
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
The polynomial $$Q(x)=(1+x+x^2+\ldots+x^{19})^2-x^{19}$$has 38 complex zeros of the form $z_k=r_k\left[\cos(2\pi\beta_k) + i\sin(2\pi\beta_k)\right]$, $k=1,2,3,\ldots,38$, with $0<\beta_1\le\beta_2\le\beta_3\le\dots\le\beta_{38}<1$ and $r_k>0$. Find $\beta_1+\beta_2+\beta_3+\beta_4+\beta_5.$

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