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 $\_\_\_\_$

Let $S = \{ z \in \mathbb { C } : \bar { z } = i z ^ { 2 } + \operatorname { Re } ( \bar { z } ) \}$. Then $\sum _ { z \in S } | z | ^ { 2 }$ is equal to
(1) $\frac { 5 } { 2 }$
(2) 4
(3) $\frac { 7 } { 2 }$
(4) 3