Please reason step by step, and put your final answer within \boxed{}.
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.
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