
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
Vertex $A$ of an equilateral triangle $\triangle ABE$ is at the origin of the Cartesian coordinate system and vertices $B$ and $C$ of square $ABCD$ are on the positive x-axis and positive y-axis respectively. The side length of square $ABCD$ is $\sqrt{2}$. The base of $\triangle ABE$ extends from vertex $A$ to the midpoint of edge $BC$ of the square. Point $F$ is the intersection of the diagonal $BD$ of the square with line segment $AE$. Determine the area of $\triangle ABF$.
**A)** $\frac{\sqrt{3}}{4}$  
**B)** $\frac{\sqrt{2}}{4}$   
**C)** $\sqrt{2}-\sqrt{3}$   
**D)** $\frac{\sqrt{3}-1}{2}$  
**E)** $\frac{1}{4}+\frac{\sqrt{3}}{4}$

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