
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
Let \( \triangle ABC \) be a triangle such that \( AB = 7 \), and let the angle bisector of \( \angle BAC \) intersect line \( BC \) at \( D \). If there exist points \( E \) and \( F \) on sides \( AC \) and \( BC \), respectively, such that lines \( AD \) and \( EF \) are parallel and divide triangle \( ABC \) into three parts of equal area, determine the number of possible integral values for \( BC \).

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