
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 $f(x) = 2\sqrt{3}\sin x\cos x - 2\cos^{2}x + 1$.
$(1)$ Find the smallest positive period of the function $f(x)$;
$(2)$ The graph of function $f(x)$ is translated to the left by $\dfrac{\pi}{3}$ units to obtain the graph of function $g(x)$. In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, 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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