
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
Given a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) that satisfies the condition: for any real numbers \( x \) and \( y \),
\[ f(2x) + f(2y) = f(x+y) f(x-y) \]

and given that \( f(\pi) = 0 \) and \( f(x) \) is not identically zero, determine the period of \( f(x) \).

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