
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 an odd prime. An integer  $x$  is called a *quadratic non-residue* if  $p$  does not divide  $x-t^2$  for any integer  $t$ . 
 
Denote by  $A$  the set of all integers  $a$  such that  $1\le a<p$ , and both  $a$  and  $4-a$  are quadratic non-residues. Calculate the remainder when the product of the elements of  $A$  is divided by  $p$ . 

*Proposed by Richard Stong and Toni Bluher*

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