
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: $$\overrightarrow{a} = (2\sin x, -\sqrt{3}\cos x), \overrightarrow{b} = (\cos x, 2\cos x),$$ and let $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$.
(1) Find the smallest positive period and the maximum value of $f(x)$.
(2) Translate the graph of $f(x)$ to the left by $\frac{\pi}{3}$ units and upward by $\sqrt{3}$ units to obtain the graph of $g(x)$. Find the analytical expression for $g(x)$.
(3) Let $h(x)$ be the derivative of $g(x)$. When $0 \leq x \leq \frac{\pi}{2}$, find the range of $h(x)$.

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