HMMT 2021 A7

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

$$x y z=\sqrt{3}+i \sqrt{5} .$$

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 $100 a+b$.