⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣤⣶⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠲⣦⡑⢖⠂⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⣴⠟⠉⠉⠳⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠘⡄⠈⢿⠆⠑⣄⢠⡀⠀⠀⠀
⠀⢦⢠⠀⠐⠒⠒⠒⠒⠒⠒⢒⡶⠞⣛⡷⡲⢓⣒⣶⣾⣷⣖⡒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⣞⡗⠒⡖⠚⠲⣺⣲⢒⣷⠒⠚⡆⠹⠄⠀
⠈⢮⢻⡀⠀⠀⠀⠀⡀⠄⣴⠋⢀⡾⠣⡞⠀⣸⣿⣿⣿⣶⣤⣝⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠁⢸⡀⢫⡟⠄⡏⢧⠀⡇⠀⠀⠀⠀
⠀⠀⢳⡽⣄⠀⠀⠀⠄⡼⠁⡰⣿⣠⠞⣡⠾⠋⠀⠉⠛⠿⢿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠃⢸⡸⠀⣸⠈⢇⡇⠀⠀⠀⠀
⠀⠀⢸⠙⢮⣳⣄⠀⣸⠁⢰⡟⣿⠃⡴⠁⠀⠀⠀⠀⠀⠀⠀⢈⡹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠸⢸⠀⢫⠂⡜⡇⠀⠀⠀⠀
⠀⠀⢸⠀⠀⠱⣝⢦⡿⣆⡿⢈⢏⡞⠀⠀⠀⠀⢀⡠⠤⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⢀⠈⠻⣿⣷⣄⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⡏⠃⠆⡇⠀⠀⠀⠀
⠀⠀⢸⠀⠀⠀⠈⠻⣿⣿⣏⡜⡜⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣤⣜⡛⠲⣤⣀⠀⠀⠈⠻⣿⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠃⡀⢠⣧⠀⠀⠀⠀
⠀⠀⢸⠀⠀⠀⠀⠤⢻⡽⠀⣿⠁⠀⠀⠀⠀⢸⣦⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣤⡙⢷⣄⢀⡄⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠟⢆⣺⣸⣿⡀⠀⠀⠀
⠀⠀⢸⠀⡀⠀⠠⢀⣺⠃⣇⡏⢀⣄⣀⠄⠀⠀⠀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣾⣿⡿⠁⠀⠻⢀⠀⠀⠀⠀⠀⠀⠀⡆⡇⢸⢃⣿⣧⣿⡏⠀⠀⡀⠀
⠀⠀⢸⣾⣤⣶⣾⣦⡿⢸⢸⠃⠸⣥⣀⠙⢦⡀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⠀⢰⢣⠏⢰⠁⠀⠀⠀⠀⢰⢣⠃⣼⣿⣿⣿⣿⣷⡿⣶⡇⠀
⠀⠀⢸⣿⣿⣿⣿⣿⠇⢸⡏⠀⢸⣿⣎⣑⣄⡈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠟⡟⢏⣾⠀⡏⠀⠀⠀⠀⡄⡜⡜⠀⣿⡿⣿⣿⣧⣿⡽⠋⠀⠀
⠀⠀⢸⣿⣿⣿⣿⡟⠀⡸⢁⣠⣿⣿⣿⣿⣦⣽⣌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡽⣜⡇⢸⠀⡀⣰⠄⣸⣼⢱⠁⢰⣿⣵⣿⢿⡿⡿⠁⠀⠀⠀
⠀⠀⢸⢻⢻⣿⣿⢀⡤⠓⠉⠀⠈⠻⢿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣷⢳⡻⢀⡇⢠⡇⡽⢠⡿⢇⣎⢸⣼⢿⠟⠁⣿⣿⡇⠀⠀⠀⠀
⠀⠀⢸⠑⣣⠾⠛⠁⠀⠀⢀⣤⣤⠶⠛⢉⠉⠻⠿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⡼⣸⢇⣯⠃⣸⠀⡞⣠⢣⢯⢃⢸⡇⣿⡿⠋⠀⣰⠟⠁⡿⡄⠀⠀⠀
⠀⠀⢸⡟⠉⠀⠀⠀⢄⣢⠞⠋⠀⠀⠈⠊⠀⢀⡴⠛⢻⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⢳⠛⡼⡞⢠⠟⢰⠹⡏⣜⡏⡜⣸⢦⡟⠀⢠⡾⠋⠀⠀⣗⡇⠀⠀⠀
⠀⣠⢿⠀⠀⠀⣴⡿⠏⠀⠀⠀⠀⠀⠀⣠⠾⠉⢀⡴⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣷⠃⣶⢳⢱⡜⠠⡆⡸⠀⡽⢠⢣⣟⣞⡀⣴⠟⠀⠀⢀⣾⡇⡇⠀⠀⠀
⠀⠁⢸⠀⢀⡼⠋⠀⠀⠀⠀⢀⣠⠴⠏⠁⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣻⢡⠏⠇⢯⠃⡳⢡⠇⠎⢣⠏⡼⡏⣾⡝⠁⠀⠀⣀⣼⠏⣷⠃⠀⠀⠀
⠀⠀⢸⣰⠟⠁⠀⠀⠀⠀⣰⣿⠂⠴⠝⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠛⣇⡞⡼⠀⡏⣎⢇⠏⠀⢐⠎⡸⠹⣴⠏⠀⠀⢀⣼⣿⣻⣾⡏⠀⠀⠀⠀
⠀⠀⣼⠃⠀⠀⠀⠀⠀⣰⣿⣯⣭⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⣞⡏⡼⣹⠁⠀⠀⢰⡏⠘⠀⠠⠎⣰⣣⡞⠉⠀⢂⣴⣿⣿⣟⡽⠑⡇⠀⠀⠀⠀
⠀⠀⢸⠀⠀⠀⠀⠀⣰⣿⠿⠛⠁⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⣾⢷⠺⢡⡟⠀⣸⢀⡏⢠⠀⡰⠁⡔⡳⠋⠀⠀⣠⣿⣿⣿⡿⣯⡀⠀⡇⠀⠀⠀⠀
⠀⠀⠈⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⡏⡜⡜⠀⠀⠃⡎⢠⠃⣰⠃⣸⣸⠇⠀⣠⠾⣛⣹⣿⣿⣏⡸⠆⠀⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡋⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠁⠀⠀⡸⢁⢏⢆⠇⣬⡇⣽⡖⣶⣿⠿⢿⣿⣿⣿⣶⡆⠀⠀⡇⠀⠀⠀⠀
⠀⠀⠸⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⢹⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣸⢠⣯⢷⡟⠷⠋⠉⠋⠀⠈⠙⠯⡻⣧⠀⠀⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡘⠀⠊⡀⠀⠀⠀⠀⠀⠀⠀⠡⢈⠘⣧⠀⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⣹⡄⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠠⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆⡆⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠶⠞⡽⠀⠀⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⢿⣸⢹⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣄⠀⠀⣿⣿⣽⡄⠀⠀⠀⠀⠀⠀⢀⡶⢺⣿⣷⣶⣶⡮⠭⠭⢤⡤⠴⣾⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠴⢿⠼⡯⢼⠤⢽⠤⠤⠤⠤⠀⠀⠤⠤⠤⠤⠤⠤⠤⠿⠷⠿⠦⠧⣿⠭⠭⠿⠤⠄⠀⠤⠤⠤⠬⡧⠬⠿⢿⠿⠿⠤⠤⠤⠤⠽⠦⡿⠤⠤⠤⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡟⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡑⣆⡇⠀⠀⡇⠀⢹⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⡟⠲⠀⣸⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⡞⡆⡞⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢸⠀⠀⠀⢳⠀⠈⡇⠀⠀⢧⠀⠀⠀⠀⠀⡼⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠚⠀⠂⠀⠀⠒⠶⠀⠀⠤⠄