
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 ellipse $E: \frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$ (with $a > b > 0$), where $F_1$ and $F_2$ are its left and right foci, respectively. A line $l$ passing through the left focus with a slope of 1 intersects the ellipse at points $A$ and $B$ such that $AF_2$, $AB$, $BF_2$ form an arithmetic sequence.

(1) Find the eccentricity of the ellipse $E$.
(2) Suppose both points $A$ and $B$ lie on the same circle centered at $P(-2, 0)$. Find the equation of the ellipse $E$.

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