
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 $f(x) = ax^3 + bx^2 + cx + d$ ($a \neq 0$), a definition is given: Let $f'(x)$ be the derivative of the function $y = f(x)$, and $f''(x)$ the derivative of $f'(x)$. If the equation $f''(x) = 0$ has real solutions $x_0$, then the point $(x_0, f(x_0))$ is called the "inflection point" of the function $y = f(x)$. A student discovered that any cubic function has an "inflection point" and a center of symmetry, and the inflection point is the center of symmetry. If $f(x) = \frac{1}{3}x^3 - \frac{1}{2}x^2 + \frac{1}{6}x + 1$, then the center of symmetry for this function is _______, and calculate $f\left( \frac{1}{2013} \right) + f\left( \frac{2}{2013} \right) + f\left( \frac{3}{2013} \right) + \ldots + f\left( \frac{2012}{2013} \right) =$ _______.

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