There are two ways to answer this question- from a syntactic/semantic point of view, and by treating this as a reference to an English idiom.
The English idiom is "one thing leads to another", which in itself suggests that a chain of events takes place, generally outside the control of the people involved.
Looking at the syntax, the word must be a pronoun or noun. This rules out other because it is a determiner: you can talk about "other problems" or "other things" but you can't use "other" on its own.
You can only use "the other" when there are only two possibilities- for example hands... "His phone was in one hand and a pack of cigarettes was in the other". As it states in the first sentence, there are young people face a lot more than two problems.
"the others" would mean "all of the other problems": this does not work because the speaker subsequently refers to "more problems" that follow on from the second one.
another can be a determiner or pronoun meaning "one more person or thing or an extra amount". This meaning would work, and the pronoun form would fit syntactically.