
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
In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are denoted as $a$, $b$, and $c$, respectively. Given that
$$
\begin{array}{l}
f(x) = \boldsymbol{m} \cdot \boldsymbol{n}, \boldsymbol{m} = (2 \cos x, 1), \\
\boldsymbol{n} = (\cos x, \sqrt{3} \sin 2x), \\
f(A) = 2, b = 1, S_{\triangle ABC} = \frac{\sqrt{3}}{2}.
\end{array}
$$
Determine the value of
$$
\frac{b + \boldsymbol{c}}{\sin B + \sin C}.
$$

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