
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
In  $\vartriangle ABC$  points  $D, E$ , and  $F$  lie on side  $\overline{BC}$  such that  $\overline{AD}$  is an angle bisector of  $\angle BAC$ ,  $\overline{AE}$  is a median, and  $\overline{AF}$  is an altitude. Given that  $AB = 154$  and  $AC = 128$ , and  $9 \times DE = EF,$  find the side length  $BC$ .

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