
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 $\frac{\sqrt{2}}{2}$, and its left focus at $F(-1,0)$. A line $l$ passing through $D(0,2)$ with a slope of $k$ intersects the ellipse at points $A$ and $B$.

(1) Find the standard equation of the ellipse $C$;
(2) Determine if there exists a fixed point $E$ on the $y$-axis such that $\overrightarrow{AE} \cdot \overrightarrow{BE}$ is always constant. If so, find the coordinates of $E$ and the constant value; if not, explain the reason.

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