
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
Consider a rectangular block with dimensions 2 cm by 1 cm by 1 cm. A dot is marked at the center of one of the longer faces (2 cm by 1 cm), and the block is resting with another longer face on a flat surface. The block is rolled over its edges in one direction such that at least two of its vertices always touch the table, and continues until the dot returns to an upward facing position. Compute the length of the path traced by the dot, presented as $k\pi$, where $k$ is a constant to be determined.

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