AMC 12 2025b 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)}\frac 14 \textbf{(B)}\frac 12 \textbf{(C)}\frac{9}{16} \textbf{(D)}\frac 34 \textbf{(E)}~1