The given reaction is an example of the Hell-Volhard-Zelinsky (HVZ) reaction, which selectively brominates the $\alpha$-carbon of carboxylic acids in the presence of Br$_2$ and red phosphorus. The reaction mechanism involves the formation of an $\alpha$-brominated carboxylic acid.
\[\text{COOH} \xrightarrow{\text{Br}_2/\text{Red P}} \text{COOH-Br} \xrightarrow{\text{H}_2\text{O}} \text{COOH-Br (Product)}\]
Thus, the product formed is the $\alpha$-bromo derivative of the given carboxylic acid.
Let \( S = \left\{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \right\} \), where
\[ A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \]Then \( n(S) \) is equal to ______.