
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
Let $F$ be the focus of the parabola $y^2 = 4x$, and let $A$ and $B$ be two points on the parabola different from the origin $O$, such that $\overrightarrow{FA} \cdot \overrightarrow{FB} = 0$. Extend $AF$ and $BF$ to intersect the parabola at points $C$ and $D$ respectively. Find the minimum area of quadrilateral $ABCD$.

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