Triangle $ABC$ has $\overline{AB} = 5, \overline{BC} = 4, \overline{CA} = 6$. Points $D$ and $E$ are on sides $AB$ and $AC$, respectively, such that $\overline{AD} = \overline{AE} = \overline{BC}$. Let $CD$ and $BE$ intersect at $F$ and let $AF$ and $DE$ intersect at $G$. The length of $FG$ can be expressed in the form $\tfrac{a\sqrt{b}}{c}$ in simplified form. What is $a + b + c$?


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