
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
Determine all positive real numbers  $a$  such that there exists a positive integer  $n$  and partition  $A_1$ ,  $A_2$ , ...,  $A_n$  of infinity sets of the set of the integers satisfying the following condition: for every set  $A_i$ , the positive difference of any pair of elements in  $A_i$  is at least  $a^i$ .

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