
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 triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $2a\cos B=c\cos B+b\cos C$.
(1) Find the measure of angle $B$;
(2) Let vector $\overrightarrow{m}=(\cos A,\cos 2A)$, $\overrightarrow{n}=(12,-5)$, and side length $a=4$, find the value of the triangle's area $S_{\triangle ABC}$ when $\overrightarrow{m} \cdot \overrightarrow{n}$ is at its maximum.

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