
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
In a rectangular parallelepiped \( A B C D A_{1} B_{1} C_{1} D_{1} \), four numbers - the lengths of the edges and the diagonal \( A C_{1} \) - form an arithmetic progression with a positive difference \( d \), and \( A A_{1} < A B < B C \). Two externally tangent spheres of identical and unknown radius \( R \) are located so that their centers lie inside the parallelepiped. The first sphere touches the faces \( A B B_{1} A_{1}, A D D_{1} A_{1}, A B C D \), and the second sphere touches the faces \( B C C_{1} B_{1} \), \( C D D_{1} C_{1} \), and \( A_{1} B_{1} C_{1} D_{1} \). Find: a) the lengths of the edges of the parallelepiped; b) the angle between the lines \( C D_{1} \) and \( A C_{1} \); c) the radius \( R \).

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