
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 vectors $\overrightarrow{a}=(5 \sqrt{3}\cos x,\cos x)$, $\overrightarrow{b}=(\sin x,2\cos x)$, and the function $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}+|\overrightarrow{b}|^{2}+\frac{3}{2}$.
(I) Find the minimum value of the function $f(x)$ when $x\in[0, \frac{\pi}{2}]$.
(II) If $f(x)=8$ when $x\in[\frac{\pi}{6}, \frac{\pi}{2}]$, find the value of the function $f(x+\frac{\pi}{8})$.

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