
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 rectangular parallelepiped \( ABCD A_{1} B_{1} C_{1} D_{1} \) with edges \( AB=4 \), \( AD=2 \), \( AA_{1}=3\sqrt{2} \), a plane passes through the diagonal \( BD_{1} \) and intersects the edge \( AA_{1} \) such that the cross-section of the parallelepiped with this plane has the smallest perimeter. Find the area of this cross-section.

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