
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 \( g(x) \) be a polynomial of degree 2010 with real coefficients, having roots \( s_1, s_2, \dots, s_{2010} \). There are exactly 1005 distinct values among \( |s_1|, |s_2|, \dots, |s_{2010}| \). Determine the minimum number of real roots that \( g(x) \) can have.

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