
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
Given a unit cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, a black and a white ant start crawling from point $A$ along the edges. Each time an ant traverses one edge, it is said to have completed one segment. The white ant's path is $A A_{1} \rightarrow A_{1} D_{1} \rightarrow \cdots$, while the black ant's path is $A B \rightarrow B B_{1} \rightarrow \cdots$. Both ants follow the rule that the $(n+2)$-th segment they crawl must be on a line that is skew to the line of the $n$-th segment. Assume that after each of the black and white ants has crawled 2008 segments, each stops at a vertex of the cube. Determine the distance between the black and white ants at this moment.

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