Question 95
Let $\omega$ denote the incircle of triangle $ABC$. The segments $BC$, $CA$, and $AB$ are tangent to $\omega$ at $D$, $E$, and $F$, respectively. Point $P$ lies on $EF$ such that segment $PD$ is perpendicular to $BC$. The line $AP$ intersects $BC$ at $Q$. The circles $\omega_1$ and $\omega_2$ pass through $B$ and $C$, respectively, and are tangent to $AQ$ at $Q$; the former meets $AB$ again at $X$, and the latter meets $AC$ again at $Y$. The line $XY$ intersects $BC$ at $Z$. Given that $AB=15$, $BC=14$, and $CA=13$, find $\lfloor XZ \cdot YZ\rfloor$.
