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