First uniform draw that is not a running maximum

Let $X_1,X_2,\dots$ be i.i.d. uniform on $(0,1)$, and let $N$ be the smallest $n$ for which $X_n$ is not the largest of $X_1,\dots,X_n$. Then $\mathbb{E}[N]=a+be$ with integers $a,b$, where $e$ is Euler's number. Find $a+b$.

$1$. $N>n$ means $X_1<\dots<X_n$, which has probability $\tfrac1{n!}$ (and is certain for $n=0,1$). By the tail-sum formula $\mathbb{E}[N]=\sum_{n\ge0}\tfrac1{n!}=e$, so $a=0$, $b=1$.