Oscar and Eric play the following game. First, they agree on a positive integer $n$, and they begin by writing its binary representation on a blackboard. They then take turns, with Oscar going first, to write a number on the blackboard in binary representation, such that the sum of all written numbers does not exceed $2n$.

The game ends when there are no valid moves left. Oscar wins if the number of $1$s on the blackboard is odd, and Eric wins if it is even.

Let $S(N)$ be the sum of all $n\le 2^N$ for which Eric can guarantee winning, assuming optimal play.

For example, the first few values of $n$ for which Eric can guarantee winning are $1,3,4,7,15,16$. Hence $S(4)=46$.
You are also given that $S(12)=54532$ and $S(1234)\equiv 690421393(\mathrm{mod}1000000007)$.

Find $S(12345678)$. Give your answer modulo $1000000007$.