
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
A sequence of polynomials \( a_{n}(x) \) is defined recursively by:
\[ 
\begin{array}{l} 
a_{0}(x) = 1, \\ 
a_{1}(x) = x^{2} + x + 1, \\ 
a_{n}(x) = (x^{n} + 1) a_{n-1}(x) - a_{n-2}(x), \text{ for all } n \geq 2. 
\end{array} 
\]

Evaluate \( a_{2010}(1) \).

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