Inspired by Pi Han Goh!

\[\large{a_n = n^2 + 500 \quad ; \quad d_n = \gcd(a_n, \ a_{n+1})}\]

For integers \(n \geq 1\), define \(a_n\) and \(d_n\) as above. Determine the largest possible value of \(d_n\).

Inspiration.

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