
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 complex numbers \( Z_{1} \) and \( Z_{2} \) satisfy \( \left|Z_{1}\right| = \left|Z_{1} + Z_{2}\right| = 3 \) and \( \left|Z_{1} - Z_{2}\right| = 3 \sqrt{3} \). Determine the value of \( \log _{3} \left| \left( Z_{1} \cdot \overline{Z_{2}} \right)^{2009} + \left( \overline{Z_{1}} Z_{2} \right)^{2009} \right| \).

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