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

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

$(2)$ Given points $A,B$ 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 $E$ and the constant value; if not, explain why.

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