
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 $E$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left focus $F_{1}$ and right focus $F_{2}$, and the focal distance $F_{1}F_{2}$ is $2$. A line passing through $F_{1}$ intersects the ellipse $E$ at points $A$ and $B$, and the perimeter of $\triangle ABF_{2}$ is $4\sqrt{3}$.  
$(1)$ Find the equation of the ellipse $E$;  
$(2)$ If the slope of line $AB$ is $2$, find the area of $\triangle ABF_{2}$.

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