Expected Maximum of Correlated Bernoullis via Indicator and Tower

Let $U \sim \operatorname{Uniform}(0,1)$ and, given $U$, let $X$ and $Y$ be conditionally i.i.d.\ $\operatorname{Bernoulli}(U)$. Define $M = \max(X, Y)$. Using the tower property and the indicator representation $M = \mathbf{1}\{X \ge 1 \text{ or } Y \ge 1\}$, find $E[M]$.