
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, the two foci of the ellipse and one endpoint of the minor axis form an equilateral triangle, and the length of the minor axis of ellipse $C$ is $2\sqrt{3}$.

(Ⅰ) Find the standard equation of ellipse $C$;

(Ⅱ) Determine whether there exists a line passing through point $P(0,2)$ that intersects ellipse $C$ at two distinct points $M$ and $N$, satisfying $\overrightarrow{OM} \cdot \overrightarrow{ON} = 2$ (where $O$ is the origin). If such a line exists, find its equation; if not, explain why.

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