I am studying commutative algebra and saw the following question in one of the tests:

What is the Krull dimension of $R=\mathbb{Q}[x^2+y+z,\ x+y^2+z,\ x+y+z^2,\ x^3+y^3,\ y^4+z^4]?$

I know that $\dim R \leq \dim\mathbb{Q}[x,\ y,\ z]=3$ and $\dim R>0$ since $R$ is not a field, but this is not very helpful.

I guess I should find a maximal chain of prime ideals, or maybe use $\dim R= \dim(R/P)+\operatorname{height}(P)$ for some prime ideal $P$, but I couldn't think of anything..

I would be grateful for any help!