Expected uniform draws until the sum passes 2

Draw independent $\text{Unif}(0,1)$ variables $X_1, X_2, \dots$ one after another and stop at the first $n$ for which $X_1+\dots+X_n>2$; call that stopping index $N$.

The expectation $\mathbb{E}[N]$ can be written as $ae^2+be$ with integers $a$ and $b$, where $e$ is the base of the natural logarithm. Find $a+b$.

$0$. Let $f(x)$ be the expected number of draws to exceed $x$; it is known that $f(x)=e^x$ for $0\le x\le1$. Conditioning on the first draw gives $f(x)=1+\int_0^1 f(x-t)\,dt$, so on $(1,2)$ we get $f'(x)=f(x)-e^{x-1}$ with $f(1)=e$. Solving yields $f(2)=e^2-e$, so $a=1$, $b=-1$.