Limiting Distribution of the given function

Limiting Distribution of the given function

Can someone please help me in finding the limiting distribution of
$$\frac{n(X_1X_2 + X_3X_4+\cdots+X_{2n-1}X_{2n})^2}{(X_1^2 +
X_2^2+\cdots+X_{2n}^2)^2}$$ where $X_i$ are iid standard normal $\forall
i\ge1$. I guess it has to be done using delta method, which I did
considering $(X_1,X_3,\ldots,X_{2n-1})$ as $(Z_1,Z_2,\ldots,Z_n)$ and
$(X_2,X_4,\ldots,X_{2n})$ as $(Y_1,Y_2,\ldots,Y_n)$ and tried applying
delta method
$$\left(\frac{1}{n}\sum_{i=1}^nY_iZ_i,\frac{1}{n}\sum_{i=1}^nY_i^2,\frac{1}{n}\sum_{i=1}^nZ_i^2\right)$$
But in doing so I am missing out on that $n$ in the numerator, as the
function $g4 should not depend on $n$. The calculations will be long, so
can someone please help me just with what I should take the function as
and on what should I apply it? I am sure, I will be able to do the
calculations after that.