
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 two points $A(x\_1, f(x\_1))$ and $B(x\_2, f(x\_2))$ on the function $f(x) = 2 \sin(\omega x + \phi) (\omega > 0, -\frac{\pi}{2} < \phi < 0)$, and the terminal side of angle $\phi$ passes through point $P(1, -\sqrt{3})$. If $|f(x\_1) - f(x\_2)| = 4$ when $|x\_1 - x\_2|$ is at its minimum value of $\frac{\pi}{3}$:
1. Find the analytical expression of the function $f(x)$;
2. Determine the range of real values for $m$ such that the equation $3[f(x)]^2 - f(x) + m = 0$ has two distinct solutions within the interval $x \in (\frac{\pi}{9}, \frac{4\pi}{9})$.

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