
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 the triangular pyramid \(S A B C\), the angle \(\angle ASB\) at vertex \(S\) is equal to \(30^\circ\), and the lateral edge \(SC\) is inclined to the plane of the face \(ASB\) at an angle of \(45^\circ\). The sum of the lengths of the lateral edges of the pyramid is 9. Find the largest possible value of the volume of the pyramid under these conditions.

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