An Egyptian Fraction Machine

A sequence is defined by $b_1=2$ and $b_{n+1}=b_n^2-b_n+1$ for $n\ge 1$. If$$\sum_{n=1}^k \frac{1}{b_n} = \frac{\alpha b_{k+1} + \beta}{\gamma b_{k+1} + \delta}$$where $\alpha$, $\beta$, $\gamma$, $\delta$ are integers, and $\gcd(\alpha,\beta,\gamma,\delta)$, find the value of $\alpha+\beta$.