
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 the origin $O$ as its center is tangent to the line: $$x- \sqrt {3}y=4.$$  
(1) Find the equation of circle $O$;  
(2) If there are two points $M$, $N$ on circle $O$ that are symmetric about the line $x+2y=0$, and $$|MN|=2 \sqrt {3},$$ find the equation of line $MN$;  
(3) Circle $O$ intersects the x-axis at points $A$, $B$, and a moving point $P$ inside the circle makes $|PA|$, $|PO|$, $|PB|$ form a geometric sequence, find the range of values for $$\overrightarrow {PA}\cdot \overrightarrow {PB}.$$

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