. For a complex number z, let  $\mathrm{Re}\left(z\right)$  denote the real part of z. let S be the set of all complex numbers z satisfying  $z^4-{\left|z\right|}^4=4i{z}^2,$ 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 $\mathrm{Re}\left(z_1\right)>0$   and  $\mathrm{Re}\left(z_2\right)<0,$ is