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

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

(II) Let points $A$ and $B$ be the intersections of the moving line $y=k(x-2)\,(k\neq 0)$ with the ellipse $C$. Determine if there exists a fixed point $E$ on the $x$-axis such that $(\overrightarrow{EA}^{2}+ \overrightarrow{EA} \cdot \overrightarrow{AB})$ is constant. If such a point exists, find its coordinates and the constant value; otherwise, explain why not.

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