
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 \(A B C D A_{1} B_{1} C_{1} D_{1}\) there is a center \(O\) of a sphere with radius 10. The sphere intersects the face \(A A_{1} D_{1} D\) creating a circle with radius 1, the face \(A_{1} B_{1} C_{1} D_{1}\) creating a circle with radius 1, and the face \(C D D_{1} C_{1}\) creating a circle with radius 3. Find the length of the segment \(O D_{1}\).

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