CZ-SK-MO 2004 Problem 2

Consider all words containing only letters $A$ and $B$ . For any positive integer $n$ , $p(n)$ denotes the number of all $n$ -letter words without four consecutive $A$ 's or three consecutive $B$ 's. Find the value of the expression $$\frac{p(2004)-p(2002)-p(1999)}{p(2001)+p(2000)}.$$