Similar to the last 6 problems, let \triangle A B C be a triangle with A B=4 and A C=\frac{7}{2}. Let \omega denote the A-excircle of \triangle A B C. Let \omega touch lines A B, A C at the points D, E, respectively. Let \Omega denote the circumcircle of \triangle A D E. Consider the line \ell parallel to B C such that \ell is tangent to \omega at a point F and such that \ell does not intersect \Omega. Let \ell intersect lines A B, A C at the points X, Y, respectively, with X Y=18 and A X=16. Let the perpendicular bisector of X Y meet the circumcircle of \triangle A X Y at P, Q, where the distance from P to F is smaller than the distance from Q to F. Let ray \overrightarrow{P F} meet \Omega for the first time at the point Z. If P Z^{2}=\frac{m}{n} for relatively prime positive integers m, n, find m+n.