
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

Edge \( A_{1} A \) of the parallelepiped \( A B C D A_{1} B_{1} C_{1} D_{1} \) is perpendicular to its face \( A B C D \). Sphere \( \Omega \) touches the edges \( B B_{1}, B_{1} C_{1}, C_{1} C, C B, C D \), and touches the edge \( C D \) at a point \( K \) such that \( C K = 9 \) and \( K D = 1 \).

a) Find the length of edge \( A_{1} A \).

b) Additionally, suppose the sphere \( \Omega \) touches the edge \( A_{1} D_{1} \). Find the volume of the parallelepiped \( A B C D A_{1} B_{1} C_{1} D_{1} \) and the radius of the sphere \( \Omega \).

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