
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 the function $f(x) = 2\sin\omega x \cdot \cos(\omega x) + (\omega > 0)$ has the smallest positive period of $4\pi$.

(1) Find the value of the positive real number $\omega$;  
(2) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, and it satisfies $2b\cos A = a\cos C + c\cos A$. Find the value of $f(A)$.

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