Final minimum of a decreasing run of uniforms

Draw i.i.d. $\text{Unif}(0,1)$ values $X_1,X_2,\dots$ one after another, and let $N$ be the first index at which $X_N$ is not the smallest value seen so far. Thus $X_1>X_2>\dots>X_{N-1}$ and $X_{N-1}$ is the minimum of the first $N$ draws. The expectation $\mathbb{E}[X_{N-1}]$ equals $a+be$ for integers $a,b$, where $e$ is Euler's number. Find $a+b$.

$2$ (since $\mathbb{E}[X_{N-1}]=3-e$). Let $f(x)$ be the expected final minimum when the current minimum is $x$. Conditioning on the next draw (above $x$ stops, below $x$ continues from a uniform point) and differentiating yields $f'(x)=f(x)+1-2x$ with $f(0)=0$, whose solution is $f(x)=2x+1-e^x$; then $f(1)=3-e$.