
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 $PQR$, the inradius is $4$ and the circumradius is $13$. If $3\cos{Q} = \cos{P} + \cos{R}$, then the area of triangle $PQR$ can be expressed as $\frac{x\sqrt{y}}{z}$, where $x, y$, and $z$ are positive integers such that $x$ and $z$ are relatively prime and $y$ is not divisible by the square of any prime. Compute $x+y+z$.

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