
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 and its focus is on the positive half of the x-axis, with the focus denoted as F, and points A, B, and C on the parabola such that $$\overrightarrow{FA} + \overrightarrow{FB} + \overrightarrow{FC} = \overrightarrow{0}$$ and $$|\overrightarrow{FA}| + |\overrightarrow{FB}| + |\overrightarrow{FC}| = 6$$, find the equation of the parabola.

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