
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 quadrilateral pyramid \(S A B C D\):

- The lateral faces \(S A B, S B C, S C D, S D A\) have areas of 9, 9, 27, and 27 respectively;
- The dihedral angles at the edges \(A B, B C, C D, D A\) are equal;
- The quadrilateral \(A B C D\) is inscribed in a circle, and its area is 36.

Find the volume of the pyramid \(S A B C D\).

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