
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 $A$ is acute, and
\[
\sin^2 A + \sin^2 B + 2 \cos A \cos B \sin C = \frac{9}{10},
\]
\[
\sin^2 B + \sin^2 C + 2 \cos B \cos C \sin A = \frac{11}{12}.
\]
Determine the value of $A$ if:
\[
\sin^2 C + \sin^2 A + 2 \cos C \cos A \sin B = \frac{1}{2}.
\]

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