
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
Let \( z=\cos \frac{2 \pi}{2011}+i \sin \frac{2 \pi}{2011} \), and let
\[ P(x)=x^{2008}+3 x^{2007}+6 x^{2006}+\ldots + \frac{2008 \cdot 2009}{2} x+\frac{2009 \cdot 2010}{2} \]
for all complex numbers \( x \). Evaluate \( P(z) P\left(z^{2}\right) P\left(z^{3}\right) \ldots P\left(z^{2010}\right) \).

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