
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 complex numbers \( z_{1}, z_{2} \) satisfy \( \left|z_{1}\right| = \left|z_{1} + z_{2}\right| = 3 \), and \( \left|z_{1} - z_{2}\right| = 3 \sqrt{3} \). Then \(\log _{3}\left|\left(z_{1} \overline{z_{2}}\right)^{2000} + \left(\overline{z_{1}} z_{2}\right)^{2000}\right|\) equals ______.

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