
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 \( x, y, z, \) and \( w \) be real numbers such that
\[
\cos x + \cos y + \cos z + \cos w = \sin x + \sin y + \sin z + \sin w = 0.
\]
Find the sum of all possible values of \( \cos 2x + \cos 2y + \cos 2z + \cos 2w. \)

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