Weighted sum of uniforms until the first rise

Let $X_1,X_2,\dots$ be i.i.d. $\text{Unif}(0,1)$. Define $N$ as the smallest $k$ such that $X_k<X_{k+1}$ (take $N=\infty$ if this never happens), and set $S=\sum_{i=1}^{N}\frac{X_i}{2^i}.$ The value of $E[S]$ can be written as $ae^{b}+c$ with integers $a,c$ and $b$ a reduced rational. Compute $a^2+c^2+4b$.

$15$. Write $S=\sum_i 2^{-i}X_i\mathbf 1\{N\ge i\}$; $N\ge i$ means $X_1\ge\dots\ge X_i$. Conditioning on $X_i=x$, the others are above $x$ in decreasing order with probability $\tfrac{(1-x)^{i-1}}{(i-1)!}$, giving $E[X_i\mathbf 1\{N\ge i\}]=\tfrac{1}{(i+1)!}$. Summing $\tfrac{2^{-i}}{(i+1)!}$ over $i\ge1$ gives $2\sqrt e-3$, so $4+9+2=15$.