⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠅⠡⡴⡁⣥⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠂⠀⠀⣼⣿⣿⣿⣿⡓⠰⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⠾⣭⣥⣿⣿⣷⠀⠀⠁⢄⢀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢣⠀⠀⣒⠁⠀⢀⠀⠀⠀⡰⠁⠀⠀⢀⠠⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢛⣿⣿⣿⣿⣿⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⣶⢮⢴⣾⣿⣿⣿⣿⣿⠢⠀⠀⠈⠢⠀⠀⠀⢤⠀⠘⠠⢇⠀⠀⠀⠀⠂⠀⠀⠠⠀⠆⣖⡁⠀⢀⠔⠉⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⢀⢣⠾⢫⠹⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣾⡿⡡⢴⣿⣿⣿⣿⡿⣿⣿⡷⠀⠀⠀⠈⢇⠀⠠⠀⠁⠀⠘⠁⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠃⠔⠁⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣆⢖⣥⡶⡜⣿⣿⣿⣿⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣺⣿⣯⠚⣤⡟⢙⣿⣿⣰⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠿⣰⠀⡀⠁⠀⡀⡨⢠⣄⠀⡰⠒⠂⠀⠀⠀⠀⠀⠀⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣴⡿⣧⢏⢺⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣺⣿⣿⣷⣵⠏⣀⢪⣿⣿⢸⢼⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⣀⡈⡨⢘⣛⣤⡤⠃⠲⡥⠆⠀⠀⠀⠠⠞⠓⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠈⣼⣿⡽⣫⢯⣿⣿⣿⡿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⠄⠈⣠⣿⣿⣿⣿⣽⠋⣨⠦⣿⣿⣿⡁⣼⠈⢿⣷⠄⠤⣦⣤⢄⡤⠀⠀⠈⠀⠈⣛⢛⣉⣖⣠⠂⢫⣥⣦⣄⢡⡍⠁⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⣿⣽⣫⣿⣿⣿⣿⣇⡝⢿⣿⢤⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⡰⠣⣪⣾⣿⠿⢿⣷⢋⡝⢁⣵⣿⡿⣽⡟⢑⣣⢰⣿⣿⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⡻⣧⠻⣿⣿⣿⣷⣄⠶⠬⣍⠀⠞⢀⠀⠀⠀⢠⠠⡄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣾⣀⣿⢡⡏⣿⣿⣿⣿⡇⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⢽⣵⣿⣿⡟⢸⡿⠃⢚⣴⡞⣿⠏⡇⡿⢃⢀⢲⣾⣿⡿⢃⠀⠀⠀⠀⠀⠀⢈⠁⠘⡙⠰⠾⠛⣉⠁⠭⠉⠉⠹⠀⡶⡀⠘⢀⠀⠀⠈⠂⠀⠁⠀⠀⠀⠀⠀⠒⠀⢠⣾⡏⣼⣿⡸⠀⣿⣿⣿⣿⣿⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⣷⣽⣿⣿⣿⠟⢡⣌⣵⣿⣿⣿⣇⡏⡄⣷⠃⠾⢙⡤⡿⣿⡓⠀⠀⠀⠁⠀⠃⠀⠀⢠⠘⠀⢰⣶⣾⣿⠐⣿⣿⣂⠂⢩⣿⠌⠄⠺⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡞⣿⣿⣷⣦⡛⢛⣿⣿⣿⢦⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⣶⣿⣿⣿⡿⠛⣿⠿⢢⣿⣿⣿⡿⠿⣿⣻⢱⣧⢟⡀⢁⡬⢁⣃⡜⠁⠪⠖⠑⣀⠀⠀⠀⠀⠎⠃⡄⠈⣿⣿⣏⣈⣼⣷⣿⣆⢈⠿⡸⠌⢠⡉⠀⠀⠋⢦⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣿⣿⣿⡏⢻⣿⣿⢸⡞⣿⣿⣿⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⣰⣿⣿⣿⠟⢁⣤⡾⢃⣴⣶⠋⣿⡟⡀⣸⢁⣣⢾⡏⣲⢨⣼⡿⠟⠉⡊⠁⢢⠦⠀⠀⢀⠀⠀⠀⠐⣀⣶⡀⠘⢟⡥⣤⣝⣿⣿⠃⠀⣿⡖⠇⠁⢠⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⢢⢺⣿⣫⣿⢸⠃⣿⠃⣿⡇⣼⠏⣿⡸⣿⢸⣿⣿⣷⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢠⣾⣿⣿⡿⡣⣨⡾⡋⡡⢫⢎⡏⢶⣿⡽⡱⣣⣿⡟⣱⠀⣿⢸⡇⡇⣬⡖⡅⠐⠋⠀⠂⠆⠠⠁⠀⠀⠂⠡⠋⢧⠀⠈⠛⠛⠉⠉⠁⠀⠤⡇⣻⡜⢁⠹⡉⠀⠠⠀⠀⠀⠀⠀⡀⢡⣶⡟⡡⢹⠇⠞⠈⡿⠀⠛⠁⠘⠠⢻⡇⣿⡏⣿⣿⢩⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠋⢠⣾⢟⠅⣞⣱⡇⣼⠇⡾⢈⣭⠶⣿⣿⢸⡏⡖⣿⠟⣥⡇⡿⣱⠓⠃⢉⠀⠐⠚⢄⡀⠀⠀⠀⠀⢁⡀⠀⠀⠀⢴⢀⣤⠀⢐⣔⡝⠘⠅⠂⠄⣡⣥⡄⠐⢆⠀⠠⢊⠀⠋⡜⣱⡫⠛⣂⣰⣦⡥⠠⣾⡩⢰⠀⣸⣧⢻⡗⣿⣿⡎⣿⣇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢤⣾⣿⣿⠟⢁⣵⣿⠃⣪⣾⠛⡿⢠⡿⡸⣑⣿⠃⣣⡿⣣⣿⠘⣾⠟⣾⣿⢸⠱⢁⡕⢃⠂⡑⠃⠀⠈⠈⠓⢀⠠⠞⢈⡘⠠⠀⠀⣚⡎⣿⡳⣸⡜⣎⣠⢃⠂⢠⠑⣹⣿⢈⢀⡥⢞⠁⢋⠚⠛⡌⣱⡞⣿⣿⣿⠇⣀⣷⣇⡇⠀⢨⡄⢼⡇⣿⣿⣇⢿⣿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⣴⣿⣿⠟⢅⣴⣿⡟⣡⣾⡿⠃⡪⡐⣼⠰⣶⡷⣶⢨⢹⡿⣸⣿⣾⢇⡇⣿⡟⡀⢞⢈⣶⣃⡾⢡⡄⠀⠂⠀⠀⡤⠺⠾⣿⣷⠾⣯⣵⣾⡿⠜⠷⢎⢫⡏⠎⠀⣺⣆⠲⠄⠂⣉⡀⠰⠿⢗⡳⠶⠔⠾⠿⣧⣿⣿⠏⢰⣸⣼⢙⠀⠐⣾⠂⣼⢳⣿⣿⣿⣻⣿⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⣾⣿⡟⠋⣠⡾⣉⡍⣼⣿⣯⢠⢆⡱⣱⡿⣱⡱⣸⢃⣿⠬⢡⣿⣿⡿⢘⣼⡿⢐⡌⢔⡟⣼⣏⣴⡟⠀⡀⠈⢀⠠⠹⣸⣧⡿⢿⡇⢠⣙⠻⣿⣷⢸⡀⠾⠷⠘⢃⢠⣍⠭⣡⡬⠉⣁⠀⠀⠀⠨⡘⢸⣷⣯⣸⠻⣿⢀⡟⢏⣇⡇⠈⣠⣿⠀⣿⣿⣿⣿⣿⣿⢋⢱⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣰⣿⣿⠿⠀⣾⢏⣾⢏⣾⣿⣿⡏⢸⡄⣱⣿⢳⡵⣱⢏⣾⢡⠂⣽⣿⡿⢡⢸⡟⣠⠟⢨⣾⢣⡎⣿⠛⡐⡼⠁⢠⣾⡟⢣⣟⢙⣷⢠⡃⡌⢻⣧⣿⣿⡞⠡⠼⠃⢃⣼⣯⣦⠜⢈⠔⠼⠃⡀⠸⣕⣀⠈⠸⣿⡇⠛⡄⠁⢹⡇⣼⣸⠇⠄⣿⡇⠁⣿⣾⣿⣿⣧⢻⢸⢸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣽⠿⣿⡏⣼⡿⣱⠿⣳⣿⣿⡿⠟⢵⣬⢰⣿⡇⠟⣴⢣⠻⢏⣴⣧⡛⣩⣶⣫⡜⠘⡱⢨⣿⢏⡦⣇⡾⣀⡜⢠⠀⠸⣿⠿⡚⠃⢾⣿⡆⣧⢻⣎⠙⢻⣽⣿⣿⣿⠈⣰⣞⢏⡡⢞⣳⡛⠀⠀⣴⣶⣿⡿⡄⠀⢻⡧⠃⣁⡄⣿⡇⡛⣿⠠⢸⣿⠀⠠⣽⢺⣿⣿⡇⣿⠸⢸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢡⣾⡟⢸⡿⣸⢟⡼⠃⣼⣿⣿⡿⠁⣿⣷⣭⡬⢭⣶⣿⣶⣿⣿⡿⢋⢿⣿⣿⡟⡤⠀⣾⢃⣚⡡⣿⣷⠉⡚⠀⠀⡄⡄⢐⡉⠑⠀⠹⢹⣿⡿⣨⡀⢻⣦⡁⢝⠻⠿⠁⣸⠹⠃⠈⠁⠦⠭⡄⠀⢰⣿⣿⣿⠂⢰⣆⠘⢁⢀⠈⢁⢹⠃⣸⡿⠀⣿⣿⠘⢸⣿⢹⣿⣿⣯⣿⠁⣾⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⢼⣿⠀⡟⣱⠫⡾⠁⣼⣿⣿⡟⠀⣼⣿⣿⡏⠖⣾⣿⡟⢡⣸⡟⣱⢸⣹⢻⣿⢃⠀⢰⢃⣾⡟⢹⡿⡇⣴⠱⠀⡘⡥⣾⢯⣻⠚⠆⠲⢸⣿⣷⠰⣧⣠⡹⣝⠢⣶⠖⢋⡁⢆⠆⢾⡇⠋⡉⠬⢠⡿⠋⢀⡄⡀⢀⣿⡄⠏⠀⠀⠠⣿⣾⡟⠇⢱⣿⠃⢸⠨⢱⣿⣿⣿⣧⣿⠠⢹⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣠⠸⢃⣦⣾⠁⠀⢅⣾⣿⣿⡟⣼⡂⢿⣿⡿⢡⡇⣿⣿⠁⣣⢏⣴⡟⣰⡆⣸⠁⠀⠀⢈⡞⢛⡀⠌⠁⢅⡧⡇⠢⡘⣷⣎⠓⢽⣷⠑⠜⠘⣿⣿⡄⠹⠇⠁⣴⣶⣎⢬⠣⣍⠼⢰⡀⢥⣶⡄⠓⠃⢀⣴⢻⠐⡔⠃⢀⠃⠄⠀⠀⡀⠽⡟⢐⠠⢸⡿⠐⡚⢰⣾⣿⣿⣿⢻⣿⡄⢸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣰⣿⣶⣿⣿⢁⠿⢏⣾⣿⣿⠟⢸⣿⣧⢌⣩⣴⢸⣿⣌⡓⢀⣵⣿⠏⠄⠛⠁⠋⡄⢠⡾⠿⠁⢋⠃⣆⠂⣼⢙⠐⠟⠛⣿⣿⣷⣦⠙⣦⢸⡆⣿⣿⣷⣠⣄⠠⡀⠈⠉⠦⠇⢈⡉⠋⠣⠀⠏⢈⣷⣿⣾⡶⢖⡌⠀⡴⠈⡐⣰⡖⠀⠃⢀⢁⡛⡀⢘⠇⣿⡇⣿⣿⣿⣿⣿⣹⣿⣇⠸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡟⢰⣿⣿⣿⣿⠃⣼⠃⣼⢏⢸⣿⢃⣿⣿⢋⣾⣿⣿⠇⣭⡭⢶⣷⣭⣵⡣⠄⢠⡾⡇⢧⡄⠁⠀⠀⠋⠀⠛⠐⣨⡀⠠⣰⣔⠙⢿⢿⡟⠠⣋⢸⡧⠙⠒⣨⡝⢩⣦⠇⠁⣤⠘⢤⡌⠀⢰⠠⠐⡄⠉⠙⠫⡁⠠⠤⠂⢤⠀⣠⢃⢻⠁⠀⠀⢀⡿⡄⠀⣸⢰⣿⠃⣿⣿⣿⣿⣿⣽⣿⣿⠀⡇⠀
⠀⠀⠀⠀⠀⠀⠀⡆⠈⠀⣿⣿⣿⣿⡏⠀⢇⡿⡇⣸⢸⡇⣸⣿⠇⢸⣿⣿⡇⣾⣿⣇⣾⣿⢿⣿⠇⢀⡎⠀⠁⣾⡇⢀⠀⠀⠈⠀⠀⣀⣾⡇⣸⣿⣿⣿⣿⡆⠀⣷⢱⠉⣿⠆⠀⠀⢹⣇⡆⢁⣾⠸⡀⠀⠉⠇⠁⡇⡆⣶⡿⡇⢀⣷⡶⠿⣆⠹⡆⠁⢸⡀⠀⠁⡇⢹⣷⠇⣶⡿⣸⣿⠀⣿⣿⣿⣿⣿⣾⣿⠀⡆⠇⠀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⣸⣿⣿⣿⠋⠀⡸⡸⠝⢰⡇⠟⢠⣿⡟⡰⣿⣿⢈⣴⢫⡻⣿⢟⣵⣷⠉⠀⠈⠀⠀⢠⡟⠒⠋⠀⠀⠀⠀⣰⠿⣀⣀⡀⠹⣿⣿⠟⠠⡀⠻⠰⠺⡏⠜⡢⡙⠖⠆⣶⡮⠈⠃⠩⣒⡟⡴⠀⢇⢏⠯⣓⡃⠼⠃⣐⢣⡠⠴⠂⡀⠀⠷⡀⢲⠁⣼⡾⢠⡟⡀⢃⣿⡄⣿⣿⣿⣿⣿⣿⣻⠀⡡⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡱⣿⣿⡿⠃⣾⣀⠣⣼⣿⡜⢟⡰⣜⠿⢃⣅⣛⣵⡿⢱⡎⡏⣏⡾⠛⢋⠀⠀⠀⠀⠁⡎⠀⠀⠐⠐⡉⠄⠐⢉⢺⣟⡛⢿⣦⣹⡏⢀⠴⡄⢡⠐⠀⢿⣤⠿⠘⠠⢜⠷⠉⠺⠖⠀⠉⠙⣄⣀⠾⣅⡍⠒⠠⠤⠌⠹⣷⣦⠃⠡⠁⠀⠀⢒⡘⠀⢹⠧⣾⣿⠀⣸⣏⢸⣿⣿⣿⣿⣿⣽⣿⠀⢧⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⢳⣿⡿⣡⡺⡋⡋⡆⣾⡿⠙⣿⢏⣶⡊⢹⡿⣁⣭⣶⢸⣳⡇⠫⠀⠀⢈⢰⠃⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢄⣤⣟⣿⣏⠀⠘⢓⡵⢃⡆⠇⢿⡄⢮⠤⠃⠔⠋⣠⡄⠄⠀⠀⢠⣁⣀⣉⡀⣤⠆⠐⠘⠋⠱⣶⡉⠓⠈⣿⡄⠀⠀⠀⡈⠠⡇⠀⢀⠀⠻⢠⠚⢻⣽⢺⣿⣿⣿⣿⣿⣿⡇⠆⠸⠀⡄
⠀⠀⠀⠀⠀⠠⢰⡏⣿⣿⣇⣿⡟⡼⣸⠅⣿⠇⢲⢏⣾⣿⡟⣬⡎⡇⣿⡟⣾⢇⠾⠿⠀⠀⠀⠏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡗⡀⢙⣦⠈⡄⠀⠸⣿⣦⣠⢶⠮⡁⢠⣴⣦⠰⣇⠀⠀⠚⠉⢛⣠⡌⢉⣉⡉⢒⣉⣽⡀⠀⢸⠂⠀⠀⠰⠁⡄⠀⠀⠀⠆⡆⢀⠷⢵⢸⣹⣿⣿⣿⣿⣿⣿⠸⠁⠂⠀⡇
⠀⠀⠀⠀⠀⠄⣿⣸⣿⡿⠩⡛⣼⢣⣿⣇⠻⢠⣾⣿⠏⣾⢣⣿⢸⢵⡿⡸⢣⠂⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⠟⢩⡴⠒⠀⣿⡆⢱⠀⣿⣿⣿⠋⣪⡞⣵⣿⣿⡟⣃⠁⠀⠥⣔⢒⣪⠙⣇⠺⡲⠸⢤⠏⠚⡇⠀⢨⠀⠀⠀⠙⠐⠃⠀⠠⠀⠸⠐⡼⡀⡿⣼⣹⣽⣿⣿⣿⣿⡿⠀⠄⡁⠀⡁
⠀⠀⠀⠀⠐⠀⢁⡏⡿⣱⢧⣰⡿⣼⣿⣿⣷⡘⡿⠁⣰⡏⢟⣅⡟⢬⠔⠀⣋⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⣉⠀⠈⠀⠀⢰⣚⢤⡯⢆⣿⡿⠃⠀⣀⢾⣿⣿⣿⣿⠋⣰⠋⡴⣇⡌⠉⠁⠘⢆⢝⣃⠈⠓⢹⣧⠀⠀⠀⠀⠀⠀⠀⠂⠁⠐⠆⠀⣱⢱⡇⠄⣿⢼⣿⣿⣿⣿⠟⡧⢈⠐⡀⠀⡄
⠀⠀⠀⠀⠀⠰⣼⣈⣵⣿⢸⣿⢣⣿⣿⣿⣿⣿⠗⠀⣴⢹⣻⢉⣾⡇⠀⣾⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠫⠈⠀⢋⣄⠀⠀⠀⡯⢧⢲⠀⣴⣾⠏⣰⠏⣼⠟⡿⢟⠕⢀⣜⡒⠒⠉⡍⠐⠒⢦⠈⢒⠍⠠⠀⠃⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠁⡀⣾⠙⣠⠈⢸⣿⣿⣿⣿⣾⡗⢀⠂⠄⠀⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⣿⣼⣿⣿⣿⡿⢡⡆⢸⣿⢸⠁⢰⣿⠀⠁⡿⠆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠋⠀⠀⠀⠰⢸⣿⡀⠀⠀⠐⡋⣡⣾⣿⣜⢓⢋⢚⡋⡖⠵⠃⢀⣿⣿⡇⠀⠉⠆⠢⢤⠎⢀⡼⢺⡟⠤⣈⣿⣿⠀⠀⡐⠀⠀⠀⠀⢀⡏⠀⢀⢀⢁⠄⠨⠀⣼⣿⣿⣿⡏⣿⡇⡀⠈⠂⠀⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢾⡇⣿⠟⠉⣿⠇⠘⠠⢼⡟⢎⡀⡸⠇⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣴⣿⣿⢻⣿⠸⢸⠨⠆⡠⣢⠮⠘⠋⣿⡗⣀⠈⠌⠙⠒⠒⠀⠀⡨⣀⠀⠀⢻⡇⠀⡆⡅⠀⠀⠀⠀⠜⡡⠀⠀⠱⣸⠀⠀⣾⣿⣿⣿⣿⡇⣿⠃⣀⠠⠁⠀⡅
⠀⠀⠀⠀⠰⢺⣿⣿⣿⢰⢼⢹⡏⠀⠀⠟⠀⠀⡎⠘⡇⣬⢰⠁⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠇⠀⠸⠿⠛⠛⢋⣿⣦⣽⣾⣜⠅⠙⡱⣫⠆⣿⡧⠟⠀⠀⢈⠀⣐⣀⠰⣀⠴⠿⢃⠘⠇⠀⠇⡆⠀⠀⠀⠀⠋⠁⠀⡃⠅⠃⢀⠘⠋⠙⡜⣿⣿⢧⡟⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡞⢸⣿⣿⣿⠈⡎⣿⠃⡀⠀⣠⣴⡆⠀⠈⢡⡏⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣼⣿⣿⣿⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢰⠈⢁⣵⣿⣿⣿⣿⠿⢋⣥⣬⣾⡧⣤⣿⡗⠀⠀⢰⠇⠠⣟⢈⠆⠘⠆⣉⢀⢲⡇⠨⠁⠀⡅⠀⠀⠀⠀⠀⠰⢀⠀⣸⣹⠇⡇⢐⠇⣸⣿⢸⢃⠀⡄⠀⠀⠀⠁
⠀⠀⠀⢰⡧⣾⣿⣿⡏⡀⠛⢉⡾⠈⣸⡏⠛⠃⠀⠀⣼⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣼⣿⣿⡿⠟⡉⢀⣴⣿⠟⠉⡀⣀⢀⣶⣷⠀⡀⡈⠆⣉⣤⣤⠂⣘⠶⣿⣿⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢆⠏⣰⡇⡰⣠⣿⡿⡹⠆⠸⠀⠀⠀⠀⠀
⠀⠀⠀⠘⢰⢻⣿⣿⡏⢸⣿⠀⠆⠀⠛⣿⠆⢠⠀⠀⠈⠂⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠁⠠⣀⣴⠿⠋⣡⠎⠀⣄⠿⢸⣿⣿⡇⢠⠣⡀⠉⠄⢢⠅⠀⠀⣂⣽⡧⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢁⣰⢀⢻⣇⢑⣿⣿⢇⣇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⣸⣿⣿⢇⢸⣿⡄⠀⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠟⠀⠑⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢉⣤⣶⠿⠋⠁⠀⢠⣏⡀⡞⠈⠈⠁⢻⣿⡇⠈⠃⠀⣙⡁⠎⠡⠀⠀⣿⢸⠃⠀⢀⠀⠀⠐⠀⠀⢀⠀⠀⠌⠀⠄⢫⠏⡁⢎⣽⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣬⢼⣿⣿⢸⢸⣿⣿⠀⠁⡇⠀⡀⢸⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⠁⠀⢀⣀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠟⢉⠠⢒⣤⡞⢁⣬⣾⠗⠀⣤⡷⠶⢸⣿⡃⢸⡄⠀⡜⢧⠰⢲⢸⠀⡟⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢢⣴⢃⣼⣿⣿⣇⣿⠊⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⡻⣼⣿⣿⠈⣾⠻⠏⡠⠰⠃⠹⢣⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣴⠋⣘⡈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣵⣾⣿⣫⡾⢟⠛⣡⣴⠿⠫⠄⠸⡌⢿⣿⠈⣷⡇⠃⠘⠀⠘⠀⠀⠡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢔⡿⡃⣾⣿⣿⡿⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠇⣿⣿⣟⡰⣷⠘⠀⠁⠀⠀⠊⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣩⠈⠈⢃⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⡋⠡⣰⠟⡋⢁⢰⠆⠀⡆⠱⢨⣿⡅⣟⠇⢈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢠⡿⡱⣹⣿⣿⢋⣵⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⠇⣿⣿⡏⢧⠿⠅⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠹⣄⠀⣸⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⣋⢠⣢⠖⡊⢔⣠⢾⡏⠏⠀⣀⣀⠄⠀⢿⣷⠨⠃⠈⠀⠀⢃⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠘⠄⢰⣿⣿⢣⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⡞⣿⣿⡇⠸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⢪⡴⠋⠔⢉⣴⣿⠃⠈⡀⣠⡞⠹⣿⡀⠀⢺⡇⠀⢀⠀⠀⢰⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡆⠈⢀⣼⣿⢏⣾⡿⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠀⣿⢿⣿⣿⡇⢀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣽⣰⣾⣿⡿⠋⠀⢁⠂⡟⠀⢸⡽⠅⠃⠨⣷⠈⢸⠀⠀⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣼⠃⢠⣾⣿⣏⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣿⣸⣿⡣⠇⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⢿⣿⠿⠁⡁⢸⣷⡿⢠⡗⠂⡇⢢⢠⢰⠄⢺⣦⠀⢌⢃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⡴⣼⣿⡟⣼⠋⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣿⣿⣽⠃⠄⡏⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⢰⣔⠈⠀⡁⢾⢟⣵⢺⡇⠁⣇⢸⡏⠠⡄⠈⣶⡆⣄⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡐⣴⣿⣯⠚⠁⠌⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢱⣿⣸⠀⠁⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠿⡘⠃⢁⣴⣿⣿⡌⣿⡆⢹⠀⣗⡂⣇⡁⣄⢨⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠏⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣿⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⢀⠱⣴⠛⠿⣿⣿⣿⣧⣄⣾⠃⡻⣧⢿⡿⠀⡒⡄⡔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠐⠀⡀⣼⣿⡏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡄⠸⡏⡇⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠌⢿⣠⣾⣿⣿⣿⣿⣿⣿⣿⣆⢿⡸⣧⠀⠇⢃⠀⠘⡀⡂⠀⠀⠀⡀⠀⠀⠀⠀⠁⢤⡎⠼⣾⣿⡿⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢱⠀⡇⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠈⢻⣿⡿⣿⣿⣿⡇⣿⣿⣿⣼⡇⣿⣇⠚⡈⡝⣦⠡⡅⠀⠀⠐⠀⠀⠀⠀⡠⢠⡆⢠⠞⣛⠿⠃⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⣟⣽⣿⣿⣿⡇⢿⣿⣿⡏⣧⢿⢷⠠⢙⢧⢿⠰⣡⠀⠀⠀⠀⠀⠀⠀⣠⢇⡘⢋⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⠿⣹⣿⣿⠃⢻⣿⣿⣷⣿⡞⠾⢎⠼⣌⡜⡜⠦⢃⠀⠁⠀⠠⠁⠖⢩⡎⡵⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀