$X\sim N(0,1)$ and $U=1 \ \text{if} \ X \ge 0 \ \text{and} \ U=0 \ \text{if} \ X<0$

Find $\text{cov}(\text{abs} (X),U)$

where $\text{abs()}$ is the absolute value function.

I did:

$\text{cov}(\text{abs}(X),U)=E(\text{abs}(X)U)-E(\text{abs}(X))E(U)=E(\text{abs}(X)U |X \ge 0)P(X \ge 0)+ E(\text{abs}(X)U | X<0)P(X<0)-E(\text{abs}(X))E(U)$

$=E(X)P(X \ge 0)-E(|X|)E(U)=-\frac{1}{2}(\sqrt{\frac{2}{\pi}})$

But the answer given is that the covariance is $0$.
Where am I going wrong?
Please help