
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
Consider all possible tetrahedrons $A B C D$ where $A B=2$, $A C=C B=7$, and $A D=D B=8$. Each of these tetrahedrons is inscribed in a cylinder such that all vertices lie on the lateral surface of the cylinder, and the edge $C D$ is parallel to the axis of the cylinder. Choose the tetrahedron for which the cylinder's radius is the smallest among all possibilities. What values can the length of $C D$ take in such a tetrahedron?

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