Consider the relaxed $8-$puzzle problem where any tile can be swapped with the blank space, not just tiles directly adjacent to the blank space. The distance to the goal state in this relaxed problem is an admissible heuristic to the original $8-$puzzle problem. Give an example state where this heuristic gives a better value than the $$ sum of Manhattan distances $$heuristic$.$

Describe an algorithm for how to compute this heuristic. $($pseudo code$)$