
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  $\mathbb N$  denote the set of all positive integers. Find all real numbers  $c$  for which there exists a function  $f:\mathbb N\to \mathbb N$  satisfying:

    
- for any  $x,a\in\mathbb N$ , the quantity  $\frac{f(x+a)-f(x)}{a}$  is an integer if and only if  $a=1$ ;
- for all  $x\in \mathbb N$ , we have  $|f(x)-cx|<2023$ .


*Proposed by Sutanay Bhattacharya*

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