⠀⠀⢀⣴⡾⡙⠂⢡⠻⣿⣟⣿⣿⣷⣿⣿⣾⣿⣷⣿⣿⣾⣿⣿⣻⣿⣿⣿⣿⣽⣿⢿⣞⣭⣷⣿⣿⢟⣯⢶⡻⣽⡟⣵⠞⠭⠳⣛⠾⡽⢿⠿⡿⠿⢛⠟⡉⠂⠀⠀
⠀⢠⣿⣿⡓⠀⠀⠀⠳⢻⣿⣿⢿⣾⣿⣿⣽⣿⣿⣽⣿⣟⣿⣽⣿⣿⣷⣿⠿⣛⣭⣾⣿⣿⢟⣫⣶⣿⡃⣯⢗⢋⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢺⣿⣿⡄⠀⠀⠀⠀⠉⢿⣻⣿⣿⣷⣿⣿⣿⣾⣿⡿⣿⣿⡿⣿⢽⣻⣼⣿⣿⣿⡟⣻⣼⣾⣿⣿⡿⣅⢫⣤⣿⣿⣆⣀⡀⢀⡀⣀⢀⣀⣤⡴⣂⠀⠀⠀⠀⠀⠀
⠀⠸⣿⣿⣷⣀⠀⠀⠀⠀⠀⠻⣯⣿⣿⣽⣷⣿⣿⣻⢿⣿⣏⣿⣷⣿⣿⣿⣻⣹⣶⣿⣿⣿⣟⣯⣷⣿⣇⠾⢯⣿⣿⣿⣿⣿⣿⣾⣷⣿⡿⠛⠁⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠈⠻⢿⣿⣷⣾⣶⣷⣶⣦⣴⣯⣿⣽⣻⣾⣷⣿⣿⣟⡿⢿⡻⣝⣮⣗⣿⣿⣿⣯⣿⣿⣻⣿⣿⢿⣯⠎⣇⠀⢻⣿⣿⡽⢿⠻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠙⠛⠋⠛⠉⠛⠙⠉⠋⠙⠈⠀⠀⢟⣳⢯⣞⣿⣮⣟⣷⣿⣷⣿⣿⢿⣻⣿⡿⡜⣷⠀⠀⠈⠙⠾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀