
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
In triangle \( ABC \), point \( S \) is the center of the inscribed circle. The area of quadrilateral \( ABCS \) is equal to four-fifths the area of triangle \( ABC \). The side lengths of triangle \( ABC \), expressed in centimeters, are all integers, and the perimeter of triangle \( ABC \) is \( 15 \text{ cm} \).

Determine the side lengths of triangle \( ABC \). Find all possible solutions.

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