Excircles Kissing a Central Circle

Triangle $ABC$ has integer side lengths with $AB=AC$. A circle $\omega$ is centered at the incenter of $\triangle ABC$. The excircle opposite to the side $BC$ is internally tangent to $\omega$, while the excircles opposite to the sides $AC$ and $AB$ are externally tangent to $\omega$. Find the minimum possible perimeter of $\triangle ABC$.