Ellina has twelve blocks, two each of red ($R$), blue ($B$), yellow ($Y$), green ($G$), orange ($O$), and purple ($P$). Call an arrangement of blocks $even$ if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement  $R$ $B$ $B$ $Y$ $G$ $G$ $Y$ $R$ $O$ $P$ $P$ $O$  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 $m+n$. (2022 AIME I Problem 9) 
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