
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 the vertex of the parabola is at the origin, its focus is on the x-axis, and the three vertices of the triangle $\triangle ABC$ all lie on the parabola. Additionally, the centroid of $\triangle ABC$ is the focus $F$ of the parabola. If the equation of the line where side $BC$ lies is $4x + y - 20 = 0$, then find the equation of the parabola.

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