Numerical Methods: Euler to Runge–Kutta · Superchallenge problem

An implicit two-step method $y_{n+1} + a_1y_n + a_0y_{n-1} = h\,b\,f(t_{n+1}, y_{n+1})$ is designed to be second-order: its three coefficients are chosen so that it reproduces every quadratic solution exactly. It is then applied to $y' = -y$ with $h = 1$. The iterates have the form $y_k = A\xi_1^{\,k} + B\xi_2^{\,k}$. Find $|\xi_1|$.