Let the complex number $z$ have a nonzero imaginary part and $|z| = 2$. It is known that on the complex plane, $1, z, z^{3}$ are collinear. Select the correct options.
(1) $z \cdot \bar{z} = 2$
(2) The imaginary part of $\frac{z^{3} - z}{z - 1}$ is 0
(3) The real part of $z$ is $-\frac{1}{2}$
(4) $z$ satisfies $z^{2} - z + 4 = 0$
(5) On the complex plane, $-2, z, z^{2}$ are collinear