Two players play a game. At the start of the game each player secretly chooses an integer; the first player from $1,...,n$ and the second player from $1,...,m$. Then they take alternate turns, starting with the first player. The player, whose turn it is, displays a set of numbers and the other player tells whether their secret number is in the set or not. The player to correctly guess a set with a single number is the winner and the game ends.

Let $p(m,n)$ be the winning probability of the first player assuming both players play optimally. For example $p(1,n)=1$ and $p(m,1)=1/m$.

You are also given $p(7,5)\approx 0.51428571$.

Find ${\displaystyle \sum _{i=0}^{20}\sum _{j=0}^{20}p(7^i,5^j)}$ and give your answer rounded to 8 digits after the decimal point.