For a complex number $z$, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^{4} - |z|^{4} = 4iz^{2}$, where $i = \sqrt{-1}$. Then the minimum possible value of $|z_{1} - z_{2}|^{2}$, where $z_{1}, z_{2} \in S$ with $\operatorname{Re}(z_{1}) > 0$ and $\operatorname{Re}(z_{2}) < 0$, is $\_\_\_\_$