⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠖⣋⠋⠉⠉⣿⠲⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⢘⡇⠀⠀⣿⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⡞⠀⠀⠀⠸⣄⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⣿⠀⠀⠀⢠⡏⠀⣠⠶⣄⠀⢻⡀⠀⠀⠈⣿⡦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠔⠚⠛⠳⣄⣀⢠⡟⠛⠓⠒⠒⠖⠛⢁⣤⣈⠻⣶⠟⠒⢲⡟⢻⣧⣤⠜⠛⠛⠓⠦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢤⣶⣶⡛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢈⣿⠳⠤⣤⣄⡀⢠⡶⢋⣬⡙⢦⣘⢛⣿⣾⡤⠼⢿⣤⣤⣤⣀⣀⡀⠀⠀⠀⠉⠙⣻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡻⢷⣶⣾⣛⡛⠉⠉⠉⢉⡿⢤⣤⣄⡀⠉⢯⡘⢿⣥⣿⣤⡿⠉⢀⣠⣤⣴⠾⣧⡀⠀⠀⠀⠉⣉⣭⣿⡿⠛⣻⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⣄⡈⠙⠛⠿⣶⣿⣍⠀⢀⣠⠜⠛⠛⠳⣿⣦⣭⣵⣿⠟⠛⠛⠛⠢⢤⣄⡀⢉⣷⣦⠶⠛⠋⢁⣀⡤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⡀⠙⠻⣿⣿⣄⠀⠀⠀⠀⠏⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⠀⣠⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⠀⠙⠻⣷⣦⡀⠀⣸⠀⠀⠀⢹⠀⠀⣠⣴⡿⠛⠁⢀⣠⠞⢋⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⣶⣌⠈⠛⢿⣯⣿⠀⠀⠀⢀⣷⣾⠟⠁⢀⣤⠞⠋⠁⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠉⣻⣦⣄⠨⠻⣷⣦⣴⡿⠋⢀⣼⣾⡋⠁⠀⢾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⣦⠀⠀⠻⣝⠿⣷⣤⣀⠉⢀⣤⣾⣿⣻⠞⠁⠀⣠⣾⠟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠈⠳⣄⠀⠈⠳⠮⠿⠛⠶⠋⠁⠿⠾⠁⠀⠠⢞⡽⢃⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⣠⠞⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣾⡲⢄⡀⠈⠉⠁⠀⠀⠀⠀⠀⠴⠞⠁⣠⠞⠁⠀⢸⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⣼⣹⢹⡶⢭⣑⡂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⢧⡙⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠳⠶⣀⣠⢯⠇⠈⡇⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠿⠾⠏⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢞⣻⢿⣷⣶⣄⠀⢉⣿⣿⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠳⣄⠀⠀⠀⢀⣺⣟⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣩⣶⣿⣏⣵⣯⢵⣿⣷⠟⠁⠙⣾⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⢿⣦⡀⠀⠘⢷⣰⡶⣛⣿⡚⢿⡾⣝⡶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⣴⠞⣩⣴⡾⣏⡵⣾⡿⣋⣵⢟⣹⠯⣤⣴⠋⣹⢋⡿⠛⠲⠶⠶⣶⣶⡶⠞⠉⢉⣠⠼⠷⡌⠳⡝⠶⣄⠈⣳⡽⢾⣙⢿⣳⣯⣛⢿⣾⣝⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣴⣟⢿⣯⣟⣩⣾⡽⢋⣵⡾⣿⡷⠋⠁⠀⣼⠁⣼⢡⡮⠵⠶⠒⠊⠉⢉⣀⡤⠶⠚⠉⠁⠀⠀⠙⣦⡙⢆⠘⣏⠉⠙⠲⣝⢷⣮⡛⢿⣷⢮⡹⢿⣶⣍⣳⣄⣤⡀⠀⠀⠀
⠀⠀⣰⣷⣿⣷⣸⣿⡟⣩⢴⣿⣫⠿⠋⠀⠀⠀⣰⠃⣰⢡⡟⢀⣤⡤⠴⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⢀⡞⠉⢳⡘⣇⠘⣆⠀⠀⠈⠳⢽⡻⣧⣽⡻⢿⡦⣍⣳⠿⣻⡿⣧⠀⠀⠀
⢀⣾⣿⡏⣨⣿⡿⣟⣿⠿⣫⠞⠁⠀⠀⠀⠀⢠⠏⢀⠇⡾⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠏⢹⡝⡄⠘⡄⠀⠀⠀⠀⠙⢯⡝⢿⣦⣽⣿⢏⣼⣿⢿⣿⣷⡀⠀
⢀⡽⢿⣟⣯⠀⢻⡿⣤⣾⠁⠀⠀⠀⠀⠀⠀⣾⠀⣾⣼⠃⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⡴⠋⠀⠈⣷⢻⠀⠻⡀⠀⠀⠀⠀⠀⠙⢦⡙⠛⢣⠞⣽⢧⣄⠈⢙⡿⣦
⠃⠀⠀⠉⢳⣟⣤⡻⣝⣾⣆⠀⠀⠀⠀⠀⣠⡇⠀⠇⡟⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢠⠏⠀⠀⠀⠀⠘⡆⣧⠀⣷⠀⠀⠀⠀⠀⠀⠀⢹⣦⡟⣼⠁⢀⣿⢺⣯⣭⣽
⡇⠀⠀⠀⣠⠟⣏⢻⠻⣼⣞⡆⠀⠀⠀⠀⢹⠀⢸⢸⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⣴⠃⠀⠀⠀⠀⠀⠀⢿⢹⠀⢸⡄⠀⠀⠀⠀⠀⢠⣯⢏⡾⣥⣾⣥⠞⠉⠀⠀⠀
⣧⣤⡤⠞⠁⠀⣿⠀⡇⠹⣜⣿⣆⠀⠀⠀⢸⠀⢸⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⠀⢠⣿⠏⡾⢹⠁⣾⠓⠤⣄⠀⠀⠀
⡇⠀⠀⠀⠀⠀⢸⠀⣷⠀⠹⣞⣟⣇⠀⠀⢸⡀⣾⣿⡤⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣧⠀⠀⠀⢠⣿⠏⡾⠁⣾⠀⡟⠀⠀⠈⠉⠓⠤
⡇⠀⠀⠀⠀⠀⠘⡇⢻⠀⠀⠸⣟⣿⡷⠒⠚⢷⣿⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠀⣿⠀⠀⢠⢿⢯⡾⠁⠀⡟⢀⡇⠀⠀⠀⠀⠀⠀
⢣⠀⠀⠀⠀⠀⠀⣧⢸⠀⠀⠀⠸⣯⢣⡀⠀⠀⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⠁⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡾⠋⠛⠛⢿⢧⡞⠁⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀
⢸⡀⠀⠀⠀⠀⠀⣿⠘⡆⠀⠀⠀⣿⠣⡗⠤⢤⣿⣿⡀⢷⡀⠀⠀⠀⣠⠴⠋⢀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡀⠀⠀⣰⣿⡟⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀
⠀⢇⠀⠀⠀⠀⠀⣹⠀⡇⠀⠀⠀⣷⠀⠹⣿⠉⠉⠉⠙⢦⡳⣄⣤⠞⠁⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢮⣿⣿⣛⣳⠃⡇⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠸⣇⠀⠀⣀⡾⢋⡼⠁⠀⠀⢠⡇⢀⡼⠁⠀⠀⠀⠀⠀⠹⡎⢠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⡇⠀⠀⠀⢰⡇⢿⠀⠀⠀⠀⠀⠀⠈
⠀⠸⣏⠉⠉⢉⡶⠋⠀⠀⠀⠀⢸⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⢷⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢹⡄⠀⠀⠀⠷⠈⠳⣄⠀⠀⣠⠴⢳
⠀⠀⢹⡏⠀⠈⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⢷⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⢀⡞
⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀
⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣻⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀