AMO-Bench 2025 Problem 7

$$\left\lfloor \frac{n!}{(n+1)(n+2)} \right\rfloor$$ represents the greatest integer less than or equal to $\frac{n!}{(n+1)(n+2)}$. Find all possible values of $$\left\lfloor \frac{n!}{(n+1)(n+2)} \right\rfloor - \left\lfloor \frac{1}{32} \cdot \frac{n!}{(n+1)(n+2)} \right\rfloor \times 32.$$