⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⢀⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⠀⣾⣿⣿⡿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣄⣠⣾⣿⣿⣿⣿⣿⢣⣿⣿⠟⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⢱⣾⣿⣿⣟⣽⠋⣾⢿⣿⣾⣿⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣛⣭⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⢸⣿⣿⣿⣿⢀⣼⢃⣿⣿⣷⡿⠇⠀⠀⠀⠀⣿⣷⣿⣿⣿⣿⣿⣿⣿⢣⣼⣿⡿⢟⣥⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠸⣿⣿⣿⣿⡿⡟⠿⣿⣿⣿⢏⣼⣿⣿⣿⣿⡄⢡⣿⣿⠟⢁⣴⣧⠀⠀⠀⣠⡀⠑⢿⣿⣿⣿⡹⣿⢃⣛⣫⠭⠂⢉⣁⣤⠭⣑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣿⣿⣿⣿⣧⢻⣌⢢⣭⣵⣾⣿⣿⣿⣿⣿⣷⠘⣿⠋⢠⣿⣿⣿⣇⠀⣴⣿⣿⣆⠈⠻⢁⣶⣶⠢⠘⡟⠁⢠⣴⠿⣫⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢸⣿⣿⣿⣿⡎⢿⣧⢻⣿⣿⣿⣿⡿⠿⠿⠿⠆⠹⠀⣾⣿⣿⡿⣿⣶⣿⡟⣩⣥⢄⠀⣾⣿⢃⣿⠗⠀⡰⢟⣥⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣿⣯⢹⢿⣿⡘⣿⣇⠻⢟⣩⣶⣾⣿⣿⣿⣿⣿⣶⣭⡛⢿⢱⣿⣿⡏⢠⣭⣍⢨⣥⡈⠍⠬⠉⢁⣠⣰⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⣿⡈⠘⣿⣷⠹⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⢿⣿⣷⡌⢿⣿⡸⡟⠀⣠⣤⣶⣌⢻⣿⣿⢏⡴⠀⠀⠀⠀⠀⠀⠂⠀⣾⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠲⢖⣮⣭⣥⣷⣦⣭⢱⣿⣇⣀⣀⣀⣀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣹⣎⢿⣿⢸⣶⣶⣆⠄⠘⣿⣿⡿⠁⡸⢛⣵⣿⣗⣄⣀⡤⡠⡪⣱⣴⣍⠀⠈⠙⢿⣿⣶⣦⢤⡴⠖⠀⠀⠀
⠀⠀⠀⠹⣿⣿⣿⣿⢇⣿⣿⣿⣿⣿⣿⣿⡄⣾⣿⡇⢸⣿⣿⣿⣿⣿⣿⡜⣿⡎⣿⣿⡏⣔⠆⡚⢛⣁⢰⣾⣿⣿⣿⣿⡿⣫⡞⠀⣌⠙⢋⠉⠀⠀⠀⠀⢋⣵⣾⠏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢹⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣧⣿⣶⣄⠀⡤⠀⣴⢲⣶⣶⡤⠠⠤⠹⠛⡟⣴⣿⡇⢀⣏⠉⠉⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠉⠛⢿⣿⡀⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡏⣿⣿⡟⡼⠿⣦⠹⣸⣿⡟⢀⣎⣾⣟⡀⣸⣿⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢻⣇⠘⢿⣿⣿⣿⣿⣿⡇⣿⣿⡁⢈⣿⣿⣿⣿⣿⣿⢱⡟⡟⢸⣿⣿⡟⠁⢰⣶⣶⣾⣿⣿⣿⠇⡋⢉⣴⣿⣷⣄⡀⠀⠀⠀⠀⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⠈⠃⣀⣤⣄⡀⠀⢀⠀⠀⠄⠀⣀⣤⣀⡼⢣⣿⣿⣰⣿⣿⣟⠀⣾⢸⣿⣿⠿⣛⣭⣦⣝⣱⣿⣿⣿⣿⣿⡿⣱⡟⣹⡿⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠠⠴⠿⠿⣿⣧⡀⠈⠙⠿⣷⣤⣴⣬⣤⣤⣬⣿⠿⠋⣤⢻⣿⣿⣿⣿⣿⣿⣿⠌⢘⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣶⡾⠿⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢷⣤⣀⣀⠈⠉⠉⠉⢉⣉⠀⢶⣇⠾⡧⠙⣿⣿⡿⢟⣫⣴⣿⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⢣⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣄⠙⢠⠁⠼⠙⠡⣾⣿⣿⣿⣿⡎⠉⠉⠉⠉⠉⢹⣿⣿⣿⢿⣿⢣⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⠿⢿⣛⢛⣫⣭⣥⣭⢅⠀⠀⡠⣤⡌⢭⣽⣿⡟⣟⡁⠀⠀⠀⠀⠀⢸⠿⣋⣵⣿⢣⣿⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣯⡭⣲⣾⢣⡿⣣⣿⡿⢛⠭⠂⠁⠠⠀⣮⠻⢿⣎⢿⣿⣷⢹⡇⢐⣴⣤⣥⣴⢰⣬⣭⣙⢁⡩⠶⢿⣿⢟⣩⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠁⠉⠉⠉⠗⠈⠀⠀⠀⠀⡀⢠⢻⣷⣌⢽⣧⣻⣿⣧⠁⢸⣿⣿⣿⣿⢸⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⢢⠹⣷⣿⣿⡇⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡿⠿⠿⠏⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢿⠛⢿⢾⡇⡙⣿⣽⡇⡼⣿⣿⣿⣿⠸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠹⠇⣿⣌⢿⡇⣗⠝⣛⡻⢛⡄⠀⠉⠛⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠎⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠟⡶⠷⠄⠀⠀⠀⠀⠀⠀⢤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠰⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⢷⣄⡈⠉⠉⠍⠀⠛⠛⠀⡐⠀⠀⠠⠀⠈⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢐⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄⠀⠙⢿⣦⠄⠀⠀⠀⠀⠀⠁⠬⣀⡀⠀⡶⠀⣤⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⡀⠈⠟⠓⡀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡝⠀⠀⠀⠠⠀⠀⠀⠀⠢⠁⡀⠀⠀⠀⠀⠈⠻⠦⠈⠛⠀⡶⠘⢐⠀⡀⣀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠂⠀⠈⠀⠉⠂⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⣠⣿⠀⠀⠀⠠⠀⠀⠁⠀⠀⣿⡆⠀⠀⢀⣄⡀⠀⠀⠀⠀⠪⠁⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⢼⣿⣿⠀⠀⠀⠀⠁⠀⠀⣤⠈⠛⠇⠀⠀⠘⢿⡇⠀⠚⠃⣔⣒⠤⠤⣤⣄⠀⢀⣒⡀⠀⣤⣤⡄⠀⠃⠀⢀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣼⣿⠃⠀⠀⣼⣿⠋⠂⣠⣿⠂⠀⠂⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⢂⠀⠀⠘⢟⡇⠀⠙⠁⠀⠊⠩⠉⠈⢌⠁⠀⠌⠈⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢨⡈⠀⣿⡿⠀⠀⢰⢿⠃⠀⠀⠈⠁⠀⠀⠀⢷⣦⣄⠀⣀⣠⣤⣅⡀⠀⠐⠶⠀⣀⠀⠁⠀⠀⢀⠀⣀⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢷⢰⣿⠇⠀⢀⡯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠿⠟⢿⠟⠿⠗⢄⠀⠀⠀⠀⠀⠀⠁⠁⠐⢻⠿⠿⠦⠄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⡿⣀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁⣠⡀⣹⡁⣠⡀⣿⡎⠛⠀⢀⠀⣠⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢈⠎⠀⠀⠰⠀⠀⠀⠀⢠⣤⣤⠀⠀⠈⣿⣿⣿⡿⠿⠿⣿⣿⣿⠇⠀⠀⠈⠀⠀⠐⠻⠿⢿⠿⢷⣦⠄⠈⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⠎⠀⢶⣆⠀⠀⠠⡄⢄⠀⠀⠉⠁⠀⠛⡌⠙⢏⣠⣶⣆⣸⣿⠏⠤⣿⠆⢠⣄⣀⡀⠀⠀⡰⠈⠉⠉⠀⠁⠑⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠏⢾⡄⢈⠋⠀⠀⠀⠀⠘⠈⢀⡀⢀⡀⠀⠠⠀⠀⠈⠍⢫⣥⣄⣀⠀⠀⠀⡀⠑⠨⠿⠏⢀⣶⣶⣦⡄⠀⠀⡀⠂⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⠀⠘⢿⣄⠀⠁⠢⠀⣀⠀⠀⠀⠁⠘⠿⣷⣦⣄⣀⠀⠀⠈⠉⠛⠿⠏⢂⣤⣤⣀⠀⠀⠸⠀⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⠻⣦⡀⠀⠀⠙⠀⠛⠆⠀⠀⠀⠀⠀⠉⠘⠿⣢⣶⣿⣿⣿⣿⣶⣍⠛⠿⣿⣷⣤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠳⣄⠀⠈⠻⡗⣠⡀⠀⠀⠀⠀⠂⢠⣦⣄⡀⠈⢰⣿⠋⠉⢻⣿⠉⠉⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠙⢷⣄⠀⠈⠙⠏⠁⢤⣄⠀⠀⠀⢌⠙⠛⢷⠸⣿⣦⣤⣾⣿⣦⣤⣿⣿⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠙⠷⠄⠈⠀⠀⠌⠛⠛⠲⠤⠀⠀⠀⠀⠀⠹⣿⣇⠘⠛⠉⣽⣿⠋⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠙⠓⠀⠈⠀⠐⠻⡶⠁⡀⠀⠄⠀⠐⠆⠀⠑⠂⢄⣀⠀⠉⠛⠒⠛⡋⠴⠶⠶⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠙⡇⠂⡀⠀⠀⠐⠀⡤⠌⠀⠀⠈⢁⠀⡀⠓⠀⣉⣠⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃⠐⠀⡈⠀⠀⠐⠢⠀⠀⠠⠀⣀⠀⠈⠉⠉⠉⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠈⣁⠀⠀⠰⠀⠀⠠⠀⠁⢉⠄⠈⠻⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣟⠂⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠢⢍⠉⢢⠀⠀⢠⢠⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⡈⠀⠀⠇⠀⠀⠁⡇⠀⠡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠀⠀⠸⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠔⠁⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⡤⠒⠒⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡌⠀⢠⣀⡀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠿⠶⠶⠶⠷⠷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀
⢣⠀⠈⠉⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⡄⣀⣀⣀⣀⣈⢽⡟⠁⠀⠀⠀⠀⠀⠀
⠀⠑⠢⠤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣄⣼⡅⠀⠀⠀⠀⠀⣠⣞⠉⠉⠈⠈⠙⠉⠘⠃⠀⠀⠀⢴⣦⣤⣶⠆⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣄⣀⣠⣠⣄⡀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠃⢀⣘⠯⠁⠀⠀⠀⠀⢘⣉⣀⣠⣤⣠⣤⣴⣶⣦⣾⢿⢶⣶⡶⡶⡶⣿⢶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡍⠉⡍⠉⠀⠀⠀⠀⣰⡀⡀⢠⣀⣰⠴⠶⠺⢉⠈⠁⣀⣠⣴⣠⣿⣽⣿⣶⢿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣵⣿⣿⢏⡮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⣶⣇⠀⠀⠀⠀⠀⠀⠛⡗⠋⠀⠀⠀⢄⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣟⣷⢹⣿⣿⣿⡷⡧⡧⣿⣿⠾⠼⠶
⠿⠾⠶⠶⠷⠿⠖⠀⠀⠀⠀⢴⣳⢐⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⣿⣿⡿⢿⠟⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣦⣺⠀⠀⡀⠀⠀⠀⠀⠀⢸⡿⢻⠳⠆⠀⠀⠀⠀⢠⣤⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⡶⠂⠀⠀⠀⣴⣦⣶⡄⣀⠀⠀⠀⠀⠀⠀
⠈⣿⡻⡏⢰⠷⠀⠀⠀⠀⠀⠚⠀⣨⣇⠀⠀⠀⠀⣠⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⢷⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣷⣶⣶⢀⣬⡉⣿⡟⣿⣧⠀⠀⠀⠀⠀
⠀⣹⣶⣇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣬⣷⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠁⠀⠙⠉⠉⣼⣿⣅⡈⢾⡟⠛⠀⠀⠀⠀⠀
⠀⣌⡿⠓⣞⡏⠀⠀⠀⠀⠀⠀⠀⣀⢠⣿⡿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡤⠀⠀⠀⠀⠀⠈⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡼⠀⢾⣽⡦⠄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡷⣾⣿⣿⣷⣿⣯⣟⣧⣷⣿⢿⣿⣿⣿⡿⣿⡛⠙⠀⠉⠛⠉⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣟⢟⣷⣿⣿⣿⣿⣿⣿⡿⡋⠁⠘⡄⢳⡀⣀⠦⢤⢴⣷⣹⣿⢿⡿⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣿⣿⣿⣯⢻⣿⣿⢿⡿⠛⠀⣻⡟⠿⠋⠙⢿⣿⣿⡿⠀⢠⣀⠼⢦⠤⠤⠔⠎⢠⣷⡟⣻⡸⡿⢮⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢩⡞⠛⢿⡿⣿⣿⣿⣯⡝⠋⢀⣴⣿⠟⠀⠀⠀⠀⣀⠉⣠⣤⡀⢏⣠⠔⠋⠀⠀⠀⠀⢠⣿⡍⢻⢳⣶⡎⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣵⣌⢿⣾⣿⢟⣵⣿⣷⣄⣙⣅⠀⠀⠀⢠⣾⣿⠗⠙⠻⣿⡇⠀⠁⠀⠀⠀⠀⠀⠈⠉⠁⠿⠗⠉⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣲⣞⡖⡵⣯⡍⠉⠈⠛⠁⠙⣫⣽⣿⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣓⣺⡗⠮⢽⡟⠀⠀⠀⠀⠀⠈⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀