Let $\theta =\sqrt{-2}$.

Define $T$ to be the set of numbers of the form $a+b\theta$, where $a$ and $b$ are integers and either $a>0$, or $a=0$ and $b>0$. For a set $S\subseteq T$ and element $z\in T$, define $p(S,z)$ to be the number of ways of choosing two distinct elements from $S$ with product either $z$ or $-z$.

For example if $S=\{1,2,4\}$ and $z=4$, there is only one valid pair of elements with product $\pm 4$, namely $1$ and $4$. Thus, in this case $p(S,z)=1$.

For another example, if $S=\{1,\theta ,1+\theta ,2-\theta \}$ and $z=2-\theta$, we have $1\cdot (2-\theta )=z$ and $\theta \cdot (1+\theta )=-z$, giving $p(S,z)=2$.

Let $A$ and $B$ be two sets satisfying the following conditions:

• $1\in A$
• $A\cap B=\mathrm{\varnothing }$
• $A\cup B=T$
• $p(A,z)=p(B,z)$ for all $z\in T$

Remarkably, these four conditions uniquely determine the sets $A$ and $B$.

Let $F_n$ be the set of the first $n$ factorials: $F_n=\{1!,2!,\dots ,n!\}$, and define $G(n)$ to be the sum of all elements of $F_n\cap A$.

You are given $G(4)=25$, $G(7)=745$, and $G(100)\equiv 709772949(\mathrm{mod}{10}^9+7)$.

Find $G({10}^8)$ and give your answer modulo ${10}^9+7$.