
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  $G$  be a graph whose vertices are the integers. Assume that any two integers are connected by a finite path in  $G$ . For two integers  $x$  and  $y$ , we denote by  $d(x, y)$  the length of the shortest path from  $x$  to  $y$ , where the length of a path is the number of edges in it. Assume that  $d(x, y) \mid x-y$  for all integers  $x, y$  and define  $S(G)=\{d(x, y) | x, y \in \mathbb{Z}\}$ . Find all possible sets  $S(G)$ .

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