
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
 $\triangle ABC$  has side lengths  $AB=20$ ,  $BC=15$ , and  $CA=7$ . Let the altitudes of  $\triangle ABC$  be  $AD$ ,  $BE$ , and  $CF$ . What is the distance between the orthocenter (intersection of the altitudes) of  $\triangle ABC$  and the incenter of  $\triangle DEF$ ?

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