
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. A natural number \( x \) is called a quadratic non-residue modulo \( p \) if for any natural number \( t \), \( p \) does not divide \( x - t^2 \). Define the set \( A = \{ a \mid a \in \mathbb{Z}, 1 \leq a < p, \text{both } a \text{ and } 4 - a \text{ are quadratic non-residues modulo } p \} \). Find the remainder when the product of all elements in \( A \) is divided by \( p \).

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