
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 the ellipse $\Gamma$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$ has an eccentricity of $\dfrac{\sqrt{3}}{2}$, the left vertex is $C$, the top vertex is $D$, and $|CD| = \sqrt{5}$  
$(1)$ Find the equation of the ellipse $\Gamma$  
$(2)$ Let $O$ be the origin, a line with slope $k$ passes through the right focus $P$ and intersects $\Gamma$ at points $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$. If $\dfrac{x_{1}x_{2}}{a^{2}} + \dfrac{y_{1}y_{2}}{b^{2}} = 0$, find the area of $\triangle AOB$.

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