Least value of $|z_1 − z_2|^2 + |z_1 − z_4|^2 + |z_2 − z_3|^2 + |z_3 − z_4|^2$ is $2$.

Question

If $z_1, z_2, z_3, z_4 \in \mathbb{C}$ satisfy $z_1 + z_2 + z_3 + z_4 = 0$ and $|z_1|^2 + |z_2|^2 + |z_3|^2 + |z_4|^2 = 1$, then the least value of $|z_1 − z_2|^2 + |z_1 − z_4|^2 + |z_2 − z_3|^2 + |z_3 − z_4|^2$ is $2$.

My work -

After some simple manipulations, I got the following inequality -

$2-2(|z_1||z_2|+|z_1||z_4|+|z_2||z_3|+|z_3||z_4|)$ $\leq |z_1 − z_2|^2 + |z_1 − z_4|^2 + |z_2 − z_3|^2 + |z_3 − z_4|^2$

Now, how do I proceed?

Thanks!