$x,yinmathbb{R}^p$ are independently drawn from the standard multivariate Gaussian distribution $N(0,I_p)$. How do you prove that the projection length $z=frac{x^T y}{|x|}$ is normally distributed?

If $x$ is fixed, then $z$ is just a linear combination of the entries of $y$, so it is normally distributed. And we can easily calculate the mean and variance, so $z|xsim N(0,1)$ for any $x$. Is this enough to conclude that $zsim N(0,1)$ unconditionally?