
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 $f(x) = 2 \sqrt{3} \sin(wx + \frac{\pi}{6}) \cos wx$ $(0 < w < 2)$, and the graph of $f(x)$ passes through the point $\left( \frac{5\pi}{12}, \frac{\sqrt{3}}{2} \right)$.  
(1) Find the value of $w$ and the smallest positive period of the function $f(x)$.  
(2) The graph of $y = f(x)$ is translated to the right by $\frac{\pi}{6}$ units to obtain the graph of the function $y = g(x)$. Given that $g\left( \frac{\alpha}{2} \right) = \frac{5\sqrt{3}}{6}$, find the value of $\cos\left(2\alpha - \frac{\pi}{3}\right)$.

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