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 $z^4-|z|^4=4 i z^2$, where $i=\sqrt{-1}$. Then the minimum possible value of $\left|z_1-z_2\right|^2$, where $z_1, z_2 \in S$ with $\operatorname{Re}\left(z_1\right)>0$ and $\operatorname{Re}\left(z_2\right)<0$, is $\_\_\_\_$