
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 increase;  
$(2)$ The range of the function $f(x)$ on the interval $\left[-\frac{\pi}{6}, \frac{\pi}{3}\right]$.

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