On elementary theories of algebraically closed groups

In paper for any algebraically closed group $G$, as well as for the class of the algebraically closed groups, we prove algorithmic undecidability of the positive $\forall^2 \exists^{24}$-theory and $\forall^3 \exists^{2}$-theory. For an arbitrary $g\in G$, we also prove the decidability of the equation of the type
$$ w(x_1, \ldots , x_n) = g, $$
where $w(x_1, \ldots , x_n)$ is a non-empty irreducible word in the unknowns $x_1,\ldots x_n\in G$.