Triangle A B C is inscribed in circle \omega with A B=5, B C=7, and A C=3. The bisector of angle A meets side \overline{B C} at D and circle \omega at a second point E. Let \gamma be the circle with diameter \overline{D E}. Circles \omega and \gamma meet at E and a second point F. Then A F^{2}=\frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.