
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 points $A$, $B$, and $C$ form a triangle with $AB=1$, $BC=\sqrt{2}$, and $AC=1$, and point $D$ lies on the extension of $AC$ such that $CD=1$, and a line parallel to $AC$ passes through $B$ and intersects the extension of $AB$ at point $E$ where $BE=1$, calculate the number of possible values for the area of the triangle with vertices $A$, $B$, $C$, $D$, and $E$.

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