
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: one is tangent to rays $\overline{DE}$, $\overline{DF}$, and segment $\overline{EF}$; the other is tangent to rays $\overline{DE}$, $\overline{DF}$, but only externally to segment $\overline{EF}$. Compute the distance between the centers of these two circles.

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