
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 $f(x) = [2\sin(x + \frac{\pi}{3}) + \sin x]\cos x - \sqrt{3}\sin^2 x$.

1. Find the smallest positive period of $f(x)$.
2. If there exists $x\_0 \in [0, \frac{5\pi}{12}]$ such that $mf(x\_0) - 2 = 0$, find the range of values for the real number $m$.
3. In acute triangle $ABC$, if $\angle B = 2\angle A$, find the range of values for $\frac{f(\frac{C}{2} - \frac{\pi}{6})}{f(\frac{B}{2} - \frac{\pi}{6})}$.

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