Let $A, B, C, D$ be points, in order, on a straight line such that $AB = BC = CD$. Let $E$ be a point closer to $B$ than $D$ such that $BE = EC = CD$ and let $F$ be the midpoint of $DE$. Let $AF$ intersect $EC$ at $G$ and let $BF$ intersect $EC$ at $H$. If $[BHC] + [GHF] = 1$, then $AD^2 = \frac{a\sqrt{b}}{c}$ where $a, b,$ and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a + b + c$.

Let's think step by step and output the final answer within \boxed{}.
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