
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
Let \(ABC\) be a triangle with \(AB=13\), \(BC=14\), \(CA=15\). Let \(H\) be the orthocenter of \(ABC\). Find the radius of the circle with nonzero radius tangent to the circumcircles of \(AHB\), \(BHC\), and \(CHA\).

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