Jan's jazzy sum

Number Theory Level pending

Suppose \(a,b\) are positive coprime integers such that \[\sum_{k = 1}^{b - 1} \left\lfloor \frac{ak}{b} \right\rfloor = 1337.\] If \(S\) is the sum of all possible values of \(a\), what are the last three digits of \(S\)?

This problem is posed by Jan J.

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