
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 ellipse $C:\dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1(a > b > 0)$ with left and right foci $F\_1$ and $F\_2$ respectively, and an eccentricity of $\dfrac{\sqrt{3}}{3}$. The line $l$ passing through $F\_2$ intersects $C$ at points $A$ and $B$. If the perimeter of $\triangle AF_{1}B$ is $4\sqrt{3}$, determine the equation of $C$.

A) $\dfrac{{{x}^{2}}}{3}+\dfrac{{{y}^{2}}}{2}=1$
B) $\dfrac{{{x}^{2}}}{3}+{{y}^{2}}=1$
C) $\dfrac{{{x}^{2}}}{12}+\dfrac{{{y}^{2}}}{8}=1$
D) $\dfrac{{{x}^{2}}}{12}+\dfrac{{{y}^{2}}}{4}=1$

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