
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 the function $f(x) = \overrightarrow{m} \cdot \overrightarrow{n}$, where vector $\overrightarrow{m} = (2\cos x, 1)$, $\overrightarrow{n} = (\cos x, \sqrt{3}\sin 2x)$, and $x \in \mathbb{R}$.
1. Find the interval of monotonic increase for $f(x)$.
2. In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$, respectively. Given that $f(A) = 2$, $b = 1$, and the area of $\triangle ABC$ is $\frac{\sqrt{3}}{2}$, find the value of $c$.

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