
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 vectors $\overrightarrow{a} = (\sin(x + \frac{\pi}{6}), 1)$ and $\overrightarrow{b} = (4, 4\cos x - \sqrt{3})$:

(I) If $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of $\sin(x + \frac{4\pi}{3})$;

(II) Let $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$. If $\alpha \in [0, \frac{\pi}{2}]$ and $f(\alpha - \frac{\pi}{6}) = 2\sqrt{3}$, find the value of $\cos\alpha$.

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