Let $\vartriangle ABC$  be a triangle with $AB = 4$ and $AC = \frac72$ . Let $\omega$ denote the $A$-excircle of $\vartriangle ABC$. Let $\omega$ touch lines $AB$, $AC$ at the points $D$, $E$, respectively. Let $\Omega$ denote the circumcircle of $\vartriangle ADE$. Consider the line $\ell$ parallel to $BC$ such that $\ell$  is tangent to $\omega$ at a point $F$ and such that $\ell$  does not intersect $\Omega$. Let $\ell$  intersect lines $AB$, $AC$ at the points $X$, $Y$ , respectively, with $XY = 18$ and $AX = 16$. Let the perpendicular bisector of $XY$ meet the circumcircle of $\vartriangle AXY$ at $P$, $Q$, where the distance from $P$ to $F$ is smaller than the distance from $Q$ to$ F$. Let ray $\overrightarrow {PF}$ meet $\Omega$ for the first time at the point $Z$. If $PZ^2 = \frac{m}{n}$ for relatively prime positive integers $m$, $n$, find $m + n$.

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