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Consider a frog that lives on two rocks $A$ and $B$ and moves from one rock to the other randomly. If it is at Rock $A$ at any time, irrespective of which rocks it occupied in the past, it jumps back to Rock $A$ with probability $2 / 3$ and instead jumps to Rock $B$ with probability $1 / 3$. Similarly, if it is at Rock $B$ at any time, irrespective of which rocks it occupied in the past, it jumps back to Rock $B$ with probability $2 / 3$ and instead jumps to Rock $A$ with probability $1/3$. After the first jump, it is at Rock $A$. What is the limiting probability that it is at Rock $A$ after a total of $n$ jumps as $n \rightarrow \infty?$

- $1 / 2 $
- $2 / 3$
- $1$
- The limit does not exist
- None of the above