
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
Find all integers \( n > 3 \) for which there exist \( n \) points \( A_1, \ldots, A_n \) in the plane and real numbers \( r_1, \ldots, r_n \) such that:

1. Any three points are never collinear.
2. For any \( i, j, k \), the area of the triangle \( A_i A_j A_k \) is equal to \( r_i + r_j + r_k \).

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