
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 equation of an ellipse $\frac{y^{2}}{a^{2}}+ \frac{x^{2}}{b^{2}}=1 (a > b > 0)$, its lower focus $F\_1$ coincides with the focus of the parabola $x^{2}=-4y$, and the eccentricity $e= \frac{ \sqrt{2}}{2}$. Line $l$ passes through $F\_1$ and intersects the ellipse at points $A$ and $B$.

1. Find the equation of the ellipse;
2. Find the equation of the circle that passes through the origin $O$, focus $F\_1$, and is tangent to the line $y=- \frac{a^{2}}{c}$ (where $c$ is the semi-focal distance of the ellipse);
3. Find the equation of line $l$ when $ \overrightarrow{F\_2A} \cdot \overrightarrow{F\_2B}= \frac{5}{4}$, and find the length of the chord cut off by line $l$ on the circle from part (2) when the slope is greater than $0$. The circle's center is in the fourth quadrant.

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