
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 marked in the center of the top face. The cube sits on a flat table and is rolled in one direction without lifting or slipping, making a few rotations by pivoting around its edges. Calculate the total length, in centimeters, of the path followed by the dot if it rolls in such a way that it initially rotates around the edge joining the top and bottom faces, then rotates around an edge on the side face until the dot returns to the top face. The length of the path is $d\pi$, where $d$ is a constant. Determine $d$.

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