
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
Consider the parabola $y^{2}=4x$. Let $A, B, C,$ and $D$ be points on the parabola such that chords $AC$ and $BD$ are perpendicular to each other and pass through the focus $F$ of the parabola. Find the minimum area of the quadrilateral formed by $A, B, C,$ and $D$.

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