
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 distance $AD = d$. We mark the distance $AB = x$ (variable) on this line segment and construct the equilateral triangle $ABC$. At point $C$, we draw a perpendicular to $BC$, and at point $D$, we draw a perpendicular to $AD$, which intersect at point $E$. Study how the area of the quadrilateral $ACED$ changes as $0 \leq x \leq d$.

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