Let $U_1, U_2, \dots$ be independent uniform$(0, 1)$ random variables, and let $X$ have the distribution
$P(X = x) = \frac{c}{x!}$, $x = 1, 2, \dots$,
where $c = \frac{1}{(e - 1)}$. Find the distribution of $Z = \min\{U_1, \dots, U_X\}$. [Hint: The distribution of $Z | X = x$ is that of the first order statistic from a sample of size $x$.]