Length of a Segment Trisected by an Inscribed Circle

Shown below is an equilateral triangle $\triangle ABC$ with side length $1$. It has an inscribed circle which is tangent to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$ at points $D$, $E$, and $F$ respectively. A line segment $\overline{PQ}$, with $P$ between $D$ and $B$, and with $Q$ between $F$ and $C$, is such that $\overline{PQ}$ is parallel to $\overline{BC}$ and the inscribed circle of $\triangle ABC$ trisects $\overline{PQ}$ (divides $\overline{PQ}$ into three congruent parts). What is the length of $\overline{PQ}$? The answer is in the form $\frac{a+\sqrt{b}}{c}$ where $a$, $b$, $c$ are positive integers, and $\gcd(a,c) = 1$. Input the value of $a+b+c$.