Eigenvalues · Hard problem

A discrete process satisfies $x_{k+1} = 0.6x_k - c\,x_{k-1}$, written as $\begin{pmatrix}x_{k+1}\\x_k\end{pmatrix} = A\begin{pmatrix}x_k\\x_{k-1}\end{pmatrix}$ with $A = \begin{pmatrix}0.6 & -c\\1 & 0\end{pmatrix}$. For $c > 0.09$, $A$'s two eigenvalues form a complex-conjugate pair, and the process is stable (every $x_k \to 0$) exactly when their modulus is less than $1$. Find the value of $c$ at which the process stops being stable.