
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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \frac{\sqrt{6}}{3}$, the distance from the origin to the line passing through points $A(0, -b)$ and $B(a, 0)$ is $\frac{\sqrt{3}}{2}$.

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

$(2)$ Given a fixed point $E(-1, 0)$, if the line $y = kx + 2 (k \neq 0)$ intersects the ellipse at points $C$ and $D$, is there a value of $k$ such that the circle with diameter $CD$ passes through point $E$? Please explain your reasoning.

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