
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  $\mathbb{N}_0$  denote the set of all nonnegative integers. Determine all functions  $f:\mathbb{N}_0\to\mathbb{N}_0$  with the following two properties:



- $0\le f(x)\le x^2$  for all  $x\in\mathbb{N}_0$
- $x-y$  divides  $f(x)-f(y)$  for all  $x,y\in\mathbb{N}_0$  with  $x>y$


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