
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 the vectors $\overrightarrow{m}=(2\cos x,1)$ and $\overrightarrow{n}=(\cos x, \sqrt {3}\sin 2x)$, with $x \in \mathbb{R}$.  
$(1)$ Find the intervals 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}
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