$|X|$ is Bernoulli$(\theta)$ and complete; $X$ has mean zero whatever $\theta$

Let X be a r.v. having the pmf $P[X=x]=\bigl(\tfrac\theta2\bigr)^{|x|}(1-\theta)^{1-|x|}$; $x=-1,0,1$, $0<\theta<1$. The complete statistics for $\theta$: