Triangle $ABC$ has sidelengths $AB=1$, $BC=\sqrt{3}$, and $AC=2$. Points $D$, $E$, and $F$ are chosen on $AB$, $BC$, and $AC$ respectively, such that $\angle EDF = \angle DFA = 90^{\circ}$. Given that the maximum possible value of $[DEF]^2$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $\gcd(a, b) = 1$, find $a + b$. (Here $[DEF]$ denotes the area of triangle $DEF$.)

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