
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
Let $f'(x)$ be the derivative of the function $y=f(x)$, and $f''(x)$ be the derivative of $f'(x)$. If the equation $f''(x)=0$ has a real solution $x_0$, then the point $(x_0, f(x_0))$ is called the "inflection point" of the function $y=f(x)$. It is known that any cubic function has both an inflection point and a center of symmetry, and the inflection point is the center of symmetry. Given $f(x)=\frac{1}{3}x^3-2x^2+\frac{8}{3}x+2$, and the general formula of the sequence $\{a_n\}$ is $a_n=n-1007$, then $\sum_{i=1}^{2017}f(a_i)=$ .

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