probability - Normalized vector of i.i.d. copies of $X$ uniformly distributed on the sphere means $X$ is normally distributed

Let $X$ be a random variable with $\mathbb{E} X^2 = 1$. Let $X_i$ be i.i.d. copies of $X$ such that
$$
\frac{1}{\sqrt{\sum X_i^2}} \left(X_1, ..., X_N\right)
$$
is uniformly distributed on $\mathbb{S}^{N-1}$. Prove that $X = \mathcal{N}(0,1)$ in distribution.