ellina has twelve blocks, two each of red ($\textbf{r}$), blue ($\textbf{b}$), yellow ($\textbf{y}$), green ($\textbf{g}$), orange ($\textbf{o}$), and purple ($\textbf{p}$). call an arrangement of blocks $\textit{even}$ if there is an even number of blocks between each pair of blocks of the same color. for example, the arrangement\[\textbf{r b b y g g y r o p p o}\]is even. ellina arranges her blocks in a row in random order. the probability that her arrangement is even is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. find $m n.$
