
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 $A, B, C$ be angles of a triangle, where angle $B$ is obtuse, and
\[
\cos^2 A + \cos^2 B - 2 \sin A \sin B \cos C = \frac{11}{8}
\]
and
\[
\cos^2 B + \cos^2 C - 2 \sin B \sin C \cos A = \frac{17}{9}.
\]
Find positive integers $p$, $q$, $r$, and $s$ for which
\[
\cos^2 C + \cos^2 A - 2 \sin C \sin A \cos B = \frac{p-q\sqrt{r}}{s},
\]
where $p+q$ and $s$ are relatively prime and $r$ is not divisible by the square of any prime.  Determine $p+q+r+s$.

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