
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 a line $l$ passing through $F\_1$ at a $45^{\circ}$ angle intersects $E$ at points $A$ and $B$, with $|AB| = \frac{4a}{3}$.
(I) Find the eccentricity of $E$.
(II) Let point $P(0, -1)$ satisfy $|PA| = |PB|$. Find the equation of $E$.

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