⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣷⣮⣵⣶⡴⡠⠶⣧⠟⣠⡞⣭⢶⣉⣶⣈⡽⣯⠿⣿⣻⣟⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⢾⡽⣯⣟⡿⣿⣶⣤⣢⣽⡞⣯⢻⡝⣮⢏⡷⣭⣛⢶⡹⢮⡽⣭⢟⡦⣄⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡿⣽⣳⣯⣟⡷⣯⣟⡳⣝⢾⡱⣏⢾⡱⣏⢾⡱⣏⢾⣹⣧⢻⡜⣯⢞⡽⣺⣝⣻⢭⣿⣟⡷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣟⣿⣻⣽⣟⣯⢿⣽⣟⡳⣎⢷⡹⣎⢷⡹⣎⢷⡹⣎⢷⡽⣾⡹⣎⢷⡹⣎⠿⣜⡳⣎⢷⣫⢶⡯⠿⢭⣿⣾⣿⢷⣦⣀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡾⡿⠿⠛⣿⡾⣽⣟⡳⣎⢷⡹⣎⢷⡹⣎⢷⡹⣎⣷⡟⣯⢳⢧⡻⣜⣧⢿⣹⢻⣜⡳⣝⢮⡳⢯⡽⣦⢿⢽⡕⣿⣿⣞⡿⣷⢦⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠰⣯⣽⢎⣿⡿⣜⡳⣝⢮⡳⣝⣮⢷⡹⣎⢷⡽⢧⡻⣜⢯⡞⣵⢯⣛⢮⡳⣏⢾⡱⣏⢾⡹⢧⡻⣭⢯⣶⣿⣷⢯⢻⣽⣯⠋⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢽⣷⢾⢘⣯⡳⣭⢳⡝⣮⢳⣽⣣⢏⡷⣹⡾⣝⢧⡻⣜⢧⡿⣝⢧⣛⢮⣳⣯⢞⡵⣫⢞⡽⣣⢟⣼⢫⣿⡟⣿⣽⢫⡷⠃⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣼⣼⢖⣾⡳⣝⣾⢫⡞⣵⡻⢧⣛⣮⢿⡟⣵⢫⡞⣵⢫⡾⣝⢮⡳⣽⢾⢍⣷⢫⡞⣵⢫⡞⣵⢫⢾⡳⣞⣗⣺⣵⠟⠒⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡅⣧⡾⣳⡝⣾⣵⣫⣞⣵⡿⣟⡽⣎⡿⣺⢵⣫⢞⣵⡟⣿⡹⣷⢿⣯⠋⣸⡜⣧⢻⡜⣧⢻⡜⣯⣿⡱⣏⢾⣿⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢷⡹⢧⣻⢗⡾⣳⣽⢎⡷⣹⢞⡹⣞⡵⣫⢞⡽⣾⣽⠖⣫⠞⠋⠀⢨⢷⡿⣜⢧⡻⣜⢧⡻⣼⢧⡻⣜⢯⣟⣇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⡿⣜⢧⣛⡷⣯⢞⣷⡹⣮⠿⣝⡯⣞⡳⣝⢾⡱⣏⡾⢹⣻⣿⣷⣾⣔⣠⠟⣽⣱⢏⡾⣱⢏⡾⣱⣯⢳⡝⣮⡷⣿⣻⠇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⣤⢴⣏⢷⣹⢳⡝⣮⣷⢻⣼⡿⣲⣽⢏⣿⡝⣾⣱⢻⡜⣧⢻⣜⡇⠈⠛⠙⠭⠁⠙⠇⣰⢧⣟⢮⣳⡝⣞⣷⢻⣧⢻⡜⣧⣿⠟⠁⠀⠀⠀⠀⠀⠀
⢻⣏⢿⡹⣏⢷⡹⣞⡼⣣⣏⢷⣹⢾⣱⣟⡷⣏⢷⢯⣾⢻⣞⡵⣫⢷⡹⣎⣿⢿⠀⠀⠀⠀⠀⠀⢀⣼⢿⣻⣼⢷⣫⠽⡺⠃⣸⢯⡳⣝⢶⢿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠓⠻⠼⠧⠟⣾⢫⣗⣮⠷⠏⠛⠉⠛⢷⣽⣻⢟⡼⣻⣎⢷⣹⣎⠷⣭⡟⣻⠀⠐⠀⠀⠀⠀⠉⠉⠙⡏⠉⢡⣤⡾⣴⡤⣜⣧⢻⣜⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⣴⢻⡼⠃⠁⠀⠀⠀⢂⠀⠀⠀⠑⢿⣎⢷⣹⣯⡞⡝⢨⡟⡶⡇⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢫⠷⠀⣈⢿⡯⣞⣷⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⣀⣴⢺⡵⣮⠋⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠈⢻⢮⣽⣿⣯⠇⠀⢹⣳⡅⠀⠁⠀⠀⠀⣄⡀⠀⠀⠀⠀⠆⠀⢀⢔⣷⡾⠕⠉⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣯⢳⣎⠷⡽⠁⠒⠒⠂⠠⣀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠹⡯⢹⡇⠁⠀⠀⠙⣷⠄⠀⠀⠀⠀⠐⠈⢂⣄⠀⠀⠈⢐⡍⡊⠄⠀⠀⢀⢷⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡗⣯⢞⠏⠀⠀⠀⡠⠀⠀⠀⠁⠢⣀⠀⠀⠀⠀⠀⠀⠀⠡⣾⠀⠀⠀⠀⢀⢨⣦⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⣠⣟⢮⡳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣻⠜⠁⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⢣⠀⢀⠄⠎⡁⠀⠙⣮⣛⣦⣀⡠⣤⣴⠂⠀⠁⠀⠀⠀⠀⢀⣼⢻⣮⢳⡽⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡍⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠣⡀⠀⠂⠀⠀⠀⡄⠁⢠⢸⠀⠀⠀⢘⣽⢖⣧⢻⣽⣺⠀⠀⠀⠀⠀⠀⣴⢻⡜⣯⢻⣧⢻⡵⡂⠀⠀⠀⠀⠀⠀⠀⠀
⢏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠑⡄⠘⠄⠀⠀⢡⠀⠀⡆⠀⠀⠀⡨⢃⡾⣜⡳⣾⣳⢆⠀⠀⠀⠀⢸⢮⡳⣝⢮⣛⣧⣿⢳⡇⠀⠀⠀⠀⠀⠀⠀⠀
⣌⡟⡄⠀⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢄⠸⡄⠰⠀⠀⠀⠠⡘⠀⠀⠀⡰⠁⠀⢿⣱⢏⡇⡟⣾⠀⠀⠀⠀⠀⣯⢳⡝⣮⢳⡿⣜⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀
⡿⢋⠕⠀⡇⠀⢸⢫⠑⡊⡭⡶⣤⣈⠠⢄⠀⠈⡆⠘⡄⠀⠀⢀⠰⠁⠀⢠⠊⠀⠀⠀⢫⣷⣫⠆⣿⡱⡇⠀⠀⠀⠀⠹⣧⢻⣼⢯⣳⣽⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡡⠋⠀⠀⡇⠀⢸⢸⡱⣗⣧⣉⣱⠶⡵⡌⠱⣄⠸⡀⠰⡀⠀⡸⠁⢀⡔⠁⠀⡀⠀⡐⢸⡟⡹⣸⢶⡹⢧⠀⠀⠀⠀⠠⢯⡿⣽⠾⣝⣯⣽⢳⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢰⠀⠀⠈⡝⡆⣩⣇⣻⣿⡠⢹⢅⠀⠈⠲⣣⠀⢣⠀⡇⡰⠂⠀⢀⠌⠀⢀⠂⢸⢸⠁⡟⣯⣝⣻⡀⠀⠀⠀⠀⠸⣝⢮⡻⣼⣟⣧⢟⡼⣳⢄⠀⠀⠀⠀⠀
⠀⠀⠀⠐⠀⠀⠀⡇⢳⠌⣉⣶⡬⢽⡿⡯⠆⠀⠀⠹⡄⠀⠣⠜⠀⡄⢀⠎⠀⠀⡌⠀⡼⢸⠡⣉⢹⢳⢷⡇⠀⠀⠀⠀⠀⢯⣳⣽⡳⣝⠾⣝⡾⣱⢏⡷⣲⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢷⢸⠘⠿⡄⣞⣩⢸⡸⡕⢤⡀⠀⠹⡀⠀⠀⢰⢁⠋⠀⠀⡈⠀⠀⡍⣼⠐⡙⡓⢫⠸⡇⠀⠀⠀⠀⠀⢸⣳⢎⡷⣭⢻⡵⡉⠋⠚⠓⠛⠚⠒⠉
⠐⠤⢀⠀⠄⠀⠀⢸⠈⡜⡐⢓⣒⡮⡇⣧⢱⡀⠈⠢⡀⢣⠀⠀⢸⠆⠀⠀⠌⠀⡀⠀⣇⡣⣕⣠⠑⡡⣾⠃⠀⠀⠀⠀⠀⠀⢯⢿⣜⢧⡻⣜⢷⡀⠀⠀⠀⠀⠀⠀
⠠⠀⠀⠀⠈⢢⠀⠈⡇⢧⠘⡄⣛⣪⢔⠸⡆⢧⠀⠀⠈⠣⣇⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠉⡉⠚⠡⠭⠆⡇⠀⠀⠀⠀⠀⠀⢸⣏⠾⣯⣳⡝⣮⣛⣆⠀⠀⣁⠀⠀
⠁⠀⠀⠀⡠⠂⠱⡀⢹⢸⣴⡎⣗⡿⣸⠂⡗⢸⡀⠀⠀⠀⠀⠢⣘⢄⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⣾⡹⣞⡽⣿⣲⡝⣮⠷⡀⠉⠀⠀