
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
In the rectangular coordinate system $(xOy)$, a circle $(O)$ with the coordinate origin as its center is tangent to the line $x-\sqrt{3}y=4$.

(I) Find the equation of circle $(O)$;

(II) Circle $(O)$ intersects the $x$-axis at points $A$ and $B$. A moving point $P$ inside the circle makes the distances $|PA|$, $|PO|$, and $|PB|$ form a geometric sequence. Find the range of values for $\overrightarrow{PA}\cdot\overrightarrow{PB}$.

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