
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 $q(x) = x^{2010} + x^{2009} + x^{2008} + \cdots + x + 1,$
and let $s(x)$ be the polynomial remainder when $q(x)$ is divided by $x^4 - x^3 + 2x^2 - x + 1$. Find the remainder when $|s(2010)|$ is divided by $1000$.

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