
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 \( q_{1} = 2 \) and for \( n \geqslant 2 \), \( q_{n} \) is the smallest integer greater than \( q_{n-1} \) such that for any \( 1 \leqslant i \leqslant j \leqslant n-1 \), \( q_{n} \) cannot be expressed as the product \( q_{i} q_{j} \). This \( q_{n} \) is then called a "quasi-prime." Determine whether 1000 is a quasi-prime, and provide a proof.

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