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 $x = x_1 + ix_2$, $y = y_1 + iy_2$, and $z = z_1 + iz_2$ for real $x_1, x_2, y_1, y_2, z_1$, and $z_2$, then:
\[
(x_1x_2 + y_1y_2 + z_1z_2)^2
\]
can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.

Let's think step by step and output the final answer within \boxed{}.
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