2020 AIME I Problem 13

Point D lies on side \overline{B C} of \triangle A B C so that \overline{A D} bisects \angle B A C. The perpendicular bisector of \overline{A D} intersects the bisectors of \angle A B C and \angle A C B in points E and F, respectively. Given that A B=4, B C=5, and C A=6, the area of \triangle A E F can be written as \frac{m \sqrt{n}}{p}, where m and p are relatively prime positive integers, and n is a positive integer not divisible by the square of any prime. Find m+n+p.