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

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