
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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 \( 100a + b \).

Present the answer in LaTex format: \boxed{Your answer}
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