
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
Given a prime number \( p \) and a natural number \( n \) such that \( p \geqslant n \geqslant 3 \), let the set \( A \) be composed of sequences of length \( n \) with elements drawn from the set \( \{1, 2, \cdots, p-1\} \). For any two sequences \((x_{1}, x_{2}, \cdots, x_{n})\) and \((y_{1}, y_{2}, \cdots, y_{n})\) from set \( A \), there must exist three distinct positive integers \( k, l, m \) such that \( x_{k} \neq y_{k}, x_{l} \neq y_{l}, x_{m} \neq y_{m} \). Find the maximum number of elements in set \( A \).

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