
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 the cubic function $f(x) = ax^3 + bx^2 + cx + d$ ($a \neq 0$), define: Let $f''(x)$ be the derivative of the derivative $y = f'(x)$ of the function $y = f(x)$. If the equation $f''(x) = 0$ has a real solution $x_0$, then the point $(x_0, f(x_0))$ is called the "well-behaved point" of the function $y = f(x)$. It has been discovered that "any cubic function has a 'well-behaved point'; any cubic function has a center of symmetry; and the 'well-behaved point' is the center of symmetry." Based on this discovery, please answer the question: If the function $g(x) = \frac{1}{3}x^3 - \frac{1}{2}x^2 + 3x - \frac{5}{12}$, then $g\left(\frac{1}{2011}\right) + g\left(\frac{2}{2011}\right) + g\left(\frac{3}{2011}\right) + g\left(\frac{4}{2011}\right) + \ldots + g\left(\frac{2010}{2011}\right) = \boxed{\text{\_\_\_\_\_\_}}$.

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