
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 a triangle \( \triangle ABC \) with side lengths 13, 14, and 15, there are 4 circles with the same radius \( r \) placed inside \( \triangle ABC \). Circle \( O_{1} \) is tangent to sides \( AB \) and \( AC \), circle \( O_{2} \) is tangent to sides \( BA \) and \( BC \), and circle \( O_{3} \) is tangent to sides \( CB \) and \( CA \). Circle \( O \) is tangent to \( O_{1}, O_{2}, \) and \( O_{3} \). Find the radius \( r \).

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