
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 $F_1$ and $F_2$ are the left and right foci of the ellipse E: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), and the line $l$ passing through $F_2$ with a slope of $-1$ intersects E at points A and B. The distances $|AF_1|$, $|AB|$, and $|BF_1|$ form an arithmetic sequence.

1. Find the eccentricity of E.
2. If 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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