
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
Inside the cube \(ABCD A_{1}B_{1}C_{1}D_{1}\) there is a sphere with center \(O\) and radius 10. The sphere intersects the face \(AA_{1}D_{1}D\) in a circle of radius 1, the face \(A_{1}B_{1}C_{1}D_{1}\) in a circle of radius 1, and the face \(CD D_{1}C_{1}\) in a circle of radius 3. Find the length of the segment \(OD_{1}\).

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