
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
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $(a\cos B+b\cos A)\cos 2C=c\cdot\cos C$.
(1) Find angle $C$;
(2) If $b=2a$, the area of $\triangle ABC$ is $S= \frac{ \sqrt{3}}{2}\sin A\cdot\sin B$, find $\sin A$ and the value of $c$.

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