
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 $f:\{1,2,\cdots, n,\cdots \} \rightarrow \mathbf{Z}$ that have the following properties:
1. If $a$ and $b$ are positive integers and $a \mid b$, then $f(a) \geq f(b)$.
2. If $a$ and $b$ are positive integers, then $f(ab) + f(a^2 + b^2) = f(a) + f(b)$.

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