⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⡆⠀⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡐⣄⡹⣿⣿⣶⣤⣼⣿⣰⣿⣷⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⡟⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣦⣄⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⣀⣀⠀⠀⠈⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⣦⣼⡉⣠⣾⣤⣿⣿⣿⣿⣿⣿⡟⢻⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⢿⣿⣿⣍⣉⣽⣿⡽⣟⣿⣿⣿⣧⠢⡉⢤⣼⣿⣿⣿⣿⣿⣿⣿⣿⡙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣟⣿⣟⣿⣻⣿⣿⣭⣿⣿⣿⣿⣿⣿⣏⡿⢷⣺⣿⣿⣿⣿⣽⣾⣻⣿⣿⣿⡇⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⣻⣿⠟⠂⡤⢃⡴⣾⡽⢋⠵⢾⣿⣯⣙⣿⡿⠓⠚⠛⠒⠢⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣷⣿⣾⣷⣿⡀⣀⢰⣾⣹⢁⡶⣉⢾⣀⣿⣾⣿⠏⠀⠀⢁⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⣿⣍⢸⣿⣿⣿⢱⣿⣿⡁⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠡⢀⠀⠀⠀⢄⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢿⣿⣿⣿⣷⣿⣿⡟⠸⠿⠯⢁⣿⣿⣿⣿⣽⣾⣽⣿⣿⣿⣶⢿⣻⢦⣄⣸⣧⣄⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⢯⡟⠚⠛⠞⢿⣳⢦⡞⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣯⠷⢾⣿⣿⣿⡿⠋⣷⣞⣟⢟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣤⣤⢭⣥⣶⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠋⠀⢨⣤⠿⠚⠉⠁⠀⣴⣿⣟⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠿⡿⢷⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠘⣯⣵⡻⣗⠶⠎⠁⢰⠔⣶⡖⢾⣟⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⡿⠉⠈⢼⡄⠀⠀⠀⠀⠀⠀⣠⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢨⣿⢻⣷⡙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⣻⣿⣿⣿⣿⣿⡿⠊⠉⢻⣿⠁⠀⠀⠘⣇⠀⠀⠀⠀⠀⢀⣠⣦⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⣴⣿⣦⣝⠣⣴⡷⣬⣛⠟⠻⢿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣼⡇⠀⠀⠀⠀⡾⠀⠀⢀⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⢯⣿⣿⣿⣿⢩⣿⣿⡿⢳⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⣈⠧⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡤⠾⣿⣿⢯⡷⣯⡿⣛⣩⠺⣿⣿⠇⣾⣿⣿⢿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣞⡿⣽⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⣀⣄⣤⣶⣶⣾⣷⣿⣿⢿⣿⣷⣿⣿⣻⢾⡿⢷⡿⣿⢻⠉⠛⠲⣬⣏⣹⣿⣿⣿⣇⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡷⣟⣿⣻⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⢀⣠⣶⢿⠿⠟⠛⠛⠛⠛⠛⠛⢹⣷⣾⣿⣿⣟⣧⠿⣽⣿⣾⡶⣿⠀⠀⢐⡿⣼⣏⣟⡷⢯⣟⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⣾⣿⣿⣿⣿⠟⠉
⠀⠀⢠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣞⡿⣽⢰⣶⣶⡷⡀⠀⠬⣟⣶⢿⣎⠻⣯⡇⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡿⠋⠁⠀⠀⠀
⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⢿⣯⡽⣼⣿⣭⣟⣽⣌⣶⣆⣻⣯⣟⣷⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣽⠟⠉⠀⠀⠀⠀⠀⠀
⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣽⣻⣿⣿⣿⣿⣿⣞⠟⠇⣫⣥⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⢿⣽⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⠿⠻⢻⣿⣿⣿⣿⣿⣿⣿⣻⣽⣾⡱⣾⣾⣿⡿⣿⣿⣿⣟⣿⣯⣿⣿⣿⣿⢿⣿⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠘⣭⣟⡿⠛⠉⠁⠈⢿⣿⣷⣿⣿⣿⣧⣇⠿⢻⣿⣿⣿⣿⣿⡿⣯⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡯⡄⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠘⢮⣽⣷⣿⡟⠷⢁⣴⣼⡋⢻⣿⣯⣿⡽⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡕⠀⢂⠂⡹⡀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣧⠀⢿⡉⣿⣇⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⡲⣎⡆⠀⠘⡄⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠘⣥⣿⡡⠙⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢳⢎⡇⠀⠀⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⡗⠸⢇⣝⠁⠀⠀⠂⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢻⣎⡇⠆⢰⡛⡇⠀⠀⠀⠀⠀⠙⠻⠻⡧⠖⠉⢳⣧⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡾⣸⣶⡏⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⣷⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⢳⡽⣧⣿⢞⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢽⡦⠀⠀⠀⠀⢂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢣⣏⢶⡹⣿⡾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡟⡄⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⣛⡼⢎⡷⣷⡹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡽⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡶⣹⡜⣏⡶⣿⣳⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⣽⡀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⣝⣳⡻⣼⢳⣝⣳⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣱⡇⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡾⣿⡿⣿⣿⣿⣿⣶⣻⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡷⣱⣤⣴⣠⣀⠈⠩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀