
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 $\triangle XYZ$ have side lengths $XY=15$, $XZ=20$, and $YZ=25$. Inside $\angle XYZ$, there are two circles: one is tangent to the rays $\overline{XY}$, $\overline{XZ}$, and the segment $\overline{YZ}$, while the other is tangent to the extension of $\overline{XY}$ beyond $Y$, $\overline{XZ}$, and $\overline{YZ}$. Compute the distance between the centers of these two circles.

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