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

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