
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 \( n \) such that there exist \( n \) distinct positive integers \( a_{1}, a_{2}, \cdots, a_{n} \) satisfying the following three conditions:
1. The least common multiple of \( a_{1}, a_{2}, \cdots, a_{n} \) is 2002.
2. For any \( 1 \leq i < j \leq n \), the greatest common divisor of \( a_{i} \) and \( a_{j} \) is greater than 1.
3. The product \( a_{1} a_{2} \cdots a_{n} \) is a perfect square and a multiple of 32. 

Additionally, for this smallest positive integer \( n \), find all \( n \)-tuples \( \left\{a_{1}, a_{2}, \cdots, a_{n}\right\} \) that satisfy conditions (1), (2), and (3).

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