
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 $x$, $y$, and $z$ be real numbers such that
\[\cos (x+\pi/4) + \cos (y+\pi/4) + \cos (z+\pi/4) = \sin (x+\pi/4) + \sin (y+\pi/4) + \sin (z+\pi/4) = 0.\] 
Find 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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