
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 that the domain of the function $y=f(x)$ is $D$, if for any $x\_1$, $x\_2\in D$, when $x\_1+x\_2=2a$, it always holds that $f(x\_1)+f(x\_2)=2b$, then the point $(a,b)$ is called the symmetric center of the graph of the function $y=f(x)$. Investigate a symmetric center of the function $f(x)=x^{3}+\sin x+2$ and use the definition of the symmetric center to find the value of $f(-1)+f(-\frac{19}{20})+\ldots+f(\frac{19}{20})+f(1)=\_\_\_\_\_\_\_\_\_\_\_\_$.

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