
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 $XYZ$, the inradius is $7$ and the circumradius is $25$. It is also given that $2\cos Y = \cos X + \cos Z$. Determine the area of triangle $XYZ$, expressing your answer as $\frac{p\sqrt{q}}{r}$, where $p, q,$ and $r$ are positive integers such that $p$ and $r$ are relatively prime and $q$ is not divisible by the square of any prime. Compute $p+q+r$.

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