Find the volume given a vector field and the flux

Question

The problem gives that $$ \int_{E} \mathbf{F} \cdot \mathbf{n} d S=1 $$ where$$ \mathbf{F}=\left(3 x,\left(x^{2}+z^{2}\right) y^{2},\left(x^{2}+y^{2}\right) z^{2}\right) $$ and asks to find the volume enclosed by $E$. My first attempt is to use the divergence theorem, but the divergence here does not reduce to a constant or something simple. I also think the squares are suggesting the usage of spherical coordinates maybe, but I do not see how. Any hints or solutions are appreciated!