
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
Find the sum of the five smallest \( \alpha \) values, where the polynomial \( Q(x) = (1+x+x^2+\ldots+x^{20})^2 - x^{15} \) has complex zeros of the form \( z_k=r_k\left[\cos(2\pi\alpha_k) + i\sin(2\pi\alpha_k)\right] \), with \( 0<\alpha_1\le\alpha_2\le\ldots\le\alpha_{34}<1 \) and \( r_k>0 \).

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