
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  $p$  is a prime and  $p\equiv 2\pmod 3$ . For  $\forall a\in\mathbb Z$ , if  $$ p\mid \prod\limits_{i=1}^p(i^3-ai-1), $$  then  $a$  is called a "GuGu" number. How many "GuGu" numbers are there in the set  $\{1,2,\cdots ,p\}?$ (We are allowed to discuss now. It is after 00:00 Feb 14 Beijing Time)

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