TopsOJ Summer 2025 - Guts Round - Problem 22

In triangle ABC with AB=65, AC=104, \angle BAC = 60^\circ, point D is constructed on \overline{BC} such that BD = 35. The circle with diameter \overline{AD} intersects \overline{AB} and \overline{AC} at points E \neq A and F \neq A, respectively. Given that \frac{[DEB]}{[DFC]} = \frac{m}{n} for relatively prime positive integers m and n, find m+n.