
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

We are given a regular prism \( K L M N K_{1} L_{1} M_{1} N_{1} \) with the base \( K L M N \). Planes \( \alpha \) and \( \beta \) are perpendicular to \( L_{1} N \) and pass through vertices \( K \) and \( N_{1} \) respectively. Let \( A \) and \( B \) be the points of intersection of planes \( \alpha \) and \( \beta \) with the diagonal \( L_{1} N \), where \( A \) is closer to \( N \) than \( B \).

a) Find the ratio \( L_{1} B: A N \).

b) Additionally, suppose there exists a sphere of radius \( \frac{1}{2} \) that touches all the lateral faces of the prism as well as the planes \( \alpha \) and \( \beta \). Find the length of segment \( L_{1} N \) and the volume of the prism \( K L M N K_{1} L_{1} M_{1} N_{1} \).

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