
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
Given a cubic function $f(x)=ax^{3}+bx^{2}+cx+d(a\neq 0)$, define the "nice point" of the function as the point $(x_{0},f(x_{0}))$ where $x_{0}$ is a real root of the equation $f''(x)=0$. It has been observed that every cubic function has a "nice point," a symmetry center, and that the "nice point" is the symmetry center. Based on this observation, find the value of $g(\frac{1}{2011})+g(\frac{2}{2011})+g(\frac{3}{2011})+g(\frac{4}{2011})+\cdots+g(\frac{2010}{2011})$ for the function $g(x)=\frac{1}{3}x^{3}-\frac{1}{2}x^{2}+3x-\frac{5}{12}$.

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