
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
Determine all functions \(f: \mathbf{N} \rightarrow \mathbf{N}\) such that
\[ 
f(a+b)=f(a)+f(b)+f(c)+f(d)
\]
for all non-negative integers \(a\), \(b\), \(c\), and \(d\) that satisfy \(2ab = c^2 + d^2\).

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