A mentally handicapped problem

Algebra Level 5

Let \(a_1,a_2,\ldots, a_{31}\) and \(b_1,b_2, \ldots, b_{31}\) be positive integers such that \(a_1< a_2<\cdots< a_{31} \leq 2015\); \( \ b_1< b_2<\ldots<b_{31} \leq 2015 \ \) and \(\ a_1+a_2+\cdots+a_{31}=b_1+b_2+\cdots+b_{31}\).

Find the maximum value of

\[S=|a_1-b_1|+|a_2-b_2|+\cdots+|a_{31}-b_{31}|.\]

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