Chain of words

A word chain of length $n$ is defined as a sequence of words w_1, w_2, \ldots, w_n such that for every 1 \leq i < n, the word w_i is a proper prefix of the word w_{i+1}.

Recall that a word $u$ of length $k$ is called a proper prefix of a word $v$ of length $l$ if l > k and the first $k$ letters of $v$ are the same as $u$.

Given a set of words S = \{s_1, s_2, \ldots, s_m\}, find the maximum length of a word chain that can be formed using some (possibly not all) words from this set.