
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 $13$. Given $3\cos{Q} = \cos{P} + 2\cos{R}$, compute the area of triangle $PQR$ and express it as $\frac{d\sqrt{e}}{f}$, where $d, e,$ and $f$ are positive integers with $d$ and $f$ being relatively prime, and $e$ not divisible by the square of any prime. Compute $d+e+f$.

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