
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 Cartesian coordinate system $xOy$, a circle with center $O$ is tangent to the line $x - \sqrt{3}y = 4$.  
(1) Find the equation of circle $O$;  
(2) Circle $O$ intersects the x-axis at points $A$ and $B$. A moving point $P$ inside the circle satisfies that $PA$, $PO$, and $PB$ form a geometric sequence. Find the equation of the trajectory of point $P$.

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