
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 $\overrightarrow{a} = (5\sqrt{3}\cos x, \cos x)$ and $\overrightarrow{b} = (\sin x, 2\cos x)$, and the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} + |\overrightarrow{b}|^2 + \frac{3}{2}$.
(I) Find the range of $f(x)$ when $x \in [\frac{\pi}{6}, \frac{\pi}{2}]$.
(II) If $f(x) = 8$ when $x \in [\frac{\pi}{6}, \frac{\pi}{2}]$, find the value of $f(x - \frac{\pi}{12})$.

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