Let $P$ be the parabola in the plane determined by the equation $y = x^2$. Suppose a circle $C$ in the plane intersects $P$ at four distinct points. If three of these points are $(-28, 784)$, $(-2, 4)$, and $(13, 169)$, find the sum of the distances from the focus of $P$ to all four of the intersection points.

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