
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 $C$: $\dfrac{x^{2}}{a^{2}}+ \dfrac{y^{2}}{b^{2}}=1$ ($a > b > 0$), with its left and right foci as $F_{1}$ and $F_{2}$, respectively, and its eccentricity as $\dfrac{\sqrt{3}}{3}$. A line 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 the ellipse $C$.

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