Standard Nim

Let $C(n,k)$ be the number of integer solutions $(x_1,\dots ,x_k)$ such that $\underset{i=1}{\overset{k}{⨁}}{x}_i=0$ and $0\le x_i<n$ for each $i$. Here, $x\oplus y$ means $x$ xor $y$.

Let $F(n)$ be the $n$-th fibonacci number: $F(0)=0,F(1)=1$, and $F(i)=F(i-1)+F(i-2)$ for $i\ge 2$.

Let $$S(n):=\sum _{i=1}^n\sum _{j=1}^{n}C(F(i),F(j)).$$ You are given $S(6)=2153296$ and $S(7)=18998620089329$.

Find $S(92)$ modulo ${10}^9+7$.

[My timing] 0.3 seconds.