If $z_1,z_2,z_3$  are complex numbers such that $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=\left|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}\right|=1,$ then $\left|z_1+z_2+z_3\right|$  is

Given $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|$ $=\left|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}\right|=1$ 

$\Rightarrow {\left|z_1\right|}^2={\left|z_2\right|}^2=|z_3{|}^2=1$

$\Rightarrow z_1{\overline{z}}_1=z_2{\overline{z}}_2=z_3{\overline{z}}_3=1$   $\left(\sin ce {\left|z\right|}^2=z\overline{z}\right)$

$\Rightarrow \frac{1}{z_1}={\overline{z}}_1,\frac{1}{z_2}={\overline{z}}_2,\frac{1}{\overline{z_3}}={\overline{z}}_3.$

$\sin ce |\frac{1}{z_1}+\frac{1}{z_3}+\frac{1}{z_3}|=1$

$$\begin{gathered} \Rightarrow \left|\overline{z_1}+\overline{z_2}+\overline{z_3}\right| =1 \\ \Rightarrow \left|\overline{z_1+z_2+z_3}\right|=1 \end{gathered}$$

$\Rightarrow \left|z_1+z_2+z_3\right|=1.$   $\left(\sin ce \left|z\right|=\left|\overline{z}\right|\right)$