
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 $f(x) = \cos x \cdot \sin \left( x + \frac{\pi}{3} \right) - \sqrt{3}\cos^2 x + \frac{\sqrt{3}}{4}, x \in \mathbb{R}$,

$(1)$ Find the smallest positive period of $f(x)$;

$(2)$ Find the maximum and minimum values of $f(x)$ on the closed interval $\left[ -\frac{\pi}{4}, \frac{\pi}{4} \right]$.

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