A frog is placed on the number line. Every step the frog jumps either $a$ units to the left or $b$ units to the right, both with $1/2$ probability.

Define $f(a,b)$ as the limit $\underset{n\to \mathrm{\infty }}{lim}\frac{c_n}{n}$ where $c_n$ is the expected number of unique numbers visited in the first $n$ steps. You are given $f(1,1)=0$ and $f(1,2)\approx 0.427050983$.

Find $f(89,97)$. Give your answer rounded to nine digits after the decimal point.