Consider the sequence $\{a_i\}_{i\geq0}$ such that $a_0=\sqrt{7}+1$ and $a_{n+1}=a_n^2-a_n+1$. The sum $\sum_{n=0}^{\infty} \frac{1}{a_n},$ can be written as $\frac{1}{\sqrt{b}}$. What is the value of $b$?

This problem is shared by Faraz M. from AMSP.