
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 the complex numbers \( z_{1} \) and \( z_{2} \) satisfy \( \left|z_{1}\right|=\left|z_{1}+z_{2}\right|=3 \) 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}
user