A rectangular sheet of grid paper with integer dimensions $w\times h$ is given. Its grid spacing is $1$.
When we cut the sheet along the grid lines into two pieces and rearrange those pieces without overlap, we can make new rectangles with different dimensions.

For example, from a sheet with dimensions $9\times 4$, we can make rectangles with dimensions $18\times 2$, $12\times 3$ and $6\times 6$ by cutting and rearranging as below:

Similarly, from a sheet with dimensions $9\times 8$, we can make rectangles with dimensions $18\times 4$ and $12\times 6$.

For a pair $w$ and $h$, let $F(w,h)$ be the number of distinct rectangles that can be made from a sheet with dimensions $w\times h$.
For example, $F(2,1)=0$, $F(2,2)=1$, $F(9,4)=3$ and $F(9,8)=2$.
Note that rectangles congruent to the initial one are not counted in $F(w,h)$.
Note also that rectangles with dimensions $w\times h$ and dimensions $h\times w$ are not considered distinct.

For an integer $N$, let $G(N)$ be the sum of $F(w,h)$ for all pairs $w$ and $h$ which satisfy $0<h\le w\le N$.
We can verify that $G(10)=55$, $G({10}^3)=971745$ and $G({10}^5)=9992617687$.

Find $G({10}^{12})$. Give your answer modulo ${10}^8$.