
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 $E$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \left(a > b > 0\right)$, whose left and right foci are $F_1$ and $F_2$ respectively, and its eccentricity is $\dfrac{1}{2}$. A line passing through point $F_1$, denoted as $I_1$, intersects the ellipse $E$ at points $A$ and $B$. Another line passing through point $F_2$, denoted as $I_2$, intersects the ellipse $E$ at points $C$ and $D$, and $AB \perp CD$. When $CD \perp x$-axis, $|CD| = 3$.

$(1)$ Find the standard equation of the ellipse $E$;

$(2)$ Find the minimum area of the quadrilateral $ABCD$.

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