
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 $C$ is obtuse. Given that:
\[
\cos^2 A + \cos^2 C + 2 \sin A \sin C \cos B = \frac{16}{9} \text{ and}
\]
\[
\cos^2 C + \cos^2 B + 2 \sin C \sin B \cos A = \frac{17}{10},
\]
find positive integers $u$, $v$, $w$, and $x$ such that:
\[
\cos^2 B + \cos^2 A + 2 \sin B \sin A \cos C = \frac{u-v\sqrt{w}}{x},
\]
where $u+v$ and $x$ are relatively prime, and $w$ is not divisible by the square of any prime. Determine $u+v+w+x$.

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