Based on optimisation technique, so we use a cut so that the volume that which we won't be cutting further is around $1-3\text{rd}$ of the total volume of the cuboid. Then it would be either $400$ or $300$. Then by further inspection, we see that we can say that we will have the following cases for the optimisation technique to work: $300-350-350$ and $400-300-300$. Therefore the smallest possible volume of the largest of the three cuboids is $\boxed{350}$.