
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 chord $BC$ of the ellipse at its midpoint $A$, find the equation of line $l$.

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