Mean of an average of squared and product terms

Let $X_1,X_2,\dots$ be i.i.d. with mean $5$, variance $20$, and finite third and fourth moments. Define $Y_n=\frac1n\sum_{j=1}^{n}\left(X_{4j-3}^2+X_{4j-2}^2\,X_{4j-1}\,X_{4j}\right).$ What is $\lim_{n\to\infty}\mathbb{E}[Y_n]$? (Enter $-1$ if the limit does not exist.)

$1170$. The $n$ bracketed terms are i.i.d., so $\mathbb{E}[Y_n]$ equals the mean of one term for every $n$. With $\mathbb{E}[X^2]=20+25=45$ and independence, that mean is $45+45\cdot5\cdot5=1170$.