
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
Complex numbers \( z_{1} \) and \( z_{2} \) satisfy
\[ \left| \left(\left|z_{1}\right| z_{1}\right) + \left(\left|z_{2}\right| z_{2}\right)\right| = \left| \left(\left|z_{1}\right| z_{2}\right) + \left(z_{1}\left|z_{2}\right|\right) \right|, \]
and the sum of the maximum and minimum values of \( \left| z_{1} - \frac{1}{z_{2}} \right| + \left| z_{2} - \frac{1}{z_{1}} \right| \) is \( 4\sqrt{3} \).

Find the value of \( \left(\left| z_{1} \overline{z_{2}} \right| + \left| z_{2} \overline{z_{1}} \right|\right)^{2016}. \)

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