
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 triangle \(ABC\), side \(BC\) is equal to 5. A circle passes through vertices \(B\) and \(C\) and intersects side \(AC\) at point \(K\), where \(CK = 3\) and \(KA = 1\). It is known that the cosine of angle \(ACB\) is \(\frac{4}{5}\). Find the ratio of the radius of this circle to the radius of the circle inscribed in triangle \(ABK\).

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