
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

Points \( M \), \( N \), and \( K \) are located on the lateral edges \( A A_{1} \), \( B B_{1} \), and \( C C_{1} \) of the triangular prism \( A B C A_{1} B_{1} C_{1} \) such that \( \frac{A M}{A A_{1}} = \frac{3}{7} \), \( \frac{B N}{B B_{1}} = \frac{2}{5} \), and \( \frac{C K}{C C_{1}} = \frac{4}{9} \). Point \( P \) lies within the prism. Find the maximum possible volume of the pyramid \( M N K P \) if the volume of the prism is 40.

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