Matrices and Determinants: JEE Main 2025 Mathematics question

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