AMC 12 2022a Problem 13

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 $3$ and $4i$, and $z_2$ has magnitude at most $1$. What integer is closest to the area of $\mathcal{R}$?

$\textbf{(A) } 13 \textbf{(B) } 14 \textbf{(C) } 15 \textbf{(D) } 16 \textbf{(E) } 17$