
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
Find all functions  $f:\mathbb{N}_0\to\mathbb{N}_0$  for which  $f(0)=0$  and
\[f(x^2-y^2)=f(x)f(y) \]
for all  $x,y\in\mathbb{N}_0$  with  $x>y$ .

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