
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 its left and right foci being $F_1$ and $F_2$ respectively, and the maximum distance from point $F_1$ to any point on the ellipse $C$ being $3$, and the eccentricity of ellipse $C$ is $\frac{1}{2}$.

$(1)$ Find the standard equation of the ellipse $C$;  
$(2)$ Determine whether there exists a line $l$ with a slope of $-1$ that intersects the circle with diameter $F_1F_2$ at points $A$ and $B$, and intersects the ellipse at points $C$ and $D$, such that $\frac{|CD|}{|AB|} = \frac{8 \sqrt{3}}{7}$. If it exists, find the equation of the line $l$; if not, explain why.

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