How to show that a r.v. is normally distributed and that a r.v. is not almost surely constant

Question

Let $L(Y) = N(0,1)$ and set for some constant a>0: $$Z=Y1_{|Y|≤a}−Y1_{|Y|>a}$$

Hence, $Y+Z=2Y1_{|Y|≤a}$, that is $$Y+Z=2Y$$ if |Y|≤a $$Y+Z=0$$ if |Y|>a. How can I show that:$$1)L(Z) = N(0,1)$$ and $$2) Y+Z$$ is not almost surely a constant?