
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 the parabola \(\Gamma: y^{2}=8x\) with the focus \(F\), a line \(l\) passes through \(F\) and intersects the parabola \(\Gamma\) at two points \(A\) and \(B\). Tangents are drawn at points \(A\) and \(B\) on the parabola \(\Gamma\) and intersect the \(y\)-axis at points \(P\) and \(Q\), respectively. Find the minimum area of the quadrilateral \(APQB\).

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