
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
Construct a circle with center \( S \) and radius \( 3 \text{ cm} \). Construct two mutually perpendicular diameters \( AC \) and \( BD \) of this circle. Construct isosceles triangles \( ABK, BCL, CDM, DAN \) such that:

- The base of each triangle is a side of the quadrilateral \( ABCD \).
- The base of each triangle is equal to the height on this side.
- No triangle overlaps the quadrilateral \( ABCD \).

From the given data, calculate the area of the polygon \( AKBLCDMN \).

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