Expected peak of an increasing run of uniforms

Draw i.i.d. $\text{Unif}(0,1)$ values $X_1, X_2, \dots$ and let $N$ be the first index at which the new value is not a record, i.e. the first $n$ with $X_n<\max\{X_1,\dots,X_n\}$. Then $X_{N-1}$ is the largest of the values drawn.

The expectation $\mathbb{E}[X_{N-1}]$ can be written as $a+be$ with integers $a,b$ ($e$ being Euler's number). Find $a+b$.

$-1$. Let $f(x)$ be the expected final maximum when the current record is $x$. The next draw falls below $x$ with probability $x$ (and the game ends at $x$), or becomes a new record $y\in(x,1)$. This gives an integral equation which, after differentiating, becomes $f'+f=2x$ with $f(1)=1$. So $f(x)=2x-2+e^{1-x}$ and $f(0)=e-2$, i.e. $a=-2$, $b=1$.