
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
How many functions \( f \) from \(\{-1005, \ldots, 1005\}\) to \(\{-2010, \ldots, 2010\}\) are there such that the following two conditions are satisfied?
- If \( a < b \) then \( f(a) < f(b) \).
- There is no \( n \) in \(\{-1005, \ldots, 1005\}\) such that \(|f(n)| = |n|\).

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