
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 an equilateral triangle \(ABC\), draw the three parabolas, each with its focus at \(A\), \(B\), and \(C\) respectively, and directrices \(BC\), \(CA\), and \(AB\) respectively. Find the area of the part of the triangle that is not covered by the interior of any of the parabolas. (Given \(AB=1\); the interior of a parabola is the part of the plane on the same side of the parabolic curve as the focus.)

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