Sixteen dots are arranged in a 4x4 grid. The distance between any two dots is defined as the minimum number of horizontal and vertical steps along the grid lines needed to travel from one dot to the other. For instance, two adjacent dots are a distance of 1 apart, and two dots at opposite corners of the grid are a distance of 6 apart.

The problem is to find the mean distance between two distinct dots in the grid, expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine the value of $m + n$.

Let's think step by step and output the final answer within \boxed{}.
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