2023 AMC 12A Problem 16

Consider the set of complex numbers z satisfying |1+z+z^2|=4. The maximum value of the imaginary part of z can be written in the form \dfrac{\sqrt{m}}{n}, where m and n are relatively prime positive integers. What is m+n?

\textbf{(A)} 20 \qquad \textbf{(B)} 21 \qquad \textbf{(C)} 22 \qquad \textbf{(D)} 23 \qquad \textbf{(E)} 24

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