
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 $f(x) = \sqrt{3}(\cos \frac{x}{2} - \sin \frac{x}{2})(\cos \frac{x}{2} + \sin \frac{x}{2}) + 2\sin \frac{x}{2}\cos \frac{x}{2}$.  
(1) Find the smallest positive period of $f(x)$.  
(2) If the graph of $f(x)$ is translated to the right by $\frac{\pi}{6}$ units to obtain the graph of function $g(x)$, find the intervals of increase for $g(x)$.

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