Question 12
Let $\triangle PQR$ be a triangle with $\angle P = 75^\circ$ and $\angle Q = 60^\circ.$ A regular hexagon $ABCDEF$ with side length $1$ is drawn inside $\triangle PQR$ so that side $\overline{AB}$ lies on $\overline{PQ},$ side $\overline{CD}$ lies on $\overline{QR},$ and one of the remaining vertices lies on $\overline{RP}.$ There are positive integers $a,$ $b,$ $c,$ and $d$ such that the area of $\triangle PQR$ can be expressed in the form $\frac{a + b\sqrt{c}}{d},$ where $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d.$