
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 \( 1 \leq a < p \), and both \( a \) and \( 4 - a \) are quadratic non-residues. Calculate the remainder when the product of the elements of \( A \) is divided by \( p \).

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