The decreasing run of uniforms

Draw $X_1, X_2, \dots$ independently from $\text{Uniform}(0,1)$. Let $N$ be the first index $n$ at which $X_n$ is not the smallest of $X_1, \dots, X_n$ (i.e. the first time a draw fails to set a new minimum). Then $X_1 > X_2 > \dots > X_{N-1}$, and $X_{N-1}$ is the smallest value seen before the run breaks. The value $\mathbb{E}[X_{N-1}]$ can be written as $a + b\,e$ for integers $a, b$ (with $e$ Euler's number). Find $a + b$.