⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⡇⠀⠙⡻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠟⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢻⠿⠃⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠉⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢳⠏⣾⣣⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣜⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡇⣿⢰⢋⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠒⠛⠛⠚⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⠙⢧⢈⣁⠀⢠⠄⠸⣤⠴⠶⠓⠒⠛⠋⠉⢿⣯⣷⡶⢾⠶⢶⣶⣶⣦⣤⣤⣀⡀⢸⠀⠀⠀⡄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢰⡎⣶⣥⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⡡⠀⠈⠀⠀⢙⠛⠋⠀⠀⠉⠙⠃⠸⠃⠀⠆⢠⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠀⢫⢋⣿⡽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠵⣠⣤⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣹⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⠀⡰⡻⣌⣻⠿⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣠⣴⣶⡤⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣟⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡇⠀⠻⢥⢎⠟⠔⡀⠙⠲⢦⣤⣤⣤⣤⡴⣶⣾⣿⡟⠛⠉⠙⠛⠉⠙⠮⣙⠒⠤⣝⡲⠤⠀⡀⠀⡰⣣⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⣠⠊⠄⢇⠀⠀⠆⢀⠈⣱⣿⢯⣯⣽⣯⠥⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣽⣾⣤⣍⢻⠛⣲⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢈⠠⡀⠀⠀⢠⡀⡞⢠⣿⡟⠀⠻⣟⢿⣿⡷⢽⣦⠀⠀⠀⢀⡴⠭⣔⣈⠉⣽⣿⣿⢀⡆⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠵⣷⠁⠀⠄⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⠉⠃⠨⠌⠉⡽⠀⠀⠀⡿⣿⣟⣿⣾⣿⣿⣿⢿⠀⠣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⢱⠙⠰⠀⡀⠀⠀⠘⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠳⡈⠊⠻⠶⠿⣿⠎⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠢⢣⠄⠈⠁⠀⠁⢠⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠂⡟⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠁⠀⠀⠀⠀⠸⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⢵⢦⡀⠉⣠⣯⡇⠀⠀⠀⠀⣸⠇⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⡀⠀⠀⠁⢣⢹⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⡙⠶⠿⠛⠋⠀⠀⠀⢀⡿⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡅⠀⢄⠀⠀⠀⠈⣆⢧⠀⠀⠀⠈⢳⣶⣶⣶⣶⣷⣤⣄⡀⠀⠀⠀⠀⣿⠃⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠐⠀⠀⠀⠀⠈⢎⢧⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⢓⡖⢀⣾⡟⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠈⠣⡳⣄⠀⠀⠀⠰⠿⠛⠛⠋⠉⠀⠘⣴⣿⠟⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⢄⡀⠀⠀⠀⠀⠨⣮⣗⠤⣄⡀⠀⠀⠀⠀⠀⣠⠼⠛⠛⠶⢯⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠉⠀⠀⠉⠢⢄⡀⠀⠀⠈⠙⠫⠄⠩⣹⢶⢲⣴⡋⠔⣲⡖⠬⢟⣿⣕⠻⢵⡒⠮⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠑⢄⠀⠀⠀⠀⠈⠉⠒⠢⠤⢀⣀⡜⢀⣴⣿⣿⣿⣶⣱⡶⣷⣾⣿⣿⡄⠀⠈⠙⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣿⣿⠟⠛⢶⡾⡿⡋⠈⠙⣿⣿⣷⡀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢽⠅⠀⠀⠨⢡⣇⢝⠀⡔⣱⣾⣿⣷⡤⠴⠀⠀⠙⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣸⢀⠠⠂⡤⡏⢫⢀⣽⣏⡽⣿⣿⣿⣿⡄⠀⠀⠀⠘⢯⠙⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⣻⣆⣠⣤⣾⣷⣾⣿⢾⣿⡿⠿⠿⢻⣿⣿⠀⠀⠀⠀⠈⢧⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀
⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⠋⠉⠉⠋⠛⠋⠉⢭⠏⠀⠉⢀⠠⣄⡗⢻⣿⠇⠀⠀⠀⠀⠀⢳⡀⠀⡔⠀⠱⢄⠀⠀⠀⠀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠱⣤⣀⣀⣠⣼⡾⣖⣲⣶⣾⣿⣯⣀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠹⣄⠸⡄⠀⠀⠳⡀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣞⣾⣿⣿⣿⣾⣿⣿⣿⣯⣿⣿⣿⣋⠙⢿⡺⠖⠂⠀⠀⠀⠀⠘⣧⠃⠀⠀⠀⠱⡄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣟⣻⣿⣵⣿⣶⣮⣽⣿⣿⠟⣿⢦⠈⢯⠀⠀⠀⠀⠀⠀⣀⠟⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢩⣽⣿⠻⣿⠛⣿⣿⢛⣽⢛⡷⣼⡟⢘⣧⠸⠉⠀⠉⠉⠉⠉⠀⠀⠂⠀⠀⠀⠀⠀⠈