
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 $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ with $a > b > 0$, one of its vertices is $A(0, b)$, the eccentricity is $\frac{\sqrt{2}}{2}$, and a line passing through point $B(0, -2)$ and the left focus $F_{1}$ intersects the ellipse at points $C$ and $D$. Let the right focus be $F_{2}$. 

(1) Find the equation of the ellipse;
(2) Calculate the area of $\triangle CDF_{2}$.

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