Let $a \in \mathbb{R}$ and $A$ be a matrix of order $3 \times 3$ such that $\det(A) = 4$ and

where $I$ is the identity matrix of order $3 \times 3$. If $\det\left((a+1)\,\text{adj}\left((a-1)A\right)\right) = 2^{m} \cdot 3^{n}, \; m, n \in \{0,1,2,\ldots,20\}$, then $m + n$ is equal to: