Let $$z$$ be a complex number with a non-zero imaginary part. If

$$ \frac{2+3 z+4 z^{2}}{2-3 z+4 z^{2}} $$

is a real number, then the value of $$|z|^{2}$$ is _________.

Let $$\bar{z}$$ denote the complex conjugate of a complex number $$z$$ and let $$i=\sqrt{-1}$$. In the set of complex numbers, the number of distinct roots of the equation

$$ \bar{z}-z^{2}=i\left(\bar{z}+z^{2}\right) $$

is _________.

For a complex number z, let Re(z) denote that real part of z. Let S be the set of all complex numbers z satisfying $${z^4} - |z{|^4} = 4i{z^2}$$, where i = $$\sqrt { - 1} $$. Then the minimum possible value of |z₁ $$-$$ z₂|², where z₁, z₂$$ \in $$S with Re(z₁) > 0 and Re(z₂) < 0 is .........

Let $$\omega \ne 1$$ be a cube root of unity. Then the minimum of the set $$\{ {\left| {a + b\omega + c{\omega ^2}} \right|^2}:a,b,c$$ distinct non-zero integers} equals ..................