$\vartriangle ABC$ is a triangle such that $BC = 13$, $CA = 14$, $AB = 15$. $D, E, F$ are points on line segments $BC$, $CA$, $AB$ respectively such that $\vartriangle AEF$, $\vartriangle BFD$, $\vartriangle CDE$ have the same inradii. Let the inradii of these three triangles be $r_1$ and the inradius of $\vartriangle DEF$ be $r_2$. It is known that $r^2_1 + r^2_2 =\frac{ 25}{2}$ and $r_1 = \frac{p}{q} \le r_2$, where $p$, $q$ are integers that are coprime. Enter $p^2 + q^2$.