
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 $\overrightarrow{m} = (\sqrt{3}\sin\frac{x}{2}, \cos^2\frac{x}{2} - \frac{1}{2})$ and $\overrightarrow{n} = (\cos\frac{x}{2}, 1)$. Define $f(x) = \overrightarrow{m} \cdot \overrightarrow{n}$. In triangle ABC, the sides opposite to angles A, B, and C are denoted as a, b, and c, respectively.  
(1) Find the interval where $f(x)$ is monotonically increasing;  
(2) If $f(A) = 1$ and $a = \sqrt{3}$, find the range of possible values for $b$.

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