Putnam 2024 B4

Let $n$ be a positive integer. Set $a_{n,0} = 1$. For $k \geq 0$, choose an integer $m_{n,k}$ uniformly at random from the set $\{1,\dots,n\}$, and let $$a_{n,k+1} = \begin{cases} a_{n,k} + 1, & \text{if $m_{n,k} > a_{n,k};$} \\ a_{n,k}, & \text{if $m_{n,k} = a_{n,k}$;} \\ a_{n,k}-1, & \text{if $m_{n,k} < a_{n,k}$.} \end{cases}$$ Let $E(n)$ be the expected value of $a_{n,n}$. Determine $\lim_{n\to \infty} E(n)/n$.