
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\frac{\sqrt{6}}{3}$, and a circle centered at the origin $O$ with the major axis of the ellipse $C$ as its radius, which is tangent to the line $2x - \sqrt{2}y + 6 = 0$.

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

$(2)$ Given points $A$ and $B$ are the two intersection points of the moving line $y = k(x - 2) (k \neq 0)$ with the ellipse $C$, is there a point $E$ on the $x$-axis such that $\overrightarrow{EA}^2 + \overrightarrow{EA} \cdot \overrightarrow{AB}$ is a constant value? If it exists, find the coordinates of point $E$ and the constant value; if not, explain why.

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