
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 $f(x)$ be a polynomial of degree 3009 with real coefficients, and let its roots be $r_1,$ $r_2,$ $\dots,$ $r_{3009}.$  There are exactly 1505 distinct values among
\[|r_1|, |r_2|, \dots, |r_{3009}|.\]What is the minimum number of real roots that $f(x)$ can have?

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