
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\omega x\cos\omega x - 2\sqrt{3}\sin^2\omega x + \sqrt{3}(\_\omega (\_ > 0)), the lines $x = \_x\_{1}$ and $x = \_x\_{2}$ are any two symmetry axes of the graph of the function $y = f(x)$, and the minimum value of $|x\_1 - x\_2|$ is $\frac{\pi}{2}$.

1. Find the value of $\omega$;
2. Find the intervals where the function $f(x)$ is increasing;
3. If $f(\alpha) = \frac{2}{3}$, find the value of $\sin(\frac{5}{6}\pi - 4\alpha)$.

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