⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣀⣠⡤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢀⡴⠞⢉⣠⡴⣶⣿⣧⡈⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢀⡴⠋⣤⢾⡏⣈⠀⠘⣿⡏⠙⠳⡄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢀⡟⢠⣼⣧⢏⣁⣇⡿⣿⢿⣧⣄⠀⢿⣸⡄⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣸⢁⠇⣿⠋⡉⠀⣍⡙⢡⣿⣇⣙⣾⣶⡍⢣⡄⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢠⡏⢸⠛⡇⠀⢦⡈⠉⣽⣿⣿⣿⣿⣷⣿⣿⡀⣧⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢸⣿⡿⡄⣧⠈⠉⠳⣬⣹⣿⡞⠋⠚⠀⣀⣥⡜⣮⠇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠸⡿⢆⣿⡇⠉⠉⣑⣿⣯⣿⣇⣛⣺⡿⠛⣿⣿⣶⠂⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣷⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢷⢨⣿⣷⢦⣾⠛⣩⣟⣽⣟⢯⣝⢿⠛⣿⠉⣷⣿⣧⡈⢧⣀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢁⣀⣀⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⢸⢶⣻⣾⣿⠉⣴⣿⡟⣿⡟⣷⣬⡛⠉⠙⣤⣿⢸⣿⣿⡷⣌⠙⢦⣄⠀⠀⠀⢞⣵⢛⣫⠉⣇⢋⣯⣍⣟⢦⣄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠸⣦⣿⣿⣽⣸⣿⣿⣿⣷⣷⣿⡿⣿⡆⢰⣿⡃⣼⣇⢹⣿⣿⣇⡀⠬⣝⠲⠶⣯⣟⡳⠛⡴⣻⣶⣆⡀⠿⣷⡝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡇⠀⠀⠀ ⠈⠉⢻⣿⣿⣷⣿⡉⠉⡏⠏⠸⠂⠉⠙⣷⢸⢹⡇⣸⣿⣧⣿⣿⡿⢿⣾⡾⢍⠓⣦⠈⡉⣛⠿⣿⣿⣿⣷⡘⡷⠇⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠃⠀⠀⠀⠀⠀ ⢸⣼⣿⣿⢿⣧⣤⣵⣰⣠⣴⠶⢾⣿⠸⣶⢿⣿⣷⣅⣾⣿⣷⣄⡛⠛⣿⣿⣬⣓⡆⣈⠛⣼⣟⣻⠶⠾⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠒⠒⠂⣠⢴⣶⣿⣻⣿⠈⣿⣿⣷⣽⣿⣿⣿⣿⢿⢠⣿⣿⣿⣿⣿⣝⡻⠿⢿⣿⣾⣿⣿⢻⣛⠧⡾⣿⣿⢷⣾⡉⢹⣿⡏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⣾⠿⣭⣿⣿⣿⣿⣦⡉⠛⠉⠛⠻⠿⠟⠛⣡⣼⣿⢻⣷⡮⢷⣻⠿⣎⢳⡺⣽⠁⣘⢮⣯⣿⣿⣽⣿⣿⣿⣷⠞⡁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⣏⣰⣷⣿⣿⣿⣿⣿⣿⣿⣄⡀⣀⣀⣀⣤⣾⣿⣿⣿⣧⣿⣧⣛⠛⣷⣈⠳⣄⣤⣸⣿⣷⣯⣳⣭⣘⣷⢟⣢⠵⠚⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣭⠭⣽⣾⣿⣿⣿⡿⡟⢡⣽⣿⣿⣟⣿⣿⡿⣿⣿⣿⣿⣿⣿⡯⢉⣚⣿⣿⣾⣺⣷⣾⡟⠿⠿⢹⡴⠟⣻⡯⠖⠋⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣋⡩⠶⣶⡞⣡⣾⣿⣿⢏⣾⣷⣿⣿⣿⣿⣿⣿⣏⣸⣿⠿⢿⣏⡻⣾⣿⣿⣿⣿⡿⠏⠙⠉⣻⣵⣶⡿⠿⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⢒⡴⢯⣤⡿⢋⡼⣻⣿⡿⣻⣿⣿⣿⡿⡿⣥⣿⣿⠟⣹⣿⣗⣀⣰⣿⢿⡟⣿⡉⠉⠀⠀⠀⠀⠀⢹⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⠞⣱⣴⠿⣫⠇⣤⠟⣠⣿⣡⠟⢠⣾⣽⣯⣾⠁⡀⣠⣿⣿⡽⣑⣻⡾⢿⣛⣿⠭⠙⠚⠛⢳⡄⠀⠀⠀⣀⣀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣾⡟⣱⣿⣥⣾⣅⣰⢏⣼⡟⠷⠏⠠⢿⣽⣯⠬⠵⠿⠾⠓⠋⠈⠉⠁⠉⠳⢯⣀⢀⠀⠀⠀⠀⠀⢻⡀⠀⣾⡟⣋⣙⡆⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢀⣾⣽⢠⣿⣡⣿⣦⣾⢃⡾⠁⠉⢛⣻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠇⠀⠀⠀⠀⠈⢷⡈⠿⢿⡿⠿⣃⣬⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡾⣸⢧⣏⣿⣿⢟⡿⣱⠏⠀⠁⣰⡿⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠷⠤⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣸⣱⣿⢻⢿⡟⣴⡟⣱⠃⠀⠀⢠⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡿⣿⡏⣼⣿⣸⢟⣴⡇⢤⣤⣖⣿⣤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⣯⣿⣷⣿⣿⣻⢿⣿⣵⣯⡍⠁⢤⣿⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⢿⣏⣿⣼⣿⢿⣼⡇⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⣯⡎⢿⣿⣻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠘⢷⣶⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⢰⠟⣯⣾⣿⠽⢄⠐⢛⠛⠛⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿
⠀⠀⠸⣆⡹⣿⡽⣆⠸⡿⠏⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⢹⡏⠉⠀⠈⢷⣇⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⢻⠄⠀⠀⡼⣿⠖⠛⠓⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠛⠙⠛⠓⠛⠛⠛⠛⠚⠛⠛⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿