
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 vectors $\overrightarrow{a}=(\sin x,-1)$, $\overrightarrow{b}=(\sqrt{3}\cos x,-\frac{1}{2})$, and the function $f(x)=(\overrightarrow{a}+\overrightarrow{b})\cdot \overrightarrow{a}-1$.
- (I) Find the interval of monotonic increase for the function $f(x)$;
- (II) In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. If $f\left(\frac{A}{2}\right)=\frac{3}{2}$ and $a=2$, find the range of values for $b+c$.

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