
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
The internal angle bisectors of triangle \(ABC\) from vertices \(A, B, C\) intersect the opposite sides at points \(D, E, F\), respectively.

What is the necessary and sufficient condition for the areas of triangles \(BDF\) and \(EDF\) to be equal?

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