The curve \(C\) has polar equation \(r=1+2 \cos \theta\). Sketch the curve for \(-\frac{2}{3} \pi \leqslant \theta\lt \frac{2}{3} \pi\).

Find the area bounded by \(C\) and the half-lines \(\theta=-\frac{1}{3} \pi, \theta=\frac{1}{3} \pi\).