
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
Given a triangle \( \triangle ABC \) with internal angles \( A, B, \) and \( C \), what is the sum of the numerator and denominator when the maximum value of \( (\sin A \cos B + \sin B \cos C + \sin C \cos A)^{2} \) is expressed in its simplest fractional form?

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