
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
Among right-angled triangles with equal perimeters, determine the one in which the largest inscribed circle can be drawn. What is the radius of the circle if the perimeter of the triangle is \( k \)?

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