
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 $f(x)=2\cos ^{2}x+ \sqrt {3}\sin 2x$, $x\in R$.
(I) Find the interval(s) where the function $f(x)$ is monotonically increasing.
(II) Stretch the horizontal coordinates of all points on the graph of $f(x)$ by a factor of $2$ without changing the vertical coordinates to obtain the graph of function $h(x)$. Then shift the graph of $h(x)$ to the right by $\frac{\pi}{3}$ units to obtain the graph of function $g(x)$. Find the analytical expression of $g(x)$ and its range over the interval $[0,\pi]$.

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