
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 the function $f(x)=\sin \left(x+ \frac{\pi}{6}\right)+2\sin \frac{2x}{2}$.

$(I)$ Find the interval of monotonic increase for the function $f(x)$;

$(II)$ Let $\triangle ABC$ have internal angles $A$, $B$, $C$ with opposite sides of lengths $a$, $b$, $c$ respectively. If $f(A)= \frac{3}{2}$ and the area of $\triangle ABC$, $S= \frac{\sqrt{3}}{2}$, with $a= \sqrt{3}$, find the value of $b+c$.

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