The complex numbers $z_1,z_2\mathrm{...............}{z}_n$ represent the vertices of a regular polygon of n sides, inscribed in a circle of units radius and $z_3+z_n=A{z}_1+\overline{A}{z}_2,\left[x\right]$ be the greatest integer  $\le x$. Then

Let $\alpha$ be an intcrior angle of the polygon

$$\begin{gathered} z_n-z_1-(z_2-z_1)e^{i\alpha }\Rightarrow z_n-(1-e^{i\alpha })z_1+e^{i\alpha }{z}_2 \\ {z}_3-z_2=(z_1-z_2)e^{-i\alpha }\Rightarrow z_3=e^{-i\alpha }{z}_1+(1-e^{-i\alpha })z_2 \\ {Z}_3+Z_n=(1-2i\sin \alpha )z_1+(1+2i\sin \alpha )z_2 \\ A=1-2i\sin \alpha \\ \left|A\right|=\sqrt{3-2\cos 2\alpha }=\sqrt{3-2\cos \left(\frac{4\pi }{n}\right)} \\ where n=16\Rightarrow \left|A\right|=\sqrt{3-\sqrt{2}} \\ n=6\Rightarrow \left|A\right|=2 \\ n=8\Rightarrow \left|A\right|=\sqrt{3} \\ n=12\Rightarrow \left|A\right|=\sqrt{2} \end{gathered}$$