If $a, b, c$ are non-zero real numbers with $c \neq 1$ such that $a^2+b^2+c^2=c$ and if $\alpha=\frac{a+i b}{1-c}$,then $a^2+b^2=$

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:

If $z = x + iy$ $(x, y \in R, x \neq -1/2)$,the number of values of $z$ satisfying $|z|^n = z^2|z|^{n-2} + z|z|^{n-2} + 1$ $(n \in N, n > 1)$ is

If $x_n = \cos \left(\frac{\pi}{4^n}\right) + i \sin \left(\frac{\pi}{4^n}\right)$,then the product $x_1 x_2 x_3 \ldots \infty$ is equal to

If $z_1 = 2 - 3i$ and the roots of the equation $z^3 + bz^2 + cz + d = 0$ are $i$,$z_1$,and $\bar{z}_1$,then $b + c + d =$

If $z$ is a non-real complex number, then the minimum value of $\frac{\operatorname{Im}(z^5)}{(\operatorname{Im} z)^5}$ is