
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 \( f(x) = \boldsymbol{m} \cdot \boldsymbol{n} \), where \( \boldsymbol{m} = (2 \cos x, 1) \) and \( \boldsymbol{n} = (\cos x, \sqrt{3} \sin 2 x) \). In \( \triangle ABC \), \(\angle A\) satisfies \( f(A) = 2 \). The side opposite \(\angle B\) is \( b = 1 \), and the area of \(\triangle ABC\) is \( S = \frac{\sqrt{3}}{2} \). Find \( \frac{b+c}{\sin B + \sin C} \).

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