An equilateral triangle $ABC$ has a side length of $7$. Point $P$ is located inside triangle $ABC$ such that $PB = 3$ and $PC = 5$. The distance between the circumcenters of triangles $ABC$ and $PBC$ can be expressed in the form $\frac{m\sqrt{n}}{p}$, where $n$ is not divisible by the square of any prime number, and $m$ and $p$ are relatively prime positive integers. Determine the value of $m + n + p$.

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