Hongcow Masters the Cyclic Shift

Hongcow's teacher heard that Hongcow had learned about the cyclic shift, and decided to set the following problem for him.

You are given a list of n strings s₁, s₂, ..., s_n contained in the list A.

A list X of strings is called stable if the following condition holds.

First, a message is defined as a concatenation of some elements of the list X. You can use an arbitrary element as many times as you want, and you may concatenate these elements in any arbitrary order. Let S_X denote the set of of all messages you can construct from the list. Of course, this set has infinite size if your list is nonempty.

Call a single message good if the following conditions hold:

• Suppose the message is the concatenation of k strings w₁, w₂, ..., w_k, where each w_i is an element of X.
• Consider the |w₁| + |w₂| + ... + |w_k| cyclic shifts of the string. Let m be the number of these cyclic shifts of the string that are elements of S_X.
• A message is good if and only if m is exactly equal to k.

The list X is called stable if and only if every element of S_X is good.

Let f(L) be 1 if L is a stable list, and 0 otherwise.

Find the sum of f(L) where L is a nonempty contiguous sublist of A (there are contiguous sublists in total).