
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
In the rectangular coordinate system $(xOy)$, an ellipse $C:\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1 (a > b > 0)$ has an eccentricity of $\frac{\sqrt{6}}{3}$. A line $l$ intersects the $x$-axis at point $E$ and intersects the ellipse $C$ at points $A$ and $B$. When line $l$ is perpendicular to the $x$-axis and point $E$ is the right focus of ellipse $C$, the length of chord $AB$ is $\frac{2\sqrt{6}}{3}$.

(1) Find the equation of the ellipse $C$.

(2) Determine whether there exists a point $E$ such that $\frac{1}{E{{A}^{2}}}+\frac{1}{E{{B}^{2}}}$ is a constant value. If such a point exists, find its coordinates and the constant value. If not, provide an explanation.

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