
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 a prime number such that $p > 3$ and $k \geq 2$. Théo and Vincent play the following game: Théo chooses a permutation $a_{1}, \ldots, a_{k}$ of the integers $1, \ldots, k$, and tells Vincent the value of $a_{i} a_{j}$ modulo $p$ for all $i \neq j$. Vincent wins if he can guess the $\left(a_{i}\right)_{1 \leq i \leq k}$. For what values of $k$ can Vincent win regardless of the permutation chosen by Théo?

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