
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
Find the number of sequences  $(a_n)_{n=1}^\infty$  of integers satisfying  $a_n \ne -1$  and
\[a_{n+2} =\frac{a_n + 2006}{a_{n+1} + 1}\]
for each  $n \in \mathbb{N}$ .

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