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