
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 an eccentricity of $\frac{\sqrt{6}}{3}$, a circle centered at the origin with the radius equal to the major axis of the ellipse intersects the line $2x - \sqrt{2}y + 6 = 0$ at a tangent point.
(I) Find the standard equation of the ellipse $C$.
(II) Let points $A$ and $B$ be the two intersections of the moving line $y = k(x - 2) (k \neq 0)$ and the ellipse $C$. Determine if there exists a point $E$ on the $x$-axis such that $\overrightarrow{EA} \cdot \overrightarrow{EB}$ is constant. If such a point exists, find its coordinates and the constant value; otherwise, explain the reason.

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