
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 $nx^{3}+2x-n=0$. Define $a_{n}=\left\lfloor(n+1)x_{n}\right\rfloor$ (for $n\geqslant 2$), where $\left\lfloor x \right\rfloor$ denotes the greatest integer less than or equal to the real number $x$. Find the value of $\dfrac {1}{1007}(a_{2}+a_{3}+\ldots+a_{2015})$.

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