
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ whose one focus coincides with the focus of $y^2 = 4\sqrt{3}x$, and the point $(\sqrt{3}, \frac{1}{2})$ lies on the ellipse $C$.

$(1)$ Find the equation of the ellipse $C$;  
$(2)$ A line $l$ with a slope of $\frac{1}{2}$ passes through the right focus $F$ of the ellipse and intersects the ellipse $C$ at points $A$ and $B$. Find the length of the chord $|AB|$.

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