
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
Given complex numbers \( z_1 \) and \( z_2 \) that satisfy \( \left|z_1\right| = \left|z_1 + z_2\right| = 3 \) and \( \left|z_1 - z_2\right| = 3\sqrt{3} \), determine:

\[ \log_{2} \left| \left(z_1 \overline{z_2}\right)^{2000} + \left(\overline{z_1} z_2\right)^{2000} \right| \]

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