
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
Determine all integers \( n > 3 \) such that there are \( n \) points \( A_{1}, A_{2}, \ldots, A_{n} \) in the plane that satisfy the following two conditions simultaneously:
(a) No three lie on the same line.
(b) There exist real numbers \( p_{1}, p_{2}, \ldots, p_{n} \) such that the area of \( \triangle A_{i} A_{j} A_{k} \) is equal to \( p_{i} + p_{j} + p_{k} \), for \( 1 \leq i < j < k \leq n \).

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