
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 $f(x)$ be a function defined on $\mathbb{R}$ such that for any real numbers $a$ and $b$, the equation $f(a+b) + f(a-b) = 2f(a)f(b)$ holds, and $f(0) \neq 0$.
1. Find the value of $f(0)$;
2. Determine the parity (odd or even) of $f(x)$;
3. If there exists a constant $c > 0$ such that $f\left(\frac{c}{2}\right) = 0$, determine whether $f(x)$ is a periodic function. If it is, identify a period of the function; if not, provide a reason.

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