Complex Numbers Equal Magnitude Sum Product

Suppose that $x$, $y$, and $z$ are complex numbers of equal magnitude that satisfy $$ x + y + z = -\frac{\sqrt{3}}{2} - i \sqrt{5} $$ and $$ xyz = \sqrt{3} + i \sqrt{5}. $$ If $x = x_1 + i x_2$, $y = y_1 + i y_2$, and $z = z_1 + i z_2$ for real $x_1$, $x_2$, $y_1$, $y_2$, $z_1$, and $z_2$, then $$ \left(x_1 x_2 + y_1 y_2 + z_1 z_2\right)^2 $$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.