
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 rectangle $ABCD$ with length $AB = \sqrt{2+\sqrt{2}}$ and width $BC = 1$, vertex $E$ of equilateral $\triangle ABE$ is in the interior of the rectangle, and $F$ is the intersection point of diagonal $BD$ and line segment $AE$. Determine the area of $\triangle ABF$.
**A)** $\frac{1}{2\sqrt{2}}$
**B)** $\frac{7}{4} - \frac{2\sqrt{3}}{3}$
**C)** $\frac{\sqrt{2+\sqrt{2}} \sqrt{3}}{2\left(\sqrt{3} + \frac{1}{\sqrt{2+\sqrt{2}}}\right)}$
**D)** $2- \sqrt{3}$
**E)** $\frac{\sqrt{2}}{2} + \frac{1}{4}$

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