
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 an ellipse C with its center of symmetry at the origin O, and its foci on the x-axis, with the left and right foci being $F_1$ and $F_2$ respectively, and the distance between $F_1$ and $F_2$ is 2. Point P (1, $\frac{3}{2}$) lies on this ellipse.  
(Ⅰ) Find the equation of ellipse C and its eccentricity $e$;  
(Ⅱ) A line $l$ passing through $F_1$ intersects the 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 centered at $F_2$ that is tangent to line $l$.

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