
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 edges of length 2 cm has a dot located a quarter distance from the center towards one edge on the top face. The cube is sitting on a flat table. It is rolled in one direction without slipping, twice completely in such a way that at least two of its vertices always touch the table, until the dot returns to be on the top face. The length, in centimeters, of the path followed by the dot is $c\pi$, where $c$ is a constant. Determine $c$.

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