
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 $ABC$, where $C$ is the right angle and $\angle BAC < 45^\circ$, let $AB = 5$. A point $P$ on $\overline{AB}$ is such that $\angle APC = 3\angle ACP$ and $CP = 2$. Find $AP$ and $BP$ and express the ratio $\frac{AP}{BP}$ in simplest form, represented as $p + q\sqrt{r}$, where $p$, $q$, $r$ are positive integers and $r$ not divisible by squares of any primes. Compute $p + q + r$.

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