
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
The polynomial 
\[P(x)=(1+x+x^2+\ldots+x^{19})^2-x^{19}\]
has 38 complex zeros of the form \(z_k=r_k\left[\cos(2\pi\alpha_k) + i\sin(2\pi\alpha_k)\right]\), \(k=1,2,3,\ldots,38\), with \(0<\alpha_1\le\alpha_2\le\alpha_3\le\dots\le\alpha_{38}<1\) and \(r_k>0\). Find \(\alpha_1+\alpha_2+\alpha_3+\alpha_4+\alpha_5.\)

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