
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 $\overrightarrow{m}=(\sqrt{3}\sin x,2)$ and $\overrightarrow{n}=(2\cos x,\cos^2 x)$, let the function $f(x)=\overrightarrow{m}\cdot\overrightarrow{n}$,  
(1) Find the range of the function $f(x)$;  
(2) In $\triangle ABC$, with angles $A$, $B$, $C$ and sides $a$, $b$, $c$ satisfying $a=2$, $f(A)=2$, $\sin B=2\sin C$, find the length of side $c$.

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