
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 $f(x) = \sin^2x + 2\sqrt{3}\sin x\cos x + 3\cos^2x - 2$, where $x \in \mathbb{R}$, find:  
(1) The smallest positive period of the function $f(x)$ and its intervals of monotonic increase;  
(2) The range of $f(x)$ in the interval $\left[-\frac{\pi}{6}, \frac{\pi}{3}\right]$.

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