Let \( C \) be the circle of minimum area enclosing the ellipse \( E:\; \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1 \) with eccentricity \( \dfrac{1}{2} \) 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 \( PQR \) is:
