
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 exists a positive integer \( n \) and \( n \) pairwise disjoint infinite sets \( A_{1}, A_{2}, \ldots, A_{n} \) satisfying \( A_{1} \cup A_{2} \cup \ldots \cup A_{n} = \mathbf{N}^{*} \), and for any two numbers \( b > c \) in each \( A_{i} \), it holds that \( b - c \geq a^{i} \).

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