Let $ABC$ be a triangle with circumradius $17$, inradius $4$, circumcircle $\Gamma$, and $A$-excircle $\Omega$. Suppose the reflection of $\Omega$ over line $BC$ is internally tangent to $\Gamma$. Compute the area of $\triangle ABC$.

Let's think step by step and output the final answer within \boxed{}.
user