
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 that the center of the ellipse $C$ is the origin $O$, its foci lie on the $x$-axis, and the distance between the left and right foci $F\_1$ and $F\_2$ is $2$, with point $P(1, \frac{3}{2})$ on the ellipse.

1. Find the equation of ellipse $C$ and its eccentricity $e$.
2. Line $l$ passing through $F\_1$ intersects ellipse $C$ at points $A$ and $B$. If the area of $\triangle AF\_2B$ is $\frac{12\sqrt{2}}{7}$, find the equation of the circle with center $F\_2$ that is tangent to line $l$.

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