
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
Let  $ABC$  be a triangle with  $AB=5$ ,  $BC=7$ ,  $CA=8$ , and circumcircle  $\omega$ . Let  $P$  be a point inside  $ABC$  such that  $PA:PB:PC=2:3:6$ . Let rays  $\overrightarrow{AP}$ ,  $\overrightarrow{BP}$ , and  $\overrightarrow{CP}$  intersect  $\omega$  again at  $X$ ,  $Y$ , and  $Z$ , respectively. The area of  $XYZ$  can be expressed in the form  $\dfrac{p\sqrt q}{r}$  where  $p$  and  $r$  are relatively prime positive integers and  $q$  is a positive integer not divisible by the square of any prime. What is  $p+q+r$ ?

*Proposed by James Lin*

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