Tower Property with a Continuous Mixing Parameter

Let $U \sim \operatorname{Uniform}(0,1)$ and, given $U = u$, let $X \sim \operatorname{Geometric}(u)$ (number of trials until first success, so $P(X{=}k \mid U{=}u) = (1-u)^{k-1}u$ for $k = 1, 2, \ldots$). Find $E[X]$ using the tower property.