What is the pdf of $\vec{Y} = \frac{\vec{X} }{\lVert \vec{X} \rVert_\infty}$ with $\vec{X}$ a random vector following a multivariate standard normal distribution (zero-mean $\vec{\mu} = 0$ and covariance-matrix $\Sigma = I$ ($I$ denotes identity matrix))?

$\lVert \vec{X} \rVert_\infty$ denotes the uniform-norm with $\lVert \vec{X} \rVert_\infty = \max(\lvert X_1 \rvert,...\lvert X_n \rvert,...\lvert X_N \rvert)$ with $X_n$ the $n$th entry of vector $\vec{X} \in \mathbb{R}^N$.