
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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with the left focus $F$, right vertex $A$, eccentricity $\frac{1}{2}$, and $|AF| = 3$.  
$(1)$ Find the equation of the ellipse;  
$(2)$ A line passing through point $A$ with slope $k (k < 0)$ intersects the ellipse at another point $B$. Point $C$ lies on the $y$-axis, and $FC \perp AB$. If the slope of line $BC$ is $-\frac{3}{10}k$, find the equation of line $AB$.

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