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

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