
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 hyperbola $Γ:\frac{{x}^{2}}{3}-\frac{{y}^{2}}{12}=1$, where $A\left(2,2\right)$ is a point on the hyperbola $Γ$.<br/>$(1)$ If an ellipse $C$ has the vertex of the hyperbola $Γ$ as its focus and the length of its major axis is $4\sqrt{3}$, find the standard equation of the ellipse $C$;<br/>$(2)$ Let $P$ be a point on the asymptote of the hyperbola $Γ$ in the first quadrant, and $Q$ be a point on the hyperbola $Γ$ such that $\overrightarrow{PA}=\overrightarrow{AQ}$. Find the area of $\triangle POQ$ where $S(O)$ is the origin.

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