
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 $\triangle DEF$ have side lengths $DE=12$, $DF=16$, and $EF=20$. Inside $\triangle DEF$, there are two circles tangent to the sides $\overline{DE}$, $\overline{DF}$, and $\overline{EF}$. These circles are tangent to each other and each tangent to two sides and the extension of the third side. Compute the distance between the centers of these two circles.

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