A complex number Z is said to be unimodular if |Z| = 1. Let, Z₁ and Z₂ are complex numbers such that $\frac{{\mathrm{Z}}_1-2{\mathrm{Z}}_2}{2-{\mathrm{Z}}_1\overline{{\overline{\mathrm{Z}}}_2}}$ is unimodular and Z₂ is not unimodular, then $\left|{\mathrm{z}}_1\right|=$?

$$\begin{gathered} \left|\frac{{\mathrm{Z}}_1-2{\mathrm{Z}}_2}{2-{\mathrm{Z}}_1\overline{{\overline{\mathrm{Z}}}_2}}\right|=1\Rightarrow \left|{\mathrm{Z}}_1-2{\mathrm{Z}}_2\right|=\left|2-{\mathrm{Z}}_1\overline{{\overline{\mathrm{Z}}}_2}\right| \\ \Rightarrow {\left|{\mathrm{Z}}_1-2{\mathrm{Z}}_2\right|}^2={\left|2-{\mathrm{Z}}_1\overline{{\overline{\mathrm{Z}}}_2}\right|}^2 \\ \Rightarrow \left({\mathrm{Z}}_1-2{\mathrm{Z}}_2\right)\left({\overline{\mathrm{Z}}}_1-2{\overline{\mathrm{Z}}}_2\right)=\left(2-{\mathrm{Z}}_1\overline{{\overline{\mathrm{Z}}}_2}\right)\left(2-{\overline{\mathrm{Z}}}_1{z}_2\right) \\ \Rightarrow {\left|z_1\right|}^2-2z_1{\overline{z}}_2-2z_2{\overline{z}}_1+4{\left|z_2\right|}^2=4-2z_1{z}_2-2z_1{z}_2+{\left|z_1\right|}^2{\left|z_2\right|}^2 \\ \Rightarrow {\left|z_1\right|}^2-4+4{\left|z_2\right|}^2-{\left|z_1\right|}^2{\left|z_2\right|}^2=0 \\ \Rightarrow \left(1-{\left|z_2\right|}^2\right)\left({\left|z_1\right|}^2-4\right)=0 \\ \because \left|z_2\right|\ne 1,{\left|z_1\right|}^2=4\Rightarrow \left|z_1\right|=2 \end{gathered}$$