
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 a parabola $y^2 = 2px$ ($p > 0$) with focus F, let P be a point on the parabola with a y-coordinate of 4 and $|PF| = 4$.

(Ⅰ) Find the equation of the parabola;
(Ⅱ) Let points A($x_1$, $y_1$) and B($x_2$, $y_2$) ($y_i \leq 0$, $i = 1, 2$) be two points on the parabola such that the angle bisector of $\angle APB$ is perpendicular to the x-axis. Find the equation of line AB that maximizes the area of $\triangle PAB$.

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