⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣟⠓⠀⡀⣿⣷⡴⣦⣴⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠕⢡⢠⣿⣿⠿⣼⢬⣻⣽⣿⡿⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⡴⣚⠽⢈⣳⠘⡱⢿⣾⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⢟⡫⢬⡵⣶⢦⣌⣯⠵⣎⣶⣻⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⠝⡉⠀⠀⠀⠀⠈⡀⣏⠿⣹⡛⡿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢶⢈⡤⠀⠀⠀⠑⢀⠀⠑⢲⣿⣮⣶⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⢿⡛⢿⢤⣀⠘⢧⣤⣤⣟⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠷⣾⣦⣽⣷⣦⣑⡬⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⢶⠴⡴⣴⠾⣉⠦⡁⠀⣽⢹⣿⣿⢷⣦⣀⣀⣀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡯⣝⢣⢏⡴⢣⠝⠈⠀⠀⠐⠨⠎⠀⠉⠯⡡⡝⠿⣿⣛⢻⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⣾⡱⣍⢞⡢⣗⡃⠀⠀⠀⠀⠀⠀⠐⠀⠃⢐⡒⠄⠐⠨⣻⣦⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢺⣷⣄⠀⠀⠀⠀⢿⣷⡞⣬⢲⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⡠⢄⠀⠀⠀⠀⠉⠛⣿⡇⢀⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⣸⣿⡽⢆⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⢞⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣦⣤⣦⣟⡸⢜⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⣿⣿⣽⣿⣆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⢔⣩⠞⡄⠠⠀⡀⠀⠀⠀⢀⢀⠄⠁⠀⠄⠈⢢⡀⠀⠀⠀⠀⠄⠂⣼⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣤⣄⡈⡘⣦⣄⣀⣀⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠙⣦⣀⣀⣀⣰⣴⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⡽⣿⣼⡇⠀⠘⢿⣛⡋⠀⠀⠀⠀⠀⠀⠀⠠⢸⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣿⣿⣿⢃⠀⣀⣼⡵⠌⠀⠀⠀⠀⠀⠀⠀⢀⢒⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣐⣦⣦⣤⣽⣿⣿⣿⣿⣿⠿⣚⠭⡓⠬⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣿⣴⣦⣤⣾⣿⢿⡿⣽⣷⣻⣽⣻⣟⢯⢣⠏⣔⡣⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢨⢻⡿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣴⣿⣿⣻⣽⣿⣾⢿⣻⣯⣷⢿⣻⣿⢏⡎⢧⢚⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣸⡹⣤⠀⠀⠀⠀⠀⠀⠀
⣿⡿⣽⣿⣻⣷⢿⣻⣯⣷⣿⢿⡟⣏⠞⡼⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀
⣿⡿⣿⣽⡿⣽⣿⣻⡷⡟⣥⢫⡜⣬⠛⣜⡱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡐⣿⣿⡿⠀⠀⠀⠀⠀⠀
⣿⣿⣻⣷⡿⣿⣷⣯⡳⡝⢦⠳⡜⢦⡛⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣷⡀⠀⠀⠀⠀
⣿⣷⣿⣷⣿⡟⣿⣿⢳⣭⠓⣯⠙⣦⢹⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢻⣾⡟⣿⡄⠀⠀⠀
⣿⣯⣷⣿⣽⣿⣿⣏⢶⡩⢞⡬⣓⢎⠧⢣⠆⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣙⠲⣬⢹⣿⣿⣶⠀
⣿⣯⣷⣿⣳⣿⣿⣎⢧⠻⣜⠲⣍⢮⡙⢦⢫⠀⠀⠀⠈⠄⠀⠀⠀⠀⡡⠋⠀⠀⠀⡙⣬⢓⢦⣋⠾⣽⣿⣦⠴⠀
⣿⣯⣷⣿⣻⣿⣿⣞⠯⣝⢮⣙⣖⣢⡝⣊⠖⣣⠀⠀⠀⠈⡘⠀⠀⡟⠁⠀⠀⠀⣪⢱⢊⡍⢶⡩⢞⣽⣻⣯⠐⠀
⣿⣿⣳⣿⢯⣿⣿⣾⣛⣼⣾⣿⣿⣿⣿⣿⣿⣦⣅⠀⠀⠀⡐⠟⠀⠀⠀⠀⣴⣼⣾⣷⣿⣿⣾⣷⣺⣟⣿⣜⠀
⣿⣿⣟⣯⣿⡿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢃⣩⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀
⣿⣯⣿⣟⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡧⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡖⠀⠀⠀⠀⠘⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠀⠀⠀⠀⠒⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠂⠀⠀⠀⠀⠘⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⠀⠀⠀⠀⢹⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢫⢿⡄⠀⠀⠀⢀⡠⠤⠄⡀⣸⡞⡄⠀⠀⣠⣳⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⢸⣏⣷⡀⠀⠀⣯⣿⣶⣮⣽⣻⣏⣷⣠⣴⣳⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣠⣸⣮⣽⣇⠠⣔⡺⣿⢿⣿⣛⠿⣿⣷⣿⣿⢿⣧⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣽⣷⣻⣵⣿⣷⣾⢿⣾⣿⣿⣿⣶⣫⣟⣿⣿⣷⣿⣵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣟⣵⣿⣿⣟⣻⣭⣾⡟⢿⣿⣯⣿⣿⣿⣾⣿⣿⣿⣽⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⢋⣿⣿⣿⣿⣿⣽⣒⠉⢉⣙⠻⠿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠓⠉⡙⠻⣟⡻⣽⣹⠀⣾⣧⣿⡿⢷⣼⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠞⢻⣿⡟⢡⣶⣷⣿⣧⢻⡳⢧⣻⠄⠋⠋⠋⠷⠜⠁⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣉⣻⠜⠉⠈⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣀⠖⢀⡴⠂⠀⠀⠀⠀⡸⢹⣿⣿⣿⣿⣿⣿⣿⣾⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣯⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣟⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣍⡒⢠⣤⣀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⣗⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣯⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⠦⣶⣖⣤⣶⣾⣿⣿⣿⣿⣿⣙⣀⣀⣀⡀⣤⣤⣼⣿⣿⣿⣿⣿⣟⡻⡽⢿⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡞⣧⣛⢮⡽⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣾⡷⠫⣝⠾⣩⢻⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣳⢺⣵⣫⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣲⣿⣿⡿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀
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⠀⠀⠀⠀⣠⣿⢿⣽⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠊⠀⠀⠀⠀⠀⠀⢀⣤⡒⢲⣤⣤⣤⣄⠀⠘⢿⣿⣿⣿⣿⣿⢿⣻⣷⣻⣿⣧⠀⠀⠀
⠀⠀⠀⣰⣿⣿⣟⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⣿⣿⠄⠀⠀⢻⣿⣿⣿⣾⣿⢿⣾⣿⣿⣿⣧⠀⠀
⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⠟⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠋⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀
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⠀⠀⠀⣽⣿⣿⣿⣿⣿⣷⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀
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⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⢀⣀⣠⡖⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀ ⢸⠊⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀