
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 $M(-1,0)$ is tangent to the line $x- \sqrt{3}y-3=0$.
1. Find the equation of circle $M$.
2. If there exist two points on the circle that are symmetric with respect to the line $mx+y+1=0$, find the value of $m$.
3. Given points $A(-2,0)$ and $B(2,0)$, and a moving point $P$ inside the circle satisfies $|PA|\cdot|PB|=|PO|^2$, find the range of values for $\overrightarrow{PA} \cdot \overrightarrow{PB}$.

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