Let ‘S’ be the set of all complex numbers ‘z’ satisfying $|z^2+z+1|=1$ , then which of the following statement is/are TRUE?

$$\begin{gathered} \begin{array}{l}|z_2+z+1|=1\\ \Rightarrow |{(z+\frac{1}{2})}^2+\frac{3}{4}|=1\\ {|z+\frac{1}{2}|}^2+\frac{3}{4}\ge |z^2+z+1|=1\end{array} \\ \begin{array}{l}\Rightarrow {|1z+1/2|}^2\ge \frac{1}{2}1/4\\ \therefore |z+\frac{1}{2}|\ge \frac{1}{2}and\\ |1z^2+z1-1|\le 1\\ \Rightarrow 0\le |z+z|\le 2\end{array} \\ \begin{array}{l}\Rightarrow |z{|}^2-|z|-2\le 0\\ \left(\right|z|-2)\left(\right|z|+1)\le 0\\ \therefore \left|z\right|\le 2\\ Bysolvingz=\frac{-1+\sqrt{7}i}{2}issolution\end{array} \end{gathered}$$