
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 $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left and right foci $F_{1}$ and $F_{2}$, and top vertex $D$, where $\triangle DF_{1}F_{2}$ is an equilateral triangle. A line $l$ passing through the focus $F_{2}$ intersects the ellipse $C$ at points $A$ and $B$, and the perimeter of $\triangle F_{1}AB$ is $8$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Find the maximum area of $\triangle F_{1}AB$ and the equation of the line $l$ at that time.

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