
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
Let \(a\) and \(b\) be real numbers. Suppose the complex numbers
\[
\begin{aligned}
z_{1} &= \sqrt{3} a - 1 + (\sqrt{3} - b) i \\
\text{and} \quad z_{2} &= 2 - \sqrt{3} a + b i
\end{aligned}
\]
have the same modulus, and \(z_{1} \overline{z_{2}}\) is a purely imaginary number. Find \(a + b\).

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