
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
The base of the pyramid \(T A B C D\) is a rectangle \(A B C D\). The height of the pyramid, which is equal to \(h\), coincides with the lateral edge \(T A\), and the lateral edge \(T C\) is inclined to the plane of the base at an angle of \(30^{\circ}\). A plane passing through the edge \(T C\) and parallel to the diagonal of the base \(B D\) makes an angle of \(60^{\circ}\) with the plane of the base. What is the minimum area of the cross-section of the pyramid by a plane passing through the diagonal of the base \(B D\)?

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