
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 $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $e = \dfrac{\sqrt{6}}{3}$, and the distance between the left focus and one endpoint of the minor axis is $\sqrt{3}$.

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

$(II)$ Given the fixed point $E(-1, 0)$, if the line $y = kx + 2$ intersects the ellipse at points $A$ and $B$. Is there a real number $k$ such that the circle with diameter $AB$ passes through point $E$? Please explain your reasoning.

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