
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 $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 \ (a > b > 0)$, the shortest distance from a point on the ellipse to the left focus is $\sqrt{6} - 2$, and the length of the major axis is $2\sqrt{6}$.  
(1) Find the standard equation of ellipse $C$;  
(2) If a line with a non-zero and finite slope passes through the right focus of ellipse $C$ and intersects the ellipse at points $A$ and $B$, is there a fixed point $E$ on the $x$-axis such that $\overrightarrow{EA} \cdot \overrightarrow{EB}$ is a constant value? If it exists, find the coordinates of point $E$ and the constant value; if it does not exist, please explain why.

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