
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 a hyperbola $E$: $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt 0,b \gt 0\right)$, with one of its asymptotes having a slope of $30^{\circ}$, and the minimum distance from one focus to a point on $E$ is $2-\sqrt{3}$. <br/>$(Ⅰ)$ Find the equation of $E$; <br/>$(Ⅱ)$ Given the line $l:y=x-2$ intersects the hyperbola $E$ at points $A$ and $B$, perpendiculars are drawn from $A$ and $B$ to line $l$ intersecting $E$ at points $D$ and $C$ respectively, find the area of quadrilateral $ABCD$.

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