2021 AMC 12B Problem 18

Let z be a complex number satisfying 12|z|^2=2|z+2|^2+|z^2+1|^2+31. What is the value of z+\frac 6z?

\textbf{(A) }-2 \qquad \textbf{(B) }-1 \qquad \textbf{(C) }\frac12\qquad \textbf{(D) }1 \qquad \textbf{(E) }4

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