
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 positive integers $n > 3$ such that there exist $n$ points $A_{1}, A_{2}, \cdots, A_{n}$ in the plane and real numbers $r_{1}, r_{2}, \cdots, r_{n}$ satisfying the following conditions:
(1) Any 3 points among $A_{1}, A_{2}, \cdots, A_{n}$ are not collinear;
(2) For each triplet of points $\left\{A_{i}, A_{j}, A_{k}\right\}$ (where $1 \leq i < j < k \leq n$), the area of triangle $\triangle A_{i} A_{j} A_{k}$, denoted by $S_{ijk}$, is equal to $r_{i} + r_{j} + r_{k}$.

(Note: This is a problem from the 36th IMO in 1995)

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