
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 \( A_1, A_2, \ldots, A_n \) on a plane and real numbers \( r_1, r_2, \ldots, r_n \) such that the following two conditions are satisfied:
(1) No three points \( A_1, A_2, \ldots, A_n \) are collinear.
(2) For every set of three points \( \{A_i, A_j, A_k\} \) with \( 1 \leq i < j < k \leq n \), the area of triangle \( A_i A_j A_k \), denoted by \( S_{ijk} \), is equal to \( r_i + r_j + r_k \).

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