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 \(r\).

(第二十五届AIME1 2007 第15题)