$[10$ pts$]$ Find $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{x}}$ and $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{y}}$ if $z$ is defined implicitly by the equation $x^3+y^3+z^3+4xyz=1.$