
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
Consider the polynomial 
$$Q(x) = \left(1 + x + x^2 + \ldots + x^{20}\right)^2 - x^{19}.$$
This polynomial has complex zeros of the form $z_k = r_k \left[\cos(2\pi \beta_k) + i\sin(2\pi \beta_k)\right]$, where $k=1,2,3,\ldots,42$, with $0 < \beta_1 \leq \beta_2 \leq \ldots \leq \beta_{42} < 1$ and $r_k > 0$. Find the sum $\beta_1 + \beta_2 + \beta_3 + \beta_4 + \beta_5.$

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