Drop it Like it's a Floor Function

Algebra Level 5

Let \(n\) be a positive real. If the number of real solutions to \[nx^2=n\lfloor x^2\rfloor+x\] is \(2014\), then the range of possible values of \(n\) is \[n\in (a,b\,]\] If \(b-a=N\), then find the value of \[\lfloor 1000N\rfloor \]

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