
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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a>b>0$) with two foci $F_1(-c, 0)$ and $F_2(c, 0)$ ($c>0$). A line passing through the point $E(\frac{a^2}{c}, 0)$ intersects the ellipse at points $A$ and $B$. Additionally, $F_1A$ is parallel to $F_2B$, and $|F_1A| = 2|F_2B|$.
(I) Find the eccentricity of the ellipse.
(II) Find the slope of line $AB$.

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