
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
The edge \(A_{1} A\) of 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 \(B B_{1}\), \(B_{1} C_{1}\), \(C_{1} C\), \(C B\), and \(C_{1} D_{1}\), and touches the edge \(C_{1} D_{1}\) at a point \(K\) such that \(C_{1} K = 9\) and \(K D_{1} = 4\).

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

b) Suppose additionally that the sphere \(\Omega\) touches the edge \(A D\). 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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