Let $-\frac{\pi}{6}<\theta<-\frac{\pi}{12}$. Suppose $\alpha_1$ and $\beta_1$ are the roots of the equation $x^2-2 x \sec \theta+1=0$ and $\alpha_2$ and $\beta_2$ are the roots of the equation $x^2+2 x \tan \theta+1=0$. If $\alpha_1>\beta_1$ and $\alpha_2>\beta_2$, then $\alpha_1+\beta_2$ equals