Tan on the floor!

Level pending

A positive integer \(n \in S\), the set of tanny integers, if and only if for some value of \(\theta \in (0,\frac{\pi}{2})\), \(\lfloor \tan^2{\theta} \rfloor + \tan{\theta} = n\). When the elements of \(S\) are arranged in increasing order, let \(N_{T_{2014}}\) denote the \(2014\)th tanny integer. Find the last three digits of \(N_{T_{2014}}\).

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