Question 15
Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB,$ respectively, with $FA = 5$ and $CD = 2.$ Point $E$ lies on side $CA$ such that $\angle DEF = 60^\circ.$ The area of triangle $DEF$ is $14\sqrt{3}.$ The two possible values of the length of side $AB$ 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 $r.$