
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 $\triangle ABC$ have side lengths $AB = 12$, $AC = 16$, and $BC = 20$. Inside $\angle BAC$, two circles are positioned, each tangent to rays $\overline{AB}$ and $\overline{AC}$, and the segment $\overline{BC}$. Compute the distance between the centers of these two circles.

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