
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 $F_2$, eccentricity as $\dfrac{\sqrt{3}}{3}$, and a line $l$ passing through $F_2$ intersects $C$ at points $A$ and $B$. If the perimeter of $\triangle AF_1B$ is $4\sqrt{3}$, then the standard equation of the ellipse $C$ is \_\_\_\_\_\_\_\_.

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