It's too complex.

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

Now , we know $\left|z_1\right|^2 = z_1.\overline{z_1} = 1$

$\therefore$ $\overline{z_1} = \frac{1}{z_1}$

Similarly , $\overline{z_2} = \frac{1}{z_2}$ and $\overline{z_3} = \frac{1}{z_3}$

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

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

By using properties of conjugate for a complex number , we get

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

Now by property of modulus of complex numbers ,

$\left|z\right| = \left|\overline{z}\right|$

Hence , $\left|z_1+z_2+z_3\right| = 1$

NOTE : $\overline{z}$ is conjugate of $z$