
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
Given finite sets \( A \) and \( B \) composed of distinct positive real numbers, and a given positive integer \( n \) (where \( n > 1 \)), both \( A \) and \( B \) each contain at least \( n \) elements. If the sum of any \( n \) distinct numbers in \( A \) belongs to \( B \), and the product of any \( n \) distinct numbers in \( B \) belongs to \( A \), find the maximum number of elements in sets \( A \) and \( B \).

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