End of the world

$\begin{aligned} S & = \displaystyle \sum^{\infty}_{a=0} \displaystyle \sum^{\infty}_{b=0} \displaystyle \sum^{\infty}_{c=0}\dfrac{a+b+c+abc} {2^a(2^{a+b}+2^{b+c}+2^{a+c})} \\3S & = \displaystyle \sum^{\infty}_{a=0} \displaystyle \sum^{\infty}_{b=0} \displaystyle \sum^{\infty}_{c=0}\dfrac{a+b+c+abc} {(2^{a+b}+2^{b+c}+2^{a+c})}×\left[ \frac{1}{2^a}+\frac{1} {2^b}+\frac{1}{2^c} \right] \\ & = \displaystyle \sum^{\infty}_{a=0} \displaystyle \sum^{\infty}_{b=0} \displaystyle \sum^{\infty}_{c=0}\dfrac{a+b+c+abc} {2^a2^b2^c } \\ & = \displaystyle \sum^{\infty}_{a=0}\frac{a}{2^a} \displaystyle \sum^{\infty}_{b=0}\frac{1}{2^b} \displaystyle \sum^{\infty}_{c=0}\frac{1}{2^c}+ \displaystyle \sum^{\infty}_{a=0}\frac{1}{2^a} \displaystyle \sum^{\infty}_{b=0}\frac{b}{2^b} \displaystyle \sum^{\infty}_{c=0}\frac{1}{2^c}+ \displaystyle \sum^{\infty}_{a=0}\frac{1}{2^a} \displaystyle \sum^{\infty}_{b=0}\frac{1}{2^b} \displaystyle \sum^{\infty}_{c=0}\frac{c}{2^c}+ \displaystyle \sum^{\infty}_{a=0}\frac{a}{2^a} \displaystyle \sum^{\infty}_{b=0}\frac{b}{2^b} \displaystyle \sum^{\infty}_{c=0}\frac{c}{2^c} \\ &= 2^3+2^3+2^3+2^3 =32 \\ & \Rightarrow S=\frac{32}{3}\end{aligned}$