Given $n\ge 2$ bowls arranged in a circle, $m\ge 2$ balls are distributed amongst them.

Initially the balls are distributed randomly: for each ball, a bowl is chosen equiprobably and independently of the other balls. After this is done, we start the following process:

1. Choose one of the $m$ balls equiprobably at random.
2. Choose a direction to move - either clockwise or anticlockwise - again equiprobably at random.
3. Move the chosen ball to the neighbouring bowl in the chosen direction.
4. Return to step 1.

This process stops when all the $m$ balls are located in the same bowl. Note that this may be after zero steps, if the balls happen to have been initially distributed all in the same bowl.

Let $F(n,m)$ be the expected number of times we move a ball before the process stops. For example, $F(2,2)=\frac{1}{2}$, $F(3,2)=\frac{4}{3}$, $F(2,3)=\frac{9}{4}$, and $F(4,5)=\frac{6875}{24}$.

Let $G(N,M)=\sum _{n=2}^N\sum _{m=2}^{M}F(n,m)$. For example, $G(3,3)=\frac{137}{12}$ and $G(4,5)=\frac{6277}{12}$. You are also given that $G(6,6)\approx 1.681521567954e4$ in scientific format with 12 significant digits after the decimal point.

Find $G(12,12)$. Give your answer in scientific format with 12 significant digits after the decimal point.