They're Not All The Same Degree!

\[\large \dfrac {x^3+y^3}{3}+xy = 2007\]

Let there be \(n\) unordered positive integral solutions to the above equation. Call these solutions \((x_1, y_1), (x_2, y_2) , \ldots, (x_n, y_n)\). Find the value of

\[\displaystyle \sum_{i=1}^n x_i y_i. \]

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