
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 cube with an edge length of \(1 \text{ cm}\) and two vertices \(A\) and \(B\) that are diagonally opposite on one face of the cube. A path of length \(n\) is defined as a sequence of \(n+1\) vertices such that each pair of consecutive vertices is \(1 \text{ cm}\) apart. Which of the following numbers is greater: the number of paths of length \(1000 \text{ cm}\) that start and end at \(A\), or the number of paths of length \(1000 \text{ cm}\) that start at \(A\) and end at \(B\)? Justify your answer.

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