Frogs can be placed on the real number line at integer locations. Given coprime positive integers $(a,b)$, each frog has the ability to make jumps of distances $a$ or $b$ in the positive direction.

Two frogs placed at $m$ and $n$, $m<n$, are attacking if the frog at $m$ can hop to $n$ with some series of jumps. For example if $(a,b)=(3,5)$, frogs placed at $0$ and $11$ are attacking as the former can make two jumps of $3$ and one jump of $5$ to reach $11$. However, frogs placed at $4$ and $11$ are non-attacking.

A non-attacking configuration is a placement of any number of frogs such that:

• one frog is placed at $0$;
• all other frogs are placed at distinct positive integers;
• no two frogs are attacking.

Define $F(a,b)$ to be sum of the integer locations of every frog, summing over all non-attacking configurations. For example if $(a,b)=(3,5)$ there are seven non-attacking configurations: $$\{0\}\{0,1\}\{0,2\}\{0,4\}\{0,7\}\{0,1,2\}\{0,2,4\}$$giving $F(3,5)=23$.

You are also given $F(5,13)=16336$.

Find $F(19,53)$.