Advanced Number Theory · Hard problem

The ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ and the circle $x^2 + y^2 = 16$ meet in four points, which are the vertices of a rectangle. The area of that rectangle is $\dfrac{m\sqrt{n}}{p}$, where $m$, $n$, $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m + n + p$.