
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 \(a\), \(b\), and \(c\) be real numbers such that
\[
\cos a + \cos b + \cos c = \sin a + \sin b + \sin c = 1.
\]
Find the sum of all possible values of \(\cos 2a + \cos 2b + \cos 2c.\)

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