
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 $xOy$, it is known that for any real number $k$, the line $(\sqrt{3}k+1)x+(k-\sqrt{3})y-(3k+\sqrt{3})=0$ always passes through a fixed point F. For an ellipse C with its center at the origin and one focus at F, the maximum distance from a point on the ellipse C to F is $2+\sqrt{3}$.
(1) Find the equation of ellipse C;
(2) Let $(m,n)$ be any point on ellipse C, and let circle O: $x^2+y^2=r^2$ ($r > 0$) have 4 distinct common points with ellipse C. Determine the positional relationship between circle O and the lines $l_1: mx+ny=1$ and $l_2: mx+ny=4$.

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