⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣴⣄⠀⠀⣠⣶⣶⡎⢛⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢛⣛⣿⣿⣕⣦⣀⣼⡿⣍⢦⣶⣿⣿⣿⣦⡉⢳⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⢰⣿⣿⣿⣿⣶⣿⠑⣻⣤⡻⣙⢿⣟⣛⢛⣿⣧⡘⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⣿⣿⣟⣴⣿⣿⠓⠲⢼⣿⣮⡿⣉⠩⣻⡗⢻⣿⣿⢸⣾⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡌⣿⣿⡏⠀⣽⣿⣐⣋⣼⠟⠷⣬⣭⣶⣿⡂⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⢠⣿⣧⢠⣿⡷⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠸⣿⣿⣿⣿⣉⠈⣡⣽⢿⣽⣿⢀⠸⣿⣈⠻⢄⣀⡀⠀⠀⠀⠀⠀⢸⣿⣷⢸⣿⣿⡄⠈⣧⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣏⢸⣿⣿⣾⣿⡟⠀⠀⠀⠀⠐⢷⣤⡤⠇⣰⣿⠇⣸⣿⣧⣌⡙⢁⢄⢈⣴⣿⡿⠷⠈⠁⠙⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⡇⣼⣿⡄⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠛⢿⣿⠀⠀⠀⣠⣤⠀⠙⠻⠿⠿⠋⠀⠀⠈⢹⣿⣿⠩⣯⣍⠏⠙⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣄⡀⠀⢏⣇⡀⠀⠉⠙⠻⢿⠁⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠟⠁⠀⢀⢠⣴⣟⡆⣤⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣠⠀⠈⢡⠢⢞⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⣸⣾⣿⣷⡒⢢⣤⣼⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣷⣶⣍⢁⣿⣿⣿⡟⠿⢟⡟⠀⠀⢶⣶⣶⣤⣿⣷⣤⣼⣿⣇⠳⢸⠌⠈⢸⠛⣕⣿⡟⠻⡣⢶⣿⣏⣤⣤⣲⠠⠀⠙⣿⠿⢻⣿⣿⣿⡗⣾⣿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠚⣿⣿⡿⠃⣠⠋⢲⣶⣶⡜⠻⡉⠭⣭⣭⢛⡻⢟⢾⡥⢘⣸⣦⣍⣸⠿⢶⢟⡛⠛⠂⠴⣍⡍⣽⣂⣗⡀⠀⠈⠲⢬⡻⠿⠟⢑⠻⠗⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡙⣍⢡⢀⡀⢀⠖⠉⠀⠀⣠⡿⡿⠷⠏⣠⣞⠤⠊⠁⠀⠀⠀⢜⠂⠝⣿⠯⠀⡠⡗⠁⠀⠀⠈⠉⠢⡀⠀⠀⠙⢿⣿⠀⠀⠀⠈⣦⣠⣶⣧⣾⣿⣀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡧⣼⠀⢀⠴⠊⠁⠀⠀⢀⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠣⠈⣿⠁⠉⠞⠀⠀⠀⠀⠀⠀⠀⠈⢐⠈⠉⠁⠈⠓⢤⡀⠀⡟⣿⣿⣿⣿⣿⡟⠀⠀⠀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⠉⡶⠁⠄⠀⠀⡀⣴⡋⠒⠉⠉⠀⠉⠉⠒⠀⢀⠀⠀⠘⣿⠀⠀⠀⢀⡤⠐⠒⠉⠉⠑⠒⠤⣑⣄⠀⠀⢀⠀⠈⡾⠀⣿⣿⣿⣿⣿⣧⠀⠀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡀⠘⣤⣀⣤⣾⡂⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠠⠀⠀⣿⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⣀⡰⢁⡴⣿⣿⣿⣿⣿⣿⠇⠀⠀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠃⠱⡄⣹⣿⣿⡇⠀⠀⠀⡔⠀⣀⡀⠐⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠔⠊⠙⠲⣄⠀⠀⠀⢻⡀⣹⣇⡞⠁⠻⣿⣿⣿⣿⣿⡇⠀⠀
⠀⠀⠀⠸⣿⣿⣿⣿⡿⠃⠀⠀⠀⣽⣿⣿⡇⠀⠀⠀⠃⠸⡿⠃⠀⠃⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢾⠀⠰⣯⣶⠘⠀⠀⠀⠀⣾⠰⣚⠀⠀⠀⠈⢹⣿⣿⣿⡟⠀⠀⠀
⠀⠀⠀⠀⢹⣿⣿⣿⠇⠀⠀⠀⣸⡿⠿⢿⣿⡀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⣇⠀⠀⠀⠀⠀⠀⠘⣄⢀⠀⠀⠀⠀⠀⠀⢰⡇⢨⡄⠀⠀⠀⢈⣿⣿⣿⡵⡻⠋⠀
⠀⠀⠀⠀⠰⣿⣿⣿⣇⠀⠀⠀⠁⠀⢀⣤⠿⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣾⡁⠋⠉⠀⠀⢠⣾⣿⣿⠋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣶⣤⠴⠚⢉⡤⡄⠀⠈⠓⠤⣀⡀⠀⠀⣀⣀⣤⠾⠛⠁⠀⠘⠀⠈⠿⣶⣄⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠈⠉⠓⠶⢤⣄⣶⣿⣿⡛⠛⠁⠀⠀
⠀⠀⠀⠀⠀⠈⠉⠉⠹⠿⠃⢀⣄⡀⠹⣠⡿⠀⠀⠀⠀⠀⣸⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠷⣶⣶⣶⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠧⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡤⣾⠁⠈⢢⢹⡀⠀⠀⢀⣴⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⡆⠀⣄⠀⡟⡏⠀⠈⢯⣤⣄⡏⠈⡇⠀⢀⣾⣿⣿⣿⠟⠁⠀⠀⠀⠈⠉⠓⠶⣤⣤⠴⠒⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠔⠉⣢⠀⡞⠛⠁⠀⠘⠁⠀
⠀⠀⢀⡟⠀⠘⣿⢇⠡⣄⣰⠟⠀⠀⠀⠀⣠⣶⣻⣿⣿⢋⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠅⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣺⢷⣂⠀⠀⠀⠀⢀⠔⠊⠀⠓⠒⣆⠀⠀⠀⠀
⠀⢀⡇⣠⡖⢲⢳⠈⠟⠀⠀⠀⠀⠀⣀⣼⣫⣿⣿⣿⣮⣎⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⠿⣵⡄⠀⠈⠛⠛⠋⣀⡀⠀⣴⠇⠀⠀⠀
⠀⠈⢉⡁⠀⢀⠇⠃⠀⠀⢀⡠⠖⣻⢉⣡⣝⡛⣻⣿⡿⢻⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢿⣿⠿⢿⣿⣿⣝⠿⡆⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀
⠀⠀⠀⠉⠉⠁⠀⠀⣠⠖⠛⠀⠀⡁⠛⠉⠀⠙⠉⠀⠀⠀⠈⠻⡟⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠟⣥⠏⠀⠀⠀⠀⠙⠻⢣⢘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⡀⢀⢨⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⠙⠒⠶⠤⠤⠀⠠⠤⠤⠞⠛⠁⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠘⢄⡣⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣾⡥⠤⠤⠾⣵⡿⠟⠁⢀⣠⠶⠒⠀⠈⠁⠒⠤⡀⠀⠺⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠙⠽⡍⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⡯⠥⠐⠒⠒⣯⠊⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢀⠴⠂⠀⠀⠀⠀⠈⠉⠳⣦⡀⠀⠛⡍⠀⠀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⣰⠃⠀⠀⠀⠀⢀⡇⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡟⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⢡⡀⠀⠀⠈⠱⡀⠀⠀⠀
⠀⢠⠃⠀⠀⠀⣄⣠⣾⠃⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢠⣿⡆⠀⠀⠀⢰⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠙⠷⣦⣤⣀⣻⡀⠀⠀
⠀⠇⣀⠴⣚⡩⠟⠁⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⣿⡇⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⡄⠈⠉⠻⣟⡄⠀
⢸⠨⠔⠋⠁⠤⣀⡀⠂⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⣿⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⠀⠹⡀⠀⠀⢸⠁⠀
⠜⠀⠀⠀⠀⠀⢀⣧⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀⣴⣿⣄⣄⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠃⠀⠑⢀⡀⠈⡇⠀
⡄⠀⠀⢠⣦⣴⣾⡃⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣿⠁⢸⣿⠀⢹⢟⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢉⡢⡏⢆
⡇⠀⣾⡿⠟⠋⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⡟⣀⡾⠻⣆⢸⡆⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⢠⣤⣀⠈⡗⠚
⠃⠀⢀⣠⣤⣤⡀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⡿⠛⠁⠀⠀⠈⠛⢿⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠀⠐⠻⣿⣷⣇⡀
⢆⢰⣿⣿⣿⣿⠇⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣿⡾⠁⠀⠀⠀⠀⠀⠀⠘⣖⣯⡜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣾⣿⣂⢻⣧⠕
⢸⠸⣿⣿⣿⣿⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡮⣪⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠘⢻⣿⢿⣿⢱⠃
⠈⣧⣿⣿⣿⣿⣆⠄⠀⠀⠀⠀⠙⠢⠤⣀⣀⣀⣀⢤⡴⠟⠉⢀⡽⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠝⠢⢄⡀⠀⠀⠀⠀⢀⣠⠖⠁⠀⠀⣠⣿⣶⣾⣿⣷⡻⠀
⠀⣸⢻⡿⠟⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀�

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⠀⠀⠀⠀⣀⣀⣠⣄⣤⣾⣿⡿⠋⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠳⢶⣚⣭⣿⣿⠿⠿⠿⠿⠿⠷⣦⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠈⠐⢄⣽⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⡇⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠿⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⣿⣷⣿⠋⠀⠀⣸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢻⡶⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⢿⣿⡟⠀⠀⢀⣿⠻⣧⡐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⢰⠀⠈⢿⣇⠀⢀⣾⠃⢀⡠⠽⠾⣦⡀⠀⠀⠀⠀⠀⠀⢀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⡸⠀⢀⠈⢿⣄⢸⡇⠚⢁⣠⣤⣤⣄⠹⣦⡀⠀⠀⠀⠀⠘⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀
⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠸⠀⠀⠹⣼⡤⠿⣿⣿⡿⠧⠚⠁⠈⠙⠦⣄⡀⠀⠀⠿⣧⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀
⠀⠀⠀⠀⠀⠀⠀⣾⠃⢈⡆⢰⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⡏⣉⢁⣷⠋⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⢷⣳⡸⣆⠈⢧⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⣀⡐⢸⡋⣸⠏⠳⠦⢽⣦⣄⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣼⣿⡵⢡⢹⠀⠈⠳⣄⠀⠀⠘⢦⡀⠀⠀⠚⠈⠙⠃⢣⢻⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⢿⡋⠀⢸⠈⢇⠀⠀⠈⠙⠒⢤⡈⢿⠒⠒⠤⢤⡤⠴⠃⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⣣⡈⡢⣀⠀⠀⠀⠀⠸⣾⣀⠀⠀⢀⣷⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣠⡭⠗⠺⠗⠮⣺⣍⠛⠶⡀⠀⠈⢿⣿⣿⣿⣿⡿⠉⠉⡩⠉⢉⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠘⠛⠷⣴⡇⠀⠤⣼⢯⡉⠈⠁⠀⠐⠀⠀⠰⡋⠣⣰⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⢳⠀⠀⠙⣄⠳⢤⡀⠀⣀⠠⢔⣊⣹⡷⣾⢧⡈⣆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⣰⣀⣀⡈⠑⠢⢄⠀⠉⣩⢝⡯⠑⢒⣹⣉⣛⢿⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠉⠉⠉⠒⢽⣆⠉⠉⠋⠉⢭⣙⠩⣽⡙⢇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠊⠀⢀⡀⠀⠀⠀⠀⢙⡆⠀⠀⠀⠀⠀⠀⠀⠙⣻⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡇⢠⠀⠀⢸⣿⣷⣴⣿⣿⠄⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣤⠀⠀⠀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡀⢸⠀⠀⠈⠻⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣟⠀⠀⠀
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢸⡇⠘⣆⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣇⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⣀⠴⠁⠀⠀⠠⠀⠀⠰⢤⣀⠀⢀⡠⠞⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⡟⣆⠀⠀⠉⠓⠒⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢯⠁⠀⠀⢿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠸⡄⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠇⠀⠰⡀⠀⣾⠀⠀⠀⣸⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢴⠇⠀⢿⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡇⠓⠠⠐⠃⢿⠆⠀⣸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⢰⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⢸⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⡞⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠙⠢⢺⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣆⡞⠁⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⣸⠀⣇⠀⠀⠀⠀⢀⠤⣠⣴
⠀⠀⠀⠀⠀⣠⠟⠀⠀⠈⠑⠢⣄⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⢀⣹⣦⣤⠤⣄⣵⡟⠛⠀
⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣋⡤⢺⡈⠑⠚⣷⣏⡿⠃⠀⠀
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠒⠈⠉⠀⠐⠒⠽⢓⢒⣿⠋⠀⠀⠀⠀
⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠹⡄⠀⠀⠀⠀
⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⢳⡀⠀⠀⠀
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢃⡀⠀⠀⣀⡤⠤⠤⠒⠒⠉⠁⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠙⢦⡀⠃⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠱⡀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢠⠇⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀
⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⣄⣀⡤⠖⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀