
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 \( g(x) = x^{3}(1-x)^{3} \). What is the value of the sum
\[
g\left(\frac{1}{2021}\right) - g\left(\frac{2}{2021}\right) + g\left(\frac{3}{2021}\right) - g\left(\frac{4}{2021}\right) + \cdots + g\left(\frac{2019}{2021}\right) - g\left(\frac{2020}{2021}\right)?
\]
A) 0
B) \(\frac{1}{2021^6}\)
C) \(\frac{2020^3}{2021^6}\)
D) \(\frac{2022^3}{2021^6}\)
E) 1

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