Enrique and Yan frequently play a game called 1D Scrabble. The game is played on a one-dimensional, infinite board using a fixed dictionary of unique words. On the first turn ($$$i = 1$$$), any word $$$w_1$$$ from the dictionary may be played. On every subsequent turn ($$$i \gt 1$$$), the played word $$$w_i$$$ must contain the previous word $$$w_{i-1}$$$ as a contiguous substring. The game concludes as soon as no valid moves remain.

Recently, Yan did not have time to play, so Enrique started playing with GePeTo. However, GePeTo started hallucinating! It consistently states that this game is trivial, arguing that there are very few combinations. Help Enrique convince GePeTo by calculating the total number of distinct possible games. Two games are distinct if their sequences of played words $$$(w_1, \ldots, w_k)$$$ are different.