
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  $p$  be a prime number. Find all subsets  $S\subseteq\mathbb Z/p\mathbb Z$  such that
1. if  $a,b\in S$ , then  $ab\in S$ , and
2. there exists an  $r\in S$  such that for all  $a\in S$ , we have  $r-a\in S\cup\{0\}$ .

*Proposed by Harun Khan*

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