A Probability Problem

Rephrased from this post by reddit user /u/phenomist, using the setup from the original 538 puzzler.

Let $X_1, \ldots, X_n$ be distributions over the set of positive integers such that $\mathbb{E}[X_i]$ is finite for each $i$.

Initially, we set $n$ variables, $t_1, \ldots, t_n$, where each $t_i$ is sampled independently from $X_i$. Then we repeat the following algorithm: if all of the $t_i$ are equal to a common value $t$, then we return the value of $t$. Otherwise, we take the index $k$ such that $t_k$ is minimal and replace $t_k$ with $t_k + x_k$, where $x_k$ is independently sampled from $X_k$. (Note that if there is a tie, it does not matter what order we resolve the tie in.)

For example, if $n = 2$ and $X_1, X_2 \sim \operatorname{Unif}\{1, 2, 3, 4, 5\}$, then this process might go $(2, 4) \rightarrow (3, 4) \rightarrow (7, 4) \rightarrow (7, 8) \rightarrow (8, 8) \implies 8$.

Show that the expected value of this process is \[ \operatorname{lcm}(d_1,\ldots,d_n)\cdot \prod_{i = 1}^n \frac{E[X_i]}{d_i} \] where $d_i = \gcd\{n\in \mathbb{Z}^+\mid \mathbb{P}[X_i = n] > 0\}$