Circular Visitor

There are $2n$ positions in a circle labelled $0$ to $2n-1$ arranged increasing in CCW order starting from $0$. A particle starts at position $0$ and moves to position $i+1$ or $i-1$ with equal probability at each step (for position $0$, either $1$ or $2n-1$, and similar with $2n-1$). Find the probability that the particle has visited every other position at least one by the time it reaches position $n$ for the first time when $n = 100$.