
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 $$Q(x) = \left(\frac{x^{24} - 1}{x-1}\right)^2 - x^{23}.$$ This polynomial has 46 complex zeros of the form \(z_k = r_k[\cos(2\pi\alpha_k) + i\sin(2\pi\alpha_k)]\), with \(0 < \alpha_1 \leq \alpha_2 \leq \ldots \leq \alpha_{46} < 1\) and \(r_k > 0\). Calculate the sum \(\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 + \alpha_5.\)

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