Eccoci.

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 $0 \lt h \le w \le N$.
We can verify that $G(10) = 55$, $G(10^3) = 971745$ and $G(10^5) = 9992617687$.

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