Given a positive integer $n$, consider the following process:

1. Place $n$ stones at position $0$ on the real line, and initialize $i=0$.
2. If there are $m$ stones at position $i$, move $\lfloor \frac{m}{2}\rfloor$ stones from position $i$ to position $i+1$, and move another $\lfloor \frac{m}{2}\rfloor$ stones from position $i$ to $i+3$.
3. Increment $i:=i+1$ and return to Step 2.

For some values of $n$ this process terminates; for others it continues indefinitely. In either case, there is a finite set of singleton stones that are left behind at each position $i$ where $m$ was odd. A singleton is lonely if it is at least distance $3$ from any other singleton. We call a positive integer $n$ sad if all singletons left behind are lonely.

The first sad integer is $1$, trivially. The second is $68$, which leaves behind singletons at positions $2,5,8,13$. The third is $90$, which leaves behind only two singletons at positions $1$ and $13$.

Define $S(k)$ to be the sum of all sad integers $n$ which leave behind singletons only at positions $0\le i<k$. For example $S(14)=159$ and $S(30)=33438$.

Find $S(80)$.