
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 $$Q(x) = \left(\frac{x^{20} - 1}{x-1}\right)^2 - x^{19}.$$ Find the sum of the smallest five angles, \( \beta_1 + \beta_2 + \beta_3 + \beta_4 + \beta_5 \), where the complex zeros of \( Q(x) \) can be expressed in the form \( z_k = s_k \left[ \cos(2\pi\beta_k) + i \sin(2\pi\beta_k) \right] \), for \( k = 1, 2, \ldots, 38 \) and \( 0 < \beta_1 \le \beta_2 \le \ldots \le \beta_{38} < 1 \).

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