
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 the polynomial $$Q(x) = \left(1 + x + x^2 + \ldots + x^{20}\right)^2 - x^{20},$$ find the sum $$\beta_1 + \beta_2 + \beta_6$$ where the complex zeros of $Q(x)$ are written in the form, $\beta_k=r_k[\cos(2\pi\beta_k)+i\sin(2\pi\beta_k)]$, with $0<\beta_1\le\beta_2\le\ldots\le\beta_{41}<1$ and $r_k>0$.

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