
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
Triangle $ABC$ has a right angle at $C$, $\angle BAC < 45^\circ$, and $AB = 6$. A point $P$ on $\overline{AB}$ is such that $\angle APC = 3\angle ACP$ and $CP = 2$. Find the ratio $\frac{AP}{BP}$ in the form $p + q\sqrt{r}$, where $p$, $q$, $r$ are positive integers and $r$ is not divisible by the square of any prime. Determine $p+q+r$.

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