
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 an array of \( n \) different positive integers \(\left(a_{1}, a_{2}, \cdots, a_{n}\right)\) such that \( a_{1} < a_{2} < \cdots < a_{n} \), and for any non-negative integer \( d \), the following condition holds:
$$
\frac{\left(a_{1}+d\right)\left(a_{2}+d\right) \cdots \left(a_{n}+d\right)}{a_{1} a_{2} \cdots a_{n}} \in \mathbf{Z}.
$$

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