
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 $\Omega$: $\frac{{x^2}}{16}+\frac{{y^2}}{12}=1$. The real axis vertex of the hyperbola $\Gamma$ is the focus of the ellipse $\Omega$, and the focal length of the hyperbola $\Gamma$ is equal to the length of the major axis of the ellipse $\Omega$.  
$(1)$ Find the standard equation of the hyperbola $\Gamma$;  
$(2)$ Let the line $l$ pass through the point $E(3,0)$ and intersect the ellipse $\Omega$ at points $A$ and $B$. Find the maximum area of $\triangle OAB$;  
$(3)$ Let the line $l: y=kx+m$ (where $k$ and $m$ are integers) intersect the ellipse $\Omega$ at two distinct points $A$ and $B$, and intersect the hyperbola $\Gamma$ at two distinct points $C$ and $D$. Determine if there exists a line $l$ such that the vector $\overrightarrow{AC}+\overrightarrow{BD}=\overrightarrow{0}$. If it exists, indicate how many such lines exist. If it does not exist, explain the reason.

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