
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 cubic function $f(x)=ax^3+bx^2+cx+d\ (a\neq 0)$, define: if the equation $f''(x)=0$ has a real solution $x_0$, then the point $(x_0, f(x_0))$ is called an "inflection point" of the function $y=f(x)$. A student found that "any cubic function has an 'inflection point'; any cubic function has a center of symmetry; and the 'inflection point' is the center of symmetry." Given this discovery, for the function $f(x)=x^3- \frac{3}{2}x^2+3x-\frac{1}{4}$, determine its center of symmetry and calculate the sum $f(\frac{1}{2013})+f(\frac{2}{2013})+f(\frac{3}{2013})+\ldots+f(\frac{2012}{2013})$.

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