
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 foci $F_1(-1,0)$ and $F_2(1,0)$, and $P$ is a point on the ellipse, and $2|F_1F_2| = |PF_1| + |PF_2|$

(1) Find the equation of ellipse $C$;  
(2) A line $l$ passing through $F_1$ intersects ellipse $C$ at points $A$ and $B$. If the area of $\triangle AF_2B$ is $\frac{12\sqrt{6}}{11}$, 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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