If $A,B$ and $\big(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1})\big)$ are non-singular matrices of the same order, then the inverse of $A\Big(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1})\Big)^{-1}B$ is equal to: