
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
$$Q(x) = \left(\frac{x^{22} - 1}{x-1}\right)^2 - x^{21}.$$
Determine all the zeros of form \( z_k = r_k[\cos(2\pi\alpha_k) + i\sin(2\pi\alpha_k)] \), where \( k = 1, 2, \ldots, 42 \), with \( 0 < \alpha_1 \le \alpha_2 \le \ldots \le \alpha_{42} < 1 \) and \( r_k > 0 \). Find 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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