
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 $C$: $\frac{x^2}{16}+\frac{y^2}{4}=1$, with the left and right foci being $F_{1}$ and $F_{2}$ respectively. Line $l$ intersects the ellipse at points $A$ and $B$, where chord $AB$ is bisected by the point $(\sqrt{3},\frac{\sqrt{3}}{2})$.  
$(1)$ Find the equation of line $l$;  
$(2)$ Find the area of $\triangle F_{1}AB$.

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