
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 $\overrightarrow{a}=(2\sin x,-\cos x)$ and $\overrightarrow{b}=(\sqrt{3}\cos x,2\cos x)$, and function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}+1$.

(I) Find the smallest positive period of function $f(x)$, and find the range of $f(x)$ when $x\in\left[\dfrac{\pi}{12},\dfrac{2\pi}{3}\right]$;

(II) Translate the graph of function $f(x)$ to the left by $\dfrac{\pi}{3}$ unit to obtain the graph of function $g(x)$. In triangle $ABC$, sides $a$, $b$, and $c$ are opposite to angles $A$, $B$, and $C$, respectively. If $g\left(\dfrac{A}{2}\right)=1$, $a=2$, and $b+c=4$, find the area of $\triangle ABC$.

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