Binary Blackboard

 Published on Saturday, 11th April 2020, 08:00 pm; Solved by 251;
Difficulty rating: 35%

Problem 711

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 $S(1234) \equiv 690421393 \pmod{1\,000\,000\,007}$.

Find $S(12\,345\,678)$. Give your answer modulo $1\,000\,000\,007$.