
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 with an eccentricity of $\frac{\sqrt{6}}{3}$, one of its foci is point $F$, and the line passing through $F$ perpendicular to the $x$-axis intersects the ellipse at points $A$ and $B$ with $|AB| = \frac{2\sqrt{3}}{3}$.

(1) Find the equation of this ellipse;

(2) The line $y = kx + 2$ intersects the ellipse at points $C$ and $D$. If a circle passing through point $E(-1, 0)$ takes line segment $CD$ as its diameter, find the value of $k$.

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