
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
A subset  $S$  of the set  $M=\{1,2,.....,p-1\}$ ,where  $p$  is a prime number of the kind
 $12n+11$ ,is *essential*,if the product  ${\Pi}_s$  of all elements of the subset
is not less than the product  $\bar{{\Pi}_s}$  of all other elements of the set.The 
**difference**  $\bigtriangleup_s=\Pi_s-\bar{{\Pi}_s}$  is called *the deviation*
of the subset  $S$ .Define the least possible remainder of division by  $p$  of the deviation of an essential subset,containing  $\frac{p-1}{2}$  elements.

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