
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 all positive integers \( n \geq 3 \) such that the following statement holds:

Given \( a_{1}, a_{2}, \cdots, a_{n} \) forming an arithmetic sequence, if \( a_{1} + 2 a_{2} + \cdots + n a_{n} \) is a rational number, then at least one of \( a_{1}, a_{2}, \cdots, a_{n} \) is a rational number.

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