
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 $f(x) = \sin \left( \frac{x}{2} \right) + \sqrt{3} \cos \left( \frac{x}{2} \right)$, where $x \in \mathbb{R}$.  
(1) Find the smallest positive period of the function $f(x)$.  
(2) Determine the intervals of monotonic increase for the function $f(x)$ on the interval $x \in [-2\pi, 2\pi]$.

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