Consider a two player game. There are $N$ balls marked $1$ to $N$. A move consists of removing a ball $n$ and all the remaining balls which are divisors of $n$ (including $1$). The players alternate the moves. The one who takes the last ball wins the game. Let us assume that both players play optimally. Find the probability that the player who starts the game wins it, given that N will be a random integer between $1$ and $\infty$. Let this probability be $x$. Give your answer as $x \times 1000000000000$ .