
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 $W$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (a > b > 0)$ with eccentricity $\frac{\sqrt{2}}{2}$, and left and right foci $F_{1}$, $F_{2}$ respectively. A line passing through $F_{2}$ and perpendicular to the x-axis intersects the ellipse $W$ at a segment of length $2\sqrt{2}$.  
$(1)$ Find the equation of the ellipse $W$;  
$(2)$ The line $y=kx (k\neq 0)$ intersects the ellipse $W$ at points $A$ and $B$. Connect $AF_{1}$ and let it intersect the ellipse $W$ at another point $C$. If the area of $\triangle ABC$ is $\frac{12\sqrt{2}}{5}$, find the equation of the line $AC$.

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