
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
Let $F_1$ and $F_2$ be the left and right foci of the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$). A line $l$ passing through $F_1$ with a slope of 1 intersects $E$ at points $A$ and $B$, and $|AF_2|$, $|AB|$, $|BF_2|$ form an arithmetic sequence.  
(1) Find the eccentricity of $E$.  
(2) Given point $P(0, -1)$ satisfies $|PA| = |PB|$, find the equation of $E$.

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