
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
Find all positive real numbers \(a\) such that there exist a positive integer \(n\) and \(n\) pairwise disjoint infinite sets \(A_{1}, A_{2}, \cdots, A_{n}\) satisfying \(A_{1} \cup A_{2} \cup \cdots \cup A_{n} = \mathbf{Z}\), and for any two numbers \(b > c\) in any \(A_{i}\), \(b - c \geqslant a^{i}\).

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