
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 $ABC$, the cosine of angle $BAC$ is equal to $\frac{1}{2}$, $AB=2$, and $AC=3$. Point $D$ lies on the extension of side $AC$ such that $C$ is between $A$ and $D$, and $CD=3$. Find the ratio of the radius of the circumcircle of triangle $ABC$ to the radius of the incircle of triangle $ABD$.

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