
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 a positive integer \( n \), let \( x_n \) be the real root of the equation \( n x^{3} + 2 x - n = 0 \). Define \( a_n = \left[ (n+1) x_n \right] \) (where \( [x] \) denotes the greatest integer less than or equal to \( x \)) for \( n = 2, 3, \ldots \). Then find \( \frac{1}{1005} \left( a_2 + a_3 + a_4 + \cdots + a_{2011} \right) \).

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