
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
For all \( n \in \mathbb{N}^{*} \), let \( p_{n} \) be the number of sequences of \( A \) and \( B \) of length \( n \) containing neither \( AAAA \) nor \( BBB \). Calculate

$$
\frac{p_{2004}-p_{2002}-p_{1999}}{p_{2000}+p_{2001}}
$$

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