
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 degree 2006 polynomial with complex roots \( c_{1}, c_{2}, \ldots, c_{2006} \), such that the set
\[ \{ |c_{1}|, |c_{2}|, \ldots, |c_{2006}| \} \]
consists of exactly 1006 distinct values. What is the minimum number of real roots of \( f(x) \)?

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