.
                        

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