The balanced reaction is:
\[ 2 \text{MnO}_4^- + 5 \text{C}_2\text{O}_4^{2-} + 16 \text{H}^+ \rightarrow 2 \text{Mn}^{2+} + 10 \text{CO}_2 + 8 \text{H}_2\text{O} \]From the stoichiometry, we see that 16 moles of \(\text{H}^+\) are required for 2 moles of \(\text{MnO}_4^-\). Therefore, 8 moles of \(\text{H}^+\) are needed for each mole of \(\text{MnO}_4^-\).
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 ______.