
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 \( X \) be the set of non-negative integers and \( f : X \to X \) a map such that \( ( f(2n+1) )^2 - ( f(2n) )^2 = 6 f(n) + 1 \) and \( f(2n) \geq f(n) \) for all \( n \) in \( X \). How many numbers in \( f(X) \) are less than 2003?

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