
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 $S$ with its vertex at the origin $O$ and the focus on the $x$-axis. A chord $AB$ passes through the focus $F$. The extensions of line segments $AO$ and $BO$ intersect the directrix at points $C$ and $D$, respectively. If the minimum area of quadrilateral $ABCD$ is 8, find the equation of the parabola.

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