Let \(u,v,w\) be real numbers satisfying \(uvw=u+v+w=1\) and \(u^2+v^2+w^2=2013\). If \[ \dfrac{1}{uv+w}+\dfrac{1}{wu+v}+\dfrac{1}{vw+u} = -\frac{a}{b}, \] where \(a\) and \(b\) are positive coprime integers, what is the value of \(a\)?

This problem is posed by Christyan Tamaro N.

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