If $\alpha$, $\beta \neq 0$, and $f(n) = \alpha^n + \beta^n$ and
$$\begin{vmatrix} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{vmatrix} = K(1-\alpha)^2(1-\beta)^2(\alpha-\beta)^2,$$
then $K$ is equal to:
(1) $\alpha\beta$
(2) $\frac{1}{\alpha\beta}$
(3) 1
(4) $-1$