
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
Given an obtuse triangle \( \triangle ABC \) with the following conditions:
1. The lengths of \( AB \), \( BC \), and \( CA \) are positive integers.
2. The lengths of \( AB \), \( BC \), and \( CA \) do not exceed 50.
3. The lengths of \( AB \), \( BC \), and \( CA \) form an arithmetic sequence with a positive common difference.

Determine the number of obtuse triangles that satisfy the above conditions, and identify the side lengths of the obtuse triangle with the largest perimeter.

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