The curve \(C\) has polar equation \(r=a(1+\sin \theta)\), where \(a\) is a positive constant and \(0 \leqslant \theta\lt 2 \pi\). Draw a sketch of \(C\).

Find the exact value of the area of the region enclosed by \(C\) and the half-lines \(\theta=\frac{1}{3} \pi\) and \(\theta=\frac{2}{3} \pi\).