
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 \(z_1, z_2\) satisfying \(\left|z_1\right|=\left|z_1+z_2\right|=3\) and \(\left|z_1-z_2\right|=3\sqrt{3}\), find the value of \(\log_3 \left| \left(z_1 \bar{z}_2\right)^{2000} + \left(\bar{z}_1 z_2\right)^{2000} \right|\).

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