2011 AIME II Question 9

Let $x_1,$ $x_2,$ $\ldots,$ $x_6$ be nonnegative real numbers such that $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 =$ $1,$ and $x_1x_3x_5 + x_2x_4x_6 \ge \frac{1}{540}.$ Let $p$ and $q$ be positive relatively prime integers such that $\frac{p}{q}$ is the maximum possible value of $$
\begin{aligned} &x_1x_2x_3 + x_2x_3x_4 \\ &\quad {}+ x_3x_4x_5 + x_4x_5x_6 \\ &\quad {}+ x_5x_6x_1 + x_6x_1x_2. \end{aligned}
$$ Find $p + q.$