Prove that mean independence implies zero conditional expectation in regression model

Question

Let $Y,X_i$ for $i=1,2,3$ be random variables and let $$Y=\beta_0+\beta_1X_1+\beta_2X_2+\beta_3X_3+U$$ s.t. $E(Y)=0$. Suppose that $$E\left(U \mid X_1X_2X_3\right)=E\left(U\mid X_1X_2\right)=E\left(U\mid X_1X_3\right)=E\left(U\mid X_2X_3\right)$$ and prove that $E\left(U\mid X_1X_2X_3\right)=0$.

Please give me a hint on how to start. Thanks.