
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 in the Cartesian coordinate system $xOy$, its center is at the origin, the left focus is $F(-\sqrt{3},0)$, and the right vertex is $D(2,0)$. Let point $A(1, \frac{1}{2})$.

$(1)$ Find the standard equation of the ellipse;  
$(2)$ Given that line $l$ intersects the ellipse and the midpoint of chord $BC$ is $A$, find the equation of line $l$;  
$(3)$ Find the area $S_{\triangle FBC}$ of $\triangle FBC$.

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