Forgetful singer restarting a countdown song

A countdown song has verses that mention $N, N-1, \dots, 1$ bottles in that order, and the song ends after the verse for $1$ bottle. Each time he sings a verse, a singer independently forgets his place with probability $\tfrac1N$; when that happens he completes the current verse and then restarts from the $N$-bottle verse. Let $K$ be the expected total number of verses he sings. The limit $\lim_{N\to\infty}K/N$ equals $ae-b$ for integers $a,b$. Compute $a^2+b^2$.

$2$. Let $q=1-\tfrac1N$. Writing first-step equations for the expected verses remaining from each verse and unrolling them gives $K=\tfrac{1-q^{N-1}}{(1-q)\,q^{N-1}}+q^{N-1}$. Since $q^{N-1}\to e^{-1}$, $K/N\to e-1$, so $a=b=1$ and $a^2+b^2=2$.