
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
Let $x \in \mathbb{R}$, and the function $f(x) = \cos x (2\sqrt{3}\sin x - \cos x) + \sin^2 x$.

(1) Find the intervals of monotonic increase for the function $f(x)$.

(2) If $f\left(\frac{\alpha}{2}\right) = \frac{1}{2}$, where $\frac{\pi}{6} < \alpha < \frac{2\pi}{3}$, find $\sin \alpha$.

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