
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 $\dfrac{\sqrt{2}}{2}$, and its left focus at $F(-1,0)$. A line $l$ passing through point $D(0,2)$ with slope $k$ intersects the ellipse at points $A$ and $B$.  
$(1)$ Find the standard equation of ellipse $C$;  
$(2)$ On the $y$-axis, is there a fixed point $E$ such that $\overrightarrow{AE} \cdot \overrightarrow{BE}$ is always a constant? If it exists, find the coordinates of point $E$ and this constant value; if not, explain why.

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