
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 isosceles  $\vartriangle ABC, AB = AC, \angle BAC$  is obtuse, and points  $E$  and  $F$  lie on sides  $AB$  and  $AC$ , respectively, so that  $AE = 10, AF = 15$ . The area of  $\vartriangle AEF$  is  $60$ , and the area of quadrilateral  $BEFC$  is  $102$ . Find  $BC$ .

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