
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 vectors $\overrightarrow{a}=(\sqrt{3}\sin x,\cos x)$ and $\overrightarrow{b}=(\cos x,\cos x)$, let $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$.

(1) Determine the smallest positive period of the function $f(x)$;

(2) Describe how the graph of this function can be obtained through translation and scaling transformations of the graph of $y=\sin x (x\in R)$;

(3) If $x\in [-\frac{\pi }{6},\frac{\pi }{3}]$ and the minimum value of the function $g(x)=f(x)+m$ is $2$, find the maximum value of the function $g(x)$ and the value of $x$ when $g(x)$ reaches its maximum.

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