
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 $\frac{x^{2}}{3}+y^{2}=1$ and the fixed point $E(-1,0)$, if the line $y=kx+2 (k \neq 0)$ intersects the ellipse at points $C$ and $D$, does there exist a value of $k$ such that a circle with $CD$ as its diameter passes through point $E$? Please explain your reasoning.

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