
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
\[ Q(x) = \left(1 + x + x^2 + \ldots + x^{19}\right)^2 - x^{19} \]
Find the sum of the smallest five distinct angles \( \beta_1, \beta_2, \beta_3, \beta_4, \beta_5 \) corresponding to the complex zeros of \( Q(x) \) in the form \( z_k = r_k \left[\cos(2\pi\beta_k) + i\sin(2\pi\beta_k)\right] \), where \( 0 < \beta_1 \le \beta_2 \le \beta_3 \le \ldots \le \beta_{38} < 1 \) and \( r_k > 0 \).

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