
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
Assume \(x\), \(y\), and \(z\) are real numbers such that:
\[ \cos x + \cos (y + \frac{\pi}{3}) + \cos (z - \frac{\pi}{3}) = \sin x + \sin (y + \frac{\pi}{3}) + \sin (z - \frac{\pi}{3}) = 0. \]
Determine the sum of all possible values of \(\cos 2x + \cos 2y + \cos 2z\).

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