Triangle Geometry and Tangent Lengths

Let $\triangle ABC$ be an acute triangle where $AB = 13$, $BC = 14$, and $AC = 15$. Let $\omega$ be the incircle of $\triangle ABC$. It is known that $AB$ is tangential to $\omega$ at $F$, and $AC$ is tangential to $\omega$ at $E$. Construct a cyclic rectangle $FUVE$ that is inscribed in $\omega$. Let $\gamma$ be the circumcircle of $\triangle ABC$ and define $M$ as the angle bisector of $A$ lying on the minor arc $\widehat{BC}$ such that $MU \cong MV$. Define $X$ to be the intersection of lines $AB$ and $MU$. Similarly, define $Y$ to be the intersection of lines $AC$ and $MV$. It is known that the individual side lengths of the quadrilateral $AXMY$ can be expressed as the ratio $q/p$ of a coprime $q$ and a prime $p$ in its simplest form. Find $q^p \bmod 100$.