
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 \( p > 3 \) be a prime and let \( a_{1}, a_{2}, \ldots, a_{\frac{p-1}{2}} \) be a permutation of \( 1, 2, \ldots, \frac{p-1}{2} \). For which \( p \) is it always possible to determine the sequence \( a_{1}, a_{2}, \ldots, a_{\frac{p-1}{2}} \) if for all \( i, j \in \left\{1, 2, \ldots, \frac{p-1}{2}\right\} \) with \( i \neq j \) the residue of \( a_{i} a_{j} \) modulo \( p \) is known?

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