
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
In a regular quadrilateral pyramid \(S A B C D\), the side of the base \(A B C D\) is equal to \(a\), and the height is \(2 a \sqrt{2}\). A plane is drawn through vertex \(A\) parallel to the diagonal \(B D\) of the base such that the angle between the line \(A B\) and this plane is \(30^{\circ}\). Find the area of the cross-section.

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