
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
As shown in the diagram, circle $O_1$ is externally tangent to circle $O_2$ at point $P$. From point $A$ on circle $O_1$, a tangent $AB$ is drawn to circle $O_2$ at point $B$. Line segment $AP$ is extended to intersect circle $O_2$ at point $C$. Given that the radii of circles $O_1$ and $O_2$ are $2$ and $1$ respectively, find the ratio $\frac{AC}{AB}$.

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