Let $\mathcal{R}$ be the region in the complex plane consisting of all complex numbers $z$ that can be written as the sum of complex numbers $z_1$ and $z_2$, where $z_1$ lies on the segment with endpoints $6$ and $8i$, and $z_2$ has magnitude at most $2$. What integer is closest to the area of $\mathcal{R}$?