⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⣿⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣂⣹⣶⣶⣶⣦ ⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⣻⣏⢺⡏⠀⣠⠆⠰⣾⣭⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⡤⡾⣾⣮⣿⣿⣿⣿⣻⣽⢯⣿ ⠀⠀⠀⠀⠀⠀⠀⠀⢿⢷⣿⣵⣿⠀⠈⠉⢠⡼⠟⠛⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⢷⢟⣟⣿⣿⣿⣿⣻⢾⣽⡾⣿⣽ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⡿⣿⣧⣤⣠⠾⠿⠀⢀⣴⣶⣿⠿⠄⠀⠀⠀⠀⣐⣶⣿⣿⣿⣿⣟⡿⡟⡟⣯⣿⣿⣿⣿⣽⣟⣷⢿⣳⡿ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⣻⢿⣯⣤⣴⣾⣿⢿⣻⡿⣿⣿⠤⢤⣶⣿⡿⣟⣿⣟⡿⣷⣿⣇⡣⣫⢿⣿⡿⢻⣿⣿⣟⡞⠋⠉⠀ ⠀⠀⢀⣀⣰⣶⣶⡦⣼⣿⣿⣾⣿⢯⣟⣿⣻⢿⣽⡾⣟⣯⣟⣿⣻⣿⣾⣻⣯⣟⣿⣿⡾⣿⣿⢷⣿⣫⠦⣿⣯⡷⢃⣿⣟⡯⣘⠀⠀⠀ ⣤⡾⢟⢶⣻⣿⣟⣿⢿⣻⣾⣽⣾⣿⣿⣾⣽⡿⣾⣽⢿⣽⡾⣷⣻⣾⡽⣿⣻⢾⣿⣿⣿⣯⣟⣽⣟⣋⣟⣿⣫⣵⣿⡟⢜⣇⡉⠀⠀⠀ ⡞⣵⣯⣿⣿⡿⣞⣯⠿⠛⠉⠀⢀⣽⣤⡀⠈⠙⠛⢯⣿⡾⣽⡷⣟⡷⣟⣯⣟⣯⡿⣷⣿⡿⣿⣿⡿⢿⣿⡫⣿⣏⣹⣧⣾⡇⣿⠀⠀⠀ ⢭⣿⢿⡿⣿⣿⠟⠁⠀⠀⢀⣴⣿⣿⠟⠁⢀⡀⠀⠀⠙⢿⣯⣟⡿⣽⡿⣽⣾⣻⣿⣽⣾⣿⣿⣫⣡⡷⣿⣿⡿⣿⣿⡷⡿⣓⡭⠀⠀⠀ ⣿⣿⣼⣿⣿⠃⠀⠀⢠⣴⣿⣿⠟⠁⠀⢀⣴⣿⣄⠀⠀⠀⢻⣿⣽⢯⣿⣳⣿⣿⣿⣷⣿⣿⣿⣿⠿⡫⣿⣿⣿⣿⠏⣾⣻⡎⠆⠀⠀⠀ ⣾⣿⣿⣿⡇⠀⣀⠀⠈⠻⣿⣿⣷⣄⣴⣿⣿⣿⣿⣷⣄⠀⠀⣿⣯⢿⣳⣟⡷⣯⣿⢾⣽⣿⣿⣿⣿⣿⣯⣿⣿⢋⢺⠵⡟⡿⣅⡀⠀⠀ ⣻⣿⣟⣾⠅⣼⣿⣦⡀⠀⠈⢻⣿⣿⣿⣿⠋⠀⠙⢿⣿⣷⡶⢼⣿⣻⣽⡾⣿⣽⢾⣻⣽⣟⣿⣽⣻⢿⣿⣿⡿⡸⡦⣺⣾⣿⣽⠀⠀⠀ ⣾⣿⣯⣿⡇⠈⠻⣿⣿⣦⣴⣿⣿⡿⣿⣿⣷⣄⠀⠀⠙⠏⠁⣼⣿⣽⣾⣻⣷⠿⣿⣯⣷⣻⣿⢾⣯⢿⣿⣽⣿⣿⣯⣳⣿⡿⠋⠁⠀⠀ ⣿⣿⣷⣻⣧⠀⠀⠈⠻⣿⣿⡿⠋⠀⠈⣻⣿⣿⡷⠀⠀⠀⢰⣿⣿⣋⣩⣿⣷⣿⢿⣳⡿⡿⣯⣿⢾⡿⣿⣿⣿⡳⠖⠋⠀⠀⠀⠀⠀⠀ ⢹⣿⣿⣽⣻⣷⡀⠀⠀⠈⠋⠀⠀⣠⣾⣿⡿⠋⠀⠀⡀⣴⣿⣯⡗⣾⢟⣿⣟⣿⣿⣿⡯⣷⣿⠻⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠈⢽⣿⣷⣻⣽⢿⣦⣄⠀⠀⠀⠻⣿⡿⠋⠀⠀⢀⣨⣿⣿⡿⣷⣾⡿⠿⣛⣏⣿⣿⣯⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠈⠛⠿⣷⣿⢯⣟⡿⣿⣶⣦⣤⣬⣤⣤⣶⣾⡿⣿⢯⣿⣿⣷⣿⣾⣯⡿⡿⣿⣿⣷⣻⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠈⣹⣿⣟⣷⢯⡿⣽⣻⢿⣽⣳⣯⢿⣽⣻⢷⣻⣞⣿⣻⢿⣾⣷⣿⢿⣻⣟⣿⣻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢸⣿⣟⣾⢯⣿⣻⣽⢯⣿⡽⣷⣻⣯⢿⣽⣻⢯⣟⣾⣻⣯⣟⣯⢿⣻⣽⡾⣯⣟⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢠⣿⡿⣞⣯⣿⣳⡿⣽⣻⢷⣻⣯⡷⣟⣯⡿⣽⣻⢯⣷⣟⡷⣯⢿⣻⣽⢷⣻⣽⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠸⠿⠟⠿⠉⠛⠳⢿⣿⣽⣻⣟⣾⣽⢿⣽⣻⣽⣟⣯⡷⣿⡽⣟⣿⡽⣯⡿⣯⣷⢿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣳⣯⢿⡾⣽⣟⣾⣯⣷⢯⣿⡽⣷⢿⣻⢷⣻⣯⣟⣷⣯⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠻⠿⠉⠉⠁⠘⠿⣻⣞⣿⣽⣻⢯⣿⣻⠾⠽⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠞⠽⠾⣻⣿⢯⣻⣽⣿⣞⣻⠿⠘⠁⠉⠉⠿⠻⠙⠛⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣯⣷⣟⣯⣻⢷⣻⢿⣷⡽⣿⢯⣷⣯⣾⣟⣽⡾⢿⣯⣳⣿⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⢿⣷⣯⡿⣯⡽⣿⣟⡽⣿⡷⣯⣟⣽⣻⣽⢿⣽⣾⣟⣻⣽⣿⢿⠳⠛⠉⠿⠟⠿⠸⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣽⣻⢷⣽⣻⢿⣯⡷⣟⣷⢯⣻⣽⡿⣯⣟⡷⣯⣻⢷⣻⣽⡿⣳⣿⣯⣞⡿⣿⢠⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣿⣟⣯⡾⣽⣻⢿⣯⣟⣯⣻⣾⣟⢯⣻⣽⢿⣯⣻⣷⡽⣿⢯⣽⣻⣿⢯⣾⣟⣿⢸⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣻⣿⣟⣻⢿⣿⣷⣾⢿⣻⣿⣞⣻⢷⣻⣽⢿⣯⣳⣽⢿⣻⣽⡿⢯⣷⣟⣿⣹⠈⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣿⣻⣷⣿⣿⡿⡿⣯⣾⣿⣷⣿⣿⢯⣿⡿⣾⣶⣤⣤⣬⣤⣦⣶⣿⡿⣟⢯⣿⣷⠿⠛⠈⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡟⣯⣿⣿⣏⣛⠿⡿⣾⣷⡿⣿⣿⣨⢀⠀⠀⠋⡿⣿⠻⠀⠀⠀⣄⣦⢿⣽⣻⣷⣿⢽⠈ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠿⠿⠻⣿⣷⡯⣿⣿⣿⣟⣿⢟⣾⡗⣯⣿⣴⡀⠀⠀⠋⡿⣿⣾⣠⠀⠀⠋⠈⠀⠀⡀⣷⣻⣽⣿⣿⢹ ⠀⠀⠀⠀⠀⠀⠋⠖⡳⣿⣿⣿⡿⢾⣿⣯⡿⡿⣳⢿⣿⣷⣿⣩⣋⣿⣿⢰⠀⠀⠀⡷⣿⣿⣻⠈⠀⠋⡿⣿⣿⠻⠈⠀⠀⣧⣻⣷⣿⣿ ⠀⠀⠁⠋⡿⣿⣳⣯⣿⣿⣽⣿⢿⣯⢾⣿⣻⣷⣯⣿⠿⣷⣻⣾⣽⣿⣼⠁⠏⠙⠀⠀⣄⣷⣿⣿⡿⣿⣿⣴⣦⣿⣿⠻⠈⡇⣿⣯⣿⣾ ⠀⠀⠀⣽⣿⣾⣺⡦⡸⡿⣿⣿⢿⣻⣽⣿⣟⣽⣻⢾⣽⣿⡾⣽⣻⣿⢼⡶⣷⣿⢿⠙⠀⠋⣿⣿⣿⣿⢻⠈⠀⡀⣦⣿⣼⠅⣾⣟⣿⣻ ⠀⠀⡀⣅⡿⡟⠵⢺⢋⣿⣿⣯⣿⣿⣿⣿⣿⣽⢾⣿⣯⡷⣟⣳⢿⣯⣿⠀⠀⣄⣷⣿⣿⣿⣿⣴⣄⣷⣿⣿⠻⠈⠀⣀⠀⡇⣿⣿⣿⣾ ⠀⠀⠀⠆⡎⣻⣾⠏⣿⣿⣿⣿⡫⠿⣿⣿⣿⣿⣷⣿⣿⣿⣳⣿⢯⣽⣿⢻⠀⠀⠀⣄⣿⣴⢀⠀⠁⠟⣿⣿⣴⢠⠀⠀⠃⣿⣿⣼⣿⣿ ⠀⠀⠀⡭⣓⡿⡷⣿⣿⡿⣿⣿⡷⣡⣫⣿⣿⣾⣽⣿⣻⣾⣽⡿⣽⡿⣟⣯⢿⠙⠀⠀⡀⢀⠁⠟⣿⣿⣴⢀⠀⠀⠁⠟⣿⣿⡿⢿⣿⢭ ⠀⠀⠀⣿⡇⣾⣧⣹⣏⣿⡫⣿⢿⡿⣿⣿⡿⣿⣷⡿⣯⣟⣯⣟⡷⣟⡷⣽⡾⣿⢯⠛⠙⠈⡀⣤⣽⢀⠀⠉⠛⠿⣯⣞⡿⣿⣿⣯⣵⡞ ⠀⠀⠀⡉⣇⢜⡟⣿⣵⣫⣿⣟⣋⣟⣽⣟⣯⣿⣿⣿⢾⣻⣿⡽⣾⣻⣷⡾⣽⢿⣽⣾⡿⣽⣾⣿⣿⣾⣽⣾⣻⢿⣿⣟⣿⣻⢶⢟⡾⣤ ⠀⠀⠀⣘⡯⣟⣿⢃⡷⣯⣿⠦⣫⣿⢷⣿⣿⡾⣿⣿⣟⣯⣻⣾⣿⣻⣿⣟⣯⣟⡾⣽⢿⣻⣿⣟⢯⣿⣾⣿⣿⣼⡦⣶⣶⣰⣀⢀⠀⠀ ⠀⠉⠋⡞⣟⣿⣿⢻⡿⣿⢿⣫⡣⣇⣿⣷⡿⣟⣿⣟⡿⣿⣶⢤⠤⣿⣿⡿⣻⢿⣿⣾⣴⣤⣯⢿⣻⣷⣿⣿⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⡿⣳⢿⣷⣟⣽⣿⣿⣿⣿⣯⡟⡟⡿⣟⣿⣿⣿⣿⣶⣐⠀⠀⠀⠀⠄⠿⣿⣶⣴⢀⠀⠿⠾⣠⣤⣧⣿⡿⣿⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⣽⣿⡾⣽⢾⣻⣿⣿⣿⣿⣟⢟⢷⣿⣿⣿⣴⣠⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠋⠛⠟⡼⢠⠉⠈⠀⣿⣵⣿⢷⢿⠀⠀⠀⠀⠀⠀⠀⠀ ⣿⢯⣽⣻⣿⣿⣿⣿⣮⣾⡾⡤⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣶⣭⣾⠰⠆⣠⠀⡏⢺⣏⣻⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀ ⣦⣶⣶⣶⣹⣂⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣟⣿⣯⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀