
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 the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$, the right focus of $C$ is $F(1,0)$, and the left and right endpoints of the major axis are $A_1$ and $A_2$, respectively, with $\overrightarrow{FA_1} \cdot \overrightarrow{FA_2} = -1$.

(Ⅰ) Find the equation of the ellipse $C$;

(Ⅱ) A line $l$ passing through the focus $F$ with a slope of $k (k \neq 0)$ intersects the ellipse $C$ at points $A$ and $B$. The perpendicular bisector of chord $AB$ intersects the $x$-axis at point $D$. Is there a point $E$ on the ellipse $C$ such that the quadrilateral $ADBE$ is a rhombus? If so, find the distance from point $E$ to the $y$-axis; if not, explain why.

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