
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \ (a > b > 0)$ with its foci $F_1$ and $F_2$ on the left and right respectively, a line passing through $F_1$ intersects the ellipse at points $A$ and $B$. If triangle $\triangle F_2AB$ forms an equilateral triangle with an area of $4\sqrt{3}$, then find the equation of the ellipse $C$.

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