
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 that have the following properties:
$$
f:\{1,2, \cdots, n, \cdots\} \rightarrow \mathbb{Z} 
$$
(1) If \(a\) and \(b\) are positive integers, and \(a\) divides \(b\), then \(f(a) \geq f(b)\).

(2) If \(a\) and \(b\) are positive integers, then
$$
f(ab)+f\left(a^{2}+b^{2}\right)=f(a)+f(b) .
$$

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