AMC 12 2023a 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 $\tfrac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

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