Probability of Gaussian vector outside of a box

Question

Given are

1. An $n$-dimensional Gaussian vector with $X\sim N(0,\Sigma)$
2. Set of intervals $[-a_i,a_i]$ with $a_i>0$ and $i=1,\dots,n$.

I am interested in an upper bound for the probability $$\Pr(\cup_i \{ |X_i|\geq a_i\} ).$$

In part, what is the relation of this probability and $\det \Sigma$?

If in general, the determinant is not sufficient to determine an upper bound: can we choose the box dimensions, i.e. the $\{a_i\}$ such that they are proportional to the determinant somehow and obtain the desired relation between the upper bound and the determinant.

I think I know how to use the Q-function if we are looking on each component of $X$ as an independent r.v. but the covariance matrix should give more information.