
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 triangle $\triangle ABC$, $\angle A = 60^\circ$ and $\angle B = 45^\circ$. A line segment $DF$, with $D$ on side $AB$, makes an angle $\angle ADF = 30^\circ$ with $AB$. The line segment bisects the area of $\triangle ABC$. Calculate the ratio $\frac{AD}{AB}$.
A) $\frac{\sqrt{6} + \sqrt{2}}{4\sqrt{2}}$
B) $\frac{\sqrt{6} - \sqrt{2}}{4\sqrt{2}}$
C) $\frac{\sqrt{6} - \sqrt{2}}{2\sqrt{2}}$
D) $\frac{2}{\sqrt{2}}$

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