The observations for each test are:
A: Br$_2$ water test indicates unsaturation (reddish orange color disappears).
B: Ceric ammonium nitrate test detects alcohols (red color appears).
C: Ferric chloride test detects phenols (blue, green, violet, or red color appears).
D: 2,4-DNP test detects aldehydes/ketones (yellow-orange precipitate).
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 ______.