
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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ with the right focus at $F$, points $A$ and $B$ are two symmetric points on the ellipse with respect to the origin $O$. When the coordinates of point $A$ are $(-1, \frac{\sqrt{14}}{2})$, the perimeter of $\triangle ABF$ is exactly $7\sqrt{2}$.  
$(1)$ Find the equation of the ellipse.  
$(2)$ Let line $l$ passing through point $F$ intersect the ellipse at points $C$ and $D$, and $CD$ is parallel to $AB$. Find the maximum area of $\triangle ACD$.

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