Leaving his house at noon, Jim walks at a constant rate of $4$ miles per hour along a $4$ mile square route returning to his house at $1$ PM. At a randomly chosen time between noon and $1$ PM, Sally chooses a random location along Jim's route and begins running at a constant rate of $7$ miles per hour along Jim's route in the same direction that Jim is walking until she completes one $4$ mile circuit of the square route.
The probability that Sally runs past Jim while he is walking is given by $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

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