
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]


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The vertices \( A, B \) and \( D_1 \) of a cube \( ABCD A_1 B_1 C_1 D_1 \) lie on the lateral surface of a cylinder, the axis of which is parallel to the line \( DC_1 \). Find the radius of the base of the cylinder, if the edge of the cube is equal to \( a \).

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