
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
The center of an ellipse is at the origin, and its left focus $F_1$ coincides with the focus of the parabola $y^2 = -4x$. A line $l$ passing through $F_1$ intersects the ellipse at points $A$ and $B$, and the parabola at points $C$ and $D$. When line $l$ is perpendicular to the x-axis, $$\frac{|CD|}{|AB|} = 2\sqrt{2}$$.
(Ⅰ) Find the equation of the ellipse;
(Ⅱ) Find the equation of the circle that passes through point $O$, point $F_1$, and is tangent to the directrix of the ellipse on the left;
(Ⅲ) Find the maximum and minimum values of $\overrightarrow{F_{2}A} \cdot \overrightarrow{F_{2}B}$.

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