A circle has the same center as an ellipse and passes through the foci $F_1$ and $F_2$ of the ellipse.  The two curves intersect in 4 points.  Let $P$ be any point of intersection.  If the major axis of the ellipse is 15 and the area of triangle $PF_1 F_2$ is 26, compute the distance between the foci.

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