
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 \( z_{1}, z_{2} \in \mathbf{C} \) such that \(\left| z_{1} \right| = \left| z_{1} + z_{2} \right| = 3 \) and \(\left| z_{1} - z_{2} \right| = 3\sqrt{3}\). Find the value of \(\log_{3} \left| \left(z_{1} \overline{z_{2}} \right)^{2000} + \left( \overline{z_{1}} z_{2} \right)^{2000} \right| \).

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