
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
In a triangle $\triangle ABC$, $\angle A = 45^\circ$ and $\angle B = 60^\circ$. A line segment $DE$, with $D$ on $AB$ and $\angle ADE = 30^\circ$, divides $\triangle ABC$ into two pieces of equal area. Determine the ratio $\frac{AD}{AB}$.
A) $\frac{\sqrt{3}}{4}$
B) $\frac{\sqrt{6} + \sqrt{2}}{4}$
C) $\frac{1}{4}$
D) $\frac{1}{2}$

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