Asymptotics for the logarithm of the number of $k$-free solutions sets in Abelian group

The set $(A_1,\dots,A_k)$ of subsets of the Abelian group $G$ is $k$-free solutions, if $x_1+\dots+x_k=0$ have not solution in the set $(A_1,\dots,A_k),$ where $x_1\in A_1,\dots,x_k\in A_k.$ Asymptotics for the logarithm of the number of $k$-free solutions sets in abelian groups is obtained.
Passcode: a six digit number N=(4!)2+(p−5)2 where p is the smallest prime such that p>600