
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
An ellipse has its center 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 intersects the parabola at points $C$ and $D$. When the line $l$ is perpendicular to the $x$-axis, $|CD| = 2 \sqrt{2} |AB|$.

1. Find the equation of the ellipse.
2. Find the equation of the circle passing through point $O$, point $F\_1$, and is tangent to the left latus rectum of the ellipse.
3. Find the maximum value of $\overrightarrow{F\_2A} \cdot \overrightarrow{F\_2B}$.

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