
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
The domain of the function \( f(x) \) is \( \mathbf{R} \). For any \( a, b \in \mathbf{R} \), it satisfies the equation \( f(a+b) + f(a-b) = 2 f(a) f(b) \), and there exists \( c > 0 \) such that \( f\left(\frac{c}{2}\right) = 0 \). Is \( f(x) \) a periodic function? If so, find one of its periods; if not, explain why.

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