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