2022 AMC 12A 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 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 16 \qquad \textbf{(E) } 17

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.