
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
On the edge \( BC \) of parallelepiped \( ABCD A_1 B_1 C_1 D_1 \), a point \( M \) is chosen. A sphere, constructed on the segment \( C_1 M \) as its diameter, touches the planes of four faces of the parallelepiped, with one such touchpoint lying on the edge \( B_1 B \). Given that \( BM = 1 \) and \( CM = 8 \), find the length of the edge \( AA_1 \), the radius of the sphere, and the volume of the parallelepiped.

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