We define a pseudo-geometric sequence to be a finite sequence $a_0,a_1,\dots ,a_n$ of positive integers, satisfying the following conditions:

• $n\ge 4$, i.e. the sequence has at least $5$ terms.
• $0<a_0<a_1<\cdots <a_n$, i.e. the sequence is strictly increasing.
• $|a_i^2-a_{i-1}{a}_{i+1}|\le 2$ for $1\le i\le n-1$.

Let $G(N)$ be the number of different pseudo-geometric sequences whose terms do not exceed $N$.
For example, $G(6)=4$, as the following $4$ sequences give a complete list:

$1,2,3,4,51,2,3,4,62,3,4,5,61,2,3,4,5,6$

Also, $G(10)=26$, $G(100)=4710$ and $G(1000)=496805$.

Find $G({10}^{18})$. Give your answer modulo $1000000007$.