Let $Re(z)$ and $Im(z)$ be the real and imaginary parts of any complex number $z$, and $\arg(z)$ denotes the principal argument of $z$. Let $z_1$ and $z_2$ be two distinct complex numbers such that $\operatorname{Re}(z_1) = |z_1 - 2| \quad \text{and} \quad \operatorname{Re}(z_2) = |z_2 - 2|.$ If $\arg(z_1 - z_2) = \frac{\pi}{6},$ then