
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 the smallest positive integer  $n$  or show no such  $n$  exists, with the following property: there are infinitely many distinct  $n$ -tuples of positive rational numbers  $(a_1, a_2, \ldots, a_n)$  such that both  $$ a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n} $$  are integers.

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