The original question was: Suppose $X,Y,Z$ are i.i.d. $\mathcal{N}(0,1)$, find a nonnegative continuous function $g$ such that $\frac{X+YZ}{g(Z)} \sim \mathcal{N}(0,1)$.
The solution says, since $E(X+YZ\mid Z=z)=0$ and $Var(X+YZ\mid Z=z)=1+z^2$, for all $z \in \mathcal{R}$. So $g=\sqrt{1+Z^2}$.
I see the calculation of the expectation and the variance, but not sure why $\frac{X+YZ}{g(Z)}$ follows a normal distribution. The solution says, since $X+YZ\mid Z=z$ follows a normal distribution for all $z$, thus $X+YZ$ is normal.