
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
The side of the base \(ABC\) of a regular prism \(ABCA_1B_1C_1\) is 1, and each of the lateral edges has a length of \(\frac{1}{\sqrt{3}}\). A right circular cylinder is positioned such that the point \(A_1\) and the midpoint \(M\) of the edge \(CC_1\) lie on its lateral surface, and the axis of the cylinder is parallel to the line \(AB_1\) and is at a distance of \(\frac{1}{4}\) from it. Find the radius of the cylinder.

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