Convergents of infinite sum

Define a sequence $b_n$ as below: $b_0=c,b_n=b_{n-1}^2-2(n\ge 1)$, where $c$ is a positive integer greater than or equal to 3.

Let infinite sum $s$ be $$s=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{\prod }_{k=0}^n{b}_k}=\frac{1}{b_0}+\frac{1}{b_0{b}_1}+\frac{1}{b_0{b}_1{b}_2}+\frac{1}{b_0{b}_1{b}_2{b}_3}+\cdots$$ It can be proved that the infinite sum is convergent and is an irrational number for all possible values of $c$.

$s$ can be represented as an infinite continued fraction and corresponding convergents are denoted by $p_n/q_n$ ($n\ge 0$, $p_n$ and $q_n$ are coprime). For example, for $c$ = 6, $s$ = 0.1715728752..., and the first several convergents are $p_0/q_0$ = 0/1, $p_1/q_1$ = 1/5, $p_2/q_2$ = 1/6, $p_3/q_3$ = 5/29 and so on.

Let $P(c,n)$ and $Q(c,n)$ be numerator and denominator of the $n$th convergents $p_n/q_n$ of $s$ with value $c$, respectively. For instance, $P(6,3)$ = 5, $Q(6,3)$ = 29. Given Fibonacci sequence fn defined as $f_1=1$, $f_2=1$, $f_n=f_{n-1}+f_{n-2}$ ($n\ge 3$). The value of this sequence is no less than 3 from the 4th item.

Let the sum $SP(m,n)=\sum _{i=4}^{m}P(f_i,n)$ and $SQ(m,n)=\sum _{i=4}^{m}Q(f_i,n)$. You are given $SP(5,10)$ = 606, $SQ(5,10)$ = 2784, $SP(10,100)\mathrm{mod}1000000007$ = 774200907, $SQ(10,100)\mathrm{mod}1000000007$ = 830200702.

Find $SP({10}^5,{10}^{18})$ and $SQ({10}^5,{10}^{18})$, both modulo 1000000007.

Answer format: [$SP({10}^5,{10}^{18})$],[$SQ({10}^5,{10}^{18})$]

Example: 774200907,830200702 for $SP(10,100)$ and $SQ(10,100)$

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Thanks to czp for the idea.