Let $A$ be a $3\times 3$ matrix and $|A| = 2$. If $\left|\text{adj}\bigl(\text{adj}(\cdots(\text{adj}\,A)\cdots)\bigr)\right| = 2^{n}$, where the adjoint is applied $2024$ times, then the remainder when $2^{n}$ is divided by $9$ is equal to: