Binomial coefficients in the denominator!

If the value of the sum

\[\displaystyle\sum_{r=0}^{1006} \frac{2012-2r}{(r+1)\dbinom{2013}{r+1}}\]

is \(S\) then find the value of \(S+\frac{\displaystyle1}{\dbinom{2013}{1006}}\).

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