
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
In a right triangle \(ABC\) with right angle at \(C\), a median \(CD\) is drawn from \(C\) to the hypotenuse. Find the distance between the centers of the inscribed circles in triangles \(ACD\) and \(BCD\), given that \(BC = 4\) and the circumradius of triangle \(ABC\) is \( \frac{5}{2} \).

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