Question 13
Triangle $ABC$ has side lengths $AB = 4,$ $BC = 5,$ and $CA = 6.$ Points $D$ and $E$ are on ray $AB$ with $AB \lt AD \lt AE.$ The point $F \neq C$ is a point of intersection of the circumcircles of $\triangle ACD$ and $\triangle EBC$ satisfying $DF = 2$ and $EF = 7.$ Then $BE$ can be expressed as $\frac{a + b\sqrt{c}}{d},$ where $a,$ $b,$ $c,$ and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d.$