
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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
\[\frac{p(2004)-p(2002)-p(1999)}{p(2001)+p(2000)}.\]

Present the answer in LaTex format: \boxed{Your answer}
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