A finite sequence of five-digit integers has the property that the thousands, hundreds, tens, and units digits of each term are, respectively, the tens of thousands, thousands, hundreds, and tens digits of the next term, and this property holds cyclically for the last term to the first term. For example, such a sequence might begin with the terms $12345$, $23456$, and $34567$ and end with the term $89012$. Let $S$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $S$?