
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 $\overrightarrow{a} = (2\sin(x + \frac{\pi}{6}), -2)$ and $\overrightarrow{b} = (2, \frac{\sqrt{3}}{2} - 2\cos x)$.  
(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 $x \in [0, \pi]$, find the range of $f(x)$.

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