
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
Given the ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \left( a > b > 0 \right)$, its eccentricity $e=\dfrac{\sqrt{6}}{3}$, and the sum of distances from a point on the ellipse to the two foci is $2\sqrt{3}$.

$(1)$ Find the equation of ellipse $C$;

$(2)$ A line $l$ passing through point $P(0,2)$ with slope $k$ intersects the ellipse $C$ at two distinct points $A$ and $B$, $O$ is the origin. If $\angle AOB$ is acute, find the range of values for the slope $k$ of line $l$.

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