
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
The function \( f(x)=\cos^{2} x + 2 \sin x \cos x - \sin^{2} x + Ax + B \) is defined over the interval \( 0 \leqslant x \leqslant \frac{3}{2} \pi \). Determine the values of \( A \) and \( B \) that minimize the maximum value \( M \) of \( f(x) \) on this interval, and provide a proof.

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