
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:

(i) Any three of the points are never collinear.

(ii) For all \( \{i, j, k\} \), 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