
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
For a prime  $p$  of the form  $12k+1$  and  $\mathbb{Z}_p=\{0,1,2,\cdots,p-1\}$ , let 
\[\mathbb{E}_p=\{(a,b) \mid a,b \in \mathbb{Z}_p,\quad  p\nmid 4a^3+27b^2\}\]
For  $(a,b), (a',b') \in \mathbb{E}_p$  we say that  $(a,b)$  and  $(a',b')$  are equivalent if there is a non zero element  $c\in \mathbb{Z}_p$  such that  $p\mid (a' -ac^4)$  and  $p\mid (b'-bc^6)$ . Find the maximal number of inequivalent elements in  $\mathbb{E}_p$ .

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