2023 AMC 10B Problem 24

What is the perimeter of the boundary of the region consisting of all points which can be expressed as (2u - 3w,\ v + 4w) with 0 \le u \le 1, 0 \le v \le 1, and 0 \le w \le 1?

\textbf{(A)} 10\sqrt{3} \qquad \textbf{(B)} 13 \qquad \textbf{(C)} 12 \qquad \textbf{(D)} 18 \qquad \textbf{(E)} 16

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