$X+Y$ is $2X$ on $|X|\le\alpha$ and exactly 0 otherwise: an atom at the origin

Let X be a zero-mean unit variance Gaussian random variable. Define the random variable Y as $Y=X$ if $|X|\le\alpha$ and $Y=-X$ if $|X|>\alpha$, $\alpha$ is some positive real number. What is the distribution of $X+Y$?