
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 sets  $ A_1, A_2, \ldots, A_n$  satisfying the following conditions:

(1) every set  $ A_i$  has infinitely many elements;
(2) every pair of distinct sets  $ A_i$  and  $ A_j$  do not share any common element
(3) the union of sets  $ A_1, A_2, \ldots, A_n$  is the set of all integers;
(4) 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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