
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 \( AA_1 \), \( BB_1 \), and \( CC_1 \) of the triangular prism \( ABC A_1 B_1 C_1 \) such that \( \frac{AM}{AA_1} = \frac{2}{3} \), \( \frac{BN}{BB_1} = \frac{3}{5} \), \( \frac{CK}{CC_1} = \frac{4}{7} \). Point \( P \) belongs to the prism. Find the maximum possible volume of the pyramid \( MNKP \), if the volume of the prism is 27.

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