Mex Sequence (noch nicht übersetzt)

In this problem $\oplus$ is used to represent the bitwise exclusive or of two numbers.
Starting with blank paper repeatedly do the following:

1. Write down the smallest positive integer $a$ which is currently not on the paper;
2. Find the smallest positive integer $b$ such that neither $b$ nor $(a \oplus b)$ is currently on the paper. Then write down both $b$ and $(a \oplus b)$.

After the first round $\{1,2,3\}$ will be written on the paper. In the second round $a=4$ and because $(4 \oplus 5)$, $(4 \oplus 6)$ and $(4 \oplus 7)$ are all already written $b$ must be $8$.

After $n$ rounds there will be $3n$ numbers on the paper. Their sum is denoted by $M(n)$.
For example, $M(10) = 642$ and $M(1000) = 5432148$.

Find $M(10^{18})$. Give your answer modulo $1\,000\,000\,007$.