
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 
\[xyz=\sqrt{3} + i\sqrt{5}.\]
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  $\tfrac{a}{b}$  for relatively prime positive integers  $a$  and  $b$ . Compute  $100a+b.$ 

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