
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

Inside a right quadrangular pyramid \( S A B C D \) with a base \( A B C D \), there is inscribed a right quadrangular prism \( K L M N K_{1} L_{1} M_{1} N_{1} \), with its base \( K L M N \) lying in the plane \( A B C \). The center of the base \( K L M N \) of the prism is located on the segment \( A C \) such that \( K L \parallel A C \) and \( K N \parallel B D \) (points \( K \) and \( B \) are on the same side relative to \( A C \)). The side length of the base of the prism is 2, and the lateral edge \( K K_{1} \) of the prism is 1. The vertices \( L_{1} \) and \( M_{1} \) of the upper base of the prism \( K L M N K_{1} L_{1} M_{1} N_{1} \) lie on the lateral faces \( S B C \) and \( S C D \) of the pyramid \( S A B C D \), respectively. A plane \( \gamma \) passes through the points \( B \), \( K_{1} \), and \( N_{1} \). Find the volumes of the parts into which the plane \( \gamma \) divides the pyramid \( S A B C D \), if the side length of the pyramid's base is \( 8 \sqrt{2} \) and its height is 4. (20 points)

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