
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 $C$ is at the origin $O$, the focus $F$ is on the positive semi-axis of $x$, and the line passing through $F$ intersects the parabola $C$ at points $A$ and $B$, determine the equation of the parabola $C$, given that $\overrightarrow{OA} \cdot \overrightarrow{OB} = -12$.

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