Problem

The diagram shows a sector of a circle, centre $O$, where $OB=OC=15$ cm. The size of angle $BOC$ is $\dfrac{2}{5}\pi$ radians. Points $A$ and $D$ on the lines $OB$ and $OC$ respectively are joined by an arc $AD$ of a circle with centre $O$. The shaded region is bounded by the arcs $AD$ and $BC$ and by the straight lines $AB$ and $DC$. It is given that the area of the shaded region is $\dfrac{209}{5}\pi\text{ cm}^2$.

Find the perimeter of the shaded region. Give your answer in terms of $\pi$.