HMMT November 2023 U35

Dorothea has a $3 \times 4$ grid of dots. She colors each dot red, blue, or dark gray. Compute the number of ways Dorothea can color the grid such that there is no rectangle whose sides are parallel to the grid lines and whose vertices all have the same color. Submit a positive integer $A$. If the correct answer is $C$ and your answer is $A$, you will receive $\left\lfloor 20\left(\min \left(\frac{A}{C}, \frac{C}{A}\right)\right)^{2}\right\rfloor$ points.