When is this an Integer?

Algebra Level 5

\[\sqrt[3] {\text{min}\left[\prod_{cyc}\left(\dfrac{ax^2+bx}{c}\right)\right]}\]

Let \(0< x\le 1000\) be a real constant. Find the number of distinct \(x\)'s such that, for positive real variables \(a,b,c\) the expression above is an integer.

Note: \(a,b,c\) are variables, \(x\) is a constant. The minimum value is taken where \(a,b,c\) vary over positive reals.

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