
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
Petya came up with 1004 monic quadratic trinomials \( f_{1}, \ldots, f_{1004} \), among the roots of which all integers from 0 to 2007 appear. Vasya considers all possible equations \( f_{i} = f_{j} \) ( \( i \neq j \) ), and for each root found, Petya pays Vasya one ruble. What is the smallest possible income for Vasya?

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