
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
Find the sum of the smallest five distinct $\beta_k$ from the polynomial $$Q(x) = \left(1 + x + x^2 + \ldots + x^{19}\right)^2 - x^{19},$$ where its zeros $\omega_k = r_k\left[\cos(2\pi\beta_k) + i\sin(2\pi\beta_k)\right]$, $k=1,2,3,\ldots,38$, are in the form of complex numbers, with $0 < \beta_1 \le \beta_2 \le \ldots \le \beta_{38} < 1$ and $r_k > 0$.

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