
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 $f(x)=2\cos \left(2x+ \frac{2\pi}{3}\right)+ \sqrt{3}\sin 2x$.

(1) Find the smallest positive period and the maximum value of the function $f(x)$;

(2) Let the three interior angles of $\triangle ABC$ be $A$, $B$, and $C$, respectively. If $f\left(\frac{C}{2}\right)=- \frac{1}{2}$ and $AC=1$, $BC=3$, find the value of $\sin A$.

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