
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 $$Q(x)=(1+x+x^2+\ldots+x^{19})^2-x^{19}.$$ The polynomial \( Q(x) \) has 38 complex zeros of the form \( z_k = r_k [\cos(2\pi\beta_k) + i\sin(2\pi\beta_k)] \), \( k=1,2,\ldots,38 \), with \( 0 < \beta_1 \le \beta_2 \le \ldots \le \beta_{38} < 1 \) and \( r_k > 0 \). Find \( \beta_1 + \beta_2 + \beta_3 + \beta_4 + \beta_5 + \beta_6 \).

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