
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 \( S A B C D \) with vertex \( S \), the side length of the base of the pyramid is \( b \), and the height of the pyramid is \( b \sqrt{2} \). A sphere inscribed in this pyramid touches the lateral face \( S A D \) at point \( K \). Find the area of the cross-section of the pyramid that passes through edge \( A B \) and point \( K \).

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