TxO Math Bowl 2024 - Team Contest - Problem 16

Cities $A$, $B$, $C$, $D$, $W$, and $L$ are located in the country of Portaland. A one-way magic portal exists in each of the cities $A$, $B$, $C$, and $D$. A portal code shows a portal’s destinations and their corresponding probability. The format of portal codes is Portal $M$: ($N$, $P$, $Q$), which represents that by taking the portal in city $M$, the probability of travelling to cities $N$, $P$, $Q$ are $\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{6}$, respectively. The portal codes of the four portals are as follows:

Portal $A:\;$ ($B$, $C$, $D$)
Portal $B:\;$ ($C$, $W$, $L$)
Portal $C:\;$ ($D$, $W$, $L$)
Portal $D:\;$ ($A$, $W$, $L$)

If Mike starts in city $A$ and continually takes portals until he reaches a city without a portal, the probability that he ends up in city $W$ can be expressed as $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. Compute $x+y$.