2025 AMC 12B Problem 22

What is the greatest possible area of the triangle in the complex plane with vertices 2z, (1+i)z, and (1-i)z, where z is a complex number satisfying |4z - 2| = 1?

\textbf{(A)} \dfrac{1}{4} \qquad \textbf{(B)} \dfrac{1}{2} \qquad \textbf{(C)} \dfrac{9}{16} \qquad \textbf{(D)} \dfrac{3}{4} \qquad \textbf{(E)} 1

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