Let $C$ be the circle of minimum area enclosing the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $\frac{1}{2}$ and foci $( \pm 2,0)$. Let PQR be a variable triangle, whose vertex $P$ is on the circle $C$ and the side QR of length 2a is parallel to the major axis of $E$ and contains the point of intersection of $E$ with the negative $y$ axis. Then the maximum area of the triangle $P Q R$ is :