In my probability and numerical analysis research I have come across the following predicament:

If we have a standard normal random variable X with CDF $ \Phi $, and PDF $ \phi $ I am interested in the asymptotic behavior of $\int_{-\infty}^{\infty} \phi(x) {\Phi(\frac{x}{a})}^{qa} dx $ where $ a,q \geq 1 $ are constants and when $ a \to \infty $?

I tried Laplace's method (quite trivially) but it did not yield good explicit results, so I thought the only possible answer to this might be the saddle point method which I barely know anything about using (in general or in probability theory) so I am writing here in the hopes of someone helping me do something with this here, I realize a similar question exists on this site here but that one got no attention (it is not mine either and I know not its owner), so my question is more concrete and I need help in applying the method of saddle point approximation in this case. I thank all helpers