
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 quadrilateral pyramid \( S A B C D \) is a square \( A B C D \), and \( S A \) is the height of the pyramid. Let \( M \) and \( N \) be the midpoints of the edges \( S C \) and \( A D \), respectively. What is the maximum possible area of the triangle \( B S A \) if \( M N = 3 \)?

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