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

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