Variance limit of an average of grouped products

Let $X_1,X_2,\dots$ be i.i.d. with mean $5$, variance $20$, and finite third and fourth moments. Define $Y_n=\frac{1}{n}\sum_{j=1}^{n}\Big(X_{4j-3}^2+X_{4j-2}^2\,X_{4j-1}\,X_{4j}\Big).$ What is $\lim_{n\to\infty}\operatorname{Var}(Y_n)$? Answer $-1$ if the limit does not exist.

$0$. The bracketed terms use disjoint blocks of the $X$s, so they are i.i.d. Their variance is finite because it involves only moments up to order four, which are finite. Hence $\operatorname{Var}(Y_n)$ is that variance divided by $n$, which tends to $0$.