
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  $AHB$ ,  $BHC$ ,  $CHA$ .

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