
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 $p(x)$ be a polynomial of degree 2010 with real coefficients, and let its roots be $s_1,$ $s_2,$ $\dots,$ $s_{2010}.$  Among these roots, there are exactly 1009 distinct values among the absolute values
\[|s_1|, |s_2|, \dots, |s_{2010}|.\] Determine the minimum number of real roots that $p(x)$ can have.

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