
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
The left and right foci of the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ are $F_1$ and $F_2$, respectively. A line $l$ perpendicular to the $x$-axis through $F_2$ intersects the ellipse $E$ at point $P$ in the first quadrant. If $|PF_1| = 5$ and $3a = b^2$, then:  
(Ⅰ) Find the equation of the ellipse $E$;  
(Ⅱ) $A$ and $B$ are two points on the ellipse $C$ located on either side of the line $l$. If the line $AB$ passes through the point $(1, -1)$ and $\angle APF_2 = \angle BPF_2$, find the equation of the line $AB$.

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