
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
Given a function \(\varphi: \mathbf{N} \rightarrow \mathbf{N}\), does there exist a function \(f\) on \(\mathbf{N}\) such that for all \(x \in \mathbf{N}\), \(f(x) > f(\varphi(x))\), and

(1) the range of \(f\) is a subset of \(\mathbf{N}\)?  
(2) the range of \(f\) is a subset of \(\mathbf{Z}\)?

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