Two to the power of two to the power of...

Let

$S=\dfrac{1}{2+1}+\dfrac{2}{2^2+1}+\dfrac{4}{2^4+1}+\dfrac{8}{2^8+1}+\dots+\dfrac{2^{2014}}{2^{2^{2014}}+1}$

The exact value of $S$ can be expressed in the form $S=\frac{1}{a}-\frac{b}{2^b-c}$ , where $a$ , $b$ and $c$ are real numbers. What are the last 3 digits of $\log_{2}(abc)$ ?