
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
A cube with edge length 3 cm is sitting on a flat surface. Initially, a dot is marked at one corner of the top face. The cube is rolled over its edges on the flat surface without lifting or slipping, ensuring that at least two of its vertices are always touching the table. After a series of rolls, the cube is restored to its original position with the dot back on the top face. Calculate the length, in centimeters, of the path followed by the dot across the surface of the cube, expressed in the form $c\pi$, where $c$ is a constant.

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