⠀⠀⠀⠀⠀⠀⠀ ⣠⠴⡗⣲⠞⠉⠩⣍⡙⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣴⠽⠒⠒⠛⠓⠂⠀⠀⠀⠉⠲⣄⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⡏⠁⠀⠀⢀⠀⠈⠳⡄⠘⢆⠀⠀⠈⠳⡈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⡏⢠⠃⠀⠀⠘⡆⠀⢇⠘⡄⠈⣎⣑⡀⠀⢱⡆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⢠⢸⠀⠀⠀⠀⣷⠀⠘⠀⢹⡄⣸⠀⠈⠙⠂⡿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⢸⡘⡄⠀⠀⠀⣸⡆⡜⠀⢨⣧⣏⠀⠢⣁⠀⢳⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠳⣷⣽⣦⣀⡴⢧⣷⡷⠻⣾⡏⢹⠓⢦⠜⡇⠘⡟⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⣿⣳⡄⠀⠸⣿⣿⠿⢺⣧⠈⡇⠘⡆⠸⡀⣷⠈⢿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⣿⠙⡧⠀⠀⠈⠁⠀⢸⣿⣦⠸⣶⣷⡀⠃⡏⣎⠀⠳⡛⢦⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠸⢿⣟⡗⣤⣠⠀⢀⡤⣾⡇⠉⠙⢒⢟⠷⢼⡑⠋⣧⡀⠘⢷⣝⢦⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣿⡟⣻⣷⠟⢁⠀⣠⡿⠃⠀⠀⢹⡔⡇⡇⠀⢈⡏⠆⢻⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⠃⠀⢸⣑⣉⣔⣽⠁⠀⠀⠀⠀⣿⢢⡇⠀⣸⢳⡘⣸⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡹⡄⠀⠸⣄⠈⠉⣿⠀⠀⠀⠀⠀⣿⡟⣡⢾⠃⣾⡵⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣹⠁⢹⣆⠀⠈⠢⣠⢻⠀⠀⠀⠀⠀⢹⢿⠁⡏⠀⢹⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⠃⠀⢸⠟⠳⡄⠀⠈⠑⢦⣀⡀⠀⡆⠈⢸⠀⠹⣄⠈⠳⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⠀⠀⣸⣦⣸⣼⣆⠀⠀⠀⠹⡽⠛⢳⣿⣼⠳⡀⠈⠓⣄⢈⠳⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠦⣔⡽⣿⣿⣿⣿⣷⣦⠴⢿⠱⠒⠺⡉⢧⠀⠙⡆⠀⠘⡆⢀⠈⢧⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠄⡼⠁⢛⡟⢀⡏⠛⠃⠀⠸⡄⠇⠀⢳⠈⢧⠀⢹⡀⠀⢹⠘⡆⢸⡆
⠀⠀⠀⠀⠀⠀⢰⢫⠋⢀⡼⠁⠀⣼⠁⢸⡇⠀⠀⠀⠀⠁⠀⠀⠸⡀⠀⢧⢰⠇⠴⣸⠁⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠸⣄⠀⡸⠁⠀⢰⡏⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⣿⠀⠀⠀⠀⡰⠋⠀
⠀⠀⠀⠀⠀⠀⠀⠙⣶⠃⠀⢠⣿⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢀⡠⠞⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⣾⠇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣾⡟⠀⠀⠀⣇⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡿⢱⠃⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⠁⠀⠤⠀⠉⡎⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠘⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀
⠀⠀⢠⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡄⠀⠀⠀⠀⠀⢠⢸⠀⠀⢸⠈⢧⠀⠀⠀⠀
⠀⠀⢘⣦⣄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠀⢰⠀⠀⠀⠀⠀⠀⢈⡀⡇⠀⢸⢀⣸⡆⠀⠀⠀
⠀⠀⠀⠑⠾⣗⠒⠢⠤⢶⣠⠾⠠⣄⡀⠀⠀⠀⣏⣀⣀⡀⠀⠀⠀⢰⠷⡦⠤⠬⣻⠏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢈⠟⠒⠒⠦⠔⣲⢮⣀⣉⠉⠀⢀⣸⠦⠷⢄⣉⣀⣀⠼⠟⠓⢲⠞⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⡴⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠙⣄⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⠃⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣠⠃⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠘⡄⠀⠀⠀⠀⠀
⠀⠀⣠⠇⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢳⡀⠀⠀⠀⠀
⠀⣴⣯⡦⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣠⡼⣷⠀⠀⠀⠀
⣴⡋⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡋⠉⠉⢽⡇⠀⠀⠀
⣧⡭⠭⢩⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⢤⣾⠃

⠀⠀⠀⠀⠀⠀⠀ ⣠⠴⡗⣲⠞⠉⠩⣍⡙⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢀⣴⠽⠒⠒⠛⠓⠂⠀⠀⠀⠉⠲⣄⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣰⡏⠁⠀⠀⢀⠀⠈⠳⡄⠘⢆⠀⠀⠈⠳⡈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢠⡏⢠⠃⠀⠀⠘⡆⠀⢇⠘⡄⠈⣎⣑⡀⠀⢱⡆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢸⢠⢸⠀⠀⠀⠀⣷⠀⠘⠀⢹⡄⣸⠀⠈⠙⠂⡿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢸⢸⡘⡄⠀⠀⠀⣸⡆⡜⠀⢨⣧⣏⠀⠢⣁⠀⢳⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠈⠳⣷⣽⣦⣀⡴⢧⣷⡷⠻⣾⡏⢹⠓⢦⠜⡇⠘⡟⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠈⣿⣳⡄⠀⠸⣿⣿⠿⢺⣧⠈⡇⠘⡆⠸⡀⣷⠈⢿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢸⣿⠙⡧⠀⠀⠈⠁⠀⢸⣿⣦⠸⣶⣷⡀⠃⡏⣎⠀⠳⡛⢦⡀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠸⢿⣟⡗⣤⣠⠀⢀⡤⣾⡇⠉⠙⢒⢟⠷⢼⡑⠋⣧⡀⠘⢷⣝⢦⡀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣿⡟⣻⣷⠟⢁⠀⣠⡿⠃⠀⠀⢹⡔⡇⡇⠀⢈⡏⠆⢻⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⠃⠀⢸⣑⣉⣔⣽⠁⠀⠀⠀⠀⣿⢢⡇⠀⣸⢳⡘⣸⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡹⡄⠀⠸⣄⠈⠉⣿⠀⠀⠀⠀⠀⣿⡟⣡⢾⠃⣾⡵⠃⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣹⠁⢹⣆⠀⠈⠢⣠⢻⠀⠀⠀⠀⠀⢹⢿⠁⡏⠀⢹⠁⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⠃⠀⢸⠟⠳⡄⠀⠈⠑⢦⣀⡀⠀⡆⠈⢸⠀⠹⣄⠈⠳⣄⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⠀⠀⣸⣦⣸⣼⣆⠀⠀⠀⠹⡽⠛⢳⣿⣼⠳⡀⠈⠓⣄⢈⠳⣄⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠦⣔⡽⣿⣿⣿⣿⣷⣦⠴⢿⠱⠒⠺⡉⢧⠀⠙⡆⠀⠘⡆⢀⠈⢧⠀ ⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠄⡼⠁⢛⡟⢀⡏⠛⠃⠀⠸⡄⠇⠀⢳⠈⢧⠀⢹⡀⠀⢹⠘⡆⢸⡆ ⠀⠀⠀⠀⠀⠀⢰⢫⠋⢀⡼⠁⠀⣼⠁⢸⡇⠀⠀⠀⠀⠁⠀⠀⠸⡀⠀⢧⢰⠇⠴⣸⠁⠀⢰⠇ ⠀⠀⠀⠀⠀⠀⠸⣄⠀⡸⠁⠀⢰⡏⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⣿⠀⠀⠀⠀⡰⠋⠀ ⠀⠀⠀⠀⠀⠀⠀⠙⣶⠃⠀⢠⣿⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢀⡠⠞⠁⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⣾⠇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠋⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣾⡟⠀⠀⠀⣇⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡿⢱⠃⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣰⠁⠀⠤⠀⠉⡎⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠘⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀ ⠀⠀⢠⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡄⠀⠀⠀⠀⠀⢠⢸⠀⠀⢸⠈⢧⠀⠀⠀⠀ ⠀⠀⢘⣦⣄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠀⢰⠀⠀⠀⠀⠀⠀⢈⡀⡇⠀⢸⢀⣸⡆⠀⠀⠀ ⠀⠀⠀⠑⠾⣗⠒⠢⠤⢶⣠⠾⠠⣄⡀⠀⠀⠀⣏⣀⣀⡀⠀⠀⠀⢰⠷⡦⠤⠬⣻⠏⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢈⠟⠒⠒⠦⠔⣲⢮⣀⣉⠉⠀⢀⣸⠦⠷⢄⣉⣀⣀⠼⠟⠓⢲⠞⠁⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⡴⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠙⣄⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢰⠃⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢷⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⣠⠃⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠘⡄⠀⠀⠀⠀⠀ ⠀⠀⣠⠇⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢳⡀⠀⠀⠀⠀ ⠀⣴⣯⡦⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣠⡼⣷⠀⠀⠀⠀ ⣴⡋⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡋⠉⠉⢽⡇⠀⠀⠀ ⣧⡭⠭⢩⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⢤⣾⠃