
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 with center at the origin $O$ is tangent to the line $y=x+2 \sqrt {2}$.
(1) Find the equation of the circle $O$;
(2) The circle $O$ intersects the $x$-axis at points $A$ and $B$. A point $P$ moves inside the circle such that 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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