
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
A line passes through the focus F of the parabola $y^2 = 2px$ ($p > 0$) and intersects the parabola at points A and B. The vector $\overrightarrow{AF} = 3\overrightarrow{FB}$. The projections of A and B onto the directrix of the parabola are D and C, respectively. If the area of trapezoid ABCD is $8\sqrt{3}$, find the equation of the parabola.

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