Tail probability of minimum to maximum ratio among $n$ i.i.d. half-normal random variables

Question

I have $n$ i.i.d.$\sim\mathcal{N}(0,1)$ random variables $X_1,\cdots,\ X_n$. For my research, I am interested in finding bounds (upper and lower) of the tail probabilities of the ratio $\frac{|X|_{(1)}}{|X|_{(n)}}$, where $|X|_{(1)}$ and $|X|_{(n)}$ are the minimum and maximum among $|X_1|,\cdots, |X_n|$, respectively. While I am able to find some bounds, I am not sure how well they fare compared to exisitng results. But I couldn't find anything in the literature about these bounds. Can anyone kindly refer to me some of the relevant literature. Kindly do not provide a full answer, and only point to refrences. Thanks in advance.