
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 one vertex at $A(0,1)$ and an eccentricity of $\frac{\sqrt{2}}{2}$. A line passing through point $B(0,-2)$ and the left focus $F_{1}$ intersects the ellipse at points $C$ and $D$, with the right focus denoted as $F_{2}$.
$(1)$ Find the equation of the ellipse;
$(2)$ Calculate the area of $\triangle CDF_{2}$.

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