
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  $ABC$  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}
user