Escape time given the number of attempts

A miner trapped underground repeatedly chooses one of three tunnels uniformly at random, independently each time. Tunnel 1 brings him back to the start after $2$ days, tunnel 2 brings him back after $4$ days, and tunnel 3 reaches the surface after $8$ days. Let $X_i$ be the days spent on the $i$-th choice, $N$ the number of choices made up to and including the escape, and $T$ the total time to escape. Write $B_n$ for $\mathbb{E}\left[\sum_{i=1}^n X_i\right]$ (no conditioning) and $A_n$ for the conditional mean $\mathbb{E}[T\mid N=n]$. Find $\lim_{n\to\infty} A_n/B_n$, or answer $-1$ if the limit does not exist.

$9/14$. Unconditionally each trip averages $14/3$ days, so $B_n=\tfrac{14}{3}n$. Knowing $N=n$ means the first $n-1$ trips used tunnels 1 or 2 (mean $3$ days each) and the last used tunnel 3, so $A_n=3(n-1)+8$. The ratio tends to $3\big/\tfrac{14}{3}=\tfrac{9}{14}$.