
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  $f$  from the set of non-negative integers to itself such that  $f(a + b) = f(a) + f(b) + f(c) + f(d)$ , whenever  $a, b, c, d$ , are non-negative integers satisfying  $2ab = c^2 + d^2$ .

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