
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
Determine the sum $\beta_1 + \beta_2 + \beta_3 + \beta_4 + \beta_5$ for the polynomial $$Q(x) = \left(1+x+x^2+\ldots+x^{12}\right)^3 - x^{12},$$where $Q(x)$ has complex zeros of the form $z_k = s_k [\cos(2\pi \beta_k) + i \sin(2\pi \beta_k)]$, $k = 1, 2, 3, ..., 36$, with $0 < \beta_1 \leq \beta_2 \leq \beta_3 \leq ... \leq \beta_{36} < 1$ and $s_k > 0$.

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