
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 an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$, where the two foci of the ellipse form an equilateral triangle with one endpoint of the minor axis, and the length of the minor axis of the ellipse $C$ is $2\sqrt{3}$. <br/>$(1)$ Find the standard equation of the ellipse $C$; <br/>$(2)$ Determine whether there exists a line $l$ passing through the point $P(0,2)$ that intersects the ellipse $C$ at two distinct points $M$ and $N$, satisfying $\overrightarrow{OM}•\overrightarrow{ON}=2$ (where $O$ is the origin). If such a line exists, find the equation of the line; if not, explain the reason.

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