
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 $\Gamma$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ with the right focus $F\left(c,0\right)\left(c \gt 0\right)$, eccentricity $\frac{{\sqrt{3}}}{2}$, a line passing through $F$ and perpendicular to the $x$-axis intersects $\Gamma$ at a point $M$ in the first quadrant. Let $O$ be the origin, and $|OM|=\frac{{\sqrt{13}}}{2}$. <br/>$(Ⅰ)$ Find the equation of the ellipse $\Gamma$; <br/>$(Ⅱ)$ Suppose a line not passing through the origin $O$ and with a slope of $\frac{1}{2}$ intersects the ellipse $\Gamma$ at points $A$ and $B$. Let $C$ and $D$ be the points symmetric to $A$ and $B$ with respect to the origin $O$. Determine if the quadrilateral $ABCD$ has a maximum area. If it does, find the maximum area; if not, explain why.

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