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Consider the transition system shown in the figure below with the initial state $s_1$. A token is initially placed at $s_1$, and it moves to $s_2$ with probability $\frac{2}{3}$, and to $s_3$ with probability $\frac{1}{3}$. From $s_2$ and $s_3$, the token always moves to $s_1$ and $s_2$ respectively. A run of the system consists of an infinite sequence of states constructed by moving the token from one state to another following the transitions forever. Assuming such a run is chosen randomly, what is the fraction of times that the state $s_2$ is expected to appear in the run?

1. $\frac{1}{7}$
2. $\frac{2}{7}$
3. $\frac{3}{7}$
4. $\frac{5}{7}$
5. None of the above