
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 \( N_{0} \) be the set of non-negative integers, and \( f: N_{0} \rightarrow N_{0} \) be a function such that \( f(0)=0 \) and for any \( n \in N_{0} \), \( [f(2n+1)]^{2} - [f(2n)]^{2} = 6f(n) + 1 \) and \( f(2n) > f(n) \). Determine how many elements in \( f(N_{0}) \) are less than 2004.


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