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