2007 AIME I Problem 15

Let A B C be an equilateral triangle, and let D and F be points on sides B C and A B, respectively, with F A=5 and C D=2. Point E lies on side C A such that \angle D E F=60^{\circ}. The area of triangle D E F is 14 \sqrt{3}. The two possible values of the length of side A B are p \pm q \sqrt{r}, where p and q are rational, and r is an integer not divisible by the square of a prime. Find r.