Let $w$ be a primitive element of $\mathbb F_{5^4}$. Let $\alpha=w^{13}$. Define, $F:\mathbb F_{5^4}\times \mathbb F_{5}\times \mathbb F_{5} \Rightarrow \mathbb F_{5} $ as, $$F(x,y,z)= Tr (\alpha x^{24}) + x^{312} + yz,$$
where Tr is the trace function from $\mathbb F_{5^4}$ to $F_{5}$ defined as $Tr(x)=x+x^5+x^{25}+x^{125}$.
What is the algebraic norm (ANF) of $F$?