The numbers $x_i,i=1,2…6\in \mathbb{R}^+$ are such that $x_1+x_2+...+x_6=1$ and $x_1 x_3 x_5+x_2 x_4 x_6\geq \frac{1}{540}$. Let $S=x_1 x_2 x_3+x_2 x_3 x_4+...+x_6 x_1 x_2$. If $max\, S=\frac{p}{q}$ , where $gcd(p,q)=1$, find $p+q$.