
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 $f(x)=\sqrt{3}\sin 2x - 2\cos^2 x + 1$.
(1) Find the range of the function $f(x)$ in the interval $[\frac{-\pi}{12}, \frac{\pi}{2})$.
(2) Suppose $\alpha, \beta \in (0, \frac{\pi}{2})$, $f(\frac{1}{2}\alpha + \frac{\pi}{12}) = \frac{10}{13}$, and $f(\frac{1}{2}\beta + \frac{\pi}{3}) = \frac{6}{5}$, find the value of $\sin(\alpha - \beta)$.

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